src/HOL/Word/Word.thy
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(*  Title:      HOL/Word/Word.thy
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    Author:     Jeremy Dawson and Gerwin Klein, NICTA
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*)
e77ea0ea7f2c * HOL-Word:
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section \<open>A type of finite bit strings\<close>
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theory Word
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imports
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  "HOL-Library.Type_Length"
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  "HOL-Library.Boolean_Algebra"
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  "HOL-Library.Bit_Operations"
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  Bits_Int
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  Bit_Comprehension
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  Bit_Lists
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  Misc_Typedef
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begin
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subsection \<open>Type definition\<close>
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quotient_type (overloaded) 'a word = int / \<open>\<lambda>k l. take_bit LENGTH('a) k = take_bit LENGTH('a::len) l\<close>
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  morphisms rep_word word_of_int by (auto intro!: equivpI reflpI sympI transpI)
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lift_definition uint :: \<open>'a::len word \<Rightarrow> int\<close>
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  is \<open>take_bit LENGTH('a)\<close> .
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lemma uint_nonnegative: "0 \<le> uint w"
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  by transfer simp
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lemma uint_bounded: "uint w < 2 ^ LENGTH('a)"
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  for w :: "'a::len word"
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  by transfer (simp add: take_bit_eq_mod)
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lemma uint_idem: "uint w mod 2 ^ LENGTH('a) = uint w"
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  for w :: "'a::len word"
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  using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial)
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lemma word_uint_eqI: "uint a = uint b \<Longrightarrow> a = b"
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  by transfer simp
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lemma word_uint_eq_iff: "a = b \<longleftrightarrow> uint a = uint b"
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  using word_uint_eqI by auto
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lemma uint_word_of_int: "uint (word_of_int k :: 'a::len word) = k mod 2 ^ LENGTH('a)"
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  by transfer (simp add: take_bit_eq_mod)
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lemma word_of_int_uint: "word_of_int (uint w) = w"
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  by transfer simp
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lemma split_word_all: "(\<And>x::'a::len word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))"
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proof
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  fix x :: "'a word"
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  assume "\<And>x. PROP P (word_of_int x)"
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  then have "PROP P (word_of_int (uint x))" .
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  then show "PROP P x" by (simp add: word_of_int_uint)
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qed
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subsection \<open>Type conversions and casting\<close>
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lemma signed_take_bit_decr_length_iff:
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  \<open>signed_take_bit (LENGTH('a::len) - Suc 0) k = signed_take_bit (LENGTH('a) - Suc 0) l
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    \<longleftrightarrow> take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
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  by (cases \<open>LENGTH('a)\<close>)
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    (simp_all add: signed_take_bit_eq_iff_take_bit_eq)
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lift_definition sint :: \<open>'a::len word \<Rightarrow> int\<close>
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  \<comment> \<open>treats the most-significant bit as a sign bit\<close>
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  is \<open>signed_take_bit (LENGTH('a) - 1)\<close>  
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  by (simp add: signed_take_bit_decr_length_iff)
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lemma sint_uint [code]:
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  \<open>sint w = signed_take_bit (LENGTH('a) - 1) (uint w)\<close>
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  for w :: \<open>'a::len word\<close>
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  by transfer simp
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lift_definition unat :: \<open>'a::len word \<Rightarrow> nat\<close>
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  is \<open>nat \<circ> take_bit LENGTH('a)\<close>
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  by transfer simp
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lemma nat_uint_eq [simp]:
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  \<open>nat (uint w) = unat w\<close>
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  by transfer simp
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lemma unat_eq_nat_uint [code]:
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  \<open>unat w = nat (uint w)\<close>
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  by simp
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lift_definition ucast :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
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  is \<open>take_bit LENGTH('a)\<close>
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  by simp
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lemma ucast_eq [code]:
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  \<open>ucast w = word_of_int (uint w)\<close>
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  by transfer simp
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lift_definition scast :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
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  is \<open>signed_take_bit (LENGTH('a) - 1)\<close>
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  by (simp flip: signed_take_bit_decr_length_iff)
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lemma scast_eq [code]:
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  \<open>scast w = word_of_int (sint w)\<close>
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  by transfer simp
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instantiation word :: (len) size
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begin
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lift_definition size_word :: \<open>'a word \<Rightarrow> nat\<close>
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  is \<open>\<lambda>_. LENGTH('a)\<close> ..
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instance ..
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end
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lemma word_size [code]:
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  \<open>size w = LENGTH('a)\<close> for w :: \<open>'a::len word\<close>
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  by (fact size_word.rep_eq)
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lemma word_size_gt_0 [iff]: "0 < size w"
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  for w :: "'a::len word"
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  by (simp add: word_size)
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lemmas lens_gt_0 = word_size_gt_0 len_gt_0
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lemma lens_not_0 [iff]:
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  \<open>size w \<noteq> 0\<close> for  w :: \<open>'a::len word\<close>
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  by auto
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lift_definition source_size :: \<open>('a::len word \<Rightarrow> 'b) \<Rightarrow> nat\<close>
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  is \<open>\<lambda>_. LENGTH('a)\<close> .
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lift_definition target_size :: \<open>('a \<Rightarrow> 'b::len word) \<Rightarrow> nat\<close>
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  is \<open>\<lambda>_. LENGTH('b)\<close> ..
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lift_definition is_up :: \<open>('a::len word \<Rightarrow> 'b::len word) \<Rightarrow> bool\<close>
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  is \<open>\<lambda>_. LENGTH('a) \<le> LENGTH('b)\<close> ..
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lift_definition is_down :: \<open>('a::len word \<Rightarrow> 'b::len word) \<Rightarrow> bool\<close>
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  is \<open>\<lambda>_. LENGTH('a) \<ge> LENGTH('b)\<close> ..
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lemma is_up_eq:
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  \<open>is_up f \<longleftrightarrow> source_size f \<le> target_size f\<close>
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  for f :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
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  by (simp add: source_size.rep_eq target_size.rep_eq is_up.rep_eq)
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lemma is_down_eq:
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  \<open>is_down f \<longleftrightarrow> target_size f \<le> source_size f\<close>
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  for f :: \<open>'a::len word \<Rightarrow> 'b::len word\<close>
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  by (simp add: source_size.rep_eq target_size.rep_eq is_down.rep_eq)
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lift_definition of_bl :: \<open>bool list \<Rightarrow> 'a::len word\<close>
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  is bl_to_bin .
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lift_definition to_bl :: \<open>'a::len word \<Rightarrow> bool list\<close>
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  is \<open>bin_to_bl LENGTH('a)\<close>
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  by (simp add: bl_to_bin_inj)
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lemma to_bl_eq:
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  \<open>to_bl w = bin_to_bl (LENGTH('a)) (uint w)\<close>
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  for w :: \<open>'a::len word\<close>
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  by transfer simp
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lift_definition word_int_case :: \<open>(int \<Rightarrow> 'b) \<Rightarrow> 'a::len word \<Rightarrow> 'b\<close>
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  is \<open>\<lambda>f. f \<circ> take_bit LENGTH('a)\<close> by simp
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lemma word_int_case_eq_uint [code]:
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  \<open>word_int_case f w = f (uint w)\<close>
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  by transfer simp
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translations
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  "case x of XCONST of_int y \<Rightarrow> b" \<rightleftharpoons> "CONST word_int_case (\<lambda>y. b) x"
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  "case x of (XCONST of_int :: 'a) y \<Rightarrow> b" \<rightharpoonup> "CONST word_int_case (\<lambda>y. b) x"
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subsection \<open>Basic code generation setup\<close>
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lift_definition Word :: \<open>int \<Rightarrow> 'a::len word\<close>
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  is id .
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lemma Word_eq_word_of_int [code_post]:
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  \<open>Word = word_of_int\<close>
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  by (simp add: fun_eq_iff Word.abs_eq)
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lemma [code abstype]:
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  \<open>Word (uint w) = w\<close>
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  by transfer simp
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lemma [code abstract]:
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  \<open>uint (word_of_int k :: 'a::len word) = take_bit LENGTH('a) k\<close>
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  by (fact uint.abs_eq)
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instantiation word :: (len) equal
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begin
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lift_definition equal_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> bool\<close>
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  is \<open>\<lambda>k l. take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
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  by simp
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instance
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  by (standard; transfer) rule
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end
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lemma [code]:
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  \<open>HOL.equal k l \<longleftrightarrow> HOL.equal (uint k) (uint l)\<close>
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  by transfer (simp add: equal)
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notation fcomp (infixl "\<circ>>" 60)
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notation scomp (infixl "\<circ>\<rightarrow>" 60)
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instantiation word :: ("{len, typerep}") random
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begin
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definition
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  "random_word i = Random.range i \<circ>\<rightarrow> (\<lambda>k. Pair (
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     let j = word_of_int (int_of_integer (integer_of_natural k)) :: 'a word
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     in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))"
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instance ..
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end
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no_notation fcomp (infixl "\<circ>>" 60)
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no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
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subsection \<open>Type-definition locale instantiations\<close>
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lemmas uint_0 = uint_nonnegative (* FIXME duplicate *)
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lemmas uint_lt = uint_bounded (* FIXME duplicate *)
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lemmas uint_mod_same = uint_idem (* FIXME duplicate *)
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definition uints :: "nat \<Rightarrow> int set"
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  \<comment> \<open>the sets of integers representing the words\<close>
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  where "uints n = range (bintrunc n)"
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definition sints :: "nat \<Rightarrow> int set"
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  where "sints n = range (sbintrunc (n - 1))"
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lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}"
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  by (simp add: uints_def range_bintrunc)
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lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}"
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  by (simp add: sints_def range_sbintrunc)
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definition unats :: "nat \<Rightarrow> nat set"
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  where "unats n = {i. i < 2 ^ n}"
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\<comment> \<open>naturals\<close>
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lemma uints_unats: "uints n = int ` unats n"
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  apply (unfold unats_def uints_num)
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  apply safe
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    apply (rule_tac image_eqI)
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     apply (erule_tac nat_0_le [symmetric])
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  by auto
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lemma unats_uints: "unats n = nat ` uints n"
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  by (auto simp: uints_unats image_iff)
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lemma td_ext_uint:
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  "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (LENGTH('a::len)))
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    (\<lambda>w::int. w mod 2 ^ LENGTH('a))"
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  apply (unfold td_ext_def')
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  apply transfer
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  apply (simp add: uints_num take_bit_eq_mod)
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  done
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interpretation word_uint:
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  td_ext
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    "uint::'a::len word \<Rightarrow> int"
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    word_of_int
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    "uints (LENGTH('a::len))"
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    "\<lambda>w. w mod 2 ^ LENGTH('a::len)"
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  by (fact td_ext_uint)
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lemmas td_uint = word_uint.td_thm
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lemmas int_word_uint = word_uint.eq_norm
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lemma td_ext_ubin:
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  "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (LENGTH('a::len)))
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    (bintrunc (LENGTH('a)))"
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  by (unfold no_bintr_alt1) (fact td_ext_uint)
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interpretation word_ubin:
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  td_ext
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    "uint::'a::len word \<Rightarrow> int"
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    word_of_int
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    "uints (LENGTH('a::len))"
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    "bintrunc (LENGTH('a::len))"
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  by (fact td_ext_ubin)
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subsection \<open>Arithmetic operations\<close>
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lift_definition word_succ :: "'a::len word \<Rightarrow> 'a word" is "\<lambda>x. x + 1"
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  by (auto simp add: bintrunc_mod2p intro: mod_add_cong)
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lift_definition word_pred :: "'a::len word \<Rightarrow> 'a word" is "\<lambda>x. x - 1"
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  by (auto simp add: bintrunc_mod2p intro: mod_diff_cong)
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instantiation word :: (len) "{neg_numeral, modulo, comm_monoid_mult, comm_ring}"
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begin
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lift_definition zero_word :: "'a word" is "0" .
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lift_definition one_word :: "'a word" is "1" .
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lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "(+)"
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  by (auto simp add: bintrunc_mod2p intro: mod_add_cong)
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lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "(-)"
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  by (auto simp add: bintrunc_mod2p intro: mod_diff_cong)
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   313
lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus
64593
50c715579715 reoriented congruence rules in non-explosive direction
haftmann
parents: 64243
diff changeset
   314
  by (auto simp add: bintrunc_mod2p intro: mod_minus_cong)
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   315
69064
5840724b1d71 Prefix form of infix with * on either side no longer needs special treatment
nipkow
parents: 68157
diff changeset
   316
lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "(*)"
64593
50c715579715 reoriented congruence rules in non-explosive direction
haftmann
parents: 64243
diff changeset
   317
  by (auto simp add: bintrunc_mod2p intro: mod_mult_cong)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   318
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   319
lift_definition divide_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   320
  is "\<lambda>a b. take_bit LENGTH('a) a div take_bit LENGTH('a) b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   321
  by simp
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   322
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   323
lift_definition modulo_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   324
  is "\<lambda>a b. take_bit LENGTH('a) a mod take_bit LENGTH('a) b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   325
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   326
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   327
instance
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61076
diff changeset
   328
  by standard (transfer, simp add: algebra_simps)+
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   329
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   330
end
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   331
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   332
lemma uint_0_eq [simp, code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   333
  \<open>uint 0 = 0\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   334
  by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   335
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   336
quickcheck_generator word
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   337
  constructors:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   338
    \<open>0 :: 'a::len word\<close>,
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   339
    \<open>numeral :: num \<Rightarrow> 'a::len word\<close>,
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   340
    \<open>uminus :: 'a word \<Rightarrow> 'a::len word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   341
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   342
lemma uint_1_eq [simp, code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   343
  \<open>uint 1 = 1\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   344
  by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   345
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   346
lemma word_div_def [code]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   347
  "a div b = word_of_int (uint a div uint b)"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   348
  by transfer rule
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   349
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   350
lemma word_mod_def [code]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   351
  "a mod b = word_of_int (uint a mod uint b)"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   352
  by transfer rule
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   353
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   354
context
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   355
  includes lifting_syntax
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   356
  notes power_transfer [transfer_rule]
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   357
begin
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   358
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   359
lemma power_transfer_word [transfer_rule]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   360
  \<open>(pcr_word ===> (=) ===> pcr_word) (^) (^)\<close>
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   361
  by transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   362
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   363
end
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   364
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   365
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   366
text \<open>Legacy theorems:\<close>
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   367
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   368
lemma word_arith_wis:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   369
  shows word_add_def [code]: "a + b = word_of_int (uint a + uint b)"
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   370
    and word_sub_wi [code]: "a - b = word_of_int (uint a - uint b)"
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   371
    and word_mult_def [code]: "a * b = word_of_int (uint a * uint b)"
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   372
    and word_minus_def [code]: "- a = word_of_int (- uint a)"
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   373
    and word_succ_alt [code]: "word_succ a = word_of_int (uint a + 1)"
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   374
    and word_pred_alt [code]: "word_pred a = word_of_int (uint a - 1)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   375
    and word_0_wi: "0 = word_of_int 0"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   376
    and word_1_wi: "1 = word_of_int 1"
71948
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   377
         apply (simp_all flip: plus_word.abs_eq minus_word.abs_eq
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   378
           times_word.abs_eq uminus_word.abs_eq
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   379
           zero_word.abs_eq one_word.abs_eq)
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   380
   apply transfer
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   381
   apply simp
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   382
  apply transfer
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   383
  apply simp
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   384
  done
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   385
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   386
lemma wi_homs:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   387
  shows wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   388
    and wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   389
    and wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   390
    and wi_hom_neg: "- word_of_int a = word_of_int (- a)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   391
    and wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   392
    and wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   393
  by (transfer, simp)+
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   394
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   395
lemmas wi_hom_syms = wi_homs [symmetric]
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   396
46013
d2f179d26133 remove some duplicate lemmas
huffman
parents: 46012
diff changeset
   397
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
46009
5cb7ef5bfef2 remove duplicate lemma lists
huffman
parents: 46001
diff changeset
   398
5cb7ef5bfef2 remove duplicate lemma lists
huffman
parents: 46001
diff changeset
   399
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   400
71954
13bb3f5cdc5b pragmatically ruled out word types of length zero: a bit string with no bits is not bit string at all
haftmann
parents: 71953
diff changeset
   401
instance word :: (len) comm_monoid_add ..
13bb3f5cdc5b pragmatically ruled out word types of length zero: a bit string with no bits is not bit string at all
haftmann
parents: 71953
diff changeset
   402
13bb3f5cdc5b pragmatically ruled out word types of length zero: a bit string with no bits is not bit string at all
haftmann
parents: 71953
diff changeset
   403
instance word :: (len) semiring_numeral ..
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   404
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   405
instance word :: (len) comm_ring_1
45810
024947a0e492 prove class instances without extra lemmas
huffman
parents: 45809
diff changeset
   406
proof
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   407
  have *: "0 < LENGTH('a)" by (rule len_gt_0)
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   408
  show "(0::'a word) \<noteq> 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   409
    by transfer (use * in \<open>auto simp add: gr0_conv_Suc\<close>)
45810
024947a0e492 prove class instances without extra lemmas
huffman
parents: 45809
diff changeset
   410
qed
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   411
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   412
lemma word_of_nat: "of_nat n = word_of_int (int n)"
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   413
  by (induct n) (auto simp add : word_of_int_hom_syms)
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   414
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   415
lemma word_of_int: "of_int = word_of_int"
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   416
  apply (rule ext)
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   417
  apply (case_tac x rule: int_diff_cases)
46013
d2f179d26133 remove some duplicate lemmas
huffman
parents: 46012
diff changeset
   418
  apply (simp add: word_of_nat wi_hom_sub)
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   419
  done
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   420
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   421
context
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   422
  includes lifting_syntax
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   423
  notes 
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   424
    transfer_rule_of_bool [transfer_rule]
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   425
    transfer_rule_numeral [transfer_rule]
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   426
    transfer_rule_of_nat [transfer_rule]
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   427
    transfer_rule_of_int [transfer_rule]
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   428
begin
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   429
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   430
lemma [transfer_rule]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   431
  "((=) ===> (pcr_word :: int \<Rightarrow> 'a::len word \<Rightarrow> bool)) of_bool of_bool"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   432
  by transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   433
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   434
lemma [transfer_rule]:
71954
13bb3f5cdc5b pragmatically ruled out word types of length zero: a bit string with no bits is not bit string at all
haftmann
parents: 71953
diff changeset
   435
  "((=) ===> (pcr_word :: int \<Rightarrow> 'a::len word \<Rightarrow> bool)) numeral numeral"
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   436
  by transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   437
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   438
lemma [transfer_rule]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   439
  "((=) ===> pcr_word) int of_nat"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   440
  by transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   441
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   442
lemma [transfer_rule]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   443
  "((=) ===> pcr_word) (\<lambda>k. k) of_int"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   444
proof -
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   445
  have "((=) ===> pcr_word) of_int of_int"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   446
    by transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   447
  then show ?thesis by (simp add: id_def)
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   448
qed
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   449
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   450
end
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   451
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   452
lemma word_of_int_eq:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   453
  "word_of_int = of_int"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   454
  by (rule ext) (transfer, rule)
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   455
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   456
definition udvd :: "'a::len word \<Rightarrow> 'a::len word \<Rightarrow> bool" (infixl "udvd" 50)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   457
  where "a udvd b = (\<exists>n\<ge>0. uint b = n * uint a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   458
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   459
context
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   460
  includes lifting_syntax
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   461
begin
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   462
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   463
lemma [transfer_rule]:
71958
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   464
  \<open>(pcr_word ===> (\<longleftrightarrow>)) even ((dvd) 2 :: 'a::len word \<Rightarrow> bool)\<close>
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   465
proof -
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   466
  have even_word_unfold: "even k \<longleftrightarrow> (\<exists>l. take_bit LENGTH('a) k = take_bit LENGTH('a) (2 * l))" (is "?P \<longleftrightarrow> ?Q")
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   467
    for k :: int
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   468
  proof
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   469
    assume ?P
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   470
    then show ?Q
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   471
      by auto
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   472
  next
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   473
    assume ?Q
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   474
    then obtain l where "take_bit LENGTH('a) k = take_bit LENGTH('a) (2 * l)" ..
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   475
    then have "even (take_bit LENGTH('a) k)"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   476
      by simp
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   477
    then show ?P
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   478
      by simp
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   479
  qed
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   480
  show ?thesis by (simp only: even_word_unfold [abs_def] dvd_def [where ?'a = "'a word", abs_def])
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   481
    transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   482
qed
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   483
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   484
end
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   485
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   486
instance word :: (len) semiring_modulo
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   487
proof
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   488
  show "a div b * b + a mod b = a" for a b :: "'a word"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   489
  proof transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   490
    fix k l :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   491
    define r :: int where "r = 2 ^ LENGTH('a)"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   492
    then have r: "take_bit LENGTH('a) k = k mod r" for k
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   493
      by (simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   494
    have "k mod r = ((k mod r) div (l mod r) * (l mod r)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   495
      + (k mod r) mod (l mod r)) mod r"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   496
      by (simp add: div_mult_mod_eq)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   497
    also have "... = (((k mod r) div (l mod r) * (l mod r)) mod r
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   498
      + (k mod r) mod (l mod r)) mod r"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   499
      by (simp add: mod_add_left_eq)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   500
    also have "... = (((k mod r) div (l mod r) * l) mod r
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   501
      + (k mod r) mod (l mod r)) mod r"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   502
      by (simp add: mod_mult_right_eq)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   503
    finally have "k mod r = ((k mod r) div (l mod r) * l
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   504
      + (k mod r) mod (l mod r)) mod r"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   505
      by (simp add: mod_simps)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   506
    with r show "take_bit LENGTH('a) (take_bit LENGTH('a) k div take_bit LENGTH('a) l * l
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   507
      + take_bit LENGTH('a) k mod take_bit LENGTH('a) l) = take_bit LENGTH('a) k"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   508
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   509
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   510
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   511
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   512
instance word :: (len) semiring_parity
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   513
proof
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   514
  show "\<not> 2 dvd (1::'a word)"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   515
    by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   516
  show even_iff_mod_2_eq_0: "2 dvd a \<longleftrightarrow> a mod 2 = 0"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   517
    for a :: "'a word"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   518
    by transfer (simp_all add: mod_2_eq_odd take_bit_Suc)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   519
  show "\<not> 2 dvd a \<longleftrightarrow> a mod 2 = 1"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   520
    for a :: "'a word"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   521
    by transfer (simp_all add: mod_2_eq_odd take_bit_Suc)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   522
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   523
71953
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   524
lemma exp_eq_zero_iff:
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   525
  \<open>2 ^ n = (0 :: 'a::len word) \<longleftrightarrow> n \<ge> LENGTH('a)\<close>
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   526
  by transfer simp
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   527
71958
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   528
lemma double_eq_zero_iff:
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   529
  \<open>2 * a = 0 \<longleftrightarrow> a = 0 \<or> a = 2 ^ (LENGTH('a) - Suc 0)\<close>
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   530
  for a :: \<open>'a::len word\<close>
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   531
proof -
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   532
  define n where \<open>n = LENGTH('a) - Suc 0\<close>
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   533
  then have *: \<open>LENGTH('a) = Suc n\<close>
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   534
    by simp
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   535
  have \<open>a = 0\<close> if \<open>2 * a = 0\<close> and \<open>a \<noteq> 2 ^ (LENGTH('a) - Suc 0)\<close>
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   536
    using that by transfer
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   537
      (auto simp add: take_bit_eq_0_iff take_bit_eq_mod *)
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   538
  moreover have \<open>2 ^ LENGTH('a) = (0 :: 'a word)\<close>
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   539
    by transfer simp
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   540
  then have \<open>2 * 2 ^ (LENGTH('a) - Suc 0) = (0 :: 'a word)\<close>
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   541
    by (simp add: *)
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   542
  ultimately show ?thesis
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   543
    by auto
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   544
qed
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   545
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   546
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   547
subsection \<open>Ordering\<close>
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   548
71954
13bb3f5cdc5b pragmatically ruled out word types of length zero: a bit string with no bits is not bit string at all
haftmann
parents: 71953
diff changeset
   549
instantiation word :: (len) linorder
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   550
begin
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   551
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   552
lift_definition less_eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   553
  is "\<lambda>a b. take_bit LENGTH('a) a \<le> take_bit LENGTH('a) b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   554
  by simp
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   555
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   556
lift_definition less_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   557
  is "\<lambda>a b. take_bit LENGTH('a) a < take_bit LENGTH('a) b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   558
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   559
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   560
instance
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   561
  by (standard; transfer) auto
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   562
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   563
end
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   564
71957
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   565
interpretation word_order: ordering_top \<open>(\<le>)\<close> \<open>(<)\<close> \<open>- 1 :: 'a::len word\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   566
  by (standard; transfer) (simp add: take_bit_eq_mod zmod_minus1)
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   567
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   568
interpretation word_coorder: ordering_top \<open>(\<ge>)\<close> \<open>(>)\<close> \<open>0 :: 'a::len word\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   569
  by (standard; transfer) simp
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   570
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   571
lemma word_le_def [code]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   572
  "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   573
  by transfer rule
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   574
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   575
lemma word_less_def [code]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   576
  "a < b \<longleftrightarrow> uint a < uint b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   577
  by transfer rule
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   578
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   579
lemma word_greater_zero_iff:
71954
13bb3f5cdc5b pragmatically ruled out word types of length zero: a bit string with no bits is not bit string at all
haftmann
parents: 71953
diff changeset
   580
  \<open>a > 0 \<longleftrightarrow> a \<noteq> 0\<close> for a :: \<open>'a::len word\<close>
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   581
  by transfer (simp add: less_le)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   582
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   583
lemma of_nat_word_eq_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   584
  \<open>of_nat m = (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m = take_bit LENGTH('a) n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   585
  by transfer (simp add: take_bit_of_nat)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   586
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   587
lemma of_nat_word_less_eq_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   588
  \<open>of_nat m \<le> (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m \<le> take_bit LENGTH('a) n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   589
  by transfer (simp add: take_bit_of_nat)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   590
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   591
lemma of_nat_word_less_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   592
  \<open>of_nat m < (of_nat n :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) m < take_bit LENGTH('a) n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   593
  by transfer (simp add: take_bit_of_nat)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   594
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   595
lemma of_nat_word_eq_0_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   596
  \<open>of_nat n = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   597
  using of_nat_word_eq_iff [where ?'a = 'a, of n 0] by (simp add: take_bit_eq_0_iff)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   598
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   599
lemma of_int_word_eq_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   600
  \<open>of_int k = (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   601
  by transfer rule
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   602
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   603
lemma of_int_word_less_eq_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   604
  \<open>of_int k \<le> (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k \<le> take_bit LENGTH('a) l\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   605
  by transfer rule
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   606
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   607
lemma of_int_word_less_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   608
  \<open>of_int k < (of_int l :: 'a::len word) \<longleftrightarrow> take_bit LENGTH('a) k < take_bit LENGTH('a) l\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   609
  by transfer rule
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   610
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   611
lemma of_int_word_eq_0_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   612
  \<open>of_int k = (0 :: 'a::len word) \<longleftrightarrow> 2 ^ LENGTH('a) dvd k\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   613
  using of_int_word_eq_iff [where ?'a = 'a, of k 0] by (simp add: take_bit_eq_0_iff)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   614
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   615
lift_definition word_sle :: \<open>'a::len word \<Rightarrow> 'a word \<Rightarrow> bool\<close>  (\<open>(_/ <=s _)\<close> [50, 51] 50)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   616
  is \<open>\<lambda>k l. signed_take_bit (LENGTH('a) - 1) k \<le> signed_take_bit (LENGTH('a) - 1) l\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   617
  by (simp flip: signed_take_bit_decr_length_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   618
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   619
lemma word_sle_eq [code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   620
  \<open>a <=s b \<longleftrightarrow> sint a \<le> sint b\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   621
  by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   622
  
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   623
lift_definition word_sless :: \<open>'a::len word \<Rightarrow> 'a word \<Rightarrow> bool\<close>  (\<open>(_/ <s _)\<close> [50, 51] 50)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   624
  is \<open>\<lambda>k l. signed_take_bit (LENGTH('a) - 1) k < signed_take_bit (LENGTH('a) - 1) l\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   625
  by (simp flip: signed_take_bit_decr_length_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   626
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   627
lemma word_sless_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   628
  \<open>x <s y \<longleftrightarrow> x <=s y \<and> x \<noteq> y\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   629
  by transfer (simp add: signed_take_bit_decr_length_iff less_le)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   630
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   631
lemma [code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   632
  \<open>a <s b \<longleftrightarrow> sint a < sint b\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   633
  by transfer simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   634
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   635
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   636
subsection \<open>Bit-wise operations\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   637
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   638
lemma word_bit_induct [case_names zero even odd]:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   639
  \<open>P a\<close> if word_zero: \<open>P 0\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   640
    and word_even: \<open>\<And>a. P a \<Longrightarrow> 0 < a \<Longrightarrow> a < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> P (2 * a)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   641
    and word_odd: \<open>\<And>a. P a \<Longrightarrow> a < 2 ^ (LENGTH('a) - 1) \<Longrightarrow> P (1 + 2 * a)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   642
  for P and a :: \<open>'a::len word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   643
proof -
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   644
  define m :: nat where \<open>m = LENGTH('a) - 1\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   645
  then have l: \<open>LENGTH('a) = Suc m\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   646
    by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   647
  define n :: nat where \<open>n = unat a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   648
  then have \<open>n < 2 ^ LENGTH('a)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   649
    by (unfold unat_def) (transfer, simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   650
  then have \<open>n < 2 * 2 ^ m\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   651
    by (simp add: l)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   652
  then have \<open>P (of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   653
  proof (induction n rule: nat_bit_induct)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   654
    case zero
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   655
    show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   656
      by simp (rule word_zero)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   657
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   658
    case (even n)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   659
    then have \<open>n < 2 ^ m\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   660
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   661
    with even.IH have \<open>P (of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   662
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   663
    moreover from \<open>n < 2 ^ m\<close> even.hyps have \<open>0 < (of_nat n :: 'a word)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   664
      by (auto simp add: word_greater_zero_iff of_nat_word_eq_0_iff l)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   665
    moreover from \<open>n < 2 ^ m\<close> have \<open>(of_nat n :: 'a word) < 2 ^ (LENGTH('a) - 1)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   666
      using of_nat_word_less_iff [where ?'a = 'a, of n \<open>2 ^ m\<close>]
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   667
      by (cases \<open>m = 0\<close>) (simp_all add: not_less take_bit_eq_self ac_simps l)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   668
    ultimately have \<open>P (2 * of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   669
      by (rule word_even)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   670
    then show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   671
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   672
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   673
    case (odd n)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   674
    then have \<open>Suc n \<le> 2 ^ m\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   675
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   676
    with odd.IH have \<open>P (of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   677
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   678
    moreover from \<open>Suc n \<le> 2 ^ m\<close> have \<open>(of_nat n :: 'a word) < 2 ^ (LENGTH('a) - 1)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   679
      using of_nat_word_less_iff [where ?'a = 'a, of n \<open>2 ^ m\<close>]
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   680
      by (cases \<open>m = 0\<close>) (simp_all add: not_less take_bit_eq_self ac_simps l)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   681
    ultimately have \<open>P (1 + 2 * of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   682
      by (rule word_odd)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   683
    then show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   684
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   685
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   686
  moreover have \<open>of_nat (nat (uint a)) = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   687
    by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   688
  ultimately show ?thesis
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   689
    by (simp add: n_def)
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   690
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   691
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   692
lemma bit_word_half_eq:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   693
  \<open>(of_bool b + a * 2) div 2 = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   694
    if \<open>a < 2 ^ (LENGTH('a) - Suc 0)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   695
    for a :: \<open>'a::len word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   696
proof (cases \<open>2 \<le> LENGTH('a::len)\<close>)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   697
  case False
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   698
  have \<open>of_bool (odd k) < (1 :: int) \<longleftrightarrow> even k\<close> for k :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   699
    by auto
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   700
  with False that show ?thesis
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   701
    by transfer (simp add: eq_iff)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   702
next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   703
  case True
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   704
  obtain n where length: \<open>LENGTH('a) = Suc n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   705
    by (cases \<open>LENGTH('a)\<close>) simp_all
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   706
  show ?thesis proof (cases b)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   707
    case False
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   708
    moreover have \<open>a * 2 div 2 = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   709
    using that proof transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   710
      fix k :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   711
      from length have \<open>k * 2 mod 2 ^ LENGTH('a) = (k mod 2 ^ n) * 2\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   712
        by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   713
      moreover assume \<open>take_bit LENGTH('a) k < take_bit LENGTH('a) (2 ^ (LENGTH('a) - Suc 0))\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   714
      with \<open>LENGTH('a) = Suc n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   715
      have \<open>k mod 2 ^ LENGTH('a) = k mod 2 ^ n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   716
        by (simp add: take_bit_eq_mod divmod_digit_0)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   717
      ultimately have \<open>take_bit LENGTH('a) (k * 2) = take_bit LENGTH('a) k * 2\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   718
        by (simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   719
      with True show \<open>take_bit LENGTH('a) (take_bit LENGTH('a) (k * 2) div take_bit LENGTH('a) 2)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   720
        = take_bit LENGTH('a) k\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   721
        by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   722
    qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   723
    ultimately show ?thesis
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   724
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   725
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   726
    case True
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   727
    moreover have \<open>(1 + a * 2) div 2 = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   728
    using that proof transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   729
      fix k :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   730
      from length have \<open>(1 + k * 2) mod 2 ^ LENGTH('a) = 1 + (k mod 2 ^ n) * 2\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   731
        using pos_zmod_mult_2 [of \<open>2 ^ n\<close> k] by (simp add: ac_simps)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   732
      moreover assume \<open>take_bit LENGTH('a) k < take_bit LENGTH('a) (2 ^ (LENGTH('a) - Suc 0))\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   733
      with \<open>LENGTH('a) = Suc n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   734
      have \<open>k mod 2 ^ LENGTH('a) = k mod 2 ^ n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   735
        by (simp add: take_bit_eq_mod divmod_digit_0)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   736
      ultimately have \<open>take_bit LENGTH('a) (1 + k * 2) = 1 + take_bit LENGTH('a) k * 2\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   737
        by (simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   738
      with True show \<open>take_bit LENGTH('a) (take_bit LENGTH('a) (1 + k * 2) div take_bit LENGTH('a) 2)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   739
        = take_bit LENGTH('a) k\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   740
        by (auto simp add: take_bit_Suc)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   741
    qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   742
    ultimately show ?thesis
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   743
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   744
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   745
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   746
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   747
lemma even_mult_exp_div_word_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   748
  \<open>even (a * 2 ^ m div 2 ^ n) \<longleftrightarrow> \<not> (
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   749
    m \<le> n \<and>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   750
    n < LENGTH('a) \<and> odd (a div 2 ^ (n - m)))\<close> for a :: \<open>'a::len word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   751
  by transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   752
    (auto simp flip: drop_bit_eq_div simp add: even_drop_bit_iff_not_bit bit_take_bit_iff,
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   753
      simp_all flip: push_bit_eq_mult add: bit_push_bit_iff_int)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   754
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   755
instantiation word :: (len) semiring_bits
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   756
begin
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   757
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   758
lift_definition bit_word :: \<open>'a word \<Rightarrow> nat \<Rightarrow> bool\<close>
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   759
  is \<open>\<lambda>k n. n < LENGTH('a) \<and> bit k n\<close>
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   760
proof
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   761
  fix k l :: int and n :: nat
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   762
  assume *: \<open>take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   763
  show \<open>n < LENGTH('a) \<and> bit k n \<longleftrightarrow> n < LENGTH('a) \<and> bit l n\<close>
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   764
  proof (cases \<open>n < LENGTH('a)\<close>)
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   765
    case True
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   766
    from * have \<open>bit (take_bit LENGTH('a) k) n \<longleftrightarrow> bit (take_bit LENGTH('a) l) n\<close>
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   767
      by simp
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   768
    then show ?thesis
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   769
      by (simp add: bit_take_bit_iff)
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   770
  next
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   771
    case False
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   772
    then show ?thesis
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   773
      by simp
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   774
  qed
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   775
qed
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   776
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   777
instance proof
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   778
  show \<open>P a\<close> if stable: \<open>\<And>a. a div 2 = a \<Longrightarrow> P a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   779
    and rec: \<open>\<And>a b. P a \<Longrightarrow> (of_bool b + 2 * a) div 2 = a \<Longrightarrow> P (of_bool b + 2 * a)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   780
  for P and a :: \<open>'a word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   781
  proof (induction a rule: word_bit_induct)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   782
    case zero
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   783
    have \<open>0 div 2 = (0::'a word)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   784
      by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   785
    with stable [of 0] show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   786
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   787
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   788
    case (even a)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   789
    with rec [of a False] show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   790
      using bit_word_half_eq [of a False] by (simp add: ac_simps)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   791
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   792
    case (odd a)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   793
    with rec [of a True] show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   794
      using bit_word_half_eq [of a True] by (simp add: ac_simps)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   795
  qed
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   796
  show \<open>bit a n \<longleftrightarrow> odd (a div 2 ^ n)\<close> for a :: \<open>'a word\<close> and n
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   797
    by transfer (simp flip: drop_bit_eq_div add: drop_bit_take_bit bit_iff_odd_drop_bit)
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   798
  show \<open>0 div a = 0\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   799
    for a :: \<open>'a word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   800
    by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   801
  show \<open>a div 1 = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   802
    for a :: \<open>'a word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   803
    by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   804
  show \<open>a mod b div b = 0\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   805
    for a b :: \<open>'a word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   806
    apply transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   807
    apply (simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   808
    apply (subst (3) mod_pos_pos_trivial [of _ \<open>2 ^ LENGTH('a)\<close>])
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   809
      apply simp_all
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   810
     apply (metis le_less mod_by_0 pos_mod_conj zero_less_numeral zero_less_power)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   811
    using pos_mod_bound [of \<open>2 ^ LENGTH('a)\<close>] apply simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   812
  proof -
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   813
    fix aa :: int and ba :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   814
    have f1: "\<And>i n. (i::int) mod 2 ^ n = 0 \<or> 0 < i mod 2 ^ n"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   815
      by (metis le_less take_bit_eq_mod take_bit_nonnegative)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   816
    have "(0::int) < 2 ^ len_of (TYPE('a)::'a itself) \<and> ba mod 2 ^ len_of (TYPE('a)::'a itself) \<noteq> 0 \<or> aa mod 2 ^ len_of (TYPE('a)::'a itself) mod (ba mod 2 ^ len_of (TYPE('a)::'a itself)) < 2 ^ len_of (TYPE('a)::'a itself)"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   817
      by (metis (no_types) mod_by_0 unique_euclidean_semiring_numeral_class.pos_mod_bound zero_less_numeral zero_less_power)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   818
    then show "aa mod 2 ^ len_of (TYPE('a)::'a itself) mod (ba mod 2 ^ len_of (TYPE('a)::'a itself)) < 2 ^ len_of (TYPE('a)::'a itself)"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   819
      using f1 by (meson le_less less_le_trans unique_euclidean_semiring_numeral_class.pos_mod_bound)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   820
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   821
  show \<open>(1 + a) div 2 = a div 2\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   822
    if \<open>even a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   823
    for a :: \<open>'a word\<close>
71953
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   824
    using that by transfer
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   825
      (auto dest: le_Suc_ex simp add: mod_2_eq_odd take_bit_Suc elim!: evenE)
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   826
  show \<open>(2 :: 'a word) ^ m div 2 ^ n = of_bool ((2 :: 'a word) ^ m \<noteq> 0 \<and> n \<le> m) * 2 ^ (m - n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   827
    for m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   828
    by transfer (simp, simp add: exp_div_exp_eq)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   829
  show "a div 2 ^ m div 2 ^ n = a div 2 ^ (m + n)"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   830
    for a :: "'a word" and m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   831
    apply transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   832
    apply (auto simp add: not_less take_bit_drop_bit ac_simps simp flip: drop_bit_eq_div)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   833
    apply (simp add: drop_bit_take_bit)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   834
    done
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   835
  show "a mod 2 ^ m mod 2 ^ n = a mod 2 ^ min m n"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   836
    for a :: "'a word" and m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   837
    by transfer (auto simp flip: take_bit_eq_mod simp add: ac_simps)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   838
  show \<open>a * 2 ^ m mod 2 ^ n = a mod 2 ^ (n - m) * 2 ^ m\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   839
    if \<open>m \<le> n\<close> for a :: "'a word" and m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   840
    using that apply transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   841
    apply (auto simp flip: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   842
           apply (auto simp flip: push_bit_eq_mult simp add: push_bit_take_bit split: split_min_lin)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   843
    done
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   844
  show \<open>a div 2 ^ n mod 2 ^ m = a mod (2 ^ (n + m)) div 2 ^ n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   845
    for a :: "'a word" and m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   846
    by transfer (auto simp add: not_less take_bit_drop_bit ac_simps simp flip: take_bit_eq_mod drop_bit_eq_div split: split_min_lin)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   847
  show \<open>even ((2 ^ m - 1) div (2::'a word) ^ n) \<longleftrightarrow> 2 ^ n = (0::'a word) \<or> m \<le> n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   848
    for m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   849
    by transfer (auto simp add: take_bit_of_mask even_mask_div_iff)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   850
  show \<open>even (a * 2 ^ m div 2 ^ n) \<longleftrightarrow> n < m \<or> (2::'a word) ^ n = 0 \<or> m \<le> n \<and> even (a div 2 ^ (n - m))\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   851
    for a :: \<open>'a word\<close> and m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   852
  proof transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   853
    show \<open>even (take_bit LENGTH('a) (k * 2 ^ m) div take_bit LENGTH('a) (2 ^ n)) \<longleftrightarrow>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   854
      n < m
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   855
      \<or> take_bit LENGTH('a) ((2::int) ^ n) = take_bit LENGTH('a) 0
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   856
      \<or> (m \<le> n \<and> even (take_bit LENGTH('a) k div take_bit LENGTH('a) (2 ^ (n - m))))\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   857
    for m n :: nat and k l :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   858
      by (auto simp flip: take_bit_eq_mod drop_bit_eq_div push_bit_eq_mult
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   859
        simp add: div_push_bit_of_1_eq_drop_bit drop_bit_take_bit drop_bit_push_bit_int [of n m])
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   860
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   861
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   862
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   863
end
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   864
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   865
instantiation word :: (len) semiring_bit_shifts
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   866
begin
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   867
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   868
lift_definition push_bit_word :: \<open>nat \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   869
  is push_bit
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   870
proof -
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   871
  show \<open>take_bit LENGTH('a) (push_bit n k) = take_bit LENGTH('a) (push_bit n l)\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   872
    if \<open>take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close> for k l :: int and n :: nat
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   873
  proof -
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   874
    from that
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   875
    have \<open>take_bit (LENGTH('a) - n) (take_bit LENGTH('a) k)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   876
      = take_bit (LENGTH('a) - n) (take_bit LENGTH('a) l)\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   877
      by simp
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   878
    moreover have \<open>min (LENGTH('a) - n) LENGTH('a) = LENGTH('a) - n\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   879
      by simp
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   880
    ultimately show ?thesis
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   881
      by (simp add: take_bit_push_bit)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   882
  qed
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   883
qed
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   884
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   885
lift_definition drop_bit_word :: \<open>nat \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   886
  is \<open>\<lambda>n. drop_bit n \<circ> take_bit LENGTH('a)\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   887
  by (simp add: take_bit_eq_mod)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   888
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   889
lift_definition take_bit_word :: \<open>nat \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   890
  is \<open>\<lambda>n. take_bit (min LENGTH('a) n)\<close>
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   891
  by (simp add: ac_simps) (simp only: flip: take_bit_take_bit)
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   892
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   893
instance proof
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   894
  show \<open>push_bit n a = a * 2 ^ n\<close> for n :: nat and a :: \<open>'a word\<close>
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   895
    by transfer (simp add: push_bit_eq_mult)
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   896
  show \<open>drop_bit n a = a div 2 ^ n\<close> for n :: nat and a :: \<open>'a word\<close>
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   897
    by transfer (simp flip: drop_bit_eq_div add: drop_bit_take_bit)
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   898
  show \<open>take_bit n a = a mod 2 ^ n\<close> for n :: nat and a :: \<open>'a word\<close>
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
   899
    by transfer (auto simp flip: take_bit_eq_mod)
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   900
qed
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   901
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   902
end
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   903
71958
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   904
lemma bit_word_eqI:
4320875eb8a1 more lemmas
haftmann
parents: 71957
diff changeset
   905
  \<open>a = b\<close> if \<open>\<And>n. n \<le> LENGTH('a) \<Longrightarrow> bit a n \<longleftrightarrow> bit b n\<close>
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   906
  for a b :: \<open>'a::len word\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   907
  using that by transfer (auto simp add: nat_less_le bit_eq_iff bit_take_bit_iff)
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   908
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   909
lemma bit_imp_le_length:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   910
  \<open>n < LENGTH('a)\<close> if \<open>bit w n\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   911
    for w :: \<open>'a::len word\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   912
  using that by transfer simp
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   913
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   914
lemma not_bit_length [simp]:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   915
  \<open>\<not> bit w LENGTH('a)\<close> for w :: \<open>'a::len word\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   916
  by transfer simp
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   917
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   918
lemma uint_take_bit_eq [code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   919
  \<open>uint (take_bit n w) = take_bit n (uint w)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   920
  by transfer (simp add: ac_simps)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   921
72027
759532ef0885 prefer canonically oriented lists of bits and more direct characterizations in definitions
haftmann
parents: 72010
diff changeset
   922
lemma take_bit_length_eq [simp]:
759532ef0885 prefer canonically oriented lists of bits and more direct characterizations in definitions
haftmann
parents: 72010
diff changeset
   923
  \<open>take_bit LENGTH('a) w = w\<close> for w :: \<open>'a::len word\<close>
759532ef0885 prefer canonically oriented lists of bits and more direct characterizations in definitions
haftmann
parents: 72010
diff changeset
   924
  by transfer simp
759532ef0885 prefer canonically oriented lists of bits and more direct characterizations in definitions
haftmann
parents: 72010
diff changeset
   925
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   926
lemma bit_word_of_int_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   927
  \<open>bit (word_of_int k :: 'a::len word) n \<longleftrightarrow> n < LENGTH('a) \<and> bit k n\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   928
  by transfer rule
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   929
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   930
lemma bit_uint_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   931
  \<open>bit (uint w) n \<longleftrightarrow> n < LENGTH('a) \<and> bit w n\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   932
    for w :: \<open>'a::len word\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   933
  by transfer (simp add: bit_take_bit_iff)
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   934
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   935
lemma bit_sint_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   936
  \<open>bit (sint w) n \<longleftrightarrow> n \<ge> LENGTH('a) \<and> bit w (LENGTH('a) - 1) \<or> bit w n\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   937
  for w :: \<open>'a::len word\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   938
  by transfer (auto simp add: bit_signed_take_bit_iff min_def le_less not_less)
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   939
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   940
lemma bit_word_ucast_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   941
  \<open>bit (ucast w :: 'b::len word) n \<longleftrightarrow> n < LENGTH('a) \<and> n < LENGTH('b) \<and> bit w n\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   942
  for w :: \<open>'a::len word\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   943
  by transfer (simp add: bit_take_bit_iff ac_simps)
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   944
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   945
lemma bit_word_scast_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   946
  \<open>bit (scast w :: 'b::len word) n \<longleftrightarrow>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   947
    n < LENGTH('b) \<and> (bit w n \<or> LENGTH('a) \<le> n \<and> bit w (LENGTH('a) - Suc 0))\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   948
  for w :: \<open>'a::len word\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   949
  by transfer (auto simp add: bit_signed_take_bit_iff le_less min_def)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   950
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   951
lift_definition shiftl1 :: \<open>'a::len word \<Rightarrow> 'a word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   952
  is \<open>(*) 2\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   953
  by (auto simp add: take_bit_eq_mod intro: mod_mult_cong)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   954
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   955
lemma shiftl1_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   956
  \<open>shiftl1 w = word_of_int (2 * uint w)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   957
  by transfer (simp add: take_bit_eq_mod mod_simps)
70191
bdc835d934b7 no need to maintain two separate type classes
haftmann
parents: 70190
diff changeset
   958
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   959
lemma shiftl1_eq_mult_2:
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   960
  \<open>shiftl1 = (*) 2\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   961
  by (rule ext, transfer) simp
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   962
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   963
lemma bit_shiftl1_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   964
  \<open>bit (shiftl1 w) n \<longleftrightarrow> 0 < n \<and> n < LENGTH('a) \<and> bit w (n - 1)\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   965
    for w :: \<open>'a::len word\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   966
  by (simp add: shiftl1_eq_mult_2 bit_double_iff exp_eq_zero_iff not_le) (simp add: ac_simps)
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   967
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   968
lift_definition shiftr1 :: \<open>'a::len word \<Rightarrow> 'a word\<close>
70191
bdc835d934b7 no need to maintain two separate type classes
haftmann
parents: 70190
diff changeset
   969
  \<comment> \<open>shift right as unsigned or as signed, ie logical or arithmetic\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   970
  is \<open>\<lambda>k. take_bit LENGTH('a) k div 2\<close> by simp
70191
bdc835d934b7 no need to maintain two separate type classes
haftmann
parents: 70190
diff changeset
   971
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   972
lemma shiftr1_eq_div_2:
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   973
  \<open>shiftr1 w = w div 2\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   974
  by transfer simp
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   975
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   976
lemma bit_shiftr1_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   977
  \<open>bit (shiftr1 w) n \<longleftrightarrow> bit w (Suc n)\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   978
  by transfer (auto simp flip: bit_Suc simp add: bit_take_bit_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   979
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   980
lemma shiftr1_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   981
  \<open>shiftr1 w = word_of_int (bin_rest (uint w))\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
   982
  by transfer simp
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
   983
71957
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   984
instantiation word :: (len) ring_bit_operations
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   985
begin
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   986
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   987
lift_definition not_word :: \<open>'a word \<Rightarrow> 'a word\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   988
  is not
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   989
  by (simp add: take_bit_not_iff)
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   990
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   991
lift_definition and_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   992
  is \<open>and\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   993
  by simp
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   994
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   995
lift_definition or_word :: \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   996
  is or
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   997
  by simp
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   998
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
   999
lift_definition xor_word ::  \<open>'a word \<Rightarrow> 'a word \<Rightarrow> 'a word\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1000
  is xor
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1001
  by simp
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1002
72082
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1003
lift_definition mask_word :: \<open>nat \<Rightarrow> 'a word\<close>
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1004
  is mask
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1005
  .
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1006
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1007
instance by (standard; transfer)
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1008
  (auto simp add: minus_eq_not_minus_1 mask_eq_exp_minus_1
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1009
    bit_not_iff bit_and_iff bit_or_iff bit_xor_iff)
71957
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1010
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1011
end
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1012
72009
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1013
context
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1014
  includes lifting_syntax
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1015
begin
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1016
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1017
lemma set_bit_word_transfer [transfer_rule]:
72009
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1018
  \<open>((=) ===> pcr_word ===> pcr_word) set_bit set_bit\<close>
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1019
  by (unfold set_bit_def) transfer_prover
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1020
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1021
lemma unset_bit_word_transfer [transfer_rule]:
72009
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1022
  \<open>((=) ===> pcr_word ===> pcr_word) unset_bit unset_bit\<close>
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1023
  by (unfold unset_bit_def) transfer_prover
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1024
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1025
lemma flip_bit_word_transfer [transfer_rule]:
72009
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1026
  \<open>((=) ===> pcr_word ===> pcr_word) flip_bit flip_bit\<close>
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1027
  by (unfold flip_bit_def) transfer_prover
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1028
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1029
end
febdd4eead56 more on single-bit operations
haftmann
parents: 72000
diff changeset
  1030
72000
379d0c207c29 separation of traditional bit operations
haftmann
parents: 71997
diff changeset
  1031
instantiation word :: (len) semiring_bit_syntax
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1032
begin
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1033
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1034
lift_definition test_bit_word :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> bool\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1035
  is \<open>\<lambda>k n. n < LENGTH('a) \<and> bit k n\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1036
proof
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1037
  fix k l :: int and n :: nat
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1038
  assume *: \<open>take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1039
  show \<open>n < LENGTH('a) \<and> bit k n \<longleftrightarrow> n < LENGTH('a) \<and> bit l n\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1040
  proof (cases \<open>n < LENGTH('a)\<close>)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1041
    case True
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1042
    from * have \<open>bit (take_bit LENGTH('a) k) n \<longleftrightarrow> bit (take_bit LENGTH('a) l) n\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1043
      by simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1044
    then show ?thesis
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1045
      by (simp add: bit_take_bit_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1046
  next
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1047
    case False
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1048
    then show ?thesis
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1049
      by simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1050
  qed
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1051
qed
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1052
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
  1053
lemma test_bit_word_eq:
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1054
  \<open>test_bit = (bit :: 'a word \<Rightarrow> _)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1055
  by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1056
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1057
lemma [code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1058
  \<open>bit w n \<longleftrightarrow> w AND push_bit n 1 \<noteq> 0\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1059
  for w :: \<open>'a::len word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1060
  apply (simp add: bit_eq_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1061
  apply (auto simp add: bit_and_iff bit_push_bit_iff bit_1_iff exp_eq_0_imp_not_bit)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1062
  done
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1063
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1064
lemma [code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1065
  \<open>test_bit w n \<longleftrightarrow> w AND push_bit n 1 \<noteq> 0\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1066
  for w :: \<open>'a::len word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1067
  apply (simp add: test_bit_word_eq bit_eq_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1068
  apply (auto simp add: bit_and_iff bit_push_bit_iff bit_1_iff exp_eq_0_imp_not_bit)
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
  1069
  done
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
  1070
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1071
lift_definition shiftl_word :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1072
  is \<open>\<lambda>k n. push_bit n k\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1073
proof -
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1074
  show \<open>take_bit LENGTH('a) (push_bit n k) = take_bit LENGTH('a) (push_bit n l)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1075
    if \<open>take_bit LENGTH('a) k = take_bit LENGTH('a) l\<close> for k l :: int and n :: nat
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1076
  proof -
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1077
    from that
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1078
    have \<open>take_bit (LENGTH('a) - n) (take_bit LENGTH('a) k)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1079
      = take_bit (LENGTH('a) - n) (take_bit LENGTH('a) l)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1080
      by simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1081
    moreover have \<open>min (LENGTH('a) - n) LENGTH('a) = LENGTH('a) - n\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1082
      by simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1083
    ultimately show ?thesis
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1084
      by (simp add: take_bit_push_bit)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1085
  qed
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1086
qed
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1087
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
  1088
lemma shiftl_word_eq:
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
  1089
  \<open>w << n = push_bit n w\<close> for w :: \<open>'a::len word\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1090
  by transfer rule
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1091
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1092
lift_definition shiftr_word :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1093
  is \<open>\<lambda>k n. drop_bit n (take_bit LENGTH('a) k)\<close> by simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1094
  
72000
379d0c207c29 separation of traditional bit operations
haftmann
parents: 71997
diff changeset
  1095
lemma shiftr_word_eq:
379d0c207c29 separation of traditional bit operations
haftmann
parents: 71997
diff changeset
  1096
  \<open>w >> n = drop_bit n w\<close> for w :: \<open>'a::len word\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1097
  by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1098
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1099
instance
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1100
  by (standard; transfer) simp_all
72000
379d0c207c29 separation of traditional bit operations
haftmann
parents: 71997
diff changeset
  1101
379d0c207c29 separation of traditional bit operations
haftmann
parents: 71997
diff changeset
  1102
end
379d0c207c29 separation of traditional bit operations
haftmann
parents: 71997
diff changeset
  1103
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1104
lemma shiftl_code [code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1105
  \<open>w << n = w * 2 ^ n\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1106
  for w :: \<open>'a::len word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1107
  by transfer (simp add: push_bit_eq_mult)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1108
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1109
lemma shiftl1_code [code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1110
  \<open>shiftl1 w = w << 1\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1111
  by transfer (simp add: push_bit_eq_mult ac_simps)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1112
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1113
lemma uint_shiftr_eq [code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1114
  \<open>uint (w >> n) = uint w div 2 ^ n\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1115
  for w :: \<open>'a::len word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1116
  by transfer (simp flip: drop_bit_eq_div add: drop_bit_take_bit min_def le_less less_diff_conv)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1117
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1118
lemma shiftr1_code [code]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1119
  \<open>shiftr1 w = w >> 1\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1120
  by transfer (simp add: drop_bit_Suc)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1121
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1122
lemma word_test_bit_def: 
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1123
  \<open>test_bit a = bin_nth (uint a)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1124
  by transfer (simp add: fun_eq_iff bit_take_bit_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1125
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1126
lemma shiftl_def:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1127
  \<open>w << n = (shiftl1 ^^ n) w\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1128
proof -
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1129
  have \<open>push_bit n = (((*) 2 ^^ n) :: int \<Rightarrow> int)\<close> for n
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1130
    by (induction n) (simp_all add: fun_eq_iff funpow_swap1, simp add: ac_simps)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1131
  then show ?thesis
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1132
    by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1133
qed
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1134
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1135
lemma shiftr_def:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1136
  \<open>w >> n = (shiftr1 ^^ n) w\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1137
proof -
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1138
  have \<open>drop_bit n = (((\<lambda>k::int. k div 2) ^^ n))\<close> for n
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1139
    by (rule sym, induction n)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1140
       (simp_all add: fun_eq_iff drop_bit_Suc flip: drop_bit_half)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1141
  then show ?thesis
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1142
    apply transfer
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1143
    apply simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1144
    apply (metis bintrunc_bintrunc rco_bintr)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1145
    done
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1146
qed
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1147
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1148
lemma bit_shiftl_word_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1149
  \<open>bit (w << m) n \<longleftrightarrow> m \<le> n \<and> n < LENGTH('a) \<and> bit w (n - m)\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1150
  for w :: \<open>'a::len word\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1151
  by (simp add: shiftl_word_eq bit_push_bit_iff exp_eq_zero_iff not_le)
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1152
71955
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1153
lemma [code]:
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1154
  \<open>push_bit n w = w << n\<close> for w :: \<open>'a::len word\<close>
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1155
  by (simp add: shiftl_word_eq)
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1156
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1157
lemma bit_shiftr_word_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1158
  \<open>bit (w >> m) n \<longleftrightarrow> bit w (m + n)\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1159
  for w :: \<open>'a::len word\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1160
  by (simp add: shiftr_word_eq bit_drop_bit_eq)
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1161
71955
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1162
lemma [code]:
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1163
  \<open>drop_bit n w = w >> n\<close> for w :: \<open>'a::len word\<close>
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1164
  by (simp add: shiftr_word_eq)
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1165
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
  1166
lemma [code]:
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1167
  \<open>take_bit n a = a AND mask n\<close> for a :: \<open>'a::len word\<close>
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
  1168
  by (fact take_bit_eq_mask)
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
  1169
72082
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1170
lemma [code]:
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1171
  \<open>mask (Suc n) = push_bit n (1 :: 'a word) OR mask n\<close>
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1172
  \<open>mask 0 = (0 :: 'a::len word)\<close>
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1173
  by (simp_all add: mask_Suc_exp push_bit_of_1)
41393ecb57ac uniform mask operation
haftmann
parents: 72079
diff changeset
  1174
71955
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1175
lemma [code_abbrev]:
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1176
  \<open>push_bit n 1 = (2 :: 'a::len word) ^ n\<close>
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1177
  by (fact push_bit_of_1)
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
  1178
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1179
lemma
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1180
  word_not_def [code]: "NOT (a::'a::len word) = word_of_int (NOT (uint a))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1181
    and word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1182
    and word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1183
    and word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
71957
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1184
  by (transfer, simp add: take_bit_not_take_bit)+
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
  1185
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1186
lemma [code abstract]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1187
  \<open>uint (v AND w) = uint v AND uint w\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1188
  by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1189
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1190
lemma [code abstract]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1191
  \<open>uint (v OR w) = uint v OR uint w\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1192
  by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1193
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1194
lemma [code abstract]:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1195
  \<open>uint (v XOR w) = uint v XOR uint w\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1196
  by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1197
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1198
lift_definition setBit :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1199
  is \<open>\<lambda>k n. set_bit n k\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1200
  by (simp add: take_bit_set_bit_eq)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1201
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1202
lemma set_Bit_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1203
  \<open>setBit w n = set_bit n w\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1204
  by transfer simp
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1205
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1206
lemma bit_setBit_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1207
  \<open>bit (setBit w m) n \<longleftrightarrow> (m = n \<and> n < LENGTH('a) \<or> bit w n)\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1208
  for w :: \<open>'a::len word\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1209
  by transfer (auto simp add: bit_set_bit_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1210
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1211
lift_definition clearBit :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1212
  is \<open>\<lambda>k n. unset_bit n k\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1213
  by (simp add: take_bit_unset_bit_eq)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1214
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1215
lemma clear_Bit_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1216
  \<open>clearBit w n = unset_bit n w\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1217
  by transfer simp
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1218
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1219
lemma bit_clearBit_iff:
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1220
  \<open>bit (clearBit w m) n \<longleftrightarrow> m \<noteq> n \<and> bit w n\<close>
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1221
  for w :: \<open>'a::len word\<close>
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1222
  by transfer (auto simp add: bit_unset_bit_iff)
71990
66beb9d92e43 explicit proofs for bit projections
haftmann
parents: 71986
diff changeset
  1223
71957
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1224
definition even_word :: \<open>'a::len word \<Rightarrow> bool\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1225
  where [code_abbrev]: \<open>even_word = even\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1226
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1227
lemma even_word_iff [code]:
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1228
  \<open>even_word a \<longleftrightarrow> a AND 1 = 0\<close>
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1229
  by (simp add: and_one_eq even_iff_mod_2_eq_zero even_word_def)
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1230
71965
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
  1231
lemma bit_word_iff_drop_bit_and [code]:
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
  1232
  \<open>bit a n \<longleftrightarrow> drop_bit n a AND 1 = 1\<close> for a :: \<open>'a::len word\<close>
d45f5d4c41bd more class operations for the sake of efficient generated code
haftmann
parents: 71958
diff changeset
  1233
  by (simp add: bit_iff_odd_drop_bit odd_iff_mod_2_eq_one and_one_eq)
71957
3e162c63371a build bit operations on word on library theory on bit operations
haftmann
parents: 71955
diff changeset
  1234
72079
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1235
lemma map_bit_range_eq_if_take_bit_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1236
  \<open>map (bit k) [0..<n] = map (bit l) [0..<n]\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1237
  if \<open>take_bit n k = take_bit n l\<close> for k l :: int
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1238
using that proof (induction n arbitrary: k l)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1239
  case 0
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1240
  then show ?case
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1241
    by simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1242
next
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1243
  case (Suc n)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1244
  from Suc.prems have \<open>take_bit n (k div 2) = take_bit n (l div 2)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1245
    by (simp add: take_bit_Suc)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1246
  then have \<open>map (bit (k div 2)) [0..<n] = map (bit (l div 2)) [0..<n]\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1247
    by (rule Suc.IH)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1248
  moreover have \<open>bit (r div 2) = bit r \<circ> Suc\<close> for r :: int
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1249
    by (simp add: fun_eq_iff bit_Suc)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1250
  moreover from Suc.prems have \<open>even k \<longleftrightarrow> even l\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1251
    by (auto simp add: take_bit_Suc elim!: evenE oddE) arith+
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1252
  ultimately show ?case
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1253
    by (simp only: map_Suc_upt upt_conv_Cons flip: list.map_comp) simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1254
qed
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1255
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1256
lemma bit_of_bl_iff:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1257
  \<open>bit (of_bl bs :: 'a word) n \<longleftrightarrow> rev bs ! n \<and> n < LENGTH('a::len) \<and> n < length bs\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1258
  by (auto simp add: of_bl_def bit_word_of_int_iff bin_nth_of_bl)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1259
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1260
lemma rev_to_bl_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1261
  \<open>rev (to_bl w) = map (bit w) [0..<LENGTH('a)]\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1262
  for w :: \<open>'a::len word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1263
  apply (rule nth_equalityI)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1264
   apply (simp add: to_bl.rep_eq)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1265
  apply (simp add: bin_nth_bl bit_word.rep_eq to_bl.rep_eq)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1266
  done
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1267
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1268
lemma of_bl_rev_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1269
  \<open>of_bl (rev bs) = horner_sum of_bool 2 bs\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1270
  apply (rule bit_word_eqI)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1271
  apply (simp add: bit_of_bl_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1272
  apply transfer
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1273
  apply (simp add: bit_horner_sum_bit_iff ac_simps)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1274
  done
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1275
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1276
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1277
subsection \<open>More shift operations\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1278
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1279
lift_definition sshiftr1 :: \<open>'a::len word \<Rightarrow> 'a word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1280
  is \<open>\<lambda>k. take_bit LENGTH('a) (signed_take_bit (LENGTH('a) - 1) k div 2)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1281
  by (simp flip: signed_take_bit_decr_length_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1282
 
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1283
lift_definition sshiftr :: \<open>'a::len word \<Rightarrow> nat \<Rightarrow> 'a word\<close>  (infixl \<open>>>>\<close> 55)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1284
  is \<open>\<lambda>k n. take_bit LENGTH('a) (drop_bit n (signed_take_bit (LENGTH('a) - 1) k))\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1285
  by (simp flip: signed_take_bit_decr_length_iff)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1286
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1287
lift_definition bshiftr1 :: \<open>bool \<Rightarrow> 'a::len word \<Rightarrow> 'a word\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1288
  is \<open>\<lambda>b k. take_bit LENGTH('a) k div 2 + of_bool b * 2 ^ (LENGTH('a) - Suc 0)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1289
  by (fact arg_cong)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1290
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1291
lemma sshiftr1_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1292
  \<open>sshiftr1 w = word_of_int (bin_rest (sint w))\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1293
  by transfer simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1294
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1295
lemma bshiftr1_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1296
  \<open>bshiftr1 b w = of_bl (b # butlast (to_bl w))\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1297
  apply transfer
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1298
  apply auto
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1299
   apply (subst bl_to_bin_app_cat [of \<open>[True]\<close>, simplified])
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1300
   apply simp
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1301
   apply (metis One_nat_def add.commute bin_bl_bin bin_last_bl_to_bin bin_rest_bl_to_bin butlast_bin_rest concat_bit_eq last.simps list.distinct(1) list.size(3) list.size(4) odd_iff_mod_2_eq_one plus_1_eq_Suc power_Suc0_right push_bit_of_1 size_bin_to_bl take_bit_eq_mod trunc_bl2bin_len)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1302
  apply (simp add: butlast_rest_bl2bin)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1303
  done
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1304
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1305
lemma sshiftr_eq:
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1306
  \<open>w >>> n = (sshiftr1 ^^ n) w\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1307
proof -
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1308
  have *: \<open>(\<lambda>k. take_bit LENGTH('a) (signed_take_bit (LENGTH('a) - Suc 0) k div 2)) ^^ Suc n =
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1309
    take_bit LENGTH('a) \<circ> drop_bit (Suc n) \<circ> signed_take_bit (LENGTH('a) - Suc 0)\<close>
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1310
    for n
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1311
    apply (induction n)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1312
     apply (auto simp add: fun_eq_iff drop_bit_Suc)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1313
    apply (metis (no_types, lifting) Suc_pred funpow_swap1 len_gt_0 sbintrunc_bintrunc sbintrunc_rest)
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1314
    done
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1315
  show ?thesis
8c355e2dd7db more consequent transferability
haftmann
parents: 72043
diff changeset
  1316
    apply transfer