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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  Bits_Int
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  Bits_Z2
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  Bit_Comprehension
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  Misc_Typedef
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  Misc_Arithmetic
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begin
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text \<open>See \<^file>\<open>Word_Examples.thy\<close> for examples.\<close>
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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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definition sint :: "'a::len word \<Rightarrow> int"
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  \<comment> \<open>treats the most-significant-bit as a sign bit\<close>
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  where sint_uint: "sint w = sbintrunc (LENGTH('a) - 1) (uint w)"
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definition unat :: "'a::len word \<Rightarrow> nat"
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  where "unat w = nat (uint w)"
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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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definition norm_sint :: "nat \<Rightarrow> int \<Rightarrow> int"
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  where "norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)"
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definition scast :: "'a::len word \<Rightarrow> 'b::len word"
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  \<comment> \<open>cast a word to a different length\<close>
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  where "scast w = word_of_int (sint w)"
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definition ucast :: "'a::len word \<Rightarrow> 'b::len word"
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  where "ucast w = word_of_int (uint w)"
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instantiation word :: (len) size
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begin
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definition word_size: "size (w :: 'a word) = LENGTH('a)"
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instance ..
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end
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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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definition source_size :: "('a::len word \<Rightarrow> 'b) \<Rightarrow> nat"
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  \<comment> \<open>whether a cast (or other) function is to a longer or shorter length\<close>
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  where [code del]: "source_size c = (let arb = undefined; x = c arb in size arb)"
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definition target_size :: "('a \<Rightarrow> 'b::len word) \<Rightarrow> nat"
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  where [code del]: "target_size c = size (c undefined)"
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definition is_up :: "('a::len word \<Rightarrow> 'b::len word) \<Rightarrow> bool"
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  where "is_up c \<longleftrightarrow> source_size c \<le> target_size c"
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definition is_down :: "('a::len word \<Rightarrow> 'b::len word) \<Rightarrow> bool"
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  where "is_down c \<longleftrightarrow> target_size c \<le> source_size c"
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definition of_bl :: "bool list \<Rightarrow> 'a::len word"
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  where "of_bl bl = word_of_int (bl_to_bin bl)"
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definition to_bl :: "'a::len word \<Rightarrow> bool list"
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  where "to_bl w = bin_to_bl (LENGTH('a)) (uint w)"
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definition word_reverse :: "'a::len word \<Rightarrow> 'a word"
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  where "word_reverse w = of_bl (rev (to_bl w))"
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definition word_int_case :: "(int \<Rightarrow> 'b) \<Rightarrow> 'a::len word \<Rightarrow> 'b"
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  where "word_int_case f w = f (uint w)"
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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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definition Word :: "int \<Rightarrow> 'a::len word"
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  where [code_post]: "Word = word_of_int"
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lemma [code abstype]: "Word (uint w) = w"
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  by (simp add: Word_def word_of_int_uint)
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declare uint_word_of_int [code abstract]
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instantiation word :: (len) equal
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begin
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definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
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  where "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"
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instance
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  by standard (simp add: equal equal_word_def word_uint_eq_iff)
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end
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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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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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lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus
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  by (auto simp add: bintrunc_mod2p intro: mod_minus_cong)
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lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "(*)"
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  by (auto simp add: bintrunc_mod2p intro: mod_mult_cong)
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lift_definition divide_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
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  is "\<lambda>a b. take_bit LENGTH('a) a div take_bit LENGTH('a) b"
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  by simp
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lift_definition modulo_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word"
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  is "\<lambda>a b. take_bit LENGTH('a) a mod take_bit LENGTH('a) b"
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  by simp
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instance
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  by standard (transfer, simp add: algebra_simps)+
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end
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lemma word_div_def [code]:
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  "a div b = word_of_int (uint a div uint b)"
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  by transfer rule
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lemma word_mod_def [code]:
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  "a mod b = word_of_int (uint a mod uint b)"
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  by transfer rule
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quickcheck_generator word
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  constructors:
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    "zero_class.zero :: ('a::len) word",
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    "numeral :: num \<Rightarrow> ('a::len) word",
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    "uminus :: ('a::len) word \<Rightarrow> ('a::len) word"
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context
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  includes lifting_syntax
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  notes power_transfer [transfer_rule]
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begin
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lemma power_transfer_word [transfer_rule]:
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  \<open>(pcr_word ===> (=) ===> pcr_word) (^) (^)\<close>
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  by transfer_prover
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end
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text \<open>Legacy theorems:\<close>
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lemma word_arith_wis [code]:
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  shows word_add_def: "a + b = word_of_int (uint a + uint b)"
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    and word_sub_wi: "a - b = word_of_int (uint a - uint b)"
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   295
    and word_mult_def: "a * b = word_of_int (uint a * uint b)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   296
    and word_minus_def: "- a = word_of_int (- uint a)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   297
    and word_succ_alt: "word_succ a = word_of_int (uint a + 1)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   298
    and word_pred_alt: "word_pred a = word_of_int (uint a - 1)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   299
    and word_0_wi: "0 = word_of_int 0"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   300
    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
   301
         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
   302
           times_word.abs_eq uminus_word.abs_eq
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   303
           zero_word.abs_eq one_word.abs_eq)
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   304
   apply transfer
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   305
   apply simp
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   306
  apply transfer
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   307
  apply simp
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   308
  done
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   309
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   310
lemma wi_homs:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   311
  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
   312
    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
   313
    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
   314
    and wi_hom_neg: "- word_of_int a = word_of_int (- a)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   315
    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
   316
    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
   317
  by (transfer, simp)+
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   318
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   319
lemmas wi_hom_syms = wi_homs [symmetric]
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   320
46013
d2f179d26133 remove some duplicate lemmas
huffman
parents: 46012
diff changeset
   321
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
46009
5cb7ef5bfef2 remove duplicate lemma lists
huffman
parents: 46001
diff changeset
   322
5cb7ef5bfef2 remove duplicate lemma lists
huffman
parents: 46001
diff changeset
   323
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
   324
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
   325
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
   326
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
   327
instance word :: (len) semiring_numeral ..
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   328
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   329
instance word :: (len) comm_ring_1
45810
024947a0e492 prove class instances without extra lemmas
huffman
parents: 45809
diff changeset
   330
proof
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   331
  have *: "0 < LENGTH('a)" by (rule len_gt_0)
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   332
  show "(0::'a word) \<noteq> 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   333
    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
   334
qed
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   335
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   336
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
   337
  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
   338
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   339
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
   340
  apply (rule ext)
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   341
  apply (case_tac x rule: int_diff_cases)
46013
d2f179d26133 remove some duplicate lemmas
huffman
parents: 46012
diff changeset
   342
  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
   343
  done
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   344
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   345
context
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   346
  includes lifting_syntax
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   347
  notes 
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   348
    transfer_rule_of_bool [transfer_rule]
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   349
    transfer_rule_numeral [transfer_rule]
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   350
    transfer_rule_of_nat [transfer_rule]
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   351
    transfer_rule_of_int [transfer_rule]
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   352
begin
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   353
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   354
lemma [transfer_rule]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   355
  "((=) ===> (pcr_word :: int \<Rightarrow> 'a::len word \<Rightarrow> bool)) of_bool of_bool"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   356
  by transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   357
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   358
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
   359
  "((=) ===> (pcr_word :: int \<Rightarrow> 'a::len word \<Rightarrow> bool)) numeral numeral"
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   360
  by transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   361
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   362
lemma [transfer_rule]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   363
  "((=) ===> pcr_word) int of_nat"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   364
  by transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   365
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   366
lemma [transfer_rule]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   367
  "((=) ===> pcr_word) (\<lambda>k. k) of_int"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   368
proof -
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   369
  have "((=) ===> pcr_word) of_int of_int"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   370
    by transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   371
  then show ?thesis by (simp add: id_def)
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   372
qed
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   373
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   374
end
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   375
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   376
lemma word_of_int_eq:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   377
  "word_of_int = of_int"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   378
  by (rule ext) (transfer, rule)
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   379
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   380
definition udvd :: "'a::len word \<Rightarrow> 'a::len word \<Rightarrow> bool" (infixl "udvd" 50)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   381
  where "a udvd b = (\<exists>n\<ge>0. uint b = n * uint a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   382
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   383
context
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   384
  includes lifting_syntax
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   385
begin
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   386
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   387
lemma [transfer_rule]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   388
  "(pcr_word ===> (\<longleftrightarrow>)) even ((dvd) 2 :: 'a::len word \<Rightarrow> bool)"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   389
proof -
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   390
  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
   391
    for k :: int
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   392
  proof
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   393
    assume ?P
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   394
    then show ?Q
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   395
      by auto
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   396
  next
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   397
    assume ?Q
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   398
    then obtain l where "take_bit LENGTH('a) k = take_bit LENGTH('a) (2 * l)" ..
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   399
    then have "even (take_bit LENGTH('a) k)"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   400
      by simp
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   401
    then show ?P
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   402
      by simp
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   403
  qed
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   404
  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
   405
    transfer_prover
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   406
qed
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   407
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   408
end
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   409
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   410
instance word :: (len) semiring_modulo
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   411
proof
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   412
  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
   413
  proof transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   414
    fix k l :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   415
    define r :: int where "r = 2 ^ LENGTH('a)"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   416
    then have r: "take_bit LENGTH('a) k = k mod r" for k
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   417
      by (simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   418
    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
   419
      + (k mod r) mod (l mod r)) mod r"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   420
      by (simp add: div_mult_mod_eq)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   421
    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
   422
      + (k mod r) mod (l mod r)) mod r"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   423
      by (simp add: mod_add_left_eq)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   424
    also have "... = (((k mod r) div (l mod r) * l) mod r
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   425
      + (k mod r) mod (l mod r)) mod r"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   426
      by (simp add: mod_mult_right_eq)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   427
    finally have "k mod r = ((k mod r) div (l mod r) * l
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   428
      + (k mod r) mod (l mod r)) mod r"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   429
      by (simp add: mod_simps)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   430
    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
   431
      + 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
   432
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   433
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   434
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   435
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   436
instance word :: (len) semiring_parity
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   437
proof
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   438
  show "\<not> 2 dvd (1::'a word)"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   439
    by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   440
  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
   441
    for a :: "'a word"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   442
    by transfer (simp_all add: mod_2_eq_odd take_bit_Suc)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   443
  show "\<not> 2 dvd a \<longleftrightarrow> a mod 2 = 1"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   444
    for a :: "'a word"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   445
    by transfer (simp_all add: mod_2_eq_odd take_bit_Suc)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   446
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   447
71953
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   448
lemma exp_eq_zero_iff:
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   449
  \<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
   450
  by transfer simp
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   451
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   452
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   453
subsection \<open>Ordering\<close>
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   454
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
   455
instantiation word :: (len) linorder
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   456
begin
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   457
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   458
lift_definition less_eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   459
  is "\<lambda>a b. take_bit LENGTH('a) a \<le> take_bit LENGTH('a) b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   460
  by simp
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   461
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   462
lift_definition less_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   463
  is "\<lambda>a b. take_bit LENGTH('a) a < take_bit LENGTH('a) b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   464
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   465
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   466
instance
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   467
  by (standard; transfer) auto
45547
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   468
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   469
end
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   470
71950
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   471
lemma word_le_def [code]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   472
  "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   473
  by transfer rule
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   474
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   475
lemma word_less_def [code]:
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   476
  "a < b \<longleftrightarrow> uint a < uint b"
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   477
  by transfer rule
c9251bc7da4e more transfer rules
haftmann
parents: 71949
diff changeset
   478
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   479
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
   480
  \<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
   481
  by transfer (simp add: less_le)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   482
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   483
lemma of_nat_word_eq_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   484
  \<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
   485
  by transfer (simp add: take_bit_of_nat)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   486
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   487
lemma of_nat_word_less_eq_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   488
  \<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
   489
  by transfer (simp add: take_bit_of_nat)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   490
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   491
lemma of_nat_word_less_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   492
  \<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
   493
  by transfer (simp add: take_bit_of_nat)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   494
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   495
lemma of_nat_word_eq_0_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   496
  \<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
   497
  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
   498
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   499
lemma of_int_word_eq_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   500
  \<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
   501
  by transfer rule
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   502
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   503
lemma of_int_word_less_eq_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   504
  \<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
   505
  by transfer rule
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   506
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   507
lemma of_int_word_less_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   508
  \<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
   509
  by transfer rule
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   510
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   511
lemma of_int_word_eq_0_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   512
  \<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
   513
  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
   514
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   515
definition word_sle :: "'a::len word \<Rightarrow> 'a word \<Rightarrow> bool"  ("(_/ <=s _)" [50, 51] 50)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   516
  where "a <=s b \<longleftrightarrow> sint a \<le> sint b"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   517
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   518
definition word_sless :: "'a::len word \<Rightarrow> 'a word \<Rightarrow> bool"  ("(_/ <s _)" [50, 51] 50)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   519
  where "x <s y \<longleftrightarrow> x <=s y \<and> x \<noteq> y"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   520
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   521
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   522
subsection \<open>Bit-wise operations\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   523
71951
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   524
lemma word_bit_induct [case_names zero even odd]:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   525
  \<open>P a\<close> if word_zero: \<open>P 0\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   526
    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
   527
    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
   528
  for P and a :: \<open>'a::len word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   529
proof -
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   530
  define m :: nat where \<open>m = LENGTH('a) - 1\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   531
  then have l: \<open>LENGTH('a) = Suc m\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   532
    by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   533
  define n :: nat where \<open>n = unat a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   534
  then have \<open>n < 2 ^ LENGTH('a)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   535
    by (unfold unat_def) (transfer, simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   536
  then have \<open>n < 2 * 2 ^ m\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   537
    by (simp add: l)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   538
  then have \<open>P (of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   539
  proof (induction n rule: nat_bit_induct)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   540
    case zero
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   541
    show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   542
      by simp (rule word_zero)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   543
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   544
    case (even n)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   545
    then have \<open>n < 2 ^ m\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   546
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   547
    with even.IH have \<open>P (of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   548
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   549
    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
   550
      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
   551
    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
   552
      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
   553
      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
   554
    ultimately have \<open>P (2 * of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   555
      by (rule word_even)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   556
    then show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   557
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   558
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   559
    case (odd n)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   560
    then have \<open>Suc n \<le> 2 ^ m\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   561
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   562
    with odd.IH have \<open>P (of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   563
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   564
    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
   565
      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
   566
      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
   567
    ultimately have \<open>P (1 + 2 * of_nat n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   568
      by (rule word_odd)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   569
    then show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   570
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   571
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   572
  moreover have \<open>of_nat (nat (uint a)) = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   573
    by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   574
  ultimately show ?thesis
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   575
    by (simp add: n_def unat_def)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   576
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   577
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   578
lemma bit_word_half_eq:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   579
  \<open>(of_bool b + a * 2) div 2 = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   580
    if \<open>a < 2 ^ (LENGTH('a) - Suc 0)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   581
    for a :: \<open>'a::len word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   582
proof (cases \<open>2 \<le> LENGTH('a::len)\<close>)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   583
  case False
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   584
  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
   585
    by auto
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   586
  with False that show ?thesis
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   587
    by transfer (simp add: eq_iff)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   588
next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   589
  case True
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   590
  obtain n where length: \<open>LENGTH('a) = Suc n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   591
    by (cases \<open>LENGTH('a)\<close>) simp_all
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   592
  show ?thesis proof (cases b)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   593
    case False
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   594
    moreover have \<open>a * 2 div 2 = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   595
    using that proof transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   596
      fix k :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   597
      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
   598
        by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   599
      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
   600
      with \<open>LENGTH('a) = Suc n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   601
      have \<open>k mod 2 ^ LENGTH('a) = k mod 2 ^ n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   602
        by (simp add: take_bit_eq_mod divmod_digit_0)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   603
      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
   604
        by (simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   605
      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
   606
        = take_bit LENGTH('a) k\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   607
        by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   608
    qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   609
    ultimately show ?thesis
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   610
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   611
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   612
    case True
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   613
    moreover have \<open>(1 + a * 2) div 2 = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   614
    using that proof transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   615
      fix k :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   616
      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
   617
        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
   618
      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
   619
      with \<open>LENGTH('a) = Suc n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   620
      have \<open>k mod 2 ^ LENGTH('a) = k mod 2 ^ n\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   621
        by (simp add: take_bit_eq_mod divmod_digit_0)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   622
      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
   623
        by (simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   624
      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
   625
        = take_bit LENGTH('a) k\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   626
        by (auto simp add: take_bit_Suc)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   627
    qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   628
    ultimately show ?thesis
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   629
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   630
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   631
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   632
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   633
lemma even_mult_exp_div_word_iff:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   634
  \<open>even (a * 2 ^ m div 2 ^ n) \<longleftrightarrow> \<not> (
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   635
    m \<le> n \<and>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   636
    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
   637
  by transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   638
    (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
   639
      simp_all flip: push_bit_eq_mult add: bit_push_bit_iff_int)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   640
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   641
instance word :: (len) semiring_bits
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   642
proof
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   643
  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
   644
    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
   645
  for P and a :: \<open>'a word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   646
  proof (induction a rule: word_bit_induct)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   647
    case zero
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   648
    have \<open>0 div 2 = (0::'a word)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   649
      by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   650
    with stable [of 0] show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   651
      by simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   652
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   653
    case (even a)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   654
    with rec [of a False] show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   655
      using bit_word_half_eq [of a False] by (simp add: ac_simps)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   656
  next
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   657
    case (odd a)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   658
    with rec [of a True] show ?case
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   659
      using bit_word_half_eq [of a True] by (simp add: ac_simps)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   660
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   661
  show \<open>0 div a = 0\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   662
    for a :: \<open>'a word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   663
    by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   664
  show \<open>a div 1 = a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   665
    for a :: \<open>'a word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   666
    by transfer simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   667
  show \<open>a mod b div b = 0\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   668
    for a b :: \<open>'a word\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   669
    apply transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   670
    apply (simp add: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   671
    apply (subst (3) mod_pos_pos_trivial [of _ \<open>2 ^ LENGTH('a)\<close>])
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   672
      apply simp_all
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   673
     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
   674
    using pos_mod_bound [of \<open>2 ^ LENGTH('a)\<close>] apply simp
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   675
  proof -
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   676
    fix aa :: int and ba :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   677
    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
   678
      by (metis le_less take_bit_eq_mod take_bit_nonnegative)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   679
    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
   680
      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
   681
    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
   682
      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
   683
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   684
  show \<open>(1 + a) div 2 = a div 2\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   685
    if \<open>even a\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   686
    for a :: \<open>'a word\<close>
71953
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   687
    using that by transfer
428609096812 more lemmas and less name space pollution
haftmann
parents: 71952
diff changeset
   688
      (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
   689
  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
   690
    for m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   691
    by transfer (simp, simp add: exp_div_exp_eq)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   692
  show "a div 2 ^ m div 2 ^ n = a div 2 ^ (m + n)"
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   693
    for a :: "'a word" and m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   694
    apply transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   695
    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
   696
    apply (simp add: drop_bit_take_bit)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   697
    done
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   698
  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
   699
    for a :: "'a word" and m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   700
    by transfer (auto simp flip: take_bit_eq_mod simp add: ac_simps)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   701
  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
   702
    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
   703
    using that apply transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   704
    apply (auto simp flip: take_bit_eq_mod)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   705
           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
   706
    done
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   707
  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
   708
    for a :: "'a word" and m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   709
    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
   710
  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
   711
    for m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   712
    by transfer (auto simp add: take_bit_of_mask even_mask_div_iff)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   713
  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
   714
    for a :: \<open>'a word\<close> and m n :: nat
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   715
  proof transfer
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   716
    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
   717
      n < m
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   718
      \<or> take_bit LENGTH('a) ((2::int) ^ n) = take_bit LENGTH('a) 0
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   719
      \<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
   720
    for m n :: nat and k l :: int
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   721
      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
   722
        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
   723
  qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   724
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   725
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   726
context
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   727
  includes lifting_syntax
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   728
begin
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   729
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   730
lemma transfer_rule_bit_word [transfer_rule]:
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   731
  \<open>((pcr_word :: int \<Rightarrow> 'a::len word \<Rightarrow> bool) ===> (=)) (\<lambda>k n. n < LENGTH('a) \<and> bit k n) bit\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   732
proof -
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   733
  let ?t = \<open>\<lambda>a n. odd (take_bit LENGTH('a) a div take_bit LENGTH('a) ((2::int) ^ n))\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   734
  have \<open>((pcr_word :: int \<Rightarrow> 'a word \<Rightarrow> bool) ===> (=)) ?t bit\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   735
    by (unfold bit_def) transfer_prover
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   736
  also have \<open>?t = (\<lambda>k n. n < LENGTH('a) \<and> bit k n)\<close>
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   737
    by (simp add: fun_eq_iff bit_take_bit_iff flip: bit_def)
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   738
  finally show ?thesis .
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   739
qed
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   740
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   741
end
ac6f9738c200 essential instance about bit structure
haftmann
parents: 71950
diff changeset
   742
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   743
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
   744
begin
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   745
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   746
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
   747
  is push_bit
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   748
proof -
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   749
  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
   750
    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
   751
  proof -
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   752
    from that
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   753
    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
   754
      = 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
   755
      by simp
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   756
    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
   757
      by simp
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   758
    ultimately show ?thesis
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   759
      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
   760
  qed
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   761
qed
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   762
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   763
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
   764
  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
   765
  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
   766
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   767
instance proof
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   768
  show \<open>push_bit n a = a * 2 ^ n\<close> for n :: nat and a :: "'a word"
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   769
    by transfer (simp add: push_bit_eq_mult)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   770
  show \<open>drop_bit n a = a div 2 ^ n\<close> for n :: nat and a :: "'a word"
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   771
    by transfer (simp flip: drop_bit_eq_div add: drop_bit_take_bit)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   772
qed
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   773
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   774
end
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   775
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
   776
definition shiftl1 :: "'a::len word \<Rightarrow> 'a word"
70191
bdc835d934b7 no need to maintain two separate type classes
haftmann
parents: 70190
diff changeset
   777
  where "shiftl1 w = word_of_int (uint w BIT False)"
bdc835d934b7 no need to maintain two separate type classes
haftmann
parents: 70190
diff changeset
   778
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   779
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
   780
  \<open>shiftl1 = (*) 2\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   781
  apply (simp add: fun_eq_iff shiftl1_def Bit_def)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   782
  apply (simp only: mult_2)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   783
  apply transfer
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   784
  apply (simp only: take_bit_add)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   785
  apply simp
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   786
  done
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   787
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
   788
definition shiftr1 :: "'a::len word \<Rightarrow> 'a word"
70191
bdc835d934b7 no need to maintain two separate type classes
haftmann
parents: 70190
diff changeset
   789
  \<comment> \<open>shift right as unsigned or as signed, ie logical or arithmetic\<close>
bdc835d934b7 no need to maintain two separate type classes
haftmann
parents: 70190
diff changeset
   790
  where "shiftr1 w = word_of_int (bin_rest (uint w))"
bdc835d934b7 no need to maintain two separate type classes
haftmann
parents: 70190
diff changeset
   791
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   792
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
   793
  \<open>shiftr1 w = w div 2\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   794
  apply (simp add: fun_eq_iff shiftr1_def)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   795
  apply transfer
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   796
  apply (auto simp add: not_le dest: less_2_cases)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   797
  done
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   798
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
   799
instantiation word :: (len) bit_operations
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   800
begin
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   801
71826
f424e164d752 modernized notation for bit operations
haftmann
parents: 71149
diff changeset
   802
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is NOT
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   803
  by (metis bin_trunc_not)
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   804
71826
f424e164d752 modernized notation for bit operations
haftmann
parents: 71149
diff changeset
   805
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is \<open>(AND)\<close>
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   806
  by (metis bin_trunc_and)
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   807
71826
f424e164d752 modernized notation for bit operations
haftmann
parents: 71149
diff changeset
   808
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is \<open>(OR)\<close>
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   809
  by (metis bin_trunc_or)
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   810
71826
f424e164d752 modernized notation for bit operations
haftmann
parents: 71149
diff changeset
   811
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is \<open>(XOR)\<close>
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   812
  by (metis bin_trunc_xor)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   813
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   814
definition word_test_bit_def: "test_bit a = bin_nth (uint a)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   815
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   816
definition word_set_bit_def: "set_bit a n x = word_of_int (bin_sc n x (uint a))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   817
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   818
definition word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   819
70175
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   820
definition "msb a \<longleftrightarrow> bin_sign (sbintrunc (LENGTH('a) - 1) (uint a)) = - 1"
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   821
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   822
definition shiftl_def: "w << n = (shiftl1 ^^ n) w"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   823
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   824
definition shiftr_def: "w >> n = (shiftr1 ^^ n) w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   825
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   826
instance ..
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   827
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   828
end
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   829
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   830
lemma test_bit_word_eq:
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   831
  \<open>test_bit w = bit w\<close> for w :: \<open>'a::len word\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   832
  apply (simp add: word_test_bit_def fun_eq_iff)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   833
  apply transfer
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   834
  apply (simp add: bit_take_bit_iff)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   835
  done
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   836
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   837
lemma lsb_word_eq:
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   838
  \<open>lsb = (odd :: 'a word \<Rightarrow> bool)\<close> for w :: \<open>'a::len word\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   839
  apply (simp add: word_lsb_def fun_eq_iff)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   840
  apply transfer
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   841
  apply simp
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   842
  done
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   843
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   844
lemma msb_word_eq:
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   845
  \<open>msb w \<longleftrightarrow> bit w (LENGTH('a) - 1)\<close> for w :: \<open>'a::len word\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   846
  apply (simp add: msb_word_def bin_sign_lem)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   847
  apply transfer
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   848
  apply (simp add: bit_take_bit_iff)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   849
  done
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   850
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   851
lemma shiftl_word_eq:
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   852
  \<open>w << n = push_bit n w\<close> for w :: \<open>'a::len word\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   853
  by (induction n) (simp_all add: shiftl_def shiftl1_eq_mult_2 push_bit_double)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   854
71955
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   855
lemma [code]:
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   856
  \<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
   857
  by (simp add: shiftl_word_eq)
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   858
71952
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   859
lemma shiftr_word_eq:
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   860
  \<open>w >> n = drop_bit n w\<close> for w :: \<open>'a::len word\<close>
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   861
  by (induction n) (simp_all add: shiftr_def shiftr1_eq_div_2 drop_bit_Suc drop_bit_half)
2efc5b8c7456 canonical bit shifts for word type, leaving duplicates as they are at the moment
haftmann
parents: 71951
diff changeset
   862
71955
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   863
lemma [code]:
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   864
  \<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
   865
  by (simp add: shiftr_word_eq)
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   866
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   867
lemma [code_abbrev]:
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   868
  \<open>push_bit n 1 = (2 :: 'a::len word) ^ n\<close>
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   869
  by (fact push_bit_of_1)
a9f913d17d00 tweak for code generation
haftmann
parents: 71954
diff changeset
   870
70175
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   871
lemma word_msb_def:
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   872
  "msb a \<longleftrightarrow> bin_sign (sint a) = - 1"
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   873
  by (simp add: msb_word_def sint_uint)
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   874
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   875
lemma [code]:
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
   876
  shows word_not_def: "NOT (a::'a::len word) = word_of_int (NOT (uint a))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   877
    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
   878
    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
   879
    and word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
71948
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   880
  by (simp_all flip: bitNOT_word.abs_eq bitAND_word.abs_eq
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
   881
     bitOR_word.abs_eq bitXOR_word.abs_eq)
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   882
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
   883
definition setBit :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   884
  where "setBit w n = set_bit w n True"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   885
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
   886
definition clearBit :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   887
  where "clearBit w n = set_bit w n False"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   888
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   889
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   890
subsection \<open>Shift operations\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   891
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   892
definition sshiftr1 :: "'a::len word \<Rightarrow> 'a word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   893
  where "sshiftr1 w = word_of_int (bin_rest (sint w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   894
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   895
definition bshiftr1 :: "bool \<Rightarrow> 'a::len word \<Rightarrow> 'a word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   896
  where "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   897
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   898
definition sshiftr :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word"  (infixl ">>>" 55)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   899
  where "w >>> n = (sshiftr1 ^^ n) w"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   900
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   901
definition mask :: "nat \<Rightarrow> 'a::len word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   902
  where "mask n = (1 << n) - 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   903
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
   904
definition revcast :: "'a::len word \<Rightarrow> 'b::len word"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   905
  where "revcast w =  of_bl (takefill False (LENGTH('b)) (to_bl w))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   906
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
   907
definition slice1 :: "nat \<Rightarrow> 'a::len word \<Rightarrow> 'b::len word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   908
  where "slice1 n w = of_bl (takefill False n (to_bl w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   909
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
   910
definition slice :: "nat \<Rightarrow> 'a::len word \<Rightarrow> 'b::len word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   911
  where "slice n w = slice1 (size w - n) w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   912
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   913
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   914
subsection \<open>Rotation\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   915
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   916
definition rotater1 :: "'a list \<Rightarrow> 'a list"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   917
  where "rotater1 ys =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   918
    (case ys of [] \<Rightarrow> [] | x # xs \<Rightarrow> last ys # butlast ys)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   919
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   920
definition rotater :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   921
  where "rotater n = rotater1 ^^ n"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   922
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
   923
definition word_rotr :: "nat \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   924
  where "word_rotr n w = of_bl (rotater n (to_bl w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   925
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
   926
definition word_rotl :: "nat \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   927
  where "word_rotl n w = of_bl (rotate n (to_bl w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   928
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
   929
definition word_roti :: "int \<Rightarrow> 'a::len word \<Rightarrow> 'a::len word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   930
  where "word_roti i w =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   931
    (if i \<ge> 0 then word_rotr (nat i) w else word_rotl (nat (- i)) w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   932
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   933
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   934
subsection \<open>Split and cat operations\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   935
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
   936
definition word_cat :: "'a::len word \<Rightarrow> 'b::len word \<Rightarrow> 'c::len word"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   937
  where "word_cat a b = word_of_int (bin_cat (uint a) (LENGTH('b)) (uint b))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   938
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
   939
definition word_split :: "'a::len word \<Rightarrow> 'b::len word \<times> 'c::len word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   940
  where "word_split a =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   941
    (case bin_split (LENGTH('c)) (uint a) of
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   942
      (u, v) \<Rightarrow> (word_of_int u, word_of_int v))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   943
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
   944
definition word_rcat :: "'a::len word list \<Rightarrow> 'b::len word"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   945
  where "word_rcat ws = word_of_int (bin_rcat (LENGTH('a)) (map uint ws))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   946
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
   947
definition word_rsplit :: "'a::len word \<Rightarrow> 'b::len word list"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   948
  where "word_rsplit w = map word_of_int (bin_rsplit (LENGTH('b)) (LENGTH('a), uint w))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   949
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
   950
abbreviation (input) max_word :: \<open>'a::len word\<close>
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67408
diff changeset
   951
  \<comment> \<open>Largest representable machine integer.\<close>
71946
4d4079159be7 replaced mere alias by abbreviation
haftmann
parents: 71945
diff changeset
   952
  where "max_word \<equiv> - 1"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   953
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   954
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   955
subsection \<open>Theorems about typedefs\<close>
46010
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
   956
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   957
lemma sint_sbintrunc': "sint (word_of_int bin :: 'a word) = sbintrunc (LENGTH('a::len) - 1) bin"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   958
  by (auto simp: sint_uint word_ubin.eq_norm sbintrunc_bintrunc_lt)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   959
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   960
lemma uint_sint: "uint w = bintrunc (LENGTH('a)) (sint w)"
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
   961
  for w :: "'a::len word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   962
  by (auto simp: sint_uint bintrunc_sbintrunc_le)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   963
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   964
lemma bintr_uint: "LENGTH('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w"
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
   965
  for w :: "'a::len word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   966
  apply (subst word_ubin.norm_Rep [symmetric])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   967
  apply (simp only: bintrunc_bintrunc_min word_size)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54854
diff changeset
   968
  apply (simp add: min.absorb2)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   969
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   970
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   971
lemma wi_bintr:
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
   972
  "LENGTH('a::len) \<le> n \<Longrightarrow>
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   973
    word_of_int (bintrunc n w) = (word_of_int w :: 'a word)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   974
  by (auto simp: word_ubin.norm_eq_iff [symmetric] min.absorb1)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   975
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   976
lemma td_ext_sbin:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   977
  "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (LENGTH('a::len)))
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   978
    (sbintrunc (LENGTH('a) - 1))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   979
  apply (unfold td_ext_def' sint_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   980
  apply (simp add : word_ubin.eq_norm)
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   981
  apply (cases "LENGTH('a)")
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   982
   apply (auto simp add : sints_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   983
  apply (rule sym [THEN trans])
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   984
   apply (rule word_ubin.Abs_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   985
  apply (simp only: bintrunc_sbintrunc)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   986
  apply (drule sym)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   987
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   988
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   989
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   990
lemma td_ext_sint:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   991
  "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (LENGTH('a::len)))
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   992
     (\<lambda>w. (w + 2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) -
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   993
         2 ^ (LENGTH('a) - 1))"
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   994
  using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   995
67408
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   996
text \<open>
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   997
  We do \<open>sint\<close> before \<open>sbin\<close>, before \<open>sint\<close> is the user version
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   998
  and interpretations do not produce thm duplicates. I.e.
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   999
  we get the name \<open>word_sint.Rep_eqD\<close>, but not \<open>word_sbin.Req_eqD\<close>,
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
  1000
  because the latter is the same thm as the former.
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
  1001
\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1002
interpretation word_sint:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1003
  td_ext
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1004
    "sint ::'a::len word \<Rightarrow> int"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1005
    word_of_int
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1006
    "sints (LENGTH('a::len))"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1007
    "\<lambda>w. (w + 2^(LENGTH('a::len) - 1)) mod 2^LENGTH('a::len) -
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1008
      2 ^ (LENGTH('a::len) - 1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1009
  by (rule td_ext_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1010
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1011
interpretation word_sbin:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1012
  td_ext
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1013
    "sint ::'a::len word \<Rightarrow> int"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1014
    word_of_int
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1015
    "sints (LENGTH('a::len))"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1016
    "sbintrunc (LENGTH('a::len) - 1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1017
  by (rule td_ext_sbin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1018
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1019
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1020
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1021
lemmas td_sint = word_sint.td
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1022
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
  1023
lemma to_bl_def': "(to_bl :: 'a::len word \<Rightarrow> bool list) = bin_to_bl (LENGTH('a)) \<circ> uint"
44762
8f9d09241a68 tuned proofs;
wenzelm
parents: 42793
diff changeset
  1024
  by (auto simp: to_bl_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1025
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1026
lemmas word_reverse_no_def [simp] =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1027
  word_reverse_def [of "numeral w"] for w
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1028
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1029
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1030
  by (fact uints_def [unfolded no_bintr_alt1])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1031
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1032
lemma word_numeral_alt: "numeral b = word_of_int (numeral b)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1033
  by (induct b, simp_all only: numeral.simps word_of_int_homs)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1034
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1035
declare word_numeral_alt [symmetric, code_abbrev]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1036
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1037
lemma word_neg_numeral_alt: "- numeral b = word_of_int (- numeral b)"
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1038
  by (simp only: word_numeral_alt wi_hom_neg)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1039
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1040
declare word_neg_numeral_alt [symmetric, code_abbrev]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1041
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1042
lemma uint_bintrunc [simp]:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1043
  "uint (numeral bin :: 'a word) =
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
  1044
    bintrunc (LENGTH('a::len)) (numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1045
  unfolding word_numeral_alt by (rule word_ubin.eq_norm)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1046
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1047
lemma uint_bintrunc_neg [simp]:
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
  1048
  "uint (- numeral bin :: 'a word) = bintrunc (LENGTH('a::len)) (- numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1049
  by (simp only: word_neg_numeral_alt word_ubin.eq_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1050
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1051
lemma sint_sbintrunc [simp]:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1052
  "sint (numeral bin :: 'a word) = sbintrunc (LENGTH('a::len) - 1) (numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1053
  by (simp only: word_numeral_alt word_sbin.eq_norm)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1054
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1055
lemma sint_sbintrunc_neg [simp]:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1056
  "sint (- numeral bin :: 'a word) = sbintrunc (LENGTH('a::len) - 1) (- numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1057
  by (simp only: word_neg_numeral_alt word_sbin.eq_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1058
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1059
lemma unat_bintrunc [simp]:
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
  1060
  "unat (numeral bin :: 'a::len word) = nat (bintrunc (LENGTH('a)) (numeral bin))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1061
  by (simp only: unat_def uint_bintrunc)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1062
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1063
lemma unat_bintrunc_neg [simp]:
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
  1064
  "unat (- numeral bin :: 'a::len word) = nat (bintrunc (LENGTH('a)) (- numeral bin))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1065
  by (simp only: unat_def uint_bintrunc_neg)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1066
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1067
lemma size_0_eq: "size w = 0 \<Longrightarrow> v = w"
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
  1068
  for v w :: "'a::len word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1069
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1070
  apply (rule word_uint.Rep_eqD)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1071
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1072
    defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1073
    apply (rule word_ubin.norm_Rep)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1074
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1075
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1076
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1077
lemma uint_ge_0 [iff]: "0 \<le> uint x"
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
  1078
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1079
  using word_uint.Rep [of x] by (simp add: uints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1080
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1081
lemma uint_lt2p [iff]: "uint x < 2 ^ LENGTH('a)"
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
  1082
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1083
  using word_uint.Rep [of x] by (simp add: uints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1084
71946
4d4079159be7 replaced mere alias by abbreviation
haftmann
parents: 71945
diff changeset
  1085
lemma word_exp_length_eq_0 [simp]:
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
  1086
  \<open>(2 :: 'a::len word) ^ LENGTH('a) = 0\<close>
71946
4d4079159be7 replaced mere alias by abbreviation
haftmann
parents: 71945
diff changeset
  1087
  by transfer (simp add: bintrunc_mod2p)
4d4079159be7 replaced mere alias by abbreviation
haftmann
parents: 71945
diff changeset
  1088
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1089
lemma sint_ge: "- (2 ^ (LENGTH('a) - 1)) \<le> sint x"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1090
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1091
  using word_sint.Rep [of x] by (simp add: sints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1092
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1093
lemma sint_lt: "sint x < 2 ^ (LENGTH('a) - 1)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1094
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1095
  using word_sint.Rep [of x] by (simp add: sints_num)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1096
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1097
lemma sign_uint_Pls [simp]: "bin_sign (uint x) = 0"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1098
  by (simp add: sign_Pls_ge_0)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1099
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1100
lemma uint_m2p_neg: "uint x - 2 ^ LENGTH('a) < 0"
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
  1101
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1102
  by (simp only: diff_less_0_iff_less uint_lt2p)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1103
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1104
lemma uint_m2p_not_non_neg: "\<not> 0 \<le> uint x - 2 ^ LENGTH('a)"
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
  1105
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1106
  by (simp only: not_le uint_m2p_neg)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1107
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1108
lemma lt2p_lem: "LENGTH('a) \<le> n \<Longrightarrow> uint w < 2 ^ n"
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
  1109
  for w :: "'a::len word"
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
  1110
  by (metis bintr_uint bintrunc_mod2p int_mod_lem zless2p)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1111
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1112
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
70749
5d06b7bb9d22 More type class generalisations. Note that linorder_antisym_conv1 and linorder_antisym_conv2 no longer exist.
paulson <lp15@cam.ac.uk>
parents: 70342
diff changeset
  1113
  by (fact uint_ge_0 [THEN leD, THEN antisym_conv1])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1114
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1115
lemma uint_nat: "uint w = int (unat w)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1116
  by (auto simp: unat_def)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1117
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
  1118
lemma uint_numeral: "uint (numeral b :: 'a::len word) = numeral b mod 2 ^ LENGTH('a)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1119
  by (simp only: word_numeral_alt int_word_uint)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1120
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
  1121
lemma uint_neg_numeral: "uint (- numeral b :: 'a::len word) = - numeral b mod 2 ^ LENGTH('a)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1122
  by (simp only: word_neg_numeral_alt int_word_uint)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1123
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
  1124
lemma unat_numeral: "unat (numeral b :: 'a::len word) = numeral b mod 2 ^ LENGTH('a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1125
  apply (unfold unat_def)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1126
  apply (clarsimp simp only: uint_numeral)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1127
  apply (rule nat_mod_distrib [THEN trans])
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1128
    apply (rule zero_le_numeral)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1129
   apply (simp_all add: nat_power_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1130
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1131
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1132
lemma sint_numeral:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1133
  "sint (numeral b :: 'a::len word) =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1134
    (numeral b +
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1135
      2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) -
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1136
      2 ^ (LENGTH('a) - 1)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1137
  unfolding word_numeral_alt by (rule int_word_sint)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1138
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1139
lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1140
  unfolding word_0_wi ..
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1141
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1142
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1143
  unfolding word_1_wi ..
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1144
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1145
lemma word_of_int_neg_1 [simp]: "word_of_int (- 1) = - 1"
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1146
  by (simp add: wi_hom_syms)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1147
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
  1148
lemma word_of_int_numeral [simp] : "(word_of_int (numeral bin) :: 'a::len word) = numeral bin"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1149
  by (simp only: word_numeral_alt)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1150
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1151
lemma word_of_int_neg_numeral [simp]:
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
  1152
  "(word_of_int (- numeral bin) :: 'a::len word) = - numeral bin"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1153
  by (simp only: word_numeral_alt wi_hom_syms)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1154
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1155
lemma word_int_case_wi:
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
  1156
  "word_int_case f (word_of_int i :: 'b word) = f (i mod 2 ^ LENGTH('b::len))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1157
  by (simp add: word_int_case_def word_uint.eq_norm)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1158
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1159
lemma word_int_split:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1160
  "P (word_int_case f x) =
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
  1161
    (\<forall>i. x = (word_of_int i :: 'b::len word) \<and> 0 \<le> i \<and> i < 2 ^ LENGTH('b) \<longrightarrow> P (f i))"
71942
d2654b30f7bd eliminated warnings
haftmann
parents: 71826
diff changeset
  1162
  by (auto simp: word_int_case_def word_uint.eq_norm)
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1163
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1164
lemma word_int_split_asm:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1165
  "P (word_int_case f x) =
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
  1166
    (\<nexists>n. x = (word_of_int n :: 'b::len word) \<and> 0 \<le> n \<and> n < 2 ^ LENGTH('b::len) \<and> \<not> P (f n))"
71942
d2654b30f7bd eliminated warnings
haftmann
parents: 71826
diff changeset
  1167
  by (auto simp: word_int_case_def word_uint.eq_norm)
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1168
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1169
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1170
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1171
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1172
lemma uint_range_size: "0 \<le> uint w \<and> uint w < 2 ^ size w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1173
  unfolding word_size by (rule uint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1174
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1175
lemma sint_range_size: "- (2 ^ (size w - Suc 0)) \<le> sint w \<and> sint w < 2 ^ (size w - Suc 0)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1176
  unfolding word_size by (rule sint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1177
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1178
lemma sint_above_size: "2 ^ (size w - 1) \<le> x \<Longrightarrow> sint w < x"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1179
  for w :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1180
  unfolding word_size by (rule less_le_trans [OF sint_lt])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1181
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1182
lemma sint_below_size: "x \<le> - (2 ^ (size w - 1)) \<Longrightarrow> x \<le> sint w"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1183
  for w :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1184
  unfolding word_size by (rule order_trans [OF _ sint_ge])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1185
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
  1186
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1187
subsection \<open>Testing bits\<close>
46010
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
  1188
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1189
lemma test_bit_eq_iff: "test_bit u = test_bit v \<longleftrightarrow> u = v"
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
  1190
  for u v :: "'a::len word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1191
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1192
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1193
lemma test_bit_size [rule_format] : "w !! n \<longrightarrow> n < size w"
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
  1194
  for w :: "'a::len word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1195
  apply (unfold word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1196
  apply (subst word_ubin.norm_Rep [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1197
  apply (simp only: nth_bintr word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1198
  apply fast
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1199
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1200
71948
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1201
lemma word_eq_iff: "x = y \<longleftrightarrow> (\<forall>n<LENGTH('a). x !! n = y !! n)" (is \<open>?P \<longleftrightarrow> ?Q\<close>)
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
  1202
  for x y :: "'a::len word"
71948
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1203
proof
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1204
  assume ?P
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1205
  then show ?Q
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1206
    by simp
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1207
next
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1208
  assume ?Q
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1209
  then have *: \<open>bit (uint x) n \<longleftrightarrow> bit (uint y) n\<close> if \<open>n < LENGTH('a)\<close> for n
71949
5b8b1183c641 dropped yet another duplicate
haftmann
parents: 71948
diff changeset
  1210
    using that by (simp add: word_test_bit_def)
71948
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1211
  show ?P
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1212
  proof (rule word_uint_eqI, rule bit_eqI, rule iffI)
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1213
    fix n
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1214
    assume \<open>bit (uint x) n\<close>
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1215
    then have \<open>n < LENGTH('a)\<close>
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1216
      by (simp add: bit_take_bit_iff uint.rep_eq)
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1217
    with * \<open>bit (uint x) n\<close>
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1218
    show \<open>bit (uint y) n\<close>
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1219
      by simp
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1220
  next
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1221
    fix n
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1222
    assume \<open>bit (uint y) n\<close>
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1223
    then have \<open>n < LENGTH('a)\<close>
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1224
      by (simp add: bit_take_bit_iff uint.rep_eq)
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1225
    with * \<open>bit (uint y) n\<close>
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1226
    show \<open>bit (uint x) n\<close>
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1227
      by simp
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1228
  qed
6ede899d26d3 fundamental construction of word type following existing transfer rules
haftmann
parents: 71947
diff changeset
  1229
qed  
46021
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
  1230
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1231
lemma word_eqI: "(\<And>n. n < size u \<longrightarrow> u !! n = v !! n) \<Longrightarrow> u = v"
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
  1232
  for u :: "'a::len word"
46021
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
  1233
  by (simp add: word_size word_eq_iff)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1234
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1235
lemma word_eqD: "u = v \<Longrightarrow> u !! x = v !! x"
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
  1236
  for u v :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1237
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1238
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1239
lemma test_bit_bin': "w !! n \<longleftrightarrow> n < size w \<and> bin_nth (uint w) n"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1240
  by (simp add: word_test_bit_def word_size nth_bintr [symmetric])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1241
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1242
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1243
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1244
lemma bin_nth_uint_imp: "bin_nth (uint w) n \<Longrightarrow> n < LENGTH('a)"
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
  1245
  for w :: "'a::len word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1246
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1247
  apply (subst word_ubin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1248
  apply assumption
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1249
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1250
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
  1251
lemma bin_nth_sint:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1252
  "LENGTH('a) \<le> n \<Longrightarrow>
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1253
    bin_nth (sint w) n = bin_nth (sint w) (LENGTH('a) - 1)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1254
  for w :: "'a::len word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1255
  apply (subst word_sbin.norm_Rep [symmetric])
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
  1256
  apply (auto simp add: nth_sbintr)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1257
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1258
67408
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
  1259
\<comment> \<open>type definitions theorem for in terms of equivalent bool list\<close>
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1260
lemma td_bl:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1261
  "type_definition
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
  1262
    (to_bl :: 'a::len word \<Rightarrow> bool list)
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1263
    of_bl
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1264
    {bl. length bl = LENGTH('a)}"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1265
  apply (unfold type_definition_def of_bl_def to_bl_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1266
  apply (simp add: word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1267
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1268
  apply (drule sym)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1269
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1270
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1271
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1272
interpretation word_bl:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1273
  type_definition
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
  1274
    "to_bl :: 'a::len word \<Rightarrow> bool list"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1275
    of_bl
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
  1276
    "{bl. length bl = LENGTH('a::len)}"
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
  1277
  by (fact td_bl)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1278
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1279
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
45538
1fffa81b9b83 eliminated slightly odd Rep' with dynamically-scoped [simplified];
wenzelm
parents: 45529
diff changeset
  1280
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1281
lemma word_size_bl: "size w = size (to_bl w)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1282
  by (auto simp: word_size)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1283
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1284
lemma to_bl_use_of_bl: "to_bl w = bl \<longleftrightarrow> w = of_bl bl \<and> length bl = length (to_bl w)"
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1285
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1286