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_Bit
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  Bits_Int
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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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typedef (overloaded) 'a word = "{(0::int) ..< 2 ^ LENGTH('a::len0)}"
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  morphisms uint Abs_word by auto
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lemma uint_nonnegative: "0 \<le> uint w"
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  using word.uint [of w] by simp
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lemma uint_bounded: "uint w < 2 ^ LENGTH('a)"
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  for w :: "'a::len0 word"
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  using word.uint [of w] by simp
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lemma uint_idem: "uint w mod 2 ^ LENGTH('a) = uint w"
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  for w :: "'a::len0 word"
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  using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial)
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lemma word_uint_eq_iff: "a = b \<longleftrightarrow> uint a = uint b"
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  by (simp add: uint_inject)
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lemma word_uint_eqI: "uint a = uint b \<Longrightarrow> a = b"
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  by (simp add: word_uint_eq_iff)
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definition word_of_int :: "int \<Rightarrow> 'a::len0 word"
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  \<comment> \<open>representation of words using unsigned or signed bins,
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    only difference in these is the type class\<close>
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  where "word_of_int k = Abs_word (k mod 2 ^ LENGTH('a))"
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lemma uint_word_of_int: "uint (word_of_int k :: 'a::len0 word) = k mod 2 ^ LENGTH('a)"
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  by (auto simp add: word_of_int_def intro: Abs_word_inverse)
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lemma word_of_int_uint: "word_of_int (uint w) = w"
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  by (simp add: word_of_int_def uint_idem uint_inverse)
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lemma split_word_all: "(\<And>x::'a::len0 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::len0 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::len0 word \<Rightarrow> 'b::len0 word"
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  where "ucast w = word_of_int (uint w)"
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instantiation word :: (len0) 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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  fixes w :: "'a::len word"
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  shows "size w \<noteq> 0"
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  and "LENGTH('a) \<noteq> 0"
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  by auto
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definition source_size :: "('a::len0 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::len0 word) \<Rightarrow> nat"
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  where [code del]: "target_size c = size (c undefined)"
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definition is_up :: "('a::len0 word \<Rightarrow> 'b::len0 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::len0 word \<Rightarrow> 'b::len0 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::len0 word"
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  where "of_bl bl = word_of_int (bl_to_bin bl)"
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definition to_bl :: "'a::len0 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::len0 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::len0 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>Correspondence relation for theorem transfer\<close>
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definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool"
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  where "cr_word = (\<lambda>x y. word_of_int x = y)"
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lemma Quotient_word:
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  "Quotient (\<lambda>x y. bintrunc (LENGTH('a)) x = bintrunc (LENGTH('a)) y)
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    word_of_int uint (cr_word :: _ \<Rightarrow> 'a::len0 word \<Rightarrow> bool)"
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  unfolding Quotient_alt_def cr_word_def
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  by (simp add: no_bintr_alt1 word_of_int_uint) (simp add: word_of_int_def Abs_word_inject)
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lemma reflp_word:
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  "reflp (\<lambda>x y. bintrunc (LENGTH('a::len0)) x = bintrunc (LENGTH('a)) y)"
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  by (simp add: reflp_def)
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setup_lifting Quotient_word reflp_word
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text \<open>TODO: The next lemma could be generated automatically.\<close>
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lemma uint_transfer [transfer_rule]:
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  "(rel_fun pcr_word (=)) (bintrunc (LENGTH('a))) (uint :: 'a::len0 word \<Rightarrow> int)"
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  unfolding rel_fun_def word.pcr_cr_eq cr_word_def
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  by (simp add: no_bintr_alt1 uint_word_of_int)
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subsection \<open>Basic code generation setup\<close>
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definition Word :: "int \<Rightarrow> 'a::len0 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 :: (len0) 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 :: ("{len0, 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::len0)))
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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 (simp add: uints_num word_of_int_def bintrunc_mod2p)
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  apply (simp add: uint_mod_same uint_0 uint_lt
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                   word.uint_inverse word.Abs_word_inverse int_mod_lem)
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  done
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interpretation word_uint:
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  td_ext
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    "uint::'a::len0 word \<Rightarrow> int"
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    word_of_int
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    "uints (LENGTH('a::len0))"
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    "\<lambda>w. w mod 2 ^ LENGTH('a::len0)"
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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::len0)))
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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::len0 word \<Rightarrow> int"
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    word_of_int
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    "uints (LENGTH('a::len0))"
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    "bintrunc (LENGTH('a::len0))"
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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::len0 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::len0 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 :: (len0) "{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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definition word_div_def: "a div b = word_of_int (uint a div uint b)"
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definition word_mod_def: "a mod b = word_of_int (uint a mod uint b)"
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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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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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    and word_mult_def: "a * b = word_of_int (uint a * uint b)"
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    and word_minus_def: "- a = word_of_int (- uint a)"
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    and word_succ_alt: "word_succ a = word_of_int (uint a + 1)"
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    and word_pred_alt: "word_pred a = word_of_int (uint a - 1)"
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    and word_0_wi: "0 = word_of_int 0"
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   298
    and word_1_wi: "1 = word_of_int 1"
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   299
  unfolding plus_word_def minus_word_def times_word_def uminus_word_def
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diff changeset
   300
  unfolding word_succ_def word_pred_def zero_word_def one_word_def
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diff changeset
   301
  by simp_all
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   302
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   303
lemma wi_homs:
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   304
  shows wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)"
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wenzelm
parents: 64593
diff changeset
   305
    and wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)"
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wenzelm
parents: 64593
diff changeset
   306
    and wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)"
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wenzelm
parents: 64593
diff changeset
   307
    and wi_hom_neg: "- word_of_int a = word_of_int (- a)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   308
    and wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)"
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wenzelm
parents: 64593
diff changeset
   309
    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
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diff changeset
   310
  by (transfer, simp)+
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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   311
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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   312
lemmas wi_hom_syms = wi_homs [symmetric]
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   313
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diff changeset
   314
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
46009
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huffman
parents: 46001
diff changeset
   315
5cb7ef5bfef2 remove duplicate lemma lists
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diff changeset
   316
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
45545
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   317
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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   318
instance word :: (len) comm_ring_1
45810
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diff changeset
   319
proof
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ac1706cdde25 clarified notation
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parents: 70183
diff changeset
   320
  have *: "0 < LENGTH('a)" by (rule len_gt_0)
65268
75f2aa8ecb12 misc tuning and modernization;
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parents: 64593
diff changeset
   321
  show "(0::'a word) \<noteq> 1"
75f2aa8ecb12 misc tuning and modernization;
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diff changeset
   322
    by transfer (use * in \<open>auto simp add: gr0_conv_Suc\<close>)
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parents: 45809
diff changeset
   323
qed
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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parents: 45544
diff changeset
   324
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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diff changeset
   325
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
   326
  by (induct n) (auto simp add : word_of_int_hom_syms)
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huffman
parents: 45544
diff changeset
   327
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   328
lemma word_of_int: "of_int = word_of_int"
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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parents: 45544
diff changeset
   329
  apply (rule ext)
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   330
  apply (case_tac x rule: int_diff_cases)
46013
d2f179d26133 remove some duplicate lemmas
huffman
parents: 46012
diff changeset
   331
  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
   332
  done
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
   333
65268
75f2aa8ecb12 misc tuning and modernization;
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diff changeset
   334
definition udvd :: "'a::len word \<Rightarrow> 'a::len word \<Rightarrow> bool" (infixl "udvd" 50)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   335
  where "a udvd b = (\<exists>n\<ge>0. uint b = n * uint a)"
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   336
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parents: 45546
diff changeset
   337
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parents: 61649
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   338
subsection \<open>Ordering\<close>
45547
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parents: 45546
diff changeset
   339
94c37f3df10f HOL-Word: removed more duplicate theorems
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parents: 45546
diff changeset
   340
instantiation word :: (len0) linorder
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parents: 45546
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   341
begin
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parents: 45546
diff changeset
   342
65268
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   343
definition word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
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   344
75f2aa8ecb12 misc tuning and modernization;
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   345
definition word_less_def: "a < b \<longleftrightarrow> uint a < uint b"
37660
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parents: 36899
diff changeset
   346
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94c37f3df10f HOL-Word: removed more duplicate theorems
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parents: 45546
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   347
instance
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61076
diff changeset
   348
  by standard (auto simp: word_less_def word_le_def)
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94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   349
94c37f3df10f HOL-Word: removed more duplicate theorems
huffman
parents: 45546
diff changeset
   350
end
94c37f3df10f HOL-Word: removed more duplicate theorems
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parents: 45546
diff changeset
   351
65268
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parents: 64593
diff changeset
   352
definition word_sle :: "'a::len word \<Rightarrow> 'a word \<Rightarrow> bool"  ("(_/ <=s _)" [50, 51] 50)
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wenzelm
parents: 64593
diff changeset
   353
  where "a <=s b \<longleftrightarrow> sint a \<le> sint b"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   354
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   355
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
   356
  where "x <s y \<longleftrightarrow> x <=s y \<and> x \<noteq> y"
37660
56e3520b68b2 one unified Word theory
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parents: 36899
diff changeset
   357
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   358
61799
4cf66f21b764 isabelle update_cartouches -c -t;
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parents: 61649
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   359
subsection \<open>Bit-wise operations\<close>
37660
56e3520b68b2 one unified Word theory
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parents: 36899
diff changeset
   360
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   361
instantiation word :: (len0) bits
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   362
begin
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   363
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   364
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   365
  by (metis bin_trunc_not)
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   366
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   367
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   368
  by (metis bin_trunc_and)
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   369
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   370
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   371
  by (metis bin_trunc_or)
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   372
47387
a0f257197741 remove now-unnecessary type annotations from lift_definition commands
huffman
parents: 47377
diff changeset
   373
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   374
  by (metis bin_trunc_xor)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   375
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   376
definition word_test_bit_def: "test_bit a = bin_nth (uint a)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   377
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   378
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
   379
70175
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   380
definition word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth LENGTH('a) f)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   381
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   382
definition word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   383
70175
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   384
definition "msb a \<longleftrightarrow> bin_sign (sbintrunc (LENGTH('a) - 1) (uint a)) = - 1"
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   385
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   386
definition shiftl1 :: "'a word \<Rightarrow> 'a word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   387
  where "shiftl1 w = word_of_int (uint w BIT False)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   388
54848
a303daddebbf syntactically tuned
haftmann
parents: 54847
diff changeset
   389
definition shiftr1 :: "'a word \<Rightarrow> 'a word"
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67408
diff changeset
   390
  \<comment> \<open>shift right as unsigned or as signed, ie logical or arithmetic\<close>
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
   391
  where "shiftr1 w = word_of_int (bin_rest (uint w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   392
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   393
definition shiftl_def: "w << n = (shiftl1 ^^ n) w"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   394
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   395
definition shiftr_def: "w >> n = (shiftr1 ^^ n) w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   396
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   397
instance ..
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   398
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   399
end
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   400
70175
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   401
lemma word_msb_def:
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   402
  "msb a \<longleftrightarrow> bin_sign (sint a) = - 1"
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   403
  by (simp add: msb_word_def sint_uint)
85fb1a585f52 eliminated type class
haftmann
parents: 70173
diff changeset
   404
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   405
lemma [code]:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   406
  shows word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   407
    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
   408
    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
   409
    and word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   410
  by (simp_all add: bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def)
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
   411
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   412
definition setBit :: "'a::len0 word \<Rightarrow> nat \<Rightarrow> 'a word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   413
  where "setBit w n = set_bit w n True"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   414
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   415
definition clearBit :: "'a::len0 word \<Rightarrow> nat \<Rightarrow> 'a word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   416
  where "clearBit w n = set_bit w n False"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   417
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   418
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   419
subsection \<open>Shift operations\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   420
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   421
definition sshiftr1 :: "'a::len word \<Rightarrow> 'a word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   422
  where "sshiftr1 w = word_of_int (bin_rest (sint w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   423
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   424
definition bshiftr1 :: "bool \<Rightarrow> 'a::len word \<Rightarrow> 'a word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   425
  where "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   426
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   427
definition sshiftr :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word"  (infixl ">>>" 55)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   428
  where "w >>> n = (sshiftr1 ^^ n) w"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   429
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   430
definition mask :: "nat \<Rightarrow> 'a::len word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   431
  where "mask n = (1 << n) - 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   432
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   433
definition revcast :: "'a::len0 word \<Rightarrow> 'b::len0 word"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   434
  where "revcast w =  of_bl (takefill False (LENGTH('b)) (to_bl w))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   435
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   436
definition slice1 :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'b::len0 word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   437
  where "slice1 n w = of_bl (takefill False n (to_bl w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   438
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   439
definition slice :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'b::len0 word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   440
  where "slice n w = slice1 (size w - n) w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   441
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   442
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   443
subsection \<open>Rotation\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   444
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   445
definition rotater1 :: "'a list \<Rightarrow> 'a list"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   446
  where "rotater1 ys =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   447
    (case ys of [] \<Rightarrow> [] | x # xs \<Rightarrow> last ys # butlast ys)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   448
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   449
definition rotater :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   450
  where "rotater n = rotater1 ^^ n"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   451
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   452
definition word_rotr :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'a::len0 word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   453
  where "word_rotr n w = of_bl (rotater n (to_bl w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   454
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   455
definition word_rotl :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'a::len0 word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   456
  where "word_rotl n w = of_bl (rotate n (to_bl w))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   457
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   458
definition word_roti :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> 'a::len0 word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   459
  where "word_roti i w =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   460
    (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
   461
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   462
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   463
subsection \<open>Split and cat operations\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   464
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   465
definition word_cat :: "'a::len0 word \<Rightarrow> 'b::len0 word \<Rightarrow> 'c::len0 word"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   466
  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
   467
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   468
definition word_split :: "'a::len0 word \<Rightarrow> 'b::len0 word \<times> 'c::len0 word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   469
  where "word_split a =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   470
    (case bin_split (LENGTH('c)) (uint a) of
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   471
      (u, v) \<Rightarrow> (word_of_int u, word_of_int v))"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   472
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   473
definition word_rcat :: "'a::len0 word list \<Rightarrow> 'b::len0 word"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   474
  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
   475
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   476
definition word_rsplit :: "'a::len0 word \<Rightarrow> 'b::len word list"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   477
  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
   478
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
   479
definition max_word :: "'a::len word"
67443
3abf6a722518 standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents: 67408
diff changeset
   480
  \<comment> \<open>Largest representable machine integer.\<close>
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   481
  where "max_word = word_of_int (2 ^ LENGTH('a) - 1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   482
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   483
lemmas of_nth_def = word_set_bits_def (* FIXME duplicate *)
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   484
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   485
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   486
subsection \<open>Theorems about typedefs\<close>
46010
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
   487
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   488
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
   489
  by (auto simp: sint_uint word_ubin.eq_norm sbintrunc_bintrunc_lt)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   490
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   491
lemma uint_sint: "uint w = bintrunc (LENGTH('a)) (sint w)"
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
   492
  for w :: "'a::len word"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   493
  by (auto simp: sint_uint bintrunc_sbintrunc_le)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   494
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   495
lemma bintr_uint: "LENGTH('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   496
  for w :: "'a::len0 word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   497
  apply (subst word_ubin.norm_Rep [symmetric])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   498
  apply (simp only: bintrunc_bintrunc_min word_size)
54863
82acc20ded73 prefer more canonical names for lemmas on min/max
haftmann
parents: 54854
diff changeset
   499
  apply (simp add: min.absorb2)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   500
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   501
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   502
lemma wi_bintr:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   503
  "LENGTH('a::len0) \<le> n \<Longrightarrow>
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   504
    word_of_int (bintrunc n w) = (word_of_int w :: 'a word)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   505
  by (auto simp: word_ubin.norm_eq_iff [symmetric] min.absorb1)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   506
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   507
lemma td_ext_sbin:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   508
  "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (LENGTH('a::len)))
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   509
    (sbintrunc (LENGTH('a) - 1))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   510
  apply (unfold td_ext_def' sint_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   511
  apply (simp add : word_ubin.eq_norm)
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   512
  apply (cases "LENGTH('a)")
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   513
   apply (auto simp add : sints_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   514
  apply (rule sym [THEN trans])
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   515
   apply (rule word_ubin.Abs_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   516
  apply (simp only: bintrunc_sbintrunc)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   517
  apply (drule sym)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   518
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   519
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   520
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   521
lemma td_ext_sint:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   522
  "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (LENGTH('a::len)))
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   523
     (\<lambda>w. (w + 2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) -
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   524
         2 ^ (LENGTH('a) - 1))"
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   525
  using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   526
67408
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   527
text \<open>
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   528
  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
   529
  and interpretations do not produce thm duplicates. I.e.
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   530
  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
   531
  because the latter is the same thm as the former.
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   532
\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   533
interpretation word_sint:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   534
  td_ext
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   535
    "sint ::'a::len word \<Rightarrow> int"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   536
    word_of_int
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   537
    "sints (LENGTH('a::len))"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   538
    "\<lambda>w. (w + 2^(LENGTH('a::len) - 1)) mod 2^LENGTH('a::len) -
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   539
      2 ^ (LENGTH('a::len) - 1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   540
  by (rule td_ext_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   541
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   542
interpretation word_sbin:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   543
  td_ext
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   544
    "sint ::'a::len word \<Rightarrow> int"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   545
    word_of_int
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   546
    "sints (LENGTH('a::len))"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   547
    "sbintrunc (LENGTH('a::len) - 1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   548
  by (rule td_ext_sbin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   549
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   550
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   551
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   552
lemmas td_sint = word_sint.td
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   553
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   554
lemma to_bl_def': "(to_bl :: 'a::len0 word \<Rightarrow> bool list) = bin_to_bl (LENGTH('a)) \<circ> uint"
44762
8f9d09241a68 tuned proofs;
wenzelm
parents: 42793
diff changeset
   555
  by (auto simp: to_bl_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   556
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   557
lemmas word_reverse_no_def [simp] =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   558
  word_reverse_def [of "numeral w"] for w
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   559
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   560
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   561
  by (fact uints_def [unfolded no_bintr_alt1])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   562
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   563
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
   564
  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
   565
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   566
declare word_numeral_alt [symmetric, code_abbrev]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   567
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   568
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
   569
  by (simp only: word_numeral_alt wi_hom_neg)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   570
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   571
declare word_neg_numeral_alt [symmetric, code_abbrev]
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   572
47372
9ab4e22dac7b configure transfer method for 'a word -> int
huffman
parents: 47168
diff changeset
   573
lemma word_numeral_transfer [transfer_rule]:
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67122
diff changeset
   574
  "(rel_fun (=) pcr_word) numeral numeral"
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67122
diff changeset
   575
  "(rel_fun (=) pcr_word) (- numeral) (- numeral)"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 55833
diff changeset
   576
  apply (simp_all add: rel_fun_def word.pcr_cr_eq cr_word_def)
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   577
  using word_numeral_alt [symmetric] word_neg_numeral_alt [symmetric] by auto
47372
9ab4e22dac7b configure transfer method for 'a word -> int
huffman
parents: 47168
diff changeset
   578
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   579
lemma uint_bintrunc [simp]:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   580
  "uint (numeral bin :: 'a word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   581
    bintrunc (LENGTH('a::len0)) (numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   582
  unfolding word_numeral_alt by (rule word_ubin.eq_norm)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   583
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   584
lemma uint_bintrunc_neg [simp]:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   585
  "uint (- numeral bin :: 'a word) = bintrunc (LENGTH('a::len0)) (- numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   586
  by (simp only: word_neg_numeral_alt word_ubin.eq_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   587
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   588
lemma sint_sbintrunc [simp]:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   589
  "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
   590
  by (simp only: word_numeral_alt word_sbin.eq_norm)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   591
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   592
lemma sint_sbintrunc_neg [simp]:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   593
  "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
   594
  by (simp only: word_neg_numeral_alt word_sbin.eq_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   595
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   596
lemma unat_bintrunc [simp]:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   597
  "unat (numeral bin :: 'a::len0 word) = nat (bintrunc (LENGTH('a)) (numeral bin))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   598
  by (simp only: unat_def uint_bintrunc)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   599
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   600
lemma unat_bintrunc_neg [simp]:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   601
  "unat (- numeral bin :: 'a::len0 word) = nat (bintrunc (LENGTH('a)) (- numeral bin))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   602
  by (simp only: unat_def uint_bintrunc_neg)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   603
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
   604
lemma size_0_eq: "size w = 0 \<Longrightarrow> v = w"
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
   605
  for v w :: "'a::len0 word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   606
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   607
  apply (rule word_uint.Rep_eqD)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   608
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   609
    defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   610
    apply (rule word_ubin.norm_Rep)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   611
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   612
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   613
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   614
lemma uint_ge_0 [iff]: "0 \<le> uint x"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   615
  for x :: "'a::len0 word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   616
  using word_uint.Rep [of x] by (simp add: uints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   617
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   618
lemma uint_lt2p [iff]: "uint x < 2 ^ LENGTH('a)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   619
  for x :: "'a::len0 word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   620
  using word_uint.Rep [of x] by (simp add: uints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   621
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   622
lemma sint_ge: "- (2 ^ (LENGTH('a) - 1)) \<le> sint x"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   623
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   624
  using word_sint.Rep [of x] by (simp add: sints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   625
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   626
lemma sint_lt: "sint x < 2 ^ (LENGTH('a) - 1)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   627
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   628
  using word_sint.Rep [of x] by (simp add: sints_num)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   629
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   630
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
   631
  by (simp add: sign_Pls_ge_0)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   632
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   633
lemma uint_m2p_neg: "uint x - 2 ^ LENGTH('a) < 0"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   634
  for x :: "'a::len0 word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   635
  by (simp only: diff_less_0_iff_less uint_lt2p)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   636
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   637
lemma uint_m2p_not_non_neg: "\<not> 0 \<le> uint x - 2 ^ LENGTH('a)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   638
  for x :: "'a::len0 word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   639
  by (simp only: not_le uint_m2p_neg)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   640
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   641
lemma lt2p_lem: "LENGTH('a) \<le> n \<Longrightarrow> uint w < 2 ^ n"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   642
  for w :: "'a::len0 word"
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   643
  by (metis bintr_uint bintrunc_mod2p int_mod_lem zless2p)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   644
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   645
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   646
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   647
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   648
lemma uint_nat: "uint w = int (unat w)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   649
  by (auto simp: unat_def)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   650
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   651
lemma uint_numeral: "uint (numeral b :: 'a::len0 word) = numeral b mod 2 ^ LENGTH('a)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   652
  by (simp only: word_numeral_alt int_word_uint)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   653
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   654
lemma uint_neg_numeral: "uint (- numeral b :: 'a::len0 word) = - numeral b mod 2 ^ LENGTH('a)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   655
  by (simp only: word_neg_numeral_alt int_word_uint)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   656
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   657
lemma unat_numeral: "unat (numeral b :: 'a::len0 word) = numeral b mod 2 ^ LENGTH('a)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   658
  apply (unfold unat_def)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   659
  apply (clarsimp simp only: uint_numeral)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   660
  apply (rule nat_mod_distrib [THEN trans])
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   661
    apply (rule zero_le_numeral)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   662
   apply (simp_all add: nat_power_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   663
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   664
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   665
lemma sint_numeral:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   666
  "sint (numeral b :: 'a::len word) =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   667
    (numeral b +
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   668
      2 ^ (LENGTH('a) - 1)) mod 2 ^ LENGTH('a) -
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   669
      2 ^ (LENGTH('a) - 1)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   670
  unfolding word_numeral_alt by (rule int_word_sint)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   671
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   672
lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   673
  unfolding word_0_wi ..
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   674
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   675
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   676
  unfolding word_1_wi ..
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   677
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   678
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
   679
  by (simp add: wi_hom_syms)
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   680
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   681
lemma word_of_int_numeral [simp] : "(word_of_int (numeral bin) :: 'a::len0 word) = numeral bin"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   682
  by (simp only: word_numeral_alt)
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   683
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   684
lemma word_of_int_neg_numeral [simp]:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   685
  "(word_of_int (- numeral bin) :: 'a::len0 word) = - numeral bin"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   686
  by (simp only: word_numeral_alt wi_hom_syms)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   687
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   688
lemma word_int_case_wi:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   689
  "word_int_case f (word_of_int i :: 'b word) = f (i mod 2 ^ LENGTH('b::len0))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   690
  by (simp add: word_int_case_def word_uint.eq_norm)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   691
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   692
lemma word_int_split:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   693
  "P (word_int_case f x) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   694
    (\<forall>i. x = (word_of_int i :: 'b::len0 word) \<and> 0 \<le> i \<and> i < 2 ^ LENGTH('b) \<longrightarrow> P (f i))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   695
  by (auto simp: word_int_case_def word_uint.eq_norm mod_pos_pos_trivial)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   696
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   697
lemma word_int_split_asm:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   698
  "P (word_int_case f x) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   699
    (\<nexists>n. x = (word_of_int n :: 'b::len0 word) \<and> 0 \<le> n \<and> n < 2 ^ LENGTH('b::len0) \<and> \<not> P (f n))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   700
  by (auto simp: word_int_case_def word_uint.eq_norm mod_pos_pos_trivial)
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   701
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   702
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   703
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
   704
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   705
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
   706
  unfolding word_size by (rule uint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   707
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   708
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
   709
  unfolding word_size by (rule sint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   710
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   711
lemma sint_above_size: "2 ^ (size w - 1) \<le> x \<Longrightarrow> sint w < x"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   712
  for w :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   713
  unfolding word_size by (rule less_le_trans [OF sint_lt])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   714
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   715
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
   716
  for w :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   717
  unfolding word_size by (rule order_trans [OF _ sint_ge])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   718
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   719
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   720
subsection \<open>Testing bits\<close>
46010
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
   721
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   722
lemma test_bit_eq_iff: "test_bit u = test_bit v \<longleftrightarrow> u = v"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   723
  for u v :: "'a::len0 word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   724
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   725
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   726
lemma test_bit_size [rule_format] : "w !! n \<longrightarrow> n < size w"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   727
  for w :: "'a::len0 word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   728
  apply (unfold word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   729
  apply (subst word_ubin.norm_Rep [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   730
  apply (simp only: nth_bintr word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   731
  apply fast
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   732
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   733
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   734
lemma word_eq_iff: "x = y \<longleftrightarrow> (\<forall>n<LENGTH('a). x !! n = y !! n)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   735
  for x y :: "'a::len0 word"
46021
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   736
  unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric]
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   737
  by (metis test_bit_size [unfolded word_size])
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   738
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   739
lemma word_eqI: "(\<And>n. n < size u \<longrightarrow> u !! n = v !! n) \<Longrightarrow> u = v"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   740
  for u :: "'a::len0 word"
46021
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   741
  by (simp add: word_size word_eq_iff)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   742
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   743
lemma word_eqD: "u = v \<Longrightarrow> u !! x = v !! x"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   744
  for u v :: "'a::len0 word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   745
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   746
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   747
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
   748
  by (simp add: word_test_bit_def word_size nth_bintr [symmetric])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   749
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   750
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   751
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   752
lemma bin_nth_uint_imp: "bin_nth (uint w) n \<Longrightarrow> n < LENGTH('a)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   753
  for w :: "'a::len0 word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   754
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   755
  apply (subst word_ubin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   756
  apply assumption
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   757
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   758
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   759
lemma bin_nth_sint:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   760
  "LENGTH('a) \<le> n \<Longrightarrow>
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   761
    bin_nth (sint w) n = bin_nth (sint w) (LENGTH('a) - 1)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   762
  for w :: "'a::len word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   763
  apply (subst word_sbin.norm_Rep [symmetric])
46057
8664713db181 remove unnecessary intermediate lemmas
huffman
parents: 46026
diff changeset
   764
  apply (auto simp add: nth_sbintr)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   765
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   766
67408
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   767
\<comment> \<open>type definitions theorem for in terms of equivalent bool list\<close>
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   768
lemma td_bl:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   769
  "type_definition
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   770
    (to_bl :: 'a::len0 word \<Rightarrow> bool list)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   771
    of_bl
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   772
    {bl. length bl = LENGTH('a)}"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   773
  apply (unfold type_definition_def of_bl_def to_bl_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   774
  apply (simp add: word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   775
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   776
  apply (drule sym)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   777
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   778
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   779
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   780
interpretation word_bl:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   781
  type_definition
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   782
    "to_bl :: 'a::len0 word \<Rightarrow> bool list"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   783
    of_bl
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   784
    "{bl. length bl = LENGTH('a::len0)}"
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
   785
  by (fact td_bl)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   786
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
   787
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
   788
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   789
lemma word_size_bl: "size w = size (to_bl w)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   790
  by (auto simp: word_size)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   791
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   792
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
   793
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   794
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   795
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   796
  by (simp add: word_reverse_def word_bl.Abs_inverse)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   797
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   798
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   799
  by (simp add: word_reverse_def word_bl.Abs_inverse)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   800
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   801
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   802
  by (metis word_rev_rev)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   803
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   804
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   805
  by simp
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   806
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   807
lemma length_bl_gt_0 [iff]: "0 < length (to_bl x)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   808
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   809
  unfolding word_bl_Rep' by (rule len_gt_0)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   810
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   811
lemma bl_not_Nil [iff]: "to_bl x \<noteq> []"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   812
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   813
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   814
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   815
lemma length_bl_neq_0 [iff]: "length (to_bl x) \<noteq> 0"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   816
  for x :: "'a::len word"
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   817
  by (fact length_bl_gt_0 [THEN gr_implies_not0])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   818
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   819
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   820
  apply (unfold to_bl_def sint_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   821
  apply (rule trans [OF _ bl_sbin_sign])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   822
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   823
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   824
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   825
lemma of_bl_drop':
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   826
  "lend = length bl - LENGTH('a::len0) \<Longrightarrow>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   827
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   828
  by (auto simp: of_bl_def trunc_bl2bin [symmetric])
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   829
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   830
lemma test_bit_of_bl:
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   831
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < LENGTH('a) \<and> n < length bl)"
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
   832
  by (auto simp add: of_bl_def word_test_bit_def word_size
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
   833
      word_ubin.eq_norm nth_bintr bin_nth_of_bl)
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   834
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   835
lemma no_of_bl: "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (LENGTH('a)) (numeral bin))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   836
  by (simp add: of_bl_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   837
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   838
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   839
  by (auto simp: word_size to_bl_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   840
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   841
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   842
  by (simp add: uint_bl word_size)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   843
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   844
lemma to_bl_of_bin: "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (LENGTH('a)) bin"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   845
  by (auto simp: uint_bl word_ubin.eq_norm word_size)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   846
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   847
lemma to_bl_numeral [simp]:
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   848
  "to_bl (numeral bin::'a::len0 word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   849
    bin_to_bl (LENGTH('a)) (numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   850
  unfolding word_numeral_alt by (rule to_bl_of_bin)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   851
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   852
lemma to_bl_neg_numeral [simp]:
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
   853
  "to_bl (- numeral bin::'a::len0 word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   854
    bin_to_bl (LENGTH('a)) (- numeral bin)"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   855
  unfolding word_neg_numeral_alt by (rule to_bl_of_bin)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   856
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   857
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   858
  by (simp add: uint_bl word_size)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   859
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   860
lemma uint_bl_bin: "bl_to_bin (bin_to_bl (LENGTH('a)) (uint x)) = uint x"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   861
  for x :: "'a::len0 word"
46011
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   862
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   863
67408
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   864
\<comment> \<open>naturals\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   865
lemma uints_unats: "uints n = int ` unats n"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   866
  apply (unfold unats_def uints_num)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   867
  apply safe
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   868
    apply (rule_tac image_eqI)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   869
     apply (erule_tac nat_0_le [symmetric])
66912
a99a7cbf0fb5 generalized lemmas cancelling real_of_int/real in (in)equalities with power; completed set of related simp rules; lemmas about floorlog/bitlen
immler
parents: 66808
diff changeset
   870
  by auto
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   871
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   872
lemma unats_uints: "unats n = nat ` uints n"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   873
  by (auto simp: uints_unats image_iff)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   874
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   875
lemmas bintr_num =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   876
  word_ubin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   877
lemmas sbintr_num =
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   878
  word_sbin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   879
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   880
lemma num_of_bintr':
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   881
  "bintrunc (LENGTH('a::len0)) (numeral a) = (numeral b) \<Longrightarrow>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   882
    numeral a = (numeral b :: 'a word)"
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   883
  unfolding bintr_num by (erule subst, simp)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   884
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   885
lemma num_of_sbintr':
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   886
  "sbintrunc (LENGTH('a::len) - 1) (numeral a) = (numeral b) \<Longrightarrow>
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   887
    numeral a = (numeral b :: 'a word)"
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   888
  unfolding sbintr_num by (erule subst, simp)
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   889
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   890
lemma num_abs_bintr:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   891
  "(numeral x :: 'a word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   892
    word_of_int (bintrunc (LENGTH('a::len0)) (numeral x))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   893
  by (simp only: word_ubin.Abs_norm word_numeral_alt)
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   894
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   895
lemma num_abs_sbintr:
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   896
  "(numeral x :: 'a word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   897
    word_of_int (sbintrunc (LENGTH('a::len) - 1) (numeral x))"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   898
  by (simp only: word_sbin.Abs_norm word_numeral_alt)
46962
5bdcdb28be83 make more word theorems respect int/bin distinction
huffman
parents: 46656
diff changeset
   899
67408
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   900
text \<open>
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   901
  \<open>cast\<close> -- note, no arg for new length, as it's determined by type of result,
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   902
  thus in \<open>cast w = w\<close>, the type means cast to length of \<open>w\<close>!
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   903
\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   904
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   905
lemma ucast_id: "ucast w = w"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   906
  by (auto simp: ucast_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   907
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   908
lemma scast_id: "scast w = w"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   909
  by (auto simp: scast_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   910
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   911
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   912
  by (auto simp: ucast_def of_bl_def uint_bl word_size)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   913
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   914
lemma nth_ucast: "(ucast w::'a::len0 word) !! n = (w !! n \<and> n < LENGTH('a))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   915
  by (simp add: ucast_def test_bit_bin word_ubin.eq_norm nth_bintr word_size)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   916
    (fast elim!: bin_nth_uint_imp)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   917
67408
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
   918
\<comment> \<open>literal u(s)cast\<close>
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   919
lemma ucast_bintr [simp]:
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
   920
  "ucast (numeral w :: 'a::len0 word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   921
    word_of_int (bintrunc (LENGTH('a)) (numeral w))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   922
  by (simp add: ucast_def)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   923
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
   924
(* TODO: neg_numeral *)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   925
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   926
lemma scast_sbintr [simp]:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   927
  "scast (numeral w ::'a::len word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   928
    word_of_int (sbintrunc (LENGTH('a) - Suc 0) (numeral w))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   929
  by (simp add: scast_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   930
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   931
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = LENGTH('a)"
46011
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   932
  unfolding source_size_def word_size Let_def ..
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   933
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   934
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = LENGTH('b)"
46011
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   935
  unfolding target_size_def word_size Let_def ..
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   936
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   937
lemma is_down: "is_down c \<longleftrightarrow> LENGTH('b) \<le> LENGTH('a)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   938
  for c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   939
  by (simp only: is_down_def source_size target_size)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   940
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   941
lemma is_up: "is_up c \<longleftrightarrow> LENGTH('a) \<le> LENGTH('b)"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   942
  for c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   943
  by (simp only: is_up_def source_size target_size)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   944
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   945
lemmas is_up_down = trans [OF is_up is_down [symmetric]]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   946
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   947
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   948
  apply (unfold is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   949
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   950
  apply (rule ext)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   951
  apply (unfold ucast_def scast_def uint_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   952
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   953
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   954
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   955
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   956
lemma word_rev_tf:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   957
  "to_bl (of_bl bl::'a::len0 word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   958
    rev (takefill False (LENGTH('a)) (rev bl))"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   959
  by (auto simp: of_bl_def uint_bl bl_bin_bl_rtf word_ubin.eq_norm word_size)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   960
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   961
lemma word_rep_drop:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   962
  "to_bl (of_bl bl::'a::len0 word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   963
    replicate (LENGTH('a) - length bl) False @
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   964
    drop (length bl - LENGTH('a)) bl"
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   965
  by (simp add: word_rev_tf takefill_alt rev_take)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   966
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   967
lemma to_bl_ucast:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   968
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) =
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   969
    replicate (LENGTH('a) - LENGTH('b)) False @
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
   970
    drop (LENGTH('b) - LENGTH('a)) (to_bl w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   971
  apply (unfold ucast_bl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   972
  apply (rule trans)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   973
   apply (rule word_rep_drop)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   974
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   975
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   976
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   977
lemma ucast_up_app [OF refl]:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   978
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   979
    to_bl (uc w) = replicate n False @ (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   980
  by (auto simp add : source_size target_size to_bl_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   981
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   982
lemma ucast_down_drop [OF refl]:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   983
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   984
    to_bl (uc w) = drop n (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   985
  by (auto simp add : source_size target_size to_bl_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   986
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   987
lemma scast_down_drop [OF refl]:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   988
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   989
    to_bl (sc w) = drop n (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   990
  apply (subgoal_tac "sc = ucast")
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   991
   apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   992
   apply simp
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   993
   apply (erule ucast_down_drop)
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   994
  apply (rule down_cast_same [symmetric])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   995
  apply (simp add : source_size target_size is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   996
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   997
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
   998
lemma sint_up_scast [OF refl]: "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   999
  apply (unfold is_up)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1000
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1001
  apply (simp add: scast_def word_sbin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1002
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1003
    prefer 3
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1004
    apply (rule word_sbin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1005
   apply (rule sbintrunc_sbintrunc_l)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1006
   defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1007
   apply (subst word_sbin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1008
   apply (rule refl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1009
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1010
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1011
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1012
lemma uint_up_ucast [OF refl]: "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1013
  apply (unfold is_up)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1014
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1015
  apply (rule bin_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1016
  apply (fold word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1017
  apply (auto simp add: nth_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1018
  apply (auto simp add: test_bit_bin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1019
  done
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1020
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1021
lemma ucast_up_ucast [OF refl]: "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1022
  apply (simp (no_asm) add: ucast_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1023
  apply (clarsimp simp add: uint_up_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1024
  done
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1025
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1026
lemma scast_up_scast [OF refl]: "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1027
  apply (simp (no_asm) add: scast_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1028
  apply (clarsimp simp add: sint_up_scast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1029
  done
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1030
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1031
lemma ucast_of_bl_up [OF refl]: "w = of_bl bl \<Longrightarrow> size bl \<le> size w \<Longrightarrow> ucast w = of_bl bl"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1032
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1033
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1034
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1035
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1036
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1037
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1038
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1039
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1040
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1041
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1042
lemma up_ucast_surj:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1043
  "is_up (ucast :: 'b::len0 word \<Rightarrow> 'a::len0 word) \<Longrightarrow>
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1044
    surj (ucast :: 'a word \<Rightarrow> 'b word)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1045
  by (rule surjI) (erule ucast_up_ucast_id)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1046
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1047
lemma up_scast_surj:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1048
  "is_up (scast :: 'b::len word \<Rightarrow> 'a::len word) \<Longrightarrow>
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1049
    surj (scast :: 'a word \<Rightarrow> 'b word)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1050
  by (rule surjI) (erule scast_up_scast_id)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1051
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1052
lemma down_scast_inj:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1053
  "is_down (scast :: 'b::len word \<Rightarrow> 'a::len word) \<Longrightarrow>
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1054
    inj_on (ucast :: 'a word \<Rightarrow> 'b word) A"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1055
  by (rule inj_on_inverseI, erule scast_down_scast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1056
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1057
lemma down_ucast_inj:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1058
  "is_down (ucast :: 'b::len0 word \<Rightarrow> 'a::len0 word) \<Longrightarrow>
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1059
    inj_on (ucast :: 'a word \<Rightarrow> 'b word) A"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1060
  by (rule inj_on_inverseI) (erule ucast_down_ucast_id)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1061
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1062
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1063
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1064
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1065
lemma ucast_down_wi [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
46646
0abbf6dd09ee remove ill-formed lemma of_bl_no; adapt proofs
huffman
parents: 46645
diff changeset
  1066
  apply (unfold is_down)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1067
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1068
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1069
  apply (erule bintrunc_bintrunc_ge)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1070
  done
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1071
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1072
lemma ucast_down_no [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1073
  unfolding word_numeral_alt by clarify (rule ucast_down_wi)
46646
0abbf6dd09ee remove ill-formed lemma of_bl_no; adapt proofs
huffman
parents: 46645
diff changeset
  1074
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1075
lemma ucast_down_bl [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
46646
0abbf6dd09ee remove ill-formed lemma of_bl_no; adapt proofs
huffman
parents: 46645
diff changeset
  1076
  unfolding of_bl_def by clarify (erule ucast_down_wi)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1077
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1078
lemmas slice_def' = slice_def [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1079
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1080
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1081
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1082
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1083
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1084
subsection \<open>Word Arithmetic\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1085
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1086
lemma word_less_alt: "a < b \<longleftrightarrow> uint a < uint b"
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1087
  by (fact word_less_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1088
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1089
lemma signed_linorder: "class.linorder word_sle word_sless"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1090
  by standard (auto simp: word_sle_def word_sless_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1091
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1092
interpretation signed: linorder "word_sle" "word_sless"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1093
  by (rule signed_linorder)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1094
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1095
lemma udvdI: "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1096
  by (auto simp: udvd_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1097
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1098
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1099
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1100
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1101
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1102
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1103
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1104
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1105
lemma word_m1_wi: "- 1 = word_of_int (- 1)"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1106
  by (simp add: word_neg_numeral_alt [of Num.One])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1107
46648
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1108
lemma word_0_bl [simp]: "of_bl [] = 0"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1109
  by (simp add: of_bl_def)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1110
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1111
lemma word_1_bl: "of_bl [True] = 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1112
  by (simp add: of_bl_def bl_to_bin_def)
46648
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1113
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1114
lemma uint_eq_0 [simp]: "uint 0 = 0"
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1115
  unfolding word_0_wi word_ubin.eq_norm by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1116
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1117
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
46648
689ebcbd6343 avoid using Int.Pls_def in proofs
huffman
parents: 46647
diff changeset
  1118
  by (simp add: of_bl_def bl_to_bin_rep_False)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1119
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1120
lemma to_bl_0 [simp]: "to_bl (0::'a::len0 word) = replicate (LENGTH('a)) False"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1121
  by (simp add: uint_bl word_size bin_to_bl_zero)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1122
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1123
lemma uint_0_iff: "uint x = 0 \<longleftrightarrow> x = 0"
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1124
  by (simp add: word_uint_eq_iff)
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1125
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1126
lemma unat_0_iff: "unat x = 0 \<longleftrightarrow> x = 0"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1127
  by (auto simp: unat_def nat_eq_iff uint_0_iff)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1128
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1129
lemma unat_0 [simp]: "unat 0 = 0"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1130
  by (auto simp: unat_def)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1131
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1132
lemma size_0_same': "size w = 0 \<Longrightarrow> w = v"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1133
  for v w :: "'a::len0 word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1134
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1135
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1136
    defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1137
    apply (rule word_uint.Rep_inverse)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1138
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1139
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1140
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1141
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1142
lemmas size_0_same = size_0_same' [unfolded word_size]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1143
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1144
lemmas unat_eq_0 = unat_0_iff
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1145
lemmas unat_eq_zero = unat_0_iff
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1146
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1147
lemma unat_gt_0: "0 < unat x \<longleftrightarrow> x \<noteq> 0"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1148
  by (auto simp: unat_0_iff [symmetric])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1149
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1150
lemma ucast_0 [simp]: "ucast 0 = 0"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1151
  by (simp add: ucast_def)
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1152
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1153
lemma sint_0 [simp]: "sint 0 = 0"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1154
  by (simp add: sint_uint)
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1155
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1156
lemma scast_0 [simp]: "scast 0 = 0"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1157
  by (simp add: scast_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1158
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58061
diff changeset
  1159
lemma sint_n1 [simp] : "sint (- 1) = - 1"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1160
  by (simp only: word_m1_wi word_sbin.eq_norm) simp
54489
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1161
03ff4d1e6784 eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents: 54225
diff changeset
  1162
lemma scast_n1 [simp]: "scast (- 1) = - 1"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1163
  by (simp add: scast_def)
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1164
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1165
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1166
  by (simp only: word_1_wi word_ubin.eq_norm) (simp add: bintrunc_minus_simps(4))
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1167
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1168
lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1169
  by (simp add: unat_def)
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1170
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1171
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1172
  by (simp add: ucast_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1173
67408
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
  1174
\<comment> \<open>now, to get the weaker results analogous to \<open>word_div\<close>/\<open>mod_def\<close>\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1175
55816
e8dd03241e86 cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents: 55415
diff changeset
  1176
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1177
subsection \<open>Transferring goals from words to ints\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1178
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1179
lemma word_ths:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1180
  shows word_succ_p1: "word_succ a = a + 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1181
    and word_pred_m1: "word_pred a = a - 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1182
    and word_pred_succ: "word_pred (word_succ a) = a"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1183
    and word_succ_pred: "word_succ (word_pred a) = a"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1184
    and word_mult_succ: "word_succ a * b = b + a * b"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
  1185
  by (transfer, simp add: algebra_simps)+
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1186
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1187
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1188
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1189
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1190
lemma uint_word_ariths:
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1191
  fixes a b :: "'a::len0 word"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1192
  shows "uint (a + b) = (uint a + uint b) mod 2 ^ LENGTH('a::len0)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1193
    and "uint (a - b) = (uint a - uint b) mod 2 ^ LENGTH('a)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1194
    and "uint (a * b) = uint a * uint b mod 2 ^ LENGTH('a)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1195
    and "uint (- a) = - uint a mod 2 ^ LENGTH('a)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1196
    and "uint (word_succ a) = (uint a + 1) mod 2 ^ LENGTH('a)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1197
    and "uint (word_pred a) = (uint a - 1) mod 2 ^ LENGTH('a)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1198
    and "uint (0 :: 'a word) = 0 mod 2 ^ LENGTH('a)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1199
    and "uint (1 :: 'a word) = 1 mod 2 ^ LENGTH('a)"
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1200
  by (simp_all add: word_arith_wis [THEN trans [OF uint_cong int_word_uint]])
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1201
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1202
lemma uint_word_arith_bintrs:
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1203
  fixes a b :: "'a::len0 word"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1204
  shows "uint (a + b) = bintrunc (LENGTH('a)) (uint a + uint b)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1205
    and "uint (a - b) = bintrunc (LENGTH('a)) (uint a - uint b)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1206
    and "uint (a * b) = bintrunc (LENGTH('a)) (uint a * uint b)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1207
    and "uint (- a) = bintrunc (LENGTH('a)) (- uint a)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1208
    and "uint (word_succ a) = bintrunc (LENGTH('a)) (uint a + 1)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1209
    and "uint (word_pred a) = bintrunc (LENGTH('a)) (uint a - 1)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1210
    and "uint (0 :: 'a word) = bintrunc (LENGTH('a)) 0"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1211
    and "uint (1 :: 'a word) = bintrunc (LENGTH('a)) 1"
55818
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1212
  by (simp_all add: uint_word_ariths bintrunc_mod2p)
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1213
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1214
lemma sint_word_ariths:
d8b2f50705d0 more precise imports;
haftmann
parents: 55817
diff changeset
  1215
  fixes a b :: "'a::len word"
70185
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1216
  shows "sint (a + b) = sbintrunc (LENGTH('a) - 1) (sint a + sint b)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1217
    and "sint (a - b) = sbintrunc (LENGTH('a) - 1) (sint a - sint b)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1218
    and "sint (a * b) = sbintrunc (LENGTH('a) - 1) (sint a * sint b)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1219
    and "sint (- a) = sbintrunc (LENGTH('a) - 1) (- sint a)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1220
    and "sint (word_succ a) = sbintrunc (LENGTH('a) - 1) (sint a + 1)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1221
    and "sint (word_pred a) = sbintrunc (LENGTH('a) - 1) (sint a - 1)"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1222
    and "sint (0 :: 'a word) = sbintrunc (LENGTH('a) - 1) 0"
ac1706cdde25 clarified notation
haftmann
parents: 70183
diff changeset
  1223
    and "sint (1 :: 'a word) = sbintrunc (LENGTH('a) - 1) 1"
64593
50c715579715 reoriented congruence rules in non-explosive direction
haftmann
parents: 64243
diff changeset
  1224
         apply (simp_all only: word_sbin.inverse_norm [symmetric])
50c715579715 reoriented congruence rules in non-explosive direction
haftmann
parents: 64243
diff changeset
  1225
         apply (simp_all add: wi_hom_syms)
50c715579715 reoriented congruence rules in non-explosive direction
haftmann
parents: 64243
diff changeset
  1226
   apply transfer apply simp
50c715579715 reoriented congruence rules in non-explosive direction
haftmann
parents: 64243
diff changeset
  1227
  apply transfer apply simp
50c715579715 reoriented congruence rules in non-explosive direction
haftmann
parents: 64243
diff changeset
  1228
  done
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1229
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1230
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1231
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1232
58410
6d46ad54a2ab explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents: 58061
diff changeset
  1233
lemma word_pred_0_n1: "word_pred 0 = word_of_int (- 1)"
47374
9475d524bafb set up and use lift_definition for word operations
huffman
parents: 47372
diff changeset
  1234
  unfolding word_pred_m1 by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1235
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1236
lemma succ_pred_no [simp]:
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1237
    "word_succ (numeral w) = numeral w + 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1238
    "word_pred (numeral w) = numeral w - 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1239
    "word_succ (- numeral w) = - numeral w + 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1240
    "word_pred (- numeral w) = - numeral w - 1"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1241
  by (simp_all add: word_succ_p1 word_pred_m1)
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1242
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1243
lemma word_sp_01 [simp]:
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1244
  "word_succ (- 1) = 0 \<and> word_succ 0 = 1 \<and> word_pred 0 = - 1 \<and> word_pred 1 = 0"
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1245
  by (simp_all add: word_succ_p1 word_pred_m1)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1246
67408
4a4c14b24800 prefer formal comments;
wenzelm
parents: 67399
diff changeset
  1247
\<comment> \<open>alternative approach to lifting arithmetic equalities\<close>
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1248
lemma word_of_int_Ex: "\<exists>y. x = word_of_int y"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1249
  by (rule_tac x="uint x" in exI) simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1250
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1251
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1252
subsection \<open>Order on fixed-length words\<close>
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1253
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1254
lemma word_zero_le [simp]: "0 \<le> y"
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1255
  for y :: "'a::len0 word"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1256
  unfolding word_le_def by auto
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1257
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1258
lemma word_m1_ge [simp] : "word_pred 0 \<ge> y" (* FIXME: delete *)
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1259
  by (simp only: word_le_def word_pred_0_n1 word_uint.eq_norm m1mod2k) auto
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1260
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1261
lemma word_n1_ge [simp]: "y \<le> -1"
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1262
  for y :: "'a::len0 word"
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1263
  by (simp only: word_le_def word_m1_wi word_uint.eq_norm m1mod2k) auto
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1264
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1265
lemmas word_not_simps [simp] =
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1266
  word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1267
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1268
lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> y"
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1269
  for y :: "'a::len0 word"
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1270
  by (simp add: less_le)
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1271
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 46962
diff changeset
  1272
lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1273
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1274
lemma word_sless_alt: "a <s b \<longleftrightarrow> sint a < sint b"
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1275
  by (auto simp add: word_sle_def word_sless_def less_le)
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1276
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1277
lemma word_le_nat_alt: "a \<le> b \<longleftrightarrow> unat a \<le> unat b"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1278
  unfolding unat_def word_le_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1279
  by (rule nat_le_eq_zle [symmetric]) simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1280
65328
2510b0ce28da misc tuning and modernization;
wenzelm
parents: 65268
diff changeset
  1281
lemma word_less_nat_alt: "a < b \<longleftrightarrow> unat a < unat b"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1282
  unfolding unat_def word_less_alt
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1283
  by (rule nat_less_eq_zless [symmetric]) simp
65268
75f2aa8ecb12 misc tuning and modernization;
wenzelm
parents: 64593
diff changeset
  1284