src/HOL/Word/Word.thy
author huffman
Sun Mar 25 20:15:39 2012 +0200 (2012-03-25)
changeset 47108 2a1953f0d20d
parent 46962 5bdcdb28be83
child 47168 8395d7d63fc8
permissions -rw-r--r--
merged fork with new numeral representation (see NEWS)
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(*  Title:      HOL/Word/Word.thy
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    Author:     Jeremy Dawson and Gerwin Klein, NICTA
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*)
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header {* A type of finite bit strings *}
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theory Word
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imports
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  Type_Length
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  Misc_Typedef
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  "~~/src/HOL/Library/Boolean_Algebra"
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  Bool_List_Representation
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uses ("~~/src/HOL/Word/Tools/smt_word.ML")
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begin
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text {* see @{text "Examples/WordExamples.thy"} for examples *}
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subsection {* Type definition *}
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typedef (open) 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}"
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  morphisms uint Abs_word by auto
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lemma uint_nonnegative:
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  "0 \<le> uint w"
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  using word.uint [of w] by simp
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lemma uint_bounded:
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  fixes w :: "'a::len0 word"
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  shows "uint w < 2 ^ len_of TYPE('a)"
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  using word.uint [of w] by simp
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lemma uint_idem:
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  fixes w :: "'a::len0 word"
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  shows "uint w mod 2 ^ len_of TYPE('a) = uint w"
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  using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial)
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definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" where
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  -- {* representation of words using unsigned or signed bins, 
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        only difference in these is the type class *}
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  "word_of_int k = Abs_word (k mod 2 ^ len_of TYPE('a))" 
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lemma uint_word_of_int:
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  "uint (word_of_int k :: 'a::len0 word) = k mod 2 ^ len_of TYPE('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:
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  "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 word_uint_eq_iff:
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  "a = b \<longleftrightarrow> uint a = uint b"
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  by (simp add: uint_inject)
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lemma word_uint_eqI:
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  "uint a = uint b \<Longrightarrow> a = b"
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  by (simp add: word_uint_eq_iff)
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subsection {* Basic code generation setup *}
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definition Word :: "int \<Rightarrow> 'a::len0 word"
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where
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  [code_post]: "Word = word_of_int"
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lemma [code abstype]:
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  "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" where
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  "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"
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instance proof
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qed (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 (Code_Numeral.int_of 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 {* Type conversions and casting *}
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definition sint :: "'a :: len word => int" where
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  -- {* treats the most-significant-bit as a sign bit *}
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  sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)"
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definition unat :: "'a :: len0 word => nat" where
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  "unat w = nat (uint w)"
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definition uints :: "nat => int set" where
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  -- "the sets of integers representing the words"
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  "uints n = range (bintrunc n)"
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definition sints :: "nat => int set" where
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  "sints n = range (sbintrunc (n - 1))"
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definition unats :: "nat => nat set" where
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  "unats n = {i. i < 2 ^ n}"
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definition norm_sint :: "nat => int => int" where
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  "norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)"
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definition scast :: "'a :: len word => 'b :: len word" where
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  -- "cast a word to a different length"
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  "scast w = word_of_int (sint w)"
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definition ucast :: "'a :: len0 word => 'b :: len0 word" where
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  "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
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  word_size: "size (w :: 'a word) = len_of TYPE('a)"
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instance ..
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end
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definition source_size :: "('a :: len0 word => 'b) => nat" where
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  -- "whether a cast (or other) function is to a longer or shorter length"
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  "source_size c = (let arb = undefined ; x = c arb in size arb)"  
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definition target_size :: "('a => 'b :: len0 word) => nat" where
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  "target_size c = size (c undefined)"
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definition is_up :: "('a :: len0 word => 'b :: len0 word) => bool" where
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  "is_up c \<longleftrightarrow> source_size c <= target_size c"
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definition is_down :: "('a :: len0 word => 'b :: len0 word) => bool" where
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  "is_down c \<longleftrightarrow> target_size c <= source_size c"
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definition of_bl :: "bool list => 'a :: len0 word" where
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  "of_bl bl = word_of_int (bl_to_bin bl)"
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definition to_bl :: "'a :: len0 word => bool list" where
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  "to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)"
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definition word_reverse :: "'a :: len0 word => 'a word" where
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  "word_reverse w = of_bl (rev (to_bl w))"
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definition word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b" where
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  "word_int_case f w = f (uint w)"
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translations
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  "case x of XCONST of_int y => b" == "CONST word_int_case (%y. b) x"
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  "case x of (XCONST of_int :: 'a) y => b" => "CONST word_int_case (%y. b) x"
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subsection {* Type-definition locale instantiations *}
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lemma word_size_gt_0 [iff]: "0 < size (w::'a::len word)"
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  by (fact xtr1 [OF word_size len_gt_0])
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lemmas lens_gt_0 = word_size_gt_0 len_gt_0
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lemmas lens_not_0 [iff] = lens_gt_0 [THEN gr_implies_not0]
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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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lemma 
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  uint_0:"0 <= uint x" and 
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  uint_lt: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
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  by (auto simp: uint [unfolded atLeastLessThan_iff])
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lemma uint_mod_same:
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  "uint x mod 2 ^ len_of TYPE('a) = uint (x::'a::len0 word)"
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  by (simp add: int_mod_eq uint_lt uint_0)
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lemma td_ext_uint: 
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  "td_ext (uint :: 'a word => int) word_of_int (uints (len_of TYPE('a::len0))) 
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    (%w::int. w mod 2 ^ len_of TYPE('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 "uint::'a::len0 word \<Rightarrow> int" 
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         word_of_int 
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         "uints (len_of TYPE('a::len0))"
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         "\<lambda>w. w mod 2 ^ len_of TYPE('a::len0)"
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  by (rule 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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lemmas td_ext_ubin = td_ext_uint 
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  [unfolded len_gt_0 no_bintr_alt1 [symmetric]]
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interpretation word_ubin:
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  td_ext "uint::'a::len0 word \<Rightarrow> int" 
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         word_of_int 
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         "uints (len_of TYPE('a::len0))"
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         "bintrunc (len_of TYPE('a::len0))"
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  by (rule td_ext_ubin)
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lemma split_word_all:
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  "(\<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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  hence "PROP P (word_of_int (uint x))" .
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  thus "PROP P x" by simp
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qed
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subsection  "Arithmetic operations"
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definition
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  word_succ :: "'a :: len0 word => 'a word"
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where
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  "word_succ a = word_of_int (uint a + 1)"
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definition
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  word_pred :: "'a :: len0 word => 'a word"
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where
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  "word_pred a = word_of_int (uint a - 1)"
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instantiation word :: (len0) "{neg_numeral, Divides.div, comm_monoid_mult, comm_ring}"
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begin
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definition
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  word_0_wi: "0 = word_of_int 0"
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definition
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  word_1_wi: "1 = word_of_int 1"
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definition
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  word_add_def: "a + b = word_of_int (uint a + uint b)"
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definition
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  word_sub_wi: "a - b = word_of_int (uint a - uint b)"
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definition
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  word_minus_def: "- a = word_of_int (- uint a)"
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definition
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  word_mult_def: "a * b = word_of_int (uint a * uint b)"
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definition
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  word_div_def: "a div b = word_of_int (uint a div uint b)"
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definition
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  word_mod_def: "a mod b = word_of_int (uint a mod uint b)"
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lemmas word_arith_wis =
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  word_add_def word_sub_wi word_mult_def word_minus_def 
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  word_succ_def word_pred_def word_0_wi word_1_wi
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lemmas arths = 
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  bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm]
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lemma wi_homs: 
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  shows
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  wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and
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  wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)" and
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  wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and
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  wi_hom_neg: "- word_of_int a = word_of_int (- a)" and
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  wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and
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  wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
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  by (auto simp: word_arith_wis arths)
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lemmas wi_hom_syms = wi_homs [symmetric]
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lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
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lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
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instance
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  by default (auto simp: split_word_all word_of_int_homs algebra_simps)
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end
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instance word :: (len) comm_ring_1
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proof
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  have "0 < len_of TYPE('a)" by (rule len_gt_0)
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  then show "(0::'a word) \<noteq> 1"
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    unfolding word_0_wi word_1_wi
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    by (auto simp add: word_ubin.norm_eq_iff [symmetric] gr0_conv_Suc)
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qed
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lemma word_of_nat: "of_nat n = word_of_int (int n)"
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  by (induct n) (auto simp add : word_of_int_hom_syms)
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lemma word_of_int: "of_int = word_of_int"
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  apply (rule ext)
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  apply (case_tac x rule: int_diff_cases)
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  apply (simp add: word_of_nat wi_hom_sub)
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  done
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definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where
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  "a udvd b = (EX n>=0. uint b = n * uint a)"
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subsection "Ordering"
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instantiation word :: (len0) linorder
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begin
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definition
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  word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
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definition
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  word_less_def: "a < b \<longleftrightarrow> uint a < uint b"
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instance
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  by default (auto simp: word_less_def word_le_def)
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end
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definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where
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  "a <=s b = (sint a <= sint b)"
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definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where
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  "(x <s y) = (x <=s y & x ~= y)"
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subsection "Bit-wise operations"
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   343
haftmann@37660
   344
instantiation word :: (len0) bits
haftmann@37660
   345
begin
haftmann@37660
   346
haftmann@37660
   347
definition
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   348
  word_and_def: 
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   349
  "(a::'a word) AND b = word_of_int (uint a AND uint b)"
haftmann@37660
   350
haftmann@37660
   351
definition
haftmann@37660
   352
  word_or_def:  
haftmann@37660
   353
  "(a::'a word) OR b = word_of_int (uint a OR uint b)"
haftmann@37660
   354
haftmann@37660
   355
definition
haftmann@37660
   356
  word_xor_def: 
haftmann@37660
   357
  "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
haftmann@37660
   358
haftmann@37660
   359
definition
haftmann@37660
   360
  word_not_def: 
haftmann@37660
   361
  "NOT (a::'a word) = word_of_int (NOT (uint a))"
haftmann@37660
   362
haftmann@37660
   363
definition
haftmann@37660
   364
  word_test_bit_def: "test_bit a = bin_nth (uint a)"
haftmann@37660
   365
haftmann@37660
   366
definition
haftmann@37660
   367
  word_set_bit_def: "set_bit a n x =
haftmann@37660
   368
   word_of_int (bin_sc n (If x 1 0) (uint a))"
haftmann@37660
   369
haftmann@37660
   370
definition
haftmann@37660
   371
  word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"
haftmann@37660
   372
haftmann@37660
   373
definition
haftmann@37660
   374
  word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1"
haftmann@37660
   375
haftmann@37660
   376
definition shiftl1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   377
  "shiftl1 w = word_of_int (uint w BIT 0)"
haftmann@37660
   378
haftmann@37660
   379
definition shiftr1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   380
  -- "shift right as unsigned or as signed, ie logical or arithmetic"
haftmann@37660
   381
  "shiftr1 w = word_of_int (bin_rest (uint w))"
haftmann@37660
   382
haftmann@37660
   383
definition
haftmann@37660
   384
  shiftl_def: "w << n = (shiftl1 ^^ n) w"
haftmann@37660
   385
haftmann@37660
   386
definition
haftmann@37660
   387
  shiftr_def: "w >> n = (shiftr1 ^^ n) w"
haftmann@37660
   388
haftmann@37660
   389
instance ..
haftmann@37660
   390
haftmann@37660
   391
end
haftmann@37660
   392
haftmann@37660
   393
instantiation word :: (len) bitss
haftmann@37660
   394
begin
haftmann@37660
   395
haftmann@37660
   396
definition
haftmann@37660
   397
  word_msb_def: 
huffman@46001
   398
  "msb a \<longleftrightarrow> bin_sign (sint a) = -1"
haftmann@37660
   399
haftmann@37660
   400
instance ..
haftmann@37660
   401
haftmann@37660
   402
end
haftmann@37660
   403
haftmann@37660
   404
definition setBit :: "'a :: len0 word => nat => 'a word" where 
haftmann@40827
   405
  "setBit w n = set_bit w n True"
haftmann@37660
   406
haftmann@37660
   407
definition clearBit :: "'a :: len0 word => nat => 'a word" where
haftmann@40827
   408
  "clearBit w n = set_bit w n False"
haftmann@37660
   409
haftmann@37660
   410
haftmann@37660
   411
subsection "Shift operations"
haftmann@37660
   412
haftmann@37660
   413
definition sshiftr1 :: "'a :: len word => 'a word" where 
haftmann@40827
   414
  "sshiftr1 w = word_of_int (bin_rest (sint w))"
haftmann@37660
   415
haftmann@37660
   416
definition bshiftr1 :: "bool => 'a :: len word => 'a word" where
haftmann@40827
   417
  "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
haftmann@37660
   418
haftmann@37660
   419
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where
haftmann@40827
   420
  "w >>> n = (sshiftr1 ^^ n) w"
haftmann@37660
   421
haftmann@37660
   422
definition mask :: "nat => 'a::len word" where
haftmann@40827
   423
  "mask n = (1 << n) - 1"
haftmann@37660
   424
haftmann@37660
   425
definition revcast :: "'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   426
  "revcast w =  of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
haftmann@37660
   427
haftmann@37660
   428
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   429
  "slice1 n w = of_bl (takefill False n (to_bl w))"
haftmann@37660
   430
haftmann@37660
   431
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   432
  "slice n w = slice1 (size w - n) w"
haftmann@37660
   433
haftmann@37660
   434
haftmann@37660
   435
subsection "Rotation"
haftmann@37660
   436
haftmann@37660
   437
definition rotater1 :: "'a list => 'a list" where
haftmann@40827
   438
  "rotater1 ys = 
haftmann@40827
   439
    (case ys of [] => [] | x # xs => last ys # butlast ys)"
haftmann@37660
   440
haftmann@37660
   441
definition rotater :: "nat => 'a list => 'a list" where
haftmann@40827
   442
  "rotater n = rotater1 ^^ n"
haftmann@37660
   443
haftmann@37660
   444
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   445
  "word_rotr n w = of_bl (rotater n (to_bl w))"
haftmann@37660
   446
haftmann@37660
   447
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   448
  "word_rotl n w = of_bl (rotate n (to_bl w))"
haftmann@37660
   449
haftmann@37660
   450
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   451
  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
haftmann@40827
   452
                    else word_rotl (nat (- i)) w)"
haftmann@37660
   453
haftmann@37660
   454
haftmann@37660
   455
subsection "Split and cat operations"
haftmann@37660
   456
haftmann@37660
   457
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where
haftmann@40827
   458
  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
haftmann@37660
   459
haftmann@37660
   460
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where
haftmann@40827
   461
  "word_split a = 
haftmann@40827
   462
   (case bin_split (len_of TYPE ('c)) (uint a) of 
haftmann@40827
   463
     (u, v) => (word_of_int u, word_of_int v))"
haftmann@37660
   464
haftmann@37660
   465
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where
haftmann@40827
   466
  "word_rcat ws = 
haftmann@37660
   467
  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
haftmann@37660
   468
haftmann@37660
   469
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where
haftmann@40827
   470
  "word_rsplit w = 
haftmann@37660
   471
  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
haftmann@37660
   472
haftmann@37660
   473
definition max_word :: "'a::len word" -- "Largest representable machine integer." where
haftmann@40827
   474
  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
haftmann@37660
   475
haftmann@37660
   476
primrec of_bool :: "bool \<Rightarrow> 'a::len word" where
haftmann@37660
   477
  "of_bool False = 0"
haftmann@37660
   478
| "of_bool True = 1"
haftmann@37660
   479
huffman@45805
   480
(* FIXME: only provide one theorem name *)
haftmann@37660
   481
lemmas of_nth_def = word_set_bits_def
haftmann@37660
   482
huffman@46010
   483
subsection {* Theorems about typedefs *}
huffman@46010
   484
haftmann@37660
   485
lemma sint_sbintrunc': 
haftmann@37660
   486
  "sint (word_of_int bin :: 'a word) = 
haftmann@37660
   487
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
haftmann@37660
   488
  unfolding sint_uint 
haftmann@37660
   489
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)
haftmann@37660
   490
haftmann@37660
   491
lemma uint_sint: 
haftmann@37660
   492
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
haftmann@37660
   493
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
haftmann@37660
   494
huffman@46057
   495
lemma bintr_uint:
huffman@46057
   496
  fixes w :: "'a::len0 word"
huffman@46057
   497
  shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w"
haftmann@37660
   498
  apply (subst word_ubin.norm_Rep [symmetric]) 
haftmann@37660
   499
  apply (simp only: bintrunc_bintrunc_min word_size)
haftmann@37660
   500
  apply (simp add: min_max.inf_absorb2)
haftmann@37660
   501
  done
haftmann@37660
   502
huffman@46057
   503
lemma wi_bintr:
huffman@46057
   504
  "len_of TYPE('a::len0) \<le> n \<Longrightarrow>
huffman@46057
   505
    word_of_int (bintrunc n w) = (word_of_int w :: 'a word)"
haftmann@37660
   506
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1)
haftmann@37660
   507
haftmann@37660
   508
lemma td_ext_sbin: 
haftmann@37660
   509
  "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 
haftmann@37660
   510
    (sbintrunc (len_of TYPE('a) - 1))"
haftmann@37660
   511
  apply (unfold td_ext_def' sint_uint)
haftmann@37660
   512
  apply (simp add : word_ubin.eq_norm)
haftmann@37660
   513
  apply (cases "len_of TYPE('a)")
haftmann@37660
   514
   apply (auto simp add : sints_def)
haftmann@37660
   515
  apply (rule sym [THEN trans])
haftmann@37660
   516
  apply (rule word_ubin.Abs_norm)
haftmann@37660
   517
  apply (simp only: bintrunc_sbintrunc)
haftmann@37660
   518
  apply (drule sym)
haftmann@37660
   519
  apply simp
haftmann@37660
   520
  done
haftmann@37660
   521
haftmann@37660
   522
lemmas td_ext_sint = td_ext_sbin 
haftmann@37660
   523
  [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]]
haftmann@37660
   524
haftmann@37660
   525
(* We do sint before sbin, before sint is the user version
haftmann@37660
   526
   and interpretations do not produce thm duplicates. I.e. 
haftmann@37660
   527
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
haftmann@37660
   528
   because the latter is the same thm as the former *)
haftmann@37660
   529
interpretation word_sint:
haftmann@37660
   530
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   531
          word_of_int 
haftmann@37660
   532
          "sints (len_of TYPE('a::len))"
haftmann@37660
   533
          "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) -
haftmann@37660
   534
               2 ^ (len_of TYPE('a::len) - 1)"
haftmann@37660
   535
  by (rule td_ext_sint)
haftmann@37660
   536
haftmann@37660
   537
interpretation word_sbin:
haftmann@37660
   538
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   539
          word_of_int 
haftmann@37660
   540
          "sints (len_of TYPE('a::len))"
haftmann@37660
   541
          "sbintrunc (len_of TYPE('a::len) - 1)"
haftmann@37660
   542
  by (rule td_ext_sbin)
haftmann@37660
   543
wenzelm@45604
   544
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
haftmann@37660
   545
haftmann@37660
   546
lemmas td_sint = word_sint.td
haftmann@37660
   547
haftmann@37660
   548
lemma to_bl_def': 
haftmann@37660
   549
  "(to_bl :: 'a :: len0 word => bool list) =
haftmann@37660
   550
    bin_to_bl (len_of TYPE('a)) o uint"
wenzelm@44762
   551
  by (auto simp: to_bl_def)
haftmann@37660
   552
huffman@47108
   553
lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w
haftmann@37660
   554
huffman@45805
   555
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
huffman@45805
   556
  by (fact uints_def [unfolded no_bintr_alt1])
huffman@45805
   557
huffman@47108
   558
lemma word_numeral_alt:
huffman@47108
   559
  "numeral b = word_of_int (numeral b)"
huffman@47108
   560
  by (induct b, simp_all only: numeral.simps word_of_int_homs)
huffman@47108
   561
huffman@47108
   562
declare word_numeral_alt [symmetric, code_abbrev]
huffman@47108
   563
huffman@47108
   564
lemma word_neg_numeral_alt:
huffman@47108
   565
  "neg_numeral b = word_of_int (neg_numeral b)"
huffman@47108
   566
  by (simp only: neg_numeral_def word_numeral_alt wi_hom_neg)
huffman@47108
   567
huffman@47108
   568
declare word_neg_numeral_alt [symmetric, code_abbrev]
huffman@47108
   569
huffman@45805
   570
lemma uint_bintrunc [simp]:
huffman@47108
   571
  "uint (numeral bin :: 'a word) = 
huffman@47108
   572
    bintrunc (len_of TYPE ('a :: len0)) (numeral bin)"
huffman@47108
   573
  unfolding word_numeral_alt by (rule word_ubin.eq_norm)
huffman@47108
   574
huffman@47108
   575
lemma uint_bintrunc_neg [simp]: "uint (neg_numeral bin :: 'a word) = 
huffman@47108
   576
    bintrunc (len_of TYPE ('a :: len0)) (neg_numeral bin)"
huffman@47108
   577
  by (simp only: word_neg_numeral_alt word_ubin.eq_norm)
haftmann@37660
   578
huffman@45805
   579
lemma sint_sbintrunc [simp]:
huffman@47108
   580
  "sint (numeral bin :: 'a word) = 
huffman@47108
   581
    sbintrunc (len_of TYPE ('a :: len) - 1) (numeral bin)"
huffman@47108
   582
  by (simp only: word_numeral_alt word_sbin.eq_norm)
huffman@47108
   583
huffman@47108
   584
lemma sint_sbintrunc_neg [simp]: "sint (neg_numeral bin :: 'a word) = 
huffman@47108
   585
    sbintrunc (len_of TYPE ('a :: len) - 1) (neg_numeral bin)"
huffman@47108
   586
  by (simp only: word_neg_numeral_alt word_sbin.eq_norm)
haftmann@37660
   587
huffman@45805
   588
lemma unat_bintrunc [simp]:
huffman@47108
   589
  "unat (numeral bin :: 'a :: len0 word) =
huffman@47108
   590
    nat (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
   591
  by (simp only: unat_def uint_bintrunc)
huffman@47108
   592
huffman@47108
   593
lemma unat_bintrunc_neg [simp]:
huffman@47108
   594
  "unat (neg_numeral bin :: 'a :: len0 word) =
huffman@47108
   595
    nat (bintrunc (len_of TYPE('a)) (neg_numeral bin))"
huffman@47108
   596
  by (simp only: unat_def uint_bintrunc_neg)
haftmann@37660
   597
haftmann@40827
   598
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
haftmann@37660
   599
  apply (unfold word_size)
haftmann@37660
   600
  apply (rule word_uint.Rep_eqD)
haftmann@37660
   601
  apply (rule box_equals)
haftmann@37660
   602
    defer
haftmann@37660
   603
    apply (rule word_ubin.norm_Rep)+
haftmann@37660
   604
  apply simp
haftmann@37660
   605
  done
haftmann@37660
   606
huffman@45805
   607
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)"
huffman@45805
   608
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   609
huffman@45805
   610
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
huffman@45805
   611
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   612
huffman@45805
   613
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)"
huffman@45805
   614
  using word_sint.Rep [of x] by (simp add: sints_num)
huffman@45805
   615
huffman@45805
   616
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)"
huffman@45805
   617
  using word_sint.Rep [of x] by (simp add: sints_num)
haftmann@37660
   618
haftmann@37660
   619
lemma sign_uint_Pls [simp]: 
huffman@46604
   620
  "bin_sign (uint x) = 0"
huffman@47108
   621
  by (simp add: sign_Pls_ge_0)
haftmann@37660
   622
huffman@45805
   623
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0"
huffman@45805
   624
  by (simp only: diff_less_0_iff_less uint_lt2p)
huffman@45805
   625
huffman@45805
   626
lemma uint_m2p_not_non_neg:
huffman@45805
   627
  "\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)"
huffman@45805
   628
  by (simp only: not_le uint_m2p_neg)
haftmann@37660
   629
haftmann@37660
   630
lemma lt2p_lem:
haftmann@40827
   631
  "len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n"
haftmann@37660
   632
  by (rule xtr8 [OF _ uint_lt2p]) simp
haftmann@37660
   633
huffman@45805
   634
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
huffman@45805
   635
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])
haftmann@37660
   636
haftmann@40827
   637
lemma uint_nat: "uint w = int (unat w)"
haftmann@37660
   638
  unfolding unat_def by auto
haftmann@37660
   639
huffman@47108
   640
lemma uint_numeral:
huffman@47108
   641
  "uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   642
  unfolding word_numeral_alt
haftmann@37660
   643
  by (simp only: int_word_uint)
haftmann@37660
   644
huffman@47108
   645
lemma uint_neg_numeral:
huffman@47108
   646
  "uint (neg_numeral b :: 'a :: len0 word) = neg_numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   647
  unfolding word_neg_numeral_alt
huffman@47108
   648
  by (simp only: int_word_uint)
huffman@47108
   649
huffman@47108
   650
lemma unat_numeral: 
huffman@47108
   651
  "unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)"
haftmann@37660
   652
  apply (unfold unat_def)
huffman@47108
   653
  apply (clarsimp simp only: uint_numeral)
haftmann@37660
   654
  apply (rule nat_mod_distrib [THEN trans])
huffman@47108
   655
    apply (rule zero_le_numeral)
haftmann@37660
   656
   apply (simp_all add: nat_power_eq)
haftmann@37660
   657
  done
haftmann@37660
   658
huffman@47108
   659
lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + 
haftmann@37660
   660
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
haftmann@37660
   661
    2 ^ (len_of TYPE('a) - 1)"
huffman@47108
   662
  unfolding word_numeral_alt by (rule int_word_sint)
huffman@47108
   663
huffman@47108
   664
lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0"
huffman@45958
   665
  unfolding word_0_wi ..
huffman@45958
   666
huffman@47108
   667
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1"
huffman@45958
   668
  unfolding word_1_wi ..
huffman@45958
   669
huffman@47108
   670
lemma word_of_int_numeral [simp] : 
huffman@47108
   671
  "(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)"
huffman@47108
   672
  unfolding word_numeral_alt ..
huffman@47108
   673
huffman@47108
   674
lemma word_of_int_neg_numeral [simp]:
huffman@47108
   675
  "(word_of_int (neg_numeral bin) :: 'a :: len0 word) = (neg_numeral bin)"
huffman@47108
   676
  unfolding neg_numeral_def word_numeral_alt wi_hom_syms ..
haftmann@37660
   677
haftmann@37660
   678
lemma word_int_case_wi: 
haftmann@37660
   679
  "word_int_case f (word_of_int i :: 'b word) = 
haftmann@37660
   680
    f (i mod 2 ^ len_of TYPE('b::len0))"
haftmann@37660
   681
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)
haftmann@37660
   682
haftmann@37660
   683
lemma word_int_split: 
haftmann@37660
   684
  "P (word_int_case f x) = 
haftmann@37660
   685
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
haftmann@37660
   686
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
haftmann@37660
   687
  unfolding word_int_case_def
haftmann@37660
   688
  by (auto simp: word_uint.eq_norm int_mod_eq')
haftmann@37660
   689
haftmann@37660
   690
lemma word_int_split_asm: 
haftmann@37660
   691
  "P (word_int_case f x) = 
haftmann@37660
   692
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
haftmann@37660
   693
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
haftmann@37660
   694
  unfolding word_int_case_def
haftmann@37660
   695
  by (auto simp: word_uint.eq_norm int_mod_eq')
huffman@45805
   696
wenzelm@45604
   697
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
wenzelm@45604
   698
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
haftmann@37660
   699
haftmann@37660
   700
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
haftmann@37660
   701
  unfolding word_size by (rule uint_range')
haftmann@37660
   702
haftmann@37660
   703
lemma sint_range_size:
haftmann@37660
   704
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
haftmann@37660
   705
  unfolding word_size by (rule sint_range')
haftmann@37660
   706
huffman@45805
   707
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x"
huffman@45805
   708
  unfolding word_size by (rule less_le_trans [OF sint_lt])
huffman@45805
   709
huffman@45805
   710
lemma sint_below_size:
huffman@45805
   711
  "x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w"
huffman@45805
   712
  unfolding word_size by (rule order_trans [OF _ sint_ge])
haftmann@37660
   713
huffman@46010
   714
subsection {* Testing bits *}
huffman@46010
   715
haftmann@37660
   716
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)"
haftmann@37660
   717
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
haftmann@37660
   718
haftmann@37660
   719
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
haftmann@37660
   720
  apply (unfold word_test_bit_def)
haftmann@37660
   721
  apply (subst word_ubin.norm_Rep [symmetric])
haftmann@37660
   722
  apply (simp only: nth_bintr word_size)
haftmann@37660
   723
  apply fast
haftmann@37660
   724
  done
haftmann@37660
   725
huffman@46021
   726
lemma word_eq_iff:
huffman@46021
   727
  fixes x y :: "'a::len0 word"
huffman@46021
   728
  shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)"
huffman@46021
   729
  unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric]
huffman@46021
   730
  by (metis test_bit_size [unfolded word_size])
huffman@46021
   731
huffman@46023
   732
lemma word_eqI [rule_format]:
haftmann@37660
   733
  fixes u :: "'a::len0 word"
haftmann@40827
   734
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
huffman@46021
   735
  by (simp add: word_size word_eq_iff)
haftmann@37660
   736
huffman@45805
   737
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x"
huffman@45805
   738
  by simp
haftmann@37660
   739
haftmann@37660
   740
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
haftmann@37660
   741
  unfolding word_test_bit_def word_size
haftmann@37660
   742
  by (simp add: nth_bintr [symmetric])
haftmann@37660
   743
haftmann@37660
   744
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
haftmann@37660
   745
huffman@46057
   746
lemma bin_nth_uint_imp:
huffman@46057
   747
  "bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)"
haftmann@37660
   748
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
haftmann@37660
   749
  apply (subst word_ubin.norm_Rep)
haftmann@37660
   750
  apply assumption
haftmann@37660
   751
  done
haftmann@37660
   752
huffman@46057
   753
lemma bin_nth_sint:
huffman@46057
   754
  fixes w :: "'a::len word"
huffman@46057
   755
  shows "len_of TYPE('a) \<le> n \<Longrightarrow>
huffman@46057
   756
    bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)"
haftmann@37660
   757
  apply (subst word_sbin.norm_Rep [symmetric])
huffman@46057
   758
  apply (auto simp add: nth_sbintr)
haftmann@37660
   759
  done
haftmann@37660
   760
haftmann@37660
   761
(* type definitions theorem for in terms of equivalent bool list *)
haftmann@37660
   762
lemma td_bl: 
haftmann@37660
   763
  "type_definition (to_bl :: 'a::len0 word => bool list) 
haftmann@37660
   764
                   of_bl  
haftmann@37660
   765
                   {bl. length bl = len_of TYPE('a)}"
haftmann@37660
   766
  apply (unfold type_definition_def of_bl_def to_bl_def)
haftmann@37660
   767
  apply (simp add: word_ubin.eq_norm)
haftmann@37660
   768
  apply safe
haftmann@37660
   769
  apply (drule sym)
haftmann@37660
   770
  apply simp
haftmann@37660
   771
  done
haftmann@37660
   772
haftmann@37660
   773
interpretation word_bl:
haftmann@37660
   774
  type_definition "to_bl :: 'a::len0 word => bool list"
haftmann@37660
   775
                  of_bl  
haftmann@37660
   776
                  "{bl. length bl = len_of TYPE('a::len0)}"
haftmann@37660
   777
  by (rule td_bl)
haftmann@37660
   778
huffman@45816
   779
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
wenzelm@45538
   780
haftmann@40827
   781
lemma word_size_bl: "size w = size (to_bl w)"
haftmann@37660
   782
  unfolding word_size by auto
haftmann@37660
   783
haftmann@37660
   784
lemma to_bl_use_of_bl:
haftmann@37660
   785
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
huffman@45816
   786
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
haftmann@37660
   787
haftmann@37660
   788
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
haftmann@37660
   789
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)
haftmann@37660
   790
haftmann@37660
   791
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
haftmann@37660
   792
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)
haftmann@37660
   793
haftmann@40827
   794
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
huffman@47108
   795
  by (metis word_rev_rev)
haftmann@37660
   796
huffman@45805
   797
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
huffman@45805
   798
  by simp
huffman@45805
   799
huffman@45805
   800
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))"
huffman@45805
   801
  unfolding word_bl_Rep' by (rule len_gt_0)
huffman@45805
   802
huffman@45805
   803
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []"
huffman@45805
   804
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
huffman@45805
   805
huffman@45805
   806
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0"
huffman@45805
   807
  by (fact length_bl_gt_0 [THEN gr_implies_not0])
haftmann@37660
   808
huffman@46001
   809
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
haftmann@37660
   810
  apply (unfold to_bl_def sint_uint)
haftmann@37660
   811
  apply (rule trans [OF _ bl_sbin_sign])
haftmann@37660
   812
  apply simp
haftmann@37660
   813
  done
haftmann@37660
   814
haftmann@37660
   815
lemma of_bl_drop': 
haftmann@40827
   816
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
haftmann@37660
   817
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
haftmann@37660
   818
  apply (unfold of_bl_def)
haftmann@37660
   819
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
haftmann@37660
   820
  done
haftmann@37660
   821
haftmann@37660
   822
lemma test_bit_of_bl:  
haftmann@37660
   823
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)"
haftmann@37660
   824
  apply (unfold of_bl_def word_test_bit_def)
haftmann@37660
   825
  apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl)
haftmann@37660
   826
  done
haftmann@37660
   827
haftmann@37660
   828
lemma no_of_bl: 
huffman@47108
   829
  "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) (numeral bin))"
huffman@47108
   830
  unfolding of_bl_def by simp
haftmann@37660
   831
haftmann@40827
   832
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
haftmann@37660
   833
  unfolding word_size to_bl_def by auto
haftmann@37660
   834
haftmann@37660
   835
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   836
  unfolding uint_bl by (simp add : word_size)
haftmann@37660
   837
haftmann@37660
   838
lemma to_bl_of_bin: 
haftmann@37660
   839
  "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
haftmann@37660
   840
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)
haftmann@37660
   841
huffman@47108
   842
lemma to_bl_numeral [simp]:
huffman@47108
   843
  "to_bl (numeral bin::'a::len0 word) =
huffman@47108
   844
    bin_to_bl (len_of TYPE('a)) (numeral bin)"
huffman@47108
   845
  unfolding word_numeral_alt by (rule to_bl_of_bin)
huffman@47108
   846
huffman@47108
   847
lemma to_bl_neg_numeral [simp]:
huffman@47108
   848
  "to_bl (neg_numeral bin::'a::len0 word) =
huffman@47108
   849
    bin_to_bl (len_of TYPE('a)) (neg_numeral bin)"
huffman@47108
   850
  unfolding word_neg_numeral_alt by (rule to_bl_of_bin)
haftmann@37660
   851
haftmann@37660
   852
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   853
  unfolding uint_bl by (simp add : word_size)
huffman@46011
   854
huffman@46011
   855
lemma uint_bl_bin:
huffman@46011
   856
  fixes x :: "'a::len0 word"
huffman@46011
   857
  shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x"
huffman@46011
   858
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])
wenzelm@45604
   859
haftmann@37660
   860
(* naturals *)
haftmann@37660
   861
lemma uints_unats: "uints n = int ` unats n"
haftmann@37660
   862
  apply (unfold unats_def uints_num)
haftmann@37660
   863
  apply safe
haftmann@37660
   864
  apply (rule_tac image_eqI)
haftmann@37660
   865
  apply (erule_tac nat_0_le [symmetric])
haftmann@37660
   866
  apply auto
haftmann@37660
   867
  apply (erule_tac nat_less_iff [THEN iffD2])
haftmann@37660
   868
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
haftmann@37660
   869
  apply (auto simp add : nat_power_eq int_power)
haftmann@37660
   870
  done
haftmann@37660
   871
haftmann@37660
   872
lemma unats_uints: "unats n = nat ` uints n"
haftmann@37660
   873
  by (auto simp add : uints_unats image_iff)
haftmann@37660
   874
huffman@46962
   875
lemmas bintr_num = word_ubin.norm_eq_iff
huffman@47108
   876
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
huffman@46962
   877
lemmas sbintr_num = word_sbin.norm_eq_iff
huffman@47108
   878
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
haftmann@37660
   879
haftmann@37660
   880
lemma num_of_bintr':
huffman@47108
   881
  "bintrunc (len_of TYPE('a :: len0)) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   882
    numeral a = (numeral b :: 'a word)"
huffman@46962
   883
  unfolding bintr_num by (erule subst, simp)
haftmann@37660
   884
haftmann@37660
   885
lemma num_of_sbintr':
huffman@47108
   886
  "sbintrunc (len_of TYPE('a :: len) - 1) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   887
    numeral a = (numeral b :: 'a word)"
huffman@46962
   888
  unfolding sbintr_num by (erule subst, simp)
huffman@46962
   889
huffman@46962
   890
lemma num_abs_bintr:
huffman@47108
   891
  "(numeral x :: 'a word) =
huffman@47108
   892
    word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))"
huffman@47108
   893
  by (simp only: word_ubin.Abs_norm word_numeral_alt)
huffman@46962
   894
huffman@46962
   895
lemma num_abs_sbintr:
huffman@47108
   896
  "(numeral x :: 'a word) =
huffman@47108
   897
    word_of_int (sbintrunc (len_of TYPE('a::len) - 1) (numeral x))"
huffman@47108
   898
  by (simp only: word_sbin.Abs_norm word_numeral_alt)
huffman@46962
   899
haftmann@37660
   900
(** cast - note, no arg for new length, as it's determined by type of result,
haftmann@37660
   901
  thus in "cast w = w, the type means cast to length of w! **)
haftmann@37660
   902
haftmann@37660
   903
lemma ucast_id: "ucast w = w"
haftmann@37660
   904
  unfolding ucast_def by auto
haftmann@37660
   905
haftmann@37660
   906
lemma scast_id: "scast w = w"
haftmann@37660
   907
  unfolding scast_def by auto
haftmann@37660
   908
haftmann@40827
   909
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
haftmann@37660
   910
  unfolding ucast_def of_bl_def uint_bl
haftmann@37660
   911
  by (auto simp add : word_size)
haftmann@37660
   912
haftmann@37660
   913
lemma nth_ucast: 
haftmann@37660
   914
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
haftmann@37660
   915
  apply (unfold ucast_def test_bit_bin)
haftmann@37660
   916
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
haftmann@37660
   917
  apply (fast elim!: bin_nth_uint_imp)
haftmann@37660
   918
  done
haftmann@37660
   919
haftmann@37660
   920
(* for literal u(s)cast *)
haftmann@37660
   921
huffman@46001
   922
lemma ucast_bintr [simp]:
huffman@47108
   923
  "ucast (numeral w ::'a::len0 word) = 
huffman@47108
   924
   word_of_int (bintrunc (len_of TYPE('a)) (numeral w))"
haftmann@37660
   925
  unfolding ucast_def by simp
huffman@47108
   926
(* TODO: neg_numeral *)
haftmann@37660
   927
huffman@46001
   928
lemma scast_sbintr [simp]:
huffman@47108
   929
  "scast (numeral w ::'a::len word) = 
huffman@47108
   930
   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (numeral w))"
haftmann@37660
   931
  unfolding scast_def by simp
haftmann@37660
   932
huffman@46011
   933
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)"
huffman@46011
   934
  unfolding source_size_def word_size Let_def ..
huffman@46011
   935
huffman@46011
   936
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)"
huffman@46011
   937
  unfolding target_size_def word_size Let_def ..
huffman@46011
   938
huffman@46011
   939
lemma is_down:
huffman@46011
   940
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
   941
  shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)"
huffman@46011
   942
  unfolding is_down_def source_size target_size ..
huffman@46011
   943
huffman@46011
   944
lemma is_up:
huffman@46011
   945
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
   946
  shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)"
huffman@46011
   947
  unfolding is_up_def source_size target_size ..
haftmann@37660
   948
wenzelm@45604
   949
lemmas is_up_down = trans [OF is_up is_down [symmetric]]
haftmann@37660
   950
huffman@45811
   951
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
haftmann@37660
   952
  apply (unfold is_down)
haftmann@37660
   953
  apply safe
haftmann@37660
   954
  apply (rule ext)
haftmann@37660
   955
  apply (unfold ucast_def scast_def uint_sint)
haftmann@37660
   956
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
   957
  apply simp
haftmann@37660
   958
  done
haftmann@37660
   959
huffman@45811
   960
lemma word_rev_tf:
huffman@45811
   961
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
   962
    rev (takefill False (len_of TYPE('a)) (rev bl))"
haftmann@37660
   963
  unfolding of_bl_def uint_bl
haftmann@37660
   964
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
haftmann@37660
   965
huffman@45811
   966
lemma word_rep_drop:
huffman@45811
   967
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
   968
    replicate (len_of TYPE('a) - length bl) False @
huffman@45811
   969
    drop (length bl - len_of TYPE('a)) bl"
huffman@45811
   970
  by (simp add: word_rev_tf takefill_alt rev_take)
haftmann@37660
   971
haftmann@37660
   972
lemma to_bl_ucast: 
haftmann@37660
   973
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
haftmann@37660
   974
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
haftmann@37660
   975
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
haftmann@37660
   976
  apply (unfold ucast_bl)
haftmann@37660
   977
  apply (rule trans)
haftmann@37660
   978
   apply (rule word_rep_drop)
haftmann@37660
   979
  apply simp
haftmann@37660
   980
  done
haftmann@37660
   981
huffman@45811
   982
lemma ucast_up_app [OF refl]:
haftmann@40827
   983
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
haftmann@37660
   984
    to_bl (uc w) = replicate n False @ (to_bl w)"
haftmann@37660
   985
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
   986
huffman@45811
   987
lemma ucast_down_drop [OF refl]:
haftmann@40827
   988
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
haftmann@37660
   989
    to_bl (uc w) = drop n (to_bl w)"
haftmann@37660
   990
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
   991
huffman@45811
   992
lemma scast_down_drop [OF refl]:
haftmann@40827
   993
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
haftmann@37660
   994
    to_bl (sc w) = drop n (to_bl w)"
haftmann@37660
   995
  apply (subgoal_tac "sc = ucast")
haftmann@37660
   996
   apply safe
haftmann@37660
   997
   apply simp
huffman@45811
   998
   apply (erule ucast_down_drop)
huffman@45811
   999
  apply (rule down_cast_same [symmetric])
haftmann@37660
  1000
  apply (simp add : source_size target_size is_down)
haftmann@37660
  1001
  done
haftmann@37660
  1002
huffman@45811
  1003
lemma sint_up_scast [OF refl]:
haftmann@40827
  1004
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
haftmann@37660
  1005
  apply (unfold is_up)
haftmann@37660
  1006
  apply safe
haftmann@37660
  1007
  apply (simp add: scast_def word_sbin.eq_norm)
haftmann@37660
  1008
  apply (rule box_equals)
haftmann@37660
  1009
    prefer 3
haftmann@37660
  1010
    apply (rule word_sbin.norm_Rep)
haftmann@37660
  1011
   apply (rule sbintrunc_sbintrunc_l)
haftmann@37660
  1012
   defer
haftmann@37660
  1013
   apply (subst word_sbin.norm_Rep)
haftmann@37660
  1014
   apply (rule refl)
haftmann@37660
  1015
  apply simp
haftmann@37660
  1016
  done
haftmann@37660
  1017
huffman@45811
  1018
lemma uint_up_ucast [OF refl]:
haftmann@40827
  1019
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
haftmann@37660
  1020
  apply (unfold is_up)
haftmann@37660
  1021
  apply safe
haftmann@37660
  1022
  apply (rule bin_eqI)
haftmann@37660
  1023
  apply (fold word_test_bit_def)
haftmann@37660
  1024
  apply (auto simp add: nth_ucast)
haftmann@37660
  1025
  apply (auto simp add: test_bit_bin)
haftmann@37660
  1026
  done
huffman@45811
  1027
huffman@45811
  1028
lemma ucast_up_ucast [OF refl]:
huffman@45811
  1029
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
haftmann@37660
  1030
  apply (simp (no_asm) add: ucast_def)
haftmann@37660
  1031
  apply (clarsimp simp add: uint_up_ucast)
haftmann@37660
  1032
  done
haftmann@37660
  1033
    
huffman@45811
  1034
lemma scast_up_scast [OF refl]:
huffman@45811
  1035
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
haftmann@37660
  1036
  apply (simp (no_asm) add: scast_def)
haftmann@37660
  1037
  apply (clarsimp simp add: sint_up_scast)
haftmann@37660
  1038
  done
haftmann@37660
  1039
    
huffman@45811
  1040
lemma ucast_of_bl_up [OF refl]:
haftmann@40827
  1041
  "w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl"
haftmann@37660
  1042
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
haftmann@37660
  1043
haftmann@37660
  1044
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
haftmann@37660
  1045
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
haftmann@37660
  1046
haftmann@37660
  1047
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
haftmann@37660
  1048
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
haftmann@37660
  1049
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
haftmann@37660
  1050
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
haftmann@37660
  1051
haftmann@37660
  1052
lemma up_ucast_surj:
haftmann@40827
  1053
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1054
   surj (ucast :: 'a word => 'b word)"
haftmann@37660
  1055
  by (rule surjI, erule ucast_up_ucast_id)
haftmann@37660
  1056
haftmann@37660
  1057
lemma up_scast_surj:
haftmann@40827
  1058
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1059
   surj (scast :: 'a word => 'b word)"
haftmann@37660
  1060
  by (rule surjI, erule scast_up_scast_id)
haftmann@37660
  1061
haftmann@37660
  1062
lemma down_scast_inj:
haftmann@40827
  1063
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1064
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1065
  by (rule inj_on_inverseI, erule scast_down_scast_id)
haftmann@37660
  1066
haftmann@37660
  1067
lemma down_ucast_inj:
haftmann@40827
  1068
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1069
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1070
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)
haftmann@37660
  1071
haftmann@37660
  1072
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
haftmann@37660
  1073
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
huffman@45811
  1074
huffman@46646
  1075
lemma ucast_down_wi [OF refl]:
huffman@46646
  1076
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
huffman@46646
  1077
  apply (unfold is_down)
haftmann@37660
  1078
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
haftmann@37660
  1079
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1080
  apply (erule bintrunc_bintrunc_ge)
haftmann@37660
  1081
  done
huffman@45811
  1082
huffman@46646
  1083
lemma ucast_down_no [OF refl]:
huffman@47108
  1084
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin"
huffman@47108
  1085
  unfolding word_numeral_alt by clarify (rule ucast_down_wi)
huffman@46646
  1086
huffman@45811
  1087
lemma ucast_down_bl [OF refl]:
huffman@45811
  1088
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
huffman@46646
  1089
  unfolding of_bl_def by clarify (erule ucast_down_wi)
haftmann@37660
  1090
haftmann@37660
  1091
lemmas slice_def' = slice_def [unfolded word_size]
haftmann@37660
  1092
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
haftmann@37660
  1093
haftmann@37660
  1094
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
haftmann@37660
  1095
haftmann@37660
  1096
haftmann@37660
  1097
subsection {* Word Arithmetic *}
haftmann@37660
  1098
haftmann@37660
  1099
lemma word_less_alt: "(a < b) = (uint a < uint b)"
huffman@46012
  1100
  unfolding word_less_def word_le_def by (simp add: less_le)
haftmann@37660
  1101
haftmann@37660
  1102
lemma signed_linorder: "class.linorder word_sle word_sless"
wenzelm@46124
  1103
  by default (unfold word_sle_def word_sless_def, auto)
haftmann@37660
  1104
haftmann@37660
  1105
interpretation signed: linorder "word_sle" "word_sless"
haftmann@37660
  1106
  by (rule signed_linorder)
haftmann@37660
  1107
haftmann@37660
  1108
lemma udvdI: 
haftmann@40827
  1109
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
haftmann@37660
  1110
  by (auto simp: udvd_def)
haftmann@37660
  1111
huffman@47108
  1112
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1113
huffman@47108
  1114
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1115
huffman@47108
  1116
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1117
huffman@47108
  1118
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1119
huffman@47108
  1120
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1121
huffman@47108
  1122
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b
haftmann@37660
  1123
huffman@46020
  1124
lemma word_1_no: "(1::'a::len0 word) = Numeral1"
huffman@47108
  1125
  by (simp add: word_numeral_alt)
haftmann@37660
  1126
haftmann@40827
  1127
lemma word_m1_wi: "-1 = word_of_int -1" 
huffman@47108
  1128
  by (rule word_neg_numeral_alt)
haftmann@37660
  1129
huffman@46648
  1130
lemma word_0_bl [simp]: "of_bl [] = 0"
huffman@46648
  1131
  unfolding of_bl_def by simp
haftmann@37660
  1132
haftmann@37660
  1133
lemma word_1_bl: "of_bl [True] = 1" 
huffman@46648
  1134
  unfolding of_bl_def by (simp add: bl_to_bin_def)
huffman@46648
  1135
huffman@46648
  1136
lemma uint_eq_0 [simp]: "uint 0 = 0"
huffman@46648
  1137
  unfolding word_0_wi word_ubin.eq_norm by simp
haftmann@37660
  1138
huffman@45995
  1139
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
huffman@46648
  1140
  by (simp add: of_bl_def bl_to_bin_rep_False)
haftmann@37660
  1141
huffman@45805
  1142
lemma to_bl_0 [simp]:
haftmann@37660
  1143
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
haftmann@37660
  1144
  unfolding uint_bl
huffman@46617
  1145
  by (simp add: word_size bin_to_bl_zero)
haftmann@37660
  1146
haftmann@37660
  1147
lemma uint_0_iff: "(uint x = 0) = (x = 0)"
haftmann@37660
  1148
  by (auto intro!: word_uint.Rep_eqD)
haftmann@37660
  1149
haftmann@37660
  1150
lemma unat_0_iff: "(unat x = 0) = (x = 0)"
haftmann@37660
  1151
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)
haftmann@37660
  1152
haftmann@37660
  1153
lemma unat_0 [simp]: "unat 0 = 0"
haftmann@37660
  1154
  unfolding unat_def by auto
haftmann@37660
  1155
haftmann@40827
  1156
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
haftmann@37660
  1157
  apply (unfold word_size)
haftmann@37660
  1158
  apply (rule box_equals)
haftmann@37660
  1159
    defer
haftmann@37660
  1160
    apply (rule word_uint.Rep_inverse)+
haftmann@37660
  1161
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1162
  apply simp
haftmann@37660
  1163
  done
haftmann@37660
  1164
huffman@45816
  1165
lemmas size_0_same = size_0_same' [unfolded word_size]
haftmann@37660
  1166
haftmann@37660
  1167
lemmas unat_eq_0 = unat_0_iff
haftmann@37660
  1168
lemmas unat_eq_zero = unat_0_iff
haftmann@37660
  1169
haftmann@37660
  1170
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
haftmann@37660
  1171
by (auto simp: unat_0_iff [symmetric])
haftmann@37660
  1172
huffman@45958
  1173
lemma ucast_0 [simp]: "ucast 0 = 0"
huffman@45995
  1174
  unfolding ucast_def by simp
huffman@45958
  1175
huffman@45958
  1176
lemma sint_0 [simp]: "sint 0 = 0"
huffman@45958
  1177
  unfolding sint_uint by simp
huffman@45958
  1178
huffman@45958
  1179
lemma scast_0 [simp]: "scast 0 = 0"
huffman@45995
  1180
  unfolding scast_def by simp
haftmann@37660
  1181
haftmann@37660
  1182
lemma sint_n1 [simp] : "sint -1 = -1"
huffman@45958
  1183
  unfolding word_m1_wi by (simp add: word_sbin.eq_norm)
huffman@45958
  1184
huffman@45958
  1185
lemma scast_n1 [simp]: "scast -1 = -1"
huffman@45958
  1186
  unfolding scast_def by simp
huffman@45958
  1187
huffman@45958
  1188
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
haftmann@37660
  1189
  unfolding word_1_wi
huffman@45995
  1190
  by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1)
huffman@45958
  1191
huffman@45958
  1192
lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
huffman@45958
  1193
  unfolding unat_def by simp
huffman@45958
  1194
huffman@45958
  1195
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
huffman@45995
  1196
  unfolding ucast_def by simp
haftmann@37660
  1197
haftmann@37660
  1198
(* now, to get the weaker results analogous to word_div/mod_def *)
haftmann@37660
  1199
haftmann@37660
  1200
lemmas word_arith_alts = 
huffman@46000
  1201
  word_sub_wi
huffman@46000
  1202
  word_arith_wis (* FIXME: duplicate *)
huffman@46000
  1203
huffman@46000
  1204
lemmas word_succ_alt = word_succ_def (* FIXME: duplicate *)
huffman@46000
  1205
lemmas word_pred_alt = word_pred_def (* FIXME: duplicate *)
haftmann@37660
  1206
haftmann@37660
  1207
subsection  "Transferring goals from words to ints"
haftmann@37660
  1208
haftmann@37660
  1209
lemma word_ths:  
haftmann@37660
  1210
  shows
haftmann@37660
  1211
  word_succ_p1:   "word_succ a = a + 1" and
haftmann@37660
  1212
  word_pred_m1:   "word_pred a = a - 1" and
haftmann@37660
  1213
  word_pred_succ: "word_pred (word_succ a) = a" and
haftmann@37660
  1214
  word_succ_pred: "word_succ (word_pred a) = a" and
haftmann@37660
  1215
  word_mult_succ: "word_succ a * b = b + a * b"
haftmann@37660
  1216
  by (rule word_uint.Abs_cases [of b],
haftmann@37660
  1217
      rule word_uint.Abs_cases [of a],
huffman@46000
  1218
      simp add: add_commute mult_commute 
huffman@46009
  1219
                ring_distribs word_of_int_homs
huffman@45995
  1220
           del: word_of_int_0 word_of_int_1)+
haftmann@37660
  1221
huffman@45816
  1222
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
huffman@45816
  1223
  by simp
haftmann@37660
  1224
haftmann@37660
  1225
lemmas uint_word_ariths = 
wenzelm@45604
  1226
  word_arith_alts [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1227
haftmann@37660
  1228
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p]
haftmann@37660
  1229
haftmann@37660
  1230
(* similar expressions for sint (arith operations) *)
haftmann@37660
  1231
lemmas sint_word_ariths = uint_word_arith_bintrs
haftmann@37660
  1232
  [THEN uint_sint [symmetric, THEN trans],
haftmann@37660
  1233
  unfolded uint_sint bintr_arith1s bintr_ariths 
wenzelm@45604
  1234
    len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep]
wenzelm@45604
  1235
wenzelm@45604
  1236
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
wenzelm@45604
  1237
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1238
haftmann@37660
  1239
lemma word_pred_0_n1: "word_pred 0 = word_of_int -1"
huffman@47108
  1240
  unfolding word_pred_def uint_eq_0 by simp
haftmann@37660
  1241
haftmann@37660
  1242
lemma succ_pred_no [simp]:
huffman@47108
  1243
  "word_succ (numeral w) = numeral w + 1"
huffman@47108
  1244
  "word_pred (numeral w) = numeral w - 1"
huffman@47108
  1245
  "word_succ (neg_numeral w) = neg_numeral w + 1"
huffman@47108
  1246
  "word_pred (neg_numeral w) = neg_numeral w - 1"
huffman@47108
  1247
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1248
haftmann@37660
  1249
lemma word_sp_01 [simp] : 
haftmann@37660
  1250
  "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0"
huffman@47108
  1251
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1252
haftmann@37660
  1253
(* alternative approach to lifting arithmetic equalities *)
haftmann@37660
  1254
lemma word_of_int_Ex:
haftmann@37660
  1255
  "\<exists>y. x = word_of_int y"
haftmann@37660
  1256
  by (rule_tac x="uint x" in exI) simp
haftmann@37660
  1257
haftmann@37660
  1258
haftmann@37660
  1259
subsection "Order on fixed-length words"
haftmann@37660
  1260
haftmann@37660
  1261
lemma word_zero_le [simp] :
haftmann@37660
  1262
  "0 <= (y :: 'a :: len0 word)"
haftmann@37660
  1263
  unfolding word_le_def by auto
haftmann@37660
  1264
  
huffman@45816
  1265
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *)
haftmann@37660
  1266
  unfolding word_le_def
haftmann@37660
  1267
  by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1268
huffman@45816
  1269
lemma word_n1_ge [simp]: "y \<le> (-1::'a::len0 word)"
huffman@45816
  1270
  unfolding word_le_def
huffman@45816
  1271
  by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1272
haftmann@37660
  1273
lemmas word_not_simps [simp] = 
haftmann@37660
  1274
  word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD]
haftmann@37660
  1275
huffman@47108
  1276
lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> (y :: 'a :: len0 word)"
huffman@47108
  1277
  by (simp add: less_le)
huffman@47108
  1278
huffman@47108
  1279
lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y
haftmann@37660
  1280
haftmann@40827
  1281
lemma word_sless_alt: "(a <s b) = (sint a < sint b)"
haftmann@37660
  1282
  unfolding word_sle_def word_sless_def
haftmann@37660
  1283
  by (auto simp add: less_le)
haftmann@37660
  1284
haftmann@37660
  1285
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)"
haftmann@37660
  1286
  unfolding unat_def word_le_def
haftmann@37660
  1287
  by (rule nat_le_eq_zle [symmetric]) simp
haftmann@37660
  1288
haftmann@37660
  1289
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)"
haftmann@37660
  1290
  unfolding unat_def word_less_alt
haftmann@37660
  1291
  by (rule nat_less_eq_zless [symmetric]) simp
haftmann@37660
  1292
  
haftmann@37660
  1293
lemma wi_less: 
haftmann@37660
  1294
  "(word_of_int n < (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1295
    (n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1296
  unfolding word_less_alt by (simp add: word_uint.eq_norm)
haftmann@37660
  1297
haftmann@37660
  1298
lemma wi_le: 
haftmann@37660
  1299
  "(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1300
    (n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1301
  unfolding word_le_def by (simp add: word_uint.eq_norm)
haftmann@37660
  1302
haftmann@37660
  1303
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)"
haftmann@37660
  1304
  apply (unfold udvd_def)
haftmann@37660
  1305
  apply safe
haftmann@37660
  1306
   apply (simp add: unat_def nat_mult_distrib)
haftmann@37660
  1307
  apply (simp add: uint_nat int_mult)
haftmann@37660
  1308
  apply (rule exI)
haftmann@37660
  1309
  apply safe
haftmann@37660
  1310
   prefer 2
haftmann@37660
  1311
   apply (erule notE)
haftmann@37660
  1312
   apply (rule refl)
haftmann@37660
  1313
  apply force
haftmann@37660
  1314
  done
haftmann@37660
  1315
haftmann@37660
  1316
lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y"
haftmann@37660
  1317
  unfolding dvd_def udvd_nat_alt by force
haftmann@37660
  1318
wenzelm@45604
  1319
lemmas unat_mono = word_less_nat_alt [THEN iffD1]
haftmann@37660
  1320
haftmann@40827
  1321
lemma unat_minus_one: "x ~= 0 \<Longrightarrow> unat (x - 1) = unat x - 1"
haftmann@37660
  1322
  apply (unfold unat_def)
haftmann@37660
  1323
  apply (simp only: int_word_uint word_arith_alts rdmods)
haftmann@37660
  1324
  apply (subgoal_tac "uint x >= 1")
haftmann@37660
  1325
   prefer 2
haftmann@37660
  1326
   apply (drule contrapos_nn)
haftmann@37660
  1327
    apply (erule word_uint.Rep_inverse' [symmetric])
haftmann@37660
  1328
   apply (insert uint_ge_0 [of x])[1]
haftmann@37660
  1329
   apply arith
haftmann@37660
  1330
  apply (rule box_equals)
haftmann@37660
  1331
    apply (rule nat_diff_distrib)
haftmann@37660
  1332
     prefer 2
haftmann@37660
  1333
     apply assumption
haftmann@37660
  1334
    apply simp
haftmann@37660
  1335
   apply (subst mod_pos_pos_trivial)
haftmann@37660
  1336
     apply arith
haftmann@37660
  1337
    apply (insert uint_lt2p [of x])[1]
haftmann@37660
  1338
    apply arith
haftmann@37660
  1339
   apply (rule refl)
haftmann@37660
  1340
  apply simp
haftmann@37660
  1341
  done
haftmann@37660
  1342
    
haftmann@40827
  1343
lemma measure_unat: "p ~= 0 \<Longrightarrow> unat (p - 1) < unat p"
haftmann@37660
  1344
  by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric])
haftmann@37660
  1345
  
wenzelm@45604
  1346
lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
wenzelm@45604
  1347
lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
haftmann@37660
  1348
haftmann@37660
  1349
lemma uint_sub_lt2p [simp]: 
haftmann@37660
  1350
  "uint (x :: 'a :: len0 word) - uint (y :: 'b :: len0 word) < 
haftmann@37660
  1351
    2 ^ len_of TYPE('a)"
haftmann@37660
  1352
  using uint_ge_0 [of y] uint_lt2p [of x] by arith
haftmann@37660
  1353
haftmann@37660
  1354
haftmann@37660
  1355
subsection "Conditions for the addition (etc) of two words to overflow"
haftmann@37660
  1356
haftmann@37660
  1357
lemma uint_add_lem: 
haftmann@37660
  1358
  "(uint x + uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1359
    (uint (x + y :: 'a :: len0 word) = uint x + uint y)"
haftmann@37660
  1360
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1361
haftmann@37660
  1362
lemma uint_mult_lem: 
haftmann@37660
  1363
  "(uint x * uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1364
    (uint (x * y :: 'a :: len0 word) = uint x * uint y)"
haftmann@37660
  1365
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1366
haftmann@37660
  1367
lemma uint_sub_lem: 
haftmann@37660
  1368
  "(uint x >= uint y) = (uint (x - y) = uint x - uint y)"
haftmann@37660
  1369
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1370
haftmann@37660
  1371
lemma uint_add_le: "uint (x + y) <= uint x + uint y"
haftmann@37660
  1372
  unfolding uint_word_ariths by (auto simp: mod_add_if_z)
haftmann@37660
  1373
haftmann@37660
  1374
lemma uint_sub_ge: "uint (x - y) >= uint x - uint y"
haftmann@37660
  1375
  unfolding uint_word_ariths by (auto simp: mod_sub_if_z)
haftmann@37660
  1376
wenzelm@45604
  1377
lemmas uint_sub_if' = trans [OF uint_word_ariths(1) mod_sub_if_z, simplified]
wenzelm@45604
  1378
lemmas uint_plus_if' = trans [OF uint_word_ariths(2) mod_add_if_z, simplified]
haftmann@37660
  1379
haftmann@37660
  1380
haftmann@37660
  1381
subsection {* Definition of uint\_arith *}
haftmann@37660
  1382
haftmann@37660
  1383
lemma word_of_int_inverse:
haftmann@40827
  1384
  "word_of_int r = a \<Longrightarrow> 0 <= r \<Longrightarrow> r < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1385
   uint (a::'a::len0 word) = r"
haftmann@37660
  1386
  apply (erule word_uint.Abs_inverse' [rotated])
haftmann@37660
  1387
  apply (simp add: uints_num)
haftmann@37660
  1388
  done
haftmann@37660
  1389
haftmann@37660
  1390
lemma uint_split:
haftmann@37660
  1391
  fixes x::"'a::len0 word"
haftmann@37660
  1392
  shows "P (uint x) = 
haftmann@37660
  1393
         (ALL i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) --> P i)"
haftmann@37660
  1394
  apply (fold word_int_case_def)
haftmann@37660
  1395
  apply (auto dest!: word_of_int_inverse simp: int_word_uint int_mod_eq'
haftmann@37660
  1396
              split: word_int_split)
haftmann@37660
  1397
  done
haftmann@37660
  1398
haftmann@37660
  1399
lemma uint_split_asm:
haftmann@37660
  1400
  fixes x::"'a::len0 word"
haftmann@37660
  1401
  shows "P (uint x) = 
haftmann@37660
  1402
         (~(EX i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) & ~ P i))"
haftmann@37660
  1403
  by (auto dest!: word_of_int_inverse 
haftmann@37660
  1404
           simp: int_word_uint int_mod_eq'
haftmann@37660
  1405
           split: uint_split)
haftmann@37660
  1406
haftmann@37660
  1407
lemmas uint_splits = uint_split uint_split_asm
haftmann@37660
  1408
haftmann@37660
  1409
lemmas uint_arith_simps = 
haftmann@37660
  1410
  word_le_def word_less_alt
haftmann@37660
  1411
  word_uint.Rep_inject [symmetric] 
haftmann@37660
  1412
  uint_sub_if' uint_plus_if'
haftmann@37660
  1413
haftmann@37660
  1414
(* use this to stop, eg, 2 ^ len_of TYPE (32) being simplified *)
haftmann@40827
  1415
lemma power_False_cong: "False \<Longrightarrow> a ^ b = c ^ d" 
haftmann@37660
  1416
  by auto
haftmann@37660
  1417
haftmann@37660
  1418
(* uint_arith_tac: reduce to arithmetic on int, try to solve by arith *)
haftmann@37660
  1419
ML {*
haftmann@37660
  1420
fun uint_arith_ss_of ss = 
haftmann@37660
  1421
  ss addsimps @{thms uint_arith_simps}
haftmann@37660
  1422
     delsimps @{thms word_uint.Rep_inject}
wenzelm@45620
  1423
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1424
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1425
haftmann@37660
  1426
fun uint_arith_tacs ctxt = 
haftmann@37660
  1427
  let
haftmann@37660
  1428
    fun arith_tac' n t =
haftmann@37660
  1429
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1430
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1431
  in 
wenzelm@42793
  1432
    [ clarify_tac ctxt 1,
wenzelm@42793
  1433
      full_simp_tac (uint_arith_ss_of (simpset_of ctxt)) 1,
wenzelm@45620
  1434
      ALLGOALS (full_simp_tac (HOL_ss |> fold Splitter.add_split @{thms uint_splits}
wenzelm@45620
  1435
                                      |> fold Simplifier.add_cong @{thms power_False_cong})),
haftmann@37660
  1436
      rewrite_goals_tac @{thms word_size}, 
haftmann@37660
  1437
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1438
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1439
                         REPEAT (dtac @{thm word_of_int_inverse} n 
haftmann@37660
  1440
                                 THEN atac n 
haftmann@37660
  1441
                                 THEN atac n)),
haftmann@37660
  1442
      TRYALL arith_tac' ]
haftmann@37660
  1443
  end
haftmann@37660
  1444
haftmann@37660
  1445
fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt))
haftmann@37660
  1446
*}
haftmann@37660
  1447
haftmann@37660
  1448
method_setup uint_arith = 
haftmann@37660
  1449
  {* Scan.succeed (SIMPLE_METHOD' o uint_arith_tac) *}
haftmann@37660
  1450
  "solving word arithmetic via integers and arith"
haftmann@37660
  1451
haftmann@37660
  1452
haftmann@37660
  1453
subsection "More on overflows and monotonicity"
haftmann@37660
  1454
haftmann@37660
  1455
lemma no_plus_overflow_uint_size: 
haftmann@37660
  1456
  "((x :: 'a :: len0 word) <= x + y) = (uint x + uint y < 2 ^ size x)"
haftmann@37660
  1457
  unfolding word_size by uint_arith
haftmann@37660
  1458
haftmann@37660
  1459
lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size]
haftmann@37660
  1460
haftmann@37660
  1461
lemma no_ulen_sub: "((x :: 'a :: len0 word) >= x - y) = (uint y <= uint x)"
haftmann@37660
  1462
  by uint_arith
haftmann@37660
  1463
haftmann@37660
  1464
lemma no_olen_add':
haftmann@37660
  1465
  fixes x :: "'a::len0 word"
haftmann@37660
  1466
  shows "(x \<le> y + x) = (uint y + uint x < 2 ^ len_of TYPE('a))"
huffman@45546
  1467
  by (simp add: add_ac no_olen_add)
haftmann@37660
  1468
wenzelm@45604
  1469
lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric]]
wenzelm@45604
  1470
wenzelm@45604
  1471
lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem]
wenzelm@45604
  1472
lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1]
wenzelm@45604
  1473
lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem]
haftmann@37660
  1474
lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def]
haftmann@37660
  1475
lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def]
wenzelm@45604
  1476
lemmas word_sub_le = word_sub_le_iff [THEN iffD2]
haftmann@37660
  1477
haftmann@37660
  1478
lemma word_less_sub1: 
haftmann@40827
  1479
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 < x) = (0 < x - 1)"
haftmann@37660
  1480
  by uint_arith
haftmann@37660
  1481
haftmann@37660
  1482
lemma word_le_sub1: 
haftmann@40827
  1483
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 <= x) = (0 <= x - 1)"
haftmann@37660
  1484
  by uint_arith
haftmann@37660
  1485
haftmann@37660
  1486
lemma sub_wrap_lt: 
haftmann@37660
  1487
  "((x :: 'a :: len0 word) < x - z) = (x < z)"
haftmann@37660
  1488
  by uint_arith
haftmann@37660
  1489
haftmann@37660
  1490
lemma sub_wrap: 
haftmann@37660
  1491
  "((x :: 'a :: len0 word) <= x - z) = (z = 0 | x < z)"
haftmann@37660
  1492
  by uint_arith
haftmann@37660
  1493
haftmann@37660
  1494
lemma plus_minus_not_NULL_ab: 
haftmann@40827
  1495
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> c ~= 0 \<Longrightarrow> x + c ~= 0"
haftmann@37660
  1496
  by uint_arith
haftmann@37660
  1497
haftmann@37660
  1498
lemma plus_minus_no_overflow_ab: 
haftmann@40827
  1499
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> x <= x + c" 
haftmann@37660
  1500
  by uint_arith
haftmann@37660
  1501
haftmann@37660
  1502
lemma le_minus': 
haftmann@40827
  1503
  "(a :: 'a :: len0 word) + c <= b \<Longrightarrow> a <= a + c \<Longrightarrow> c <= b - a"
haftmann@37660
  1504
  by uint_arith
haftmann@37660
  1505
haftmann@37660
  1506
lemma le_plus': 
haftmann@40827
  1507
  "(a :: 'a :: len0 word) <= b \<Longrightarrow> c <= b - a \<Longrightarrow> a + c <= b"
haftmann@37660
  1508
  by uint_arith
haftmann@37660
  1509
haftmann@37660
  1510
lemmas le_plus = le_plus' [rotated]
haftmann@37660
  1511
huffman@46011
  1512
lemmas le_minus = leD [THEN thin_rl, THEN le_minus'] (* FIXME *)
haftmann@37660
  1513
haftmann@37660
  1514
lemma word_plus_mono_right: 
haftmann@40827
  1515
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= x + z \<Longrightarrow> x + y <= x + z"
haftmann@37660
  1516
  by uint_arith
haftmann@37660
  1517
haftmann@37660
  1518
lemma word_less_minus_cancel: 
haftmann@40827
  1519
  "y - x < z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) < z"
haftmann@37660
  1520
  by uint_arith
haftmann@37660
  1521
haftmann@37660
  1522
lemma word_less_minus_mono_left: 
haftmann@40827
  1523
  "(y :: 'a :: len0 word) < z \<Longrightarrow> x <= y \<Longrightarrow> y - x < z - x"
haftmann@37660
  1524
  by uint_arith
haftmann@37660
  1525
haftmann@37660
  1526
lemma word_less_minus_mono:  
haftmann@40827
  1527
  "a < c \<Longrightarrow> d < b \<Longrightarrow> a - b < a \<Longrightarrow> c - d < c 
haftmann@40827
  1528
  \<Longrightarrow> a - b < c - (d::'a::len word)"
haftmann@37660
  1529
  by uint_arith
haftmann@37660
  1530
haftmann@37660
  1531
lemma word_le_minus_cancel: 
haftmann@40827
  1532
  "y - x <= z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) <= z"
haftmann@37660
  1533
  by uint_arith
haftmann@37660
  1534
haftmann@37660
  1535
lemma word_le_minus_mono_left: 
haftmann@40827
  1536
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= y \<Longrightarrow> y - x <= z - x"
haftmann@37660
  1537
  by uint_arith
haftmann@37660
  1538
haftmann@37660
  1539
lemma word_le_minus_mono:  
haftmann@40827
  1540
  "a <= c \<Longrightarrow> d <= b \<Longrightarrow> a - b <= a \<Longrightarrow> c - d <= c 
haftmann@40827
  1541
  \<Longrightarrow> a - b <= c - (d::'a::len word)"
haftmann@37660
  1542
  by uint_arith
haftmann@37660
  1543
haftmann@37660
  1544
lemma plus_le_left_cancel_wrap: 
haftmann@40827
  1545
  "(x :: 'a :: len0 word) + y' < x \<Longrightarrow> x + y < x \<Longrightarrow> (x + y' < x + y) = (y' < y)"
haftmann@37660
  1546
  by uint_arith
haftmann@37660
  1547
haftmann@37660
  1548
lemma plus_le_left_cancel_nowrap: 
haftmann@40827
  1549
  "(x :: 'a :: len0 word) <= x + y' \<Longrightarrow> x <= x + y \<Longrightarrow> 
haftmann@37660
  1550
    (x + y' < x + y) = (y' < y)" 
haftmann@37660
  1551
  by uint_arith
haftmann@37660
  1552
haftmann@37660
  1553
lemma word_plus_mono_right2: 
haftmann@40827
  1554
  "(a :: 'a :: len0 word) <= a + b \<Longrightarrow> c <= b \<Longrightarrow> a <= a + c"
haftmann@37660
  1555
  by uint_arith
haftmann@37660
  1556
haftmann@37660
  1557
lemma word_less_add_right: 
haftmann@40827
  1558
  "(x :: 'a :: len0 word) < y - z \<Longrightarrow> z <= y \<Longrightarrow> x + z < y"
haftmann@37660
  1559
  by uint_arith
haftmann@37660
  1560
haftmann@37660
  1561
lemma word_less_sub_right: 
haftmann@40827
  1562
  "(x :: 'a :: len0 word) < y + z \<Longrightarrow> y <= x \<Longrightarrow> x - y < z"
haftmann@37660
  1563
  by uint_arith
haftmann@37660
  1564
haftmann@37660
  1565
lemma word_le_plus_either: 
haftmann@40827
  1566
  "(x :: 'a :: len0 word) <= y | x <= z \<Longrightarrow> y <= y + z \<Longrightarrow> x <= y + z"
haftmann@37660
  1567
  by uint_arith
haftmann@37660
  1568
haftmann@37660
  1569
lemma word_less_nowrapI: 
haftmann@40827
  1570
  "(x :: 'a :: len0 word) < z - k \<Longrightarrow> k <= z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
haftmann@37660
  1571
  by uint_arith
haftmann@37660
  1572
haftmann@40827
  1573
lemma inc_le: "(i :: 'a :: len word) < m \<Longrightarrow> i + 1 <= m"
haftmann@37660
  1574
  by uint_arith
haftmann@37660
  1575
haftmann@37660
  1576
lemma inc_i: 
haftmann@40827
  1577
  "(1 :: 'a :: len word) <= i \<Longrightarrow> i < m \<Longrightarrow> 1 <= (i + 1) & i + 1 <= m"
haftmann@37660
  1578
  by uint_arith
haftmann@37660
  1579
haftmann@37660
  1580
lemma udvd_incr_lem:
haftmann@40827
  1581
  "up < uq \<Longrightarrow> up = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1582
    uq = ua + n' * uint K \<Longrightarrow> up + uint K <= uq"
haftmann@37660
  1583
  apply clarsimp
haftmann@37660
  1584
  apply (drule less_le_mult)
haftmann@37660
  1585
  apply safe
haftmann@37660
  1586
  done
haftmann@37660
  1587
haftmann@37660
  1588
lemma udvd_incr': 
haftmann@40827
  1589
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1590
    uint q = ua + n' * uint K \<Longrightarrow> p + K <= q" 
haftmann@37660
  1591
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1592
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1593
  apply (erule uint_add_le [THEN order_trans])
haftmann@37660
  1594
  done
haftmann@37660
  1595
haftmann@37660
  1596
lemma udvd_decr': 
haftmann@40827
  1597
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1598
    uint q = ua + n' * uint K \<Longrightarrow> p <= q - K"
haftmann@37660
  1599
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1600
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1601
  apply (drule le_diff_eq [THEN iffD2])
haftmann@37660
  1602
  apply (erule order_trans)
haftmann@37660
  1603
  apply (rule uint_sub_ge)
haftmann@37660
  1604
  done
haftmann@37660
  1605
huffman@45816
  1606
lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, unfolded add_0_left]
huffman@45816
  1607
lemmas udvd_incr0 = udvd_incr' [where ua=0, unfolded add_0_left]
huffman@45816
  1608
lemmas udvd_decr0 = udvd_decr' [where ua=0, unfolded add_0_left]
haftmann@37660
  1609
haftmann@37660
  1610
lemma udvd_minus_le': 
haftmann@40827
  1611
  "xy < k \<Longrightarrow> z udvd xy \<Longrightarrow> z udvd k \<Longrightarrow> xy <= k - z"
haftmann@37660
  1612
  apply (unfold udvd_def)
haftmann@37660
  1613
  apply clarify
haftmann@37660
  1614
  apply (erule (2) udvd_decr0)
haftmann@37660
  1615
  done
haftmann@37660
  1616
huffman@45284
  1617
ML {* Delsimprocs [@{simproc linordered_ring_less_cancel_factor}] *}
haftmann@37660
  1618
haftmann@37660
  1619
lemma udvd_incr2_K: 
haftmann@40827
  1620
  "p < a + s \<Longrightarrow> a <= a + s \<Longrightarrow> K udvd s \<Longrightarrow> K udvd p - a \<Longrightarrow> a <= p \<Longrightarrow> 
haftmann@40827
  1621
    0 < K \<Longrightarrow> p <= p + K & p + K <= a + s"
haftmann@37660
  1622
  apply (unfold udvd_def)
haftmann@37660
  1623
  apply clarify
haftmann@37660
  1624
  apply (simp add: uint_arith_simps split: split_if_asm)
haftmann@37660
  1625
   prefer 2 
haftmann@37660
  1626
   apply (insert uint_range' [of s])[1]
haftmann@37660
  1627
   apply arith
haftmann@37660
  1628
  apply (drule add_commute [THEN xtr1])
haftmann@37660
  1629
  apply (simp add: diff_less_eq [symmetric])
haftmann@37660
  1630
  apply (drule less_le_mult)
haftmann@37660
  1631
   apply arith
haftmann@37660
  1632
  apply simp
haftmann@37660
  1633
  done
haftmann@37660
  1634
huffman@45284
  1635
ML {* Addsimprocs [@{simproc linordered_ring_less_cancel_factor}] *}
haftmann@37660
  1636
haftmann@37660
  1637
(* links with rbl operations *)
haftmann@37660
  1638
lemma word_succ_rbl:
haftmann@40827
  1639
  "to_bl w = bl \<Longrightarrow> to_bl (word_succ w) = (rev (rbl_succ (rev bl)))"
haftmann@37660
  1640
  apply (unfold word_succ_def)
haftmann@37660
  1641
  apply clarify
haftmann@37660
  1642
  apply (simp add: to_bl_of_bin)
huffman@46654
  1643
  apply (simp add: to_bl_def rbl_succ)
haftmann@37660
  1644
  done
haftmann@37660
  1645
haftmann@37660
  1646
lemma word_pred_rbl:
haftmann@40827
  1647
  "to_bl w = bl \<Longrightarrow> to_bl (word_pred w) = (rev (rbl_pred (rev bl)))"
haftmann@37660
  1648
  apply (unfold word_pred_def)
haftmann@37660
  1649
  apply clarify
haftmann@37660
  1650
  apply (simp add: to_bl_of_bin)
huffman@46654
  1651
  apply (simp add: to_bl_def rbl_pred)
haftmann@37660
  1652
  done
haftmann@37660
  1653
haftmann@37660
  1654
lemma word_add_rbl:
haftmann@40827
  1655
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1656
    to_bl (v + w) = (rev (rbl_add (rev vbl) (rev wbl)))"
haftmann@37660
  1657
  apply (unfold word_add_def)
haftmann@37660
  1658
  apply clarify
haftmann@37660
  1659
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1660
  apply (simp add: to_bl_def rbl_add)
haftmann@37660
  1661
  done
haftmann@37660
  1662
haftmann@37660
  1663
lemma word_mult_rbl:
haftmann@40827
  1664
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1665
    to_bl (v * w) = (rev (rbl_mult (rev vbl) (rev wbl)))"
haftmann@37660
  1666
  apply (unfold word_mult_def)
haftmann@37660
  1667
  apply clarify
haftmann@37660
  1668
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1669
  apply (simp add: to_bl_def rbl_mult)
haftmann@37660
  1670
  done
haftmann@37660
  1671
haftmann@37660
  1672
lemma rtb_rbl_ariths:
haftmann@37660
  1673
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys"
haftmann@37660
  1674
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys"
haftmann@40827
  1675
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs"
haftmann@40827
  1676
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs"
haftmann@37660
  1677
  by (auto simp: rev_swap [symmetric] word_succ_rbl 
haftmann@37660
  1678
                 word_pred_rbl word_mult_rbl word_add_rbl)
haftmann@37660
  1679
haftmann@37660
  1680
haftmann@37660
  1681
subsection "Arithmetic type class instantiations"
haftmann@37660
  1682
haftmann@37660
  1683
lemmas word_le_0_iff [simp] =
haftmann@37660
  1684
  word_zero_le [THEN leD, THEN linorder_antisym_conv1]
haftmann@37660
  1685
haftmann@37660
  1686
lemma word_of_int_nat: 
haftmann@40827
  1687
  "0 <= x \<Longrightarrow> word_of_int x = of_nat (nat x)"
haftmann@37660
  1688
  by (simp add: of_nat_nat word_of_int)
haftmann@37660
  1689
huffman@46603
  1690
(* note that iszero_def is only for class comm_semiring_1_cancel,
huffman@46603
  1691
   which requires word length >= 1, ie 'a :: len word *) 
huffman@46603
  1692
lemma iszero_word_no [simp]:
huffman@47108
  1693
  "iszero (numeral bin :: 'a :: len word) = 
huffman@47108
  1694
    iszero (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
  1695
  using word_ubin.norm_eq_iff [where 'a='a, of "numeral bin" 0]
huffman@46603
  1696
  by (simp add: iszero_def [symmetric])
huffman@47108
  1697
    
huffman@47108
  1698
text {* Use @{text iszero} to simplify equalities between word numerals. *}
huffman@47108
  1699
huffman@47108
  1700
lemmas word_eq_numeral_iff_iszero [simp] =
huffman@47108
  1701
  eq_numeral_iff_iszero [where 'a="'a::len word"]
huffman@46603
  1702
haftmann@37660
  1703
haftmann@37660
  1704
subsection "Word and nat"
haftmann@37660
  1705
huffman@45811
  1706
lemma td_ext_unat [OF refl]:
haftmann@40827
  1707
  "n = len_of TYPE ('a :: len) \<Longrightarrow> 
haftmann@37660
  1708
    td_ext (unat :: 'a word => nat) of_nat 
haftmann@37660
  1709
    (unats n) (%i. i mod 2 ^ n)"
haftmann@37660
  1710
  apply (unfold td_ext_def' unat_def word_of_nat unats_uints)
haftmann@37660
  1711
  apply (auto intro!: imageI simp add : word_of_int_hom_syms)
haftmann@37660
  1712
  apply (erule word_uint.Abs_inverse [THEN arg_cong])
haftmann@37660
  1713
  apply (simp add: int_word_uint nat_mod_distrib nat_power_eq)
haftmann@37660
  1714
  done
haftmann@37660
  1715
wenzelm@45604
  1716
lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm]
haftmann@37660
  1717
haftmann@37660
  1718
interpretation word_unat:
haftmann@37660
  1719
  td_ext "unat::'a::len word => nat" 
haftmann@37660
  1720
         of_nat 
haftmann@37660
  1721
         "unats (len_of TYPE('a::len))"
haftmann@37660
  1722
         "%i. i mod 2 ^ len_of TYPE('a::len)"
haftmann@37660
  1723
  by (rule td_ext_unat)
haftmann@37660
  1724
haftmann@37660
  1725
lemmas td_unat = word_unat.td_thm
haftmann@37660
  1726
haftmann@37660
  1727
lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq]
haftmann@37660
  1728
haftmann@40827
  1729
lemma unat_le: "y <= unat (z :: 'a :: len word) \<Longrightarrow> y : unats (len_of TYPE ('a))"
haftmann@37660
  1730
  apply (unfold unats_def)
haftmann@37660
  1731
  apply clarsimp
haftmann@37660
  1732
  apply (rule xtrans, rule unat_lt2p, assumption) 
haftmann@37660
  1733
  done
haftmann@37660
  1734
haftmann@37660
  1735
lemma word_nchotomy:
haftmann@37660
  1736
  "ALL w. EX n. (w :: 'a :: len word) = of_nat n & n < 2 ^ len_of TYPE ('a)"
haftmann@37660
  1737
  apply (rule allI)
haftmann@37660
  1738
  apply (rule word_unat.Abs_cases)
haftmann@37660
  1739
  apply (unfold unats_def)
haftmann@37660
  1740
  apply auto
haftmann@37660
  1741
  done
haftmann@37660
  1742
haftmann@37660
  1743
lemma of_nat_eq:
haftmann@37660
  1744
  fixes w :: "'a::len word"
haftmann@37660
  1745
  shows "(of_nat n = w) = (\<exists>q. n = unat w + q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1746
  apply (rule trans)
haftmann@37660
  1747
   apply (rule word_unat.inverse_norm)
haftmann@37660
  1748
  apply (rule iffI)
haftmann@37660
  1749
   apply (rule mod_eqD)
haftmann@37660
  1750
   apply simp
haftmann@37660
  1751
  apply clarsimp
haftmann@37660
  1752
  done
haftmann@37660
  1753
haftmann@37660
  1754
lemma of_nat_eq_size: 
haftmann@37660
  1755
  "(of_nat n = w) = (EX q. n = unat w + q * 2 ^ size w)"
haftmann@37660
  1756
  unfolding word_size by (rule of_nat_eq)
haftmann@37660
  1757
haftmann@37660
  1758
lemma of_nat_0:
haftmann@37660
  1759
  "(of_nat m = (0::'a::len word)) = (\<exists>q. m = q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1760
  by (simp add: of_nat_eq)
haftmann@37660
  1761
huffman@45805
  1762
lemma of_nat_2p [simp]:
huffman@45805
  1763
  "of_nat (2 ^ len_of TYPE('a)) = (0::'a::len word)"
huffman@45805
  1764
  by (fact mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]])
haftmann@37660
  1765
haftmann@40827
  1766
lemma of_nat_gt_0: "of_nat k ~= 0 \<Longrightarrow> 0 < k"
haftmann@37660
  1767
  by (cases k) auto
haftmann@37660
  1768
haftmann@37660
  1769
lemma of_nat_neq_0: 
haftmann@40827
  1770
  "0 < k \<Longrightarrow> k < 2 ^ len_of TYPE ('a :: len) \<Longrightarrow> of_nat k ~= (0 :: 'a word)"
haftmann@37660
  1771
  by (clarsimp simp add : of_nat_0)
haftmann@37660
  1772
haftmann@37660
  1773
lemma Abs_fnat_hom_add:
haftmann@37660
  1774
  "of_nat a + of_nat b = of_nat (a + b)"
haftmann@37660
  1775
  by simp
haftmann@37660
  1776
haftmann@37660
  1777
lemma Abs_fnat_hom_mult:
haftmann@37660
  1778
  "of_nat a * of_nat b = (of_nat (a * b) :: 'a :: len word)"
huffman@46013
  1779
  by (simp add: word_of_nat wi_hom_mult zmult_int)
haftmann@37660
  1780
haftmann@37660
  1781
lemma Abs_fnat_hom_Suc:
haftmann@37660
  1782
  "word_succ (of_nat a) = of_nat (Suc a)"
huffman@46013
  1783
  by (simp add: word_of_nat wi_hom_succ add_ac)
haftmann@37660
  1784
haftmann@37660
  1785
lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0"
huffman@45995
  1786
  by simp
haftmann@37660
  1787
haftmann@37660
  1788
lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)"
huffman@45995
  1789
  by simp
haftmann@37660
  1790
haftmann@37660
  1791
lemmas Abs_fnat_homs = 
haftmann@37660
  1792
  Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc 
haftmann@37660
  1793
  Abs_fnat_hom_0 Abs_fnat_hom_1
haftmann@37660
  1794
haftmann@37660
  1795
lemma word_arith_nat_add:
haftmann@37660
  1796
  "a + b = of_nat (unat a + unat b)" 
haftmann@37660
  1797
  by simp
haftmann@37660
  1798
haftmann@37660
  1799
lemma word_arith_nat_mult:
haftmann@37660
  1800
  "a * b = of_nat (unat a * unat b)"
huffman@45995
  1801
  by (simp add: of_nat_mult)
haftmann@37660
  1802
    
haftmann@37660
  1803
lemma word_arith_nat_Suc:
haftmann@37660
  1804
  "word_succ a = of_nat (Suc (unat a))"
haftmann@37660
  1805
  by (subst Abs_fnat_hom_Suc [symmetric]) simp
haftmann@37660
  1806
haftmann@37660
  1807
lemma word_arith_nat_div:
haftmann@37660
  1808
  "a div b = of_nat (unat a div unat b)"
haftmann@37660
  1809
  by (simp add: word_div_def word_of_nat zdiv_int uint_nat)
haftmann@37660
  1810
haftmann@37660
  1811
lemma word_arith_nat_mod:
haftmann@37660
  1812
  "a mod b = of_nat (unat a mod unat b)"
haftmann@37660
  1813
  by (simp add: word_mod_def word_of_nat zmod_int uint_nat)
haftmann@37660
  1814
haftmann@37660
  1815
lemmas word_arith_nat_defs =
haftmann@37660
  1816
  word_arith_nat_add word_arith_nat_mult
haftmann@37660
  1817
  word_arith_nat_Suc Abs_fnat_hom_0
haftmann@37660
  1818
  Abs_fnat_hom_1 word_arith_nat_div
haftmann@37660
  1819
  word_arith_nat_mod 
haftmann@37660
  1820
huffman@45816
  1821
lemma unat_cong: "x = y \<Longrightarrow> unat x = unat y"
huffman@45816
  1822
  by simp
haftmann@37660
  1823
  
haftmann@37660
  1824
lemmas unat_word_ariths = word_arith_nat_defs
wenzelm@45604
  1825
  [THEN trans [OF unat_cong unat_of_nat]]
haftmann@37660
  1826
haftmann@37660
  1827
lemmas word_sub_less_iff = word_sub_le_iff
huffman@45816
  1828
  [unfolded linorder_not_less [symmetric] Not_eq_iff]
haftmann@37660
  1829
haftmann@37660
  1830
lemma unat_add_lem: 
haftmann@37660
  1831
  "(unat x + unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1832
    (unat (x + y :: 'a :: len word) = unat x + unat y)"
haftmann@37660
  1833
  unfolding unat_word_ariths
haftmann@37660
  1834
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1835
haftmann@37660
  1836
lemma unat_mult_lem: 
haftmann@37660
  1837
  "(unat x * unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1838
    (unat (x * y :: 'a :: len word) = unat x * unat y)"
haftmann@37660
  1839
  unfolding unat_word_ariths
haftmann@37660
  1840
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1841
wenzelm@45604
  1842
lemmas unat_plus_if' = trans [OF unat_word_ariths(1) mod_nat_add, simplified]
haftmann@37660
  1843
haftmann@37660
  1844
lemma le_no_overflow: 
haftmann@40827
  1845
  "x <= b \<Longrightarrow> a <= a + b \<Longrightarrow> x <= a + (b :: 'a :: len0 word)"
haftmann@37660
  1846
  apply (erule order_trans)
haftmann@37660
  1847
  apply (erule olen_add_eqv [THEN iffD1])
haftmann@37660
  1848
  done
haftmann@37660
  1849
wenzelm@45604
  1850
lemmas un_ui_le = trans [OF word_le_nat_alt [symmetric] word_le_def]
haftmann@37660
  1851
haftmann@37660
  1852
lemma unat_sub_if_size:
haftmann@37660
  1853
  "unat (x - y) = (if unat y <= unat x 
haftmann@37660
  1854
   then unat x - unat y 
haftmann@37660
  1855
   else unat x + 2 ^ size x - unat y)"
haftmann@37660
  1856
  apply (unfold word_size)
haftmann@37660
  1857
  apply (simp add: un_ui_le)
haftmann@37660
  1858
  apply (auto simp add: unat_def uint_sub_if')
haftmann@37660
  1859
   apply (rule nat_diff_distrib)
haftmann@37660
  1860
    prefer 3
haftmann@37660
  1861
    apply (simp add: algebra_simps)
haftmann@37660
  1862
    apply (rule nat_diff_distrib [THEN trans])
haftmann@37660
  1863
      prefer 3
haftmann@37660
  1864
      apply (subst nat_add_distrib)
haftmann@37660
  1865
        prefer 3
haftmann@37660
  1866
        apply (simp add: nat_power_eq)
haftmann@37660
  1867
       apply auto
haftmann@37660
  1868
  apply uint_arith
haftmann@37660
  1869
  done
haftmann@37660
  1870
haftmann@37660
  1871
lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size]
haftmann@37660
  1872
haftmann@37660
  1873
lemma unat_div: "unat ((x :: 'a :: len word) div y) = unat x div unat y"
haftmann@37660
  1874
  apply (simp add : unat_word_ariths)
haftmann@37660
  1875
  apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1876
  apply (rule div_le_dividend)
haftmann@37660
  1877
  done
haftmann@37660
  1878
haftmann@37660
  1879
lemma unat_mod: "unat ((x :: 'a :: len word) mod y) = unat x mod unat y"
haftmann@37660
  1880
  apply (clarsimp simp add : unat_word_ariths)
haftmann@37660
  1881
  apply (cases "unat y")
haftmann@37660
  1882
   prefer 2
haftmann@37660
  1883
   apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1884
   apply (rule mod_le_divisor)
haftmann@37660
  1885
   apply auto
haftmann@37660
  1886
  done
haftmann@37660
  1887
haftmann@37660
  1888
lemma uint_div: "uint ((x :: 'a :: len word) div y) = uint x div uint y"
haftmann@37660
  1889
  unfolding uint_nat by (simp add : unat_div zdiv_int)
haftmann@37660
  1890
haftmann@37660
  1891
lemma uint_mod: "uint ((x :: 'a :: len word) mod y) = uint x mod uint y"
haftmann@37660
  1892
  unfolding uint_nat by (simp add : unat_mod zmod_int)
haftmann@37660
  1893
haftmann@37660
  1894
haftmann@37660
  1895
subsection {* Definition of unat\_arith tactic *}
haftmann@37660
  1896
haftmann@37660
  1897
lemma unat_split:
haftmann@37660
  1898
  fixes x::"'a::len word"
haftmann@37660
  1899
  shows "P (unat x) = 
haftmann@37660
  1900
         (ALL n. of_nat n = x & n < 2^len_of TYPE('a) --> P n)"
haftmann@37660
  1901
  by (auto simp: unat_of_nat)
haftmann@37660
  1902
haftmann@37660
  1903
lemma unat_split_asm:
haftmann@37660
  1904
  fixes x::"'a::len word"
haftmann@37660
  1905
  shows "P (unat x) = 
haftmann@37660
  1906
         (~(EX n. of_nat n = x & n < 2^len_of TYPE('a) & ~ P n))"
haftmann@37660
  1907
  by (auto simp: unat_of_nat)
haftmann@37660
  1908
haftmann@37660
  1909
lemmas of_nat_inverse = 
haftmann@37660
  1910
  word_unat.Abs_inverse' [rotated, unfolded unats_def, simplified]
haftmann@37660
  1911
haftmann@37660
  1912
lemmas unat_splits = unat_split unat_split_asm
haftmann@37660
  1913
haftmann@37660
  1914
lemmas unat_arith_simps =
haftmann@37660
  1915
  word_le_nat_alt word_less_nat_alt
haftmann@37660
  1916
  word_unat.Rep_inject [symmetric]
haftmann@37660
  1917
  unat_sub_if' unat_plus_if' unat_div unat_mod
haftmann@37660
  1918
haftmann@37660
  1919
(* unat_arith_tac: tactic to reduce word arithmetic to nat, 
haftmann@37660
  1920
   try to solve via arith *)
haftmann@37660
  1921
ML {*
haftmann@37660
  1922
fun unat_arith_ss_of ss = 
haftmann@37660
  1923
  ss addsimps @{thms unat_arith_simps}
haftmann@37660
  1924
     delsimps @{thms word_unat.Rep_inject}
wenzelm@45620
  1925
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1926
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1927
haftmann@37660
  1928
fun unat_arith_tacs ctxt =   
haftmann@37660
  1929
  let
haftmann@37660
  1930
    fun arith_tac' n t =
haftmann@37660
  1931
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1932
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1933
  in 
wenzelm@42793
  1934
    [ clarify_tac ctxt 1,
wenzelm@42793
  1935
      full_simp_tac (unat_arith_ss_of (simpset_of ctxt)) 1,
wenzelm@45620
  1936
      ALLGOALS (full_simp_tac (HOL_ss |> fold Splitter.add_split @{thms unat_splits}
wenzelm@45620
  1937
                                      |> fold Simplifier.add_cong @{thms power_False_cong})),
haftmann@37660
  1938
      rewrite_goals_tac @{thms word_size}, 
haftmann@37660
  1939
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1940
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1941
                         REPEAT (dtac @{thm of_nat_inverse} n THEN atac n)),
haftmann@37660
  1942
      TRYALL arith_tac' ] 
haftmann@37660
  1943
  end
haftmann@37660
  1944
haftmann@37660
  1945
fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt))
haftmann@37660
  1946
*}
haftmann@37660
  1947
haftmann@37660
  1948
method_setup unat_arith = 
haftmann@37660
  1949
  {* Scan.succeed (SIMPLE_METHOD' o unat_arith_tac) *}
haftmann@37660
  1950
  "solving word arithmetic via natural numbers and arith"
haftmann@37660
  1951
haftmann@37660
  1952
lemma no_plus_overflow_unat_size: 
haftmann@37660
  1953
  "((x :: 'a :: len word) <= x + y) = (unat x + unat y < 2 ^ size x)" 
haftmann@37660
  1954
  unfolding word_size by unat_arith
haftmann@37660
  1955
haftmann@37660
  1956
lemmas no_olen_add_nat = no_plus_overflow_unat_size [unfolded word_size]
haftmann@37660
  1957
wenzelm@45604
  1958
lemmas unat_plus_simple = trans [OF no_olen_add_nat unat_add_lem]
haftmann@37660
  1959
haftmann@37660
  1960
lemma word_div_mult: 
haftmann@40827
  1961
  "(0 :: 'a :: len word) < y \<Longrightarrow> unat x * unat y < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1962
    x * y div y = x"
haftmann@37660
  1963
  apply unat_arith
haftmann@37660
  1964
  apply clarsimp
haftmann@37660
  1965
  apply (subst unat_mult_lem [THEN iffD1])
haftmann@37660
  1966
  apply auto
haftmann@37660
  1967
  done
haftmann@37660
  1968
haftmann@40827
  1969
lemma div_lt': "(i :: 'a :: len word) <= k div x \<Longrightarrow> 
haftmann@37660
  1970
    unat i * unat x < 2 ^ len_of TYPE('a)"
haftmann@37660
  1971
  apply unat_arith
haftmann@37660
  1972
  apply clarsimp
haftmann@37660
  1973
  apply (drule mult_le_mono1)
haftmann@37660
  1974
  apply (erule order_le_less_trans)
haftmann@37660
  1975
  apply (rule xtr7 [OF unat_lt2p div_mult_le])
haftmann@37660
  1976
  done
haftmann@37660
  1977
haftmann@37660
  1978
lemmas div_lt'' = order_less_imp_le [THEN div_lt']
haftmann@37660
  1979
haftmann@40827
  1980
lemma div_lt_mult: "(i :: 'a :: len word) < k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x < k"
haftmann@37660
  1981
  apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  1982
  apply (simp add: unat_arith_simps)
haftmann@37660
  1983
  apply (drule (1) mult_less_mono1)
haftmann@37660
  1984
  apply (erule order_less_le_trans)
haftmann@37660
  1985
  apply (rule div_mult_le)
haftmann@37660
  1986
  done
haftmann@37660
  1987
haftmann@37660
  1988
lemma div_le_mult: 
haftmann@40827
  1989
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x <= k"
haftmann@37660
  1990
  apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  1991
  apply (simp add: unat_arith_simps)
haftmann@37660
  1992
  apply (drule mult_le_mono1)
haftmann@37660
  1993
  apply (erule order_trans)
haftmann@37660
  1994
  apply (rule div_mult_le)
haftmann@37660
  1995
  done
haftmann@37660
  1996
haftmann@37660
  1997
lemma div_lt_uint': 
haftmann@40827
  1998
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> uint i * uint x < 2 ^ len_of TYPE('a)"
haftmann@37660
  1999
  apply (unfold uint_nat)
haftmann@37660
  2000
  apply (drule div_lt')
haftmann@37660
  2001
  apply (simp add: zmult_int zless_nat_eq_int_zless [symmetric] 
haftmann@37660
  2002
                   nat_power_eq)
haftmann@37660
  2003
  done
haftmann@37660
  2004
haftmann@37660
  2005
lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint']
haftmann@37660
  2006
haftmann@37660
  2007
lemma word_le_exists': 
haftmann@40827
  2008
  "(x :: 'a :: len0 word) <= y \<Longrightarrow> 
haftmann@37660
  2009
    (EX z. y = x + z & uint x + uint z < 2 ^ len_of TYPE('a))"
haftmann@37660
  2010
  apply (rule exI)
haftmann@37660
  2011
  apply (rule conjI)
haftmann@37660
  2012
  apply (rule zadd_diff_inverse)
haftmann@37660
  2013
  apply uint_arith
haftmann@37660
  2014
  done
haftmann@37660
  2015
haftmann@37660
  2016
lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab]
haftmann@37660
  2017
haftmann@37660
  2018
lemmas plus_minus_no_overflow =
haftmann@37660
  2019
  order_less_imp_le [THEN plus_minus_no_overflow_ab]
haftmann@37660
  2020
  
haftmann@37660
  2021
lemmas mcs = word_less_minus_cancel word_less_minus_mono_left
haftmann@37660
  2022
  word_le_minus_cancel word_le_minus_mono_left
haftmann@37660
  2023
wenzelm@45604
  2024
lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2025
lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2026
lemmas word_plus_mcs = word_diff_ls [where y = "v + x", unfolded add_diff_cancel] for v x
haftmann@37660
  2027
haftmann@37660
  2028
lemmas le_unat_uoi = unat_le [THEN word_unat.Abs_inverse]
haftmann@37660
  2029
haftmann@37660
  2030
lemmas thd = refl [THEN [2] split_div_lemma [THEN iffD2], THEN conjunct1]
haftmann@37660
  2031
haftmann@37660
  2032
lemma thd1:
haftmann@37660
  2033
  "a div b * b \<le> (a::nat)"
haftmann@37660
  2034
  using gt_or_eq_0 [of b]
haftmann@37660
  2035
  apply (rule disjE)
haftmann@37660
  2036
   apply (erule xtr4 [OF thd mult_commute])
haftmann@37660
  2037
  apply clarsimp
haftmann@37660
  2038
  done
haftmann@37660
  2039
wenzelm@45604
  2040
lemmas uno_simps [THEN le_unat_uoi] = mod_le_divisor div_le_dividend thd1 
haftmann@37660
  2041
haftmann@37660
  2042
lemma word_mod_div_equality:
haftmann@37660
  2043
  "(n div b) * b + (n mod b) = (n :: 'a :: len word)"
haftmann@37660
  2044
  apply (unfold word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2045
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2046
  apply (erule disjE)
haftmann@37660
  2047
   apply (simp add: mod_div_equality uno_simps)
haftmann@37660
  2048
  apply simp
haftmann@37660
  2049
  done
haftmann@37660
  2050
haftmann@37660
  2051
lemma word_div_mult_le: "a div b * b <= (a::'a::len word)"
haftmann@37660
  2052
  apply (unfold word_le_nat_alt word_arith_nat_defs)
haftmann@37660
  2053
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2054
  apply (erule disjE)
haftmann@37660
  2055
   apply (simp add: div_mult_le uno_simps)
haftmann@37660
  2056
  apply simp
haftmann@37660
  2057
  done
haftmann@37660
  2058
haftmann@40827
  2059
lemma word_mod_less_divisor: "0 < n \<Longrightarrow> m mod n < (n :: 'a :: len word)"
haftmann@37660
  2060
  apply (simp only: word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2061
  apply (clarsimp simp add : uno_simps)
haftmann@37660
  2062
  done
haftmann@37660
  2063
haftmann@37660
  2064
lemma word_of_int_power_hom: 
haftmann@37660
  2065
  "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a :: len word)"
huffman@45995
  2066
  by (induct n) (simp_all add: wi_hom_mult [symmetric])
haftmann@37660
  2067
haftmann@37660
  2068
lemma word_arith_power_alt: 
haftmann@37660
  2069
  "a ^ n = (word_of_int (uint a ^ n) :: 'a :: len word)"
haftmann@37660
  2070
  by (simp add : word_of_int_power_hom [symmetric])
haftmann@37660
  2071
haftmann@37660
  2072
lemma of_bl_length_less: 
haftmann@40827
  2073
  "length x = k \<Longrightarrow> k < len_of TYPE('a) \<Longrightarrow> (of_bl x :: 'a :: len word) < 2 ^ k"
huffman@47108
  2074
  apply (unfold of_bl_def word_less_alt word_numeral_alt)
haftmann@37660
  2075
  apply safe
haftmann@37660
  2076
  apply (simp (no_asm) add: word_of_int_power_hom word_uint.eq_norm 
huffman@47108
  2077
                       del: word_of_int_numeral)
haftmann@37660
  2078
  apply (simp add: mod_pos_pos_trivial)
haftmann@37660
  2079
  apply (subst mod_pos_pos_trivial)
haftmann@37660
  2080
    apply (rule bl_to_bin_ge0)
haftmann@37660
  2081
   apply (rule order_less_trans)
haftmann@37660
  2082
    apply (rule bl_to_bin_lt2p)
haftmann@37660
  2083
   apply simp
huffman@46646
  2084
  apply (rule bl_to_bin_lt2p)
haftmann@37660
  2085
  done
haftmann@37660
  2086
haftmann@37660
  2087
haftmann@37660
  2088
subsection "Cardinality, finiteness of set of words"
haftmann@37660
  2089
huffman@45809
  2090
instance word :: (len0) finite
huffman@45809
  2091
  by (default, simp add: type_definition.univ [OF type_definition_word])
huffman@45809
  2092
huffman@45809
  2093
lemma card_word: "CARD('a::len0 word) = 2 ^ len_of TYPE('a)"
huffman@45809
  2094
  by (simp add: type_definition.card [OF type_definition_word] nat_power_eq)
haftmann@37660
  2095
haftmann@37660
  2096
lemma card_word_size: 
huffman@45809
  2097
  "card (UNIV :: 'a :: len0 word set) = (2 ^ size (x :: 'a word))"
haftmann@37660
  2098
unfolding word_size by (rule card_word)
haftmann@37660
  2099
haftmann@37660
  2100
haftmann@37660
  2101
subsection {* Bitwise Operations on Words *}
haftmann@37660
  2102
haftmann@37660
  2103
lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or
haftmann@37660
  2104
  
haftmann@37660
  2105
(* following definitions require both arithmetic and bit-wise word operations *)
haftmann@37660
  2106
haftmann@37660
  2107
(* to get word_no_log_defs from word_log_defs, using bin_log_bintrs *)
haftmann@37660
  2108
lemmas wils1 = bin_log_bintrs [THEN word_ubin.norm_eq_iff [THEN iffD1],
wenzelm@45604
  2109
  folded word_ubin.eq_norm, THEN eq_reflection]
haftmann@37660
  2110
haftmann@37660
  2111
(* the binary operations only *)
huffman@46013
  2112
(* BH: why is this needed? *)
haftmann@37660
  2113
lemmas word_log_binary_defs = 
haftmann@37660
  2114
  word_and_def word_or_def word_xor_def
haftmann@37660
  2115
huffman@46011
  2116
lemma word_wi_log_defs:
huffman@46011
  2117
  "NOT word_of_int a = word_of_int (NOT a)"
huffman@46011
  2118
  "word_of_int a AND word_of_int b = word_of_int (a AND b)"
huffman@46011
  2119
  "word_of_int a OR word_of_int b = word_of_int (a OR b)"
huffman@46011
  2120
  "word_of_int a XOR word_of_int b = word_of_int (a XOR b)"
huffman@46013
  2121
  unfolding word_log_defs wils1 by simp_all
huffman@46011
  2122
huffman@46011
  2123
lemma word_no_log_defs [simp]:
huffman@47108
  2124
  "NOT (numeral a) = word_of_int (NOT (numeral a))"
huffman@47108
  2125
  "NOT (neg_numeral a) = word_of_int (NOT (neg_numeral a))"
huffman@47108
  2126
  "numeral a AND numeral b = word_of_int (numeral a AND numeral b)"
huffman@47108
  2127
  "numeral a AND neg_numeral b = word_of_int (numeral a AND neg_numeral b)"
huffman@47108
  2128
  "neg_numeral a AND numeral b = word_of_int (neg_numeral a AND numeral b)"
huffman@47108
  2129
  "neg_numeral a AND neg_numeral b = word_of_int (neg_numeral a AND neg_numeral b)"
huffman@47108
  2130
  "numeral a OR numeral b = word_of_int (numeral a OR numeral b)"
huffman@47108
  2131
  "numeral a OR neg_numeral b = word_of_int (numeral a OR neg_numeral b)"
huffman@47108
  2132
  "neg_numeral a OR numeral b = word_of_int (neg_numeral a OR numeral b)"
huffman@47108
  2133
  "neg_numeral a OR neg_numeral b = word_of_int (neg_numeral a OR neg_numeral b)"
huffman@47108
  2134
  "numeral a XOR numeral b = word_of_int (numeral a XOR numeral b)"
huffman@47108
  2135
  "numeral a XOR neg_numeral b = word_of_int (numeral a XOR neg_numeral b)"
huffman@47108
  2136
  "neg_numeral a XOR numeral b = word_of_int (neg_numeral a XOR numeral b)"
huffman@47108
  2137
  "neg_numeral a XOR neg_numeral b = word_of_int (neg_numeral a XOR neg_numeral b)"
huffman@47108
  2138
  unfolding word_numeral_alt word_neg_numeral_alt word_wi_log_defs by simp_all
haftmann@37660
  2139
huffman@46064
  2140
text {* Special cases for when one of the arguments equals 1. *}
huffman@46064
  2141
huffman@46064
  2142
lemma word_bitwise_1_simps [simp]:
huffman@46064
  2143
  "NOT (1::'a::len0 word) = -2"
huffman@47108
  2144
  "1 AND numeral b = word_of_int (1 AND numeral b)"
huffman@47108
  2145
  "1 AND neg_numeral b = word_of_int (1 AND neg_numeral b)"
huffman@47108
  2146
  "numeral a AND 1 = word_of_int (numeral a AND 1)"
huffman@47108
  2147
  "neg_numeral a AND 1 = word_of_int (neg_numeral a AND 1)"
huffman@47108
  2148
  "1 OR numeral b = word_of_int (1 OR numeral b)"
huffman@47108
  2149
  "1 OR neg_numeral b = word_of_int (1 OR neg_numeral b)"
huffman@47108
  2150
  "numeral a OR 1 = word_of_int (numeral a OR 1)"
huffman@47108
  2151
  "neg_numeral a OR 1 = word_of_int (neg_numeral a OR 1)"
huffman@47108
  2152
  "1 XOR numeral b = word_of_int (1 XOR numeral b)"
huffman@47108
  2153
  "1 XOR neg_numeral b = word_of_int (1 XOR neg_numeral b)"
huffman@47108
  2154
  "numeral a XOR 1 = word_of_int (numeral a XOR 1)"
huffman@47108
  2155
  "neg_numeral a XOR 1 = word_of_int (neg_numeral a XOR 1)"
huffman@46064
  2156
  unfolding word_1_no word_no_log_defs by simp_all
huffman@46064
  2157
haftmann@37660
  2158
lemma uint_or: "uint (x OR y) = (uint x) OR (uint y)"
huffman@45550
  2159
  by (simp add: word_or_def word_wi_log_defs word_ubin.eq_norm
haftmann@37660
  2160
                bin_trunc_ao(2) [symmetric])
haftmann@37660
  2161
haftmann@37660
  2162
lemma uint_and: "uint (x AND y) = (uint x) AND (uint y)"
huffman@45550
  2163
  by (simp add: word_and_def word_wi_log_defs word_ubin.eq_norm
haftmann@37660
  2164
                bin_trunc_ao(1) [symmetric]) 
haftmann@37660
  2165
haftmann@37660
  2166
lemma word_ops_nth_size:
haftmann@40827
  2167
  "n < size (x::'a::len0 word) \<Longrightarrow> 
haftmann@37660
  2168
    (x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2169
    (x AND y) !! n = (x !! n & y !! n) & 
haftmann@37660
  2170
    (x XOR y) !! n = (x !! n ~= y !! n) & 
haftmann@37660
  2171
    (NOT x) !! n = (~ x !! n)"
huffman@45550
  2172
  unfolding word_size word_test_bit_def word_log_defs
haftmann@37660
  2173
  by (clarsimp simp add : word_ubin.eq_norm nth_bintr bin_nth_ops)
haftmann@37660
  2174
haftmann@37660
  2175
lemma word_ao_nth:
haftmann@37660
  2176
  fixes x :: "'a::len0 word"
haftmann@37660
  2177
  shows "(x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2178
         (x AND y) !! n = (x !! n & y !! n)"
haftmann@37660
  2179
  apply (cases "n < size x")
haftmann@37660
  2180
   apply (drule_tac y = "y" in word_ops_nth_size)
haftmann@37660
  2181
   apply simp
haftmann@37660
  2182
  apply (simp add : test_bit_bin word_size)
haftmann@37660
  2183
  done
haftmann@37660
  2184
huffman@46023
  2185
lemma test_bit_wi [simp]:
huffman@46023
  2186
  "(word_of_int x::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a) \<and> bin_nth x n"
huffman@46023
  2187
  unfolding word_test_bit_def
huffman@46023
  2188
  by (simp add: nth_bintr [symmetric] word_ubin.eq_norm)
huffman@46023
  2189
huffman@47108
  2190
lemma test_bit_numeral [simp]:
huffman@47108
  2191
  "(numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
huffman@47108
  2192
    n < len_of TYPE('a) \<and> bin_nth (numeral w) n"
huffman@47108
  2193
  unfolding word_numeral_alt test_bit_wi ..
huffman@47108
  2194
huffman@47108
  2195
lemma test_bit_neg_numeral [simp]:
huffman@47108
  2196
  "(neg_numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
huffman@47108
  2197
    n < len_of TYPE('a) \<and> bin_nth (neg_numeral w) n"
huffman@47108
  2198
  unfolding word_neg_numeral_alt test_bit_wi ..
huffman@46023
  2199
huffman@46172
  2200
lemma test_bit_1 [simp]: "(1::'a::len word) !! n \<longleftrightarrow> n = 0"
huffman@46172
  2201
  unfolding word_1_wi test_bit_wi by auto
huffman@46172
  2202
  
huffman@46023
  2203
lemma nth_0 [simp]: "~ (0::'a::len0 word) !! n"
huffman@46023
  2204
  unfolding word_test_bit_def by simp
huffman@46023
  2205
huffman@47108
  2206
lemma nth_minus1 [simp]: "(-1::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a)"
huffman@47108
  2207
  unfolding word_test_bit_def by (simp add: nth_bintr)
huffman@47108
  2208
haftmann@37660
  2209
(* get from commutativity, associativity etc of int_and etc
haftmann@37660
  2210
  to same for word_and etc *)
haftmann@37660
  2211
haftmann@37660
  2212
lemmas bwsimps = 
huffman@46013
  2213
  wi_hom_add
haftmann@37660
  2214
  word_wi_log_defs
haftmann@37660
  2215
haftmann@37660
  2216
lemma word_bw_assocs:
haftmann@37660
  2217
  fixes x :: "'a::len0 word"
haftmann@37660
  2218
  shows
haftmann@37660
  2219
  "(x AND y) AND z = x AND y AND z"
haftmann@37660
  2220
  "(x OR y) OR z = x OR y OR z"
haftmann@37660
  2221
  "(x XOR y) XOR z = x XOR y XOR z"
huffman@46022
  2222
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2223
  
haftmann@37660
  2224
lemma word_bw_comms:
haftmann@37660
  2225
  fixes x :: "'a::len0 word"
haftmann@37660
  2226
  shows
haftmann@37660
  2227
  "x AND y = y AND x"
haftmann@37660
  2228
  "x OR y = y OR x"
haftmann@37660
  2229
  "x XOR y = y XOR x"
huffman@46022
  2230
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2231
  
haftmann@37660
  2232
lemma word_bw_lcs:
haftmann@37660
  2233
  fixes x :: "'a::len0 word"
haftmann@37660
  2234
  shows
haftmann@37660
  2235
  "y AND x AND z = x AND y AND z"
haftmann@37660
  2236
  "y OR x OR z = x OR y OR z"
haftmann@37660
  2237
  "y XOR x XOR z = x XOR y XOR z"
huffman@46022
  2238
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2239
haftmann@37660
  2240
lemma word_log_esimps [simp]:
haftmann@37660
  2241
  fixes x :: "'a::len0 word"
haftmann@37660
  2242
  shows
haftmann@37660
  2243
  "x AND 0 = 0"
haftmann@37660
  2244
  "x AND -1 = x"
haftmann@37660
  2245
  "x OR 0 = x"
haftmann@37660
  2246
  "x OR -1 = -1"
haftmann@37660
  2247
  "x XOR 0 = x"
haftmann@37660
  2248
  "x XOR -1 = NOT x"
haftmann@37660
  2249
  "0 AND x = 0"
haftmann@37660
  2250
  "-1 AND x = x"
haftmann@37660
  2251
  "0 OR x = x"
haftmann@37660
  2252
  "-1 OR x = -1"
haftmann@37660
  2253
  "0 XOR x = x"
haftmann@37660
  2254
  "-1 XOR x = NOT x"
huffman@46023
  2255
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2256
haftmann@37660
  2257
lemma word_not_dist:
haftmann@37660
  2258
  fixes x :: "'a::len0 word"
haftmann@37660
  2259
  shows
haftmann@37660
  2260
  "NOT (x OR y) = NOT x AND NOT y"
haftmann@37660
  2261
  "NOT (x AND y) = NOT x OR NOT y"
huffman@46022
  2262
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2263
haftmann@37660
  2264
lemma word_bw_same:
haftmann@37660
  2265
  fixes x :: "'a::len0 word"
haftmann@37660
  2266
  shows
haftmann@37660
  2267
  "x AND x = x"
haftmann@37660
  2268
  "x OR x = x"
haftmann@37660
  2269
  "x XOR x = 0"
huffman@46023
  2270
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2271
haftmann@37660
  2272
lemma word_ao_absorbs [simp]:
haftmann@37660
  2273
  fixes x :: "'a::len0 word"
haftmann@37660
  2274
  shows
haftmann@37660
  2275
  "x AND (y OR x) = x"
haftmann@37660
  2276
  "x OR y AND x = x"
haftmann@37660
  2277
  "x AND (x OR y) = x"
haftmann@37660
  2278
  "y AND x OR x = x"
haftmann@37660
  2279
  "(y OR x) AND x = x"
haftmann@37660
  2280
  "x OR x AND y = x"
haftmann@37660
  2281
  "(x OR y) AND x = x"
haftmann@37660
  2282
  "x AND y OR x = x"
huffman@46022
  2283
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2284
haftmann@37660
  2285
lemma word_not_not [simp]:
haftmann@37660
  2286
  "NOT NOT (x::'a::len0 word) = x"
huffman@46022
  2287
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2288
haftmann@37660
  2289
lemma word_ao_dist:
haftmann@37660
  2290
  fixes x :: "'a::len0 word"
haftmann@37660
  2291
  shows "(x OR y) AND z = x AND z OR y AND z"
huffman@46022
  2292
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2293
haftmann@37660
  2294
lemma word_oa_dist:
haftmann@37660
  2295
  fixes x :: "'a::len0 word"
haftmann@37660
  2296
  shows "x AND y OR z = (x OR z) AND (y OR z)"
huffman@46022
  2297
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2298
haftmann@37660
  2299
lemma word_add_not [simp]: 
haftmann@37660
  2300
  fixes x :: "'a::len0 word"
haftmann@37660
  2301
  shows "x + NOT x = -1"
haftmann@37660
  2302
  using word_of_int_Ex [where x=x] 
huffman@46656
  2303
  by (auto simp: bwsimps bin_add_not [unfolded Min_def])
haftmann@37660
  2304
haftmann@37660
  2305
lemma word_plus_and_or [simp]:
haftmann@37660
  2306
  fixes x :: "'a::len0 word"
haftmann@37660
  2307
  shows "(x AND y) + (x OR y) = x + y"
haftmann@37660
  2308
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2309
        word_of_int_Ex [where x=y] 
haftmann@37660
  2310
  by (auto simp: bwsimps plus_and_or)
haftmann@37660
  2311
haftmann@37660
  2312
lemma leoa:   
haftmann@37660
  2313
  fixes x :: "'a::len0 word"
haftmann@40827
  2314
  shows "(w = (x OR y)) \<Longrightarrow> (y = (w AND y))" by auto
haftmann@37660
  2315
lemma leao: 
haftmann@37660
  2316
  fixes x' :: "'a::len0 word"
haftmann@40827
  2317
  shows "(w' = (x' AND y')) \<Longrightarrow> (x' = (x' OR w'))" by auto 
haftmann@37660
  2318
haftmann@37660
  2319
lemmas word_ao_equiv = leao [COMP leoa [COMP iffI]]
haftmann@37660
  2320
haftmann@37660
  2321
lemma le_word_or2: "x <= x OR (y::'a::len0 word)"
haftmann@37660
  2322
  unfolding word_le_def uint_or
haftmann@37660
  2323
  by (auto intro: le_int_or) 
haftmann@37660
  2324
wenzelm@45604
  2325
lemmas le_word_or1 = xtr3 [OF word_bw_comms (2) le_word_or2]
wenzelm@45604
  2326
lemmas word_and_le1 = xtr3 [OF word_ao_absorbs (4) [symmetric] le_word_or2]
wenzelm@45604
  2327
lemmas word_and_le2 = xtr3 [OF word_ao_absorbs (8) [symmetric] le_word_or2]
haftmann@37660
  2328
haftmann@37660
  2329
lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" 
huffman@45550
  2330
  unfolding to_bl_def word_log_defs bl_not_bin
huffman@45550
  2331
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2332
haftmann@37660
  2333
lemma bl_word_xor: "to_bl (v XOR w) = map2 op ~= (to_bl v) (to_bl w)" 
haftmann@37660
  2334
  unfolding to_bl_def word_log_defs bl_xor_bin
huffman@45550
  2335
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2336
haftmann@37660
  2337
lemma bl_word_or: "to_bl (v OR w) = map2 op | (to_bl v) (to_bl w)" 
huffman@45550
  2338
  unfolding to_bl_def word_log_defs bl_or_bin
huffman@45550
  2339
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2340
haftmann@37660
  2341
lemma bl_word_and: "to_bl (v AND w) = map2 op & (to_bl v) (to_bl w)" 
huffman@45550
  2342
  unfolding to_bl_def word_log_defs bl_and_bin
huffman@45550
  2343
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2344
haftmann@37660
  2345
lemma word_lsb_alt: "lsb (w::'a::len0 word) = test_bit w 0"
haftmann@37660
  2346
  by (auto simp: word_test_bit_def word_lsb_def)
haftmann@37660
  2347
huffman@45805
  2348
lemma word_lsb_1_0 [simp]: "lsb (1::'a::len word) & ~ lsb (0::'b::len0 word)"
huffman@45550
  2349
  unfolding word_lsb_def uint_eq_0 uint_1 by simp
haftmann@37660
  2350
haftmann@37660
  2351
lemma word_lsb_last: "lsb (w::'a::len word) = last (to_bl w)"
haftmann@37660
  2352
  apply (unfold word_lsb_def uint_bl bin_to_bl_def) 
haftmann@37660
  2353
  apply (rule_tac bin="uint w" in bin_exhaust)
haftmann@37660
  2354
  apply (cases "size w")
haftmann@37660
  2355
   apply auto
haftmann@37660
  2356
   apply (auto simp add: bin_to_bl_aux_alt)
haftmann@37660
  2357
  done
haftmann@37660
  2358
haftmann@37660
  2359
lemma word_lsb_int: "lsb w = (uint w mod 2 = 1)"
huffman@45529
  2360
  unfolding word_lsb_def bin_last_def by auto
haftmann@37660
  2361
haftmann@37660
  2362
lemma word_msb_sint: "msb w = (sint w < 0)" 
huffman@46604
  2363
  unfolding word_msb_def sign_Min_lt_0 ..
haftmann@37660
  2364
huffman@46173
  2365
lemma msb_word_of_int:
huffman@46173
  2366
  "msb (word_of_int x::'a::len word) = bin_nth x (len_of TYPE('a) - 1)"
huffman@46173
  2367
  unfolding word_msb_def by (simp add: word_sbin.eq_norm bin_sign_lem)
huffman@46173
  2368
huffman@47108
  2369
lemma word_msb_numeral [simp]:
huffman@47108
  2370
  "msb (numeral w::'a::len word) = bin_nth (numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2371
  unfolding word_numeral_alt by (rule msb_word_of_int)
huffman@47108
  2372
huffman@47108
  2373
lemma word_msb_neg_numeral [simp]:
huffman@47108
  2374
  "msb (neg_numeral w::'a::len word) = bin_nth (neg_numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2375
  unfolding word_neg_numeral_alt by (rule msb_word_of_int)
huffman@46173
  2376
huffman@46173
  2377
lemma word_msb_0 [simp]: "\<not> msb (0::'a::len word)"
huffman@46173
  2378
  unfolding word_msb_def by simp
huffman@46173
  2379
huffman@46173
  2380
lemma word_msb_1 [simp]: "msb (1::'a::len word) \<longleftrightarrow> len_of TYPE('a) = 1"
huffman@46173
  2381
  unfolding word_1_wi msb_word_of_int eq_iff [where 'a=nat]
huffman@46173
  2382
  by (simp add: Suc_le_eq)
huffman@45811
  2383
huffman@45811
  2384
lemma word_msb_nth:
huffman@45811
  2385
  "msb (w::'a::len word) = bin_nth (uint w) (len_of TYPE('a) - 1)"
huffman@46023
  2386
  unfolding word_msb_def sint_uint by (simp add: bin_sign_lem)
haftmann@37660
  2387
haftmann@37660
  2388
lemma word_msb_alt: "msb (w::'a::len word) = hd (to_bl w)"
haftmann@37660
  2389
  apply (unfold word_msb_nth uint_bl)
haftmann@37660
  2390
  apply (subst hd_conv_nth)
haftmann@37660
  2391
  apply (rule length_greater_0_conv [THEN iffD1])
haftmann@37660
  2392
   apply simp
haftmann@37660
  2393
  apply (simp add : nth_bin_to_bl word_size)
haftmann@37660
  2394
  done
haftmann@37660
  2395
huffman@45805
  2396
lemma word_set_nth [simp]:
haftmann@37660
  2397
  "set_bit w n (test_bit w n) = (w::'a::len0 word)"
haftmann@37660
  2398
  unfolding word_test_bit_def word_set_bit_def by auto
haftmann@37660
  2399
haftmann@37660
  2400
lemma bin_nth_uint':
haftmann@37660
  2401
  "bin_nth (uint w) n = (rev (bin_to_bl (size w) (uint w)) ! n & n < size w)"
haftmann@37660
  2402
  apply (unfold word_size)
haftmann@37660
  2403
  apply (safe elim!: bin_nth_uint_imp)
haftmann@37660
  2404
   apply (frule bin_nth_uint_imp)
haftmann@37660
  2405
   apply (fast dest!: bin_nth_bl)+
haftmann@37660
  2406
  done
haftmann@37660
  2407
haftmann@37660
  2408
lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size]
haftmann@37660
  2409
haftmann@37660
  2410
lemma test_bit_bl: "w !! n = (rev (to_bl w) ! n & n < size w)"
haftmann@37660
  2411
  unfolding to_bl_def word_test_bit_def word_size
haftmann@37660
  2412
  by (rule bin_nth_uint)
haftmann@37660
  2413
haftmann@40827
  2414
lemma to_bl_nth: "n < size w \<Longrightarrow> to_bl w ! n = w !! (size w - Suc n)"
haftmann@37660
  2415
  apply (unfold test_bit_bl)
haftmann@37660
  2416
  apply clarsimp
haftmann@37660
  2417
  apply (rule trans)
haftmann@37660
  2418
   apply (rule nth_rev_alt)
haftmann@37660
  2419
   apply (auto simp add: word_size)
haftmann@37660
  2420
  done
haftmann@37660
  2421
haftmann@37660
  2422
lemma test_bit_set: 
haftmann@37660
  2423
  fixes w :: "'a::len0 word"
haftmann@37660
  2424
  shows "(set_bit w n x) !! n = (n < size w & x)"
haftmann@37660
  2425
  unfolding word_size word_test_bit_def word_set_bit_def
haftmann@37660
  2426
  by (clarsimp simp add : word_ubin.eq_norm nth_bintr)
haftmann@37660
  2427
haftmann@37660
  2428
lemma test_bit_set_gen: 
haftmann@37660
  2429
  fixes w :: "'a::len0 word"
haftmann@37660
  2430
  shows "test_bit (set_bit w n x) m = 
haftmann@37660
  2431
         (if m = n then n < size w & x else test_bit w m)"
haftmann@37660
  2432
  apply (unfold word_size word_test_bit_def word_set_bit_def)
haftmann@37660
  2433
  apply (clarsimp simp add: word_ubin.eq_norm nth_bintr bin_nth_sc_gen)
haftmann@37660
  2434
  apply (auto elim!: test_bit_size [unfolded word_size]
haftmann@37660
  2435
              simp add: word_test_bit_def [symmetric])
haftmann@37660
  2436
  done
haftmann@37660
  2437
haftmann@37660
  2438
lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs"
haftmann@37660
  2439
  unfolding of_bl_def bl_to_bin_rep_F by auto
haftmann@37660
  2440
  
huffman@45811
  2441
lemma msb_nth:
haftmann@37660
  2442
  fixes w :: "'a::len word"
huffman@45811
  2443
  shows "msb w = w !! (len_of TYPE('a) - 1)"
huffman@45811
  2444
  unfolding word_msb_nth word_test_bit_def by simp
haftmann@37660
  2445
wenzelm@45604
  2446
lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2447
lemmas msb1 = msb0 [where i = 0]
haftmann@37660
  2448
lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]]
haftmann@37660
  2449
wenzelm@45604
  2450
lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2451
lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt]
haftmann@37660
  2452
huffman@45811
  2453
lemma td_ext_nth [OF refl refl refl, unfolded word_size]:
haftmann@40827
  2454
  "n = size (w::'a::len0 word) \<Longrightarrow> ofn = set_bits \<Longrightarrow> [w, ofn g] = l \<Longrightarrow> 
haftmann@37660
  2455
    td_ext test_bit ofn {f. ALL i. f i --> i < n} (%h i. h i & i < n)"
haftmann@37660
  2456
  apply (unfold word_size td_ext_def')
wenzelm@46008
  2457
  apply safe
haftmann@37660
  2458
     apply (rule_tac [3] ext)
haftmann@37660
  2459
     apply (rule_tac [4] ext)
haftmann@37660
  2460
     apply (unfold word_size of_nth_def test_bit_bl)
haftmann@37660
  2461
     apply safe
haftmann@37660
  2462
       defer
haftmann@37660
  2463
       apply (clarsimp simp: word_bl.Abs_inverse)+
haftmann@37660
  2464
  apply (rule word_bl.Rep_inverse')
haftmann@37660
  2465
  apply (rule sym [THEN trans])
haftmann@37660
  2466
  apply (rule bl_of_nth_nth)
haftmann@37660
  2467
  apply simp
haftmann@37660
  2468
  apply (rule bl_of_nth_inj)
haftmann@37660
  2469
  apply (clarsimp simp add : test_bit_bl word_size)
haftmann@37660
  2470
  done
haftmann@37660
  2471
haftmann@37660
  2472
interpretation test_bit:
haftmann@37660
  2473
  td_ext "op !! :: 'a::len0 word => nat => bool"
haftmann@37660
  2474
         set_bits
haftmann@37660
  2475
         "{f. \<forall>i. f i \<longrightarrow> i < len_of TYPE('a::len0)}"
haftmann@37660
  2476
         "(\<lambda>h i. h i \<and> i < len_of TYPE('a::len0))"
haftmann@37660
  2477
  by (rule td_ext_nth)
haftmann@37660
  2478
haftmann@37660
  2479
lemmas td_nth = test_bit.td_thm
haftmann@37660
  2480
huffman@45805
  2481
lemma word_set_set_same [simp]:
haftmann@37660
  2482
  fixes w :: "'a::len0 word"
haftmann@37660
  2483
  shows "set_bit (set_bit w n x) n y = set_bit w n y" 
haftmann@37660
  2484
  by (rule word_eqI) (simp add : test_bit_set_gen word_size)
haftmann@37660
  2485
    
haftmann@37660
  2486
lemma word_set_set_diff: 
haftmann@37660
  2487
  fixes w :: "'a::len0 word"
haftmann@37660
  2488
  assumes "m ~= n"
haftmann@37660
  2489
  shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x" 
wenzelm@41550
  2490
  by (rule word_eqI) (clarsimp simp add: test_bit_set_gen word_size assms)
huffman@46001
  2491
haftmann@37660
  2492
lemma nth_sint: 
haftmann@37660
  2493
  fixes w :: "'a::len word"
haftmann@37660
  2494
  defines "l \<equiv> len_of TYPE ('a)"
haftmann@37660
  2495
  shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
haftmann@37660
  2496
  unfolding sint_uint l_def
haftmann@37660
  2497
  by (clarsimp simp add: nth_sbintr word_test_bit_def [symmetric])
haftmann@37660
  2498
huffman@47108
  2499
lemma word_lsb_numeral [simp]:
huffman@47108
  2500
  "lsb (numeral bin :: 'a :: len word) = (bin_last (numeral bin) = 1)"
huffman@47108
  2501
  unfolding word_lsb_alt test_bit_numeral by simp
huffman@47108
  2502
huffman@47108
  2503
lemma word_lsb_neg_numeral [simp]:
huffman@47108
  2504
  "lsb (neg_numeral bin :: 'a :: len word) = (bin_last (neg_numeral bin) = 1)"
huffman@47108
  2505
  unfolding word_lsb_alt test_bit_neg_numeral by simp
haftmann@37660
  2506
huffman@46173
  2507
lemma set_bit_word_of_int:
huffman@46173
  2508
  "set_bit (word_of_int x) n b = word_of_int (bin_sc n (if b then 1 else 0) x)"
huffman@46173
  2509
  unfolding word_set_bit_def
huffman@46173
  2510
  apply (rule word_eqI)
huffman@46173
  2511
  apply (simp add: word_size bin_nth_sc_gen word_ubin.eq_norm nth_bintr)
huffman@46173
  2512
  done
huffman@46173
  2513
huffman@47108
  2514
lemma word_set_numeral [simp]:
huffman@47108
  2515
  "set_bit (numeral bin::'a::len0 word) n b = 
huffman@47108
  2516
    word_of_int (bin_sc n (if b then 1 else 0) (numeral bin))"
huffman@47108
  2517
  unfolding word_numeral_alt by (rule set_bit_word_of_int)
huffman@47108
  2518
huffman@47108
  2519
lemma word_set_neg_numeral [simp]:
huffman@47108
  2520
  "set_bit (neg_numeral bin::'a::len0 word) n b = 
huffman@47108
  2521
    word_of_int (bin_sc n (if b then 1 else 0) (neg_numeral bin))"
huffman@47108
  2522
  unfolding word_neg_numeral_alt by (rule set_bit_word_of_int)
huffman@46173
  2523
huffman@46173
  2524
lemma word_set_bit_0 [simp]:
huffman@46173
  2525
  "set_bit 0 n b = word_of_int (bin_sc n (if b then 1 else 0) 0)"
huffman@46173
  2526
  unfolding word_0_wi by (rule set_bit_word_of_int)
huffman@46173
  2527
huffman@46173
  2528
lemma word_set_bit_1 [simp]:
huffman@46173
  2529
  "set_bit 1 n b = word_of_int (bin_sc n (if b then 1 else 0) 1)"
huffman@46173
  2530
  unfolding word_1_wi by (rule set_bit_word_of_int)
haftmann@37660
  2531
huffman@45805
  2532
lemma setBit_no [simp]:
huffman@47108
  2533
  "setBit (numeral bin) n = word_of_int (bin_sc n 1 (numeral bin))"
huffman@45805
  2534
  by (simp add: setBit_def)
huffman@45805
  2535
huffman@45805
  2536
lemma clearBit_no [simp]:
huffman@47108
  2537
  "clearBit (numeral bin) n = word_of_int (bin_sc n 0 (numeral bin))"
huffman@45805
  2538
  by (simp add: clearBit_def)
haftmann@37660
  2539
haftmann@37660
  2540
lemma to_bl_n1: 
haftmann@37660
  2541
  "to_bl (-1::'a::len0 word) = replicate (len_of TYPE ('a)) True"
haftmann@37660
  2542
  apply (rule word_bl.Abs_inverse')
haftmann@37660
  2543
   apply simp
haftmann@37660
  2544
  apply (rule word_eqI)
huffman@45805
  2545
  apply (clarsimp simp add: word_size)
haftmann@37660
  2546
  apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size)
haftmann@37660
  2547
  done
haftmann@37660
  2548
huffman@45805
  2549
lemma word_msb_n1 [simp]: "msb (-1::'a::len word)"
wenzelm@41550
  2550
  unfolding word_msb_alt to_bl_n1 by simp
haftmann@37660
  2551
haftmann@37660
  2552
lemma word_set_nth_iff: 
haftmann@37660
  2553
  "(set_bit w n b = w) = (w !! n = b | n >= size (w::'a::len0 word))"
haftmann@37660
  2554
  apply (rule iffI)
haftmann@37660
  2555
   apply (rule disjCI)
haftmann@37660
  2556
   apply (drule word_eqD)
haftmann@37660
  2557
   apply (erule sym [THEN trans])
haftmann@37660
  2558
   apply (simp add: test_bit_set)
haftmann@37660
  2559
  apply (erule disjE)
haftmann@37660
  2560
   apply clarsimp
haftmann@37660
  2561
  apply (rule word_eqI)
haftmann@37660
  2562
  apply (clarsimp simp add : test_bit_set_gen)
haftmann@37660
  2563
  apply (drule test_bit_size)
haftmann@37660
  2564
  apply force
haftmann@37660
  2565
  done
haftmann@37660
  2566
huffman@45811
  2567
lemma test_bit_2p:
huffman@45811
  2568
  "(word_of_int (2 ^ n)::'a::len word) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a)"
huffman@45811
  2569
  unfolding word_test_bit_def
haftmann@37660
  2570
  by (auto simp add: word_ubin.eq_norm nth_bintr nth_2p_bin)
haftmann@37660
  2571
haftmann@37660
  2572
lemma nth_w2p:
haftmann@37660
  2573
  "((2\<Colon>'a\<Colon>len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a\<Colon>len)"
haftmann@37660
  2574
  unfolding test_bit_2p [symmetric] word_of_int [symmetric]
haftmann@37660
  2575
  by (simp add:  of_int_power)
haftmann@37660
  2576
haftmann@37660
  2577
lemma uint_2p: 
haftmann@40827
  2578
  "(0::'a::len word) < 2 ^ n \<Longrightarrow> uint (2 ^ n::'a::len word) = 2 ^ n"
haftmann@37660
  2579
  apply (unfold word_arith_power_alt)
haftmann@37660
  2580
  apply (case_tac "len_of TYPE ('a)")
haftmann@37660
  2581
   apply clarsimp
haftmann@37660
  2582
  apply (case_tac "nat")
haftmann@37660
  2583
   apply clarsimp
haftmann@37660
  2584
   apply (case_tac "n")
huffman@46001
  2585
    apply clarsimp
huffman@46001
  2586
   apply clarsimp
haftmann@37660
  2587
  apply (drule word_gt_0 [THEN iffD1])
wenzelm@46124
  2588
  apply (safe intro!: word_eqI bin_nth_lem)
huffman@46001
  2589
     apply (auto simp add: test_bit_2p nth_2p_bin word_test_bit_def [symmetric])
haftmann@37660
  2590
  done
haftmann@37660
  2591
haftmann@37660
  2592
lemma word_of_int_2p: "(word_of_int (2 ^ n) :: 'a :: len word) = 2 ^ n" 
haftmann@37660
  2593
  apply (unfold word_arith_power_alt)
haftmann@37660
  2594
  apply (case_tac "len_of TYPE ('a)")
haftmann@37660
  2595
   apply clarsimp
haftmann@37660
  2596
  apply (case_tac "nat")
haftmann@37660
  2597
   apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 
haftmann@37660
  2598
   apply (rule box_equals) 
haftmann@37660
  2599
     apply (rule_tac [2] bintr_ariths (1))+ 
huffman@47108
  2600
   apply simp
huffman@46001
  2601
  apply simp
haftmann@37660
  2602
  done
haftmann@37660
  2603
haftmann@40827
  2604
lemma bang_is_le: "x !! m \<Longrightarrow> 2 ^ m <= (x :: 'a :: len word)" 
haftmann@37660
  2605
  apply (rule xtr3) 
haftmann@37660
  2606
  apply (rule_tac [2] y = "x" in le_word_or2)
haftmann@37660
  2607
  apply (rule word_eqI)
haftmann@37660
  2608
  apply (auto simp add: word_ao_nth nth_w2p word_size)
haftmann@37660
  2609
  done
haftmann@37660
  2610
haftmann@37660
  2611
lemma word_clr_le: 
haftmann@37660
  2612
  fixes w :: "'a::len0 word"
haftmann@37660
  2613
  shows "w >= set_bit w n False"
haftmann@37660
  2614
  apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm)
haftmann@37660
  2615
  apply simp
haftmann@37660
  2616
  apply (rule order_trans)
haftmann@37660
  2617
   apply (rule bintr_bin_clr_le)
haftmann@37660
  2618
  apply simp
haftmann@37660
  2619
  done
haftmann@37660
  2620
haftmann@37660
  2621
lemma word_set_ge: 
haftmann@37660
  2622
  fixes w :: "'a::len word"
haftmann@37660
  2623
  shows "w <= set_bit w n True"
haftmann@37660
  2624
  apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm)
haftmann@37660
  2625
  apply simp
haftmann@37660
  2626
  apply (rule order_trans [OF _ bintr_bin_set_ge])
haftmann@37660
  2627
  apply simp
haftmann@37660
  2628
  done
haftmann@37660
  2629
haftmann@37660
  2630
haftmann@37660
  2631
subsection {* Shifting, Rotating, and Splitting Words *}
haftmann@37660