src/HOL/Word/Word.thy
author huffman
Fri Apr 06 14:40:00 2012 +0200 (2012-04-06)
changeset 47387 a0f257197741
parent 47377 360d7ed4cc0f
child 47521 69f95ac85c3d
permissions -rw-r--r--
remove now-unnecessary type annotations from lift_definition commands
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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 {* Correspondence relation for theorem transfer *}
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definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool"
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  where "cr_word \<equiv> (\<lambda>x y. word_of_int x = y)"
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lemma Quotient_word:
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  "Quotient (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y)
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    word_of_int uint (cr_word :: _ \<Rightarrow> 'a::len0 word \<Rightarrow> bool)"
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  unfolding Quotient_alt_def cr_word_def
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  by (simp add: word_ubin.norm_eq_iff)
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lemma reflp_word:
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  "reflp (\<lambda>x y. bintrunc (len_of TYPE('a::len0)) x = bintrunc (len_of TYPE('a)) y)"
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  by (simp add: reflp_def)
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local_setup {*
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  Lifting_Setup.setup_lifting_infr @{thm Quotient_word} @{thm reflp_word}
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*}
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text {* TODO: The next several lemmas could be generated automatically. *}
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lemma bi_total_cr_word [transfer_rule]: "bi_total cr_word"
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  using Quotient_word reflp_word by (rule Quotient_bi_total)
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lemma right_unique_cr_word [transfer_rule]: "right_unique cr_word"
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  using Quotient_word by (rule Quotient_right_unique)
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lemma word_eq_transfer [transfer_rule]:
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  "(fun_rel cr_word (fun_rel cr_word op =))
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    (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y)
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    (op = :: 'a::len0 word \<Rightarrow> 'a word \<Rightarrow> bool)"
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  using Quotient_word by (rule Quotient_rel_eq_transfer)
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lemma word_of_int_transfer [transfer_rule]:
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  "(fun_rel op = cr_word) (\<lambda>x. x) word_of_int"
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  using Quotient_word reflp_word by (rule Quotient_id_abs_transfer)
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lemma uint_transfer [transfer_rule]:
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  "(fun_rel cr_word op =) (bintrunc (len_of TYPE('a)))
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    (uint :: 'a::len0 word \<Rightarrow> int)"
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  unfolding fun_rel_def cr_word_def by (simp add: word_ubin.eq_norm)
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subsection  "Arithmetic operations"
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lift_definition word_succ :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x + 1"
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  by (metis bintr_ariths(6))
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lift_definition word_pred :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x - 1"
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  by (metis bintr_ariths(7))
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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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lift_definition zero_word :: "'a word" is "0" .
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lift_definition one_word :: "'a word" is "1" .
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lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op +"
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  by (metis bintr_ariths(2))
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lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op -"
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  by (metis bintr_ariths(3))
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lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus
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  by (metis bintr_ariths(5))
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lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op *"
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  by (metis bintr_ariths(4))
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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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instance
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  by default (transfer, simp add: algebra_simps)+
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end
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text {* Legacy theorems: *}
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lemma word_arith_wis: shows
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  word_add_def: "a + b = word_of_int (uint a + uint b)" and
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  word_sub_wi: "a - b = word_of_int (uint a - uint b)" and
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  word_mult_def: "a * b = word_of_int (uint a * uint b)" and
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  word_minus_def: "- a = word_of_int (- uint a)" and
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  word_succ_alt: "word_succ a = word_of_int (uint a + 1)" and
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  word_pred_alt: "word_pred a = word_of_int (uint a - 1)" and
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  word_0_wi: "0 = word_of_int 0" and
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  word_1_wi: "1 = word_of_int 1"
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  unfolding plus_word_def minus_word_def times_word_def uminus_word_def
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  unfolding word_succ_def word_pred_def zero_word_def one_word_def
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  by simp_all
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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: 
huffman@45545
   330
  shows
huffman@45545
   331
  wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and
huffman@46013
   332
  wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)" and
huffman@45545
   333
  wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and
huffman@45545
   334
  wi_hom_neg: "- word_of_int a = word_of_int (- a)" and
huffman@46000
   335
  wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and
huffman@46000
   336
  wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
huffman@47374
   337
  by (transfer, simp)+
huffman@45545
   338
huffman@45545
   339
lemmas wi_hom_syms = wi_homs [symmetric]
huffman@45545
   340
huffman@46013
   341
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
huffman@46009
   342
huffman@46009
   343
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
huffman@45545
   344
huffman@45545
   345
instance word :: (len) comm_ring_1
huffman@45810
   346
proof
huffman@45810
   347
  have "0 < len_of TYPE('a)" by (rule len_gt_0)
huffman@45810
   348
  then show "(0::'a word) \<noteq> 1"
huffman@47372
   349
    by - (transfer, auto simp add: gr0_conv_Suc)
huffman@45810
   350
qed
huffman@45545
   351
huffman@45545
   352
lemma word_of_nat: "of_nat n = word_of_int (int n)"
huffman@45545
   353
  by (induct n) (auto simp add : word_of_int_hom_syms)
huffman@45545
   354
huffman@45545
   355
lemma word_of_int: "of_int = word_of_int"
huffman@45545
   356
  apply (rule ext)
huffman@45545
   357
  apply (case_tac x rule: int_diff_cases)
huffman@46013
   358
  apply (simp add: word_of_nat wi_hom_sub)
huffman@45545
   359
  done
huffman@45545
   360
haftmann@37660
   361
definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where
haftmann@40827
   362
  "a udvd b = (EX n>=0. uint b = n * uint a)"
haftmann@37660
   363
huffman@45547
   364
huffman@45547
   365
subsection "Ordering"
huffman@45547
   366
huffman@45547
   367
instantiation word :: (len0) linorder
huffman@45547
   368
begin
huffman@45547
   369
haftmann@37660
   370
definition
haftmann@37660
   371
  word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
haftmann@37660
   372
haftmann@37660
   373
definition
huffman@47108
   374
  word_less_def: "a < b \<longleftrightarrow> uint a < uint b"
haftmann@37660
   375
huffman@45547
   376
instance
huffman@45547
   377
  by default (auto simp: word_less_def word_le_def)
huffman@45547
   378
huffman@45547
   379
end
huffman@45547
   380
haftmann@37660
   381
definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where
haftmann@40827
   382
  "a <=s b = (sint a <= sint b)"
haftmann@37660
   383
haftmann@37660
   384
definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where
haftmann@40827
   385
  "(x <s y) = (x <=s y & x ~= y)"
haftmann@37660
   386
haftmann@37660
   387
haftmann@37660
   388
subsection "Bit-wise operations"
haftmann@37660
   389
haftmann@37660
   390
instantiation word :: (len0) bits
haftmann@37660
   391
begin
haftmann@37660
   392
huffman@47387
   393
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT
huffman@47374
   394
  by (metis bin_trunc_not)
huffman@47374
   395
huffman@47387
   396
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND
huffman@47374
   397
  by (metis bin_trunc_and)
huffman@47374
   398
huffman@47387
   399
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR
huffman@47374
   400
  by (metis bin_trunc_or)
huffman@47374
   401
huffman@47387
   402
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR
huffman@47374
   403
  by (metis bin_trunc_xor)
haftmann@37660
   404
haftmann@37660
   405
definition
haftmann@37660
   406
  word_test_bit_def: "test_bit a = bin_nth (uint a)"
haftmann@37660
   407
haftmann@37660
   408
definition
haftmann@37660
   409
  word_set_bit_def: "set_bit a n x =
haftmann@37660
   410
   word_of_int (bin_sc n (If x 1 0) (uint a))"
haftmann@37660
   411
haftmann@37660
   412
definition
haftmann@37660
   413
  word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"
haftmann@37660
   414
haftmann@37660
   415
definition
haftmann@37660
   416
  word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1"
haftmann@37660
   417
haftmann@37660
   418
definition shiftl1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   419
  "shiftl1 w = word_of_int (uint w BIT 0)"
haftmann@37660
   420
haftmann@37660
   421
definition shiftr1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   422
  -- "shift right as unsigned or as signed, ie logical or arithmetic"
haftmann@37660
   423
  "shiftr1 w = word_of_int (bin_rest (uint w))"
haftmann@37660
   424
haftmann@37660
   425
definition
haftmann@37660
   426
  shiftl_def: "w << n = (shiftl1 ^^ n) w"
haftmann@37660
   427
haftmann@37660
   428
definition
haftmann@37660
   429
  shiftr_def: "w >> n = (shiftr1 ^^ n) w"
haftmann@37660
   430
haftmann@37660
   431
instance ..
haftmann@37660
   432
haftmann@37660
   433
end
haftmann@37660
   434
huffman@47374
   435
lemma shows
huffman@47374
   436
  word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" and
huffman@47374
   437
  word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" and
huffman@47374
   438
  word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" and
huffman@47374
   439
  word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
huffman@47374
   440
  unfolding bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def
huffman@47374
   441
  by simp_all
huffman@47374
   442
haftmann@37660
   443
instantiation word :: (len) bitss
haftmann@37660
   444
begin
haftmann@37660
   445
haftmann@37660
   446
definition
haftmann@37660
   447
  word_msb_def: 
huffman@46001
   448
  "msb a \<longleftrightarrow> bin_sign (sint a) = -1"
haftmann@37660
   449
haftmann@37660
   450
instance ..
haftmann@37660
   451
haftmann@37660
   452
end
haftmann@37660
   453
haftmann@37660
   454
definition setBit :: "'a :: len0 word => nat => 'a word" where 
haftmann@40827
   455
  "setBit w n = set_bit w n True"
haftmann@37660
   456
haftmann@37660
   457
definition clearBit :: "'a :: len0 word => nat => 'a word" where
haftmann@40827
   458
  "clearBit w n = set_bit w n False"
haftmann@37660
   459
haftmann@37660
   460
haftmann@37660
   461
subsection "Shift operations"
haftmann@37660
   462
haftmann@37660
   463
definition sshiftr1 :: "'a :: len word => 'a word" where 
haftmann@40827
   464
  "sshiftr1 w = word_of_int (bin_rest (sint w))"
haftmann@37660
   465
haftmann@37660
   466
definition bshiftr1 :: "bool => 'a :: len word => 'a word" where
haftmann@40827
   467
  "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
haftmann@37660
   468
haftmann@37660
   469
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where
haftmann@40827
   470
  "w >>> n = (sshiftr1 ^^ n) w"
haftmann@37660
   471
haftmann@37660
   472
definition mask :: "nat => 'a::len word" where
haftmann@40827
   473
  "mask n = (1 << n) - 1"
haftmann@37660
   474
haftmann@37660
   475
definition revcast :: "'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   476
  "revcast w =  of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
haftmann@37660
   477
haftmann@37660
   478
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   479
  "slice1 n w = of_bl (takefill False n (to_bl w))"
haftmann@37660
   480
haftmann@37660
   481
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   482
  "slice n w = slice1 (size w - n) w"
haftmann@37660
   483
haftmann@37660
   484
haftmann@37660
   485
subsection "Rotation"
haftmann@37660
   486
haftmann@37660
   487
definition rotater1 :: "'a list => 'a list" where
haftmann@40827
   488
  "rotater1 ys = 
haftmann@40827
   489
    (case ys of [] => [] | x # xs => last ys # butlast ys)"
haftmann@37660
   490
haftmann@37660
   491
definition rotater :: "nat => 'a list => 'a list" where
haftmann@40827
   492
  "rotater n = rotater1 ^^ n"
haftmann@37660
   493
haftmann@37660
   494
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   495
  "word_rotr n w = of_bl (rotater n (to_bl w))"
haftmann@37660
   496
haftmann@37660
   497
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   498
  "word_rotl n w = of_bl (rotate n (to_bl w))"
haftmann@37660
   499
haftmann@37660
   500
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   501
  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
haftmann@40827
   502
                    else word_rotl (nat (- i)) w)"
haftmann@37660
   503
haftmann@37660
   504
haftmann@37660
   505
subsection "Split and cat operations"
haftmann@37660
   506
haftmann@37660
   507
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where
haftmann@40827
   508
  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
haftmann@37660
   509
haftmann@37660
   510
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where
haftmann@40827
   511
  "word_split a = 
haftmann@40827
   512
   (case bin_split (len_of TYPE ('c)) (uint a) of 
haftmann@40827
   513
     (u, v) => (word_of_int u, word_of_int v))"
haftmann@37660
   514
haftmann@37660
   515
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where
haftmann@40827
   516
  "word_rcat ws = 
haftmann@37660
   517
  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
haftmann@37660
   518
haftmann@37660
   519
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where
haftmann@40827
   520
  "word_rsplit w = 
haftmann@37660
   521
  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
haftmann@37660
   522
haftmann@37660
   523
definition max_word :: "'a::len word" -- "Largest representable machine integer." where
haftmann@40827
   524
  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
haftmann@37660
   525
haftmann@37660
   526
primrec of_bool :: "bool \<Rightarrow> 'a::len word" where
haftmann@37660
   527
  "of_bool False = 0"
haftmann@37660
   528
| "of_bool True = 1"
haftmann@37660
   529
huffman@45805
   530
(* FIXME: only provide one theorem name *)
haftmann@37660
   531
lemmas of_nth_def = word_set_bits_def
haftmann@37660
   532
huffman@46010
   533
subsection {* Theorems about typedefs *}
huffman@46010
   534
haftmann@37660
   535
lemma sint_sbintrunc': 
haftmann@37660
   536
  "sint (word_of_int bin :: 'a word) = 
haftmann@37660
   537
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
haftmann@37660
   538
  unfolding sint_uint 
haftmann@37660
   539
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)
haftmann@37660
   540
haftmann@37660
   541
lemma uint_sint: 
haftmann@37660
   542
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
haftmann@37660
   543
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
haftmann@37660
   544
huffman@46057
   545
lemma bintr_uint:
huffman@46057
   546
  fixes w :: "'a::len0 word"
huffman@46057
   547
  shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w"
haftmann@37660
   548
  apply (subst word_ubin.norm_Rep [symmetric]) 
haftmann@37660
   549
  apply (simp only: bintrunc_bintrunc_min word_size)
haftmann@37660
   550
  apply (simp add: min_max.inf_absorb2)
haftmann@37660
   551
  done
haftmann@37660
   552
huffman@46057
   553
lemma wi_bintr:
huffman@46057
   554
  "len_of TYPE('a::len0) \<le> n \<Longrightarrow>
huffman@46057
   555
    word_of_int (bintrunc n w) = (word_of_int w :: 'a word)"
haftmann@37660
   556
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1)
haftmann@37660
   557
haftmann@37660
   558
lemma td_ext_sbin: 
haftmann@37660
   559
  "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 
haftmann@37660
   560
    (sbintrunc (len_of TYPE('a) - 1))"
haftmann@37660
   561
  apply (unfold td_ext_def' sint_uint)
haftmann@37660
   562
  apply (simp add : word_ubin.eq_norm)
haftmann@37660
   563
  apply (cases "len_of TYPE('a)")
haftmann@37660
   564
   apply (auto simp add : sints_def)
haftmann@37660
   565
  apply (rule sym [THEN trans])
haftmann@37660
   566
  apply (rule word_ubin.Abs_norm)
haftmann@37660
   567
  apply (simp only: bintrunc_sbintrunc)
haftmann@37660
   568
  apply (drule sym)
haftmann@37660
   569
  apply simp
haftmann@37660
   570
  done
haftmann@37660
   571
haftmann@37660
   572
lemmas td_ext_sint = td_ext_sbin 
haftmann@37660
   573
  [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]]
haftmann@37660
   574
haftmann@37660
   575
(* We do sint before sbin, before sint is the user version
haftmann@37660
   576
   and interpretations do not produce thm duplicates. I.e. 
haftmann@37660
   577
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
haftmann@37660
   578
   because the latter is the same thm as the former *)
haftmann@37660
   579
interpretation word_sint:
haftmann@37660
   580
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   581
          word_of_int 
haftmann@37660
   582
          "sints (len_of TYPE('a::len))"
haftmann@37660
   583
          "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) -
haftmann@37660
   584
               2 ^ (len_of TYPE('a::len) - 1)"
haftmann@37660
   585
  by (rule td_ext_sint)
haftmann@37660
   586
haftmann@37660
   587
interpretation word_sbin:
haftmann@37660
   588
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   589
          word_of_int 
haftmann@37660
   590
          "sints (len_of TYPE('a::len))"
haftmann@37660
   591
          "sbintrunc (len_of TYPE('a::len) - 1)"
haftmann@37660
   592
  by (rule td_ext_sbin)
haftmann@37660
   593
wenzelm@45604
   594
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
haftmann@37660
   595
haftmann@37660
   596
lemmas td_sint = word_sint.td
haftmann@37660
   597
haftmann@37660
   598
lemma to_bl_def': 
haftmann@37660
   599
  "(to_bl :: 'a :: len0 word => bool list) =
haftmann@37660
   600
    bin_to_bl (len_of TYPE('a)) o uint"
wenzelm@44762
   601
  by (auto simp: to_bl_def)
haftmann@37660
   602
huffman@47108
   603
lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w
haftmann@37660
   604
huffman@45805
   605
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
huffman@45805
   606
  by (fact uints_def [unfolded no_bintr_alt1])
huffman@45805
   607
huffman@47108
   608
lemma word_numeral_alt:
huffman@47108
   609
  "numeral b = word_of_int (numeral b)"
huffman@47108
   610
  by (induct b, simp_all only: numeral.simps word_of_int_homs)
huffman@47108
   611
huffman@47108
   612
declare word_numeral_alt [symmetric, code_abbrev]
huffman@47108
   613
huffman@47108
   614
lemma word_neg_numeral_alt:
huffman@47108
   615
  "neg_numeral b = word_of_int (neg_numeral b)"
huffman@47108
   616
  by (simp only: neg_numeral_def word_numeral_alt wi_hom_neg)
huffman@47108
   617
huffman@47108
   618
declare word_neg_numeral_alt [symmetric, code_abbrev]
huffman@47108
   619
huffman@47372
   620
lemma word_numeral_transfer [transfer_rule]:
huffman@47372
   621
  "(fun_rel op = cr_word) numeral numeral"
huffman@47372
   622
  "(fun_rel op = cr_word) neg_numeral neg_numeral"
huffman@47372
   623
  unfolding fun_rel_def cr_word_def word_numeral_alt word_neg_numeral_alt
huffman@47372
   624
  by simp_all
huffman@47372
   625
huffman@45805
   626
lemma uint_bintrunc [simp]:
huffman@47108
   627
  "uint (numeral bin :: 'a word) = 
huffman@47108
   628
    bintrunc (len_of TYPE ('a :: len0)) (numeral bin)"
huffman@47108
   629
  unfolding word_numeral_alt by (rule word_ubin.eq_norm)
huffman@47108
   630
huffman@47108
   631
lemma uint_bintrunc_neg [simp]: "uint (neg_numeral bin :: 'a word) = 
huffman@47108
   632
    bintrunc (len_of TYPE ('a :: len0)) (neg_numeral bin)"
huffman@47108
   633
  by (simp only: word_neg_numeral_alt word_ubin.eq_norm)
haftmann@37660
   634
huffman@45805
   635
lemma sint_sbintrunc [simp]:
huffman@47108
   636
  "sint (numeral bin :: 'a word) = 
huffman@47108
   637
    sbintrunc (len_of TYPE ('a :: len) - 1) (numeral bin)"
huffman@47108
   638
  by (simp only: word_numeral_alt word_sbin.eq_norm)
huffman@47108
   639
huffman@47108
   640
lemma sint_sbintrunc_neg [simp]: "sint (neg_numeral bin :: 'a word) = 
huffman@47108
   641
    sbintrunc (len_of TYPE ('a :: len) - 1) (neg_numeral bin)"
huffman@47108
   642
  by (simp only: word_neg_numeral_alt word_sbin.eq_norm)
haftmann@37660
   643
huffman@45805
   644
lemma unat_bintrunc [simp]:
huffman@47108
   645
  "unat (numeral bin :: 'a :: len0 word) =
huffman@47108
   646
    nat (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
   647
  by (simp only: unat_def uint_bintrunc)
huffman@47108
   648
huffman@47108
   649
lemma unat_bintrunc_neg [simp]:
huffman@47108
   650
  "unat (neg_numeral bin :: 'a :: len0 word) =
huffman@47108
   651
    nat (bintrunc (len_of TYPE('a)) (neg_numeral bin))"
huffman@47108
   652
  by (simp only: unat_def uint_bintrunc_neg)
haftmann@37660
   653
haftmann@40827
   654
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
haftmann@37660
   655
  apply (unfold word_size)
haftmann@37660
   656
  apply (rule word_uint.Rep_eqD)
haftmann@37660
   657
  apply (rule box_equals)
haftmann@37660
   658
    defer
haftmann@37660
   659
    apply (rule word_ubin.norm_Rep)+
haftmann@37660
   660
  apply simp
haftmann@37660
   661
  done
haftmann@37660
   662
huffman@45805
   663
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)"
huffman@45805
   664
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   665
huffman@45805
   666
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
huffman@45805
   667
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   668
huffman@45805
   669
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)"
huffman@45805
   670
  using word_sint.Rep [of x] by (simp add: sints_num)
huffman@45805
   671
huffman@45805
   672
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)"
huffman@45805
   673
  using word_sint.Rep [of x] by (simp add: sints_num)
haftmann@37660
   674
haftmann@37660
   675
lemma sign_uint_Pls [simp]: 
huffman@46604
   676
  "bin_sign (uint x) = 0"
huffman@47108
   677
  by (simp add: sign_Pls_ge_0)
haftmann@37660
   678
huffman@45805
   679
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0"
huffman@45805
   680
  by (simp only: diff_less_0_iff_less uint_lt2p)
huffman@45805
   681
huffman@45805
   682
lemma uint_m2p_not_non_neg:
huffman@45805
   683
  "\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)"
huffman@45805
   684
  by (simp only: not_le uint_m2p_neg)
haftmann@37660
   685
haftmann@37660
   686
lemma lt2p_lem:
haftmann@40827
   687
  "len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n"
haftmann@37660
   688
  by (rule xtr8 [OF _ uint_lt2p]) simp
haftmann@37660
   689
huffman@45805
   690
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
huffman@45805
   691
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])
haftmann@37660
   692
haftmann@40827
   693
lemma uint_nat: "uint w = int (unat w)"
haftmann@37660
   694
  unfolding unat_def by auto
haftmann@37660
   695
huffman@47108
   696
lemma uint_numeral:
huffman@47108
   697
  "uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   698
  unfolding word_numeral_alt
haftmann@37660
   699
  by (simp only: int_word_uint)
haftmann@37660
   700
huffman@47108
   701
lemma uint_neg_numeral:
huffman@47108
   702
  "uint (neg_numeral b :: 'a :: len0 word) = neg_numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   703
  unfolding word_neg_numeral_alt
huffman@47108
   704
  by (simp only: int_word_uint)
huffman@47108
   705
huffman@47108
   706
lemma unat_numeral: 
huffman@47108
   707
  "unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)"
haftmann@37660
   708
  apply (unfold unat_def)
huffman@47108
   709
  apply (clarsimp simp only: uint_numeral)
haftmann@37660
   710
  apply (rule nat_mod_distrib [THEN trans])
huffman@47108
   711
    apply (rule zero_le_numeral)
haftmann@37660
   712
   apply (simp_all add: nat_power_eq)
haftmann@37660
   713
  done
haftmann@37660
   714
huffman@47108
   715
lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + 
haftmann@37660
   716
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
haftmann@37660
   717
    2 ^ (len_of TYPE('a) - 1)"
huffman@47108
   718
  unfolding word_numeral_alt by (rule int_word_sint)
huffman@47108
   719
huffman@47108
   720
lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0"
huffman@45958
   721
  unfolding word_0_wi ..
huffman@45958
   722
huffman@47108
   723
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1"
huffman@45958
   724
  unfolding word_1_wi ..
huffman@45958
   725
huffman@47108
   726
lemma word_of_int_numeral [simp] : 
huffman@47108
   727
  "(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)"
huffman@47108
   728
  unfolding word_numeral_alt ..
huffman@47108
   729
huffman@47108
   730
lemma word_of_int_neg_numeral [simp]:
huffman@47108
   731
  "(word_of_int (neg_numeral bin) :: 'a :: len0 word) = (neg_numeral bin)"
huffman@47108
   732
  unfolding neg_numeral_def word_numeral_alt wi_hom_syms ..
haftmann@37660
   733
haftmann@37660
   734
lemma word_int_case_wi: 
haftmann@37660
   735
  "word_int_case f (word_of_int i :: 'b word) = 
haftmann@37660
   736
    f (i mod 2 ^ len_of TYPE('b::len0))"
haftmann@37660
   737
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)
haftmann@37660
   738
haftmann@37660
   739
lemma word_int_split: 
haftmann@37660
   740
  "P (word_int_case f x) = 
haftmann@37660
   741
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
haftmann@37660
   742
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
haftmann@37660
   743
  unfolding word_int_case_def
haftmann@37660
   744
  by (auto simp: word_uint.eq_norm int_mod_eq')
haftmann@37660
   745
haftmann@37660
   746
lemma word_int_split_asm: 
haftmann@37660
   747
  "P (word_int_case f x) = 
haftmann@37660
   748
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
haftmann@37660
   749
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
haftmann@37660
   750
  unfolding word_int_case_def
haftmann@37660
   751
  by (auto simp: word_uint.eq_norm int_mod_eq')
huffman@45805
   752
wenzelm@45604
   753
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
wenzelm@45604
   754
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
haftmann@37660
   755
haftmann@37660
   756
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
haftmann@37660
   757
  unfolding word_size by (rule uint_range')
haftmann@37660
   758
haftmann@37660
   759
lemma sint_range_size:
haftmann@37660
   760
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
haftmann@37660
   761
  unfolding word_size by (rule sint_range')
haftmann@37660
   762
huffman@45805
   763
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x"
huffman@45805
   764
  unfolding word_size by (rule less_le_trans [OF sint_lt])
huffman@45805
   765
huffman@45805
   766
lemma sint_below_size:
huffman@45805
   767
  "x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w"
huffman@45805
   768
  unfolding word_size by (rule order_trans [OF _ sint_ge])
haftmann@37660
   769
huffman@46010
   770
subsection {* Testing bits *}
huffman@46010
   771
haftmann@37660
   772
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)"
haftmann@37660
   773
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
haftmann@37660
   774
haftmann@37660
   775
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
haftmann@37660
   776
  apply (unfold word_test_bit_def)
haftmann@37660
   777
  apply (subst word_ubin.norm_Rep [symmetric])
haftmann@37660
   778
  apply (simp only: nth_bintr word_size)
haftmann@37660
   779
  apply fast
haftmann@37660
   780
  done
haftmann@37660
   781
huffman@46021
   782
lemma word_eq_iff:
huffman@46021
   783
  fixes x y :: "'a::len0 word"
huffman@46021
   784
  shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)"
huffman@46021
   785
  unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric]
huffman@46021
   786
  by (metis test_bit_size [unfolded word_size])
huffman@46021
   787
huffman@46023
   788
lemma word_eqI [rule_format]:
haftmann@37660
   789
  fixes u :: "'a::len0 word"
haftmann@40827
   790
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
huffman@46021
   791
  by (simp add: word_size word_eq_iff)
haftmann@37660
   792
huffman@45805
   793
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x"
huffman@45805
   794
  by simp
haftmann@37660
   795
haftmann@37660
   796
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
haftmann@37660
   797
  unfolding word_test_bit_def word_size
haftmann@37660
   798
  by (simp add: nth_bintr [symmetric])
haftmann@37660
   799
haftmann@37660
   800
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
haftmann@37660
   801
huffman@46057
   802
lemma bin_nth_uint_imp:
huffman@46057
   803
  "bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)"
haftmann@37660
   804
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
haftmann@37660
   805
  apply (subst word_ubin.norm_Rep)
haftmann@37660
   806
  apply assumption
haftmann@37660
   807
  done
haftmann@37660
   808
huffman@46057
   809
lemma bin_nth_sint:
huffman@46057
   810
  fixes w :: "'a::len word"
huffman@46057
   811
  shows "len_of TYPE('a) \<le> n \<Longrightarrow>
huffman@46057
   812
    bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)"
haftmann@37660
   813
  apply (subst word_sbin.norm_Rep [symmetric])
huffman@46057
   814
  apply (auto simp add: nth_sbintr)
haftmann@37660
   815
  done
haftmann@37660
   816
haftmann@37660
   817
(* type definitions theorem for in terms of equivalent bool list *)
haftmann@37660
   818
lemma td_bl: 
haftmann@37660
   819
  "type_definition (to_bl :: 'a::len0 word => bool list) 
haftmann@37660
   820
                   of_bl  
haftmann@37660
   821
                   {bl. length bl = len_of TYPE('a)}"
haftmann@37660
   822
  apply (unfold type_definition_def of_bl_def to_bl_def)
haftmann@37660
   823
  apply (simp add: word_ubin.eq_norm)
haftmann@37660
   824
  apply safe
haftmann@37660
   825
  apply (drule sym)
haftmann@37660
   826
  apply simp
haftmann@37660
   827
  done
haftmann@37660
   828
haftmann@37660
   829
interpretation word_bl:
haftmann@37660
   830
  type_definition "to_bl :: 'a::len0 word => bool list"
haftmann@37660
   831
                  of_bl  
haftmann@37660
   832
                  "{bl. length bl = len_of TYPE('a::len0)}"
haftmann@37660
   833
  by (rule td_bl)
haftmann@37660
   834
huffman@45816
   835
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
wenzelm@45538
   836
haftmann@40827
   837
lemma word_size_bl: "size w = size (to_bl w)"
haftmann@37660
   838
  unfolding word_size by auto
haftmann@37660
   839
haftmann@37660
   840
lemma to_bl_use_of_bl:
haftmann@37660
   841
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
huffman@45816
   842
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
haftmann@37660
   843
haftmann@37660
   844
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
haftmann@37660
   845
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)
haftmann@37660
   846
haftmann@37660
   847
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
haftmann@37660
   848
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)
haftmann@37660
   849
haftmann@40827
   850
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
huffman@47108
   851
  by (metis word_rev_rev)
haftmann@37660
   852
huffman@45805
   853
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
huffman@45805
   854
  by simp
huffman@45805
   855
huffman@45805
   856
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))"
huffman@45805
   857
  unfolding word_bl_Rep' by (rule len_gt_0)
huffman@45805
   858
huffman@45805
   859
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []"
huffman@45805
   860
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
huffman@45805
   861
huffman@45805
   862
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0"
huffman@45805
   863
  by (fact length_bl_gt_0 [THEN gr_implies_not0])
haftmann@37660
   864
huffman@46001
   865
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
haftmann@37660
   866
  apply (unfold to_bl_def sint_uint)
haftmann@37660
   867
  apply (rule trans [OF _ bl_sbin_sign])
haftmann@37660
   868
  apply simp
haftmann@37660
   869
  done
haftmann@37660
   870
haftmann@37660
   871
lemma of_bl_drop': 
haftmann@40827
   872
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
haftmann@37660
   873
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
haftmann@37660
   874
  apply (unfold of_bl_def)
haftmann@37660
   875
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
haftmann@37660
   876
  done
haftmann@37660
   877
haftmann@37660
   878
lemma test_bit_of_bl:  
haftmann@37660
   879
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)"
haftmann@37660
   880
  apply (unfold of_bl_def word_test_bit_def)
haftmann@37660
   881
  apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl)
haftmann@37660
   882
  done
haftmann@37660
   883
haftmann@37660
   884
lemma no_of_bl: 
huffman@47108
   885
  "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) (numeral bin))"
huffman@47108
   886
  unfolding of_bl_def by simp
haftmann@37660
   887
haftmann@40827
   888
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
haftmann@37660
   889
  unfolding word_size to_bl_def by auto
haftmann@37660
   890
haftmann@37660
   891
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   892
  unfolding uint_bl by (simp add : word_size)
haftmann@37660
   893
haftmann@37660
   894
lemma to_bl_of_bin: 
haftmann@37660
   895
  "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
haftmann@37660
   896
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)
haftmann@37660
   897
huffman@47108
   898
lemma to_bl_numeral [simp]:
huffman@47108
   899
  "to_bl (numeral bin::'a::len0 word) =
huffman@47108
   900
    bin_to_bl (len_of TYPE('a)) (numeral bin)"
huffman@47108
   901
  unfolding word_numeral_alt by (rule to_bl_of_bin)
huffman@47108
   902
huffman@47108
   903
lemma to_bl_neg_numeral [simp]:
huffman@47108
   904
  "to_bl (neg_numeral bin::'a::len0 word) =
huffman@47108
   905
    bin_to_bl (len_of TYPE('a)) (neg_numeral bin)"
huffman@47108
   906
  unfolding word_neg_numeral_alt by (rule to_bl_of_bin)
haftmann@37660
   907
haftmann@37660
   908
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   909
  unfolding uint_bl by (simp add : word_size)
huffman@46011
   910
huffman@46011
   911
lemma uint_bl_bin:
huffman@46011
   912
  fixes x :: "'a::len0 word"
huffman@46011
   913
  shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x"
huffman@46011
   914
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])
wenzelm@45604
   915
haftmann@37660
   916
(* naturals *)
haftmann@37660
   917
lemma uints_unats: "uints n = int ` unats n"
haftmann@37660
   918
  apply (unfold unats_def uints_num)
haftmann@37660
   919
  apply safe
haftmann@37660
   920
  apply (rule_tac image_eqI)
haftmann@37660
   921
  apply (erule_tac nat_0_le [symmetric])
haftmann@37660
   922
  apply auto
haftmann@37660
   923
  apply (erule_tac nat_less_iff [THEN iffD2])
haftmann@37660
   924
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
haftmann@37660
   925
  apply (auto simp add : nat_power_eq int_power)
haftmann@37660
   926
  done
haftmann@37660
   927
haftmann@37660
   928
lemma unats_uints: "unats n = nat ` uints n"
haftmann@37660
   929
  by (auto simp add : uints_unats image_iff)
haftmann@37660
   930
huffman@46962
   931
lemmas bintr_num = word_ubin.norm_eq_iff
huffman@47108
   932
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
huffman@46962
   933
lemmas sbintr_num = word_sbin.norm_eq_iff
huffman@47108
   934
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
haftmann@37660
   935
haftmann@37660
   936
lemma num_of_bintr':
huffman@47108
   937
  "bintrunc (len_of TYPE('a :: len0)) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   938
    numeral a = (numeral b :: 'a word)"
huffman@46962
   939
  unfolding bintr_num by (erule subst, simp)
haftmann@37660
   940
haftmann@37660
   941
lemma num_of_sbintr':
huffman@47108
   942
  "sbintrunc (len_of TYPE('a :: len) - 1) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   943
    numeral a = (numeral b :: 'a word)"
huffman@46962
   944
  unfolding sbintr_num by (erule subst, simp)
huffman@46962
   945
huffman@46962
   946
lemma num_abs_bintr:
huffman@47108
   947
  "(numeral x :: 'a word) =
huffman@47108
   948
    word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))"
huffman@47108
   949
  by (simp only: word_ubin.Abs_norm word_numeral_alt)
huffman@46962
   950
huffman@46962
   951
lemma num_abs_sbintr:
huffman@47108
   952
  "(numeral x :: 'a word) =
huffman@47108
   953
    word_of_int (sbintrunc (len_of TYPE('a::len) - 1) (numeral x))"
huffman@47108
   954
  by (simp only: word_sbin.Abs_norm word_numeral_alt)
huffman@46962
   955
haftmann@37660
   956
(** cast - note, no arg for new length, as it's determined by type of result,
haftmann@37660
   957
  thus in "cast w = w, the type means cast to length of w! **)
haftmann@37660
   958
haftmann@37660
   959
lemma ucast_id: "ucast w = w"
haftmann@37660
   960
  unfolding ucast_def by auto
haftmann@37660
   961
haftmann@37660
   962
lemma scast_id: "scast w = w"
haftmann@37660
   963
  unfolding scast_def by auto
haftmann@37660
   964
haftmann@40827
   965
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
haftmann@37660
   966
  unfolding ucast_def of_bl_def uint_bl
haftmann@37660
   967
  by (auto simp add : word_size)
haftmann@37660
   968
haftmann@37660
   969
lemma nth_ucast: 
haftmann@37660
   970
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
haftmann@37660
   971
  apply (unfold ucast_def test_bit_bin)
haftmann@37660
   972
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
haftmann@37660
   973
  apply (fast elim!: bin_nth_uint_imp)
haftmann@37660
   974
  done
haftmann@37660
   975
haftmann@37660
   976
(* for literal u(s)cast *)
haftmann@37660
   977
huffman@46001
   978
lemma ucast_bintr [simp]:
huffman@47108
   979
  "ucast (numeral w ::'a::len0 word) = 
huffman@47108
   980
   word_of_int (bintrunc (len_of TYPE('a)) (numeral w))"
haftmann@37660
   981
  unfolding ucast_def by simp
huffman@47108
   982
(* TODO: neg_numeral *)
haftmann@37660
   983
huffman@46001
   984
lemma scast_sbintr [simp]:
huffman@47108
   985
  "scast (numeral w ::'a::len word) = 
huffman@47108
   986
   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (numeral w))"
haftmann@37660
   987
  unfolding scast_def by simp
haftmann@37660
   988
huffman@46011
   989
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)"
huffman@46011
   990
  unfolding source_size_def word_size Let_def ..
huffman@46011
   991
huffman@46011
   992
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)"
huffman@46011
   993
  unfolding target_size_def word_size Let_def ..
huffman@46011
   994
huffman@46011
   995
lemma is_down:
huffman@46011
   996
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
   997
  shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)"
huffman@46011
   998
  unfolding is_down_def source_size target_size ..
huffman@46011
   999
huffman@46011
  1000
lemma is_up:
huffman@46011
  1001
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
  1002
  shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)"
huffman@46011
  1003
  unfolding is_up_def source_size target_size ..
haftmann@37660
  1004
wenzelm@45604
  1005
lemmas is_up_down = trans [OF is_up is_down [symmetric]]
haftmann@37660
  1006
huffman@45811
  1007
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
haftmann@37660
  1008
  apply (unfold is_down)
haftmann@37660
  1009
  apply safe
haftmann@37660
  1010
  apply (rule ext)
haftmann@37660
  1011
  apply (unfold ucast_def scast_def uint_sint)
haftmann@37660
  1012
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1013
  apply simp
haftmann@37660
  1014
  done
haftmann@37660
  1015
huffman@45811
  1016
lemma word_rev_tf:
huffman@45811
  1017
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
  1018
    rev (takefill False (len_of TYPE('a)) (rev bl))"
haftmann@37660
  1019
  unfolding of_bl_def uint_bl
haftmann@37660
  1020
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
haftmann@37660
  1021
huffman@45811
  1022
lemma word_rep_drop:
huffman@45811
  1023
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
  1024
    replicate (len_of TYPE('a) - length bl) False @
huffman@45811
  1025
    drop (length bl - len_of TYPE('a)) bl"
huffman@45811
  1026
  by (simp add: word_rev_tf takefill_alt rev_take)
haftmann@37660
  1027
haftmann@37660
  1028
lemma to_bl_ucast: 
haftmann@37660
  1029
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
haftmann@37660
  1030
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
haftmann@37660
  1031
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
haftmann@37660
  1032
  apply (unfold ucast_bl)
haftmann@37660
  1033
  apply (rule trans)
haftmann@37660
  1034
   apply (rule word_rep_drop)
haftmann@37660
  1035
  apply simp
haftmann@37660
  1036
  done
haftmann@37660
  1037
huffman@45811
  1038
lemma ucast_up_app [OF refl]:
haftmann@40827
  1039
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
haftmann@37660
  1040
    to_bl (uc w) = replicate n False @ (to_bl w)"
haftmann@37660
  1041
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
  1042
huffman@45811
  1043
lemma ucast_down_drop [OF refl]:
haftmann@40827
  1044
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
haftmann@37660
  1045
    to_bl (uc w) = drop n (to_bl w)"
haftmann@37660
  1046
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
  1047
huffman@45811
  1048
lemma scast_down_drop [OF refl]:
haftmann@40827
  1049
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
haftmann@37660
  1050
    to_bl (sc w) = drop n (to_bl w)"
haftmann@37660
  1051
  apply (subgoal_tac "sc = ucast")
haftmann@37660
  1052
   apply safe
haftmann@37660
  1053
   apply simp
huffman@45811
  1054
   apply (erule ucast_down_drop)
huffman@45811
  1055
  apply (rule down_cast_same [symmetric])
haftmann@37660
  1056
  apply (simp add : source_size target_size is_down)
haftmann@37660
  1057
  done
haftmann@37660
  1058
huffman@45811
  1059
lemma sint_up_scast [OF refl]:
haftmann@40827
  1060
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
haftmann@37660
  1061
  apply (unfold is_up)
haftmann@37660
  1062
  apply safe
haftmann@37660
  1063
  apply (simp add: scast_def word_sbin.eq_norm)
haftmann@37660
  1064
  apply (rule box_equals)
haftmann@37660
  1065
    prefer 3
haftmann@37660
  1066
    apply (rule word_sbin.norm_Rep)
haftmann@37660
  1067
   apply (rule sbintrunc_sbintrunc_l)
haftmann@37660
  1068
   defer
haftmann@37660
  1069
   apply (subst word_sbin.norm_Rep)
haftmann@37660
  1070
   apply (rule refl)
haftmann@37660
  1071
  apply simp
haftmann@37660
  1072
  done
haftmann@37660
  1073
huffman@45811
  1074
lemma uint_up_ucast [OF refl]:
haftmann@40827
  1075
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
haftmann@37660
  1076
  apply (unfold is_up)
haftmann@37660
  1077
  apply safe
haftmann@37660
  1078
  apply (rule bin_eqI)
haftmann@37660
  1079
  apply (fold word_test_bit_def)
haftmann@37660
  1080
  apply (auto simp add: nth_ucast)
haftmann@37660
  1081
  apply (auto simp add: test_bit_bin)
haftmann@37660
  1082
  done
huffman@45811
  1083
huffman@45811
  1084
lemma ucast_up_ucast [OF refl]:
huffman@45811
  1085
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
haftmann@37660
  1086
  apply (simp (no_asm) add: ucast_def)
haftmann@37660
  1087
  apply (clarsimp simp add: uint_up_ucast)
haftmann@37660
  1088
  done
haftmann@37660
  1089
    
huffman@45811
  1090
lemma scast_up_scast [OF refl]:
huffman@45811
  1091
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
haftmann@37660
  1092
  apply (simp (no_asm) add: scast_def)
haftmann@37660
  1093
  apply (clarsimp simp add: sint_up_scast)
haftmann@37660
  1094
  done
haftmann@37660
  1095
    
huffman@45811
  1096
lemma ucast_of_bl_up [OF refl]:
haftmann@40827
  1097
  "w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl"
haftmann@37660
  1098
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
haftmann@37660
  1099
haftmann@37660
  1100
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
haftmann@37660
  1101
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
haftmann@37660
  1102
haftmann@37660
  1103
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
haftmann@37660
  1104
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
haftmann@37660
  1105
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
haftmann@37660
  1106
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
haftmann@37660
  1107
haftmann@37660
  1108
lemma up_ucast_surj:
haftmann@40827
  1109
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1110
   surj (ucast :: 'a word => 'b word)"
haftmann@37660
  1111
  by (rule surjI, erule ucast_up_ucast_id)
haftmann@37660
  1112
haftmann@37660
  1113
lemma up_scast_surj:
haftmann@40827
  1114
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1115
   surj (scast :: 'a word => 'b word)"
haftmann@37660
  1116
  by (rule surjI, erule scast_up_scast_id)
haftmann@37660
  1117
haftmann@37660
  1118
lemma down_scast_inj:
haftmann@40827
  1119
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1120
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1121
  by (rule inj_on_inverseI, erule scast_down_scast_id)
haftmann@37660
  1122
haftmann@37660
  1123
lemma down_ucast_inj:
haftmann@40827
  1124
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1125
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1126
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)
haftmann@37660
  1127
haftmann@37660
  1128
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
haftmann@37660
  1129
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
huffman@45811
  1130
huffman@46646
  1131
lemma ucast_down_wi [OF refl]:
huffman@46646
  1132
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
huffman@46646
  1133
  apply (unfold is_down)
haftmann@37660
  1134
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
haftmann@37660
  1135
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1136
  apply (erule bintrunc_bintrunc_ge)
haftmann@37660
  1137
  done
huffman@45811
  1138
huffman@46646
  1139
lemma ucast_down_no [OF refl]:
huffman@47108
  1140
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin"
huffman@47108
  1141
  unfolding word_numeral_alt by clarify (rule ucast_down_wi)
huffman@46646
  1142
huffman@45811
  1143
lemma ucast_down_bl [OF refl]:
huffman@45811
  1144
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
huffman@46646
  1145
  unfolding of_bl_def by clarify (erule ucast_down_wi)
haftmann@37660
  1146
haftmann@37660
  1147
lemmas slice_def' = slice_def [unfolded word_size]
haftmann@37660
  1148
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
haftmann@37660
  1149
haftmann@37660
  1150
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
haftmann@37660
  1151
haftmann@37660
  1152
haftmann@37660
  1153
subsection {* Word Arithmetic *}
haftmann@37660
  1154
haftmann@37660
  1155
lemma word_less_alt: "(a < b) = (uint a < uint b)"
huffman@46012
  1156
  unfolding word_less_def word_le_def by (simp add: less_le)
haftmann@37660
  1157
haftmann@37660
  1158
lemma signed_linorder: "class.linorder word_sle word_sless"
wenzelm@46124
  1159
  by default (unfold word_sle_def word_sless_def, auto)
haftmann@37660
  1160
haftmann@37660
  1161
interpretation signed: linorder "word_sle" "word_sless"
haftmann@37660
  1162
  by (rule signed_linorder)
haftmann@37660
  1163
haftmann@37660
  1164
lemma udvdI: 
haftmann@40827
  1165
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
haftmann@37660
  1166
  by (auto simp: udvd_def)
haftmann@37660
  1167
huffman@47108
  1168
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1169
huffman@47108
  1170
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1171
huffman@47108
  1172
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1173
huffman@47108
  1174
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1175
huffman@47108
  1176
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1177
huffman@47108
  1178
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b
haftmann@37660
  1179
huffman@46020
  1180
lemma word_1_no: "(1::'a::len0 word) = Numeral1"
huffman@47108
  1181
  by (simp add: word_numeral_alt)
haftmann@37660
  1182
haftmann@40827
  1183
lemma word_m1_wi: "-1 = word_of_int -1" 
huffman@47108
  1184
  by (rule word_neg_numeral_alt)
haftmann@37660
  1185
huffman@46648
  1186
lemma word_0_bl [simp]: "of_bl [] = 0"
huffman@46648
  1187
  unfolding of_bl_def by simp
haftmann@37660
  1188
haftmann@37660
  1189
lemma word_1_bl: "of_bl [True] = 1" 
huffman@46648
  1190
  unfolding of_bl_def by (simp add: bl_to_bin_def)
huffman@46648
  1191
huffman@46648
  1192
lemma uint_eq_0 [simp]: "uint 0 = 0"
huffman@46648
  1193
  unfolding word_0_wi word_ubin.eq_norm by simp
haftmann@37660
  1194
huffman@45995
  1195
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
huffman@46648
  1196
  by (simp add: of_bl_def bl_to_bin_rep_False)
haftmann@37660
  1197
huffman@45805
  1198
lemma to_bl_0 [simp]:
haftmann@37660
  1199
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
haftmann@37660
  1200
  unfolding uint_bl
huffman@46617
  1201
  by (simp add: word_size bin_to_bl_zero)
haftmann@37660
  1202
haftmann@37660
  1203
lemma uint_0_iff: "(uint x = 0) = (x = 0)"
haftmann@37660
  1204
  by (auto intro!: word_uint.Rep_eqD)
haftmann@37660
  1205
haftmann@37660
  1206
lemma unat_0_iff: "(unat x = 0) = (x = 0)"
haftmann@37660
  1207
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)
haftmann@37660
  1208
haftmann@37660
  1209
lemma unat_0 [simp]: "unat 0 = 0"
haftmann@37660
  1210
  unfolding unat_def by auto
haftmann@37660
  1211
haftmann@40827
  1212
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
haftmann@37660
  1213
  apply (unfold word_size)
haftmann@37660
  1214
  apply (rule box_equals)
haftmann@37660
  1215
    defer
haftmann@37660
  1216
    apply (rule word_uint.Rep_inverse)+
haftmann@37660
  1217
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1218
  apply simp
haftmann@37660
  1219
  done
haftmann@37660
  1220
huffman@45816
  1221
lemmas size_0_same = size_0_same' [unfolded word_size]
haftmann@37660
  1222
haftmann@37660
  1223
lemmas unat_eq_0 = unat_0_iff
haftmann@37660
  1224
lemmas unat_eq_zero = unat_0_iff
haftmann@37660
  1225
haftmann@37660
  1226
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
haftmann@37660
  1227
by (auto simp: unat_0_iff [symmetric])
haftmann@37660
  1228
huffman@45958
  1229
lemma ucast_0 [simp]: "ucast 0 = 0"
huffman@45995
  1230
  unfolding ucast_def by simp
huffman@45958
  1231
huffman@45958
  1232
lemma sint_0 [simp]: "sint 0 = 0"
huffman@45958
  1233
  unfolding sint_uint by simp
huffman@45958
  1234
huffman@45958
  1235
lemma scast_0 [simp]: "scast 0 = 0"
huffman@45995
  1236
  unfolding scast_def by simp
haftmann@37660
  1237
haftmann@37660
  1238
lemma sint_n1 [simp] : "sint -1 = -1"
huffman@45958
  1239
  unfolding word_m1_wi by (simp add: word_sbin.eq_norm)
huffman@45958
  1240
huffman@45958
  1241
lemma scast_n1 [simp]: "scast -1 = -1"
huffman@45958
  1242
  unfolding scast_def by simp
huffman@45958
  1243
huffman@45958
  1244
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
haftmann@37660
  1245
  unfolding word_1_wi
huffman@45995
  1246
  by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1)
huffman@45958
  1247
huffman@45958
  1248
lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
huffman@45958
  1249
  unfolding unat_def by simp
huffman@45958
  1250
huffman@45958
  1251
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
huffman@45995
  1252
  unfolding ucast_def by simp
haftmann@37660
  1253
haftmann@37660
  1254
(* now, to get the weaker results analogous to word_div/mod_def *)
haftmann@37660
  1255
haftmann@37660
  1256
lemmas word_arith_alts = 
huffman@46000
  1257
  word_sub_wi
huffman@46000
  1258
  word_arith_wis (* FIXME: duplicate *)
huffman@46000
  1259
haftmann@37660
  1260
subsection  "Transferring goals from words to ints"
haftmann@37660
  1261
haftmann@37660
  1262
lemma word_ths:  
haftmann@37660
  1263
  shows
haftmann@37660
  1264
  word_succ_p1:   "word_succ a = a + 1" and
haftmann@37660
  1265
  word_pred_m1:   "word_pred a = a - 1" and
haftmann@37660
  1266
  word_pred_succ: "word_pred (word_succ a) = a" and
haftmann@37660
  1267
  word_succ_pred: "word_succ (word_pred a) = a" and
haftmann@37660
  1268
  word_mult_succ: "word_succ a * b = b + a * b"
huffman@47374
  1269
  by (transfer, simp add: algebra_simps)+
haftmann@37660
  1270
huffman@45816
  1271
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
huffman@45816
  1272
  by simp
haftmann@37660
  1273
haftmann@37660
  1274
lemmas uint_word_ariths = 
wenzelm@45604
  1275
  word_arith_alts [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1276
haftmann@37660
  1277
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p]
haftmann@37660
  1278
haftmann@37660
  1279
(* similar expressions for sint (arith operations) *)
haftmann@37660
  1280
lemmas sint_word_ariths = uint_word_arith_bintrs
haftmann@37660
  1281
  [THEN uint_sint [symmetric, THEN trans],
haftmann@37660
  1282
  unfolded uint_sint bintr_arith1s bintr_ariths 
wenzelm@45604
  1283
    len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep]
wenzelm@45604
  1284
wenzelm@45604
  1285
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
wenzelm@45604
  1286
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1287
haftmann@37660
  1288
lemma word_pred_0_n1: "word_pred 0 = word_of_int -1"
huffman@47374
  1289
  unfolding word_pred_m1 by simp
haftmann@37660
  1290
haftmann@37660
  1291
lemma succ_pred_no [simp]:
huffman@47108
  1292
  "word_succ (numeral w) = numeral w + 1"
huffman@47108
  1293
  "word_pred (numeral w) = numeral w - 1"
huffman@47108
  1294
  "word_succ (neg_numeral w) = neg_numeral w + 1"
huffman@47108
  1295
  "word_pred (neg_numeral w) = neg_numeral w - 1"
huffman@47108
  1296
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1297
haftmann@37660
  1298
lemma word_sp_01 [simp] : 
haftmann@37660
  1299
  "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0"
huffman@47108
  1300
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1301
haftmann@37660
  1302
(* alternative approach to lifting arithmetic equalities *)
haftmann@37660
  1303
lemma word_of_int_Ex:
haftmann@37660
  1304
  "\<exists>y. x = word_of_int y"
haftmann@37660
  1305
  by (rule_tac x="uint x" in exI) simp
haftmann@37660
  1306
haftmann@37660
  1307
haftmann@37660
  1308
subsection "Order on fixed-length words"
haftmann@37660
  1309
haftmann@37660
  1310
lemma word_zero_le [simp] :
haftmann@37660
  1311
  "0 <= (y :: 'a :: len0 word)"
haftmann@37660
  1312
  unfolding word_le_def by auto
haftmann@37660
  1313
  
huffman@45816
  1314
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *)
haftmann@37660
  1315
  unfolding word_le_def
haftmann@37660
  1316
  by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1317
huffman@45816
  1318
lemma word_n1_ge [simp]: "y \<le> (-1::'a::len0 word)"
huffman@45816
  1319
  unfolding word_le_def
huffman@45816
  1320
  by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1321
haftmann@37660
  1322
lemmas word_not_simps [simp] = 
haftmann@37660
  1323
  word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD]
haftmann@37660
  1324
huffman@47108
  1325
lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> (y :: 'a :: len0 word)"
huffman@47108
  1326
  by (simp add: less_le)
huffman@47108
  1327
huffman@47108
  1328
lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y
haftmann@37660
  1329
haftmann@40827
  1330
lemma word_sless_alt: "(a <s b) = (sint a < sint b)"
haftmann@37660
  1331
  unfolding word_sle_def word_sless_def
haftmann@37660
  1332
  by (auto simp add: less_le)
haftmann@37660
  1333
haftmann@37660
  1334
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)"
haftmann@37660
  1335
  unfolding unat_def word_le_def
haftmann@37660
  1336
  by (rule nat_le_eq_zle [symmetric]) simp
haftmann@37660
  1337
haftmann@37660
  1338
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)"
haftmann@37660
  1339
  unfolding unat_def word_less_alt
haftmann@37660
  1340
  by (rule nat_less_eq_zless [symmetric]) simp
haftmann@37660
  1341
  
haftmann@37660
  1342
lemma wi_less: 
haftmann@37660
  1343
  "(word_of_int n < (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1344
    (n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1345
  unfolding word_less_alt by (simp add: word_uint.eq_norm)
haftmann@37660
  1346
haftmann@37660
  1347
lemma wi_le: 
haftmann@37660
  1348
  "(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1349
    (n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1350
  unfolding word_le_def by (simp add: word_uint.eq_norm)
haftmann@37660
  1351
haftmann@37660
  1352
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)"
haftmann@37660
  1353
  apply (unfold udvd_def)
haftmann@37660
  1354
  apply safe
haftmann@37660
  1355
   apply (simp add: unat_def nat_mult_distrib)
haftmann@37660
  1356
  apply (simp add: uint_nat int_mult)
haftmann@37660
  1357
  apply (rule exI)
haftmann@37660
  1358
  apply safe
haftmann@37660
  1359
   prefer 2
haftmann@37660
  1360
   apply (erule notE)
haftmann@37660
  1361
   apply (rule refl)
haftmann@37660
  1362
  apply force
haftmann@37660
  1363
  done
haftmann@37660
  1364
haftmann@37660
  1365
lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y"
haftmann@37660
  1366
  unfolding dvd_def udvd_nat_alt by force
haftmann@37660
  1367
wenzelm@45604
  1368
lemmas unat_mono = word_less_nat_alt [THEN iffD1]
haftmann@37660
  1369
haftmann@40827
  1370
lemma unat_minus_one: "x ~= 0 \<Longrightarrow> unat (x - 1) = unat x - 1"
haftmann@37660
  1371
  apply (unfold unat_def)
haftmann@37660
  1372
  apply (simp only: int_word_uint word_arith_alts rdmods)
haftmann@37660
  1373
  apply (subgoal_tac "uint x >= 1")
haftmann@37660
  1374
   prefer 2
haftmann@37660
  1375
   apply (drule contrapos_nn)
haftmann@37660
  1376
    apply (erule word_uint.Rep_inverse' [symmetric])
haftmann@37660
  1377
   apply (insert uint_ge_0 [of x])[1]
haftmann@37660
  1378
   apply arith
haftmann@37660
  1379
  apply (rule box_equals)
haftmann@37660
  1380
    apply (rule nat_diff_distrib)
haftmann@37660
  1381
     prefer 2
haftmann@37660
  1382
     apply assumption
haftmann@37660
  1383
    apply simp
haftmann@37660
  1384
   apply (subst mod_pos_pos_trivial)
haftmann@37660
  1385
     apply arith
haftmann@37660
  1386
    apply (insert uint_lt2p [of x])[1]
haftmann@37660
  1387
    apply arith
haftmann@37660
  1388
   apply (rule refl)
haftmann@37660
  1389
  apply simp
haftmann@37660
  1390
  done
haftmann@37660
  1391
    
haftmann@40827
  1392
lemma measure_unat: "p ~= 0 \<Longrightarrow> unat (p - 1) < unat p"
haftmann@37660
  1393
  by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric])
haftmann@37660
  1394
  
wenzelm@45604
  1395
lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
wenzelm@45604
  1396
lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
haftmann@37660
  1397
haftmann@37660
  1398
lemma uint_sub_lt2p [simp]: 
haftmann@37660
  1399
  "uint (x :: 'a :: len0 word) - uint (y :: 'b :: len0 word) < 
haftmann@37660
  1400
    2 ^ len_of TYPE('a)"
haftmann@37660
  1401
  using uint_ge_0 [of y] uint_lt2p [of x] by arith
haftmann@37660
  1402
haftmann@37660
  1403
haftmann@37660
  1404
subsection "Conditions for the addition (etc) of two words to overflow"
haftmann@37660
  1405
haftmann@37660
  1406
lemma uint_add_lem: 
haftmann@37660
  1407
  "(uint x + uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1408
    (uint (x + y :: 'a :: len0 word) = uint x + uint y)"
haftmann@37660
  1409
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1410
haftmann@37660
  1411
lemma uint_mult_lem: 
haftmann@37660
  1412
  "(uint x * uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1413
    (uint (x * y :: 'a :: len0 word) = uint x * uint y)"
haftmann@37660
  1414
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1415
haftmann@37660
  1416
lemma uint_sub_lem: 
haftmann@37660
  1417
  "(uint x >= uint y) = (uint (x - y) = uint x - uint y)"
haftmann@37660
  1418
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1419
haftmann@37660
  1420
lemma uint_add_le: "uint (x + y) <= uint x + uint y"
haftmann@37660
  1421
  unfolding uint_word_ariths by (auto simp: mod_add_if_z)
haftmann@37660
  1422
haftmann@37660
  1423
lemma uint_sub_ge: "uint (x - y) >= uint x - uint y"
haftmann@37660
  1424
  unfolding uint_word_ariths by (auto simp: mod_sub_if_z)
haftmann@37660
  1425
wenzelm@45604
  1426
lemmas uint_sub_if' = trans [OF uint_word_ariths(1) mod_sub_if_z, simplified]
wenzelm@45604
  1427
lemmas uint_plus_if' = trans [OF uint_word_ariths(2) mod_add_if_z, simplified]
haftmann@37660
  1428
haftmann@37660
  1429
haftmann@37660
  1430
subsection {* Definition of uint\_arith *}
haftmann@37660
  1431
haftmann@37660
  1432
lemma word_of_int_inverse:
haftmann@40827
  1433
  "word_of_int r = a \<Longrightarrow> 0 <= r \<Longrightarrow> r < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1434
   uint (a::'a::len0 word) = r"
haftmann@37660
  1435
  apply (erule word_uint.Abs_inverse' [rotated])
haftmann@37660
  1436
  apply (simp add: uints_num)
haftmann@37660
  1437
  done
haftmann@37660
  1438
haftmann@37660
  1439
lemma uint_split:
haftmann@37660
  1440
  fixes x::"'a::len0 word"
haftmann@37660
  1441
  shows "P (uint x) = 
haftmann@37660
  1442
         (ALL i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) --> P i)"
haftmann@37660
  1443
  apply (fold word_int_case_def)
haftmann@37660
  1444
  apply (auto dest!: word_of_int_inverse simp: int_word_uint int_mod_eq'
haftmann@37660
  1445
              split: word_int_split)
haftmann@37660
  1446
  done
haftmann@37660
  1447
haftmann@37660
  1448
lemma uint_split_asm:
haftmann@37660
  1449
  fixes x::"'a::len0 word"
haftmann@37660
  1450
  shows "P (uint x) = 
haftmann@37660
  1451
         (~(EX i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) & ~ P i))"
haftmann@37660
  1452
  by (auto dest!: word_of_int_inverse 
haftmann@37660
  1453
           simp: int_word_uint int_mod_eq'
haftmann@37660
  1454
           split: uint_split)
haftmann@37660
  1455
haftmann@37660
  1456
lemmas uint_splits = uint_split uint_split_asm
haftmann@37660
  1457
haftmann@37660
  1458
lemmas uint_arith_simps = 
haftmann@37660
  1459
  word_le_def word_less_alt
haftmann@37660
  1460
  word_uint.Rep_inject [symmetric] 
haftmann@37660
  1461
  uint_sub_if' uint_plus_if'
haftmann@37660
  1462
haftmann@37660
  1463
(* use this to stop, eg, 2 ^ len_of TYPE (32) being simplified *)
haftmann@40827
  1464
lemma power_False_cong: "False \<Longrightarrow> a ^ b = c ^ d" 
haftmann@37660
  1465
  by auto
haftmann@37660
  1466
haftmann@37660
  1467
(* uint_arith_tac: reduce to arithmetic on int, try to solve by arith *)
haftmann@37660
  1468
ML {*
haftmann@37660
  1469
fun uint_arith_ss_of ss = 
haftmann@37660
  1470
  ss addsimps @{thms uint_arith_simps}
haftmann@37660
  1471
     delsimps @{thms word_uint.Rep_inject}
wenzelm@45620
  1472
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1473
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1474
haftmann@37660
  1475
fun uint_arith_tacs ctxt = 
haftmann@37660
  1476
  let
haftmann@37660
  1477
    fun arith_tac' n t =
haftmann@37660
  1478
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1479
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1480
  in 
wenzelm@42793
  1481
    [ clarify_tac ctxt 1,
wenzelm@42793
  1482
      full_simp_tac (uint_arith_ss_of (simpset_of ctxt)) 1,
wenzelm@45620
  1483
      ALLGOALS (full_simp_tac (HOL_ss |> fold Splitter.add_split @{thms uint_splits}
wenzelm@45620
  1484
                                      |> fold Simplifier.add_cong @{thms power_False_cong})),
haftmann@37660
  1485
      rewrite_goals_tac @{thms word_size}, 
haftmann@37660
  1486
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1487
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1488
                         REPEAT (dtac @{thm word_of_int_inverse} n 
haftmann@37660
  1489
                                 THEN atac n 
haftmann@37660
  1490
                                 THEN atac n)),
haftmann@37660
  1491
      TRYALL arith_tac' ]
haftmann@37660
  1492
  end
haftmann@37660
  1493
haftmann@37660
  1494
fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt))
haftmann@37660
  1495
*}
haftmann@37660
  1496
haftmann@37660
  1497
method_setup uint_arith = 
haftmann@37660
  1498
  {* Scan.succeed (SIMPLE_METHOD' o uint_arith_tac) *}
haftmann@37660
  1499
  "solving word arithmetic via integers and arith"
haftmann@37660
  1500
haftmann@37660
  1501
haftmann@37660
  1502
subsection "More on overflows and monotonicity"
haftmann@37660
  1503
haftmann@37660
  1504
lemma no_plus_overflow_uint_size: 
haftmann@37660
  1505
  "((x :: 'a :: len0 word) <= x + y) = (uint x + uint y < 2 ^ size x)"
haftmann@37660
  1506
  unfolding word_size by uint_arith
haftmann@37660
  1507
haftmann@37660
  1508
lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size]
haftmann@37660
  1509
haftmann@37660
  1510
lemma no_ulen_sub: "((x :: 'a :: len0 word) >= x - y) = (uint y <= uint x)"
haftmann@37660
  1511
  by uint_arith
haftmann@37660
  1512
haftmann@37660
  1513
lemma no_olen_add':
haftmann@37660
  1514
  fixes x :: "'a::len0 word"
haftmann@37660
  1515
  shows "(x \<le> y + x) = (uint y + uint x < 2 ^ len_of TYPE('a))"
huffman@45546
  1516
  by (simp add: add_ac no_olen_add)
haftmann@37660
  1517
wenzelm@45604
  1518
lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric]]
wenzelm@45604
  1519
wenzelm@45604
  1520
lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem]
wenzelm@45604
  1521
lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1]
wenzelm@45604
  1522
lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem]
haftmann@37660
  1523
lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def]
haftmann@37660
  1524
lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def]
wenzelm@45604
  1525
lemmas word_sub_le = word_sub_le_iff [THEN iffD2]
haftmann@37660
  1526
haftmann@37660
  1527
lemma word_less_sub1: 
haftmann@40827
  1528
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 < x) = (0 < x - 1)"
haftmann@37660
  1529
  by uint_arith
haftmann@37660
  1530
haftmann@37660
  1531
lemma word_le_sub1: 
haftmann@40827
  1532
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 <= x) = (0 <= x - 1)"
haftmann@37660
  1533
  by uint_arith
haftmann@37660
  1534
haftmann@37660
  1535
lemma sub_wrap_lt: 
haftmann@37660
  1536
  "((x :: 'a :: len0 word) < x - z) = (x < z)"
haftmann@37660
  1537
  by uint_arith
haftmann@37660
  1538
haftmann@37660
  1539
lemma sub_wrap: 
haftmann@37660
  1540
  "((x :: 'a :: len0 word) <= x - z) = (z = 0 | x < z)"
haftmann@37660
  1541
  by uint_arith
haftmann@37660
  1542
haftmann@37660
  1543
lemma plus_minus_not_NULL_ab: 
haftmann@40827
  1544
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> c ~= 0 \<Longrightarrow> x + c ~= 0"
haftmann@37660
  1545
  by uint_arith
haftmann@37660
  1546
haftmann@37660
  1547
lemma plus_minus_no_overflow_ab: 
haftmann@40827
  1548
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> x <= x + c" 
haftmann@37660
  1549
  by uint_arith
haftmann@37660
  1550
haftmann@37660
  1551
lemma le_minus': 
haftmann@40827
  1552
  "(a :: 'a :: len0 word) + c <= b \<Longrightarrow> a <= a + c \<Longrightarrow> c <= b - a"
haftmann@37660
  1553
  by uint_arith
haftmann@37660
  1554
haftmann@37660
  1555
lemma le_plus': 
haftmann@40827
  1556
  "(a :: 'a :: len0 word) <= b \<Longrightarrow> c <= b - a \<Longrightarrow> a + c <= b"
haftmann@37660
  1557
  by uint_arith
haftmann@37660
  1558
haftmann@37660
  1559
lemmas le_plus = le_plus' [rotated]
haftmann@37660
  1560
huffman@46011
  1561
lemmas le_minus = leD [THEN thin_rl, THEN le_minus'] (* FIXME *)
haftmann@37660
  1562
haftmann@37660
  1563
lemma word_plus_mono_right: 
haftmann@40827
  1564
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= x + z \<Longrightarrow> x + y <= x + z"
haftmann@37660
  1565
  by uint_arith
haftmann@37660
  1566
haftmann@37660
  1567
lemma word_less_minus_cancel: 
haftmann@40827
  1568
  "y - x < z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) < z"
haftmann@37660
  1569
  by uint_arith
haftmann@37660
  1570
haftmann@37660
  1571
lemma word_less_minus_mono_left: 
haftmann@40827
  1572
  "(y :: 'a :: len0 word) < z \<Longrightarrow> x <= y \<Longrightarrow> y - x < z - x"
haftmann@37660
  1573
  by uint_arith
haftmann@37660
  1574
haftmann@37660
  1575
lemma word_less_minus_mono:  
haftmann@40827
  1576
  "a < c \<Longrightarrow> d < b \<Longrightarrow> a - b < a \<Longrightarrow> c - d < c 
haftmann@40827
  1577
  \<Longrightarrow> a - b < c - (d::'a::len word)"
haftmann@37660
  1578
  by uint_arith
haftmann@37660
  1579
haftmann@37660
  1580
lemma word_le_minus_cancel: 
haftmann@40827
  1581
  "y - x <= z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) <= z"
haftmann@37660
  1582
  by uint_arith
haftmann@37660
  1583
haftmann@37660
  1584
lemma word_le_minus_mono_left: 
haftmann@40827
  1585
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= y \<Longrightarrow> y - x <= z - x"
haftmann@37660
  1586
  by uint_arith
haftmann@37660
  1587
haftmann@37660
  1588
lemma word_le_minus_mono:  
haftmann@40827
  1589
  "a <= c \<Longrightarrow> d <= b \<Longrightarrow> a - b <= a \<Longrightarrow> c - d <= c 
haftmann@40827
  1590
  \<Longrightarrow> a - b <= c - (d::'a::len word)"
haftmann@37660
  1591
  by uint_arith
haftmann@37660
  1592
haftmann@37660
  1593
lemma plus_le_left_cancel_wrap: 
haftmann@40827
  1594
  "(x :: 'a :: len0 word) + y' < x \<Longrightarrow> x + y < x \<Longrightarrow> (x + y' < x + y) = (y' < y)"
haftmann@37660
  1595
  by uint_arith
haftmann@37660
  1596
haftmann@37660
  1597
lemma plus_le_left_cancel_nowrap: 
haftmann@40827
  1598
  "(x :: 'a :: len0 word) <= x + y' \<Longrightarrow> x <= x + y \<Longrightarrow> 
haftmann@37660
  1599
    (x + y' < x + y) = (y' < y)" 
haftmann@37660
  1600
  by uint_arith
haftmann@37660
  1601
haftmann@37660
  1602
lemma word_plus_mono_right2: 
haftmann@40827
  1603
  "(a :: 'a :: len0 word) <= a + b \<Longrightarrow> c <= b \<Longrightarrow> a <= a + c"
haftmann@37660
  1604
  by uint_arith
haftmann@37660
  1605
haftmann@37660
  1606
lemma word_less_add_right: 
haftmann@40827
  1607
  "(x :: 'a :: len0 word) < y - z \<Longrightarrow> z <= y \<Longrightarrow> x + z < y"
haftmann@37660
  1608
  by uint_arith
haftmann@37660
  1609
haftmann@37660
  1610
lemma word_less_sub_right: 
haftmann@40827
  1611
  "(x :: 'a :: len0 word) < y + z \<Longrightarrow> y <= x \<Longrightarrow> x - y < z"
haftmann@37660
  1612
  by uint_arith
haftmann@37660
  1613
haftmann@37660
  1614
lemma word_le_plus_either: 
haftmann@40827
  1615
  "(x :: 'a :: len0 word) <= y | x <= z \<Longrightarrow> y <= y + z \<Longrightarrow> x <= y + z"
haftmann@37660
  1616
  by uint_arith
haftmann@37660
  1617
haftmann@37660
  1618
lemma word_less_nowrapI: 
haftmann@40827
  1619
  "(x :: 'a :: len0 word) < z - k \<Longrightarrow> k <= z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
haftmann@37660
  1620
  by uint_arith
haftmann@37660
  1621
haftmann@40827
  1622
lemma inc_le: "(i :: 'a :: len word) < m \<Longrightarrow> i + 1 <= m"
haftmann@37660
  1623
  by uint_arith
haftmann@37660
  1624
haftmann@37660
  1625
lemma inc_i: 
haftmann@40827
  1626
  "(1 :: 'a :: len word) <= i \<Longrightarrow> i < m \<Longrightarrow> 1 <= (i + 1) & i + 1 <= m"
haftmann@37660
  1627
  by uint_arith
haftmann@37660
  1628
haftmann@37660
  1629
lemma udvd_incr_lem:
haftmann@40827
  1630
  "up < uq \<Longrightarrow> up = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1631
    uq = ua + n' * uint K \<Longrightarrow> up + uint K <= uq"
haftmann@37660
  1632
  apply clarsimp
haftmann@37660
  1633
  apply (drule less_le_mult)
haftmann@37660
  1634
  apply safe
haftmann@37660
  1635
  done
haftmann@37660
  1636
haftmann@37660
  1637
lemma udvd_incr': 
haftmann@40827
  1638
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1639
    uint q = ua + n' * uint K \<Longrightarrow> p + K <= q" 
haftmann@37660
  1640
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1641
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1642
  apply (erule uint_add_le [THEN order_trans])
haftmann@37660
  1643
  done
haftmann@37660
  1644
haftmann@37660
  1645
lemma udvd_decr': 
haftmann@40827
  1646
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1647
    uint q = ua + n' * uint K \<Longrightarrow> p <= q - K"
haftmann@37660
  1648
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1649
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1650
  apply (drule le_diff_eq [THEN iffD2])
haftmann@37660
  1651
  apply (erule order_trans)
haftmann@37660
  1652
  apply (rule uint_sub_ge)
haftmann@37660
  1653
  done
haftmann@37660
  1654
huffman@45816
  1655
lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, unfolded add_0_left]
huffman@45816
  1656
lemmas udvd_incr0 = udvd_incr' [where ua=0, unfolded add_0_left]
huffman@45816
  1657
lemmas udvd_decr0 = udvd_decr' [where ua=0, unfolded add_0_left]
haftmann@37660
  1658
haftmann@37660
  1659
lemma udvd_minus_le': 
haftmann@40827
  1660
  "xy < k \<Longrightarrow> z udvd xy \<Longrightarrow> z udvd k \<Longrightarrow> xy <= k - z"
haftmann@37660
  1661
  apply (unfold udvd_def)
haftmann@37660
  1662
  apply clarify
haftmann@37660
  1663
  apply (erule (2) udvd_decr0)
haftmann@37660
  1664
  done
haftmann@37660
  1665
huffman@45284
  1666
ML {* Delsimprocs [@{simproc linordered_ring_less_cancel_factor}] *}
haftmann@37660
  1667
haftmann@37660
  1668
lemma udvd_incr2_K: 
haftmann@40827
  1669
  "p < a + s \<Longrightarrow> a <= a + s \<Longrightarrow> K udvd s \<Longrightarrow> K udvd p - a \<Longrightarrow> a <= p \<Longrightarrow> 
haftmann@40827
  1670
    0 < K \<Longrightarrow> p <= p + K & p + K <= a + s"
haftmann@37660
  1671
  apply (unfold udvd_def)
haftmann@37660
  1672
  apply clarify
haftmann@37660
  1673
  apply (simp add: uint_arith_simps split: split_if_asm)
haftmann@37660
  1674
   prefer 2 
haftmann@37660
  1675
   apply (insert uint_range' [of s])[1]
haftmann@37660
  1676
   apply arith
haftmann@37660
  1677
  apply (drule add_commute [THEN xtr1])
haftmann@37660
  1678
  apply (simp add: diff_less_eq [symmetric])
haftmann@37660
  1679
  apply (drule less_le_mult)
haftmann@37660
  1680
   apply arith
haftmann@37660
  1681
  apply simp
haftmann@37660
  1682
  done
haftmann@37660
  1683
huffman@45284
  1684
ML {* Addsimprocs [@{simproc linordered_ring_less_cancel_factor}] *}
haftmann@37660
  1685
haftmann@37660
  1686
(* links with rbl operations *)
haftmann@37660
  1687
lemma word_succ_rbl:
haftmann@40827
  1688
  "to_bl w = bl \<Longrightarrow> to_bl (word_succ w) = (rev (rbl_succ (rev bl)))"
haftmann@37660
  1689
  apply (unfold word_succ_def)
haftmann@37660
  1690
  apply clarify
haftmann@37660
  1691
  apply (simp add: to_bl_of_bin)
huffman@46654
  1692
  apply (simp add: to_bl_def rbl_succ)
haftmann@37660
  1693
  done
haftmann@37660
  1694
haftmann@37660
  1695
lemma word_pred_rbl:
haftmann@40827
  1696
  "to_bl w = bl \<Longrightarrow> to_bl (word_pred w) = (rev (rbl_pred (rev bl)))"
haftmann@37660
  1697
  apply (unfold word_pred_def)
haftmann@37660
  1698
  apply clarify
haftmann@37660
  1699
  apply (simp add: to_bl_of_bin)
huffman@46654
  1700
  apply (simp add: to_bl_def rbl_pred)
haftmann@37660
  1701
  done
haftmann@37660
  1702
haftmann@37660
  1703
lemma word_add_rbl:
haftmann@40827
  1704
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1705
    to_bl (v + w) = (rev (rbl_add (rev vbl) (rev wbl)))"
haftmann@37660
  1706
  apply (unfold word_add_def)
haftmann@37660
  1707
  apply clarify
haftmann@37660
  1708
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1709
  apply (simp add: to_bl_def rbl_add)
haftmann@37660
  1710
  done
haftmann@37660
  1711
haftmann@37660
  1712
lemma word_mult_rbl:
haftmann@40827
  1713
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1714
    to_bl (v * w) = (rev (rbl_mult (rev vbl) (rev wbl)))"
haftmann@37660
  1715
  apply (unfold word_mult_def)
haftmann@37660
  1716
  apply clarify
haftmann@37660
  1717
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1718
  apply (simp add: to_bl_def rbl_mult)
haftmann@37660
  1719
  done
haftmann@37660
  1720
haftmann@37660
  1721
lemma rtb_rbl_ariths:
haftmann@37660
  1722
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys"
haftmann@37660
  1723
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys"
haftmann@40827
  1724
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs"
haftmann@40827
  1725
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs"
haftmann@37660
  1726
  by (auto simp: rev_swap [symmetric] word_succ_rbl 
haftmann@37660
  1727
                 word_pred_rbl word_mult_rbl word_add_rbl)
haftmann@37660
  1728
haftmann@37660
  1729
haftmann@37660
  1730
subsection "Arithmetic type class instantiations"
haftmann@37660
  1731
haftmann@37660
  1732
lemmas word_le_0_iff [simp] =
haftmann@37660
  1733
  word_zero_le [THEN leD, THEN linorder_antisym_conv1]
haftmann@37660
  1734
haftmann@37660
  1735
lemma word_of_int_nat: 
haftmann@40827
  1736
  "0 <= x \<Longrightarrow> word_of_int x = of_nat (nat x)"
haftmann@37660
  1737
  by (simp add: of_nat_nat word_of_int)
haftmann@37660
  1738
huffman@46603
  1739
(* note that iszero_def is only for class comm_semiring_1_cancel,
huffman@46603
  1740
   which requires word length >= 1, ie 'a :: len word *) 
huffman@46603
  1741
lemma iszero_word_no [simp]:
huffman@47108
  1742
  "iszero (numeral bin :: 'a :: len word) = 
huffman@47108
  1743
    iszero (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
  1744
  using word_ubin.norm_eq_iff [where 'a='a, of "numeral bin" 0]
huffman@46603
  1745
  by (simp add: iszero_def [symmetric])
huffman@47108
  1746
    
huffman@47108
  1747
text {* Use @{text iszero} to simplify equalities between word numerals. *}
huffman@47108
  1748
huffman@47108
  1749
lemmas word_eq_numeral_iff_iszero [simp] =
huffman@47108
  1750
  eq_numeral_iff_iszero [where 'a="'a::len word"]
huffman@46603
  1751
haftmann@37660
  1752
haftmann@37660
  1753
subsection "Word and nat"
haftmann@37660
  1754
huffman@45811
  1755
lemma td_ext_unat [OF refl]:
haftmann@40827
  1756
  "n = len_of TYPE ('a :: len) \<Longrightarrow> 
haftmann@37660
  1757
    td_ext (unat :: 'a word => nat) of_nat 
haftmann@37660
  1758
    (unats n) (%i. i mod 2 ^ n)"
haftmann@37660
  1759
  apply (unfold td_ext_def' unat_def word_of_nat unats_uints)
haftmann@37660
  1760
  apply (auto intro!: imageI simp add : word_of_int_hom_syms)
haftmann@37660
  1761
  apply (erule word_uint.Abs_inverse [THEN arg_cong])
haftmann@37660
  1762
  apply (simp add: int_word_uint nat_mod_distrib nat_power_eq)
haftmann@37660
  1763
  done
haftmann@37660
  1764
wenzelm@45604
  1765
lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm]
haftmann@37660
  1766
haftmann@37660
  1767
interpretation word_unat:
haftmann@37660
  1768
  td_ext "unat::'a::len word => nat" 
haftmann@37660
  1769
         of_nat 
haftmann@37660
  1770
         "unats (len_of TYPE('a::len))"
haftmann@37660
  1771
         "%i. i mod 2 ^ len_of TYPE('a::len)"
haftmann@37660
  1772
  by (rule td_ext_unat)
haftmann@37660
  1773
haftmann@37660
  1774
lemmas td_unat = word_unat.td_thm
haftmann@37660
  1775
haftmann@37660
  1776
lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq]
haftmann@37660
  1777
haftmann@40827
  1778
lemma unat_le: "y <= unat (z :: 'a :: len word) \<Longrightarrow> y : unats (len_of TYPE ('a))"
haftmann@37660
  1779
  apply (unfold unats_def)
haftmann@37660
  1780
  apply clarsimp
haftmann@37660
  1781
  apply (rule xtrans, rule unat_lt2p, assumption) 
haftmann@37660
  1782
  done
haftmann@37660
  1783
haftmann@37660
  1784
lemma word_nchotomy:
haftmann@37660
  1785
  "ALL w. EX n. (w :: 'a :: len word) = of_nat n & n < 2 ^ len_of TYPE ('a)"
haftmann@37660
  1786
  apply (rule allI)
haftmann@37660
  1787
  apply (rule word_unat.Abs_cases)
haftmann@37660
  1788
  apply (unfold unats_def)
haftmann@37660
  1789
  apply auto
haftmann@37660
  1790
  done
haftmann@37660
  1791
haftmann@37660
  1792
lemma of_nat_eq:
haftmann@37660
  1793
  fixes w :: "'a::len word"
haftmann@37660
  1794
  shows "(of_nat n = w) = (\<exists>q. n = unat w + q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1795
  apply (rule trans)
haftmann@37660
  1796
   apply (rule word_unat.inverse_norm)
haftmann@37660
  1797
  apply (rule iffI)
haftmann@37660
  1798
   apply (rule mod_eqD)
haftmann@37660
  1799
   apply simp
haftmann@37660
  1800
  apply clarsimp
haftmann@37660
  1801
  done
haftmann@37660
  1802
haftmann@37660
  1803
lemma of_nat_eq_size: 
haftmann@37660
  1804
  "(of_nat n = w) = (EX q. n = unat w + q * 2 ^ size w)"
haftmann@37660
  1805
  unfolding word_size by (rule of_nat_eq)
haftmann@37660
  1806
haftmann@37660
  1807
lemma of_nat_0:
haftmann@37660
  1808
  "(of_nat m = (0::'a::len word)) = (\<exists>q. m = q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1809
  by (simp add: of_nat_eq)
haftmann@37660
  1810
huffman@45805
  1811
lemma of_nat_2p [simp]:
huffman@45805
  1812
  "of_nat (2 ^ len_of TYPE('a)) = (0::'a::len word)"
huffman@45805
  1813
  by (fact mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]])
haftmann@37660
  1814
haftmann@40827
  1815
lemma of_nat_gt_0: "of_nat k ~= 0 \<Longrightarrow> 0 < k"
haftmann@37660
  1816
  by (cases k) auto
haftmann@37660
  1817
haftmann@37660
  1818
lemma of_nat_neq_0: 
haftmann@40827
  1819
  "0 < k \<Longrightarrow> k < 2 ^ len_of TYPE ('a :: len) \<Longrightarrow> of_nat k ~= (0 :: 'a word)"
haftmann@37660
  1820
  by (clarsimp simp add : of_nat_0)
haftmann@37660
  1821
haftmann@37660
  1822
lemma Abs_fnat_hom_add:
haftmann@37660
  1823
  "of_nat a + of_nat b = of_nat (a + b)"
haftmann@37660
  1824
  by simp
haftmann@37660
  1825
haftmann@37660
  1826
lemma Abs_fnat_hom_mult:
haftmann@37660
  1827
  "of_nat a * of_nat b = (of_nat (a * b) :: 'a :: len word)"
huffman@46013
  1828
  by (simp add: word_of_nat wi_hom_mult zmult_int)
haftmann@37660
  1829
haftmann@37660
  1830
lemma Abs_fnat_hom_Suc:
haftmann@37660
  1831
  "word_succ (of_nat a) = of_nat (Suc a)"
huffman@46013
  1832
  by (simp add: word_of_nat wi_hom_succ add_ac)
haftmann@37660
  1833
haftmann@37660
  1834
lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0"
huffman@45995
  1835
  by simp
haftmann@37660
  1836
haftmann@37660
  1837
lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)"
huffman@45995
  1838
  by simp
haftmann@37660
  1839
haftmann@37660
  1840
lemmas Abs_fnat_homs = 
haftmann@37660
  1841
  Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc 
haftmann@37660
  1842
  Abs_fnat_hom_0 Abs_fnat_hom_1
haftmann@37660
  1843
haftmann@37660
  1844
lemma word_arith_nat_add:
haftmann@37660
  1845
  "a + b = of_nat (unat a + unat b)" 
haftmann@37660
  1846
  by simp
haftmann@37660
  1847
haftmann@37660
  1848
lemma word_arith_nat_mult:
haftmann@37660
  1849
  "a * b = of_nat (unat a * unat b)"
huffman@45995
  1850
  by (simp add: of_nat_mult)
haftmann@37660
  1851
    
haftmann@37660
  1852
lemma word_arith_nat_Suc:
haftmann@37660
  1853
  "word_succ a = of_nat (Suc (unat a))"
haftmann@37660
  1854
  by (subst Abs_fnat_hom_Suc [symmetric]) simp
haftmann@37660
  1855
haftmann@37660
  1856
lemma word_arith_nat_div:
haftmann@37660
  1857
  "a div b = of_nat (unat a div unat b)"
haftmann@37660
  1858
  by (simp add: word_div_def word_of_nat zdiv_int uint_nat)
haftmann@37660
  1859
haftmann@37660
  1860
lemma word_arith_nat_mod:
haftmann@37660
  1861
  "a mod b = of_nat (unat a mod unat b)"
haftmann@37660
  1862
  by (simp add: word_mod_def word_of_nat zmod_int uint_nat)
haftmann@37660
  1863
haftmann@37660
  1864
lemmas word_arith_nat_defs =
haftmann@37660
  1865
  word_arith_nat_add word_arith_nat_mult
haftmann@37660
  1866
  word_arith_nat_Suc Abs_fnat_hom_0
haftmann@37660
  1867
  Abs_fnat_hom_1 word_arith_nat_div
haftmann@37660
  1868
  word_arith_nat_mod 
haftmann@37660
  1869
huffman@45816
  1870
lemma unat_cong: "x = y \<Longrightarrow> unat x = unat y"
huffman@45816
  1871
  by simp
haftmann@37660
  1872
  
haftmann@37660
  1873
lemmas unat_word_ariths = word_arith_nat_defs
wenzelm@45604
  1874
  [THEN trans [OF unat_cong unat_of_nat]]
haftmann@37660
  1875
haftmann@37660
  1876
lemmas word_sub_less_iff = word_sub_le_iff
huffman@45816
  1877
  [unfolded linorder_not_less [symmetric] Not_eq_iff]
haftmann@37660
  1878
haftmann@37660
  1879
lemma unat_add_lem: 
haftmann@37660
  1880
  "(unat x + unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1881
    (unat (x + y :: 'a :: len word) = unat x + unat y)"
haftmann@37660
  1882
  unfolding unat_word_ariths
haftmann@37660
  1883
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1884
haftmann@37660
  1885
lemma unat_mult_lem: 
haftmann@37660
  1886
  "(unat x * unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1887
    (unat (x * y :: 'a :: len word) = unat x * unat y)"
haftmann@37660
  1888
  unfolding unat_word_ariths
haftmann@37660
  1889
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1890
wenzelm@45604
  1891
lemmas unat_plus_if' = trans [OF unat_word_ariths(1) mod_nat_add, simplified]
haftmann@37660
  1892
haftmann@37660
  1893
lemma le_no_overflow: 
haftmann@40827
  1894
  "x <= b \<Longrightarrow> a <= a + b \<Longrightarrow> x <= a + (b :: 'a :: len0 word)"
haftmann@37660
  1895
  apply (erule order_trans)
haftmann@37660
  1896
  apply (erule olen_add_eqv [THEN iffD1])
haftmann@37660
  1897
  done
haftmann@37660
  1898
wenzelm@45604
  1899
lemmas un_ui_le = trans [OF word_le_nat_alt [symmetric] word_le_def]
haftmann@37660
  1900
haftmann@37660
  1901
lemma unat_sub_if_size:
haftmann@37660
  1902
  "unat (x - y) = (if unat y <= unat x 
haftmann@37660
  1903
   then unat x - unat y 
haftmann@37660
  1904
   else unat x + 2 ^ size x - unat y)"
haftmann@37660
  1905
  apply (unfold word_size)
haftmann@37660
  1906
  apply (simp add: un_ui_le)
haftmann@37660
  1907
  apply (auto simp add: unat_def uint_sub_if')
haftmann@37660
  1908
   apply (rule nat_diff_distrib)
haftmann@37660
  1909
    prefer 3
haftmann@37660
  1910
    apply (simp add: algebra_simps)
haftmann@37660
  1911
    apply (rule nat_diff_distrib [THEN trans])
haftmann@37660
  1912
      prefer 3
haftmann@37660
  1913
      apply (subst nat_add_distrib)
haftmann@37660
  1914
        prefer 3
haftmann@37660
  1915
        apply (simp add: nat_power_eq)
haftmann@37660
  1916
       apply auto
haftmann@37660
  1917
  apply uint_arith
haftmann@37660
  1918
  done
haftmann@37660
  1919
haftmann@37660
  1920
lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size]
haftmann@37660
  1921
haftmann@37660
  1922
lemma unat_div: "unat ((x :: 'a :: len word) div y) = unat x div unat y"
haftmann@37660
  1923
  apply (simp add : unat_word_ariths)
haftmann@37660
  1924
  apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1925
  apply (rule div_le_dividend)
haftmann@37660
  1926
  done
haftmann@37660
  1927
haftmann@37660
  1928
lemma unat_mod: "unat ((x :: 'a :: len word) mod y) = unat x mod unat y"
haftmann@37660
  1929
  apply (clarsimp simp add : unat_word_ariths)
haftmann@37660
  1930
  apply (cases "unat y")
haftmann@37660
  1931
   prefer 2
haftmann@37660
  1932
   apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1933
   apply (rule mod_le_divisor)
haftmann@37660
  1934
   apply auto
haftmann@37660
  1935
  done
haftmann@37660
  1936
haftmann@37660
  1937
lemma uint_div: "uint ((x :: 'a :: len word) div y) = uint x div uint y"
haftmann@37660
  1938
  unfolding uint_nat by (simp add : unat_div zdiv_int)
haftmann@37660
  1939
haftmann@37660
  1940
lemma uint_mod: "uint ((x :: 'a :: len word) mod y) = uint x mod uint y"
haftmann@37660
  1941
  unfolding uint_nat by (simp add : unat_mod zmod_int)
haftmann@37660
  1942
haftmann@37660
  1943
haftmann@37660
  1944
subsection {* Definition of unat\_arith tactic *}
haftmann@37660
  1945
haftmann@37660
  1946
lemma unat_split:
haftmann@37660
  1947
  fixes x::"'a::len word"
haftmann@37660
  1948
  shows "P (unat x) = 
haftmann@37660
  1949
         (ALL n. of_nat n = x & n < 2^len_of TYPE('a) --> P n)"
haftmann@37660
  1950
  by (auto simp: unat_of_nat)
haftmann@37660
  1951
haftmann@37660
  1952
lemma unat_split_asm:
haftmann@37660
  1953
  fixes x::"'a::len word"
haftmann@37660
  1954
  shows "P (unat x) = 
haftmann@37660
  1955
         (~(EX n. of_nat n = x & n < 2^len_of TYPE('a) & ~ P n))"
haftmann@37660
  1956
  by (auto simp: unat_of_nat)
haftmann@37660
  1957
haftmann@37660
  1958
lemmas of_nat_inverse = 
haftmann@37660
  1959
  word_unat.Abs_inverse' [rotated, unfolded unats_def, simplified]
haftmann@37660
  1960
haftmann@37660
  1961
lemmas unat_splits = unat_split unat_split_asm
haftmann@37660
  1962
haftmann@37660
  1963
lemmas unat_arith_simps =
haftmann@37660
  1964
  word_le_nat_alt word_less_nat_alt
haftmann@37660
  1965
  word_unat.Rep_inject [symmetric]
haftmann@37660
  1966
  unat_sub_if' unat_plus_if' unat_div unat_mod
haftmann@37660
  1967
haftmann@37660
  1968
(* unat_arith_tac: tactic to reduce word arithmetic to nat, 
haftmann@37660
  1969
   try to solve via arith *)
haftmann@37660
  1970
ML {*
haftmann@37660
  1971
fun unat_arith_ss_of ss = 
haftmann@37660
  1972
  ss addsimps @{thms unat_arith_simps}
haftmann@37660
  1973
     delsimps @{thms word_unat.Rep_inject}
wenzelm@45620
  1974
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1975
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1976
haftmann@37660
  1977
fun unat_arith_tacs ctxt =   
haftmann@37660
  1978
  let
haftmann@37660
  1979
    fun arith_tac' n t =
haftmann@37660
  1980
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1981
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1982
  in 
wenzelm@42793
  1983
    [ clarify_tac ctxt 1,
wenzelm@42793
  1984
      full_simp_tac (unat_arith_ss_of (simpset_of ctxt)) 1,
wenzelm@45620
  1985
      ALLGOALS (full_simp_tac (HOL_ss |> fold Splitter.add_split @{thms unat_splits}
wenzelm@45620
  1986
                                      |> fold Simplifier.add_cong @{thms power_False_cong})),
haftmann@37660
  1987
      rewrite_goals_tac @{thms word_size}, 
haftmann@37660
  1988
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1989
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1990
                         REPEAT (dtac @{thm of_nat_inverse} n THEN atac n)),
haftmann@37660
  1991
      TRYALL arith_tac' ] 
haftmann@37660
  1992
  end
haftmann@37660
  1993
haftmann@37660
  1994
fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt))
haftmann@37660
  1995
*}
haftmann@37660
  1996
haftmann@37660
  1997
method_setup unat_arith = 
haftmann@37660
  1998
  {* Scan.succeed (SIMPLE_METHOD' o unat_arith_tac) *}
haftmann@37660
  1999
  "solving word arithmetic via natural numbers and arith"
haftmann@37660
  2000
haftmann@37660
  2001
lemma no_plus_overflow_unat_size: 
haftmann@37660
  2002
  "((x :: 'a :: len word) <= x + y) = (unat x + unat y < 2 ^ size x)" 
haftmann@37660
  2003
  unfolding word_size by unat_arith
haftmann@37660
  2004
haftmann@37660
  2005
lemmas no_olen_add_nat = no_plus_overflow_unat_size [unfolded word_size]
haftmann@37660
  2006
wenzelm@45604
  2007
lemmas unat_plus_simple = trans [OF no_olen_add_nat unat_add_lem]
haftmann@37660
  2008
haftmann@37660
  2009
lemma word_div_mult: 
haftmann@40827
  2010
  "(0 :: 'a :: len word) < y \<Longrightarrow> unat x * unat y < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  2011
    x * y div y = x"
haftmann@37660
  2012
  apply unat_arith
haftmann@37660
  2013
  apply clarsimp
haftmann@37660
  2014
  apply (subst unat_mult_lem [THEN iffD1])
haftmann@37660
  2015
  apply auto
haftmann@37660
  2016
  done
haftmann@37660
  2017
haftmann@40827
  2018
lemma div_lt': "(i :: 'a :: len word) <= k div x \<Longrightarrow> 
haftmann@37660
  2019
    unat i * unat x < 2 ^ len_of TYPE('a)"
haftmann@37660
  2020
  apply unat_arith
haftmann@37660
  2021
  apply clarsimp
haftmann@37660
  2022
  apply (drule mult_le_mono1)
haftmann@37660
  2023
  apply (erule order_le_less_trans)
haftmann@37660
  2024
  apply (rule xtr7 [OF unat_lt2p div_mult_le])
haftmann@37660
  2025
  done
haftmann@37660
  2026
haftmann@37660
  2027
lemmas div_lt'' = order_less_imp_le [THEN div_lt']
haftmann@37660
  2028
haftmann@40827
  2029
lemma div_lt_mult: "(i :: 'a :: len word) < k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x < k"
haftmann@37660
  2030
  apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  2031
  apply (simp add: unat_arith_simps)
haftmann@37660
  2032
  apply (drule (1) mult_less_mono1)
haftmann@37660
  2033
  apply (erule order_less_le_trans)
haftmann@37660
  2034
  apply (rule div_mult_le)
haftmann@37660
  2035
  done
haftmann@37660
  2036
haftmann@37660
  2037
lemma div_le_mult: 
haftmann@40827
  2038
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x <= k"
haftmann@37660
  2039
  apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  2040
  apply (simp add: unat_arith_simps)
haftmann@37660
  2041
  apply (drule mult_le_mono1)
haftmann@37660
  2042
  apply (erule order_trans)
haftmann@37660
  2043
  apply (rule div_mult_le)
haftmann@37660
  2044
  done
haftmann@37660
  2045
haftmann@37660
  2046
lemma div_lt_uint': 
haftmann@40827
  2047
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> uint i * uint x < 2 ^ len_of TYPE('a)"
haftmann@37660
  2048
  apply (unfold uint_nat)
haftmann@37660
  2049
  apply (drule div_lt')
haftmann@37660
  2050
  apply (simp add: zmult_int zless_nat_eq_int_zless [symmetric] 
haftmann@37660
  2051
                   nat_power_eq)
haftmann@37660
  2052
  done
haftmann@37660
  2053
haftmann@37660
  2054
lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint']
haftmann@37660
  2055
haftmann@37660
  2056
lemma word_le_exists': 
haftmann@40827
  2057
  "(x :: 'a :: len0 word) <= y \<Longrightarrow> 
haftmann@37660
  2058
    (EX z. y = x + z & uint x + uint z < 2 ^ len_of TYPE('a))"
haftmann@37660
  2059
  apply (rule exI)
haftmann@37660
  2060
  apply (rule conjI)
haftmann@37660
  2061
  apply (rule zadd_diff_inverse)
haftmann@37660
  2062
  apply uint_arith
haftmann@37660
  2063
  done
haftmann@37660
  2064
haftmann@37660
  2065
lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab]
haftmann@37660
  2066
haftmann@37660
  2067
lemmas plus_minus_no_overflow =
haftmann@37660
  2068
  order_less_imp_le [THEN plus_minus_no_overflow_ab]
haftmann@37660
  2069
  
haftmann@37660
  2070
lemmas mcs = word_less_minus_cancel word_less_minus_mono_left
haftmann@37660
  2071
  word_le_minus_cancel word_le_minus_mono_left
haftmann@37660
  2072
wenzelm@45604
  2073
lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2074
lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2075
lemmas word_plus_mcs = word_diff_ls [where y = "v + x", unfolded add_diff_cancel] for v x
haftmann@37660
  2076
haftmann@37660
  2077
lemmas le_unat_uoi = unat_le [THEN word_unat.Abs_inverse]
haftmann@37660
  2078
haftmann@37660
  2079
lemmas thd = refl [THEN [2] split_div_lemma [THEN iffD2], THEN conjunct1]
haftmann@37660
  2080
haftmann@37660
  2081
lemma thd1:
haftmann@37660
  2082
  "a div b * b \<le> (a::nat)"
haftmann@37660
  2083
  using gt_or_eq_0 [of b]
haftmann@37660
  2084
  apply (rule disjE)
haftmann@37660
  2085
   apply (erule xtr4 [OF thd mult_commute])
haftmann@37660
  2086
  apply clarsimp
haftmann@37660
  2087
  done
haftmann@37660
  2088
wenzelm@45604
  2089
lemmas uno_simps [THEN le_unat_uoi] = mod_le_divisor div_le_dividend thd1 
haftmann@37660
  2090
haftmann@37660
  2091
lemma word_mod_div_equality:
haftmann@37660
  2092
  "(n div b) * b + (n mod b) = (n :: 'a :: len word)"
haftmann@37660
  2093
  apply (unfold word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2094
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2095
  apply (erule disjE)
haftmann@37660
  2096
   apply (simp add: mod_div_equality uno_simps)
haftmann@37660
  2097
  apply simp
haftmann@37660
  2098
  done
haftmann@37660
  2099
haftmann@37660
  2100
lemma word_div_mult_le: "a div b * b <= (a::'a::len word)"
haftmann@37660
  2101
  apply (unfold word_le_nat_alt word_arith_nat_defs)
haftmann@37660
  2102
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2103
  apply (erule disjE)
haftmann@37660
  2104
   apply (simp add: div_mult_le uno_simps)
haftmann@37660
  2105
  apply simp
haftmann@37660
  2106
  done
haftmann@37660
  2107
haftmann@40827
  2108
lemma word_mod_less_divisor: "0 < n \<Longrightarrow> m mod n < (n :: 'a :: len word)"
haftmann@37660
  2109
  apply (simp only: word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2110
  apply (clarsimp simp add : uno_simps)
haftmann@37660
  2111
  done
haftmann@37660
  2112
haftmann@37660
  2113
lemma word_of_int_power_hom: 
haftmann@37660
  2114
  "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a :: len word)"
huffman@45995
  2115
  by (induct n) (simp_all add: wi_hom_mult [symmetric])
haftmann@37660
  2116
haftmann@37660
  2117
lemma word_arith_power_alt: 
haftmann@37660
  2118
  "a ^ n = (word_of_int (uint a ^ n) :: 'a :: len word)"
haftmann@37660
  2119
  by (simp add : word_of_int_power_hom [symmetric])
haftmann@37660
  2120
haftmann@37660
  2121
lemma of_bl_length_less: 
haftmann@40827
  2122
  "length x = k \<Longrightarrow> k < len_of TYPE('a) \<Longrightarrow> (of_bl x :: 'a :: len word) < 2 ^ k"
huffman@47108
  2123
  apply (unfold of_bl_def word_less_alt word_numeral_alt)
haftmann@37660
  2124
  apply safe
haftmann@37660
  2125
  apply (simp (no_asm) add: word_of_int_power_hom word_uint.eq_norm 
huffman@47108
  2126
                       del: word_of_int_numeral)
haftmann@37660
  2127
  apply (simp add: mod_pos_pos_trivial)
haftmann@37660
  2128
  apply (subst mod_pos_pos_trivial)
haftmann@37660
  2129
    apply (rule bl_to_bin_ge0)
haftmann@37660
  2130
   apply (rule order_less_trans)
haftmann@37660
  2131
    apply (rule bl_to_bin_lt2p)
haftmann@37660
  2132
   apply simp
huffman@46646
  2133
  apply (rule bl_to_bin_lt2p)
haftmann@37660
  2134
  done
haftmann@37660
  2135
haftmann@37660
  2136
haftmann@37660
  2137
subsection "Cardinality, finiteness of set of words"
haftmann@37660
  2138
huffman@45809
  2139
instance word :: (len0) finite
huffman@45809
  2140
  by (default, simp add: type_definition.univ [OF type_definition_word])
huffman@45809
  2141
huffman@45809
  2142
lemma card_word: "CARD('a::len0 word) = 2 ^ len_of TYPE('a)"
huffman@45809
  2143
  by (simp add: type_definition.card [OF type_definition_word] nat_power_eq)
haftmann@37660
  2144
haftmann@37660
  2145
lemma card_word_size: 
huffman@45809
  2146
  "card (UNIV :: 'a :: len0 word set) = (2 ^ size (x :: 'a word))"
haftmann@37660
  2147
unfolding word_size by (rule card_word)
haftmann@37660
  2148
haftmann@37660
  2149
haftmann@37660
  2150
subsection {* Bitwise Operations on Words *}
haftmann@37660
  2151
haftmann@37660
  2152
lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or
haftmann@37660
  2153
  
haftmann@37660
  2154
(* following definitions require both arithmetic and bit-wise word operations *)
haftmann@37660
  2155
haftmann@37660
  2156
(* to get word_no_log_defs from word_log_defs, using bin_log_bintrs *)
haftmann@37660
  2157
lemmas wils1 = bin_log_bintrs [THEN word_ubin.norm_eq_iff [THEN iffD1],
wenzelm@45604
  2158
  folded word_ubin.eq_norm, THEN eq_reflection]
haftmann@37660
  2159
haftmann@37660
  2160
(* the binary operations only *)
huffman@46013
  2161
(* BH: why is this needed? *)
haftmann@37660
  2162
lemmas word_log_binary_defs = 
haftmann@37660
  2163
  word_and_def word_or_def word_xor_def
haftmann@37660
  2164
huffman@46011
  2165
lemma word_wi_log_defs:
huffman@46011
  2166
  "NOT word_of_int a = word_of_int (NOT a)"
huffman@46011
  2167
  "word_of_int a AND word_of_int b = word_of_int (a AND b)"
huffman@46011
  2168
  "word_of_int a OR word_of_int b = word_of_int (a OR b)"
huffman@46011
  2169
  "word_of_int a XOR word_of_int b = word_of_int (a XOR b)"
huffman@47374
  2170
  by (transfer, rule refl)+
huffman@47372
  2171
huffman@46011
  2172
lemma word_no_log_defs [simp]:
huffman@47108
  2173
  "NOT (numeral a) = word_of_int (NOT (numeral a))"
huffman@47108
  2174
  "NOT (neg_numeral a) = word_of_int (NOT (neg_numeral a))"
huffman@47108
  2175
  "numeral a AND numeral b = word_of_int (numeral a AND numeral b)"
huffman@47108
  2176
  "numeral a AND neg_numeral b = word_of_int (numeral a AND neg_numeral b)"
huffman@47108
  2177
  "neg_numeral a AND numeral b = word_of_int (neg_numeral a AND numeral b)"
huffman@47108
  2178
  "neg_numeral a AND neg_numeral b = word_of_int (neg_numeral a AND neg_numeral b)"
huffman@47108
  2179
  "numeral a OR numeral b = word_of_int (numeral a OR numeral b)"
huffman@47108
  2180
  "numeral a OR neg_numeral b = word_of_int (numeral a OR neg_numeral b)"
huffman@47108
  2181
  "neg_numeral a OR numeral b = word_of_int (neg_numeral a OR numeral b)"
huffman@47108
  2182
  "neg_numeral a OR neg_numeral b = word_of_int (neg_numeral a OR neg_numeral b)"
huffman@47108
  2183
  "numeral a XOR numeral b = word_of_int (numeral a XOR numeral b)"
huffman@47108
  2184
  "numeral a XOR neg_numeral b = word_of_int (numeral a XOR neg_numeral b)"
huffman@47108
  2185
  "neg_numeral a XOR numeral b = word_of_int (neg_numeral a XOR numeral b)"
huffman@47108
  2186
  "neg_numeral a XOR neg_numeral b = word_of_int (neg_numeral a XOR neg_numeral b)"
huffman@47372
  2187
  by (transfer, rule refl)+
haftmann@37660
  2188
huffman@46064
  2189
text {* Special cases for when one of the arguments equals 1. *}
huffman@46064
  2190
huffman@46064
  2191
lemma word_bitwise_1_simps [simp]:
huffman@46064
  2192
  "NOT (1::'a::len0 word) = -2"
huffman@47108
  2193
  "1 AND numeral b = word_of_int (1 AND numeral b)"
huffman@47108
  2194
  "1 AND neg_numeral b = word_of_int (1 AND neg_numeral b)"
huffman@47108
  2195
  "numeral a AND 1 = word_of_int (numeral a AND 1)"
huffman@47108
  2196
  "neg_numeral a AND 1 = word_of_int (neg_numeral a AND 1)"
huffman@47108
  2197
  "1 OR numeral b = word_of_int (1 OR numeral b)"
huffman@47108
  2198
  "1 OR neg_numeral b = word_of_int (1 OR neg_numeral b)"
huffman@47108
  2199
  "numeral a OR 1 = word_of_int (numeral a OR 1)"
huffman@47108
  2200
  "neg_numeral a OR 1 = word_of_int (neg_numeral a OR 1)"
huffman@47108
  2201
  "1 XOR numeral b = word_of_int (1 XOR numeral b)"
huffman@47108
  2202
  "1 XOR neg_numeral b = word_of_int (1 XOR neg_numeral b)"
huffman@47108
  2203
  "numeral a XOR 1 = word_of_int (numeral a XOR 1)"
huffman@47108
  2204
  "neg_numeral a XOR 1 = word_of_int (neg_numeral a XOR 1)"
huffman@47372
  2205
  by (transfer, simp)+
huffman@46064
  2206
haftmann@37660
  2207
lemma uint_or: "uint (x OR y) = (uint x) OR (uint y)"
huffman@47372
  2208
  by (transfer, simp add: bin_trunc_ao)
haftmann@37660
  2209
haftmann@37660
  2210
lemma uint_and: "uint (x AND y) = (uint x) AND (uint y)"
huffman@47372
  2211
  by (transfer, simp add: bin_trunc_ao)
huffman@47372
  2212
huffman@47372
  2213
lemma test_bit_wi [simp]:
huffman@47372
  2214
  "(word_of_int x::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a) \<and> bin_nth x n"
huffman@47372
  2215
  unfolding word_test_bit_def
huffman@47372
  2216
  by (simp add: word_ubin.eq_norm nth_bintr)
huffman@47372
  2217
huffman@47372
  2218
lemma word_test_bit_transfer [transfer_rule]:
huffman@47372
  2219
  "(fun_rel cr_word (fun_rel op = op =))
huffman@47372
  2220
    (\<lambda>x n. n < len_of TYPE('a) \<and> bin_nth x n) (test_bit :: 'a::len0 word \<Rightarrow> _)"
huffman@47372
  2221
  unfolding fun_rel_def cr_word_def by simp
haftmann@37660
  2222
haftmann@37660
  2223
lemma word_ops_nth_size:
haftmann@40827
  2224
  "n < size (x::'a::len0 word) \<Longrightarrow> 
haftmann@37660
  2225
    (x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2226
    (x AND y) !! n = (x !! n & y !! n) & 
haftmann@37660
  2227
    (x XOR y) !! n = (x !! n ~= y !! n) & 
haftmann@37660
  2228
    (NOT x) !! n = (~ x !! n)"
huffman@47372
  2229
  unfolding word_size by transfer (simp add: bin_nth_ops)
haftmann@37660
  2230
haftmann@37660
  2231
lemma word_ao_nth:
haftmann@37660
  2232
  fixes x :: "'a::len0 word"
haftmann@37660
  2233
  shows "(x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2234
         (x AND y) !! n = (x !! n & y !! n)"
huffman@47372
  2235
  by transfer (auto simp add: bin_nth_ops)
huffman@46023
  2236
huffman@47108
  2237
lemma test_bit_numeral [simp]:
huffman@47108
  2238
  "(numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
huffman@47108
  2239
    n < len_of TYPE('a) \<and> bin_nth (numeral w) n"
huffman@47372
  2240
  by transfer (rule refl)
huffman@47108
  2241
huffman@47108
  2242
lemma test_bit_neg_numeral [simp]:
huffman@47108
  2243
  "(neg_numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
huffman@47108
  2244
    n < len_of TYPE('a) \<and> bin_nth (neg_numeral w) n"
huffman@47372
  2245
  by transfer (rule refl)
huffman@46023
  2246
huffman@46172
  2247
lemma test_bit_1 [simp]: "(1::'a::len word) !! n \<longleftrightarrow> n = 0"
huffman@47372
  2248
  by transfer auto
huffman@46172
  2249
  
huffman@46023
  2250
lemma nth_0 [simp]: "~ (0::'a::len0 word) !! n"
huffman@47372
  2251
  by transfer simp
huffman@46023
  2252
huffman@47108
  2253
lemma nth_minus1 [simp]: "(-1::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a)"
huffman@47372
  2254
  by transfer simp
huffman@47108
  2255
haftmann@37660
  2256
(* get from commutativity, associativity etc of int_and etc
haftmann@37660
  2257
  to same for word_and etc *)
haftmann@37660
  2258
haftmann@37660
  2259
lemmas bwsimps = 
huffman@46013
  2260
  wi_hom_add
haftmann@37660
  2261
  word_wi_log_defs
haftmann@37660
  2262
haftmann@37660
  2263
lemma word_bw_assocs:
haftmann@37660
  2264
  fixes x :: "'a::len0 word"
haftmann@37660
  2265
  shows
haftmann@37660
  2266
  "(x AND y) AND z = x AND y AND z"
haftmann@37660
  2267
  "(x OR y) OR z = x OR y OR z"
haftmann@37660
  2268
  "(x XOR y) XOR z = x XOR y XOR z"
huffman@46022
  2269
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2270
  
haftmann@37660
  2271
lemma word_bw_comms:
haftmann@37660
  2272
  fixes x :: "'a::len0 word"
haftmann@37660
  2273
  shows
haftmann@37660
  2274
  "x AND y = y AND x"
haftmann@37660
  2275
  "x OR y = y OR x"
haftmann@37660
  2276
  "x XOR y = y XOR x"
huffman@46022
  2277
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2278
  
haftmann@37660
  2279
lemma word_bw_lcs:
haftmann@37660
  2280
  fixes x :: "'a::len0 word"
haftmann@37660
  2281
  shows
haftmann@37660
  2282
  "y AND x AND z = x AND y AND z"
haftmann@37660
  2283
  "y OR x OR z = x OR y OR z"
haftmann@37660
  2284
  "y XOR x XOR z = x XOR y XOR z"
huffman@46022
  2285
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2286
haftmann@37660
  2287
lemma word_log_esimps [simp]:
haftmann@37660
  2288
  fixes x :: "'a::len0 word"
haftmann@37660
  2289
  shows
haftmann@37660
  2290
  "x AND 0 = 0"
haftmann@37660
  2291
  "x AND -1 = x"
haftmann@37660
  2292
  "x OR 0 = x"
haftmann@37660
  2293
  "x OR -1 = -1"
haftmann@37660
  2294
  "x XOR 0 = x"
haftmann@37660
  2295
  "x XOR -1 = NOT x"
haftmann@37660
  2296
  "0 AND x = 0"
haftmann@37660
  2297
  "-1 AND x = x"
haftmann@37660
  2298
  "0 OR x = x"
haftmann@37660
  2299
  "-1 OR x = -1"
haftmann@37660
  2300
  "0 XOR x = x"
haftmann@37660
  2301
  "-1 XOR x = NOT x"
huffman@46023
  2302
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2303
haftmann@37660
  2304
lemma word_not_dist:
haftmann@37660
  2305
  fixes x :: "'a::len0 word"
haftmann@37660
  2306
  shows
haftmann@37660
  2307
  "NOT (x OR y) = NOT x AND NOT y"
haftmann@37660
  2308
  "NOT (x AND y) = NOT x OR NOT y"
huffman@46022
  2309
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2310
haftmann@37660
  2311
lemma word_bw_same:
haftmann@37660
  2312
  fixes x :: "'a::len0 word"
haftmann@37660
  2313
  shows
haftmann@37660
  2314
  "x AND x = x"
haftmann@37660
  2315
  "x OR x = x"
haftmann@37660
  2316
  "x XOR x = 0"
huffman@46023
  2317
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2318
haftmann@37660
  2319
lemma word_ao_absorbs [simp]:
haftmann@37660
  2320
  fixes x :: "'a::len0 word"
haftmann@37660
  2321
  shows
haftmann@37660
  2322
  "x AND (y OR x) = x"
haftmann@37660
  2323
  "x OR y AND x = x"
haftmann@37660
  2324
  "x AND (x OR y) = x"
haftmann@37660
  2325
  "y AND x OR x = x"
haftmann@37660
  2326
  "(y OR x) AND x = x"
haftmann@37660
  2327
  "x OR x AND y = x"
haftmann@37660
  2328
  "(x OR y) AND x = x"
haftmann@37660
  2329
  "x AND y OR x = x"
huffman@46022
  2330
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2331
haftmann@37660
  2332
lemma word_not_not [simp]:
haftmann@37660
  2333
  "NOT NOT (x::'a::len0 word) = x"
huffman@46022
  2334
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2335
haftmann@37660
  2336
lemma word_ao_dist:
haftmann@37660
  2337
  fixes x :: "'a::len0 word"
haftmann@37660
  2338
  shows "(x OR y) AND z = x AND z OR y AND z"
huffman@46022
  2339
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2340
haftmann@37660
  2341
lemma word_oa_dist:
haftmann@37660
  2342
  fixes x :: "'a::len0 word"
haftmann@37660
  2343
  shows "x AND y OR z = (x OR z) AND (y OR z)"
huffman@46022
  2344
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2345
haftmann@37660
  2346
lemma word_add_not [simp]: 
haftmann@37660
  2347
  fixes x :: "'a::len0 word"
haftmann@37660
  2348
  shows "x + NOT x = -1"
huffman@47372
  2349
  by transfer (simp add: bin_add_not)
haftmann@37660
  2350
haftmann@37660
  2351
lemma word_plus_and_or [simp]:
haftmann@37660
  2352
  fixes x :: "'a::len0 word"
haftmann@37660
  2353
  shows "(x AND y) + (x OR y) = x + y"
huffman@47372
  2354
  by transfer (simp add: plus_and_or)
haftmann@37660
  2355
haftmann@37660
  2356
lemma leoa:   
haftmann@37660
  2357
  fixes x :: "'a::len0 word"
haftmann@40827
  2358
  shows "(w = (x OR y)) \<Longrightarrow> (y = (w AND y))" by auto
haftmann@37660
  2359
lemma leao: 
haftmann@37660
  2360
  fixes x' :: "'a::len0 word"
haftmann@40827
  2361
  shows "(w' = (x' AND y')) \<Longrightarrow> (x' = (x' OR w'))" by auto 
haftmann@37660
  2362
haftmann@37660
  2363
lemmas word_ao_equiv = leao [COMP leoa [COMP iffI]]
haftmann@37660
  2364
haftmann@37660
  2365
lemma le_word_or2: "x <= x OR (y::'a::len0 word)"
haftmann@37660
  2366
  unfolding word_le_def uint_or
haftmann@37660
  2367
  by (auto intro: le_int_or) 
haftmann@37660
  2368
wenzelm@45604
  2369
lemmas le_word_or1 = xtr3 [OF word_bw_comms (2) le_word_or2]
wenzelm@45604
  2370
lemmas word_and_le1 = xtr3 [OF word_ao_absorbs (4) [symmetric] le_word_or2]
wenzelm@45604
  2371
lemmas word_and_le2 = xtr3 [OF word_ao_absorbs (8) [symmetric] le_word_or2]
haftmann@37660
  2372
haftmann@37660
  2373
lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" 
huffman@45550
  2374
  unfolding to_bl_def word_log_defs bl_not_bin
huffman@45550
  2375
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2376
haftmann@37660
  2377
lemma bl_word_xor: "to_bl (v XOR w) = map2 op ~= (to_bl v) (to_bl w)" 
haftmann@37660
  2378
  unfolding to_bl_def word_log_defs bl_xor_bin
huffman@45550
  2379
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2380
haftmann@37660
  2381
lemma bl_word_or: "to_bl (v OR w) = map2 op | (to_bl v) (to_bl w)" 
huffman@45550
  2382
  unfolding to_bl_def word_log_defs bl_or_bin
huffman@45550
  2383
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2384
haftmann@37660
  2385
lemma bl_word_and: "to_bl (v AND w) = map2 op & (to_bl v) (to_bl w)" 
huffman@45550
  2386
  unfolding to_bl_def word_log_defs bl_and_bin
huffman@45550
  2387
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2388
haftmann@37660
  2389
lemma word_lsb_alt: "lsb (w::'a::len0 word) = test_bit w 0"
haftmann@37660
  2390
  by (auto simp: word_test_bit_def word_lsb_def)
haftmann@37660
  2391
huffman@45805
  2392
lemma word_lsb_1_0 [simp]: "lsb (1::'a::len word) & ~ lsb (0::'b::len0 word)"
huffman@45550
  2393
  unfolding word_lsb_def uint_eq_0 uint_1 by simp
haftmann@37660
  2394
haftmann@37660
  2395
lemma word_lsb_last: "lsb (w::'a::len word) = last (to_bl w)"
haftmann@37660
  2396
  apply (unfold word_lsb_def uint_bl bin_to_bl_def) 
haftmann@37660
  2397
  apply (rule_tac bin="uint w" in bin_exhaust)
haftmann@37660
  2398
  apply (cases "size w")
haftmann@37660
  2399
   apply auto
haftmann@37660
  2400
   apply (auto simp add: bin_to_bl_aux_alt)
haftmann@37660
  2401
  done
haftmann@37660
  2402
haftmann@37660
  2403
lemma word_lsb_int: "lsb w = (uint w mod 2 = 1)"
huffman@45529
  2404
  unfolding word_lsb_def bin_last_def by auto
haftmann@37660
  2405
haftmann@37660
  2406
lemma word_msb_sint: "msb w = (sint w < 0)" 
huffman@46604
  2407
  unfolding word_msb_def sign_Min_lt_0 ..
haftmann@37660
  2408
huffman@46173
  2409
lemma msb_word_of_int:
huffman@46173
  2410
  "msb (word_of_int x::'a::len word) = bin_nth x (len_of TYPE('a) - 1)"
huffman@46173
  2411
  unfolding word_msb_def by (simp add: word_sbin.eq_norm bin_sign_lem)
huffman@46173
  2412
huffman@47108
  2413
lemma word_msb_numeral [simp]:
huffman@47108
  2414
  "msb (numeral w::'a::len word) = bin_nth (numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2415
  unfolding word_numeral_alt by (rule msb_word_of_int)
huffman@47108
  2416
huffman@47108
  2417
lemma word_msb_neg_numeral [simp]:
huffman@47108
  2418
  "msb (neg_numeral w::'a::len word) = bin_nth (neg_numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2419
  unfolding word_neg_numeral_alt by (rule msb_word_of_int)
huffman@46173
  2420
huffman@46173
  2421
lemma word_msb_0 [simp]: "\<not> msb (0::'a::len word)"
huffman@46173
  2422
  unfolding word_msb_def by simp
huffman@46173
  2423
huffman@46173
  2424
lemma word_msb_1 [simp]: "msb (1::'a::len word) \<longleftrightarrow> len_of TYPE('a) = 1"
huffman@46173
  2425
  unfolding word_1_wi msb_word_of_int eq_iff [where 'a=nat]
huffman@46173
  2426
  by (simp add: Suc_le_eq)
huffman@45811
  2427
huffman@45811
  2428
lemma word_msb_nth:
huffman@45811
  2429
  "msb (w::'a::len word) = bin_nth (uint w) (len_of TYPE('a) - 1)"
huffman@46023
  2430
  unfolding word_msb_def sint_uint by (simp add: bin_sign_lem)
haftmann@37660
  2431
haftmann@37660
  2432
lemma word_msb_alt: "msb (w::'a::len word) = hd (to_bl w)"
haftmann@37660
  2433
  apply (unfold word_msb_nth uint_bl)
haftmann@37660
  2434
  apply (subst hd_conv_nth)
haftmann@37660
  2435
  apply (rule length_greater_0_conv [THEN iffD1])
haftmann@37660
  2436
   apply simp
haftmann@37660
  2437
  apply (simp add : nth_bin_to_bl word_size)
haftmann@37660
  2438
  done
haftmann@37660
  2439
huffman@45805
  2440
lemma word_set_nth [simp]:
haftmann@37660
  2441
  "set_bit w n (test_bit w n) = (w::'a::len0 word)"
haftmann@37660
  2442
  unfolding word_test_bit_def word_set_bit_def by auto
haftmann@37660
  2443
haftmann@37660
  2444
lemma bin_nth_uint':
haftmann@37660
  2445
  "bin_nth (uint w) n = (rev (bin_to_bl (size w) (uint w)) ! n & n < size w)"
haftmann@37660
  2446
  apply (unfold word_size)
haftmann@37660
  2447
  apply (safe elim!: bin_nth_uint_imp)
haftmann@37660
  2448
   apply (frule bin_nth_uint_imp)
haftmann@37660
  2449
   apply (fast dest!: bin_nth_bl)+
haftmann@37660
  2450
  done
haftmann@37660
  2451
haftmann@37660
  2452
lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size]
haftmann@37660
  2453
haftmann@37660
  2454
lemma test_bit_bl: "w !! n = (rev (to_bl w) ! n & n < size w)"
haftmann@37660
  2455
  unfolding to_bl_def word_test_bit_def word_size
haftmann@37660
  2456
  by (rule bin_nth_uint)
haftmann@37660
  2457
haftmann@40827
  2458
lemma to_bl_nth: "n < size w \<Longrightarrow> to_bl w ! n = w !! (size w - Suc n)"
haftmann@37660
  2459
  apply (unfold test_bit_bl)
haftmann@37660
  2460
  apply clarsimp
haftmann@37660
  2461
  apply (rule trans)
haftmann@37660
  2462
   apply (rule nth_rev_alt)
haftmann@37660
  2463
   apply (auto simp add: word_size)
haftmann@37660
  2464
  done
haftmann@37660
  2465
haftmann@37660
  2466
lemma test_bit_set: 
haftmann@37660
  2467
  fixes w :: "'a::len0 word"
haftmann@37660
  2468
  shows "(set_bit w n x) !! n = (n < size w & x)"
haftmann@37660
  2469
  unfolding word_size word_test_bit_def word_set_bit_def
haftmann@37660
  2470
  by (clarsimp simp add : word_ubin.eq_norm nth_bintr)
haftmann@37660
  2471
haftmann@37660
  2472
lemma test_bit_set_gen: 
haftmann@37660
  2473
  fixes w :: "'a::len0 word"
haftmann@37660
  2474
  shows "test_bit (set_bit w n x) m = 
haftmann@37660
  2475
         (if m = n then n < size w & x else test_bit w m)"
haftmann@37660
  2476
  apply (unfold word_size word_test_bit_def word_set_bit_def)
haftmann@37660
  2477
  apply (clarsimp simp add: word_ubin.eq_norm nth_bintr bin_nth_sc_gen)
haftmann@37660
  2478
  apply (auto elim!: test_bit_size [unfolded word_size]
haftmann@37660
  2479
              simp add: word_test_bit_def [symmetric])
haftmann@37660
  2480
  done
haftmann@37660
  2481
haftmann@37660
  2482
lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs"
haftmann@37660
  2483
  unfolding of_bl_def bl_to_bin_rep_F by auto
haftmann@37660
  2484
  
huffman@45811
  2485
lemma msb_nth:
haftmann@37660
  2486
  fixes w :: "'a::len word"
huffman@45811
  2487
  shows "msb w = w !! (len_of TYPE('a) - 1)"
huffman@45811
  2488
  unfolding word_msb_nth word_test_bit_def by simp
haftmann@37660
  2489
wenzelm@45604
  2490
lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2491
lemmas msb1 = msb0 [where i = 0]
haftmann@37660
  2492
lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]]
haftmann@37660
  2493
wenzelm@45604
  2494
lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2495
lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt]
haftmann@37660
  2496
huffman@45811
  2497
lemma td_ext_nth [OF refl refl refl, unfolded word_size]:
haftmann@40827
  2498
  "n = size (w::'a::len0 word) \<Longrightarrow> ofn = set_bits \<Longrightarrow> [w, ofn g] = l \<Longrightarrow> 
haftmann@37660
  2499
    td_ext test_bit ofn {f. ALL i. f i --> i < n} (%h i. h i & i < n)"
haftmann@37660
  2500
  apply (unfold word_size td_ext_def')
wenzelm@46008
  2501
  apply safe
haftmann@37660
  2502
     apply (rule_tac [3] ext)
haftmann@37660
  2503
     apply (rule_tac [4] ext)
haftmann@37660
  2504
     apply (unfold word_size of_nth_def test_bit_bl)
haftmann@37660
  2505
     apply safe
haftmann@37660
  2506
       defer
haftmann@37660
  2507
       apply (clarsimp simp: word_bl.Abs_inverse)+
haftmann@37660
  2508
  apply (rule word_bl.Rep_inverse')
haftmann@37660
  2509
  apply (rule sym [THEN trans])
haftmann@37660
  2510
  apply (rule bl_of_nth_nth)
haftmann@37660
  2511
  apply simp
haftmann@37660
  2512
  apply (rule bl_of_nth_inj)
haftmann@37660
  2513
  apply (clarsimp simp add : test_bit_bl word_size)
haftmann@37660
  2514
  done
haftmann@37660
  2515
haftmann@37660
  2516
interpretation test_bit:
haftmann@37660
  2517
  td_ext "op !! :: 'a::len0 word => nat => bool"
haftmann@37660
  2518
         set_bits
haftmann@37660
  2519
         "{f. \<forall>i. f i \<longrightarrow> i < len_of TYPE('a::len0)}"
haftmann@37660
  2520
         "(\<lambda>h i. h i \<and> i < len_of TYPE('a::len0))"
haftmann@37660
  2521
  by (rule td_ext_nth)
haftmann@37660
  2522
haftmann@37660
  2523
lemmas td_nth = test_bit.td_thm
haftmann@37660
  2524
huffman@45805
  2525
lemma word_set_set_same [simp]:
haftmann@37660
  2526
  fixes w :: "'a::len0 word"
haftmann@37660
  2527
  shows "set_bit (set_bit w n x) n y = set_bit w n y" 
haftmann@37660
  2528
  by (rule word_eqI) (simp add : test_bit_set_gen word_size)
haftmann@37660
  2529
    
haftmann@37660
  2530
lemma word_set_set_diff: 
haftmann@37660
  2531
  fixes w :: "'a::len0 word"
haftmann@37660
  2532
  assumes "m ~= n"
haftmann@37660
  2533
  shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x" 
wenzelm@41550
  2534
  by (rule word_eqI) (clarsimp simp add: test_bit_set_gen word_size assms)
huffman@46001
  2535
haftmann@37660
  2536
lemma nth_sint: 
haftmann@37660
  2537
  fixes w :: "'a::len word"
haftmann@37660
  2538
  defines "l \<equiv> len_of TYPE ('a)"
haftmann@37660
  2539
  shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
haftmann@37660
  2540
  unfolding sint_uint l_def
haftmann@37660
  2541
  by (clarsimp simp add: nth_sbintr word_test_bit_def [symmetric])
haftmann@37660
  2542
huffman@47108
  2543
lemma word_lsb_numeral [simp]:
huffman@47108
  2544
  "lsb (numeral bin :: 'a :: len word) = (bin_last (numeral bin) = 1)"
huffman@47108
  2545
  unfolding word_lsb_alt test_bit_numeral by simp
huffman@47108
  2546
huffman@47108
  2547
lemma word_lsb_neg_numeral [simp]:
huffman@47108
  2548
  "lsb (neg_numeral bin :: 'a :: len word) = (bin_last (neg_numeral bin) = 1)"
huffman@47108
  2549
  unfolding word_lsb_alt test_bit_neg_numeral by simp
haftmann@37660
  2550
huffman@46173
  2551
lemma set_bit_word_of_int:
huffman@46173
  2552
  "set_bit (word_of_int x) n b = word_of_int (bin_sc n (if b then 1 else 0) x)"
huffman@46173
  2553
  unfolding word_set_bit_def
huffman@46173
  2554
  apply (rule word_eqI)
huffman@46173
  2555
  apply (simp add: word_size bin_nth_sc_gen word_ubin.eq_norm nth_bintr)
huffman@46173
  2556
  done
huffman@46173
  2557
huffman@47108
  2558
lemma word_set_numeral [simp]:
huffman@47108
  2559
  "set_bit (numeral bin::'a::len0 word) n b = 
huffman@47108
  2560
    word_of_int (bin_sc n (if b then 1 else 0) (numeral bin))"
huffman@47108
  2561
  unfolding word_numeral_alt by (rule set_bit_word_of_int)
huffman@47108
  2562
huffman@47108
  2563
lemma word_set_neg_numeral [simp]:
huffman@47108
  2564
  "set_bit (neg_numeral bin::'a::len0 word) n b = 
huffman@47108
  2565
    word_of_int (bin_sc n (if b then 1 else 0) (neg_numeral bin))"
huffman@47108
  2566
  unfolding word_neg_numeral_alt by (rule set_bit_word_of_int)
huffman@46173
  2567
huffman@46173
  2568
lemma word_set_bit_0 [simp]:
huffman@46173
  2569
  "set_bit 0 n b = word_of_int (bin_sc n (if b then 1 else 0) 0)"
huffman@46173
  2570
  unfolding word_0_wi by (rule set_bit_word_of_int)
huffman@46173
  2571
huffman@46173
  2572
lemma word_set_bit_1 [simp]:
huffman@46173
  2573
  "set_bit 1 n b = word_of_int (bin_sc n (if b then 1 else 0) 1)"
huffman@46173
  2574
  unfolding word_1_wi by (rule set_bit_word_of_int)
haftmann@37660
  2575
huffman@45805
  2576
lemma setBit_no [simp]:
huffman@47108
  2577
  "setBit (numeral bin) n = word_of_int (bin_sc n 1 (numeral bin))"
huffman@45805
  2578
  by (simp add: setBit_def)
huffman@45805
  2579
huffman@45805
  2580
lemma clearBit_no [simp]:
huffman@47108
  2581
  "clearBit (numeral bin) n = word_of_int (bin_sc n 0 (numeral bin))"
huffman@45805
  2582
  by (simp add: clearBit_def)
haftmann@37660
  2583
haftmann@37660
  2584
lemma to_bl_n1: 
haftmann@37660
  2585
  "to_bl (-1::'a::len0 word) = replicate (len_of TYPE ('a)) True"
haftmann@37660
  2586
  apply (rule word_bl.Abs_inverse')
haftmann@37660
  2587
   apply simp
haftmann@37660
  2588
  apply (rule word_eqI)
huffman@45805
  2589
  apply (clarsimp simp add: word_size)
haftmann@37660
  2590
  apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size)
haftmann@37660
  2591
  done
haftmann@37660
  2592
huffman@45805
  2593
lemma word_msb_n1 [simp]: "msb (-1::'a::len word)"
wenzelm@41550
  2594
  unfolding word_msb_alt to_bl_n1 by simp
haftmann@37660
  2595
haftmann@37660
  2596
lemma word_set_nth_iff: 
haftmann@37660
  2597
  "(set_bit w n b = w) = (w !! n = b | n >= size (w::'a::len0 word))"
haftmann@37660
  2598
  apply (rule iffI)
haftmann@37660
  2599
   apply (rule disjCI)
haftmann@37660
  2600
   apply (drule word_eqD)
haftmann@37660
  2601
   apply (erule sym [THEN trans])
haftmann@37660
  2602
   apply (simp add: test_bit_set)
haftmann@37660
  2603
  apply (erule disjE)
haftmann@37660
  2604
   apply clarsimp
haftmann@37660
  2605
  apply (rule word_eqI)
haftmann@37660
  2606
  apply (clarsimp simp add : test_bit_set_gen)
haftmann@37660
  2607
  apply (drule test_bit_size)
haftmann@37660
  2608
  apply force
haftmann@37660
  2609
  done
haftmann@37660
  2610
huffman@45811
  2611
lemma test_bit_2p:
huffman@45811
  2612
  "(word_of_int (2 ^ n)::'a::len word) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a)"
huffman@45811
  2613
  unfolding word_test_bit_def
haftmann@37660
  2614
  by (auto simp add: word_ubin.eq_norm nth_bintr nth_2p_bin)
haftmann@37660
  2615
haftmann@37660
  2616
lemma nth_w2p:
haftmann@37660
  2617
  "((2\<Colon>'a\<Colon>len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a\<Colon>len)"
haftmann@37660
  2618
  unfolding test_bit_2p [symmetric] word_of_int [symmetric]
haftmann@37660
  2619
  by (simp add:  of_int_power)
haftmann@37660
  2620
haftmann@37660
  2621
lemma uint_2p: 
haftmann@40827
  2622
  "(0::'a::len word) < 2 ^ n \<Longrightarrow> uint (2 ^ n::'a::len word) = 2 ^ n"
haftmann@37660
  2623
  apply (unfold word_arith_power_alt)