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permissions  rwrr 
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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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"~~/src/HOL/Library/Boolean_Algebra" 
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Bits_Bit 
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Bool_List_Representation 
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Misc_Typedef 
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Word_Miscellaneous 
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begin 
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text {* See @{file "Examples/WordExamples.thy"} for examples. *} 
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subsection {* Type definition *} 

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typedef '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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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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definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" 
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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 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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then have "PROP P (word_of_int (uint x))" . 
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find_theorems word_of_int uint 
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then show "PROP P x" by (simp add: word_of_int_uint) 
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qed 
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subsection {* Type conversions and casting *} 
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definition sint :: "'a::len word \<Rightarrow> int" 
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where 
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 {* treats the mostsignificantbit 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 \<Rightarrow> nat" 
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where 
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"unat w = nat (uint w)" 
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definition uints :: "nat \<Rightarrow> int set" 
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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 \<Rightarrow> int set" 
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where 
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"sints n = range (sbintrunc (n  1))" 
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lemma uints_num: 
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"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: 
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"sints n = {i.  (2 ^ (n  1)) \<le> i \<and> i < 2 ^ (n  1)}" 
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by (simp add: sints_def range_sbintrunc) 
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definition unats :: "nat \<Rightarrow> nat set" 
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where 
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"unats n = {i. i < 2 ^ n}" 
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definition norm_sint :: "nat \<Rightarrow> int \<Rightarrow> int" 
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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 \<Rightarrow> 'b::len word" 
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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 \<Rightarrow> 'b::len0 word" 
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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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lemma word_size_gt_0 [iff]: 
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"0 < size (w::'a::len word)" 
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by (simp add: word_size) 
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lemmas lens_gt_0 = word_size_gt_0 len_gt_0 
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lemma lens_not_0 [iff]: 
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shows "size (w::'a::len word) \<noteq> 0" 
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and "len_of TYPE('a::len) \<noteq> 0" 
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by auto 
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definition source_size :: "('a::len0 word \<Rightarrow> 'b) \<Rightarrow> nat" 
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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 \<Rightarrow> 'b::len0 word) \<Rightarrow> nat" 
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where 
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"target_size c = size (c undefined)" 
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definition is_up :: "('a::len0 word \<Rightarrow> 'b::len0 word) \<Rightarrow> bool" 
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where 
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"is_up c \<longleftrightarrow> source_size c \<le> target_size c" 
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definition is_down :: "('a :: len0 word \<Rightarrow> 'b :: len0 word) \<Rightarrow> bool" 
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where 
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"is_down c \<longleftrightarrow> target_size c \<le> source_size c" 
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definition of_bl :: "bool list \<Rightarrow> 'a::len0 word" 
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where 
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"of_bl bl = word_of_int (bl_to_bin bl)" 
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definition to_bl :: "'a::len0 word \<Rightarrow> bool list" 
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where 
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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 \<Rightarrow> 'a word" 
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where 
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"word_reverse w = of_bl (rev (to_bl w))" 
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definition word_int_case :: "(int \<Rightarrow> 'b) \<Rightarrow> 'a::len0 word \<Rightarrow> 'b" 
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where 
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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 {* Correspondence relation for theorem transfer *} 
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definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool" 
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where 
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"cr_word = (\<lambda>x y. word_of_int x = y)" 
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lemma Quotient_word: 
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"Quotient (\<lambda>x y. bintrunc (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: no_bintr_alt1 word_of_int_uint) (simp add: word_of_int_def Abs_word_inject) 
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lemma reflp_word: 
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"reflp (\<lambda>x y. bintrunc (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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setup_lifting (no_code) Quotient_word reflp_word 
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text {* TODO: The next lemma could be generated automatically. *} 
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lemma uint_transfer [transfer_rule]: 
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"(fun_rel pcr_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 word.pcr_cr_eq cr_word_def 
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by (simp add: no_bintr_alt1 uint_word_of_int) 
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subsection {* 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" 
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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 (int_of_integer (integer_of_natural k)) :: 'a word 
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in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))" 
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instance .. 
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end 
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no_notation fcomp (infixl "\<circ>>" 60) 
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no_notation scomp (infixl "\<circ>\<rightarrow>" 60) 
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subsection {* Typedefinition locale instantiations *} 
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lemmas uint_0 = uint_nonnegative (* FIXME duplicate *) 
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lemmas uint_lt = uint_bounded (* FIXME duplicate *) 
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lemmas uint_mod_same = uint_idem (* FIXME duplicate *) 
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lemma td_ext_uint: 
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"td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0))) 
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(\<lambda>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 (fact td_ext_uint) 
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lemmas td_uint = word_uint.td_thm 
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lemmas int_word_uint = word_uint.eq_norm 
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lemma td_ext_ubin: 
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"td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0))) 
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(bintrunc (len_of TYPE('a)))" 
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by (unfold no_bintr_alt1) (fact td_ext_uint) 
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interpretation word_ubin: 
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td_ext "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 (fact td_ext_ubin) 
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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}" 
37660  292 
begin 
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294 
lift_definition zero_word :: "'a word" is "0" . 
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295 

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296 
lift_definition one_word :: "'a word" is "1" . 
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297 

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lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op +" 
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299 
by (metis bintr_ariths(2)) 
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300 

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301 
lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op " 
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302 
by (metis bintr_ariths(3)) 
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303 

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304 
lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus 
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305 
by (metis bintr_ariths(5)) 
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306 

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307 
lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op *" 
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308 
by (metis bintr_ariths(4)) 
37660  309 

310 
definition 

311 
word_div_def: "a div b = word_of_int (uint a div uint b)" 

312 

313 
definition 

314 
word_mod_def: "a mod b = word_of_int (uint a mod uint b)" 

315 

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316 
instance 
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317 
by default (transfer, simp add: algebra_simps)+ 
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318 

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319 
end 
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320 

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321 
text {* Legacy theorems: *} 
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322 

47611  323 
lemma word_arith_wis [code]: shows 
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324 
word_add_def: "a + b = word_of_int (uint a + uint b)" and 
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325 
word_sub_wi: "a  b = word_of_int (uint a  uint b)" and 
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326 
word_mult_def: "a * b = word_of_int (uint a * uint b)" and 
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327 
word_minus_def: " a = word_of_int ( uint a)" and 
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328 
word_succ_alt: "word_succ a = word_of_int (uint a + 1)" and 
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329 
word_pred_alt: "word_pred a = word_of_int (uint a  1)" and 
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330 
word_0_wi: "0 = word_of_int 0" and 
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331 
word_1_wi: "1 = word_of_int 1" 
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332 
unfolding plus_word_def minus_word_def times_word_def uminus_word_def 
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333 
unfolding word_succ_def word_pred_def zero_word_def one_word_def 
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334 
by simp_all 
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335 

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336 
lemmas arths = 
45604  337 
bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm] 
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338 

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339 
lemma wi_homs: 
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340 
shows 
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341 
wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and 
46013  342 
wi_hom_sub: "word_of_int a  word_of_int b = word_of_int (a  b)" and 
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343 
wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and 
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344 
wi_hom_neg: " word_of_int a = word_of_int ( a)" and 
46000  345 
wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and 
346 
wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a  1)" 

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347 
by (transfer, simp)+ 
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348 

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349 
lemmas wi_hom_syms = wi_homs [symmetric] 
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350 

46013  351 
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi 
46009  352 

353 
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric] 

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354 

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355 
instance word :: (len) comm_ring_1 
45810  356 
proof 
357 
have "0 < len_of TYPE('a)" by (rule len_gt_0) 

358 
then show "(0::'a word) \<noteq> 1" 

47372  359 
by  (transfer, auto simp add: gr0_conv_Suc) 
45810  360 
qed 
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361 

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362 
lemma word_of_nat: "of_nat n = word_of_int (int n)" 
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363 
by (induct n) (auto simp add : word_of_int_hom_syms) 
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364 

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365 
lemma word_of_int: "of_int = word_of_int" 
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366 
apply (rule ext) 
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367 
apply (case_tac x rule: int_diff_cases) 
46013  368 
apply (simp add: word_of_nat wi_hom_sub) 
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369 
done 
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370 

54848  371 
definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) 
372 
where 

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373 
"a udvd b = (EX n>=0. uint b = n * uint a)" 
37660  374 

45547  375 

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376 
subsection {* Ordering *} 
45547  377 

378 
instantiation word :: (len0) linorder 

379 
begin 

380 

37660  381 
definition 
382 
word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b" 

383 

384 
definition 

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385 
word_less_def: "a < b \<longleftrightarrow> uint a < uint b" 
37660  386 

45547  387 
instance 
388 
by default (auto simp: word_less_def word_le_def) 

389 

390 
end 

391 

54848  392 
definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) 
393 
where 

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394 
"a <=s b = (sint a <= sint b)" 
37660  395 

54848  396 
definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) 
397 
where 

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398 
"(x <s y) = (x <=s y & x ~= y)" 
37660  399 

400 

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401 
subsection {* Bitwise operations *} 
37660  402 

403 
instantiation word :: (len0) bits 

404 
begin 

405 

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406 
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT 
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407 
by (metis bin_trunc_not) 
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408 

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409 
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND 
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410 
by (metis bin_trunc_and) 
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411 

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412 
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR 
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413 
by (metis bin_trunc_or) 
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414 

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415 
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR 
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416 
by (metis bin_trunc_xor) 
37660  417 

418 
definition 

419 
word_test_bit_def: "test_bit a = bin_nth (uint a)" 

420 

421 
definition 

422 
word_set_bit_def: "set_bit a n x = 

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423 
word_of_int (bin_sc n x (uint a))" 
37660  424 

425 
definition 

426 
word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)" 

427 

428 
definition 

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429 
word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a)" 
37660  430 

54848  431 
definition shiftl1 :: "'a word \<Rightarrow> 'a word" 
432 
where 

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433 
"shiftl1 w = word_of_int (uint w BIT False)" 
37660  434 

54848  435 
definition shiftr1 :: "'a word \<Rightarrow> 'a word" 
436 
where 

37660  437 
 "shift right as unsigned or as signed, ie logical or arithmetic" 
438 
"shiftr1 w = word_of_int (bin_rest (uint w))" 

439 

440 
definition 

441 
shiftl_def: "w << n = (shiftl1 ^^ n) w" 

442 

443 
definition 

444 
shiftr_def: "w >> n = (shiftr1 ^^ n) w" 

445 

446 
instance .. 

447 

448 
end 

449 

47611  450 
lemma [code]: shows 
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451 
word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" and 
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452 
word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" and 
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453 
word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" and 
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454 
word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)" 
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455 
unfolding bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def 
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456 
by simp_all 
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457 

37660  458 
instantiation word :: (len) bitss 
459 
begin 

460 

461 
definition 

462 
word_msb_def: 

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463 
"msb a \<longleftrightarrow> bin_sign (sint a) = 1" 
37660  464 

465 
instance .. 

466 

467 
end 

468 

54848  469 
definition setBit :: "'a :: len0 word => nat => 'a word" 
470 
where 

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471 
"setBit w n = set_bit w n True" 
37660  472 

54848  473 
definition clearBit :: "'a :: len0 word => nat => 'a word" 
474 
where 

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475 
"clearBit w n = set_bit w n False" 
37660  476 

477 

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478 
subsection {* Shift operations *} 
37660  479 

54848  480 
definition sshiftr1 :: "'a :: len word => 'a word" 
481 
where 

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482 
"sshiftr1 w = word_of_int (bin_rest (sint w))" 
37660  483 

54848  484 
definition bshiftr1 :: "bool => 'a :: len word => 'a word" 
485 
where 

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486 
"bshiftr1 b w = of_bl (b # butlast (to_bl w))" 
37660  487 

54848  488 
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) 
489 
where 

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490 
"w >>> n = (sshiftr1 ^^ n) w" 
37660  491 

54848  492 
definition mask :: "nat => 'a::len word" 
493 
where 

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494 
"mask n = (1 << n)  1" 
37660  495 

54848  496 
definition revcast :: "'a :: len0 word => 'b :: len0 word" 
497 
where 

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498 
"revcast w = of_bl (takefill False (len_of TYPE('b)) (to_bl w))" 
37660  499 

54848  500 
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" 
501 
where 

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502 
"slice1 n w = of_bl (takefill False n (to_bl w))" 
37660  503 

54848  504 
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" 
505 
where 

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506 
"slice n w = slice1 (size w  n) w" 
37660  507 

508 

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509 
subsection {* Rotation *} 
37660  510 

54848  511 
definition rotater1 :: "'a list => 'a list" 
512 
where 

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513 
"rotater1 ys = 
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514 
(case ys of [] => []  x # xs => last ys # butlast ys)" 
37660  515 

54848  516 
definition rotater :: "nat => 'a list => 'a list" 
517 
where 

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518 
"rotater n = rotater1 ^^ n" 
37660  519 

54848  520 
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" 
521 
where 

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522 
"word_rotr n w = of_bl (rotater n (to_bl w))" 
37660  523 

54848  524 
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" 
525 
where 

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526 
"word_rotl n w = of_bl (rotate n (to_bl w))" 
37660  527 

54848  528 
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" 
529 
where 

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530 
"word_roti i w = (if i >= 0 then word_rotr (nat i) w 
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531 
else word_rotl (nat ( i)) w)" 
37660  532 

533 

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534 
subsection {* Split and cat operations *} 
37660  535 

54848  536 
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" 
537 
where 

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538 
"word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))" 
37660  539 

54848  540 
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" 
541 
where 

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542 
"word_split a = 
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543 
(case bin_split (len_of TYPE ('c)) (uint a) of 
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544 
(u, v) => (word_of_int u, word_of_int v))" 
37660  545 

54848  546 
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" 
547 
where 

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548 
"word_rcat ws = 
37660  549 
word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))" 
550 

54848  551 
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" 
552 
where 

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553 
"word_rsplit w = 
37660  554 
map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))" 
555 

54848  556 
definition max_word :: "'a::len word"  "Largest representable machine integer." 
557 
where 

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558 
"max_word = word_of_int (2 ^ len_of TYPE('a)  1)" 
37660  559 

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560 
lemmas of_nth_def = word_set_bits_def (* FIXME duplicate *) 
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561 

37660  562 

46010  563 
subsection {* Theorems about typedefs *} 
564 

37660  565 
lemma sint_sbintrunc': 
566 
"sint (word_of_int bin :: 'a word) = 

567 
(sbintrunc (len_of TYPE ('a :: len)  1) bin)" 

568 
unfolding sint_uint 

569 
by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt) 

570 

571 
lemma uint_sint: 

572 
"uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))" 

573 
unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le) 

574 

46057  575 
lemma bintr_uint: 
576 
fixes w :: "'a::len0 word" 

577 
shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w" 

37660  578 
apply (subst word_ubin.norm_Rep [symmetric]) 
579 
apply (simp only: bintrunc_bintrunc_min word_size) 

54863
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580 
apply (simp add: min.absorb2) 
37660  581 
done 
582 

46057  583 
lemma wi_bintr: 
584 
"len_of TYPE('a::len0) \<le> n \<Longrightarrow> 

585 
word_of_int (bintrunc n w) = (word_of_int w :: 'a word)" 

54863
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586 
by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min.absorb1) 
37660  587 

588 
lemma td_ext_sbin: 

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589 
"td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len))) 
37660  590 
(sbintrunc (len_of TYPE('a)  1))" 
591 
apply (unfold td_ext_def' sint_uint) 

592 
apply (simp add : word_ubin.eq_norm) 

593 
apply (cases "len_of TYPE('a)") 

594 
apply (auto simp add : sints_def) 

595 
apply (rule sym [THEN trans]) 

596 
apply (rule word_ubin.Abs_norm) 

597 
apply (simp only: bintrunc_sbintrunc) 

598 
apply (drule sym) 

599 
apply simp 

600 
done 

601 

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602 
lemma td_ext_sint: 
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603 
"td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len))) 
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604 
(\<lambda>w. (w + 2 ^ (len_of TYPE('a)  1)) mod 2 ^ len_of TYPE('a)  
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605 
2 ^ (len_of TYPE('a)  1))" 
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606 
using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2) 
37660  607 

608 
(* We do sint before sbin, before sint is the user version 

609 
and interpretations do not produce thm duplicates. I.e. 

610 
we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, 

611 
because the latter is the same thm as the former *) 

612 
interpretation word_sint: 

613 
td_ext "sint ::'a::len word => int" 

614 
word_of_int 

615 
"sints (len_of TYPE('a::len))" 

616 
"%w. (w + 2^(len_of TYPE('a::len)  1)) mod 2^len_of TYPE('a::len)  

617 
2 ^ (len_of TYPE('a::len)  1)" 

618 
by (rule td_ext_sint) 

619 

620 
interpretation word_sbin: 

621 
td_ext "sint ::'a::len word => int" 

622 
word_of_int 

623 
"sints (len_of TYPE('a::len))" 

624 
"sbintrunc (len_of TYPE('a::len)  1)" 

625 
by (rule td_ext_sbin) 

626 

45604  627 
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm] 
37660  628 

629 
lemmas td_sint = word_sint.td 

630 

631 
lemma to_bl_def': 

632 
"(to_bl :: 'a :: len0 word => bool list) = 

633 
bin_to_bl (len_of TYPE('a)) o uint" 

44762  634 
by (auto simp: to_bl_def) 
37660  635 

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636 
lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w 
37660  637 

45805  638 
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)" 
639 
by (fact uints_def [unfolded no_bintr_alt1]) 

640 

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641 
lemma word_numeral_alt: 
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642 
"numeral b = word_of_int (numeral b)" 
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643 
by (induct b, simp_all only: numeral.simps word_of_int_homs) 
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644 

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645 
declare word_numeral_alt [symmetric, code_abbrev] 
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646 

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647 
lemma word_neg_numeral_alt: 
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648 
" numeral b = word_of_int ( numeral b)" 
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649 
by (simp only: word_numeral_alt wi_hom_neg) 
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650 

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651 
declare word_neg_numeral_alt [symmetric, code_abbrev] 
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652 

47372  653 
lemma word_numeral_transfer [transfer_rule]: 
51375
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654 
"(fun_rel op = pcr_word) numeral numeral" 
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655 
"(fun_rel op = pcr_word) ( numeral) ( numeral)" 
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656 
apply (simp_all add: fun_rel_def word.pcr_cr_eq cr_word_def) 
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657 
using word_numeral_alt [symmetric] word_neg_numeral_alt [symmetric] by blast+ 
47372  658 

45805  659 
lemma uint_bintrunc [simp]: 
47108
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660 
"uint (numeral bin :: 'a word) = 
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661 
bintrunc (len_of TYPE ('a :: len0)) (numeral bin)" 
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662 
unfolding word_numeral_alt by (rule word_ubin.eq_norm) 
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663 

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664 
lemma uint_bintrunc_neg [simp]: "uint ( numeral bin :: 'a word) = 
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665 
bintrunc (len_of TYPE ('a :: len0)) ( numeral bin)" 
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666 
by (simp only: word_neg_numeral_alt word_ubin.eq_norm) 
37660  667 

45805  668 
lemma sint_sbintrunc [simp]: 
47108
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669 
"sint (numeral bin :: 'a word) = 
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670 
sbintrunc (len_of TYPE ('a :: len)  1) (numeral bin)" 
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671 
by (simp only: word_numeral_alt word_sbin.eq_norm) 
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changeset

672 

54489
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673 
lemma sint_sbintrunc_neg [simp]: "sint ( numeral bin :: 'a word) = 
03ff4d1e6784
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674 
sbintrunc (len_of TYPE ('a :: len)  1) ( numeral bin)" 
47108
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changeset

675 
by (simp only: word_neg_numeral_alt word_sbin.eq_norm) 
37660  676 

45805  677 
lemma unat_bintrunc [simp]: 
47108
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678 
"unat (numeral bin :: 'a :: len0 word) = 
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679 
nat (bintrunc (len_of TYPE('a)) (numeral bin))" 
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680 
by (simp only: unat_def uint_bintrunc) 
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changeset

681 

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682 
lemma unat_bintrunc_neg [simp]: 
54489
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683 
"unat ( numeral bin :: 'a :: len0 word) = 
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684 
nat (bintrunc (len_of TYPE('a)) ( numeral bin))" 
47108
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changeset

685 
by (simp only: unat_def uint_bintrunc_neg) 
37660  686 

40827
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687 
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w" 
37660  688 
apply (unfold word_size) 
689 
apply (rule word_uint.Rep_eqD) 

690 
apply (rule box_equals) 

691 
defer 

692 
apply (rule word_ubin.norm_Rep)+ 

693 
apply simp 

694 
done 

695 

45805  696 
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)" 
697 
using word_uint.Rep [of x] by (simp add: uints_num) 

698 

699 
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)" 

700 
using word_uint.Rep [of x] by (simp add: uints_num) 

701 

702 
lemma sint_ge: " (2 ^ (len_of TYPE('a)  1)) \<le> sint (x::'a::len word)" 

703 
using word_sint.Rep [of x] by (simp add: sints_num) 

704 

705 
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a)  1)" 

706 
using word_sint.Rep [of x] by (simp add: sints_num) 

37660  707 

708 
lemma sign_uint_Pls [simp]: 

46604
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diff
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709 
"bin_sign (uint x) = 0" 
47108
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changeset

710 
by (simp add: sign_Pls_ge_0) 
37660  711 

45805  712 
lemma uint_m2p_neg: "uint (x::'a::len0 word)  2 ^ len_of TYPE('a) < 0" 
713 
by (simp only: diff_less_0_iff_less uint_lt2p) 

714 

715 
lemma uint_m2p_not_non_neg: 

716 
"\<not> 0 \<le> uint (x::'a::len0 word)  2 ^ len_of TYPE('a)" 

717 
by (simp only: not_le uint_m2p_neg) 

37660  718 

719 
lemma lt2p_lem: 

55816
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720 
"len_of TYPE('a) \<le> n \<Longrightarrow> uint (w :: 'a::len0 word) < 2 ^ n" 
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changeset

721 
by (metis bintr_uint bintrunc_mod2p int_mod_lem zless2p) 
37660  722 

45805  723 
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0" 
724 
by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1]) 

37660  725 

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726 
lemma uint_nat: "uint w = int (unat w)" 
37660  727 
unfolding unat_def by auto 
728 

47108
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729 
lemma uint_numeral: 
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730 
"uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)" 
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731 
unfolding word_numeral_alt 
37660  732 
by (simp only: int_word_uint) 
733 

47108
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734 
lemma uint_neg_numeral: 
54489
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735 
"uint ( numeral b :: 'a :: len0 word) =  numeral b mod 2 ^ len_of TYPE('a)" 
47108
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changeset

736 
unfolding word_neg_numeral_alt 
2a1953f0d20d
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changeset

737 
by (simp only: int_word_uint) 
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changeset

738 

2a1953f0d20d
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739 
lemma unat_numeral: 
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740 
"unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)" 
37660  741 
apply (unfold unat_def) 
47108
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742 
apply (clarsimp simp only: uint_numeral) 
37660  743 
apply (rule nat_mod_distrib [THEN trans]) 
47108
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744 
apply (rule zero_le_numeral) 
37660  745 
apply (simp_all add: nat_power_eq) 
746 
done 

747 

47108
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748 
lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + 
37660  749 
2 ^ (len_of TYPE('a)  1)) mod 2 ^ len_of TYPE('a)  
750 
2 ^ (len_of TYPE('a)  1)" 

47108
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751 
unfolding word_numeral_alt by (rule int_word_sint) 
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changeset

752 

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753 
lemma word_of_int_0 [simp, code_post]: 
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754 
"word_of_int 0 = 0" 
45958  755 
unfolding word_0_wi .. 
756 

55816
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757 
lemma word_of_int_1 [simp, code_post]: 
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758 
"word_of_int 1 = 1" 
45958  759 
unfolding word_1_wi .. 
760 

54489
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761 
lemma word_of_int_neg_1 [simp]: "word_of_int ( 1) =  1" 
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762 
by (simp add: wi_hom_syms) 
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changeset

763 

47108
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changeset

764 
lemma word_of_int_numeral [simp] : 
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diff
changeset

765 
"(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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diff
changeset

766 
unfolding word_numeral_alt .. 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset

767 

2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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diff
changeset

768 
lemma word_of_int_neg_numeral [simp]: 
54489
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eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

769 
"(word_of_int ( numeral bin) :: 'a :: len0 word) = ( numeral bin)" 
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

770 
unfolding word_numeral_alt wi_hom_syms .. 
37660  771 

772 
lemma word_int_case_wi: 

773 
"word_int_case f (word_of_int i :: 'b word) = 

774 
f (i mod 2 ^ len_of TYPE('b::len0))" 

775 
unfolding word_int_case_def by (simp add: word_uint.eq_norm) 

776 

777 
lemma word_int_split: 

778 
"P (word_int_case f x) = 

779 
(ALL i. x = (word_of_int i :: 'b :: len0 word) & 

780 
0 <= i & i < 2 ^ len_of TYPE('b) > P (f i))" 

781 
unfolding word_int_case_def 

55816
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55415
diff
changeset

782 
by (auto simp: word_uint.eq_norm mod_pos_pos_trivial) 
37660  783 

784 
lemma word_int_split_asm: 

785 
"P (word_int_case f x) = 

786 
(~ (EX n. x = (word_of_int n :: 'b::len0 word) & 

787 
0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))" 

788 
unfolding word_int_case_def 

55816
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cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset

789 
by (auto simp: word_uint.eq_norm mod_pos_pos_trivial) 
45805  790 

45604  791 
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq] 
792 
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq] 

37660  793 

794 
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w" 

795 
unfolding word_size by (rule uint_range') 

796 

797 
lemma sint_range_size: 

798 
" (2 ^ (size w  Suc 0)) <= sint w & sint w < 2 ^ (size w  Suc 0)" 

799 
unfolding word_size by (rule sint_range') 

800 

45805  801 
lemma sint_above_size: "2 ^ (size (w::'a::len word)  1) \<le> x \<Longrightarrow> sint w < x" 
802 
unfolding word_size by (rule less_le_trans [OF sint_lt]) 

803 

804 
lemma sint_below_size: 

805 
"x \<le>  (2 ^ (size (w::'a::len word)  1)) \<Longrightarrow> x \<le> sint w" 

806 
unfolding word_size by (rule order_trans [OF _ sint_ge]) 

37660  807 

55816
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cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset

808 

46010  809 
subsection {* Testing bits *} 
810 

37660  811 
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)" 
812 
unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) 

813 

814 
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n > n < size w" 

815 
apply (unfold word_test_bit_def) 

816 
apply (subst word_ubin.norm_Rep [symmetric]) 

817 
apply (simp only: nth_bintr word_size) 

818 
apply fast 

819 
done 

820 

46021  821 
lemma word_eq_iff: 
822 
fixes x y :: "'a::len0 word" 

823 
shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)" 

824 
unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric] 

825 
by (metis test_bit_size [unfolded word_size]) 

826 

46023
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset

827 
lemma word_eqI [rule_format]: 
37660  828 
fixes u :: "'a::len0 word" 
40827
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code preprocessor setup for numerals on word type;
haftmann
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39910
diff
changeset

829 
shows "(ALL n. n < size u > u !! n = v !! n) \<Longrightarrow> u = v" 
46021  830 
by (simp add: word_size word_eq_iff) 
37660  831 

45805  832 
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x" 
833 
by simp 

37660  834 

835 
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)" 

836 
unfolding word_test_bit_def word_size 

837 
by (simp add: nth_bintr [symmetric]) 

838 

839 
lemmas test_bit_bin = test_bit_bin' [unfolded word_size] 

840 

46057  841 
lemma bin_nth_uint_imp: 
842 
"bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)" 

37660  843 
apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) 
844 
apply (subst word_ubin.norm_Rep) 

845 
apply assumption 

846 
done 

847 

46057  848 
lemma bin_nth_sint: 
849 
fixes w :: "'a::len word" 

850 
shows "len_of TYPE('a) \<le> n \<Longrightarrow> 

851 
bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a)  1)" 

37660  852 
apply (subst word_sbin.norm_Rep [symmetric]) 
46057  853 
apply (auto simp add: nth_sbintr) 
37660  854 
done 
855 

856 
(* type definitions theorem for in terms of equivalent bool list *) 

857 
lemma td_bl: 

858 
"type_definition (to_bl :: 'a::len0 word => bool list) 

859 
of_bl 

860 
{bl. length bl = len_of TYPE('a)}" 

861 
apply (unfold type_definition_def of_bl_def to_bl_def) 

862 
apply (simp add: word_ubin.eq_norm) 

863 
apply safe 

864 
apply (drule sym) 

865 
apply simp 

866 
done 

867 

868 
interpretation word_bl: 

869 
type_definition "to_bl :: 'a::len0 word => bool list" 

870 
of_bl 

871 
"{bl. length bl = len_of TYPE('a::len0)}" 

55816
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cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset

872 
by (fact td_bl) 
37660  873 

45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

874 
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff] 
45538
1fffa81b9b83
eliminated slightly odd Rep' with dynamicallyscoped [simplified];
wenzelm
parents:
45529
diff
changeset

875 

40827
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code preprocessor setup for numerals on word type;
haftmann
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39910
diff
changeset

876 
lemma word_size_bl: "size w = size (to_bl w)" 
37660  877 
unfolding word_size by auto 
878 

879 
lemma to_bl_use_of_bl: 

880 
"(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))" 

45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

881 
by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq]) 
37660  882 

883 
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" 

884 
unfolding word_reverse_def by (simp add: word_bl.Abs_inverse) 

885 

886 
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" 

887 
unfolding word_reverse_def by (simp add : word_bl.Abs_inverse) 

888 

40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

889 
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w" 
47108
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merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

890 
by (metis word_rev_rev) 
37660  891 

45805  892 
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u" 
893 
by simp 

894 

895 
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))" 

896 
unfolding word_bl_Rep' by (rule len_gt_0) 

897 

898 
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []" 

899 
by (fact length_bl_gt_0 [unfolded length_greater_0_conv]) 

900 

901 
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0" 

902 
by (fact length_bl_gt_0 [THEN gr_implies_not0]) 

37660  903 

46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset

904 
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = 1)" 
37660  905 
apply (unfold to_bl_def sint_uint) 
906 
apply (rule trans [OF _ bl_sbin_sign]) 

907 
apply simp 

908 
done 

909 

910 
lemma of_bl_drop': 

40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

911 
"lend = length bl  len_of TYPE ('a :: len0) \<Longrightarrow> 
37660  912 
of_bl (drop lend bl) = (of_bl bl :: 'a word)" 
913 
apply (unfold of_bl_def) 

914 
apply (clarsimp simp add : trunc_bl2bin [symmetric]) 

915 
done 

916 

917 
lemma test_bit_of_bl: 

918 
"(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)" 

919 
apply (unfold of_bl_def word_test_bit_def) 

920 
apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) 

921 
done 

922 

923 
lemma no_of_bl: 

47108
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merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

924 
"(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) (numeral bin))" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

925 
unfolding of_bl_def by simp 
37660  926 

40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

927 
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)" 
37660  928 
unfolding word_size to_bl_def by auto 
929 

930 
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" 

931 
unfolding uint_bl by (simp add : word_size) 

932 

933 
lemma to_bl_of_bin: 

934 
"to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" 

935 
unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size) 

936 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

937 
lemma to_bl_numeral [simp]: 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46962
diff
changeset

938 
"to_bl (numeral bin::'a::len0 word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

939 
bin_to_bl (len_of TYPE('a)) (numeral bin)" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

940 
unfolding word_numeral_alt by (rule to_bl_of_bin) 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

941 

2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

942 
lemma to_bl_neg_numeral [simp]: 
54489
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

943 
"to_bl ( numeral bin::'a::len0 word) = 
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

944 
bin_to_bl (len_of TYPE('a)) ( numeral bin)" 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

945 
unfolding word_neg_numeral_alt by (rule to_bl_of_bin) 
37660  946 

947 
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" 

948 
unfolding uint_bl by (simp add : word_size) 

46011  949 

950 
lemma uint_bl_bin: 

951 
fixes x :: "'a::len0 word" 

952 
shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x" 

953 
by (rule trans [OF bin_bl_bin word_ubin.norm_Rep]) 

45604  954 

37660  955 
(* naturals *) 
956 
lemma uints_unats: "uints n = int ` unats n" 

957 
apply (unfold unats_def uints_num) 

958 
apply safe 

959 
apply (rule_tac image_eqI) 

960 
apply (erule_tac nat_0_le [symmetric]) 

961 
apply auto 

962 
apply (erule_tac nat_less_iff [THEN iffD2]) 

963 
apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) 

964 
apply (auto simp add : nat_power_eq int_power) 

965 
done 

966 

967 
lemma unats_uints: "unats n = nat ` uints n" 

968 
by (auto simp add : uints_unats image_iff) 

969 

46962
5bdcdb28be83
make more word theorems respect int/bin distinction
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parents:
46656
diff
changeset

970 
lemmas bintr_num = word_ubin.norm_eq_iff 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

971 
[of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

972 
lemmas sbintr_num = word_sbin.norm_eq_iff 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

973 
[of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b 
37660  974 

975 
lemma num_of_bintr': 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

976 
"bintrunc (len_of TYPE('a :: len0)) (numeral a) = (numeral b) \<Longrightarrow> 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

977 
numeral a = (numeral b :: 'a word)" 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

978 
unfolding bintr_num by (erule subst, simp) 
37660  979 

980 
lemma num_of_sbintr': 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

981 
"sbintrunc (len_of TYPE('a :: len)  1) (numeral a) = (numeral b) \<Longrightarrow> 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

982 
numeral a = (numeral b :: 'a word)" 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

983 
unfolding sbintr_num by (erule subst, simp) 
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

984 

5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

985 
lemma num_abs_bintr: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

986 
"(numeral x :: 'a word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

987 
word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

988 
by (simp only: word_ubin.Abs_norm word_numeral_alt) 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

989 

5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

990 
lemma num_abs_sbintr: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

991 
"(numeral x :: 'a word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

992 
word_of_int (sbintrunc (len_of TYPE('a::len)  1) (numeral x))" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

993 
by (simp only: word_sbin.Abs_norm word_numeral_alt) 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

994 

37660  995 
(** cast  note, no arg for new length, as it's determined by type of result, 
996 
thus in "cast w = w, the type means cast to length of w! **) 

997 

998 
lemma ucast_id: "ucast w = w" 

999 
unfolding ucast_def by auto 

1000 

1001 
lemma scast_id: "scast w = w" 

1002 
unfolding scast_def by auto 

1003 

40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1004 
lemma ucast_bl: "ucast w = of_bl (to_bl w)" 
37660  1005 
unfolding ucast_def of_bl_def uint_bl 
1006 
by (auto simp add : word_size) 

1007 

1008 
lemma nth_ucast: 

1009 
"(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))" 

1010 
apply (unfold ucast_def test_bit_bin) 

1011 
apply (simp add: word_ubin.eq_norm nth_bintr word_size) 

1012 
apply (fast elim!: bin_nth_uint_imp) 

1013 
done 

1014 

1015 
(* for literal u(s)cast *) 

1016 

46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset

1017 
lemma ucast_bintr [simp]: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1018 
"ucast (numeral w ::'a::len0 word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1019 
word_of_int (bintrunc (len_of TYPE('a)) (numeral w))" 
37660  1020 
unfolding ucast_def by simp 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1021 
(* TODO: neg_numeral *) 
37660  1022 

46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset

1023 
lemma scast_sbintr [simp]: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1024 
"scast (numeral w ::'a::len word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1025 
word_of_int (sbintrunc (len_of TYPE('a)  Suc 0) (numeral w))" 
37660  1026 
unfolding scast_def by simp 
1027 

46011  1028 
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)" 
1029 
unfolding source_size_def word_size Let_def .. 

1030 

1031 
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)" 

1032 
unfolding target_size_def word_size Let_def .. 

1033 

1034 
lemma is_down: 

1035 
fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word" 

1036 
shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)" 

1037 
unfolding is_down_def source_size target_size .. 

1038 

1039 
lemma is_up: 

1040 
fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word" 

1041 
shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)" 

1042 
unfolding is_up_def source_size target_size .. 

37660  1043 

45604  1044 
lemmas is_up_down = trans [OF is_up is_down [symmetric]] 
37660  1045 

45811  1046 
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast" 
37660  1047 
apply (unfold is_down) 
1048 
apply safe 

1049 
apply (rule ext) 

1050 
apply (unfold ucast_def scast_def uint_sint) 

1051 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1052 
apply simp 

1053 
done 

1054 

45811  1055 
lemma word_rev_tf: 
1056 
"to_bl (of_bl bl::'a::len0 word) = 

1057 
rev (takefill False (len_of TYPE('a)) (rev bl))" 

37660  1058 
unfolding of_bl_def uint_bl 
1059 
by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size) 

1060 

45811  1061 
lemma word_rep_drop: 
1062 
"to_bl (of_bl bl::'a::len0 word) = 

1063 
replicate (len_of TYPE('a)  length bl) False @ 

1064 
drop (length bl  len_of TYPE('a)) bl" 

1065 
by (simp add: word_rev_tf takefill_alt rev_take) 

37660  1066 

1067 
lemma to_bl_ucast: 

1068 
"to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 

1069 
replicate (len_of TYPE('a)  len_of TYPE('b)) False @ 

1070 
drop (len_of TYPE('b)  len_of TYPE('a)) (to_bl w)" 

1071 
apply (unfold ucast_bl) 

1072 
apply (rule trans) 

1073 
apply (rule word_rep_drop) 

1074 
apply simp 

1075 
done 

1076 

45811  1077 
lemma ucast_up_app [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1078 
"uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
37660  1079 
to_bl (uc w) = replicate n False @ (to_bl w)" 
1080 
by (auto simp add : source_size target_size to_bl_ucast) 

1081 

45811  1082 
lemma ucast_down_drop [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1083 
"uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
37660  1084 
to_bl (uc w) = drop n (to_bl w)" 
1085 
by (auto simp add : source_size target_size to_bl_ucast) 

1086 

45811  1087 
lemma scast_down_drop [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1088 
"sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
37660  1089 
to_bl (sc w) = drop n (to_bl w)" 
1090 
apply (subgoal_tac "sc = ucast") 

1091 
apply safe 

1092 
apply simp 

45811  1093 
apply (erule ucast_down_drop) 
1094 
apply (rule down_cast_same [symmetric]) 

37660  1095 
apply (simp add : source_size target_size is_down) 
1096 
done 

1097 

45811  1098 
lemma sint_up_scast [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1099 
"sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w" 
37660  1100 
apply (unfold is_up) 
1101 
apply safe 

1102 
apply (simp add: scast_def word_sbin.eq_norm) 

1103 
apply (rule box_equals) 

1104 
prefer 3 

1105 
apply (rule word_sbin.norm_Rep) 

1106 
apply (rule sbintrunc_sbintrunc_l) 

1107 
defer 

1108 
apply (subst word_sbin.norm_Rep) 

1109 
apply (rule refl) 

1110 
apply simp 

1111 
done 

1112 

45811  1113 
lemma uint_up_ucast [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1114 
"uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w" 
37660  1115 
apply (unfold is_up) 
1116 
apply safe 

1117 
apply (rule bin_eqI) 

1118 
apply (fold word_test_bit_def) 

1119 
apply (auto simp add: nth_ucast) 

1120 
apply (auto simp add: test_bit_bin) 

1121 
done 

45811  1122 

1123 
lemma ucast_up_ucast [OF refl]: 

1124 
"uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w" 

37660  1125 
apply (simp (no_asm) add: ucast_def) 
1126 
apply (clarsimp simp add: uint_up_ucast) 

1127 
done 

1128 

45811  1129 
lemma scast_up_scast [OF refl]: 
1130 
"sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w" 

37660  1131 
apply (simp (no_asm) add: scast_def) 
1132 
apply (clarsimp simp add: sint_up_scast) 

1133 
done 

1134 

45811  1135 
lemma ucast_of_bl_up [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1136 
"w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl" 
37660  1137 
by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) 
1138 

1139 
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] 

1140 
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] 

1141 

1142 
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] 

1143 
lemmas isdus = is_up_down [where c = "scast", THEN iffD2] 

1144 
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] 

1145 
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] 

1146 

1147 
lemma up_ucast_surj: 

40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1148 
"is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660  1149 
surj (ucast :: 'a word => 'b word)" 
1150 
by (rule surjI, erule ucast_up_ucast_id) 

1151 

1152 
lemma up_scast_surj: 

40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1153 
"is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660  1154 
surj (scast :: 'a word => 'b word)" 
1155 
by (rule surjI, erule scast_up_scast_id) 

1156 

1157 
lemma down_scast_inj: 

40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1158 
"is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660  1159 
inj_on (ucast :: 'a word => 'b word) A" 
1160 
by (rule inj_on_inverseI, erule scast_down_scast_id) 

1161 

1162 
lemma down_ucast_inj: 

40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1163 
"is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660  1164 
inj_on (ucast :: 'a word => 'b word) A" 
1165 
by (rule inj_on_inverseI, erule ucast_down_ucast_id) 

1166 

1167 
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" 

1168 
by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) 

45811  1169 

46646  1170 
lemma ucast_down_wi [OF refl]: 
1171 
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x" 

1172 
apply (unfold is_down) 

37660  1173 
apply (clarsimp simp add: ucast_def word_ubin.eq_norm) 
1174 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1175 
apply (erule bintrunc_bintrunc_ge) 

1176 
done 

45811  1177 

46646  1178 
lemma ucast_down_no [OF refl]: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1179 
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1180 
unfolding word_numeral_alt by clarify (rule ucast_down_wi) 
46646  1181 

45811  1182 
lemma ucast_down_bl [OF refl]: 
1183 
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl" 

46646  1184 
unfolding of_bl_def by clarify (erule ucast_down_wi) 
37660  1185 

1186 
lemmas slice_def' = slice_def [unfolded word_size] 

1187 
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] 

1188 

1189 
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def 

1190 

1191 

1192 
subsection {* Word Arithmetic *} 

1193 

1194 
lemma word_less_alt: "(a < b) = (uint a < uint b)" 

46012  1195 
unfolding word_less_def word_le_def by (simp add: less_le) 
37660  1196 

1197 
lemma signed_linorder: "class.linorder word_sle word_sless" 

46124  1198 
by default (unfold word_sle_def word_sless_def, auto) 
37660  1199 

1200 
interpretation signed: linorder "word_sle" "word_sless" 

1201 
by (rule signed_linorder) 

1202 

1203 
lemma udvdI: 

40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1204 
"0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b" 
37660  1205 
by (auto simp: udvd_def) 
1206 

47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1207 
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1208 

2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1209 
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1210 

2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1211 
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1212 

2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1213 
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1214 

2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1215 
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1216 

2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1217 
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b 
37660  1218 

54489
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

1219 
lemma word_m1_wi: " 1 = word_of_int ( 1)" 
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

1220 
using word_neg_numeral_alt [of Num.One] by simp 
37660  1221 

46648  1222 
lemma word_0_bl [simp]: "of_bl [] = 0" 
1223 
unfolding of_bl_def by simp 

37660  1224 

1225 
lemma word_1_bl: "of_bl [True] = 1" 

46648  1226 
unfolding of_bl_def by (simp add: bl_to_bin_def) 
1227 

1228 
lemma uint_eq_0 [simp]: "uint 0 = 0" 

1229 
unfolding word_0_wi word_ubin.eq_norm by simp 

37660  1230 

45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset

1231 
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0" 
46648  1232 
by (simp add: of_bl_def bl_to_bin_rep_False) 
37660  1233 

45805  1234 
lemma to_bl_0 [simp]: 
37660  1235 
"to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" 
1236 
unfolding uint_bl 

46617
8c5d10d41391
make bool list functions respect int/bin distinction
huffman
parents:
46604
diff
changeset

1237 
by (simp add: word_size bin_to_bl_zero) 
37660  1238 

1239 
lemma uint_0_iff: "(uint x = 0) = (x = 0)" 

1240 
by (auto intro!: word_uint.Rep_eqD) 

1241 

1242 
lemma unat_0_iff: "(unat x = 0) = (x = 0)" 

1243 
unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff) 

1244 

1245 
lemma unat_0 [simp]: "unat 0 = 0" 

1246 
unfolding unat_def by auto 

1247 

40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1248 
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)" 
37660  1249 
apply (unfold word_size) 
1250 
apply (rule box_equals) 

1251 
defer 

1252 
apply (rule word_uint.Rep_inverse)+ 

1253 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1254 
apply simp 

1255 
done 

1256 

45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

1257 
lemmas size_0_same = size_0_same' [unfolded word_size] 
37660  1258 

1259 
lemmas unat_eq_0 = unat_0_iff 

1260 
lemmas unat_eq_zero = unat_0_iff 

1261 

1262 
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)" 

1263 
by (auto simp: unat_0_iff [symmetric]) 

1264 

45958  1265 
lemma ucast_0 [simp]: "ucast 0 = 0" 
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset

1266 
unfolding ucast_def by simp 
45958  1267 

1268 
lemma sint_0 [simp]: "sint 0 = 0" 

1269 
unfolding sint_uint by simp 

1270 

1271 
lemma scast_0 [simp]: "scast 0 = 0" 

45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset

1272 
unfolding scast_def by simp 
37660  1273 

1274 
lemma sint_n1 [simp] : "sint 1 = 1" 

54489
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

1275 
unfolding word_m1_wi word_sbin.eq_norm by simp 
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

1276 

03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

1277 
lemma scast_n1 [simp]: "scast ( 1) =  1" 
45958  1278 
unfolding scast_def by simp 
1279 

1280 
lemma uint_1 [simp]: "uint (1::'a::len word) = 1" 

37660  1281 
unfolding word_1_wi 
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset

1282 
by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1) 
45958  1283 

1284 
lemma unat_1 [simp]: "unat (1::'a::len word) = 1" 

1285 
unfolding unat_def by simp 

1286 

1287 
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1" 

45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset

1288 
unfolding ucast_def by simp 
37660  1289 

1290 
(* now, to get the weaker results analogous to word_div/mod_def *) 

1291 

1292 
lemmas word_arith_alts = 

46000  1293 
word_sub_wi 
1294 
word_arith_wis (* FIXME: duplicate *) 

1295 

55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset

1296 

e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset

1297 
subsection {* Transferring goals from words to ints *} 
37660  1298 

1299 
lemma word_ths: 

1300 
shows 

1301 
word_succ_p1: "word_succ a = a + 1" and 

1302 
word_pred_m1: "word_pred a = a  1" and 

1303 
word_pred_succ: "word_pred (word_succ a) = a" and 

1304 
word_succ_pred: "word_succ (word_pred a) = a" and 

1305 
word_mult_succ: "word_succ a * b = b + a * b" 

47374
9475d524bafb
set up and use lift_definition for word operations
huffman
parents:
47372
diff
changeset

1306 
by (transfer, simp add: algebra_simps)+ 
37660  1307 

45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

1308 
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y" 
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

1309 
by simp 
37660  1310 

1311 
lemmas uint_word_ariths = 
