author  wenzelm 
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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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Misc_Typedef 
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"~~/src/HOL/Library/Boolean_Algebra" 
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Bool_List_Representation 
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uses 
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("~~/src/HOL/Word/Tools/smt_word.ML") 
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("~~/src/HOL/Word/Tools/word_lib.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 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 => 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 {* Typedefinition 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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223 

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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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226 
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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231 
qed 
37660  232 

47372  233 
subsection {* Correspondence relation for theorem transfer *} 
234 

235 
definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool" 

236 
where "cr_word \<equiv> (\<lambda>x y. word_of_int x = y)" 

237 

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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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243 

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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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247 

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setup_lifting(no_code) Quotient_word reflp_word 
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249 

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text {* TODO: The next lemma could be generated automatically. *} 
47372  251 

252 
lemma uint_transfer [transfer_rule]: 

253 
"(fun_rel cr_word op =) (bintrunc (len_of TYPE('a))) 

254 
(uint :: 'a::len0 word \<Rightarrow> int)" 

255 
unfolding fun_rel_def cr_word_def by (simp add: word_ubin.eq_norm) 

256 

37660  257 
subsection "Arithmetic operations" 
258 

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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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261 

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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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264 

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instantiation word :: (len0) "{neg_numeral, Divides.div, comm_monoid_mult, comm_ring}" 
37660  266 
begin 
267 

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lift_definition zero_word :: "'a word" is "0" . 
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269 

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lift_definition one_word :: "'a word" is "1" . 
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271 

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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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274 

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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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277 

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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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280 

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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)) 
37660  283 

284 
definition 

285 
word_div_def: "a div b = word_of_int (uint a div uint b)" 

286 

287 
definition 

288 
word_mod_def: "a mod b = word_of_int (uint a mod uint b)" 

289 

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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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294 

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text {* Legacy theorems: *} 
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47611  297 
lemma word_arith_wis [code]: 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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308 
by simp_all 
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309 

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lemmas arths = 
45604  311 
bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm] 
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312 

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lemma wi_homs: 
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shows 
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wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and 
46013  316 
wi_hom_sub: "word_of_int a  word_of_int b = word_of_int (a  b)" and 
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wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and 
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wi_hom_neg: " word_of_int a = word_of_int ( a)" and 
46000  319 
wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and 
320 
wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a  1)" 

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321 
by (transfer, simp)+ 
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322 

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lemmas wi_hom_syms = wi_homs [symmetric] 
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46013  325 
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi 
46009  326 

327 
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric] 

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328 

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instance word :: (len) comm_ring_1 
45810  330 
proof 
331 
have "0 < len_of TYPE('a)" by (rule len_gt_0) 

332 
then show "(0::'a word) \<noteq> 1" 

47372  333 
by  (transfer, auto simp add: gr0_conv_Suc) 
45810  334 
qed 
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335 

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lemma word_of_nat: "of_nat n = word_of_int (int n)" 
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by (induct n) (auto simp add : word_of_int_hom_syms) 
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338 

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lemma word_of_int: "of_int = word_of_int" 
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340 
apply (rule ext) 
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apply (case_tac x rule: int_diff_cases) 
46013  342 
apply (simp add: word_of_nat wi_hom_sub) 
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done 
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344 

37660  345 
definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where 
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"a udvd b = (EX n>=0. uint b = n * uint a)" 
37660  347 

45547  348 

349 
subsection "Ordering" 

350 

351 
instantiation word :: (len0) linorder 

352 
begin 

353 

37660  354 
definition 
355 
word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b" 

356 

357 
definition 

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word_less_def: "a < b \<longleftrightarrow> uint a < uint b" 
37660  359 

45547  360 
instance 
361 
by default (auto simp: word_less_def word_le_def) 

362 

363 
end 

364 

37660  365 
definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where 
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"a <=s b = (sint a <= sint b)" 
37660  367 

368 
definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where 

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"(x <s y) = (x <=s y & x ~= y)" 
37660  370 

371 

372 
subsection "Bitwise operations" 

373 

374 
instantiation word :: (len0) bits 

375 
begin 

376 

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lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT 
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378 
by (metis bin_trunc_not) 
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379 

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lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND 
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381 
by (metis bin_trunc_and) 
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382 

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lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR 
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384 
by (metis bin_trunc_or) 
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385 

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lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR 
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387 
by (metis bin_trunc_xor) 
37660  388 

389 
definition 

390 
word_test_bit_def: "test_bit a = bin_nth (uint a)" 

391 

392 
definition 

393 
word_set_bit_def: "set_bit a n x = 

394 
word_of_int (bin_sc n (If x 1 0) (uint a))" 

395 

396 
definition 

397 
word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)" 

398 

399 
definition 

400 
word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1" 

401 

402 
definition shiftl1 :: "'a word \<Rightarrow> 'a word" where 

403 
"shiftl1 w = word_of_int (uint w BIT 0)" 

404 

405 
definition shiftr1 :: "'a word \<Rightarrow> 'a word" where 

406 
 "shift right as unsigned or as signed, ie logical or arithmetic" 

407 
"shiftr1 w = word_of_int (bin_rest (uint w))" 

408 

409 
definition 

410 
shiftl_def: "w << n = (shiftl1 ^^ n) w" 

411 

412 
definition 

413 
shiftr_def: "w >> n = (shiftr1 ^^ n) w" 

414 

415 
instance .. 

416 

417 
end 

418 

47611  419 
lemma [code]: shows 
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420 
word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" and 
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word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" and 
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422 
word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" and 
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423 
word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)" 
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424 
unfolding bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def 
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425 
by simp_all 
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426 

37660  427 
instantiation word :: (len) bitss 
428 
begin 

429 

430 
definition 

431 
word_msb_def: 

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432 
"msb a \<longleftrightarrow> bin_sign (sint a) = 1" 
37660  433 

434 
instance .. 

435 

436 
end 

437 

438 
definition setBit :: "'a :: len0 word => nat => 'a word" where 

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439 
"setBit w n = set_bit w n True" 
37660  440 

441 
definition clearBit :: "'a :: len0 word => nat => 'a word" where 

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442 
"clearBit w n = set_bit w n False" 
37660  443 

444 

445 
subsection "Shift operations" 

446 

447 
definition sshiftr1 :: "'a :: len word => 'a word" where 

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448 
"sshiftr1 w = word_of_int (bin_rest (sint w))" 
37660  449 

450 
definition bshiftr1 :: "bool => 'a :: len word => 'a word" where 

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451 
"bshiftr1 b w = of_bl (b # butlast (to_bl w))" 
37660  452 

453 
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where 

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454 
"w >>> n = (sshiftr1 ^^ n) w" 
37660  455 

456 
definition mask :: "nat => 'a::len word" where 

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457 
"mask n = (1 << n)  1" 
37660  458 

459 
definition revcast :: "'a :: len0 word => 'b :: len0 word" where 

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460 
"revcast w = of_bl (takefill False (len_of TYPE('b)) (to_bl w))" 
37660  461 

462 
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where 

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463 
"slice1 n w = of_bl (takefill False n (to_bl w))" 
37660  464 

465 
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where 

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466 
"slice n w = slice1 (size w  n) w" 
37660  467 

468 

469 
subsection "Rotation" 

470 

471 
definition rotater1 :: "'a list => 'a list" where 

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472 
"rotater1 ys = 
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473 
(case ys of [] => []  x # xs => last ys # butlast ys)" 
37660  474 

475 
definition rotater :: "nat => 'a list => 'a list" where 

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476 
"rotater n = rotater1 ^^ n" 
37660  477 

478 
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where 

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479 
"word_rotr n w = of_bl (rotater n (to_bl w))" 
37660  480 

481 
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where 

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482 
"word_rotl n w = of_bl (rotate n (to_bl w))" 
37660  483 

484 
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where 

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485 
"word_roti i w = (if i >= 0 then word_rotr (nat i) w 
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486 
else word_rotl (nat ( i)) w)" 
37660  487 

488 

489 
subsection "Split and cat operations" 

490 

491 
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where 

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492 
"word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))" 
37660  493 

494 
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where 

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495 
"word_split a = 
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496 
(case bin_split (len_of TYPE ('c)) (uint a) of 
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497 
(u, v) => (word_of_int u, word_of_int v))" 
37660  498 

499 
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where 

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500 
"word_rcat ws = 
37660  501 
word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))" 
502 

503 
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where 

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504 
"word_rsplit w = 
37660  505 
map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))" 
506 

507 
definition max_word :: "'a::len word"  "Largest representable machine integer." where 

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508 
"max_word = word_of_int (2 ^ len_of TYPE('a)  1)" 
37660  509 

510 
primrec of_bool :: "bool \<Rightarrow> 'a::len word" where 

511 
"of_bool False = 0" 

512 
 "of_bool True = 1" 

513 

45805  514 
(* FIXME: only provide one theorem name *) 
37660  515 
lemmas of_nth_def = word_set_bits_def 
516 

46010  517 
subsection {* Theorems about typedefs *} 
518 

37660  519 
lemma sint_sbintrunc': 
520 
"sint (word_of_int bin :: 'a word) = 

521 
(sbintrunc (len_of TYPE ('a :: len)  1) bin)" 

522 
unfolding sint_uint 

523 
by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt) 

524 

525 
lemma uint_sint: 

526 
"uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))" 

527 
unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le) 

528 

46057  529 
lemma bintr_uint: 
530 
fixes w :: "'a::len0 word" 

531 
shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w" 

37660  532 
apply (subst word_ubin.norm_Rep [symmetric]) 
533 
apply (simp only: bintrunc_bintrunc_min word_size) 

534 
apply (simp add: min_max.inf_absorb2) 

535 
done 

536 

46057  537 
lemma wi_bintr: 
538 
"len_of TYPE('a::len0) \<le> n \<Longrightarrow> 

539 
word_of_int (bintrunc n w) = (word_of_int w :: 'a word)" 

37660  540 
by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1) 
541 

542 
lemma td_ext_sbin: 

543 
"td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 

544 
(sbintrunc (len_of TYPE('a)  1))" 

545 
apply (unfold td_ext_def' sint_uint) 

546 
apply (simp add : word_ubin.eq_norm) 

547 
apply (cases "len_of TYPE('a)") 

548 
apply (auto simp add : sints_def) 

549 
apply (rule sym [THEN trans]) 

550 
apply (rule word_ubin.Abs_norm) 

551 
apply (simp only: bintrunc_sbintrunc) 

552 
apply (drule sym) 

553 
apply simp 

554 
done 

555 

556 
lemmas td_ext_sint = td_ext_sbin 

557 
[simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]] 

558 

559 
(* We do sint before sbin, before sint is the user version 

560 
and interpretations do not produce thm duplicates. I.e. 

561 
we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, 

562 
because the latter is the same thm as the former *) 

563 
interpretation word_sint: 

564 
td_ext "sint ::'a::len word => int" 

565 
word_of_int 

566 
"sints (len_of TYPE('a::len))" 

567 
"%w. (w + 2^(len_of TYPE('a::len)  1)) mod 2^len_of TYPE('a::len)  

568 
2 ^ (len_of TYPE('a::len)  1)" 

569 
by (rule td_ext_sint) 

570 

571 
interpretation word_sbin: 

572 
td_ext "sint ::'a::len word => int" 

573 
word_of_int 

574 
"sints (len_of TYPE('a::len))" 

575 
"sbintrunc (len_of TYPE('a::len)  1)" 

576 
by (rule td_ext_sbin) 

577 

45604  578 
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm] 
37660  579 

580 
lemmas td_sint = word_sint.td 

581 

582 
lemma to_bl_def': 

583 
"(to_bl :: 'a :: len0 word => bool list) = 

584 
bin_to_bl (len_of TYPE('a)) o uint" 

44762  585 
by (auto simp: to_bl_def) 
37660  586 

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587 
lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w 
37660  588 

45805  589 
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)" 
590 
by (fact uints_def [unfolded no_bintr_alt1]) 

591 

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592 
lemma word_numeral_alt: 
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593 
"numeral b = word_of_int (numeral b)" 
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594 
by (induct b, simp_all only: numeral.simps word_of_int_homs) 
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595 

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596 
declare word_numeral_alt [symmetric, code_abbrev] 
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597 

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598 
lemma word_neg_numeral_alt: 
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599 
"neg_numeral b = word_of_int (neg_numeral b)" 
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600 
by (simp only: neg_numeral_def word_numeral_alt wi_hom_neg) 
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601 

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602 
declare word_neg_numeral_alt [symmetric, code_abbrev] 
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603 

47372  604 
lemma word_numeral_transfer [transfer_rule]: 
605 
"(fun_rel op = cr_word) numeral numeral" 

606 
"(fun_rel op = cr_word) neg_numeral neg_numeral" 

607 
unfolding fun_rel_def cr_word_def word_numeral_alt word_neg_numeral_alt 

608 
by simp_all 

609 

45805  610 
lemma uint_bintrunc [simp]: 
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611 
"uint (numeral bin :: 'a word) = 
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612 
bintrunc (len_of TYPE ('a :: len0)) (numeral bin)" 
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613 
unfolding word_numeral_alt by (rule word_ubin.eq_norm) 
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614 

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615 
lemma uint_bintrunc_neg [simp]: "uint (neg_numeral bin :: 'a word) = 
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616 
bintrunc (len_of TYPE ('a :: len0)) (neg_numeral bin)" 
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617 
by (simp only: word_neg_numeral_alt word_ubin.eq_norm) 
37660  618 

45805  619 
lemma sint_sbintrunc [simp]: 
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620 
"sint (numeral bin :: 'a word) = 
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621 
sbintrunc (len_of TYPE ('a :: len)  1) (numeral bin)" 
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622 
by (simp only: word_numeral_alt word_sbin.eq_norm) 
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623 

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624 
lemma sint_sbintrunc_neg [simp]: "sint (neg_numeral bin :: 'a word) = 
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625 
sbintrunc (len_of TYPE ('a :: len)  1) (neg_numeral bin)" 
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626 
by (simp only: word_neg_numeral_alt word_sbin.eq_norm) 
37660  627 

45805  628 
lemma unat_bintrunc [simp]: 
47108
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629 
"unat (numeral bin :: 'a :: len0 word) = 
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630 
nat (bintrunc (len_of TYPE('a)) (numeral bin))" 
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631 
by (simp only: unat_def uint_bintrunc) 
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632 

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633 
lemma unat_bintrunc_neg [simp]: 
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634 
"unat (neg_numeral bin :: 'a :: len0 word) = 
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635 
nat (bintrunc (len_of TYPE('a)) (neg_numeral bin))" 
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636 
by (simp only: unat_def uint_bintrunc_neg) 
37660  637 

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638 
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w" 
37660  639 
apply (unfold word_size) 
640 
apply (rule word_uint.Rep_eqD) 

641 
apply (rule box_equals) 

642 
defer 

643 
apply (rule word_ubin.norm_Rep)+ 

644 
apply simp 

645 
done 

646 

45805  647 
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)" 
648 
using word_uint.Rep [of x] by (simp add: uints_num) 

649 

650 
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)" 

651 
using word_uint.Rep [of x] by (simp add: uints_num) 

652 

653 
lemma sint_ge: " (2 ^ (len_of TYPE('a)  1)) \<le> sint (x::'a::len word)" 

654 
using word_sint.Rep [of x] by (simp add: sints_num) 

655 

656 
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a)  1)" 

657 
using word_sint.Rep [of x] by (simp add: sints_num) 

37660  658 

659 
lemma sign_uint_Pls [simp]: 

46604
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660 
"bin_sign (uint x) = 0" 
47108
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661 
by (simp add: sign_Pls_ge_0) 
37660  662 

45805  663 
lemma uint_m2p_neg: "uint (x::'a::len0 word)  2 ^ len_of TYPE('a) < 0" 
664 
by (simp only: diff_less_0_iff_less uint_lt2p) 

665 

666 
lemma uint_m2p_not_non_neg: 

667 
"\<not> 0 \<le> uint (x::'a::len0 word)  2 ^ len_of TYPE('a)" 

668 
by (simp only: not_le uint_m2p_neg) 

37660  669 

670 
lemma lt2p_lem: 

40827
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671 
"len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n" 
37660  672 
by (rule xtr8 [OF _ uint_lt2p]) simp 
673 

45805  674 
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0" 
675 
by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1]) 

37660  676 

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677 
lemma uint_nat: "uint w = int (unat w)" 
37660  678 
unfolding unat_def by auto 
679 

47108
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680 
lemma uint_numeral: 
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681 
"uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)" 
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682 
unfolding word_numeral_alt 
37660  683 
by (simp only: int_word_uint) 
684 

47108
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685 
lemma uint_neg_numeral: 
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686 
"uint (neg_numeral b :: 'a :: len0 word) = neg_numeral b mod 2 ^ len_of TYPE('a)" 
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687 
unfolding word_neg_numeral_alt 
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688 
by (simp only: int_word_uint) 
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689 

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690 
lemma unat_numeral: 
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691 
"unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)" 
37660  692 
apply (unfold unat_def) 
47108
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changeset

693 
apply (clarsimp simp only: uint_numeral) 
37660  694 
apply (rule nat_mod_distrib [THEN trans]) 
47108
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changeset

695 
apply (rule zero_le_numeral) 
37660  696 
apply (simp_all add: nat_power_eq) 
697 
done 

698 

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699 
lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + 
37660  700 
2 ^ (len_of TYPE('a)  1)) mod 2 ^ len_of TYPE('a)  
701 
2 ^ (len_of TYPE('a)  1)" 

47108
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changeset

702 
unfolding word_numeral_alt by (rule int_word_sint) 
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changeset

703 

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704 
lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0" 
45958  705 
unfolding word_0_wi .. 
706 

47108
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707 
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1" 
45958  708 
unfolding word_1_wi .. 
709 

47108
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710 
lemma word_of_int_numeral [simp] : 
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711 
"(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)" 
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changeset

712 
unfolding word_numeral_alt .. 
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changeset

713 

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714 
lemma word_of_int_neg_numeral [simp]: 
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715 
"(word_of_int (neg_numeral bin) :: 'a :: len0 word) = (neg_numeral bin)" 
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changeset

716 
unfolding neg_numeral_def word_numeral_alt wi_hom_syms .. 
37660  717 

718 
lemma word_int_case_wi: 

719 
"word_int_case f (word_of_int i :: 'b word) = 

720 
f (i mod 2 ^ len_of TYPE('b::len0))" 

721 
unfolding word_int_case_def by (simp add: word_uint.eq_norm) 

722 

723 
lemma word_int_split: 

724 
"P (word_int_case f x) = 

725 
(ALL i. x = (word_of_int i :: 'b :: len0 word) & 

726 
0 <= i & i < 2 ^ len_of TYPE('b) > P (f i))" 

727 
unfolding word_int_case_def 

728 
by (auto simp: word_uint.eq_norm int_mod_eq') 

729 

730 
lemma word_int_split_asm: 

731 
"P (word_int_case f x) = 

732 
(~ (EX n. x = (word_of_int n :: 'b::len0 word) & 

733 
0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))" 

734 
unfolding word_int_case_def 

735 
by (auto simp: word_uint.eq_norm int_mod_eq') 

45805  736 

45604  737 
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq] 
738 
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq] 

37660  739 

740 
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w" 

741 
unfolding word_size by (rule uint_range') 

742 

743 
lemma sint_range_size: 

744 
" (2 ^ (size w  Suc 0)) <= sint w & sint w < 2 ^ (size w  Suc 0)" 

745 
unfolding word_size by (rule sint_range') 

746 

45805  747 
lemma sint_above_size: "2 ^ (size (w::'a::len word)  1) \<le> x \<Longrightarrow> sint w < x" 
748 
unfolding word_size by (rule less_le_trans [OF sint_lt]) 

749 

750 
lemma sint_below_size: 

751 
"x \<le>  (2 ^ (size (w::'a::len word)  1)) \<Longrightarrow> x \<le> sint w" 

752 
unfolding word_size by (rule order_trans [OF _ sint_ge]) 

37660  753 

46010  754 
subsection {* Testing bits *} 
755 

37660  756 
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)" 
757 
unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) 

758 

759 
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n > n < size w" 

760 
apply (unfold word_test_bit_def) 

761 
apply (subst word_ubin.norm_Rep [symmetric]) 

762 
apply (simp only: nth_bintr word_size) 

763 
apply fast 

764 
done 

765 

46021  766 
lemma word_eq_iff: 
767 
fixes x y :: "'a::len0 word" 

768 
shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)" 

769 
unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric] 

770 
by (metis test_bit_size [unfolded word_size]) 

771 

46023
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46022
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changeset

772 
lemma word_eqI [rule_format]: 
37660  773 
fixes u :: "'a::len0 word" 
40827
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774 
shows "(ALL n. n < size u > u !! n = v !! n) \<Longrightarrow> u = v" 
46021  775 
by (simp add: word_size word_eq_iff) 
37660  776 

45805  777 
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x" 
778 
by simp 

37660  779 

780 
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)" 

781 
unfolding word_test_bit_def word_size 

782 
by (simp add: nth_bintr [symmetric]) 

783 

784 
lemmas test_bit_bin = test_bit_bin' [unfolded word_size] 

785 

46057  786 
lemma bin_nth_uint_imp: 
787 
"bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)" 

37660  788 
apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) 
789 
apply (subst word_ubin.norm_Rep) 

790 
apply assumption 

791 
done 

792 

46057  793 
lemma bin_nth_sint: 
794 
fixes w :: "'a::len word" 

795 
shows "len_of TYPE('a) \<le> n \<Longrightarrow> 

796 
bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a)  1)" 

37660  797 
apply (subst word_sbin.norm_Rep [symmetric]) 
46057  798 
apply (auto simp add: nth_sbintr) 
37660  799 
done 
800 

801 
(* type definitions theorem for in terms of equivalent bool list *) 

802 
lemma td_bl: 

803 
"type_definition (to_bl :: 'a::len0 word => bool list) 

804 
of_bl 

805 
{bl. length bl = len_of TYPE('a)}" 

806 
apply (unfold type_definition_def of_bl_def to_bl_def) 

807 
apply (simp add: word_ubin.eq_norm) 

808 
apply safe 

809 
apply (drule sym) 

810 
apply simp 

811 
done 

812 

813 
interpretation word_bl: 

814 
type_definition "to_bl :: 'a::len0 word => bool list" 

815 
of_bl 

816 
"{bl. length bl = len_of TYPE('a::len0)}" 

817 
by (rule td_bl) 

818 

45816
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diff
changeset

819 
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

820 

40827
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changeset

821 
lemma word_size_bl: "size w = size (to_bl w)" 
37660  822 
unfolding word_size by auto 
823 

824 
lemma to_bl_use_of_bl: 

825 
"(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))" 

45816
6a04efd99f25
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huffman
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diff
changeset

826 
by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq]) 
37660  827 

828 
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" 

829 
unfolding word_reverse_def by (simp add: word_bl.Abs_inverse) 

830 

831 
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" 

832 
unfolding word_reverse_def by (simp add : word_bl.Abs_inverse) 

833 

40827
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diff
changeset

834 
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w" 
47108
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diff
changeset

835 
by (metis word_rev_rev) 
37660  836 

45805  837 
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u" 
838 
by simp 

839 

840 
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))" 

841 
unfolding word_bl_Rep' by (rule len_gt_0) 

842 

843 
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []" 

844 
by (fact length_bl_gt_0 [unfolded length_greater_0_conv]) 

845 

846 
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0" 

847 
by (fact length_bl_gt_0 [THEN gr_implies_not0]) 

37660  848 

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

849 
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = 1)" 
37660  850 
apply (unfold to_bl_def sint_uint) 
851 
apply (rule trans [OF _ bl_sbin_sign]) 

852 
apply simp 

853 
done 

854 

855 
lemma of_bl_drop': 

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

856 
"lend = length bl  len_of TYPE ('a :: len0) \<Longrightarrow> 
37660  857 
of_bl (drop lend bl) = (of_bl bl :: 'a word)" 
858 
apply (unfold of_bl_def) 

859 
apply (clarsimp simp add : trunc_bl2bin [symmetric]) 

860 
done 

861 

862 
lemma test_bit_of_bl: 

863 
"(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)" 

864 
apply (unfold of_bl_def word_test_bit_def) 

865 
apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) 

866 
done 

867 

868 
lemma no_of_bl: 

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

869 
"(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

870 
unfolding of_bl_def by simp 
37660  871 

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

872 
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)" 
37660  873 
unfolding word_size to_bl_def by auto 
874 

875 
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" 

876 
unfolding uint_bl by (simp add : word_size) 

877 

878 
lemma to_bl_of_bin: 

879 
"to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" 

880 
unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size) 

881 

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

882 
lemma to_bl_numeral [simp]: 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

883 
"to_bl (numeral bin::'a::len0 word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

884 
bin_to_bl (len_of TYPE('a)) (numeral bin)" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

885 
unfolding word_numeral_alt by (rule to_bl_of_bin) 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

886 

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

887 
lemma to_bl_neg_numeral [simp]: 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

888 
"to_bl (neg_numeral bin::'a::len0 word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

889 
bin_to_bl (len_of TYPE('a)) (neg_numeral bin)" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

890 
unfolding word_neg_numeral_alt by (rule to_bl_of_bin) 
37660  891 

892 
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" 

893 
unfolding uint_bl by (simp add : word_size) 

46011  894 

895 
lemma uint_bl_bin: 

896 
fixes x :: "'a::len0 word" 

897 
shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x" 

898 
by (rule trans [OF bin_bl_bin word_ubin.norm_Rep]) 

45604  899 

37660  900 
(* naturals *) 
901 
lemma uints_unats: "uints n = int ` unats n" 

902 
apply (unfold unats_def uints_num) 

903 
apply safe 

904 
apply (rule_tac image_eqI) 

905 
apply (erule_tac nat_0_le [symmetric]) 

906 
apply auto 

907 
apply (erule_tac nat_less_iff [THEN iffD2]) 

908 
apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) 

909 
apply (auto simp add : nat_power_eq int_power) 

910 
done 

911 

912 
lemma unats_uints: "unats n = nat ` uints n" 

913 
by (auto simp add : uints_unats image_iff) 

914 

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

915 
lemmas bintr_num = word_ubin.norm_eq_iff 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

916 
[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

917 
lemmas sbintr_num = word_sbin.norm_eq_iff 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

918 
[of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b 
37660  919 

920 
lemma num_of_bintr': 

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

921 
"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

922 
numeral a = (numeral b :: 'a word)" 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

923 
unfolding bintr_num by (erule subst, simp) 
37660  924 

925 
lemma num_of_sbintr': 

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

926 
"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

927 
numeral a = (numeral b :: 'a word)" 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

928 
unfolding sbintr_num by (erule subst, simp) 
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

929 

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

930 
lemma num_abs_bintr: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

931 
"(numeral x :: 'a word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

932 
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

933 
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

934 

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

935 
lemma num_abs_sbintr: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

936 
"(numeral x :: 'a word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

937 
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

938 
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

939 

37660  940 
(** cast  note, no arg for new length, as it's determined by type of result, 
941 
thus in "cast w = w, the type means cast to length of w! **) 

942 

943 
lemma ucast_id: "ucast w = w" 

944 
unfolding ucast_def by auto 

945 

946 
lemma scast_id: "scast w = w" 

947 
unfolding scast_def by auto 

948 

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

949 
lemma ucast_bl: "ucast w = of_bl (to_bl w)" 
37660  950 
unfolding ucast_def of_bl_def uint_bl 
951 
by (auto simp add : word_size) 

952 

953 
lemma nth_ucast: 

954 
"(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))" 

955 
apply (unfold ucast_def test_bit_bin) 

956 
apply (simp add: word_ubin.eq_norm nth_bintr word_size) 

957 
apply (fast elim!: bin_nth_uint_imp) 

958 
done 

959 

960 
(* for literal u(s)cast *) 

961 

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

962 
lemma ucast_bintr [simp]: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

963 
"ucast (numeral w ::'a::len0 word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

964 
word_of_int (bintrunc (len_of TYPE('a)) (numeral w))" 
37660  965 
unfolding ucast_def by simp 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

966 
(* TODO: neg_numeral *) 
37660  967 

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

968 
lemma scast_sbintr [simp]: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

969 
"scast (numeral w ::'a::len word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

970 
word_of_int (sbintrunc (len_of TYPE('a)  Suc 0) (numeral w))" 
37660  971 
unfolding scast_def by simp 
972 

46011  973 
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)" 
974 
unfolding source_size_def word_size Let_def .. 

975 

976 
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)" 

977 
unfolding target_size_def word_size Let_def .. 

978 

979 
lemma is_down: 

980 
fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word" 

981 
shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)" 

982 
unfolding is_down_def source_size target_size .. 

983 

984 
lemma is_up: 

985 
fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word" 

986 
shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)" 

987 
unfolding is_up_def source_size target_size .. 

37660  988 

45604  989 
lemmas is_up_down = trans [OF is_up is_down [symmetric]] 
37660  990 

45811  991 
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast" 
37660  992 
apply (unfold is_down) 
993 
apply safe 

994 
apply (rule ext) 

995 
apply (unfold ucast_def scast_def uint_sint) 

996 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

997 
apply simp 

998 
done 

999 

45811  1000 
lemma word_rev_tf: 
1001 
"to_bl (of_bl bl::'a::len0 word) = 

1002 
rev (takefill False (len_of TYPE('a)) (rev bl))" 

37660  1003 
unfolding of_bl_def uint_bl 
1004 
by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size) 

1005 

45811  1006 
lemma word_rep_drop: 
1007 
"to_bl (of_bl bl::'a::len0 word) = 

1008 
replicate (len_of TYPE('a)  length bl) False @ 

1009 
drop (length bl  len_of TYPE('a)) bl" 

1010 
by (simp add: word_rev_tf takefill_alt rev_take) 

37660  1011 

1012 
lemma to_bl_ucast: 

1013 
"to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 

1014 
replicate (len_of TYPE('a)  len_of TYPE('b)) False @ 

1015 
drop (len_of TYPE('b)  len_of TYPE('a)) (to_bl w)" 

1016 
apply (unfold ucast_bl) 

1017 
apply (rule trans) 

1018 
apply (rule word_rep_drop) 

1019 
apply simp 

1020 
done 

1021 

45811  1022 
lemma ucast_up_app [OF refl]: 
40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1023 
"uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
37660  1024 
to_bl (uc w) = replicate n False @ (to_bl w)" 
1025 
by (auto simp add : source_size target_size to_bl_ucast) 

1026 

45811  1027 
lemma ucast_down_drop [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1028 
"uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
37660  1029 
to_bl (uc w) = drop n (to_bl w)" 
1030 
by (auto simp add : source_size target_size to_bl_ucast) 

1031 

45811  1032 
lemma scast_down_drop [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1033 
"sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
37660  1034 
to_bl (sc w) = drop n (to_bl w)" 
1035 
apply (subgoal_tac "sc = ucast") 

1036 
apply safe 

1037 
apply simp 

45811  1038 
apply (erule ucast_down_drop) 
1039 
apply (rule down_cast_same [symmetric]) 

37660  1040 
apply (simp add : source_size target_size is_down) 
1041 
done 

1042 

45811  1043 
lemma sint_up_scast [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1044 
"sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w" 
37660  1045 
apply (unfold is_up) 
1046 
apply safe 

1047 
apply (simp add: scast_def word_sbin.eq_norm) 

1048 
apply (rule box_equals) 

1049 
prefer 3 

1050 
apply (rule word_sbin.norm_Rep) 

1051 
apply (rule sbintrunc_sbintrunc_l) 

1052 
defer 

1053 
apply (subst word_sbin.norm_Rep) 

1054 
apply (rule refl) 

1055 
apply simp 

1056 
done 

1057 

45811  1058 
lemma uint_up_ucast [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1059 
"uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w" 
37660  1060 
apply (unfold is_up) 
1061 
apply safe 

1062 
apply (rule bin_eqI) 

1063 
apply (fold word_test_bit_def) 

1064 
apply (auto simp add: nth_ucast) 

1065 
apply (auto simp add: test_bit_bin) 

1066 
done 

45811  1067 

1068 
lemma ucast_up_ucast [OF refl]: 

1069 
"uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w" 

37660  1070 
apply (simp (no_asm) add: ucast_def) 
1071 
apply (clarsimp simp add: uint_up_ucast) 

1072 
done 

1073 

45811  1074 
lemma scast_up_scast [OF refl]: 
1075 
"sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w" 

37660  1076 
apply (simp (no_asm) add: scast_def) 
1077 
apply (clarsimp simp add: sint_up_scast) 

1078 
done 

1079 

45811  1080 
lemma ucast_of_bl_up [OF refl]: 
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset

1081 
"w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl" 
37660  1082 
by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) 
1083 

1084 
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] 

1085 
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] 

1086 

1087 
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] 

1088 
lemmas isdus = is_up_down [where c = "scast", THEN iffD2] 

1089 
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] 

1090 
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] 

1091 

1092 
lemma up_ucast_surj: 

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

1093 
"is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660  1094 
surj (ucast :: 'a word => 'b word)" 
1095 
by (rule surjI, erule ucast_up_ucast_id) 

1096 

1097 
lemma up_scast_surj: 

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

1098 
"is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660  1099 
surj (scast :: 'a word => 'b word)" 
1100 
by (rule surjI, erule scast_up_scast_id) 

1101 

1102 
lemma down_scast_inj: 

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

1103 
"is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660  1104 
inj_on (ucast :: 'a word => 'b word) A" 
1105 
by (rule inj_on_inverseI, erule scast_down_scast_id) 

1106 

1107 
lemma down_ucast_inj: 

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

1108 
"is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660  1109 
inj_on (ucast :: 'a word => 'b word) A" 
1110 
by (rule inj_on_inverseI, erule ucast_down_ucast_id) 

1111 

1112 
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" 

1113 
by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) 

45811  1114 

46646  1115 
lemma ucast_down_wi [OF refl]: 
1116 
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x" 

1117 
apply (unfold is_down) 

37660  1118 
apply (clarsimp simp add: ucast_def word_ubin.eq_norm) 
1119 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1120 
apply (erule bintrunc_bintrunc_ge) 

1121 
done 

45811  1122 

46646  1123 
lemma ucast_down_no [OF refl]: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1124 
"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

1125 
unfolding word_numeral_alt by clarify (rule ucast_down_wi) 
46646  1126 

45811  1127 
lemma ucast_down_bl [OF refl]: 
1128 
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl" 

46646  1129 
unfolding of_bl_def by clarify (erule ucast_down_wi) 
37660  1130 

1131 
lemmas slice_def' = slice_def [unfolded word_size] 

1132 
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] 

1133 

1134 
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def 

1135 

1136 

1137 
subsection {* Word Arithmetic *} 

1138 

1139 
lemma word_less_alt: "(a < b) = (uint a < uint b)" 

46012  1140 
unfolding word_less_def word_le_def by (simp add: less_le) 
37660  1141 

1142 
lemma signed_linorder: "class.linorder word_sle word_sless" 

46124  1143 
by default (unfold word_sle_def word_sless_def, auto) 
37660  1144 

1145 
interpretation signed: linorder "word_sle" "word_sless" 

1146 
by (rule signed_linorder) 

1147 

1148 
lemma udvdI: 

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

1149 
"0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b" 
37660  1150 
by (auto simp: udvd_def) 
1151 

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

1152 
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

1153 

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

1154 
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

1155 

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

1156 
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

1157 

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

1158 
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

1159 

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

1160 
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

1161 

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

1162 
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b 
37660  1163 

46020  1164 
lemma word_1_no: "(1::'a::len0 word) = Numeral1" 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1165 
by (simp add: word_numeral_alt) 
37660  1166 

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

1167 
lemma word_m1_wi: "1 = word_of_int 1" 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1168 
by (rule word_neg_numeral_alt) 
37660  1169 

46648  1170 
lemma word_0_bl [simp]: "of_bl [] = 0" 
1171 
unfolding of_bl_def by simp 

37660  1172 

1173 
lemma word_1_bl: "of_bl [True] = 1" 

46648  1174 
unfolding of_bl_def by (simp add: bl_to_bin_def) 
1175 

1176 
lemma uint_eq_0 [simp]: "uint 0 = 0" 

1177 
unfolding word_0_wi word_ubin.eq_norm by simp 

37660  1178 

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

1179 
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0" 
46648  1180 
by (simp add: of_bl_def bl_to_bin_rep_False) 
37660  1181 

45805  1182 
lemma to_bl_0 [simp]: 
37660  1183 
"to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" 
1184 
unfolding uint_bl 

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

1185 
by (simp add: word_size bin_to_bl_zero) 
37660  1186 

1187 
lemma uint_0_iff: "(uint x = 0) = (x = 0)" 

1188 
by (auto intro!: word_uint.Rep_eqD) 

1189 

1190 
lemma unat_0_iff: "(unat x = 0) = (x = 0)" 

1191 
unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff) 

1192 

1193 
lemma unat_0 [simp]: "unat 0 = 0" 

1194 
unfolding unat_def by auto 

1195 

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

1196 
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)" 
37660  1197 
apply (unfold word_size) 
1198 
apply (rule box_equals) 

1199 
defer 

1200 
apply (rule word_uint.Rep_inverse)+ 

1201 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1202 
apply simp 

1203 
done 

1204 

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

1205 
lemmas size_0_same = size_0_same' [unfolded word_size] 
37660  1206 

1207 
lemmas unat_eq_0 = unat_0_iff 

1208 
lemmas unat_eq_zero = unat_0_iff 

1209 

1210 
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)" 

1211 
by (auto simp: unat_0_iff [symmetric]) 

1212 

45958  1213 
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

1214 
unfolding ucast_def by simp 
45958  1215 

1216 
lemma sint_0 [simp]: "sint 0 = 0" 

1217 
unfolding sint_uint by simp 

1218 

1219 
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

1220 
unfolding scast_def by simp 
37660  1221 

1222 
lemma sint_n1 [simp] : "sint 1 = 1" 

45958  1223 
unfolding word_m1_wi by (simp add: word_sbin.eq_norm) 
1224 

1225 
lemma scast_n1 [simp]: "scast 1 = 1" 

1226 
unfolding scast_def by simp 

1227 

1228 
lemma uint_1 [simp]: "uint (1::'a::len word) = 1" 

37660  1229 
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

1230 
by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1) 
45958  1231 

1232 
lemma unat_1 [simp]: "unat (1::'a::len word) = 1" 

1233 
unfolding unat_def by simp 

1234 

1235 
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

1236 
unfolding ucast_def by simp 
37660  1237 

1238 
(* now, to get the weaker results analogous to word_div/mod_def *) 

1239 

1240 
lemmas word_arith_alts = 

46000  1241 
word_sub_wi 
1242 
word_arith_wis (* FIXME: duplicate *) 

1243 

37660  1244 
subsection "Transferring goals from words to ints" 
1245 

1246 
lemma word_ths: 

1247 
shows 

1248 
word_succ_p1: "word_succ a = a + 1" and 

1249 
word_pred_m1: "word_pred a = a  1" and 

1250 
word_pred_succ: "word_pred (word_succ a) = a" and 

1251 
word_succ_pred: "word_succ (word_pred a) = a" and 

1252 
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

1253 
by (transfer, simp add: algebra_simps)+ 
37660  1254 

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

1255 
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y" 
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

1256 
by simp 
37660  1257 

1258 
lemmas uint_word_ariths = 

45604  1259 
word_arith_alts [THEN trans [OF uint_cong int_word_uint]] 
37660  1260 

1261 
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p] 

1262 

1263 
(* similar expressions for sint (arith operations) *) 

1264 
lemmas sint_word_ariths = uint_word_arith_bintrs 

1265 
[THEN uint_sint [symmetric, THEN trans], 

1266 
unfolded uint_sint bintr_arith1s bintr_ariths 

45604  1267 
len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep] 
1268 

1269 
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]] 

1270 
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]] 

37660  1271 

1272 
lemma word_pred_0_n1: "word_pred 0 = word_of_int 1" 

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

1273 
unfolding word_pred_m1 by simp 
37660  1274 

1275 
lemma succ_pred_no [simp]: 

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

1276 
"word_succ (numeral w) = numeral w + 1" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1277 
"word_pred (numeral w) = numeral w  1" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1278 
"word_succ (neg_numeral w) = neg_numeral w + 1" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1279 
"word_pred (neg_numeral w) = neg_numeral w  1" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1280 
unfolding word_succ_p1 word_pred_m1 by simp_all 
37660  1281 

1282 
lemma word_sp_01 [simp] : 

1283 
"word_succ 1 = 0 & word_succ 0 = 1 & word_pred 0 = 1 & word_pred 1 = 0" 

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

1284 
unfolding word_succ_p1 word_pred_m1 by simp_all 
37660  1285 

1286 
(* alternative approach to lifting arithmetic equalities *) 

1287 
lemma word_of_int_Ex: 

1288 
"\<exists>y. x = word_of_int y" 

1289 
by (rule_tac x="uint x" in exI) simp 

1290 

1291 

1292 
subsection "Order on fixedlength words" 

1293 

1294 
lemma word_zero_le [simp] : 

1295 
"0 <= (y :: 'a :: len0 word)" 

1296 
unfolding word_le_def by auto 

1297 

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

1298 
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *) 
37660  1299 
unfolding word_le_def 
1300 
by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto 

1301 

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

1302 
lemma word_n1_ge [simp]: "y \<le> (1::'a::len0 word)" 
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

1303 
unfolding word_le_def 
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

1304 
by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto 
37660  1305 

1306 
lemmas word_not_simps [simp] = 

1307 
word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD] 

1308 

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

1309 
lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> (y :: 'a :: len0 word)" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1310 
by (simp add: less_le) 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1311 

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

1312 
lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y 
37660  1313 

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

1314 
lemma word_sless_alt: "(a <s b) = (sint a < sint b)" 
37660  1315 
unfolding word_sle_def word_sless_def 
1316 
by (auto simp add: less_le) 

1317 

1318 
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)" 

1319 
unfolding unat_def word_le_def 

1320 
by (rule nat_le_eq_zle [symmetric]) simp 

