author  huffman 
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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 ("~~/src/HOL/Word/Tools/smt_word.ML") 
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begin 
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text {* see @{text "Examples/WordExamples.thy"} for examples *} 

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subsection {* Type definition *} 

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typedef (open) 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}" 
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morphisms uint Abs_word by auto 
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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 w = Abs_word (bintrunc (len_of TYPE ('a)) w)" 

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lemma uint_word_of_int [code]: "uint (word_of_int w \<Colon> 'a\<Colon>len0 word) = w mod 2 ^ len_of TYPE('a)" 

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by (auto simp add: word_of_int_def bintrunc_mod2p intro: Abs_word_inverse) 

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code_datatype word_of_int 

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subsection {* Random instance *} 
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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 (max i (2 ^ len_of TYPE('a))) \<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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syntax 

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of_int :: "int => 'a" 

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translations 

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"case x of CONST of_int 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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(* FIXME: delete *) 
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lemmas atLeastLessThan_alt = atLeastLessThan_def [unfolded 
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atLeast_def lessThan_def Collect_conj_eq [symmetric]] 
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lemma mod_in_reps: "m > 0 \<Longrightarrow> y mod m : {0::int ..< m}" 
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by auto 
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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 [simplified]) 
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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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lemma int_word_uint: 
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"uint (word_of_int x::'a::len0 word) = x mod 2 ^ len_of TYPE('a)" 

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by (fact td_ext_uint [THEN td_ext.eq_norm]) 

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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 td_ext_ubin = td_ext_uint 
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[simplified len_gt_0 no_bintr_alt1 [symmetric]] 
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interpretation word_ubin: 
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td_ext "uint::'a::len0 word \<Rightarrow> int" 
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word_of_int 
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"uints (len_of TYPE('a::len0))" 
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"bintrunc (len_of TYPE('a::len0))" 
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by (rule td_ext_ubin) 
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lemma split_word_all: 
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"(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))" 
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proof 
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fix x :: "'a word" 
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assume "\<And>x. PROP P (word_of_int x)" 
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hence "PROP P (word_of_int (uint x))" . 
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thus "PROP P x" by simp 
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qed 
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subsection "Arithmetic operations" 

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definition 
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word_succ :: "'a :: len0 word => 'a word" 
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where 
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"word_succ a = word_of_int (Int.succ (uint a))" 
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200 

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definition 
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word_pred :: "'a :: len0 word => 'a word" 
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where 
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"word_pred a = word_of_int (Int.pred (uint a))" 
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instantiation word :: (len0) "{number, Divides.div, comm_monoid_mult, comm_ring}" 
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begin 
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209 
definition 

210 
word_0_wi: "0 = word_of_int 0" 

211 

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definition 

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word_1_wi: "1 = word_of_int 1" 

214 

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definition 

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word_add_def: "a + b = word_of_int (uint a + uint b)" 

217 

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definition 

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word_sub_wi: "a  b = word_of_int (uint a  uint b)" 

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definition 

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word_minus_def: " a = word_of_int ( uint a)" 

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definition 

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word_mult_def: "a * b = word_of_int (uint a * uint b)" 

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definition 

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

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definition 

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

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definition 

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word_number_of_def: "number_of w = word_of_int w" 

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lemmas word_arith_wis = 
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word_add_def word_mult_def word_minus_def 
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word_succ_def word_pred_def word_0_wi word_1_wi 
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239 

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lemmas arths = 
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bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm] 
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242 

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lemma wi_homs: 
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shows 
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wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and 
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wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and 
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wi_hom_neg: " word_of_int a = word_of_int ( a)" and 
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248 
wi_hom_succ: "word_succ (word_of_int a) = word_of_int (Int.succ a)" and 
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wi_hom_pred: "word_pred (word_of_int a) = word_of_int (Int.pred a)" 
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250 
by (auto simp: word_arith_wis arths) 
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251 

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lemmas wi_hom_syms = wi_homs [symmetric] 
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253 

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lemma word_sub_def: "a  b = a +  (b :: 'a :: len0 word)" 
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255 
unfolding word_sub_wi diff_minus 
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256 
by (simp only : word_uint.Rep_inverse wi_hom_syms) 
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257 

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lemma word_of_int_sub_hom: 
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259 
"(word_of_int a)  word_of_int b = word_of_int (a  b)" 
45805  260 
by (simp add: word_sub_wi arths) 
45545
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261 

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lemmas new_word_of_int_homs = 
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word_of_int_sub_hom wi_homs word_0_wi word_1_wi 
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264 

45604  265 
lemmas new_word_of_int_hom_syms = new_word_of_int_homs [symmetric] 
45545
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266 

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lemmas word_of_int_hom_syms = 
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new_word_of_int_hom_syms [unfolded succ_def pred_def] 
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269 

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lemmas word_of_int_homs = 
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new_word_of_int_homs [unfolded succ_def pred_def] 
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45805  273 
(* FIXME: provide only one copy of these theorems! *) 
274 
lemmas word_of_int_add_hom = wi_hom_add 

275 
lemmas word_of_int_mult_hom = wi_hom_mult 

276 
lemmas word_of_int_minus_hom = wi_hom_neg 

277 
lemmas word_of_int_succ_hom = wi_hom_succ [unfolded succ_def] 

278 
lemmas word_of_int_pred_hom = wi_hom_pred [unfolded pred_def] 

279 
lemmas word_of_int_0_hom = word_0_wi 

280 
lemmas word_of_int_1_hom = word_1_wi 

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282 
instance 
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by default (auto simp: split_word_all word_of_int_homs algebra_simps) 
37660  284 

285 
end 

286 

45545
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lemma word_zero_neq_one: "0 < len_of TYPE ('a :: len0) \<Longrightarrow> (0 :: 'a word) ~= 1" 
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288 
unfolding word_arith_wis 
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289 
by (auto simp add: word_ubin.norm_eq_iff [symmetric] gr0_conv_Suc) 
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290 

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lemmas lenw1_zero_neq_one = len_gt_0 [THEN word_zero_neq_one] 
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292 

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293 
instance word :: (len) comm_ring_1 
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by (intro_classes) (simp add: lenw1_zero_neq_one) 
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295 

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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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298 

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lemma word_of_int: "of_int = word_of_int" 
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apply (rule ext) 
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apply (case_tac x rule: int_diff_cases) 
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apply (simp add: word_of_nat word_of_int_sub_hom) 
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303 
done 
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304 

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305 
lemma word_number_of_eq: 
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"number_of w = (of_int w :: 'a :: len word)" 
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307 
unfolding word_number_of_def word_of_int by auto 
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308 

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309 
instance word :: (len) number_ring 
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310 
by (intro_classes) (simp add : word_number_of_eq) 
37660  311 

312 
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  314 

45547  315 

316 
subsection "Ordering" 

317 

318 
instantiation word :: (len0) linorder 

319 
begin 

320 

37660  321 
definition 
322 
word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b" 

323 

324 
definition 

325 
word_less_def: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> (y \<Colon> 'a word)" 

326 

45547  327 
instance 
328 
by default (auto simp: word_less_def word_le_def) 

329 

330 
end 

331 

37660  332 
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  334 

335 
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  337 

338 

339 
subsection "Bitwise operations" 

340 

341 
instantiation word :: (len0) bits 

342 
begin 

343 

344 
definition 

345 
word_and_def: 

346 
"(a::'a word) AND b = word_of_int (uint a AND uint b)" 

347 

348 
definition 

349 
word_or_def: 

350 
"(a::'a word) OR b = word_of_int (uint a OR uint b)" 

351 

352 
definition 

353 
word_xor_def: 

354 
"(a::'a word) XOR b = word_of_int (uint a XOR uint b)" 

355 

356 
definition 

357 
word_not_def: 

358 
"NOT (a::'a word) = word_of_int (NOT (uint a))" 

359 

360 
definition 

361 
word_test_bit_def: "test_bit a = bin_nth (uint a)" 

362 

363 
definition 

364 
word_set_bit_def: "set_bit a n x = 

365 
word_of_int (bin_sc n (If x 1 0) (uint a))" 

366 

367 
definition 

368 
word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)" 

369 

370 
definition 

371 
word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1" 

372 

373 
definition shiftl1 :: "'a word \<Rightarrow> 'a word" where 

374 
"shiftl1 w = word_of_int (uint w BIT 0)" 

375 

376 
definition shiftr1 :: "'a word \<Rightarrow> 'a word" where 

377 
 "shift right as unsigned or as signed, ie logical or arithmetic" 

378 
"shiftr1 w = word_of_int (bin_rest (uint w))" 

379 

380 
definition 

381 
shiftl_def: "w << n = (shiftl1 ^^ n) w" 

382 

383 
definition 

384 
shiftr_def: "w >> n = (shiftr1 ^^ n) w" 

385 

386 
instance .. 

387 

388 
end 

389 

390 
instantiation word :: (len) bitss 

391 
begin 

392 

393 
definition 

394 
word_msb_def: 

395 
"msb a \<longleftrightarrow> bin_sign (sint a) = Int.Min" 

396 

397 
instance .. 

398 

399 
end 

400 

37667  401 
lemma [code]: 
402 
"msb a \<longleftrightarrow> bin_sign (sint a) = ( 1 :: int)" 

403 
by (simp only: word_msb_def Min_def) 

404 

37660  405 
definition setBit :: "'a :: len0 word => nat => 'a word" where 
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"setBit w n = set_bit w n True" 
37660  407 

408 
definition clearBit :: "'a :: len0 word => nat => 'a word" where 

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409 
"clearBit w n = set_bit w n False" 
37660  410 

411 

412 
subsection "Shift operations" 

413 

414 
definition sshiftr1 :: "'a :: len word => 'a word" where 

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"sshiftr1 w = word_of_int (bin_rest (sint w))" 
37660  416 

417 
definition bshiftr1 :: "bool => 'a :: len word => 'a word" where 

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"bshiftr1 b w = of_bl (b # butlast (to_bl w))" 
37660  419 

420 
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where 

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"w >>> n = (sshiftr1 ^^ n) w" 
37660  422 

423 
definition mask :: "nat => 'a::len word" where 

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"mask n = (1 << n)  1" 
37660  425 

426 
definition revcast :: "'a :: len0 word => 'b :: len0 word" where 

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427 
"revcast w = of_bl (takefill False (len_of TYPE('b)) (to_bl w))" 
37660  428 

429 
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where 

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430 
"slice1 n w = of_bl (takefill False n (to_bl w))" 
37660  431 

432 
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where 

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433 
"slice n w = slice1 (size w  n) w" 
37660  434 

435 

436 
subsection "Rotation" 

437 

438 
definition rotater1 :: "'a list => 'a list" where 

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"rotater1 ys = 
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440 
(case ys of [] => []  x # xs => last ys # butlast ys)" 
37660  441 

442 
definition rotater :: "nat => 'a list => 'a list" where 

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443 
"rotater n = rotater1 ^^ n" 
37660  444 

445 
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where 

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446 
"word_rotr n w = of_bl (rotater n (to_bl w))" 
37660  447 

448 
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where 

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449 
"word_rotl n w = of_bl (rotate n (to_bl w))" 
37660  450 

451 
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where 

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"word_roti i w = (if i >= 0 then word_rotr (nat i) w 
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453 
else word_rotl (nat ( i)) w)" 
37660  454 

455 

456 
subsection "Split and cat operations" 

457 

458 
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where 

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459 
"word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))" 
37660  460 

461 
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where 

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462 
"word_split a = 
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463 
(case bin_split (len_of TYPE ('c)) (uint a) of 
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464 
(u, v) => (word_of_int u, word_of_int v))" 
37660  465 

466 
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where 

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467 
"word_rcat ws = 
37660  468 
word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))" 
469 

470 
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where 

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471 
"word_rsplit w = 
37660  472 
map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))" 
473 

474 
definition max_word :: "'a::len word"  "Largest representable machine integer." where 

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475 
"max_word = word_of_int (2 ^ len_of TYPE('a)  1)" 
37660  476 

477 
primrec of_bool :: "bool \<Rightarrow> 'a::len word" where 

478 
"of_bool False = 0" 

479 
 "of_bool True = 1" 

480 

45805  481 
(* FIXME: only provide one theorem name *) 
37660  482 
lemmas of_nth_def = word_set_bits_def 
483 

484 
lemma sint_sbintrunc': 

485 
"sint (word_of_int bin :: 'a word) = 

486 
(sbintrunc (len_of TYPE ('a :: len)  1) bin)" 

487 
unfolding sint_uint 

488 
by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt) 

489 

490 
lemma uint_sint: 

491 
"uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))" 

492 
unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le) 

493 

494 
lemma bintr_uint': 

40827
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495 
"n >= size w \<Longrightarrow> bintrunc n (uint w) = uint w" 
37660  496 
apply (unfold word_size) 
497 
apply (subst word_ubin.norm_Rep [symmetric]) 

498 
apply (simp only: bintrunc_bintrunc_min word_size) 

499 
apply (simp add: min_max.inf_absorb2) 

500 
done 

501 

502 
lemma wi_bintr': 

40827
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503 
"wb = word_of_int bin \<Longrightarrow> n >= size wb \<Longrightarrow> 
37660  504 
word_of_int (bintrunc n bin) = wb" 
505 
unfolding word_size 

506 
by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1) 

507 

508 
lemmas bintr_uint = bintr_uint' [unfolded word_size] 

509 
lemmas wi_bintr = wi_bintr' [unfolded word_size] 

510 

511 
lemma td_ext_sbin: 

512 
"td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 

513 
(sbintrunc (len_of TYPE('a)  1))" 

514 
apply (unfold td_ext_def' sint_uint) 

515 
apply (simp add : word_ubin.eq_norm) 

516 
apply (cases "len_of TYPE('a)") 

517 
apply (auto simp add : sints_def) 

518 
apply (rule sym [THEN trans]) 

519 
apply (rule word_ubin.Abs_norm) 

520 
apply (simp only: bintrunc_sbintrunc) 

521 
apply (drule sym) 

522 
apply simp 

523 
done 

524 

525 
lemmas td_ext_sint = td_ext_sbin 

526 
[simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]] 

527 

528 
(* We do sint before sbin, before sint is the user version 

529 
and interpretations do not produce thm duplicates. I.e. 

530 
we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, 

531 
because the latter is the same thm as the former *) 

532 
interpretation word_sint: 

533 
td_ext "sint ::'a::len word => int" 

534 
word_of_int 

535 
"sints (len_of TYPE('a::len))" 

536 
"%w. (w + 2^(len_of TYPE('a::len)  1)) mod 2^len_of TYPE('a::len)  

537 
2 ^ (len_of TYPE('a::len)  1)" 

538 
by (rule td_ext_sint) 

539 

540 
interpretation word_sbin: 

541 
td_ext "sint ::'a::len word => int" 

542 
word_of_int 

543 
"sints (len_of TYPE('a::len))" 

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

545 
by (rule td_ext_sbin) 

546 

45604  547 
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm] 
37660  548 

549 
lemmas td_sint = word_sint.td 

550 

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551 
lemma word_number_of_alt [code_unfold_post]: 
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552 
"number_of b = word_of_int (number_of b)" 
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553 
by (simp add: number_of_eq word_number_of_def) 
37660  554 

555 
lemma word_no_wi: "number_of = word_of_int" 

44762  556 
by (auto simp: word_number_of_def) 
37660  557 

558 
lemma to_bl_def': 

559 
"(to_bl :: 'a :: len0 word => bool list) = 

560 
bin_to_bl (len_of TYPE('a)) o uint" 

44762  561 
by (auto simp: to_bl_def) 
37660  562 

45604  563 
lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w"] for w 
37660  564 

45805  565 
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)" 
566 
by (fact uints_def [unfolded no_bintr_alt1]) 

567 

568 
lemma uint_bintrunc [simp]: 

569 
"uint (number_of bin :: 'a word) = 

37660  570 
number_of (bintrunc (len_of TYPE ('a :: len0)) bin)" 
571 
unfolding word_number_of_def number_of_eq 

572 
by (auto intro: word_ubin.eq_norm) 

573 

45805  574 
lemma sint_sbintrunc [simp]: 
575 
"sint (number_of bin :: 'a word) = 

37660  576 
number_of (sbintrunc (len_of TYPE ('a :: len)  1) bin)" 
577 
unfolding word_number_of_def number_of_eq 

578 
by (subst word_sbin.eq_norm) simp 

579 

45805  580 
lemma unat_bintrunc [simp]: 
37660  581 
"unat (number_of bin :: 'a :: len0 word) = 
582 
number_of (bintrunc (len_of TYPE('a)) bin)" 

583 
unfolding unat_def nat_number_of_def 

584 
by (simp only: uint_bintrunc) 

585 

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586 
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w" 
37660  587 
apply (unfold word_size) 
588 
apply (rule word_uint.Rep_eqD) 

589 
apply (rule box_equals) 

590 
defer 

591 
apply (rule word_ubin.norm_Rep)+ 

592 
apply simp 

593 
done 

594 

45805  595 
(* TODO: remove uint_lem and sint_lem *) 
37660  596 
lemmas uint_lem = word_uint.Rep [unfolded uints_num mem_Collect_eq] 
597 
lemmas sint_lem = word_sint.Rep [unfolded sints_num mem_Collect_eq] 

45805  598 

599 
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)" 

600 
using word_uint.Rep [of x] by (simp add: uints_num) 

601 

602 
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)" 

603 
using word_uint.Rep [of x] by (simp add: uints_num) 

604 

605 
lemma sint_ge: " (2 ^ (len_of TYPE('a)  1)) \<le> sint (x::'a::len word)" 

606 
using word_sint.Rep [of x] by (simp add: sints_num) 

607 

608 
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a)  1)" 

609 
using word_sint.Rep [of x] by (simp add: sints_num) 

37660  610 

611 
lemma sign_uint_Pls [simp]: 

612 
"bin_sign (uint x) = Int.Pls" 

613 
by (simp add: sign_Pls_ge_0 number_of_eq) 

614 

45805  615 
lemma uint_m2p_neg: "uint (x::'a::len0 word)  2 ^ len_of TYPE('a) < 0" 
616 
by (simp only: diff_less_0_iff_less uint_lt2p) 

617 

618 
lemma uint_m2p_not_non_neg: 

619 
"\<not> 0 \<le> uint (x::'a::len0 word)  2 ^ len_of TYPE('a)" 

620 
by (simp only: not_le uint_m2p_neg) 

37660  621 

622 
lemma lt2p_lem: 

40827
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623 
"len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n" 
37660  624 
by (rule xtr8 [OF _ uint_lt2p]) simp 
625 

45805  626 
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0" 
627 
by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1]) 

37660  628 

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629 
lemma uint_nat: "uint w = int (unat w)" 
37660  630 
unfolding unat_def by auto 
631 

632 
lemma uint_number_of: 

633 
"uint (number_of b :: 'a :: len0 word) = number_of b mod 2 ^ len_of TYPE('a)" 

634 
unfolding word_number_of_alt 

635 
by (simp only: int_word_uint) 

636 

637 
lemma unat_number_of: 

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638 
"bin_sign b = Int.Pls \<Longrightarrow> 
37660  639 
unat (number_of b::'a::len0 word) = number_of b mod 2 ^ len_of TYPE ('a)" 
640 
apply (unfold unat_def) 

641 
apply (clarsimp simp only: uint_number_of) 

642 
apply (rule nat_mod_distrib [THEN trans]) 

643 
apply (erule sign_Pls_ge_0 [THEN iffD1]) 

644 
apply (simp_all add: nat_power_eq) 

645 
done 

646 

647 
lemma sint_number_of: "sint (number_of b :: 'a :: len word) = (number_of b + 

648 
2 ^ (len_of TYPE('a)  1)) mod 2 ^ len_of TYPE('a)  

649 
2 ^ (len_of TYPE('a)  1)" 

650 
unfolding word_number_of_alt by (rule int_word_sint) 

651 

652 
lemma word_of_int_bin [simp] : 

653 
"(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)" 

654 
unfolding word_number_of_alt by auto 

655 

656 
lemma word_int_case_wi: 

657 
"word_int_case f (word_of_int i :: 'b word) = 

658 
f (i mod 2 ^ len_of TYPE('b::len0))" 

659 
unfolding word_int_case_def by (simp add: word_uint.eq_norm) 

660 

661 
lemma word_int_split: 

662 
"P (word_int_case f x) = 

663 
(ALL i. x = (word_of_int i :: 'b :: len0 word) & 

664 
0 <= i & i < 2 ^ len_of TYPE('b) > P (f i))" 

665 
unfolding word_int_case_def 

666 
by (auto simp: word_uint.eq_norm int_mod_eq') 

667 

668 
lemma word_int_split_asm: 

669 
"P (word_int_case f x) = 

670 
(~ (EX n. x = (word_of_int n :: 'b::len0 word) & 

671 
0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))" 

672 
unfolding word_int_case_def 

673 
by (auto simp: word_uint.eq_norm int_mod_eq') 

45805  674 

675 
(* FIXME: uint_range' is an exact duplicate of uint_lem *) 

45604  676 
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq] 
677 
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq] 

37660  678 

679 
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w" 

680 
unfolding word_size by (rule uint_range') 

681 

682 
lemma sint_range_size: 

683 
" (2 ^ (size w  Suc 0)) <= sint w & sint w < 2 ^ (size w  Suc 0)" 

684 
unfolding word_size by (rule sint_range') 

685 

45805  686 
lemma sint_above_size: "2 ^ (size (w::'a::len word)  1) \<le> x \<Longrightarrow> sint w < x" 
687 
unfolding word_size by (rule less_le_trans [OF sint_lt]) 

688 

689 
lemma sint_below_size: 

690 
"x \<le>  (2 ^ (size (w::'a::len word)  1)) \<Longrightarrow> x \<le> sint w" 

691 
unfolding word_size by (rule order_trans [OF _ sint_ge]) 

37660  692 

693 
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)" 

694 
unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) 

695 

696 
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n > n < size w" 

697 
apply (unfold word_test_bit_def) 

698 
apply (subst word_ubin.norm_Rep [symmetric]) 

699 
apply (simp only: nth_bintr word_size) 

700 
apply fast 

701 
done 

702 

703 
lemma word_eqI [rule_format] : 

704 
fixes u :: "'a::len0 word" 

40827
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705 
shows "(ALL n. n < size u > u !! n = v !! n) \<Longrightarrow> u = v" 
37660  706 
apply (rule test_bit_eq_iff [THEN iffD1]) 
707 
apply (rule ext) 

708 
apply (erule allE) 

709 
apply (erule impCE) 

710 
prefer 2 

711 
apply assumption 

712 
apply (auto dest!: test_bit_size simp add: word_size) 

713 
done 

714 

45805  715 
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x" 
716 
by simp 

37660  717 

718 
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)" 

719 
unfolding word_test_bit_def word_size 

720 
by (simp add: nth_bintr [symmetric]) 

721 

722 
lemmas test_bit_bin = test_bit_bin' [unfolded word_size] 

723 

724 
lemma bin_nth_uint_imp': "bin_nth (uint w) n > n < size w" 

725 
apply (unfold word_size) 

726 
apply (rule impI) 

727 
apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) 

728 
apply (subst word_ubin.norm_Rep) 

729 
apply assumption 

730 
done 

731 

732 
lemma bin_nth_sint': 

733 
"n >= size w > bin_nth (sint w) n = bin_nth (sint w) (size w  1)" 

734 
apply (rule impI) 

735 
apply (subst word_sbin.norm_Rep [symmetric]) 

736 
apply (simp add : nth_sbintr word_size) 

737 
apply auto 

738 
done 

739 

740 
lemmas bin_nth_uint_imp = bin_nth_uint_imp' [rule_format, unfolded word_size] 

741 
lemmas bin_nth_sint = bin_nth_sint' [rule_format, unfolded word_size] 

742 

743 
(* type definitions theorem for in terms of equivalent bool list *) 

744 
lemma td_bl: 

745 
"type_definition (to_bl :: 'a::len0 word => bool list) 

746 
of_bl 

747 
{bl. length bl = len_of TYPE('a)}" 

748 
apply (unfold type_definition_def of_bl_def to_bl_def) 

749 
apply (simp add: word_ubin.eq_norm) 

750 
apply safe 

751 
apply (drule sym) 

752 
apply simp 

753 
done 

754 

755 
interpretation word_bl: 

756 
type_definition "to_bl :: 'a::len0 word => bool list" 

757 
of_bl 

758 
"{bl. length bl = len_of TYPE('a::len0)}" 

759 
by (rule td_bl) 

760 

45538
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761 
lemmas word_bl_Rep' = word_bl.Rep [simplified, iff] 
1fffa81b9b83
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diff
changeset

762 

40827
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763 
lemma word_size_bl: "size w = size (to_bl w)" 
37660  764 
unfolding word_size by auto 
765 

766 
lemma to_bl_use_of_bl: 

767 
"(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))" 

44890
22f665a2e91c
new fastforce replacing fastsimp  less confusing name
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parents:
44821
diff
changeset

768 
by (fastforce elim!: word_bl.Abs_inverse [simplified]) 
37660  769 

770 
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" 

771 
unfolding word_reverse_def by (simp add: word_bl.Abs_inverse) 

772 

773 
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" 

774 
unfolding word_reverse_def by (simp add : word_bl.Abs_inverse) 

775 

40827
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haftmann
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diff
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776 
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w" 
37660  777 
by auto 
778 

45805  779 
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u" 
780 
by simp 

781 

782 
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))" 

783 
unfolding word_bl_Rep' by (rule len_gt_0) 

784 

785 
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []" 

786 
by (fact length_bl_gt_0 [unfolded length_greater_0_conv]) 

787 

788 
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0" 

789 
by (fact length_bl_gt_0 [THEN gr_implies_not0]) 

37660  790 

791 
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = Int.Min)" 

792 
apply (unfold to_bl_def sint_uint) 

793 
apply (rule trans [OF _ bl_sbin_sign]) 

794 
apply simp 

795 
done 

796 

797 
lemma of_bl_drop': 

40827
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diff
changeset

798 
"lend = length bl  len_of TYPE ('a :: len0) \<Longrightarrow> 
37660  799 
of_bl (drop lend bl) = (of_bl bl :: 'a word)" 
800 
apply (unfold of_bl_def) 

801 
apply (clarsimp simp add : trunc_bl2bin [symmetric]) 

802 
done 

803 

45805  804 
lemma of_bl_no: "of_bl bl = number_of (bl_to_bin bl)" 
805 
by (fact of_bl_def [folded word_number_of_def]) 

37660  806 

807 
lemma test_bit_of_bl: 

808 
"(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)" 

809 
apply (unfold of_bl_def word_test_bit_def) 

810 
apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) 

811 
done 

812 

813 
lemma no_of_bl: 

814 
"(number_of bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) bin)" 

815 
unfolding word_size of_bl_no by (simp add : word_number_of_def) 

816 

40827
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code preprocessor setup for numerals on word type;
haftmann
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diff
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817 
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)" 
37660  818 
unfolding word_size to_bl_def by auto 
819 

820 
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" 

821 
unfolding uint_bl by (simp add : word_size) 

822 

823 
lemma to_bl_of_bin: 

824 
"to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" 

825 
unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size) 

826 

45805  827 
lemma to_bl_no_bin [simp]: 
828 
"to_bl (number_of bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" 

829 
by (fact to_bl_of_bin [folded word_number_of_def]) 

37660  830 

831 
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" 

832 
unfolding uint_bl by (simp add : word_size) 

833 

45604  834 
lemmas uint_bl_bin [simp] = trans [OF bin_bl_bin word_ubin.norm_Rep] 
835 

45805  836 
(* FIXME: the next two lemmas should be unnecessary, because the lhs 
837 
terms should never occur in practice *) 

838 
lemma num_AB_u [simp]: "number_of (uint x) = x" 

839 
unfolding word_number_of_def by (rule word_uint.Rep_inverse) 

840 

841 
lemma num_AB_s [simp]: "number_of (sint x) = x" 

842 
unfolding word_number_of_def by (rule word_sint.Rep_inverse) 

37660  843 

844 
(* naturals *) 

845 
lemma uints_unats: "uints n = int ` unats n" 

846 
apply (unfold unats_def uints_num) 

847 
apply safe 

848 
apply (rule_tac image_eqI) 

849 
apply (erule_tac nat_0_le [symmetric]) 

850 
apply auto 

851 
apply (erule_tac nat_less_iff [THEN iffD2]) 

852 
apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) 

853 
apply (auto simp add : nat_power_eq int_power) 

854 
done 

855 

856 
lemma unats_uints: "unats n = nat ` uints n" 

857 
by (auto simp add : uints_unats image_iff) 

858 

45604  859 
lemmas bintr_num = word_ubin.norm_eq_iff [symmetric, folded word_number_of_def] 
860 
lemmas sbintr_num = word_sbin.norm_eq_iff [symmetric, folded word_number_of_def] 

861 

862 
lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def] 

863 
lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def] 

37660  864 

865 
(* don't add these to simpset, since may want bintrunc n w to be simplified; 

866 
may want these in reverse, but loop as simp rules, so use following *) 

867 

868 
lemma num_of_bintr': 

40827
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code preprocessor setup for numerals on word type;
haftmann
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39910
diff
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869 
"bintrunc (len_of TYPE('a :: len0)) a = b \<Longrightarrow> 
37660  870 
number_of a = (number_of b :: 'a word)" 
871 
apply safe 

872 
apply (rule_tac num_of_bintr [symmetric]) 

873 
done 

874 

875 
lemma num_of_sbintr': 

40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
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diff
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876 
"sbintrunc (len_of TYPE('a :: len)  1) a = b \<Longrightarrow> 
37660  877 
number_of a = (number_of b :: 'a word)" 
878 
apply safe 

879 
apply (rule_tac num_of_sbintr [symmetric]) 

880 
done 

881 

45604  882 
lemmas num_abs_bintr = sym [THEN trans, OF num_of_bintr word_number_of_def] 
883 
lemmas num_abs_sbintr = sym [THEN trans, OF num_of_sbintr word_number_of_def] 

37660  884 

885 
(** cast  note, no arg for new length, as it's determined by type of result, 

886 
thus in "cast w = w, the type means cast to length of w! **) 

887 

888 
lemma ucast_id: "ucast w = w" 

889 
unfolding ucast_def by auto 

890 

891 
lemma scast_id: "scast w = w" 

892 
unfolding scast_def by auto 

893 

40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
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diff
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894 
lemma ucast_bl: "ucast w = of_bl (to_bl w)" 
37660  895 
unfolding ucast_def of_bl_def uint_bl 
896 
by (auto simp add : word_size) 

897 

898 
lemma nth_ucast: 

899 
"(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))" 

900 
apply (unfold ucast_def test_bit_bin) 

901 
apply (simp add: word_ubin.eq_norm nth_bintr word_size) 

902 
apply (fast elim!: bin_nth_uint_imp) 

903 
done 

904 

905 
(* for literal u(s)cast *) 

906 

907 
lemma ucast_bintr [simp]: 

908 
"ucast (number_of w ::'a::len0 word) = 

909 
number_of (bintrunc (len_of TYPE('a)) w)" 

910 
unfolding ucast_def by simp 

911 

912 
lemma scast_sbintr [simp]: 

913 
"scast (number_of w ::'a::len word) = 

914 
number_of (sbintrunc (len_of TYPE('a)  Suc 0) w)" 

915 
unfolding scast_def by simp 

916 

917 
lemmas source_size = source_size_def [unfolded Let_def word_size] 

918 
lemmas target_size = target_size_def [unfolded Let_def word_size] 

919 
lemmas is_down = is_down_def [unfolded source_size target_size] 

920 
lemmas is_up = is_up_def [unfolded source_size target_size] 

921 

45604  922 
lemmas is_up_down = trans [OF is_up is_down [symmetric]] 
37660  923 

40827
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haftmann
parents:
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924 
lemma down_cast_same': "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast" 
37660  925 
apply (unfold is_down) 
926 
apply safe 

927 
apply (rule ext) 

928 
apply (unfold ucast_def scast_def uint_sint) 

929 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

930 
apply simp 

931 
done 

932 

933 
lemma word_rev_tf': 

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

934 
"r = to_bl (of_bl bl) \<Longrightarrow> r = rev (takefill False (length r) (rev bl))" 
37660  935 
unfolding of_bl_def uint_bl 
936 
by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size) 

937 

45604  938 
lemmas word_rev_tf = refl [THEN word_rev_tf', unfolded word_bl_Rep'] 
37660  939 

940 
lemmas word_rep_drop = word_rev_tf [simplified takefill_alt, 

941 
simplified, simplified rev_take, simplified] 

942 

943 
lemma to_bl_ucast: 

944 
"to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 

945 
replicate (len_of TYPE('a)  len_of TYPE('b)) False @ 

946 
drop (len_of TYPE('b)  len_of TYPE('a)) (to_bl w)" 

947 
apply (unfold ucast_bl) 

948 
apply (rule trans) 

949 
apply (rule word_rep_drop) 

950 
apply simp 

951 
done 

952 

953 
lemma ucast_up_app': 

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

954 
"uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
37660  955 
to_bl (uc w) = replicate n False @ (to_bl w)" 
956 
by (auto simp add : source_size target_size to_bl_ucast) 

957 

958 
lemma ucast_down_drop': 

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

959 
"uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
37660  960 
to_bl (uc w) = drop n (to_bl w)" 
961 
by (auto simp add : source_size target_size to_bl_ucast) 

962 

963 
lemma scast_down_drop': 

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

964 
"sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
37660  965 
to_bl (sc w) = drop n (to_bl w)" 
966 
apply (subgoal_tac "sc = ucast") 

967 
apply safe 

968 
apply simp 

969 
apply (erule refl [THEN ucast_down_drop']) 

970 
apply (rule refl [THEN down_cast_same', symmetric]) 

971 
apply (simp add : source_size target_size is_down) 

972 
done 

973 

974 
lemma sint_up_scast': 

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

975 
"sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w" 
37660  976 
apply (unfold is_up) 
977 
apply safe 

978 
apply (simp add: scast_def word_sbin.eq_norm) 

979 
apply (rule box_equals) 

980 
prefer 3 

981 
apply (rule word_sbin.norm_Rep) 

982 
apply (rule sbintrunc_sbintrunc_l) 

983 
defer 

984 
apply (subst word_sbin.norm_Rep) 

985 
apply (rule refl) 

986 
apply simp 

987 
done 

988 

989 
lemma uint_up_ucast': 

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

990 
"uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w" 
37660  991 
apply (unfold is_up) 
992 
apply safe 

993 
apply (rule bin_eqI) 

994 
apply (fold word_test_bit_def) 

995 
apply (auto simp add: nth_ucast) 

996 
apply (auto simp add: test_bit_bin) 

997 
done 

998 

999 
lemmas down_cast_same = refl [THEN down_cast_same'] 

1000 
lemmas ucast_up_app = refl [THEN ucast_up_app'] 

1001 
lemmas ucast_down_drop = refl [THEN ucast_down_drop'] 

1002 
lemmas scast_down_drop = refl [THEN scast_down_drop'] 

1003 
lemmas uint_up_ucast = refl [THEN uint_up_ucast'] 

1004 
lemmas sint_up_scast = refl [THEN sint_up_scast'] 

1005 

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

1006 
lemma ucast_up_ucast': "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w" 
37660  1007 
apply (simp (no_asm) add: ucast_def) 
1008 
apply (clarsimp simp add: uint_up_ucast) 

1009 
done 

1010 

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

1011 
lemma scast_up_scast': "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w" 
37660  1012 
apply (simp (no_asm) add: scast_def) 
1013 
apply (clarsimp simp add: sint_up_scast) 

1014 
done 

1015 

1016 
lemma ucast_of_bl_up': 

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

1017 
"w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl" 
37660  1018 
by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) 
1019 

1020 
lemmas ucast_up_ucast = refl [THEN ucast_up_ucast'] 

1021 
lemmas scast_up_scast = refl [THEN scast_up_scast'] 

1022 
lemmas ucast_of_bl_up = refl [THEN ucast_of_bl_up'] 

1023 

1024 
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] 

1025 
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] 

1026 

1027 
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] 

1028 
lemmas isdus = is_up_down [where c = "scast", THEN iffD2] 

1029 
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] 

1030 
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] 

1031 

1032 
lemma up_ucast_surj: 

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

1033 
"is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660  1034 
surj (ucast :: 'a word => 'b word)" 
1035 
by (rule surjI, erule ucast_up_ucast_id) 

1036 

1037 
lemma up_scast_surj: 

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

1038 
"is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660  1039 
surj (scast :: 'a word => 'b word)" 
1040 
by (rule surjI, erule scast_up_scast_id) 

1041 

1042 
lemma down_scast_inj: 

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

1043 
"is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660  1044 
inj_on (ucast :: 'a word => 'b word) A" 
1045 
by (rule inj_on_inverseI, erule scast_down_scast_id) 

1046 

1047 
lemma down_ucast_inj: 

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

1048 
"is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660  1049 
inj_on (ucast :: 'a word => 'b word) A" 
1050 
by (rule inj_on_inverseI, erule ucast_down_ucast_id) 

1051 

1052 
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" 

1053 
by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) 

1054 

1055 
lemma ucast_down_no': 

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

1056 
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (number_of bin) = number_of bin" 
37660  1057 
apply (unfold word_number_of_def is_down) 
1058 
apply (clarsimp simp add: ucast_def word_ubin.eq_norm) 

1059 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1060 
apply (erule bintrunc_bintrunc_ge) 

1061 
done 

1062 

1063 
lemmas ucast_down_no = ucast_down_no' [OF refl] 

1064 

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

1065 
lemma ucast_down_bl': "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl" 
37660  1066 
unfolding of_bl_no by clarify (erule ucast_down_no) 
1067 

1068 
lemmas ucast_down_bl = ucast_down_bl' [OF refl] 

1069 

1070 
lemmas slice_def' = slice_def [unfolded word_size] 

1071 
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] 

1072 

1073 
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def 

1074 
lemmas word_log_bin_defs = word_log_defs 

1075 

1076 
text {* Executable equality *} 

1077 

38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38527
diff
changeset

1078 
instantiation word :: (len0) equal 
24333  1079 
begin 
1080 

38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38527
diff
changeset

1081 
definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where 
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38527
diff
changeset

1082 
"equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)" 
37660  1083 

1084 
instance proof 

38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38527
diff
changeset

1085 
qed (simp add: equal equal_word_def) 
37660  1086 

1087 
end 

1088 

1089 

1090 
subsection {* Word Arithmetic *} 

1091 

1092 
lemma word_less_alt: "(a < b) = (uint a < uint b)" 

1093 
unfolding word_less_def word_le_def 

1094 
by (auto simp del: word_uint.Rep_inject 

1095 
simp: word_uint.Rep_inject [symmetric]) 

1096 

1097 
lemma signed_linorder: "class.linorder word_sle word_sless" 

1098 
proof 

1099 
qed (unfold word_sle_def word_sless_def, auto) 

1100 

1101 
interpretation signed: linorder "word_sle" "word_sless" 

1102 
by (rule signed_linorder) 

1103 

1104 
lemma udvdI: 

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

1105 
"0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b" 
37660  1106 
by (auto simp: udvd_def) 
1107 

45604  1108 
lemmas word_div_no [simp] = word_div_def [of "number_of a" "number_of b"] for a b 
1109 

1110 
lemmas word_mod_no [simp] = word_mod_def [of "number_of a" "number_of b"] for a b 

1111 

1112 
lemmas word_less_no [simp] = word_less_def [of "number_of a" "number_of b"] for a b 

1113 

1114 
lemmas word_le_no [simp] = word_le_def [of "number_of a" "number_of b"] for a b 

1115 

1116 
lemmas word_sless_no [simp] = word_sless_def [of "number_of a" "number_of b"] for a b 

1117 

1118 
lemmas word_sle_no [simp] = word_sle_def [of "number_of a" "number_of b"] for a b 

37660  1119 

1120 
(* following two are available in class number_ring, 

1121 
but convenient to have them here here; 

1122 
note  the number_ring versions, numeral_0_eq_0 and numeral_1_eq_1 

1123 
are in the default simpset, so to use the automatic simplifications for 

1124 
(eg) sint (number_of bin) on sint 1, must do 

1125 
(simp add: word_1_no del: numeral_1_eq_1) 

1126 
*) 

1127 
lemmas word_0_wi_Pls = word_0_wi [folded Pls_def] 

1128 
lemmas word_0_no = word_0_wi_Pls [folded word_no_wi] 

1129 

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

1130 
lemma int_one_bin: "(1 :: int) = (Int.Pls BIT 1)" 
37660  1131 
unfolding Pls_def Bit_def by auto 
1132 

1133 
lemma word_1_no: 

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

1134 
"(1 :: 'a :: len0 word) = number_of (Int.Pls BIT 1)" 
37660  1135 
unfolding word_1_wi word_number_of_def int_one_bin by auto 
1136 

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

1137 
lemma word_m1_wi: "1 = word_of_int 1" 
37660  1138 
by (rule word_number_of_alt) 
1139 

1140 
lemma word_m1_wi_Min: "1 = word_of_int Int.Min" 

1141 
by (simp add: word_m1_wi number_of_eq) 

1142 

45805  1143 
lemma word_0_bl [simp]: "of_bl [] = 0" 
37660  1144 
unfolding word_0_wi of_bl_def by (simp add : Pls_def) 
1145 

1146 
lemma word_1_bl: "of_bl [True] = 1" 

1147 
unfolding word_1_wi of_bl_def 

1148 
by (simp add : bl_to_bin_def Bit_def Pls_def) 

1149 

1150 
lemma uint_eq_0 [simp] : "(uint 0 = 0)" 

1151 
unfolding word_0_wi 

1152 
by (simp add: word_ubin.eq_norm Pls_def [symmetric]) 

1153 

1154 
lemma of_bl_0 [simp] : "of_bl (replicate n False) = 0" 

1155 
by (simp add : word_0_wi of_bl_def bl_to_bin_rep_False Pls_def) 

1156 

45805  1157 
lemma to_bl_0 [simp]: 
37660  1158 
"to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" 
1159 
unfolding uint_bl 

1160 
by (simp add : word_size bin_to_bl_Pls Pls_def [symmetric]) 

1161 

1162 
lemma uint_0_iff: "(uint x = 0) = (x = 0)" 

1163 
by (auto intro!: word_uint.Rep_eqD) 

1164 

1165 
lemma unat_0_iff: "(unat x = 0) = (x = 0)" 

1166 
unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff) 

1167 

1168 
lemma unat_0 [simp]: "unat 0 = 0" 

1169 
unfolding unat_def by auto 

1170 

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

1171 
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)" 
37660  1172 
apply (unfold word_size) 
1173 
apply (rule box_equals) 

1174 
defer 

1175 
apply (rule word_uint.Rep_inverse)+ 

1176 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1177 
apply simp 

1178 
done 

1179 

1180 
lemmas size_0_same = size_0_same' [folded word_size] 

1181 

1182 
lemmas unat_eq_0 = unat_0_iff 

1183 
lemmas unat_eq_zero = unat_0_iff 

1184 

1185 
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)" 

1186 
by (auto simp: unat_0_iff [symmetric]) 

1187 

1188 
lemma ucast_0 [simp] : "ucast 0 = 0" 

1189 
unfolding ucast_def 

1190 
by simp (simp add: word_0_wi) 

1191 

1192 
lemma sint_0 [simp] : "sint 0 = 0" 

1193 
unfolding sint_uint 

1194 
by (simp add: Pls_def [symmetric]) 

1195 

1196 
lemma scast_0 [simp] : "scast 0 = 0" 

1197 
apply (unfold scast_def) 

1198 
apply simp 

1199 
apply (simp add: word_0_wi) 

1200 
done 

1201 

1202 
lemma sint_n1 [simp] : "sint 1 = 1" 

1203 
apply (unfold word_m1_wi_Min) 

1204 
apply (simp add: word_sbin.eq_norm) 

1205 
apply (unfold Min_def number_of_eq) 

1206 
apply simp 

1207 
done 

1208 

1209 
lemma scast_n1 [simp] : "scast 1 = 1" 

1210 
apply (unfold scast_def sint_n1) 

1211 
apply (unfold word_number_of_alt) 

1212 
apply (rule refl) 

1213 
done 

1214 

1215 
lemma uint_1 [simp] : "uint (1 :: 'a :: len word) = 1" 

1216 
unfolding word_1_wi 

1217 
by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) 

1218 

1219 
lemma unat_1 [simp] : "unat (1 :: 'a :: len word) = 1" 

1220 
by (unfold unat_def uint_1) auto 

1221 

1222 
lemma ucast_1 [simp] : "ucast (1 :: 'a :: len word) = 1" 

1223 
unfolding ucast_def word_1_wi 

1224 
by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) 

1225 

1226 
(* now, to get the weaker results analogous to word_div/mod_def *) 

1227 

1228 
lemmas word_arith_alts = 

45604  1229 
word_sub_wi [unfolded succ_def pred_def] 
1230 
word_arith_wis [unfolded succ_def pred_def] 

37660  1231 

1232 
lemmas word_succ_alt = word_arith_alts (5) 

1233 
lemmas word_pred_alt = word_arith_alts (6) 

1234 

1235 
subsection "Transferring goals from words to ints" 

1236 

1237 
lemma word_ths: 

1238 
shows 

1239 
word_succ_p1: "word_succ a = a + 1" and 

1240 
word_pred_m1: "word_pred a = a  1" and 

1241 
word_pred_succ: "word_pred (word_succ a) = a" and 

1242 
word_succ_pred: "word_succ (word_pred a) = a" and 

1243 
word_mult_succ: "word_succ a * b = b + a * b" 

1244 
by (rule word_uint.Abs_cases [of b], 

1245 
rule word_uint.Abs_cases [of a], 

1246 
simp add: pred_def succ_def add_commute mult_commute 

1247 
ring_distribs new_word_of_int_homs)+ 

1248 

1249 
lemmas uint_cong = arg_cong [where f = uint] 

1250 

1251 
lemmas uint_word_ariths = 

45604  1252 
word_arith_alts [THEN trans [OF uint_cong int_word_uint]] 
37660  1253 

1254 
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p] 

1255 

1256 
(* similar expressions for sint (arith operations) *) 

1257 
lemmas sint_word_ariths = uint_word_arith_bintrs 

1258 
[THEN uint_sint [symmetric, THEN trans], 

1259 
unfolded uint_sint bintr_arith1s bintr_ariths 

45604  1260 
len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep] 
1261 

1262 
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]] 

1263 
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]] 

37660  1264 

1265 
lemma word_pred_0_n1: "word_pred 0 = word_of_int 1" 

45550
73a4f31d41c4
Word.thy: reduce usage of numeralrepresentationdependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset

1266 
unfolding word_pred_def uint_eq_0 pred_def by simp 
37660  1267 

1268 
lemma word_pred_0_Min: "word_pred 0 = word_of_int Int.Min" 

1269 
by (simp add: word_pred_0_n1 number_of_eq) 

1270 

1271 
lemma word_m1_Min: " 1 = word_of_int Int.Min" 

1272 
unfolding Min_def by (simp only: word_of_int_hom_syms) 

1273 

1274 
lemma succ_pred_no [simp]: 

1275 
"word_succ (number_of bin) = number_of (Int.succ bin) & 

1276 
word_pred (number_of bin) = number_of (Int.pred bin)" 

1277 
unfolding word_number_of_def by (simp add : new_word_of_int_homs) 

1278 

1279 
lemma word_sp_01 [simp] : 

1280 
"word_succ 1 = 0 & word_succ 0 = 1 & word_pred 0 = 1 & word_pred 1 = 0" 

1281 
by (unfold word_0_no word_1_no) auto 

1282 

1283 
(* alternative approach to lifting arithmetic equalities *) 

1284 
lemma word_of_int_Ex: 

1285 
"\<exists>y. x = word_of_int y" 

1286 
by (rule_tac x="uint x" in exI) simp 

1287 

1288 

1289 
subsection "Order on fixedlength words" 

1290 

1291 
lemma word_zero_le [simp] : 

1292 
"0 <= (y :: 'a :: len0 word)" 

1293 
unfolding word_le_def by auto 

1294 

1295 
lemma word_m1_ge [simp] : "word_pred 0 >= y" 

1296 
unfolding word_le_def 

1297 
by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto 

1298 

1299 
lemmas word_n1_ge [simp] = word_m1_ge [simplified word_sp_01] 

1300 

1301 
lemmas word_not_simps [simp] = 

1302 
word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD] 

1303 

1304 
lemma word_gt_0: "0 < y = (0 ~= (y :: 'a :: len0 word))" 

1305 
unfolding word_less_def by auto 

1306 

45604  1307 
lemmas word_gt_0_no [simp] = word_gt_0 [of "number_of y"] for y 
37660  1308 

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

1309 
lemma word_sless_alt: "(a <s b) = (sint a < sint b)" 
37660  1310 
unfolding word_sle_def word_sless_def 
1311 
by (auto simp add: less_le) 

1312 

1313 
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)" 

1314 
unfolding unat_def word_le_def 

1315 
by (rule nat_le_eq_zle [symmetric]) simp 

1316 

1317 
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)" 

1318 
unfolding unat_def word_less_alt 

1319 
by (rule nat_less_eq_zless [symmetric]) simp 

1320 

1321 
lemma wi_less: 

1322 
"(word_of_int n < (word_of_int m :: 'a :: len0 word)) = 

1323 
(n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))" 

1324 
unfolding word_less_alt by (simp add: word_uint.eq_norm) 

1325 

1326 
lemma wi_le: 

1327 
"(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = 

1328 
(n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))" 

1329 
unfolding word_le_def by (simp add: word_uint.eq_norm) 

1330 

1331 
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)" 

1332 
apply (unfold udvd_def) 

1333 
apply safe 

1334 
apply (simp add: unat_def nat_mult_distrib) 

1335 
apply (simp add: uint_nat int_mult) 

1336 
apply (rule exI) 

1337 
apply safe 

1338 
prefer 2 

1339 
apply (erule notE) 

1340 
apply (rule refl) 

1341 
apply force 

1342 
done 

1343 

1344 
lemma udvd_iff_dvd: "x udvd y <> unat x dvd unat y" 

1345 
unfolding dvd_def udvd_nat_alt by 