author  haftmann 
Wed, 30 Jun 2010 17:12:38 +0200  
changeset 37660  56e3520b68b2 
parent 36899  bcd6fce5bf06 
child 37667  41acc0fa6b6c 
permissions  rwrr 
29628  1 
(* Title: HOL/Word/Word.thy 
37660  2 
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 Type_Length Misc_Typedef Boolean_Algebra Bool_List_Representation 
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uses ("~~/src/HOL/Tools/SMT/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 word) '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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notation fcomp (infixl "o>" 60) 

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notation scomp (infixl "o\<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))) o\<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 "o>" 60) 

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no_notation scomp (infixl "o\<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 "Arithmetic operations" 

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instantiation word :: (len0) "{number, uminus, minus, plus, one, zero, times, Divides.div, ord, bit}" 

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begin 

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definition 

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word_0_wi: "0 = word_of_int 0" 

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definition 

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word_1_wi: "1 = word_of_int 1" 

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definition 

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

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definition 

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

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definition 

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word_minus_def: " a = word_of_int ( uint a)" 

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definition 

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

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definition 

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

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definition 

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

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definition 

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word_number_of_def: "number_of w = word_of_int w" 

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definition 

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word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b" 

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definition 

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

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definition 

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word_and_def: 

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

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definition 

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word_or_def: 

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

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definition 

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word_xor_def: 

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

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definition 

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word_not_def: 

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

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instance .. 

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end 

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

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"a udvd b == EX n>=0. uint b = n * uint a" 

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

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"a <=s b == sint a <= sint b" 

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

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

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subsection "Bitwise operations" 

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instantiation word :: (len0) bits 

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begin 

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definition 

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

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definition 

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word_set_bit_def: "set_bit a n x = 

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

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definition 

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

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definition 

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

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

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

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

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

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

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definition 

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

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definition 

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

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instance .. 

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end 

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instantiation word :: (len) bitss 

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begin 

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definition 

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word_msb_def: 

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

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instance .. 

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end 

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

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"setBit w n == set_bit w n True" 

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

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"clearBit w n == set_bit w n False" 

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subsection "Shift operations" 

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

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

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

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

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

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

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

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

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

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

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

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"slice1 n w == of_bl (takefill False n (to_bl w))" 

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

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"slice n w == slice1 (size w  n) w" 

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subsection "Rotation" 

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

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"rotater1 ys == 

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case ys of [] => []  x # xs => last ys # butlast ys" 

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

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"rotater n == rotater1 ^^ n" 

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

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"word_rotr n w == of_bl (rotater n (to_bl w))" 

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

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"word_rotl n w == of_bl (rotate n (to_bl w))" 

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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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else word_rotl (nat ( i)) w" 

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subsection "Split and cat operations" 

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

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

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

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"word_split a == 

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case bin_split (len_of TYPE ('c)) (uint a) of 

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(u, v) => (word_of_int u, word_of_int v)" 

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

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"word_rcat ws == 

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word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))" 

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

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"word_rsplit w == 

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map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))" 

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

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"max_word \<equiv> word_of_int (2 ^ len_of TYPE('a)  1)" 

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

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"of_bool False = 0" 

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 "of_bool True = 1" 

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lemmas of_nth_def = word_set_bits_def 

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lemmas word_size_gt_0 [iff] = 

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xtr1 [OF word_size len_gt_0, standard] 

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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, standard] 

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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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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 ==> y mod m : {0::int ..< m}" 

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unfolding atLeastLessThan_alt 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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lemmas int_word_uint = td_ext_uint [THEN td_ext.eq_norm, standard] 

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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 sint_sbintrunc': 

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

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

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unfolding sint_uint 

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by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt) 

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lemma uint_sint: 

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

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unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le) 

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lemma bintr_uint': 

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"n >= size w ==> bintrunc n (uint w) = uint w" 

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apply (unfold word_size) 

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apply (subst word_ubin.norm_Rep [symmetric]) 

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apply (simp only: bintrunc_bintrunc_min word_size) 

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apply (simp add: min_max.inf_absorb2) 

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done 

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lemma wi_bintr': 

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"wb = word_of_int bin ==> n >= size wb ==> 

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word_of_int (bintrunc n bin) = wb" 

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unfolding word_size 

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

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lemmas bintr_uint = bintr_uint' [unfolded word_size] 

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lemmas wi_bintr = wi_bintr' [unfolded word_size] 

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lemma td_ext_sbin: 

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

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

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apply (unfold td_ext_def' sint_uint) 

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apply (simp add : word_ubin.eq_norm) 

410 
apply (cases "len_of TYPE('a)") 

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apply (auto simp add : sints_def) 

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apply (rule sym [THEN trans]) 

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apply (rule word_ubin.Abs_norm) 

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apply (simp only: bintrunc_sbintrunc) 

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apply (drule sym) 

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apply simp 

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done 

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lemmas td_ext_sint = td_ext_sbin 

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[simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]] 

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

423 
and interpretations do not produce thm duplicates. I.e. 

424 
we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, 

425 
because the latter is the same thm as the former *) 

426 
interpretation word_sint: 

427 
td_ext "sint ::'a::len word => int" 

428 
word_of_int 

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

430 
"%w. (w + 2^(len_of TYPE('a::len)  1)) mod 2^len_of TYPE('a::len)  

431 
2 ^ (len_of TYPE('a::len)  1)" 

432 
by (rule td_ext_sint) 

433 

434 
interpretation word_sbin: 

435 
td_ext "sint ::'a::len word => int" 

436 
word_of_int 

437 
"sints (len_of TYPE('a::len))" 

438 
"sbintrunc (len_of TYPE('a::len)  1)" 

439 
by (rule td_ext_sbin) 

440 

441 
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm, standard] 

442 

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lemmas td_sint = word_sint.td 

444 

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lemma word_number_of_alt: "number_of b == word_of_int (number_of b)" 

446 
unfolding word_number_of_def by (simp add: number_of_eq) 

447 

448 
lemma word_no_wi: "number_of = word_of_int" 

449 
by (auto simp: word_number_of_def intro: ext) 

450 

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lemma to_bl_def': 

452 
"(to_bl :: 'a :: len0 word => bool list) = 

453 
bin_to_bl (len_of TYPE('a)) o uint" 

454 
by (auto simp: to_bl_def intro: ext) 

455 

456 
lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w", standard] 

457 

458 
lemmas uints_mod = uints_def [unfolded no_bintr_alt1] 

459 

460 
lemma uint_bintrunc: "uint (number_of bin :: 'a word) = 

461 
number_of (bintrunc (len_of TYPE ('a :: len0)) bin)" 

462 
unfolding word_number_of_def number_of_eq 

463 
by (auto intro: word_ubin.eq_norm) 

464 

465 
lemma sint_sbintrunc: "sint (number_of bin :: 'a word) = 

466 
number_of (sbintrunc (len_of TYPE ('a :: len)  1) bin)" 

467 
unfolding word_number_of_def number_of_eq 

468 
by (subst word_sbin.eq_norm) simp 

469 

470 
lemma unat_bintrunc: 

471 
"unat (number_of bin :: 'a :: len0 word) = 

472 
number_of (bintrunc (len_of TYPE('a)) bin)" 

473 
unfolding unat_def nat_number_of_def 

474 
by (simp only: uint_bintrunc) 

475 

476 
(* WARNING  these may not always be helpful *) 

477 
declare 

478 
uint_bintrunc [simp] 

479 
sint_sbintrunc [simp] 

480 
unat_bintrunc [simp] 

481 

482 
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 ==> v = w" 

483 
apply (unfold word_size) 

484 
apply (rule word_uint.Rep_eqD) 

485 
apply (rule box_equals) 

486 
defer 

487 
apply (rule word_ubin.norm_Rep)+ 

488 
apply simp 

489 
done 

490 

491 
lemmas uint_lem = word_uint.Rep [unfolded uints_num mem_Collect_eq] 

492 
lemmas sint_lem = word_sint.Rep [unfolded sints_num mem_Collect_eq] 

493 
lemmas uint_ge_0 [iff] = uint_lem [THEN conjunct1, standard] 

494 
lemmas uint_lt2p [iff] = uint_lem [THEN conjunct2, standard] 

495 
lemmas sint_ge = sint_lem [THEN conjunct1, standard] 

496 
lemmas sint_lt = sint_lem [THEN conjunct2, standard] 

497 

498 
lemma sign_uint_Pls [simp]: 

499 
"bin_sign (uint x) = Int.Pls" 

500 
by (simp add: sign_Pls_ge_0 number_of_eq) 

501 

502 
lemmas uint_m2p_neg = iffD2 [OF diff_less_0_iff_less uint_lt2p, standard] 

503 
lemmas uint_m2p_not_non_neg = 

504 
iffD2 [OF linorder_not_le uint_m2p_neg, standard] 

505 

506 
lemma lt2p_lem: 

507 
"len_of TYPE('a) <= n ==> uint (w :: 'a :: len0 word) < 2 ^ n" 

508 
by (rule xtr8 [OF _ uint_lt2p]) simp 

509 

510 
lemmas uint_le_0_iff [simp] = 

511 
uint_ge_0 [THEN leD, THEN linorder_antisym_conv1, standard] 

512 

513 
lemma uint_nat: "uint w == int (unat w)" 

514 
unfolding unat_def by auto 

515 

516 
lemma uint_number_of: 

517 
"uint (number_of b :: 'a :: len0 word) = number_of b mod 2 ^ len_of TYPE('a)" 

518 
unfolding word_number_of_alt 

519 
by (simp only: int_word_uint) 

520 

521 
lemma unat_number_of: 

522 
"bin_sign b = Int.Pls ==> 

523 
unat (number_of b::'a::len0 word) = number_of b mod 2 ^ len_of TYPE ('a)" 

524 
apply (unfold unat_def) 

525 
apply (clarsimp simp only: uint_number_of) 

526 
apply (rule nat_mod_distrib [THEN trans]) 

527 
apply (erule sign_Pls_ge_0 [THEN iffD1]) 

528 
apply (simp_all add: nat_power_eq) 

529 
done 

530 

531 
lemma sint_number_of: "sint (number_of b :: 'a :: len word) = (number_of b + 

532 
2 ^ (len_of TYPE('a)  1)) mod 2 ^ len_of TYPE('a)  

533 
2 ^ (len_of TYPE('a)  1)" 

534 
unfolding word_number_of_alt by (rule int_word_sint) 

535 

536 
lemma word_of_int_bin [simp] : 

537 
"(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)" 

538 
unfolding word_number_of_alt by auto 

539 

540 
lemma word_int_case_wi: 

541 
"word_int_case f (word_of_int i :: 'b word) = 

542 
f (i mod 2 ^ len_of TYPE('b::len0))" 

543 
unfolding word_int_case_def by (simp add: word_uint.eq_norm) 

544 

545 
lemma word_int_split: 

546 
"P (word_int_case f x) = 

547 
(ALL i. x = (word_of_int i :: 'b :: len0 word) & 

548 
0 <= i & i < 2 ^ len_of TYPE('b) > P (f i))" 

549 
unfolding word_int_case_def 

550 
by (auto simp: word_uint.eq_norm int_mod_eq') 

551 

552 
lemma word_int_split_asm: 

553 
"P (word_int_case f x) = 

554 
(~ (EX n. x = (word_of_int n :: 'b::len0 word) & 

555 
0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))" 

556 
unfolding word_int_case_def 

557 
by (auto simp: word_uint.eq_norm int_mod_eq') 

558 

559 
lemmas uint_range' = 

560 
word_uint.Rep [unfolded uints_num mem_Collect_eq, standard] 

561 
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def 

562 
sints_num mem_Collect_eq, standard] 

563 

564 
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w" 

565 
unfolding word_size by (rule uint_range') 

566 

567 
lemma sint_range_size: 

568 
" (2 ^ (size w  Suc 0)) <= sint w & sint w < 2 ^ (size w  Suc 0)" 

569 
unfolding word_size by (rule sint_range') 

570 

571 
lemmas sint_above_size = sint_range_size 

572 
[THEN conjunct2, THEN [2] xtr8, folded One_nat_def, standard] 

573 

574 
lemmas sint_below_size = sint_range_size 

575 
[THEN conjunct1, THEN [2] order_trans, folded One_nat_def, standard] 

576 

577 
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)" 

578 
unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) 

579 

580 
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n > n < size w" 

581 
apply (unfold word_test_bit_def) 

582 
apply (subst word_ubin.norm_Rep [symmetric]) 

583 
apply (simp only: nth_bintr word_size) 

584 
apply fast 

585 
done 

586 

587 
lemma word_eqI [rule_format] : 

588 
fixes u :: "'a::len0 word" 

589 
shows "(ALL n. n < size u > u !! n = v !! n) ==> u = v" 

590 
apply (rule test_bit_eq_iff [THEN iffD1]) 

591 
apply (rule ext) 

592 
apply (erule allE) 

593 
apply (erule impCE) 

594 
prefer 2 

595 
apply assumption 

596 
apply (auto dest!: test_bit_size simp add: word_size) 

597 
done 

598 

599 
lemmas word_eqD = test_bit_eq_iff [THEN iffD2, THEN fun_cong, standard] 

600 

601 
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)" 

602 
unfolding word_test_bit_def word_size 

603 
by (simp add: nth_bintr [symmetric]) 

604 

605 
lemmas test_bit_bin = test_bit_bin' [unfolded word_size] 

606 

607 
lemma bin_nth_uint_imp': "bin_nth (uint w) n > n < size w" 

608 
apply (unfold word_size) 

609 
apply (rule impI) 

610 
apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) 

611 
apply (subst word_ubin.norm_Rep) 

612 
apply assumption 

613 
done 

614 

615 
lemma bin_nth_sint': 

616 
"n >= size w > bin_nth (sint w) n = bin_nth (sint w) (size w  1)" 

617 
apply (rule impI) 

618 
apply (subst word_sbin.norm_Rep [symmetric]) 

619 
apply (simp add : nth_sbintr word_size) 

620 
apply auto 

621 
done 

622 

623 
lemmas bin_nth_uint_imp = bin_nth_uint_imp' [rule_format, unfolded word_size] 

624 
lemmas bin_nth_sint = bin_nth_sint' [rule_format, unfolded word_size] 

625 

626 
(* type definitions theorem for in terms of equivalent bool list *) 

627 
lemma td_bl: 

628 
"type_definition (to_bl :: 'a::len0 word => bool list) 

629 
of_bl 

630 
{bl. length bl = len_of TYPE('a)}" 

631 
apply (unfold type_definition_def of_bl_def to_bl_def) 

632 
apply (simp add: word_ubin.eq_norm) 

633 
apply safe 

634 
apply (drule sym) 

635 
apply simp 

636 
done 

637 

638 
interpretation word_bl: 

639 
type_definition "to_bl :: 'a::len0 word => bool list" 

640 
of_bl 

641 
"{bl. length bl = len_of TYPE('a::len0)}" 

642 
by (rule td_bl) 

643 

644 
lemma word_size_bl: "size w == size (to_bl w)" 

645 
unfolding word_size by auto 

646 

647 
lemma to_bl_use_of_bl: 

648 
"(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))" 

649 
by (fastsimp elim!: word_bl.Abs_inverse [simplified]) 

650 

651 
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" 

652 
unfolding word_reverse_def by (simp add: word_bl.Abs_inverse) 

653 

654 
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" 

655 
unfolding word_reverse_def by (simp add : word_bl.Abs_inverse) 

656 

657 
lemma word_rev_gal: "word_reverse w = u ==> word_reverse u = w" 

658 
by auto 

659 

660 
lemmas word_rev_gal' = sym [THEN word_rev_gal, symmetric, standard] 

661 

662 
lemmas length_bl_gt_0 [iff] = xtr1 [OF word_bl.Rep' len_gt_0, standard] 

663 
lemmas bl_not_Nil [iff] = 

664 
length_bl_gt_0 [THEN length_greater_0_conv [THEN iffD1], standard] 

665 
lemmas length_bl_neq_0 [iff] = length_bl_gt_0 [THEN gr_implies_not0] 

666 

667 
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = Int.Min)" 

668 
apply (unfold to_bl_def sint_uint) 

669 
apply (rule trans [OF _ bl_sbin_sign]) 

670 
apply simp 

671 
done 

672 

673 
lemma of_bl_drop': 

674 
"lend = length bl  len_of TYPE ('a :: len0) ==> 

675 
of_bl (drop lend bl) = (of_bl bl :: 'a word)" 

676 
apply (unfold of_bl_def) 

677 
apply (clarsimp simp add : trunc_bl2bin [symmetric]) 

678 
done 

679 

680 
lemmas of_bl_no = of_bl_def [folded word_number_of_def] 

681 

682 
lemma test_bit_of_bl: 

683 
"(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)" 

684 
apply (unfold of_bl_def word_test_bit_def) 

685 
apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) 

686 
done 

687 

688 
lemma no_of_bl: 

689 
"(number_of bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) bin)" 

690 
unfolding word_size of_bl_no by (simp add : word_number_of_def) 

691 

692 
lemma uint_bl: "to_bl w == bin_to_bl (size w) (uint w)" 

693 
unfolding word_size to_bl_def by auto 

694 

695 
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" 

696 
unfolding uint_bl by (simp add : word_size) 

697 

698 
lemma to_bl_of_bin: 

699 
"to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" 

700 
unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size) 

701 

702 
lemmas to_bl_no_bin [simp] = to_bl_of_bin [folded word_number_of_def] 

703 

704 
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" 

705 
unfolding uint_bl by (simp add : word_size) 

706 

707 
lemmas uint_bl_bin [simp] = trans [OF bin_bl_bin word_ubin.norm_Rep, standard] 

708 

709 
lemmas num_AB_u [simp] = word_uint.Rep_inverse 

710 
[unfolded o_def word_number_of_def [symmetric], standard] 

711 
lemmas num_AB_s [simp] = word_sint.Rep_inverse 

712 
[unfolded o_def word_number_of_def [symmetric], standard] 

713 

714 
(* naturals *) 

715 
lemma uints_unats: "uints n = int ` unats n" 

716 
apply (unfold unats_def uints_num) 

717 
apply safe 

718 
apply (rule_tac image_eqI) 

719 
apply (erule_tac nat_0_le [symmetric]) 

720 
apply auto 

721 
apply (erule_tac nat_less_iff [THEN iffD2]) 

722 
apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) 

723 
apply (auto simp add : nat_power_eq int_power) 

724 
done 

725 

726 
lemma unats_uints: "unats n = nat ` uints n" 

727 
by (auto simp add : uints_unats image_iff) 

728 

729 
lemmas bintr_num = word_ubin.norm_eq_iff 

730 
[symmetric, folded word_number_of_def, standard] 

731 
lemmas sbintr_num = word_sbin.norm_eq_iff 

732 
[symmetric, folded word_number_of_def, standard] 

733 

734 
lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def, standard] 

735 
lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def, standard]; 

736 

737 
(* don't add these to simpset, since may want bintrunc n w to be simplified; 

738 
may want these in reverse, but loop as simp rules, so use following *) 

739 

740 
lemma num_of_bintr': 

741 
"bintrunc (len_of TYPE('a :: len0)) a = b ==> 

742 
number_of a = (number_of b :: 'a word)" 

743 
apply safe 

744 
apply (rule_tac num_of_bintr [symmetric]) 

745 
done 

746 

747 
lemma num_of_sbintr': 

748 
"sbintrunc (len_of TYPE('a :: len)  1) a = b ==> 

749 
number_of a = (number_of b :: 'a word)" 

750 
apply safe 

751 
apply (rule_tac num_of_sbintr [symmetric]) 

752 
done 

753 

754 
lemmas num_abs_bintr = sym [THEN trans, 

755 
OF num_of_bintr word_number_of_def, standard] 

756 
lemmas num_abs_sbintr = sym [THEN trans, 

757 
OF num_of_sbintr word_number_of_def, standard] 

758 

759 
(** cast  note, no arg for new length, as it's determined by type of result, 

760 
thus in "cast w = w, the type means cast to length of w! **) 

761 

762 
lemma ucast_id: "ucast w = w" 

763 
unfolding ucast_def by auto 

764 

765 
lemma scast_id: "scast w = w" 

766 
unfolding scast_def by auto 

767 

768 
lemma ucast_bl: "ucast w == of_bl (to_bl w)" 

769 
unfolding ucast_def of_bl_def uint_bl 

770 
by (auto simp add : word_size) 

771 

772 
lemma nth_ucast: 

773 
"(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))" 

774 
apply (unfold ucast_def test_bit_bin) 

775 
apply (simp add: word_ubin.eq_norm nth_bintr word_size) 

776 
apply (fast elim!: bin_nth_uint_imp) 

777 
done 

778 

779 
(* for literal u(s)cast *) 

780 

781 
lemma ucast_bintr [simp]: 

782 
"ucast (number_of w ::'a::len0 word) = 

783 
number_of (bintrunc (len_of TYPE('a)) w)" 

784 
unfolding ucast_def by simp 

785 

786 
lemma scast_sbintr [simp]: 

787 
"scast (number_of w ::'a::len word) = 

788 
number_of (sbintrunc (len_of TYPE('a)  Suc 0) w)" 

789 
unfolding scast_def by simp 

790 

791 
lemmas source_size = source_size_def [unfolded Let_def word_size] 

792 
lemmas target_size = target_size_def [unfolded Let_def word_size] 

793 
lemmas is_down = is_down_def [unfolded source_size target_size] 

794 
lemmas is_up = is_up_def [unfolded source_size target_size] 

795 

796 
lemmas is_up_down = trans [OF is_up is_down [symmetric], standard] 

797 

798 
lemma down_cast_same': "uc = ucast ==> is_down uc ==> uc = scast" 

799 
apply (unfold is_down) 

800 
apply safe 

801 
apply (rule ext) 

802 
apply (unfold ucast_def scast_def uint_sint) 

803 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

804 
apply simp 

805 
done 

806 

807 
lemma word_rev_tf': 

808 
"r = to_bl (of_bl bl) ==> r = rev (takefill False (length r) (rev bl))" 

809 
unfolding of_bl_def uint_bl 

810 
by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size) 

811 

812 
lemmas word_rev_tf = refl [THEN word_rev_tf', unfolded word_bl.Rep', standard] 

813 

814 
lemmas word_rep_drop = word_rev_tf [simplified takefill_alt, 

815 
simplified, simplified rev_take, simplified] 

816 

817 
lemma to_bl_ucast: 

818 
"to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 

819 
replicate (len_of TYPE('a)  len_of TYPE('b)) False @ 

820 
drop (len_of TYPE('b)  len_of TYPE('a)) (to_bl w)" 

821 
apply (unfold ucast_bl) 

822 
apply (rule trans) 

823 
apply (rule word_rep_drop) 

824 
apply simp 

825 
done 

826 

827 
lemma ucast_up_app': 

828 
"uc = ucast ==> source_size uc + n = target_size uc ==> 

829 
to_bl (uc w) = replicate n False @ (to_bl w)" 

830 
by (auto simp add : source_size target_size to_bl_ucast) 

831 

832 
lemma ucast_down_drop': 

833 
"uc = ucast ==> source_size uc = target_size uc + n ==> 

834 
to_bl (uc w) = drop n (to_bl w)" 

835 
by (auto simp add : source_size target_size to_bl_ucast) 

836 

837 
lemma scast_down_drop': 

838 
"sc = scast ==> source_size sc = target_size sc + n ==> 

839 
to_bl (sc w) = drop n (to_bl w)" 

840 
apply (subgoal_tac "sc = ucast") 

841 
apply safe 

842 
apply simp 

843 
apply (erule refl [THEN ucast_down_drop']) 

844 
apply (rule refl [THEN down_cast_same', symmetric]) 

845 
apply (simp add : source_size target_size is_down) 

846 
done 

847 

848 
lemma sint_up_scast': 

849 
"sc = scast ==> is_up sc ==> sint (sc w) = sint w" 

850 
apply (unfold is_up) 

851 
apply safe 

852 
apply (simp add: scast_def word_sbin.eq_norm) 

853 
apply (rule box_equals) 

854 
prefer 3 

855 
apply (rule word_sbin.norm_Rep) 

856 
apply (rule sbintrunc_sbintrunc_l) 

857 
defer 

858 
apply (subst word_sbin.norm_Rep) 

859 
apply (rule refl) 

860 
apply simp 

861 
done 

862 

863 
lemma uint_up_ucast': 

864 
"uc = ucast ==> is_up uc ==> uint (uc w) = uint w" 

865 
apply (unfold is_up) 

866 
apply safe 

867 
apply (rule bin_eqI) 

868 
apply (fold word_test_bit_def) 

869 
apply (auto simp add: nth_ucast) 

870 
apply (auto simp add: test_bit_bin) 

871 
done 

872 

873 
lemmas down_cast_same = refl [THEN down_cast_same'] 

874 
lemmas ucast_up_app = refl [THEN ucast_up_app'] 

875 
lemmas ucast_down_drop = refl [THEN ucast_down_drop'] 

876 
lemmas scast_down_drop = refl [THEN scast_down_drop'] 

877 
lemmas uint_up_ucast = refl [THEN uint_up_ucast'] 

878 
lemmas sint_up_scast = refl [THEN sint_up_scast'] 

879 

880 
lemma ucast_up_ucast': "uc = ucast ==> is_up uc ==> ucast (uc w) = ucast w" 

881 
apply (simp (no_asm) add: ucast_def) 

882 
apply (clarsimp simp add: uint_up_ucast) 

883 
done 

884 

885 
lemma scast_up_scast': "sc = scast ==> is_up sc ==> scast (sc w) = scast w" 

886 
apply (simp (no_asm) add: scast_def) 

887 
apply (clarsimp simp add: sint_up_scast) 

888 
done 

889 

890 
lemma ucast_of_bl_up': 

891 
"w = of_bl bl ==> size bl <= size w ==> ucast w = of_bl bl" 

892 
by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) 

893 

894 
lemmas ucast_up_ucast = refl [THEN ucast_up_ucast'] 

895 
lemmas scast_up_scast = refl [THEN scast_up_scast'] 

896 
lemmas ucast_of_bl_up = refl [THEN ucast_of_bl_up'] 

897 

898 
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] 

899 
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] 

900 

901 
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] 

902 
lemmas isdus = is_up_down [where c = "scast", THEN iffD2] 

903 
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] 

904 
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] 

905 

906 
lemma up_ucast_surj: 

907 
"is_up (ucast :: 'b::len0 word => 'a::len0 word) ==> 

908 
surj (ucast :: 'a word => 'b word)" 

909 
by (rule surjI, erule ucast_up_ucast_id) 

910 

911 
lemma up_scast_surj: 

912 
"is_up (scast :: 'b::len word => 'a::len word) ==> 

913 
surj (scast :: 'a word => 'b word)" 

914 
by (rule surjI, erule scast_up_scast_id) 

915 

916 
lemma down_scast_inj: 

917 
"is_down (scast :: 'b::len word => 'a::len word) ==> 

918 
inj_on (ucast :: 'a word => 'b word) A" 

919 
by (rule inj_on_inverseI, erule scast_down_scast_id) 

920 

921 
lemma down_ucast_inj: 

922 
"is_down (ucast :: 'b::len0 word => 'a::len0 word) ==> 

923 
inj_on (ucast :: 'a word => 'b word) A" 

924 
by (rule inj_on_inverseI, erule ucast_down_ucast_id) 

925 

926 
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" 

927 
by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) 

928 

929 
lemma ucast_down_no': 

930 
"uc = ucast ==> is_down uc ==> uc (number_of bin) = number_of bin" 

931 
apply (unfold word_number_of_def is_down) 

932 
apply (clarsimp simp add: ucast_def word_ubin.eq_norm) 

933 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

934 
apply (erule bintrunc_bintrunc_ge) 

935 
done 

936 

937 
lemmas ucast_down_no = ucast_down_no' [OF refl] 

938 

939 
lemma ucast_down_bl': "uc = ucast ==> is_down uc ==> uc (of_bl bl) = of_bl bl" 

940 
unfolding of_bl_no by clarify (erule ucast_down_no) 

941 

942 
lemmas ucast_down_bl = ucast_down_bl' [OF refl] 

943 

944 
lemmas slice_def' = slice_def [unfolded word_size] 

945 
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] 

946 

947 
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def 

948 
lemmas word_log_bin_defs = word_log_defs 

949 

950 
text {* Executable equality *} 

951 

952 
instantiation word :: ("{len0}") eq 

24333  953 
begin 
954 

37660  955 
definition eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where 
956 
"eq_word k l \<longleftrightarrow> HOL.eq (uint k) (uint l)" 

957 

958 
instance proof 

959 
qed (simp add: eq eq_word_def) 

960 

961 
end 

962 

963 

964 
subsection {* Word Arithmetic *} 

965 

966 
lemma word_less_alt: "(a < b) = (uint a < uint b)" 

967 
unfolding word_less_def word_le_def 

968 
by (auto simp del: word_uint.Rep_inject 

969 
simp: word_uint.Rep_inject [symmetric]) 

970 

971 
lemma signed_linorder: "class.linorder word_sle word_sless" 

972 
proof 

973 
qed (unfold word_sle_def word_sless_def, auto) 

974 

975 
interpretation signed: linorder "word_sle" "word_sless" 

976 
by (rule signed_linorder) 

977 

978 
lemmas word_arith_wis = 

979 
word_add_def word_mult_def word_minus_def 

980 
word_succ_def word_pred_def word_0_wi word_1_wi 

981 

982 
lemma udvdI: 

983 
"0 \<le> n ==> uint b = n * uint a ==> a udvd b" 

984 
by (auto simp: udvd_def) 

985 

986 
lemmas word_div_no [simp] = 

987 
word_div_def [of "number_of a" "number_of b", standard] 

988 

989 
lemmas word_mod_no [simp] = 

990 
word_mod_def [of "number_of a" "number_of b", standard] 

991 

992 
lemmas word_less_no [simp] = 

993 
word_less_def [of "number_of a" "number_of b", standard] 

994 

995 
lemmas word_le_no [simp] = 

996 
word_le_def [of "number_of a" "number_of b", standard] 

997 

998 
lemmas word_sless_no [simp] = 

999 
word_sless_def [of "number_of a" "number_of b", standard] 

1000 

1001 
lemmas word_sle_no [simp] = 

1002 
word_sle_def [of "number_of a" "number_of b", standard] 

1003 

1004 
(* following two are available in class number_ring, 

1005 
but convenient to have them here here; 

1006 
note  the number_ring versions, numeral_0_eq_0 and numeral_1_eq_1 

1007 
are in the default simpset, so to use the automatic simplifications for 

1008 
(eg) sint (number_of bin) on sint 1, must do 

1009 
(simp add: word_1_no del: numeral_1_eq_1) 

1010 
*) 

1011 
lemmas word_0_wi_Pls = word_0_wi [folded Pls_def] 

1012 
lemmas word_0_no = word_0_wi_Pls [folded word_no_wi] 

1013 

1014 
lemma int_one_bin: "(1 :: int) == (Int.Pls BIT 1)" 

1015 
unfolding Pls_def Bit_def by auto 

1016 

1017 
lemma word_1_no: 

1018 
"(1 :: 'a :: len0 word) == number_of (Int.Pls BIT 1)" 

1019 
unfolding word_1_wi word_number_of_def int_one_bin by auto 

1020 

1021 
lemma word_m1_wi: "1 == word_of_int 1" 

1022 
by (rule word_number_of_alt) 

1023 

1024 
lemma word_m1_wi_Min: "1 = word_of_int Int.Min" 

1025 
by (simp add: word_m1_wi number_of_eq) 

1026 

1027 
lemma word_0_bl: "of_bl [] = 0" 

1028 
unfolding word_0_wi of_bl_def by (simp add : Pls_def) 

1029 

1030 
lemma word_1_bl: "of_bl [True] = 1" 

1031 
unfolding word_1_wi of_bl_def 

1032 
by (simp add : bl_to_bin_def Bit_def Pls_def) 

1033 

1034 
lemma uint_eq_0 [simp] : "(uint 0 = 0)" 

1035 
unfolding word_0_wi 

1036 
by (simp add: word_ubin.eq_norm Pls_def [symmetric]) 

1037 

1038 
lemma of_bl_0 [simp] : "of_bl (replicate n False) = 0" 

1039 
by (simp add : word_0_wi of_bl_def bl_to_bin_rep_False Pls_def) 

1040 

1041 
lemma to_bl_0: 

1042 
"to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" 

1043 
unfolding uint_bl 

1044 
by (simp add : word_size bin_to_bl_Pls Pls_def [symmetric]) 

1045 

1046 
lemma uint_0_iff: "(uint x = 0) = (x = 0)" 

1047 
by (auto intro!: word_uint.Rep_eqD) 

1048 

1049 
lemma unat_0_iff: "(unat x = 0) = (x = 0)" 

1050 
unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff) 

1051 

1052 
lemma unat_0 [simp]: "unat 0 = 0" 

1053 
unfolding unat_def by auto 

1054 

1055 
lemma size_0_same': "size w = 0 ==> w = (v :: 'a :: len0 word)" 

1056 
apply (unfold word_size) 

1057 
apply (rule box_equals) 

1058 
defer 

1059 
apply (rule word_uint.Rep_inverse)+ 

1060 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1061 
apply simp 

1062 
done 

1063 

1064 
lemmas size_0_same = size_0_same' [folded word_size] 

1065 

1066 
lemmas unat_eq_0 = unat_0_iff 

1067 
lemmas unat_eq_zero = unat_0_iff 

1068 

1069 
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)" 

1070 
by (auto simp: unat_0_iff [symmetric]) 

1071 

1072 
lemma ucast_0 [simp] : "ucast 0 = 0" 

1073 
unfolding ucast_def 

1074 
by simp (simp add: word_0_wi) 

1075 

1076 
lemma sint_0 [simp] : "sint 0 = 0" 

1077 
unfolding sint_uint 

1078 
by (simp add: Pls_def [symmetric]) 

1079 

1080 
lemma scast_0 [simp] : "scast 0 = 0" 

1081 
apply (unfold scast_def) 

1082 
apply simp 

1083 
apply (simp add: word_0_wi) 

1084 
done 

1085 

1086 
lemma sint_n1 [simp] : "sint 1 = 1" 

1087 
apply (unfold word_m1_wi_Min) 

1088 
apply (simp add: word_sbin.eq_norm) 

1089 
apply (unfold Min_def number_of_eq) 

1090 
apply simp 

1091 
done 

1092 

1093 
lemma scast_n1 [simp] : "scast 1 = 1" 

1094 
apply (unfold scast_def sint_n1) 

1095 
apply (unfold word_number_of_alt) 

1096 
apply (rule refl) 

1097 
done 

1098 

1099 
lemma uint_1 [simp] : "uint (1 :: 'a :: len word) = 1" 

1100 
unfolding word_1_wi 

1101 
by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) 

1102 

1103 
lemma unat_1 [simp] : "unat (1 :: 'a :: len word) = 1" 

1104 
by (unfold unat_def uint_1) auto 

1105 

1106 
lemma ucast_1 [simp] : "ucast (1 :: 'a :: len word) = 1" 

1107 
unfolding ucast_def word_1_wi 

1108 
by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) 

1109 

1110 
(* abstraction preserves the operations 

1111 
(the definitions tell this for bins in range uint) *) 

1112 

1113 
lemmas arths = 

1114 
bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], 

1115 
folded word_ubin.eq_norm, standard] 

1116 

1117 
lemma wi_homs: 

1118 
shows 

1119 
wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and 

1120 
wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and 

1121 
wi_hom_neg: " word_of_int a = word_of_int ( a)" and 

1122 
wi_hom_succ: "word_succ (word_of_int a) = word_of_int (Int.succ a)" and 

1123 
wi_hom_pred: "word_pred (word_of_int a) = word_of_int (Int.pred a)" 

1124 
by (auto simp: word_arith_wis arths) 

1125 

1126 
lemmas wi_hom_syms = wi_homs [symmetric] 

1127 

1128 
lemma word_sub_def: "a  b == a +  (b :: 'a :: len0 word)" 

1129 
unfolding word_sub_wi diff_def 

1130 
by (simp only : word_uint.Rep_inverse wi_hom_syms) 

1131 

1132 
lemmas word_diff_minus = word_sub_def [THEN meta_eq_to_obj_eq, standard] 

1133 

1134 
lemma word_of_int_sub_hom: 

1135 
"(word_of_int a)  word_of_int b = word_of_int (a  b)" 

1136 
unfolding word_sub_def diff_def by (simp only : wi_homs) 

1137 

1138 
lemmas new_word_of_int_homs = 

1139 
word_of_int_sub_hom wi_homs word_0_wi word_1_wi 

1140 

1141 
lemmas new_word_of_int_hom_syms = new_word_of_int_homs [symmetric, standard] 

1142 

1143 
lemmas word_of_int_hom_syms = 

1144 
new_word_of_int_hom_syms [unfolded succ_def pred_def] 

1145 

1146 
lemmas word_of_int_homs = 

1147 
new_word_of_int_homs [unfolded succ_def pred_def] 

1148 

1149 
lemmas word_of_int_add_hom = word_of_int_homs (2) 

1150 
lemmas word_of_int_mult_hom = word_of_int_homs (3) 

1151 
lemmas word_of_int_minus_hom = word_of_int_homs (4) 

1152 
lemmas word_of_int_succ_hom = word_of_int_homs (5) 

1153 
lemmas word_of_int_pred_hom = word_of_int_homs (6) 

1154 
lemmas word_of_int_0_hom = word_of_int_homs (7) 

1155 
lemmas word_of_int_1_hom = word_of_int_homs (8) 

1156 

1157 
(* now, to get the weaker results analogous to word_div/mod_def *) 

1158 

1159 
lemmas word_arith_alts = 

1160 
word_sub_wi [unfolded succ_def pred_def, standard] 

1161 
word_arith_wis [unfolded succ_def pred_def, standard] 

1162 

1163 
lemmas word_sub_alt = word_arith_alts (1) 

1164 
lemmas word_add_alt = word_arith_alts (2) 

1165 
lemmas word_mult_alt = word_arith_alts (3) 

1166 
lemmas word_minus_alt = word_arith_alts (4) 

1167 
lemmas word_succ_alt = word_arith_alts (5) 

1168 
lemmas word_pred_alt = word_arith_alts (6) 

1169 
lemmas word_0_alt = word_arith_alts (7) 

1170 
lemmas word_1_alt = word_arith_alts (8) 

1171 

1172 
subsection "Transferring goals from words to ints" 

1173 

1174 
lemma word_ths: 

1175 
shows 

1176 
word_succ_p1: "word_succ a = a + 1" and 

1177 
word_pred_m1: "word_pred a = a  1" and 

1178 
word_pred_succ: "word_pred (word_succ a) = a" and 

1179 
word_succ_pred: "word_succ (word_pred a) = a" and 

1180 
word_mult_succ: "word_succ a * b = b + a * b" 

1181 
by (rule word_uint.Abs_cases [of b], 

1182 
rule word_uint.Abs_cases [of a], 

1183 
simp add: pred_def succ_def add_commute mult_commute 

1184 
ring_distribs new_word_of_int_homs)+ 

1185 

1186 
lemmas uint_cong = arg_cong [where f = uint] 

1187 

1188 
lemmas uint_word_ariths = 

1189 
word_arith_alts [THEN trans [OF uint_cong int_word_uint], standard] 

1190 

1191 
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p] 

1192 

1193 
(* similar expressions for sint (arith operations) *) 

1194 
lemmas sint_word_ariths = uint_word_arith_bintrs 

1195 
[THEN uint_sint [symmetric, THEN trans], 

1196 
unfolded uint_sint bintr_arith1s bintr_ariths 

1197 
len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep, standard] 

1198 

1199 
lemmas uint_div_alt = word_div_def 

1200 
[THEN trans [OF uint_cong int_word_uint], standard] 

1201 
lemmas uint_mod_alt = word_mod_def 

1202 
[THEN trans [OF uint_cong int_word_uint], standard] 

1203 

1204 
lemma word_pred_0_n1: "word_pred 0 = word_of_int 1" 

1205 
unfolding word_pred_def number_of_eq 

1206 
by (simp add : pred_def word_no_wi) 

1207 

1208 
lemma word_pred_0_Min: "word_pred 0 = word_of_int Int.Min" 

1209 
by (simp add: word_pred_0_n1 number_of_eq) 

1210 

1211 
lemma word_m1_Min: " 1 = word_of_int Int.Min" 

1212 
unfolding Min_def by (simp only: word_of_int_hom_syms) 

1213 

1214 
lemma succ_pred_no [simp]: 

1215 
"word_succ (number_of bin) = number_of (Int.succ bin) & 

1216 
word_pred (number_of bin) = number_of (Int.pred bin)" 

1217 
unfolding word_number_of_def by (simp add : new_word_of_int_homs) 

1218 

1219 
lemma word_sp_01 [simp] : 

1220 
"word_succ 1 = 0 & word_succ 0 = 1 & word_pred 0 = 1 & word_pred 1 = 0" 

1221 
by (unfold word_0_no word_1_no) auto 

1222 

1223 
(* alternative approach to lifting arithmetic equalities *) 

1224 
lemma word_of_int_Ex: 

1225 
"\<exists>y. x = word_of_int y" 

1226 
by (rule_tac x="uint x" in exI) simp 

1227 

1228 
lemma word_arith_eqs: 

1229 
fixes a :: "'a::len0 word" 

1230 
fixes b :: "'a::len0 word" 

1231 
shows 

1232 
word_add_0: "0 + a = a" and 

1233 
word_add_0_right: "a + 0 = a" and 

1234 
word_mult_1: "1 * a = a" and 

1235 
word_mult_1_right: "a * 1 = a" and 

1236 
word_add_commute: "a + b = b + a" and 

1237 
word_add_assoc: "a + b + c = a + (b + c)" and 

1238 
word_add_left_commute: "a + (b + c) = b + (a + c)" and 

1239 
word_mult_commute: "a * b = b * a" and 

1240 
word_mult_assoc: "a * b * c = a * (b * c)" and 

1241 
word_mult_left_commute: "a * (b * c) = b * (a * c)" and 

1242 
word_left_distrib: "(a + b) * c = a * c + b * c" and 

1243 
word_right_distrib: "a * (b + c) = a * b + a * c" and 

1244 
word_left_minus: " a + a = 0" and 

1245 
word_diff_0_right: "a  0 = a" and 

1246 
word_diff_self: "a  a = 0" 

1247 
using word_of_int_Ex [of a] 

1248 
word_of_int_Ex [of b] 

1249 
word_of_int_Ex [of c] 

1250 
by (auto simp: word_of_int_hom_syms [symmetric] 

1251 
zadd_0_right add_commute add_assoc add_left_commute 

1252 
mult_commute mult_assoc mult_left_commute 

1253 
left_distrib right_distrib) 

1254 

1255 
lemmas word_add_ac = word_add_commute word_add_assoc word_add_left_commute 

1256 
lemmas word_mult_ac = word_mult_commute word_mult_assoc word_mult_left_commute 

1257 

1258 
lemmas word_plus_ac0 = word_add_0 word_add_0_right word_add_ac 

1259 
lemmas word_times_ac1 = word_mult_1 word_mult_1_right word_mult_ac 

1260 

1261 

1262 
subsection "Order on fixedlength words" 

1263 

1264 
lemma word_order_trans: "x <= y ==> y <= z ==> x <= (z :: 'a :: len0 word)" 

1265 
unfolding word_le_def by auto 

1266 

1267 
lemma word_order_refl: "z <= (z :: 'a :: len0 word)" 

1268 
unfolding word_le_def by auto 

1269 

1270 
lemma word_order_antisym: "x <= y ==> y <= x ==> x = (y :: 'a :: len0 word)" 

1271 
unfolding word_le_def by (auto intro!: word_uint.Rep_eqD) 

1272 

1273 
lemma word_order_linear: 

1274 
"y <= x  x <= (y :: 'a :: len0 word)" 

1275 
unfolding word_le_def by auto 

1276 

1277 
lemma word_zero_le [simp] : 

1278 
"0 <= (y :: 'a :: len0 word)" 

1279 
unfolding word_le_def by auto 

1280 

1281 
instance word :: (len0) semigroup_add 

1282 
by intro_classes (simp add: word_add_assoc) 

1283 

1284 
instance word :: (len0) linorder 

1285 
by intro_classes (auto simp: word_less_def word_le_def) 

1286 

1287 
instance word :: (len0) ring 

1288 
by intro_classes 

1289 
(auto simp: word_arith_eqs word_diff_minus 

1290 
word_diff_self [unfolded word_diff_minus]) 

1291 

1292 
lemma word_m1_ge [simp] : "word_pred 0 >= y" 

1293 
unfolding word_le_def 

1294 
by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto 

1295 

1296 
lemmas word_n1_ge [simp] = word_m1_ge [simplified word_sp_01] 

1297 

1298 
lemmas word_not_simps [simp] = 

1299 
word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD] 

1300 

1301 
lemma word_gt_0: "0 < y = (0 ~= (y :: 'a :: len0 word))" 

1302 
unfolding word_less_def by auto 

1303 

1304 
lemmas word_gt_0_no [simp] = word_gt_0 [of "number_of y", standard] 

1305 

1306 
lemma word_sless_alt: "(a <s b) == (sint a < sint b)" 

1307 
unfolding word_sle_def word_sless_def 

1308 
by (auto simp add: less_le) 

1309 

1310 
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)" 

1311 
unfolding unat_def word_le_def 

1312 
by (rule nat_le_eq_zle [symmetric]) simp 

1313 

1314 
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)" 

1315 
unfolding unat_def word_less_alt 

1316 
by (rule nat_less_eq_zless [symmetric]) simp 

1317 

1318 
lemma wi_less: 

1319 
"(word_of_int n < (word_of_int m :: 'a :: len0 word)) = 

1320 
(n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))" 

1321 
unfolding word_less_alt by (simp add: word_uint.eq_norm) 

1322 

1323 
lemma wi_le: 

1324 
"(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = 

1325 
(n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))" 

1326 
unfolding word_le_def by (simp add: word_uint.eq_norm) 

1327 

1328 
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)" 

1329 
apply (unfold udvd_def) 

1330 
apply safe 

1331 
apply (simp add: unat_def nat_mult_distrib) 

1332 
apply (simp add: uint_nat int_mult) 

1333 
apply (rule exI) 

1334 
apply safe 

1335 
prefer 2 

1336 
apply (erule notE) 

1337 
apply (rule refl) 

1338 
apply force 

1339 
done 

1340 

1341 
lemma udvd_iff_dvd: "x udvd y <> unat x dvd unat y" 

1342 
unfolding dvd_def udvd_nat_alt by force 

1343 

1344 
lemmas unat_mono = word_less_nat_alt [THEN iffD1, standard] 

1345 

1346 
lemma word_zero_neq_one: "0 < len_of TYPE ('a :: len0) ==> (0 :: 'a word) ~= 1"; 

1347 
unfolding word_arith_wis 

1348 
by (auto simp add: word_ubin.norm_eq_iff [symmetric] gr0_conv_Suc) 

1349 

1350 
lemmas lenw1_zero_neq_one = len_gt_0 [THEN word_zero_neq_one] 

1351 

1352 
lemma no_no [simp] : "number_of (number_of b) = number_of b" 

1353 
by (simp add: number_of_eq) 

1354 

1355 
lemma unat_minus_one: "x ~= 0 ==> unat (x  1) = unat x  1" 

1356 
apply (unfold unat_def) 

1357 
apply (simp only: int_word_uint word_arith_alts rdmods) 

1358 
apply (subgoal_tac "uint x >= 1") 

1359 
prefer 2 

1360 
apply (drule contrapos_nn) 

1361 
apply (erule word_uint.Rep_inverse' [symmetric]) 

1362 
apply (insert uint_ge_0 [of x])[1] 

1363 
apply arith 

1364 
apply (rule box_equals) 

1365 
apply (rule nat_diff_distrib) 

1366 
prefer 2 

1367 
apply assumption 

1368 
apply simp 

1369 
apply (subst mod_pos_pos_trivial) 

1370 
apply arith 

1371 
apply (insert uint_lt2p [of x])[1] 

1372 
apply arith 

56e3520b68b2
one unified Word theory
haftmann</ 