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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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section \<open>A type of finite bit strings\<close> 
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theory Word 
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imports 
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"~~/src/HOL/Library/Type_Length" 
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"~~/src/HOL/Library/Boolean_Algebra" 
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Bits_Bit 
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Bool_List_Representation 
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Misc_Typedef 
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Word_Miscellaneous 
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begin 
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text \<open>See \<^file>\<open>Examples/WordExamples.thy\<close> for examples.\<close> 
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subsection \<open>Type definition\<close> 

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typedef (overloaded) 'a word = "{(0::int) ..< 2 ^ len_of TYPE('a::len0)}" 
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morphisms uint Abs_word by auto 
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lemma uint_nonnegative: "0 \<le> uint w" 
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using word.uint [of w] by simp 
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lemma uint_bounded: "uint w < 2 ^ len_of TYPE('a)" 
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for w :: "'a::len0 word" 

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using word.uint [of w] by simp 
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lemma uint_idem: "uint w mod 2 ^ len_of TYPE('a) = uint w" 
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for w :: "'a::len0 word" 

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using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial) 
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lemma word_uint_eq_iff: "a = b \<longleftrightarrow> uint a = uint b" 
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by (simp add: uint_inject) 
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lemma word_uint_eqI: "uint a = uint b \<Longrightarrow> a = b" 
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by (simp add: word_uint_eq_iff) 
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definition word_of_int :: "int \<Rightarrow> 'a::len0 word" 
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\<comment> \<open>representation of words using unsigned or signed bins, 
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only difference in these is the type class\<close> 

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where "word_of_int k = Abs_word (k mod 2 ^ len_of TYPE('a))" 
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lemma uint_word_of_int: "uint (word_of_int k :: 'a::len0 word) = k mod 2 ^ len_of TYPE('a)" 

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by (auto simp add: word_of_int_def intro: Abs_word_inverse) 
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lemma word_of_int_uint: "word_of_int (uint w) = w" 
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by (simp add: word_of_int_def uint_idem uint_inverse) 
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lemma split_word_all: "(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))" 
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proof 
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fix x :: "'a word" 
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assume "\<And>x. PROP P (word_of_int x)" 
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then have "PROP P (word_of_int (uint x))" . 
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then show "PROP P x" by (simp add: word_of_int_uint) 
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qed 
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subsection \<open>Type conversions and casting\<close> 
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definition sint :: "'a::len word \<Rightarrow> int" 
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\<comment> \<open>treats the mostsignificantbit as a sign bit\<close> 
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where sint_uint: "sint w = sbintrunc (len_of TYPE('a)  1) (uint w)" 
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definition unat :: "'a::len0 word \<Rightarrow> nat" 
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where "unat w = nat (uint w)" 
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definition uints :: "nat \<Rightarrow> int set" 
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\<comment> "the sets of integers representing the words" 
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where "uints n = range (bintrunc n)" 
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definition sints :: "nat \<Rightarrow> int set" 
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where "sints n = range (sbintrunc (n  1))" 
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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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definition unats :: "nat \<Rightarrow> nat set" 
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where "unats n = {i. i < 2 ^ n}" 
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definition norm_sint :: "nat \<Rightarrow> int \<Rightarrow> int" 
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where "norm_sint n w = (w + 2 ^ (n  1)) mod 2 ^ n  2 ^ (n  1)" 
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definition scast :: "'a::len word \<Rightarrow> 'b::len word" 
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\<comment> "cast a word to a different length" 
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where "scast w = word_of_int (sint w)" 
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definition ucast :: "'a::len0 word \<Rightarrow> 'b::len0 word" 
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where "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 word_size: "size (w :: 'a word) = len_of TYPE('a)" 
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instance .. 
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end 
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lemma word_size_gt_0 [iff]: "0 < size w" 
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for w :: "'a::len word" 

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by (simp add: word_size) 
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lemmas lens_gt_0 = word_size_gt_0 len_gt_0 
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lemma lens_not_0 [iff]: 
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fixes w :: "'a::len word" 
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shows "size w \<noteq> 0" 

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and "len_of TYPE('a) \<noteq> 0" 

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by auto 
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definition source_size :: "('a::len0 word \<Rightarrow> 'b) \<Rightarrow> nat" 
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\<comment> "whether a cast (or other) function is to a longer or shorter length" 
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where [code del]: "source_size c = (let arb = undefined; x = c arb in size arb)" 
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definition target_size :: "('a \<Rightarrow> 'b::len0 word) \<Rightarrow> nat" 
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where [code del]: "target_size c = size (c undefined)" 
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definition is_up :: "('a::len0 word \<Rightarrow> 'b::len0 word) \<Rightarrow> bool" 
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where "is_up c \<longleftrightarrow> source_size c \<le> target_size c" 
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definition is_down :: "('a::len0 word \<Rightarrow> 'b::len0 word) \<Rightarrow> bool" 

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where "is_down c \<longleftrightarrow> target_size c \<le> source_size c" 

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definition of_bl :: "bool list \<Rightarrow> 'a::len0 word" 
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where "of_bl bl = word_of_int (bl_to_bin bl)" 
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definition to_bl :: "'a::len0 word \<Rightarrow> bool list" 
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where "to_bl w = bin_to_bl (len_of TYPE('a)) (uint w)" 
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definition word_reverse :: "'a::len0 word \<Rightarrow> 'a word" 
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where "word_reverse w = of_bl (rev (to_bl w))" 
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definition word_int_case :: "(int \<Rightarrow> 'b) \<Rightarrow> 'a::len0 word \<Rightarrow> 'b" 

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where "word_int_case f w = f (uint w)" 

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translations 
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"case x of XCONST of_int y \<Rightarrow> b" \<rightleftharpoons> "CONST word_int_case (\<lambda>y. b) x" 
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"case x of (XCONST of_int :: 'a) y \<Rightarrow> b" \<rightharpoonup> "CONST word_int_case (\<lambda>y. b) x" 

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subsection \<open>Correspondence relation for theorem transfer\<close> 
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definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool" 
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where "cr_word = (\<lambda>x y. word_of_int x = y)" 
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lemma Quotient_word: 
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"Quotient (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y) 
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word_of_int uint (cr_word :: _ \<Rightarrow> 'a::len0 word \<Rightarrow> bool)" 
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unfolding Quotient_alt_def cr_word_def 
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by (simp add: no_bintr_alt1 word_of_int_uint) (simp add: word_of_int_def Abs_word_inject) 
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lemma reflp_word: 
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"reflp (\<lambda>x y. bintrunc (len_of TYPE('a::len0)) x = bintrunc (len_of TYPE('a)) y)" 
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by (simp add: reflp_def) 
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setup_lifting Quotient_word reflp_word 
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text \<open>TODO: The next lemma could be generated automatically.\<close> 
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lemma uint_transfer [transfer_rule]: 
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"(rel_fun pcr_word op =) (bintrunc (len_of TYPE('a))) (uint :: 'a::len0 word \<Rightarrow> int)" 
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unfolding rel_fun_def word.pcr_cr_eq cr_word_def 
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by (simp add: no_bintr_alt1 uint_word_of_int) 
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subsection \<open>Basic code generation setup\<close> 
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definition Word :: "int \<Rightarrow> 'a::len0 word" 
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where [code_post]: "Word = word_of_int" 
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lemma [code abstype]: "Word (uint w) = w" 

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by (simp add: Word_def word_of_int_uint) 
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declare uint_word_of_int [code abstract] 
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instantiation word :: (len0) equal 
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begin 
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definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" 
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where "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)" 
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instance 

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by standard (simp add: equal equal_word_def word_uint_eq_iff) 

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end 
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notation fcomp (infixl "\<circ>>" 60) 
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notation scomp (infixl "\<circ>\<rightarrow>" 60) 
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instantiation word :: ("{len0, typerep}") random 
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begin 
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definition 
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"random_word i = Random.range i \<circ>\<rightarrow> (\<lambda>k. Pair ( 
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let j = word_of_int (int_of_integer (integer_of_natural k)) :: 'a word 
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in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))" 
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instance .. 
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end 
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no_notation fcomp (infixl "\<circ>>" 60) 
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no_notation scomp (infixl "\<circ>\<rightarrow>" 60) 
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61799  212 
subsection \<open>Typedefinition locale instantiations\<close> 
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lemmas uint_0 = uint_nonnegative (* FIXME duplicate *) 
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lemmas uint_lt = uint_bounded (* FIXME duplicate *) 
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lemmas uint_mod_same = uint_idem (* FIXME duplicate *) 
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65268  218 
lemma td_ext_uint: 
219 
"td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0))) 

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(\<lambda>w::int. w mod 2 ^ len_of TYPE('a))" 
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apply (unfold td_ext_def') 
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apply (simp add: uints_num word_of_int_def bintrunc_mod2p) 
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apply (simp add: uint_mod_same uint_0 uint_lt 
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word.uint_inverse word.Abs_word_inverse int_mod_lem) 
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done 
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interpretation word_uint: 
65268  228 
td_ext 
229 
"uint::'a::len0 word \<Rightarrow> int" 

230 
word_of_int 

231 
"uints (len_of TYPE('a::len0))" 

232 
"\<lambda>w. w mod 2 ^ len_of TYPE('a::len0)" 

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by (fact td_ext_uint) 
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lemmas td_uint = word_uint.td_thm 
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lemmas int_word_uint = word_uint.eq_norm 
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lemma td_ext_ubin: 
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"td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0))) 
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(bintrunc (len_of TYPE('a)))" 
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by (unfold no_bintr_alt1) (fact td_ext_uint) 
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interpretation word_ubin: 
65268  244 
td_ext 
245 
"uint::'a::len0 word \<Rightarrow> int" 

246 
word_of_int 

247 
"uints (len_of TYPE('a::len0))" 

248 
"bintrunc (len_of TYPE('a::len0))" 

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by (fact td_ext_ubin) 
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61799  252 
subsection \<open>Arithmetic operations\<close> 
37660  253 

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lift_definition word_succ :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x + 1" 
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by (auto simp add: bintrunc_mod2p intro: mod_add_cong) 
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lift_definition word_pred :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x  1" 
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by (auto simp add: bintrunc_mod2p intro: mod_diff_cong) 
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instantiation word :: (len0) "{neg_numeral, modulo, comm_monoid_mult, comm_ring}" 
37660  261 
begin 
262 

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lift_definition zero_word :: "'a word" is "0" . 
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lift_definition one_word :: "'a word" is "1" . 
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lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op +" 
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by (auto simp add: bintrunc_mod2p intro: mod_add_cong) 
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lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op " 
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by (auto simp add: bintrunc_mod2p intro: mod_diff_cong) 
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lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus 
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by (auto simp add: bintrunc_mod2p intro: mod_minus_cong) 
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lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op *" 
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by (auto simp add: bintrunc_mod2p intro: mod_mult_cong) 
37660  278 

279 
definition 

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word_div_def: "a div b = word_of_int (uint a div uint b)" 
37660  281 

282 
definition 

283 
word_mod_def: "a mod b = word_of_int (uint a mod uint b)" 

284 

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instance 
61169  286 
by standard (transfer, simp add: algebra_simps)+ 
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end 
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61799  290 
text \<open>Legacy theorems:\<close> 
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65268  292 
lemma word_arith_wis [code]: 
293 
shows word_add_def: "a + b = word_of_int (uint a + uint b)" 

294 
and word_sub_wi: "a  b = word_of_int (uint a  uint b)" 

295 
and word_mult_def: "a * b = word_of_int (uint a * uint b)" 

296 
and word_minus_def: " a = word_of_int ( uint a)" 

297 
and word_succ_alt: "word_succ a = word_of_int (uint a + 1)" 

298 
and word_pred_alt: "word_pred a = word_of_int (uint a  1)" 

299 
and word_0_wi: "0 = word_of_int 0" 

300 
and word_1_wi: "1 = word_of_int 1" 

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unfolding plus_word_def minus_word_def times_word_def uminus_word_def 
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unfolding word_succ_def word_pred_def zero_word_def one_word_def 
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by simp_all 
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65268  305 
lemma wi_homs: 
306 
shows wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" 

307 
and wi_hom_sub: "word_of_int a  word_of_int b = word_of_int (a  b)" 

308 
and wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" 

309 
and wi_hom_neg: " word_of_int a = word_of_int ( a)" 

310 
and wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" 

311 
and wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a  1)" 

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by (transfer, simp)+ 
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lemmas wi_hom_syms = wi_homs [symmetric] 
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46013  316 
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi 
46009  317 

318 
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric] 

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instance word :: (len) comm_ring_1 
45810  321 
proof 
65268  322 
have *: "0 < len_of TYPE('a)" by (rule len_gt_0) 
323 
show "(0::'a word) \<noteq> 1" 

324 
by transfer (use * in \<open>auto simp add: gr0_conv_Suc\<close>) 

45810  325 
qed 
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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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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) 
46013  333 
apply (simp add: word_of_nat wi_hom_sub) 
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done 
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65268  336 
definition udvd :: "'a::len word \<Rightarrow> 'a::len word \<Rightarrow> bool" (infixl "udvd" 50) 
337 
where "a udvd b = (\<exists>n\<ge>0. uint b = n * uint a)" 

37660  338 

45547  339 

61799  340 
subsection \<open>Ordering\<close> 
45547  341 

342 
instantiation word :: (len0) linorder 

343 
begin 

344 

65268  345 
definition word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b" 
346 

347 
definition word_less_def: "a < b \<longleftrightarrow> uint a < uint b" 

37660  348 

45547  349 
instance 
61169  350 
by standard (auto simp: word_less_def word_le_def) 
45547  351 

352 
end 

353 

65268  354 
definition word_sle :: "'a::len word \<Rightarrow> 'a word \<Rightarrow> bool" ("(_/ <=s _)" [50, 51] 50) 
355 
where "a <=s b \<longleftrightarrow> sint a \<le> sint b" 

356 

357 
definition word_sless :: "'a::len word \<Rightarrow> 'a word \<Rightarrow> bool" ("(_/ <s _)" [50, 51] 50) 

358 
where "x <s y \<longleftrightarrow> x <=s y \<and> x \<noteq> y" 

37660  359 

360 

61799  361 
subsection \<open>Bitwise operations\<close> 
37660  362 

363 
instantiation word :: (len0) bits 

364 
begin 

365 

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lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT 
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by (metis bin_trunc_not) 
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lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND 
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by (metis bin_trunc_and) 
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lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR 
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by (metis bin_trunc_or) 
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lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR 
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by (metis bin_trunc_xor) 
37660  377 

65268  378 
definition word_test_bit_def: "test_bit a = bin_nth (uint a)" 
379 

380 
definition word_set_bit_def: "set_bit a n x = word_of_int (bin_sc n x (uint a))" 

381 

382 
definition word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE('a)) f)" 

383 

384 
definition word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a)" 

37660  385 

54848  386 
definition shiftl1 :: "'a word \<Rightarrow> 'a word" 
65268  387 
where "shiftl1 w = word_of_int (uint w BIT False)" 
37660  388 

54848  389 
definition shiftr1 :: "'a word \<Rightarrow> 'a word" 
390 
where 

61799  391 
\<comment> "shift right as unsigned or as signed, ie logical or arithmetic" 
37660  392 
"shiftr1 w = word_of_int (bin_rest (uint w))" 
393 

65268  394 
definition shiftl_def: "w << n = (shiftl1 ^^ n) w" 
395 

396 
definition shiftr_def: "w >> n = (shiftr1 ^^ n) w" 

37660  397 

398 
instance .. 

399 

400 
end 

401 

65268  402 
lemma [code]: 
403 
shows word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" 

404 
and word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" 

405 
and word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" 

406 
and word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)" 

407 
by (simp_all add: bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def) 

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408 

37660  409 
instantiation word :: (len) bitss 
410 
begin 

411 

65268  412 
definition word_msb_def: "msb a \<longleftrightarrow> bin_sign (sint a) = 1" 
37660  413 

414 
instance .. 

415 

416 
end 

417 

65268  418 
definition setBit :: "'a::len0 word \<Rightarrow> nat \<Rightarrow> 'a word" 
419 
where "setBit w n = set_bit w n True" 

420 

421 
definition clearBit :: "'a::len0 word \<Rightarrow> nat \<Rightarrow> 'a word" 

422 
where "clearBit w n = set_bit w n False" 

37660  423 

424 

61799  425 
subsection \<open>Shift operations\<close> 
37660  426 

65268  427 
definition sshiftr1 :: "'a::len word \<Rightarrow> 'a word" 
428 
where "sshiftr1 w = word_of_int (bin_rest (sint w))" 

429 

430 
definition bshiftr1 :: "bool \<Rightarrow> 'a::len word \<Rightarrow> 'a word" 

431 
where "bshiftr1 b w = of_bl (b # butlast (to_bl w))" 

432 

433 
definition sshiftr :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word" (infixl ">>>" 55) 

434 
where "w >>> n = (sshiftr1 ^^ n) w" 

435 

436 
definition mask :: "nat \<Rightarrow> 'a::len word" 

437 
where "mask n = (1 << n)  1" 

438 

439 
definition revcast :: "'a::len0 word \<Rightarrow> 'b::len0 word" 

440 
where "revcast w = of_bl (takefill False (len_of TYPE('b)) (to_bl w))" 

441 

442 
definition slice1 :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'b::len0 word" 

443 
where "slice1 n w = of_bl (takefill False n (to_bl w))" 

444 

445 
definition slice :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'b::len0 word" 

446 
where "slice n w = slice1 (size w  n) w" 

37660  447 

448 

61799  449 
subsection \<open>Rotation\<close> 
37660  450 

65268  451 
definition rotater1 :: "'a list \<Rightarrow> 'a list" 
452 
where "rotater1 ys = 

453 
(case ys of [] \<Rightarrow> []  x # xs \<Rightarrow> last ys # butlast ys)" 

454 

455 
definition rotater :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" 

456 
where "rotater n = rotater1 ^^ n" 

457 

458 
definition word_rotr :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'a::len0 word" 

459 
where "word_rotr n w = of_bl (rotater n (to_bl w))" 

460 

461 
definition word_rotl :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'a::len0 word" 

462 
where "word_rotl n w = of_bl (rotate n (to_bl w))" 

463 

464 
definition word_roti :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> 'a::len0 word" 

465 
where "word_roti i w = 

466 
(if i \<ge> 0 then word_rotr (nat i) w else word_rotl (nat ( i)) w)" 

37660  467 

468 

61799  469 
subsection \<open>Split and cat operations\<close> 
37660  470 

65268  471 
definition word_cat :: "'a::len0 word \<Rightarrow> 'b::len0 word \<Rightarrow> 'c::len0 word" 
472 
where "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE('b)) (uint b))" 

473 

474 
definition word_split :: "'a::len0 word \<Rightarrow> 'b::len0 word \<times> 'c::len0 word" 

475 
where "word_split a = 

476 
(case bin_split (len_of TYPE('c)) (uint a) of 

477 
(u, v) \<Rightarrow> (word_of_int u, word_of_int v))" 

478 

479 
definition word_rcat :: "'a::len0 word list \<Rightarrow> 'b::len0 word" 

480 
where "word_rcat ws = word_of_int (bin_rcat (len_of TYPE('a)) (map uint ws))" 

481 

482 
definition word_rsplit :: "'a::len0 word \<Rightarrow> 'b::len word list" 

483 
where "word_rsplit w = map word_of_int (bin_rsplit (len_of TYPE('b)) (len_of TYPE('a), uint w))" 

484 

485 
definition max_word :: "'a::len word" \<comment> "Largest representable machine integer." 

486 
where "max_word = word_of_int (2 ^ len_of TYPE('a)  1)" 

37660  487 

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lemmas of_nth_def = word_set_bits_def (* FIXME duplicate *) 
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489 

37660  490 

61799  491 
subsection \<open>Theorems about typedefs\<close> 
46010  492 

65268  493 
lemma sint_sbintrunc': "sint (word_of_int bin :: 'a word) = sbintrunc (len_of TYPE('a::len)  1) bin" 
494 
by (auto simp: sint_uint word_ubin.eq_norm sbintrunc_bintrunc_lt) 

495 

496 
lemma uint_sint: "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a::len word))" 

497 
by (auto simp: sint_uint bintrunc_sbintrunc_le) 

498 

499 
lemma bintr_uint: "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w" 

500 
for w :: "'a::len0 word" 

501 
apply (subst word_ubin.norm_Rep [symmetric]) 

37660  502 
apply (simp only: bintrunc_bintrunc_min word_size) 
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503 
apply (simp add: min.absorb2) 
37660  504 
done 
505 

46057  506 
lemma wi_bintr: 
507 
"len_of TYPE('a::len0) \<le> n \<Longrightarrow> 

508 
word_of_int (bintrunc n w) = (word_of_int w :: 'a word)" 

65268  509 
by (auto simp: word_ubin.norm_eq_iff [symmetric] min.absorb1) 
510 

511 
lemma td_ext_sbin: 

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

37660  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]) 

65268  519 
apply (rule word_ubin.Abs_norm) 
37660  520 
apply (simp only: bintrunc_sbintrunc) 
521 
apply (drule sym) 

522 
apply simp 

523 
done 

524 

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525 
lemma td_ext_sint: 
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526 
"td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len))) 
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527 
(\<lambda>w. (w + 2 ^ (len_of TYPE('a)  1)) mod 2 ^ len_of TYPE('a)  
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528 
2 ^ (len_of TYPE('a)  1))" 
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529 
using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2) 
37660  530 

531 
(* We do sint before sbin, before sint is the user version 

65268  532 
and interpretations do not produce thm duplicates. I.e. 
37660  533 
we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, 
534 
because the latter is the same thm as the former *) 

535 
interpretation word_sint: 

65268  536 
td_ext 
537 
"sint ::'a::len word \<Rightarrow> int" 

538 
word_of_int 

539 
"sints (len_of TYPE('a::len))" 

540 
"\<lambda>w. (w + 2^(len_of TYPE('a::len)  1)) mod 2^len_of TYPE('a::len)  

541 
2 ^ (len_of TYPE('a::len)  1)" 

37660  542 
by (rule td_ext_sint) 
543 

544 
interpretation word_sbin: 

65268  545 
td_ext 
546 
"sint ::'a::len word \<Rightarrow> int" 

547 
word_of_int 

548 
"sints (len_of TYPE('a::len))" 

549 
"sbintrunc (len_of TYPE('a::len)  1)" 

37660  550 
by (rule td_ext_sbin) 
551 

45604  552 
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm] 
37660  553 

554 
lemmas td_sint = word_sint.td 

555 

65268  556 
lemma to_bl_def': "(to_bl :: 'a::len0 word \<Rightarrow> bool list) = bin_to_bl (len_of TYPE('a)) \<circ> uint" 
44762  557 
by (auto simp: to_bl_def) 
37660  558 

65268  559 
lemmas word_reverse_no_def [simp] = 
560 
word_reverse_def [of "numeral w"] for w 

37660  561 

45805  562 
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)" 
563 
by (fact uints_def [unfolded no_bintr_alt1]) 

564 

65268  565 
lemma word_numeral_alt: "numeral b = word_of_int (numeral b)" 
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566 
by (induct b, simp_all only: numeral.simps word_of_int_homs) 
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567 

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568 
declare word_numeral_alt [symmetric, code_abbrev] 
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569 

65268  570 
lemma word_neg_numeral_alt: " numeral b = word_of_int ( numeral b)" 
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571 
by (simp only: word_numeral_alt wi_hom_neg) 
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572 

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573 
declare word_neg_numeral_alt [symmetric, code_abbrev] 
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574 

47372  575 
lemma word_numeral_transfer [transfer_rule]: 
55945  576 
"(rel_fun op = pcr_word) numeral numeral" 
577 
"(rel_fun op = pcr_word) ( numeral) ( numeral)" 

578 
apply (simp_all add: rel_fun_def word.pcr_cr_eq cr_word_def) 

65268  579 
using word_numeral_alt [symmetric] word_neg_numeral_alt [symmetric] by auto 
47372  580 

45805  581 
lemma uint_bintrunc [simp]: 
65268  582 
"uint (numeral bin :: 'a word) = 
583 
bintrunc (len_of TYPE('a::len0)) (numeral bin)" 

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584 
unfolding word_numeral_alt by (rule word_ubin.eq_norm) 
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585 

65268  586 
lemma uint_bintrunc_neg [simp]: 
587 
"uint ( numeral bin :: 'a word) = bintrunc (len_of TYPE('a::len0)) ( numeral bin)" 

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588 
by (simp only: word_neg_numeral_alt word_ubin.eq_norm) 
37660  589 

45805  590 
lemma sint_sbintrunc [simp]: 
65268  591 
"sint (numeral bin :: 'a word) = sbintrunc (len_of TYPE('a::len)  1) (numeral bin)" 
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592 
by (simp only: word_numeral_alt word_sbin.eq_norm) 
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593 

65268  594 
lemma sint_sbintrunc_neg [simp]: 
595 
"sint ( numeral bin :: 'a word) = sbintrunc (len_of TYPE('a::len)  1) ( numeral bin)" 

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596 
by (simp only: word_neg_numeral_alt word_sbin.eq_norm) 
37660  597 

45805  598 
lemma unat_bintrunc [simp]: 
65268  599 
"unat (numeral bin :: 'a::len0 word) = nat (bintrunc (len_of TYPE('a)) (numeral bin))" 
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600 
by (simp only: unat_def uint_bintrunc) 
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601 

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602 
lemma unat_bintrunc_neg [simp]: 
65268  603 
"unat ( numeral bin :: 'a::len0 word) = nat (bintrunc (len_of TYPE('a)) ( numeral bin))" 
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604 
by (simp only: unat_def uint_bintrunc_neg) 
37660  605 

65268  606 
lemma size_0_eq: "size (w :: 'a::len0 word) = 0 \<Longrightarrow> v = w" 
37660  607 
apply (unfold word_size) 
608 
apply (rule word_uint.Rep_eqD) 

609 
apply (rule box_equals) 

610 
defer 

611 
apply (rule word_ubin.norm_Rep)+ 

612 
apply simp 

613 
done 

614 

65268  615 
lemma uint_ge_0 [iff]: "0 \<le> uint x" 
616 
for x :: "'a::len0 word" 

45805  617 
using word_uint.Rep [of x] by (simp add: uints_num) 
618 

65268  619 
lemma uint_lt2p [iff]: "uint x < 2 ^ len_of TYPE('a)" 
620 
for x :: "'a::len0 word" 

45805  621 
using word_uint.Rep [of x] by (simp add: uints_num) 
622 

65268  623 
lemma sint_ge: " (2 ^ (len_of TYPE('a)  1)) \<le> sint x" 
624 
for x :: "'a::len word" 

45805  625 
using word_sint.Rep [of x] by (simp add: sints_num) 
626 

65268  627 
lemma sint_lt: "sint x < 2 ^ (len_of TYPE('a)  1)" 
628 
for x :: "'a::len word" 

45805  629 
using word_sint.Rep [of x] by (simp add: sints_num) 
37660  630 

65268  631 
lemma sign_uint_Pls [simp]: "bin_sign (uint x) = 0" 
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632 
by (simp add: sign_Pls_ge_0) 
37660  633 

65268  634 
lemma uint_m2p_neg: "uint x  2 ^ len_of TYPE('a) < 0" 
635 
for x :: "'a::len0 word" 

45805  636 
by (simp only: diff_less_0_iff_less uint_lt2p) 
637 

65268  638 
lemma uint_m2p_not_non_neg: "\<not> 0 \<le> uint x  2 ^ len_of TYPE('a)" 
639 
for x :: "'a::len0 word" 

45805  640 
by (simp only: not_le uint_m2p_neg) 
37660  641 

65268  642 
lemma lt2p_lem: "len_of TYPE('a) \<le> n \<Longrightarrow> uint w < 2 ^ n" 
643 
for w :: "'a::len0 word" 

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644 
by (metis bintr_uint bintrunc_mod2p int_mod_lem zless2p) 
37660  645 

45805  646 
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0" 
647 
by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1]) 

37660  648 

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649 
lemma uint_nat: "uint w = int (unat w)" 
65268  650 
by (auto simp: unat_def) 
651 

652 
lemma uint_numeral: "uint (numeral b :: 'a::len0 word) = numeral b mod 2 ^ len_of TYPE('a)" 

653 
by (simp only: word_numeral_alt int_word_uint) 

654 

655 
lemma uint_neg_numeral: "uint ( numeral b :: 'a::len0 word) =  numeral b mod 2 ^ len_of TYPE('a)" 

656 
by (simp only: word_neg_numeral_alt int_word_uint) 

657 

658 
lemma unat_numeral: "unat (numeral b :: 'a::len0 word) = numeral b mod 2 ^ len_of TYPE('a)" 

37660  659 
apply (unfold unat_def) 
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660 
apply (clarsimp simp only: uint_numeral) 
37660  661 
apply (rule nat_mod_distrib [THEN trans]) 
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662 
apply (rule zero_le_numeral) 
37660  663 
apply (simp_all add: nat_power_eq) 
664 
done 

665 

65268  666 
lemma sint_numeral: 
667 
"sint (numeral b :: 'a::len word) = 

668 
(numeral b + 

669 
2 ^ (len_of TYPE('a)  1)) mod 2 ^ len_of TYPE('a)  

670 
2 ^ (len_of TYPE('a)  1)" 

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671 
unfolding word_numeral_alt by (rule int_word_sint) 
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672 

65268  673 
lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0" 
45958  674 
unfolding word_0_wi .. 
675 

65268  676 
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1" 
45958  677 
unfolding word_1_wi .. 
678 

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679 
lemma word_of_int_neg_1 [simp]: "word_of_int ( 1) =  1" 
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680 
by (simp add: wi_hom_syms) 
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681 

65268  682 
lemma word_of_int_numeral [simp] : "(word_of_int (numeral bin) :: 'a::len0 word) = numeral bin" 
683 
by (simp only: word_numeral_alt) 

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684 

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685 
lemma word_of_int_neg_numeral [simp]: 
65268  686 
"(word_of_int ( numeral bin) :: 'a::len0 word) =  numeral bin" 
687 
by (simp only: word_numeral_alt wi_hom_syms) 

688 

689 
lemma word_int_case_wi: 

690 
"word_int_case f (word_of_int i :: 'b word) = f (i mod 2 ^ len_of TYPE('b::len0))" 

691 
by (simp add: word_int_case_def word_uint.eq_norm) 

692 

693 
lemma word_int_split: 

694 
"P (word_int_case f x) = 

695 
(\<forall>i. x = (word_of_int i :: 'b::len0 word) \<and> 0 \<le> i \<and> i < 2 ^ len_of TYPE('b) \<longrightarrow> P (f i))" 

696 
by (auto simp: word_int_case_def word_uint.eq_norm mod_pos_pos_trivial) 

697 

698 
lemma word_int_split_asm: 

699 
"P (word_int_case f x) = 

700 
(\<nexists>n. x = (word_of_int n :: 'b::len0 word) \<and> 0 \<le> n \<and> n < 2 ^ len_of TYPE('b::len0) \<and> \<not> P (f n))" 

701 
by (auto simp: word_int_case_def word_uint.eq_norm mod_pos_pos_trivial) 

45805  702 

45604  703 
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq] 
704 
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq] 

37660  705 

65268  706 
lemma uint_range_size: "0 \<le> uint w \<and> uint w < 2 ^ size w" 
37660  707 
unfolding word_size by (rule uint_range') 
708 

65268  709 
lemma sint_range_size: " (2 ^ (size w  Suc 0)) \<le> sint w \<and> sint w < 2 ^ (size w  Suc 0)" 
37660  710 
unfolding word_size by (rule sint_range') 
711 

65268  712 
lemma sint_above_size: "2 ^ (size w  1) \<le> x \<Longrightarrow> sint w < x" 
713 
for w :: "'a::len word" 

45805  714 
unfolding word_size by (rule less_le_trans [OF sint_lt]) 
715 

65268  716 
lemma sint_below_size: "x \<le>  (2 ^ (size w  1)) \<Longrightarrow> x \<le> sint w" 
717 
for w :: "'a::len word" 

45805  718 
unfolding word_size by (rule order_trans [OF _ sint_ge]) 
37660  719 

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720 

61799  721 
subsection \<open>Testing bits\<close> 
46010  722 

65268  723 
lemma test_bit_eq_iff: "test_bit u = test_bit v \<longleftrightarrow> u = v" 
724 
for u v :: "'a::len0 word" 

37660  725 
unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) 
726 

65268  727 
lemma test_bit_size [rule_format] : "w !! n \<longrightarrow> n < size w" 
728 
for w :: "'a::len0 word" 

37660  729 
apply (unfold word_test_bit_def) 
730 
apply (subst word_ubin.norm_Rep [symmetric]) 

731 
apply (simp only: nth_bintr word_size) 

732 
apply fast 

733 
done 

734 

65268  735 
lemma word_eq_iff: "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)" 
736 
for x y :: "'a::len0 word" 

46021  737 
unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric] 
738 
by (metis test_bit_size [unfolded word_size]) 

739 

65268  740 
lemma word_eqI: "(\<And>n. n < size u \<longrightarrow> u !! n = v !! n) \<Longrightarrow> u = v" 
741 
for u :: "'a::len0 word" 

46021  742 
by (simp add: word_size word_eq_iff) 
37660  743 

65268  744 
lemma word_eqD: "u = v \<Longrightarrow> u !! x = v !! x" 
745 
for u v :: "'a::len0 word" 

45805  746 
by simp 
37660  747 

65268  748 
lemma test_bit_bin': "w !! n \<longleftrightarrow> n < size w \<and> bin_nth (uint w) n" 
749 
by (simp add: word_test_bit_def word_size nth_bintr [symmetric]) 

37660  750 

751 
lemmas test_bit_bin = test_bit_bin' [unfolded word_size] 

752 

65268  753 
lemma bin_nth_uint_imp: "bin_nth (uint w) n \<Longrightarrow> n < len_of TYPE('a)" 
754 
for w :: "'a::len0 word" 

37660  755 
apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) 
756 
apply (subst word_ubin.norm_Rep) 

757 
apply assumption 

758 
done 

759 

46057  760 
lemma bin_nth_sint: 
65268  761 
"len_of TYPE('a) \<le> n \<Longrightarrow> bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a)  1)" 
762 
for w :: "'a::len word" 

37660  763 
apply (subst word_sbin.norm_Rep [symmetric]) 
46057  764 
apply (auto simp add: nth_sbintr) 
37660  765 
done 
766 

767 
(* type definitions theorem for in terms of equivalent bool list *) 

65268  768 
lemma td_bl: 
769 
"type_definition 

770 
(to_bl :: 'a::len0 word \<Rightarrow> bool list) 

771 
of_bl 

772 
{bl. length bl = len_of TYPE('a)}" 

37660  773 
apply (unfold type_definition_def of_bl_def to_bl_def) 
774 
apply (simp add: word_ubin.eq_norm) 

775 
apply safe 

776 
apply (drule sym) 

777 
apply simp 

778 
done 

779 

780 
interpretation word_bl: 

65268  781 
type_definition 
782 
"to_bl :: 'a::len0 word \<Rightarrow> bool list" 

783 
of_bl 

784 
"{bl. length bl = len_of TYPE('a::len0)}" 

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

785 
by (fact td_bl) 
37660  786 

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

787 
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff] 
45538
1fffa81b9b83
eliminated slightly odd Rep' with dynamicallyscoped [simplified];
wenzelm
parents:
45529
diff
changeset

788 

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

789 
lemma word_size_bl: "size w = size (to_bl w)" 
65268  790 
by (auto simp: word_size) 
791 

792 
lemma to_bl_use_of_bl: "to_bl w = bl \<longleftrightarrow> w = of_bl bl \<and> length bl = length (to_bl w)" 

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

793 
by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq]) 
37660  794 

795 
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" 

65268  796 
by (simp add: word_reverse_def word_bl.Abs_inverse) 
37660  797 

798 
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" 

65268  799 
by (simp add: word_reverse_def word_bl.Abs_inverse) 
37660  800 

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

801 
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w" 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

802 
by (metis word_rev_rev) 
37660  803 

45805  804 
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u" 
805 
by simp 

806 

65268  807 
lemma length_bl_gt_0 [iff]: "0 < length (to_bl x)" 
808 
for x :: "'a::len word" 

45805  809 
unfolding word_bl_Rep' by (rule len_gt_0) 
810 

65268  811 
lemma bl_not_Nil [iff]: "to_bl x \<noteq> []" 
812 
for x :: "'a::len word" 

45805  813 
by (fact length_bl_gt_0 [unfolded length_greater_0_conv]) 
814 

65268  815 
lemma length_bl_neq_0 [iff]: "length (to_bl x) \<noteq> 0" 
816 
for x :: "'a::len word" 

45805  817 
by (fact length_bl_gt_0 [THEN gr_implies_not0]) 
37660  818 

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

819 
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = 1)" 
37660  820 
apply (unfold to_bl_def sint_uint) 
821 
apply (rule trans [OF _ bl_sbin_sign]) 

822 
apply simp 

823 
done 

824 

65268  825 
lemma of_bl_drop': 
826 
"lend = length bl  len_of TYPE('a::len0) \<Longrightarrow> 

37660  827 
of_bl (drop lend bl) = (of_bl bl :: 'a word)" 
65268  828 
by (auto simp: of_bl_def trunc_bl2bin [symmetric]) 
829 

830 
lemma test_bit_of_bl: 

37660  831 
"(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)" 
65268  832 
by (auto simp add: of_bl_def word_test_bit_def word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) 
833 

834 
lemma no_of_bl: "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE('a)) (numeral bin))" 

835 
by (simp add: of_bl_def) 

37660  836 

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

837 
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)" 
65268  838 
by (auto simp: word_size to_bl_def) 
37660  839 

840 
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" 

65268  841 
by (simp add: uint_bl word_size) 
842 

843 
lemma to_bl_of_bin: "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" 

844 
by (auto simp: uint_bl word_ubin.eq_norm word_size) 

37660  845 

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

846 
lemma to_bl_numeral [simp]: 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

847 
"to_bl (numeral bin::'a::len0 word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

848 
bin_to_bl (len_of TYPE('a)) (numeral bin)" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

849 
unfolding word_numeral_alt by (rule to_bl_of_bin) 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

850 

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

851 
lemma to_bl_neg_numeral [simp]: 
54489
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

852 
"to_bl ( numeral bin::'a::len0 word) = 
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

853 
bin_to_bl (len_of TYPE('a)) ( numeral bin)" 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

854 
unfolding word_neg_numeral_alt by (rule to_bl_of_bin) 
37660  855 

856 
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" 

65268  857 
by (simp add: uint_bl word_size) 
858 

859 
lemma uint_bl_bin: "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x" 

860 
for x :: "'a::len0 word" 

46011  861 
by (rule trans [OF bin_bl_bin word_ubin.norm_Rep]) 
45604  862 

37660  863 
(* naturals *) 
864 
lemma uints_unats: "uints n = int ` unats n" 

865 
apply (unfold unats_def uints_num) 

866 
apply safe 

65268  867 
apply (rule_tac image_eqI) 
868 
apply (erule_tac nat_0_le [symmetric]) 

869 
apply auto 

870 
apply (erule_tac nat_less_iff [THEN iffD2]) 

871 
apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) 

872 
apply (auto simp: nat_power_eq) 

37660  873 
done 
874 

875 
lemma unats_uints: "unats n = nat ` uints n" 

65268  876 
by (auto simp: uints_unats image_iff) 
877 

878 
lemmas bintr_num = 

879 
word_ubin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b 

880 
lemmas sbintr_num = 

881 
word_sbin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b 

37660  882 

883 
lemma num_of_bintr': 

65268  884 
"bintrunc (len_of TYPE('a::len0)) (numeral a) = (numeral b) \<Longrightarrow> 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

885 
numeral a = (numeral b :: 'a word)" 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

886 
unfolding bintr_num by (erule subst, simp) 
37660  887 

888 
lemma num_of_sbintr': 

65268  889 
"sbintrunc (len_of TYPE('a::len)  1) (numeral a) = (numeral b) \<Longrightarrow> 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

890 
numeral a = (numeral b :: 'a word)" 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

891 
unfolding sbintr_num by (erule subst, simp) 
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

892 

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

893 
lemma num_abs_bintr: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

894 
"(numeral x :: 'a word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

895 
word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

896 
by (simp only: word_ubin.Abs_norm word_numeral_alt) 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

897 

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

898 
lemma num_abs_sbintr: 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

899 
"(numeral x :: 'a word) = 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

900 
word_of_int (sbintrunc (len_of TYPE('a::len)  1) (numeral x))" 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

901 
by (simp only: word_sbin.Abs_norm word_numeral_alt) 
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset

902 

37660  903 
(** cast  note, no arg for new length, as it's determined by type of result, 
904 
thus in "cast w = w, the type means cast to length of w! **) 

905 

906 
lemma ucast_id: "ucast w = w" 

65268  907 
by (auto simp: ucast_def) 
37660  908 

909 
lemma scast_id: "scast w = w" 

65268  910 
by (auto simp: scast_def) 
37660  911 

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

912 
lemma ucast_bl: "ucast w = of_bl (to_bl w)" 
65268  913 
by (auto simp: ucast_def of_bl_def uint_bl word_size) 
914 

915 
lemma nth_ucast: "(ucast w::'a::len0 word) !! n = (w !! n \<and> n < len_of TYPE('a))" 

916 
by (simp add: ucast_def test_bit_bin word_ubin.eq_norm nth_bintr word_size) 

917 
(fast elim!: bin_nth_uint_imp) 

37660  918 

919 
(* for literal u(s)cast *) 

920 

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

921 
lemma ucast_bintr [simp]: 
65268  922 
"ucast (numeral w ::'a::len0 word) = word_of_int (bintrunc (len_of TYPE('a)) (numeral w))" 
923 
by (simp add: ucast_def) 

924 

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

925 
(* TODO: neg_numeral *) 
37660  926 

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

927 
lemma scast_sbintr [simp]: 
65268  928 
"scast (numeral w ::'a::len word) = 
929 
word_of_int (sbintrunc (len_of TYPE('a)  Suc 0) (numeral w))" 

930 
by (simp add: scast_def) 

37660  931 

46011  932 
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)" 
933 
unfolding source_size_def word_size Let_def .. 

934 

935 
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)" 

936 
unfolding target_size_def word_size Let_def .. 

937 

65268  938 
lemma is_down: "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)" 
939 
for c :: "'a::len0 word \<Rightarrow> 'b::len0 word" 

940 
by (simp only: is_down_def source_size target_size) 

941 

942 
lemma is_up: "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)" 

943 
for c :: "'a::len0 word \<Rightarrow> 'b::len0 word" 

944 
by (simp only: is_up_def source_size target_size) 

37660  945 

45604  946 
lemmas is_up_down = trans [OF is_up is_down [symmetric]] 
37660  947 

45811  948 
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast" 
37660  949 
apply (unfold is_down) 
950 
apply safe 

951 
apply (rule ext) 

952 
apply (unfold ucast_def scast_def uint_sint) 

953 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

954 
apply simp 

955 
done 

956 

45811  957 
lemma word_rev_tf: 
958 
"to_bl (of_bl bl::'a::len0 word) = 

959 
rev (takefill False (len_of TYPE('a)) (rev bl))" 

65268  960 
by (auto simp: of_bl_def uint_bl bl_bin_bl_rtf word_ubin.eq_norm word_size) 
37660  961 

45811  962 
lemma word_rep_drop: 
963 
"to_bl (of_bl bl::'a::len0 word) = 

964 
replicate (len_of TYPE('a)  length bl) False @ 

965 
drop (length bl  len_of TYPE('a)) bl" 

966 
by (simp add: word_rev_tf takefill_alt rev_take) 

37660  967 

65268  968 
lemma to_bl_ucast: 
969 
"to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 

970 
replicate (len_of TYPE('a)  len_of TYPE('b)) False @ 

971 
drop (len_of TYPE('b)  len_of TYPE('a)) (to_bl w)" 

37660  972 
apply (unfold ucast_bl) 
973 
apply (rule trans) 

974 
apply (rule word_rep_drop) 

975 
apply simp 

976 
done 

977 

45811  978 
lemma ucast_up_app [OF refl]: 
65268  979 
"uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
37660  980 
to_bl (uc w) = replicate n False @ (to_bl w)" 
981 
by (auto simp add : source_size target_size to_bl_ucast) 

982 

45811  983 
lemma ucast_down_drop [OF refl]: 
65268  984 
"uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
37660  985 
to_bl (uc w) = drop n (to_bl w)" 
986 
by (auto simp add : source_size target_size to_bl_ucast) 

987 

45811  988 
lemma scast_down_drop [OF refl]: 
65268  989 
"sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
37660  990 
to_bl (sc w) = drop n (to_bl w)" 
991 
apply (subgoal_tac "sc = ucast") 

992 
apply safe 

993 
apply simp 

45811  994 
apply (erule ucast_down_drop) 
995 
apply (rule down_cast_same [symmetric]) 

37660  996 
apply (simp add : source_size target_size is_down) 
997 
done 

998 

65268  999 
lemma sint_up_scast [OF refl]: "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w" 
37660  1000 
apply (unfold is_up) 
1001 
apply safe 

1002 
apply (simp add: scast_def word_sbin.eq_norm) 

1003 
apply (rule box_equals) 

1004 
prefer 3 

1005 
apply (rule word_sbin.norm_Rep) 

1006 
apply (rule sbintrunc_sbintrunc_l) 

1007 
defer 

1008 
apply (subst word_sbin.norm_Rep) 

1009 
apply (rule refl) 

1010 
apply simp 

1011 
done 

1012 

65268  1013 
lemma uint_up_ucast [OF refl]: "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w" 
37660  1014 
apply (unfold is_up) 
1015 
apply safe 

1016 
apply (rule bin_eqI) 

1017 
apply (fold word_test_bit_def) 

1018 
apply (auto simp add: nth_ucast) 

1019 
apply (auto simp add: test_bit_bin) 

1020 
done 

45811  1021 

65268  1022 
lemma ucast_up_ucast [OF refl]: "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w" 
37660  1023 
apply (simp (no_asm) add: ucast_def) 
1024 
apply (clarsimp simp add: uint_up_ucast) 

1025 
done 

65268  1026 

1027 
lemma scast_up_scast [OF refl]: "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w" 

37660  1028 
apply (simp (no_asm) add: scast_def) 
1029 
apply (clarsimp simp add: sint_up_scast) 

1030 
done 

65268  1031 

1032 
lemma ucast_of_bl_up [OF refl]: "w = of_bl bl \<Longrightarrow> size bl \<le> size w \<Longrightarrow> ucast w = of_bl bl" 

37660  1033 
by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) 
1034 

1035 
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] 

1036 
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] 

1037 

1038 
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] 

1039 
lemmas isdus = is_up_down [where c = "scast", THEN iffD2] 

1040 
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] 

1041 
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] 

1042 

1043 
lemma up_ucast_surj: 

65268  1044 
"is_up (ucast :: 'b::len0 word \<Rightarrow> 'a::len0 word) \<Longrightarrow> 
1045 
surj (ucast :: 'a word \<Rightarrow> 'b word)" 

1046 
by (rule surjI) (erule ucast_up_ucast_id) 

37660  1047 

1048 
lemma up_scast_surj: 

65268  1049 
"is_up (scast :: 'b::len word \<Rightarrow> 'a::len word) \<Longrightarrow> 
1050 
surj (scast :: 'a word \<Rightarrow> 'b word)" 

1051 
by (rule surjI) (erule scast_up_scast_id) 

37660  1052 

1053 
lemma down_scast_inj: 

65268  1054 
"is_down (scast :: 'b::len word \<Rightarrow> 'a::len word) \<Longrightarrow> 
1055 
inj_on (ucast :: 'a word \<Rightarrow> 'b word) A" 

37660  1056 
by (rule inj_on_inverseI, erule scast_down_scast_id) 
1057 

1058 
lemma down_ucast_inj: 

65268  1059 
"is_down (ucast :: 'b::len0 word \<Rightarrow> 'a::len0 word) \<Longrightarrow> 
1060 
inj_on (ucast :: 'a word \<Rightarrow> 'b word) A" 

1061 
by (rule inj_on_inverseI) (erule ucast_down_ucast_id) 

37660  1062 

1063 
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" 

1064 
by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) 

45811  1065 

65268  1066 
lemma ucast_down_wi [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x" 
46646  1067 
apply (unfold is_down) 
37660  1068 
apply (clarsimp simp add: ucast_def word_ubin.eq_norm) 
1069 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1070 
apply (erule bintrunc_bintrunc_ge) 

1071 
done 

45811  1072 

65268  1073 
lemma ucast_down_no [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin" 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1074 
unfolding word_numeral_alt by clarify (rule ucast_down_wi) 
46646  1075 

65268  1076 
lemma ucast_down_bl [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl" 
46646  1077 
unfolding of_bl_def by clarify (erule ucast_down_wi) 
37660  1078 

1079 
lemmas slice_def' = slice_def [unfolded word_size] 

1080 
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] 

1081 

1082 
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def 

1083 

1084 

61799  1085 
subsection \<open>Word Arithmetic\<close> 
37660  1086 

65268  1087 
lemma word_less_alt: "a < b \<longleftrightarrow> uint a < uint b" 
55818  1088 
by (fact word_less_def) 
37660  1089 

1090 
lemma signed_linorder: "class.linorder word_sle word_sless" 

65268  1091 
by standard (auto simp: word_sle_def word_sless_def) 
37660  1092 

1093 
interpretation signed: linorder "word_sle" "word_sless" 

1094 
by (rule signed_linorder) 

1095 

65268  1096 
lemma udvdI: "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b" 
37660  1097 
by (auto simp: udvd_def) 
1098 

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

1099 
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1100 
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1101 
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1102 
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1103 
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1104 
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b 
37660  1105 

65268  1106 
lemma word_m1_wi: " 1 = word_of_int ( 1)" 
1107 
by (simp add: word_neg_numeral_alt [of Num.One]) 

37660  1108 

46648  1109 
lemma word_0_bl [simp]: "of_bl [] = 0" 
65268  1110 
by (simp add: of_bl_def) 
1111 

1112 
lemma word_1_bl: "of_bl [True] = 1" 

1113 
by (simp add: of_bl_def bl_to_bin_def) 

46648  1114 

1115 
lemma uint_eq_0 [simp]: "uint 0 = 0" 

1116 
unfolding word_0_wi word_ubin.eq_norm by simp 

37660  1117 

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

1118 
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0" 
46648  1119 
by (simp add: of_bl_def bl_to_bin_rep_False) 
37660  1120 

65268  1121 
lemma to_bl_0 [simp]: "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" 
1122 
by (simp add: uint_bl word_size bin_to_bl_zero) 

1123 

1124 
lemma uint_0_iff: "uint x = 0 \<longleftrightarrow> x = 0" 

55818  1125 
by (simp add: word_uint_eq_iff) 
1126 

65268  1127 
lemma unat_0_iff: "unat x = 0 \<longleftrightarrow> x = 0" 
1128 
by (auto simp: unat_def nat_eq_iff uint_0_iff) 

1129 

1130 
lemma unat_0 [simp]: "unat 0 = 0" 

1131 
by (auto simp: unat_def) 

1132 

1133 
lemma size_0_same': "size w = 0 \<Longrightarrow> w = v" 

1134 
for v w :: "'a::len0 word" 

37660  1135 
apply (unfold word_size) 
1136 
apply (rule box_equals) 

1137 
defer 

1138 
apply (rule word_uint.Rep_inverse)+ 

1139 
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 

1140 
apply simp 

1141 
done 

1142 

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

1143 
lemmas size_0_same = size_0_same' [unfolded word_size] 
37660  1144 

1145 
lemmas unat_eq_0 = unat_0_iff 

1146 
lemmas unat_eq_zero = unat_0_iff 

1147 

65268  1148 
lemma unat_gt_0: "0 < unat x \<longleftrightarrow> x \<noteq> 0" 
1149 
by (auto simp: unat_0_iff [symmetric]) 

37660  1150 

45958  1151 
lemma ucast_0 [simp]: "ucast 0 = 0" 
65268  1152 
by (simp add: ucast_def) 
45958  1153 

1154 
lemma sint_0 [simp]: "sint 0 = 0" 

65268  1155 
by (simp add: sint_uint) 
45958  1156 

1157 
lemma scast_0 [simp]: "scast 0 = 0" 

65268  1158 
by (simp add: scast_def) 
37660  1159 

58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58061
diff
changeset

1160 
lemma sint_n1 [simp] : "sint ( 1) =  1" 
65268  1161 
by (simp only: word_m1_wi word_sbin.eq_norm) simp 
54489
03ff4d1e6784
eliminiated neg_numeral in favour of  (numeral _)
haftmann
parents:
54225
diff
changeset

1162 

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

1163 
lemma scast_n1 [simp]: "scast ( 1) =  1" 
65268  1164 
by (simp add: scast_def) 
45958  1165 

1166 
lemma uint_1 [simp]: "uint (1::'a::len word) = 1" 

55818  1167 
by (simp only: word_1_wi word_ubin.eq_norm) (simp add: bintrunc_minus_simps(4)) 
45958  1168 

1169 
lemma unat_1 [simp]: "unat (1::'a::len word) = 1" 

65268  1170 
by (simp add: unat_def) 
45958  1171 

1172 
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1" 

65268  1173 
by (simp add: ucast_def) 
37660  1174 

1175 
(* now, to get the weaker results analogous to word_div/mod_def *) 

1176 

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

1177 

61799  1178 
subsection \<open>Transferring goals from words to ints\<close> 
37660  1179 

65268  1180 
lemma word_ths: 
1181 
shows word_succ_p1: "word_succ a = a + 1" 

1182 
and word_pred_m1: "word_pred a = a  1" 

1183 
and word_pred_succ: "word_pred (word_succ a) = a" 

1184 
and word_succ_pred: "word_succ (word_pred a) = a" 

1185 
and word_mult_succ: "word_succ a * b = b + a * b" 

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

1186 
by (transfer, simp add: algebra_simps)+ 
37660  1187 

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

1188 
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y" 
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

1189 
by simp 
37660  1190 

55818  1191 
lemma uint_word_ariths: 
1192 
fixes a b :: "'a::len0 word" 

1193 
shows "uint (a + b) = (uint a + uint b) mod 2 ^ len_of TYPE('a::len0)" 

1194 
and "uint (a  b) = (uint a  uint b) mod 2 ^ len_of TYPE('a)" 

1195 
and "uint (a * b) = uint a * uint b mod 2 ^ len_of TYPE('a)" 

1196 
and "uint ( a) =  uint a mod 2 ^ len_of TYPE('a)" 

1197 
and "uint (word_succ a) = (uint a + 1) mod 2 ^ len_of TYPE('a)" 

1198 
and "uint (word_pred a) = (uint a  1) mod 2 ^ len_of TYPE('a)" 

1199 
and "uint (0 :: 'a word) = 0 mod 2 ^ len_of TYPE('a)" 

1200 
and "uint (1 :: 'a word) = 1 mod 2 ^ len_of TYPE('a)" 

1201 
by (simp_all add: word_arith_wis [THEN trans [OF uint_cong int_word_uint]]) 

1202 

1203 
lemma uint_word_arith_bintrs: 

1204 
fixes a b :: "'a::len0 word" 

1205 
shows "uint (a + b) = bintrunc (len_of TYPE('a)) (uint a + uint b)" 

1206 
and "uint (a  b) = bintrunc (len_of TYPE('a)) (uint a  uint b)" 

1207 
and "uint (a * b) = bintrunc (len_of TYPE('a)) (uint a * uint b)" 

1208 
and "uint ( a) = bintrunc (len_of TYPE('a)) ( uint a)" 

1209 
and "uint (word_succ a) = bintrunc (len_of TYPE('a)) (uint a + 1)" 

1210 
and "uint (word_pred a) = bintrunc (len_of TYPE('a)) (uint a  1)" 

1211 
and "uint (0 :: 'a word) = bintrunc (len_of TYPE('a)) 0" 

1212 
and "uint (1 :: 'a word) = bintrunc (len_of TYPE('a)) 1" 

1213 
by (simp_all add: uint_word_ariths bintrunc_mod2p) 

1214 

1215 
lemma sint_word_ariths: 

1216 
fixes a b :: "'a::len word" 

1217 
shows "sint (a + b) = sbintrunc (len_of TYPE('a)  1) (sint a + sint b)" 

1218 
and "sint (a  b) = sbintrunc (len_of TYPE('a)  1) (sint a  sint b)" 

1219 
and "sint (a * b) = sbintrunc (len_of TYPE('a)  1) (sint a * sint b)" 

1220 
and "sint ( a) = sbintrunc (len_of TYPE('a)  1) ( sint a)" 

1221 
and "sint (word_succ a) = sbintrunc (len_of TYPE('a)  1) (sint a + 1)" 

1222 
and "sint (word_pred a) = sbintrunc (len_of TYPE('a)  1) (sint a  1)" 

1223 
and "sint (0 :: 'a word) = sbintrunc (len_of TYPE('a)  1) 0" 

1224 
and "sint (1 :: 'a word) = sbintrunc (len_of TYPE('a)  1) 1" 

64593
50c715579715
reoriented congruence rules in nonexplosive direction
haftmann
parents:
64243
diff
changeset

1225 
apply (simp_all only: word_sbin.inverse_norm [symmetric]) 
50c715579715
reoriented congruence rules in nonexplosive direction
haftmann
parents:
64243
diff
changeset

1226 
apply (simp_all add: wi_hom_syms) 
50c715579715
reoriented congruence rules in nonexplosive direction
haftmann
parents:
64243
diff
changeset

1227 
apply transfer apply simp 
50c715579715
reoriented congruence rules in nonexplosive direction
haftmann
parents:
64243
diff
changeset

1228 
apply transfer apply simp 
50c715579715
reoriented congruence rules in nonexplosive direction
haftmann
parents:
64243
diff
changeset

1229 
done 
45604  1230 

1231 
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]] 

1232 
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]] 

37660  1233 

58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58061
diff
changeset

1234 
lemma word_pred_0_n1: "word_pred 0 = word_of_int ( 1)" 
47374
9475d524bafb
set up and use lift_definition for word operations
huffman
parents:
47372
diff
changeset

1235 
unfolding word_pred_m1 by simp 
37660  1236 

1237 
lemma succ_pred_no [simp]: 

65268  1238 
"word_succ (numeral w) = numeral w + 1" 
1239 
"word_pred (numeral w) = numeral w  1" 

1240 
"word_succ ( numeral w) =  numeral w + 1" 

1241 
"word_pred ( numeral w) =  numeral w  1" 

1242 
by (simp_all add: word_succ_p1 word_pred_m1) 

1243 

1244 
lemma word_sp_01 [simp]: 

1245 
"word_succ ( 1) = 0 \<and> word_succ 0 = 1 \<and> word_pred 0 =  1 \<and> word_pred 1 = 0" 

1246 
by (simp_all add: word_succ_p1 word_pred_m1) 

37660  1247 

1248 
(* alternative approach to lifting arithmetic equalities *) 

65268  1249 
lemma word_of_int_Ex: "\<exists>y. x = word_of_int y" 
37660  1250 
by (rule_tac x="uint x" in exI) simp 
1251 

1252 

61799  1253 
subsection \<open>Order on fixedlength words\<close> 
37660  1254 

1255 
lemma word_zero_le [simp] : 

65268  1256 
"0 <= (y :: 'a::len0 word)" 
37660  1257 
unfolding word_le_def by auto 
65268  1258 

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

1259 
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *) 
37660  1260 
unfolding word_le_def 
1261 
by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto 

1262 

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

1263 
lemma word_n1_ge [simp]: "y \<le> (1::'a::len0 word)" 
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

1264 
unfolding word_le_def 
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset

1265 
by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto 
37660  1266 

65268  1267 
lemmas word_not_simps [simp] = 
37660  1268 
word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD] 
1269 

65268  1270 
lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> (y :: 'a::len0 word)" 
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1271 
by (simp add: less_le) 
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset

1272 

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

1273 
lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y 
37660  1274 

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

1275 
lemma word_sless_alt: "(a <s b) = (sint a < sint b)" 
37660  1276 
unfolding word_sle_def word_sless_def 
1277 
by (auto simp add: less_le) 

1278 

1279 
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)" 

1280 
unfolding unat_def word_le_def 

1281 
by (rule nat_le_eq_zle [symmetric]) simp 

1282 

1283 
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)" 

1284 
unfolding unat_def word_less_alt 

1285 
by (rule nat_less_eq_zless [symmetric]) simp 

65268  1286 

1287 
lemma wi_less: 

1288 
"(word_of_int n < (word_of_int m :: 'a::len0 word)) = 

37660  1289 
(n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))" 
1290 
unfolding word_less_alt by (simp add: word_uint.eq_norm) 

1291 

65268  1292 
lemma wi_le: 