author | haftmann |
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(* Title: HOL/Word/Word.thy |
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Author: Jeremy Dawson and Gerwin Klein, NICTA |
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*) |
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header {* A type of finite bit strings *} |
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theory Word |
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imports |
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Type_Length |
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"~~/src/HOL/Library/Boolean_Algebra" |
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Bits_Bit |
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Bool_List_Representation |
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Misc_Typedef |
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Word_Miscellaneous |
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begin |
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text {* See @{file "Examples/WordExamples.thy"} for examples. *} |
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subsection {* Type definition *} |
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typedef 'a word = "{(0::int) ..< 2 ^ len_of TYPE('a::len0)}" |
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morphisms uint Abs_word by auto |
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lemma uint_nonnegative: |
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"0 \<le> uint w" |
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using word.uint [of w] by simp |
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lemma uint_bounded: |
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fixes w :: "'a::len0 word" |
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shows "uint w < 2 ^ len_of TYPE('a)" |
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using word.uint [of w] by simp |
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lemma uint_idem: |
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fixes w :: "'a::len0 word" |
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shows "uint w mod 2 ^ len_of TYPE('a) = uint w" |
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using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial) |
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lemma word_uint_eq_iff: |
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"a = b \<longleftrightarrow> uint a = uint b" |
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by (simp add: uint_inject) |
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lemma word_uint_eqI: |
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"uint a = uint b \<Longrightarrow> a = b" |
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by (simp add: word_uint_eq_iff) |
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definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" |
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where |
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-- {* representation of words using unsigned or signed bins, |
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only difference in these is the type class *} |
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"word_of_int k = Abs_word (k mod 2 ^ len_of TYPE('a))" |
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lemma uint_word_of_int: |
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"uint (word_of_int k :: 'a::len0 word) = k mod 2 ^ len_of TYPE('a)" |
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by (auto simp add: word_of_int_def intro: Abs_word_inverse) |
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lemma word_of_int_uint: |
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"word_of_int (uint w) = w" |
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by (simp add: word_of_int_def uint_idem uint_inverse) |
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lemma split_word_all: |
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"(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))" |
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proof |
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fix x :: "'a word" |
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assume "\<And>x. PROP P (word_of_int x)" |
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then have "PROP P (word_of_int (uint x))" . |
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find_theorems word_of_int uint |
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then show "PROP P x" by (simp add: word_of_int_uint) |
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qed |
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subsection {* Type conversions and casting *} |
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definition sint :: "'a::len word \<Rightarrow> int" |
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where |
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-- {* treats the most-significant-bit as a sign bit *} |
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sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)" |
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definition unat :: "'a::len0 word \<Rightarrow> nat" |
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where |
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"unat w = nat (uint w)" |
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definition uints :: "nat \<Rightarrow> int set" |
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where |
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-- "the sets of integers representing the words" |
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"uints n = range (bintrunc n)" |
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definition sints :: "nat \<Rightarrow> int set" |
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where |
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"sints n = range (sbintrunc (n - 1))" |
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lemma uints_num: |
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"uints n = {i. 0 \<le> i \<and> i < 2 ^ n}" |
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by (simp add: uints_def range_bintrunc) |
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lemma sints_num: |
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"sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}" |
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by (simp add: sints_def range_sbintrunc) |
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definition unats :: "nat \<Rightarrow> nat set" |
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where |
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"unats n = {i. i < 2 ^ n}" |
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definition norm_sint :: "nat \<Rightarrow> int \<Rightarrow> int" |
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where |
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"norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)" |
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definition scast :: "'a::len word \<Rightarrow> 'b::len word" |
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where |
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-- "cast a word to a different length" |
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"scast w = word_of_int (sint w)" |
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definition ucast :: "'a::len0 word \<Rightarrow> 'b::len0 word" |
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where |
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"ucast w = word_of_int (uint w)" |
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instantiation word :: (len0) size |
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begin |
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definition |
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word_size: "size (w :: 'a word) = len_of TYPE('a)" |
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instance .. |
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end |
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lemma word_size_gt_0 [iff]: |
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"0 < size (w::'a::len word)" |
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by (simp add: word_size) |
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lemmas lens_gt_0 = word_size_gt_0 len_gt_0 |
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lemma lens_not_0 [iff]: |
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shows "size (w::'a::len word) \<noteq> 0" |
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and "len_of TYPE('a::len) \<noteq> 0" |
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by auto |
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definition source_size :: "('a::len0 word \<Rightarrow> 'b) \<Rightarrow> nat" |
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where |
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-- "whether a cast (or other) function is to a longer or shorter length" |
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"source_size c = (let arb = undefined ; x = c arb in size arb)" |
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141 |
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definition target_size :: "('a \<Rightarrow> 'b::len0 word) \<Rightarrow> nat" |
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where |
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"target_size c = size (c undefined)" |
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145 |
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definition is_up :: "('a::len0 word \<Rightarrow> 'b::len0 word) \<Rightarrow> bool" |
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where |
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"is_up c \<longleftrightarrow> source_size c \<le> target_size c" |
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149 |
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definition is_down :: "('a :: len0 word \<Rightarrow> 'b :: len0 word) \<Rightarrow> bool" |
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where |
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"is_down c \<longleftrightarrow> target_size c \<le> source_size c" |
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153 |
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definition of_bl :: "bool list \<Rightarrow> 'a::len0 word" |
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155 |
where |
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"of_bl bl = word_of_int (bl_to_bin bl)" |
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157 |
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definition to_bl :: "'a::len0 word \<Rightarrow> bool list" |
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where |
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"to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)" |
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161 |
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definition word_reverse :: "'a::len0 word \<Rightarrow> 'a word" |
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163 |
where |
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"word_reverse w = of_bl (rev (to_bl w))" |
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definition word_int_case :: "(int \<Rightarrow> 'b) \<Rightarrow> 'a::len0 word \<Rightarrow> 'b" |
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where |
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"word_int_case f w = f (uint w)" |
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169 |
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170 |
translations |
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"case x of XCONST of_int y => b" == "CONST word_int_case (%y. b) x" |
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"case x of (XCONST of_int :: 'a) y => b" => "CONST word_int_case (%y. b) x" |
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173 |
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174 |
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subsection {* Correspondence relation for theorem transfer *} |
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definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool" |
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where |
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"cr_word = (\<lambda>x y. word_of_int x = y)" |
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lemma Quotient_word: |
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"Quotient (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y) |
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word_of_int uint (cr_word :: _ \<Rightarrow> 'a::len0 word \<Rightarrow> bool)" |
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unfolding Quotient_alt_def cr_word_def |
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by (simp add: no_bintr_alt1 word_of_int_uint) (simp add: word_of_int_def Abs_word_inject) |
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186 |
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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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190 |
|
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191 |
setup_lifting (no_code) Quotient_word reflp_word |
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192 |
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text {* TODO: The next lemma could be generated automatically. *} |
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194 |
|
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lemma uint_transfer [transfer_rule]: |
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"(fun_rel pcr_word op =) (bintrunc (len_of TYPE('a))) |
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(uint :: 'a::len0 word \<Rightarrow> int)" |
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unfolding fun_rel_def word.pcr_cr_eq cr_word_def |
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199 |
by (simp add: no_bintr_alt1 uint_word_of_int) |
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200 |
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201 |
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subsection {* Basic code generation setup *} |
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203 |
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definition Word :: "int \<Rightarrow> 'a::len0 word" |
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where |
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[code_post]: "Word = word_of_int" |
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207 |
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lemma [code abstype]: |
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"Word (uint w) = w" |
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by (simp add: Word_def word_of_int_uint) |
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211 |
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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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218 |
where |
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"equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)" |
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220 |
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instance proof |
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qed (simp add: equal equal_word_def word_uint_eq_iff) |
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223 |
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224 |
end |
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225 |
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226 |
notation fcomp (infixl "\<circ>>" 60) |
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227 |
notation scomp (infixl "\<circ>\<rightarrow>" 60) |
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228 |
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instantiation word :: ("{len0, typerep}") random |
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230 |
begin |
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231 |
|
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232 |
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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236 |
|
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237 |
instance .. |
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239 |
end |
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241 |
no_notation fcomp (infixl "\<circ>>" 60) |
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242 |
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
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243 |
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subsection {* Type-definition locale instantiations *} |
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246 |
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lemmas uint_0 = uint_nonnegative (* FIXME duplicate *) |
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lemmas uint_lt = uint_bounded (* FIXME duplicate *) |
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lemmas uint_mod_same = uint_idem (* FIXME duplicate *) |
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lemma td_ext_uint: |
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"td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0))) |
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(\<lambda>w::int. w mod 2 ^ len_of TYPE('a))" |
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apply (unfold td_ext_def') |
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apply (simp add: uints_num word_of_int_def bintrunc_mod2p) |
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apply (simp add: uint_mod_same uint_0 uint_lt |
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word.uint_inverse word.Abs_word_inverse int_mod_lem) |
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done |
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interpretation word_uint: |
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td_ext "uint::'a::len0 word \<Rightarrow> int" |
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word_of_int |
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"uints (len_of TYPE('a::len0))" |
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"\<lambda>w. w mod 2 ^ len_of TYPE('a::len0)" |
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by (fact td_ext_uint) |
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266 |
|
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267 |
lemmas td_uint = word_uint.td_thm |
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lemmas int_word_uint = word_uint.eq_norm |
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269 |
|
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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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272 |
(bintrunc (len_of TYPE('a)))" |
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273 |
by (unfold no_bintr_alt1) (fact td_ext_uint) |
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274 |
|
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275 |
interpretation word_ubin: |
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276 |
td_ext "uint::'a::len0 word \<Rightarrow> int" |
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277 |
word_of_int |
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278 |
"uints (len_of TYPE('a::len0))" |
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279 |
"bintrunc (len_of TYPE('a::len0))" |
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280 |
by (fact td_ext_ubin) |
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281 |
|
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282 |
|
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283 |
subsection {* Arithmetic operations *} |
37660 | 284 |
|
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lift_definition word_succ :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x + 1" |
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286 |
by (metis bintr_ariths(6)) |
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287 |
|
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288 |
lift_definition word_pred :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x - 1" |
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289 |
by (metis bintr_ariths(7)) |
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290 |
|
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instantiation word :: (len0) "{neg_numeral, Divides.div, comm_monoid_mult, comm_ring}" |
37660 | 292 |
begin |
293 |
||
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lift_definition zero_word :: "'a word" is "0" . |
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295 |
|
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lift_definition one_word :: "'a word" is "1" . |
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297 |
|
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298 |
lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op +" |
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299 |
by (metis bintr_ariths(2)) |
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300 |
|
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301 |
lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op -" |
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302 |
by (metis bintr_ariths(3)) |
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303 |
|
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304 |
lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus |
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305 |
by (metis bintr_ariths(5)) |
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306 |
|
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307 |
lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op *" |
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308 |
by (metis bintr_ariths(4)) |
37660 | 309 |
|
310 |
definition |
|
311 |
word_div_def: "a div b = word_of_int (uint a div uint b)" |
|
312 |
||
313 |
definition |
|
314 |
word_mod_def: "a mod b = word_of_int (uint a mod uint b)" |
|
315 |
||
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316 |
instance |
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by default (transfer, simp add: algebra_simps)+ |
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318 |
|
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319 |
end |
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320 |
|
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321 |
text {* Legacy theorems: *} |
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322 |
|
47611 | 323 |
lemma word_arith_wis [code]: shows |
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324 |
word_add_def: "a + b = word_of_int (uint a + uint b)" and |
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325 |
word_sub_wi: "a - b = word_of_int (uint a - uint b)" and |
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326 |
word_mult_def: "a * b = word_of_int (uint a * uint b)" and |
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327 |
word_minus_def: "- a = word_of_int (- uint a)" and |
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328 |
word_succ_alt: "word_succ a = word_of_int (uint a + 1)" and |
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329 |
word_pred_alt: "word_pred a = word_of_int (uint a - 1)" and |
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330 |
word_0_wi: "0 = word_of_int 0" and |
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331 |
word_1_wi: "1 = word_of_int 1" |
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332 |
unfolding plus_word_def minus_word_def times_word_def uminus_word_def |
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333 |
unfolding word_succ_def word_pred_def zero_word_def one_word_def |
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334 |
by simp_all |
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335 |
|
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336 |
lemmas arths = |
45604 | 337 |
bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm] |
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338 |
|
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339 |
lemma wi_homs: |
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340 |
shows |
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341 |
wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and |
46013 | 342 |
wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)" and |
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343 |
wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and |
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344 |
wi_hom_neg: "- word_of_int a = word_of_int (- a)" and |
46000 | 345 |
wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and |
346 |
wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)" |
|
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347 |
by (transfer, simp)+ |
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348 |
|
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349 |
lemmas wi_hom_syms = wi_homs [symmetric] |
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350 |
|
46013 | 351 |
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi |
46009 | 352 |
|
353 |
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric] |
|
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354 |
|
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355 |
instance word :: (len) comm_ring_1 |
45810 | 356 |
proof |
357 |
have "0 < len_of TYPE('a)" by (rule len_gt_0) |
|
358 |
then show "(0::'a word) \<noteq> 1" |
|
47372 | 359 |
by - (transfer, auto simp add: gr0_conv_Suc) |
45810 | 360 |
qed |
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361 |
|
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362 |
lemma word_of_nat: "of_nat n = word_of_int (int n)" |
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363 |
by (induct n) (auto simp add : word_of_int_hom_syms) |
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364 |
|
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365 |
lemma word_of_int: "of_int = word_of_int" |
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366 |
apply (rule ext) |
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|
367 |
apply (case_tac x rule: int_diff_cases) |
46013 | 368 |
apply (simp add: word_of_nat wi_hom_sub) |
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369 |
done |
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370 |
|
54848 | 371 |
definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) |
372 |
where |
|
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373 |
"a udvd b = (EX n>=0. uint b = n * uint a)" |
37660 | 374 |
|
45547 | 375 |
|
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376 |
subsection {* Ordering *} |
45547 | 377 |
|
378 |
instantiation word :: (len0) linorder |
|
379 |
begin |
|
380 |
||
37660 | 381 |
definition |
382 |
word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b" |
|
383 |
||
384 |
definition |
|
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385 |
word_less_def: "a < b \<longleftrightarrow> uint a < uint b" |
37660 | 386 |
|
45547 | 387 |
instance |
388 |
by default (auto simp: word_less_def word_le_def) |
|
389 |
||
390 |
end |
|
391 |
||
54848 | 392 |
definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) |
393 |
where |
|
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394 |
"a <=s b = (sint a <= sint b)" |
37660 | 395 |
|
54848 | 396 |
definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) |
397 |
where |
|
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398 |
"(x <s y) = (x <=s y & x ~= y)" |
37660 | 399 |
|
400 |
||
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401 |
subsection {* Bit-wise operations *} |
37660 | 402 |
|
403 |
instantiation word :: (len0) bits |
|
404 |
begin |
|
405 |
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406 |
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT |
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407 |
by (metis bin_trunc_not) |
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408 |
|
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409 |
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND |
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410 |
by (metis bin_trunc_and) |
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411 |
|
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412 |
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR |
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413 |
by (metis bin_trunc_or) |
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414 |
|
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415 |
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR |
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416 |
by (metis bin_trunc_xor) |
37660 | 417 |
|
418 |
definition |
|
419 |
word_test_bit_def: "test_bit a = bin_nth (uint a)" |
|
420 |
||
421 |
definition |
|
422 |
word_set_bit_def: "set_bit a n x = |
|
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423 |
word_of_int (bin_sc n x (uint a))" |
37660 | 424 |
|
425 |
definition |
|
426 |
word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)" |
|
427 |
||
428 |
definition |
|
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|
429 |
word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a)" |
37660 | 430 |
|
54848 | 431 |
definition shiftl1 :: "'a word \<Rightarrow> 'a word" |
432 |
where |
|
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|
433 |
"shiftl1 w = word_of_int (uint w BIT False)" |
37660 | 434 |
|
54848 | 435 |
definition shiftr1 :: "'a word \<Rightarrow> 'a word" |
436 |
where |
|
37660 | 437 |
-- "shift right as unsigned or as signed, ie logical or arithmetic" |
438 |
"shiftr1 w = word_of_int (bin_rest (uint w))" |
|
439 |
||
440 |
definition |
|
441 |
shiftl_def: "w << n = (shiftl1 ^^ n) w" |
|
442 |
||
443 |
definition |
|
444 |
shiftr_def: "w >> n = (shiftr1 ^^ n) w" |
|
445 |
||
446 |
instance .. |
|
447 |
||
448 |
end |
|
449 |
||
47611 | 450 |
lemma [code]: shows |
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changeset
|
451 |
word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" and |
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changeset
|
452 |
word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" and |
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changeset
|
453 |
word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" and |
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changeset
|
454 |
word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)" |
9475d524bafb
set up and use lift_definition for word operations
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changeset
|
455 |
unfolding bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def |
9475d524bafb
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parents:
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diff
changeset
|
456 |
by simp_all |
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parents:
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changeset
|
457 |
|
37660 | 458 |
instantiation word :: (len) bitss |
459 |
begin |
|
460 |
||
461 |
definition |
|
462 |
word_msb_def: |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
463 |
"msb a \<longleftrightarrow> bin_sign (sint a) = -1" |
37660 | 464 |
|
465 |
instance .. |
|
466 |
||
467 |
end |
|
468 |
||
54848 | 469 |
definition setBit :: "'a :: len0 word => nat => 'a word" |
470 |
where |
|
40827
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changeset
|
471 |
"setBit w n = set_bit w n True" |
37660 | 472 |
|
54848 | 473 |
definition clearBit :: "'a :: len0 word => nat => 'a word" |
474 |
where |
|
40827
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haftmann
parents:
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changeset
|
475 |
"clearBit w n = set_bit w n False" |
37660 | 476 |
|
477 |
||
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changeset
|
478 |
subsection {* Shift operations *} |
37660 | 479 |
|
54848 | 480 |
definition sshiftr1 :: "'a :: len word => 'a word" |
481 |
where |
|
40827
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haftmann
parents:
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diff
changeset
|
482 |
"sshiftr1 w = word_of_int (bin_rest (sint w))" |
37660 | 483 |
|
54848 | 484 |
definition bshiftr1 :: "bool => 'a :: len word => 'a word" |
485 |
where |
|
40827
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haftmann
parents:
39910
diff
changeset
|
486 |
"bshiftr1 b w = of_bl (b # butlast (to_bl w))" |
37660 | 487 |
|
54848 | 488 |
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) |
489 |
where |
|
40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
490 |
"w >>> n = (sshiftr1 ^^ n) w" |
37660 | 491 |
|
54848 | 492 |
definition mask :: "nat => 'a::len word" |
493 |
where |
|
40827
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haftmann
parents:
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diff
changeset
|
494 |
"mask n = (1 << n) - 1" |
37660 | 495 |
|
54848 | 496 |
definition revcast :: "'a :: len0 word => 'b :: len0 word" |
497 |
where |
|
40827
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changeset
|
498 |
"revcast w = of_bl (takefill False (len_of TYPE('b)) (to_bl w))" |
37660 | 499 |
|
54848 | 500 |
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" |
501 |
where |
|
40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
502 |
"slice1 n w = of_bl (takefill False n (to_bl w))" |
37660 | 503 |
|
54848 | 504 |
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" |
505 |
where |
|
40827
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haftmann
parents:
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diff
changeset
|
506 |
"slice n w = slice1 (size w - n) w" |
37660 | 507 |
|
508 |
||
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diff
changeset
|
509 |
subsection {* Rotation *} |
37660 | 510 |
|
54848 | 511 |
definition rotater1 :: "'a list => 'a list" |
512 |
where |
|
40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
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diff
changeset
|
513 |
"rotater1 ys = |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
514 |
(case ys of [] => [] | x # xs => last ys # butlast ys)" |
37660 | 515 |
|
54848 | 516 |
definition rotater :: "nat => 'a list => 'a list" |
517 |
where |
|
40827
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haftmann
parents:
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diff
changeset
|
518 |
"rotater n = rotater1 ^^ n" |
37660 | 519 |
|
54848 | 520 |
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" |
521 |
where |
|
40827
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haftmann
parents:
39910
diff
changeset
|
522 |
"word_rotr n w = of_bl (rotater n (to_bl w))" |
37660 | 523 |
|
54848 | 524 |
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" |
525 |
where |
|
40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
526 |
"word_rotl n w = of_bl (rotate n (to_bl w))" |
37660 | 527 |
|
54848 | 528 |
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" |
529 |
where |
|
40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
530 |
"word_roti i w = (if i >= 0 then word_rotr (nat i) w |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
531 |
else word_rotl (nat (- i)) w)" |
37660 | 532 |
|
533 |
||
55816
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haftmann
parents:
55415
diff
changeset
|
534 |
subsection {* Split and cat operations *} |
37660 | 535 |
|
54848 | 536 |
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" |
537 |
where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
538 |
"word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))" |
37660 | 539 |
|
54848 | 540 |
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" |
541 |
where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
542 |
"word_split a = |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
543 |
(case bin_split (len_of TYPE ('c)) (uint a) of |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
544 |
(u, v) => (word_of_int u, word_of_int v))" |
37660 | 545 |
|
54848 | 546 |
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" |
547 |
where |
|
40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
548 |
"word_rcat ws = |
37660 | 549 |
word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))" |
550 |
||
54848 | 551 |
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" |
552 |
where |
|
40827
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code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
553 |
"word_rsplit w = |
37660 | 554 |
map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))" |
555 |
||
54848 | 556 |
definition max_word :: "'a::len word" -- "Largest representable machine integer." |
557 |
where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
558 |
"max_word = word_of_int (2 ^ len_of TYPE('a) - 1)" |
37660 | 559 |
|
55816
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cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
560 |
lemmas of_nth_def = word_set_bits_def (* FIXME duplicate *) |
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cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
561 |
|
37660 | 562 |
|
46010 | 563 |
subsection {* Theorems about typedefs *} |
564 |
||
37660 | 565 |
lemma sint_sbintrunc': |
566 |
"sint (word_of_int bin :: 'a word) = |
|
567 |
(sbintrunc (len_of TYPE ('a :: len) - 1) bin)" |
|
568 |
unfolding sint_uint |
|
569 |
by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt) |
|
570 |
||
571 |
lemma uint_sint: |
|
572 |
"uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))" |
|
573 |
unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le) |
|
574 |
||
46057 | 575 |
lemma bintr_uint: |
576 |
fixes w :: "'a::len0 word" |
|
577 |
shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w" |
|
37660 | 578 |
apply (subst word_ubin.norm_Rep [symmetric]) |
579 |
apply (simp only: bintrunc_bintrunc_min word_size) |
|
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54854
diff
changeset
|
580 |
apply (simp add: min.absorb2) |
37660 | 581 |
done |
582 |
||
46057 | 583 |
lemma wi_bintr: |
584 |
"len_of TYPE('a::len0) \<le> n \<Longrightarrow> |
|
585 |
word_of_int (bintrunc n w) = (word_of_int w :: 'a word)" |
|
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54854
diff
changeset
|
586 |
by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min.absorb1) |
37660 | 587 |
|
588 |
lemma td_ext_sbin: |
|
55816
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cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
589 |
"td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len))) |
37660 | 590 |
(sbintrunc (len_of TYPE('a) - 1))" |
591 |
apply (unfold td_ext_def' sint_uint) |
|
592 |
apply (simp add : word_ubin.eq_norm) |
|
593 |
apply (cases "len_of TYPE('a)") |
|
594 |
apply (auto simp add : sints_def) |
|
595 |
apply (rule sym [THEN trans]) |
|
596 |
apply (rule word_ubin.Abs_norm) |
|
597 |
apply (simp only: bintrunc_sbintrunc) |
|
598 |
apply (drule sym) |
|
599 |
apply simp |
|
600 |
done |
|
601 |
||
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
602 |
lemma td_ext_sint: |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
603 |
"td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len))) |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
604 |
(\<lambda>w. (w + 2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) - |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
605 |
2 ^ (len_of TYPE('a) - 1))" |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
606 |
using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2) |
37660 | 607 |
|
608 |
(* We do sint before sbin, before sint is the user version |
|
609 |
and interpretations do not produce thm duplicates. I.e. |
|
610 |
we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, |
|
611 |
because the latter is the same thm as the former *) |
|
612 |
interpretation word_sint: |
|
613 |
td_ext "sint ::'a::len word => int" |
|
614 |
word_of_int |
|
615 |
"sints (len_of TYPE('a::len))" |
|
616 |
"%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) - |
|
617 |
2 ^ (len_of TYPE('a::len) - 1)" |
|
618 |
by (rule td_ext_sint) |
|
619 |
||
620 |
interpretation word_sbin: |
|
621 |
td_ext "sint ::'a::len word => int" |
|
622 |
word_of_int |
|
623 |
"sints (len_of TYPE('a::len))" |
|
624 |
"sbintrunc (len_of TYPE('a::len) - 1)" |
|
625 |
by (rule td_ext_sbin) |
|
626 |
||
45604 | 627 |
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm] |
37660 | 628 |
|
629 |
lemmas td_sint = word_sint.td |
|
630 |
||
631 |
lemma to_bl_def': |
|
632 |
"(to_bl :: 'a :: len0 word => bool list) = |
|
633 |
bin_to_bl (len_of TYPE('a)) o uint" |
|
44762 | 634 |
by (auto simp: to_bl_def) |
37660 | 635 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
636 |
lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w |
37660 | 637 |
|
45805 | 638 |
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)" |
639 |
by (fact uints_def [unfolded no_bintr_alt1]) |
|
640 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
641 |
lemma word_numeral_alt: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
642 |
"numeral b = word_of_int (numeral b)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
643 |
by (induct b, simp_all only: numeral.simps word_of_int_homs) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
644 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
645 |
declare word_numeral_alt [symmetric, code_abbrev] |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
646 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
647 |
lemma word_neg_numeral_alt: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
648 |
"- numeral b = word_of_int (- numeral b)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
649 |
by (simp only: word_numeral_alt wi_hom_neg) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
650 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
651 |
declare word_neg_numeral_alt [symmetric, code_abbrev] |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
652 |
|
47372 | 653 |
lemma word_numeral_transfer [transfer_rule]: |
51375
d9e62d9c98de
patch Isabelle ditribution to conform to changes regarding the parametricity
kuncar
parents:
51286
diff
changeset
|
654 |
"(fun_rel op = pcr_word) numeral numeral" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
655 |
"(fun_rel op = pcr_word) (- numeral) (- numeral)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
656 |
apply (simp_all add: fun_rel_def word.pcr_cr_eq cr_word_def) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
657 |
using word_numeral_alt [symmetric] word_neg_numeral_alt [symmetric] by blast+ |
47372 | 658 |
|
45805 | 659 |
lemma uint_bintrunc [simp]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
660 |
"uint (numeral bin :: 'a word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
661 |
bintrunc (len_of TYPE ('a :: len0)) (numeral bin)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
662 |
unfolding word_numeral_alt by (rule word_ubin.eq_norm) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
663 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
664 |
lemma uint_bintrunc_neg [simp]: "uint (- numeral bin :: 'a word) = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
665 |
bintrunc (len_of TYPE ('a :: len0)) (- numeral bin)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
666 |
by (simp only: word_neg_numeral_alt word_ubin.eq_norm) |
37660 | 667 |
|
45805 | 668 |
lemma sint_sbintrunc [simp]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
669 |
"sint (numeral bin :: 'a word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
670 |
sbintrunc (len_of TYPE ('a :: len) - 1) (numeral bin)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
671 |
by (simp only: word_numeral_alt word_sbin.eq_norm) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
672 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
673 |
lemma sint_sbintrunc_neg [simp]: "sint (- numeral bin :: 'a word) = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
674 |
sbintrunc (len_of TYPE ('a :: len) - 1) (- numeral bin)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
675 |
by (simp only: word_neg_numeral_alt word_sbin.eq_norm) |
37660 | 676 |
|
45805 | 677 |
lemma unat_bintrunc [simp]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
678 |
"unat (numeral bin :: 'a :: len0 word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
679 |
nat (bintrunc (len_of TYPE('a)) (numeral bin))" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
680 |
by (simp only: unat_def uint_bintrunc) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
681 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
682 |
lemma unat_bintrunc_neg [simp]: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
683 |
"unat (- numeral bin :: 'a :: len0 word) = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
684 |
nat (bintrunc (len_of TYPE('a)) (- numeral bin))" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
685 |
by (simp only: unat_def uint_bintrunc_neg) |
37660 | 686 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
687 |
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w" |
37660 | 688 |
apply (unfold word_size) |
689 |
apply (rule word_uint.Rep_eqD) |
|
690 |
apply (rule box_equals) |
|
691 |
defer |
|
692 |
apply (rule word_ubin.norm_Rep)+ |
|
693 |
apply simp |
|
694 |
done |
|
695 |
||
45805 | 696 |
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)" |
697 |
using word_uint.Rep [of x] by (simp add: uints_num) |
|
698 |
||
699 |
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)" |
|
700 |
using word_uint.Rep [of x] by (simp add: uints_num) |
|
701 |
||
702 |
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)" |
|
703 |
using word_sint.Rep [of x] by (simp add: sints_num) |
|
704 |
||
705 |
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)" |
|
706 |
using word_sint.Rep [of x] by (simp add: sints_num) |
|
37660 | 707 |
|
708 |
lemma sign_uint_Pls [simp]: |
|
46604
9f9e85264e4d
make uses of bin_sign respect int/bin distinction
huffman
parents:
46603
diff
changeset
|
709 |
"bin_sign (uint x) = 0" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
710 |
by (simp add: sign_Pls_ge_0) |
37660 | 711 |
|
45805 | 712 |
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0" |
713 |
by (simp only: diff_less_0_iff_less uint_lt2p) |
|
714 |
||
715 |
lemma uint_m2p_not_non_neg: |
|
716 |
"\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)" |
|
717 |
by (simp only: not_le uint_m2p_neg) |
|
37660 | 718 |
|
719 |
lemma lt2p_lem: |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
720 |
"len_of TYPE('a) \<le> n \<Longrightarrow> uint (w :: 'a::len0 word) < 2 ^ n" |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
721 |
by (metis bintr_uint bintrunc_mod2p int_mod_lem zless2p) |
37660 | 722 |
|
45805 | 723 |
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0" |
724 |
by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1]) |
|
37660 | 725 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
726 |
lemma uint_nat: "uint w = int (unat w)" |
37660 | 727 |
unfolding unat_def by auto |
728 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
729 |
lemma uint_numeral: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
730 |
"uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
731 |
unfolding word_numeral_alt |
37660 | 732 |
by (simp only: int_word_uint) |
733 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
734 |
lemma uint_neg_numeral: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
735 |
"uint (- numeral b :: 'a :: len0 word) = - numeral b mod 2 ^ len_of TYPE('a)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
736 |
unfolding word_neg_numeral_alt |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
737 |
by (simp only: int_word_uint) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
738 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
739 |
lemma unat_numeral: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
740 |
"unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)" |
37660 | 741 |
apply (unfold unat_def) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
742 |
apply (clarsimp simp only: uint_numeral) |
37660 | 743 |
apply (rule nat_mod_distrib [THEN trans]) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
744 |
apply (rule zero_le_numeral) |
37660 | 745 |
apply (simp_all add: nat_power_eq) |
746 |
done |
|
747 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
748 |
lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + |
37660 | 749 |
2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) - |
750 |
2 ^ (len_of TYPE('a) - 1)" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
751 |
unfolding word_numeral_alt by (rule int_word_sint) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
752 |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
753 |
lemma word_of_int_0 [simp, code_post]: |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
754 |
"word_of_int 0 = 0" |
45958 | 755 |
unfolding word_0_wi .. |
756 |
||
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
757 |
lemma word_of_int_1 [simp, code_post]: |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
758 |
"word_of_int 1 = 1" |
45958 | 759 |
unfolding word_1_wi .. |
760 |
||
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
761 |
lemma word_of_int_neg_1 [simp]: "word_of_int (- 1) = - 1" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
762 |
by (simp add: wi_hom_syms) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
763 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
764 |
lemma word_of_int_numeral [simp] : |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
765 |
"(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
766 |
unfolding word_numeral_alt .. |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
767 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
768 |
lemma word_of_int_neg_numeral [simp]: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
769 |
"(word_of_int (- numeral bin) :: 'a :: len0 word) = (- numeral bin)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
770 |
unfolding word_numeral_alt wi_hom_syms .. |
37660 | 771 |
|
772 |
lemma word_int_case_wi: |
|
773 |
"word_int_case f (word_of_int i :: 'b word) = |
|
774 |
f (i mod 2 ^ len_of TYPE('b::len0))" |
|
775 |
unfolding word_int_case_def by (simp add: word_uint.eq_norm) |
|
776 |
||
777 |
lemma word_int_split: |
|
778 |
"P (word_int_case f x) = |
|
779 |
(ALL i. x = (word_of_int i :: 'b :: len0 word) & |
|
780 |
0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))" |
|
781 |
unfolding word_int_case_def |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
782 |
by (auto simp: word_uint.eq_norm mod_pos_pos_trivial) |
37660 | 783 |
|
784 |
lemma word_int_split_asm: |
|
785 |
"P (word_int_case f x) = |
|
786 |
(~ (EX n. x = (word_of_int n :: 'b::len0 word) & |
|
787 |
0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))" |
|
788 |
unfolding word_int_case_def |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
789 |
by (auto simp: word_uint.eq_norm mod_pos_pos_trivial) |
45805 | 790 |
|
45604 | 791 |
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq] |
792 |
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq] |
|
37660 | 793 |
|
794 |
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w" |
|
795 |
unfolding word_size by (rule uint_range') |
|
796 |
||
797 |
lemma sint_range_size: |
|
798 |
"- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)" |
|
799 |
unfolding word_size by (rule sint_range') |
|
800 |
||
45805 | 801 |
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x" |
802 |
unfolding word_size by (rule less_le_trans [OF sint_lt]) |
|
803 |
||
804 |
lemma sint_below_size: |
|
805 |
"x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w" |
|
806 |
unfolding word_size by (rule order_trans [OF _ sint_ge]) |
|
37660 | 807 |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
808 |
|
46010 | 809 |
subsection {* Testing bits *} |
810 |
||
37660 | 811 |
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)" |
812 |
unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) |
|
813 |
||
814 |
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w" |
|
815 |
apply (unfold word_test_bit_def) |
|
816 |
apply (subst word_ubin.norm_Rep [symmetric]) |
|
817 |
apply (simp only: nth_bintr word_size) |
|
818 |
apply fast |
|
819 |
done |
|
820 |
||
46021 | 821 |
lemma word_eq_iff: |
822 |
fixes x y :: "'a::len0 word" |
|
823 |
shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)" |
|
824 |
unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric] |
|
825 |
by (metis test_bit_size [unfolded word_size]) |
|
826 |
||
46023
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset
|
827 |
lemma word_eqI [rule_format]: |
37660 | 828 |
fixes u :: "'a::len0 word" |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
829 |
shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v" |
46021 | 830 |
by (simp add: word_size word_eq_iff) |
37660 | 831 |
|
45805 | 832 |
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x" |
833 |
by simp |
|
37660 | 834 |
|
835 |
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)" |
|
836 |
unfolding word_test_bit_def word_size |
|
837 |
by (simp add: nth_bintr [symmetric]) |
|
838 |
||
839 |
lemmas test_bit_bin = test_bit_bin' [unfolded word_size] |
|
840 |
||
46057 | 841 |
lemma bin_nth_uint_imp: |
842 |
"bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)" |
|
37660 | 843 |
apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) |
844 |
apply (subst word_ubin.norm_Rep) |
|
845 |
apply assumption |
|
846 |
done |
|
847 |
||
46057 | 848 |
lemma bin_nth_sint: |
849 |
fixes w :: "'a::len word" |
|
850 |
shows "len_of TYPE('a) \<le> n \<Longrightarrow> |
|
851 |
bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)" |
|
37660 | 852 |
apply (subst word_sbin.norm_Rep [symmetric]) |
46057 | 853 |
apply (auto simp add: nth_sbintr) |
37660 | 854 |
done |
855 |
||
856 |
(* type definitions theorem for in terms of equivalent bool list *) |
|
857 |
lemma td_bl: |
|
858 |
"type_definition (to_bl :: 'a::len0 word => bool list) |
|
859 |
of_bl |
|
860 |
{bl. length bl = len_of TYPE('a)}" |
|
861 |
apply (unfold type_definition_def of_bl_def to_bl_def) |
|
862 |
apply (simp add: word_ubin.eq_norm) |
|
863 |
apply safe |
|
864 |
apply (drule sym) |
|
865 |
apply simp |
|
866 |
done |
|
867 |
||
868 |
interpretation word_bl: |
|
869 |
type_definition "to_bl :: 'a::len0 word => bool list" |
|
870 |
of_bl |
|
871 |
"{bl. length bl = len_of TYPE('a::len0)}" |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
872 |
by (fact td_bl) |
37660 | 873 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
874 |
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff] |
45538
1fffa81b9b83
eliminated slightly odd Rep' with dynamically-scoped [simplified];
wenzelm
parents:
45529
diff
changeset
|
875 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
876 |
lemma word_size_bl: "size w = size (to_bl w)" |
37660 | 877 |
unfolding word_size by auto |
878 |
||
879 |
lemma to_bl_use_of_bl: |
|
880 |
"(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))" |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
881 |
by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq]) |
37660 | 882 |
|
883 |
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" |
|
884 |
unfolding word_reverse_def by (simp add: word_bl.Abs_inverse) |
|
885 |
||
886 |
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" |
|
887 |
unfolding word_reverse_def by (simp add : word_bl.Abs_inverse) |
|
888 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
889 |
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
890 |
by (metis word_rev_rev) |
37660 | 891 |
|
45805 | 892 |
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u" |
893 |
by simp |
|
894 |
||
895 |
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))" |
|
896 |
unfolding word_bl_Rep' by (rule len_gt_0) |
|
897 |
||
898 |
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []" |
|
899 |
by (fact length_bl_gt_0 [unfolded length_greater_0_conv]) |
|
900 |
||
901 |
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0" |
|
902 |
by (fact length_bl_gt_0 [THEN gr_implies_not0]) |
|
37660 | 903 |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
904 |
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)" |
37660 | 905 |
apply (unfold to_bl_def sint_uint) |
906 |
apply (rule trans [OF _ bl_sbin_sign]) |
|
907 |
apply simp |
|
908 |
done |
|
909 |
||
910 |
lemma of_bl_drop': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
911 |
"lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> |
37660 | 912 |
of_bl (drop lend bl) = (of_bl bl :: 'a word)" |
913 |
apply (unfold of_bl_def) |
|
914 |
apply (clarsimp simp add : trunc_bl2bin [symmetric]) |
|
915 |
done |
|
916 |
||
917 |
lemma test_bit_of_bl: |
|
918 |
"(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)" |
|
919 |
apply (unfold of_bl_def word_test_bit_def) |
|
920 |
apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) |
|
921 |
done |
|
922 |
||
923 |
lemma no_of_bl: |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
924 |
"(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) (numeral bin))" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
925 |
unfolding of_bl_def by simp |
37660 | 926 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
927 |
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)" |
37660 | 928 |
unfolding word_size to_bl_def by auto |
929 |
||
930 |
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" |
|
931 |
unfolding uint_bl by (simp add : word_size) |
|
932 |
||
933 |
lemma to_bl_of_bin: |
|
934 |
"to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" |
|
935 |
unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size) |
|
936 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
937 |
lemma to_bl_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
938 |
"to_bl (numeral bin::'a::len0 word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
939 |
bin_to_bl (len_of TYPE('a)) (numeral bin)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
940 |
unfolding word_numeral_alt by (rule to_bl_of_bin) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
941 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
942 |
lemma to_bl_neg_numeral [simp]: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
943 |
"to_bl (- numeral bin::'a::len0 word) = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
944 |
bin_to_bl (len_of TYPE('a)) (- numeral bin)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
945 |
unfolding word_neg_numeral_alt by (rule to_bl_of_bin) |
37660 | 946 |
|
947 |
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" |
|
948 |
unfolding uint_bl by (simp add : word_size) |
|
46011 | 949 |
|
950 |
lemma uint_bl_bin: |
|
951 |
fixes x :: "'a::len0 word" |
|
952 |
shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x" |
|
953 |
by (rule trans [OF bin_bl_bin word_ubin.norm_Rep]) |
|
45604 | 954 |
|
37660 | 955 |
(* naturals *) |
956 |
lemma uints_unats: "uints n = int ` unats n" |
|
957 |
apply (unfold unats_def uints_num) |
|
958 |
apply safe |
|
959 |
apply (rule_tac image_eqI) |
|
960 |
apply (erule_tac nat_0_le [symmetric]) |
|
961 |
apply auto |
|
962 |
apply (erule_tac nat_less_iff [THEN iffD2]) |
|
963 |
apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) |
|
964 |
apply (auto simp add : nat_power_eq int_power) |
|
965 |
done |
|
966 |
||
967 |
lemma unats_uints: "unats n = nat ` uints n" |
|
968 |
by (auto simp add : uints_unats image_iff) |
|
969 |
||
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
970 |
lemmas bintr_num = word_ubin.norm_eq_iff |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
971 |
[of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b |
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
972 |
lemmas sbintr_num = word_sbin.norm_eq_iff |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
973 |
[of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b |
37660 | 974 |
|
975 |
lemma num_of_bintr': |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
976 |
"bintrunc (len_of TYPE('a :: len0)) (numeral a) = (numeral b) \<Longrightarrow> |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
977 |
numeral a = (numeral b :: 'a word)" |
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
978 |
unfolding bintr_num by (erule subst, simp) |
37660 | 979 |
|
980 |
lemma num_of_sbintr': |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
981 |
"sbintrunc (len_of TYPE('a :: len) - 1) (numeral a) = (numeral b) \<Longrightarrow> |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
982 |
numeral a = (numeral b :: 'a word)" |
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
983 |
unfolding sbintr_num by (erule subst, simp) |
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
984 |
|
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
985 |
lemma num_abs_bintr: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
986 |
"(numeral x :: 'a word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
987 |
word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
988 |
by (simp only: word_ubin.Abs_norm word_numeral_alt) |
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
989 |
|
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
990 |
lemma num_abs_sbintr: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
991 |
"(numeral x :: 'a word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
992 |
word_of_int (sbintrunc (len_of TYPE('a::len) - 1) (numeral x))" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
993 |
by (simp only: word_sbin.Abs_norm word_numeral_alt) |
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
994 |
|
37660 | 995 |
(** cast - note, no arg for new length, as it's determined by type of result, |
996 |
thus in "cast w = w, the type means cast to length of w! **) |
|
997 |
||
998 |
lemma ucast_id: "ucast w = w" |
|
999 |
unfolding ucast_def by auto |
|
1000 |
||
1001 |
lemma scast_id: "scast w = w" |
|
1002 |
unfolding scast_def by auto |
|
1003 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1004 |
lemma ucast_bl: "ucast w = of_bl (to_bl w)" |
37660 | 1005 |
unfolding ucast_def of_bl_def uint_bl |
1006 |
by (auto simp add : word_size) |
|
1007 |
||
1008 |
lemma nth_ucast: |
|
1009 |
"(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))" |
|
1010 |
apply (unfold ucast_def test_bit_bin) |
|
1011 |
apply (simp add: word_ubin.eq_norm nth_bintr word_size) |
|
1012 |
apply (fast elim!: bin_nth_uint_imp) |
|
1013 |
done |
|
1014 |
||
1015 |
(* for literal u(s)cast *) |
|
1016 |
||
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
1017 |
lemma ucast_bintr [simp]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1018 |
"ucast (numeral w ::'a::len0 word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1019 |
word_of_int (bintrunc (len_of TYPE('a)) (numeral w))" |
37660 | 1020 |
unfolding ucast_def by simp |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1021 |
(* TODO: neg_numeral *) |
37660 | 1022 |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
1023 |
lemma scast_sbintr [simp]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1024 |
"scast (numeral w ::'a::len word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1025 |
word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (numeral w))" |
37660 | 1026 |
unfolding scast_def by simp |
1027 |
||
46011 | 1028 |
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)" |
1029 |
unfolding source_size_def word_size Let_def .. |
|
1030 |
||
1031 |
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)" |
|
1032 |
unfolding target_size_def word_size Let_def .. |
|
1033 |
||
1034 |
lemma is_down: |
|
1035 |
fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word" |
|
1036 |
shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)" |
|
1037 |
unfolding is_down_def source_size target_size .. |
|
1038 |
||
1039 |
lemma is_up: |
|
1040 |
fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word" |
|
1041 |
shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)" |
|
1042 |
unfolding is_up_def source_size target_size .. |
|
37660 | 1043 |
|
45604 | 1044 |
lemmas is_up_down = trans [OF is_up is_down [symmetric]] |
37660 | 1045 |
|
45811 | 1046 |
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast" |
37660 | 1047 |
apply (unfold is_down) |
1048 |
apply safe |
|
1049 |
apply (rule ext) |
|
1050 |
apply (unfold ucast_def scast_def uint_sint) |
|
1051 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
1052 |
apply simp |
|
1053 |
done |
|
1054 |
||
45811 | 1055 |
lemma word_rev_tf: |
1056 |
"to_bl (of_bl bl::'a::len0 word) = |
|
1057 |
rev (takefill False (len_of TYPE('a)) (rev bl))" |
|
37660 | 1058 |
unfolding of_bl_def uint_bl |
1059 |
by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size) |
|
1060 |
||
45811 | 1061 |
lemma word_rep_drop: |
1062 |
"to_bl (of_bl bl::'a::len0 word) = |
|
1063 |
replicate (len_of TYPE('a) - length bl) False @ |
|
1064 |
drop (length bl - len_of TYPE('a)) bl" |
|
1065 |
by (simp add: word_rev_tf takefill_alt rev_take) |
|
37660 | 1066 |
|
1067 |
lemma to_bl_ucast: |
|
1068 |
"to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = |
|
1069 |
replicate (len_of TYPE('a) - len_of TYPE('b)) False @ |
|
1070 |
drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)" |
|
1071 |
apply (unfold ucast_bl) |
|
1072 |
apply (rule trans) |
|
1073 |
apply (rule word_rep_drop) |
|
1074 |
apply simp |
|
1075 |
done |
|
1076 |
||
45811 | 1077 |
lemma ucast_up_app [OF refl]: |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1078 |
"uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> |
37660 | 1079 |
to_bl (uc w) = replicate n False @ (to_bl w)" |
1080 |
by (auto simp add : source_size target_size to_bl_ucast) |
|
1081 |
||
45811 | 1082 |
lemma ucast_down_drop [OF refl]: |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1083 |
"uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> |
37660 | 1084 |
to_bl (uc w) = drop n (to_bl w)" |
1085 |
by (auto simp add : source_size target_size to_bl_ucast) |
|
1086 |
||
45811 | 1087 |
lemma scast_down_drop [OF refl]: |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1088 |
"sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> |
37660 | 1089 |
to_bl (sc w) = drop n (to_bl w)" |
1090 |
apply (subgoal_tac "sc = ucast") |
|
1091 |
apply safe |
|
1092 |
apply simp |
|
45811 | 1093 |
apply (erule ucast_down_drop) |
1094 |
apply (rule down_cast_same [symmetric]) |
|
37660 | 1095 |
apply (simp add : source_size target_size is_down) |
1096 |
done |
|
1097 |
||
45811 | 1098 |
lemma sint_up_scast [OF refl]: |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1099 |
"sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w" |
37660 | 1100 |
apply (unfold is_up) |
1101 |
apply safe |
|
1102 |
apply (simp add: scast_def word_sbin.eq_norm) |
|
1103 |
apply (rule box_equals) |
|
1104 |
prefer 3 |
|
1105 |
apply (rule word_sbin.norm_Rep) |
|
1106 |
apply (rule sbintrunc_sbintrunc_l) |
|
1107 |
defer |
|
1108 |
apply (subst word_sbin.norm_Rep) |
|
1109 |
apply (rule refl) |
|
1110 |
apply simp |
|
1111 |
done |
|
1112 |
||
45811 | 1113 |
lemma uint_up_ucast [OF refl]: |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1114 |
"uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w" |
37660 | 1115 |
apply (unfold is_up) |
1116 |
apply safe |
|
1117 |
apply (rule bin_eqI) |
|
1118 |
apply (fold word_test_bit_def) |
|
1119 |
apply (auto simp add: nth_ucast) |
|
1120 |
apply (auto simp add: test_bit_bin) |
|
1121 |
done |
|
45811 | 1122 |
|
1123 |
lemma ucast_up_ucast [OF refl]: |
|
1124 |
"uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w" |
|
37660 | 1125 |
apply (simp (no_asm) add: ucast_def) |
1126 |
apply (clarsimp simp add: uint_up_ucast) |
|
1127 |
done |
|
1128 |
||
45811 | 1129 |
lemma scast_up_scast [OF refl]: |
1130 |
"sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w" |
|
37660 | 1131 |
apply (simp (no_asm) add: scast_def) |
1132 |
apply (clarsimp simp add: sint_up_scast) |
|
1133 |
done |
|
1134 |
||
45811 | 1135 |
lemma ucast_of_bl_up [OF refl]: |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1136 |
"w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl" |
37660 | 1137 |
by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) |
1138 |
||
1139 |
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] |
|
1140 |
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] |
|
1141 |
||
1142 |
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] |
|
1143 |
lemmas isdus = is_up_down [where c = "scast", THEN iffD2] |
|
1144 |
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] |
|
1145 |
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] |
|
1146 |
||
1147 |
lemma up_ucast_surj: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1148 |
"is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> |
37660 | 1149 |
surj (ucast :: 'a word => 'b word)" |
1150 |
by (rule surjI, erule ucast_up_ucast_id) |
|
1151 |
||
1152 |
lemma up_scast_surj: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1153 |
"is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> |
37660 | 1154 |
surj (scast :: 'a word => 'b word)" |
1155 |
by (rule surjI, erule scast_up_scast_id) |
|
1156 |
||
1157 |
lemma down_scast_inj: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1158 |
"is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> |
37660 | 1159 |
inj_on (ucast :: 'a word => 'b word) A" |
1160 |
by (rule inj_on_inverseI, erule scast_down_scast_id) |
|
1161 |
||
1162 |
lemma down_ucast_inj: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1163 |
"is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> |
37660 | 1164 |
inj_on (ucast :: 'a word => 'b word) A" |
1165 |
by (rule inj_on_inverseI, erule ucast_down_ucast_id) |
|
1166 |
||
1167 |
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" |
|
1168 |
by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) |
|
45811 | 1169 |
|
46646 | 1170 |
lemma ucast_down_wi [OF refl]: |
1171 |
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x" |
|
1172 |
apply (unfold is_down) |
|
37660 | 1173 |
apply (clarsimp simp add: ucast_def word_ubin.eq_norm) |
1174 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
1175 |
apply (erule bintrunc_bintrunc_ge) |
|
1176 |
done |
|
45811 | 1177 |
|
46646 | 1178 |
lemma ucast_down_no [OF refl]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46962
diff
changeset
|
1179 |
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1180 |
unfolding word_numeral_alt by clarify (rule ucast_down_wi) |
46646 | 1181 |
|
45811 | 1182 |
lemma ucast_down_bl [OF refl]: |
1183 |
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl" |
|
46646 | 1184 |
unfolding of_bl_def by clarify (erule ucast_down_wi) |
37660 | 1185 |
|
1186 |
lemmas slice_def' = slice_def [unfolded word_size] |
|
1187 |
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] |
|
1188 |
||
1189 |
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def |
|
1190 |
||
1191 |
||
1192 |
subsection {* Word Arithmetic *} |
|
1193 |
||
1194 |
lemma word_less_alt: "(a < b) = (uint a < uint b)" |
|
55818 | 1195 |
by (fact word_less_def) |
37660 | 1196 |
|
1197 |
lemma signed_linorder: "class.linorder word_sle word_sless" |
|
46124 | 1198 |
by default (unfold word_sle_def word_sless_def, auto) |
37660 | 1199 |
|
1200 |
interpretation signed: linorder "word_sle" "word_sless" |
|
1201 |
by (rule signed_linorder) |
|
1202 |
||
1203 |
lemma udvdI: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1204 |
"0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b" |
37660 | 1205 |
by (auto simp: udvd_def) |
1206 |
||
47108
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parents:
46962
diff
changeset
|
1207 |
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
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diff
changeset
|
1208 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset
|
1209 |
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset
|
1210 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset
|
1211 |
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1212 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset
|
1213 |
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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diff
changeset
|
1214 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46962
diff
changeset
|
1215 |
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1216 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
46962
diff
changeset
|
1217 |
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b |
37660 | 1218 |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
1219 |
lemma word_m1_wi: "- 1 = word_of_int (- 1)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
1220 |
using word_neg_numeral_alt [of Num.One] by simp |
37660 | 1221 |
|
46648 | 1222 |
lemma word_0_bl [simp]: "of_bl [] = 0" |
1223 |
unfolding of_bl_def by simp |
|
37660 | 1224 |
|
1225 |
lemma word_1_bl: "of_bl [True] = 1" |
|
46648 | 1226 |
unfolding of_bl_def by (simp add: bl_to_bin_def) |
1227 |
||
1228 |
lemma uint_eq_0 [simp]: "uint 0 = 0" |
|
1229 |
unfolding word_0_wi word_ubin.eq_norm by simp |
|
37660 | 1230 |
|
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
1231 |
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0" |
46648 | 1232 |
by (simp add: of_bl_def bl_to_bin_rep_False) |
37660 | 1233 |
|
45805 | 1234 |
lemma to_bl_0 [simp]: |
37660 | 1235 |
"to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" |
1236 |
unfolding uint_bl |
|
46617
8c5d10d41391
make bool list functions respect int/bin distinction
huffman
parents:
46604
diff
changeset
|
1237 |
by (simp add: word_size bin_to_bl_zero) |
37660 | 1238 |
|
55818 | 1239 |
lemma uint_0_iff: |
1240 |
"uint x = 0 \<longleftrightarrow> x = 0" |
|
1241 |
by (simp add: word_uint_eq_iff) |
|
1242 |
||
1243 |
lemma unat_0_iff: |
|
1244 |
"unat x = 0 \<longleftrightarrow> x = 0" |
|
37660 | 1245 |
unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff) |
1246 |
||
55818 | 1247 |
lemma unat_0 [simp]: |
1248 |
"unat 0 = 0" |
|
37660 | 1249 |
unfolding unat_def by auto |
1250 |
||
55818 | 1251 |
lemma size_0_same': |
1252 |
"size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)" |
|
37660 | 1253 |
apply (unfold word_size) |
1254 |
apply (rule box_equals) |
|
1255 |
defer |
|
1256 |
apply (rule word_uint.Rep_inverse)+ |
|
1257 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
1258 |
apply simp |
|
1259 |
done |
|
1260 |
||
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1261 |
lemmas size_0_same = size_0_same' [unfolded word_size] |
37660 | 1262 |
|
1263 |
lemmas unat_eq_0 = unat_0_iff |
|
1264 |
lemmas unat_eq_zero = unat_0_iff |
|
1265 |
||
1266 |
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)" |
|
1267 |
by (auto simp: unat_0_iff [symmetric]) |
|
1268 |
||
45958 | 1269 |
lemma ucast_0 [simp]: "ucast 0 = 0" |
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
1270 |
unfolding ucast_def by simp |
45958 | 1271 |
|
1272 |
lemma sint_0 [simp]: "sint 0 = 0" |
|
1273 |
unfolding sint_uint by simp |
|
1274 |
||
1275 |
lemma scast_0 [simp]: "scast 0 = 0" |
|
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
1276 |
unfolding scast_def by simp |
37660 | 1277 |
|
1278 |
lemma sint_n1 [simp] : "sint -1 = -1" |
|
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
1279 |
unfolding word_m1_wi word_sbin.eq_norm by simp |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
1280 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
1281 |
lemma scast_n1 [simp]: "scast (- 1) = - 1" |
45958 | 1282 |
unfolding scast_def by simp |
1283 |
||
1284 |
lemma uint_1 [simp]: "uint (1::'a::len word) = 1" |
|
55818 | 1285 |
by (simp only: word_1_wi word_ubin.eq_norm) (simp add: bintrunc_minus_simps(4)) |
45958 | 1286 |
|
1287 |
lemma unat_1 [simp]: "unat (1::'a::len word) = 1" |
|
1288 |
unfolding unat_def by simp |
|
1289 |
||
1290 |
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1" |
|
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
1291 |
unfolding ucast_def by simp |
37660 | 1292 |
|
1293 |
(* now, to get the weaker results analogous to word_div/mod_def *) |
|
1294 |
||
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1295 |
|
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1296 |
subsection {* Transferring goals from words to ints *} |
37660 | 1297 |
|
1298 |
lemma word_ths: |
|
1299 |
shows |
|
1300 |
word_succ_p1: "word_succ a = a + 1" and |
|
1301 |
word_pred_m1: "word_pred a = a - 1" and |
|
1302 |
word_pred_succ: "word_pred (word_succ a) = a" and |
|
1303 |
word_succ_pred: "word_succ (word_pred a) = a" and |
|
1304 |
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
|
1305 |
by (transfer, simp add: algebra_simps)+ |
37660 | 1306 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1307 |
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y" |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1308 |
by simp |
37660 | 1309 |
|
55818 | 1310 |
lemma uint_word_ariths: |
1311 |
fixes a b :: "'a::len0 word" |
|