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(* Title: HOL/Word/Word.thy |
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Author: Jeremy Dawson and Gerwin Klein, NICTA |
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*) |
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section \<open>A type of finite bit strings\<close> |
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theory Word |
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imports |
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"~~/src/HOL/Library/Type_Length" |
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"~~/src/HOL/Library/Boolean_Algebra" |
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Bits_Bit |
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Bool_List_Representation |
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Misc_Typedef |
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Word_Miscellaneous |
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begin |
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text \<open>See \<^file>\<open>Examples/WordExamples.thy\<close> for examples.\<close> |
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subsection \<open>Type definition\<close> |
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typedef (overloaded) 'a word = "{(0::int) ..< 2 ^ len_of TYPE('a::len0)}" |
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morphisms uint Abs_word by auto |
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lemma uint_nonnegative: "0 \<le> uint w" |
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using word.uint [of w] by simp |
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lemma uint_bounded: "uint w < 2 ^ len_of TYPE('a)" |
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for w :: "'a::len0 word" |
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using word.uint [of w] by simp |
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lemma uint_idem: "uint w mod 2 ^ len_of TYPE('a) = uint w" |
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for w :: "'a::len0 word" |
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using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial) |
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lemma word_uint_eq_iff: "a = b \<longleftrightarrow> uint a = uint b" |
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by (simp add: uint_inject) |
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lemma word_uint_eqI: "uint a = uint b \<Longrightarrow> a = b" |
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by (simp add: word_uint_eq_iff) |
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definition word_of_int :: "int \<Rightarrow> 'a::len0 word" |
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\<comment> \<open>representation of words using unsigned or signed bins, |
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only difference in these is the type class\<close> |
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where "word_of_int k = Abs_word (k mod 2 ^ len_of TYPE('a))" |
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lemma uint_word_of_int: "uint (word_of_int k :: 'a::len0 word) = k mod 2 ^ len_of TYPE('a)" |
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by (auto simp add: word_of_int_def intro: Abs_word_inverse) |
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lemma word_of_int_uint: "word_of_int (uint w) = w" |
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by (simp add: word_of_int_def uint_idem uint_inverse) |
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lemma split_word_all: "(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))" |
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proof |
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fix x :: "'a word" |
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assume "\<And>x. PROP P (word_of_int x)" |
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then have "PROP P (word_of_int (uint x))" . |
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then show "PROP P x" by (simp add: word_of_int_uint) |
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qed |
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subsection \<open>Type conversions and casting\<close> |
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definition sint :: "'a::len word \<Rightarrow> int" |
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\<comment> \<open>treats the most-significant-bit as a sign bit\<close> |
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where sint_uint: "sint w = sbintrunc (len_of TYPE('a) - 1) (uint w)" |
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definition unat :: "'a::len0 word \<Rightarrow> nat" |
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where "unat w = nat (uint w)" |
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definition uints :: "nat \<Rightarrow> int set" |
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\<comment> "the sets of integers representing the words" |
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where "uints n = range (bintrunc n)" |
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definition sints :: "nat \<Rightarrow> int set" |
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where "sints n = range (sbintrunc (n - 1))" |
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lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}" |
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by (simp add: uints_def range_bintrunc) |
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lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}" |
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by (simp add: sints_def range_sbintrunc) |
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definition unats :: "nat \<Rightarrow> nat set" |
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where "unats n = {i. i < 2 ^ n}" |
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definition norm_sint :: "nat \<Rightarrow> int \<Rightarrow> int" |
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where "norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)" |
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definition scast :: "'a::len word \<Rightarrow> 'b::len word" |
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\<comment> "cast a word to a different length" |
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where "scast w = word_of_int (sint w)" |
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definition ucast :: "'a::len0 word \<Rightarrow> 'b::len0 word" |
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where "ucast w = word_of_int (uint w)" |
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instantiation word :: (len0) size |
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begin |
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definition word_size: "size (w :: 'a word) = len_of TYPE('a)" |
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instance .. |
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end |
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lemma word_size_gt_0 [iff]: "0 < size w" |
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for w :: "'a::len word" |
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by (simp add: word_size) |
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lemmas lens_gt_0 = word_size_gt_0 len_gt_0 |
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lemma lens_not_0 [iff]: |
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fixes w :: "'a::len word" |
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shows "size w \<noteq> 0" |
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and "len_of TYPE('a) \<noteq> 0" |
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by auto |
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definition source_size :: "('a::len0 word \<Rightarrow> 'b) \<Rightarrow> nat" |
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\<comment> "whether a cast (or other) function is to a longer or shorter length" |
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where [code del]: "source_size c = (let arb = undefined; x = c arb in size arb)" |
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definition target_size :: "('a \<Rightarrow> 'b::len0 word) \<Rightarrow> nat" |
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where [code del]: "target_size c = size (c undefined)" |
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definition is_up :: "('a::len0 word \<Rightarrow> 'b::len0 word) \<Rightarrow> bool" |
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where "is_up c \<longleftrightarrow> source_size c \<le> target_size c" |
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definition is_down :: "('a::len0 word \<Rightarrow> 'b::len0 word) \<Rightarrow> bool" |
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where "is_down c \<longleftrightarrow> target_size c \<le> source_size c" |
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definition of_bl :: "bool list \<Rightarrow> 'a::len0 word" |
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where "of_bl bl = word_of_int (bl_to_bin bl)" |
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definition to_bl :: "'a::len0 word \<Rightarrow> bool list" |
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where "to_bl w = bin_to_bl (len_of TYPE('a)) (uint w)" |
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definition word_reverse :: "'a::len0 word \<Rightarrow> 'a word" |
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where "word_reverse w = of_bl (rev (to_bl w))" |
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definition word_int_case :: "(int \<Rightarrow> 'b) \<Rightarrow> 'a::len0 word \<Rightarrow> 'b" |
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where "word_int_case f w = f (uint w)" |
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translations |
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"case x of XCONST of_int y \<Rightarrow> b" \<rightleftharpoons> "CONST word_int_case (\<lambda>y. b) x" |
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"case x of (XCONST of_int :: 'a) y \<Rightarrow> b" \<rightharpoonup> "CONST word_int_case (\<lambda>y. b) x" |
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subsection \<open>Correspondence relation for theorem transfer\<close> |
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definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool" |
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where "cr_word = (\<lambda>x y. word_of_int x = y)" |
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lemma Quotient_word: |
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"Quotient (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y) |
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word_of_int uint (cr_word :: _ \<Rightarrow> 'a::len0 word \<Rightarrow> bool)" |
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unfolding Quotient_alt_def cr_word_def |
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by (simp add: no_bintr_alt1 word_of_int_uint) (simp add: word_of_int_def Abs_word_inject) |
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lemma reflp_word: |
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"reflp (\<lambda>x y. bintrunc (len_of TYPE('a::len0)) x = bintrunc (len_of TYPE('a)) y)" |
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by (simp add: reflp_def) |
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setup_lifting Quotient_word reflp_word |
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text \<open>TODO: The next lemma could be generated automatically.\<close> |
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lemma uint_transfer [transfer_rule]: |
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"(rel_fun pcr_word op =) (bintrunc (len_of TYPE('a))) (uint :: 'a::len0 word \<Rightarrow> int)" |
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unfolding rel_fun_def word.pcr_cr_eq cr_word_def |
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by (simp add: no_bintr_alt1 uint_word_of_int) |
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subsection \<open>Basic code generation setup\<close> |
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definition Word :: "int \<Rightarrow> 'a::len0 word" |
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where [code_post]: "Word = word_of_int" |
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lemma [code abstype]: "Word (uint w) = w" |
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by (simp add: Word_def word_of_int_uint) |
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declare uint_word_of_int [code abstract] |
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instantiation word :: (len0) equal |
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begin |
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definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" |
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where "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)" |
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instance |
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by standard (simp add: equal equal_word_def word_uint_eq_iff) |
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end |
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notation fcomp (infixl "\<circ>>" 60) |
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notation scomp (infixl "\<circ>\<rightarrow>" 60) |
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instantiation word :: ("{len0, typerep}") random |
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begin |
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definition |
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"random_word i = Random.range i \<circ>\<rightarrow> (\<lambda>k. Pair ( |
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let j = word_of_int (int_of_integer (integer_of_natural k)) :: 'a word |
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in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))" |
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instance .. |
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end |
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no_notation fcomp (infixl "\<circ>>" 60) |
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no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
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subsection \<open>Type-definition locale instantiations\<close> |
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213 |
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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: |
65268 | 228 |
td_ext |
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"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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234 |
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lemmas td_uint = word_uint.td_thm |
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lemmas int_word_uint = word_uint.eq_norm |
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lemma td_ext_ubin: |
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"td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0))) |
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(bintrunc (len_of TYPE('a)))" |
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by (unfold no_bintr_alt1) (fact td_ext_uint) |
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interpretation word_ubin: |
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td_ext |
245 |
"uint::'a::len0 word \<Rightarrow> int" |
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word_of_int |
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"uints (len_of TYPE('a::len0))" |
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"bintrunc (len_of TYPE('a::len0))" |
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by (fact td_ext_ubin) |
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250 |
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subsection \<open>Arithmetic operations\<close> |
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lift_definition word_succ :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x + 1" |
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by (auto simp add: bintrunc_mod2p intro: mod_add_cong) |
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lift_definition word_pred :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x - 1" |
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by (auto simp add: bintrunc_mod2p intro: mod_diff_cong) |
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instantiation word :: (len0) "{neg_numeral, modulo, comm_monoid_mult, comm_ring}" |
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begin |
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lift_definition zero_word :: "'a word" is "0" . |
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lift_definition one_word :: "'a word" is "1" . |
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lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op +" |
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by (auto simp add: bintrunc_mod2p intro: mod_add_cong) |
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lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op -" |
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by (auto simp add: bintrunc_mod2p intro: mod_diff_cong) |
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lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus |
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by (auto simp add: bintrunc_mod2p intro: mod_minus_cong) |
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lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op *" |
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by (auto simp add: bintrunc_mod2p intro: mod_mult_cong) |
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|
65328 | 279 |
definition word_div_def: "a div b = word_of_int (uint a div uint b)" |
280 |
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281 |
definition word_mod_def: "a mod b = word_of_int (uint a mod uint b)" |
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instance |
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by standard (transfer, simp add: algebra_simps)+ |
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end |
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text \<open>Legacy theorems:\<close> |
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|
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lemma word_arith_wis [code]: |
291 |
shows word_add_def: "a + b = word_of_int (uint a + uint b)" |
|
292 |
and word_sub_wi: "a - b = word_of_int (uint a - uint b)" |
|
293 |
and word_mult_def: "a * b = word_of_int (uint a * uint b)" |
|
294 |
and word_minus_def: "- a = word_of_int (- uint a)" |
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295 |
and word_succ_alt: "word_succ a = word_of_int (uint a + 1)" |
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296 |
and word_pred_alt: "word_pred a = word_of_int (uint a - 1)" |
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297 |
and word_0_wi: "0 = word_of_int 0" |
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298 |
and word_1_wi: "1 = word_of_int 1" |
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unfolding plus_word_def minus_word_def times_word_def uminus_word_def |
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unfolding word_succ_def word_pred_def zero_word_def one_word_def |
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by simp_all |
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302 |
|
65268 | 303 |
lemma wi_homs: |
304 |
shows wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" |
|
305 |
and wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)" |
|
306 |
and wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" |
|
307 |
and wi_hom_neg: "- word_of_int a = word_of_int (- a)" |
|
308 |
and wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" |
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309 |
and wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)" |
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by (transfer, simp)+ |
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311 |
|
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lemmas wi_hom_syms = wi_homs [symmetric] |
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|
46013 | 314 |
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi |
46009 | 315 |
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316 |
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric] |
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317 |
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318 |
instance word :: (len) comm_ring_1 |
45810 | 319 |
proof |
65268 | 320 |
have *: "0 < len_of TYPE('a)" by (rule len_gt_0) |
321 |
show "(0::'a word) \<noteq> 1" |
|
322 |
by transfer (use * in \<open>auto simp add: gr0_conv_Suc\<close>) |
|
45810 | 323 |
qed |
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324 |
|
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325 |
lemma word_of_nat: "of_nat n = word_of_int (int n)" |
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326 |
by (induct n) (auto simp add : word_of_int_hom_syms) |
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327 |
|
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328 |
lemma word_of_int: "of_int = word_of_int" |
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329 |
apply (rule ext) |
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330 |
apply (case_tac x rule: int_diff_cases) |
46013 | 331 |
apply (simp add: word_of_nat wi_hom_sub) |
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332 |
done |
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|
333 |
|
65268 | 334 |
definition udvd :: "'a::len word \<Rightarrow> 'a::len word \<Rightarrow> bool" (infixl "udvd" 50) |
335 |
where "a udvd b = (\<exists>n\<ge>0. uint b = n * uint a)" |
|
37660 | 336 |
|
45547 | 337 |
|
61799 | 338 |
subsection \<open>Ordering\<close> |
45547 | 339 |
|
340 |
instantiation word :: (len0) linorder |
|
341 |
begin |
|
342 |
||
65268 | 343 |
definition word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b" |
344 |
||
345 |
definition word_less_def: "a < b \<longleftrightarrow> uint a < uint b" |
|
37660 | 346 |
|
45547 | 347 |
instance |
61169 | 348 |
by standard (auto simp: word_less_def word_le_def) |
45547 | 349 |
|
350 |
end |
|
351 |
||
65268 | 352 |
definition word_sle :: "'a::len word \<Rightarrow> 'a word \<Rightarrow> bool" ("(_/ <=s _)" [50, 51] 50) |
353 |
where "a <=s b \<longleftrightarrow> sint a \<le> sint b" |
|
354 |
||
355 |
definition word_sless :: "'a::len word \<Rightarrow> 'a word \<Rightarrow> bool" ("(_/ <s _)" [50, 51] 50) |
|
356 |
where "x <s y \<longleftrightarrow> x <=s y \<and> x \<noteq> y" |
|
37660 | 357 |
|
358 |
||
61799 | 359 |
subsection \<open>Bit-wise operations\<close> |
37660 | 360 |
|
361 |
instantiation word :: (len0) bits |
|
362 |
begin |
|
363 |
||
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|
364 |
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT |
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|
365 |
by (metis bin_trunc_not) |
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|
366 |
|
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|
367 |
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND |
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|
368 |
by (metis bin_trunc_and) |
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|
369 |
|
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|
370 |
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR |
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|
371 |
by (metis bin_trunc_or) |
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|
372 |
|
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|
373 |
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR |
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|
374 |
by (metis bin_trunc_xor) |
37660 | 375 |
|
65268 | 376 |
definition word_test_bit_def: "test_bit a = bin_nth (uint a)" |
377 |
||
378 |
definition word_set_bit_def: "set_bit a n x = word_of_int (bin_sc n x (uint a))" |
|
379 |
||
380 |
definition word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE('a)) f)" |
|
381 |
||
382 |
definition word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a)" |
|
37660 | 383 |
|
54848 | 384 |
definition shiftl1 :: "'a word \<Rightarrow> 'a word" |
65268 | 385 |
where "shiftl1 w = word_of_int (uint w BIT False)" |
37660 | 386 |
|
54848 | 387 |
definition shiftr1 :: "'a word \<Rightarrow> 'a word" |
61799 | 388 |
\<comment> "shift right as unsigned or as signed, ie logical or arithmetic" |
65328 | 389 |
where "shiftr1 w = word_of_int (bin_rest (uint w))" |
37660 | 390 |
|
65268 | 391 |
definition shiftl_def: "w << n = (shiftl1 ^^ n) w" |
392 |
||
393 |
definition shiftr_def: "w >> n = (shiftr1 ^^ n) w" |
|
37660 | 394 |
|
395 |
instance .. |
|
396 |
||
397 |
end |
|
398 |
||
65268 | 399 |
lemma [code]: |
400 |
shows word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" |
|
401 |
and word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" |
|
402 |
and word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" |
|
403 |
and word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)" |
|
404 |
by (simp_all add: bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def) |
|
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|
405 |
|
37660 | 406 |
instantiation word :: (len) bitss |
407 |
begin |
|
408 |
||
65268 | 409 |
definition word_msb_def: "msb a \<longleftrightarrow> bin_sign (sint a) = -1" |
37660 | 410 |
|
411 |
instance .. |
|
412 |
||
413 |
end |
|
414 |
||
65268 | 415 |
definition setBit :: "'a::len0 word \<Rightarrow> nat \<Rightarrow> 'a word" |
416 |
where "setBit w n = set_bit w n True" |
|
417 |
||
418 |
definition clearBit :: "'a::len0 word \<Rightarrow> nat \<Rightarrow> 'a word" |
|
419 |
where "clearBit w n = set_bit w n False" |
|
37660 | 420 |
|
421 |
||
61799 | 422 |
subsection \<open>Shift operations\<close> |
37660 | 423 |
|
65268 | 424 |
definition sshiftr1 :: "'a::len word \<Rightarrow> 'a word" |
425 |
where "sshiftr1 w = word_of_int (bin_rest (sint w))" |
|
426 |
||
427 |
definition bshiftr1 :: "bool \<Rightarrow> 'a::len word \<Rightarrow> 'a word" |
|
428 |
where "bshiftr1 b w = of_bl (b # butlast (to_bl w))" |
|
429 |
||
430 |
definition sshiftr :: "'a::len word \<Rightarrow> nat \<Rightarrow> 'a word" (infixl ">>>" 55) |
|
431 |
where "w >>> n = (sshiftr1 ^^ n) w" |
|
432 |
||
433 |
definition mask :: "nat \<Rightarrow> 'a::len word" |
|
434 |
where "mask n = (1 << n) - 1" |
|
435 |
||
436 |
definition revcast :: "'a::len0 word \<Rightarrow> 'b::len0 word" |
|
437 |
where "revcast w = of_bl (takefill False (len_of TYPE('b)) (to_bl w))" |
|
438 |
||
439 |
definition slice1 :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'b::len0 word" |
|
440 |
where "slice1 n w = of_bl (takefill False n (to_bl w))" |
|
441 |
||
442 |
definition slice :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'b::len0 word" |
|
443 |
where "slice n w = slice1 (size w - n) w" |
|
37660 | 444 |
|
445 |
||
61799 | 446 |
subsection \<open>Rotation\<close> |
37660 | 447 |
|
65268 | 448 |
definition rotater1 :: "'a list \<Rightarrow> 'a list" |
449 |
where "rotater1 ys = |
|
450 |
(case ys of [] \<Rightarrow> [] | x # xs \<Rightarrow> last ys # butlast ys)" |
|
451 |
||
452 |
definition rotater :: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" |
|
453 |
where "rotater n = rotater1 ^^ n" |
|
454 |
||
455 |
definition word_rotr :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'a::len0 word" |
|
456 |
where "word_rotr n w = of_bl (rotater n (to_bl w))" |
|
457 |
||
458 |
definition word_rotl :: "nat \<Rightarrow> 'a::len0 word \<Rightarrow> 'a::len0 word" |
|
459 |
where "word_rotl n w = of_bl (rotate n (to_bl w))" |
|
460 |
||
461 |
definition word_roti :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> 'a::len0 word" |
|
462 |
where "word_roti i w = |
|
463 |
(if i \<ge> 0 then word_rotr (nat i) w else word_rotl (nat (- i)) w)" |
|
37660 | 464 |
|
465 |
||
61799 | 466 |
subsection \<open>Split and cat operations\<close> |
37660 | 467 |
|
65268 | 468 |
definition word_cat :: "'a::len0 word \<Rightarrow> 'b::len0 word \<Rightarrow> 'c::len0 word" |
469 |
where "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE('b)) (uint b))" |
|
470 |
||
471 |
definition word_split :: "'a::len0 word \<Rightarrow> 'b::len0 word \<times> 'c::len0 word" |
|
472 |
where "word_split a = |
|
473 |
(case bin_split (len_of TYPE('c)) (uint a) of |
|
474 |
(u, v) \<Rightarrow> (word_of_int u, word_of_int v))" |
|
475 |
||
476 |
definition word_rcat :: "'a::len0 word list \<Rightarrow> 'b::len0 word" |
|
477 |
where "word_rcat ws = word_of_int (bin_rcat (len_of TYPE('a)) (map uint ws))" |
|
478 |
||
479 |
definition word_rsplit :: "'a::len0 word \<Rightarrow> 'b::len word list" |
|
480 |
where "word_rsplit w = map word_of_int (bin_rsplit (len_of TYPE('b)) (len_of TYPE('a), uint w))" |
|
481 |
||
65328 | 482 |
definition max_word :: "'a::len word" |
483 |
\<comment> "Largest representable machine integer." |
|
65268 | 484 |
where "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)" |
37660 | 485 |
|
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|
486 |
lemmas of_nth_def = word_set_bits_def (* FIXME duplicate *) |
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|
487 |
|
37660 | 488 |
|
61799 | 489 |
subsection \<open>Theorems about typedefs\<close> |
46010 | 490 |
|
65268 | 491 |
lemma sint_sbintrunc': "sint (word_of_int bin :: 'a word) = sbintrunc (len_of TYPE('a::len) - 1) bin" |
492 |
by (auto simp: sint_uint word_ubin.eq_norm sbintrunc_bintrunc_lt) |
|
493 |
||
65328 | 494 |
lemma uint_sint: "uint w = bintrunc (len_of TYPE('a)) (sint w)" |
495 |
for w :: "'a::len word" |
|
65268 | 496 |
by (auto simp: sint_uint bintrunc_sbintrunc_le) |
497 |
||
498 |
lemma bintr_uint: "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w" |
|
499 |
for w :: "'a::len0 word" |
|
500 |
apply (subst word_ubin.norm_Rep [symmetric]) |
|
37660 | 501 |
apply (simp only: bintrunc_bintrunc_min word_size) |
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54854
diff
changeset
|
502 |
apply (simp add: min.absorb2) |
37660 | 503 |
done |
504 |
||
46057 | 505 |
lemma wi_bintr: |
506 |
"len_of TYPE('a::len0) \<le> n \<Longrightarrow> |
|
507 |
word_of_int (bintrunc n w) = (word_of_int w :: 'a word)" |
|
65268 | 508 |
by (auto simp: word_ubin.norm_eq_iff [symmetric] min.absorb1) |
509 |
||
510 |
lemma td_ext_sbin: |
|
511 |
"td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len))) |
|
37660 | 512 |
(sbintrunc (len_of TYPE('a) - 1))" |
513 |
apply (unfold td_ext_def' sint_uint) |
|
514 |
apply (simp add : word_ubin.eq_norm) |
|
515 |
apply (cases "len_of TYPE('a)") |
|
516 |
apply (auto simp add : sints_def) |
|
517 |
apply (rule sym [THEN trans]) |
|
65268 | 518 |
apply (rule word_ubin.Abs_norm) |
37660 | 519 |
apply (simp only: bintrunc_sbintrunc) |
520 |
apply (drule sym) |
|
521 |
apply simp |
|
522 |
done |
|
523 |
||
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|
524 |
lemma td_ext_sint: |
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|
525 |
"td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len))) |
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|
526 |
(\<lambda>w. (w + 2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) - |
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|
527 |
2 ^ (len_of TYPE('a) - 1))" |
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|
528 |
using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2) |
37660 | 529 |
|
530 |
(* We do sint before sbin, before sint is the user version |
|
65268 | 531 |
and interpretations do not produce thm duplicates. I.e. |
37660 | 532 |
we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, |
533 |
because the latter is the same thm as the former *) |
|
534 |
interpretation word_sint: |
|
65268 | 535 |
td_ext |
536 |
"sint ::'a::len word \<Rightarrow> int" |
|
537 |
word_of_int |
|
538 |
"sints (len_of TYPE('a::len))" |
|
539 |
"\<lambda>w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) - |
|
540 |
2 ^ (len_of TYPE('a::len) - 1)" |
|
37660 | 541 |
by (rule td_ext_sint) |
542 |
||
543 |
interpretation word_sbin: |
|
65268 | 544 |
td_ext |
545 |
"sint ::'a::len word \<Rightarrow> int" |
|
546 |
word_of_int |
|
547 |
"sints (len_of TYPE('a::len))" |
|
548 |
"sbintrunc (len_of TYPE('a::len) - 1)" |
|
37660 | 549 |
by (rule td_ext_sbin) |
550 |
||
45604 | 551 |
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm] |
37660 | 552 |
|
553 |
lemmas td_sint = word_sint.td |
|
554 |
||
65268 | 555 |
lemma to_bl_def': "(to_bl :: 'a::len0 word \<Rightarrow> bool list) = bin_to_bl (len_of TYPE('a)) \<circ> uint" |
44762 | 556 |
by (auto simp: to_bl_def) |
37660 | 557 |
|
65268 | 558 |
lemmas word_reverse_no_def [simp] = |
559 |
word_reverse_def [of "numeral w"] for w |
|
37660 | 560 |
|
45805 | 561 |
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)" |
562 |
by (fact uints_def [unfolded no_bintr_alt1]) |
|
563 |
||
65268 | 564 |
lemma word_numeral_alt: "numeral b = word_of_int (numeral b)" |
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|
565 |
by (induct b, simp_all only: numeral.simps word_of_int_homs) |
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|
566 |
|
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|
567 |
declare word_numeral_alt [symmetric, code_abbrev] |
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|
568 |
|
65268 | 569 |
lemma word_neg_numeral_alt: "- numeral b = word_of_int (- numeral b)" |
54489
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54225
diff
changeset
|
570 |
by (simp only: word_numeral_alt wi_hom_neg) |
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|
571 |
|
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|
572 |
declare word_neg_numeral_alt [symmetric, code_abbrev] |
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|
573 |
|
47372 | 574 |
lemma word_numeral_transfer [transfer_rule]: |
55945 | 575 |
"(rel_fun op = pcr_word) numeral numeral" |
576 |
"(rel_fun op = pcr_word) (- numeral) (- numeral)" |
|
577 |
apply (simp_all add: rel_fun_def word.pcr_cr_eq cr_word_def) |
|
65268 | 578 |
using word_numeral_alt [symmetric] word_neg_numeral_alt [symmetric] by auto |
47372 | 579 |
|
45805 | 580 |
lemma uint_bintrunc [simp]: |
65268 | 581 |
"uint (numeral bin :: 'a word) = |
582 |
bintrunc (len_of TYPE('a::len0)) (numeral bin)" |
|
47108
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changeset
|
583 |
unfolding word_numeral_alt by (rule word_ubin.eq_norm) |
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|
584 |
|
65268 | 585 |
lemma uint_bintrunc_neg [simp]: |
586 |
"uint (- numeral bin :: 'a word) = bintrunc (len_of TYPE('a::len0)) (- numeral bin)" |
|
47108
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changeset
|
587 |
by (simp only: word_neg_numeral_alt word_ubin.eq_norm) |
37660 | 588 |
|
45805 | 589 |
lemma sint_sbintrunc [simp]: |
65268 | 590 |
"sint (numeral bin :: 'a word) = sbintrunc (len_of TYPE('a::len) - 1) (numeral bin)" |
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|
591 |
by (simp only: word_numeral_alt word_sbin.eq_norm) |
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|
592 |
|
65268 | 593 |
lemma sint_sbintrunc_neg [simp]: |
594 |
"sint (- numeral bin :: 'a word) = sbintrunc (len_of TYPE('a::len) - 1) (- numeral bin)" |
|
47108
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changeset
|
595 |
by (simp only: word_neg_numeral_alt word_sbin.eq_norm) |
37660 | 596 |
|
45805 | 597 |
lemma unat_bintrunc [simp]: |
65268 | 598 |
"unat (numeral bin :: 'a::len0 word) = nat (bintrunc (len_of TYPE('a)) (numeral bin))" |
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changeset
|
599 |
by (simp only: unat_def uint_bintrunc) |
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|
600 |
|
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|
601 |
lemma unat_bintrunc_neg [simp]: |
65268 | 602 |
"unat (- numeral bin :: 'a::len0 word) = nat (bintrunc (len_of TYPE('a)) (- numeral bin))" |
47108
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changeset
|
603 |
by (simp only: unat_def uint_bintrunc_neg) |
37660 | 604 |
|
65328 | 605 |
lemma size_0_eq: "size w = 0 \<Longrightarrow> v = w" |
606 |
for v w :: "'a::len0 word" |
|
37660 | 607 |
apply (unfold word_size) |
608 |
apply (rule word_uint.Rep_eqD) |
|
609 |
apply (rule box_equals) |
|
610 |
defer |
|
611 |
apply (rule word_ubin.norm_Rep)+ |
|
612 |
apply simp |
|
613 |
done |
|
614 |
||
65268 | 615 |
lemma uint_ge_0 [iff]: "0 \<le> uint x" |
616 |
for x :: "'a::len0 word" |
|
45805 | 617 |
using word_uint.Rep [of x] by (simp add: uints_num) |
618 |
||
65268 | 619 |
lemma uint_lt2p [iff]: "uint x < 2 ^ len_of TYPE('a)" |
620 |
for x :: "'a::len0 word" |
|
45805 | 621 |
using word_uint.Rep [of x] by (simp add: uints_num) |
622 |
||
65268 | 623 |
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint x" |
624 |
for x :: "'a::len word" |
|
45805 | 625 |
using word_sint.Rep [of x] by (simp add: sints_num) |
626 |
||
65268 | 627 |
lemma sint_lt: "sint x < 2 ^ (len_of TYPE('a) - 1)" |
628 |
for x :: "'a::len word" |
|
45805 | 629 |
using word_sint.Rep [of x] by (simp add: sints_num) |
37660 | 630 |
|
65268 | 631 |
lemma sign_uint_Pls [simp]: "bin_sign (uint x) = 0" |
47108
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merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
632 |
by (simp add: sign_Pls_ge_0) |
37660 | 633 |
|
65268 | 634 |
lemma uint_m2p_neg: "uint x - 2 ^ len_of TYPE('a) < 0" |
635 |
for x :: "'a::len0 word" |
|
45805 | 636 |
by (simp only: diff_less_0_iff_less uint_lt2p) |
637 |
||
65268 | 638 |
lemma uint_m2p_not_non_neg: "\<not> 0 \<le> uint x - 2 ^ len_of TYPE('a)" |
639 |
for x :: "'a::len0 word" |
|
45805 | 640 |
by (simp only: not_le uint_m2p_neg) |
37660 | 641 |
|
65268 | 642 |
lemma lt2p_lem: "len_of TYPE('a) \<le> n \<Longrightarrow> uint w < 2 ^ n" |
643 |
for w :: "'a::len0 word" |
|
55816
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cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
644 |
by (metis bintr_uint bintrunc_mod2p int_mod_lem zless2p) |
37660 | 645 |
|
45805 | 646 |
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0" |
647 |
by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1]) |
|
37660 | 648 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
649 |
lemma uint_nat: "uint w = int (unat w)" |
65268 | 650 |
by (auto simp: unat_def) |
651 |
||
652 |
lemma uint_numeral: "uint (numeral b :: 'a::len0 word) = numeral b mod 2 ^ len_of TYPE('a)" |
|
653 |
by (simp only: word_numeral_alt int_word_uint) |
|
654 |
||
655 |
lemma uint_neg_numeral: "uint (- numeral b :: 'a::len0 word) = - numeral b mod 2 ^ len_of TYPE('a)" |
|
656 |
by (simp only: word_neg_numeral_alt int_word_uint) |
|
657 |
||
658 |
lemma unat_numeral: "unat (numeral b :: 'a::len0 word) = numeral b mod 2 ^ len_of TYPE('a)" |
|
37660 | 659 |
apply (unfold unat_def) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
660 |
apply (clarsimp simp only: uint_numeral) |
37660 | 661 |
apply (rule nat_mod_distrib [THEN trans]) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
662 |
apply (rule zero_le_numeral) |
37660 | 663 |
apply (simp_all add: nat_power_eq) |
664 |
done |
|
665 |
||
65268 | 666 |
lemma sint_numeral: |
667 |
"sint (numeral b :: 'a::len word) = |
|
668 |
(numeral b + |
|
669 |
2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) - |
|
670 |
2 ^ (len_of TYPE('a) - 1)" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
671 |
unfolding word_numeral_alt by (rule int_word_sint) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
672 |
|
65268 | 673 |
lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0" |
45958 | 674 |
unfolding word_0_wi .. |
675 |
||
65268 | 676 |
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1" |
45958 | 677 |
unfolding word_1_wi .. |
678 |
||
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
679 |
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
|
680 |
by (simp add: wi_hom_syms) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
681 |
|
65268 | 682 |
lemma word_of_int_numeral [simp] : "(word_of_int (numeral bin) :: 'a::len0 word) = numeral bin" |
683 |
by (simp only: word_numeral_alt) |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
684 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
685 |
lemma word_of_int_neg_numeral [simp]: |
65268 | 686 |
"(word_of_int (- numeral bin) :: 'a::len0 word) = - numeral bin" |
687 |
by (simp only: word_numeral_alt wi_hom_syms) |
|
688 |
||
689 |
lemma word_int_case_wi: |
|
690 |
"word_int_case f (word_of_int i :: 'b word) = f (i mod 2 ^ len_of TYPE('b::len0))" |
|
691 |
by (simp add: word_int_case_def word_uint.eq_norm) |
|
692 |
||
693 |
lemma word_int_split: |
|
694 |
"P (word_int_case f x) = |
|
695 |
(\<forall>i. x = (word_of_int i :: 'b::len0 word) \<and> 0 \<le> i \<and> i < 2 ^ len_of TYPE('b) \<longrightarrow> P (f i))" |
|
696 |
by (auto simp: word_int_case_def word_uint.eq_norm mod_pos_pos_trivial) |
|
697 |
||
698 |
lemma word_int_split_asm: |
|
699 |
"P (word_int_case f x) = |
|
700 |
(\<nexists>n. x = (word_of_int n :: 'b::len0 word) \<and> 0 \<le> n \<and> n < 2 ^ len_of TYPE('b::len0) \<and> \<not> P (f n))" |
|
701 |
by (auto simp: word_int_case_def word_uint.eq_norm mod_pos_pos_trivial) |
|
45805 | 702 |
|
45604 | 703 |
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq] |
704 |
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq] |
|
37660 | 705 |
|
65268 | 706 |
lemma uint_range_size: "0 \<le> uint w \<and> uint w < 2 ^ size w" |
37660 | 707 |
unfolding word_size by (rule uint_range') |
708 |
||
65268 | 709 |
lemma sint_range_size: "- (2 ^ (size w - Suc 0)) \<le> sint w \<and> sint w < 2 ^ (size w - Suc 0)" |
37660 | 710 |
unfolding word_size by (rule sint_range') |
711 |
||
65268 | 712 |
lemma sint_above_size: "2 ^ (size w - 1) \<le> x \<Longrightarrow> sint w < x" |
713 |
for w :: "'a::len word" |
|
45805 | 714 |
unfolding word_size by (rule less_le_trans [OF sint_lt]) |
715 |
||
65268 | 716 |
lemma sint_below_size: "x \<le> - (2 ^ (size w - 1)) \<Longrightarrow> x \<le> sint w" |
717 |
for w :: "'a::len word" |
|
45805 | 718 |
unfolding word_size by (rule order_trans [OF _ sint_ge]) |
37660 | 719 |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
720 |
|
61799 | 721 |
subsection \<open>Testing bits\<close> |
46010 | 722 |
|
65268 | 723 |
lemma test_bit_eq_iff: "test_bit u = test_bit v \<longleftrightarrow> u = v" |
724 |
for u v :: "'a::len0 word" |
|
37660 | 725 |
unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) |
726 |
||
65268 | 727 |
lemma test_bit_size [rule_format] : "w !! n \<longrightarrow> n < size w" |
728 |
for w :: "'a::len0 word" |
|
37660 | 729 |
apply (unfold word_test_bit_def) |
730 |
apply (subst word_ubin.norm_Rep [symmetric]) |
|
731 |
apply (simp only: nth_bintr word_size) |
|
732 |
apply fast |
|
733 |
done |
|
734 |
||
65268 | 735 |
lemma word_eq_iff: "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)" |
736 |
for x y :: "'a::len0 word" |
|
46021 | 737 |
unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric] |
738 |
by (metis test_bit_size [unfolded word_size]) |
|
739 |
||
65268 | 740 |
lemma word_eqI: "(\<And>n. n < size u \<longrightarrow> u !! n = v !! n) \<Longrightarrow> u = v" |
741 |
for u :: "'a::len0 word" |
|
46021 | 742 |
by (simp add: word_size word_eq_iff) |
37660 | 743 |
|
65268 | 744 |
lemma word_eqD: "u = v \<Longrightarrow> u !! x = v !! x" |
745 |
for u v :: "'a::len0 word" |
|
45805 | 746 |
by simp |
37660 | 747 |
|
65268 | 748 |
lemma test_bit_bin': "w !! n \<longleftrightarrow> n < size w \<and> bin_nth (uint w) n" |
749 |
by (simp add: word_test_bit_def word_size nth_bintr [symmetric]) |
|
37660 | 750 |
|
751 |
lemmas test_bit_bin = test_bit_bin' [unfolded word_size] |
|
752 |
||
65268 | 753 |
lemma bin_nth_uint_imp: "bin_nth (uint w) n \<Longrightarrow> n < len_of TYPE('a)" |
754 |
for w :: "'a::len0 word" |
|
37660 | 755 |
apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) |
756 |
apply (subst word_ubin.norm_Rep) |
|
757 |
apply assumption |
|
758 |
done |
|
759 |
||
46057 | 760 |
lemma bin_nth_sint: |
65328 | 761 |
"len_of TYPE('a) \<le> n \<Longrightarrow> |
762 |
bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)" |
|
65268 | 763 |
for w :: "'a::len word" |
37660 | 764 |
apply (subst word_sbin.norm_Rep [symmetric]) |
46057 | 765 |
apply (auto simp add: nth_sbintr) |
37660 | 766 |
done |
767 |
||
768 |
(* type definitions theorem for in terms of equivalent bool list *) |
|
65268 | 769 |
lemma td_bl: |
770 |
"type_definition |
|
771 |
(to_bl :: 'a::len0 word \<Rightarrow> bool list) |
|
772 |
of_bl |
|
773 |
{bl. length bl = len_of TYPE('a)}" |
|
37660 | 774 |
apply (unfold type_definition_def of_bl_def to_bl_def) |
775 |
apply (simp add: word_ubin.eq_norm) |
|
776 |
apply safe |
|
777 |
apply (drule sym) |
|
778 |
apply simp |
|
779 |
done |
|
780 |
||
781 |
interpretation word_bl: |
|
65268 | 782 |
type_definition |
783 |
"to_bl :: 'a::len0 word \<Rightarrow> bool list" |
|
784 |
of_bl |
|
785 |
"{bl. length bl = len_of TYPE('a::len0)}" |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
786 |
by (fact td_bl) |
37660 | 787 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
788 |
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
|
789 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
790 |
lemma word_size_bl: "size w = size (to_bl w)" |
65268 | 791 |
by (auto simp: word_size) |
792 |
||
793 |
lemma to_bl_use_of_bl: "to_bl w = bl \<longleftrightarrow> w = of_bl bl \<and> length bl = length (to_bl w)" |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
794 |
by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq]) |
37660 | 795 |
|
796 |
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" |
|
65268 | 797 |
by (simp add: word_reverse_def word_bl.Abs_inverse) |
37660 | 798 |
|
799 |
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" |
|
65268 | 800 |
by (simp add: word_reverse_def word_bl.Abs_inverse) |
37660 | 801 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
802 |
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
|
803 |
by (metis word_rev_rev) |
37660 | 804 |
|
45805 | 805 |
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u" |
806 |
by simp |
|
807 |
||
65268 | 808 |
lemma length_bl_gt_0 [iff]: "0 < length (to_bl x)" |
809 |
for x :: "'a::len word" |
|
45805 | 810 |
unfolding word_bl_Rep' by (rule len_gt_0) |
811 |
||
65268 | 812 |
lemma bl_not_Nil [iff]: "to_bl x \<noteq> []" |
813 |
for x :: "'a::len word" |
|
45805 | 814 |
by (fact length_bl_gt_0 [unfolded length_greater_0_conv]) |
815 |
||
65268 | 816 |
lemma length_bl_neq_0 [iff]: "length (to_bl x) \<noteq> 0" |
817 |
for x :: "'a::len word" |
|
45805 | 818 |
by (fact length_bl_gt_0 [THEN gr_implies_not0]) |
37660 | 819 |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
820 |
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)" |
37660 | 821 |
apply (unfold to_bl_def sint_uint) |
822 |
apply (rule trans [OF _ bl_sbin_sign]) |
|
823 |
apply simp |
|
824 |
done |
|
825 |
||
65268 | 826 |
lemma of_bl_drop': |
827 |
"lend = length bl - len_of TYPE('a::len0) \<Longrightarrow> |
|
37660 | 828 |
of_bl (drop lend bl) = (of_bl bl :: 'a word)" |
65268 | 829 |
by (auto simp: of_bl_def trunc_bl2bin [symmetric]) |
830 |
||
831 |
lemma test_bit_of_bl: |
|
37660 | 832 |
"(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)" |
65328 | 833 |
by (auto simp add: of_bl_def word_test_bit_def word_size |
834 |
word_ubin.eq_norm nth_bintr bin_nth_of_bl) |
|
65268 | 835 |
|
836 |
lemma no_of_bl: "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE('a)) (numeral bin))" |
|
837 |
by (simp add: of_bl_def) |
|
37660 | 838 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
839 |
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)" |
65268 | 840 |
by (auto simp: word_size to_bl_def) |
37660 | 841 |
|
842 |
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" |
|
65268 | 843 |
by (simp add: uint_bl word_size) |
844 |
||
845 |
lemma to_bl_of_bin: "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" |
|
846 |
by (auto simp: uint_bl word_ubin.eq_norm word_size) |
|
37660 | 847 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
848 |
lemma to_bl_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
849 |
"to_bl (numeral bin::'a::len0 word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
850 |
bin_to_bl (len_of TYPE('a)) (numeral bin)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
851 |
unfolding word_numeral_alt by (rule to_bl_of_bin) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
852 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
853 |
lemma to_bl_neg_numeral [simp]: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
854 |
"to_bl (- numeral bin::'a::len0 word) = |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
855 |
bin_to_bl (len_of TYPE('a)) (- numeral bin)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
856 |
unfolding word_neg_numeral_alt by (rule to_bl_of_bin) |
37660 | 857 |
|
858 |
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" |
|
65268 | 859 |
by (simp add: uint_bl word_size) |
860 |
||
861 |
lemma uint_bl_bin: "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x" |
|
862 |
for x :: "'a::len0 word" |
|
46011 | 863 |
by (rule trans [OF bin_bl_bin word_ubin.norm_Rep]) |
45604 | 864 |
|
37660 | 865 |
(* naturals *) |
866 |
lemma uints_unats: "uints n = int ` unats n" |
|
867 |
apply (unfold unats_def uints_num) |
|
868 |
apply safe |
|
65268 | 869 |
apply (rule_tac image_eqI) |
870 |
apply (erule_tac nat_0_le [symmetric]) |
|
871 |
apply auto |
|
872 |
apply (erule_tac nat_less_iff [THEN iffD2]) |
|
873 |
apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) |
|
874 |
apply (auto simp: nat_power_eq) |
|
37660 | 875 |
done |
876 |
||
877 |
lemma unats_uints: "unats n = nat ` uints n" |
|
65268 | 878 |
by (auto simp: uints_unats image_iff) |
879 |
||
880 |
lemmas bintr_num = |
|
881 |
word_ubin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b |
|
882 |
lemmas sbintr_num = |
|
883 |
word_sbin.norm_eq_iff [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b |
|
37660 | 884 |
|
885 |
lemma num_of_bintr': |
|
65268 | 886 |
"bintrunc (len_of TYPE('a::len0)) (numeral a) = (numeral b) \<Longrightarrow> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
887 |
numeral a = (numeral b :: 'a word)" |
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
888 |
unfolding bintr_num by (erule subst, simp) |
37660 | 889 |
|
890 |
lemma num_of_sbintr': |
|
65268 | 891 |
"sbintrunc (len_of TYPE('a::len) - 1) (numeral a) = (numeral b) \<Longrightarrow> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
892 |
numeral a = (numeral b :: 'a word)" |
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
893 |
unfolding sbintr_num by (erule subst, simp) |
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
894 |
|
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
895 |
lemma num_abs_bintr: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
896 |
"(numeral x :: 'a word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
897 |
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
|
898 |
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
|
899 |
|
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
900 |
lemma num_abs_sbintr: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
901 |
"(numeral x :: 'a word) = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
902 |
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
|
903 |
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
|
904 |
|
37660 | 905 |
(** cast - note, no arg for new length, as it's determined by type of result, |
906 |
thus in "cast w = w, the type means cast to length of w! **) |
|
907 |
||
908 |
lemma ucast_id: "ucast w = w" |
|
65268 | 909 |
by (auto simp: ucast_def) |
37660 | 910 |
|
911 |
lemma scast_id: "scast w = w" |
|
65268 | 912 |
by (auto simp: scast_def) |
37660 | 913 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
914 |
lemma ucast_bl: "ucast w = of_bl (to_bl w)" |
65268 | 915 |
by (auto simp: ucast_def of_bl_def uint_bl word_size) |
916 |
||
917 |
lemma nth_ucast: "(ucast w::'a::len0 word) !! n = (w !! n \<and> n < len_of TYPE('a))" |
|
918 |
by (simp add: ucast_def test_bit_bin word_ubin.eq_norm nth_bintr word_size) |
|
919 |
(fast elim!: bin_nth_uint_imp) |
|
37660 | 920 |
|
921 |
(* for literal u(s)cast *) |
|
922 |
||
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
923 |
lemma ucast_bintr [simp]: |
65328 | 924 |
"ucast (numeral w :: 'a::len0 word) = |
925 |
word_of_int (bintrunc (len_of TYPE('a)) (numeral w))" |
|
65268 | 926 |
by (simp add: ucast_def) |
927 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
928 |
(* TODO: neg_numeral *) |
37660 | 929 |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
930 |
lemma scast_sbintr [simp]: |
65268 | 931 |
"scast (numeral w ::'a::len word) = |
932 |
word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (numeral w))" |
|
933 |
by (simp add: scast_def) |
|
37660 | 934 |
|
46011 | 935 |
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)" |
936 |
unfolding source_size_def word_size Let_def .. |
|
937 |
||
938 |
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)" |
|
939 |
unfolding target_size_def word_size Let_def .. |
|
940 |
||
65268 | 941 |
lemma is_down: "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)" |
942 |
for c :: "'a::len0 word \<Rightarrow> 'b::len0 word" |
|
943 |
by (simp only: is_down_def source_size target_size) |
|
944 |
||
945 |
lemma is_up: "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)" |
|
946 |
for c :: "'a::len0 word \<Rightarrow> 'b::len0 word" |
|
947 |
by (simp only: is_up_def source_size target_size) |
|
37660 | 948 |
|
45604 | 949 |
lemmas is_up_down = trans [OF is_up is_down [symmetric]] |
37660 | 950 |
|
45811 | 951 |
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast" |
37660 | 952 |
apply (unfold is_down) |
953 |
apply safe |
|
954 |
apply (rule ext) |
|
955 |
apply (unfold ucast_def scast_def uint_sint) |
|
956 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
957 |
apply simp |
|
958 |
done |
|
959 |
||
45811 | 960 |
lemma word_rev_tf: |
961 |
"to_bl (of_bl bl::'a::len0 word) = |
|
962 |
rev (takefill False (len_of TYPE('a)) (rev bl))" |
|
65268 | 963 |
by (auto simp: of_bl_def uint_bl bl_bin_bl_rtf word_ubin.eq_norm word_size) |
37660 | 964 |
|
45811 | 965 |
lemma word_rep_drop: |
966 |
"to_bl (of_bl bl::'a::len0 word) = |
|
967 |
replicate (len_of TYPE('a) - length bl) False @ |
|
968 |
drop (length bl - len_of TYPE('a)) bl" |
|
969 |
by (simp add: word_rev_tf takefill_alt rev_take) |
|
37660 | 970 |
|
65268 | 971 |
lemma to_bl_ucast: |
972 |
"to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = |
|
973 |
replicate (len_of TYPE('a) - len_of TYPE('b)) False @ |
|
974 |
drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)" |
|
37660 | 975 |
apply (unfold ucast_bl) |
976 |
apply (rule trans) |
|
977 |
apply (rule word_rep_drop) |
|
978 |
apply simp |
|
979 |
done |
|
980 |
||
45811 | 981 |
lemma ucast_up_app [OF refl]: |
65268 | 982 |
"uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> |
37660 | 983 |
to_bl (uc w) = replicate n False @ (to_bl w)" |
984 |
by (auto simp add : source_size target_size to_bl_ucast) |
|
985 |
||
45811 | 986 |
lemma ucast_down_drop [OF refl]: |
65268 | 987 |
"uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> |
37660 | 988 |
to_bl (uc w) = drop n (to_bl w)" |
989 |
by (auto simp add : source_size target_size to_bl_ucast) |
|
990 |
||
45811 | 991 |
lemma scast_down_drop [OF refl]: |
65268 | 992 |
"sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> |
37660 | 993 |
to_bl (sc w) = drop n (to_bl w)" |
994 |
apply (subgoal_tac "sc = ucast") |
|
995 |
apply safe |
|
996 |
apply simp |
|
45811 | 997 |
apply (erule ucast_down_drop) |
998 |
apply (rule down_cast_same [symmetric]) |
|
37660 | 999 |
apply (simp add : source_size target_size is_down) |
1000 |
done |
|
1001 |
||
65268 | 1002 |
lemma sint_up_scast [OF refl]: "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w" |
37660 | 1003 |
apply (unfold is_up) |
1004 |
apply safe |
|
1005 |
apply (simp add: scast_def word_sbin.eq_norm) |
|
1006 |
apply (rule box_equals) |
|
1007 |
prefer 3 |
|
1008 |
apply (rule word_sbin.norm_Rep) |
|
1009 |
apply (rule sbintrunc_sbintrunc_l) |
|
1010 |
defer |
|
1011 |
apply (subst word_sbin.norm_Rep) |
|
1012 |
apply (rule refl) |
|
1013 |
apply simp |
|
1014 |
done |
|
1015 |
||
65268 | 1016 |
lemma uint_up_ucast [OF refl]: "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w" |
37660 | 1017 |
apply (unfold is_up) |
1018 |
apply safe |
|
1019 |
apply (rule bin_eqI) |
|
1020 |
apply (fold word_test_bit_def) |
|
1021 |
apply (auto simp add: nth_ucast) |
|
1022 |
apply (auto simp add: test_bit_bin) |
|
1023 |
done |
|
45811 | 1024 |
|
65268 | 1025 |
lemma ucast_up_ucast [OF refl]: "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w" |
37660 | 1026 |
apply (simp (no_asm) add: ucast_def) |
1027 |
apply (clarsimp simp add: uint_up_ucast) |
|
1028 |
done |
|
65268 | 1029 |
|
1030 |
lemma scast_up_scast [OF refl]: "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w" |
|
37660 | 1031 |
apply (simp (no_asm) add: scast_def) |
1032 |
apply (clarsimp simp add: sint_up_scast) |
|
1033 |
done |
|
65268 | 1034 |
|
1035 |
lemma ucast_of_bl_up [OF refl]: "w = of_bl bl \<Longrightarrow> size bl \<le> size w \<Longrightarrow> ucast w = of_bl bl" |
|
37660 | 1036 |
by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) |
1037 |
||
1038 |
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] |
|
1039 |
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] |
|
1040 |
||
1041 |
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] |
|
1042 |
lemmas isdus = is_up_down [where c = "scast", THEN iffD2] |
|
1043 |
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] |
|
1044 |
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] |
|
1045 |
||
1046 |
lemma up_ucast_surj: |
|
65268 | 1047 |
"is_up (ucast :: 'b::len0 word \<Rightarrow> 'a::len0 word) \<Longrightarrow> |
1048 |
surj (ucast :: 'a word \<Rightarrow> 'b word)" |
|
1049 |
by (rule surjI) (erule ucast_up_ucast_id) |
|
37660 | 1050 |
|
1051 |
lemma up_scast_surj: |
|
65268 | 1052 |
"is_up (scast :: 'b::len word \<Rightarrow> 'a::len word) \<Longrightarrow> |
1053 |
surj (scast :: 'a word \<Rightarrow> 'b word)" |
|
1054 |
by (rule surjI) (erule scast_up_scast_id) |
|
37660 | 1055 |
|
1056 |
lemma down_scast_inj: |
|
65268 | 1057 |
"is_down (scast :: 'b::len word \<Rightarrow> 'a::len word) \<Longrightarrow> |
1058 |
inj_on (ucast :: 'a word \<Rightarrow> 'b word) A" |
|
37660 | 1059 |
by (rule inj_on_inverseI, erule scast_down_scast_id) |
1060 |
||
1061 |
lemma down_ucast_inj: |
|
65268 | 1062 |
"is_down (ucast :: 'b::len0 word \<Rightarrow> 'a::len0 word) \<Longrightarrow> |
1063 |
inj_on (ucast :: 'a word \<Rightarrow> 'b word) A" |
|
1064 |
by (rule inj_on_inverseI) (erule ucast_down_ucast_id) |
|
37660 | 1065 |
|
1066 |
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" |
|
1067 |
by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) |
|
45811 | 1068 |
|
65268 | 1069 |
lemma ucast_down_wi [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x" |
46646 | 1070 |
apply (unfold is_down) |
37660 | 1071 |
apply (clarsimp simp add: ucast_def word_ubin.eq_norm) |
1072 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
1073 |
apply (erule bintrunc_bintrunc_ge) |
|
1074 |
done |
|
45811 | 1075 |
|
65268 | 1076 |
lemma ucast_down_no [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1077 |
unfolding word_numeral_alt by clarify (rule ucast_down_wi) |
46646 | 1078 |
|
65268 | 1079 |
lemma ucast_down_bl [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl" |
46646 | 1080 |
unfolding of_bl_def by clarify (erule ucast_down_wi) |
37660 | 1081 |
|
1082 |
lemmas slice_def' = slice_def [unfolded word_size] |
|
1083 |
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] |
|
1084 |
||
1085 |
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def |
|
1086 |
||
1087 |
||
61799 | 1088 |
subsection \<open>Word Arithmetic\<close> |
37660 | 1089 |
|
65268 | 1090 |
lemma word_less_alt: "a < b \<longleftrightarrow> uint a < uint b" |
55818 | 1091 |
by (fact word_less_def) |
37660 | 1092 |
|
1093 |
lemma signed_linorder: "class.linorder word_sle word_sless" |
|
65268 | 1094 |
by standard (auto simp: word_sle_def word_sless_def) |
37660 | 1095 |
|
1096 |
interpretation signed: linorder "word_sle" "word_sless" |
|
1097 |
by (rule signed_linorder) |
|
1098 |
||
65268 | 1099 |
lemma udvdI: "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b" |
37660 | 1100 |
by (auto simp: udvd_def) |
1101 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1102 |
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1103 |
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1104 |
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
|
1105 |
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1106 |
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
|
1107 |
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b |
37660 | 1108 |
|
65268 | 1109 |
lemma word_m1_wi: "- 1 = word_of_int (- 1)" |
1110 |
by (simp add: word_neg_numeral_alt [of Num.One]) |
|
37660 | 1111 |
|
46648 | 1112 |
lemma word_0_bl [simp]: "of_bl [] = 0" |
65268 | 1113 |
by (simp add: of_bl_def) |
1114 |
||
1115 |
lemma word_1_bl: "of_bl [True] = 1" |
|
1116 |
by (simp add: of_bl_def bl_to_bin_def) |
|
46648 | 1117 |
|
1118 |
lemma uint_eq_0 [simp]: "uint 0 = 0" |
|
1119 |
unfolding word_0_wi word_ubin.eq_norm by simp |
|
37660 | 1120 |
|
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
1121 |
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0" |
46648 | 1122 |
by (simp add: of_bl_def bl_to_bin_rep_False) |
37660 | 1123 |
|
65268 | 1124 |
lemma to_bl_0 [simp]: "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" |
1125 |
by (simp add: uint_bl word_size bin_to_bl_zero) |
|
1126 |
||
1127 |
lemma uint_0_iff: "uint x = 0 \<longleftrightarrow> x = 0" |
|
55818 | 1128 |
by (simp add: word_uint_eq_iff) |
1129 |
||
65268 | 1130 |
lemma unat_0_iff: "unat x = 0 \<longleftrightarrow> x = 0" |
1131 |
by (auto simp: unat_def nat_eq_iff uint_0_iff) |
|
1132 |
||
1133 |
lemma unat_0 [simp]: "unat 0 = 0" |
|
1134 |
by (auto simp: unat_def) |
|
1135 |
||
1136 |
lemma size_0_same': "size w = 0 \<Longrightarrow> w = v" |
|
1137 |
for v w :: "'a::len0 word" |
|
37660 | 1138 |
apply (unfold word_size) |
1139 |
apply (rule box_equals) |
|
1140 |
defer |
|
1141 |
apply (rule word_uint.Rep_inverse)+ |
|
1142 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
1143 |
apply simp |
|
1144 |
done |
|
1145 |
||
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1146 |
lemmas size_0_same = size_0_same' [unfolded word_size] |
37660 | 1147 |
|
1148 |
lemmas unat_eq_0 = unat_0_iff |
|
1149 |
lemmas unat_eq_zero = unat_0_iff |
|
1150 |
||
65268 | 1151 |
lemma unat_gt_0: "0 < unat x \<longleftrightarrow> x \<noteq> 0" |
1152 |
by (auto simp: unat_0_iff [symmetric]) |
|
37660 | 1153 |
|
45958 | 1154 |
lemma ucast_0 [simp]: "ucast 0 = 0" |
65268 | 1155 |
by (simp add: ucast_def) |
45958 | 1156 |
|
1157 |
lemma sint_0 [simp]: "sint 0 = 0" |
|
65268 | 1158 |
by (simp add: sint_uint) |
45958 | 1159 |
|
1160 |
lemma scast_0 [simp]: "scast 0 = 0" |
|
65268 | 1161 |
by (simp add: scast_def) |
37660 | 1162 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58061
diff
changeset
|
1163 |
lemma sint_n1 [simp] : "sint (- 1) = - 1" |
65268 | 1164 |
by (simp only: word_m1_wi word_sbin.eq_norm) simp |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
1165 |
|
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
1166 |
lemma scast_n1 [simp]: "scast (- 1) = - 1" |
65268 | 1167 |
by (simp add: scast_def) |
45958 | 1168 |
|
1169 |
lemma uint_1 [simp]: "uint (1::'a::len word) = 1" |
|
55818 | 1170 |
by (simp only: word_1_wi word_ubin.eq_norm) (simp add: bintrunc_minus_simps(4)) |
45958 | 1171 |
|
1172 |
lemma unat_1 [simp]: "unat (1::'a::len word) = 1" |
|
65268 | 1173 |
by (simp add: unat_def) |
45958 | 1174 |
|
1175 |
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1" |
|
65268 | 1176 |
by (simp add: ucast_def) |
37660 | 1177 |
|
1178 |
(* now, to get the weaker results analogous to word_div/mod_def *) |
|
1179 |
||
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1180 |
|
61799 | 1181 |
subsection \<open>Transferring goals from words to ints\<close> |
37660 | 1182 |
|
65268 | 1183 |
lemma word_ths: |
1184 |
shows word_succ_p1: "word_succ a = a + 1" |
|
1185 |
and word_pred_m1: "word_pred a = a - 1" |
|
1186 |
and word_pred_succ: "word_pred (word_succ a) = a" |
|
1187 |
and word_succ_pred: "word_succ (word_pred a) = a" |
|
1188 |
and word_mult_succ: "word_succ a * b = b + a * b" |
|
47374
9475d524bafb
set up and use lift_definition for word operations
huffman
parents:
47372
diff
changeset
|
1189 |
by (transfer, simp add: algebra_simps)+ |
37660 | 1190 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1191 |
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y" |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1192 |
by simp |
37660 | 1193 |
|
55818 | 1194 |
lemma uint_word_ariths: |
1195 |
fixes a b :: "'a::len0 word" |
|
1196 |
shows "uint (a + b) = (uint a + uint b) mod 2 ^ len_of TYPE('a::len0)" |
|
1197 |
and "uint (a - b) = (uint a - uint b) mod 2 ^ len_of TYPE('a)" |
|
1198 |
and "uint (a * b) = uint a * uint b mod 2 ^ len_of TYPE('a)" |
|
1199 |
and "uint (- a) = - uint a mod 2 ^ len_of TYPE('a)" |
|
1200 |
and "uint (word_succ a) = (uint a + 1) mod 2 ^ len_of TYPE('a)" |
|
1201 |
and "uint (word_pred a) = (uint a - 1) mod 2 ^ len_of TYPE('a)" |
|
1202 |
and "uint (0 :: 'a word) = 0 mod 2 ^ len_of TYPE('a)" |
|
1203 |
and "uint (1 :: 'a word) = 1 mod 2 ^ len_of TYPE('a)" |
|
1204 |
by (simp_all add: word_arith_wis [THEN trans [OF uint_cong int_word_uint]]) |
|
1205 |
||
1206 |
lemma uint_word_arith_bintrs: |
|
1207 |
fixes a b :: "'a::len0 word" |
|
1208 |
shows "uint (a + b) = bintrunc (len_of TYPE('a)) (uint a + uint b)" |
|
1209 |
and "uint (a - b) = bintrunc (len_of TYPE('a)) (uint a - uint b)" |
|
1210 |
and "uint (a * b) = bintrunc (len_of TYPE('a)) (uint a * uint b)" |
|
1211 |
and "uint (- a) = bintrunc (len_of TYPE('a)) (- uint a)" |
|
1212 |
and "uint (word_succ a) = bintrunc (len_of TYPE('a)) (uint a + 1)" |
|
1213 |
and "uint (word_pred a) = bintrunc (len_of TYPE('a)) (uint a - 1)" |
|
1214 |
and "uint (0 :: 'a word) = bintrunc (len_of TYPE('a)) 0" |
|
1215 |
and "uint (1 :: 'a word) = bintrunc (len_of TYPE('a)) 1" |
|
1216 |
by (simp_all add: uint_word_ariths bintrunc_mod2p) |
|
1217 |
||
1218 |
lemma sint_word_ariths: |
|
1219 |
fixes a b :: "'a::len word" |
|
1220 |
shows "sint (a + b) = sbintrunc (len_of TYPE('a) - 1) (sint a + sint b)" |
|
1221 |
and "sint (a - b) = sbintrunc (len_of TYPE('a) - 1) (sint a - sint b)" |
|
1222 |
and "sint (a * b) = sbintrunc (len_of TYPE('a) - 1) (sint a * sint b)" |
|
1223 |
and "sint (- a) = sbintrunc (len_of TYPE('a) - 1) (- sint a)" |
|
1224 |
and "sint (word_succ a) = sbintrunc (len_of TYPE('a) - 1) (sint a + 1)" |
|
1225 |
and "sint (word_pred a) = sbintrunc (len_of TYPE('a) - 1) (sint a - 1)" |
|
1226 |
and "sint (0 :: 'a word) = sbintrunc (len_of TYPE('a) - 1) 0" |
|
1227 |
and "sint (1 :: 'a word) = sbintrunc (len_of TYPE('a) - 1) 1" |
|
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
1228 |
apply (simp_all only: word_sbin.inverse_norm [symmetric]) |
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
1229 |
apply (simp_all add: wi_hom_syms) |
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
1230 |
apply transfer apply simp |
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
1231 |
apply transfer apply simp |
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
1232 |
done |
45604 | 1233 |
|
1234 |
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]] |
|
1235 |
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]] |
|
37660 | 1236 |
|
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58061
diff
changeset
|
1237 |
lemma word_pred_0_n1: "word_pred 0 = word_of_int (- 1)" |
47374
9475d524bafb
set up and use lift_definition for word operations
huffman
parents:
47372
diff
changeset
|
1238 |
unfolding word_pred_m1 by simp |
37660 | 1239 |
|
1240 |
lemma succ_pred_no [simp]: |
|
65268 | 1241 |
"word_succ (numeral w) = numeral w + 1" |
1242 |
"word_pred (numeral w) = numeral w - 1" |
|
1243 |
"word_succ (- numeral w) = - numeral w + 1" |
|
1244 |
"word_pred (- numeral w) = - numeral w - 1" |
|
1245 |
by (simp_all add: word_succ_p1 word_pred_m1) |
|
1246 |
||
1247 |
lemma word_sp_01 [simp]: |
|
1248 |
"word_succ (- 1) = 0 \<and> word_succ 0 = 1 \<and> word_pred 0 = - 1 \<and> word_pred 1 = 0" |
|
1249 |
by (simp_all add: word_succ_p1 word_pred_m1) |
|
37660 | 1250 |
|
1251 |
(* alternative approach to lifting arithmetic equalities *) |
|
65268 | 1252 |
lemma word_of_int_Ex: "\<exists>y. x = word_of_int y" |
37660 | 1253 |
by (rule_tac x="uint x" in exI) simp |
1254 |
||
1255 |
||
61799 | 1256 |
subsection \<open>Order on fixed-length words\<close> |
37660 | 1257 |
|
65328 | 1258 |
lemma word_zero_le [simp]: "0 \<le> y" |
1259 |
for y :: "'a::len0 word" |
|
37660 | 1260 |
unfolding word_le_def by auto |
65268 | 1261 |
|
65328 | 1262 |
lemma word_m1_ge [simp] : "word_pred 0 \<ge> y" (* FIXME: delete *) |
1263 |
by (simp only: word_le_def word_pred_0_n1 word_uint.eq_norm m1mod2k) auto |
|
1264 |
||
1265 |
lemma word_n1_ge [simp]: "y \<le> -1" |
|
1266 |
for y :: "'a::len0 word" |
|
1267 |
by (simp only: word_le_def word_m1_wi word_uint.eq_norm m1mod2k) auto |
|
37660 | 1268 |
|
65268 | 1269 |
lemmas word_not_simps [simp] = |
37660 | 1270 |
word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD] |
1271 |
||
65328 | 1272 |
lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> y" |
1273 |
for y :: "'a::len0 word" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1274 |
by (simp add: less_le) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1275 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1276 |
lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y |
37660 | 1277 |
|
65328 | 1278 |
lemma word_sless_alt: "a <s b \<longleftrightarrow> sint a < sint b" |
1279 |
by (auto simp add: word_sle_def word_sless_def less_le) |
|
1280 |
||
1281 |
lemma word_le_nat_alt: "a \<le> b \<longleftrightarrow> unat a \<le> unat b" |
|
37660 | 1282 |
unfolding unat_def word_le_def |
1283 |
by (rule nat_le_eq_zle [symmetric]) simp |
|
1284 |
||
65328 | 1285 |
lemma word_less_nat_alt: "a < b \<longleftrightarrow> unat a < unat b" |
37660 | 1286 |
unfolding unat_def word_less_alt |
1287 |
by (rule nat_less_eq_zless [symmetric]) simp |
|
65268 | 1288 |
|
1289 |
lemma wi_less: |
|
1290 |
"(word_of_int n < (word_of_int m :: 'a::len0 word)) = |
|
37660 | 1291 |
(n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))" |
1292 |
unfolding word_less_alt by (simp add: word_uint.eq_norm) |
|
1293 |
||
65268 | 1294 |
lemma wi_le: |
65328 | 1295 |
"(word_of_int n \<le> (word_of_int m :: 'a::len0 word)) = |
1296 |
(n mod 2 ^ len_of TYPE('a) \<le> m mod 2 ^ len_of TYPE('a))" |
|
37660 | 1297 |
unfolding word_le_def by (simp add: word_uint.eq_norm) |
1298 |
||
65328 | 1299 |
lemma udvd_nat_alt: "a udvd b \<longleftrightarrow> (\<exists>n\<ge>0. unat b = n * unat a)" |
37660 | 1300 |
apply (unfold udvd_def) |
1301 |
apply safe |
|
1302 |
apply (simp add: unat_def nat_mult_distrib) |
|
65328 | 1303 |
apply (simp add: uint_nat) |
37660 | 1304 |
apply (rule exI) |
1305 |
apply safe |
|
1306 |
prefer 2 |
|
1307 |
apply (erule notE) |
|
1308 |
apply (rule refl) |
|
1309 |
apply force |
|
1310 |
done |
|
1311 |
||
61941 | 1312 |
lemma udvd_iff_dvd: "x udvd y \<longleftrightarrow> unat x dvd unat y" |
37660 | 1313 |
unfolding dvd_def udvd_nat_alt by force |
1314 |
||
45604 | 1315 |
lemmas unat_mono = word_less_nat_alt [THEN iffD1] |
37660 | 1316 |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1317 |
lemma unat_minus_one: |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1318 |
assumes "w \<noteq> 0" |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1319 |
shows "unat (w - 1) = unat w - 1" |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1320 |
proof - |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1321 |
have "0 \<le> uint w" by (fact uint_nonnegative) |
65328 | 1322 |
moreover from assms have "0 \<noteq> uint w" |
1323 |
by (simp add: uint_0_iff) |
|
1324 |
ultimately have "1 \<le> uint w" |
|
1325 |
by arith |
|
1326 |
from uint_lt2p [of w] have "uint w - 1 < 2 ^ len_of TYPE('a)" |
|
1327 |
by arith |
|
61799 | 1328 |
with \<open>1 \<le> uint w\<close> have "(uint w - 1) mod 2 ^ len_of TYPE('a) = uint w - 1" |
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1329 |
by (auto intro: mod_pos_pos_trivial) |
61799 | 1330 |
with \<open>1 \<le> uint w\<close> have "nat ((uint w - 1) mod 2 ^ len_of TYPE('a)) = nat (uint w) - 1" |
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1331 |
by auto |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1332 |
then show ?thesis |
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
1333 |
by (simp only: unat_def int_word_uint word_arith_wis mod_diff_right_eq) |
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1334 |
qed |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1335 |
|
65328 | 1336 |
lemma measure_unat: "p \<noteq> 0 \<Longrightarrow> unat (p - 1) < unat p" |
37660 | 1337 |
by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric]) |
65268 | 1338 |
|
45604 | 1339 |
lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0] |
1340 |
lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0] |
|
37660 | 1341 |
|
65328 | 1342 |
lemma uint_sub_lt2p [simp]: "uint x - uint y < 2 ^ len_of TYPE('a)" |
1343 |
for x :: "'a::len0 word" and y :: "'b::len0 word" |
|
37660 | 1344 |
using uint_ge_0 [of y] uint_lt2p [of x] by arith |
1345 |
||
1346 |
||
61799 | 1347 |
subsection \<open>Conditions for the addition (etc) of two words to overflow\<close> |
37660 | 1348 |
|
65268 | 1349 |
lemma uint_add_lem: |
1350 |
"(uint x + uint y < 2 ^ len_of TYPE('a)) = |
|
65328 | 1351 |
(uint (x + y) = uint x + uint y)" |
1352 |
for x y :: "'a::len0 word" |
|
37660 | 1353 |
by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem]) |
1354 |
||
65268 | 1355 |
lemma uint_mult_lem: |
1356 |
"(uint x * uint y < 2 ^ len_of TYPE('a)) = |
|
65328 | 1357 |
(uint (x * y) = uint x * uint y)" |
1358 |
for x y :: "'a::len0 word" |
|
37660 | 1359 |
by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem]) |
1360 |
||
65328 | 1361 |
lemma uint_sub_lem: "uint x \<ge> uint y \<longleftrightarrow> uint (x - y) = uint x - uint y" |
1362 |
by (auto simp: uint_word_ariths intro!: trans [OF _ int_mod_lem]) |
|
1363 |
||
1364 |
lemma uint_add_le: "uint (x + y) \<le> uint x + uint y" |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1365 |
unfolding uint_word_ariths by (metis uint_add_ge0 zmod_le_nonneg_dividend) |
37660 | 1366 |
|
65328 | 1367 |
lemma uint_sub_ge: "uint (x - y) \<ge> uint x - uint y" |
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1368 |
unfolding uint_word_ariths by (metis int_mod_ge uint_sub_lt2p zless2p) |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1369 |
|
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1370 |
lemma mod_add_if_z: |
65328 | 1371 |
"x < z \<Longrightarrow> y < z \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> z \<Longrightarrow> |
1372 |
(x + y) mod z = (if x + y < z then x + y else x + y - z)" |
|
1373 |
for x y z :: int |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1374 |
by (auto intro: int_mod_eq) |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1375 |
|
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1376 |
lemma uint_plus_if': |
65328 | 1377 |
"uint (a + b) = |
1378 |
(if uint a + uint b < 2 ^ len_of TYPE('a) then uint a + uint b |
|
1379 |
else uint a + uint b - 2 ^ len_of TYPE('a))" |
|
1380 |
for a b :: "'a::len0 word" |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1381 |
using mod_add_if_z [of "uint a" _ "uint b"] by (simp add: uint_word_ariths) |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1382 |
|
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1383 |
lemma mod_sub_if_z: |
65328 | 1384 |
"x < z \<Longrightarrow> y < z \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> 0 \<le> z \<Longrightarrow> |
1385 |
(x - y) mod z = (if y \<le> x then x - y else x - y + z)" |
|
1386 |
for x y z :: int |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1387 |
by (auto intro: int_mod_eq) |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1388 |
|
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1389 |
lemma uint_sub_if': |
65328 | 1390 |
"uint (a - b) = |
1391 |
(if uint b \<le> uint a then uint a - uint b |
|
1392 |
else uint a - uint b + 2 ^ len_of TYPE('a))" |
|
1393 |
for a b :: "'a::len0 word" |
|
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1394 |
using mod_sub_if_z [of "uint a" _ "uint b"] by (simp add: uint_word_ariths) |
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1395 |
|
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1396 |
|
61799 | 1397 |
subsection \<open>Definition of \<open>uint_arith\<close>\<close> |
37660 | 1398 |
|
1399 |
lemma word_of_int_inverse: |
|
65328 | 1400 |
"word_of_int r = a \<Longrightarrow> 0 \<le> r \<Longrightarrow> r < 2 ^ len_of TYPE('a) \<Longrightarrow> uint a = r" |
1401 |
for a :: "'a::len0 word" |
|
37660 | 1402 |
apply (erule word_uint.Abs_inverse' [rotated]) |
1403 |
apply (simp add: uints_num) |
|
1404 |
done |
|
1405 |
||
1406 |
lemma uint_split: |
|
65328 | 1407 |
"P (uint x) = (\<forall>i. word_of_int i = x \<and> 0 \<le> i \<and> i < 2^len_of TYPE('a) \<longrightarrow> P i)" |
1408 |
for x :: "'a::len0 word" |
|
37660 | 1409 |
apply (fold word_int_case_def) |
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
1410 |
apply (auto dest!: word_of_int_inverse simp: int_word_uint mod_pos_pos_trivial |
65328 | 1411 |
split: word_int_split) |
37660 | 1412 |
done |
1413 |
||
1414 |
lemma uint_split_asm: |
|
65328 | 1415 |
"P (uint x) = (\<nexists>i. word_of_int i = x \<and> 0 \<le> i \<and> i < 2^len_of TYPE('a) \<and> \<not> P i)" |
1416 |
for x :: "'a::len0 word" |
|
65268 | 1417 |
by (auto dest!: word_of_int_inverse |
65328 | 1418 |
simp: int_word_uint mod_pos_pos_trivial |
1419 |
split: uint_split) |
|
37660 | 1420 |
|
1421 |
lemmas uint_splits = uint_split uint_split_asm |
|
1422 |
||
65268 | 1423 |
lemmas uint_arith_simps = |
37660 | 1424 |
word_le_def word_less_alt |
65268 | 1425 |
word_uint.Rep_inject [symmetric] |
37660 | 1426 |
uint_sub_if' uint_plus_if' |
1427 |
||
65268 | 1428 |
(* use this to stop, eg, 2 ^ len_of TYPE(32) being simplified *) |
1429 |
lemma power_False_cong: "False \<Longrightarrow> a ^ b = c ^ d" |
|
37660 | 1430 |
by auto |
1431 |
||
1432 |
(* uint_arith_tac: reduce to arithmetic on int, try to solve by arith *) |
|
61799 | 1433 |
ML \<open> |
65268 | 1434 |
fun uint_arith_simpset ctxt = |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1435 |
ctxt addsimps @{thms uint_arith_simps} |
37660 | 1436 |
delsimps @{thms word_uint.Rep_inject} |
62390 | 1437 |
|> fold Splitter.add_split @{thms if_split_asm} |
45620
f2a587696afb
modernized some old-style infix operations, which were left over from the time of ML proof scripts;
wenzelm
parents:
45604
diff
changeset
|
1438 |
|> fold Simplifier.add_cong @{thms power_False_cong} |
37660 | 1439 |
|
65268 | 1440 |
fun uint_arith_tacs ctxt = |
37660 | 1441 |
let |
1442 |
fun arith_tac' n t = |
|
59657
2441a80fb6c1
eliminated unused arith "verbose" flag -- tools that need options can use the context;
wenzelm
parents:
59498
diff
changeset
|
1443 |
Arith_Data.arith_tac ctxt n t |
37660 | 1444 |
handle Cooper.COOPER _ => Seq.empty; |
65268 | 1445 |
in |
42793 | 1446 |
[ clarify_tac ctxt 1, |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1447 |
full_simp_tac (uint_arith_simpset ctxt) 1, |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1448 |
ALLGOALS (full_simp_tac |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1449 |
(put_simpset HOL_ss ctxt |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1450 |
|> fold Splitter.add_split @{thms uint_splits} |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1451 |
|> fold Simplifier.add_cong @{thms power_False_cong})), |
65268 | 1452 |
rewrite_goals_tac ctxt @{thms word_size}, |
59498
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
59487
diff
changeset
|
1453 |
ALLGOALS (fn n => REPEAT (resolve_tac ctxt [allI, impI] n) THEN |
60754 | 1454 |
REPEAT (eresolve_tac ctxt [conjE] n) THEN |
65268 | 1455 |
REPEAT (dresolve_tac ctxt @{thms word_of_int_inverse} n |
1456 |
THEN assume_tac ctxt n |
|
58963
26bf09b95dda
proper context for assume_tac (atac remains as fall-back without context);
wenzelm
parents:
58874
diff
changeset
|
1457 |
THEN assume_tac ctxt n)), |
37660 | 1458 |
TRYALL arith_tac' ] |
1459 |
end |
|
1460 |
||
1461 |
fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt)) |
|
61799 | 1462 |
\<close> |
37660 | 1463 |
|
65268 | 1464 |
method_setup uint_arith = |
61799 | 1465 |
\<open>Scan.succeed (SIMPLE_METHOD' o uint_arith_tac)\<close> |
37660 | 1466 |
"solving word arithmetic via integers and arith" |
1467 |
||
1468 |
||
61799 | 1469 |
subsection \<open>More on overflows and monotonicity\<close> |
37660 | 1470 |
|
65328 | 1471 |
lemma no_plus_overflow_uint_size: "x \<le> x + y \<longleftrightarrow> uint x + uint y < 2 ^ size x" |
1472 |
for x y :: "'a::len0 word" |
|
37660 | 1473 |
unfolding word_size by uint_arith |
1474 |
||
1475 |
lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size] |
|
1476 |
||
65328 | 1477 |
lemma no_ulen_sub: "x \<ge> x - y \<longleftrightarrow> uint y \<le> uint x" |
1478 |
for x y :: "'a::len0 word" |
|
37660 | 1479 |
by uint_arith |
1480 |
||
65328 | 1481 |
lemma no_olen_add': "x \<le> y + x \<longleftrightarrow> uint y + uint x < 2 ^ len_of TYPE('a)" |
1482 |
for x y :: "'a::len0 word" |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1483 |
by (simp add: ac_simps no_olen_add) |
37660 | 1484 |
|
45604 | 1485 |
lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric]] |
1486 |
||
1487 |
lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem] |
|
1488 |
lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1] |
|
1489 |
lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem] |
|
37660 | 1490 |
lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def] |
1491 |
lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def] |
|
45604 | 1492 |
lemmas word_sub_le = word_sub_le_iff [THEN iffD2] |
37660 | 1493 |
|
65328 | 1494 |
lemma word_less_sub1: "x \<noteq> 0 \<Longrightarrow> 1 < x \<longleftrightarrow> 0 < x - 1" |
1495 |
for x :: "'a::len word" |
|
37660 | 1496 |
by uint_arith |
1497 |
||
65328 | 1498 |
lemma word_le_sub1: "x \<noteq> 0 \<Longrightarrow> 1 \<le> x \<longleftrightarrow> 0 \<le> x - 1" |
1499 |
for x :: "'a::len word" |
|
37660 | 1500 |
by uint_arith |
1501 |
||
65328 | 1502 |
lemma sub_wrap_lt: "x < x - z \<longleftrightarrow> x < z" |
1503 |
for x z :: "'a::len0 word" |
|
37660 | 1504 |
by uint_arith |
1505 |
||
65328 | 1506 |
lemma sub_wrap: "x \<le> x - z \<longleftrightarrow> z = 0 \<or> x < z" |
1507 |
for x z :: "'a::len0 word" |
|
37660 | 1508 |
by uint_arith |
1509 |
||
65328 | 1510 |
lemma plus_minus_not_NULL_ab: "x \<le> ab - c \<Longrightarrow> c \<le> ab \<Longrightarrow> c \<noteq> 0 \<Longrightarrow> x + c \<noteq> 0" |
1511 |
for x ab c :: "'a::len0 word" |
|
37660 | 1512 |
by uint_arith |
1513 |
||
65328 | 1514 |
lemma plus_minus_no_overflow_ab: "x \<le> ab - c \<Longrightarrow> c \<le> ab \<Longrightarrow> x \<le> x + c" |
1515 |
for x ab c :: "'a::len0 word" |
|
37660 | 1516 |
by uint_arith |
1517 |
||
65328 | 1518 |
lemma le_minus': "a + c \<le> b \<Longrightarrow> a \<le> a + c \<Longrightarrow> c \<le> b - a" |
1519 |
for a b c :: "'a::len0 word" |
|
37660 | 1520 |
by uint_arith |
1521 |
||
65328 | 1522 |
lemma le_plus': "a \<le> b \<Longrightarrow> c \<le> b - a \<Longrightarrow> a + c \<le> b" |
1523 |
for a b c :: "'a::len0 word" |
|
37660 | 1524 |
by uint_arith |
1525 |
||
1526 |
lemmas le_plus = le_plus' [rotated] |
|
1527 |
||
46011 | 1528 |
lemmas le_minus = leD [THEN thin_rl, THEN le_minus'] (* FIXME *) |
37660 | 1529 |
|
65328 | 1530 |
lemma word_plus_mono_right: "y \<le> z \<Longrightarrow> x \<le> x + z \<Longrightarrow> x + y \<le> x + z" |
1531 |
for x y z :: "'a::len0 word" |
|
37660 | 1532 |
by uint_arith |
1533 |
||
65328 | 1534 |
lemma word_less_minus_cancel: "y - x < z - x \<Longrightarrow> x \<le> z \<Longrightarrow> y < z" |
1535 |
for x y z :: "'a::len0 word" |
|
37660 | 1536 |
by uint_arith |
1537 |
||
65328 | 1538 |
lemma word_less_minus_mono_left: "y < z \<Longrightarrow> x \<le> y \<Longrightarrow> y - x < z - x" |
1539 |
for x y z :: "'a::len0 word" |
|
37660 | 1540 |
by uint_arith |
1541 |
||
65328 | 1542 |
lemma word_less_minus_mono: "a < c \<Longrightarrow> d < b \<Longrightarrow> a - b < a \<Longrightarrow> c - d < c \<Longrightarrow> a - b < c - d" |
1543 |
for a b c d :: "'a::len word" |
|
37660 | 1544 |
by uint_arith |
1545 |
||
65328 | 1546 |
lemma word_le_minus_cancel: "y - x \<le> z - x \<Longrightarrow> x \<le> z \<Longrightarrow> y \<le> z" |
1547 |
for x y z :: "'a::len0 word" |
|
37660 | 1548 |
by uint_arith |
1549 |
||
65328 | 1550 |
lemma word_le_minus_mono_left: "y \<le> z \<Longrightarrow> x \<le> y \<Longrightarrow> y - x \<le> z - x" |
1551 |
for x y z :: "'a::len0 word" |
|
37660 | 1552 |
by uint_arith |
1553 |
||
65268 | 1554 |
lemma word_le_minus_mono: |
65328 | 1555 |
"a \<le> c \<Longrightarrow> d \<le> b \<Longrightarrow> a - b \<le> a \<Longrightarrow> c - d \<le> c \<Longrightarrow> a - b \<le> c - d" |
1556 |
for a b c d :: "'a::len word" |
|
37660 | 1557 |
by uint_arith |
1558 |
||
65328 | 1559 |
lemma plus_le_left_cancel_wrap: "x + y' < x \<Longrightarrow> x + y < x \<Longrightarrow> x + y' < x + y \<longleftrightarrow> y' < y" |
1560 |
for x y y' :: "'a::len0 word" |
|
37660 | 1561 |
by uint_arith |
1562 |
||
65328 | 1563 |
lemma plus_le_left_cancel_nowrap: "x \<le> x + y' \<Longrightarrow> x \<le> x + y \<Longrightarrow> x + y' < x + y \<longleftrightarrow> y' < y" |
1564 |
for x y y' :: "'a::len0 word" |
|
37660 | 1565 |
by uint_arith |
1566 |
||
65328 | 1567 |
lemma word_plus_mono_right2: "a \<le> a + b \<Longrightarrow> c \<le> b \<Longrightarrow> a \<le> a + c" |
1568 |
for a b c :: "'a::len0 word" |
|
1569 |
by uint_arith |
|
1570 |
||
1571 |
lemma word_less_add_right: "x < y - z \<Longrightarrow> z \<le> y \<Longrightarrow> x + z < y" |
|
1572 |
for x y z :: "'a::len0 word" |
|
37660 | 1573 |
by uint_arith |
1574 |
||
65328 | 1575 |
lemma word_less_sub_right: "x < y + z \<Longrightarrow> y \<le> x \<Longrightarrow> x - y < z" |
1576 |
for x y z :: "'a::len0 word" |
|
37660 | 1577 |
by uint_arith |
1578 |
||
65328 | 1579 |
lemma word_le_plus_either: "x \<le> y \<or> x \<le> z \<Longrightarrow> y \<le> y + z \<Longrightarrow> x \<le> y + z" |
1580 |
for x y z :: "'a::len0 word" |
|
37660 | 1581 |
by uint_arith |
1582 |
||
65328 | 1583 |
lemma word_less_nowrapI: "x < z - k \<Longrightarrow> k \<le> z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k" |
1584 |
for x z k :: "'a::len0 word" |
|
37660 | 1585 |
by uint_arith |
1586 |
||
65328 | 1587 |
lemma inc_le: "i < m \<Longrightarrow> i + 1 \<le> m" |
1588 |
for i m :: "'a::len word" |
|
37660 | 1589 |
by uint_arith |
1590 |
||
65328 | 1591 |
lemma inc_i: "1 \<le> i \<Longrightarrow> i < m \<Longrightarrow> 1 \<le> i + 1 \<and> i + 1 \<le> m" |
1592 |
for i m :: "'a::len word" |
|
37660 | 1593 |
by uint_arith |
1594 |
||
1595 |
lemma udvd_incr_lem: |
|
65268 | 1596 |
"up < uq \<Longrightarrow> up = ua + n * uint K \<Longrightarrow> |
65328 | 1597 |
uq = ua + n' * uint K \<Longrightarrow> up + uint K \<le> uq" |
37660 | 1598 |
apply clarsimp |
1599 |
apply (drule less_le_mult) |
|
65328 | 1600 |
apply safe |
37660 | 1601 |
done |
1602 |
||
65268 | 1603 |
lemma udvd_incr': |
1604 |
"p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> |
|
65328 | 1605 |
uint q = ua + n' * uint K \<Longrightarrow> p + K \<le> q" |
37660 | 1606 |
apply (unfold word_less_alt word_le_def) |
1607 |
apply (drule (2) udvd_incr_lem) |
|
1608 |
apply (erule uint_add_le [THEN order_trans]) |
|
1609 |
done |
|
1610 |
||
65268 | 1611 |
lemma udvd_decr': |
1612 |
"p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> |
|
65328 | 1613 |
uint q = ua + n' * uint K \<Longrightarrow> p \<le> q - K" |
37660 | 1614 |
apply (unfold word_less_alt word_le_def) |
1615 |
apply (drule (2) udvd_incr_lem) |
|
1616 |
apply (drule le_diff_eq [THEN iffD2]) |
|
1617 |
apply (erule order_trans) |
|
1618 |
apply (rule uint_sub_ge) |
|
1619 |
done |
|
1620 |
||
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1621 |
lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, unfolded add_0_left] |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1622 |
lemmas udvd_incr0 = udvd_incr' [where ua=0, unfolded add_0_left] |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1623 |
lemmas udvd_decr0 = udvd_decr' [where ua=0, unfolded add_0_left] |
37660 | 1624 |
|
65328 | 1625 |
lemma udvd_minus_le': "xy < k \<Longrightarrow> z udvd xy \<Longrightarrow> z udvd k \<Longrightarrow> xy \<le> k - z" |
37660 | 1626 |
apply (unfold udvd_def) |
1627 |
apply clarify |
|
1628 |
apply (erule (2) udvd_decr0) |
|
1629 |
done |
|
1630 |
||
65268 | 1631 |
lemma udvd_incr2_K: |
65328 | 1632 |
"p < a + s \<Longrightarrow> a \<le> a + s \<Longrightarrow> K udvd s \<Longrightarrow> K udvd p - a \<Longrightarrow> a \<le> p \<Longrightarrow> |
1633 |
0 < K \<Longrightarrow> p \<le> p + K \<and> p + K \<le> a + s" |
|
1634 |
supply [[simproc del: linordered_ring_less_cancel_factor]] |
|
37660 | 1635 |
apply (unfold udvd_def) |
1636 |
apply clarify |
|
62390 | 1637 |
apply (simp add: uint_arith_simps split: if_split_asm) |
65268 | 1638 |
prefer 2 |
37660 | 1639 |
apply (insert uint_range' [of s])[1] |
1640 |
apply arith |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
1641 |
apply (drule add.commute [THEN xtr1]) |
37660 | 1642 |
apply (simp add: diff_less_eq [symmetric]) |
1643 |
apply (drule less_le_mult) |
|
1644 |
apply arith |
|
1645 |
apply simp |
|
1646 |
done |
|
1647 |
||
1648 |
(* links with rbl operations *) |
|
65328 | 1649 |
lemma word_succ_rbl: "to_bl w = bl \<Longrightarrow> to_bl (word_succ w) = rev (rbl_succ (rev bl))" |
37660 | 1650 |
apply (unfold word_succ_def) |
1651 |
apply clarify |
|
1652 |
apply (simp add: to_bl_of_bin) |
|
46654 | 1653 |
apply (simp add: to_bl_def rbl_succ) |
37660 | 1654 |
done |
1655 |
||
65328 | 1656 |
lemma word_pred_rbl: "to_bl w = bl \<Longrightarrow> to_bl (word_pred w) = rev (rbl_pred (rev bl))" |
37660 | 1657 |
apply (unfold word_pred_def) |
1658 |
apply clarify |
|
1659 |
apply (simp add: to_bl_of_bin) |
|
46654 | 1660 |
apply (simp add: to_bl_def rbl_pred) |
37660 | 1661 |
done |
1662 |
||
1663 |
lemma word_add_rbl: |
|
65268 | 1664 |
"to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> |
65328 | 1665 |
to_bl (v + w) = rev (rbl_add (rev vbl) (rev wbl))" |
37660 | 1666 |
apply (unfold word_add_def) |
1667 |
apply clarify |
|
1668 |
apply (simp add: to_bl_of_bin) |
|
1669 |
apply (simp add: to_bl_def rbl_add) |
|
1670 |
done |
|
1671 |
||
1672 |
lemma word_mult_rbl: |
|
65268 | 1673 |
"to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> |
65328 | 1674 |
to_bl (v * w) = rev (rbl_mult (rev vbl) (rev wbl))" |
37660 | 1675 |
apply (unfold word_mult_def) |
1676 |
apply clarify |
|
1677 |
apply (simp add: to_bl_of_bin) |
|
1678 |
apply (simp add: to_bl_def rbl_mult) |
|
1679 |
done |
|
1680 |
||
1681 |
lemma rtb_rbl_ariths: |
|
1682 |
"rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys" |
|
1683 |
"rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys" |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1684 |
"rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs" |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1685 |
"rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs" |
65328 | 1686 |
by (auto simp: rev_swap [symmetric] word_succ_rbl word_pred_rbl word_mult_rbl word_add_rbl) |
37660 | 1687 |
|
1688 |
||
61799 | 1689 |
subsection \<open>Arithmetic type class instantiations\<close> |
37660 | 1690 |
|
1691 |
lemmas word_le_0_iff [simp] = |
|
1692 |
word_zero_le [THEN leD, THEN linorder_antisym_conv1] |
|
1693 |
||
65328 | 1694 |
lemma word_of_int_nat: "0 \<le> x \<Longrightarrow> word_of_int x = of_nat (nat x)" |
1695 |
by (simp add: word_of_int) |
|
37660 | 1696 |
|
46603 | 1697 |
(* note that iszero_def is only for class comm_semiring_1_cancel, |
65268 | 1698 |
which requires word length >= 1, ie 'a::len word *) |
46603 | 1699 |
lemma iszero_word_no [simp]: |
65268 | 1700 |
"iszero (numeral bin :: 'a::len word) = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1701 |
iszero (bintrunc (len_of TYPE('a)) (numeral bin))" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1702 |
using word_ubin.norm_eq_iff [where 'a='a, of "numeral bin" 0] |
46603 | 1703 |
by (simp add: iszero_def [symmetric]) |
65268 | 1704 |
|
61799 | 1705 |
text \<open>Use \<open>iszero\<close> to simplify equalities between word numerals.\<close> |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1706 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1707 |
lemmas word_eq_numeral_iff_iszero [simp] = |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
1708 |
eq_numeral_iff_iszero [where 'a="'a::len word"] |
46603 | 1709 |
|
37660 | 1710 |
|
61799 | 1711 |
subsection \<open>Word and nat\<close> |
37660 | 1712 |
|
45811 | 1713 |
lemma td_ext_unat [OF refl]: |
65268 | 1714 |
"n = len_of TYPE('a::len) \<Longrightarrow> |
65328 | 1715 |
td_ext (unat :: 'a word \<Rightarrow> nat) of_nat (unats n) (\<lambda>i. i mod 2 ^ n)" |
37660 | 1716 |
apply (unfold td_ext_def' unat_def word_of_nat unats_uints) |
1717 |
apply (auto intro!: imageI simp add : word_of_int_hom_syms) |
|
65328 | 1718 |
apply (erule word_uint.Abs_inverse [THEN arg_cong]) |
37660 | 1719 |
apply (simp add: int_word_uint nat_mod_distrib nat_power_eq) |
1720 |
done |
|
1721 |
||
45604 | 1722 |
lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm] |
37660 | 1723 |
|
1724 |
interpretation word_unat: |
|
65328 | 1725 |
td_ext |
1726 |
"unat::'a::len word \<Rightarrow> nat" |
|
1727 |
of_nat |
|
1728 |
"unats (len_of TYPE('a::len))" |
|
1729 |
"\<lambda>i. i mod 2 ^ len_of TYPE('a::len)" |
|
37660 | 1730 |
by (rule td_ext_unat) |
1731 |
||
1732 |
lemmas td_unat = word_unat.td_thm |
|
1733 |
||
1734 |
lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq] |
|
1735 |
||
65328 | 1736 |
lemma unat_le: "y \<le> unat z \<Longrightarrow> y \<in> unats (len_of TYPE('a))" |
1737 |
for z :: "'a::len word" |
|
37660 | 1738 |
apply (unfold unats_def) |
1739 |
apply clarsimp |
|
65268 | 1740 |
apply (rule xtrans, rule unat_lt2p, assumption) |
37660 | 1741 |
done |
1742 |
||
65328 | 1743 |
lemma word_nchotomy: "\<forall>w :: 'a::len word. \<exists>n. w = of_nat n \<and> n < 2 ^ len_of TYPE('a)" |
37660 | 1744 |
apply (rule allI) |
1745 |
apply (rule word_unat.Abs_cases) |
|
1746 |
apply (unfold unats_def) |
|
1747 |
apply auto |
|
1748 |
done |
|
1749 |
||
65328 | 1750 |
lemma of_nat_eq: "of_nat n = w \<longleftrightarrow> (\<exists>q. n = unat w + q * 2 ^ len_of TYPE('a))" |
1751 |
for w :: "'a::len word" |
|
37660 | 1752 |
apply (rule trans) |
1753 |
apply (rule word_unat.inverse_norm) |
|
1754 |
apply (rule iffI) |
|
1755 |
apply (rule mod_eqD) |
|
1756 |
apply simp |
|
1757 |
apply clarsimp |
|
1758 |
done |
|
1759 |
||
65328 | 1760 |
lemma of_nat_eq_size: "of_nat n = w \<longleftrightarrow> (\<exists>q. n = unat w + q * 2 ^ size w)" |
37660 | 1761 |
unfolding word_size by (rule of_nat_eq) |
1762 |
||
65328 | 1763 |
lemma of_nat_0: "of_nat m = (0::'a::len word) \<longleftrightarrow> (\<exists>q. m = q * 2 ^ len_of TYPE('a))" |
37660 | 1764 |
by (simp add: of_nat_eq) |
1765 |
||
65328 | 1766 |
lemma of_nat_2p [simp]: "of_nat (2 ^ len_of TYPE('a)) = (0::'a::len word)" |
45805 | 1767 |
by (fact mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]]) |
37660 | 1768 |
|
65328 | 1769 |
lemma of_nat_gt_0: "of_nat k \<noteq> 0 \<Longrightarrow> 0 < k" |
37660 | 1770 |
by (cases k) auto |
1771 |
||
65328 | 1772 |
lemma of_nat_neq_0: "0 < k \<Longrightarrow> k < 2 ^ len_of TYPE('a::len) \<Longrightarrow> of_nat k \<noteq> (0 :: 'a word)" |
1773 |
by (auto simp add : of_nat_0) |
|
1774 |
||
1775 |
lemma Abs_fnat_hom_add: "of_nat a + of_nat b = of_nat (a + b)" |
|
37660 | 1776 |
by simp |
1777 |
||
65328 | 1778 |
lemma Abs_fnat_hom_mult: "of_nat a * of_nat b = (of_nat (a * b) :: 'a::len word)" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
1779 |
by (simp add: word_of_nat wi_hom_mult) |
37660 | 1780 |
|
65328 | 1781 |
lemma Abs_fnat_hom_Suc: "word_succ (of_nat a) = of_nat (Suc a)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
1782 |
by (simp add: word_of_nat wi_hom_succ ac_simps) |
37660 | 1783 |
|
1784 |
lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0" |
|
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
1785 |
by simp |
37660 | 1786 |
|
1787 |
lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)" |
|
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
1788 |
by simp |
37660 | 1789 |
|
65268 | 1790 |
lemmas Abs_fnat_homs = |
1791 |
Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc |
|
37660 | 1792 |
Abs_fnat_hom_0 Abs_fnat_hom_1 |
1793 |
||
65328 | 1794 |
lemma word_arith_nat_add: "a + b = of_nat (unat a + unat b)" |
1795 |
by simp |
|
1796 |
||
1797 |
lemma word_arith_nat_mult: "a * b = of_nat (unat a * unat b)" |
|
37660 | 1798 |
by simp |
1799 |
||
65328 | 1800 |
lemma word_arith_nat_Suc: "word_succ a = of_nat (Suc (unat a))" |
37660 | 1801 |
by (subst Abs_fnat_hom_Suc [symmetric]) simp |
1802 |
||
65328 | 1803 |
lemma word_arith_nat_div: "a div b = of_nat (unat a div unat b)" |
37660 | 1804 |
by (simp add: word_div_def word_of_nat zdiv_int uint_nat) |
1805 |
||
65328 | 1806 |
lemma word_arith_nat_mod: "a mod b = of_nat (unat a mod unat b)" |
37660 | 1807 |
by (simp add: word_mod_def word_of_nat zmod_int uint_nat) |
1808 |
||
1809 |
lemmas word_arith_nat_defs = |
|
1810 |
word_arith_nat_add word_arith_nat_mult |
|
1811 |
word_arith_nat_Suc Abs_fnat_hom_0 |
|
1812 |
Abs_fnat_hom_1 word_arith_nat_div |
|
65268 | 1813 |
word_arith_nat_mod |
37660 | 1814 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1815 |
lemma unat_cong: "x = y \<Longrightarrow> unat x = unat y" |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1816 |
by simp |
65268 | 1817 |
|
37660 | 1818 |
lemmas unat_word_ariths = word_arith_nat_defs |
45604 | 1819 |
[THEN trans [OF unat_cong unat_of_nat]] |
37660 | 1820 |
|
1821 |
lemmas word_sub_less_iff = word_sub_le_iff |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
1822 |
[unfolded linorder_not_less [symmetric] Not_eq_iff] |
37660 | 1823 |
|
65268 | 1824 |
lemma unat_add_lem: |
65328 | 1825 |
"unat x + unat y < 2 ^ len_of TYPE('a) \<longleftrightarrow> unat (x + y) = unat x + unat y" |
1826 |
for x y :: "'a::len word" |
|
1827 |
by (auto simp: unat_word_ariths intro!: trans [OF _ nat_mod_lem]) |
|
37660 | 1828 |
|
65268 | 1829 |
lemma unat_mult_lem: |
65328 | 1830 |
"unat x * unat y < 2 ^ len_of TYPE('a) \<longleftrightarrow> |
1831 |
unat (x * y :: 'a::len word) = unat x * unat y" |
|
1832 |
by (auto simp: unat_word_ariths intro!: trans [OF _ nat_mod_lem]) |
|
1833 |
||
1834 |
lemmas unat_plus_if' = |
|
1835 |
trans [OF unat_word_ariths(1) mod_nat_add, simplified] |
|
1836 |
||
1837 |
lemma le_no_overflow: "x \<le> b \<Longrightarrow> a \<le> a + b \<Longrightarrow> x \<le> a + b" |
|
1838 |
for a b x :: "'a::len0 word" |
|
37660 | 1839 |
apply (erule order_trans) |
1840 |
apply (erule olen_add_eqv [THEN iffD1]) |
|
1841 |
done |
|
1842 |
||
65328 | 1843 |
lemmas un_ui_le = |
1844 |
trans [OF word_le_nat_alt [symmetric] word_le_def] |
|
37660 | 1845 |
|
1846 |
lemma unat_sub_if_size: |
|
65328 | 1847 |
"unat (x - y) = |
1848 |
(if unat y \<le> unat x |
|
1849 |
then unat x - unat y |
|
1850 |
else unat x + 2 ^ size x - unat y)" |
|
37660 | 1851 |
apply (unfold word_size) |
1852 |
apply (simp add: un_ui_le) |
|
1853 |
apply (auto simp add: unat_def uint_sub_if') |
|
1854 |
apply (rule nat_diff_distrib) |
|
1855 |
prefer 3 |
|
1856 |
apply (simp add: algebra_simps) |
|
1857 |
apply (rule nat_diff_distrib [THEN trans]) |
|
1858 |
prefer 3 |
|
1859 |
apply (subst nat_add_distrib) |
|
1860 |
prefer 3 |
|
1861 |
apply (simp add: nat_power_eq) |
|
1862 |
apply auto |
|
1863 |
apply uint_arith |
|
1864 |
done |
|
1865 |
||
1866 |
lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size] |
|
1867 |
||
65328 | 1868 |
lemma unat_div: "unat (x div y) = unat x div unat y" |
1869 |
for x y :: " 'a::len word" |
|
37660 | 1870 |
apply (simp add : unat_word_ariths) |
1871 |
apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq']) |
|
1872 |
apply (rule div_le_dividend) |
|
1873 |
done |
|
1874 |
||
65328 | 1875 |
lemma unat_mod: "unat (x mod y) = unat x mod unat y" |
1876 |
for x y :: "'a::len word" |
|
37660 | 1877 |
apply (clarsimp simp add : unat_word_ariths) |
1878 |
apply (cases "unat y") |
|
1879 |
prefer 2 |
|
1880 |
apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq']) |
|
1881 |
apply (rule mod_le_divisor) |
|
1882 |
apply auto |
|
1883 |
done |
|
1884 |
||
65328 | 1885 |
lemma uint_div: "uint (x div y) = uint x div uint y" |
1886 |
for x y :: "'a::len word" |
|
1887 |
by (simp add: uint_nat unat_div zdiv_int) |
|
1888 |
||
1889 |
lemma uint_mod: "uint (x mod y) = uint x mod uint y" |
|
1890 |
for x y :: "'a::len word" |
|
1891 |
by (simp add: uint_nat unat_mod zmod_int) |
|
37660 | 1892 |
|
1893 |
||
61799 | 1894 |
subsection \<open>Definition of \<open>unat_arith\<close> tactic\<close> |
37660 | 1895 |
|
65328 | 1896 |
lemma unat_split: "P (unat x) \<longleftrightarrow> (\<forall>n. of_nat n = x \<and> n < 2^len_of TYPE('a) \<longrightarrow> P n)" |
1897 |
for x :: "'a::len word" |
|
37660 | 1898 |
by (auto simp: unat_of_nat) |
1899 |
||
65328 | 1900 |
lemma unat_split_asm: "P (unat x) \<longleftrightarrow> (\<nexists>n. of_nat n = x \<and> n < 2^len_of TYPE('a) \<and> \<not> P n)" |
1901 |
for x :: "'a::len word" |
|
37660 | 1902 |
by (auto simp: unat_of_nat) |
1903 |
||
65268 | 1904 |
lemmas of_nat_inverse = |
37660 | 1905 |
word_unat.Abs_inverse' [rotated, unfolded unats_def, simplified] |
1906 |
||
1907 |
lemmas unat_splits = unat_split unat_split_asm |
|
1908 |
||
1909 |
lemmas unat_arith_simps = |
|
1910 |
word_le_nat_alt word_less_nat_alt |
|
1911 |
word_unat.Rep_inject [symmetric] |
|
1912 |
unat_sub_if' unat_plus_if' unat_div unat_mod |
|
1913 |
||
65268 | 1914 |
(* unat_arith_tac: tactic to reduce word arithmetic to nat, |
37660 | 1915 |
try to solve via arith *) |
61799 | 1916 |
ML \<open> |
65268 | 1917 |
fun unat_arith_simpset ctxt = |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1918 |
ctxt addsimps @{thms unat_arith_simps} |
37660 | 1919 |
delsimps @{thms word_unat.Rep_inject} |
62390 | 1920 |
|> fold Splitter.add_split @{thms if_split_asm} |
45620
f2a587696afb
modernized some old-style infix operations, which were left over from the time of ML proof scripts;
wenzelm
parents:
45604
diff
changeset
|
1921 |
|> fold Simplifier.add_cong @{thms power_False_cong} |
37660 | 1922 |
|
65268 | 1923 |
fun unat_arith_tacs ctxt = |
37660 | 1924 |
let |
1925 |
fun arith_tac' n t = |
|
59657
2441a80fb6c1
eliminated unused arith "verbose" flag -- tools that need options can use the context;
wenzelm
parents:
59498
diff
changeset
|
1926 |
Arith_Data.arith_tac ctxt n t |
37660 | 1927 |
handle Cooper.COOPER _ => Seq.empty; |
65268 | 1928 |
in |
42793 | 1929 |
[ clarify_tac ctxt 1, |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1930 |
full_simp_tac (unat_arith_simpset ctxt) 1, |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1931 |
ALLGOALS (full_simp_tac |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1932 |
(put_simpset HOL_ss ctxt |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1933 |
|> fold Splitter.add_split @{thms unat_splits} |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
51375
diff
changeset
|
1934 |
|> fold Simplifier.add_cong @{thms power_False_cong})), |
65268 | 1935 |
rewrite_goals_tac ctxt @{thms word_size}, |
60754 | 1936 |
ALLGOALS (fn n => REPEAT (resolve_tac ctxt [allI, impI] n) THEN |
1937 |
REPEAT (eresolve_tac ctxt [conjE] n) THEN |
|
1938 |
REPEAT (dresolve_tac ctxt @{thms of_nat_inverse} n THEN assume_tac ctxt n)), |
|
65268 | 1939 |
TRYALL arith_tac' ] |
37660 | 1940 |
end |
1941 |
||
1942 |
fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt)) |
|
61799 | 1943 |
\<close> |
37660 | 1944 |
|
65268 | 1945 |
method_setup unat_arith = |
61799 | 1946 |
\<open>Scan.succeed (SIMPLE_METHOD' o unat_arith_tac)\<close> |
37660 | 1947 |
"solving word arithmetic via natural numbers and arith" |
1948 |
||
65328 | 1949 |
lemma no_plus_overflow_unat_size: "x \<le> x + y \<longleftrightarrow> unat x + unat y < 2 ^ size x" |
1950 |
for x y :: "'a::len word" |
|
37660 | 1951 |
unfolding word_size by unat_arith |
1952 |
||
65328 | 1953 |
lemmas no_olen_add_nat = |
1954 |
no_plus_overflow_unat_size [unfolded word_size] |
|
1955 |
||
1956 |
lemmas unat_plus_simple = |
|
1957 |
trans [OF no_olen_add_nat unat_add_lem] |
|
1958 |
||
1959 |
lemma word_div_mult: "0 < y \<Longrightarrow> unat x * unat y < 2 ^ len_of TYPE('a) \<Longrightarrow> x * y div y = x" |
|
1960 |
for x y :: "'a::len word" |
|
37660 | 1961 |
apply unat_arith |
1962 |
apply clarsimp |
|
1963 |
apply (subst unat_mult_lem [THEN iffD1]) |
|
65328 | 1964 |
apply auto |
37660 | 1965 |
done |
1966 |
||
65328 | 1967 |
lemma div_lt': "i \<le> k div x \<Longrightarrow> unat i * unat x < 2 ^ len_of TYPE('a)" |
1968 |
for i k x :: "'a::len word" |
|
37660 | 1969 |
apply unat_arith |
1970 |
apply clarsimp |
|
1971 |
apply (drule mult_le_mono1) |
|
1972 |
apply (erule order_le_less_trans) |
|
1973 |
apply (rule xtr7 [OF unat_lt2p div_mult_le]) |
|
1974 |
done |
|
1975 |
||
1976 |
lemmas div_lt'' = order_less_imp_le [THEN div_lt'] |
|
1977 |
||
65328 | 1978 |
lemma div_lt_mult: "i < k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x < k" |
1979 |
for i k x :: "'a::len word" |
|
37660 | 1980 |
apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]]) |
1981 |
apply (simp add: unat_arith_simps) |
|
1982 |
apply (drule (1) mult_less_mono1) |
|
1983 |
apply (erule order_less_le_trans) |
|
1984 |
apply (rule div_mult_le) |
|
1985 |
done |
|
1986 |
||
65328 | 1987 |
lemma div_le_mult: "i \<le> k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x \<le> k" |
1988 |
for i k x :: "'a::len word" |
|
37660 | 1989 |
apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]]) |
1990 |
apply (simp add: unat_arith_simps) |
|
1991 |
apply (drule mult_le_mono1) |
|
1992 |
apply (erule order_trans) |
|
1993 |
apply (rule div_mult_le) |
|
1994 |
done |
|
1995 |
||
65328 | 1996 |
lemma div_lt_uint': "i \<le> k div x \<Longrightarrow> uint i * uint x < 2 ^ len_of TYPE('a)" |
1997 |
for i k x :: "'a::len word" |
|
37660 | 1998 |
apply (unfold uint_nat) |
1999 |
apply (drule div_lt') |
|
65328 | 2000 |
apply (metis of_nat_less_iff of_nat_mult of_nat_numeral of_nat_power) |
2001 |
done |
|
37660 | 2002 |
|
2003 |
lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint'] |
|
2004 |
||
65328 | 2005 |
lemma word_le_exists': "x \<le> y \<Longrightarrow> (\<exists>z. y = x + z \<and> uint x + uint z < 2 ^ len_of TYPE('a))" |
2006 |
for x y z :: "'a::len0 word" |
|
37660 | 2007 |
apply (rule exI) |
2008 |
apply (rule conjI) |
|
65328 | 2009 |
apply (rule zadd_diff_inverse) |
37660 | 2010 |
apply uint_arith |
2011 |
done |
|
2012 |
||
2013 |
lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab] |
|
2014 |
||
2015 |
lemmas plus_minus_no_overflow = |
|
2016 |
order_less_imp_le [THEN plus_minus_no_overflow_ab] |
|
65268 | 2017 |
|
37660 | 2018 |
lemmas mcs = word_less_minus_cancel word_less_minus_mono_left |
2019 |
word_le_minus_cancel word_le_minus_mono_left |
|
2020 |
||
45604 | 2021 |
lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel] for w x |
2022 |
lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel] for w x |
|
2023 |
lemmas word_plus_mcs = word_diff_ls [where y = "v + x", unfolded add_diff_cancel] for v x |
|
37660 | 2024 |
|
2025 |
lemmas le_unat_uoi = unat_le [THEN word_unat.Abs_inverse] |
|
2026 |
||
2027 |
lemmas thd = refl [THEN [2] split_div_lemma [THEN iffD2], THEN conjunct1] |
|
2028 |
||
65268 | 2029 |
lemmas uno_simps [THEN le_unat_uoi] = mod_le_divisor div_le_dividend dtle |
37660 | 2030 |
|
65328 | 2031 |
lemma word_mod_div_equality: "(n div b) * b + (n mod b) = n" |
2032 |
for n b :: "'a::len word" |
|
37660 | 2033 |
apply (unfold word_less_nat_alt word_arith_nat_defs) |
2034 |
apply (cut_tac y="unat b" in gt_or_eq_0) |
|
2035 |
apply (erule disjE) |
|
64242
93c6f0da5c70
more standardized theorem names for facts involving the div and mod identity
haftmann
parents:
63950
diff
changeset
|
2036 |
apply (simp only: div_mult_mod_eq uno_simps Word.word_unat.Rep_inverse) |
37660 | 2037 |
apply simp |
2038 |
done |
|
2039 |
||
65328 | 2040 |
lemma word_div_mult_le: "a div b * b \<le> a" |
2041 |
for a b :: "'a::len word" |
|
37660 | 2042 |
apply (unfold word_le_nat_alt word_arith_nat_defs) |
2043 |
apply (cut_tac y="unat b" in gt_or_eq_0) |
|
2044 |
apply (erule disjE) |
|
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
2045 |
apply (simp only: div_mult_le uno_simps Word.word_unat.Rep_inverse) |
37660 | 2046 |
apply simp |
2047 |
done |
|
2048 |
||
65328 | 2049 |
lemma word_mod_less_divisor: "0 < n \<Longrightarrow> m mod n < n" |
2050 |
for m n :: "'a::len word" |
|
37660 | 2051 |
apply (simp only: word_less_nat_alt word_arith_nat_defs) |
65328 | 2052 |
apply (auto simp: uno_simps) |
37660 | 2053 |
done |
2054 |
||
65328 | 2055 |
lemma word_of_int_power_hom: "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a::len word)" |
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
2056 |
by (induct n) (simp_all add: wi_hom_mult [symmetric]) |
37660 | 2057 |
|
65328 | 2058 |
lemma word_arith_power_alt: "a ^ n = (word_of_int (uint a ^ n) :: 'a::len word)" |
37660 | 2059 |
by (simp add : word_of_int_power_hom [symmetric]) |
2060 |
||
65268 | 2061 |
lemma of_bl_length_less: |
2062 |
"length x = k \<Longrightarrow> k < len_of TYPE('a) \<Longrightarrow> (of_bl x :: 'a::len word) < 2 ^ k" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2063 |
apply (unfold of_bl_def word_less_alt word_numeral_alt) |
37660 | 2064 |
apply safe |
65268 | 2065 |
apply (simp (no_asm) add: word_of_int_power_hom word_uint.eq_norm |
65328 | 2066 |
del: word_of_int_numeral) |
37660 | 2067 |
apply (simp add: mod_pos_pos_trivial) |
2068 |
apply (subst mod_pos_pos_trivial) |
|
2069 |
apply (rule bl_to_bin_ge0) |
|
2070 |
apply (rule order_less_trans) |
|
2071 |
apply (rule bl_to_bin_lt2p) |
|
2072 |
apply simp |
|
46646 | 2073 |
apply (rule bl_to_bin_lt2p) |
37660 | 2074 |
done |
2075 |
||
2076 |
||
61799 | 2077 |
subsection \<open>Cardinality, finiteness of set of words\<close> |
37660 | 2078 |
|
45809
2bee94cbae72
finite class instance for word type; remove unused lemmas
huffman
parents:
45808
diff
changeset
|
2079 |
instance word :: (len0) finite |
61169 | 2080 |
by standard (simp add: type_definition.univ [OF type_definition_word]) |
45809
2bee94cbae72
finite class instance for word type; remove unused lemmas
huffman
parents:
45808
diff
changeset
|
2081 |
|
2bee94cbae72
finite class instance for word type; remove unused lemmas
huffman
parents:
45808
diff
changeset
|
2082 |
lemma card_word: "CARD('a::len0 word) = 2 ^ len_of TYPE('a)" |
2bee94cbae72
finite class instance for word type; remove unused lemmas
huffman
parents:
45808
diff
changeset
|
2083 |
by (simp add: type_definition.card [OF type_definition_word] nat_power_eq) |
37660 | 2084 |
|
65328 | 2085 |
lemma card_word_size: "card (UNIV :: 'a word set) = (2 ^ size x)" |
2086 |
for x :: "'a::len0 word" |
|
2087 |
unfolding word_size by (rule card_word) |
|
37660 | 2088 |
|
2089 |
||
61799 | 2090 |
subsection \<open>Bitwise Operations on Words\<close> |
37660 | 2091 |
|
2092 |
lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or |
|
65268 | 2093 |
|
37660 | 2094 |
(* following definitions require both arithmetic and bit-wise word operations *) |
2095 |
||
2096 |
(* to get word_no_log_defs from word_log_defs, using bin_log_bintrs *) |
|
2097 |
lemmas wils1 = bin_log_bintrs [THEN word_ubin.norm_eq_iff [THEN iffD1], |
|
45604 | 2098 |
folded word_ubin.eq_norm, THEN eq_reflection] |
37660 | 2099 |
|
2100 |
(* the binary operations only *) |
|
46013 | 2101 |
(* BH: why is this needed? *) |
65268 | 2102 |
lemmas word_log_binary_defs = |
37660 | 2103 |
word_and_def word_or_def word_xor_def |
2104 |
||
46011 | 2105 |
lemma word_wi_log_defs: |
2106 |
"NOT word_of_int a = word_of_int (NOT a)" |
|
2107 |
"word_of_int a AND word_of_int b = word_of_int (a AND b)" |
|
2108 |
"word_of_int a OR word_of_int b = word_of_int (a OR b)" |
|
2109 |
"word_of_int a XOR word_of_int b = word_of_int (a XOR b)" |
|
47374
9475d524bafb
set up and use lift_definition for word operations
huffman
parents:
47372
diff
changeset
|
2110 |
by (transfer, rule refl)+ |
47372 | 2111 |
|
46011 | 2112 |
lemma word_no_log_defs [simp]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2113 |
"NOT (numeral a) = word_of_int (NOT (numeral a))" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2114 |
"NOT (- numeral a) = word_of_int (NOT (- numeral a))" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2115 |
"numeral a AND numeral b = word_of_int (numeral a AND numeral b)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2116 |
"numeral a AND - numeral b = word_of_int (numeral a AND - numeral b)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2117 |
"- numeral a AND numeral b = word_of_int (- numeral a AND numeral b)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2118 |
"- numeral a AND - numeral b = word_of_int (- numeral a AND - numeral b)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2119 |
"numeral a OR numeral b = word_of_int (numeral a OR numeral b)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2120 |
"numeral a OR - numeral b = word_of_int (numeral a OR - numeral b)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2121 |
"- numeral a OR numeral b = word_of_int (- numeral a OR numeral b)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2122 |
"- numeral a OR - numeral b = word_of_int (- numeral a OR - numeral b)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2123 |
"numeral a XOR numeral b = word_of_int (numeral a XOR numeral b)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2124 |
"numeral a XOR - numeral b = word_of_int (numeral a XOR - numeral b)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2125 |
"- numeral a XOR numeral b = word_of_int (- numeral a XOR numeral b)" |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2126 |
"- numeral a XOR - numeral b = word_of_int (- numeral a XOR - numeral b)" |
47372 | 2127 |
by (transfer, rule refl)+ |
37660 | 2128 |
|
61799 | 2129 |
text \<open>Special cases for when one of the arguments equals 1.\<close> |
46064
88ef116e0522
add simp rules for bitwise word operations with 1
huffman
parents:
46057
diff
changeset
|
2130 |
|
88ef116e0522
add simp rules for bitwise word operations with 1
huffman
parents:
46057
diff
changeset
|
2131 |
lemma word_bitwise_1_simps [simp]: |
88ef116e0522
add simp rules for bitwise word operations with 1
huffman
parents:
46057
diff
changeset
|
2132 |
"NOT (1::'a::len0 word) = -2" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2133 |
"1 AND numeral b = word_of_int (1 AND numeral b)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2134 |
"1 AND - numeral b = word_of_int (1 AND - numeral b)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2135 |
"numeral a AND 1 = word_of_int (numeral a AND 1)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2136 |
"- numeral a AND 1 = word_of_int (- numeral a AND 1)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2137 |
"1 OR numeral b = word_of_int (1 OR numeral b)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2138 |
"1 OR - numeral b = word_of_int (1 OR - numeral b)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2139 |
"numeral a OR 1 = word_of_int (numeral a OR 1)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2140 |
"- numeral a OR 1 = word_of_int (- numeral a OR 1)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2141 |
"1 XOR numeral b = word_of_int (1 XOR numeral b)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2142 |
"1 XOR - numeral b = word_of_int (1 XOR - numeral b)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2143 |
"numeral a XOR 1 = word_of_int (numeral a XOR 1)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2144 |
"- numeral a XOR 1 = word_of_int (- numeral a XOR 1)" |
47372 | 2145 |
by (transfer, simp)+ |
46064
88ef116e0522
add simp rules for bitwise word operations with 1
huffman
parents:
46057
diff
changeset
|
2146 |
|
61799 | 2147 |
text \<open>Special cases for when one of the arguments equals -1.\<close> |
56979 | 2148 |
|
2149 |
lemma word_bitwise_m1_simps [simp]: |
|
2150 |
"NOT (-1::'a::len0 word) = 0" |
|
2151 |
"(-1::'a::len0 word) AND x = x" |
|
2152 |
"x AND (-1::'a::len0 word) = x" |
|
2153 |
"(-1::'a::len0 word) OR x = -1" |
|
2154 |
"x OR (-1::'a::len0 word) = -1" |
|
2155 |
" (-1::'a::len0 word) XOR x = NOT x" |
|
2156 |
"x XOR (-1::'a::len0 word) = NOT x" |
|
2157 |
by (transfer, simp)+ |
|
2158 |
||
65328 | 2159 |
lemma uint_or: "uint (x OR y) = uint x OR uint y" |
47372 | 2160 |
by (transfer, simp add: bin_trunc_ao) |
37660 | 2161 |
|
65328 | 2162 |
lemma uint_and: "uint (x AND y) = uint x AND uint y" |
47372 | 2163 |
by (transfer, simp add: bin_trunc_ao) |
2164 |
||
2165 |
lemma test_bit_wi [simp]: |
|
65328 | 2166 |
"(word_of_int x :: 'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a) \<and> bin_nth x n" |
2167 |
by (simp add: word_test_bit_def word_ubin.eq_norm nth_bintr) |
|
47372 | 2168 |
|
2169 |
lemma word_test_bit_transfer [transfer_rule]: |
|
55945 | 2170 |
"(rel_fun pcr_word (rel_fun op = op =)) |
47372 | 2171 |
(\<lambda>x n. n < len_of TYPE('a) \<and> bin_nth x n) (test_bit :: 'a::len0 word \<Rightarrow> _)" |
55945 | 2172 |
unfolding rel_fun_def word.pcr_cr_eq cr_word_def by simp |
37660 | 2173 |
|
2174 |
lemma word_ops_nth_size: |
|
65328 | 2175 |
"n < size x \<Longrightarrow> |
2176 |
(x OR y) !! n = (x !! n | y !! n) \<and> |
|
2177 |
(x AND y) !! n = (x !! n \<and> y !! n) \<and> |
|
2178 |
(x XOR y) !! n = (x !! n \<noteq> y !! n) \<and> |
|
2179 |
(NOT x) !! n = (\<not> x !! n)" |
|
2180 |
for x :: "'a::len0 word" |
|
47372 | 2181 |
unfolding word_size by transfer (simp add: bin_nth_ops) |
37660 | 2182 |
|
2183 |
lemma word_ao_nth: |
|
65328 | 2184 |
"(x OR y) !! n = (x !! n | y !! n) \<and> |
2185 |
(x AND y) !! n = (x !! n \<and> y !! n)" |
|
2186 |
for x :: "'a::len0 word" |
|
47372 | 2187 |
by transfer (auto simp add: bin_nth_ops) |
46023
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset
|
2188 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2189 |
lemma test_bit_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2190 |
"(numeral w :: 'a::len0 word) !! n \<longleftrightarrow> |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2191 |
n < len_of TYPE('a) \<and> bin_nth (numeral w) n" |
47372 | 2192 |
by transfer (rule refl) |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2193 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2194 |
lemma test_bit_neg_numeral [simp]: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2195 |
"(- numeral w :: 'a::len0 word) !! n \<longleftrightarrow> |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2196 |
n < len_of TYPE('a) \<and> bin_nth (- numeral w) n" |
47372 | 2197 |
by transfer (rule refl) |
46023
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset
|
2198 |
|
65328 | 2199 |
lemma test_bit_1 [simp]: "(1 :: 'a::len word) !! n \<longleftrightarrow> n = 0" |
47372 | 2200 |
by transfer auto |
65268 | 2201 |
|
65328 | 2202 |
lemma nth_0 [simp]: "\<not> (0 :: 'a::len0 word) !! n" |
47372 | 2203 |
by transfer simp |
46023
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset
|
2204 |
|
65328 | 2205 |
lemma nth_minus1 [simp]: "(-1 :: 'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a)" |
47372 | 2206 |
by transfer simp |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2207 |
|
37660 | 2208 |
(* get from commutativity, associativity etc of int_and etc |
2209 |
to same for word_and etc *) |
|
2210 |
||
65268 | 2211 |
lemmas bwsimps = |
46013 | 2212 |
wi_hom_add |
37660 | 2213 |
word_wi_log_defs |
2214 |
||
2215 |
lemma word_bw_assocs: |
|
2216 |
"(x AND y) AND z = x AND y AND z" |
|
2217 |
"(x OR y) OR z = x OR y OR z" |
|
2218 |
"(x XOR y) XOR z = x XOR y XOR z" |
|
65328 | 2219 |
for x :: "'a::len0 word" |
46022 | 2220 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
65268 | 2221 |
|
37660 | 2222 |
lemma word_bw_comms: |
2223 |
"x AND y = y AND x" |
|
2224 |
"x OR y = y OR x" |
|
2225 |
"x XOR y = y XOR x" |
|
65328 | 2226 |
for x :: "'a::len0 word" |
46022 | 2227 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
65268 | 2228 |
|
37660 | 2229 |
lemma word_bw_lcs: |
2230 |
"y AND x AND z = x AND y AND z" |
|
2231 |
"y OR x OR z = x OR y OR z" |
|
2232 |
"y XOR x XOR z = x XOR y XOR z" |
|
65328 | 2233 |
for x :: "'a::len0 word" |
46022 | 2234 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
37660 | 2235 |
|
2236 |
lemma word_log_esimps [simp]: |
|
2237 |
"x AND 0 = 0" |
|
2238 |
"x AND -1 = x" |
|
2239 |
"x OR 0 = x" |
|
2240 |
"x OR -1 = -1" |
|
2241 |
"x XOR 0 = x" |
|
2242 |
"x XOR -1 = NOT x" |
|
2243 |
"0 AND x = 0" |
|
2244 |
"-1 AND x = x" |
|
2245 |
"0 OR x = x" |
|
2246 |
"-1 OR x = -1" |
|
2247 |
"0 XOR x = x" |
|
2248 |
"-1 XOR x = NOT x" |
|
65328 | 2249 |
for x :: "'a::len0 word" |
46023
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset
|
2250 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
37660 | 2251 |
|
2252 |
lemma word_not_dist: |
|
2253 |
"NOT (x OR y) = NOT x AND NOT y" |
|
2254 |
"NOT (x AND y) = NOT x OR NOT y" |
|
65328 | 2255 |
for x :: "'a::len0 word" |
46022 | 2256 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
37660 | 2257 |
|
2258 |
lemma word_bw_same: |
|
2259 |
"x AND x = x" |
|
2260 |
"x OR x = x" |
|
2261 |
"x XOR x = 0" |
|
65328 | 2262 |
for x :: "'a::len0 word" |
46023
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset
|
2263 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
37660 | 2264 |
|
2265 |
lemma word_ao_absorbs [simp]: |
|
2266 |
"x AND (y OR x) = x" |
|
2267 |
"x OR y AND x = x" |
|
2268 |
"x AND (x OR y) = x" |
|
2269 |
"y AND x OR x = x" |
|
2270 |
"(y OR x) AND x = x" |
|
2271 |
"x OR x AND y = x" |
|
2272 |
"(x OR y) AND x = x" |
|
2273 |
"x AND y OR x = x" |
|
65328 | 2274 |
for x :: "'a::len0 word" |
46022 | 2275 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
37660 | 2276 |
|
65328 | 2277 |
lemma word_not_not [simp]: "NOT NOT x = x" |
2278 |
for x :: "'a::len0 word" |
|
46022 | 2279 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
37660 | 2280 |
|
65328 | 2281 |
lemma word_ao_dist: "(x OR y) AND z = x AND z OR y AND z" |
2282 |
for x :: "'a::len0 word" |
|
46022 | 2283 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
37660 | 2284 |
|
65328 | 2285 |
lemma word_oa_dist: "x AND y OR z = (x OR z) AND (y OR z)" |
2286 |
for x :: "'a::len0 word" |
|
2287 |
by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size]) |
|
2288 |
||
2289 |
lemma word_add_not [simp]: "x + NOT x = -1" |
|
2290 |
for x :: "'a::len0 word" |
|
47372 | 2291 |
by transfer (simp add: bin_add_not) |
37660 | 2292 |
|
65328 | 2293 |
lemma word_plus_and_or [simp]: "(x AND y) + (x OR y) = x + y" |
2294 |
for x :: "'a::len0 word" |
|
47372 | 2295 |
by transfer (simp add: plus_and_or) |
37660 | 2296 |
|
65328 | 2297 |
lemma leoa: "w = x OR y \<Longrightarrow> y = w AND y" |
2298 |
for x :: "'a::len0 word" |
|
2299 |
by auto |
|
2300 |
||
2301 |
lemma leao: "w' = x' AND y' \<Longrightarrow> x' = x' OR w'" |
|
2302 |
for x' :: "'a::len0 word" |
|
2303 |
by auto |
|
2304 |
||
2305 |
lemma word_ao_equiv: "w = w OR w' \<longleftrightarrow> w' = w AND w'" |
|
2306 |
for w w' :: "'a::len0 word" |
|
48196 | 2307 |
by (auto intro: leoa leao) |
37660 | 2308 |
|
65328 | 2309 |
lemma le_word_or2: "x \<le> x OR y" |
2310 |
for x y :: "'a::len0 word" |
|
2311 |
by (auto simp: word_le_def uint_or intro: le_int_or) |
|
37660 | 2312 |
|
45604 | 2313 |
lemmas le_word_or1 = xtr3 [OF word_bw_comms (2) le_word_or2] |
2314 |
lemmas word_and_le1 = xtr3 [OF word_ao_absorbs (4) [symmetric] le_word_or2] |
|
2315 |
lemmas word_and_le2 = xtr3 [OF word_ao_absorbs (8) [symmetric] le_word_or2] |
|
37660 | 2316 |
|
65268 | 2317 |
lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" |
45550
73a4f31d41c4
Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset
|
2318 |
unfolding to_bl_def word_log_defs bl_not_bin |
73a4f31d41c4
Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset
|
2319 |
by (simp add: word_ubin.eq_norm) |
37660 | 2320 |
|
65328 | 2321 |
lemma bl_word_xor: "to_bl (v XOR w) = map2 op \<noteq> (to_bl v) (to_bl w)" |
37660 | 2322 |
unfolding to_bl_def word_log_defs bl_xor_bin |
45550
73a4f31d41c4
Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset
|
2323 |
by (simp add: word_ubin.eq_norm) |
37660 | 2324 |
|
65328 | 2325 |
lemma bl_word_or: "to_bl (v OR w) = map2 op \<or> (to_bl v) (to_bl w)" |
45550
73a4f31d41c4
Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset
|
2326 |
unfolding to_bl_def word_log_defs bl_or_bin |
73a4f31d41c4
Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset
|
2327 |
by (simp add: word_ubin.eq_norm) |
37660 | 2328 |
|
65328 | 2329 |
lemma bl_word_and: "to_bl (v AND w) = map2 op \<and> (to_bl v) (to_bl w)" |
45550
73a4f31d41c4
Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset
|
2330 |
unfolding to_bl_def word_log_defs bl_and_bin |
73a4f31d41c4
Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset
|
2331 |
by (simp add: word_ubin.eq_norm) |
37660 | 2332 |
|
2333 |
lemma word_lsb_alt: "lsb (w::'a::len0 word) = test_bit w 0" |
|
2334 |
by (auto simp: word_test_bit_def word_lsb_def) |
|
2335 |
||
65328 | 2336 |
lemma word_lsb_1_0 [simp]: "lsb (1::'a::len word) \<and> \<not> lsb (0::'b::len0 word)" |
45550
73a4f31d41c4
Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset
|
2337 |
unfolding word_lsb_def uint_eq_0 uint_1 by simp |
37660 | 2338 |
|
2339 |
lemma word_lsb_last: "lsb (w::'a::len word) = last (to_bl w)" |
|
65268 | 2340 |
apply (unfold word_lsb_def uint_bl bin_to_bl_def) |
37660 | 2341 |
apply (rule_tac bin="uint w" in bin_exhaust) |
2342 |
apply (cases "size w") |
|
2343 |
apply auto |
|
2344 |
apply (auto simp add: bin_to_bl_aux_alt) |
|
2345 |
done |
|
2346 |
||
65328 | 2347 |
lemma word_lsb_int: "lsb w \<longleftrightarrow> uint w mod 2 = 1" |
2348 |
by (auto simp: word_lsb_def bin_last_def) |
|
2349 |
||
2350 |
lemma word_msb_sint: "msb w \<longleftrightarrow> sint w < 0" |
|
2351 |
by (simp only: word_msb_def sign_Min_lt_0) |
|
2352 |
||
2353 |
lemma msb_word_of_int: "msb (word_of_int x::'a::len word) = bin_nth x (len_of TYPE('a) - 1)" |
|
2354 |
by (simp add: word_msb_def word_sbin.eq_norm bin_sign_lem) |
|
46173
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2355 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2356 |
lemma word_msb_numeral [simp]: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2357 |
"msb (numeral w::'a::len word) = bin_nth (numeral w) (len_of TYPE('a) - 1)" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2358 |
unfolding word_numeral_alt by (rule msb_word_of_int) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2359 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2360 |
lemma word_msb_neg_numeral [simp]: |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2361 |
"msb (- numeral w::'a::len word) = bin_nth (- numeral w) (len_of TYPE('a) - 1)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2362 |
unfolding word_neg_numeral_alt by (rule msb_word_of_int) |
46173
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2363 |
|
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2364 |
lemma word_msb_0 [simp]: "\<not> msb (0::'a::len word)" |
65328 | 2365 |
by (simp add: word_msb_def) |
46173
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2366 |
|
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2367 |
lemma word_msb_1 [simp]: "msb (1::'a::len word) \<longleftrightarrow> len_of TYPE('a) = 1" |
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2368 |
unfolding word_1_wi msb_word_of_int eq_iff [where 'a=nat] |
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2369 |
by (simp add: Suc_le_eq) |
45811 | 2370 |
|
65328 | 2371 |
lemma word_msb_nth: "msb w = bin_nth (uint w) (len_of TYPE('a) - 1)" |
2372 |
for w :: "'a::len word" |
|
2373 |
by (simp add: word_msb_def sint_uint bin_sign_lem) |
|
2374 |
||
2375 |
lemma word_msb_alt: "msb w = hd (to_bl w)" |
|
2376 |
for w :: "'a::len word" |
|
37660 | 2377 |
apply (unfold word_msb_nth uint_bl) |
2378 |
apply (subst hd_conv_nth) |
|
65328 | 2379 |
apply (rule length_greater_0_conv [THEN iffD1]) |
37660 | 2380 |
apply simp |
2381 |
apply (simp add : nth_bin_to_bl word_size) |
|
2382 |
done |
|
2383 |
||
65328 | 2384 |
lemma word_set_nth [simp]: "set_bit w n (test_bit w n) = w" |
2385 |
for w :: "'a::len0 word" |
|
2386 |
by (auto simp: word_test_bit_def word_set_bit_def) |
|
2387 |
||
2388 |
lemma bin_nth_uint': "bin_nth (uint w) n \<longleftrightarrow> rev (bin_to_bl (size w) (uint w)) ! n \<and> n < size w" |
|
37660 | 2389 |
apply (unfold word_size) |
2390 |
apply (safe elim!: bin_nth_uint_imp) |
|
2391 |
apply (frule bin_nth_uint_imp) |
|
2392 |
apply (fast dest!: bin_nth_bl)+ |
|
2393 |
done |
|
2394 |
||
2395 |
lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size] |
|
2396 |
||
65328 | 2397 |
lemma test_bit_bl: "w !! n \<longleftrightarrow> rev (to_bl w) ! n \<and> n < size w" |
2398 |
unfolding to_bl_def word_test_bit_def word_size by (rule bin_nth_uint) |
|
37660 | 2399 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
2400 |
lemma to_bl_nth: "n < size w \<Longrightarrow> to_bl w ! n = w !! (size w - Suc n)" |
37660 | 2401 |
apply (unfold test_bit_bl) |
2402 |
apply clarsimp |
|
2403 |
apply (rule trans) |
|
2404 |
apply (rule nth_rev_alt) |
|
2405 |
apply (auto simp add: word_size) |
|
2406 |
done |
|
2407 |
||
65328 | 2408 |
lemma test_bit_set: "(set_bit w n x) !! n \<longleftrightarrow> n < size w \<and> x" |
2409 |
for w :: "'a::len0 word" |
|
2410 |
by (auto simp: word_size word_test_bit_def word_set_bit_def word_ubin.eq_norm nth_bintr) |
|
37660 | 2411 |
|
65268 | 2412 |
lemma test_bit_set_gen: |
65328 | 2413 |
"test_bit (set_bit w n x) m = (if m = n then n < size w \<and> x else test_bit w m)" |
2414 |
for w :: "'a::len0 word" |
|
37660 | 2415 |
apply (unfold word_size word_test_bit_def word_set_bit_def) |
2416 |
apply (clarsimp simp add: word_ubin.eq_norm nth_bintr bin_nth_sc_gen) |
|
2417 |
apply (auto elim!: test_bit_size [unfolded word_size] |
|
65328 | 2418 |
simp add: word_test_bit_def [symmetric]) |
37660 | 2419 |
done |
2420 |
||
2421 |
lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs" |
|
2422 |
unfolding of_bl_def bl_to_bin_rep_F by auto |
|
65268 | 2423 |
|
65328 | 2424 |
lemma msb_nth: "msb w = w !! (len_of TYPE('a) - 1)" |
2425 |
for w :: "'a::len word" |
|
2426 |
by (simp add: word_msb_nth word_test_bit_def) |
|
37660 | 2427 |
|
45604 | 2428 |
lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]] |
37660 | 2429 |
lemmas msb1 = msb0 [where i = 0] |
2430 |
lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]] |
|
2431 |
||
45604 | 2432 |
lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]] |
37660 | 2433 |
lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt] |
2434 |
||
45811 | 2435 |
lemma td_ext_nth [OF refl refl refl, unfolded word_size]: |
65328 | 2436 |
"n = size w \<Longrightarrow> ofn = set_bits \<Longrightarrow> [w, ofn g] = l \<Longrightarrow> |
2437 |
td_ext test_bit ofn {f. \<forall>i. f i \<longrightarrow> i < n} (\<lambda>h i. h i \<and> i < n)" |
|
2438 |
for w :: "'a::len0 word" |
|
37660 | 2439 |
apply (unfold word_size td_ext_def') |
46008
c296c75f4cf4
reverted some changes for set->predicate transition, according to "hg log -u berghofe -r Isabelle2007:Isabelle2008";
wenzelm
parents:
46001
diff
changeset
|
2440 |
apply safe |
37660 | 2441 |
apply (rule_tac [3] ext) |
2442 |
apply (rule_tac [4] ext) |
|
2443 |
apply (unfold word_size of_nth_def test_bit_bl) |
|
2444 |
apply safe |
|
2445 |
defer |
|
2446 |
apply (clarsimp simp: word_bl.Abs_inverse)+ |
|
2447 |
apply (rule word_bl.Rep_inverse') |
|
2448 |
apply (rule sym [THEN trans]) |
|
65328 | 2449 |
apply (rule bl_of_nth_nth) |
37660 | 2450 |
apply simp |
2451 |
apply (rule bl_of_nth_inj) |
|
2452 |
apply (clarsimp simp add : test_bit_bl word_size) |
|
2453 |
done |
|
2454 |
||
2455 |
interpretation test_bit: |
|
65328 | 2456 |
td_ext |
2457 |
"op !! :: 'a::len0 word \<Rightarrow> nat \<Rightarrow> bool" |
|
2458 |
set_bits |
|
2459 |
"{f. \<forall>i. f i \<longrightarrow> i < len_of TYPE('a::len0)}" |
|
2460 |
"(\<lambda>h i. h i \<and> i < len_of TYPE('a::len0))" |
|
37660 | 2461 |
by (rule td_ext_nth) |
2462 |
||
2463 |
lemmas td_nth = test_bit.td_thm |
|
2464 |
||
65328 | 2465 |
lemma word_set_set_same [simp]: "set_bit (set_bit w n x) n y = set_bit w n y" |
2466 |
for w :: "'a::len0 word" |
|
37660 | 2467 |
by (rule word_eqI) (simp add : test_bit_set_gen word_size) |
65268 | 2468 |
|
2469 |
lemma word_set_set_diff: |
|
37660 | 2470 |
fixes w :: "'a::len0 word" |
65328 | 2471 |
assumes "m \<noteq> n" |
65268 | 2472 |
shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x" |
65328 | 2473 |
by (rule word_eqI) (auto simp: test_bit_set_gen word_size assms) |
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2474 |
|
65268 | 2475 |
lemma nth_sint: |
37660 | 2476 |
fixes w :: "'a::len word" |
65268 | 2477 |
defines "l \<equiv> len_of TYPE('a)" |
37660 | 2478 |
shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))" |
2479 |
unfolding sint_uint l_def |
|
65328 | 2480 |
by (auto simp: nth_sbintr word_test_bit_def [symmetric]) |
37660 | 2481 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2482 |
lemma word_lsb_numeral [simp]: |
65268 | 2483 |
"lsb (numeral bin :: 'a::len word) \<longleftrightarrow> bin_last (numeral bin)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2484 |
unfolding word_lsb_alt test_bit_numeral by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2485 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2486 |
lemma word_lsb_neg_numeral [simp]: |
65268 | 2487 |
"lsb (- numeral bin :: 'a::len word) \<longleftrightarrow> bin_last (- numeral bin)" |
65328 | 2488 |
by (simp add: word_lsb_alt) |
2489 |
||
2490 |
lemma set_bit_word_of_int: "set_bit (word_of_int x) n b = word_of_int (bin_sc n b x)" |
|
46173
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2491 |
unfolding word_set_bit_def |
65328 | 2492 |
by (rule word_eqI)(simp add: word_size bin_nth_sc_gen word_ubin.eq_norm nth_bintr) |
46173
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2493 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2494 |
lemma word_set_numeral [simp]: |
65268 | 2495 |
"set_bit (numeral bin::'a::len0 word) n b = |
54847
d6cf9a5b9be9
prefer plain bool over dedicated type for binary digits
haftmann
parents:
54743
diff
changeset
|
2496 |
word_of_int (bin_sc n b (numeral bin))" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2497 |
unfolding word_numeral_alt by (rule set_bit_word_of_int) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2498 |
|
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2499 |
lemma word_set_neg_numeral [simp]: |
65268 | 2500 |
"set_bit (- numeral bin::'a::len0 word) n b = |
54847
d6cf9a5b9be9
prefer plain bool over dedicated type for binary digits
haftmann
parents:
54743
diff
changeset
|
2501 |
word_of_int (bin_sc n b (- numeral bin))" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2502 |
unfolding word_neg_numeral_alt by (rule set_bit_word_of_int) |
46173
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2503 |
|
65328 | 2504 |
lemma word_set_bit_0 [simp]: "set_bit 0 n b = word_of_int (bin_sc n b 0)" |
46173
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2505 |
unfolding word_0_wi by (rule set_bit_word_of_int) |
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2506 |
|
65328 | 2507 |
lemma word_set_bit_1 [simp]: "set_bit 1 n b = word_of_int (bin_sc n b 1)" |
46173
5cc700033194
add simp rules for set_bit and msb applied to 0 and 1
huffman
parents:
46172
diff
changeset
|
2508 |
unfolding word_1_wi by (rule set_bit_word_of_int) |
37660 | 2509 |
|
65328 | 2510 |
lemma setBit_no [simp]: "setBit (numeral bin) n = word_of_int (bin_sc n True (numeral bin))" |
45805 | 2511 |
by (simp add: setBit_def) |
2512 |
||
2513 |
lemma clearBit_no [simp]: |
|
54847
d6cf9a5b9be9
prefer plain bool over dedicated type for binary digits
haftmann
parents:
54743
diff
changeset
|
2514 |
"clearBit (numeral bin) n = word_of_int (bin_sc n False (numeral bin))" |
45805 | 2515 |
by (simp add: clearBit_def) |
37660 | 2516 |
|
65328 | 2517 |
lemma to_bl_n1: "to_bl (-1::'a::len0 word) = replicate (len_of TYPE('a)) True" |
37660 | 2518 |
apply (rule word_bl.Abs_inverse') |
2519 |
apply simp |
|
2520 |
apply (rule word_eqI) |
|
45805 | 2521 |
apply (clarsimp simp add: word_size) |
37660 | 2522 |
apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size) |
2523 |
done |
|
2524 |
||
45805 | 2525 |
lemma word_msb_n1 [simp]: "msb (-1::'a::len word)" |
41550 | 2526 |
unfolding word_msb_alt to_bl_n1 by simp |
37660 | 2527 |
|
65328 | 2528 |
lemma word_set_nth_iff: "set_bit w n b = w \<longleftrightarrow> w !! n = b \<or> n \<ge> size w" |
2529 |
for w :: "'a::len0 word" |
|
37660 | 2530 |
apply (rule iffI) |
2531 |
apply (rule disjCI) |
|
2532 |
apply (drule word_eqD) |
|
2533 |
apply (erule sym [THEN trans]) |
|
2534 |
apply (simp add: test_bit_set) |
|
2535 |
apply (erule disjE) |
|
2536 |
apply clarsimp |
|
2537 |
apply (rule word_eqI) |
|
2538 |
apply (clarsimp simp add : test_bit_set_gen) |
|
2539 |
apply (drule test_bit_size) |
|
2540 |
apply force |
|
2541 |
done |
|
2542 |
||
65328 | 2543 |
lemma test_bit_2p: "(word_of_int (2 ^ n)::'a::len word) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a)" |
2544 |
by (auto simp: word_test_bit_def word_ubin.eq_norm nth_bintr nth_2p_bin) |
|
2545 |
||
2546 |
lemma nth_w2p: "((2::'a::len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a::len)" |
|
2547 |
by (simp add: test_bit_2p [symmetric] word_of_int [symmetric]) |
|
2548 |
||
2549 |
lemma uint_2p: "(0::'a::len word) < 2 ^ n \<Longrightarrow> uint (2 ^ n::'a::len word) = 2 ^ n" |
|
37660 | 2550 |
apply (unfold word_arith_power_alt) |
65268 | 2551 |
apply (case_tac "len_of TYPE('a)") |
37660 | 2552 |
apply clarsimp |
2553 |
apply (case_tac "nat") |
|
2554 |
apply clarsimp |
|
2555 |
apply (case_tac "n") |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2556 |
apply clarsimp |
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2557 |
apply clarsimp |
37660 | 2558 |
apply (drule word_gt_0 [THEN iffD1]) |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2559 |
apply (safe intro!: word_eqI) |
65328 | 2560 |
apply (auto simp add: nth_2p_bin) |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2561 |
apply (erule notE) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2562 |
apply (simp (no_asm_use) add: uint_word_of_int word_size) |
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54225
diff
changeset
|
2563 |
apply (subst mod_pos_pos_trivial) |
65328 | 2564 |
apply simp |
2565 |
apply (rule power_strict_increasing) |
|
2566 |
apply simp_all |
|
37660 | 2567 |
done |
2568 |
||
65268 | 2569 |
lemma word_of_int_2p: "(word_of_int (2 ^ n) :: 'a::len word) = 2 ^ n" |
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
2570 |
by (induct n) (simp_all add: wi_hom_syms) |
37660 | 2571 |
|
65328 | 2572 |
lemma bang_is_le: "x !! m \<Longrightarrow> 2 ^ m \<le> x" |
2573 |
for x :: "'a::len word" |
|
65268 | 2574 |
apply (rule xtr3) |
65328 | 2575 |
apply (rule_tac [2] y = "x" in le_word_or2) |
37660 | 2576 |
apply (rule word_eqI) |
2577 |
apply (auto simp add: word_ao_nth nth_w2p word_size) |
|
2578 |
done |
|
2579 |
||
65328 | 2580 |
lemma word_clr_le: "w \<ge> set_bit w n False" |
2581 |
for w :: "'a::len0 word" |
|
37660 | 2582 |
apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm) |
2583 |
apply (rule order_trans) |
|
2584 |
apply (rule bintr_bin_clr_le) |
|
2585 |
apply simp |
|
2586 |
done |
|
2587 |
||
65328 | 2588 |
lemma word_set_ge: "w \<le> set_bit w n True" |
2589 |
for w :: "'a::len word" |
|
37660 | 2590 |
apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm) |
2591 |
apply (rule order_trans [OF _ bintr_bin_set_ge]) |
|
2592 |
apply simp |
|
2593 |
done |
|
2594 |
||
2595 |
||
61799 | 2596 |
subsection \<open>Shifting, Rotating, and Splitting Words\<close> |
37660 | 2597 |
|
54847
d6cf9a5b9be9
prefer plain bool over dedicated type for binary digits
haftmann
parents:
54743
diff
changeset
|
2598 |
lemma shiftl1_wi [simp]: "shiftl1 (word_of_int w) = word_of_int (w BIT False)" |
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2599 |
unfolding shiftl1_def |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2600 |
apply (simp add: word_ubin.norm_eq_iff [symmetric] word_ubin.eq_norm) |
37660 | 2601 |
apply (subst refl [THEN bintrunc_BIT_I, symmetric]) |
2602 |
apply (subst bintrunc_bintrunc_min) |
|
2603 |
apply simp |
|
2604 |
done |
|
2605 |
||
65328 | 2606 |
lemma shiftl1_numeral [simp]: "shiftl1 (numeral w) = numeral (Num.Bit0 w)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2607 |
unfolding word_numeral_alt shiftl1_wi by simp |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2608 |
|
65328 | 2609 |
lemma shiftl1_neg_numeral [simp]: "shiftl1 (- numeral w) = - numeral (Num.Bit0 w)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2610 |
unfolding word_neg_numeral_alt shiftl1_wi by simp |
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2611 |
|
37660 | 2612 |
lemma shiftl1_0 [simp] : "shiftl1 0 = 0" |
65328 | 2613 |
by (simp add: shiftl1_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
|
2614 |
|
54847
d6cf9a5b9be9
prefer plain bool over dedicated type for binary digits
haftmann
parents:
54743
diff
changeset
|
2615 |
lemma shiftl1_def_u: "shiftl1 w = word_of_int (uint w BIT False)" |
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2616 |
by (simp only: shiftl1_def) (* FIXME: duplicate *) |
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2617 |
|
54847
d6cf9a5b9be9
prefer plain bool over dedicated type for binary digits
haftmann
parents:
54743
diff
changeset
|
2618 |
lemma shiftl1_def_s: "shiftl1 w = word_of_int (sint w BIT False)" |
65328 | 2619 |
by (simp add: shiftl1_def Bit_B0 wi_hom_syms) |
37660 | 2620 |
|
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
2621 |
lemma shiftr1_0 [simp]: "shiftr1 0 = 0" |
65328 | 2622 |
by (simp add: shiftr1_def) |
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
2623 |
|
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
2624 |
lemma sshiftr1_0 [simp]: "sshiftr1 0 = 0" |
65328 | 2625 |
by (simp add: sshiftr1_def) |
2626 |
||
2627 |
lemma sshiftr1_n1 [simp]: "sshiftr1 (- 1) = - 1" |
|
2628 |
by (simp add: sshiftr1_def) |
|
2629 |
||
2630 |
lemma shiftl_0 [simp]: "(0::'a::len0 word) << n = 0" |
|
2631 |
by (induct n) (auto simp: shiftl_def) |
|
2632 |
||
2633 |
lemma shiftr_0 [simp]: "(0::'a::len0 word) >> n = 0" |
|
2634 |
by (induct n) (auto simp: shiftr_def) |
|
2635 |
||
2636 |
lemma sshiftr_0 [simp]: "0 >>> n = 0" |
|
2637 |
by (induct n) (auto simp: sshiftr_def) |
|
2638 |
||
2639 |
lemma sshiftr_n1 [simp]: "-1 >>> n = -1" |
|
2640 |
by (induct n) (auto simp: sshiftr_def) |
|
2641 |
||
2642 |
lemma nth_shiftl1: "shiftl1 w !! n \<longleftrightarrow> n < size w \<and> n > 0 \<and> w !! (n - 1)" |
|
37660 | 2643 |
apply (unfold shiftl1_def word_test_bit_def) |
2644 |
apply (simp add: nth_bintr word_ubin.eq_norm word_size) |
|
2645 |
apply (cases n) |
|
2646 |
apply auto |
|
2647 |
done |
|
2648 |
||
65328 | 2649 |
lemma nth_shiftl': "(w << m) !! n \<longleftrightarrow> n < size w \<and> n >= m \<and> w !! (n - m)" |
2650 |
for w :: "'a::len0 word" |
|
37660 | 2651 |
apply (unfold shiftl_def) |
65328 | 2652 |
apply (induct m arbitrary: n) |
37660 | 2653 |
apply (force elim!: test_bit_size) |
2654 |
apply (clarsimp simp add : nth_shiftl1 word_size) |
|
2655 |
apply arith |
|
2656 |
done |
|
2657 |
||
65268 | 2658 |
lemmas nth_shiftl = nth_shiftl' [unfolded word_size] |
37660 | 2659 |
|
2660 |
lemma nth_shiftr1: "shiftr1 w !! n = w !! Suc n" |
|
2661 |
apply (unfold shiftr1_def word_test_bit_def) |
|
2662 |
apply (simp add: nth_bintr word_ubin.eq_norm) |
|
2663 |
apply safe |
|
2664 |
apply (drule bin_nth.Suc [THEN iffD2, THEN bin_nth_uint_imp]) |
|
2665 |
apply simp |
|
2666 |
done |
|
2667 |
||
65328 | 2668 |
lemma nth_shiftr: "(w >> m) !! n = w !! (n + m)" |
2669 |
for w :: "'a::len0 word" |
|
37660 | 2670 |
apply (unfold shiftr_def) |
65328 | 2671 |
apply (induct "m" arbitrary: n) |
2672 |
apply (auto simp add: nth_shiftr1) |
|
37660 | 2673 |
done |
65268 | 2674 |
|
37660 | 2675 |
(* see paper page 10, (1), (2), shiftr1_def is of the form of (1), |
2676 |
where f (ie bin_rest) takes normal arguments to normal results, |
|
2677 |
thus we get (2) from (1) *) |
|
2678 |
||
65268 | 2679 |
lemma uint_shiftr1: "uint (shiftr1 w) = bin_rest (uint w)" |
37660 | 2680 |
apply (unfold shiftr1_def word_ubin.eq_norm bin_rest_trunc_i) |
2681 |
apply (subst bintr_uint [symmetric, OF order_refl]) |
|
2682 |
apply (simp only : bintrunc_bintrunc_l) |
|
65268 | 2683 |
apply simp |
37660 | 2684 |
done |
2685 |
||
65328 | 2686 |
lemma nth_sshiftr1: "sshiftr1 w !! n = (if n = size w - 1 then w !! n else w !! Suc n)" |
37660 | 2687 |
apply (unfold sshiftr1_def word_test_bit_def) |
65328 | 2688 |
apply (simp add: nth_bintr word_ubin.eq_norm bin_nth.Suc [symmetric] word_size |
2689 |
del: bin_nth.simps) |
|
37660 | 2690 |
apply (simp add: nth_bintr uint_sint del : bin_nth.simps) |
2691 |
apply (auto simp add: bin_nth_sint) |
|
2692 |
done |
|
2693 |
||
65268 | 2694 |
lemma nth_sshiftr [rule_format] : |
65328 | 2695 |
"\<forall>n. sshiftr w m !! n = |
2696 |
(n < size w \<and> (if n + m \<ge> size w then w !! (size w - 1) else w !! (n + m)))" |
|
37660 | 2697 |
apply (unfold sshiftr_def) |
65328 | 2698 |
apply (induct_tac m) |
37660 | 2699 |
apply (simp add: test_bit_bl) |
2700 |
apply (clarsimp simp add: nth_sshiftr1 word_size) |
|
2701 |
apply safe |
|
2702 |
apply arith |
|
2703 |
apply arith |
|
2704 |
apply (erule thin_rl) |
|
2705 |
apply (case_tac n) |
|
2706 |
apply safe |
|
2707 |
apply simp |
|
2708 |
apply simp |
|
2709 |
apply (erule thin_rl) |
|
2710 |
apply (case_tac n) |
|
2711 |
apply safe |
|
2712 |
apply simp |
|
2713 |
apply simp |
|
2714 |
apply arith+ |
|
2715 |
done |
|
65268 | 2716 |
|
37660 | 2717 |
lemma shiftr1_div_2: "uint (shiftr1 w) = uint w div 2" |
45529
0e1037d4e049
remove redundant lemmas bin_last_mod and bin_rest_div, use bin_last_def and bin_rest_def instead
huffman
parents:
45528
diff
changeset
|
2718 |
apply (unfold shiftr1_def bin_rest_def) |
37660 | 2719 |
apply (rule word_uint.Abs_inverse) |
2720 |
apply (simp add: uints_num pos_imp_zdiv_nonneg_iff) |
|
2721 |
apply (rule xtr7) |
|
2722 |
prefer 2 |
|
2723 |
apply (rule zdiv_le_dividend) |
|
2724 |
apply auto |
|
2725 |
done |
|
2726 |
||
2727 |
lemma sshiftr1_div_2: "sint (sshiftr1 w) = sint w div 2" |
|
45529
0e1037d4e049
remove redundant lemmas bin_last_mod and bin_rest_div, use bin_last_def and bin_rest_def instead
huffman
parents:
45528
diff
changeset
|
2728 |
apply (unfold sshiftr1_def bin_rest_def [symmetric]) |
37660 | 2729 |
apply (simp add: word_sbin.eq_norm) |
2730 |
apply (rule trans) |
|
2731 |
defer |
|
2732 |
apply (subst word_sbin.norm_Rep [symmetric]) |
|
2733 |
apply (rule refl) |
|
2734 |
apply (subst word_sbin.norm_Rep [symmetric]) |
|
2735 |
apply (unfold One_nat_def) |
|
2736 |
apply (rule sbintrunc_rest) |
|
2737 |
done |
|
2738 |
||
2739 |
lemma shiftr_div_2n: "uint (shiftr w n) = uint w div 2 ^ n" |
|
2740 |
apply (unfold shiftr_def) |
|
65328 | 2741 |
apply (induct n) |
37660 | 2742 |
apply simp |
65328 | 2743 |
apply (simp add: shiftr1_div_2 mult.commute zdiv_zmult2_eq [symmetric]) |
37660 | 2744 |
done |
2745 |
||
2746 |
lemma sshiftr_div_2n: "sint (sshiftr w n) = sint w div 2 ^ n" |
|
2747 |
apply (unfold sshiftr_def) |
|
65328 | 2748 |
apply (induct n) |
37660 | 2749 |
apply simp |
65328 | 2750 |
apply (simp add: sshiftr1_div_2 mult.commute zdiv_zmult2_eq [symmetric]) |
37660 | 2751 |
done |
2752 |
||
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
2753 |
|
61799 | 2754 |
subsubsection \<open>shift functions in terms of lists of bools\<close> |
37660 | 2755 |
|
65268 | 2756 |
lemmas bshiftr1_numeral [simp] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2757 |
bshiftr1_def [where w="numeral w", unfolded to_bl_numeral] for w |
37660 | 2758 |
|
2759 |
lemma bshiftr1_bl: "to_bl (bshiftr1 b w) = b # butlast (to_bl w)" |
|
2760 |
unfolding bshiftr1_def by (rule word_bl.Abs_inverse) simp |
|
2761 |
||
2762 |
lemma shiftl1_of_bl: "shiftl1 (of_bl bl) = of_bl (bl @ [False])" |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2763 |
by (simp add: of_bl_def bl_to_bin_append) |
37660 | 2764 |
|
2765 |
lemma shiftl1_bl: "shiftl1 (w::'a::len0 word) = of_bl (to_bl w @ [False])" |
|
2766 |
proof - |
|
65328 | 2767 |
have "shiftl1 w = shiftl1 (of_bl (to_bl w))" |
2768 |
by simp |
|
2769 |
also have "\<dots> = of_bl (to_bl w @ [False])" |
|
2770 |
by (rule shiftl1_of_bl) |
|
37660 | 2771 |
finally show ?thesis . |
2772 |
qed |
|
2773 |
||
65328 | 2774 |
lemma bl_shiftl1: "to_bl (shiftl1 w) = tl (to_bl w) @ [False]" |
2775 |
for w :: "'a::len word" |
|
2776 |
by (simp add: shiftl1_bl word_rep_drop drop_Suc drop_Cons') (fast intro!: Suc_leI) |
|
37660 | 2777 |
|
45807 | 2778 |
(* Generalized version of bl_shiftl1. Maybe this one should replace it? *) |
65328 | 2779 |
lemma bl_shiftl1': "to_bl (shiftl1 w) = tl (to_bl w @ [False])" |
2780 |
by (simp add: shiftl1_bl word_rep_drop drop_Suc del: drop_append) |
|
45807 | 2781 |
|
37660 | 2782 |
lemma shiftr1_bl: "shiftr1 w = of_bl (butlast (to_bl w))" |
2783 |
apply (unfold shiftr1_def uint_bl of_bl_def) |
|
2784 |
apply (simp add: butlast_rest_bin word_size) |
|
2785 |
apply (simp add: bin_rest_trunc [symmetric, unfolded One_nat_def]) |
|
2786 |
done |
|
2787 |
||
65328 | 2788 |
lemma bl_shiftr1: "to_bl (shiftr1 w) = False # butlast (to_bl w)" |
2789 |
for w :: "'a::len word" |
|
2790 |
by (simp add: shiftr1_bl word_rep_drop len_gt_0 [THEN Suc_leI]) |
|
37660 | 2791 |
|
45807 | 2792 |
(* Generalized version of bl_shiftr1. Maybe this one should replace it? *) |
65328 | 2793 |
lemma bl_shiftr1': "to_bl (shiftr1 w) = butlast (False # to_bl w)" |
45807 | 2794 |
apply (rule word_bl.Abs_inverse') |
65328 | 2795 |
apply (simp del: butlast.simps) |
45807 | 2796 |
apply (simp add: shiftr1_bl of_bl_def) |
2797 |
done |
|
2798 |
||
65328 | 2799 |
lemma shiftl1_rev: "shiftl1 w = word_reverse (shiftr1 (word_reverse w))" |
37660 | 2800 |
apply (unfold word_reverse_def) |
2801 |
apply (rule word_bl.Rep_inverse' [symmetric]) |
|
45807 | 2802 |
apply (simp add: bl_shiftl1' bl_shiftr1' word_bl.Abs_inverse) |
37660 | 2803 |
apply (cases "to_bl w") |
2804 |
apply auto |
|
2805 |
done |
|
2806 |
||
65328 | 2807 |
lemma shiftl_rev: "shiftl w n = word_reverse (shiftr (word_reverse w) n)" |
2808 |
by (induct n) (auto simp add: shiftl_def shiftr_def shiftl1_rev) |
|
37660 | 2809 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
2810 |
lemma rev_shiftl: "word_reverse w << n = word_reverse (w >> n)" |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
2811 |
by (simp add: shiftl_rev) |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
2812 |
|
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
2813 |
lemma shiftr_rev: "w >> n = word_reverse (word_reverse w << n)" |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
2814 |
by (simp add: rev_shiftl) |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
2815 |
|
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
2816 |
lemma rev_shiftr: "word_reverse w >> n = word_reverse (w << n)" |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
2817 |
by (simp add: shiftr_rev) |
37660 | 2818 |
|
65328 | 2819 |
lemma bl_sshiftr1: "to_bl (sshiftr1 w) = hd (to_bl w) # butlast (to_bl w)" |
2820 |
for w :: "'a::len word" |
|
37660 | 2821 |
apply (unfold sshiftr1_def uint_bl word_size) |
2822 |
apply (simp add: butlast_rest_bin word_ubin.eq_norm) |
|
2823 |
apply (simp add: sint_uint) |
|
2824 |
apply (rule nth_equalityI) |
|
2825 |
apply clarsimp |
|
2826 |
apply clarsimp |
|
2827 |
apply (case_tac i) |
|
2828 |
apply (simp_all add: hd_conv_nth length_0_conv [symmetric] |
|
65328 | 2829 |
nth_bin_to_bl bin_nth.Suc [symmetric] nth_sbintr |
2830 |
del: bin_nth.Suc) |
|
37660 | 2831 |
apply force |
2832 |
apply (rule impI) |
|
2833 |
apply (rule_tac f = "bin_nth (uint w)" in arg_cong) |
|
2834 |
apply simp |
|
2835 |
done |
|
2836 |
||
65328 | 2837 |
lemma drop_shiftr: "drop n (to_bl (w >> n)) = take (size w - n) (to_bl w)" |
2838 |
for w :: "'a::len word" |
|
37660 | 2839 |
apply (unfold shiftr_def) |
2840 |
apply (induct n) |
|
2841 |
prefer 2 |
|
2842 |
apply (simp add: drop_Suc bl_shiftr1 butlast_drop [symmetric]) |
|
2843 |
apply (rule butlast_take [THEN trans]) |
|
65328 | 2844 |
apply (auto simp: word_size) |
37660 | 2845 |
done |
2846 |
||
65328 | 2847 |
lemma drop_sshiftr: "drop n (to_bl (w >>> n)) = take (size w - n) (to_bl w)" |
2848 |
for w :: "'a::len word" |
|
37660 | 2849 |
apply (unfold sshiftr_def) |
2850 |
apply (induct n) |
|
2851 |
prefer 2 |
|
2852 |
apply (simp add: drop_Suc bl_sshiftr1 butlast_drop [symmetric]) |
|
2853 |
apply (rule butlast_take [THEN trans]) |
|
65328 | 2854 |
apply (auto simp: word_size) |
37660 | 2855 |
done |
2856 |
||
65328 | 2857 |
lemma take_shiftr: "n \<le> size w \<Longrightarrow> take n (to_bl (w >> n)) = replicate n False" |
37660 | 2858 |
apply (unfold shiftr_def) |
2859 |
apply (induct n) |
|
2860 |
prefer 2 |
|
45807 | 2861 |
apply (simp add: bl_shiftr1' length_0_conv [symmetric] word_size) |
37660 | 2862 |
apply (rule take_butlast [THEN trans]) |
65328 | 2863 |
apply (auto simp: word_size) |
37660 | 2864 |
done |
2865 |
||
2866 |
lemma take_sshiftr' [rule_format] : |
|
65328 | 2867 |
"n \<le> size w \<longrightarrow> hd (to_bl (w >>> n)) = hd (to_bl w) \<and> |
65268 | 2868 |
take n (to_bl (w >>> n)) = replicate n (hd (to_bl w))" |
65328 | 2869 |
for w :: "'a::len word" |
37660 | 2870 |
apply (unfold sshiftr_def) |
2871 |
apply (induct n) |
|
2872 |
prefer 2 |
|
2873 |
apply (simp add: bl_sshiftr1) |
|
2874 |
apply (rule impI) |
|
2875 |
apply (rule take_butlast [THEN trans]) |
|
65328 | 2876 |
apply (auto simp: word_size) |
37660 | 2877 |
done |
2878 |
||
45604 | 2879 |
lemmas hd_sshiftr = take_sshiftr' [THEN conjunct1] |
2880 |
lemmas take_sshiftr = take_sshiftr' [THEN conjunct2] |
|
37660 | 2881 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
2882 |
lemma atd_lem: "take n xs = t \<Longrightarrow> drop n xs = d \<Longrightarrow> xs = t @ d" |
37660 | 2883 |
by (auto intro: append_take_drop_id [symmetric]) |
2884 |
||
2885 |
lemmas bl_shiftr = atd_lem [OF take_shiftr drop_shiftr] |
|
2886 |
lemmas bl_sshiftr = atd_lem [OF take_sshiftr drop_sshiftr] |
|
2887 |
||
2888 |
lemma shiftl_of_bl: "of_bl bl << n = of_bl (bl @ replicate n False)" |
|
65328 | 2889 |
by (induct n) (auto simp: shiftl_def shiftl1_of_bl replicate_app_Cons_same) |
2890 |
||
2891 |
lemma shiftl_bl: "w << n = of_bl (to_bl w @ replicate n False)" |
|
2892 |
for w :: "'a::len0 word" |
|
37660 | 2893 |
proof - |
65328 | 2894 |
have "w << n = of_bl (to_bl w) << n" |
2895 |
by simp |
|
2896 |
also have "\<dots> = of_bl (to_bl w @ replicate n False)" |
|
2897 |
by (rule shiftl_of_bl) |
|
37660 | 2898 |
finally show ?thesis . |
2899 |
qed |
|
2900 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2901 |
lemmas shiftl_numeral [simp] = shiftl_def [where w="numeral w"] for w |
37660 | 2902 |
|
65328 | 2903 |
lemma bl_shiftl: "to_bl (w << n) = drop n (to_bl w) @ replicate (min (size w) n) False" |
37660 | 2904 |
by (simp add: shiftl_bl word_rep_drop word_size) |
2905 |
||
65328 | 2906 |
lemma shiftl_zero_size: "size x \<le> n \<Longrightarrow> x << n = 0" |
2907 |
for x :: "'a::len0 word" |
|
37660 | 2908 |
apply (unfold word_size) |
2909 |
apply (rule word_eqI) |
|
2910 |
apply (clarsimp simp add: shiftl_bl word_size test_bit_of_bl nth_append) |
|
2911 |
done |
|
2912 |
||
65268 | 2913 |
(* note - the following results use 'a::len word < number_ring *) |
2914 |
||
65328 | 2915 |
lemma shiftl1_2t: "shiftl1 w = 2 * w" |
2916 |
for w :: "'a::len word" |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2917 |
by (simp add: shiftl1_def Bit_def wi_hom_mult [symmetric]) |
37660 | 2918 |
|
65328 | 2919 |
lemma shiftl1_p: "shiftl1 w = w + w" |
2920 |
for w :: "'a::len word" |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
2921 |
by (simp add: shiftl1_2t) |
37660 | 2922 |
|
65328 | 2923 |
lemma shiftl_t2n: "shiftl w n = 2 ^ n * w" |
2924 |
for w :: "'a::len word" |
|
2925 |
by (induct n) (auto simp: shiftl_def shiftl1_2t) |
|
37660 | 2926 |
|
2927 |
lemma shiftr1_bintr [simp]: |
|
65268 | 2928 |
"(shiftr1 (numeral w) :: 'a::len0 word) = |
2929 |
word_of_int (bin_rest (bintrunc (len_of TYPE('a)) (numeral w)))" |
|
65328 | 2930 |
unfolding shiftr1_def word_numeral_alt by (simp add: word_ubin.eq_norm) |
46962
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
2931 |
|
5bdcdb28be83
make more word theorems respect int/bin distinction
huffman
parents:
46656
diff
changeset
|
2932 |
lemma sshiftr1_sbintr [simp]: |
65268 | 2933 |
"(sshiftr1 (numeral w) :: 'a::len word) = |
2934 |
word_of_int (bin_rest (sbintrunc (len_of TYPE('a) - 1) (numeral w)))" |
|
65328 | 2935 |
unfolding sshiftr1_def word_numeral_alt by (simp add: word_sbin.eq_norm) |
37660 | 2936 |
|
46057 | 2937 |
lemma shiftr_no [simp]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2938 |
(* FIXME: neg_numeral *) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2939 |
"(numeral w::'a::len0 word) >> n = word_of_int |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2940 |
((bin_rest ^^ n) (bintrunc (len_of TYPE('a)) (numeral w)))" |
65328 | 2941 |
by (rule word_eqI) (auto simp: nth_shiftr nth_rest_power_bin nth_bintr word_size) |
37660 | 2942 |
|
46057 | 2943 |
lemma sshiftr_no [simp]: |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2944 |
(* FIXME: neg_numeral *) |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2945 |
"(numeral w::'a::len word) >>> n = word_of_int |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
2946 |
((bin_rest ^^ n) (sbintrunc (len_of TYPE('a) - 1) (numeral w)))" |
37660 | 2947 |
apply (rule word_eqI) |
2948 |
apply (auto simp: nth_sshiftr nth_rest_power_bin nth_sbintr word_size) |
|
2949 |
apply (subgoal_tac "na + n = len_of TYPE('a) - Suc 0", simp, simp)+ |
|
2950 |
done |
|
2951 |
||
45811 | 2952 |
lemma shiftr1_bl_of: |
2953 |
"length bl \<le> len_of TYPE('a) \<Longrightarrow> |
|
2954 |
shiftr1 (of_bl bl::'a::len0 word) = of_bl (butlast bl)" |
|
65328 | 2955 |
by (clarsimp simp: shiftr1_def of_bl_def butlast_rest_bl2bin word_ubin.eq_norm trunc_bl2bin) |
37660 | 2956 |
|
45811 | 2957 |
lemma shiftr_bl_of: |
2958 |
"length bl \<le> len_of TYPE('a) \<Longrightarrow> |
|
2959 |
(of_bl bl::'a::len0 word) >> n = of_bl (take (length bl - n) bl)" |
|
37660 | 2960 |
apply (unfold shiftr_def) |
2961 |
apply (induct n) |
|
2962 |
apply clarsimp |
|
2963 |
apply clarsimp |
|
2964 |
apply (subst shiftr1_bl_of) |
|
2965 |
apply simp |
|
2966 |
apply (simp add: butlast_take) |
|
2967 |
done |
|
2968 |
||
65328 | 2969 |
lemma shiftr_bl: "x >> n \<equiv> of_bl (take (len_of TYPE('a) - n) (to_bl x))" |
2970 |
for x :: "'a::len0 word" |
|
45811 | 2971 |
using shiftr_bl_of [where 'a='a, of "to_bl x"] by simp |
2972 |
||
65328 | 2973 |
lemma msb_shift: "msb w \<longleftrightarrow> (w >> (len_of TYPE('a) - 1)) \<noteq> 0" |
2974 |
for w :: "'a::len word" |
|
37660 | 2975 |
apply (unfold shiftr_bl word_msb_alt) |
2976 |
apply (simp add: word_size Suc_le_eq take_Suc) |
|
2977 |
apply (cases "hd (to_bl w)") |
|
65328 | 2978 |
apply (auto simp: word_1_bl of_bl_rep_False [where n=1 and bs="[]", simplified]) |
37660 | 2979 |
done |
2980 |
||
65328 | 2981 |
lemma zip_replicate: "n \<ge> length ys \<Longrightarrow> zip (replicate n x) ys = map (\<lambda>y. (x, y)) ys" |
2982 |
apply (induct ys arbitrary: n) |
|
2983 |
apply simp_all |
|
2984 |
apply (case_tac n) |
|
2985 |
apply simp_all |
|
57492
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
56979
diff
changeset
|
2986 |
done |
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
56979
diff
changeset
|
2987 |
|
37660 | 2988 |
lemma align_lem_or [rule_format] : |
65328 | 2989 |
"\<forall>x m. length x = n + m \<longrightarrow> length y = n + m \<longrightarrow> |
2990 |
drop m x = replicate n False \<longrightarrow> take m y = replicate m False \<longrightarrow> |
|
37660 | 2991 |
map2 op | x y = take m x @ drop m y" |
65328 | 2992 |
apply (induct y) |
37660 | 2993 |
apply force |
2994 |
apply clarsimp |
|
65328 | 2995 |
apply (case_tac x) |
2996 |
apply force |
|
2997 |
apply (case_tac m) |
|
2998 |
apply auto |
|
59807 | 2999 |
apply (drule_tac t="length xs" for xs in sym) |
65328 | 3000 |
apply (auto simp: map2_def zip_replicate o_def) |
37660 | 3001 |
done |
3002 |
||
3003 |
lemma align_lem_and [rule_format] : |
|
65328 | 3004 |
"\<forall>x m. length x = n + m \<longrightarrow> length y = n + m \<longrightarrow> |
3005 |
drop m x = replicate n False \<longrightarrow> take m y = replicate m False \<longrightarrow> |
|
3006 |
map2 op \<and> x y = replicate (n + m) False" |
|
3007 |
apply (induct y) |
|
37660 | 3008 |
apply force |
3009 |
apply clarsimp |
|
65328 | 3010 |
apply (case_tac x) |
3011 |
apply force |
|
3012 |
apply (case_tac m) |
|
3013 |
apply auto |
|
59807 | 3014 |
apply (drule_tac t="length xs" for xs in sym) |
65328 | 3015 |
apply (auto simp: map2_def zip_replicate o_def map_replicate_const) |
37660 | 3016 |
done |
3017 |
||
45811 | 3018 |
lemma aligned_bl_add_size [OF refl]: |
65328 | 3019 |
"size x - n = m \<Longrightarrow> n \<le> size x \<Longrightarrow> drop m (to_bl x) = replicate n False \<Longrightarrow> |
65268 | 3020 |
take m (to_bl y) = replicate m False \<Longrightarrow> |
37660 | 3021 |
to_bl (x + y) = take m (to_bl x) @ drop m (to_bl y)" |
3022 |
apply (subgoal_tac "x AND y = 0") |
|
3023 |
prefer 2 |
|
3024 |
apply (rule word_bl.Rep_eqD) |
|
45805 | 3025 |
apply (simp add: bl_word_and) |
37660 | 3026 |
apply (rule align_lem_and [THEN trans]) |
3027 |
apply (simp_all add: word_size)[5] |
|
3028 |
apply simp |
|
3029 |
apply (subst word_plus_and_or [symmetric]) |
|
3030 |
apply (simp add : bl_word_or) |
|
3031 |
apply (rule align_lem_or) |
|
3032 |
apply (simp_all add: word_size) |
|
3033 |
done |
|
3034 |
||
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
3035 |
|
61799 | 3036 |
subsubsection \<open>Mask\<close> |
37660 | 3037 |
|
65328 | 3038 |
lemma nth_mask [OF refl, simp]: "m = mask n \<Longrightarrow> test_bit m i \<longleftrightarrow> i < n \<and> i < size m" |
37660 | 3039 |
apply (unfold mask_def test_bit_bl) |
3040 |
apply (simp only: word_1_bl [symmetric] shiftl_of_bl) |
|
3041 |
apply (clarsimp simp add: word_size) |
|
46645
573aff6b9b0a
adapt lemma mask_lem to respect int/bin distinction
huffman
parents:
46618
diff
changeset
|
3042 |
apply (simp only: of_bl_def mask_lem word_of_int_hom_syms add_diff_cancel2) |
573aff6b9b0a
adapt lemma mask_lem to respect int/bin distinction
huffman
parents:
46618
diff
changeset
|
3043 |
apply (fold of_bl_def) |
37660 | 3044 |
apply (simp add: word_1_bl) |
3045 |
apply (rule test_bit_of_bl [THEN trans, unfolded test_bit_bl word_size]) |
|
3046 |
apply auto |
|
3047 |
done |
|
3048 |
||
3049 |
lemma mask_bl: "mask n = of_bl (replicate n True)" |
|
3050 |
by (auto simp add : test_bit_of_bl word_size intro: word_eqI) |
|
3051 |
||
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58061
diff
changeset
|
3052 |
lemma mask_bin: "mask n = word_of_int (bintrunc n (- 1))" |
37660 | 3053 |
by (auto simp add: nth_bintr word_size intro: word_eqI) |
3054 |
||
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
3055 |
lemma and_mask_bintr: "w AND mask n = word_of_int (bintrunc n (uint w))" |
37660 | 3056 |
apply (rule word_eqI) |
3057 |
apply (simp add: nth_bintr word_size word_ops_nth_size) |
|
3058 |
apply (auto simp add: test_bit_bin) |
|
3059 |
done |
|
3060 |
||
45811 | 3061 |
lemma and_mask_wi: "word_of_int i AND mask n = word_of_int (bintrunc n i)" |
46023
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset
|
3062 |
by (auto simp add: nth_bintr word_size word_ops_nth_size word_eq_iff) |
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset
|
3063 |
|
65328 | 3064 |
lemma and_mask_wi': |
3065 |
"word_of_int i AND mask n = (word_of_int (bintrunc (min LENGTH('a) n) i) :: 'a::len word)" |
|
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
3066 |
by (auto simp add: nth_bintr word_size word_ops_nth_size word_eq_iff) |
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
3067 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3068 |
lemma and_mask_no: "numeral i AND mask n = word_of_int (bintrunc n (numeral i))" |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3069 |
unfolding word_numeral_alt by (rule and_mask_wi) |
37660 | 3070 |
|
3071 |
lemma bl_and_mask': |
|
65268 | 3072 |
"to_bl (w AND mask n :: 'a::len word) = |
3073 |
replicate (len_of TYPE('a) - n) False @ |
|
37660 | 3074 |
drop (len_of TYPE('a) - n) (to_bl w)" |
3075 |
apply (rule nth_equalityI) |
|
3076 |
apply simp |
|
3077 |
apply (clarsimp simp add: to_bl_nth word_size) |
|
3078 |
apply (simp add: word_size word_ops_nth_size) |
|
3079 |
apply (auto simp add: word_size test_bit_bl nth_append nth_rev) |
|
3080 |
done |
|
3081 |
||
45811 | 3082 |
lemma and_mask_mod_2p: "w AND mask n = word_of_int (uint w mod 2 ^ n)" |
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
3083 |
by (simp only: and_mask_bintr bintrunc_mod2p) |
37660 | 3084 |
|
3085 |
lemma and_mask_lt_2p: "uint (w AND mask n) < 2 ^ n" |
|
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
3086 |
apply (simp add: and_mask_bintr word_ubin.eq_norm) |
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
3087 |
apply (simp add: bintrunc_mod2p) |
37660 | 3088 |
apply (rule xtr8) |
3089 |
prefer 2 |
|
3090 |
apply (rule pos_mod_bound) |
|
65328 | 3091 |
apply auto |
37660 | 3092 |
done |
3093 |
||
45811 | 3094 |
lemma eq_mod_iff: "0 < (n::int) \<Longrightarrow> b = b mod n \<longleftrightarrow> 0 \<le> b \<and> b < n" |
3095 |
by (simp add: int_mod_lem eq_sym_conv) |
|
37660 | 3096 |
|
61941 | 3097 |
lemma mask_eq_iff: "(w AND mask n) = w \<longleftrightarrow> uint w < 2 ^ n" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3098 |
apply (simp add: and_mask_bintr) |
37660 | 3099 |
apply (simp add: word_ubin.inverse_norm) |
3100 |
apply (simp add: eq_mod_iff bintrunc_mod2p min_def) |
|
3101 |
apply (fast intro!: lt2p_lem) |
|
3102 |
done |
|
3103 |
||
65328 | 3104 |
lemma and_mask_dvd: "2 ^ n dvd uint w \<longleftrightarrow> w AND mask n = 0" |
37660 | 3105 |
apply (simp add: dvd_eq_mod_eq_0 and_mask_mod_2p) |
65328 | 3106 |
apply (simp add: word_uint.norm_eq_iff [symmetric] word_of_int_homs del: word_of_int_0) |
37660 | 3107 |
apply (subst word_uint.norm_Rep [symmetric]) |
3108 |
apply (simp only: bintrunc_bintrunc_min bintrunc_mod2p [symmetric] min_def) |
|
3109 |
apply auto |
|
3110 |
done |
|
3111 |
||
65328 | 3112 |
lemma and_mask_dvd_nat: "2 ^ n dvd unat w \<longleftrightarrow> w AND mask n = 0" |
37660 | 3113 |
apply (unfold unat_def) |
3114 |
apply (rule trans [OF _ and_mask_dvd]) |
|
65268 | 3115 |
apply (unfold dvd_def) |
3116 |
apply auto |
|
65328 | 3117 |
apply (drule uint_ge_0 [THEN nat_int.Abs_inverse' [simplified], symmetric]) |
3118 |
apply simp |
|
3119 |
apply (simp add: nat_mult_distrib nat_power_eq) |
|
37660 | 3120 |
done |
3121 |
||
65328 | 3122 |
lemma word_2p_lem: "n < size w \<Longrightarrow> w < 2 ^ n = (uint w < 2 ^ n)" |
3123 |
for w :: "'a::len word" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3124 |
apply (unfold word_size word_less_alt word_numeral_alt) |
65328 | 3125 |
apply (auto simp add: word_of_int_power_hom word_uint.eq_norm mod_pos_pos_trivial |
3126 |
simp del: word_of_int_numeral) |
|
37660 | 3127 |
done |
3128 |
||
65328 | 3129 |
lemma less_mask_eq: "x < 2 ^ n \<Longrightarrow> x AND mask n = x" |
3130 |
for x :: "'a::len word" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3131 |
apply (unfold word_less_alt word_numeral_alt) |
65328 | 3132 |
apply (clarsimp simp add: and_mask_mod_2p word_of_int_power_hom word_uint.eq_norm |
3133 |
simp del: word_of_int_numeral) |
|
37660 | 3134 |
apply (drule xtr8 [rotated]) |
65328 | 3135 |
apply (rule int_mod_le) |
3136 |
apply (auto simp add : mod_pos_pos_trivial) |
|
37660 | 3137 |
done |
3138 |
||
45604 | 3139 |
lemmas mask_eq_iff_w2p = trans [OF mask_eq_iff word_2p_lem [symmetric]] |
3140 |
||
3141 |
lemmas and_mask_less' = iffD2 [OF word_2p_lem and_mask_lt_2p, simplified word_size] |
|
37660 | 3142 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3143 |
lemma and_mask_less_size: "n < size x \<Longrightarrow> x AND mask n < 2^n" |
37660 | 3144 |
unfolding word_size by (erule and_mask_less') |
3145 |
||
65328 | 3146 |
lemma word_mod_2p_is_mask [OF refl]: "c = 2 ^ n \<Longrightarrow> c > 0 \<Longrightarrow> x mod c = x AND mask n" |
3147 |
for c x :: "'a::len word" |
|
3148 |
by (auto simp: word_mod_def uint_2p and_mask_mod_2p) |
|
37660 | 3149 |
|
3150 |
lemma mask_eqs: |
|
3151 |
"(a AND mask n) + b AND mask n = a + b AND mask n" |
|
3152 |
"a + (b AND mask n) AND mask n = a + b AND mask n" |
|
3153 |
"(a AND mask n) - b AND mask n = a - b AND mask n" |
|
3154 |
"a - (b AND mask n) AND mask n = a - b AND mask n" |
|
3155 |
"a * (b AND mask n) AND mask n = a * b AND mask n" |
|
3156 |
"(b AND mask n) * a AND mask n = b * a AND mask n" |
|
3157 |
"(a AND mask n) + (b AND mask n) AND mask n = a + b AND mask n" |
|
3158 |
"(a AND mask n) - (b AND mask n) AND mask n = a - b AND mask n" |
|
3159 |
"(a AND mask n) * (b AND mask n) AND mask n = a * b AND mask n" |
|
3160 |
"- (a AND mask n) AND mask n = - a AND mask n" |
|
3161 |
"word_succ (a AND mask n) AND mask n = word_succ a AND mask n" |
|
3162 |
"word_pred (a AND mask n) AND mask n = word_pred a AND mask n" |
|
3163 |
using word_of_int_Ex [where x=a] word_of_int_Ex [where x=b] |
|
65328 | 3164 |
by (auto simp: and_mask_wi' word_of_int_homs word.abs_eq_iff bintrunc_mod2p mod_simps) |
3165 |
||
3166 |
lemma mask_power_eq: "(x AND mask n) ^ k AND mask n = x ^ k AND mask n" |
|
37660 | 3167 |
using word_of_int_Ex [where x=x] |
65328 | 3168 |
by (auto simp: and_mask_wi' word_of_int_power_hom word.abs_eq_iff bintrunc_mod2p mod_simps) |
37660 | 3169 |
|
3170 |
||
61799 | 3171 |
subsubsection \<open>Revcast\<close> |
37660 | 3172 |
|
3173 |
lemmas revcast_def' = revcast_def [simplified] |
|
3174 |
lemmas revcast_def'' = revcast_def' [simplified word_size] |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3175 |
lemmas revcast_no_def [simp] = revcast_def' [where w="numeral w", unfolded word_size] for w |
37660 | 3176 |
|
65268 | 3177 |
lemma to_bl_revcast: |
65328 | 3178 |
"to_bl (revcast w :: 'a::len0 word) = takefill False (len_of TYPE('a)) (to_bl w)" |
37660 | 3179 |
apply (unfold revcast_def' word_size) |
3180 |
apply (rule word_bl.Abs_inverse) |
|
3181 |
apply simp |
|
3182 |
done |
|
3183 |
||
65268 | 3184 |
lemma revcast_rev_ucast [OF refl refl refl]: |
3185 |
"cs = [rc, uc] \<Longrightarrow> rc = revcast (word_reverse w) \<Longrightarrow> uc = ucast w \<Longrightarrow> |
|
37660 | 3186 |
rc = word_reverse uc" |
3187 |
apply (unfold ucast_def revcast_def' Let_def word_reverse_def) |
|
65328 | 3188 |
apply (auto simp: to_bl_of_bin takefill_bintrunc) |
3189 |
apply (simp add: word_bl.Abs_inverse word_size) |
|
37660 | 3190 |
done |
3191 |
||
45811 | 3192 |
lemma revcast_ucast: "revcast w = word_reverse (ucast (word_reverse w))" |
3193 |
using revcast_rev_ucast [of "word_reverse w"] by simp |
|
3194 |
||
3195 |
lemma ucast_revcast: "ucast w = word_reverse (revcast (word_reverse w))" |
|
3196 |
by (fact revcast_rev_ucast [THEN word_rev_gal']) |
|
3197 |
||
3198 |
lemma ucast_rev_revcast: "ucast (word_reverse w) = word_reverse (revcast w)" |
|
3199 |
by (fact revcast_ucast [THEN word_rev_gal']) |
|
37660 | 3200 |
|
3201 |
||
65328 | 3202 |
text "linking revcast and cast via shift" |
37660 | 3203 |
|
3204 |
lemmas wsst_TYs = source_size target_size word_size |
|
3205 |
||
45811 | 3206 |
lemma revcast_down_uu [OF refl]: |
65328 | 3207 |
"rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = ucast (w >> n)" |
3208 |
for w :: "'a::len word" |
|
37660 | 3209 |
apply (simp add: revcast_def') |
3210 |
apply (rule word_bl.Rep_inverse') |
|
3211 |
apply (rule trans, rule ucast_down_drop) |
|
3212 |
prefer 2 |
|
3213 |
apply (rule trans, rule drop_shiftr) |
|
3214 |
apply (auto simp: takefill_alt wsst_TYs) |
|
3215 |
done |
|
3216 |
||
45811 | 3217 |
lemma revcast_down_us [OF refl]: |
65328 | 3218 |
"rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = ucast (w >>> n)" |
3219 |
for w :: "'a::len word" |
|
37660 | 3220 |
apply (simp add: revcast_def') |
3221 |
apply (rule word_bl.Rep_inverse') |
|
3222 |
apply (rule trans, rule ucast_down_drop) |
|
3223 |
prefer 2 |
|
3224 |
apply (rule trans, rule drop_sshiftr) |
|
3225 |
apply (auto simp: takefill_alt wsst_TYs) |
|
3226 |
done |
|
3227 |
||
45811 | 3228 |
lemma revcast_down_su [OF refl]: |
65328 | 3229 |
"rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = scast (w >> n)" |
3230 |
for w :: "'a::len word" |
|
37660 | 3231 |
apply (simp add: revcast_def') |
3232 |
apply (rule word_bl.Rep_inverse') |
|
3233 |
apply (rule trans, rule scast_down_drop) |
|
3234 |
prefer 2 |
|
3235 |
apply (rule trans, rule drop_shiftr) |
|
3236 |
apply (auto simp: takefill_alt wsst_TYs) |
|
3237 |
done |
|
3238 |
||
45811 | 3239 |
lemma revcast_down_ss [OF refl]: |
65328 | 3240 |
"rc = revcast \<Longrightarrow> source_size rc = target_size rc + n \<Longrightarrow> rc w = scast (w >>> n)" |
3241 |
for w :: "'a::len word" |
|
37660 | 3242 |
apply (simp add: revcast_def') |
3243 |
apply (rule word_bl.Rep_inverse') |
|
3244 |
apply (rule trans, rule scast_down_drop) |
|
3245 |
prefer 2 |
|
3246 |
apply (rule trans, rule drop_sshiftr) |
|
3247 |
apply (auto simp: takefill_alt wsst_TYs) |
|
3248 |
done |
|
3249 |
||
45811 | 3250 |
(* FIXME: should this also be [OF refl] ? *) |
65268 | 3251 |
lemma cast_down_rev: |
65328 | 3252 |
"uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> uc w = revcast (w << n)" |
3253 |
for w :: "'a::len word" |
|
37660 | 3254 |
apply (unfold shiftl_rev) |
3255 |
apply clarify |
|
3256 |
apply (simp add: revcast_rev_ucast) |
|
3257 |
apply (rule word_rev_gal') |
|
3258 |
apply (rule trans [OF _ revcast_rev_ucast]) |
|
3259 |
apply (rule revcast_down_uu [symmetric]) |
|
3260 |
apply (auto simp add: wsst_TYs) |
|
3261 |
done |
|
3262 |
||
45811 | 3263 |
lemma revcast_up [OF refl]: |
65268 | 3264 |
"rc = revcast \<Longrightarrow> source_size rc + n = target_size rc \<Longrightarrow> |
3265 |
rc w = (ucast w :: 'a::len word) << n" |
|
37660 | 3266 |
apply (simp add: revcast_def') |
3267 |
apply (rule word_bl.Rep_inverse') |
|
3268 |
apply (simp add: takefill_alt) |
|
3269 |
apply (rule bl_shiftl [THEN trans]) |
|
3270 |
apply (subst ucast_up_app) |
|
65328 | 3271 |
apply (auto simp add: wsst_TYs) |
37660 | 3272 |
done |
3273 |
||
65268 | 3274 |
lemmas rc1 = revcast_up [THEN |
37660 | 3275 |
revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]] |
65268 | 3276 |
lemmas rc2 = revcast_down_uu [THEN |
37660 | 3277 |
revcast_rev_ucast [symmetric, THEN trans, THEN word_rev_gal, symmetric]] |
3278 |
||
3279 |
lemmas ucast_up = |
|
3280 |
rc1 [simplified rev_shiftr [symmetric] revcast_ucast [symmetric]] |
|
65268 | 3281 |
lemmas ucast_down = |
37660 | 3282 |
rc2 [simplified rev_shiftr revcast_ucast [symmetric]] |
3283 |
||
3284 |
||
61799 | 3285 |
subsubsection \<open>Slices\<close> |
37660 | 3286 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3287 |
lemma slice1_no_bin [simp]: |
65268 | 3288 |
"slice1 n (numeral w :: 'b word) = of_bl (takefill False n (bin_to_bl (len_of TYPE('b::len0)) (numeral w)))" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3289 |
by (simp add: slice1_def) (* TODO: neg_numeral *) |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3290 |
|
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3291 |
lemma slice_no_bin [simp]: |
65268 | 3292 |
"slice n (numeral w :: 'b word) = of_bl (takefill False (len_of TYPE('b::len0) - n) |
3293 |
(bin_to_bl (len_of TYPE('b::len0)) (numeral w)))" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3294 |
by (simp add: slice_def word_size) (* TODO: neg_numeral *) |
37660 | 3295 |
|
3296 |
lemma slice1_0 [simp] : "slice1 n 0 = 0" |
|
45805 | 3297 |
unfolding slice1_def by simp |
37660 | 3298 |
|
3299 |
lemma slice_0 [simp] : "slice n 0 = 0" |
|
3300 |
unfolding slice_def by auto |
|
3301 |
||
3302 |
lemma slice_take': "slice n w = of_bl (take (size w - n) (to_bl w))" |
|
3303 |
unfolding slice_def' slice1_def |
|
3304 |
by (simp add : takefill_alt word_size) |
|
3305 |
||
3306 |
lemmas slice_take = slice_take' [unfolded word_size] |
|
3307 |
||
65268 | 3308 |
\<comment> "shiftr to a word of the same size is just slice, |
37660 | 3309 |
slice is just shiftr then ucast" |
45604 | 3310 |
lemmas shiftr_slice = trans [OF shiftr_bl [THEN meta_eq_to_obj_eq] slice_take [symmetric]] |
37660 | 3311 |
|
3312 |
lemma slice_shiftr: "slice n w = ucast (w >> n)" |
|
3313 |
apply (unfold slice_take shiftr_bl) |
|
3314 |
apply (rule ucast_of_bl_up [symmetric]) |
|
3315 |
apply (simp add: word_size) |
|
3316 |
done |
|
3317 |
||
65336 | 3318 |
lemma nth_slice: "(slice n w :: 'a::len0 word) !! m = (w !! (m + n) \<and> m < len_of TYPE('a))" |
3319 |
by (simp add: slice_shiftr nth_ucast nth_shiftr) |
|
37660 | 3320 |
|
65268 | 3321 |
lemma slice1_down_alt': |
3322 |
"sl = slice1 n w \<Longrightarrow> fs = size sl \<Longrightarrow> fs + k = n \<Longrightarrow> |
|
37660 | 3323 |
to_bl sl = takefill False fs (drop k (to_bl w))" |
65336 | 3324 |
by (auto simp: slice1_def word_size of_bl_def uint_bl |
3325 |
word_ubin.eq_norm bl_bin_bl_rep_drop drop_takefill) |
|
37660 | 3326 |
|
65268 | 3327 |
lemma slice1_up_alt': |
3328 |
"sl = slice1 n w \<Longrightarrow> fs = size sl \<Longrightarrow> fs = n + k \<Longrightarrow> |
|
37660 | 3329 |
to_bl sl = takefill False fs (replicate k False @ (to_bl w))" |
3330 |
apply (unfold slice1_def word_size of_bl_def uint_bl) |
|
65336 | 3331 |
apply (clarsimp simp: word_ubin.eq_norm bl_bin_bl_rep_drop takefill_append [symmetric]) |
3332 |
apply (rule_tac f = "\<lambda>k. takefill False (len_of TYPE('a)) |
|
37660 | 3333 |
(replicate k False @ bin_to_bl (len_of TYPE('b)) (uint w))" in arg_cong) |
3334 |
apply arith |
|
3335 |
done |
|
65268 | 3336 |
|
37660 | 3337 |
lemmas sd1 = slice1_down_alt' [OF refl refl, unfolded word_size] |
3338 |
lemmas su1 = slice1_up_alt' [OF refl refl, unfolded word_size] |
|
3339 |
lemmas slice1_down_alt = le_add_diff_inverse [THEN sd1] |
|
65268 | 3340 |
lemmas slice1_up_alts = |
3341 |
le_add_diff_inverse [symmetric, THEN su1] |
|
37660 | 3342 |
le_add_diff_inverse2 [symmetric, THEN su1] |
3343 |
||
3344 |
lemma ucast_slice1: "ucast w = slice1 (size w) w" |
|
65336 | 3345 |
by (simp add: slice1_def ucast_bl takefill_same' word_size) |
37660 | 3346 |
|
3347 |
lemma ucast_slice: "ucast w = slice 0 w" |
|
65336 | 3348 |
by (simp add: slice_def ucast_slice1) |
37660 | 3349 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3350 |
lemma slice_id: "slice 0 t = t" |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3351 |
by (simp only: ucast_slice [symmetric] ucast_id) |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3352 |
|
65336 | 3353 |
lemma revcast_slice1 [OF refl]: "rc = revcast w \<Longrightarrow> slice1 (size rc) w = rc" |
3354 |
by (simp add: slice1_def revcast_def' word_size) |
|
37660 | 3355 |
|
65268 | 3356 |
lemma slice1_tf_tf': |
3357 |
"to_bl (slice1 n w :: 'a::len0 word) = |
|
65336 | 3358 |
rev (takefill False (len_of TYPE('a)) (rev (takefill False n (to_bl w))))" |
37660 | 3359 |
unfolding slice1_def by (rule word_rev_tf) |
3360 |
||
45604 | 3361 |
lemmas slice1_tf_tf = slice1_tf_tf' [THEN word_bl.Rep_inverse', symmetric] |
37660 | 3362 |
|
3363 |
lemma rev_slice1: |
|
65268 | 3364 |
"n + k = len_of TYPE('a) + len_of TYPE('b) \<Longrightarrow> |
65336 | 3365 |
slice1 n (word_reverse w :: 'b::len0 word) = |
3366 |
word_reverse (slice1 k w :: 'a::len0 word)" |
|
37660 | 3367 |
apply (unfold word_reverse_def slice1_tf_tf) |
3368 |
apply (rule word_bl.Rep_inverse') |
|
3369 |
apply (rule rev_swap [THEN iffD1]) |
|
3370 |
apply (rule trans [symmetric]) |
|
65336 | 3371 |
apply (rule tf_rev) |
37660 | 3372 |
apply (simp add: word_bl.Abs_inverse) |
3373 |
apply (simp add: word_bl.Abs_inverse) |
|
3374 |
done |
|
3375 |
||
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3376 |
lemma rev_slice: |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3377 |
"n + k + len_of TYPE('a::len0) = len_of TYPE('b::len0) \<Longrightarrow> |
65336 | 3378 |
slice n (word_reverse (w::'b word)) = word_reverse (slice k w :: 'a word)" |
37660 | 3379 |
apply (unfold slice_def word_size) |
3380 |
apply (rule rev_slice1) |
|
3381 |
apply arith |
|
3382 |
done |
|
3383 |
||
65268 | 3384 |
lemmas sym_notr = |
37660 | 3385 |
not_iff [THEN iffD2, THEN not_sym, THEN not_iff [THEN iffD1]] |
3386 |
||
61799 | 3387 |
\<comment> \<open>problem posed by TPHOLs referee: |
3388 |
criterion for overflow of addition of signed integers\<close> |
|
37660 | 3389 |
|
3390 |
lemma sofl_test: |
|
65336 | 3391 |
"(sint x + sint y = sint (x + y)) = |
3392 |
((((x + y) XOR x) AND ((x + y) XOR y)) >> (size x - 1) = 0)" |
|
3393 |
for x y :: "'a::len word" |
|
37660 | 3394 |
apply (unfold word_size) |
65268 | 3395 |
apply (cases "len_of TYPE('a)", simp) |
37660 | 3396 |
apply (subst msb_shift [THEN sym_notr]) |
3397 |
apply (simp add: word_ops_msb) |
|
3398 |
apply (simp add: word_msb_sint) |
|
3399 |
apply safe |
|
3400 |
apply simp_all |
|
3401 |
apply (unfold sint_word_ariths) |
|
3402 |
apply (unfold word_sbin.set_iff_norm [symmetric] sints_num) |
|
3403 |
apply safe |
|
65336 | 3404 |
apply (insert sint_range' [where x=x]) |
3405 |
apply (insert sint_range' [where x=y]) |
|
3406 |
defer |
|
3407 |
apply (simp (no_asm), arith) |
|
37660 | 3408 |
apply (simp (no_asm), arith) |
65336 | 3409 |
defer |
3410 |
defer |
|
37660 | 3411 |
apply (simp (no_asm), arith) |
3412 |
apply (simp (no_asm), arith) |
|
65336 | 3413 |
apply (rule notI [THEN notnotD], |
3414 |
drule leI not_le_imp_less, |
|
3415 |
drule sbintrunc_inc sbintrunc_dec, |
|
3416 |
simp)+ |
|
37660 | 3417 |
done |
3418 |
||
3419 |
||
61799 | 3420 |
subsection \<open>Split and cat\<close> |
37660 | 3421 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3422 |
lemmas word_split_bin' = word_split_def |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3423 |
lemmas word_cat_bin' = word_cat_def |
37660 | 3424 |
|
3425 |
lemma word_rsplit_no: |
|
65268 | 3426 |
"(word_rsplit (numeral bin :: 'b::len0 word) :: 'a word list) = |
3427 |
map word_of_int (bin_rsplit (len_of TYPE('a::len)) |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3428 |
(len_of TYPE('b), bintrunc (len_of TYPE('b)) (numeral bin)))" |
65336 | 3429 |
by (simp add: word_rsplit_def word_ubin.eq_norm) |
37660 | 3430 |
|
3431 |
lemmas word_rsplit_no_cl [simp] = word_rsplit_no |
|
3432 |
[unfolded bin_rsplitl_def bin_rsplit_l [symmetric]] |
|
3433 |
||
3434 |
lemma test_bit_cat: |
|
65336 | 3435 |
"wc = word_cat a b \<Longrightarrow> wc !! n = (n < size wc \<and> |
37660 | 3436 |
(if n < size b then b !! n else a !! (n - size b)))" |
65336 | 3437 |
apply (auto simp: word_cat_bin' test_bit_bin word_ubin.eq_norm nth_bintr bin_nth_cat word_size) |
37660 | 3438 |
apply (erule bin_nth_uint_imp) |
3439 |
done |
|
3440 |
||
3441 |
lemma word_cat_bl: "word_cat a b = of_bl (to_bl a @ to_bl b)" |
|
65336 | 3442 |
by (simp add: of_bl_def to_bl_def word_cat_bin' bl_to_bin_app_cat) |
37660 | 3443 |
|
3444 |
lemma of_bl_append: |
|
65268 | 3445 |
"(of_bl (xs @ ys) :: 'a::len word) = of_bl xs * 2^(length ys) + of_bl ys" |
65336 | 3446 |
apply (simp add: of_bl_def bl_to_bin_app_cat bin_cat_num) |
46009 | 3447 |
apply (simp add: word_of_int_power_hom [symmetric] word_of_int_hom_syms) |
37660 | 3448 |
done |
3449 |
||
65336 | 3450 |
lemma of_bl_False [simp]: "of_bl (False#xs) = of_bl xs" |
3451 |
by (rule word_eqI) (auto simp: test_bit_of_bl nth_append) |
|
3452 |
||
3453 |
lemma of_bl_True [simp]: "(of_bl (True # xs) :: 'a::len word) = 2^length xs + of_bl xs" |
|
3454 |
by (subst of_bl_append [where xs="[True]", simplified]) (simp add: word_1_bl) |
|
3455 |
||
3456 |
lemma of_bl_Cons: "of_bl (x#xs) = of_bool x * 2^length xs + of_bl xs" |
|
45805 | 3457 |
by (cases x) simp_all |
37660 | 3458 |
|
65336 | 3459 |
lemma split_uint_lem: "bin_split n (uint w) = (a, b) \<Longrightarrow> |
3460 |
a = bintrunc (len_of TYPE('a) - n) a \<and> b = bintrunc (len_of TYPE('a)) b" |
|
3461 |
for w :: "'a::len0 word" |
|
37660 | 3462 |
apply (frule word_ubin.norm_Rep [THEN ssubst]) |
3463 |
apply (drule bin_split_trunc1) |
|
3464 |
apply (drule sym [THEN trans]) |
|
65336 | 3465 |
apply assumption |
37660 | 3466 |
apply safe |
3467 |
done |
|
3468 |
||
65268 | 3469 |
lemma word_split_bl': |
3470 |
"std = size c - size b \<Longrightarrow> (word_split c = (a, b)) \<Longrightarrow> |
|
65336 | 3471 |
(a = of_bl (take std (to_bl c)) \<and> b = of_bl (drop std (to_bl c)))" |
37660 | 3472 |
apply (unfold word_split_bin') |
3473 |
apply safe |
|
3474 |
defer |
|
3475 |
apply (clarsimp split: prod.splits) |
|
57492
74bf65a1910a
Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents:
56979
diff
changeset
|
3476 |
apply hypsubst_thin |
37660 | 3477 |
apply (drule word_ubin.norm_Rep [THEN ssubst]) |
3478 |
apply (drule split_bintrunc) |
|
65336 | 3479 |
apply (simp add: of_bl_def bl2bin_drop word_size |
3480 |
word_ubin.norm_eq_iff [symmetric] min_def del: word_ubin.norm_Rep) |
|
37660 | 3481 |
apply (clarsimp split: prod.splits) |
3482 |
apply (frule split_uint_lem [THEN conjunct1]) |
|
3483 |
apply (unfold word_size) |
|
65336 | 3484 |
apply (cases "len_of TYPE('a) \<ge> len_of TYPE('b)") |
37660 | 3485 |
defer |
46001
0b562d564d5f
redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents:
46000
diff
changeset
|
3486 |
apply simp |
37660 | 3487 |
apply (simp add : of_bl_def to_bl_def) |
3488 |
apply (subst bin_split_take1 [symmetric]) |
|
3489 |
prefer 2 |
|
3490 |
apply assumption |
|
3491 |
apply simp |
|
3492 |
apply (erule thin_rl) |
|
3493 |
apply (erule arg_cong [THEN trans]) |
|
3494 |
apply (simp add : word_ubin.norm_eq_iff [symmetric]) |
|
3495 |
done |
|
3496 |
||
65268 | 3497 |
lemma word_split_bl: "std = size c - size b \<Longrightarrow> |
65336 | 3498 |
(a = of_bl (take std (to_bl c)) \<and> b = of_bl (drop std (to_bl c))) \<longleftrightarrow> |
37660 | 3499 |
word_split c = (a, b)" |
3500 |
apply (rule iffI) |
|
3501 |
defer |
|
3502 |
apply (erule (1) word_split_bl') |
|
3503 |
apply (case_tac "word_split c") |
|
65336 | 3504 |
apply (auto simp add: word_size) |
37660 | 3505 |
apply (frule word_split_bl' [rotated]) |
65336 | 3506 |
apply (auto simp add: word_size) |
37660 | 3507 |
done |
3508 |
||
3509 |
lemma word_split_bl_eq: |
|
65336 | 3510 |
"(word_split c :: ('c::len0 word \<times> 'd::len0 word)) = |
3511 |
(of_bl (take (len_of TYPE('a::len) - len_of TYPE('d::len0)) (to_bl c)), |
|
3512 |
of_bl (drop (len_of TYPE('a) - len_of TYPE('d)) (to_bl c)))" |
|
3513 |
for c :: "'a::len word" |
|
37660 | 3514 |
apply (rule word_split_bl [THEN iffD1]) |
65336 | 3515 |
apply (unfold word_size) |
3516 |
apply (rule refl conjI)+ |
|
37660 | 3517 |
done |
3518 |
||
61799 | 3519 |
\<comment> "keep quantifiers for use in simplification" |
37660 | 3520 |
lemma test_bit_split': |
65336 | 3521 |
"word_split c = (a, b) \<longrightarrow> |
3522 |
(\<forall>n m. |
|
3523 |
b !! n = (n < size b \<and> c !! n) \<and> |
|
3524 |
a !! m = (m < size a \<and> c !! (m + size b)))" |
|
37660 | 3525 |
apply (unfold word_split_bin' test_bit_bin) |
3526 |
apply (clarify) |
|
3527 |
apply (clarsimp simp: word_ubin.eq_norm nth_bintr word_size split: prod.splits) |
|
3528 |
apply (drule bin_nth_split) |
|
3529 |
apply safe |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
3530 |
apply (simp_all add: add.commute) |
37660 | 3531 |
apply (erule bin_nth_uint_imp)+ |
3532 |
done |
|
3533 |
||
3534 |
lemma test_bit_split: |
|
3535 |
"word_split c = (a, b) \<Longrightarrow> |
|
65336 | 3536 |
(\<forall>n::nat. b !! n \<longleftrightarrow> n < size b \<and> c !! n) \<and> |
3537 |
(\<forall>m::nat. a !! m \<longleftrightarrow> m < size a \<and> c !! (m + size b))" |
|
37660 | 3538 |
by (simp add: test_bit_split') |
3539 |
||
65336 | 3540 |
lemma test_bit_split_eq: |
3541 |
"word_split c = (a, b) \<longleftrightarrow> |
|
3542 |
((\<forall>n::nat. b !! n = (n < size b \<and> c !! n)) \<and> |
|
3543 |
(\<forall>m::nat. a !! m = (m < size a \<and> c !! (m + size b))))" |
|
37660 | 3544 |
apply (rule_tac iffI) |
3545 |
apply (rule_tac conjI) |
|
3546 |
apply (erule test_bit_split [THEN conjunct1]) |
|
3547 |
apply (erule test_bit_split [THEN conjunct2]) |
|
3548 |
apply (case_tac "word_split c") |
|
3549 |
apply (frule test_bit_split) |
|
3550 |
apply (erule trans) |
|
65336 | 3551 |
apply (fastforce intro!: word_eqI simp add: word_size) |
37660 | 3552 |
done |
3553 |
||
65268 | 3554 |
\<comment> \<open>this odd result is analogous to \<open>ucast_id\<close>, |
61799 | 3555 |
result to the length given by the result type\<close> |
37660 | 3556 |
|
3557 |
lemma word_cat_id: "word_cat a b = b" |
|
65336 | 3558 |
by (simp add: word_cat_bin' word_ubin.inverse_norm) |
37660 | 3559 |
|
61799 | 3560 |
\<comment> "limited hom result" |
37660 | 3561 |
lemma word_cat_hom: |
65336 | 3562 |
"len_of TYPE('a::len0) \<le> len_of TYPE('b::len0) + len_of TYPE('c::len0) \<Longrightarrow> |
3563 |
(word_cat (word_of_int w :: 'b word) (b :: 'c word) :: 'a word) = |
|
3564 |
word_of_int (bin_cat w (size b) (uint b))" |
|
3565 |
by (auto simp: word_cat_def word_size word_ubin.norm_eq_iff [symmetric] |
|
54863
82acc20ded73
prefer more canonical names for lemmas on min/max
haftmann
parents:
54854
diff
changeset
|
3566 |
word_ubin.eq_norm bintr_cat min.absorb1) |
65336 | 3567 |
|
3568 |
lemma word_cat_split_alt: "size w \<le> size u + size v \<Longrightarrow> word_split w = (u, v) \<Longrightarrow> word_cat u v = w" |
|
37660 | 3569 |
apply (rule word_eqI) |
3570 |
apply (drule test_bit_split) |
|
3571 |
apply (clarsimp simp add : test_bit_cat word_size) |
|
3572 |
apply safe |
|
3573 |
apply arith |
|
3574 |
done |
|
3575 |
||
45604 | 3576 |
lemmas word_cat_split_size = sym [THEN [2] word_cat_split_alt [symmetric]] |
37660 | 3577 |
|
3578 |
||
61799 | 3579 |
subsubsection \<open>Split and slice\<close> |
37660 | 3580 |
|
65336 | 3581 |
lemma split_slices: "word_split w = (u, v) \<Longrightarrow> u = slice (size v) w \<and> v = slice 0 w" |
37660 | 3582 |
apply (drule test_bit_split) |
3583 |
apply (rule conjI) |
|
3584 |
apply (rule word_eqI, clarsimp simp: nth_slice word_size)+ |
|
3585 |
done |
|
3586 |
||
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3587 |
lemma slice_cat1 [OF refl]: |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3588 |
"wc = word_cat a b \<Longrightarrow> size wc >= size a + size b \<Longrightarrow> slice (size b) wc = a" |
37660 | 3589 |
apply safe |
3590 |
apply (rule word_eqI) |
|
3591 |
apply (simp add: nth_slice test_bit_cat word_size) |
|
3592 |
done |
|
3593 |
||
3594 |
lemmas slice_cat2 = trans [OF slice_id word_cat_id] |
|
3595 |
||
3596 |
lemma cat_slices: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3597 |
"a = slice n c \<Longrightarrow> b = slice 0 c \<Longrightarrow> n = size b \<Longrightarrow> |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3598 |
size a + size b >= size c \<Longrightarrow> word_cat a b = c" |
37660 | 3599 |
apply safe |
3600 |
apply (rule word_eqI) |
|
3601 |
apply (simp add: nth_slice test_bit_cat word_size) |
|
3602 |
apply safe |
|
3603 |
apply arith |
|
3604 |
done |
|
3605 |
||
3606 |
lemma word_split_cat_alt: |
|
65336 | 3607 |
"w = word_cat u v \<Longrightarrow> size u + size v \<le> size w \<Longrightarrow> word_split w = (u, v)" |
59807 | 3608 |
apply (case_tac "word_split w") |
37660 | 3609 |
apply (rule trans, assumption) |
3610 |
apply (drule test_bit_split) |
|
3611 |
apply safe |
|
3612 |
apply (rule word_eqI, clarsimp simp: test_bit_cat word_size)+ |
|
3613 |
done |
|
3614 |
||
3615 |
lemmas word_cat_bl_no_bin [simp] = |
|
65336 | 3616 |
word_cat_bl [where a="numeral a" and b="numeral b", unfolded to_bl_numeral] |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3617 |
for a b (* FIXME: negative numerals, 0 and 1 *) |
37660 | 3618 |
|
3619 |
lemmas word_split_bl_no_bin [simp] = |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3620 |
word_split_bl_eq [where c="numeral c", unfolded to_bl_numeral] for c |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
3621 |
|
65336 | 3622 |
text \<open> |
3623 |
This odd result arises from the fact that the statement of the |
|
3624 |
result implies that the decoded words are of the same type, |
|
3625 |
and therefore of the same length, as the original word.\<close> |
|
37660 | 3626 |
|
3627 |
lemma word_rsplit_same: "word_rsplit w = [w]" |
|
65336 | 3628 |
by (simp add: word_rsplit_def bin_rsplit_all) |
3629 |
||
3630 |
lemma word_rsplit_empty_iff_size: "word_rsplit w = [] \<longleftrightarrow> size w = 0" |
|
3631 |
by (simp add: word_rsplit_def bin_rsplit_def word_size bin_rsplit_aux_simp_alt Let_def |
|
3632 |
split: prod.split) |
|
37660 | 3633 |
|
3634 |
lemma test_bit_rsplit: |
|
65268 | 3635 |
"sw = word_rsplit w \<Longrightarrow> m < size (hd sw :: 'a::len word) \<Longrightarrow> |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3636 |
k < length sw \<Longrightarrow> (rev sw ! k) !! m = (w !! (k * size (hd sw) + m))" |
37660 | 3637 |
apply (unfold word_rsplit_def word_test_bit_def) |
3638 |
apply (rule trans) |
|
65336 | 3639 |
apply (rule_tac f = "\<lambda>x. bin_nth x m" in arg_cong) |
37660 | 3640 |
apply (rule nth_map [symmetric]) |
3641 |
apply simp |
|
3642 |
apply (rule bin_nth_rsplit) |
|
3643 |
apply simp_all |
|
3644 |
apply (simp add : word_size rev_map) |
|
3645 |
apply (rule trans) |
|
3646 |
defer |
|
3647 |
apply (rule map_ident [THEN fun_cong]) |
|
3648 |
apply (rule refl [THEN map_cong]) |
|
3649 |
apply (simp add : word_ubin.eq_norm) |
|
3650 |
apply (erule bin_rsplit_size_sign [OF len_gt_0 refl]) |
|
3651 |
done |
|
3652 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3653 |
lemma word_rcat_bl: "word_rcat wl = of_bl (concat (map to_bl wl))" |
65336 | 3654 |
by (auto simp: word_rcat_def to_bl_def' of_bl_def bin_rcat_bl) |
3655 |
||
3656 |
lemma size_rcat_lem': "size (concat (map to_bl wl)) = length wl * size (hd wl)" |
|
3657 |
by (induct wl) (auto simp: word_size) |
|
37660 | 3658 |
|
3659 |
lemmas size_rcat_lem = size_rcat_lem' [unfolded word_size] |
|
3660 |
||
45604 | 3661 |
lemmas td_gal_lt_len = len_gt_0 [THEN td_gal_lt] |
37660 | 3662 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3663 |
lemma nth_rcat_lem: |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3664 |
"n < length (wl::'a word list) * len_of TYPE('a::len) \<Longrightarrow> |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3665 |
rev (concat (map to_bl wl)) ! n = |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3666 |
rev (to_bl (rev wl ! (n div len_of TYPE('a)))) ! (n mod len_of TYPE('a))" |
65336 | 3667 |
apply (induct wl) |
37660 | 3668 |
apply clarsimp |
3669 |
apply (clarsimp simp add : nth_append size_rcat_lem) |
|
65268 | 3670 |
apply (simp (no_asm_use) only: mult_Suc [symmetric] |
64243 | 3671 |
td_gal_lt_len less_Suc_eq_le minus_div_mult_eq_mod [symmetric]) |
37660 | 3672 |
apply clarsimp |
3673 |
done |
|
3674 |
||
3675 |
lemma test_bit_rcat: |
|
65268 | 3676 |
"sw = size (hd wl :: 'a::len word) \<Longrightarrow> rc = word_rcat wl \<Longrightarrow> rc !! n = |
65336 | 3677 |
(n < size rc \<and> n div sw < size wl \<and> (rev wl) ! (n div sw) !! (n mod sw))" |
37660 | 3678 |
apply (unfold word_rcat_bl word_size) |
65336 | 3679 |
apply (clarsimp simp add: test_bit_of_bl size_rcat_lem word_size td_gal_lt_len) |
37660 | 3680 |
apply safe |
65336 | 3681 |
apply (auto simp: test_bit_bl word_size td_gal_lt_len [THEN iffD2, THEN nth_rcat_lem]) |
37660 | 3682 |
done |
3683 |
||
65336 | 3684 |
lemma foldl_eq_foldr: "foldl op + x xs = foldr op + (x # xs) 0" |
3685 |
for x :: "'a::comm_monoid_add" |
|
3686 |
by (induct xs arbitrary: x) (auto simp: add.assoc) |
|
37660 | 3687 |
|
3688 |
lemmas test_bit_cong = arg_cong [where f = "test_bit", THEN fun_cong] |
|
3689 |
||
65268 | 3690 |
lemmas test_bit_rsplit_alt = |
3691 |
trans [OF nth_rev_alt [THEN test_bit_cong] |
|
37660 | 3692 |
test_bit_rsplit [OF refl asm_rl diff_Suc_less]] |
3693 |
||
61799 | 3694 |
\<comment> "lazy way of expressing that u and v, and su and sv, have same types" |
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3695 |
lemma word_rsplit_len_indep [OF refl refl refl refl]: |
65268 | 3696 |
"[u,v] = p \<Longrightarrow> [su,sv] = q \<Longrightarrow> word_rsplit u = su \<Longrightarrow> |
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
3697 |
word_rsplit v = sv \<Longrightarrow> length su = length sv" |
65336 | 3698 |
by (auto simp: word_rsplit_def bin_rsplit_len_indep) |
37660 | 3699 |
|
65268 | 3700 |
lemma length_word_rsplit_size: |
3701 |
"n = len_of TYPE('a::len) \<Longrightarrow> |
|
65336 | 3702 |
length (word_rsplit w :: 'a word list) \<le> m \<longleftrightarrow> size w \<le> m * n" |
3703 |
by (auto simp: word_rsplit_def word_size bin_rsplit_len_le) |
|
37660 | 3704 |
|
65268 | 3705 |
lemmas length_word_rsplit_lt_size = |
37660 | 3706 |
length_word_rsplit_size [unfolded Not_eq_iff linorder_not_less [symmetric]] |
3707 |
||
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3708 |
lemma length_word_rsplit_exp_size: |
65268 | 3709 |
"n = len_of TYPE('a::len) \<Longrightarrow> |
37660 | 3710 |
length (word_rsplit w :: 'a word list) = (size w + n - 1) div n" |
65336 | 3711 |
by (auto simp: word_rsplit_def word_size bin_rsplit_len) |
37660 | 3712 |
|
65268 | 3713 |
lemma length_word_rsplit_even_size: |
3714 |
"n = len_of TYPE('a::len) \<Longrightarrow> size w = m * n \<Longrightarrow> |
|
37660 | 3715 |
length (word_rsplit w :: 'a word list) = m" |
65336 | 3716 |
by (auto simp: length_word_rsplit_exp_size given_quot_alt) |
37660 | 3717 |
|
3718 |
lemmas length_word_rsplit_exp_size' = refl [THEN length_word_rsplit_exp_size] |
|
3719 |
||
3720 |
(* alternative proof of word_rcat_rsplit *) |
|
65268 | 3721 |
lemmas tdle = iffD2 [OF split_div_lemma refl, THEN conjunct1] |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
3722 |
lemmas dtle = xtr4 [OF tdle mult.commute] |
37660 | 3723 |
|
3724 |
lemma word_rcat_rsplit: "word_rcat (word_rsplit w) = w" |
|
3725 |
apply (rule word_eqI) |
|
65336 | 3726 |
apply (clarsimp simp: test_bit_rcat word_size) |
37660 | 3727 |
apply (subst refl [THEN test_bit_rsplit]) |
65268 | 3728 |
apply (simp_all add: word_size |
37660 | 3729 |
refl [THEN length_word_rsplit_size [simplified not_less [symmetric], simplified]]) |
3730 |
apply safe |
|
3731 |
apply (erule xtr7, rule len_gt_0 [THEN dtle])+ |
|
3732 |
done |
|
3733 |
||
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3734 |
lemma size_word_rsplit_rcat_size: |
65336 | 3735 |
"word_rcat ws = frcw \<Longrightarrow> size frcw = length ws * len_of TYPE('a) |
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3736 |
\<Longrightarrow> length (word_rsplit frcw::'a word list) = length ws" |
65336 | 3737 |
for ws :: "'a::len word list" and frcw :: "'b::len0 word" |
3738 |
apply (clarsimp simp: word_size length_word_rsplit_exp_size') |
|
37660 | 3739 |
apply (fast intro: given_quot_alt) |
3740 |
done |
|
3741 |
||
3742 |
lemma msrevs: |
|
65336 | 3743 |
"0 < n \<Longrightarrow> (k * n + m) div n = m div n + k" |
3744 |
"(k * n + m) mod n = m mod n" |
|
3745 |
for n :: nat |
|
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57492
diff
changeset
|
3746 |
by (auto simp: add.commute) |
37660 | 3747 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3748 |
lemma word_rsplit_rcat_size [OF refl]: |
65336 | 3749 |
"word_rcat ws = frcw \<Longrightarrow> |
65268 | 3750 |
size frcw = length ws * len_of TYPE('a) \<Longrightarrow> word_rsplit frcw = ws" |
65336 | 3751 |
for ws :: "'a::len word list" |
37660 | 3752 |
apply (frule size_word_rsplit_rcat_size, assumption) |
3753 |
apply (clarsimp simp add : word_size) |
|
3754 |
apply (rule nth_equalityI, assumption) |
|
3755 |
apply clarsimp |
|
46023
fad87bb608fc
restate some lemmas to respect int/bin distinction
huffman
parents:
46022
diff
changeset
|
3756 |
apply (rule word_eqI [rule_format]) |
37660 | 3757 |
apply (rule trans) |
3758 |
apply (rule test_bit_rsplit_alt) |
|
3759 |
apply (clarsimp simp: word_size)+ |
|
3760 |
apply (rule trans) |
|
65336 | 3761 |
apply (rule test_bit_rcat [OF refl refl]) |
55818 | 3762 |
apply (simp add: word_size) |
37660 | 3763 |
apply (subst nth_rev) |
3764 |
apply arith |
|
41550 | 3765 |
apply (simp add: le0 [THEN [2] xtr7, THEN diff_Suc_less]) |
37660 | 3766 |
apply safe |
41550 | 3767 |
apply (simp add: diff_mult_distrib) |
37660 | 3768 |
apply (rule mpl_lem) |
65336 | 3769 |
apply (cases "size ws") |
3770 |
apply simp_all |
|
37660 | 3771 |
done |
3772 |
||
3773 |
||
61799 | 3774 |
subsection \<open>Rotation\<close> |
37660 | 3775 |
|
3776 |
lemmas rotater_0' [simp] = rotater_def [where n = "0", simplified] |
|
3777 |
||
3778 |
lemmas word_rot_defs = word_roti_def word_rotr_def word_rotl_def |
|
3779 |
||
65336 | 3780 |
lemma rotate_eq_mod: "m mod length xs = n mod length xs \<Longrightarrow> rotate m xs = rotate n xs" |
37660 | 3781 |
apply (rule box_equals) |
3782 |
defer |
|
3783 |
apply (rule rotate_conv_mod [symmetric])+ |
|
3784 |
apply simp |
|
3785 |
done |
|
3786 |
||
65268 | 3787 |
lemmas rotate_eqs = |
37660 | 3788 |
trans [OF rotate0 [THEN fun_cong] id_apply] |
65268 | 3789 |
rotate_rotate [symmetric] |
45604 | 3790 |
rotate_id |
65268 | 3791 |
rotate_conv_mod |
37660 | 3792 |
rotate_eq_mod |
3793 |
||
3794 |
||
61799 | 3795 |
subsubsection \<open>Rotation of list to right\<close> |
37660 | 3796 |
|
3797 |
lemma rotate1_rl': "rotater1 (l @ [a]) = a # l" |
|
65336 | 3798 |
by (cases l) (auto simp: rotater1_def) |
37660 | 3799 |
|
3800 |
lemma rotate1_rl [simp] : "rotater1 (rotate1 l) = l" |
|
3801 |
apply (unfold rotater1_def) |
|
3802 |
apply (cases "l") |
|
65336 | 3803 |
apply (case_tac [2] "list") |
3804 |
apply auto |
|
37660 | 3805 |
done |
3806 |
||
3807 |
lemma rotate1_lr [simp] : "rotate1 (rotater1 l) = l" |
|
65336 | 3808 |
by (cases l) (auto simp: rotater1_def) |
37660 | 3809 |
|
3810 |
lemma rotater1_rev': "rotater1 (rev xs) = rev (rotate1 xs)" |
|
65336 | 3811 |
by (cases "xs") (simp add: rotater1_def, simp add: rotate1_rl') |
65268 | 3812 |
|
37660 | 3813 |
lemma rotater_rev': "rotater n (rev xs) = rev (rotate n xs)" |
65336 | 3814 |
by (induct n) (auto simp: rotater_def intro: rotater1_rev') |
37660 | 3815 |
|
45816
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3816 |
lemma rotater_rev: "rotater n ys = rev (rotate n (rev ys))" |
6a04efd99f25
replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents:
45811
diff
changeset
|
3817 |
using rotater_rev' [where xs = "rev ys"] by simp |
37660 | 3818 |
|
65268 | 3819 |
lemma rotater_drop_take: |
3820 |
"rotater n xs = |
|
65336 | 3821 |
drop (length xs - n mod length xs) xs @ |
3822 |
take (length xs - n mod length xs) xs" |
|
3823 |
by (auto simp: rotater_rev rotate_drop_take rev_take rev_drop) |
|
3824 |
||
3825 |
lemma rotater_Suc [simp]: "rotater (Suc n) xs = rotater1 (rotater n xs)" |
|
37660 | 3826 |
unfolding rotater_def by auto |
3827 |
||
3828 |
lemma rotate_inv_plus [rule_format] : |
|
65336 | 3829 |
"\<forall>k. k = m + n \<longrightarrow> rotater k (rotate n xs) = rotater m xs \<and> |
3830 |
rotate k (rotater n xs) = rotate m xs \<and> |
|
3831 |
rotater n (rotate k xs) = rotate m xs \<and> |
|
37660 | 3832 |
rotate n (rotater k xs) = rotater m xs" |
65336 | 3833 |
by (induct n) (auto simp: rotater_def rotate_def intro: funpow_swap1 [THEN trans]) |
65268 | 3834 |
|
37660 | 3835 |
lemmas rotate_inv_rel = le_add_diff_inverse2 [symmetric, THEN rotate_inv_plus] |
3836 |
||
3837 |
lemmas rotate_inv_eq = order_refl [THEN rotate_inv_rel, simplified] |
|
3838 |
||
45604 | 3839 |
lemmas rotate_lr [simp] = rotate_inv_eq [THEN conjunct1] |
3840 |
lemmas rotate_rl [simp] = rotate_inv_eq [THEN conjunct2, THEN conjunct1] |
|
37660 | 3841 |
|
65336 | 3842 |
lemma rotate_gal: "rotater n xs = ys \<longleftrightarrow> rotate n ys = xs" |
37660 | 3843 |
by auto |
3844 |
||
65336 | 3845 |
lemma rotate_gal': "ys = rotater n xs \<longleftrightarrow> xs = rotate n ys" |
37660 | 3846 |
by auto |
3847 |
||
65336 | 3848 |
lemma length_rotater [simp]: "length (rotater n xs) = length xs" |
37660 | 3849 |
by (simp add : rotater_rev) |
3850 |
||
65336 | 3851 |
lemma restrict_to_left: "x = y \<Longrightarrow> x = z \<longleftrightarrow> y = z" |
3852 |
by simp |
|
38527 | 3853 |
|
65268 | 3854 |
lemmas rrs0 = rotate_eqs [THEN restrict_to_left, |
45604 | 3855 |
simplified rotate_gal [symmetric] rotate_gal' [symmetric]] |
37660 | 3856 |
lemmas rrs1 = rrs0 [THEN refl [THEN rev_iffD1]] |
45604 | 3857 |
lemmas rotater_eqs = rrs1 [simplified length_rotater] |
37660 | 3858 |
lemmas rotater_0 = rotater_eqs (1) |
3859 |
lemmas rotater_add = rotater_eqs (2) |
|
3860 |
||
3861 |
||
61799 | 3862 |
subsubsection \<open>map, map2, commuting with rotate(r)\<close> |
37660 | 3863 |
|
65336 | 3864 |
lemma butlast_map: "xs \<noteq> [] \<Longrightarrow> butlast (map f xs) = map f (butlast xs)" |
37660 | 3865 |
by (induct xs) auto |
3866 |
||
65268 | 3867 |
lemma rotater1_map: "rotater1 (map f xs) = map f (rotater1 xs)" |
65336 | 3868 |
by (cases xs) (auto simp: rotater1_def last_map butlast_map) |
3869 |
||
3870 |
lemma rotater_map: "rotater n (map f xs) = map f (rotater n xs)" |
|
3871 |
by (induct n) (auto simp: rotater_def rotater1_map) |
|
37660 | 3872 |
|
3873 |
lemma but_last_zip [rule_format] : |
|
65336 | 3874 |
"\<forall>ys. length xs = length ys \<longrightarrow> xs \<noteq> [] \<longrightarrow> |
3875 |
last (zip xs ys) = (last xs, last ys) \<and> |
|
65268 | 3876 |
butlast (zip xs ys) = zip (butlast xs) (butlast ys)" |
65336 | 3877 |
apply (induct xs) |
3878 |
apply auto |
|
37660 | 3879 |
apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+ |
3880 |
done |
|
3881 |
||
3882 |
lemma but_last_map2 [rule_format] : |
|
65336 | 3883 |
"\<forall>ys. length xs = length ys \<longrightarrow> xs \<noteq> [] \<longrightarrow> |
3884 |
last (map2 f xs ys) = f (last xs) (last ys) \<and> |
|
65268 | 3885 |
butlast (map2 f xs ys) = map2 f (butlast xs) (butlast ys)" |
65336 | 3886 |
apply (induct xs) |
3887 |
apply auto |
|
37660 | 3888 |
apply (unfold map2_def) |
3889 |
apply ((case_tac ys, auto simp: neq_Nil_conv)[1])+ |
|
3890 |
done |
|
3891 |
||
3892 |
lemma rotater1_zip: |
|
65268 | 3893 |
"length xs = length ys \<Longrightarrow> |
3894 |
rotater1 (zip xs ys) = zip (rotater1 xs) (rotater1 ys)" |
|
37660 | 3895 |
apply (unfold rotater1_def) |
65336 | 3896 |
apply (cases xs) |
37660 | 3897 |
apply auto |
3898 |
apply ((case_tac ys, auto simp: neq_Nil_conv but_last_zip)[1])+ |
|
3899 |
done |
|
3900 |
||
3901 |
lemma rotater1_map2: |
|
65268 | 3902 |
"length xs = length ys \<Longrightarrow> |
3903 |
rotater1 (map2 f xs ys) = map2 f (rotater1 xs) (rotater1 ys)" |
|
65336 | 3904 |
by (simp add: map2_def rotater1_map rotater1_zip) |
37660 | 3905 |
|
65268 | 3906 |
lemmas lrth = |
3907 |
box_equals [OF asm_rl length_rotater [symmetric] |
|
3908 |
length_rotater [symmetric], |
|
37660 | 3909 |
THEN rotater1_map2] |
3910 |
||
65268 | 3911 |
lemma rotater_map2: |
3912 |
"length xs = length ys \<Longrightarrow> |
|
3913 |
rotater n (map2 f xs ys) = map2 f (rotater n xs) (rotater n ys)" |
|
37660 | 3914 |
by (induct n) (auto intro!: lrth) |
3915 |
||
3916 |
lemma rotate1_map2: |
|
65268 | 3917 |
"length xs = length ys \<Longrightarrow> |
3918 |
rotate1 (map2 f xs ys) = map2 f (rotate1 xs) (rotate1 ys)" |
|
65336 | 3919 |
by (cases xs; cases ys) (auto simp: map2_def) |
37660 | 3920 |
|
65268 | 3921 |
lemmas lth = box_equals [OF asm_rl length_rotate [symmetric] |
37660 | 3922 |
length_rotate [symmetric], THEN rotate1_map2] |
3923 |
||
65268 | 3924 |
lemma rotate_map2: |
3925 |
"length xs = length ys \<Longrightarrow> |
|
3926 |
rotate n (map2 f xs ys) = map2 f (rotate n xs) (rotate n ys)" |
|
37660 | 3927 |
by (induct n) (auto intro!: lth) |
3928 |
||
3929 |
||
61799 | 3930 |
\<comment> "corresponding equalities for word rotation" |
37660 | 3931 |
|
65336 | 3932 |
lemma to_bl_rotl: "to_bl (word_rotl n w) = rotate n (to_bl w)" |
37660 | 3933 |
by (simp add: word_bl.Abs_inverse' word_rotl_def) |
3934 |
||
3935 |
lemmas blrs0 = rotate_eqs [THEN to_bl_rotl [THEN trans]] |
|
3936 |
||
3937 |
lemmas word_rotl_eqs = |
|
45538
1fffa81b9b83
eliminated slightly odd Rep' with dynamically-scoped [simplified];
wenzelm
parents:
45529
diff
changeset
|
3938 |
blrs0 [simplified word_bl_Rep' word_bl.Rep_inject to_bl_rotl [symmetric]] |
37660 | 3939 |
|
65336 | 3940 |
lemma to_bl_rotr: "to_bl (word_rotr n w) = rotater n (to_bl w)" |
37660 | 3941 |
by (simp add: word_bl.Abs_inverse' word_rotr_def) |
3942 |
||
3943 |
lemmas brrs0 = rotater_eqs [THEN to_bl_rotr [THEN trans]] |
|
3944 |
||
3945 |
lemmas word_rotr_eqs = |
|
45538
1fffa81b9b83
eliminated slightly odd Rep' with dynamically-scoped [simplified];
wenzelm
parents:
45529
diff
changeset
|
3946 |
brrs0 [simplified word_bl_Rep' word_bl.Rep_inject to_bl_rotr [symmetric]] |
37660 | 3947 |
|
3948 |
declare word_rotr_eqs (1) [simp] |
|
3949 |
declare word_rotl_eqs (1) [simp] |
|
3950 |
||
65336 | 3951 |
lemma word_rot_rl [simp]: "word_rotl k (word_rotr k v) = v" |
3952 |
and word_rot_lr [simp]: "word_rotr k (word_rotl k v) = v" |
|
37660 | 3953 |
by (auto simp add: to_bl_rotr to_bl_rotl word_bl.Rep_inject [symmetric]) |
3954 |
||
65336 | 3955 |
lemma word_rot_gal: "word_rotr n v = w \<longleftrightarrow> word_rotl n w = v" |
3956 |
and word_rot_gal': "w = word_rotr n v \<longleftrightarrow> v = word_rotl n w" |
|
3957 |
by (auto simp: to_bl_rotr to_bl_rotl word_bl.Rep_inject [symmetric] dest: sym) |
|
3958 |
||
3959 |
lemma word_rotr_rev: "word_rotr n w = word_reverse (word_rotl n (word_reverse w))" |
|
3960 |
by (simp only: word_bl.Rep_inject [symmetric] to_bl_word_rev to_bl_rotr to_bl_rotl rotater_rev) |
|
65268 | 3961 |
|
37660 | 3962 |
lemma word_roti_0 [simp]: "word_roti 0 w = w" |
65336 | 3963 |
by (auto simp: word_rot_defs) |
37660 | 3964 |
|
3965 |
lemmas abl_cong = arg_cong [where f = "of_bl"] |
|
3966 |
||
65336 | 3967 |
lemma word_roti_add: "word_roti (m + n) w = word_roti m (word_roti n w)" |
37660 | 3968 |
proof - |
65336 | 3969 |
have rotater_eq_lem: "\<And>m n xs. m = n \<Longrightarrow> rotater m xs = rotater n xs" |
37660 | 3970 |
by auto |
3971 |
||
65336 | 3972 |
have rotate_eq_lem: "\<And>m n xs. m = n \<Longrightarrow> rotate m xs = rotate n xs" |
37660 | 3973 |
by auto |
3974 |
||
65268 | 3975 |
note rpts [symmetric] = |
37660 | 3976 |
rotate_inv_plus [THEN conjunct1] |
3977 |
rotate_inv_plus [THEN conjunct2, THEN conjunct1] |
|
3978 |
rotate_inv_plus [THEN conjunct2, THEN conjunct2, THEN conjunct1] |
|
3979 |
rotate_inv_plus [THEN conjunct2, THEN conjunct2, THEN conjunct2] |
|
3980 |
||
3981 |
note rrp = trans [symmetric, OF rotate_rotate rotate_eq_lem] |
|
3982 |
note rrrp = trans [symmetric, OF rotater_add [symmetric] rotater_eq_lem] |
|
3983 |
||
3984 |
show ?thesis |
|
65336 | 3985 |
apply (unfold word_rot_defs) |
3986 |
apply (simp only: split: if_split) |
|
3987 |
apply (safe intro!: abl_cong) |
|
3988 |
apply (simp_all only: to_bl_rotl [THEN word_bl.Rep_inverse'] |
|
3989 |
to_bl_rotl |
|
3990 |
to_bl_rotr [THEN word_bl.Rep_inverse'] |
|
3991 |
to_bl_rotr) |
|
3992 |
apply (rule rrp rrrp rpts, |
|
3993 |
simp add: nat_add_distrib [symmetric] |
|
3994 |
nat_diff_distrib [symmetric])+ |
|
3995 |
done |
|
37660 | 3996 |
qed |
65268 | 3997 |
|
37660 | 3998 |
lemma word_roti_conv_mod': "word_roti n w = word_roti (n mod int (size w)) w" |
3999 |
apply (unfold word_rot_defs) |
|
4000 |
apply (cut_tac y="size w" in gt_or_eq_0) |
|
4001 |
apply (erule disjE) |
|
4002 |
apply simp_all |
|
4003 |
apply (safe intro!: abl_cong) |
|
4004 |
apply (rule rotater_eqs) |
|
4005 |
apply (simp add: word_size nat_mod_distrib) |
|
4006 |
apply (simp add: rotater_add [symmetric] rotate_gal [symmetric]) |
|
4007 |
apply (rule rotater_eqs) |
|
4008 |
apply (simp add: word_size nat_mod_distrib) |
|
62348 | 4009 |
apply (rule of_nat_eq_0_iff [THEN iffD1]) |
4010 |
apply (auto simp add: not_le mod_eq_0_iff_dvd zdvd_int nat_add_distrib [symmetric]) |
|
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
4011 |
using mod_mod_trivial mod_eq_dvd_iff |
62348 | 4012 |
apply blast |
37660 | 4013 |
done |
4014 |
||
4015 |
lemmas word_roti_conv_mod = word_roti_conv_mod' [unfolded word_size] |
|
4016 |
||
4017 |
||
61799 | 4018 |
subsubsection \<open>"Word rotation commutes with bit-wise operations\<close> |
37660 | 4019 |
|
4020 |
(* using locale to not pollute lemma namespace *) |
|
65268 | 4021 |
locale word_rotate |
37660 | 4022 |
begin |
4023 |
||
4024 |
lemmas word_rot_defs' = to_bl_rotl to_bl_rotr |
|
4025 |
||
4026 |
lemmas blwl_syms [symmetric] = bl_word_not bl_word_and bl_word_or bl_word_xor |
|
4027 |
||
45538
1fffa81b9b83
eliminated slightly odd Rep' with dynamically-scoped [simplified];
wenzelm
parents:
45529
diff
changeset
|
4028 |
lemmas lbl_lbl = trans [OF word_bl_Rep' word_bl_Rep' [symmetric]] |
37660 | 4029 |
|
4030 |
lemmas ths_map2 [OF lbl_lbl] = rotate_map2 rotater_map2 |
|
4031 |
||
45604 | 4032 |
lemmas ths_map [where xs = "to_bl v"] = rotate_map rotater_map for v |
37660 | 4033 |
|
4034 |
lemmas th1s [simplified word_rot_defs' [symmetric]] = ths_map2 ths_map |
|
4035 |
||
4036 |
lemma word_rot_logs: |
|
4037 |
"word_rotl n (NOT v) = NOT word_rotl n v" |
|
4038 |
"word_rotr n (NOT v) = NOT word_rotr n v" |
|
4039 |
"word_rotl n (x AND y) = word_rotl n x AND word_rotl n y" |
|
4040 |
"word_rotr n (x AND y) = word_rotr n x AND word_rotr n y" |
|
4041 |
"word_rotl n (x OR y) = word_rotl n x OR word_rotl n y" |
|
4042 |
"word_rotr n (x OR y) = word_rotr n x OR word_rotr n y" |
|
4043 |
"word_rotl n (x XOR y) = word_rotl n x XOR word_rotl n y" |
|
65268 | 4044 |
"word_rotr n (x XOR y) = word_rotr n x XOR word_rotr n y" |
37660 | 4045 |
by (rule word_bl.Rep_eqD, |
4046 |
rule word_rot_defs' [THEN trans], |
|
4047 |
simp only: blwl_syms [symmetric], |
|
65268 | 4048 |
rule th1s [THEN trans], |
37660 | 4049 |
rule refl)+ |
4050 |
end |
|
4051 |
||
4052 |
lemmas word_rot_logs = word_rotate.word_rot_logs |
|
4053 |
||
4054 |
lemmas bl_word_rotl_dt = trans [OF to_bl_rotl rotate_drop_take, |
|
45604 | 4055 |
simplified word_bl_Rep'] |
37660 | 4056 |
|
4057 |
lemmas bl_word_rotr_dt = trans [OF to_bl_rotr rotater_drop_take, |
|
45604 | 4058 |
simplified word_bl_Rep'] |
37660 | 4059 |
|
65268 | 4060 |
lemma bl_word_roti_dt': |
4061 |
"n = nat ((- i) mod int (size (w :: 'a::len word))) \<Longrightarrow> |
|
37660 | 4062 |
to_bl (word_roti i w) = drop n (to_bl w) @ take n (to_bl w)" |
4063 |
apply (unfold word_roti_def) |
|
4064 |
apply (simp add: bl_word_rotl_dt bl_word_rotr_dt word_size) |
|
4065 |
apply safe |
|
4066 |
apply (simp add: zmod_zminus1_eq_if) |
|
4067 |
apply safe |
|
4068 |
apply (simp add: nat_mult_distrib) |
|
65268 | 4069 |
apply (simp add: nat_diff_distrib [OF pos_mod_sign pos_mod_conj |
37660 | 4070 |
[THEN conjunct2, THEN order_less_imp_le]] |
4071 |
nat_mod_distrib) |
|
4072 |
apply (simp add: nat_mod_distrib) |
|
4073 |
done |
|
4074 |
||
4075 |
lemmas bl_word_roti_dt = bl_word_roti_dt' [unfolded word_size] |
|
4076 |
||
65268 | 4077 |
lemmas word_rotl_dt = bl_word_rotl_dt [THEN word_bl.Rep_inverse' [symmetric]] |
45604 | 4078 |
lemmas word_rotr_dt = bl_word_rotr_dt [THEN word_bl.Rep_inverse' [symmetric]] |
4079 |
lemmas word_roti_dt = bl_word_roti_dt [THEN word_bl.Rep_inverse' [symmetric]] |
|
37660 | 4080 |
|
65336 | 4081 |
lemma word_rotx_0 [simp] : "word_rotr i 0 = 0 \<and> word_rotl i 0 = 0" |
4082 |
by (simp add: word_rotr_dt word_rotl_dt replicate_add [symmetric]) |
|
37660 | 4083 |
|
4084 |
lemma word_roti_0' [simp] : "word_roti n 0 = 0" |
|
65336 | 4085 |
by (auto simp: word_roti_def) |
37660 | 4086 |
|
65268 | 4087 |
lemmas word_rotr_dt_no_bin' [simp] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4088 |
word_rotr_dt [where w="numeral w", unfolded to_bl_numeral] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4089 |
(* FIXME: negative numerals, 0 and 1 *) |
37660 | 4090 |
|
65268 | 4091 |
lemmas word_rotl_dt_no_bin' [simp] = |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4092 |
word_rotl_dt [where w="numeral w", unfolded to_bl_numeral] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4093 |
(* FIXME: negative numerals, 0 and 1 *) |
37660 | 4094 |
|
4095 |
declare word_roti_def [simp] |
|
4096 |
||
4097 |
||
61799 | 4098 |
subsection \<open>Maximum machine word\<close> |
37660 | 4099 |
|
4100 |
lemma word_int_cases: |
|
65336 | 4101 |
fixes x :: "'a::len0 word" |
4102 |
obtains n where "x = word_of_int n" and "0 \<le> n" and "n < 2^len_of TYPE('a)" |
|
37660 | 4103 |
by (cases x rule: word_uint.Abs_cases) (simp add: uints_num) |
4104 |
||
4105 |
lemma word_nat_cases [cases type: word]: |
|
65336 | 4106 |
fixes x :: "'a::len word" |
4107 |
obtains n where "x = of_nat n" and "n < 2^len_of TYPE('a)" |
|
37660 | 4108 |
by (cases x rule: word_unat.Abs_cases) (simp add: unats_def) |
4109 |
||
46124 | 4110 |
lemma max_word_eq: "(max_word::'a::len word) = 2^len_of TYPE('a) - 1" |
37660 | 4111 |
by (simp add: max_word_def word_of_int_hom_syms word_of_int_2p) |
4112 |
||
46124 | 4113 |
lemma max_word_max [simp,intro!]: "n \<le> max_word" |
37660 | 4114 |
by (cases n rule: word_int_cases) |
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
4115 |
(simp add: max_word_def word_le_def int_word_uint mod_pos_pos_trivial del: minus_mod_self1) |
65268 | 4116 |
|
46124 | 4117 |
lemma word_of_int_2p_len: "word_of_int (2 ^ len_of TYPE('a)) = (0::'a::len0 word)" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4118 |
by (subst word_uint.Abs_norm [symmetric]) simp |
37660 | 4119 |
|
65336 | 4120 |
lemma word_pow_0: "(2::'a::len word) ^ len_of TYPE('a) = 0" |
37660 | 4121 |
proof - |
4122 |
have "word_of_int (2 ^ len_of TYPE('a)) = (0::'a word)" |
|
4123 |
by (rule word_of_int_2p_len) |
|
65336 | 4124 |
then show ?thesis by (simp add: word_of_int_2p) |
37660 | 4125 |
qed |
4126 |
||
4127 |
lemma max_word_wrap: "x + 1 = 0 \<Longrightarrow> x = max_word" |
|
4128 |
apply (simp add: max_word_eq) |
|
4129 |
apply uint_arith |
|
65336 | 4130 |
apply (auto simp: word_pow_0) |
37660 | 4131 |
done |
4132 |
||
65336 | 4133 |
lemma max_word_minus: "max_word = (-1::'a::len word)" |
37660 | 4134 |
proof - |
65336 | 4135 |
have "-1 + 1 = (0::'a word)" |
4136 |
by simp |
|
4137 |
then show ?thesis |
|
4138 |
by (rule max_word_wrap [symmetric]) |
|
37660 | 4139 |
qed |
4140 |
||
65336 | 4141 |
lemma max_word_bl [simp]: "to_bl (max_word::'a::len word) = replicate (len_of TYPE('a)) True" |
37660 | 4142 |
by (subst max_word_minus to_bl_n1)+ simp |
4143 |
||
65336 | 4144 |
lemma max_test_bit [simp]: "(max_word::'a::len word) !! n \<longleftrightarrow> n < len_of TYPE('a)" |
4145 |
by (auto simp: test_bit_bl word_size) |
|
4146 |
||
4147 |
lemma word_and_max [simp]: "x AND max_word = x" |
|
37660 | 4148 |
by (rule word_eqI) (simp add: word_ops_nth_size word_size) |
4149 |
||
65336 | 4150 |
lemma word_or_max [simp]: "x OR max_word = max_word" |
37660 | 4151 |
by (rule word_eqI) (simp add: word_ops_nth_size word_size) |
4152 |
||
65336 | 4153 |
lemma word_ao_dist2: "x AND (y OR z) = x AND y OR x AND z" |
4154 |
for x y z :: "'a::len0 word" |
|
37660 | 4155 |
by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
4156 |
||
65336 | 4157 |
lemma word_oa_dist2: "x OR y AND z = (x OR y) AND (x OR z)" |
4158 |
for x y z :: "'a::len0 word" |
|
37660 | 4159 |
by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
4160 |
||
65336 | 4161 |
lemma word_and_not [simp]: "x AND NOT x = 0" |
4162 |
for x :: "'a::len0 word" |
|
37660 | 4163 |
by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
4164 |
||
65336 | 4165 |
lemma word_or_not [simp]: "x OR NOT x = max_word" |
37660 | 4166 |
by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
4167 |
||
65336 | 4168 |
lemma word_boolean: "boolean (op AND) (op OR) bitNOT 0 max_word" |
37660 | 4169 |
apply (rule boolean.intro) |
4170 |
apply (rule word_bw_assocs) |
|
4171 |
apply (rule word_bw_assocs) |
|
4172 |
apply (rule word_bw_comms) |
|
4173 |
apply (rule word_bw_comms) |
|
4174 |
apply (rule word_ao_dist2) |
|
4175 |
apply (rule word_oa_dist2) |
|
4176 |
apply (rule word_and_max) |
|
4177 |
apply (rule word_log_esimps) |
|
4178 |
apply (rule word_and_not) |
|
4179 |
apply (rule word_or_not) |
|
4180 |
done |
|
4181 |
||
65336 | 4182 |
interpretation word_bool_alg: boolean "op AND" "op OR" bitNOT 0 max_word |
37660 | 4183 |
by (rule word_boolean) |
4184 |
||
65336 | 4185 |
lemma word_xor_and_or: "x XOR y = x AND NOT y OR NOT x AND y" |
4186 |
for x y :: "'a::len0 word" |
|
37660 | 4187 |
by (rule word_eqI) (auto simp add: word_ops_nth_size word_size) |
4188 |
||
65336 | 4189 |
interpretation word_bool_alg: boolean_xor "op AND" "op OR" bitNOT 0 max_word "op XOR" |
37660 | 4190 |
apply (rule boolean_xor.intro) |
4191 |
apply (rule word_boolean) |
|
4192 |
apply (rule boolean_xor_axioms.intro) |
|
4193 |
apply (rule word_xor_and_or) |
|
4194 |
done |
|
4195 |
||
65336 | 4196 |
lemma shiftr_x_0 [iff]: "x >> 0 = x" |
4197 |
for x :: "'a::len0 word" |
|
37660 | 4198 |
by (simp add: shiftr_bl) |
4199 |
||
65336 | 4200 |
lemma shiftl_x_0 [simp]: "x << 0 = x" |
4201 |
for x :: "'a::len word" |
|
37660 | 4202 |
by (simp add: shiftl_t2n) |
4203 |
||
65336 | 4204 |
lemma shiftl_1 [simp]: "(1::'a::len word) << n = 2^n" |
37660 | 4205 |
by (simp add: shiftl_t2n) |
4206 |
||
65336 | 4207 |
lemma uint_lt_0 [simp]: "uint x < 0 = False" |
37660 | 4208 |
by (simp add: linorder_not_less) |
4209 |
||
65336 | 4210 |
lemma shiftr1_1 [simp]: "shiftr1 (1::'a::len word) = 0" |
45995
b16070689726
declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents:
45958
diff
changeset
|
4211 |
unfolding shiftr1_def by simp |
37660 | 4212 |
|
65336 | 4213 |
lemma shiftr_1[simp]: "(1::'a::len word) >> n = (if n = 0 then 1 else 0)" |
37660 | 4214 |
by (induct n) (auto simp: shiftr_def) |
4215 |
||
65336 | 4216 |
lemma word_less_1 [simp]: "x < 1 \<longleftrightarrow> x = 0" |
4217 |
for x :: "'a::len word" |
|
37660 | 4218 |
by (simp add: word_less_nat_alt unat_0_iff) |
4219 |
||
4220 |
lemma to_bl_mask: |
|
65268 | 4221 |
"to_bl (mask n :: 'a::len word) = |
4222 |
replicate (len_of TYPE('a) - n) False @ |
|
37660 | 4223 |
replicate (min (len_of TYPE('a)) n) True" |
4224 |
by (simp add: mask_bl word_rep_drop min_def) |
|
4225 |
||
4226 |
lemma map_replicate_True: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
4227 |
"n = length xs \<Longrightarrow> |
65336 | 4228 |
map (\<lambda>(x,y). x \<and> y) (zip xs (replicate n True)) = xs" |
37660 | 4229 |
by (induct xs arbitrary: n) auto |
4230 |
||
4231 |
lemma map_replicate_False: |
|
65336 | 4232 |
"n = length xs \<Longrightarrow> map (\<lambda>(x,y). x \<and> y) |
37660 | 4233 |
(zip xs (replicate n False)) = replicate n False" |
4234 |
by (induct xs arbitrary: n) auto |
|
4235 |
||
4236 |
lemma bl_and_mask: |
|
4237 |
fixes w :: "'a::len word" |
|
65336 | 4238 |
and n :: nat |
37660 | 4239 |
defines "n' \<equiv> len_of TYPE('a) - n" |
65336 | 4240 |
shows "to_bl (w AND mask n) = replicate n' False @ drop n' (to_bl w)" |
65268 | 4241 |
proof - |
37660 | 4242 |
note [simp] = map_replicate_True map_replicate_False |
65336 | 4243 |
have "to_bl (w AND mask n) = map2 op \<and> (to_bl w) (to_bl (mask n::'a::len word))" |
37660 | 4244 |
by (simp add: bl_word_and) |
65336 | 4245 |
also have "to_bl w = take n' (to_bl w) @ drop n' (to_bl w)" |
4246 |
by simp |
|
4247 |
also have "map2 op \<and> \<dots> (to_bl (mask n::'a::len word)) = |
|
4248 |
replicate n' False @ drop n' (to_bl w)" |
|
4249 |
unfolding to_bl_mask n'_def map2_def by (subst zip_append) auto |
|
4250 |
finally show ?thesis . |
|
37660 | 4251 |
qed |
4252 |
||
4253 |
lemma drop_rev_takefill: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
4254 |
"length xs \<le> n \<Longrightarrow> |
37660 | 4255 |
drop (n - length xs) (rev (takefill False n (rev xs))) = xs" |
4256 |
by (simp add: takefill_alt rev_take) |
|
4257 |
||
65336 | 4258 |
lemma map_nth_0 [simp]: "map (op !! (0::'a::len0 word)) xs = replicate (length xs) False" |
37660 | 4259 |
by (induct xs) auto |
4260 |
||
4261 |
lemma uint_plus_if_size: |
|
65268 | 4262 |
"uint (x + y) = |
65336 | 4263 |
(if uint x + uint y < 2^size x |
4264 |
then uint x + uint y |
|
4265 |
else uint x + uint y - 2^size x)" |
|
4266 |
by (simp add: word_arith_wis int_word_uint mod_add_if_z word_size) |
|
37660 | 4267 |
|
4268 |
lemma unat_plus_if_size: |
|
65268 | 4269 |
"unat (x + (y::'a::len word)) = |
65336 | 4270 |
(if unat x + unat y < 2^size x |
4271 |
then unat x + unat y |
|
4272 |
else unat x + unat y - 2^size x)" |
|
37660 | 4273 |
apply (subst word_arith_nat_defs) |
4274 |
apply (subst unat_of_nat) |
|
4275 |
apply (simp add: mod_nat_add word_size) |
|
4276 |
done |
|
4277 |
||
65336 | 4278 |
lemma word_neq_0_conv: "w \<noteq> 0 \<longleftrightarrow> 0 < w" |
4279 |
for w :: "'a::len word" |
|
4280 |
by (simp add: word_gt_0) |
|
4281 |
||
4282 |
lemma max_lt: "unat (max a b div c) = unat (max a b) div unat c" |
|
4283 |
for c :: "'a::len word" |
|
55818 | 4284 |
by (fact unat_div) |
37660 | 4285 |
|
4286 |
lemma uint_sub_if_size: |
|
65268 | 4287 |
"uint (x - y) = |
65336 | 4288 |
(if uint y \<le> uint x |
4289 |
then uint x - uint y |
|
4290 |
else uint x - uint y + 2^size x)" |
|
4291 |
by (simp add: word_arith_wis int_word_uint mod_sub_if_z word_size) |
|
4292 |
||
4293 |
lemma unat_sub: "b \<le> a \<Longrightarrow> unat (a - b) = unat a - unat b" |
|
37660 | 4294 |
by (simp add: unat_def uint_sub_if_size word_le_def nat_diff_distrib) |
4295 |
||
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4296 |
lemmas word_less_sub1_numberof [simp] = word_less_sub1 [of "numeral w"] for w |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4297 |
lemmas word_le_sub1_numberof [simp] = word_le_sub1 [of "numeral w"] for w |
65268 | 4298 |
|
65336 | 4299 |
lemma word_of_int_minus: "word_of_int (2^len_of TYPE('a) - i) = (word_of_int (-i)::'a::len word)" |
37660 | 4300 |
proof - |
65336 | 4301 |
have *: "2^len_of TYPE('a) - i = -i + 2^len_of TYPE('a)" |
4302 |
by simp |
|
37660 | 4303 |
show ?thesis |
65336 | 4304 |
apply (subst *) |
37660 | 4305 |
apply (subst word_uint.Abs_norm [symmetric], subst mod_add_self2) |
4306 |
apply simp |
|
4307 |
done |
|
4308 |
qed |
|
65268 | 4309 |
|
4310 |
lemmas word_of_int_inj = |
|
37660 | 4311 |
word_uint.Abs_inject [unfolded uints_num, simplified] |
4312 |
||
65336 | 4313 |
lemma word_le_less_eq: "x \<le> y \<longleftrightarrow> x = y \<or> x < y" |
4314 |
for x y :: "'z::len word" |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4315 |
by (auto simp add: order_class.le_less) |
37660 | 4316 |
|
4317 |
lemma mod_plus_cong: |
|
65336 | 4318 |
fixes b b' :: int |
4319 |
assumes 1: "b = b'" |
|
4320 |
and 2: "x mod b' = x' mod b'" |
|
4321 |
and 3: "y mod b' = y' mod b'" |
|
4322 |
and 4: "x' + y' = z'" |
|
37660 | 4323 |
shows "(x + y) mod b = z' mod b'" |
4324 |
proof - |
|
4325 |
from 1 2[symmetric] 3[symmetric] have "(x + y) mod b = (x' mod b' + y' mod b') mod b'" |
|
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
4326 |
by (simp add: mod_add_eq) |
37660 | 4327 |
also have "\<dots> = (x' + y') mod b'" |
64593
50c715579715
reoriented congruence rules in non-explosive direction
haftmann
parents:
64243
diff
changeset
|
4328 |
by (simp add: mod_add_eq) |
65336 | 4329 |
finally show ?thesis |
4330 |
by (simp add: 4) |
|
37660 | 4331 |
qed |
4332 |
||
4333 |
lemma mod_minus_cong: |
|
65336 | 4334 |
fixes b b' :: int |
4335 |
assumes "b = b'" |
|
4336 |
and "x mod b' = x' mod b'" |
|
4337 |
and "y mod b' = y' mod b'" |
|
4338 |
and "x' - y' = z'" |
|
37660 | 4339 |
shows "(x - y) mod b = z' mod b'" |
65336 | 4340 |
using assms [symmetric] by (auto intro: mod_diff_cong) |
4341 |
||
4342 |
lemma word_induct_less: "\<lbrakk>P 0; \<And>n. \<lbrakk>n < m; P n\<rbrakk> \<Longrightarrow> P (1 + n)\<rbrakk> \<Longrightarrow> P m" |
|
4343 |
for P :: "'a::len word \<Rightarrow> bool" |
|
37660 | 4344 |
apply (cases m) |
4345 |
apply atomize |
|
4346 |
apply (erule rev_mp)+ |
|
4347 |
apply (rule_tac x=m in spec) |
|
4348 |
apply (induct_tac n) |
|
4349 |
apply simp |
|
4350 |
apply clarsimp |
|
4351 |
apply (erule impE) |
|
4352 |
apply clarsimp |
|
4353 |
apply (erule_tac x=n in allE) |
|
4354 |
apply (erule impE) |
|
4355 |
apply (simp add: unat_arith_simps) |
|
4356 |
apply (clarsimp simp: unat_of_nat) |
|
4357 |
apply simp |
|
4358 |
apply (erule_tac x="of_nat na" in allE) |
|
4359 |
apply (erule impE) |
|
4360 |
apply (simp add: unat_arith_simps) |
|
4361 |
apply (clarsimp simp: unat_of_nat) |
|
4362 |
apply simp |
|
4363 |
done |
|
65268 | 4364 |
|
65336 | 4365 |
lemma word_induct: "\<lbrakk>P 0; \<And>n. P n \<Longrightarrow> P (1 + n)\<rbrakk> \<Longrightarrow> P m" |
4366 |
for P :: "'a::len word \<Rightarrow> bool" |
|
4367 |
by (erule word_induct_less) simp |
|
4368 |
||
4369 |
lemma word_induct2 [induct type]: "\<lbrakk>P 0; \<And>n. \<lbrakk>1 + n \<noteq> 0; P n\<rbrakk> \<Longrightarrow> P (1 + n)\<rbrakk> \<Longrightarrow> P n" |
|
4370 |
for P :: "'b::len word \<Rightarrow> bool" |
|
4371 |
apply (rule word_induct) |
|
4372 |
apply simp |
|
4373 |
apply (case_tac "1 + n = 0") |
|
4374 |
apply auto |
|
37660 | 4375 |
done |
4376 |
||
55816
e8dd03241e86
cursory polishing: tuned proofs, tuned symbols, tuned headings
haftmann
parents:
55415
diff
changeset
|
4377 |
|
61799 | 4378 |
subsection \<open>Recursion combinator for words\<close> |
46010 | 4379 |
|
54848 | 4380 |
definition word_rec :: "'a \<Rightarrow> ('b::len word \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'b word \<Rightarrow> 'a" |
65336 | 4381 |
where "word_rec forZero forSuc n = rec_nat forZero (forSuc \<circ> of_nat) (unat n)" |
37660 | 4382 |
|
4383 |
lemma word_rec_0: "word_rec z s 0 = z" |
|
4384 |
by (simp add: word_rec_def) |
|
4385 |
||
65268 | 4386 |
lemma word_rec_Suc: |
37660 | 4387 |
"1 + n \<noteq> (0::'a::len word) \<Longrightarrow> word_rec z s (1 + n) = s n (word_rec z s n)" |
4388 |
apply (simp add: word_rec_def unat_word_ariths) |
|
4389 |
apply (subst nat_mod_eq') |
|
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61424
diff
changeset
|
4390 |
apply (metis Suc_eq_plus1_left Suc_lessI of_nat_2p unat_1 unat_lt2p word_arith_nat_add) |
37660 | 4391 |
apply simp |
4392 |
done |
|
4393 |
||
65268 | 4394 |
lemma word_rec_Pred: |
37660 | 4395 |
"n \<noteq> 0 \<Longrightarrow> word_rec z s n = s (n - 1) (word_rec z s (n - 1))" |
4396 |
apply (rule subst[where t="n" and s="1 + (n - 1)"]) |
|
4397 |
apply simp |
|
4398 |
apply (subst word_rec_Suc) |
|
4399 |
apply simp |
|
4400 |
apply simp |
|
4401 |
done |
|
4402 |
||
65336 | 4403 |
lemma word_rec_in: "f (word_rec z (\<lambda>_. f) n) = word_rec (f z) (\<lambda>_. f) n" |
37660 | 4404 |
by (induct n) (simp_all add: word_rec_0 word_rec_Suc) |
4405 |
||
65336 | 4406 |
lemma word_rec_in2: "f n (word_rec z f n) = word_rec (f 0 z) (f \<circ> op + 1) n" |
37660 | 4407 |
by (induct n) (simp_all add: word_rec_0 word_rec_Suc) |
4408 |
||
65268 | 4409 |
lemma word_rec_twice: |
37660 | 4410 |
"m \<le> n \<Longrightarrow> word_rec z f n = word_rec (word_rec z f (n - m)) (f \<circ> op + (n - m)) m" |
65336 | 4411 |
apply (erule rev_mp) |
4412 |
apply (rule_tac x=z in spec) |
|
4413 |
apply (rule_tac x=f in spec) |
|
4414 |
apply (induct n) |
|
4415 |
apply (simp add: word_rec_0) |
|
4416 |
apply clarsimp |
|
4417 |
apply (rule_tac t="1 + n - m" and s="1 + (n - m)" in subst) |
|
4418 |
apply simp |
|
4419 |
apply (case_tac "1 + (n - m) = 0") |
|
4420 |
apply (simp add: word_rec_0) |
|
4421 |
apply (rule_tac f = "word_rec a b" for a b in arg_cong) |
|
4422 |
apply (rule_tac t="m" and s="m + (1 + (n - m))" in subst) |
|
4423 |
apply simp |
|
4424 |
apply (simp (no_asm_use)) |
|
4425 |
apply (simp add: word_rec_Suc word_rec_in2) |
|
4426 |
apply (erule impE) |
|
4427 |
apply uint_arith |
|
4428 |
apply (drule_tac x="x \<circ> op + 1" in spec) |
|
4429 |
apply (drule_tac x="x 0 xa" in spec) |
|
37660 | 4430 |
apply simp |
65336 | 4431 |
apply (rule_tac t="\<lambda>a. x (1 + (n - m + a))" and s="\<lambda>a. x (1 + (n - m) + a)" in subst) |
4432 |
apply (clarsimp simp add: fun_eq_iff) |
|
4433 |
apply (rule_tac t="(1 + (n - m + xb))" and s="1 + (n - m) + xb" in subst) |
|
4434 |
apply simp |
|
4435 |
apply (rule refl) |
|
4436 |
apply (rule refl) |
|
4437 |
done |
|
37660 | 4438 |
|
4439 |
lemma word_rec_id: "word_rec z (\<lambda>_. id) n = z" |
|
4440 |
by (induct n) (auto simp add: word_rec_0 word_rec_Suc) |
|
4441 |
||
4442 |
lemma word_rec_id_eq: "\<forall>m < n. f m = id \<Longrightarrow> word_rec z f n = z" |
|
65336 | 4443 |
apply (erule rev_mp) |
4444 |
apply (induct n) |
|
4445 |
apply (auto simp add: word_rec_0 word_rec_Suc) |
|
4446 |
apply (drule spec, erule mp) |
|
4447 |
apply uint_arith |
|
4448 |
apply (drule_tac x=n in spec, erule impE) |
|
4449 |
apply uint_arith |
|
4450 |
apply simp |
|
4451 |
done |
|
37660 | 4452 |
|
65268 | 4453 |
lemma word_rec_max: |
58410
6d46ad54a2ab
explicit separation of signed and unsigned numerals using existing lexical categories num and xnum
haftmann
parents:
58061
diff
changeset
|
4454 |
"\<forall>m\<ge>n. m \<noteq> - 1 \<longrightarrow> f m = id \<Longrightarrow> word_rec z f (- 1) = word_rec z f n" |
65336 | 4455 |
apply (subst word_rec_twice[where n="-1" and m="-1 - n"]) |
4456 |
apply simp |
|
4457 |
apply simp |
|
4458 |
apply (rule word_rec_id_eq) |
|
4459 |
apply clarsimp |
|
4460 |
apply (drule spec, rule mp, erule mp) |
|
4461 |
apply (rule word_plus_mono_right2[OF _ order_less_imp_le]) |
|
4462 |
prefer 2 |
|
4463 |
apply assumption |
|
4464 |
apply simp |
|
4465 |
apply (erule contrapos_pn) |
|
4466 |
apply simp |
|
4467 |
apply (drule arg_cong[where f="\<lambda>x. x - n"]) |
|
4468 |
apply simp |
|
4469 |
done |
|
4470 |
||
4471 |
lemma unatSuc: "1 + n \<noteq> (0::'a::len word) \<Longrightarrow> unat (1 + n) = Suc (unat n)" |
|
37660 | 4472 |
by unat_arith |
4473 |
||
4474 |
declare bin_to_bl_def [simp] |
|
4475 |
||
53080 | 4476 |
ML_file "Tools/word_lib.ML" |
58061 | 4477 |
ML_file "Tools/smt_word.ML" |
36899
bcd6fce5bf06
layered SMT setup, adapted SMT clients, added further tests, made Z3 proof abstraction configurable
boehmes
parents:
35049
diff
changeset
|
4478 |
|
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4479 |
hide_const (open) Word |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46962
diff
changeset
|
4480 |
|
41060
4199fdcfa3c0
moved smt_word.ML into the directory of the Word library
boehmes
parents:
40827
diff
changeset
|
4481 |
end |