author | huffman |
Sat, 10 Dec 2011 08:29:19 +0100 | |
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parent 45804 | 3a3e4c58083c |
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permissions | -rw-r--r-- |
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(* Title: HOL/Word/Word.thy |
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Author: Jeremy Dawson and Gerwin Klein, NICTA |
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*) |
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header {* A type of finite bit strings *} |
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theory Word |
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imports |
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Type_Length |
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Misc_Typedef |
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"~~/src/HOL/Library/Boolean_Algebra" |
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Bool_List_Representation |
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uses ("~~/src/HOL/Word/Tools/smt_word.ML") |
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begin |
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text {* see @{text "Examples/WordExamples.thy"} for examples *} |
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subsection {* Type definition *} |
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typedef (open) 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}" |
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morphisms uint Abs_word by auto |
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definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" where |
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-- {* representation of words using unsigned or signed bins, |
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only difference in these is the type class *} |
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"word_of_int w = Abs_word (bintrunc (len_of TYPE ('a)) w)" |
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lemma uint_word_of_int [code]: "uint (word_of_int w \<Colon> 'a\<Colon>len0 word) = w mod 2 ^ len_of TYPE('a)" |
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by (auto simp add: word_of_int_def bintrunc_mod2p intro: Abs_word_inverse) |
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code_datatype word_of_int |
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subsection {* Random instance *} |
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notation fcomp (infixl "\<circ>>" 60) |
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notation scomp (infixl "\<circ>\<rightarrow>" 60) |
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instantiation word :: ("{len0, typerep}") random |
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begin |
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definition |
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"random_word i = Random.range (max i (2 ^ len_of TYPE('a))) \<circ>\<rightarrow> (\<lambda>k. Pair ( |
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let j = word_of_int (Code_Numeral.int_of k) :: 'a word |
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in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))" |
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instance .. |
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end |
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no_notation fcomp (infixl "\<circ>>" 60) |
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no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
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subsection {* Type conversions and casting *} |
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definition sint :: "'a :: len word => int" where |
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-- {* treats the most-significant-bit as a sign bit *} |
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sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)" |
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definition unat :: "'a :: len0 word => nat" where |
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"unat w = nat (uint w)" |
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definition uints :: "nat => int set" where |
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-- "the sets of integers representing the words" |
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"uints n = range (bintrunc n)" |
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definition sints :: "nat => int set" where |
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"sints n = range (sbintrunc (n - 1))" |
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definition unats :: "nat => nat set" where |
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"unats n = {i. i < 2 ^ n}" |
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definition norm_sint :: "nat => int => int" where |
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"norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)" |
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definition scast :: "'a :: len word => 'b :: len word" where |
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-- "cast a word to a different length" |
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"scast w = word_of_int (sint w)" |
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definition ucast :: "'a :: len0 word => 'b :: len0 word" where |
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"ucast w = word_of_int (uint w)" |
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instantiation word :: (len0) size |
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begin |
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definition |
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word_size: "size (w :: 'a word) = len_of TYPE('a)" |
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instance .. |
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end |
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definition source_size :: "('a :: len0 word => 'b) => nat" where |
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-- "whether a cast (or other) function is to a longer or shorter length" |
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"source_size c = (let arb = undefined ; x = c arb in size arb)" |
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definition target_size :: "('a => 'b :: len0 word) => nat" where |
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"target_size c = size (c undefined)" |
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definition is_up :: "('a :: len0 word => 'b :: len0 word) => bool" where |
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"is_up c \<longleftrightarrow> source_size c <= target_size c" |
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definition is_down :: "('a :: len0 word => 'b :: len0 word) => bool" where |
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"is_down c \<longleftrightarrow> target_size c <= source_size c" |
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definition of_bl :: "bool list => 'a :: len0 word" where |
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"of_bl bl = word_of_int (bl_to_bin bl)" |
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definition to_bl :: "'a :: len0 word => bool list" where |
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"to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)" |
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definition word_reverse :: "'a :: len0 word => 'a word" where |
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"word_reverse w = of_bl (rev (to_bl w))" |
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definition word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b" where |
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"word_int_case f w = f (uint w)" |
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syntax |
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of_int :: "int => 'a" |
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translations |
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"case x of CONST of_int y => b" == "CONST word_int_case (%y. b) x" |
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subsection {* Type-definition locale instantiations *} |
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lemma word_size_gt_0 [iff]: "0 < size (w::'a::len word)" |
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by (fact xtr1 [OF word_size len_gt_0]) |
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lemmas lens_gt_0 = word_size_gt_0 len_gt_0 |
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lemmas lens_not_0 [iff] = lens_gt_0 [THEN gr_implies_not0] |
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lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}" |
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by (simp add: uints_def range_bintrunc) |
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lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}" |
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by (simp add: sints_def range_sbintrunc) |
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(* FIXME: delete *) |
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lemmas atLeastLessThan_alt = atLeastLessThan_def [unfolded |
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atLeast_def lessThan_def Collect_conj_eq [symmetric]] |
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lemma mod_in_reps: "m > 0 \<Longrightarrow> y mod m : {0::int ..< m}" |
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by auto |
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lemma |
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uint_0:"0 <= uint x" and |
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uint_lt: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)" |
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by (auto simp: uint [simplified]) |
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lemma uint_mod_same: |
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"uint x mod 2 ^ len_of TYPE('a) = uint (x::'a::len0 word)" |
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by (simp add: int_mod_eq uint_lt uint_0) |
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lemma td_ext_uint: |
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"td_ext (uint :: 'a word => int) word_of_int (uints (len_of TYPE('a::len0))) |
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(%w::int. w mod 2 ^ len_of TYPE('a))" |
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apply (unfold td_ext_def') |
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apply (simp add: uints_num word_of_int_def bintrunc_mod2p) |
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apply (simp add: uint_mod_same uint_0 uint_lt |
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word.uint_inverse word.Abs_word_inverse int_mod_lem) |
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done |
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lemma int_word_uint: |
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"uint (word_of_int x::'a::len0 word) = x mod 2 ^ len_of TYPE('a)" |
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by (fact td_ext_uint [THEN td_ext.eq_norm]) |
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interpretation word_uint: |
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td_ext "uint::'a::len0 word \<Rightarrow> int" |
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word_of_int |
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"uints (len_of TYPE('a::len0))" |
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"\<lambda>w. w mod 2 ^ len_of TYPE('a::len0)" |
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by (rule td_ext_uint) |
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lemmas td_uint = word_uint.td_thm |
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|
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lemmas td_ext_ubin = td_ext_uint |
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[simplified len_gt_0 no_bintr_alt1 [symmetric]] |
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interpretation word_ubin: |
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td_ext "uint::'a::len0 word \<Rightarrow> int" |
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word_of_int |
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"uints (len_of TYPE('a::len0))" |
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"bintrunc (len_of TYPE('a::len0))" |
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by (rule td_ext_ubin) |
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lemma split_word_all: |
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"(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))" |
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proof |
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fix x :: "'a word" |
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assume "\<And>x. PROP P (word_of_int x)" |
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hence "PROP P (word_of_int (uint x))" . |
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thus "PROP P x" by simp |
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qed |
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subsection "Arithmetic operations" |
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definition |
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word_succ :: "'a :: len0 word => 'a word" |
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where |
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"word_succ a = word_of_int (Int.succ (uint a))" |
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definition |
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word_pred :: "'a :: len0 word => 'a word" |
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where |
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"word_pred a = word_of_int (Int.pred (uint a))" |
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instantiation word :: (len0) "{number, Divides.div, comm_monoid_mult, comm_ring}" |
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begin |
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definition |
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word_0_wi: "0 = word_of_int 0" |
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definition |
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word_1_wi: "1 = word_of_int 1" |
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definition |
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word_add_def: "a + b = word_of_int (uint a + uint b)" |
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definition |
|
219 |
word_sub_wi: "a - b = word_of_int (uint a - uint b)" |
|
220 |
||
221 |
definition |
|
222 |
word_minus_def: "- a = word_of_int (- uint a)" |
|
223 |
||
224 |
definition |
|
225 |
word_mult_def: "a * b = word_of_int (uint a * uint b)" |
|
226 |
||
227 |
definition |
|
228 |
word_div_def: "a div b = word_of_int (uint a div uint b)" |
|
229 |
||
230 |
definition |
|
231 |
word_mod_def: "a mod b = word_of_int (uint a mod uint b)" |
|
232 |
||
233 |
definition |
|
234 |
word_number_of_def: "number_of w = word_of_int w" |
|
235 |
||
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236 |
lemmas word_arith_wis = |
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|
237 |
word_add_def word_mult_def word_minus_def |
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|
238 |
word_succ_def word_pred_def word_0_wi word_1_wi |
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|
239 |
|
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|
240 |
lemmas arths = |
45604 | 241 |
bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm] |
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|
242 |
|
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|
243 |
lemma wi_homs: |
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|
244 |
shows |
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|
245 |
wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and |
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|
246 |
wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and |
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|
247 |
wi_hom_neg: "- word_of_int a = word_of_int (- a)" and |
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|
248 |
wi_hom_succ: "word_succ (word_of_int a) = word_of_int (Int.succ a)" and |
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|
249 |
wi_hom_pred: "word_pred (word_of_int a) = word_of_int (Int.pred a)" |
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|
250 |
by (auto simp: word_arith_wis arths) |
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|
251 |
|
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|
252 |
lemmas wi_hom_syms = wi_homs [symmetric] |
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|
253 |
|
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|
254 |
lemma word_sub_def: "a - b = a + - (b :: 'a :: len0 word)" |
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|
255 |
unfolding word_sub_wi diff_minus |
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|
256 |
by (simp only : word_uint.Rep_inverse wi_hom_syms) |
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|
257 |
|
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|
258 |
lemma word_of_int_sub_hom: |
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|
259 |
"(word_of_int a) - word_of_int b = word_of_int (a - b)" |
45805 | 260 |
by (simp add: word_sub_wi arths) |
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|
261 |
|
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|
262 |
lemmas new_word_of_int_homs = |
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|
263 |
word_of_int_sub_hom wi_homs word_0_wi word_1_wi |
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|
264 |
|
45604 | 265 |
lemmas new_word_of_int_hom_syms = new_word_of_int_homs [symmetric] |
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|
266 |
|
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|
267 |
lemmas word_of_int_hom_syms = |
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|
268 |
new_word_of_int_hom_syms [unfolded succ_def pred_def] |
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|
269 |
|
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|
270 |
lemmas word_of_int_homs = |
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|
271 |
new_word_of_int_homs [unfolded succ_def pred_def] |
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|
272 |
|
45805 | 273 |
(* FIXME: provide only one copy of these theorems! *) |
274 |
lemmas word_of_int_add_hom = wi_hom_add |
|
275 |
lemmas word_of_int_mult_hom = wi_hom_mult |
|
276 |
lemmas word_of_int_minus_hom = wi_hom_neg |
|
277 |
lemmas word_of_int_succ_hom = wi_hom_succ [unfolded succ_def] |
|
278 |
lemmas word_of_int_pred_hom = wi_hom_pred [unfolded pred_def] |
|
279 |
lemmas word_of_int_0_hom = word_0_wi |
|
280 |
lemmas word_of_int_1_hom = word_1_wi |
|
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|
281 |
|
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|
282 |
instance |
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|
283 |
by default (auto simp: split_word_all word_of_int_homs algebra_simps) |
37660 | 284 |
|
285 |
end |
|
286 |
||
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|
287 |
lemma word_zero_neq_one: "0 < len_of TYPE ('a :: len0) \<Longrightarrow> (0 :: 'a word) ~= 1" |
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|
288 |
unfolding word_arith_wis |
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|
289 |
by (auto simp add: word_ubin.norm_eq_iff [symmetric] gr0_conv_Suc) |
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|
290 |
|
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|
291 |
lemmas lenw1_zero_neq_one = len_gt_0 [THEN word_zero_neq_one] |
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292 |
|
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|
293 |
instance word :: (len) comm_ring_1 |
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|
294 |
by (intro_classes) (simp add: lenw1_zero_neq_one) |
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|
295 |
|
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|
296 |
lemma word_of_nat: "of_nat n = word_of_int (int n)" |
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|
297 |
by (induct n) (auto simp add : word_of_int_hom_syms) |
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|
298 |
|
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|
299 |
lemma word_of_int: "of_int = word_of_int" |
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|
300 |
apply (rule ext) |
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|
301 |
apply (case_tac x rule: int_diff_cases) |
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|
302 |
apply (simp add: word_of_nat word_of_int_sub_hom) |
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|
303 |
done |
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|
304 |
|
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|
305 |
lemma word_number_of_eq: |
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|
306 |
"number_of w = (of_int w :: 'a :: len word)" |
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|
307 |
unfolding word_number_of_def word_of_int by auto |
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|
308 |
|
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|
309 |
instance word :: (len) number_ring |
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|
310 |
by (intro_classes) (simp add : word_number_of_eq) |
37660 | 311 |
|
312 |
definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where |
|
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313 |
"a udvd b = (EX n>=0. uint b = n * uint a)" |
37660 | 314 |
|
45547 | 315 |
|
316 |
subsection "Ordering" |
|
317 |
||
318 |
instantiation word :: (len0) linorder |
|
319 |
begin |
|
320 |
||
37660 | 321 |
definition |
322 |
word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b" |
|
323 |
||
324 |
definition |
|
325 |
word_less_def: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> (y \<Colon> 'a word)" |
|
326 |
||
45547 | 327 |
instance |
328 |
by default (auto simp: word_less_def word_le_def) |
|
329 |
||
330 |
end |
|
331 |
||
37660 | 332 |
definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where |
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|
333 |
"a <=s b = (sint a <= sint b)" |
37660 | 334 |
|
335 |
definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where |
|
40827
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|
336 |
"(x <s y) = (x <=s y & x ~= y)" |
37660 | 337 |
|
338 |
||
339 |
subsection "Bit-wise operations" |
|
340 |
||
341 |
instantiation word :: (len0) bits |
|
342 |
begin |
|
343 |
||
344 |
definition |
|
345 |
word_and_def: |
|
346 |
"(a::'a word) AND b = word_of_int (uint a AND uint b)" |
|
347 |
||
348 |
definition |
|
349 |
word_or_def: |
|
350 |
"(a::'a word) OR b = word_of_int (uint a OR uint b)" |
|
351 |
||
352 |
definition |
|
353 |
word_xor_def: |
|
354 |
"(a::'a word) XOR b = word_of_int (uint a XOR uint b)" |
|
355 |
||
356 |
definition |
|
357 |
word_not_def: |
|
358 |
"NOT (a::'a word) = word_of_int (NOT (uint a))" |
|
359 |
||
360 |
definition |
|
361 |
word_test_bit_def: "test_bit a = bin_nth (uint a)" |
|
362 |
||
363 |
definition |
|
364 |
word_set_bit_def: "set_bit a n x = |
|
365 |
word_of_int (bin_sc n (If x 1 0) (uint a))" |
|
366 |
||
367 |
definition |
|
368 |
word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)" |
|
369 |
||
370 |
definition |
|
371 |
word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1" |
|
372 |
||
373 |
definition shiftl1 :: "'a word \<Rightarrow> 'a word" where |
|
374 |
"shiftl1 w = word_of_int (uint w BIT 0)" |
|
375 |
||
376 |
definition shiftr1 :: "'a word \<Rightarrow> 'a word" where |
|
377 |
-- "shift right as unsigned or as signed, ie logical or arithmetic" |
|
378 |
"shiftr1 w = word_of_int (bin_rest (uint w))" |
|
379 |
||
380 |
definition |
|
381 |
shiftl_def: "w << n = (shiftl1 ^^ n) w" |
|
382 |
||
383 |
definition |
|
384 |
shiftr_def: "w >> n = (shiftr1 ^^ n) w" |
|
385 |
||
386 |
instance .. |
|
387 |
||
388 |
end |
|
389 |
||
390 |
instantiation word :: (len) bitss |
|
391 |
begin |
|
392 |
||
393 |
definition |
|
394 |
word_msb_def: |
|
395 |
"msb a \<longleftrightarrow> bin_sign (sint a) = Int.Min" |
|
396 |
||
397 |
instance .. |
|
398 |
||
399 |
end |
|
400 |
||
37667 | 401 |
lemma [code]: |
402 |
"msb a \<longleftrightarrow> bin_sign (sint a) = (- 1 :: int)" |
|
403 |
by (simp only: word_msb_def Min_def) |
|
404 |
||
37660 | 405 |
definition setBit :: "'a :: len0 word => nat => 'a word" where |
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|
406 |
"setBit w n = set_bit w n True" |
37660 | 407 |
|
408 |
definition clearBit :: "'a :: len0 word => nat => 'a word" where |
|
40827
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|
409 |
"clearBit w n = set_bit w n False" |
37660 | 410 |
|
411 |
||
412 |
subsection "Shift operations" |
|
413 |
||
414 |
definition sshiftr1 :: "'a :: len word => 'a word" where |
|
40827
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|
415 |
"sshiftr1 w = word_of_int (bin_rest (sint w))" |
37660 | 416 |
|
417 |
definition bshiftr1 :: "bool => 'a :: len word => 'a word" where |
|
40827
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|
418 |
"bshiftr1 b w = of_bl (b # butlast (to_bl w))" |
37660 | 419 |
|
420 |
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where |
|
40827
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|
421 |
"w >>> n = (sshiftr1 ^^ n) w" |
37660 | 422 |
|
423 |
definition mask :: "nat => 'a::len word" where |
|
40827
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|
424 |
"mask n = (1 << n) - 1" |
37660 | 425 |
|
426 |
definition revcast :: "'a :: len0 word => 'b :: len0 word" where |
|
40827
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|
427 |
"revcast w = of_bl (takefill False (len_of TYPE('b)) (to_bl w))" |
37660 | 428 |
|
429 |
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where |
|
40827
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|
430 |
"slice1 n w = of_bl (takefill False n (to_bl w))" |
37660 | 431 |
|
432 |
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where |
|
40827
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|
433 |
"slice n w = slice1 (size w - n) w" |
37660 | 434 |
|
435 |
||
436 |
subsection "Rotation" |
|
437 |
||
438 |
definition rotater1 :: "'a list => 'a list" where |
|
40827
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haftmann
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|
439 |
"rotater1 ys = |
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parents:
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changeset
|
440 |
(case ys of [] => [] | x # xs => last ys # butlast ys)" |
37660 | 441 |
|
442 |
definition rotater :: "nat => 'a list => 'a list" where |
|
40827
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|
443 |
"rotater n = rotater1 ^^ n" |
37660 | 444 |
|
445 |
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where |
|
40827
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changeset
|
446 |
"word_rotr n w = of_bl (rotater n (to_bl w))" |
37660 | 447 |
|
448 |
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
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diff
changeset
|
449 |
"word_rotl n w = of_bl (rotate n (to_bl w))" |
37660 | 450 |
|
451 |
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
452 |
"word_roti i w = (if i >= 0 then word_rotr (nat i) w |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
453 |
else word_rotl (nat (- i)) w)" |
37660 | 454 |
|
455 |
||
456 |
subsection "Split and cat operations" |
|
457 |
||
458 |
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
459 |
"word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))" |
37660 | 460 |
|
461 |
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
462 |
"word_split a = |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
463 |
(case bin_split (len_of TYPE ('c)) (uint a) of |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
464 |
(u, v) => (word_of_int u, word_of_int v))" |
37660 | 465 |
|
466 |
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
467 |
"word_rcat ws = |
37660 | 468 |
word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))" |
469 |
||
470 |
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
471 |
"word_rsplit w = |
37660 | 472 |
map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))" |
473 |
||
474 |
definition max_word :: "'a::len word" -- "Largest representable machine integer." where |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
475 |
"max_word = word_of_int (2 ^ len_of TYPE('a) - 1)" |
37660 | 476 |
|
477 |
primrec of_bool :: "bool \<Rightarrow> 'a::len word" where |
|
478 |
"of_bool False = 0" |
|
479 |
| "of_bool True = 1" |
|
480 |
||
45805 | 481 |
(* FIXME: only provide one theorem name *) |
37660 | 482 |
lemmas of_nth_def = word_set_bits_def |
483 |
||
484 |
lemma sint_sbintrunc': |
|
485 |
"sint (word_of_int bin :: 'a word) = |
|
486 |
(sbintrunc (len_of TYPE ('a :: len) - 1) bin)" |
|
487 |
unfolding sint_uint |
|
488 |
by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt) |
|
489 |
||
490 |
lemma uint_sint: |
|
491 |
"uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))" |
|
492 |
unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le) |
|
493 |
||
494 |
lemma bintr_uint': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
495 |
"n >= size w \<Longrightarrow> bintrunc n (uint w) = uint w" |
37660 | 496 |
apply (unfold word_size) |
497 |
apply (subst word_ubin.norm_Rep [symmetric]) |
|
498 |
apply (simp only: bintrunc_bintrunc_min word_size) |
|
499 |
apply (simp add: min_max.inf_absorb2) |
|
500 |
done |
|
501 |
||
502 |
lemma wi_bintr': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
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39910
diff
changeset
|
503 |
"wb = word_of_int bin \<Longrightarrow> n >= size wb \<Longrightarrow> |
37660 | 504 |
word_of_int (bintrunc n bin) = wb" |
505 |
unfolding word_size |
|
506 |
by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1) |
|
507 |
||
508 |
lemmas bintr_uint = bintr_uint' [unfolded word_size] |
|
509 |
lemmas wi_bintr = wi_bintr' [unfolded word_size] |
|
510 |
||
511 |
lemma td_ext_sbin: |
|
512 |
"td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) |
|
513 |
(sbintrunc (len_of TYPE('a) - 1))" |
|
514 |
apply (unfold td_ext_def' sint_uint) |
|
515 |
apply (simp add : word_ubin.eq_norm) |
|
516 |
apply (cases "len_of TYPE('a)") |
|
517 |
apply (auto simp add : sints_def) |
|
518 |
apply (rule sym [THEN trans]) |
|
519 |
apply (rule word_ubin.Abs_norm) |
|
520 |
apply (simp only: bintrunc_sbintrunc) |
|
521 |
apply (drule sym) |
|
522 |
apply simp |
|
523 |
done |
|
524 |
||
525 |
lemmas td_ext_sint = td_ext_sbin |
|
526 |
[simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]] |
|
527 |
||
528 |
(* We do sint before sbin, before sint is the user version |
|
529 |
and interpretations do not produce thm duplicates. I.e. |
|
530 |
we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD, |
|
531 |
because the latter is the same thm as the former *) |
|
532 |
interpretation word_sint: |
|
533 |
td_ext "sint ::'a::len word => int" |
|
534 |
word_of_int |
|
535 |
"sints (len_of TYPE('a::len))" |
|
536 |
"%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) - |
|
537 |
2 ^ (len_of TYPE('a::len) - 1)" |
|
538 |
by (rule td_ext_sint) |
|
539 |
||
540 |
interpretation word_sbin: |
|
541 |
td_ext "sint ::'a::len word => int" |
|
542 |
word_of_int |
|
543 |
"sints (len_of TYPE('a::len))" |
|
544 |
"sbintrunc (len_of TYPE('a::len) - 1)" |
|
545 |
by (rule td_ext_sbin) |
|
546 |
||
45604 | 547 |
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm] |
37660 | 548 |
|
549 |
lemmas td_sint = word_sint.td |
|
550 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
551 |
lemma word_number_of_alt [code_unfold_post]: |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
552 |
"number_of b = word_of_int (number_of b)" |
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
553 |
by (simp add: number_of_eq word_number_of_def) |
37660 | 554 |
|
555 |
lemma word_no_wi: "number_of = word_of_int" |
|
44762 | 556 |
by (auto simp: word_number_of_def) |
37660 | 557 |
|
558 |
lemma to_bl_def': |
|
559 |
"(to_bl :: 'a :: len0 word => bool list) = |
|
560 |
bin_to_bl (len_of TYPE('a)) o uint" |
|
44762 | 561 |
by (auto simp: to_bl_def) |
37660 | 562 |
|
45604 | 563 |
lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w"] for w |
37660 | 564 |
|
45805 | 565 |
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)" |
566 |
by (fact uints_def [unfolded no_bintr_alt1]) |
|
567 |
||
568 |
lemma uint_bintrunc [simp]: |
|
569 |
"uint (number_of bin :: 'a word) = |
|
37660 | 570 |
number_of (bintrunc (len_of TYPE ('a :: len0)) bin)" |
571 |
unfolding word_number_of_def number_of_eq |
|
572 |
by (auto intro: word_ubin.eq_norm) |
|
573 |
||
45805 | 574 |
lemma sint_sbintrunc [simp]: |
575 |
"sint (number_of bin :: 'a word) = |
|
37660 | 576 |
number_of (sbintrunc (len_of TYPE ('a :: len) - 1) bin)" |
577 |
unfolding word_number_of_def number_of_eq |
|
578 |
by (subst word_sbin.eq_norm) simp |
|
579 |
||
45805 | 580 |
lemma unat_bintrunc [simp]: |
37660 | 581 |
"unat (number_of bin :: 'a :: len0 word) = |
582 |
number_of (bintrunc (len_of TYPE('a)) bin)" |
|
583 |
unfolding unat_def nat_number_of_def |
|
584 |
by (simp only: uint_bintrunc) |
|
585 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
586 |
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w" |
37660 | 587 |
apply (unfold word_size) |
588 |
apply (rule word_uint.Rep_eqD) |
|
589 |
apply (rule box_equals) |
|
590 |
defer |
|
591 |
apply (rule word_ubin.norm_Rep)+ |
|
592 |
apply simp |
|
593 |
done |
|
594 |
||
45805 | 595 |
(* TODO: remove uint_lem and sint_lem *) |
37660 | 596 |
lemmas uint_lem = word_uint.Rep [unfolded uints_num mem_Collect_eq] |
597 |
lemmas sint_lem = word_sint.Rep [unfolded sints_num mem_Collect_eq] |
|
45805 | 598 |
|
599 |
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)" |
|
600 |
using word_uint.Rep [of x] by (simp add: uints_num) |
|
601 |
||
602 |
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)" |
|
603 |
using word_uint.Rep [of x] by (simp add: uints_num) |
|
604 |
||
605 |
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)" |
|
606 |
using word_sint.Rep [of x] by (simp add: sints_num) |
|
607 |
||
608 |
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)" |
|
609 |
using word_sint.Rep [of x] by (simp add: sints_num) |
|
37660 | 610 |
|
611 |
lemma sign_uint_Pls [simp]: |
|
612 |
"bin_sign (uint x) = Int.Pls" |
|
613 |
by (simp add: sign_Pls_ge_0 number_of_eq) |
|
614 |
||
45805 | 615 |
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0" |
616 |
by (simp only: diff_less_0_iff_less uint_lt2p) |
|
617 |
||
618 |
lemma uint_m2p_not_non_neg: |
|
619 |
"\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)" |
|
620 |
by (simp only: not_le uint_m2p_neg) |
|
37660 | 621 |
|
622 |
lemma lt2p_lem: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
623 |
"len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n" |
37660 | 624 |
by (rule xtr8 [OF _ uint_lt2p]) simp |
625 |
||
45805 | 626 |
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0" |
627 |
by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1]) |
|
37660 | 628 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
629 |
lemma uint_nat: "uint w = int (unat w)" |
37660 | 630 |
unfolding unat_def by auto |
631 |
||
632 |
lemma uint_number_of: |
|
633 |
"uint (number_of b :: 'a :: len0 word) = number_of b mod 2 ^ len_of TYPE('a)" |
|
634 |
unfolding word_number_of_alt |
|
635 |
by (simp only: int_word_uint) |
|
636 |
||
637 |
lemma unat_number_of: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
638 |
"bin_sign b = Int.Pls \<Longrightarrow> |
37660 | 639 |
unat (number_of b::'a::len0 word) = number_of b mod 2 ^ len_of TYPE ('a)" |
640 |
apply (unfold unat_def) |
|
641 |
apply (clarsimp simp only: uint_number_of) |
|
642 |
apply (rule nat_mod_distrib [THEN trans]) |
|
643 |
apply (erule sign_Pls_ge_0 [THEN iffD1]) |
|
644 |
apply (simp_all add: nat_power_eq) |
|
645 |
done |
|
646 |
||
647 |
lemma sint_number_of: "sint (number_of b :: 'a :: len word) = (number_of b + |
|
648 |
2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) - |
|
649 |
2 ^ (len_of TYPE('a) - 1)" |
|
650 |
unfolding word_number_of_alt by (rule int_word_sint) |
|
651 |
||
652 |
lemma word_of_int_bin [simp] : |
|
653 |
"(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)" |
|
654 |
unfolding word_number_of_alt by auto |
|
655 |
||
656 |
lemma word_int_case_wi: |
|
657 |
"word_int_case f (word_of_int i :: 'b word) = |
|
658 |
f (i mod 2 ^ len_of TYPE('b::len0))" |
|
659 |
unfolding word_int_case_def by (simp add: word_uint.eq_norm) |
|
660 |
||
661 |
lemma word_int_split: |
|
662 |
"P (word_int_case f x) = |
|
663 |
(ALL i. x = (word_of_int i :: 'b :: len0 word) & |
|
664 |
0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))" |
|
665 |
unfolding word_int_case_def |
|
666 |
by (auto simp: word_uint.eq_norm int_mod_eq') |
|
667 |
||
668 |
lemma word_int_split_asm: |
|
669 |
"P (word_int_case f x) = |
|
670 |
(~ (EX n. x = (word_of_int n :: 'b::len0 word) & |
|
671 |
0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))" |
|
672 |
unfolding word_int_case_def |
|
673 |
by (auto simp: word_uint.eq_norm int_mod_eq') |
|
45805 | 674 |
|
675 |
(* FIXME: uint_range' is an exact duplicate of uint_lem *) |
|
45604 | 676 |
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq] |
677 |
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq] |
|
37660 | 678 |
|
679 |
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w" |
|
680 |
unfolding word_size by (rule uint_range') |
|
681 |
||
682 |
lemma sint_range_size: |
|
683 |
"- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)" |
|
684 |
unfolding word_size by (rule sint_range') |
|
685 |
||
45805 | 686 |
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x" |
687 |
unfolding word_size by (rule less_le_trans [OF sint_lt]) |
|
688 |
||
689 |
lemma sint_below_size: |
|
690 |
"x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w" |
|
691 |
unfolding word_size by (rule order_trans [OF _ sint_ge]) |
|
37660 | 692 |
|
693 |
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)" |
|
694 |
unfolding word_test_bit_def by (simp add: bin_nth_eq_iff) |
|
695 |
||
696 |
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w" |
|
697 |
apply (unfold word_test_bit_def) |
|
698 |
apply (subst word_ubin.norm_Rep [symmetric]) |
|
699 |
apply (simp only: nth_bintr word_size) |
|
700 |
apply fast |
|
701 |
done |
|
702 |
||
703 |
lemma word_eqI [rule_format] : |
|
704 |
fixes u :: "'a::len0 word" |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
705 |
shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v" |
37660 | 706 |
apply (rule test_bit_eq_iff [THEN iffD1]) |
707 |
apply (rule ext) |
|
708 |
apply (erule allE) |
|
709 |
apply (erule impCE) |
|
710 |
prefer 2 |
|
711 |
apply assumption |
|
712 |
apply (auto dest!: test_bit_size simp add: word_size) |
|
713 |
done |
|
714 |
||
45805 | 715 |
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x" |
716 |
by simp |
|
37660 | 717 |
|
718 |
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)" |
|
719 |
unfolding word_test_bit_def word_size |
|
720 |
by (simp add: nth_bintr [symmetric]) |
|
721 |
||
722 |
lemmas test_bit_bin = test_bit_bin' [unfolded word_size] |
|
723 |
||
724 |
lemma bin_nth_uint_imp': "bin_nth (uint w) n --> n < size w" |
|
725 |
apply (unfold word_size) |
|
726 |
apply (rule impI) |
|
727 |
apply (rule nth_bintr [THEN iffD1, THEN conjunct1]) |
|
728 |
apply (subst word_ubin.norm_Rep) |
|
729 |
apply assumption |
|
730 |
done |
|
731 |
||
732 |
lemma bin_nth_sint': |
|
733 |
"n >= size w --> bin_nth (sint w) n = bin_nth (sint w) (size w - 1)" |
|
734 |
apply (rule impI) |
|
735 |
apply (subst word_sbin.norm_Rep [symmetric]) |
|
736 |
apply (simp add : nth_sbintr word_size) |
|
737 |
apply auto |
|
738 |
done |
|
739 |
||
740 |
lemmas bin_nth_uint_imp = bin_nth_uint_imp' [rule_format, unfolded word_size] |
|
741 |
lemmas bin_nth_sint = bin_nth_sint' [rule_format, unfolded word_size] |
|
742 |
||
743 |
(* type definitions theorem for in terms of equivalent bool list *) |
|
744 |
lemma td_bl: |
|
745 |
"type_definition (to_bl :: 'a::len0 word => bool list) |
|
746 |
of_bl |
|
747 |
{bl. length bl = len_of TYPE('a)}" |
|
748 |
apply (unfold type_definition_def of_bl_def to_bl_def) |
|
749 |
apply (simp add: word_ubin.eq_norm) |
|
750 |
apply safe |
|
751 |
apply (drule sym) |
|
752 |
apply simp |
|
753 |
done |
|
754 |
||
755 |
interpretation word_bl: |
|
756 |
type_definition "to_bl :: 'a::len0 word => bool list" |
|
757 |
of_bl |
|
758 |
"{bl. length bl = len_of TYPE('a::len0)}" |
|
759 |
by (rule td_bl) |
|
760 |
||
45538
1fffa81b9b83
eliminated slightly odd Rep' with dynamically-scoped [simplified];
wenzelm
parents:
45529
diff
changeset
|
761 |
lemmas word_bl_Rep' = word_bl.Rep [simplified, iff] |
1fffa81b9b83
eliminated slightly odd Rep' with dynamically-scoped [simplified];
wenzelm
parents:
45529
diff
changeset
|
762 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
763 |
lemma word_size_bl: "size w = size (to_bl w)" |
37660 | 764 |
unfolding word_size by auto |
765 |
||
766 |
lemma to_bl_use_of_bl: |
|
767 |
"(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))" |
|
44890
22f665a2e91c
new fastforce replacing fastsimp - less confusing name
nipkow
parents:
44821
diff
changeset
|
768 |
by (fastforce elim!: word_bl.Abs_inverse [simplified]) |
37660 | 769 |
|
770 |
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)" |
|
771 |
unfolding word_reverse_def by (simp add: word_bl.Abs_inverse) |
|
772 |
||
773 |
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w" |
|
774 |
unfolding word_reverse_def by (simp add : word_bl.Abs_inverse) |
|
775 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
776 |
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w" |
37660 | 777 |
by auto |
778 |
||
45805 | 779 |
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u" |
780 |
by simp |
|
781 |
||
782 |
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))" |
|
783 |
unfolding word_bl_Rep' by (rule len_gt_0) |
|
784 |
||
785 |
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []" |
|
786 |
by (fact length_bl_gt_0 [unfolded length_greater_0_conv]) |
|
787 |
||
788 |
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0" |
|
789 |
by (fact length_bl_gt_0 [THEN gr_implies_not0]) |
|
37660 | 790 |
|
791 |
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = Int.Min)" |
|
792 |
apply (unfold to_bl_def sint_uint) |
|
793 |
apply (rule trans [OF _ bl_sbin_sign]) |
|
794 |
apply simp |
|
795 |
done |
|
796 |
||
797 |
lemma of_bl_drop': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
798 |
"lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> |
37660 | 799 |
of_bl (drop lend bl) = (of_bl bl :: 'a word)" |
800 |
apply (unfold of_bl_def) |
|
801 |
apply (clarsimp simp add : trunc_bl2bin [symmetric]) |
|
802 |
done |
|
803 |
||
45805 | 804 |
lemma of_bl_no: "of_bl bl = number_of (bl_to_bin bl)" |
805 |
by (fact of_bl_def [folded word_number_of_def]) |
|
37660 | 806 |
|
807 |
lemma test_bit_of_bl: |
|
808 |
"(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)" |
|
809 |
apply (unfold of_bl_def word_test_bit_def) |
|
810 |
apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl) |
|
811 |
done |
|
812 |
||
813 |
lemma no_of_bl: |
|
814 |
"(number_of bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) bin)" |
|
815 |
unfolding word_size of_bl_no by (simp add : word_number_of_def) |
|
816 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
817 |
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)" |
37660 | 818 |
unfolding word_size to_bl_def by auto |
819 |
||
820 |
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w" |
|
821 |
unfolding uint_bl by (simp add : word_size) |
|
822 |
||
823 |
lemma to_bl_of_bin: |
|
824 |
"to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" |
|
825 |
unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size) |
|
826 |
||
45805 | 827 |
lemma to_bl_no_bin [simp]: |
828 |
"to_bl (number_of bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin" |
|
829 |
by (fact to_bl_of_bin [folded word_number_of_def]) |
|
37660 | 830 |
|
831 |
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w" |
|
832 |
unfolding uint_bl by (simp add : word_size) |
|
833 |
||
45604 | 834 |
lemmas uint_bl_bin [simp] = trans [OF bin_bl_bin word_ubin.norm_Rep] |
835 |
||
45805 | 836 |
(* FIXME: the next two lemmas should be unnecessary, because the lhs |
837 |
terms should never occur in practice *) |
|
838 |
lemma num_AB_u [simp]: "number_of (uint x) = x" |
|
839 |
unfolding word_number_of_def by (rule word_uint.Rep_inverse) |
|
840 |
||
841 |
lemma num_AB_s [simp]: "number_of (sint x) = x" |
|
842 |
unfolding word_number_of_def by (rule word_sint.Rep_inverse) |
|
37660 | 843 |
|
844 |
(* naturals *) |
|
845 |
lemma uints_unats: "uints n = int ` unats n" |
|
846 |
apply (unfold unats_def uints_num) |
|
847 |
apply safe |
|
848 |
apply (rule_tac image_eqI) |
|
849 |
apply (erule_tac nat_0_le [symmetric]) |
|
850 |
apply auto |
|
851 |
apply (erule_tac nat_less_iff [THEN iffD2]) |
|
852 |
apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1]) |
|
853 |
apply (auto simp add : nat_power_eq int_power) |
|
854 |
done |
|
855 |
||
856 |
lemma unats_uints: "unats n = nat ` uints n" |
|
857 |
by (auto simp add : uints_unats image_iff) |
|
858 |
||
45604 | 859 |
lemmas bintr_num = word_ubin.norm_eq_iff [symmetric, folded word_number_of_def] |
860 |
lemmas sbintr_num = word_sbin.norm_eq_iff [symmetric, folded word_number_of_def] |
|
861 |
||
862 |
lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def] |
|
863 |
lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def] |
|
37660 | 864 |
|
865 |
(* don't add these to simpset, since may want bintrunc n w to be simplified; |
|
866 |
may want these in reverse, but loop as simp rules, so use following *) |
|
867 |
||
868 |
lemma num_of_bintr': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
869 |
"bintrunc (len_of TYPE('a :: len0)) a = b \<Longrightarrow> |
37660 | 870 |
number_of a = (number_of b :: 'a word)" |
871 |
apply safe |
|
872 |
apply (rule_tac num_of_bintr [symmetric]) |
|
873 |
done |
|
874 |
||
875 |
lemma num_of_sbintr': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
876 |
"sbintrunc (len_of TYPE('a :: len) - 1) a = b \<Longrightarrow> |
37660 | 877 |
number_of a = (number_of b :: 'a word)" |
878 |
apply safe |
|
879 |
apply (rule_tac num_of_sbintr [symmetric]) |
|
880 |
done |
|
881 |
||
45604 | 882 |
lemmas num_abs_bintr = sym [THEN trans, OF num_of_bintr word_number_of_def] |
883 |
lemmas num_abs_sbintr = sym [THEN trans, OF num_of_sbintr word_number_of_def] |
|
37660 | 884 |
|
885 |
(** cast - note, no arg for new length, as it's determined by type of result, |
|
886 |
thus in "cast w = w, the type means cast to length of w! **) |
|
887 |
||
888 |
lemma ucast_id: "ucast w = w" |
|
889 |
unfolding ucast_def by auto |
|
890 |
||
891 |
lemma scast_id: "scast w = w" |
|
892 |
unfolding scast_def by auto |
|
893 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
894 |
lemma ucast_bl: "ucast w = of_bl (to_bl w)" |
37660 | 895 |
unfolding ucast_def of_bl_def uint_bl |
896 |
by (auto simp add : word_size) |
|
897 |
||
898 |
lemma nth_ucast: |
|
899 |
"(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))" |
|
900 |
apply (unfold ucast_def test_bit_bin) |
|
901 |
apply (simp add: word_ubin.eq_norm nth_bintr word_size) |
|
902 |
apply (fast elim!: bin_nth_uint_imp) |
|
903 |
done |
|
904 |
||
905 |
(* for literal u(s)cast *) |
|
906 |
||
907 |
lemma ucast_bintr [simp]: |
|
908 |
"ucast (number_of w ::'a::len0 word) = |
|
909 |
number_of (bintrunc (len_of TYPE('a)) w)" |
|
910 |
unfolding ucast_def by simp |
|
911 |
||
912 |
lemma scast_sbintr [simp]: |
|
913 |
"scast (number_of w ::'a::len word) = |
|
914 |
number_of (sbintrunc (len_of TYPE('a) - Suc 0) w)" |
|
915 |
unfolding scast_def by simp |
|
916 |
||
917 |
lemmas source_size = source_size_def [unfolded Let_def word_size] |
|
918 |
lemmas target_size = target_size_def [unfolded Let_def word_size] |
|
919 |
lemmas is_down = is_down_def [unfolded source_size target_size] |
|
920 |
lemmas is_up = is_up_def [unfolded source_size target_size] |
|
921 |
||
45604 | 922 |
lemmas is_up_down = trans [OF is_up is_down [symmetric]] |
37660 | 923 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
924 |
lemma down_cast_same': "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast" |
37660 | 925 |
apply (unfold is_down) |
926 |
apply safe |
|
927 |
apply (rule ext) |
|
928 |
apply (unfold ucast_def scast_def uint_sint) |
|
929 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
930 |
apply simp |
|
931 |
done |
|
932 |
||
933 |
lemma word_rev_tf': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
934 |
"r = to_bl (of_bl bl) \<Longrightarrow> r = rev (takefill False (length r) (rev bl))" |
37660 | 935 |
unfolding of_bl_def uint_bl |
936 |
by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size) |
|
937 |
||
45604 | 938 |
lemmas word_rev_tf = refl [THEN word_rev_tf', unfolded word_bl_Rep'] |
37660 | 939 |
|
940 |
lemmas word_rep_drop = word_rev_tf [simplified takefill_alt, |
|
941 |
simplified, simplified rev_take, simplified] |
|
942 |
||
943 |
lemma to_bl_ucast: |
|
944 |
"to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = |
|
945 |
replicate (len_of TYPE('a) - len_of TYPE('b)) False @ |
|
946 |
drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)" |
|
947 |
apply (unfold ucast_bl) |
|
948 |
apply (rule trans) |
|
949 |
apply (rule word_rep_drop) |
|
950 |
apply simp |
|
951 |
done |
|
952 |
||
953 |
lemma ucast_up_app': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
954 |
"uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> |
37660 | 955 |
to_bl (uc w) = replicate n False @ (to_bl w)" |
956 |
by (auto simp add : source_size target_size to_bl_ucast) |
|
957 |
||
958 |
lemma ucast_down_drop': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
959 |
"uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> |
37660 | 960 |
to_bl (uc w) = drop n (to_bl w)" |
961 |
by (auto simp add : source_size target_size to_bl_ucast) |
|
962 |
||
963 |
lemma scast_down_drop': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
964 |
"sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> |
37660 | 965 |
to_bl (sc w) = drop n (to_bl w)" |
966 |
apply (subgoal_tac "sc = ucast") |
|
967 |
apply safe |
|
968 |
apply simp |
|
969 |
apply (erule refl [THEN ucast_down_drop']) |
|
970 |
apply (rule refl [THEN down_cast_same', symmetric]) |
|
971 |
apply (simp add : source_size target_size is_down) |
|
972 |
done |
|
973 |
||
974 |
lemma sint_up_scast': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
975 |
"sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w" |
37660 | 976 |
apply (unfold is_up) |
977 |
apply safe |
|
978 |
apply (simp add: scast_def word_sbin.eq_norm) |
|
979 |
apply (rule box_equals) |
|
980 |
prefer 3 |
|
981 |
apply (rule word_sbin.norm_Rep) |
|
982 |
apply (rule sbintrunc_sbintrunc_l) |
|
983 |
defer |
|
984 |
apply (subst word_sbin.norm_Rep) |
|
985 |
apply (rule refl) |
|
986 |
apply simp |
|
987 |
done |
|
988 |
||
989 |
lemma uint_up_ucast': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
990 |
"uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w" |
37660 | 991 |
apply (unfold is_up) |
992 |
apply safe |
|
993 |
apply (rule bin_eqI) |
|
994 |
apply (fold word_test_bit_def) |
|
995 |
apply (auto simp add: nth_ucast) |
|
996 |
apply (auto simp add: test_bit_bin) |
|
997 |
done |
|
998 |
||
999 |
lemmas down_cast_same = refl [THEN down_cast_same'] |
|
1000 |
lemmas ucast_up_app = refl [THEN ucast_up_app'] |
|
1001 |
lemmas ucast_down_drop = refl [THEN ucast_down_drop'] |
|
1002 |
lemmas scast_down_drop = refl [THEN scast_down_drop'] |
|
1003 |
lemmas uint_up_ucast = refl [THEN uint_up_ucast'] |
|
1004 |
lemmas sint_up_scast = refl [THEN sint_up_scast'] |
|
1005 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1006 |
lemma ucast_up_ucast': "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w" |
37660 | 1007 |
apply (simp (no_asm) add: ucast_def) |
1008 |
apply (clarsimp simp add: uint_up_ucast) |
|
1009 |
done |
|
1010 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1011 |
lemma scast_up_scast': "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w" |
37660 | 1012 |
apply (simp (no_asm) add: scast_def) |
1013 |
apply (clarsimp simp add: sint_up_scast) |
|
1014 |
done |
|
1015 |
||
1016 |
lemma ucast_of_bl_up': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1017 |
"w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl" |
37660 | 1018 |
by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI) |
1019 |
||
1020 |
lemmas ucast_up_ucast = refl [THEN ucast_up_ucast'] |
|
1021 |
lemmas scast_up_scast = refl [THEN scast_up_scast'] |
|
1022 |
lemmas ucast_of_bl_up = refl [THEN ucast_of_bl_up'] |
|
1023 |
||
1024 |
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id] |
|
1025 |
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id] |
|
1026 |
||
1027 |
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2] |
|
1028 |
lemmas isdus = is_up_down [where c = "scast", THEN iffD2] |
|
1029 |
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id] |
|
1030 |
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id] |
|
1031 |
||
1032 |
lemma up_ucast_surj: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1033 |
"is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> |
37660 | 1034 |
surj (ucast :: 'a word => 'b word)" |
1035 |
by (rule surjI, erule ucast_up_ucast_id) |
|
1036 |
||
1037 |
lemma up_scast_surj: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1038 |
"is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> |
37660 | 1039 |
surj (scast :: 'a word => 'b word)" |
1040 |
by (rule surjI, erule scast_up_scast_id) |
|
1041 |
||
1042 |
lemma down_scast_inj: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1043 |
"is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> |
37660 | 1044 |
inj_on (ucast :: 'a word => 'b word) A" |
1045 |
by (rule inj_on_inverseI, erule scast_down_scast_id) |
|
1046 |
||
1047 |
lemma down_ucast_inj: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1048 |
"is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> |
37660 | 1049 |
inj_on (ucast :: 'a word => 'b word) A" |
1050 |
by (rule inj_on_inverseI, erule ucast_down_ucast_id) |
|
1051 |
||
1052 |
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w" |
|
1053 |
by (rule word_bl.Rep_eqD) (simp add: word_rep_drop) |
|
1054 |
||
1055 |
lemma ucast_down_no': |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1056 |
"uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (number_of bin) = number_of bin" |
37660 | 1057 |
apply (unfold word_number_of_def is_down) |
1058 |
apply (clarsimp simp add: ucast_def word_ubin.eq_norm) |
|
1059 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
1060 |
apply (erule bintrunc_bintrunc_ge) |
|
1061 |
done |
|
1062 |
||
1063 |
lemmas ucast_down_no = ucast_down_no' [OF refl] |
|
1064 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1065 |
lemma ucast_down_bl': "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl" |
37660 | 1066 |
unfolding of_bl_no by clarify (erule ucast_down_no) |
1067 |
||
1068 |
lemmas ucast_down_bl = ucast_down_bl' [OF refl] |
|
1069 |
||
1070 |
lemmas slice_def' = slice_def [unfolded word_size] |
|
1071 |
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong] |
|
1072 |
||
1073 |
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def |
|
1074 |
lemmas word_log_bin_defs = word_log_defs |
|
1075 |
||
1076 |
text {* Executable equality *} |
|
1077 |
||
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38527
diff
changeset
|
1078 |
instantiation word :: (len0) equal |
24333 | 1079 |
begin |
1080 |
||
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38527
diff
changeset
|
1081 |
definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where |
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38527
diff
changeset
|
1082 |
"equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)" |
37660 | 1083 |
|
1084 |
instance proof |
|
38857
97775f3e8722
renamed class/constant eq to equal; tuned some instantiations
haftmann
parents:
38527
diff
changeset
|
1085 |
qed (simp add: equal equal_word_def) |
37660 | 1086 |
|
1087 |
end |
|
1088 |
||
1089 |
||
1090 |
subsection {* Word Arithmetic *} |
|
1091 |
||
1092 |
lemma word_less_alt: "(a < b) = (uint a < uint b)" |
|
1093 |
unfolding word_less_def word_le_def |
|
1094 |
by (auto simp del: word_uint.Rep_inject |
|
1095 |
simp: word_uint.Rep_inject [symmetric]) |
|
1096 |
||
1097 |
lemma signed_linorder: "class.linorder word_sle word_sless" |
|
1098 |
proof |
|
1099 |
qed (unfold word_sle_def word_sless_def, auto) |
|
1100 |
||
1101 |
interpretation signed: linorder "word_sle" "word_sless" |
|
1102 |
by (rule signed_linorder) |
|
1103 |
||
1104 |
lemma udvdI: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1105 |
"0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b" |
37660 | 1106 |
by (auto simp: udvd_def) |
1107 |
||
45604 | 1108 |
lemmas word_div_no [simp] = word_div_def [of "number_of a" "number_of b"] for a b |
1109 |
||
1110 |
lemmas word_mod_no [simp] = word_mod_def [of "number_of a" "number_of b"] for a b |
|
1111 |
||
1112 |
lemmas word_less_no [simp] = word_less_def [of "number_of a" "number_of b"] for a b |
|
1113 |
||
1114 |
lemmas word_le_no [simp] = word_le_def [of "number_of a" "number_of b"] for a b |
|
1115 |
||
1116 |
lemmas word_sless_no [simp] = word_sless_def [of "number_of a" "number_of b"] for a b |
|
1117 |
||
1118 |
lemmas word_sle_no [simp] = word_sle_def [of "number_of a" "number_of b"] for a b |
|
37660 | 1119 |
|
1120 |
(* following two are available in class number_ring, |
|
1121 |
but convenient to have them here here; |
|
1122 |
note - the number_ring versions, numeral_0_eq_0 and numeral_1_eq_1 |
|
1123 |
are in the default simpset, so to use the automatic simplifications for |
|
1124 |
(eg) sint (number_of bin) on sint 1, must do |
|
1125 |
(simp add: word_1_no del: numeral_1_eq_1) |
|
1126 |
*) |
|
1127 |
lemmas word_0_wi_Pls = word_0_wi [folded Pls_def] |
|
1128 |
lemmas word_0_no = word_0_wi_Pls [folded word_no_wi] |
|
1129 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1130 |
lemma int_one_bin: "(1 :: int) = (Int.Pls BIT 1)" |
37660 | 1131 |
unfolding Pls_def Bit_def by auto |
1132 |
||
1133 |
lemma word_1_no: |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1134 |
"(1 :: 'a :: len0 word) = number_of (Int.Pls BIT 1)" |
37660 | 1135 |
unfolding word_1_wi word_number_of_def int_one_bin by auto |
1136 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1137 |
lemma word_m1_wi: "-1 = word_of_int -1" |
37660 | 1138 |
by (rule word_number_of_alt) |
1139 |
||
1140 |
lemma word_m1_wi_Min: "-1 = word_of_int Int.Min" |
|
1141 |
by (simp add: word_m1_wi number_of_eq) |
|
1142 |
||
45805 | 1143 |
lemma word_0_bl [simp]: "of_bl [] = 0" |
37660 | 1144 |
unfolding word_0_wi of_bl_def by (simp add : Pls_def) |
1145 |
||
1146 |
lemma word_1_bl: "of_bl [True] = 1" |
|
1147 |
unfolding word_1_wi of_bl_def |
|
1148 |
by (simp add : bl_to_bin_def Bit_def Pls_def) |
|
1149 |
||
1150 |
lemma uint_eq_0 [simp] : "(uint 0 = 0)" |
|
1151 |
unfolding word_0_wi |
|
1152 |
by (simp add: word_ubin.eq_norm Pls_def [symmetric]) |
|
1153 |
||
1154 |
lemma of_bl_0 [simp] : "of_bl (replicate n False) = 0" |
|
1155 |
by (simp add : word_0_wi of_bl_def bl_to_bin_rep_False Pls_def) |
|
1156 |
||
45805 | 1157 |
lemma to_bl_0 [simp]: |
37660 | 1158 |
"to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False" |
1159 |
unfolding uint_bl |
|
1160 |
by (simp add : word_size bin_to_bl_Pls Pls_def [symmetric]) |
|
1161 |
||
1162 |
lemma uint_0_iff: "(uint x = 0) = (x = 0)" |
|
1163 |
by (auto intro!: word_uint.Rep_eqD) |
|
1164 |
||
1165 |
lemma unat_0_iff: "(unat x = 0) = (x = 0)" |
|
1166 |
unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff) |
|
1167 |
||
1168 |
lemma unat_0 [simp]: "unat 0 = 0" |
|
1169 |
unfolding unat_def by auto |
|
1170 |
||
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1171 |
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)" |
37660 | 1172 |
apply (unfold word_size) |
1173 |
apply (rule box_equals) |
|
1174 |
defer |
|
1175 |
apply (rule word_uint.Rep_inverse)+ |
|
1176 |
apply (rule word_ubin.norm_eq_iff [THEN iffD1]) |
|
1177 |
apply simp |
|
1178 |
done |
|
1179 |
||
1180 |
lemmas size_0_same = size_0_same' [folded word_size] |
|
1181 |
||
1182 |
lemmas unat_eq_0 = unat_0_iff |
|
1183 |
lemmas unat_eq_zero = unat_0_iff |
|
1184 |
||
1185 |
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)" |
|
1186 |
by (auto simp: unat_0_iff [symmetric]) |
|
1187 |
||
1188 |
lemma ucast_0 [simp] : "ucast 0 = 0" |
|
1189 |
unfolding ucast_def |
|
1190 |
by simp (simp add: word_0_wi) |
|
1191 |
||
1192 |
lemma sint_0 [simp] : "sint 0 = 0" |
|
1193 |
unfolding sint_uint |
|
1194 |
by (simp add: Pls_def [symmetric]) |
|
1195 |
||
1196 |
lemma scast_0 [simp] : "scast 0 = 0" |
|
1197 |
apply (unfold scast_def) |
|
1198 |
apply simp |
|
1199 |
apply (simp add: word_0_wi) |
|
1200 |
done |
|
1201 |
||
1202 |
lemma sint_n1 [simp] : "sint -1 = -1" |
|
1203 |
apply (unfold word_m1_wi_Min) |
|
1204 |
apply (simp add: word_sbin.eq_norm) |
|
1205 |
apply (unfold Min_def number_of_eq) |
|
1206 |
apply simp |
|
1207 |
done |
|
1208 |
||
1209 |
lemma scast_n1 [simp] : "scast -1 = -1" |
|
1210 |
apply (unfold scast_def sint_n1) |
|
1211 |
apply (unfold word_number_of_alt) |
|
1212 |
apply (rule refl) |
|
1213 |
done |
|
1214 |
||
1215 |
lemma uint_1 [simp] : "uint (1 :: 'a :: len word) = 1" |
|
1216 |
unfolding word_1_wi |
|
1217 |
by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) |
|
1218 |
||
1219 |
lemma unat_1 [simp] : "unat (1 :: 'a :: len word) = 1" |
|
1220 |
by (unfold unat_def uint_1) auto |
|
1221 |
||
1222 |
lemma ucast_1 [simp] : "ucast (1 :: 'a :: len word) = 1" |
|
1223 |
unfolding ucast_def word_1_wi |
|
1224 |
by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps) |
|
1225 |
||
1226 |
(* now, to get the weaker results analogous to word_div/mod_def *) |
|
1227 |
||
1228 |
lemmas word_arith_alts = |
|
45604 | 1229 |
word_sub_wi [unfolded succ_def pred_def] |
1230 |
word_arith_wis [unfolded succ_def pred_def] |
|
37660 | 1231 |
|
1232 |
lemmas word_succ_alt = word_arith_alts (5) |
|
1233 |
lemmas word_pred_alt = word_arith_alts (6) |
|
1234 |
||
1235 |
subsection "Transferring goals from words to ints" |
|
1236 |
||
1237 |
lemma word_ths: |
|
1238 |
shows |
|
1239 |
word_succ_p1: "word_succ a = a + 1" and |
|
1240 |
word_pred_m1: "word_pred a = a - 1" and |
|
1241 |
word_pred_succ: "word_pred (word_succ a) = a" and |
|
1242 |
word_succ_pred: "word_succ (word_pred a) = a" and |
|
1243 |
word_mult_succ: "word_succ a * b = b + a * b" |
|
1244 |
by (rule word_uint.Abs_cases [of b], |
|
1245 |
rule word_uint.Abs_cases [of a], |
|
1246 |
simp add: pred_def succ_def add_commute mult_commute |
|
1247 |
ring_distribs new_word_of_int_homs)+ |
|
1248 |
||
1249 |
lemmas uint_cong = arg_cong [where f = uint] |
|
1250 |
||
1251 |
lemmas uint_word_ariths = |
|
45604 | 1252 |
word_arith_alts [THEN trans [OF uint_cong int_word_uint]] |
37660 | 1253 |
|
1254 |
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p] |
|
1255 |
||
1256 |
(* similar expressions for sint (arith operations) *) |
|
1257 |
lemmas sint_word_ariths = uint_word_arith_bintrs |
|
1258 |
[THEN uint_sint [symmetric, THEN trans], |
|
1259 |
unfolded uint_sint bintr_arith1s bintr_ariths |
|
45604 | 1260 |
len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep] |
1261 |
||
1262 |
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]] |
|
1263 |
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]] |
|
37660 | 1264 |
|
1265 |
lemma word_pred_0_n1: "word_pred 0 = word_of_int -1" |
|
45550
73a4f31d41c4
Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents:
45549
diff
changeset
|
1266 |
unfolding word_pred_def uint_eq_0 pred_def by simp |
37660 | 1267 |
|
1268 |
lemma word_pred_0_Min: "word_pred 0 = word_of_int Int.Min" |
|
1269 |
by (simp add: word_pred_0_n1 number_of_eq) |
|
1270 |
||
1271 |
lemma word_m1_Min: "- 1 = word_of_int Int.Min" |
|
1272 |
unfolding Min_def by (simp only: word_of_int_hom_syms) |
|
1273 |
||
1274 |
lemma succ_pred_no [simp]: |
|
1275 |
"word_succ (number_of bin) = number_of (Int.succ bin) & |
|
1276 |
word_pred (number_of bin) = number_of (Int.pred bin)" |
|
1277 |
unfolding word_number_of_def by (simp add : new_word_of_int_homs) |
|
1278 |
||
1279 |
lemma word_sp_01 [simp] : |
|
1280 |
"word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0" |
|
1281 |
by (unfold word_0_no word_1_no) auto |
|
1282 |
||
1283 |
(* alternative approach to lifting arithmetic equalities *) |
|
1284 |
lemma word_of_int_Ex: |
|
1285 |
"\<exists>y. x = word_of_int y" |
|
1286 |
by (rule_tac x="uint x" in exI) simp |
|
1287 |
||
1288 |
||
1289 |
subsection "Order on fixed-length words" |
|
1290 |
||
1291 |
lemma word_zero_le [simp] : |
|
1292 |
"0 <= (y :: 'a :: len0 word)" |
|
1293 |
unfolding word_le_def by auto |
|
1294 |
||
1295 |
lemma word_m1_ge [simp] : "word_pred 0 >= y" |
|
1296 |
unfolding word_le_def |
|
1297 |
by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto |
|
1298 |
||
1299 |
lemmas word_n1_ge [simp] = word_m1_ge [simplified word_sp_01] |
|
1300 |
||
1301 |
lemmas word_not_simps [simp] = |
|
1302 |
word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD] |
|
1303 |
||
1304 |
lemma word_gt_0: "0 < y = (0 ~= (y :: 'a :: len0 word))" |
|
1305 |
unfolding word_less_def by auto |
|
1306 |
||
45604 | 1307 |
lemmas word_gt_0_no [simp] = word_gt_0 [of "number_of y"] for y |
37660 | 1308 |
|
40827
abbc05c20e24
code preprocessor setup for numerals on word type;
haftmann
parents:
39910
diff
changeset
|
1309 |
lemma word_sless_alt: "(a <s b) = (sint a < sint b)" |
37660 | 1310 |
unfolding word_sle_def word_sless_def |
1311 |
by (auto simp add: less_le) |
|
1312 |
||
1313 |
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)" |
|
1314 |
unfolding unat_def word_le_def |
|
1315 |
by (rule nat_le_eq_zle [symmetric]) simp |
|
1316 |
||
1317 |
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)" |
|
1318 |
unfolding unat_def word_less_alt |
|
1319 |
by (rule nat_less_eq_zless [symmetric]) simp |
|
1320 |
||
1321 |
lemma wi_less: |
|
1322 |
"(word_of_int n < (word_of_int m :: 'a :: len0 word)) = |
|
1323 |
(n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))" |
|
1324 |
unfolding word_less_alt by (simp add: word_uint.eq_norm) |
|
1325 |
||
1326 |
lemma wi_le: |
|
1327 |
"(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = |
|
1328 |
(n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))" |
|
1329 |
unfolding word_le_def by (simp add: word_uint.eq_norm) |
|
1330 |
||
1331 |
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)" |
|
1332 |
apply (unfold udvd_def) |
|
1333 |
apply safe |
|
1334 |
apply (simp add: unat_def nat_mult_distrib) |
|
1335 |
apply (simp add: uint_nat int_mult) |
|
1336 |
apply (rule exI) |
|
1337 |
apply safe |
|
1338 |
prefer 2 |
|
1339 |
apply (erule notE) |
|
1340 |
apply (rule refl) |
|
1341 |
apply force |
|
1342 |
done |
|
1343 |
||
1344 |
lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y" |
|
1345 |
unfolding dvd_def udvd_nat_alt by |