src/HOL/Word/Word.thy
author huffman
Wed Dec 28 07:58:17 2011 +0100 (2011-12-28)
changeset 46010 ebbc2d5cd720
parent 46009 5cb7ef5bfef2
child 46011 96a5f44c22da
permissions -rw-r--r--
add section headings
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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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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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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 [unfolded atLeastLessThan_iff])
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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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lemmas td_ext_ubin = td_ext_uint 
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  [unfolded 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 (uint a + 1)"
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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 (uint a - 1)"
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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
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  word_sub_wi: "a - b = word_of_int (uint a - uint b)"
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definition
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  word_minus_def: "- a = word_of_int (- uint a)"
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definition
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  word_mult_def: "a * b = word_of_int (uint a * uint b)"
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definition
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  word_div_def: "a div b = word_of_int (uint a div uint b)"
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definition
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  word_mod_def: "a mod b = word_of_int (uint a mod uint b)"
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definition
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  word_number_of_def: "number_of w = word_of_int w"
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lemmas word_arith_wis = 
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  word_add_def word_mult_def word_minus_def 
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  word_succ_def word_pred_def word_0_wi word_1_wi
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lemmas arths = 
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  bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm]
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lemma wi_homs: 
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  shows
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  wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and
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  wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and
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  wi_hom_neg: "- word_of_int a = word_of_int (- a)" and
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  wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and
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  wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
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  by (auto simp: word_arith_wis arths)
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lemmas wi_hom_syms = wi_homs [symmetric]
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lemma word_of_int_sub_hom:
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  "(word_of_int a) - word_of_int b = word_of_int (a - b)"
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  by (simp add: word_sub_wi arths)
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lemmas word_of_int_homs =
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  word_of_int_sub_hom wi_homs word_0_wi word_1_wi
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lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
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(* FIXME: provide only one copy of these theorems! *)
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lemmas word_of_int_add_hom = wi_hom_add
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lemmas word_of_int_mult_hom = wi_hom_mult
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lemmas word_of_int_minus_hom = wi_hom_neg
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lemmas word_of_int_succ_hom = wi_hom_succ
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lemmas word_of_int_pred_hom = wi_hom_pred
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lemmas word_of_int_0_hom = word_0_wi
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lemmas word_of_int_1_hom = word_1_wi
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instance
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  by default (auto simp: split_word_all word_of_int_homs algebra_simps)
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end
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instance word :: (len) comm_ring_1
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proof
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  have "0 < len_of TYPE('a)" by (rule len_gt_0)
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  then show "(0::'a word) \<noteq> 1"
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    unfolding word_0_wi word_1_wi
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    by (auto simp add: word_ubin.norm_eq_iff [symmetric] gr0_conv_Suc)
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qed
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lemma word_of_nat: "of_nat n = word_of_int (int n)"
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  by (induct n) (auto simp add : word_of_int_hom_syms)
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lemma word_of_int: "of_int = word_of_int"
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  apply (rule ext)
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  apply (case_tac x rule: int_diff_cases)
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  apply (simp add: word_of_nat word_of_int_sub_hom)
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  done
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instance word :: (len) number_ring
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  by (default, simp add: word_number_of_def word_of_int)
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definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where
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  "a udvd b = (EX n>=0. uint b = n * uint a)"
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subsection "Ordering"
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instantiation word :: (len0) linorder
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begin
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definition
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  word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
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definition
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  word_less_def: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> (y \<Colon> 'a word)"
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instance
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  by default (auto simp: word_less_def word_le_def)
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end
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definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where
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  "a <=s b = (sint a <= sint b)"
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definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where
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  "(x <s y) = (x <=s y & x ~= y)"
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subsection "Bit-wise operations"
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instantiation word :: (len0) bits
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begin
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definition
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  word_and_def: 
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  "(a::'a word) AND b = word_of_int (uint a AND uint b)"
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definition
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  word_or_def:  
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  "(a::'a word) OR b = word_of_int (uint a OR uint b)"
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definition
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  word_xor_def: 
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  "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
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definition
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  word_not_def: 
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  "NOT (a::'a word) = word_of_int (NOT (uint a))"
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definition
haftmann@37660
   340
  word_test_bit_def: "test_bit a = bin_nth (uint a)"
haftmann@37660
   341
haftmann@37660
   342
definition
haftmann@37660
   343
  word_set_bit_def: "set_bit a n x =
haftmann@37660
   344
   word_of_int (bin_sc n (If x 1 0) (uint a))"
haftmann@37660
   345
haftmann@37660
   346
definition
haftmann@37660
   347
  word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"
haftmann@37660
   348
haftmann@37660
   349
definition
haftmann@37660
   350
  word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1"
haftmann@37660
   351
haftmann@37660
   352
definition shiftl1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   353
  "shiftl1 w = word_of_int (uint w BIT 0)"
haftmann@37660
   354
haftmann@37660
   355
definition shiftr1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   356
  -- "shift right as unsigned or as signed, ie logical or arithmetic"
haftmann@37660
   357
  "shiftr1 w = word_of_int (bin_rest (uint w))"
haftmann@37660
   358
haftmann@37660
   359
definition
haftmann@37660
   360
  shiftl_def: "w << n = (shiftl1 ^^ n) w"
haftmann@37660
   361
haftmann@37660
   362
definition
haftmann@37660
   363
  shiftr_def: "w >> n = (shiftr1 ^^ n) w"
haftmann@37660
   364
haftmann@37660
   365
instance ..
haftmann@37660
   366
haftmann@37660
   367
end
haftmann@37660
   368
haftmann@37660
   369
instantiation word :: (len) bitss
haftmann@37660
   370
begin
haftmann@37660
   371
haftmann@37660
   372
definition
haftmann@37660
   373
  word_msb_def: 
huffman@46001
   374
  "msb a \<longleftrightarrow> bin_sign (sint a) = -1"
haftmann@37660
   375
haftmann@37660
   376
instance ..
haftmann@37660
   377
haftmann@37660
   378
end
haftmann@37660
   379
haftmann@37660
   380
definition setBit :: "'a :: len0 word => nat => 'a word" where 
haftmann@40827
   381
  "setBit w n = set_bit w n True"
haftmann@37660
   382
haftmann@37660
   383
definition clearBit :: "'a :: len0 word => nat => 'a word" where
haftmann@40827
   384
  "clearBit w n = set_bit w n False"
haftmann@37660
   385
haftmann@37660
   386
haftmann@37660
   387
subsection "Shift operations"
haftmann@37660
   388
haftmann@37660
   389
definition sshiftr1 :: "'a :: len word => 'a word" where 
haftmann@40827
   390
  "sshiftr1 w = word_of_int (bin_rest (sint w))"
haftmann@37660
   391
haftmann@37660
   392
definition bshiftr1 :: "bool => 'a :: len word => 'a word" where
haftmann@40827
   393
  "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
haftmann@37660
   394
haftmann@37660
   395
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where
haftmann@40827
   396
  "w >>> n = (sshiftr1 ^^ n) w"
haftmann@37660
   397
haftmann@37660
   398
definition mask :: "nat => 'a::len word" where
haftmann@40827
   399
  "mask n = (1 << n) - 1"
haftmann@37660
   400
haftmann@37660
   401
definition revcast :: "'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   402
  "revcast w =  of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
haftmann@37660
   403
haftmann@37660
   404
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   405
  "slice1 n w = of_bl (takefill False n (to_bl w))"
haftmann@37660
   406
haftmann@37660
   407
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   408
  "slice n w = slice1 (size w - n) w"
haftmann@37660
   409
haftmann@37660
   410
haftmann@37660
   411
subsection "Rotation"
haftmann@37660
   412
haftmann@37660
   413
definition rotater1 :: "'a list => 'a list" where
haftmann@40827
   414
  "rotater1 ys = 
haftmann@40827
   415
    (case ys of [] => [] | x # xs => last ys # butlast ys)"
haftmann@37660
   416
haftmann@37660
   417
definition rotater :: "nat => 'a list => 'a list" where
haftmann@40827
   418
  "rotater n = rotater1 ^^ n"
haftmann@37660
   419
haftmann@37660
   420
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   421
  "word_rotr n w = of_bl (rotater n (to_bl w))"
haftmann@37660
   422
haftmann@37660
   423
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   424
  "word_rotl n w = of_bl (rotate n (to_bl w))"
haftmann@37660
   425
haftmann@37660
   426
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   427
  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
haftmann@40827
   428
                    else word_rotl (nat (- i)) w)"
haftmann@37660
   429
haftmann@37660
   430
haftmann@37660
   431
subsection "Split and cat operations"
haftmann@37660
   432
haftmann@37660
   433
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where
haftmann@40827
   434
  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
haftmann@37660
   435
haftmann@37660
   436
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where
haftmann@40827
   437
  "word_split a = 
haftmann@40827
   438
   (case bin_split (len_of TYPE ('c)) (uint a) of 
haftmann@40827
   439
     (u, v) => (word_of_int u, word_of_int v))"
haftmann@37660
   440
haftmann@37660
   441
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where
haftmann@40827
   442
  "word_rcat ws = 
haftmann@37660
   443
  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
haftmann@37660
   444
haftmann@37660
   445
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where
haftmann@40827
   446
  "word_rsplit w = 
haftmann@37660
   447
  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
haftmann@37660
   448
haftmann@37660
   449
definition max_word :: "'a::len word" -- "Largest representable machine integer." where
haftmann@40827
   450
  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
haftmann@37660
   451
haftmann@37660
   452
primrec of_bool :: "bool \<Rightarrow> 'a::len word" where
haftmann@37660
   453
  "of_bool False = 0"
haftmann@37660
   454
| "of_bool True = 1"
haftmann@37660
   455
huffman@45805
   456
(* FIXME: only provide one theorem name *)
haftmann@37660
   457
lemmas of_nth_def = word_set_bits_def
haftmann@37660
   458
huffman@46010
   459
subsection {* Theorems about typedefs *}
huffman@46010
   460
haftmann@37660
   461
lemma sint_sbintrunc': 
haftmann@37660
   462
  "sint (word_of_int bin :: 'a word) = 
haftmann@37660
   463
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
haftmann@37660
   464
  unfolding sint_uint 
haftmann@37660
   465
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)
haftmann@37660
   466
haftmann@37660
   467
lemma uint_sint: 
haftmann@37660
   468
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
haftmann@37660
   469
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
haftmann@37660
   470
haftmann@37660
   471
lemma bintr_uint': 
haftmann@40827
   472
  "n >= size w \<Longrightarrow> bintrunc n (uint w) = uint w"
haftmann@37660
   473
  apply (unfold word_size)
haftmann@37660
   474
  apply (subst word_ubin.norm_Rep [symmetric]) 
haftmann@37660
   475
  apply (simp only: bintrunc_bintrunc_min word_size)
haftmann@37660
   476
  apply (simp add: min_max.inf_absorb2)
haftmann@37660
   477
  done
haftmann@37660
   478
haftmann@37660
   479
lemma wi_bintr': 
haftmann@40827
   480
  "wb = word_of_int bin \<Longrightarrow> n >= size wb \<Longrightarrow> 
haftmann@37660
   481
    word_of_int (bintrunc n bin) = wb"
haftmann@37660
   482
  unfolding word_size
haftmann@37660
   483
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1)
haftmann@37660
   484
haftmann@37660
   485
lemmas bintr_uint = bintr_uint' [unfolded word_size]
haftmann@37660
   486
lemmas wi_bintr = wi_bintr' [unfolded word_size]
haftmann@37660
   487
haftmann@37660
   488
lemma td_ext_sbin: 
haftmann@37660
   489
  "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 
haftmann@37660
   490
    (sbintrunc (len_of TYPE('a) - 1))"
haftmann@37660
   491
  apply (unfold td_ext_def' sint_uint)
haftmann@37660
   492
  apply (simp add : word_ubin.eq_norm)
haftmann@37660
   493
  apply (cases "len_of TYPE('a)")
haftmann@37660
   494
   apply (auto simp add : sints_def)
haftmann@37660
   495
  apply (rule sym [THEN trans])
haftmann@37660
   496
  apply (rule word_ubin.Abs_norm)
haftmann@37660
   497
  apply (simp only: bintrunc_sbintrunc)
haftmann@37660
   498
  apply (drule sym)
haftmann@37660
   499
  apply simp
haftmann@37660
   500
  done
haftmann@37660
   501
haftmann@37660
   502
lemmas td_ext_sint = td_ext_sbin 
haftmann@37660
   503
  [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]]
haftmann@37660
   504
haftmann@37660
   505
(* We do sint before sbin, before sint is the user version
haftmann@37660
   506
   and interpretations do not produce thm duplicates. I.e. 
haftmann@37660
   507
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
haftmann@37660
   508
   because the latter is the same thm as the former *)
haftmann@37660
   509
interpretation word_sint:
haftmann@37660
   510
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   511
          word_of_int 
haftmann@37660
   512
          "sints (len_of TYPE('a::len))"
haftmann@37660
   513
          "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) -
haftmann@37660
   514
               2 ^ (len_of TYPE('a::len) - 1)"
haftmann@37660
   515
  by (rule td_ext_sint)
haftmann@37660
   516
haftmann@37660
   517
interpretation word_sbin:
haftmann@37660
   518
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   519
          word_of_int 
haftmann@37660
   520
          "sints (len_of TYPE('a::len))"
haftmann@37660
   521
          "sbintrunc (len_of TYPE('a::len) - 1)"
haftmann@37660
   522
  by (rule td_ext_sbin)
haftmann@37660
   523
wenzelm@45604
   524
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
haftmann@37660
   525
haftmann@37660
   526
lemmas td_sint = word_sint.td
haftmann@37660
   527
haftmann@40827
   528
lemma word_number_of_alt [code_unfold_post]:
haftmann@40827
   529
  "number_of b = word_of_int (number_of b)"
haftmann@40827
   530
  by (simp add: number_of_eq word_number_of_def)
haftmann@37660
   531
haftmann@37660
   532
lemma word_no_wi: "number_of = word_of_int"
wenzelm@44762
   533
  by (auto simp: word_number_of_def)
haftmann@37660
   534
haftmann@37660
   535
lemma to_bl_def': 
haftmann@37660
   536
  "(to_bl :: 'a :: len0 word => bool list) =
haftmann@37660
   537
    bin_to_bl (len_of TYPE('a)) o uint"
wenzelm@44762
   538
  by (auto simp: to_bl_def)
haftmann@37660
   539
wenzelm@45604
   540
lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w"] for w
haftmann@37660
   541
huffman@45805
   542
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
huffman@45805
   543
  by (fact uints_def [unfolded no_bintr_alt1])
huffman@45805
   544
huffman@45805
   545
lemma uint_bintrunc [simp]:
huffman@45805
   546
  "uint (number_of bin :: 'a word) =
huffman@46001
   547
    bintrunc (len_of TYPE ('a :: len0)) (number_of bin)"
huffman@46001
   548
  unfolding word_number_of_alt by (rule word_ubin.eq_norm)
haftmann@37660
   549
huffman@45805
   550
lemma sint_sbintrunc [simp]:
huffman@45805
   551
  "sint (number_of bin :: 'a word) =
huffman@46001
   552
    sbintrunc (len_of TYPE ('a :: len) - 1) (number_of bin)"
huffman@46001
   553
  unfolding word_number_of_alt by (rule word_sbin.eq_norm)
haftmann@37660
   554
huffman@45805
   555
lemma unat_bintrunc [simp]:
haftmann@37660
   556
  "unat (number_of bin :: 'a :: len0 word) =
huffman@46001
   557
    nat (bintrunc (len_of TYPE('a)) (number_of bin))"
haftmann@37660
   558
  unfolding unat_def nat_number_of_def 
haftmann@37660
   559
  by (simp only: uint_bintrunc)
haftmann@37660
   560
haftmann@40827
   561
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
haftmann@37660
   562
  apply (unfold word_size)
haftmann@37660
   563
  apply (rule word_uint.Rep_eqD)
haftmann@37660
   564
  apply (rule box_equals)
haftmann@37660
   565
    defer
haftmann@37660
   566
    apply (rule word_ubin.norm_Rep)+
haftmann@37660
   567
  apply simp
haftmann@37660
   568
  done
haftmann@37660
   569
huffman@45805
   570
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)"
huffman@45805
   571
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   572
huffman@45805
   573
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
huffman@45805
   574
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   575
huffman@45805
   576
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)"
huffman@45805
   577
  using word_sint.Rep [of x] by (simp add: sints_num)
huffman@45805
   578
huffman@45805
   579
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)"
huffman@45805
   580
  using word_sint.Rep [of x] by (simp add: sints_num)
haftmann@37660
   581
haftmann@37660
   582
lemma sign_uint_Pls [simp]: 
haftmann@37660
   583
  "bin_sign (uint x) = Int.Pls"
haftmann@37660
   584
  by (simp add: sign_Pls_ge_0 number_of_eq)
haftmann@37660
   585
huffman@45805
   586
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0"
huffman@45805
   587
  by (simp only: diff_less_0_iff_less uint_lt2p)
huffman@45805
   588
huffman@45805
   589
lemma uint_m2p_not_non_neg:
huffman@45805
   590
  "\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)"
huffman@45805
   591
  by (simp only: not_le uint_m2p_neg)
haftmann@37660
   592
haftmann@37660
   593
lemma lt2p_lem:
haftmann@40827
   594
  "len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n"
haftmann@37660
   595
  by (rule xtr8 [OF _ uint_lt2p]) simp
haftmann@37660
   596
huffman@45805
   597
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
huffman@45805
   598
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])
haftmann@37660
   599
haftmann@40827
   600
lemma uint_nat: "uint w = int (unat w)"
haftmann@37660
   601
  unfolding unat_def by auto
haftmann@37660
   602
haftmann@37660
   603
lemma uint_number_of:
haftmann@37660
   604
  "uint (number_of b :: 'a :: len0 word) = number_of b mod 2 ^ len_of TYPE('a)"
haftmann@37660
   605
  unfolding word_number_of_alt
haftmann@37660
   606
  by (simp only: int_word_uint)
haftmann@37660
   607
haftmann@37660
   608
lemma unat_number_of: 
haftmann@40827
   609
  "bin_sign b = Int.Pls \<Longrightarrow> 
haftmann@37660
   610
  unat (number_of b::'a::len0 word) = number_of b mod 2 ^ len_of TYPE ('a)"
haftmann@37660
   611
  apply (unfold unat_def)
haftmann@37660
   612
  apply (clarsimp simp only: uint_number_of)
haftmann@37660
   613
  apply (rule nat_mod_distrib [THEN trans])
haftmann@37660
   614
    apply (erule sign_Pls_ge_0 [THEN iffD1])
haftmann@37660
   615
   apply (simp_all add: nat_power_eq)
haftmann@37660
   616
  done
haftmann@37660
   617
haftmann@37660
   618
lemma sint_number_of: "sint (number_of b :: 'a :: len word) = (number_of b + 
haftmann@37660
   619
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
haftmann@37660
   620
    2 ^ (len_of TYPE('a) - 1)"
haftmann@37660
   621
  unfolding word_number_of_alt by (rule int_word_sint)
haftmann@37660
   622
huffman@45995
   623
lemma word_of_int_0 [simp]: "word_of_int 0 = 0"
huffman@45958
   624
  unfolding word_0_wi ..
huffman@45958
   625
huffman@45995
   626
lemma word_of_int_1 [simp]: "word_of_int 1 = 1"
huffman@45958
   627
  unfolding word_1_wi ..
huffman@45958
   628
haftmann@37660
   629
lemma word_of_int_bin [simp] : 
haftmann@37660
   630
  "(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)"
huffman@46001
   631
  unfolding word_number_of_alt ..
haftmann@37660
   632
haftmann@37660
   633
lemma word_int_case_wi: 
haftmann@37660
   634
  "word_int_case f (word_of_int i :: 'b word) = 
haftmann@37660
   635
    f (i mod 2 ^ len_of TYPE('b::len0))"
haftmann@37660
   636
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)
haftmann@37660
   637
haftmann@37660
   638
lemma word_int_split: 
haftmann@37660
   639
  "P (word_int_case f x) = 
haftmann@37660
   640
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
haftmann@37660
   641
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
haftmann@37660
   642
  unfolding word_int_case_def
haftmann@37660
   643
  by (auto simp: word_uint.eq_norm int_mod_eq')
haftmann@37660
   644
haftmann@37660
   645
lemma word_int_split_asm: 
haftmann@37660
   646
  "P (word_int_case f x) = 
haftmann@37660
   647
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
haftmann@37660
   648
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
haftmann@37660
   649
  unfolding word_int_case_def
haftmann@37660
   650
  by (auto simp: word_uint.eq_norm int_mod_eq')
huffman@45805
   651
wenzelm@45604
   652
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
wenzelm@45604
   653
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
haftmann@37660
   654
haftmann@37660
   655
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
haftmann@37660
   656
  unfolding word_size by (rule uint_range')
haftmann@37660
   657
haftmann@37660
   658
lemma sint_range_size:
haftmann@37660
   659
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
haftmann@37660
   660
  unfolding word_size by (rule sint_range')
haftmann@37660
   661
huffman@45805
   662
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x"
huffman@45805
   663
  unfolding word_size by (rule less_le_trans [OF sint_lt])
huffman@45805
   664
huffman@45805
   665
lemma sint_below_size:
huffman@45805
   666
  "x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w"
huffman@45805
   667
  unfolding word_size by (rule order_trans [OF _ sint_ge])
haftmann@37660
   668
huffman@46010
   669
subsection {* Testing bits *}
huffman@46010
   670
haftmann@37660
   671
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)"
haftmann@37660
   672
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
haftmann@37660
   673
haftmann@37660
   674
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
haftmann@37660
   675
  apply (unfold word_test_bit_def)
haftmann@37660
   676
  apply (subst word_ubin.norm_Rep [symmetric])
haftmann@37660
   677
  apply (simp only: nth_bintr word_size)
haftmann@37660
   678
  apply fast
haftmann@37660
   679
  done
haftmann@37660
   680
haftmann@37660
   681
lemma word_eqI [rule_format] : 
haftmann@37660
   682
  fixes u :: "'a::len0 word"
haftmann@40827
   683
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
haftmann@37660
   684
  apply (rule test_bit_eq_iff [THEN iffD1])
haftmann@37660
   685
  apply (rule ext)
haftmann@37660
   686
  apply (erule allE)
haftmann@37660
   687
  apply (erule impCE)
haftmann@37660
   688
   prefer 2
haftmann@37660
   689
   apply assumption
haftmann@37660
   690
  apply (auto dest!: test_bit_size simp add: word_size)
haftmann@37660
   691
  done
haftmann@37660
   692
huffman@45805
   693
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x"
huffman@45805
   694
  by simp
haftmann@37660
   695
haftmann@37660
   696
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
haftmann@37660
   697
  unfolding word_test_bit_def word_size
haftmann@37660
   698
  by (simp add: nth_bintr [symmetric])
haftmann@37660
   699
haftmann@37660
   700
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
haftmann@37660
   701
haftmann@37660
   702
lemma bin_nth_uint_imp': "bin_nth (uint w) n --> n < size w"
haftmann@37660
   703
  apply (unfold word_size)
haftmann@37660
   704
  apply (rule impI)
haftmann@37660
   705
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
haftmann@37660
   706
  apply (subst word_ubin.norm_Rep)
haftmann@37660
   707
  apply assumption
haftmann@37660
   708
  done
haftmann@37660
   709
haftmann@37660
   710
lemma bin_nth_sint': 
haftmann@37660
   711
  "n >= size w --> bin_nth (sint w) n = bin_nth (sint w) (size w - 1)"
haftmann@37660
   712
  apply (rule impI)
haftmann@37660
   713
  apply (subst word_sbin.norm_Rep [symmetric])
haftmann@37660
   714
  apply (simp add : nth_sbintr word_size)
haftmann@37660
   715
  apply auto
haftmann@37660
   716
  done
haftmann@37660
   717
haftmann@37660
   718
lemmas bin_nth_uint_imp = bin_nth_uint_imp' [rule_format, unfolded word_size]
haftmann@37660
   719
lemmas bin_nth_sint = bin_nth_sint' [rule_format, unfolded word_size]
haftmann@37660
   720
haftmann@37660
   721
(* type definitions theorem for in terms of equivalent bool list *)
haftmann@37660
   722
lemma td_bl: 
haftmann@37660
   723
  "type_definition (to_bl :: 'a::len0 word => bool list) 
haftmann@37660
   724
                   of_bl  
haftmann@37660
   725
                   {bl. length bl = len_of TYPE('a)}"
haftmann@37660
   726
  apply (unfold type_definition_def of_bl_def to_bl_def)
haftmann@37660
   727
  apply (simp add: word_ubin.eq_norm)
haftmann@37660
   728
  apply safe
haftmann@37660
   729
  apply (drule sym)
haftmann@37660
   730
  apply simp
haftmann@37660
   731
  done
haftmann@37660
   732
haftmann@37660
   733
interpretation word_bl:
haftmann@37660
   734
  type_definition "to_bl :: 'a::len0 word => bool list"
haftmann@37660
   735
                  of_bl  
haftmann@37660
   736
                  "{bl. length bl = len_of TYPE('a::len0)}"
haftmann@37660
   737
  by (rule td_bl)
haftmann@37660
   738
huffman@45816
   739
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
wenzelm@45538
   740
haftmann@40827
   741
lemma word_size_bl: "size w = size (to_bl w)"
haftmann@37660
   742
  unfolding word_size by auto
haftmann@37660
   743
haftmann@37660
   744
lemma to_bl_use_of_bl:
haftmann@37660
   745
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
huffman@45816
   746
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
haftmann@37660
   747
haftmann@37660
   748
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
haftmann@37660
   749
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)
haftmann@37660
   750
haftmann@37660
   751
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
haftmann@37660
   752
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)
haftmann@37660
   753
haftmann@40827
   754
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
haftmann@37660
   755
  by auto
haftmann@37660
   756
huffman@45805
   757
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
huffman@45805
   758
  by simp
huffman@45805
   759
huffman@45805
   760
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))"
huffman@45805
   761
  unfolding word_bl_Rep' by (rule len_gt_0)
huffman@45805
   762
huffman@45805
   763
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []"
huffman@45805
   764
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
huffman@45805
   765
huffman@45805
   766
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0"
huffman@45805
   767
  by (fact length_bl_gt_0 [THEN gr_implies_not0])
haftmann@37660
   768
huffman@46001
   769
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
haftmann@37660
   770
  apply (unfold to_bl_def sint_uint)
haftmann@37660
   771
  apply (rule trans [OF _ bl_sbin_sign])
haftmann@37660
   772
  apply simp
haftmann@37660
   773
  done
haftmann@37660
   774
haftmann@37660
   775
lemma of_bl_drop': 
haftmann@40827
   776
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
haftmann@37660
   777
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
haftmann@37660
   778
  apply (unfold of_bl_def)
haftmann@37660
   779
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
haftmann@37660
   780
  done
haftmann@37660
   781
huffman@45805
   782
lemma of_bl_no: "of_bl bl = number_of (bl_to_bin bl)"
huffman@45805
   783
  by (fact of_bl_def [folded word_number_of_def])
haftmann@37660
   784
haftmann@37660
   785
lemma test_bit_of_bl:  
haftmann@37660
   786
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)"
haftmann@37660
   787
  apply (unfold of_bl_def word_test_bit_def)
haftmann@37660
   788
  apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl)
haftmann@37660
   789
  done
haftmann@37660
   790
haftmann@37660
   791
lemma no_of_bl: 
haftmann@37660
   792
  "(number_of bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) bin)"
haftmann@37660
   793
  unfolding word_size of_bl_no by (simp add : word_number_of_def)
haftmann@37660
   794
haftmann@40827
   795
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
haftmann@37660
   796
  unfolding word_size to_bl_def by auto
haftmann@37660
   797
haftmann@37660
   798
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   799
  unfolding uint_bl by (simp add : word_size)
haftmann@37660
   800
haftmann@37660
   801
lemma to_bl_of_bin: 
haftmann@37660
   802
  "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
haftmann@37660
   803
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)
haftmann@37660
   804
huffman@45805
   805
lemma to_bl_no_bin [simp]:
huffman@45805
   806
  "to_bl (number_of bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
huffman@45805
   807
  by (fact to_bl_of_bin [folded word_number_of_def])
haftmann@37660
   808
haftmann@37660
   809
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   810
  unfolding uint_bl by (simp add : word_size)
haftmann@37660
   811
  
wenzelm@45604
   812
lemmas uint_bl_bin [simp] = trans [OF bin_bl_bin word_ubin.norm_Rep]
wenzelm@45604
   813
huffman@45805
   814
(* FIXME: the next two lemmas should be unnecessary, because the lhs
huffman@45805
   815
terms should never occur in practice *)
huffman@45805
   816
lemma num_AB_u [simp]: "number_of (uint x) = x"
huffman@45805
   817
  unfolding word_number_of_def by (rule word_uint.Rep_inverse)
huffman@45805
   818
huffman@45805
   819
lemma num_AB_s [simp]: "number_of (sint x) = x"
huffman@45805
   820
  unfolding word_number_of_def by (rule word_sint.Rep_inverse)
haftmann@37660
   821
haftmann@37660
   822
(* naturals *)
haftmann@37660
   823
lemma uints_unats: "uints n = int ` unats n"
haftmann@37660
   824
  apply (unfold unats_def uints_num)
haftmann@37660
   825
  apply safe
haftmann@37660
   826
  apply (rule_tac image_eqI)
haftmann@37660
   827
  apply (erule_tac nat_0_le [symmetric])
haftmann@37660
   828
  apply auto
haftmann@37660
   829
  apply (erule_tac nat_less_iff [THEN iffD2])
haftmann@37660
   830
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
haftmann@37660
   831
  apply (auto simp add : nat_power_eq int_power)
haftmann@37660
   832
  done
haftmann@37660
   833
haftmann@37660
   834
lemma unats_uints: "unats n = nat ` uints n"
haftmann@37660
   835
  by (auto simp add : uints_unats image_iff)
haftmann@37660
   836
wenzelm@45604
   837
lemmas bintr_num = word_ubin.norm_eq_iff [symmetric, folded word_number_of_def]
wenzelm@45604
   838
lemmas sbintr_num = word_sbin.norm_eq_iff [symmetric, folded word_number_of_def]
wenzelm@45604
   839
wenzelm@45604
   840
lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def]
wenzelm@45604
   841
lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def]
haftmann@37660
   842
    
haftmann@37660
   843
(* don't add these to simpset, since may want bintrunc n w to be simplified;
haftmann@37660
   844
  may want these in reverse, but loop as simp rules, so use following *)
haftmann@37660
   845
haftmann@37660
   846
lemma num_of_bintr':
haftmann@40827
   847
  "bintrunc (len_of TYPE('a :: len0)) a = b \<Longrightarrow> 
haftmann@37660
   848
    number_of a = (number_of b :: 'a word)"
haftmann@37660
   849
  apply safe
haftmann@37660
   850
  apply (rule_tac num_of_bintr [symmetric])
haftmann@37660
   851
  done
haftmann@37660
   852
haftmann@37660
   853
lemma num_of_sbintr':
haftmann@40827
   854
  "sbintrunc (len_of TYPE('a :: len) - 1) a = b \<Longrightarrow> 
haftmann@37660
   855
    number_of a = (number_of b :: 'a word)"
haftmann@37660
   856
  apply safe
haftmann@37660
   857
  apply (rule_tac num_of_sbintr [symmetric])
haftmann@37660
   858
  done
haftmann@37660
   859
wenzelm@45604
   860
lemmas num_abs_bintr = sym [THEN trans, OF num_of_bintr word_number_of_def]
wenzelm@45604
   861
lemmas num_abs_sbintr = sym [THEN trans, OF num_of_sbintr word_number_of_def]
haftmann@37660
   862
  
haftmann@37660
   863
(** cast - note, no arg for new length, as it's determined by type of result,
haftmann@37660
   864
  thus in "cast w = w, the type means cast to length of w! **)
haftmann@37660
   865
haftmann@37660
   866
lemma ucast_id: "ucast w = w"
haftmann@37660
   867
  unfolding ucast_def by auto
haftmann@37660
   868
haftmann@37660
   869
lemma scast_id: "scast w = w"
haftmann@37660
   870
  unfolding scast_def by auto
haftmann@37660
   871
haftmann@40827
   872
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
haftmann@37660
   873
  unfolding ucast_def of_bl_def uint_bl
haftmann@37660
   874
  by (auto simp add : word_size)
haftmann@37660
   875
haftmann@37660
   876
lemma nth_ucast: 
haftmann@37660
   877
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
haftmann@37660
   878
  apply (unfold ucast_def test_bit_bin)
haftmann@37660
   879
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
haftmann@37660
   880
  apply (fast elim!: bin_nth_uint_imp)
haftmann@37660
   881
  done
haftmann@37660
   882
haftmann@37660
   883
(* for literal u(s)cast *)
haftmann@37660
   884
huffman@46001
   885
lemma ucast_bintr [simp]:
haftmann@37660
   886
  "ucast (number_of w ::'a::len0 word) = 
huffman@46001
   887
   word_of_int (bintrunc (len_of TYPE('a)) (number_of w))"
haftmann@37660
   888
  unfolding ucast_def by simp
haftmann@37660
   889
huffman@46001
   890
lemma scast_sbintr [simp]:
haftmann@37660
   891
  "scast (number_of w ::'a::len word) = 
huffman@46001
   892
   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (number_of w))"
haftmann@37660
   893
  unfolding scast_def by simp
haftmann@37660
   894
haftmann@37660
   895
lemmas source_size = source_size_def [unfolded Let_def word_size]
haftmann@37660
   896
lemmas target_size = target_size_def [unfolded Let_def word_size]
haftmann@37660
   897
lemmas is_down = is_down_def [unfolded source_size target_size]
haftmann@37660
   898
lemmas is_up = is_up_def [unfolded source_size target_size]
haftmann@37660
   899
wenzelm@45604
   900
lemmas is_up_down = trans [OF is_up is_down [symmetric]]
haftmann@37660
   901
huffman@45811
   902
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
haftmann@37660
   903
  apply (unfold is_down)
haftmann@37660
   904
  apply safe
haftmann@37660
   905
  apply (rule ext)
haftmann@37660
   906
  apply (unfold ucast_def scast_def uint_sint)
haftmann@37660
   907
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
   908
  apply simp
haftmann@37660
   909
  done
haftmann@37660
   910
huffman@45811
   911
lemma word_rev_tf:
huffman@45811
   912
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
   913
    rev (takefill False (len_of TYPE('a)) (rev bl))"
haftmann@37660
   914
  unfolding of_bl_def uint_bl
haftmann@37660
   915
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
haftmann@37660
   916
huffman@45811
   917
lemma word_rep_drop:
huffman@45811
   918
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
   919
    replicate (len_of TYPE('a) - length bl) False @
huffman@45811
   920
    drop (length bl - len_of TYPE('a)) bl"
huffman@45811
   921
  by (simp add: word_rev_tf takefill_alt rev_take)
haftmann@37660
   922
haftmann@37660
   923
lemma to_bl_ucast: 
haftmann@37660
   924
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
haftmann@37660
   925
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
haftmann@37660
   926
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
haftmann@37660
   927
  apply (unfold ucast_bl)
haftmann@37660
   928
  apply (rule trans)
haftmann@37660
   929
   apply (rule word_rep_drop)
haftmann@37660
   930
  apply simp
haftmann@37660
   931
  done
haftmann@37660
   932
huffman@45811
   933
lemma ucast_up_app [OF refl]:
haftmann@40827
   934
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
haftmann@37660
   935
    to_bl (uc w) = replicate n False @ (to_bl w)"
haftmann@37660
   936
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
   937
huffman@45811
   938
lemma ucast_down_drop [OF refl]:
haftmann@40827
   939
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
haftmann@37660
   940
    to_bl (uc w) = drop n (to_bl w)"
haftmann@37660
   941
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
   942
huffman@45811
   943
lemma scast_down_drop [OF refl]:
haftmann@40827
   944
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
haftmann@37660
   945
    to_bl (sc w) = drop n (to_bl w)"
haftmann@37660
   946
  apply (subgoal_tac "sc = ucast")
haftmann@37660
   947
   apply safe
haftmann@37660
   948
   apply simp
huffman@45811
   949
   apply (erule ucast_down_drop)
huffman@45811
   950
  apply (rule down_cast_same [symmetric])
haftmann@37660
   951
  apply (simp add : source_size target_size is_down)
haftmann@37660
   952
  done
haftmann@37660
   953
huffman@45811
   954
lemma sint_up_scast [OF refl]:
haftmann@40827
   955
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
haftmann@37660
   956
  apply (unfold is_up)
haftmann@37660
   957
  apply safe
haftmann@37660
   958
  apply (simp add: scast_def word_sbin.eq_norm)
haftmann@37660
   959
  apply (rule box_equals)
haftmann@37660
   960
    prefer 3
haftmann@37660
   961
    apply (rule word_sbin.norm_Rep)
haftmann@37660
   962
   apply (rule sbintrunc_sbintrunc_l)
haftmann@37660
   963
   defer
haftmann@37660
   964
   apply (subst word_sbin.norm_Rep)
haftmann@37660
   965
   apply (rule refl)
haftmann@37660
   966
  apply simp
haftmann@37660
   967
  done
haftmann@37660
   968
huffman@45811
   969
lemma uint_up_ucast [OF refl]:
haftmann@40827
   970
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
haftmann@37660
   971
  apply (unfold is_up)
haftmann@37660
   972
  apply safe
haftmann@37660
   973
  apply (rule bin_eqI)
haftmann@37660
   974
  apply (fold word_test_bit_def)
haftmann@37660
   975
  apply (auto simp add: nth_ucast)
haftmann@37660
   976
  apply (auto simp add: test_bit_bin)
haftmann@37660
   977
  done
huffman@45811
   978
huffman@45811
   979
lemma ucast_up_ucast [OF refl]:
huffman@45811
   980
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
haftmann@37660
   981
  apply (simp (no_asm) add: ucast_def)
haftmann@37660
   982
  apply (clarsimp simp add: uint_up_ucast)
haftmann@37660
   983
  done
haftmann@37660
   984
    
huffman@45811
   985
lemma scast_up_scast [OF refl]:
huffman@45811
   986
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
haftmann@37660
   987
  apply (simp (no_asm) add: scast_def)
haftmann@37660
   988
  apply (clarsimp simp add: sint_up_scast)
haftmann@37660
   989
  done
haftmann@37660
   990
    
huffman@45811
   991
lemma ucast_of_bl_up [OF refl]:
haftmann@40827
   992
  "w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl"
haftmann@37660
   993
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
haftmann@37660
   994
haftmann@37660
   995
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
haftmann@37660
   996
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
haftmann@37660
   997
haftmann@37660
   998
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
haftmann@37660
   999
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
haftmann@37660
  1000
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
haftmann@37660
  1001
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
haftmann@37660
  1002
haftmann@37660
  1003
lemma up_ucast_surj:
haftmann@40827
  1004
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1005
   surj (ucast :: 'a word => 'b word)"
haftmann@37660
  1006
  by (rule surjI, erule ucast_up_ucast_id)
haftmann@37660
  1007
haftmann@37660
  1008
lemma up_scast_surj:
haftmann@40827
  1009
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1010
   surj (scast :: 'a word => 'b word)"
haftmann@37660
  1011
  by (rule surjI, erule scast_up_scast_id)
haftmann@37660
  1012
haftmann@37660
  1013
lemma down_scast_inj:
haftmann@40827
  1014
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1015
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1016
  by (rule inj_on_inverseI, erule scast_down_scast_id)
haftmann@37660
  1017
haftmann@37660
  1018
lemma down_ucast_inj:
haftmann@40827
  1019
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1020
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1021
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)
haftmann@37660
  1022
haftmann@37660
  1023
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
haftmann@37660
  1024
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
huffman@45811
  1025
huffman@45811
  1026
lemma ucast_down_no [OF refl]:
haftmann@40827
  1027
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (number_of bin) = number_of bin"
haftmann@37660
  1028
  apply (unfold word_number_of_def is_down)
haftmann@37660
  1029
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
haftmann@37660
  1030
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1031
  apply (erule bintrunc_bintrunc_ge)
haftmann@37660
  1032
  done
huffman@45811
  1033
huffman@45811
  1034
lemma ucast_down_bl [OF refl]:
huffman@45811
  1035
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
haftmann@37660
  1036
  unfolding of_bl_no by clarify (erule ucast_down_no)
haftmann@37660
  1037
haftmann@37660
  1038
lemmas slice_def' = slice_def [unfolded word_size]
haftmann@37660
  1039
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
haftmann@37660
  1040
haftmann@37660
  1041
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
haftmann@37660
  1042
lemmas word_log_bin_defs = word_log_defs
haftmann@37660
  1043
haftmann@37660
  1044
text {* Executable equality *}
haftmann@37660
  1045
haftmann@38857
  1046
instantiation word :: (len0) equal
kleing@24333
  1047
begin
kleing@24333
  1048
haftmann@38857
  1049
definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where
haftmann@38857
  1050
  "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"
haftmann@37660
  1051
haftmann@37660
  1052
instance proof
haftmann@38857
  1053
qed (simp add: equal equal_word_def)
haftmann@37660
  1054
haftmann@37660
  1055
end
haftmann@37660
  1056
haftmann@37660
  1057
haftmann@37660
  1058
subsection {* Word Arithmetic *}
haftmann@37660
  1059
haftmann@37660
  1060
lemma word_less_alt: "(a < b) = (uint a < uint b)"
haftmann@37660
  1061
  unfolding word_less_def word_le_def
haftmann@37660
  1062
  by (auto simp del: word_uint.Rep_inject 
haftmann@37660
  1063
           simp: word_uint.Rep_inject [symmetric])
haftmann@37660
  1064
haftmann@37660
  1065
lemma signed_linorder: "class.linorder word_sle word_sless"
haftmann@37660
  1066
proof
haftmann@37660
  1067
qed (unfold word_sle_def word_sless_def, auto)
haftmann@37660
  1068
haftmann@37660
  1069
interpretation signed: linorder "word_sle" "word_sless"
haftmann@37660
  1070
  by (rule signed_linorder)
haftmann@37660
  1071
haftmann@37660
  1072
lemma udvdI: 
haftmann@40827
  1073
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
haftmann@37660
  1074
  by (auto simp: udvd_def)
haftmann@37660
  1075
wenzelm@45604
  1076
lemmas word_div_no [simp] = word_div_def [of "number_of a" "number_of b"] for a b
wenzelm@45604
  1077
wenzelm@45604
  1078
lemmas word_mod_no [simp] = word_mod_def [of "number_of a" "number_of b"] for a b
wenzelm@45604
  1079
wenzelm@45604
  1080
lemmas word_less_no [simp] = word_less_def [of "number_of a" "number_of b"] for a b
wenzelm@45604
  1081
wenzelm@45604
  1082
lemmas word_le_no [simp] = word_le_def [of "number_of a" "number_of b"] for a b
wenzelm@45604
  1083
wenzelm@45604
  1084
lemmas word_sless_no [simp] = word_sless_def [of "number_of a" "number_of b"] for a b
wenzelm@45604
  1085
wenzelm@45604
  1086
lemmas word_sle_no [simp] = word_sle_def [of "number_of a" "number_of b"] for a b
haftmann@37660
  1087
haftmann@37660
  1088
(* following two are available in class number_ring, 
haftmann@37660
  1089
  but convenient to have them here here;
haftmann@37660
  1090
  note - the number_ring versions, numeral_0_eq_0 and numeral_1_eq_1
haftmann@37660
  1091
  are in the default simpset, so to use the automatic simplifications for
haftmann@37660
  1092
  (eg) sint (number_of bin) on sint 1, must do
haftmann@37660
  1093
  (simp add: word_1_no del: numeral_1_eq_1) 
haftmann@37660
  1094
  *)
huffman@45958
  1095
lemma word_0_wi_Pls: "0 = word_of_int Int.Pls"
huffman@45958
  1096
  by (simp only: Pls_def word_0_wi)
huffman@45958
  1097
huffman@45958
  1098
lemma word_0_no: "(0::'a::len0 word) = Numeral0"
huffman@45995
  1099
  by (simp add: word_number_of_alt)
haftmann@37660
  1100
haftmann@40827
  1101
lemma int_one_bin: "(1 :: int) = (Int.Pls BIT 1)"
haftmann@37660
  1102
  unfolding Pls_def Bit_def by auto
haftmann@37660
  1103
haftmann@37660
  1104
lemma word_1_no: 
haftmann@40827
  1105
  "(1 :: 'a :: len0 word) = number_of (Int.Pls BIT 1)"
haftmann@37660
  1106
  unfolding word_1_wi word_number_of_def int_one_bin by auto
haftmann@37660
  1107
haftmann@40827
  1108
lemma word_m1_wi: "-1 = word_of_int -1" 
haftmann@37660
  1109
  by (rule word_number_of_alt)
haftmann@37660
  1110
haftmann@37660
  1111
lemma word_m1_wi_Min: "-1 = word_of_int Int.Min"
haftmann@37660
  1112
  by (simp add: word_m1_wi number_of_eq)
haftmann@37660
  1113
huffman@45805
  1114
lemma word_0_bl [simp]: "of_bl [] = 0" 
huffman@45995
  1115
  unfolding of_bl_def by (simp add: Pls_def)
haftmann@37660
  1116
haftmann@37660
  1117
lemma word_1_bl: "of_bl [True] = 1" 
huffman@45995
  1118
  unfolding of_bl_def
huffman@45995
  1119
  by (simp add: bl_to_bin_def Bit_def Pls_def)
haftmann@37660
  1120
haftmann@37660
  1121
lemma uint_eq_0 [simp] : "(uint 0 = 0)" 
haftmann@37660
  1122
  unfolding word_0_wi
haftmann@37660
  1123
  by (simp add: word_ubin.eq_norm Pls_def [symmetric])
haftmann@37660
  1124
huffman@45995
  1125
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
huffman@45995
  1126
  by (simp add: of_bl_def bl_to_bin_rep_False Pls_def)
haftmann@37660
  1127
huffman@45805
  1128
lemma to_bl_0 [simp]:
haftmann@37660
  1129
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
haftmann@37660
  1130
  unfolding uint_bl
haftmann@37660
  1131
  by (simp add : word_size bin_to_bl_Pls Pls_def [symmetric])
haftmann@37660
  1132
haftmann@37660
  1133
lemma uint_0_iff: "(uint x = 0) = (x = 0)"
haftmann@37660
  1134
  by (auto intro!: word_uint.Rep_eqD)
haftmann@37660
  1135
haftmann@37660
  1136
lemma unat_0_iff: "(unat x = 0) = (x = 0)"
haftmann@37660
  1137
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)
haftmann@37660
  1138
haftmann@37660
  1139
lemma unat_0 [simp]: "unat 0 = 0"
haftmann@37660
  1140
  unfolding unat_def by auto
haftmann@37660
  1141
haftmann@40827
  1142
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
haftmann@37660
  1143
  apply (unfold word_size)
haftmann@37660
  1144
  apply (rule box_equals)
haftmann@37660
  1145
    defer
haftmann@37660
  1146
    apply (rule word_uint.Rep_inverse)+
haftmann@37660
  1147
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1148
  apply simp
haftmann@37660
  1149
  done
haftmann@37660
  1150
huffman@45816
  1151
lemmas size_0_same = size_0_same' [unfolded word_size]
haftmann@37660
  1152
haftmann@37660
  1153
lemmas unat_eq_0 = unat_0_iff
haftmann@37660
  1154
lemmas unat_eq_zero = unat_0_iff
haftmann@37660
  1155
haftmann@37660
  1156
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
haftmann@37660
  1157
by (auto simp: unat_0_iff [symmetric])
haftmann@37660
  1158
huffman@45958
  1159
lemma ucast_0 [simp]: "ucast 0 = 0"
huffman@45995
  1160
  unfolding ucast_def by simp
huffman@45958
  1161
huffman@45958
  1162
lemma sint_0 [simp]: "sint 0 = 0"
huffman@45958
  1163
  unfolding sint_uint by simp
huffman@45958
  1164
huffman@45958
  1165
lemma scast_0 [simp]: "scast 0 = 0"
huffman@45995
  1166
  unfolding scast_def by simp
haftmann@37660
  1167
haftmann@37660
  1168
lemma sint_n1 [simp] : "sint -1 = -1"
huffman@45958
  1169
  unfolding word_m1_wi by (simp add: word_sbin.eq_norm)
huffman@45958
  1170
huffman@45958
  1171
lemma scast_n1 [simp]: "scast -1 = -1"
huffman@45958
  1172
  unfolding scast_def by simp
huffman@45958
  1173
huffman@45958
  1174
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
haftmann@37660
  1175
  unfolding word_1_wi
huffman@45995
  1176
  by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1)
huffman@45958
  1177
huffman@45958
  1178
lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
huffman@45958
  1179
  unfolding unat_def by simp
huffman@45958
  1180
huffman@45958
  1181
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
huffman@45995
  1182
  unfolding ucast_def by simp
haftmann@37660
  1183
haftmann@37660
  1184
(* now, to get the weaker results analogous to word_div/mod_def *)
haftmann@37660
  1185
haftmann@37660
  1186
lemmas word_arith_alts = 
huffman@46000
  1187
  word_sub_wi
huffman@46000
  1188
  word_arith_wis (* FIXME: duplicate *)
huffman@46000
  1189
huffman@46000
  1190
lemmas word_succ_alt = word_succ_def (* FIXME: duplicate *)
huffman@46000
  1191
lemmas word_pred_alt = word_pred_def (* FIXME: duplicate *)
haftmann@37660
  1192
haftmann@37660
  1193
subsection  "Transferring goals from words to ints"
haftmann@37660
  1194
haftmann@37660
  1195
lemma word_ths:  
haftmann@37660
  1196
  shows
haftmann@37660
  1197
  word_succ_p1:   "word_succ a = a + 1" and
haftmann@37660
  1198
  word_pred_m1:   "word_pred a = a - 1" and
haftmann@37660
  1199
  word_pred_succ: "word_pred (word_succ a) = a" and
haftmann@37660
  1200
  word_succ_pred: "word_succ (word_pred a) = a" and
haftmann@37660
  1201
  word_mult_succ: "word_succ a * b = b + a * b"
haftmann@37660
  1202
  by (rule word_uint.Abs_cases [of b],
haftmann@37660
  1203
      rule word_uint.Abs_cases [of a],
huffman@46000
  1204
      simp add: add_commute mult_commute 
huffman@46009
  1205
                ring_distribs word_of_int_homs
huffman@45995
  1206
           del: word_of_int_0 word_of_int_1)+
haftmann@37660
  1207
huffman@45816
  1208
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
huffman@45816
  1209
  by simp
haftmann@37660
  1210
haftmann@37660
  1211
lemmas uint_word_ariths = 
wenzelm@45604
  1212
  word_arith_alts [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1213
haftmann@37660
  1214
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p]
haftmann@37660
  1215
haftmann@37660
  1216
(* similar expressions for sint (arith operations) *)
haftmann@37660
  1217
lemmas sint_word_ariths = uint_word_arith_bintrs
haftmann@37660
  1218
  [THEN uint_sint [symmetric, THEN trans],
haftmann@37660
  1219
  unfolded uint_sint bintr_arith1s bintr_ariths 
wenzelm@45604
  1220
    len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep]
wenzelm@45604
  1221
wenzelm@45604
  1222
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
wenzelm@45604
  1223
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1224
haftmann@37660
  1225
lemma word_pred_0_n1: "word_pred 0 = word_of_int -1"
huffman@45550
  1226
  unfolding word_pred_def uint_eq_0 pred_def by simp
haftmann@37660
  1227
haftmann@37660
  1228
lemma word_pred_0_Min: "word_pred 0 = word_of_int Int.Min"
haftmann@37660
  1229
  by (simp add: word_pred_0_n1 number_of_eq)
haftmann@37660
  1230
haftmann@37660
  1231
lemma word_m1_Min: "- 1 = word_of_int Int.Min"
haftmann@37660
  1232
  unfolding Min_def by (simp only: word_of_int_hom_syms)
haftmann@37660
  1233
haftmann@37660
  1234
lemma succ_pred_no [simp]:
haftmann@37660
  1235
  "word_succ (number_of bin) = number_of (Int.succ bin) & 
haftmann@37660
  1236
    word_pred (number_of bin) = number_of (Int.pred bin)"
huffman@46000
  1237
  unfolding word_number_of_def Int.succ_def Int.pred_def
huffman@46009
  1238
  by (simp add: word_of_int_homs)
haftmann@37660
  1239
haftmann@37660
  1240
lemma word_sp_01 [simp] : 
haftmann@37660
  1241
  "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0"
huffman@45847
  1242
  by (unfold word_0_no word_1_no) (auto simp: BIT_simps)
haftmann@37660
  1243
haftmann@37660
  1244
(* alternative approach to lifting arithmetic equalities *)
haftmann@37660
  1245
lemma word_of_int_Ex:
haftmann@37660
  1246
  "\<exists>y. x = word_of_int y"
haftmann@37660
  1247
  by (rule_tac x="uint x" in exI) simp
haftmann@37660
  1248
haftmann@37660
  1249
haftmann@37660
  1250
subsection "Order on fixed-length words"
haftmann@37660
  1251
haftmann@37660
  1252
lemma word_zero_le [simp] :
haftmann@37660
  1253
  "0 <= (y :: 'a :: len0 word)"
haftmann@37660
  1254
  unfolding word_le_def by auto
haftmann@37660
  1255
  
huffman@45816
  1256
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *)
haftmann@37660
  1257
  unfolding word_le_def
haftmann@37660
  1258
  by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1259
huffman@45816
  1260
lemma word_n1_ge [simp]: "y \<le> (-1::'a::len0 word)"
huffman@45816
  1261
  unfolding word_le_def
huffman@45816
  1262
  by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1263
haftmann@37660
  1264
lemmas word_not_simps [simp] = 
haftmann@37660
  1265
  word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD]
haftmann@37660
  1266
haftmann@37660
  1267
lemma word_gt_0: "0 < y = (0 ~= (y :: 'a :: len0 word))"
haftmann@37660
  1268
  unfolding word_less_def by auto
haftmann@37660
  1269
wenzelm@45604
  1270
lemmas word_gt_0_no [simp] = word_gt_0 [of "number_of y"] for y
haftmann@37660
  1271
haftmann@40827
  1272
lemma word_sless_alt: "(a <s b) = (sint a < sint b)"
haftmann@37660
  1273
  unfolding word_sle_def word_sless_def
haftmann@37660
  1274
  by (auto simp add: less_le)
haftmann@37660
  1275
haftmann@37660
  1276
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)"
haftmann@37660
  1277
  unfolding unat_def word_le_def
haftmann@37660
  1278
  by (rule nat_le_eq_zle [symmetric]) simp
haftmann@37660
  1279
haftmann@37660
  1280
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)"
haftmann@37660
  1281
  unfolding unat_def word_less_alt
haftmann@37660
  1282
  by (rule nat_less_eq_zless [symmetric]) simp
haftmann@37660
  1283
  
haftmann@37660
  1284
lemma wi_less: 
haftmann@37660
  1285
  "(word_of_int n < (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1286
    (n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1287
  unfolding word_less_alt by (simp add: word_uint.eq_norm)
haftmann@37660
  1288
haftmann@37660
  1289
lemma wi_le: 
haftmann@37660
  1290
  "(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1291
    (n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1292
  unfolding word_le_def by (simp add: word_uint.eq_norm)
haftmann@37660
  1293
haftmann@37660
  1294
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)"
haftmann@37660
  1295
  apply (unfold udvd_def)
haftmann@37660
  1296
  apply safe
haftmann@37660
  1297
   apply (simp add: unat_def nat_mult_distrib)
haftmann@37660
  1298
  apply (simp add: uint_nat int_mult)
haftmann@37660
  1299
  apply (rule exI)
haftmann@37660
  1300
  apply safe
haftmann@37660
  1301
   prefer 2
haftmann@37660
  1302
   apply (erule notE)
haftmann@37660
  1303
   apply (rule refl)
haftmann@37660
  1304
  apply force
haftmann@37660
  1305
  done
haftmann@37660
  1306
haftmann@37660
  1307
lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y"
haftmann@37660
  1308
  unfolding dvd_def udvd_nat_alt by force
haftmann@37660
  1309
wenzelm@45604
  1310
lemmas unat_mono = word_less_nat_alt [THEN iffD1]
haftmann@37660
  1311
haftmann@40827
  1312
lemma unat_minus_one: "x ~= 0 \<Longrightarrow> unat (x - 1) = unat x - 1"
haftmann@37660
  1313
  apply (unfold unat_def)
haftmann@37660
  1314
  apply (simp only: int_word_uint word_arith_alts rdmods)
haftmann@37660
  1315
  apply (subgoal_tac "uint x >= 1")
haftmann@37660
  1316
   prefer 2
haftmann@37660
  1317
   apply (drule contrapos_nn)
haftmann@37660
  1318
    apply (erule word_uint.Rep_inverse' [symmetric])
haftmann@37660
  1319
   apply (insert uint_ge_0 [of x])[1]
haftmann@37660
  1320
   apply arith
haftmann@37660
  1321
  apply (rule box_equals)
haftmann@37660
  1322
    apply (rule nat_diff_distrib)
haftmann@37660
  1323
     prefer 2
haftmann@37660
  1324
     apply assumption
haftmann@37660
  1325
    apply simp
haftmann@37660
  1326
   apply (subst mod_pos_pos_trivial)
haftmann@37660
  1327
     apply arith
haftmann@37660
  1328
    apply (insert uint_lt2p [of x])[1]
haftmann@37660
  1329
    apply arith
haftmann@37660
  1330
   apply (rule refl)
haftmann@37660
  1331
  apply simp
haftmann@37660
  1332
  done
haftmann@37660
  1333
    
haftmann@40827
  1334
lemma measure_unat: "p ~= 0 \<Longrightarrow> unat (p - 1) < unat p"
haftmann@37660
  1335
  by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric])
haftmann@37660
  1336
  
wenzelm@45604
  1337
lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
wenzelm@45604
  1338
lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
haftmann@37660
  1339
haftmann@37660
  1340
lemma uint_sub_lt2p [simp]: 
haftmann@37660
  1341
  "uint (x :: 'a :: len0 word) - uint (y :: 'b :: len0 word) < 
haftmann@37660
  1342
    2 ^ len_of TYPE('a)"
haftmann@37660
  1343
  using uint_ge_0 [of y] uint_lt2p [of x] by arith
haftmann@37660
  1344
haftmann@37660
  1345
haftmann@37660
  1346
subsection "Conditions for the addition (etc) of two words to overflow"
haftmann@37660
  1347
haftmann@37660
  1348
lemma uint_add_lem: 
haftmann@37660
  1349
  "(uint x + uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1350
    (uint (x + y :: 'a :: len0 word) = uint x + uint y)"
haftmann@37660
  1351
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1352
haftmann@37660
  1353
lemma uint_mult_lem: 
haftmann@37660
  1354
  "(uint x * uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1355
    (uint (x * y :: 'a :: len0 word) = uint x * uint y)"
haftmann@37660
  1356
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1357
haftmann@37660
  1358
lemma uint_sub_lem: 
haftmann@37660
  1359
  "(uint x >= uint y) = (uint (x - y) = uint x - uint y)"
haftmann@37660
  1360
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1361
haftmann@37660
  1362
lemma uint_add_le: "uint (x + y) <= uint x + uint y"
haftmann@37660
  1363
  unfolding uint_word_ariths by (auto simp: mod_add_if_z)
haftmann@37660
  1364
haftmann@37660
  1365
lemma uint_sub_ge: "uint (x - y) >= uint x - uint y"
haftmann@37660
  1366
  unfolding uint_word_ariths by (auto simp: mod_sub_if_z)
haftmann@37660
  1367
wenzelm@45604
  1368
lemmas uint_sub_if' = trans [OF uint_word_ariths(1) mod_sub_if_z, simplified]
wenzelm@45604
  1369
lemmas uint_plus_if' = trans [OF uint_word_ariths(2) mod_add_if_z, simplified]
haftmann@37660
  1370
haftmann@37660
  1371
haftmann@37660
  1372
subsection {* Definition of uint\_arith *}
haftmann@37660
  1373
haftmann@37660
  1374
lemma word_of_int_inverse:
haftmann@40827
  1375
  "word_of_int r = a \<Longrightarrow> 0 <= r \<Longrightarrow> r < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1376
   uint (a::'a::len0 word) = r"
haftmann@37660
  1377
  apply (erule word_uint.Abs_inverse' [rotated])
haftmann@37660
  1378
  apply (simp add: uints_num)
haftmann@37660
  1379
  done
haftmann@37660
  1380
haftmann@37660
  1381
lemma uint_split:
haftmann@37660
  1382
  fixes x::"'a::len0 word"
haftmann@37660
  1383
  shows "P (uint x) = 
haftmann@37660
  1384
         (ALL i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) --> P i)"
haftmann@37660
  1385
  apply (fold word_int_case_def)
haftmann@37660
  1386
  apply (auto dest!: word_of_int_inverse simp: int_word_uint int_mod_eq'
haftmann@37660
  1387
              split: word_int_split)
haftmann@37660
  1388
  done
haftmann@37660
  1389
haftmann@37660
  1390
lemma uint_split_asm:
haftmann@37660
  1391
  fixes x::"'a::len0 word"
haftmann@37660
  1392
  shows "P (uint x) = 
haftmann@37660
  1393
         (~(EX i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) & ~ P i))"
haftmann@37660
  1394
  by (auto dest!: word_of_int_inverse 
haftmann@37660
  1395
           simp: int_word_uint int_mod_eq'
haftmann@37660
  1396
           split: uint_split)
haftmann@37660
  1397
haftmann@37660
  1398
lemmas uint_splits = uint_split uint_split_asm
haftmann@37660
  1399
haftmann@37660
  1400
lemmas uint_arith_simps = 
haftmann@37660
  1401
  word_le_def word_less_alt
haftmann@37660
  1402
  word_uint.Rep_inject [symmetric] 
haftmann@37660
  1403
  uint_sub_if' uint_plus_if'
haftmann@37660
  1404
haftmann@37660
  1405
(* use this to stop, eg, 2 ^ len_of TYPE (32) being simplified *)
haftmann@40827
  1406
lemma power_False_cong: "False \<Longrightarrow> a ^ b = c ^ d" 
haftmann@37660
  1407
  by auto
haftmann@37660
  1408
haftmann@37660
  1409
(* uint_arith_tac: reduce to arithmetic on int, try to solve by arith *)
haftmann@37660
  1410
ML {*
haftmann@37660
  1411
fun uint_arith_ss_of ss = 
haftmann@37660
  1412
  ss addsimps @{thms uint_arith_simps}
haftmann@37660
  1413
     delsimps @{thms word_uint.Rep_inject}
wenzelm@45620
  1414
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1415
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1416
haftmann@37660
  1417
fun uint_arith_tacs ctxt = 
haftmann@37660
  1418
  let
haftmann@37660
  1419
    fun arith_tac' n t =
haftmann@37660
  1420
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1421
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1422
  in 
wenzelm@42793
  1423
    [ clarify_tac ctxt 1,
wenzelm@42793
  1424
      full_simp_tac (uint_arith_ss_of (simpset_of ctxt)) 1,
wenzelm@45620
  1425
      ALLGOALS (full_simp_tac (HOL_ss |> fold Splitter.add_split @{thms uint_splits}
wenzelm@45620
  1426
                                      |> fold Simplifier.add_cong @{thms power_False_cong})),
haftmann@37660
  1427
      rewrite_goals_tac @{thms word_size}, 
haftmann@37660
  1428
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1429
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1430
                         REPEAT (dtac @{thm word_of_int_inverse} n 
haftmann@37660
  1431
                                 THEN atac n 
haftmann@37660
  1432
                                 THEN atac n)),
haftmann@37660
  1433
      TRYALL arith_tac' ]
haftmann@37660
  1434
  end
haftmann@37660
  1435
haftmann@37660
  1436
fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt))
haftmann@37660
  1437
*}
haftmann@37660
  1438
haftmann@37660
  1439
method_setup uint_arith = 
haftmann@37660
  1440
  {* Scan.succeed (SIMPLE_METHOD' o uint_arith_tac) *}
haftmann@37660
  1441
  "solving word arithmetic via integers and arith"
haftmann@37660
  1442
haftmann@37660
  1443
haftmann@37660
  1444
subsection "More on overflows and monotonicity"
haftmann@37660
  1445
haftmann@37660
  1446
lemma no_plus_overflow_uint_size: 
haftmann@37660
  1447
  "((x :: 'a :: len0 word) <= x + y) = (uint x + uint y < 2 ^ size x)"
haftmann@37660
  1448
  unfolding word_size by uint_arith
haftmann@37660
  1449
haftmann@37660
  1450
lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size]
haftmann@37660
  1451
haftmann@37660
  1452
lemma no_ulen_sub: "((x :: 'a :: len0 word) >= x - y) = (uint y <= uint x)"
haftmann@37660
  1453
  by uint_arith
haftmann@37660
  1454
haftmann@37660
  1455
lemma no_olen_add':
haftmann@37660
  1456
  fixes x :: "'a::len0 word"
haftmann@37660
  1457
  shows "(x \<le> y + x) = (uint y + uint x < 2 ^ len_of TYPE('a))"
huffman@45546
  1458
  by (simp add: add_ac no_olen_add)
haftmann@37660
  1459
wenzelm@45604
  1460
lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric]]
wenzelm@45604
  1461
wenzelm@45604
  1462
lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem]
wenzelm@45604
  1463
lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1]
wenzelm@45604
  1464
lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem]
haftmann@37660
  1465
lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def]
haftmann@37660
  1466
lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def]
wenzelm@45604
  1467
lemmas word_sub_le = word_sub_le_iff [THEN iffD2]
haftmann@37660
  1468
haftmann@37660
  1469
lemma word_less_sub1: 
haftmann@40827
  1470
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 < x) = (0 < x - 1)"
haftmann@37660
  1471
  by uint_arith
haftmann@37660
  1472
haftmann@37660
  1473
lemma word_le_sub1: 
haftmann@40827
  1474
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 <= x) = (0 <= x - 1)"
haftmann@37660
  1475
  by uint_arith
haftmann@37660
  1476
haftmann@37660
  1477
lemma sub_wrap_lt: 
haftmann@37660
  1478
  "((x :: 'a :: len0 word) < x - z) = (x < z)"
haftmann@37660
  1479
  by uint_arith
haftmann@37660
  1480
haftmann@37660
  1481
lemma sub_wrap: 
haftmann@37660
  1482
  "((x :: 'a :: len0 word) <= x - z) = (z = 0 | x < z)"
haftmann@37660
  1483
  by uint_arith
haftmann@37660
  1484
haftmann@37660
  1485
lemma plus_minus_not_NULL_ab: 
haftmann@40827
  1486
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> c ~= 0 \<Longrightarrow> x + c ~= 0"
haftmann@37660
  1487
  by uint_arith
haftmann@37660
  1488
haftmann@37660
  1489
lemma plus_minus_no_overflow_ab: 
haftmann@40827
  1490
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> x <= x + c" 
haftmann@37660
  1491
  by uint_arith
haftmann@37660
  1492
haftmann@37660
  1493
lemma le_minus': 
haftmann@40827
  1494
  "(a :: 'a :: len0 word) + c <= b \<Longrightarrow> a <= a + c \<Longrightarrow> c <= b - a"
haftmann@37660
  1495
  by uint_arith
haftmann@37660
  1496
haftmann@37660
  1497
lemma le_plus': 
haftmann@40827
  1498
  "(a :: 'a :: len0 word) <= b \<Longrightarrow> c <= b - a \<Longrightarrow> a + c <= b"
haftmann@37660
  1499
  by uint_arith
haftmann@37660
  1500
haftmann@37660
  1501
lemmas le_plus = le_plus' [rotated]
haftmann@37660
  1502
wenzelm@45604
  1503
lemmas le_minus = leD [THEN thin_rl, THEN le_minus']
haftmann@37660
  1504
haftmann@37660
  1505
lemma word_plus_mono_right: 
haftmann@40827
  1506
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= x + z \<Longrightarrow> x + y <= x + z"
haftmann@37660
  1507
  by uint_arith
haftmann@37660
  1508
haftmann@37660
  1509
lemma word_less_minus_cancel: 
haftmann@40827
  1510
  "y - x < z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) < z"
haftmann@37660
  1511
  by uint_arith
haftmann@37660
  1512
haftmann@37660
  1513
lemma word_less_minus_mono_left: 
haftmann@40827
  1514
  "(y :: 'a :: len0 word) < z \<Longrightarrow> x <= y \<Longrightarrow> y - x < z - x"
haftmann@37660
  1515
  by uint_arith
haftmann@37660
  1516
haftmann@37660
  1517
lemma word_less_minus_mono:  
haftmann@40827
  1518
  "a < c \<Longrightarrow> d < b \<Longrightarrow> a - b < a \<Longrightarrow> c - d < c 
haftmann@40827
  1519
  \<Longrightarrow> a - b < c - (d::'a::len word)"
haftmann@37660
  1520
  by uint_arith
haftmann@37660
  1521
haftmann@37660
  1522
lemma word_le_minus_cancel: 
haftmann@40827
  1523
  "y - x <= z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) <= z"
haftmann@37660
  1524
  by uint_arith
haftmann@37660
  1525
haftmann@37660
  1526
lemma word_le_minus_mono_left: 
haftmann@40827
  1527
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= y \<Longrightarrow> y - x <= z - x"
haftmann@37660
  1528
  by uint_arith
haftmann@37660
  1529
haftmann@37660
  1530
lemma word_le_minus_mono:  
haftmann@40827
  1531
  "a <= c \<Longrightarrow> d <= b \<Longrightarrow> a - b <= a \<Longrightarrow> c - d <= c 
haftmann@40827
  1532
  \<Longrightarrow> a - b <= c - (d::'a::len word)"
haftmann@37660
  1533
  by uint_arith
haftmann@37660
  1534
haftmann@37660
  1535
lemma plus_le_left_cancel_wrap: 
haftmann@40827
  1536
  "(x :: 'a :: len0 word) + y' < x \<Longrightarrow> x + y < x \<Longrightarrow> (x + y' < x + y) = (y' < y)"
haftmann@37660
  1537
  by uint_arith
haftmann@37660
  1538
haftmann@37660
  1539
lemma plus_le_left_cancel_nowrap: 
haftmann@40827
  1540
  "(x :: 'a :: len0 word) <= x + y' \<Longrightarrow> x <= x + y \<Longrightarrow> 
haftmann@37660
  1541
    (x + y' < x + y) = (y' < y)" 
haftmann@37660
  1542
  by uint_arith
haftmann@37660
  1543
haftmann@37660
  1544
lemma word_plus_mono_right2: 
haftmann@40827
  1545
  "(a :: 'a :: len0 word) <= a + b \<Longrightarrow> c <= b \<Longrightarrow> a <= a + c"
haftmann@37660
  1546
  by uint_arith
haftmann@37660
  1547
haftmann@37660
  1548
lemma word_less_add_right: 
haftmann@40827
  1549
  "(x :: 'a :: len0 word) < y - z \<Longrightarrow> z <= y \<Longrightarrow> x + z < y"
haftmann@37660
  1550
  by uint_arith
haftmann@37660
  1551
haftmann@37660
  1552
lemma word_less_sub_right: 
haftmann@40827
  1553
  "(x :: 'a :: len0 word) < y + z \<Longrightarrow> y <= x \<Longrightarrow> x - y < z"
haftmann@37660
  1554
  by uint_arith
haftmann@37660
  1555
haftmann@37660
  1556
lemma word_le_plus_either: 
haftmann@40827
  1557
  "(x :: 'a :: len0 word) <= y | x <= z \<Longrightarrow> y <= y + z \<Longrightarrow> x <= y + z"
haftmann@37660
  1558
  by uint_arith
haftmann@37660
  1559
haftmann@37660
  1560
lemma word_less_nowrapI: 
haftmann@40827
  1561
  "(x :: 'a :: len0 word) < z - k \<Longrightarrow> k <= z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
haftmann@37660
  1562
  by uint_arith
haftmann@37660
  1563
haftmann@40827
  1564
lemma inc_le: "(i :: 'a :: len word) < m \<Longrightarrow> i + 1 <= m"
haftmann@37660
  1565
  by uint_arith
haftmann@37660
  1566
haftmann@37660
  1567
lemma inc_i: 
haftmann@40827
  1568
  "(1 :: 'a :: len word) <= i \<Longrightarrow> i < m \<Longrightarrow> 1 <= (i + 1) & i + 1 <= m"
haftmann@37660
  1569
  by uint_arith
haftmann@37660
  1570
haftmann@37660
  1571
lemma udvd_incr_lem:
haftmann@40827
  1572
  "up < uq \<Longrightarrow> up = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1573
    uq = ua + n' * uint K \<Longrightarrow> up + uint K <= uq"
haftmann@37660
  1574
  apply clarsimp
haftmann@37660
  1575
  apply (drule less_le_mult)
haftmann@37660
  1576
  apply safe
haftmann@37660
  1577
  done
haftmann@37660
  1578
haftmann@37660
  1579
lemma udvd_incr': 
haftmann@40827
  1580
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1581
    uint q = ua + n' * uint K \<Longrightarrow> p + K <= q" 
haftmann@37660
  1582
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1583
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1584
  apply (erule uint_add_le [THEN order_trans])
haftmann@37660
  1585
  done
haftmann@37660
  1586
haftmann@37660
  1587
lemma udvd_decr': 
haftmann@40827
  1588
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1589
    uint q = ua + n' * uint K \<Longrightarrow> p <= q - K"
haftmann@37660
  1590
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1591
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1592
  apply (drule le_diff_eq [THEN iffD2])
haftmann@37660
  1593
  apply (erule order_trans)
haftmann@37660
  1594
  apply (rule uint_sub_ge)
haftmann@37660
  1595
  done
haftmann@37660
  1596
huffman@45816
  1597
lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, unfolded add_0_left]
huffman@45816
  1598
lemmas udvd_incr0 = udvd_incr' [where ua=0, unfolded add_0_left]
huffman@45816
  1599
lemmas udvd_decr0 = udvd_decr' [where ua=0, unfolded add_0_left]
haftmann@37660
  1600
haftmann@37660
  1601
lemma udvd_minus_le': 
haftmann@40827
  1602
  "xy < k \<Longrightarrow> z udvd xy \<Longrightarrow> z udvd k \<Longrightarrow> xy <= k - z"
haftmann@37660
  1603
  apply (unfold udvd_def)
haftmann@37660
  1604
  apply clarify
haftmann@37660
  1605
  apply (erule (2) udvd_decr0)
haftmann@37660
  1606
  done
haftmann@37660
  1607
huffman@45284
  1608
ML {* Delsimprocs [@{simproc linordered_ring_less_cancel_factor}] *}
haftmann@37660
  1609
haftmann@37660
  1610
lemma udvd_incr2_K: 
haftmann@40827
  1611
  "p < a + s \<Longrightarrow> a <= a + s \<Longrightarrow> K udvd s \<Longrightarrow> K udvd p - a \<Longrightarrow> a <= p \<Longrightarrow> 
haftmann@40827
  1612
    0 < K \<Longrightarrow> p <= p + K & p + K <= a + s"
haftmann@37660
  1613
  apply (unfold udvd_def)
haftmann@37660
  1614
  apply clarify
haftmann@37660
  1615
  apply (simp add: uint_arith_simps split: split_if_asm)
haftmann@37660
  1616
   prefer 2 
haftmann@37660
  1617
   apply (insert uint_range' [of s])[1]
haftmann@37660
  1618
   apply arith
haftmann@37660
  1619
  apply (drule add_commute [THEN xtr1])
haftmann@37660
  1620
  apply (simp add: diff_less_eq [symmetric])
haftmann@37660
  1621
  apply (drule less_le_mult)
haftmann@37660
  1622
   apply arith
haftmann@37660
  1623
  apply simp
haftmann@37660
  1624
  done
haftmann@37660
  1625
huffman@45284
  1626
ML {* Addsimprocs [@{simproc linordered_ring_less_cancel_factor}] *}
haftmann@37660
  1627
haftmann@37660
  1628
(* links with rbl operations *)
haftmann@37660
  1629
lemma word_succ_rbl:
haftmann@40827
  1630
  "to_bl w = bl \<Longrightarrow> to_bl (word_succ w) = (rev (rbl_succ (rev bl)))"
haftmann@37660
  1631
  apply (unfold word_succ_def)
haftmann@37660
  1632
  apply clarify
haftmann@37660
  1633
  apply (simp add: to_bl_of_bin)
huffman@46000
  1634
  apply (simp add: to_bl_def rbl_succ Int.succ_def)
haftmann@37660
  1635
  done
haftmann@37660
  1636
haftmann@37660
  1637
lemma word_pred_rbl:
haftmann@40827
  1638
  "to_bl w = bl \<Longrightarrow> to_bl (word_pred w) = (rev (rbl_pred (rev bl)))"
haftmann@37660
  1639
  apply (unfold word_pred_def)
haftmann@37660
  1640
  apply clarify
haftmann@37660
  1641
  apply (simp add: to_bl_of_bin)
huffman@46000
  1642
  apply (simp add: to_bl_def rbl_pred Int.pred_def)
haftmann@37660
  1643
  done
haftmann@37660
  1644
haftmann@37660
  1645
lemma word_add_rbl:
haftmann@40827
  1646
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1647
    to_bl (v + w) = (rev (rbl_add (rev vbl) (rev wbl)))"
haftmann@37660
  1648
  apply (unfold word_add_def)
haftmann@37660
  1649
  apply clarify
haftmann@37660
  1650
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1651
  apply (simp add: to_bl_def rbl_add)
haftmann@37660
  1652
  done
haftmann@37660
  1653
haftmann@37660
  1654
lemma word_mult_rbl:
haftmann@40827
  1655
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1656
    to_bl (v * w) = (rev (rbl_mult (rev vbl) (rev wbl)))"
haftmann@37660
  1657
  apply (unfold word_mult_def)
haftmann@37660
  1658
  apply clarify
haftmann@37660
  1659
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1660
  apply (simp add: to_bl_def rbl_mult)
haftmann@37660
  1661
  done
haftmann@37660
  1662
haftmann@37660
  1663
lemma rtb_rbl_ariths:
haftmann@37660
  1664
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys"
haftmann@37660
  1665
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys"
haftmann@40827
  1666
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs"
haftmann@40827
  1667
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs"
haftmann@37660
  1668
  by (auto simp: rev_swap [symmetric] word_succ_rbl 
haftmann@37660
  1669
                 word_pred_rbl word_mult_rbl word_add_rbl)
haftmann@37660
  1670
haftmann@37660
  1671
haftmann@37660
  1672
subsection "Arithmetic type class instantiations"
haftmann@37660
  1673
haftmann@37660
  1674
(* note that iszero_def is only for class comm_semiring_1_cancel,
haftmann@37660
  1675
   which requires word length >= 1, ie 'a :: len word *) 
haftmann@37660
  1676
lemma zero_bintrunc:
haftmann@37660
  1677
  "iszero (number_of x :: 'a :: len word) = 
haftmann@37660
  1678
    (bintrunc (len_of TYPE('a)) x = Int.Pls)"
haftmann@37660
  1679
  apply (unfold iszero_def word_0_wi word_no_wi)
haftmann@37660
  1680
  apply (rule word_ubin.norm_eq_iff [symmetric, THEN trans])
haftmann@37660
  1681
  apply (simp add : Pls_def [symmetric])
haftmann@37660
  1682
  done
haftmann@37660
  1683
haftmann@37660
  1684
lemmas word_le_0_iff [simp] =
haftmann@37660
  1685
  word_zero_le [THEN leD, THEN linorder_antisym_conv1]
haftmann@37660
  1686
haftmann@37660
  1687
lemma word_of_int_nat: 
haftmann@40827
  1688
  "0 <= x \<Longrightarrow> word_of_int x = of_nat (nat x)"
haftmann@37660
  1689
  by (simp add: of_nat_nat word_of_int)
haftmann@37660
  1690
haftmann@37660
  1691
lemma iszero_word_no [simp] : 
haftmann@37660
  1692
  "iszero (number_of bin :: 'a :: len word) = 
huffman@46001
  1693
    iszero (bintrunc (len_of TYPE('a)) (number_of bin))"
haftmann@37660
  1694
  apply (simp add: zero_bintrunc number_of_is_id)
haftmann@37660
  1695
  apply (unfold iszero_def Pls_def)
haftmann@37660
  1696
  apply (rule refl)
haftmann@37660
  1697
  done
haftmann@37660
  1698
    
haftmann@37660
  1699
haftmann@37660
  1700
subsection "Word and nat"
haftmann@37660
  1701
huffman@45811
  1702
lemma td_ext_unat [OF refl]:
haftmann@40827
  1703
  "n = len_of TYPE ('a :: len) \<Longrightarrow> 
haftmann@37660
  1704
    td_ext (unat :: 'a word => nat) of_nat 
haftmann@37660
  1705
    (unats n) (%i. i mod 2 ^ n)"
haftmann@37660
  1706
  apply (unfold td_ext_def' unat_def word_of_nat unats_uints)
haftmann@37660
  1707
  apply (auto intro!: imageI simp add : word_of_int_hom_syms)
haftmann@37660
  1708
  apply (erule word_uint.Abs_inverse [THEN arg_cong])
haftmann@37660
  1709
  apply (simp add: int_word_uint nat_mod_distrib nat_power_eq)
haftmann@37660
  1710
  done
haftmann@37660
  1711
wenzelm@45604
  1712
lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm]
haftmann@37660
  1713
haftmann@37660
  1714
interpretation word_unat:
haftmann@37660
  1715
  td_ext "unat::'a::len word => nat" 
haftmann@37660
  1716
         of_nat 
haftmann@37660
  1717
         "unats (len_of TYPE('a::len))"
haftmann@37660
  1718
         "%i. i mod 2 ^ len_of TYPE('a::len)"
haftmann@37660
  1719
  by (rule td_ext_unat)
haftmann@37660
  1720
haftmann@37660
  1721
lemmas td_unat = word_unat.td_thm
haftmann@37660
  1722
haftmann@37660
  1723
lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq]
haftmann@37660
  1724
haftmann@40827
  1725
lemma unat_le: "y <= unat (z :: 'a :: len word) \<Longrightarrow> y : unats (len_of TYPE ('a))"
haftmann@37660
  1726
  apply (unfold unats_def)
haftmann@37660
  1727
  apply clarsimp
haftmann@37660
  1728
  apply (rule xtrans, rule unat_lt2p, assumption) 
haftmann@37660
  1729
  done
haftmann@37660
  1730
haftmann@37660
  1731
lemma word_nchotomy:
haftmann@37660
  1732
  "ALL w. EX n. (w :: 'a :: len word) = of_nat n & n < 2 ^ len_of TYPE ('a)"
haftmann@37660
  1733
  apply (rule allI)
haftmann@37660
  1734
  apply (rule word_unat.Abs_cases)
haftmann@37660
  1735
  apply (unfold unats_def)
haftmann@37660
  1736
  apply auto
haftmann@37660
  1737
  done
haftmann@37660
  1738
haftmann@37660
  1739
lemma of_nat_eq:
haftmann@37660
  1740
  fixes w :: "'a::len word"
haftmann@37660
  1741
  shows "(of_nat n = w) = (\<exists>q. n = unat w + q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1742
  apply (rule trans)
haftmann@37660
  1743
   apply (rule word_unat.inverse_norm)
haftmann@37660
  1744
  apply (rule iffI)
haftmann@37660
  1745
   apply (rule mod_eqD)
haftmann@37660
  1746
   apply simp
haftmann@37660
  1747
  apply clarsimp
haftmann@37660
  1748
  done
haftmann@37660
  1749
haftmann@37660
  1750
lemma of_nat_eq_size: 
haftmann@37660
  1751
  "(of_nat n = w) = (EX q. n = unat w + q * 2 ^ size w)"
haftmann@37660
  1752
  unfolding word_size by (rule of_nat_eq)
haftmann@37660
  1753
haftmann@37660
  1754
lemma of_nat_0:
haftmann@37660
  1755
  "(of_nat m = (0::'a::len word)) = (\<exists>q. m = q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1756
  by (simp add: of_nat_eq)
haftmann@37660
  1757
huffman@45805
  1758
lemma of_nat_2p [simp]:
huffman@45805
  1759
  "of_nat (2 ^ len_of TYPE('a)) = (0::'a::len word)"
huffman@45805
  1760
  by (fact mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]])
haftmann@37660
  1761
haftmann@40827
  1762
lemma of_nat_gt_0: "of_nat k ~= 0 \<Longrightarrow> 0 < k"
haftmann@37660
  1763
  by (cases k) auto
haftmann@37660
  1764
haftmann@37660
  1765
lemma of_nat_neq_0: 
haftmann@40827
  1766
  "0 < k \<Longrightarrow> k < 2 ^ len_of TYPE ('a :: len) \<Longrightarrow> of_nat k ~= (0 :: 'a word)"
haftmann@37660
  1767
  by (clarsimp simp add : of_nat_0)
haftmann@37660
  1768
haftmann@37660
  1769
lemma Abs_fnat_hom_add:
haftmann@37660
  1770
  "of_nat a + of_nat b = of_nat (a + b)"
haftmann@37660
  1771
  by simp
haftmann@37660
  1772
haftmann@37660
  1773
lemma Abs_fnat_hom_mult:
haftmann@37660
  1774
  "of_nat a * of_nat b = (of_nat (a * b) :: 'a :: len word)"
haftmann@37660
  1775
  by (simp add: word_of_nat word_of_int_mult_hom zmult_int)
haftmann@37660
  1776
haftmann@37660
  1777
lemma Abs_fnat_hom_Suc:
haftmann@37660
  1778
  "word_succ (of_nat a) = of_nat (Suc a)"
haftmann@37660
  1779
  by (simp add: word_of_nat word_of_int_succ_hom add_ac)
haftmann@37660
  1780
haftmann@37660
  1781
lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0"
huffman@45995
  1782
  by simp
haftmann@37660
  1783
haftmann@37660
  1784
lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)"
huffman@45995
  1785
  by simp
haftmann@37660
  1786
haftmann@37660
  1787
lemmas Abs_fnat_homs = 
haftmann@37660
  1788
  Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc 
haftmann@37660
  1789
  Abs_fnat_hom_0 Abs_fnat_hom_1
haftmann@37660
  1790
haftmann@37660
  1791
lemma word_arith_nat_add:
haftmann@37660
  1792
  "a + b = of_nat (unat a + unat b)" 
haftmann@37660
  1793
  by simp
haftmann@37660
  1794
haftmann@37660
  1795
lemma word_arith_nat_mult:
haftmann@37660
  1796
  "a * b = of_nat (unat a * unat b)"
huffman@45995
  1797
  by (simp add: of_nat_mult)
haftmann@37660
  1798
    
haftmann@37660
  1799
lemma word_arith_nat_Suc:
haftmann@37660
  1800
  "word_succ a = of_nat (Suc (unat a))"
haftmann@37660
  1801
  by (subst Abs_fnat_hom_Suc [symmetric]) simp
haftmann@37660
  1802
haftmann@37660
  1803
lemma word_arith_nat_div:
haftmann@37660
  1804
  "a div b = of_nat (unat a div unat b)"
haftmann@37660
  1805
  by (simp add: word_div_def word_of_nat zdiv_int uint_nat)
haftmann@37660
  1806
haftmann@37660
  1807
lemma word_arith_nat_mod:
haftmann@37660
  1808
  "a mod b = of_nat (unat a mod unat b)"
haftmann@37660
  1809
  by (simp add: word_mod_def word_of_nat zmod_int uint_nat)
haftmann@37660
  1810
haftmann@37660
  1811
lemmas word_arith_nat_defs =
haftmann@37660
  1812
  word_arith_nat_add word_arith_nat_mult
haftmann@37660
  1813
  word_arith_nat_Suc Abs_fnat_hom_0
haftmann@37660
  1814
  Abs_fnat_hom_1 word_arith_nat_div
haftmann@37660
  1815
  word_arith_nat_mod 
haftmann@37660
  1816
huffman@45816
  1817
lemma unat_cong: "x = y \<Longrightarrow> unat x = unat y"
huffman@45816
  1818
  by simp
haftmann@37660
  1819
  
haftmann@37660
  1820
lemmas unat_word_ariths = word_arith_nat_defs
wenzelm@45604
  1821
  [THEN trans [OF unat_cong unat_of_nat]]
haftmann@37660
  1822
haftmann@37660
  1823
lemmas word_sub_less_iff = word_sub_le_iff
huffman@45816
  1824
  [unfolded linorder_not_less [symmetric] Not_eq_iff]
haftmann@37660
  1825
haftmann@37660
  1826
lemma unat_add_lem: 
haftmann@37660
  1827
  "(unat x + unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1828
    (unat (x + y :: 'a :: len word) = unat x + unat y)"
haftmann@37660
  1829
  unfolding unat_word_ariths
haftmann@37660
  1830
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1831
haftmann@37660
  1832
lemma unat_mult_lem: 
haftmann@37660
  1833
  "(unat x * unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1834
    (unat (x * y :: 'a :: len word) = unat x * unat y)"
haftmann@37660
  1835
  unfolding unat_word_ariths
haftmann@37660
  1836
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1837
wenzelm@45604
  1838
lemmas unat_plus_if' = trans [OF unat_word_ariths(1) mod_nat_add, simplified]
haftmann@37660
  1839
haftmann@37660
  1840
lemma le_no_overflow: 
haftmann@40827
  1841
  "x <= b \<Longrightarrow> a <= a + b \<Longrightarrow> x <= a + (b :: 'a :: len0 word)"
haftmann@37660
  1842
  apply (erule order_trans)
haftmann@37660
  1843
  apply (erule olen_add_eqv [THEN iffD1])
haftmann@37660
  1844
  done
haftmann@37660
  1845
wenzelm@45604
  1846
lemmas un_ui_le = trans [OF word_le_nat_alt [symmetric] word_le_def]
haftmann@37660
  1847
haftmann@37660
  1848
lemma unat_sub_if_size:
haftmann@37660
  1849
  "unat (x - y) = (if unat y <= unat x 
haftmann@37660
  1850
   then unat x - unat y 
haftmann@37660
  1851
   else unat x + 2 ^ size x - unat y)"
haftmann@37660
  1852
  apply (unfold word_size)
haftmann@37660
  1853
  apply (simp add: un_ui_le)
haftmann@37660
  1854
  apply (auto simp add: unat_def uint_sub_if')
haftmann@37660
  1855
   apply (rule nat_diff_distrib)
haftmann@37660
  1856
    prefer 3
haftmann@37660
  1857
    apply (simp add: algebra_simps)
haftmann@37660
  1858
    apply (rule nat_diff_distrib [THEN trans])
haftmann@37660
  1859
      prefer 3
haftmann@37660
  1860
      apply (subst nat_add_distrib)
haftmann@37660
  1861
        prefer 3
haftmann@37660
  1862
        apply (simp add: nat_power_eq)
haftmann@37660
  1863
       apply auto
haftmann@37660
  1864
  apply uint_arith
haftmann@37660
  1865
  done
haftmann@37660
  1866
haftmann@37660
  1867
lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size]
haftmann@37660
  1868
haftmann@37660
  1869
lemma unat_div: "unat ((x :: 'a :: len word) div y) = unat x div unat y"
haftmann@37660
  1870
  apply (simp add : unat_word_ariths)
haftmann@37660
  1871
  apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1872
  apply (rule div_le_dividend)
haftmann@37660
  1873
  done
haftmann@37660
  1874
haftmann@37660
  1875
lemma unat_mod: "unat ((x :: 'a :: len word) mod y) = unat x mod unat y"
haftmann@37660
  1876
  apply (clarsimp simp add : unat_word_ariths)
haftmann@37660
  1877
  apply (cases "unat y")
haftmann@37660
  1878
   prefer 2
haftmann@37660
  1879
   apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1880
   apply (rule mod_le_divisor)
haftmann@37660
  1881
   apply auto
haftmann@37660
  1882
  done
haftmann@37660
  1883
haftmann@37660
  1884
lemma uint_div: "uint ((x :: 'a :: len word) div y) = uint x div uint y"
haftmann@37660
  1885
  unfolding uint_nat by (simp add : unat_div zdiv_int)
haftmann@37660
  1886
haftmann@37660
  1887
lemma uint_mod: "uint ((x :: 'a :: len word) mod y) = uint x mod uint y"
haftmann@37660
  1888
  unfolding uint_nat by (simp add : unat_mod zmod_int)
haftmann@37660
  1889
haftmann@37660
  1890
haftmann@37660
  1891
subsection {* Definition of unat\_arith tactic *}
haftmann@37660
  1892
haftmann@37660
  1893
lemma unat_split:
haftmann@37660
  1894
  fixes x::"'a::len word"
haftmann@37660
  1895
  shows "P (unat x) = 
haftmann@37660
  1896
         (ALL n. of_nat n = x & n < 2^len_of TYPE('a) --> P n)"
haftmann@37660
  1897
  by (auto simp: unat_of_nat)
haftmann@37660
  1898
haftmann@37660
  1899
lemma unat_split_asm:
haftmann@37660
  1900
  fixes x::"'a::len word"
haftmann@37660
  1901
  shows "P (unat x) = 
haftmann@37660
  1902
         (~(EX n. of_nat n = x & n < 2^len_of TYPE('a) & ~ P n))"
haftmann@37660
  1903
  by (auto simp: unat_of_nat)
haftmann@37660
  1904
haftmann@37660
  1905
lemmas of_nat_inverse = 
haftmann@37660
  1906
  word_unat.Abs_inverse' [rotated, unfolded unats_def, simplified]
haftmann@37660
  1907
haftmann@37660
  1908
lemmas unat_splits = unat_split unat_split_asm
haftmann@37660
  1909
haftmann@37660
  1910
lemmas unat_arith_simps =
haftmann@37660
  1911
  word_le_nat_alt word_less_nat_alt
haftmann@37660
  1912
  word_unat.Rep_inject [symmetric]
haftmann@37660
  1913
  unat_sub_if' unat_plus_if' unat_div unat_mod
haftmann@37660
  1914
haftmann@37660
  1915
(* unat_arith_tac: tactic to reduce word arithmetic to nat, 
haftmann@37660
  1916
   try to solve via arith *)
haftmann@37660
  1917
ML {*
haftmann@37660
  1918
fun unat_arith_ss_of ss = 
haftmann@37660
  1919
  ss addsimps @{thms unat_arith_simps}
haftmann@37660
  1920
     delsimps @{thms word_unat.Rep_inject}
wenzelm@45620
  1921
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1922
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1923
haftmann@37660
  1924
fun unat_arith_tacs ctxt =   
haftmann@37660
  1925
  let
haftmann@37660
  1926
    fun arith_tac' n t =
haftmann@37660
  1927
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1928
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1929
  in 
wenzelm@42793
  1930
    [ clarify_tac ctxt 1,
wenzelm@42793
  1931
      full_simp_tac (unat_arith_ss_of (simpset_of ctxt)) 1,
wenzelm@45620
  1932
      ALLGOALS (full_simp_tac (HOL_ss |> fold Splitter.add_split @{thms unat_splits}
wenzelm@45620
  1933
                                      |> fold Simplifier.add_cong @{thms power_False_cong})),
haftmann@37660
  1934
      rewrite_goals_tac @{thms word_size}, 
haftmann@37660
  1935
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1936
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1937
                         REPEAT (dtac @{thm of_nat_inverse} n THEN atac n)),
haftmann@37660
  1938
      TRYALL arith_tac' ] 
haftmann@37660
  1939
  end
haftmann@37660
  1940
haftmann@37660
  1941
fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt))
haftmann@37660
  1942
*}
haftmann@37660
  1943
haftmann@37660
  1944
method_setup unat_arith = 
haftmann@37660
  1945
  {* Scan.succeed (SIMPLE_METHOD' o unat_arith_tac) *}
haftmann@37660
  1946
  "solving word arithmetic via natural numbers and arith"
haftmann@37660
  1947
haftmann@37660
  1948
lemma no_plus_overflow_unat_size: 
haftmann@37660
  1949
  "((x :: 'a :: len word) <= x + y) = (unat x + unat y < 2 ^ size x)" 
haftmann@37660
  1950
  unfolding word_size by unat_arith
haftmann@37660
  1951
haftmann@37660
  1952
lemmas no_olen_add_nat = no_plus_overflow_unat_size [unfolded word_size]
haftmann@37660
  1953
wenzelm@45604
  1954
lemmas unat_plus_simple = trans [OF no_olen_add_nat unat_add_lem]
haftmann@37660
  1955
haftmann@37660
  1956
lemma word_div_mult: 
haftmann@40827
  1957
  "(0 :: 'a :: len word) < y \<Longrightarrow> unat x * unat y < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1958
    x * y div y = x"
haftmann@37660
  1959
  apply unat_arith
haftmann@37660
  1960
  apply clarsimp
haftmann@37660
  1961
  apply (subst unat_mult_lem [THEN iffD1])
haftmann@37660
  1962
  apply auto
haftmann@37660
  1963
  done
haftmann@37660
  1964
haftmann@40827
  1965
lemma div_lt': "(i :: 'a :: len word) <= k div x \<Longrightarrow> 
haftmann@37660
  1966
    unat i * unat x < 2 ^ len_of TYPE('a)"
haftmann@37660
  1967
  apply unat_arith
haftmann@37660
  1968
  apply clarsimp
haftmann@37660
  1969
  apply (drule mult_le_mono1)
haftmann@37660
  1970
  apply (erule order_le_less_trans)
haftmann@37660
  1971
  apply (rule xtr7 [OF unat_lt2p div_mult_le])
haftmann@37660
  1972
  done
haftmann@37660
  1973
haftmann@37660
  1974
lemmas div_lt'' = order_less_imp_le [THEN div_lt']
haftmann@37660
  1975
haftmann@40827
  1976
lemma div_lt_mult: "(i :: 'a :: len word) < k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x < k"
haftmann@37660
  1977
  apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  1978
  apply (simp add: unat_arith_simps)
haftmann@37660
  1979
  apply (drule (1) mult_less_mono1)
haftmann@37660
  1980
  apply (erule order_less_le_trans)
haftmann@37660
  1981
  apply (rule div_mult_le)
haftmann@37660
  1982
  done
haftmann@37660
  1983
haftmann@37660
  1984
lemma div_le_mult: 
haftmann@40827
  1985
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x <= k"
haftmann@37660
  1986
  apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  1987
  apply (simp add: unat_arith_simps)
haftmann@37660
  1988
  apply (drule mult_le_mono1)
haftmann@37660
  1989
  apply (erule order_trans)
haftmann@37660
  1990
  apply (rule div_mult_le)
haftmann@37660
  1991
  done
haftmann@37660
  1992
haftmann@37660
  1993
lemma div_lt_uint': 
haftmann@40827
  1994
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> uint i * uint x < 2 ^ len_of TYPE('a)"
haftmann@37660
  1995
  apply (unfold uint_nat)
haftmann@37660
  1996
  apply (drule div_lt')
haftmann@37660
  1997
  apply (simp add: zmult_int zless_nat_eq_int_zless [symmetric] 
haftmann@37660
  1998
                   nat_power_eq)
haftmann@37660
  1999
  done
haftmann@37660
  2000
haftmann@37660
  2001
lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint']
haftmann@37660
  2002
haftmann@37660
  2003
lemma word_le_exists': 
haftmann@40827
  2004
  "(x :: 'a :: len0 word) <= y \<Longrightarrow> 
haftmann@37660
  2005
    (EX z. y = x + z & uint x + uint z < 2 ^ len_of TYPE('a))"
haftmann@37660
  2006
  apply (rule exI)
haftmann@37660
  2007
  apply (rule conjI)
haftmann@37660
  2008
  apply (rule zadd_diff_inverse)
haftmann@37660
  2009
  apply uint_arith
haftmann@37660
  2010
  done
haftmann@37660
  2011
haftmann@37660
  2012
lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab]
haftmann@37660
  2013
haftmann@37660
  2014
lemmas plus_minus_no_overflow =
haftmann@37660
  2015
  order_less_imp_le [THEN plus_minus_no_overflow_ab]
haftmann@37660
  2016
  
haftmann@37660
  2017
lemmas mcs = word_less_minus_cancel word_less_minus_mono_left
haftmann@37660
  2018
  word_le_minus_cancel word_le_minus_mono_left
haftmann@37660
  2019
wenzelm@45604
  2020
lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2021
lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2022
lemmas word_plus_mcs = word_diff_ls [where y = "v + x", unfolded add_diff_cancel] for v x
haftmann@37660
  2023
haftmann@37660
  2024
lemmas le_unat_uoi = unat_le [THEN word_unat.Abs_inverse]
haftmann@37660
  2025
haftmann@37660
  2026
lemmas thd = refl [THEN [2] split_div_lemma [THEN iffD2], THEN conjunct1]
haftmann@37660
  2027
haftmann@37660
  2028
lemma thd1:
haftmann@37660
  2029
  "a div b * b \<le> (a::nat)"
haftmann@37660
  2030
  using gt_or_eq_0 [of b]
haftmann@37660
  2031
  apply (rule disjE)
haftmann@37660
  2032
   apply (erule xtr4 [OF thd mult_commute])
haftmann@37660
  2033
  apply clarsimp
haftmann@37660
  2034
  done
haftmann@37660
  2035
wenzelm@45604
  2036
lemmas uno_simps [THEN le_unat_uoi] = mod_le_divisor div_le_dividend thd1 
haftmann@37660
  2037
haftmann@37660
  2038
lemma word_mod_div_equality:
haftmann@37660
  2039
  "(n div b) * b + (n mod b) = (n :: 'a :: len word)"
haftmann@37660
  2040
  apply (unfold word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2041
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2042
  apply (erule disjE)
haftmann@37660
  2043
   apply (simp add: mod_div_equality uno_simps)
haftmann@37660
  2044
  apply simp
haftmann@37660
  2045
  done
haftmann@37660
  2046
haftmann@37660
  2047
lemma word_div_mult_le: "a div b * b <= (a::'a::len word)"
haftmann@37660
  2048
  apply (unfold word_le_nat_alt word_arith_nat_defs)
haftmann@37660
  2049
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2050
  apply (erule disjE)
haftmann@37660
  2051
   apply (simp add: div_mult_le uno_simps)
haftmann@37660
  2052
  apply simp
haftmann@37660
  2053
  done
haftmann@37660
  2054
haftmann@40827
  2055
lemma word_mod_less_divisor: "0 < n \<Longrightarrow> m mod n < (n :: 'a :: len word)"
haftmann@37660
  2056
  apply (simp only: word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2057
  apply (clarsimp simp add : uno_simps)
haftmann@37660
  2058
  done
haftmann@37660
  2059
haftmann@37660
  2060
lemma word_of_int_power_hom: 
haftmann@37660
  2061
  "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a :: len word)"
huffman@45995
  2062
  by (induct n) (simp_all add: wi_hom_mult [symmetric])
haftmann@37660
  2063
haftmann@37660
  2064
lemma word_arith_power_alt: 
haftmann@37660
  2065
  "a ^ n = (word_of_int (uint a ^ n) :: 'a :: len word)"
haftmann@37660
  2066
  by (simp add : word_of_int_power_hom [symmetric])
haftmann@37660
  2067
haftmann@37660
  2068
lemma of_bl_length_less: 
haftmann@40827
  2069
  "length x = k \<Longrightarrow> k < len_of TYPE('a) \<Longrightarrow> (of_bl x :: 'a :: len word) < 2 ^ k"
haftmann@37660
  2070
  apply (unfold of_bl_no [unfolded word_number_of_def]
haftmann@37660
  2071
                word_less_alt word_number_of_alt)
haftmann@37660
  2072
  apply safe
haftmann@37660
  2073
  apply (simp (no_asm) add: word_of_int_power_hom word_uint.eq_norm 
haftmann@37660
  2074
                       del: word_of_int_bin)
haftmann@37660
  2075
  apply (simp add: mod_pos_pos_trivial)
haftmann@37660
  2076
  apply (subst mod_pos_pos_trivial)
haftmann@37660
  2077
    apply (rule bl_to_bin_ge0)
haftmann@37660
  2078
   apply (rule order_less_trans)
haftmann@37660
  2079
    apply (rule bl_to_bin_lt2p)
haftmann@37660
  2080
   apply simp
haftmann@37660
  2081
  apply (rule bl_to_bin_lt2p)    
haftmann@37660
  2082
  done
haftmann@37660
  2083
haftmann@37660
  2084
haftmann@37660
  2085
subsection "Cardinality, finiteness of set of words"
haftmann@37660
  2086
huffman@45809
  2087
instance word :: (len0) finite
huffman@45809
  2088
  by (default, simp add: type_definition.univ [OF type_definition_word])
huffman@45809
  2089
huffman@45809
  2090
lemma card_word: "CARD('a::len0 word) = 2 ^ len_of TYPE('a)"
huffman@45809
  2091
  by (simp add: type_definition.card [OF type_definition_word] nat_power_eq)
haftmann@37660
  2092
haftmann@37660
  2093
lemma card_word_size: 
huffman@45809
  2094
  "card (UNIV :: 'a :: len0 word set) = (2 ^ size (x :: 'a word))"
haftmann@37660
  2095
unfolding word_size by (rule card_word)
haftmann@37660
  2096
haftmann@37660
  2097
haftmann@37660
  2098
subsection {* Bitwise Operations on Words *}
haftmann@37660
  2099
haftmann@37660
  2100
lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or
haftmann@37660
  2101
  
haftmann@37660
  2102
(* following definitions require both arithmetic and bit-wise word operations *)
haftmann@37660
  2103
haftmann@37660
  2104
(* to get word_no_log_defs from word_log_defs, using bin_log_bintrs *)
haftmann@37660
  2105
lemmas wils1 = bin_log_bintrs [THEN word_ubin.norm_eq_iff [THEN iffD1],
wenzelm@45604
  2106
  folded word_ubin.eq_norm, THEN eq_reflection]
haftmann@37660
  2107
haftmann@37660
  2108
(* the binary operations only *)
haftmann@37660
  2109
lemmas word_log_binary_defs = 
haftmann@37660
  2110
  word_and_def word_or_def word_xor_def
haftmann@37660
  2111
haftmann@37660
  2112
lemmas word_no_log_defs [simp] = 
haftmann@37660
  2113
  word_not_def  [where a="number_of a", 
wenzelm@45604
  2114
                 unfolded word_no_wi wils1, folded word_no_wi]
haftmann@37660
  2115
  word_log_binary_defs [where a="number_of a" and b="number_of b",
wenzelm@45604
  2116
                        unfolded word_no_wi wils1, folded word_no_wi]
wenzelm@45604
  2117
  for a b
haftmann@37660
  2118
haftmann@37660
  2119
lemmas word_wi_log_defs = word_no_log_defs [unfolded word_no_wi]
haftmann@37660
  2120
haftmann@37660
  2121
lemma uint_or: "uint (x OR y) = (uint x) OR (uint y)"
huffman@45550
  2122
  by (simp add: word_or_def word_wi_log_defs word_ubin.eq_norm
haftmann@37660
  2123
                bin_trunc_ao(2) [symmetric])
haftmann@37660
  2124
haftmann@37660
  2125
lemma uint_and: "uint (x AND y) = (uint x) AND (uint y)"
huffman@45550
  2126
  by (simp add: word_and_def word_wi_log_defs word_ubin.eq_norm
haftmann@37660
  2127
                bin_trunc_ao(1) [symmetric]) 
haftmann@37660
  2128
haftmann@37660
  2129
lemma word_ops_nth_size:
haftmann@40827
  2130
  "n < size (x::'a::len0 word) \<Longrightarrow> 
haftmann@37660
  2131
    (x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2132
    (x AND y) !! n = (x !! n & y !! n) & 
haftmann@37660
  2133
    (x XOR y) !! n = (x !! n ~= y !! n) & 
haftmann@37660
  2134
    (NOT x) !! n = (~ x !! n)"
huffman@45550
  2135
  unfolding word_size word_test_bit_def word_log_defs
haftmann@37660
  2136
  by (clarsimp simp add : word_ubin.eq_norm nth_bintr bin_nth_ops)
haftmann@37660
  2137
haftmann@37660
  2138
lemma word_ao_nth:
haftmann@37660
  2139
  fixes x :: "'a::len0 word"
haftmann@37660
  2140
  shows "(x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2141
         (x AND y) !! n = (x !! n & y !! n)"
haftmann@37660
  2142
  apply (cases "n < size x")
haftmann@37660
  2143
   apply (drule_tac y = "y" in word_ops_nth_size)
haftmann@37660
  2144
   apply simp
haftmann@37660
  2145
  apply (simp add : test_bit_bin word_size)
haftmann@37660
  2146
  done
haftmann@37660
  2147
haftmann@37660
  2148
(* get from commutativity, associativity etc of int_and etc
haftmann@37660
  2149
  to same for word_and etc *)
haftmann@37660
  2150
haftmann@37660
  2151
lemmas bwsimps = 
haftmann@37660
  2152
  word_of_int_homs(2) 
haftmann@37660
  2153
  word_0_wi_Pls
haftmann@37660
  2154
  word_m1_wi_Min
haftmann@37660
  2155
  word_wi_log_defs
haftmann@37660
  2156
haftmann@37660
  2157
lemma word_bw_assocs:
haftmann@37660
  2158
  fixes x :: "'a::len0 word"
haftmann@37660
  2159
  shows
haftmann@37660
  2160
  "(x AND y) AND z = x AND y AND z"
haftmann@37660
  2161
  "(x OR y) OR z = x OR y OR z"
haftmann@37660
  2162
  "(x XOR y) XOR z = x XOR y XOR z"
haftmann@37660
  2163
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2164
        word_of_int_Ex [where x=y] 
haftmann@37660
  2165
        word_of_int_Ex [where x=z]
haftmann@37660
  2166
  by (auto simp: bwsimps bbw_assocs)
haftmann@37660
  2167
  
haftmann@37660
  2168
lemma word_bw_comms:
haftmann@37660
  2169
  fixes x :: "'a::len0 word"
haftmann@37660
  2170
  shows
haftmann@37660
  2171
  "x AND y = y AND x"
haftmann@37660
  2172
  "x OR y = y OR x"
haftmann@37660
  2173
  "x XOR y = y XOR x"
haftmann@37660
  2174
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2175
        word_of_int_Ex [where x=y] 
haftmann@37660
  2176
  by (auto simp: bwsimps bin_ops_comm)
haftmann@37660
  2177
  
haftmann@37660
  2178
lemma word_bw_lcs:
haftmann@37660
  2179
  fixes x :: "'a::len0 word"
haftmann@37660
  2180
  shows
haftmann@37660
  2181
  "y AND x AND z = x AND y AND z"
haftmann@37660
  2182
  "y OR x OR z = x OR y OR z"
haftmann@37660
  2183
  "y XOR x XOR z = x XOR y XOR z"
haftmann@37660
  2184
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2185
        word_of_int_Ex [where x=y] 
haftmann@37660
  2186
        word_of_int_Ex [where x=z]
haftmann@37660
  2187
  by (auto simp: bwsimps)
haftmann@37660
  2188
haftmann@37660
  2189
lemma word_log_esimps [simp]:
haftmann@37660
  2190
  fixes x :: "'a::len0 word"
haftmann@37660
  2191
  shows
haftmann@37660
  2192
  "x AND 0 = 0"
haftmann@37660
  2193
  "x AND -1 = x"
haftmann@37660
  2194
  "x OR 0 = x"
haftmann@37660
  2195
  "x OR -1 = -1"
haftmann@37660
  2196
  "x XOR 0 = x"
haftmann@37660
  2197
  "x XOR -1 = NOT x"
haftmann@37660
  2198
  "0 AND x = 0"
haftmann@37660
  2199
  "-1 AND x = x"
haftmann@37660
  2200
  "0 OR x = x"
haftmann@37660
  2201
  "-1 OR x = -1"
haftmann@37660
  2202
  "0 XOR x = x"
haftmann@37660
  2203
  "-1 XOR x = NOT x"
haftmann@37660
  2204
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2205
  by (auto simp: bwsimps)
haftmann@37660
  2206
haftmann@37660
  2207
lemma word_not_dist:
haftmann@37660
  2208
  fixes x :: "'a::len0 word"
haftmann@37660
  2209
  shows
haftmann@37660
  2210
  "NOT (x OR y) = NOT x AND NOT y"
haftmann@37660
  2211
  "NOT (x AND y) = NOT x OR NOT y"
haftmann@37660
  2212
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2213
        word_of_int_Ex [where x=y] 
haftmann@37660
  2214
  by (auto simp: bwsimps bbw_not_dist)
haftmann@37660
  2215
haftmann@37660
  2216
lemma word_bw_same:
haftmann@37660
  2217
  fixes x :: "'a::len0 word"
haftmann@37660
  2218
  shows
haftmann@37660
  2219
  "x AND x = x"
haftmann@37660
  2220
  "x OR x = x"
haftmann@37660
  2221
  "x XOR x = 0"
haftmann@37660
  2222
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2223
  by (auto simp: bwsimps)
haftmann@37660
  2224
haftmann@37660
  2225
lemma word_ao_absorbs [simp]:
haftmann@37660
  2226
  fixes x :: "'a::len0 word"
haftmann@37660
  2227
  shows
haftmann@37660
  2228
  "x AND (y OR x) = x"
haftmann@37660
  2229
  "x OR y AND x = x"
haftmann@37660
  2230
  "x AND (x OR y) = x"
haftmann@37660
  2231
  "y AND x OR x = x"
haftmann@37660
  2232
  "(y OR x) AND x = x"
haftmann@37660
  2233
  "x OR x AND y = x"
haftmann@37660
  2234
  "(x OR y) AND x = x"
haftmann@37660
  2235
  "x AND y OR x = x"
haftmann@37660
  2236
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2237
        word_of_int_Ex [where x=y] 
haftmann@37660
  2238
  by (auto simp: bwsimps)
haftmann@37660
  2239
haftmann@37660
  2240
lemma word_not_not [simp]:
haftmann@37660
  2241
  "NOT NOT (x::'a::len0 word) = x"
haftmann@37660
  2242
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2243
  by (auto simp: bwsimps)
haftmann@37660
  2244
haftmann@37660
  2245
lemma word_ao_dist:
haftmann@37660
  2246
  fixes x :: "'a::len0 word"
haftmann@37660
  2247
  shows "(x OR y) AND z = x AND z OR y AND z"
haftmann@37660
  2248
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2249
        word_of_int_Ex [where x=y] 
haftmann@37660
  2250
        word_of_int_Ex [where x=z]   
wenzelm@41550
  2251
  by (auto simp: bwsimps bbw_ao_dist)
haftmann@37660
  2252
haftmann@37660
  2253
lemma word_oa_dist:
haftmann@37660
  2254
  fixes x :: "'a::len0 word"
haftmann@37660
  2255
  shows "x AND y OR z = (x OR z) AND (y OR z)"
haftmann@37660
  2256
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2257
        word_of_int_Ex [where x=y] 
haftmann@37660
  2258
        word_of_int_Ex [where x=z]   
wenzelm@41550
  2259
  by (auto simp: bwsimps bbw_oa_dist)
haftmann@37660
  2260
haftmann@37660
  2261
lemma word_add_not [simp]: 
haftmann@37660
  2262
  fixes x :: "'a::len0 word"
haftmann@37660
  2263
  shows "x + NOT x = -1"
haftmann@37660
  2264
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2265
  by (auto simp: bwsimps bin_add_not)
haftmann@37660
  2266
haftmann@37660
  2267
lemma word_plus_and_or [simp]:
haftmann@37660
  2268
  fixes x :: "'a::len0 word"
haftmann@37660
  2269
  shows "(x AND y) + (x OR y) = x + y"
haftmann@37660
  2270
  using word_of_int_Ex [where x=x] 
haftmann@37660
  2271
        word_of_int_Ex [where x=y] 
haftmann@37660
  2272
  by (auto simp: bwsimps plus_and_or)
haftmann@37660
  2273
haftmann@37660
  2274
lemma leoa:   
haftmann@37660
  2275
  fixes x :: "'a::len0 word"
haftmann@40827
  2276
  shows "(w = (x OR y)) \<Longrightarrow> (y = (w AND y))" by auto
haftmann@37660
  2277
lemma leao: 
haftmann@37660
  2278
  fixes x' :: "'a::len0 word"
haftmann@40827
  2279
  shows "(w' = (x' AND y')) \<Longrightarrow> (x' = (x' OR w'))" by auto 
haftmann@37660
  2280
haftmann@37660
  2281
lemmas word_ao_equiv = leao [COMP leoa [COMP iffI]]
haftmann@37660
  2282
haftmann@37660
  2283
lemma le_word_or2: "x <= x OR (y::'a::len0 word)"
haftmann@37660
  2284
  unfolding word_le_def uint_or
haftmann@37660
  2285
  by (auto intro: le_int_or) 
haftmann@37660
  2286
wenzelm@45604
  2287
lemmas le_word_or1 = xtr3 [OF word_bw_comms (2) le_word_or2]
wenzelm@45604
  2288
lemmas word_and_le1 = xtr3 [OF word_ao_absorbs (4) [symmetric] le_word_or2]
wenzelm@45604
  2289
lemmas word_and_le2 = xtr3 [OF word_ao_absorbs (8) [symmetric] le_word_or2]
haftmann@37660
  2290
haftmann@37660
  2291
lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" 
huffman@45550
  2292
  unfolding to_bl_def word_log_defs bl_not_bin
huffman@45550
  2293
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2294
haftmann@37660
  2295
lemma bl_word_xor: "to_bl (v XOR w) = map2 op ~= (to_bl v) (to_bl w)" 
haftmann@37660
  2296
  unfolding to_bl_def word_log_defs bl_xor_bin
huffman@45550
  2297
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2298
haftmann@37660
  2299
lemma bl_word_or: "to_bl (v OR w) = map2 op | (to_bl v) (to_bl w)" 
huffman@45550
  2300
  unfolding to_bl_def word_log_defs bl_or_bin
huffman@45550
  2301
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2302
haftmann@37660
  2303
lemma bl_word_and: "to_bl (v AND w) = map2 op & (to_bl v) (to_bl w)" 
huffman@45550
  2304
  unfolding to_bl_def word_log_defs bl_and_bin
huffman@45550
  2305
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2306
haftmann@37660
  2307
lemma word_lsb_alt: "lsb (w::'a::len0 word) = test_bit w 0"
haftmann@37660
  2308
  by (auto simp: word_test_bit_def word_lsb_def)
haftmann@37660
  2309
huffman@45805
  2310
lemma word_lsb_1_0 [simp]: "lsb (1::'a::len word) & ~ lsb (0::'b::len0 word)"
huffman@45550
  2311
  unfolding word_lsb_def uint_eq_0 uint_1 by simp
haftmann@37660
  2312
haftmann@37660
  2313
lemma word_lsb_last: "lsb (w::'a::len word) = last (to_bl w)"
haftmann@37660
  2314
  apply (unfold word_lsb_def uint_bl bin_to_bl_def) 
haftmann@37660
  2315
  apply (rule_tac bin="uint w" in bin_exhaust)
haftmann@37660
  2316
  apply (cases "size w")
haftmann@37660
  2317
   apply auto
haftmann@37660
  2318
   apply (auto simp add: bin_to_bl_aux_alt)
haftmann@37660
  2319
  done
haftmann@37660
  2320
haftmann@37660
  2321
lemma word_lsb_int: "lsb w = (uint w mod 2 = 1)"
huffman@45529
  2322
  unfolding word_lsb_def bin_last_def by auto
haftmann@37660
  2323
haftmann@37660
  2324
lemma word_msb_sint: "msb w = (sint w < 0)" 
haftmann@37660
  2325
  unfolding word_msb_def
haftmann@37660
  2326
  by (simp add : sign_Min_lt_0 number_of_is_id)
haftmann@37660
  2327
huffman@45805
  2328
lemma word_msb_no [simp]:
huffman@45805
  2329
  "msb (number_of bin :: 'a::len word) = bin_nth bin (len_of TYPE('a) - 1)"
huffman@45811
  2330
  unfolding word_msb_def word_number_of_def
huffman@45811
  2331
  by (clarsimp simp add: word_sbin.eq_norm bin_sign_lem)
huffman@45811
  2332
huffman@45811
  2333
lemma word_msb_nth:
huffman@45811
  2334
  "msb (w::'a::len word) = bin_nth (uint w) (len_of TYPE('a) - 1)"
haftmann@37660
  2335
  apply (rule trans [OF _ word_msb_no])
haftmann@37660
  2336
  apply (simp add : word_number_of_def)
haftmann@37660
  2337
  done
haftmann@37660
  2338
haftmann@37660
  2339
lemma word_msb_alt: "msb (w::'a::len word) = hd (to_bl w)"
haftmann@37660
  2340
  apply (unfold word_msb_nth uint_bl)
haftmann@37660
  2341
  apply (subst hd_conv_nth)
haftmann@37660
  2342
  apply (rule length_greater_0_conv [THEN iffD1])
haftmann@37660
  2343
   apply simp
haftmann@37660
  2344
  apply (simp add : nth_bin_to_bl word_size)
haftmann@37660
  2345
  done
haftmann@37660
  2346
huffman@45805
  2347
lemma word_set_nth [simp]:
haftmann@37660
  2348
  "set_bit w n (test_bit w n) = (w::'a::len0 word)"
haftmann@37660
  2349
  unfolding word_test_bit_def word_set_bit_def by auto
haftmann@37660
  2350
haftmann@37660
  2351
lemma bin_nth_uint':
haftmann@37660
  2352
  "bin_nth (uint w) n = (rev (bin_to_bl (size w) (uint w)) ! n & n < size w)"
haftmann@37660
  2353
  apply (unfold word_size)
haftmann@37660
  2354
  apply (safe elim!: bin_nth_uint_imp)
haftmann@37660
  2355
   apply (frule bin_nth_uint_imp)
haftmann@37660
  2356
   apply (fast dest!: bin_nth_bl)+
haftmann@37660
  2357
  done
haftmann@37660
  2358
haftmann@37660
  2359
lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size]
haftmann@37660
  2360
haftmann@37660
  2361
lemma test_bit_bl: "w !! n = (rev (to_bl w) ! n & n < size w)"
haftmann@37660
  2362
  unfolding to_bl_def word_test_bit_def word_size
haftmann@37660
  2363
  by (rule bin_nth_uint)
haftmann@37660
  2364
haftmann@40827
  2365
lemma to_bl_nth: "n < size w \<Longrightarrow> to_bl w ! n = w !! (size w - Suc n)"
haftmann@37660
  2366
  apply (unfold test_bit_bl)
haftmann@37660
  2367
  apply clarsimp
haftmann@37660
  2368
  apply (rule trans)
haftmann@37660
  2369
   apply (rule nth_rev_alt)
haftmann@37660
  2370
   apply (auto simp add: word_size)
haftmann@37660
  2371
  done
haftmann@37660
  2372
haftmann@37660
  2373
lemma test_bit_set: 
haftmann@37660
  2374
  fixes w :: "'a::len0 word"
haftmann@37660
  2375
  shows "(set_bit w n x) !! n = (n < size w & x)"
haftmann@37660
  2376
  unfolding word_size word_test_bit_def word_set_bit_def
haftmann@37660
  2377
  by (clarsimp simp add : word_ubin.eq_norm nth_bintr)
haftmann@37660
  2378
haftmann@37660
  2379
lemma test_bit_set_gen: 
haftmann@37660
  2380
  fixes w :: "'a::len0 word"
haftmann@37660
  2381
  shows "test_bit (set_bit w n x) m = 
haftmann@37660
  2382
         (if m = n then n < size w & x else test_bit w m)"
haftmann@37660
  2383
  apply (unfold word_size word_test_bit_def word_set_bit_def)
haftmann@37660
  2384
  apply (clarsimp simp add: word_ubin.eq_norm nth_bintr bin_nth_sc_gen)
haftmann@37660
  2385
  apply (auto elim!: test_bit_size [unfolded word_size]
haftmann@37660
  2386
              simp add: word_test_bit_def [symmetric])
haftmann@37660
  2387
  done
haftmann@37660
  2388
haftmann@37660
  2389
lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs"
haftmann@37660
  2390
  unfolding of_bl_def bl_to_bin_rep_F by auto
haftmann@37660
  2391
  
huffman@45811
  2392
lemma msb_nth:
haftmann@37660
  2393
  fixes w :: "'a::len word"
huffman@45811
  2394
  shows "msb w = w !! (len_of TYPE('a) - 1)"
huffman@45811
  2395
  unfolding word_msb_nth word_test_bit_def by simp
haftmann@37660
  2396
wenzelm@45604
  2397
lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2398
lemmas msb1 = msb0 [where i = 0]
haftmann@37660
  2399
lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]]
haftmann@37660
  2400
wenzelm@45604
  2401
lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2402
lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt]
haftmann@37660
  2403
huffman@45811
  2404
lemma td_ext_nth [OF refl refl refl, unfolded word_size]:
haftmann@40827
  2405
  "n = size (w::'a::len0 word) \<Longrightarrow> ofn = set_bits \<Longrightarrow> [w, ofn g] = l \<Longrightarrow> 
haftmann@37660
  2406
    td_ext test_bit ofn {f. ALL i. f i --> i < n} (%h i. h i & i < n)"
haftmann@37660
  2407
  apply (unfold word_size td_ext_def')
haftmann@37660
  2408
  apply (safe del: subset_antisym)
haftmann@37660
  2409
     apply (rule_tac [3] ext)
haftmann@37660
  2410
     apply (rule_tac [4] ext)
haftmann@37660
  2411
     apply (unfold word_size of_nth_def test_bit_bl)
haftmann@37660
  2412
     apply safe
haftmann@37660
  2413
       defer
haftmann@37660
  2414
       apply (clarsimp simp: word_bl.Abs_inverse)+
haftmann@37660
  2415
  apply (rule word_bl.Rep_inverse')
haftmann@37660
  2416
  apply (rule sym [THEN trans])
haftmann@37660
  2417
  apply (rule bl_of_nth_nth)
haftmann@37660
  2418
  apply simp
haftmann@37660
  2419
  apply (rule bl_of_nth_inj)
haftmann@37660
  2420
  apply (clarsimp simp add : test_bit_bl word_size)
haftmann@37660
  2421
  done
haftmann@37660
  2422
haftmann@37660
  2423
interpretation test_bit:
haftmann@37660
  2424
  td_ext "op !! :: 'a::len0 word => nat => bool"
haftmann@37660
  2425
         set_bits
haftmann@37660
  2426
         "{f. \<forall>i. f i \<longrightarrow> i < len_of TYPE('a::len0)}"
haftmann@37660
  2427
         "(\<lambda>h i. h i \<and> i < len_of TYPE('a::len0))"
haftmann@37660
  2428
  by (rule td_ext_nth)
haftmann@37660
  2429
haftmann@37660
  2430
lemmas td_nth = test_bit.td_thm
haftmann@37660
  2431
huffman@45805
  2432
lemma word_set_set_same [simp]:
haftmann@37660
  2433
  fixes w :: "'a::len0 word"
haftmann@37660
  2434
  shows "set_bit (set_bit w n x) n y = set_bit w n y" 
haftmann@37660
  2435
  by (rule word_eqI) (simp add : test_bit_set_gen word_size)
haftmann@37660
  2436
    
haftmann@37660
  2437
lemma word_set_set_diff: 
haftmann@37660
  2438
  fixes w :: "'a::len0 word"
haftmann@37660
  2439
  assumes "m ~= n"
haftmann@37660
  2440
  shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x" 
wenzelm@41550
  2441
  by (rule word_eqI) (clarsimp simp add: test_bit_set_gen word_size assms)
huffman@46001
  2442
huffman@46001
  2443
lemma test_bit_wi [simp]:
huffman@46001
  2444
  "(word_of_int x::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a) \<and> bin_nth x n"
huffman@46001
  2445
  unfolding word_test_bit_def
huffman@46001
  2446
  by (simp add: nth_bintr [symmetric] word_ubin.eq_norm)
huffman@46001
  2447
huffman@45811
  2448
lemma test_bit_no [simp]:
huffman@45811
  2449
  "(number_of bin :: 'a::len0 word) !! n \<equiv> n < len_of TYPE('a) \<and> bin_nth bin n"
haftmann@37660
  2450
  unfolding word_test_bit_def word_number_of_def word_size
haftmann@37660
  2451
  by (simp add : nth_bintr [symmetric] word_ubin.eq_norm)
haftmann@37660
  2452
huffman@45805
  2453
lemma nth_0 [simp]: "~ (0::'a::len0 word) !! n"
haftmann@37660
  2454
  unfolding test_bit_no word_0_no by auto
haftmann@37660
  2455
haftmann@37660
  2456
lemma nth_sint: 
haftmann@37660
  2457
  fixes w :: "'a::len word"
haftmann@37660
  2458
  defines "l \<equiv> len_of TYPE ('a)"
haftmann@37660
  2459
  shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
haftmann@37660
  2460
  unfolding sint_uint l_def
haftmann@37660
  2461
  by (clarsimp simp add: nth_sbintr word_test_bit_def [symmetric])
haftmann@37660
  2462
huffman@45805
  2463
lemma word_lsb_no [simp]:
haftmann@37660
  2464
  "lsb (number_of bin :: 'a :: len word) = (bin_last bin = 1)"
haftmann@37660
  2465
  unfolding word_lsb_alt test_bit_no by auto
haftmann@37660
  2466
huffman@45805
  2467
lemma word_set_no [simp]:
haftmann@37660
  2468
  "set_bit (number_of bin::'a::len0 word) n b = 
huffman@46001
  2469
    word_of_int (bin_sc n (if b then 1 else 0) (number_of bin))"
huffman@46001
  2470
  unfolding word_set_bit_def
haftmann@37660
  2471
  apply (rule word_eqI)
huffman@46001
  2472
  apply (simp add: word_size bin_nth_sc_gen nth_bintr)
haftmann@37660
  2473
  done
haftmann@37660
  2474
huffman@45805
  2475
lemma setBit_no [simp]:
huffman@46001
  2476
  "setBit (number_of bin) n = word_of_int (bin_sc n 1 (number_of bin))"
huffman@45805
  2477
  by (simp add: setBit_def)
huffman@45805
  2478
huffman@45805
  2479
lemma clearBit_no [simp]:
huffman@46001
  2480
  "clearBit (number_of bin) n = word_of_int (bin_sc n 0 (number_of bin))"
huffman@45805
  2481
  by (simp add: clearBit_def)
haftmann@37660
  2482
haftmann@37660
  2483
lemma to_bl_n1: 
haftmann@37660
  2484
  "to_bl (-1::'a::len0 word) = replicate (len_of TYPE ('a)) True"
haftmann@37660
  2485
  apply (rule word_bl.Abs_inverse')
haftmann@37660
  2486
   apply simp
haftmann@37660
  2487
  apply (rule word_eqI)
huffman@45805
  2488
  apply (clarsimp simp add: word_size)
haftmann@37660
  2489
  apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size)
haftmann@37660
  2490
  done
haftmann@37660
  2491
huffman@45805
  2492
lemma word_msb_n1 [simp]: "msb (-1::'a::len word)"
wenzelm@41550
  2493
  unfolding word_msb_alt to_bl_n1 by simp
haftmann@37660
  2494
haftmann@37660
  2495
lemma word_set_nth_iff: 
haftmann@37660
  2496
  "(set_bit w n b = w) = (w !! n = b | n >= size (w::'a::len0 word))"
haftmann@37660
  2497
  apply (rule iffI)
haftmann@37660
  2498
   apply (rule disjCI)
haftmann@37660
  2499
   apply (drule word_eqD)
haftmann@37660
  2500
   apply (erule sym [THEN trans])
haftmann@37660
  2501
   apply (simp add: test_bit_set)
haftmann@37660
  2502
  apply (erule disjE)
haftmann@37660
  2503
   apply clarsimp
haftmann@37660
  2504
  apply (rule word_eqI)
haftmann@37660
  2505
  apply (clarsimp simp add : test_bit_set_gen)
haftmann@37660
  2506
  apply (drule test_bit_size)
haftmann@37660
  2507
  apply force
haftmann@37660
  2508
  done
haftmann@37660
  2509
huffman@45811
  2510
lemma test_bit_2p:
huffman@45811
  2511
  "(word_of_int (2 ^ n)::'a::len word) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a)"
huffman@45811
  2512
  unfolding word_test_bit_def
haftmann@37660
  2513
  by (auto simp add: word_ubin.eq_norm nth_bintr nth_2p_bin)
haftmann@37660
  2514
haftmann@37660
  2515
lemma nth_w2p:
haftmann@37660
  2516
  "((2\<Colon>'a\<Colon>len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a\<Colon>len)"
haftmann@37660
  2517
  unfolding test_bit_2p [symmetric] word_of_int [symmetric]
haftmann@37660
  2518
  by (simp add:  of_int_power)
haftmann@37660
  2519
haftmann@37660
  2520
lemma uint_2p: 
haftmann@40827
  2521
  "(0::'a::len word) < 2 ^ n \<Longrightarrow> uint (2 ^ n::'a::len word) = 2 ^ n"
haftmann@37660
  2522
  apply (unfold word_arith_power_alt)
haftmann@37660
  2523
  apply (case_tac "len_of TYPE ('a)")
haftmann@37660
  2524
   apply clarsimp
haftmann@37660
  2525
  apply (case_tac "nat")
haftmann@37660
  2526
   apply clarsimp
haftmann@37660
  2527
   apply (case_tac "n")
huffman@46001
  2528
    apply clarsimp
huffman@46001
  2529
   apply clarsimp
haftmann@37660
  2530
  apply (drule word_gt_0 [THEN iffD1])
haftmann@37660
  2531
  apply (safe intro!: word_eqI bin_nth_lem ext)
huffman@46001
  2532
     apply (auto simp add: test_bit_2p nth_2p_bin word_test_bit_def [symmetric])
haftmann@37660
  2533
  done
haftmann@37660
  2534
haftmann@37660
  2535
lemma word_of_int_2p: "(word_of_int (2 ^ n) :: 'a :: len word) = 2 ^ n" 
haftmann@37660
  2536
  apply (unfold word_arith_power_alt)
haftmann@37660
  2537
  apply (case_tac "len_of TYPE ('a)")
haftmann@37660
  2538
   apply clarsimp
haftmann@37660
  2539
  apply (case_tac "nat")
haftmann@37660
  2540
   apply (rule word_ubin.norm_eq_iff [THEN iffD1]) 
haftmann@37660
  2541
   apply (rule box_equals) 
haftmann@37660
  2542
     apply (rule_tac [2] bintr_ariths (1))+ 
haftmann@37660
  2543
   apply (clarsimp simp add : number_of_is_id)
huffman@46001
  2544
  apply simp
haftmann@37660
  2545
  done
haftmann@37660
  2546
haftmann@40827
  2547
lemma bang_is_le: "x !! m \<Longrightarrow> 2 ^ m <= (x :: 'a :: len word)" 
haftmann@37660
  2548
  apply (rule xtr3) 
haftmann@37660
  2549
  apply (rule_tac [2] y = "x" in le_word_or2)
haftmann@37660
  2550
  apply (rule word_eqI)
haftmann@37660
  2551
  apply (auto simp add: word_ao_nth nth_w2p word_size)
haftmann@37660
  2552
  done
haftmann@37660
  2553
haftmann@37660
  2554
lemma word_clr_le: 
haftmann@37660
  2555
  fixes w :: "'a::len0 word"
haftmann@37660
  2556
  shows "w >= set_bit w n False"
haftmann@37660
  2557
  apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm)
haftmann@37660
  2558
  apply simp
haftmann@37660
  2559
  apply (rule order_trans)
haftmann@37660
  2560
   apply (rule bintr_bin_clr_le)
haftmann@37660
  2561
  apply simp
haftmann@37660
  2562
  done
haftmann@37660
  2563
haftmann@37660
  2564
lemma word_set_ge: 
haftmann@37660
  2565
  fixes w :: "'a::len word"
haftmann@37660
  2566
  shows "w <= set_bit w n True"
haftmann@37660
  2567
  apply (unfold word_set_bit_def word_le_def word_ubin.eq_norm)
haftmann@37660
  2568
  apply simp
haftmann@37660
  2569
  apply (rule order_trans [OF _ bintr_bin_set_ge])
haftmann@37660
  2570
  apply simp
haftmann@37660
  2571
  done
haftmann@37660
  2572
haftmann@37660
  2573
haftmann@37660
  2574
subsection {* Shifting, Rotating, and Splitting Words *}
haftmann@37660
  2575
huffman@46001
  2576
lemma shiftl1_wi [simp]: "shiftl1 (word_of_int w) = word_of_int (w BIT 0)"
huffman@46001
  2577
  unfolding shiftl1_def
huffman@46001
  2578
  apply (simp only: word_ubin.norm_eq_iff [symmetric] word_ubin.eq_norm)
haftmann@37660
  2579
  apply (subst refl [THEN bintrunc_BIT_I, symmetric])
haftmann@37660
  2580
  apply (subst bintrunc_bintrunc_min)
haftmann@37660
  2581
  apply simp
haftmann@37660
  2582
  done
haftmann@37660
  2583
huffman@46001
  2584
lemma shiftl1_number [simp] :
huffman@46001
  2585
  "shiftl1 (number_of w) = number_of (Int.Bit0 w)"
huffman@46001
  2586
  unfolding word_number_of_alt shiftl1_wi by simp
huffman@46001
  2587
haftmann@37660
  2588
lemma shiftl1_0 [simp] : "shiftl1 0 = 0"
huffman@46001
  2589
  unfolding shiftl1_def by simp
huffman@46001
  2590
huffman@46001
  2591
lemma shiftl1_def_u: "shiftl1 w = word_of_int (uint w BIT 0)"
huffman@46001
  2592
  by (simp only: shiftl1_def) (* FIXME: duplicate *)
huffman@46001
  2593
huffman@46001
  2594
lemma shiftl1_def_s: "shiftl1 w = word_of_int (sint w BIT 0)"
huffman@46001
  2595
  unfolding shiftl1_def Bit_B0 wi_hom_syms by simp
haftmann@37660
  2596
huffman@45995
  2597
lemma shiftr1_0 [simp]: "shiftr1 0 = 0"
huffman@45995
  2598
  unfolding shiftr1_def by simp
huffman@45995
  2599
huffman@45995
  2600
lemma sshiftr1_0 [simp]: "sshiftr1 0 = 0"
huffman@45995
  2601
  unfolding sshiftr1_def by simp
haftmann@37660
  2602
haftmann@37660
  2603
lemma sshiftr1_n1 [simp] : "sshiftr1 -1 = -1"
huffman@46001
  2604
  unfolding sshiftr1_def by simp
haftmann@37660
  2605
haftmann@37660
  2606
lemma shiftl_0 [simp] : "(0::'a::len0 word) << n = 0"
haftmann@37660
  2607
  unfolding shiftl_def by (induct n) auto
haftmann@37660
  2608
haftmann@37660
  2609
lemma shiftr_0 [simp] : "(0::'a::len0 word) >> n = 0"
haftmann@37660
  2610
  unfolding shiftr_def by (induct n) auto
haftmann@37660
  2611
haftmann@37660
  2612
lemma sshiftr_0 [simp] : "0 >>> n = 0"
haftmann@37660
  2613
  unfolding sshiftr_def by (induct n) auto
haftmann@37660
  2614
haftmann@37660
  2615
lemma sshiftr_n1 [simp] : "-1 >>> n = -1"
haftmann@37660
  2616
  unfolding sshiftr_def by (induct n) auto
haftmann@37660
  2617
haftmann@37660
  2618
lemma nth_shiftl1: "shiftl1 w !! n = (n < size w & n > 0 & w !! (n - 1))"
haftmann@37660
  2619
  apply (unfold shiftl1_def word_test_bit_def)
haftmann@37660
  2620
  apply (simp add: nth_bintr word_ubin.eq_norm word_size)
haftmann@37660
  2621
  apply (cases n)
haftmann@37660
  2622
   apply auto
haftmann@37660
  2623
  done
haftmann@37660
  2624
haftmann@37660
  2625
lemma nth_shiftl' [rule_format]:
haftmann@37660
  2626
  "ALL n. ((w::'a::len0 word) << m) !! n = (n < size w & n >= m & w !! (n - m))"
haftmann@37660
  2627
  apply (unfold shiftl_def)
haftmann@37660
  2628
  apply (induct "m")
haftmann@37660
  2629
   apply (force elim!: test_bit_size)
haftmann@37660
  2630
  apply (clarsimp simp add : nth_shiftl1 word_size)
haftmann@37660
  2631
  apply arith
haftmann@37660
  2632
  done
haftmann@37660
  2633
haftmann@37660
  2634
lemmas nth_shiftl = nth_shiftl' [unfolded word_size] 
haftmann@37660
  2635
haftmann@37660