src/HOL/Word/Word.thy
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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 word) 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}"
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  morphisms uint Abs_word by auto
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definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" where
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  -- {* representation of words using unsigned or signed bins, 
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        only difference in these is the type class *}
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  "word_of_int w = Abs_word (bintrunc (len_of TYPE ('a)) w)" 
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lemma uint_word_of_int [code]: "uint (word_of_int w \<Colon> 'a\<Colon>len0 word) = w mod 2 ^ len_of TYPE('a)"
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  by (auto simp add: word_of_int_def bintrunc_mod2p intro: Abs_word_inverse)
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code_datatype word_of_int
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subsection {* Random instance *}
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notation fcomp (infixl "\<circ>>" 60)
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notation scomp (infixl "\<circ>\<rightarrow>" 60)
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instantiation word :: ("{len0, typerep}") random
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begin
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definition
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  "random_word i = Random.range (max i (2 ^ len_of TYPE('a))) \<circ>\<rightarrow> (\<lambda>k. Pair (
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     let j = word_of_int (Code_Numeral.int_of k) :: 'a word
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     in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))"
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instance ..
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end
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no_notation fcomp (infixl "\<circ>>" 60)
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no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
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subsection {* Type conversions and casting *}
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definition sint :: "'a :: len word => int" where
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  -- {* treats the most-significant-bit as a sign bit *}
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  sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)"
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definition unat :: "'a :: len0 word => nat" where
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  "unat w = nat (uint w)"
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definition uints :: "nat => int set" where
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  -- "the sets of integers representing the words"
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  "uints n = range (bintrunc n)"
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definition sints :: "nat => int set" where
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  "sints n = range (sbintrunc (n - 1))"
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definition unats :: "nat => nat set" where
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  "unats n = {i. i < 2 ^ n}"
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definition norm_sint :: "nat => int => int" where
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  "norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)"
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definition scast :: "'a :: len word => 'b :: len word" where
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  -- "cast a word to a different length"
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  "scast w = word_of_int (sint w)"
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definition ucast :: "'a :: len0 word => 'b :: len0 word" where
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  "ucast w = word_of_int (uint w)"
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instantiation word :: (len0) size
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begin
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definition
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  word_size: "size (w :: 'a word) = len_of TYPE('a)"
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instance ..
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end
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definition source_size :: "('a :: len0 word => 'b) => nat" where
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  -- "whether a cast (or other) function is to a longer or shorter length"
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  "source_size c = (let arb = undefined ; x = c arb in size arb)"  
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definition target_size :: "('a => 'b :: len0 word) => nat" where
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  "target_size c = size (c undefined)"
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definition is_up :: "('a :: len0 word => 'b :: len0 word) => bool" where
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  "is_up c \<longleftrightarrow> source_size c <= target_size c"
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definition is_down :: "('a :: len0 word => 'b :: len0 word) => bool" where
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  "is_down c \<longleftrightarrow> target_size c <= source_size c"
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definition of_bl :: "bool list => 'a :: len0 word" where
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  "of_bl bl = word_of_int (bl_to_bin bl)"
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definition to_bl :: "'a :: len0 word => bool list" where
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  "to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)"
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definition word_reverse :: "'a :: len0 word => 'a word" where
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  "word_reverse w = of_bl (rev (to_bl w))"
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definition word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b" where
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  "word_int_case f w = f (uint w)"
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syntax
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  of_int :: "int => 'a"
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translations
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  "case x of CONST of_int y => b" == "CONST word_int_case (%y. b) x"
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subsection {* Type-definition locale instantiations *}
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lemmas word_size_gt_0 [iff] = 
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  xtr1 [OF word_size len_gt_0, standard]
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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, standard]
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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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lemmas atLeastLessThan_alt = atLeastLessThan_def [unfolded 
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  atLeast_def lessThan_def Collect_conj_eq [symmetric]]
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lemma mod_in_reps: "m > 0 \<Longrightarrow> y mod m : {0::int ..< m}"
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  unfolding atLeastLessThan_alt by auto
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lemma 
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  uint_0:"0 <= uint x" and 
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  uint_lt: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
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  by (auto simp: uint [simplified])
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lemma uint_mod_same:
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  "uint x mod 2 ^ len_of TYPE('a) = uint (x::'a::len0 word)"
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  by (simp add: int_mod_eq uint_lt uint_0)
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lemma td_ext_uint: 
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  "td_ext (uint :: 'a word => int) word_of_int (uints (len_of TYPE('a::len0))) 
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    (%w::int. w mod 2 ^ len_of TYPE('a))"
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  apply (unfold td_ext_def')
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  apply (simp add: uints_num word_of_int_def bintrunc_mod2p)
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  apply (simp add: uint_mod_same uint_0 uint_lt
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                   word.uint_inverse word.Abs_word_inverse int_mod_lem)
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  done
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lemmas int_word_uint = td_ext_uint [THEN td_ext.eq_norm, standard]
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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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  [simplified len_gt_0 no_bintr_alt1 [symmetric]]
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interpretation word_ubin:
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  td_ext "uint::'a::len0 word \<Rightarrow> int" 
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         word_of_int 
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         "uints (len_of TYPE('a::len0))"
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         "bintrunc (len_of TYPE('a::len0))"
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  by (rule td_ext_ubin)
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lemma split_word_all:
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  "(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))"
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proof
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  fix x :: "'a word"
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  assume "\<And>x. PROP P (word_of_int x)"
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  hence "PROP P (word_of_int (uint x))" .
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  thus "PROP P x" by simp
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qed
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subsection  "Arithmetic operations"
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definition
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  word_succ :: "'a :: len0 word => 'a word"
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where
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  "word_succ a = word_of_int (Int.succ (uint a))"
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definition
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  word_pred :: "'a :: len0 word => 'a word"
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where
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  "word_pred a = word_of_int (Int.pred (uint a))"
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instantiation word :: (len0) "{number, Divides.div, ord, 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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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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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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26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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lemmas arths = 
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  bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1],
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                folded word_ubin.eq_norm, standard]
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26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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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 (Int.succ a)" and
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  wi_hom_pred: "word_pred (word_of_int a) = word_of_int (Int.pred a)"
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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_sub_def: "a - b = a + - (b :: 'a :: len0 word)"
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  unfolding word_sub_wi diff_minus
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  by (simp only : word_uint.Rep_inverse wi_hom_syms)
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lemmas word_diff_minus = word_sub_def [standard]
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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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  unfolding word_sub_def diff_minus by (simp only : wi_homs)
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lemmas new_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 new_word_of_int_hom_syms = new_word_of_int_homs [symmetric, standard]
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lemmas word_of_int_hom_syms =
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  new_word_of_int_hom_syms [unfolded succ_def pred_def]
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26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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lemmas word_of_int_homs =
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  new_word_of_int_homs [unfolded succ_def pred_def]
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lemmas word_of_int_add_hom = word_of_int_homs (2)
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lemmas word_of_int_mult_hom = word_of_int_homs (3)
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lemmas word_of_int_minus_hom = word_of_int_homs (4)
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lemmas word_of_int_succ_hom = word_of_int_homs (5)
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lemmas word_of_int_pred_hom = word_of_int_homs (6)
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lemmas word_of_int_0_hom = word_of_int_homs (7)
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lemmas word_of_int_1_hom = word_of_int_homs (8)
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26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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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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lemma word_zero_neq_one: "0 < len_of TYPE ('a :: len0) \<Longrightarrow> (0 :: 'a word) ~= 1"
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  unfolding word_arith_wis
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  by (auto simp add: word_ubin.norm_eq_iff [symmetric] gr0_conv_Suc)
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26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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lemmas lenw1_zero_neq_one = len_gt_0 [THEN word_zero_neq_one]
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26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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instance word :: (len) comm_ring_1
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  by (intro_classes) (simp add: lenw1_zero_neq_one)
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26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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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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26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
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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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lemma word_number_of_eq: 
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  "number_of w = (of_int w :: 'a :: len word)"
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  unfolding word_number_of_def word_of_int by auto
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instance word :: (len) number_ring
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  by (intro_classes) (simp add : word_number_of_eq)
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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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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
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  word_test_bit_def: "test_bit a = bin_nth (uint a)"
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definition
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  word_set_bit_def: "set_bit a n x =
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   word_of_int (bin_sc n (If x 1 0) (uint a))"
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definition
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  word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"
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definition
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  word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1"
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definition shiftl1 :: "'a word \<Rightarrow> 'a word" where
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  "shiftl1 w = word_of_int (uint w BIT 0)"
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definition shiftr1 :: "'a word \<Rightarrow> 'a word" where
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  -- "shift right as unsigned or as signed, ie logical or arithmetic"
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  "shiftr1 w = word_of_int (bin_rest (uint w))"
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definition
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  shiftl_def: "w << n = (shiftl1 ^^ n) w"
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definition
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  shiftr_def: "w >> n = (shiftr1 ^^ n) w"
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instance ..
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end
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instantiation word :: (len) bitss
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begin
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definition
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  word_msb_def: 
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  "msb a \<longleftrightarrow> bin_sign (sint a) = Int.Min"
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instance ..
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end
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lemma [code]:
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  "msb a \<longleftrightarrow> bin_sign (sint a) = (- 1 :: int)"
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  by (simp only: word_msb_def Min_def)
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definition setBit :: "'a :: len0 word => nat => 'a word" where 
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  "setBit w n = set_bit w n True"
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definition clearBit :: "'a :: len0 word => nat => 'a word" where
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  "clearBit w n = set_bit w n False"
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subsection "Shift operations"
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definition sshiftr1 :: "'a :: len word => 'a word" where 
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  "sshiftr1 w = word_of_int (bin_rest (sint w))"
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   403
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definition bshiftr1 :: "bool => 'a :: len word => 'a word" where
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  "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
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definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where
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  "w >>> n = (sshiftr1 ^^ n) w"
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   409
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definition mask :: "nat => 'a::len word" where
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  "mask n = (1 << n) - 1"
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definition revcast :: "'a :: len0 word => 'b :: len0 word" where
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  "revcast w =  of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
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   415
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definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where
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  "slice1 n w = of_bl (takefill False n (to_bl w))"
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   418
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definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where
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  "slice n w = slice1 (size w - n) w"
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subsection "Rotation"
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   424
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definition rotater1 :: "'a list => 'a list" where
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  "rotater1 ys = 
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    (case ys of [] => [] | x # xs => last ys # butlast ys)"
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definition rotater :: "nat => 'a list => 'a list" where
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  "rotater n = rotater1 ^^ n"
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   432
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where
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  "word_rotr n w = of_bl (rotater n (to_bl w))"
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   434
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   435
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where
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  "word_rotl n w = of_bl (rotate n (to_bl w))"
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   437
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   438
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where
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  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
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                    else word_rotl (nat (- i)) w)"
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subsection "Split and cat operations"
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definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where
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  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
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   447
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   448
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where
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  "word_split a = 
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   (case bin_split (len_of TYPE ('c)) (uint a) of 
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     (u, v) => (word_of_int u, word_of_int v))"
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   452
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   453
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where
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  "word_rcat ws = 
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  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
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   456
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   457
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where
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  "word_rsplit w = 
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  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
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   460
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   461
definition max_word :: "'a::len word" -- "Largest representable machine integer." where
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  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
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   463
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primrec of_bool :: "bool \<Rightarrow> 'a::len word" where
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  "of_bool False = 0"
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| "of_bool True = 1"
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   467
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   468
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   469
lemmas of_nth_def = word_set_bits_def
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   470
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   471
lemma sint_sbintrunc': 
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   472
  "sint (word_of_int bin :: 'a word) = 
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diff changeset
   473
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   474
  unfolding sint_uint 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   475
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   476
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   477
lemma uint_sint: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   478
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   479
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   480
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   481
lemma bintr_uint': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   482
  "n >= size w \<Longrightarrow> bintrunc n (uint w) = uint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   483
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   484
  apply (subst word_ubin.norm_Rep [symmetric]) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   485
  apply (simp only: bintrunc_bintrunc_min word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   486
  apply (simp add: min_max.inf_absorb2)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   487
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   488
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   489
lemma wi_bintr': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   490
  "wb = word_of_int bin \<Longrightarrow> n >= size wb \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   491
    word_of_int (bintrunc n bin) = wb"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   492
  unfolding word_size
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   493
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   494
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   495
lemmas bintr_uint = bintr_uint' [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   496
lemmas wi_bintr = wi_bintr' [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   497
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   498
lemma td_ext_sbin: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   499
  "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   500
    (sbintrunc (len_of TYPE('a) - 1))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   501
  apply (unfold td_ext_def' sint_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   502
  apply (simp add : word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   503
  apply (cases "len_of TYPE('a)")
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   504
   apply (auto simp add : sints_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   505
  apply (rule sym [THEN trans])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   506
  apply (rule word_ubin.Abs_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   507
  apply (simp only: bintrunc_sbintrunc)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   508
  apply (drule sym)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   509
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   510
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   511
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   512
lemmas td_ext_sint = td_ext_sbin 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   513
  [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   514
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   515
(* We do sint before sbin, before sint is the user version
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   516
   and interpretations do not produce thm duplicates. I.e. 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   517
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   518
   because the latter is the same thm as the former *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   519
interpretation word_sint:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   520
  td_ext "sint ::'a::len word => int" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   521
          word_of_int 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   522
          "sints (len_of TYPE('a::len))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   523
          "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) -
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   524
               2 ^ (len_of TYPE('a::len) - 1)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   525
  by (rule td_ext_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   526
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   527
interpretation word_sbin:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   528
  td_ext "sint ::'a::len word => int" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   529
          word_of_int 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   530
          "sints (len_of TYPE('a::len))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   531
          "sbintrunc (len_of TYPE('a::len) - 1)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   532
  by (rule td_ext_sbin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   533
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   534
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   535
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   536
lemmas td_sint = word_sint.td
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   537
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   538
lemma word_number_of_alt [code_unfold_post]:
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   539
  "number_of b = word_of_int (number_of b)"
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   540
  by (simp add: number_of_eq word_number_of_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   541
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   542
lemma word_no_wi: "number_of = word_of_int"
44762
8f9d09241a68 tuned proofs;
wenzelm
parents: 42793
diff changeset
   543
  by (auto simp: word_number_of_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   544
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   545
lemma to_bl_def': 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   546
  "(to_bl :: 'a :: len0 word => bool list) =
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   547
    bin_to_bl (len_of TYPE('a)) o uint"
44762
8f9d09241a68 tuned proofs;
wenzelm
parents: 42793
diff changeset
   548
  by (auto simp: to_bl_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   549
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   550
lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w", standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   551
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   552
lemmas uints_mod = uints_def [unfolded no_bintr_alt1]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   553
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   554
lemma uint_bintrunc: "uint (number_of bin :: 'a word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   555
    number_of (bintrunc (len_of TYPE ('a :: len0)) bin)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   556
  unfolding word_number_of_def number_of_eq
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   557
  by (auto intro: word_ubin.eq_norm) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   558
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   559
lemma sint_sbintrunc: "sint (number_of bin :: 'a word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   560
    number_of (sbintrunc (len_of TYPE ('a :: len) - 1) bin)" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   561
  unfolding word_number_of_def number_of_eq
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   562
  by (subst word_sbin.eq_norm) simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   563
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   564
lemma unat_bintrunc: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   565
  "unat (number_of bin :: 'a :: len0 word) =
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   566
    number_of (bintrunc (len_of TYPE('a)) bin)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   567
  unfolding unat_def nat_number_of_def 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   568
  by (simp only: uint_bintrunc)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   569
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   570
(* WARNING - these may not always be helpful *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   571
declare 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   572
  uint_bintrunc [simp] 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   573
  sint_sbintrunc [simp] 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   574
  unat_bintrunc [simp]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   575
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   576
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   577
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   578
  apply (rule word_uint.Rep_eqD)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   579
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   580
    defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   581
    apply (rule word_ubin.norm_Rep)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   582
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   583
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   584
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   585
lemmas uint_lem = word_uint.Rep [unfolded uints_num mem_Collect_eq]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   586
lemmas sint_lem = word_sint.Rep [unfolded sints_num mem_Collect_eq]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   587
lemmas uint_ge_0 [iff] = uint_lem [THEN conjunct1, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   588
lemmas uint_lt2p [iff] = uint_lem [THEN conjunct2, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   589
lemmas sint_ge = sint_lem [THEN conjunct1, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   590
lemmas sint_lt = sint_lem [THEN conjunct2, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   591
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   592
lemma sign_uint_Pls [simp]: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   593
  "bin_sign (uint x) = Int.Pls"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   594
  by (simp add: sign_Pls_ge_0 number_of_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   595
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   596
lemmas uint_m2p_neg = iffD2 [OF diff_less_0_iff_less uint_lt2p, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   597
lemmas uint_m2p_not_non_neg = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   598
  iffD2 [OF linorder_not_le uint_m2p_neg, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   599
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   600
lemma lt2p_lem:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   601
  "len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   602
  by (rule xtr8 [OF _ uint_lt2p]) simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   603
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   604
lemmas uint_le_0_iff [simp] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   605
  uint_ge_0 [THEN leD, THEN linorder_antisym_conv1, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   606
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   607
lemma uint_nat: "uint w = int (unat w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   608
  unfolding unat_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   609
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   610
lemma uint_number_of:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   611
  "uint (number_of b :: 'a :: len0 word) = number_of b mod 2 ^ len_of TYPE('a)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   612
  unfolding word_number_of_alt
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   613
  by (simp only: int_word_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   614
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   615
lemma unat_number_of: 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   616
  "bin_sign b = Int.Pls \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   617
  unat (number_of b::'a::len0 word) = number_of b mod 2 ^ len_of TYPE ('a)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   618
  apply (unfold unat_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   619
  apply (clarsimp simp only: uint_number_of)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   620
  apply (rule nat_mod_distrib [THEN trans])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   621
    apply (erule sign_Pls_ge_0 [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   622
   apply (simp_all add: nat_power_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   623
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   624
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   625
lemma sint_number_of: "sint (number_of b :: 'a :: len word) = (number_of b + 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   626
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   627
    2 ^ (len_of TYPE('a) - 1)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   628
  unfolding word_number_of_alt by (rule int_word_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   629
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   630
lemma word_of_int_bin [simp] : 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   631
  "(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   632
  unfolding word_number_of_alt by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   633
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   634
lemma word_int_case_wi: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   635
  "word_int_case f (word_of_int i :: 'b word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   636
    f (i mod 2 ^ len_of TYPE('b::len0))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   637
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   638
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   639
lemma word_int_split: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   640
  "P (word_int_case f x) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   641
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   642
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   643
  unfolding word_int_case_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   644
  by (auto simp: word_uint.eq_norm int_mod_eq')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   645
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   646
lemma word_int_split_asm: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   647
  "P (word_int_case f x) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   648
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   649
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   650
  unfolding word_int_case_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   651
  by (auto simp: word_uint.eq_norm int_mod_eq')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   652
  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   653
lemmas uint_range' =
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   654
  word_uint.Rep [unfolded uints_num mem_Collect_eq, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   655
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   656
  sints_num mem_Collect_eq, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   657
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   658
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   659
  unfolding word_size by (rule uint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   661
lemma sint_range_size:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   662
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   663
  unfolding word_size by (rule sint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   664
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   665
lemmas sint_above_size = sint_range_size
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   666
  [THEN conjunct2, THEN [2] xtr8, folded One_nat_def, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   667
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   668
lemmas sint_below_size = sint_range_size
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   669
  [THEN conjunct1, THEN [2] order_trans, folded One_nat_def, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   670
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   671
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   672
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   673
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   674
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   675
  apply (unfold word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   676
  apply (subst word_ubin.norm_Rep [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   677
  apply (simp only: nth_bintr word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   678
  apply fast
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   679
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   680
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   681
lemma word_eqI [rule_format] : 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   682
  fixes u :: "'a::len0 word"
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   683
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   684
  apply (rule test_bit_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   685
  apply (rule ext)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   686
  apply (erule allE)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   687
  apply (erule impCE)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   688
   prefer 2
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   689
   apply assumption
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   690
  apply (auto dest!: test_bit_size simp add: word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   691
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   692
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   693
lemmas word_eqD = test_bit_eq_iff [THEN iffD2, THEN fun_cong, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   694
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   695
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   696
  unfolding word_test_bit_def word_size
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   697
  by (simp add: nth_bintr [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   698
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   699
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   700
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   701
lemma bin_nth_uint_imp': "bin_nth (uint w) n --> n < size w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   702
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   703
  apply (rule impI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   704
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   705
  apply (subst word_ubin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   706
  apply assumption
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   707
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   708
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   709
lemma bin_nth_sint': 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   710
  "n >= size w --> bin_nth (sint w) n = bin_nth (sint w) (size w - 1)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   711
  apply (rule impI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   712
  apply (subst word_sbin.norm_Rep [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   713
  apply (simp add : nth_sbintr word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   714
  apply auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   715
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   716
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   717
lemmas bin_nth_uint_imp = bin_nth_uint_imp' [rule_format, unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   718
lemmas bin_nth_sint = bin_nth_sint' [rule_format, unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   719
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   720
(* type definitions theorem for in terms of equivalent bool list *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   721
lemma td_bl: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   722
  "type_definition (to_bl :: 'a::len0 word => bool list) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   723
                   of_bl  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   724
                   {bl. length bl = len_of TYPE('a)}"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   725
  apply (unfold type_definition_def of_bl_def to_bl_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   726
  apply (simp add: word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   727
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   728
  apply (drule sym)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   729
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   730
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   731
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   732
interpretation word_bl:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   733
  type_definition "to_bl :: 'a::len0 word => bool list"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   734
                  of_bl  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   735
                  "{bl. length bl = len_of TYPE('a::len0)}"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   736
  by (rule td_bl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   737
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   738
lemma word_size_bl: "size w = size (to_bl w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   739
  unfolding word_size by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   740
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   741
lemma to_bl_use_of_bl:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   742
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44821
diff changeset
   743
  by (fastforce elim!: word_bl.Abs_inverse [simplified])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   744
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   745
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   746
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   747
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   748
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   749
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   750
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   751
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   752
  by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   753
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   754
lemmas word_rev_gal' = sym [THEN word_rev_gal, symmetric, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   755
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   756
lemmas length_bl_gt_0 [iff] = xtr1 [OF word_bl.Rep' len_gt_0, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   757
lemmas bl_not_Nil [iff] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   758
  length_bl_gt_0 [THEN length_greater_0_conv [THEN iffD1], standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   759
lemmas length_bl_neq_0 [iff] = length_bl_gt_0 [THEN gr_implies_not0]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   760
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   761
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = Int.Min)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   762
  apply (unfold to_bl_def sint_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   763
  apply (rule trans [OF _ bl_sbin_sign])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   764
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   765
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   766
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   767
lemma of_bl_drop': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   768
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   769
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   770
  apply (unfold of_bl_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   771
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   772
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   773
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   774
lemmas of_bl_no = of_bl_def [folded word_number_of_def]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   775
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   776
lemma test_bit_of_bl:  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   777
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   778
  apply (unfold of_bl_def word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   779
  apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   780
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   781
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   782
lemma no_of_bl: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   783
  "(number_of bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) bin)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   784
  unfolding word_size of_bl_no by (simp add : word_number_of_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   785
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   786
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   787
  unfolding word_size to_bl_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   788
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   789
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   790
  unfolding uint_bl by (simp add : word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   791
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   792
lemma to_bl_of_bin: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   793
  "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   794
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   795
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   796
lemmas to_bl_no_bin [simp] = to_bl_of_bin [folded word_number_of_def]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   797
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   798
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   799
  unfolding uint_bl by (simp add : word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   800
  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   801
lemmas uint_bl_bin [simp] = trans [OF bin_bl_bin word_ubin.norm_Rep, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   802
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   803
lemmas num_AB_u [simp] = word_uint.Rep_inverse 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   804
  [unfolded o_def word_number_of_def [symmetric], standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   805
lemmas num_AB_s [simp] = word_sint.Rep_inverse 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   806
  [unfolded o_def word_number_of_def [symmetric], standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   807
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   808
(* naturals *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   809
lemma uints_unats: "uints n = int ` unats n"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   810
  apply (unfold unats_def uints_num)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   811
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   812
  apply (rule_tac image_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   813
  apply (erule_tac nat_0_le [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   814
  apply auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   815
  apply (erule_tac nat_less_iff [THEN iffD2])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   816
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   817
  apply (auto simp add : nat_power_eq int_power)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   818
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   819
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   820
lemma unats_uints: "unats n = nat ` uints n"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   821
  by (auto simp add : uints_unats image_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   822
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   823
lemmas bintr_num = word_ubin.norm_eq_iff 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   824
  [symmetric, folded word_number_of_def, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   825
lemmas sbintr_num = word_sbin.norm_eq_iff 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   826
  [symmetric, folded word_number_of_def, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   827
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   828
lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def, standard]
45284
ae78a4ffa81d use simproc_setup for cancellation simprocs, to get proper name bindings
huffman
parents: 44938
diff changeset
   829
lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def, standard]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   830
    
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   831
(* don't add these to simpset, since may want bintrunc n w to be simplified;
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   832
  may want these in reverse, but loop as simp rules, so use following *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   833
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   834
lemma num_of_bintr':
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   835
  "bintrunc (len_of TYPE('a :: len0)) a = b \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   836
    number_of a = (number_of b :: 'a word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   837
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   838
  apply (rule_tac num_of_bintr [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   839
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   840
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   841
lemma num_of_sbintr':
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   842
  "sbintrunc (len_of TYPE('a :: len) - 1) a = b \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   843
    number_of a = (number_of b :: 'a word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   844
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   845
  apply (rule_tac num_of_sbintr [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   846
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   847
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   848
lemmas num_abs_bintr = sym [THEN trans,
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   849
  OF num_of_bintr word_number_of_def, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   850
lemmas num_abs_sbintr = sym [THEN trans,
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   851
  OF num_of_sbintr word_number_of_def, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   852
  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   853
(** cast - note, no arg for new length, as it's determined by type of result,
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   854
  thus in "cast w = w, the type means cast to length of w! **)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   855
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   856
lemma ucast_id: "ucast w = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   857
  unfolding ucast_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   858
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   859
lemma scast_id: "scast w = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   860
  unfolding scast_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   861
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   862
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   863
  unfolding ucast_def of_bl_def uint_bl
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   864
  by (auto simp add : word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   865
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   866
lemma nth_ucast: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   867
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   868
  apply (unfold ucast_def test_bit_bin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   869
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   870
  apply (fast elim!: bin_nth_uint_imp)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   871
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   872
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   873
(* for literal u(s)cast *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   874
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   875
lemma ucast_bintr [simp]: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   876
  "ucast (number_of w ::'a::len0 word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   877
   number_of (bintrunc (len_of TYPE('a)) w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   878
  unfolding ucast_def by simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   879
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   880
lemma scast_sbintr [simp]: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   881
  "scast (number_of w ::'a::len word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   882
   number_of (sbintrunc (len_of TYPE('a) - Suc 0) w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   883
  unfolding scast_def by simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   884
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   885
lemmas source_size = source_size_def [unfolded Let_def word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   886
lemmas target_size = target_size_def [unfolded Let_def word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   887
lemmas is_down = is_down_def [unfolded source_size target_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   888
lemmas is_up = is_up_def [unfolded source_size target_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   889
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   890
lemmas is_up_down =  trans [OF is_up is_down [symmetric], standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   891
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   892
lemma down_cast_same': "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   893
  apply (unfold is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   894
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   895
  apply (rule ext)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   896
  apply (unfold ucast_def scast_def uint_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   897
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   898
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   899
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   900
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   901
lemma word_rev_tf': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   902
  "r = to_bl (of_bl bl) \<Longrightarrow> r = rev (takefill False (length r) (rev bl))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   903
  unfolding of_bl_def uint_bl
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   904
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   905
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   906
lemmas word_rev_tf = refl [THEN word_rev_tf', unfolded word_bl.Rep', standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   907
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   908
lemmas word_rep_drop = word_rev_tf [simplified takefill_alt,
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   909
  simplified, simplified rev_take, simplified]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   910
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   911
lemma to_bl_ucast: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   912
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   913
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   914
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   915
  apply (unfold ucast_bl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   916
  apply (rule trans)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   917
   apply (rule word_rep_drop)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   918
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   919
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   920
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   921
lemma ucast_up_app': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   922
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   923
    to_bl (uc w) = replicate n False @ (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   924
  by (auto simp add : source_size target_size to_bl_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   925
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   926
lemma ucast_down_drop': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   927
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   928
    to_bl (uc w) = drop n (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   929
  by (auto simp add : source_size target_size to_bl_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   930
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   931
lemma scast_down_drop': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   932
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   933
    to_bl (sc w) = drop n (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   934
  apply (subgoal_tac "sc = ucast")
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   935
   apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   936
   apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   937
   apply (erule refl [THEN ucast_down_drop'])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   938
  apply (rule refl [THEN down_cast_same', symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   939
  apply (simp add : source_size target_size is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   940
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   941
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   942
lemma sint_up_scast': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   943
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   944
  apply (unfold is_up)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   945
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   946
  apply (simp add: scast_def word_sbin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   947
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   948
    prefer 3
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   949
    apply (rule word_sbin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   950
   apply (rule sbintrunc_sbintrunc_l)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   951
   defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   952
   apply (subst word_sbin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   953
   apply (rule refl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   954
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   955
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   956
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   957
lemma uint_up_ucast':
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   958
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   959
  apply (unfold is_up)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   960
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   961
  apply (rule bin_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   962
  apply (fold word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   963
  apply (auto simp add: nth_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   964
  apply (auto simp add: test_bit_bin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   965
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   966
    
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   967
lemmas down_cast_same = refl [THEN down_cast_same']
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   968
lemmas ucast_up_app = refl [THEN ucast_up_app']
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   969
lemmas ucast_down_drop = refl [THEN ucast_down_drop']
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   970
lemmas scast_down_drop = refl [THEN scast_down_drop']
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   971
lemmas uint_up_ucast = refl [THEN uint_up_ucast']
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   972
lemmas sint_up_scast = refl [THEN sint_up_scast']
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   973
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   974
lemma ucast_up_ucast': "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   975
  apply (simp (no_asm) add: ucast_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   976
  apply (clarsimp simp add: uint_up_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   977
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   978
    
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   979
lemma scast_up_scast': "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   980
  apply (simp (no_asm) add: scast_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   981
  apply (clarsimp simp add: sint_up_scast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   982
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   983
    
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   984
lemma ucast_of_bl_up': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   985
  "w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   986
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   987
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   988
lemmas ucast_up_ucast = refl [THEN ucast_up_ucast']
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   989
lemmas scast_up_scast = refl [THEN scast_up_scast']
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   990
lemmas ucast_of_bl_up = refl [THEN ucast_of_bl_up']
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   991
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   992
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   993
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   994
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   995
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   996
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   997
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   998
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   999
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1000
lemma up_ucast_surj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1001
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1002
   surj (ucast :: 'a word => 'b word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1003
  by (rule surjI, erule ucast_up_ucast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1004
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1005
lemma up_scast_surj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1006
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1007
   surj (scast :: 'a word => 'b word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1008
  by (rule surjI, erule scast_up_scast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1009
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1010
lemma down_scast_inj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1011
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1012
   inj_on (ucast :: 'a word => 'b word) A"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1013
  by (rule inj_on_inverseI, erule scast_down_scast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1014
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1015
lemma down_ucast_inj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1016
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1017
   inj_on (ucast :: 'a word => 'b word) A"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1018
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1019
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1020
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1021
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1022
  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1023
lemma ucast_down_no': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1024
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (number_of bin) = number_of bin"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1025
  apply (unfold word_number_of_def is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1026
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1027
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1028
  apply (erule bintrunc_bintrunc_ge)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1029
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1030
    
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1031
lemmas ucast_down_no = ucast_down_no' [OF refl]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1032
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1033
lemma ucast_down_bl': "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1034
  unfolding of_bl_no by clarify (erule ucast_down_no)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1035
    
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1036
lemmas ucast_down_bl = ucast_down_bl' [OF refl]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1037
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1038
lemmas slice_def' = slice_def [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1039
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1040
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1041
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1042
lemmas word_log_bin_defs = word_log_defs
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1043
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1044
text {* Executable equality *}
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1045
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38527
diff changeset
  1046
instantiation word :: (len0) equal
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1047
begin
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1048
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38527
diff changeset
  1049
definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38527
diff changeset
  1050
  "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1051
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1052
instance proof
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38527
diff changeset
  1053
qed (simp add: equal equal_word_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1054
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1055
end
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1056
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1057
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1058
subsection {* Word Arithmetic *}
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1059
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1060
lemma word_less_alt: "(a < b) = (uint a < uint b)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1061
  unfolding word_less_def word_le_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1062
  by (auto simp del: word_uint.Rep_inject 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1063
           simp: word_uint.Rep_inject [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1064
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1065
lemma signed_linorder: "class.linorder word_sle word_sless"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1066
proof
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1067
qed (unfold word_sle_def word_sless_def, auto)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1068
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1069
interpretation signed: linorder "word_sle" "word_sless"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1070
  by (rule signed_linorder)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1071
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1072
lemma udvdI: 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1073
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1074
  by (auto simp: udvd_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1075
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1076
lemmas word_div_no [simp] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1077
  word_div_def [of "number_of a" "number_of b", standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1078
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1079
lemmas word_mod_no [simp] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1080
  word_mod_def [of "number_of a" "number_of b", standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1081
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1082
lemmas word_less_no [simp] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1083
  word_less_def [of "number_of a" "number_of b", standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1084
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1085
lemmas word_le_no [simp] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1086
  word_le_def [of "number_of a" "number_of b", standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1087
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1088
lemmas word_sless_no [simp] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1089
  word_sless_def [of "number_of a" "number_of b", standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1090
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1091
lemmas word_sle_no [simp] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1092
  word_sle_def [of "number_of a" "number_of b", standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1093
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1094
(* following two are available in class number_ring, 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1095
  but convenient to have them here here;
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1096
  note - the number_ring versions, numeral_0_eq_0 and numeral_1_eq_1
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1097
  are in the default simpset, so to use the automatic simplifications for
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1098
  (eg) sint (number_of bin) on sint 1, must do
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1099
  (simp add: word_1_no del: numeral_1_eq_1) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1100
  *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1101
lemmas word_0_wi_Pls = word_0_wi [folded Pls_def]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1102
lemmas word_0_no = word_0_wi_Pls [folded word_no_wi]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1103
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1104
lemma int_one_bin: "(1 :: int) = (Int.Pls BIT 1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1105
  unfolding Pls_def Bit_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1106
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1107
lemma word_1_no: 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1108
  "(1 :: 'a :: len0 word) = number_of (Int.Pls BIT 1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1109
  unfolding word_1_wi word_number_of_def int_one_bin by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1110
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1111
lemma word_m1_wi: "-1 = word_of_int -1" 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1112
  by (rule word_number_of_alt)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1113
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1114
lemma word_m1_wi_Min: "-1 = word_of_int Int.Min"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1115
  by (simp add: word_m1_wi number_of_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1116
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1117
lemma word_0_bl: "of_bl [] = 0" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1118
  unfolding word_0_wi of_bl_def by (simp add : Pls_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1119
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1120
lemma word_1_bl: "of_bl [True] = 1" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1121
  unfolding word_1_wi of_bl_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1122
  by (simp add : bl_to_bin_def Bit_def Pls_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1123
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1124
lemma uint_eq_0 [simp] : "(uint 0 = 0)" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1125
  unfolding word_0_wi
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1126
  by (simp add: word_ubin.eq_norm Pls_def [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1127
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1128
lemma of_bl_0 [simp] : "of_bl (replicate n False) = 0"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1129
  by (simp add : word_0_wi of_bl_def bl_to_bin_rep_False Pls_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1130
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1131
lemma to_bl_0: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1132
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1133
  unfolding uint_bl
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1134
  by (simp add : word_size bin_to_bl_Pls Pls_def [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1135
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1136
lemma uint_0_iff: "(uint x = 0) = (x = 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1137
  by (auto intro!: word_uint.Rep_eqD)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1138
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1139
lemma unat_0_iff: "(unat x = 0) = (x = 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1140
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1141
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1142
lemma unat_0 [simp]: "unat 0 = 0"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1143
  unfolding unat_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1144
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1145
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1146
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1147
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1148
    defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1149
    apply (rule word_uint.Rep_inverse)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1150
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1151
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1152
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1153
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1154
lemmas size_0_same = size_0_same' [folded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1155
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1156
lemmas unat_eq_0 = unat_0_iff
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1157
lemmas unat_eq_zero = unat_0_iff
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1158
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1159
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1160
by (auto simp: unat_0_iff [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1161
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1162
lemma ucast_0 [simp] : "ucast 0 = 0"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1163
unfolding ucast_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1164
by simp (simp add: word_0_wi)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1165
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1166
lemma sint_0 [simp] : "sint 0 = 0"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1167
unfolding sint_uint
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1168
by (simp add: Pls_def [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1169
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1170
lemma scast_0 [simp] : "scast 0 = 0"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1171
apply (unfold scast_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1172
apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1173
apply (simp add: word_0_wi)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1174
done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1175
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1176
lemma sint_n1 [simp] : "sint -1 = -1"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1177
apply (unfold word_m1_wi_Min)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1178
apply (simp add: word_sbin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1179
apply (unfold Min_def number_of_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1180
apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1181
done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1182
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1183
lemma scast_n1 [simp] : "scast -1 = -1"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1184
  apply (unfold scast_def sint_n1)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1185
  apply (unfold word_number_of_alt)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1186
  apply (rule refl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1187
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1188
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1189
lemma uint_1 [simp] : "uint (1 :: 'a :: len word) = 1"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1190
  unfolding word_1_wi
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1191
  by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1192
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1193
lemma unat_1 [simp] : "unat (1 :: 'a :: len word) = 1"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1194
  by (unfold unat_def uint_1) auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1195
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1196
lemma ucast_1 [simp] : "ucast (1 :: 'a :: len word) = 1"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1197
  unfolding ucast_def word_1_wi
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1198
  by (simp add: word_ubin.eq_norm int_one_bin bintrunc_minus_simps)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1199
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1200
(* now, to get the weaker results analogous to word_div/mod_def *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1201
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1202
lemmas word_arith_alts = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1203
  word_sub_wi [unfolded succ_def pred_def, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1204
  word_arith_wis [unfolded succ_def pred_def, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1205
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
  1206
(* FIXME: Lots of duplicate lemmas *)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1207
lemmas word_sub_alt = word_arith_alts (1)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1208
lemmas word_add_alt = word_arith_alts (2)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1209
lemmas word_mult_alt = word_arith_alts (3)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1210
lemmas word_minus_alt = word_arith_alts (4)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1211
lemmas word_succ_alt = word_arith_alts (5)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1212
lemmas word_pred_alt = word_arith_alts (6)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1213
lemmas word_0_alt = word_arith_alts (7)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1214
lemmas word_1_alt = word_arith_alts (8)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1215
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1216
subsection  "Transferring goals from words to ints"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1217
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1218
lemma word_ths:  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1219
  shows
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1220
  word_succ_p1:   "word_succ a = a + 1" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1221
  word_pred_m1:   "word_pred a = a - 1" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1222
  word_pred_succ: "word_pred (word_succ a) = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1223
  word_succ_pred: "word_succ (word_pred a) = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1224
  word_mult_succ: "word_succ a * b = b + a * b"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1225
  by (rule word_uint.Abs_cases [of b],
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1226
      rule word_uint.Abs_cases [of a],
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1227
      simp add: pred_def succ_def add_commute mult_commute 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1228
                ring_distribs new_word_of_int_homs)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1229
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1230
lemmas uint_cong = arg_cong [where f = uint]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1231
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1232
lemmas uint_word_ariths = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1233
  word_arith_alts [THEN trans [OF uint_cong int_word_uint], standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1234
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1235
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1236
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1237
(* similar expressions for sint (arith operations) *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1238
lemmas sint_word_ariths = uint_word_arith_bintrs
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1239
  [THEN uint_sint [symmetric, THEN trans],
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1240
  unfolded uint_sint bintr_arith1s bintr_ariths 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1241
    len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep, standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1242
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1243
lemmas uint_div_alt = word_div_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1244
  [THEN trans [OF uint_cong int_word_uint], standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1245
lemmas uint_mod_alt = word_mod_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1246
  [THEN trans [OF uint_cong int_word_uint], standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1247
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1248
lemma word_pred_0_n1: "word_pred 0 = word_of_int -1"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1249
  unfolding word_pred_def number_of_eq
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1250
  by (simp add : pred_def word_no_wi)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1251
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1252
lemma word_pred_0_Min: "word_pred 0 = word_of_int Int.Min"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1253
  by (simp add: word_pred_0_n1 number_of_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1254
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1255
lemma word_m1_Min: "- 1 = word_of_int Int.Min"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1256
  unfolding Min_def by (simp only: word_of_int_hom_syms)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1257
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1258
lemma succ_pred_no [simp]:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1259
  "word_succ (number_of bin) = number_of (Int.succ bin) & 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1260
    word_pred (number_of bin) = number_of (Int.pred bin)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1261
  unfolding word_number_of_def by (simp add : new_word_of_int_homs)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1262
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1263
lemma word_sp_01 [simp] : 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1264
  "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1265
  by (unfold word_0_no word_1_no) auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1266
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1267
(* alternative approach to lifting arithmetic equalities *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1268
lemma word_of_int_Ex:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1269
  "\<exists>y. x = word_of_int y"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1270
  by (rule_tac x="uint x" in exI) simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1271
45545
26aebb8ac9c1 Word.thy: rearrange to instantiate arithmetic classes together with arithmetic operations
huffman
parents: 45544
diff changeset
  1272
(* FIXME: redundant theorems *)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1273
lemma word_arith_eqs:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1274
  fixes a :: "'a::len0 word"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1275
  fixes b :: "'a::len0 word"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1276
  shows
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1277
  word_add_0: "0 + a = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1278
  word_add_0_right: "a + 0 = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1279
  word_mult_1: "1 * a = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1280
  word_mult_1_right: "a * 1 = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1281
  word_add_commute: "a + b = b + a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1282
  word_add_assoc: "a + b + c = a + (b + c)" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1283
  word_add_left_commute: "a + (b + c) = b + (a + c)" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1284
  word_mult_commute: "a * b = b * a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1285
  word_mult_assoc: "a * b * c = a * (b * c)" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1286
  word_mult_left_commute: "a * (b * c) = b * (a * c)" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1287
  word_left_distrib: "(a + b) * c = a * c + b * c" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1288
  word_right_distrib: "a * (b + c) = a * b + a * c" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1289
  word_left_minus: "- a + a = 0" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1290
  word_diff_0_right: "a - 0 = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1291
  word_diff_self: "a - a = 0"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1292
  using word_of_int_Ex [of a] 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1293
        word_of_int_Ex [of b] 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1294
        word_of_int_Ex [of c]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1295
  by (auto simp: word_of_int_hom_syms [symmetric]
44821
a92f65e174cf avoid using legacy theorem names
huffman
parents: 44762
diff changeset
  1296
                 add_0_right add_commute add_assoc add_left_commute
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1297
                 mult_commute mult_assoc mult_left_commute
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1298
                 left_distrib right_distrib)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1299
  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1300
lemmas word_add_ac = word_add_commute word_add_assoc word_add_left_commute
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1301
lemmas word_mult_ac = word_mult_commute word_mult_assoc word_mult_left_commute
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1302
  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1303
lemmas word_plus_ac0 = word_add_0 word_add_0_right word_add_ac
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1304
lemmas word_times_ac1 = word_mult_1 word_mult_1_right word_mult_ac
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1305
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1306
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1307
subsection "Order on fixed-length words"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1308
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1309
lemma word_order_trans: "x <= y \<Longrightarrow> y <= z \<Longrightarrow> x <= (z :: 'a :: len0 word)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1310
  unfolding word_le_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1311
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1312
lemma word_order_refl: "z <= (z :: 'a :: len0 word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1313
  unfolding word_le_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1314
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1315
lemma word_order_antisym: "x <= y \<Longrightarrow> y <= x \<Longrightarrow> x = (y :: 'a :: len0 word)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1316
  unfolding word_le_def by (auto intro!: word_uint.Rep_eqD)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1317
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1318
lemma word_order_linear:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1319
  "y <= x | x <= (y :: 'a :: len0 word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1320
  unfolding word_le_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1321
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1322
lemma word_zero_le [simp] :
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1323
  "0 <= (y :: 'a :: len0 word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1324
  unfolding word_le_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1325
  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1326
instance word :: (len0) linorder
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1327
  by intro_classes (auto simp: word_less_def word_le_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1328
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1329
lemma word_m1_ge [simp] : "word_pred 0 >= y"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1330
  unfolding word_le_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1331
  by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1332
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1333
lemmas word_n1_ge [simp]  = word_m1_ge [simplified word_sp_01]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1334
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1335
lemmas word_not_simps [simp] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1336
  word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1337
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1338
lemma word_gt_0: "0 < y = (0 ~= (y :: 'a :: len0 word))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1339
  unfolding word_less_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1340
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1341
lemmas word_gt_0_no [simp] = word_gt_0 [of "number_of y", standard]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1342
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1343
lemma word_sless_alt: "(a <s b) = (sint a < sint b)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1344
  unfolding word_sle_def word_sless_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1345
  by (auto simp add: less_le)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1346
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1347
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1348
  unfolding unat_def word_le_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1349
  by (rule nat_le_eq_zle [symmetric]) simp
56e3520b68b2 one unified Word theory
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