src/HOL/Word/Word.thy
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add lemma word_eq_iff
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(*  Title:      HOL/Word/Word.thy
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    Author: Jeremy Dawson and Gerwin Klein, NICTA
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*)
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header {* A type of finite bit strings *}
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theory Word
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imports
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  Type_Length
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  Misc_Typedef
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  "~~/src/HOL/Library/Boolean_Algebra"
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  Bool_List_Representation
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uses ("~~/src/HOL/Word/Tools/smt_word.ML")
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begin
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text {* see @{text "Examples/WordExamples.thy"} for examples *}
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subsection {* Type definition *}
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typedef (open) 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}"
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  morphisms uint Abs_word by auto
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definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" where
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  -- {* representation of words using unsigned or signed bins, 
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        only difference in these is the type class *}
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  "word_of_int w = Abs_word (bintrunc (len_of TYPE ('a)) w)" 
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lemma uint_word_of_int [code]: "uint (word_of_int w \<Colon> 'a\<Colon>len0 word) = w mod 2 ^ len_of TYPE('a)"
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  by (auto simp add: word_of_int_def bintrunc_mod2p intro: Abs_word_inverse)
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code_datatype word_of_int
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subsection {* Random instance *}
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notation fcomp (infixl "\<circ>>" 60)
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notation scomp (infixl "\<circ>\<rightarrow>" 60)
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instantiation word :: ("{len0, typerep}") random
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begin
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definition
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  "random_word i = Random.range (max i (2 ^ len_of TYPE('a))) \<circ>\<rightarrow> (\<lambda>k. Pair (
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     let j = word_of_int (Code_Numeral.int_of k) :: 'a word
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     in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))"
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instance ..
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end
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no_notation fcomp (infixl "\<circ>>" 60)
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no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
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subsection {* Type conversions and casting *}
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definition sint :: "'a :: len word => int" where
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  -- {* treats the most-significant-bit as a sign bit *}
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  sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)"
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definition unat :: "'a :: len0 word => nat" where
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  "unat w = nat (uint w)"
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definition uints :: "nat => int set" where
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  -- "the sets of integers representing the words"
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  "uints n = range (bintrunc n)"
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definition sints :: "nat => int set" where
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  "sints n = range (sbintrunc (n - 1))"
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definition unats :: "nat => nat set" where
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  "unats n = {i. i < 2 ^ n}"
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definition norm_sint :: "nat => int => int" where
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  "norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)"
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definition scast :: "'a :: len word => 'b :: len word" where
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  -- "cast a word to a different length"
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  "scast w = word_of_int (sint w)"
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definition ucast :: "'a :: len0 word => 'b :: len0 word" where
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  "ucast w = word_of_int (uint w)"
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instantiation word :: (len0) size
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begin
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definition
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  word_size: "size (w :: 'a word) = len_of TYPE('a)"
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instance ..
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end
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definition source_size :: "('a :: len0 word => 'b) => nat" where
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  -- "whether a cast (or other) function is to a longer or shorter length"
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  "source_size c = (let arb = undefined ; x = c arb in size arb)"  
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definition target_size :: "('a => 'b :: len0 word) => nat" where
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  "target_size c = size (c undefined)"
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definition is_up :: "('a :: len0 word => 'b :: len0 word) => bool" where
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  "is_up c \<longleftrightarrow> source_size c <= target_size c"
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definition is_down :: "('a :: len0 word => 'b :: len0 word) => bool" where
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  "is_down c \<longleftrightarrow> target_size c <= source_size c"
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definition of_bl :: "bool list => 'a :: len0 word" where
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  "of_bl bl = word_of_int (bl_to_bin bl)"
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definition to_bl :: "'a :: len0 word => bool list" where
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  "to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)"
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definition word_reverse :: "'a :: len0 word => 'a word" where
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  "word_reverse w = of_bl (rev (to_bl w))"
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definition word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b" where
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  "word_int_case f w = f (uint w)"
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translations
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  "case x of CONST of_int y => b" == "CONST word_int_case (%y. b) x"
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subsection {* Type-definition locale instantiations *}
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lemma word_size_gt_0 [iff]: "0 < size (w::'a::len word)"
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  by (fact xtr1 [OF word_size len_gt_0])
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lemmas lens_gt_0 = word_size_gt_0 len_gt_0
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lemmas lens_not_0 [iff] = lens_gt_0 [THEN gr_implies_not0]
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lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}"
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  by (simp add: uints_def range_bintrunc)
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lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}"
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  by (simp add: sints_def range_sbintrunc)
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lemma 
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  uint_0:"0 <= uint x" and 
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  uint_lt: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
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  by (auto simp: uint [unfolded atLeastLessThan_iff])
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lemma uint_mod_same:
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  "uint x mod 2 ^ len_of TYPE('a) = uint (x::'a::len0 word)"
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  by (simp add: int_mod_eq uint_lt uint_0)
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lemma td_ext_uint: 
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  "td_ext (uint :: 'a word => int) word_of_int (uints (len_of TYPE('a::len0))) 
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    (%w::int. w mod 2 ^ len_of TYPE('a))"
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  apply (unfold td_ext_def')
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  apply (simp add: uints_num word_of_int_def bintrunc_mod2p)
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  apply (simp add: uint_mod_same uint_0 uint_lt
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                   word.uint_inverse word.Abs_word_inverse int_mod_lem)
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  done
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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 int_word_uint = word_uint.eq_norm
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lemmas td_ext_ubin = td_ext_uint 
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  [unfolded len_gt_0 no_bintr_alt1 [symmetric]]
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interpretation word_ubin:
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  td_ext "uint::'a::len0 word \<Rightarrow> int" 
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         word_of_int 
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         "uints (len_of TYPE('a::len0))"
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         "bintrunc (len_of TYPE('a::len0))"
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  by (rule td_ext_ubin)
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lemma split_word_all:
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  "(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))"
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proof
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  fix x :: "'a word"
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  assume "\<And>x. PROP P (word_of_int x)"
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  hence "PROP P (word_of_int (uint x))" .
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  thus "PROP P x" by simp
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qed
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subsection  "Arithmetic operations"
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definition
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  word_succ :: "'a :: len0 word => 'a word"
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where
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  "word_succ a = word_of_int (uint a + 1)"
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definition
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  word_pred :: "'a :: len0 word => 'a word"
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where
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  "word_pred a = word_of_int (uint a - 1)"
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instantiation word :: (len0) "{number, Divides.div, comm_monoid_mult, comm_ring}"
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begin
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definition
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  word_0_wi: "0 = word_of_int 0"
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definition
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  word_1_wi: "1 = word_of_int 1"
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definition
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  word_add_def: "a + b = word_of_int (uint a + uint b)"
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definition
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  word_sub_wi: "a - b = word_of_int (uint a - uint b)"
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definition
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  word_minus_def: "- a = word_of_int (- uint a)"
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definition
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  word_mult_def: "a * b = word_of_int (uint a * uint b)"
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definition
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  word_div_def: "a div b = word_of_int (uint a div uint b)"
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definition
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  word_mod_def: "a mod b = word_of_int (uint a mod uint b)"
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definition
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  word_number_of_def: "number_of w = word_of_int w"
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lemmas word_arith_wis =
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  word_add_def word_sub_wi word_mult_def word_minus_def 
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  word_succ_def word_pred_def word_0_wi word_1_wi
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lemmas arths = 
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  bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm]
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lemma wi_homs: 
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  shows
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  wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and
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  wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)" and
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  wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and
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  wi_hom_neg: "- word_of_int a = word_of_int (- a)" and
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  wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and
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  wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
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  by (auto simp: word_arith_wis arths)
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lemmas wi_hom_syms = wi_homs [symmetric]
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lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
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lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
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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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instance word :: (len) comm_ring_1
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proof
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  have "0 < len_of TYPE('a)" by (rule len_gt_0)
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  then show "(0::'a word) \<noteq> 1"
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    unfolding word_0_wi word_1_wi
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    by (auto simp add: word_ubin.norm_eq_iff [symmetric] gr0_conv_Suc)
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qed
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lemma word_of_nat: "of_nat n = word_of_int (int n)"
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  by (induct n) (auto simp add : word_of_int_hom_syms)
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lemma word_of_int: "of_int = word_of_int"
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  apply (rule ext)
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  apply (case_tac x rule: int_diff_cases)
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  apply (simp add: word_of_nat wi_hom_sub)
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  done
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instance word :: (len) number_ring
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  by (default, simp add: word_number_of_def word_of_int)
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definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where
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  "a udvd b = (EX n>=0. uint b = n * uint a)"
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subsection "Ordering"
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instantiation word :: (len0) linorder
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begin
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definition
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  word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
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definition
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  word_less_def: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> (y \<Colon> 'a word)"
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instance
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  by default (auto simp: word_less_def word_le_def)
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end
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definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where
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  "a <=s b = (sint a <= sint b)"
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definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where
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  "(x <s y) = (x <=s y & x ~= y)"
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subsection "Bit-wise operations"
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instantiation word :: (len0) bits
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begin
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definition
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  word_and_def: 
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  "(a::'a word) AND b = word_of_int (uint a AND uint b)"
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definition
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  word_or_def:  
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  "(a::'a word) OR b = word_of_int (uint a OR uint b)"
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definition
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  word_xor_def: 
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  "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
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definition
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  word_not_def: 
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  "NOT (a::'a word) = word_of_int (NOT (uint a))"
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definition
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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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   355
  word_msb_def: 
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  "msb a \<longleftrightarrow> bin_sign (sint a) = -1"
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instance ..
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end
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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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   367
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   368
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   369
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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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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   376
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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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   379
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definition mask :: "nat => 'a::len word" where
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  "mask n = (1 << n) - 1"
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   382
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   383
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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   385
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   386
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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   388
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   389
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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   391
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   392
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   393
subsection "Rotation"
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   394
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   395
definition rotater1 :: "'a list => 'a list" where
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  "rotater1 ys = 
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   397
    (case ys of [] => [] | x # xs => last ys # butlast ys)"
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   398
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   399
definition rotater :: "nat => 'a list => 'a list" where
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   400
  "rotater n = rotater1 ^^ n"
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   401
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   402
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where
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   403
  "word_rotr n w = of_bl (rotater n (to_bl w))"
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   404
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   405
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where
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   406
  "word_rotl n w = of_bl (rotate n (to_bl w))"
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   407
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   408
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where
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   409
  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
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   410
                    else word_rotl (nat (- i)) w)"
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   411
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   412
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   413
subsection "Split and cat operations"
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   414
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   415
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where
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   416
  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
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   417
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   418
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where
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  "word_split a = 
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   420
   (case bin_split (len_of TYPE ('c)) (uint a) of 
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   421
     (u, v) => (word_of_int u, word_of_int v))"
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   422
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   423
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where
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   424
  "word_rcat ws = 
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   425
  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
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   426
56e3520b68b2 one unified Word theory
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   427
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where
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   428
  "word_rsplit w = 
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   429
  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
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diff changeset
   430
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   431
definition max_word :: "'a::len word" -- "Largest representable machine integer." where
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   432
  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
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   433
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   434
primrec of_bool :: "bool \<Rightarrow> 'a::len word" where
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   435
  "of_bool False = 0"
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   436
| "of_bool True = 1"
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   437
45805
3c609e8785f2 tidied Word.thy;
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   438
(* FIXME: only provide one theorem name *)
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   439
lemmas of_nth_def = word_set_bits_def
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diff changeset
   440
46010
ebbc2d5cd720 add section headings
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   441
subsection {* Theorems about typedefs *}
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   442
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   443
lemma sint_sbintrunc': 
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   444
  "sint (word_of_int bin :: 'a word) = 
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   445
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
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   446
  unfolding sint_uint 
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   447
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)
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   448
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   449
lemma uint_sint: 
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   450
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
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   451
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
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   452
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   453
lemma bintr_uint': 
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   454
  "n >= size w \<Longrightarrow> bintrunc n (uint w) = uint w"
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   455
  apply (unfold word_size)
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diff changeset
   456
  apply (subst word_ubin.norm_Rep [symmetric]) 
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   457
  apply (simp only: bintrunc_bintrunc_min word_size)
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diff changeset
   458
  apply (simp add: min_max.inf_absorb2)
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diff changeset
   459
  done
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diff changeset
   460
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   461
lemma wi_bintr': 
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   462
  "wb = word_of_int bin \<Longrightarrow> n >= size wb \<Longrightarrow> 
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   463
    word_of_int (bintrunc n bin) = wb"
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diff changeset
   464
  unfolding word_size
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diff changeset
   465
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1)
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diff changeset
   466
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diff changeset
   467
lemmas bintr_uint = bintr_uint' [unfolded word_size]
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diff changeset
   468
lemmas wi_bintr = wi_bintr' [unfolded word_size]
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haftmann
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diff changeset
   469
56e3520b68b2 one unified Word theory
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   470
lemma td_ext_sbin: 
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   471
  "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 
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   472
    (sbintrunc (len_of TYPE('a) - 1))"
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diff changeset
   473
  apply (unfold td_ext_def' sint_uint)
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diff changeset
   474
  apply (simp add : word_ubin.eq_norm)
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haftmann
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diff changeset
   475
  apply (cases "len_of TYPE('a)")
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diff changeset
   476
   apply (auto simp add : sints_def)
56e3520b68b2 one unified Word theory
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diff changeset
   477
  apply (rule sym [THEN trans])
56e3520b68b2 one unified Word theory
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diff changeset
   478
  apply (rule word_ubin.Abs_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   479
  apply (simp only: bintrunc_sbintrunc)
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haftmann
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diff changeset
   480
  apply (drule sym)
56e3520b68b2 one unified Word theory
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diff changeset
   481
  apply simp
56e3520b68b2 one unified Word theory
haftmann
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diff changeset
   482
  done
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haftmann
parents: 36899
diff changeset
   483
56e3520b68b2 one unified Word theory
haftmann
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diff changeset
   484
lemmas td_ext_sint = td_ext_sbin 
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haftmann
parents: 36899
diff changeset
   485
  [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   486
56e3520b68b2 one unified Word theory
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diff changeset
   487
(* We do sint before sbin, before sint is the user version
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haftmann
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diff changeset
   488
   and interpretations do not produce thm duplicates. I.e. 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   489
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   490
   because the latter is the same thm as the former *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   491
interpretation word_sint:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   492
  td_ext "sint ::'a::len word => int" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   493
          word_of_int 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   494
          "sints (len_of TYPE('a::len))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   495
          "%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
   496
               2 ^ (len_of TYPE('a::len) - 1)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   497
  by (rule td_ext_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   498
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   499
interpretation word_sbin:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   500
  td_ext "sint ::'a::len word => int" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   501
          word_of_int 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   502
          "sints (len_of TYPE('a::len))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   503
          "sbintrunc (len_of TYPE('a::len) - 1)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   504
  by (rule td_ext_sbin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   505
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   506
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   507
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   508
lemmas td_sint = word_sint.td
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   509
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   510
lemma word_number_of_alt [code_unfold_post]:
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   511
  "number_of b = word_of_int (number_of b)"
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   512
  by (simp add: number_of_eq word_number_of_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   513
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   514
lemma word_no_wi: "number_of = word_of_int"
44762
8f9d09241a68 tuned proofs;
wenzelm
parents: 42793
diff changeset
   515
  by (auto simp: word_number_of_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   516
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   517
lemma to_bl_def': 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   518
  "(to_bl :: 'a :: len0 word => bool list) =
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   519
    bin_to_bl (len_of TYPE('a)) o uint"
44762
8f9d09241a68 tuned proofs;
wenzelm
parents: 42793
diff changeset
   520
  by (auto simp: to_bl_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   521
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   522
lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w"] for w
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   523
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   524
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   525
  by (fact uints_def [unfolded no_bintr_alt1])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   526
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   527
lemma uint_bintrunc [simp]:
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   528
  "uint (number_of bin :: 'a word) =
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   529
    bintrunc (len_of TYPE ('a :: len0)) (number_of bin)"
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   530
  unfolding word_number_of_alt by (rule word_ubin.eq_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   531
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   532
lemma sint_sbintrunc [simp]:
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   533
  "sint (number_of bin :: 'a word) =
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   534
    sbintrunc (len_of TYPE ('a :: len) - 1) (number_of bin)"
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   535
  unfolding word_number_of_alt by (rule word_sbin.eq_norm)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   536
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   537
lemma unat_bintrunc [simp]:
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   538
  "unat (number_of bin :: 'a :: len0 word) =
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   539
    nat (bintrunc (len_of TYPE('a)) (number_of bin))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   540
  unfolding unat_def nat_number_of_def 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   541
  by (simp only: uint_bintrunc)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   542
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   543
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   544
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   545
  apply (rule word_uint.Rep_eqD)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   546
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   547
    defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   548
    apply (rule word_ubin.norm_Rep)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   549
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   550
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   551
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   552
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   553
  using word_uint.Rep [of x] by (simp add: uints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   554
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   555
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   556
  using word_uint.Rep [of x] by (simp add: uints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   557
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   558
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   559
  using word_sint.Rep [of x] by (simp add: sints_num)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   560
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   561
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   562
  using word_sint.Rep [of x] by (simp add: sints_num)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   563
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   564
lemma sign_uint_Pls [simp]: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   565
  "bin_sign (uint x) = Int.Pls"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   566
  by (simp add: sign_Pls_ge_0 number_of_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   567
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   568
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   569
  by (simp only: diff_less_0_iff_less uint_lt2p)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   570
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   571
lemma uint_m2p_not_non_neg:
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   572
  "\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   573
  by (simp only: not_le uint_m2p_neg)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   574
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   575
lemma lt2p_lem:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   576
  "len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   577
  by (rule xtr8 [OF _ uint_lt2p]) simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   578
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   579
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   580
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   581
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   582
lemma uint_nat: "uint w = int (unat w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   583
  unfolding unat_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   584
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   585
lemma uint_number_of:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   586
  "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
   587
  unfolding word_number_of_alt
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   588
  by (simp only: int_word_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   589
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   590
lemma unat_number_of: 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   591
  "bin_sign b = Int.Pls \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   592
  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
   593
  apply (unfold unat_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   594
  apply (clarsimp simp only: uint_number_of)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   595
  apply (rule nat_mod_distrib [THEN trans])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   596
    apply (erule sign_Pls_ge_0 [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   597
   apply (simp_all add: nat_power_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   598
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   599
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   600
lemma sint_number_of: "sint (number_of b :: 'a :: len word) = (number_of b + 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   601
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   602
    2 ^ (len_of TYPE('a) - 1)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   603
  unfolding word_number_of_alt by (rule int_word_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   604
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
   605
lemma word_of_int_0 [simp]: "word_of_int 0 = 0"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   606
  unfolding word_0_wi ..
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   607
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
   608
lemma word_of_int_1 [simp]: "word_of_int 1 = 1"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   609
  unfolding word_1_wi ..
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
   610
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   611
lemma word_of_int_bin [simp] : 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   612
  "(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)"
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   613
  unfolding word_number_of_alt ..
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   614
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   615
lemma word_int_case_wi: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   616
  "word_int_case f (word_of_int i :: 'b word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   617
    f (i mod 2 ^ len_of TYPE('b::len0))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   618
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   619
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   620
lemma word_int_split: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   621
  "P (word_int_case f x) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   622
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   623
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   624
  unfolding word_int_case_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   625
  by (auto simp: word_uint.eq_norm int_mod_eq')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   626
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   627
lemma word_int_split_asm: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   628
  "P (word_int_case f x) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   629
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   630
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   631
  unfolding word_int_case_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   632
  by (auto simp: word_uint.eq_norm int_mod_eq')
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   633
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   634
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   635
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   636
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   637
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   638
  unfolding word_size by (rule uint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   639
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   640
lemma sint_range_size:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   641
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   642
  unfolding word_size by (rule sint_range')
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   643
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   644
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   645
  unfolding word_size by (rule less_le_trans [OF sint_lt])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   646
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   647
lemma sint_below_size:
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   648
  "x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   649
  unfolding word_size by (rule order_trans [OF _ sint_ge])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   650
46010
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
   651
subsection {* Testing bits *}
ebbc2d5cd720 add section headings
huffman
parents: 46009
diff changeset
   652
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   653
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
   654
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   655
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   656
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   657
  apply (unfold word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   658
  apply (subst word_ubin.norm_Rep [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   659
  apply (simp only: nth_bintr word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   660
  apply fast
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   661
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   662
46021
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   663
lemma word_eq_iff:
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   664
  fixes x y :: "'a::len0 word"
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   665
  shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)"
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   666
  unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric]
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   667
  by (metis test_bit_size [unfolded word_size])
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   668
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   669
lemma word_eqI:
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   670
  fixes u :: "'a::len0 word"
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   671
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
46021
272c63f83398 add lemma word_eq_iff
huffman
parents: 46020
diff changeset
   672
  by (simp add: word_size word_eq_iff)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   673
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   674
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   675
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   676
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   677
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   678
  unfolding word_test_bit_def word_size
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   679
  by (simp add: nth_bintr [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   680
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   681
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   682
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   683
lemma bin_nth_uint_imp': "bin_nth (uint w) n --> n < size w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   684
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   685
  apply (rule impI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   686
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   687
  apply (subst word_ubin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   688
  apply assumption
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   689
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   690
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   691
lemma bin_nth_sint': 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   692
  "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
   693
  apply (rule impI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   694
  apply (subst word_sbin.norm_Rep [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   695
  apply (simp add : nth_sbintr word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   696
  apply auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   697
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   698
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   699
lemmas bin_nth_uint_imp = bin_nth_uint_imp' [rule_format, unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   700
lemmas bin_nth_sint = bin_nth_sint' [rule_format, unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   701
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   702
(* type definitions theorem for in terms of equivalent bool list *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   703
lemma td_bl: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   704
  "type_definition (to_bl :: 'a::len0 word => bool list) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   705
                   of_bl  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   706
                   {bl. length bl = len_of TYPE('a)}"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   707
  apply (unfold type_definition_def of_bl_def to_bl_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   708
  apply (simp add: word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   709
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   710
  apply (drule sym)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   711
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   712
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   713
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   714
interpretation word_bl:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   715
  type_definition "to_bl :: 'a::len0 word => bool list"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   716
                  of_bl  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   717
                  "{bl. length bl = len_of TYPE('a::len0)}"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   718
  by (rule td_bl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   719
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
   720
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
45538
1fffa81b9b83 eliminated slightly odd Rep' with dynamically-scoped [simplified];
wenzelm
parents: 45529
diff changeset
   721
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   722
lemma word_size_bl: "size w = size (to_bl w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   723
  unfolding word_size by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   724
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   725
lemma to_bl_use_of_bl:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   726
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
   727
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   728
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   729
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   730
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   731
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   732
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   733
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   734
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   735
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   736
  by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   737
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   738
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   739
  by simp
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   740
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   741
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   742
  unfolding word_bl_Rep' by (rule len_gt_0)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   743
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   744
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   745
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   746
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   747
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   748
  by (fact length_bl_gt_0 [THEN gr_implies_not0])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   749
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   750
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   751
  apply (unfold to_bl_def sint_uint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   752
  apply (rule trans [OF _ bl_sbin_sign])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   753
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   754
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   755
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   756
lemma of_bl_drop': 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   757
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   758
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   759
  apply (unfold of_bl_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   760
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   761
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   762
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   763
lemma of_bl_no: "of_bl bl = number_of (bl_to_bin bl)"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   764
  by (fact of_bl_def [folded word_number_of_def])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   765
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   766
lemma test_bit_of_bl:  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   767
  "(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
   768
  apply (unfold of_bl_def word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   769
  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
   770
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   771
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   772
lemma no_of_bl: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   773
  "(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
   774
  unfolding word_size of_bl_no by (simp add : word_number_of_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   775
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   776
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   777
  unfolding word_size to_bl_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   778
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   779
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   780
  unfolding uint_bl by (simp add : word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   781
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   782
lemma to_bl_of_bin: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   783
  "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
   784
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   785
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   786
lemma to_bl_no_bin [simp]:
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   787
  "to_bl (number_of bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   788
  by (fact to_bl_of_bin [folded word_number_of_def])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   789
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   790
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   791
  unfolding uint_bl by (simp add : word_size)
46011
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   792
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   793
lemma uint_bl_bin:
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   794
  fixes x :: "'a::len0 word"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   795
  shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   796
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   797
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   798
(* FIXME: the next two lemmas should be unnecessary, because the lhs
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   799
terms should never occur in practice *)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   800
lemma num_AB_u [simp]: "number_of (uint x) = x"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   801
  unfolding word_number_of_def by (rule word_uint.Rep_inverse)
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   802
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   803
lemma num_AB_s [simp]: "number_of (sint x) = x"
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
   804
  unfolding word_number_of_def by (rule word_sint.Rep_inverse)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   805
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   806
(* naturals *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   807
lemma uints_unats: "uints n = int ` unats n"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   808
  apply (unfold unats_def uints_num)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   809
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   810
  apply (rule_tac image_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   811
  apply (erule_tac nat_0_le [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   812
  apply auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   813
  apply (erule_tac nat_less_iff [THEN iffD2])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   814
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   815
  apply (auto simp add : nat_power_eq int_power)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   816
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   817
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   818
lemma unats_uints: "unats n = nat ` uints n"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   819
  by (auto simp add : uints_unats image_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   820
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   821
lemmas bintr_num = word_ubin.norm_eq_iff [symmetric, folded word_number_of_def]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   822
lemmas sbintr_num = word_sbin.norm_eq_iff [symmetric, folded word_number_of_def]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   823
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   824
lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   825
lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   826
    
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   827
(* 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
   828
  may want these in reverse, but loop as simp rules, so use following *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   829
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   830
lemma num_of_bintr':
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   831
  "bintrunc (len_of TYPE('a :: len0)) a = b \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   832
    number_of a = (number_of b :: 'a word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   833
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   834
  apply (rule_tac num_of_bintr [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   835
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   836
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   837
lemma num_of_sbintr':
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   838
  "sbintrunc (len_of TYPE('a :: len) - 1) a = b \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   839
    number_of a = (number_of b :: 'a word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   840
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   841
  apply (rule_tac num_of_sbintr [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   842
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   843
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   844
lemmas num_abs_bintr = sym [THEN trans, OF num_of_bintr word_number_of_def]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   845
lemmas num_abs_sbintr = sym [THEN trans, OF num_of_sbintr word_number_of_def]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   846
  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   847
(** 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
   848
  thus in "cast w = w, the type means cast to length of w! **)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   849
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   850
lemma ucast_id: "ucast w = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   851
  unfolding ucast_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   852
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   853
lemma scast_id: "scast w = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   854
  unfolding scast_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   855
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   856
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   857
  unfolding ucast_def of_bl_def uint_bl
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   858
  by (auto simp add : word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   859
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   860
lemma nth_ucast: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   861
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   862
  apply (unfold ucast_def test_bit_bin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   863
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   864
  apply (fast elim!: bin_nth_uint_imp)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   865
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   866
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   867
(* for literal u(s)cast *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   868
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   869
lemma ucast_bintr [simp]:
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   870
  "ucast (number_of w ::'a::len0 word) = 
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   871
   word_of_int (bintrunc (len_of TYPE('a)) (number_of w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   872
  unfolding ucast_def by simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   873
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   874
lemma scast_sbintr [simp]:
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   875
  "scast (number_of w ::'a::len word) = 
46001
0b562d564d5f redefine some binary operations on integers work on abstract numerals instead of Int.Pls and Int.Min
huffman
parents: 46000
diff changeset
   876
   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (number_of w))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   877
  unfolding scast_def by simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   878
46011
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   879
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   880
  unfolding source_size_def word_size Let_def ..
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   881
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   882
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   883
  unfolding target_size_def word_size Let_def ..
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   884
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   885
lemma is_down:
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   886
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   887
  shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   888
  unfolding is_down_def source_size target_size ..
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   889
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   890
lemma is_up:
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   891
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   892
  shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)"
96a5f44c22da replace 'lemmas' with explicit 'lemma'
huffman
parents: 46010
diff changeset
   893
  unfolding is_up_def source_size target_size ..
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   894
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
   895
lemmas is_up_down = trans [OF is_up is_down [symmetric]]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   896
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   897
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   898
  apply (unfold is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   899
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   900
  apply (rule ext)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   901
  apply (unfold ucast_def scast_def uint_sint)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   902
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   903
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   904
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   905
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   906
lemma word_rev_tf:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   907
  "to_bl (of_bl bl::'a::len0 word) =
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   908
    rev (takefill False (len_of TYPE('a)) (rev bl))"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   909
  unfolding of_bl_def uint_bl
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   910
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   911
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   912
lemma word_rep_drop:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   913
  "to_bl (of_bl bl::'a::len0 word) =
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   914
    replicate (len_of TYPE('a) - length bl) False @
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   915
    drop (length bl - len_of TYPE('a)) bl"
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   916
  by (simp add: word_rev_tf takefill_alt rev_take)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   917
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   918
lemma to_bl_ucast: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   919
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   920
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   921
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   922
  apply (unfold ucast_bl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   923
  apply (rule trans)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   924
   apply (rule word_rep_drop)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   925
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   926
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   927
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   928
lemma ucast_up_app [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   929
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   930
    to_bl (uc w) = replicate n False @ (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   931
  by (auto simp add : source_size target_size to_bl_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   932
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   933
lemma ucast_down_drop [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   934
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   935
    to_bl (uc w) = drop n (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   936
  by (auto simp add : source_size target_size to_bl_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   937
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   938
lemma scast_down_drop [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   939
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   940
    to_bl (sc w) = drop n (to_bl w)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   941
  apply (subgoal_tac "sc = ucast")
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   942
   apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   943
   apply simp
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   944
   apply (erule ucast_down_drop)
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   945
  apply (rule down_cast_same [symmetric])
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   946
  apply (simp add : source_size target_size is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   947
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   948
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   949
lemma sint_up_scast [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   950
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   951
  apply (unfold is_up)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   952
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   953
  apply (simp add: scast_def word_sbin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   954
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   955
    prefer 3
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   956
    apply (rule word_sbin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   957
   apply (rule sbintrunc_sbintrunc_l)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   958
   defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   959
   apply (subst word_sbin.norm_Rep)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   960
   apply (rule refl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   961
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   962
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   963
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   964
lemma uint_up_ucast [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   965
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   966
  apply (unfold is_up)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   967
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   968
  apply (rule bin_eqI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   969
  apply (fold word_test_bit_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   970
  apply (auto simp add: nth_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   971
  apply (auto simp add: test_bit_bin)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   972
  done
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   973
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   974
lemma ucast_up_ucast [OF refl]:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   975
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   976
  apply (simp (no_asm) add: ucast_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   977
  apply (clarsimp simp add: uint_up_ucast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   978
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   979
    
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   980
lemma scast_up_scast [OF refl]:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   981
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   982
  apply (simp (no_asm) add: scast_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   983
  apply (clarsimp simp add: sint_up_scast)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   984
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   985
    
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
   986
lemma ucast_of_bl_up [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   987
  "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
   988
  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
   989
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   990
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   991
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   992
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   993
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   994
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   995
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   996
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   997
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
   998
lemma up_ucast_surj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
   999
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1000
   surj (ucast :: 'a word => 'b word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1001
  by (rule surjI, erule ucast_up_ucast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1002
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1003
lemma up_scast_surj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1004
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1005
   surj (scast :: 'a word => 'b word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1006
  by (rule surjI, erule scast_up_scast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1007
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1008
lemma down_scast_inj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1009
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1010
   inj_on (ucast :: 'a word => 'b word) A"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1011
  by (rule inj_on_inverseI, erule scast_down_scast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1012
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1013
lemma down_ucast_inj:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1014
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1015
   inj_on (ucast :: 'a word => 'b word) A"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1016
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1017
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1018
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1019
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1020
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1021
lemma ucast_down_no [OF refl]:
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1022
  "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
  1023
  apply (unfold word_number_of_def is_down)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1024
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1025
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1026
  apply (erule bintrunc_bintrunc_ge)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1027
  done
45811
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1028
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1029
lemma ucast_down_bl [OF refl]:
f506015ca2dc replace many uses of 'lemmas' with 'lemma';
huffman
parents: 45810
diff changeset
  1030
  "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
  1031
  unfolding of_bl_no by clarify (erule ucast_down_no)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1032
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1033
lemmas slice_def' = slice_def [unfolded word_size]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1034
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1035
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1036
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
  1037
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1038
text {* Executable equality *}
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1039
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38527
diff changeset
  1040
instantiation word :: (len0) equal
24333
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1041
begin
e77ea0ea7f2c * HOL-Word:
kleing
parents:
diff changeset
  1042
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38527
diff changeset
  1043
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
  1044
  "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1045
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1046
instance proof
38857
97775f3e8722 renamed class/constant eq to equal; tuned some instantiations
haftmann
parents: 38527
diff changeset
  1047
qed (simp add: equal equal_word_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1048
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1049
end
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1050
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1051
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1052
subsection {* Word Arithmetic *}
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1053
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1054
lemma word_less_alt: "(a < b) = (uint a < uint b)"
46012
8a070c62b548 simplify proof
huffman
parents: 46011
diff changeset
  1055
  unfolding word_less_def word_le_def by (simp add: less_le)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1056
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1057
lemma signed_linorder: "class.linorder word_sle word_sless"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1058
proof
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1059
qed (unfold word_sle_def word_sless_def, auto)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1060
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1061
interpretation signed: linorder "word_sle" "word_sless"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1062
  by (rule signed_linorder)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1063
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1064
lemma udvdI: 
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1065
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1066
  by (auto simp: udvd_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1067
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1068
lemmas word_div_no [simp] = word_div_def [of "number_of a" "number_of b"] for a b
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1069
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1070
lemmas word_mod_no [simp] = word_mod_def [of "number_of a" "number_of b"] for a b
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1071
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1072
lemmas word_less_no [simp] = word_less_def [of "number_of a" "number_of b"] for a b
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1073
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1074
lemmas word_le_no [simp] = word_le_def [of "number_of a" "number_of b"] for a b
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1075
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1076
lemmas word_sless_no [simp] = word_sless_def [of "number_of a" "number_of b"] for a b
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1077
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1078
lemmas word_sle_no [simp] = word_sle_def [of "number_of a" "number_of b"] for a b
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1079
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1080
(* following two are available in class number_ring, 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1081
  but convenient to have them here here;
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1082
  note - the number_ring versions, numeral_0_eq_0 and numeral_1_eq_1
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1083
  are in the default simpset, so to use the automatic simplifications for
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1084
  (eg) sint (number_of bin) on sint 1, must do
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1085
  (simp add: word_1_no del: numeral_1_eq_1) 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1086
  *)
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1087
lemma word_0_wi_Pls: "0 = word_of_int Int.Pls"
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1088
  by (simp only: Pls_def word_0_wi)
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1089
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1090
lemma word_0_no: "(0::'a::len0 word) = Numeral0"
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1091
  by (simp add: word_number_of_alt)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1092
46020
0a29b51f0b0d restate lemma word_1_no in terms of Numeral1
huffman
parents: 46013
diff changeset
  1093
lemma word_1_no: "(1::'a::len0 word) = Numeral1"
0a29b51f0b0d restate lemma word_1_no in terms of Numeral1
huffman
parents: 46013
diff changeset
  1094
  by (simp add: word_number_of_alt)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1095
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1096
lemma word_m1_wi: "-1 = word_of_int -1" 
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1097
  by (rule word_number_of_alt)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1098
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1099
lemma word_m1_wi_Min: "-1 = word_of_int Int.Min"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1100
  by (simp add: word_m1_wi number_of_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1101
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1102
lemma word_0_bl [simp]: "of_bl [] = 0" 
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1103
  unfolding of_bl_def by (simp add: Pls_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1104
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1105
lemma word_1_bl: "of_bl [True] = 1" 
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1106
  unfolding of_bl_def
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1107
  by (simp add: bl_to_bin_def Bit_def Pls_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1108
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1109
lemma uint_eq_0 [simp] : "(uint 0 = 0)" 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1110
  unfolding word_0_wi
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1111
  by (simp add: word_ubin.eq_norm Pls_def [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1112
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1113
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1114
  by (simp add: of_bl_def bl_to_bin_rep_False Pls_def)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1115
45805
3c609e8785f2 tidied Word.thy;
huffman
parents: 45804
diff changeset
  1116
lemma to_bl_0 [simp]:
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1117
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1118
  unfolding uint_bl
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1119
  by (simp add : word_size bin_to_bl_Pls Pls_def [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1120
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1121
lemma uint_0_iff: "(uint x = 0) = (x = 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1122
  by (auto intro!: word_uint.Rep_eqD)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1123
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1124
lemma unat_0_iff: "(unat x = 0) = (x = 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1125
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1126
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1127
lemma unat_0 [simp]: "unat 0 = 0"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1128
  unfolding unat_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1129
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1130
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1131
  apply (unfold word_size)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1132
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1133
    defer
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1134
    apply (rule word_uint.Rep_inverse)+
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1135
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1136
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1137
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1138
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1139
lemmas size_0_same = size_0_same' [unfolded word_size]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1140
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1141
lemmas unat_eq_0 = unat_0_iff
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1142
lemmas unat_eq_zero = unat_0_iff
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1143
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1144
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1145
by (auto simp: unat_0_iff [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1146
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1147
lemma ucast_0 [simp]: "ucast 0 = 0"
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1148
  unfolding ucast_def by simp
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1149
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1150
lemma sint_0 [simp]: "sint 0 = 0"
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1151
  unfolding sint_uint by simp
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1152
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1153
lemma scast_0 [simp]: "scast 0 = 0"
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1154
  unfolding scast_def by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1155
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1156
lemma sint_n1 [simp] : "sint -1 = -1"
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1157
  unfolding word_m1_wi by (simp add: word_sbin.eq_norm)
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1158
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1159
lemma scast_n1 [simp]: "scast -1 = -1"
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1160
  unfolding scast_def by simp
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1161
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1162
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1163
  unfolding word_1_wi
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1164
  by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1)
45958
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1165
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1166
lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1167
  unfolding unat_def by simp
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1168
c28235388c43 simplify some proofs
huffman
parents: 45957
diff changeset
  1169
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1170
  unfolding ucast_def by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1171
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1172
(* now, to get the weaker results analogous to word_div/mod_def *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1173
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1174
lemmas word_arith_alts = 
46000
871bdab23f5c remove some uses of Int.succ and Int.pred
huffman
parents: 45998
diff changeset
  1175
  word_sub_wi
871bdab23f5c remove some uses of Int.succ and Int.pred
huffman
parents: 45998
diff changeset
  1176
  word_arith_wis (* FIXME: duplicate *)
871bdab23f5c remove some uses of Int.succ and Int.pred
huffman
parents: 45998
diff changeset
  1177
871bdab23f5c remove some uses of Int.succ and Int.pred
huffman
parents: 45998
diff changeset
  1178
lemmas word_succ_alt = word_succ_def (* FIXME: duplicate *)
871bdab23f5c remove some uses of Int.succ and Int.pred
huffman
parents: 45998
diff changeset
  1179
lemmas word_pred_alt = word_pred_def (* FIXME: duplicate *)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1180
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1181
subsection  "Transferring goals from words to ints"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1182
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1183
lemma word_ths:  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1184
  shows
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1185
  word_succ_p1:   "word_succ a = a + 1" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1186
  word_pred_m1:   "word_pred a = a - 1" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1187
  word_pred_succ: "word_pred (word_succ a) = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1188
  word_succ_pred: "word_succ (word_pred a) = a" and
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1189
  word_mult_succ: "word_succ a * b = b + a * b"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1190
  by (rule word_uint.Abs_cases [of b],
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1191
      rule word_uint.Abs_cases [of a],
46000
871bdab23f5c remove some uses of Int.succ and Int.pred
huffman
parents: 45998
diff changeset
  1192
      simp add: add_commute mult_commute 
46009
5cb7ef5bfef2 remove duplicate lemma lists
huffman
parents: 46001
diff changeset
  1193
                ring_distribs word_of_int_homs
45995
b16070689726 declare word_of_int_{0,1} [simp], for consistency with word_of_int_bin
huffman
parents: 45958
diff changeset
  1194
           del: word_of_int_0 word_of_int_1)+
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1195
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1196
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1197
  by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1198
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1199
lemmas uint_word_ariths = 
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1200
  word_arith_alts [THEN trans [OF uint_cong int_word_uint]]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1201
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1202
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1203
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1204
(* similar expressions for sint (arith operations) *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1205
lemmas sint_word_ariths = uint_word_arith_bintrs
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1206
  [THEN uint_sint [symmetric, THEN trans],
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1207
  unfolded uint_sint bintr_arith1s bintr_ariths 
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1208
    len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1209
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1210
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1211
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1212
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1213
lemma word_pred_0_n1: "word_pred 0 = word_of_int -1"
45550
73a4f31d41c4 Word.thy: reduce usage of numeral-representation-dependent thms like number_of_is_id in proofs
huffman
parents: 45549
diff changeset
  1214
  unfolding word_pred_def uint_eq_0 pred_def by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1215
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1216
lemma word_pred_0_Min: "word_pred 0 = word_of_int Int.Min"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1217
  by (simp add: word_pred_0_n1 number_of_eq)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1218
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1219
lemma word_m1_Min: "- 1 = word_of_int Int.Min"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1220
  unfolding Min_def by (simp only: word_of_int_hom_syms)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1221
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1222
lemma succ_pred_no [simp]:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1223
  "word_succ (number_of bin) = number_of (Int.succ bin) & 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1224
    word_pred (number_of bin) = number_of (Int.pred bin)"
46000
871bdab23f5c remove some uses of Int.succ and Int.pred
huffman
parents: 45998
diff changeset
  1225
  unfolding word_number_of_def Int.succ_def Int.pred_def
46009
5cb7ef5bfef2 remove duplicate lemma lists
huffman
parents: 46001
diff changeset
  1226
  by (simp add: word_of_int_homs)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1227
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1228
lemma word_sp_01 [simp] : 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1229
  "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0"
46020
0a29b51f0b0d restate lemma word_1_no in terms of Numeral1
huffman
parents: 46013
diff changeset
  1230
  unfolding word_0_no word_1_no by simp
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1231
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1232
(* alternative approach to lifting arithmetic equalities *)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1233
lemma word_of_int_Ex:
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1234
  "\<exists>y. x = word_of_int y"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1235
  by (rule_tac x="uint x" in exI) simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1236
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1237
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1238
subsection "Order on fixed-length words"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1239
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1240
lemma word_zero_le [simp] :
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1241
  "0 <= (y :: 'a :: len0 word)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1242
  unfolding word_le_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1243
  
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1244
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *)
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1245
  unfolding word_le_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1246
  by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1247
45816
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1248
lemma word_n1_ge [simp]: "y \<le> (-1::'a::len0 word)"
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1249
  unfolding word_le_def
6a04efd99f25 replace more uses of 'lemmas' with explicit 'lemma';
huffman
parents: 45811
diff changeset
  1250
  by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1251
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1252
lemmas word_not_simps [simp] = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1253
  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
  1254
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1255
lemma word_gt_0: "0 < y = (0 ~= (y :: 'a :: len0 word))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1256
  unfolding word_less_def by auto
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1257
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1258
lemmas word_gt_0_no [simp] = word_gt_0 [of "number_of y"] for y
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1259
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1260
lemma word_sless_alt: "(a <s b) = (sint a < sint b)"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1261
  unfolding word_sle_def word_sless_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1262
  by (auto simp add: less_le)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1263
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1264
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1265
  unfolding unat_def word_le_def
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1266
  by (rule nat_le_eq_zle [symmetric]) simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1267
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1268
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1269
  unfolding unat_def word_less_alt
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1270
  by (rule nat_less_eq_zless [symmetric]) simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1271
  
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1272
lemma wi_less: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1273
  "(word_of_int n < (word_of_int m :: 'a :: len0 word)) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1274
    (n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1275
  unfolding word_less_alt by (simp add: word_uint.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1276
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1277
lemma wi_le: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1278
  "(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1279
    (n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1280
  unfolding word_le_def by (simp add: word_uint.eq_norm)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1281
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1282
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1283
  apply (unfold udvd_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1284
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1285
   apply (simp add: unat_def nat_mult_distrib)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1286
  apply (simp add: uint_nat int_mult)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1287
  apply (rule exI)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1288
  apply safe
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1289
   prefer 2
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1290
   apply (erule notE)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1291
   apply (rule refl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1292
  apply force
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1293
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1294
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1295
lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1296
  unfolding dvd_def udvd_nat_alt by force
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1297
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1298
lemmas unat_mono = word_less_nat_alt [THEN iffD1]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1299
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1300
lemma unat_minus_one: "x ~= 0 \<Longrightarrow> unat (x - 1) = unat x - 1"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1301
  apply (unfold unat_def)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1302
  apply (simp only: int_word_uint word_arith_alts rdmods)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1303
  apply (subgoal_tac "uint x >= 1")
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1304
   prefer 2
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1305
   apply (drule contrapos_nn)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1306
    apply (erule word_uint.Rep_inverse' [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1307
   apply (insert uint_ge_0 [of x])[1]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1308
   apply arith
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1309
  apply (rule box_equals)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1310
    apply (rule nat_diff_distrib)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1311
     prefer 2
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1312
     apply assumption
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1313
    apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1314
   apply (subst mod_pos_pos_trivial)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1315
     apply arith
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1316
    apply (insert uint_lt2p [of x])[1]
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1317
    apply arith
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1318
   apply (rule refl)
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1319
  apply simp
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1320
  done
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1321
    
40827
abbc05c20e24 code preprocessor setup for numerals on word type;
haftmann
parents: 39910
diff changeset
  1322
lemma measure_unat: "p ~= 0 \<Longrightarrow> unat (p - 1) < unat p"
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1323
  by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric])
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1324
  
45604
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1325
lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
29cf40fe8daf eliminated obsolete "standard";
wenzelm
parents: 45550
diff changeset
  1326
lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
37660
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1327
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1328
lemma uint_sub_lt2p [simp]: 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1329
  "uint (x :: 'a :: len0 word) - uint (y :: 'b :: len0 word) < 
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1330
    2 ^ len_of TYPE('a)"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1331
  using uint_ge_0 [of y] uint_lt2p [of x] by arith
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1332
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1333
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1334
subsection "Conditions for the addition (etc) of two words to overflow"
56e3520b68b2 one unified Word theory
haftmann
parents: 36899
diff changeset
  1335
56e3520b68b2 one unified Word theory
haftmann
parents: