src/HOL/Word/Word.thy
author wenzelm
Sat Dec 14 17:28:05 2013 +0100 (2013-12-14)
changeset 54742 7a86358a3c0b
parent 54489 03ff4d1e6784
child 54743 b9ae4a2f615b
permissions -rw-r--r--
proper context for basic Simplifier operations: rewrite_rule, rewrite_goals_rule, rewrite_goals_tac etc.;
clarified tool context in some boundary cases;
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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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  "~~/src/HOL/Library/Boolean_Algebra"
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  Bit_Bit
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  Bool_List_Representation
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  Misc_Typedef
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  Word_Miscellaneous
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begin
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text {* see @{text "Examples/WordExamples.thy"} for examples *}
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subsection {* Type definition *}
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typedef 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}"
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  morphisms uint Abs_word by auto
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lemma uint_nonnegative:
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  "0 \<le> uint w"
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  using word.uint [of w] by simp
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lemma uint_bounded:
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  fixes w :: "'a::len0 word"
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  shows "uint w < 2 ^ len_of TYPE('a)"
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  using word.uint [of w] by simp
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lemma uint_idem:
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  fixes w :: "'a::len0 word"
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  shows "uint w mod 2 ^ len_of TYPE('a) = uint w"
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  using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial)
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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 k = Abs_word (k mod 2 ^ len_of TYPE('a))" 
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lemma uint_word_of_int:
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  "uint (word_of_int k :: 'a::len0 word) = k mod 2 ^ len_of TYPE('a)"
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  by (auto simp add: word_of_int_def intro: Abs_word_inverse)
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lemma word_of_int_uint:
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  "word_of_int (uint w) = w"
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  by (simp add: word_of_int_def uint_idem uint_inverse)
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lemma word_uint_eq_iff:
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  "a = b \<longleftrightarrow> uint a = uint b"
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  by (simp add: uint_inject)
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lemma word_uint_eqI:
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  "uint a = uint b \<Longrightarrow> a = b"
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  by (simp add: word_uint_eq_iff)
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subsection {* Basic code generation setup *}
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definition Word :: "int \<Rightarrow> 'a::len0 word"
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where
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  [code_post]: "Word = word_of_int"
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lemma [code abstype]:
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  "Word (uint w) = w"
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  by (simp add: Word_def word_of_int_uint)
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declare uint_word_of_int [code abstract]
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instantiation word :: (len0) equal
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begin
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definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where
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  "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"
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instance proof
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qed (simp add: equal equal_word_def word_uint_eq_iff)
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end
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notation fcomp (infixl "\<circ>>" 60)
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notation scomp (infixl "\<circ>\<rightarrow>" 60)
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instantiation word :: ("{len0, typerep}") random
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begin
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definition
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  "random_word i = Random.range i \<circ>\<rightarrow> (\<lambda>k. Pair (
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     let j = word_of_int (int_of_integer (integer_of_natural k)) :: 'a word
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     in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))"
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instance ..
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end
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no_notation fcomp (infixl "\<circ>>" 60)
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no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
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subsection {* 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 XCONST of_int y => b" == "CONST word_int_case (%y. b) x"
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  "case x of (XCONST of_int :: 'a) 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 {* Correspondence relation for theorem transfer *}
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definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool"
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  where "cr_word \<equiv> (\<lambda>x y. word_of_int x = y)"
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lemma Quotient_word:
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  "Quotient (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y)
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    word_of_int uint (cr_word :: _ \<Rightarrow> 'a::len0 word \<Rightarrow> bool)"
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  unfolding Quotient_alt_def cr_word_def
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  by (simp add: word_ubin.norm_eq_iff)
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lemma reflp_word:
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  "reflp (\<lambda>x y. bintrunc (len_of TYPE('a::len0)) x = bintrunc (len_of TYPE('a)) y)"
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  by (simp add: reflp_def)
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setup_lifting(no_code) Quotient_word reflp_word
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text {* TODO: The next lemma could be generated automatically. *}
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lemma uint_transfer [transfer_rule]:
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  "(fun_rel pcr_word op =) (bintrunc (len_of TYPE('a)))
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    (uint :: 'a::len0 word \<Rightarrow> int)"
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  unfolding fun_rel_def word.pcr_cr_eq cr_word_def by (simp add: word_ubin.eq_norm)
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subsection  "Arithmetic operations"
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lift_definition word_succ :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x + 1"
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  by (metis bintr_ariths(6))
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lift_definition word_pred :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x - 1"
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  by (metis bintr_ariths(7))
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instantiation word :: (len0) "{neg_numeral, Divides.div, comm_monoid_mult, comm_ring}"
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begin
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lift_definition zero_word :: "'a word" is "0" .
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lift_definition one_word :: "'a word" is "1" .
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lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op +"
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  by (metis bintr_ariths(2))
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lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op -"
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  by (metis bintr_ariths(3))
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lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus
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  by (metis bintr_ariths(5))
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lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op *"
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  by (metis bintr_ariths(4))
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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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instance
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  by default (transfer, simp add: algebra_simps)+
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end
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text {* Legacy theorems: *}
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lemma word_arith_wis [code]: shows
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  word_add_def: "a + b = word_of_int (uint a + uint b)" and
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  word_sub_wi: "a - b = word_of_int (uint a - uint b)" and
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  word_mult_def: "a * b = word_of_int (uint a * uint b)" and
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  word_minus_def: "- a = word_of_int (- uint a)" and
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  word_succ_alt: "word_succ a = word_of_int (uint a + 1)" and
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  word_pred_alt: "word_pred a = word_of_int (uint a - 1)" and
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  word_0_wi: "0 = word_of_int 0" and
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  word_1_wi: "1 = word_of_int 1"
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  unfolding plus_word_def minus_word_def times_word_def uminus_word_def
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  unfolding word_succ_def word_pred_def zero_word_def one_word_def
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  by simp_all
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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 (transfer, simp)+
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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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instance word :: (len) comm_ring_1
huffman@45810
   329
proof
huffman@45810
   330
  have "0 < len_of TYPE('a)" by (rule len_gt_0)
huffman@45810
   331
  then show "(0::'a word) \<noteq> 1"
huffman@47372
   332
    by - (transfer, auto simp add: gr0_conv_Suc)
huffman@45810
   333
qed
huffman@45545
   334
huffman@45545
   335
lemma word_of_nat: "of_nat n = word_of_int (int n)"
huffman@45545
   336
  by (induct n) (auto simp add : word_of_int_hom_syms)
huffman@45545
   337
huffman@45545
   338
lemma word_of_int: "of_int = word_of_int"
huffman@45545
   339
  apply (rule ext)
huffman@45545
   340
  apply (case_tac x rule: int_diff_cases)
huffman@46013
   341
  apply (simp add: word_of_nat wi_hom_sub)
huffman@45545
   342
  done
huffman@45545
   343
haftmann@37660
   344
definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where
haftmann@40827
   345
  "a udvd b = (EX n>=0. uint b = n * uint a)"
haftmann@37660
   346
huffman@45547
   347
huffman@45547
   348
subsection "Ordering"
huffman@45547
   349
huffman@45547
   350
instantiation word :: (len0) linorder
huffman@45547
   351
begin
huffman@45547
   352
haftmann@37660
   353
definition
haftmann@37660
   354
  word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
haftmann@37660
   355
haftmann@37660
   356
definition
huffman@47108
   357
  word_less_def: "a < b \<longleftrightarrow> uint a < uint b"
haftmann@37660
   358
huffman@45547
   359
instance
huffman@45547
   360
  by default (auto simp: word_less_def word_le_def)
huffman@45547
   361
huffman@45547
   362
end
huffman@45547
   363
haftmann@37660
   364
definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where
haftmann@40827
   365
  "a <=s b = (sint a <= sint b)"
haftmann@37660
   366
haftmann@37660
   367
definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where
haftmann@40827
   368
  "(x <s y) = (x <=s y & x ~= y)"
haftmann@37660
   369
haftmann@37660
   370
haftmann@37660
   371
subsection "Bit-wise operations"
haftmann@37660
   372
haftmann@37660
   373
instantiation word :: (len0) bits
haftmann@37660
   374
begin
haftmann@37660
   375
huffman@47387
   376
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT
huffman@47374
   377
  by (metis bin_trunc_not)
huffman@47374
   378
huffman@47387
   379
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND
huffman@47374
   380
  by (metis bin_trunc_and)
huffman@47374
   381
huffman@47387
   382
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR
huffman@47374
   383
  by (metis bin_trunc_or)
huffman@47374
   384
huffman@47387
   385
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR
huffman@47374
   386
  by (metis bin_trunc_xor)
haftmann@37660
   387
haftmann@37660
   388
definition
haftmann@37660
   389
  word_test_bit_def: "test_bit a = bin_nth (uint a)"
haftmann@37660
   390
haftmann@37660
   391
definition
haftmann@37660
   392
  word_set_bit_def: "set_bit a n x =
haftmann@37660
   393
   word_of_int (bin_sc n (If x 1 0) (uint a))"
haftmann@37660
   394
haftmann@37660
   395
definition
haftmann@37660
   396
  word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"
haftmann@37660
   397
haftmann@37660
   398
definition
haftmann@37660
   399
  word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1"
haftmann@37660
   400
haftmann@37660
   401
definition shiftl1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   402
  "shiftl1 w = word_of_int (uint w BIT 0)"
haftmann@37660
   403
haftmann@37660
   404
definition shiftr1 :: "'a word \<Rightarrow> 'a word" where
haftmann@37660
   405
  -- "shift right as unsigned or as signed, ie logical or arithmetic"
haftmann@37660
   406
  "shiftr1 w = word_of_int (bin_rest (uint w))"
haftmann@37660
   407
haftmann@37660
   408
definition
haftmann@37660
   409
  shiftl_def: "w << n = (shiftl1 ^^ n) w"
haftmann@37660
   410
haftmann@37660
   411
definition
haftmann@37660
   412
  shiftr_def: "w >> n = (shiftr1 ^^ n) w"
haftmann@37660
   413
haftmann@37660
   414
instance ..
haftmann@37660
   415
haftmann@37660
   416
end
haftmann@37660
   417
huffman@47611
   418
lemma [code]: shows
huffman@47374
   419
  word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" and
huffman@47374
   420
  word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" and
huffman@47374
   421
  word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" and
huffman@47374
   422
  word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
huffman@47374
   423
  unfolding bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def
huffman@47374
   424
  by simp_all
huffman@47374
   425
haftmann@37660
   426
instantiation word :: (len) bitss
haftmann@37660
   427
begin
haftmann@37660
   428
haftmann@37660
   429
definition
haftmann@37660
   430
  word_msb_def: 
huffman@46001
   431
  "msb a \<longleftrightarrow> bin_sign (sint a) = -1"
haftmann@37660
   432
haftmann@37660
   433
instance ..
haftmann@37660
   434
haftmann@37660
   435
end
haftmann@37660
   436
haftmann@37660
   437
definition setBit :: "'a :: len0 word => nat => 'a word" where 
haftmann@40827
   438
  "setBit w n = set_bit w n True"
haftmann@37660
   439
haftmann@37660
   440
definition clearBit :: "'a :: len0 word => nat => 'a word" where
haftmann@40827
   441
  "clearBit w n = set_bit w n False"
haftmann@37660
   442
haftmann@37660
   443
haftmann@37660
   444
subsection "Shift operations"
haftmann@37660
   445
haftmann@37660
   446
definition sshiftr1 :: "'a :: len word => 'a word" where 
haftmann@40827
   447
  "sshiftr1 w = word_of_int (bin_rest (sint w))"
haftmann@37660
   448
haftmann@37660
   449
definition bshiftr1 :: "bool => 'a :: len word => 'a word" where
haftmann@40827
   450
  "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
haftmann@37660
   451
haftmann@37660
   452
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where
haftmann@40827
   453
  "w >>> n = (sshiftr1 ^^ n) w"
haftmann@37660
   454
haftmann@37660
   455
definition mask :: "nat => 'a::len word" where
haftmann@40827
   456
  "mask n = (1 << n) - 1"
haftmann@37660
   457
haftmann@37660
   458
definition revcast :: "'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   459
  "revcast w =  of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
haftmann@37660
   460
haftmann@37660
   461
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   462
  "slice1 n w = of_bl (takefill False n (to_bl w))"
haftmann@37660
   463
haftmann@37660
   464
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where
haftmann@40827
   465
  "slice n w = slice1 (size w - n) w"
haftmann@37660
   466
haftmann@37660
   467
haftmann@37660
   468
subsection "Rotation"
haftmann@37660
   469
haftmann@37660
   470
definition rotater1 :: "'a list => 'a list" where
haftmann@40827
   471
  "rotater1 ys = 
haftmann@40827
   472
    (case ys of [] => [] | x # xs => last ys # butlast ys)"
haftmann@37660
   473
haftmann@37660
   474
definition rotater :: "nat => 'a list => 'a list" where
haftmann@40827
   475
  "rotater n = rotater1 ^^ n"
haftmann@37660
   476
haftmann@37660
   477
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   478
  "word_rotr n w = of_bl (rotater n (to_bl w))"
haftmann@37660
   479
haftmann@37660
   480
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   481
  "word_rotl n w = of_bl (rotate n (to_bl w))"
haftmann@37660
   482
haftmann@37660
   483
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where
haftmann@40827
   484
  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
haftmann@40827
   485
                    else word_rotl (nat (- i)) w)"
haftmann@37660
   486
haftmann@37660
   487
haftmann@37660
   488
subsection "Split and cat operations"
haftmann@37660
   489
haftmann@37660
   490
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where
haftmann@40827
   491
  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
haftmann@37660
   492
haftmann@37660
   493
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where
haftmann@40827
   494
  "word_split a = 
haftmann@40827
   495
   (case bin_split (len_of TYPE ('c)) (uint a) of 
haftmann@40827
   496
     (u, v) => (word_of_int u, word_of_int v))"
haftmann@37660
   497
haftmann@37660
   498
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where
haftmann@40827
   499
  "word_rcat ws = 
haftmann@37660
   500
  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
haftmann@37660
   501
haftmann@37660
   502
definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where
haftmann@40827
   503
  "word_rsplit w = 
haftmann@37660
   504
  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
haftmann@37660
   505
haftmann@37660
   506
definition max_word :: "'a::len word" -- "Largest representable machine integer." where
haftmann@40827
   507
  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
haftmann@37660
   508
huffman@45805
   509
(* FIXME: only provide one theorem name *)
haftmann@37660
   510
lemmas of_nth_def = word_set_bits_def
haftmann@37660
   511
huffman@46010
   512
subsection {* Theorems about typedefs *}
huffman@46010
   513
haftmann@37660
   514
lemma sint_sbintrunc': 
haftmann@37660
   515
  "sint (word_of_int bin :: 'a word) = 
haftmann@37660
   516
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
haftmann@37660
   517
  unfolding sint_uint 
haftmann@37660
   518
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)
haftmann@37660
   519
haftmann@37660
   520
lemma uint_sint: 
haftmann@37660
   521
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
haftmann@37660
   522
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
haftmann@37660
   523
huffman@46057
   524
lemma bintr_uint:
huffman@46057
   525
  fixes w :: "'a::len0 word"
huffman@46057
   526
  shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w"
haftmann@37660
   527
  apply (subst word_ubin.norm_Rep [symmetric]) 
haftmann@37660
   528
  apply (simp only: bintrunc_bintrunc_min word_size)
haftmann@37660
   529
  apply (simp add: min_max.inf_absorb2)
haftmann@37660
   530
  done
haftmann@37660
   531
huffman@46057
   532
lemma wi_bintr:
huffman@46057
   533
  "len_of TYPE('a::len0) \<le> n \<Longrightarrow>
huffman@46057
   534
    word_of_int (bintrunc n w) = (word_of_int w :: 'a word)"
haftmann@37660
   535
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1)
haftmann@37660
   536
haftmann@37660
   537
lemma td_ext_sbin: 
haftmann@37660
   538
  "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 
haftmann@37660
   539
    (sbintrunc (len_of TYPE('a) - 1))"
haftmann@37660
   540
  apply (unfold td_ext_def' sint_uint)
haftmann@37660
   541
  apply (simp add : word_ubin.eq_norm)
haftmann@37660
   542
  apply (cases "len_of TYPE('a)")
haftmann@37660
   543
   apply (auto simp add : sints_def)
haftmann@37660
   544
  apply (rule sym [THEN trans])
haftmann@37660
   545
  apply (rule word_ubin.Abs_norm)
haftmann@37660
   546
  apply (simp only: bintrunc_sbintrunc)
haftmann@37660
   547
  apply (drule sym)
haftmann@37660
   548
  apply simp
haftmann@37660
   549
  done
haftmann@37660
   550
haftmann@37660
   551
lemmas td_ext_sint = td_ext_sbin 
haftmann@37660
   552
  [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]]
haftmann@37660
   553
haftmann@37660
   554
(* We do sint before sbin, before sint is the user version
haftmann@37660
   555
   and interpretations do not produce thm duplicates. I.e. 
haftmann@37660
   556
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
haftmann@37660
   557
   because the latter is the same thm as the former *)
haftmann@37660
   558
interpretation word_sint:
haftmann@37660
   559
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   560
          word_of_int 
haftmann@37660
   561
          "sints (len_of TYPE('a::len))"
haftmann@37660
   562
          "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) -
haftmann@37660
   563
               2 ^ (len_of TYPE('a::len) - 1)"
haftmann@37660
   564
  by (rule td_ext_sint)
haftmann@37660
   565
haftmann@37660
   566
interpretation word_sbin:
haftmann@37660
   567
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   568
          word_of_int 
haftmann@37660
   569
          "sints (len_of TYPE('a::len))"
haftmann@37660
   570
          "sbintrunc (len_of TYPE('a::len) - 1)"
haftmann@37660
   571
  by (rule td_ext_sbin)
haftmann@37660
   572
wenzelm@45604
   573
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
haftmann@37660
   574
haftmann@37660
   575
lemmas td_sint = word_sint.td
haftmann@37660
   576
haftmann@37660
   577
lemma to_bl_def': 
haftmann@37660
   578
  "(to_bl :: 'a :: len0 word => bool list) =
haftmann@37660
   579
    bin_to_bl (len_of TYPE('a)) o uint"
wenzelm@44762
   580
  by (auto simp: to_bl_def)
haftmann@37660
   581
huffman@47108
   582
lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w
haftmann@37660
   583
huffman@45805
   584
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
huffman@45805
   585
  by (fact uints_def [unfolded no_bintr_alt1])
huffman@45805
   586
huffman@47108
   587
lemma word_numeral_alt:
huffman@47108
   588
  "numeral b = word_of_int (numeral b)"
huffman@47108
   589
  by (induct b, simp_all only: numeral.simps word_of_int_homs)
huffman@47108
   590
huffman@47108
   591
declare word_numeral_alt [symmetric, code_abbrev]
huffman@47108
   592
huffman@47108
   593
lemma word_neg_numeral_alt:
haftmann@54489
   594
  "- numeral b = word_of_int (- numeral b)"
haftmann@54489
   595
  by (simp only: word_numeral_alt wi_hom_neg)
huffman@47108
   596
huffman@47108
   597
declare word_neg_numeral_alt [symmetric, code_abbrev]
huffman@47108
   598
huffman@47372
   599
lemma word_numeral_transfer [transfer_rule]:
kuncar@51375
   600
  "(fun_rel op = pcr_word) numeral numeral"
haftmann@54489
   601
  "(fun_rel op = pcr_word) (- numeral) (- numeral)"
haftmann@54489
   602
  apply (simp_all add: fun_rel_def word.pcr_cr_eq cr_word_def)
haftmann@54489
   603
  using word_numeral_alt [symmetric] word_neg_numeral_alt [symmetric] by blast+
huffman@47372
   604
huffman@45805
   605
lemma uint_bintrunc [simp]:
huffman@47108
   606
  "uint (numeral bin :: 'a word) = 
huffman@47108
   607
    bintrunc (len_of TYPE ('a :: len0)) (numeral bin)"
huffman@47108
   608
  unfolding word_numeral_alt by (rule word_ubin.eq_norm)
huffman@47108
   609
haftmann@54489
   610
lemma uint_bintrunc_neg [simp]: "uint (- numeral bin :: 'a word) = 
haftmann@54489
   611
    bintrunc (len_of TYPE ('a :: len0)) (- numeral bin)"
huffman@47108
   612
  by (simp only: word_neg_numeral_alt word_ubin.eq_norm)
haftmann@37660
   613
huffman@45805
   614
lemma sint_sbintrunc [simp]:
huffman@47108
   615
  "sint (numeral bin :: 'a word) = 
huffman@47108
   616
    sbintrunc (len_of TYPE ('a :: len) - 1) (numeral bin)"
huffman@47108
   617
  by (simp only: word_numeral_alt word_sbin.eq_norm)
huffman@47108
   618
haftmann@54489
   619
lemma sint_sbintrunc_neg [simp]: "sint (- numeral bin :: 'a word) = 
haftmann@54489
   620
    sbintrunc (len_of TYPE ('a :: len) - 1) (- numeral bin)"
huffman@47108
   621
  by (simp only: word_neg_numeral_alt word_sbin.eq_norm)
haftmann@37660
   622
huffman@45805
   623
lemma unat_bintrunc [simp]:
huffman@47108
   624
  "unat (numeral bin :: 'a :: len0 word) =
huffman@47108
   625
    nat (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
   626
  by (simp only: unat_def uint_bintrunc)
huffman@47108
   627
huffman@47108
   628
lemma unat_bintrunc_neg [simp]:
haftmann@54489
   629
  "unat (- numeral bin :: 'a :: len0 word) =
haftmann@54489
   630
    nat (bintrunc (len_of TYPE('a)) (- numeral bin))"
huffman@47108
   631
  by (simp only: unat_def uint_bintrunc_neg)
haftmann@37660
   632
haftmann@40827
   633
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
haftmann@37660
   634
  apply (unfold word_size)
haftmann@37660
   635
  apply (rule word_uint.Rep_eqD)
haftmann@37660
   636
  apply (rule box_equals)
haftmann@37660
   637
    defer
haftmann@37660
   638
    apply (rule word_ubin.norm_Rep)+
haftmann@37660
   639
  apply simp
haftmann@37660
   640
  done
haftmann@37660
   641
huffman@45805
   642
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)"
huffman@45805
   643
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   644
huffman@45805
   645
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
huffman@45805
   646
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   647
huffman@45805
   648
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)"
huffman@45805
   649
  using word_sint.Rep [of x] by (simp add: sints_num)
huffman@45805
   650
huffman@45805
   651
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)"
huffman@45805
   652
  using word_sint.Rep [of x] by (simp add: sints_num)
haftmann@37660
   653
haftmann@37660
   654
lemma sign_uint_Pls [simp]: 
huffman@46604
   655
  "bin_sign (uint x) = 0"
huffman@47108
   656
  by (simp add: sign_Pls_ge_0)
haftmann@37660
   657
huffman@45805
   658
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0"
huffman@45805
   659
  by (simp only: diff_less_0_iff_less uint_lt2p)
huffman@45805
   660
huffman@45805
   661
lemma uint_m2p_not_non_neg:
huffman@45805
   662
  "\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)"
huffman@45805
   663
  by (simp only: not_le uint_m2p_neg)
haftmann@37660
   664
haftmann@37660
   665
lemma lt2p_lem:
haftmann@40827
   666
  "len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n"
haftmann@37660
   667
  by (rule xtr8 [OF _ uint_lt2p]) simp
haftmann@37660
   668
huffman@45805
   669
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
huffman@45805
   670
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])
haftmann@37660
   671
haftmann@40827
   672
lemma uint_nat: "uint w = int (unat w)"
haftmann@37660
   673
  unfolding unat_def by auto
haftmann@37660
   674
huffman@47108
   675
lemma uint_numeral:
huffman@47108
   676
  "uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   677
  unfolding word_numeral_alt
haftmann@37660
   678
  by (simp only: int_word_uint)
haftmann@37660
   679
huffman@47108
   680
lemma uint_neg_numeral:
haftmann@54489
   681
  "uint (- numeral b :: 'a :: len0 word) = - numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   682
  unfolding word_neg_numeral_alt
huffman@47108
   683
  by (simp only: int_word_uint)
huffman@47108
   684
huffman@47108
   685
lemma unat_numeral: 
huffman@47108
   686
  "unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)"
haftmann@37660
   687
  apply (unfold unat_def)
huffman@47108
   688
  apply (clarsimp simp only: uint_numeral)
haftmann@37660
   689
  apply (rule nat_mod_distrib [THEN trans])
huffman@47108
   690
    apply (rule zero_le_numeral)
haftmann@37660
   691
   apply (simp_all add: nat_power_eq)
haftmann@37660
   692
  done
haftmann@37660
   693
huffman@47108
   694
lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + 
haftmann@37660
   695
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
haftmann@37660
   696
    2 ^ (len_of TYPE('a) - 1)"
huffman@47108
   697
  unfolding word_numeral_alt by (rule int_word_sint)
huffman@47108
   698
huffman@47108
   699
lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0"
huffman@45958
   700
  unfolding word_0_wi ..
huffman@45958
   701
huffman@47108
   702
lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1"
huffman@45958
   703
  unfolding word_1_wi ..
huffman@45958
   704
haftmann@54489
   705
lemma word_of_int_neg_1 [simp]: "word_of_int (- 1) = - 1"
haftmann@54489
   706
  by (simp add: wi_hom_syms)
haftmann@54489
   707
huffman@47108
   708
lemma word_of_int_numeral [simp] : 
huffman@47108
   709
  "(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)"
huffman@47108
   710
  unfolding word_numeral_alt ..
huffman@47108
   711
huffman@47108
   712
lemma word_of_int_neg_numeral [simp]:
haftmann@54489
   713
  "(word_of_int (- numeral bin) :: 'a :: len0 word) = (- numeral bin)"
haftmann@54489
   714
  unfolding word_numeral_alt wi_hom_syms ..
haftmann@37660
   715
haftmann@37660
   716
lemma word_int_case_wi: 
haftmann@37660
   717
  "word_int_case f (word_of_int i :: 'b word) = 
haftmann@37660
   718
    f (i mod 2 ^ len_of TYPE('b::len0))"
haftmann@37660
   719
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)
haftmann@37660
   720
haftmann@37660
   721
lemma word_int_split: 
haftmann@37660
   722
  "P (word_int_case f x) = 
haftmann@37660
   723
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
haftmann@37660
   724
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
haftmann@37660
   725
  unfolding word_int_case_def
haftmann@37660
   726
  by (auto simp: word_uint.eq_norm int_mod_eq')
haftmann@37660
   727
haftmann@37660
   728
lemma word_int_split_asm: 
haftmann@37660
   729
  "P (word_int_case f x) = 
haftmann@37660
   730
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
haftmann@37660
   731
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
haftmann@37660
   732
  unfolding word_int_case_def
haftmann@37660
   733
  by (auto simp: word_uint.eq_norm int_mod_eq')
huffman@45805
   734
wenzelm@45604
   735
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
wenzelm@45604
   736
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
haftmann@37660
   737
haftmann@37660
   738
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
haftmann@37660
   739
  unfolding word_size by (rule uint_range')
haftmann@37660
   740
haftmann@37660
   741
lemma sint_range_size:
haftmann@37660
   742
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
haftmann@37660
   743
  unfolding word_size by (rule sint_range')
haftmann@37660
   744
huffman@45805
   745
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x"
huffman@45805
   746
  unfolding word_size by (rule less_le_trans [OF sint_lt])
huffman@45805
   747
huffman@45805
   748
lemma sint_below_size:
huffman@45805
   749
  "x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w"
huffman@45805
   750
  unfolding word_size by (rule order_trans [OF _ sint_ge])
haftmann@37660
   751
huffman@46010
   752
subsection {* Testing bits *}
huffman@46010
   753
haftmann@37660
   754
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)"
haftmann@37660
   755
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
haftmann@37660
   756
haftmann@37660
   757
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
haftmann@37660
   758
  apply (unfold word_test_bit_def)
haftmann@37660
   759
  apply (subst word_ubin.norm_Rep [symmetric])
haftmann@37660
   760
  apply (simp only: nth_bintr word_size)
haftmann@37660
   761
  apply fast
haftmann@37660
   762
  done
haftmann@37660
   763
huffman@46021
   764
lemma word_eq_iff:
huffman@46021
   765
  fixes x y :: "'a::len0 word"
huffman@46021
   766
  shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)"
huffman@46021
   767
  unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric]
huffman@46021
   768
  by (metis test_bit_size [unfolded word_size])
huffman@46021
   769
huffman@46023
   770
lemma word_eqI [rule_format]:
haftmann@37660
   771
  fixes u :: "'a::len0 word"
haftmann@40827
   772
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
huffman@46021
   773
  by (simp add: word_size word_eq_iff)
haftmann@37660
   774
huffman@45805
   775
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x"
huffman@45805
   776
  by simp
haftmann@37660
   777
haftmann@37660
   778
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
haftmann@37660
   779
  unfolding word_test_bit_def word_size
haftmann@37660
   780
  by (simp add: nth_bintr [symmetric])
haftmann@37660
   781
haftmann@37660
   782
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
haftmann@37660
   783
huffman@46057
   784
lemma bin_nth_uint_imp:
huffman@46057
   785
  "bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)"
haftmann@37660
   786
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
haftmann@37660
   787
  apply (subst word_ubin.norm_Rep)
haftmann@37660
   788
  apply assumption
haftmann@37660
   789
  done
haftmann@37660
   790
huffman@46057
   791
lemma bin_nth_sint:
huffman@46057
   792
  fixes w :: "'a::len word"
huffman@46057
   793
  shows "len_of TYPE('a) \<le> n \<Longrightarrow>
huffman@46057
   794
    bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)"
haftmann@37660
   795
  apply (subst word_sbin.norm_Rep [symmetric])
huffman@46057
   796
  apply (auto simp add: nth_sbintr)
haftmann@37660
   797
  done
haftmann@37660
   798
haftmann@37660
   799
(* type definitions theorem for in terms of equivalent bool list *)
haftmann@37660
   800
lemma td_bl: 
haftmann@37660
   801
  "type_definition (to_bl :: 'a::len0 word => bool list) 
haftmann@37660
   802
                   of_bl  
haftmann@37660
   803
                   {bl. length bl = len_of TYPE('a)}"
haftmann@37660
   804
  apply (unfold type_definition_def of_bl_def to_bl_def)
haftmann@37660
   805
  apply (simp add: word_ubin.eq_norm)
haftmann@37660
   806
  apply safe
haftmann@37660
   807
  apply (drule sym)
haftmann@37660
   808
  apply simp
haftmann@37660
   809
  done
haftmann@37660
   810
haftmann@37660
   811
interpretation word_bl:
haftmann@37660
   812
  type_definition "to_bl :: 'a::len0 word => bool list"
haftmann@37660
   813
                  of_bl  
haftmann@37660
   814
                  "{bl. length bl = len_of TYPE('a::len0)}"
haftmann@37660
   815
  by (rule td_bl)
haftmann@37660
   816
huffman@45816
   817
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
wenzelm@45538
   818
haftmann@40827
   819
lemma word_size_bl: "size w = size (to_bl w)"
haftmann@37660
   820
  unfolding word_size by auto
haftmann@37660
   821
haftmann@37660
   822
lemma to_bl_use_of_bl:
haftmann@37660
   823
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
huffman@45816
   824
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
haftmann@37660
   825
haftmann@37660
   826
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
haftmann@37660
   827
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)
haftmann@37660
   828
haftmann@37660
   829
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
haftmann@37660
   830
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)
haftmann@37660
   831
haftmann@40827
   832
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
huffman@47108
   833
  by (metis word_rev_rev)
haftmann@37660
   834
huffman@45805
   835
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
huffman@45805
   836
  by simp
huffman@45805
   837
huffman@45805
   838
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))"
huffman@45805
   839
  unfolding word_bl_Rep' by (rule len_gt_0)
huffman@45805
   840
huffman@45805
   841
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []"
huffman@45805
   842
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
huffman@45805
   843
huffman@45805
   844
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0"
huffman@45805
   845
  by (fact length_bl_gt_0 [THEN gr_implies_not0])
haftmann@37660
   846
huffman@46001
   847
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
haftmann@37660
   848
  apply (unfold to_bl_def sint_uint)
haftmann@37660
   849
  apply (rule trans [OF _ bl_sbin_sign])
haftmann@37660
   850
  apply simp
haftmann@37660
   851
  done
haftmann@37660
   852
haftmann@37660
   853
lemma of_bl_drop': 
haftmann@40827
   854
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
haftmann@37660
   855
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
haftmann@37660
   856
  apply (unfold of_bl_def)
haftmann@37660
   857
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
haftmann@37660
   858
  done
haftmann@37660
   859
haftmann@37660
   860
lemma test_bit_of_bl:  
haftmann@37660
   861
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)"
haftmann@37660
   862
  apply (unfold of_bl_def word_test_bit_def)
haftmann@37660
   863
  apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl)
haftmann@37660
   864
  done
haftmann@37660
   865
haftmann@37660
   866
lemma no_of_bl: 
huffman@47108
   867
  "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) (numeral bin))"
huffman@47108
   868
  unfolding of_bl_def by simp
haftmann@37660
   869
haftmann@40827
   870
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
haftmann@37660
   871
  unfolding word_size to_bl_def by auto
haftmann@37660
   872
haftmann@37660
   873
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   874
  unfolding uint_bl by (simp add : word_size)
haftmann@37660
   875
haftmann@37660
   876
lemma to_bl_of_bin: 
haftmann@37660
   877
  "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
haftmann@37660
   878
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)
haftmann@37660
   879
huffman@47108
   880
lemma to_bl_numeral [simp]:
huffman@47108
   881
  "to_bl (numeral bin::'a::len0 word) =
huffman@47108
   882
    bin_to_bl (len_of TYPE('a)) (numeral bin)"
huffman@47108
   883
  unfolding word_numeral_alt by (rule to_bl_of_bin)
huffman@47108
   884
huffman@47108
   885
lemma to_bl_neg_numeral [simp]:
haftmann@54489
   886
  "to_bl (- numeral bin::'a::len0 word) =
haftmann@54489
   887
    bin_to_bl (len_of TYPE('a)) (- numeral bin)"
huffman@47108
   888
  unfolding word_neg_numeral_alt by (rule to_bl_of_bin)
haftmann@37660
   889
haftmann@37660
   890
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   891
  unfolding uint_bl by (simp add : word_size)
huffman@46011
   892
huffman@46011
   893
lemma uint_bl_bin:
huffman@46011
   894
  fixes x :: "'a::len0 word"
huffman@46011
   895
  shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x"
huffman@46011
   896
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])
wenzelm@45604
   897
haftmann@37660
   898
(* naturals *)
haftmann@37660
   899
lemma uints_unats: "uints n = int ` unats n"
haftmann@37660
   900
  apply (unfold unats_def uints_num)
haftmann@37660
   901
  apply safe
haftmann@37660
   902
  apply (rule_tac image_eqI)
haftmann@37660
   903
  apply (erule_tac nat_0_le [symmetric])
haftmann@37660
   904
  apply auto
haftmann@37660
   905
  apply (erule_tac nat_less_iff [THEN iffD2])
haftmann@37660
   906
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
haftmann@37660
   907
  apply (auto simp add : nat_power_eq int_power)
haftmann@37660
   908
  done
haftmann@37660
   909
haftmann@37660
   910
lemma unats_uints: "unats n = nat ` uints n"
haftmann@37660
   911
  by (auto simp add : uints_unats image_iff)
haftmann@37660
   912
huffman@46962
   913
lemmas bintr_num = word_ubin.norm_eq_iff
huffman@47108
   914
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
huffman@46962
   915
lemmas sbintr_num = word_sbin.norm_eq_iff
huffman@47108
   916
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
haftmann@37660
   917
haftmann@37660
   918
lemma num_of_bintr':
huffman@47108
   919
  "bintrunc (len_of TYPE('a :: len0)) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   920
    numeral a = (numeral b :: 'a word)"
huffman@46962
   921
  unfolding bintr_num by (erule subst, simp)
haftmann@37660
   922
haftmann@37660
   923
lemma num_of_sbintr':
huffman@47108
   924
  "sbintrunc (len_of TYPE('a :: len) - 1) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   925
    numeral a = (numeral b :: 'a word)"
huffman@46962
   926
  unfolding sbintr_num by (erule subst, simp)
huffman@46962
   927
huffman@46962
   928
lemma num_abs_bintr:
huffman@47108
   929
  "(numeral x :: 'a word) =
huffman@47108
   930
    word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))"
huffman@47108
   931
  by (simp only: word_ubin.Abs_norm word_numeral_alt)
huffman@46962
   932
huffman@46962
   933
lemma num_abs_sbintr:
huffman@47108
   934
  "(numeral x :: 'a word) =
huffman@47108
   935
    word_of_int (sbintrunc (len_of TYPE('a::len) - 1) (numeral x))"
huffman@47108
   936
  by (simp only: word_sbin.Abs_norm word_numeral_alt)
huffman@46962
   937
haftmann@37660
   938
(** cast - note, no arg for new length, as it's determined by type of result,
haftmann@37660
   939
  thus in "cast w = w, the type means cast to length of w! **)
haftmann@37660
   940
haftmann@37660
   941
lemma ucast_id: "ucast w = w"
haftmann@37660
   942
  unfolding ucast_def by auto
haftmann@37660
   943
haftmann@37660
   944
lemma scast_id: "scast w = w"
haftmann@37660
   945
  unfolding scast_def by auto
haftmann@37660
   946
haftmann@40827
   947
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
haftmann@37660
   948
  unfolding ucast_def of_bl_def uint_bl
haftmann@37660
   949
  by (auto simp add : word_size)
haftmann@37660
   950
haftmann@37660
   951
lemma nth_ucast: 
haftmann@37660
   952
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
haftmann@37660
   953
  apply (unfold ucast_def test_bit_bin)
haftmann@37660
   954
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
haftmann@37660
   955
  apply (fast elim!: bin_nth_uint_imp)
haftmann@37660
   956
  done
haftmann@37660
   957
haftmann@37660
   958
(* for literal u(s)cast *)
haftmann@37660
   959
huffman@46001
   960
lemma ucast_bintr [simp]:
huffman@47108
   961
  "ucast (numeral w ::'a::len0 word) = 
huffman@47108
   962
   word_of_int (bintrunc (len_of TYPE('a)) (numeral w))"
haftmann@37660
   963
  unfolding ucast_def by simp
huffman@47108
   964
(* TODO: neg_numeral *)
haftmann@37660
   965
huffman@46001
   966
lemma scast_sbintr [simp]:
huffman@47108
   967
  "scast (numeral w ::'a::len word) = 
huffman@47108
   968
   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (numeral w))"
haftmann@37660
   969
  unfolding scast_def by simp
haftmann@37660
   970
huffman@46011
   971
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)"
huffman@46011
   972
  unfolding source_size_def word_size Let_def ..
huffman@46011
   973
huffman@46011
   974
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)"
huffman@46011
   975
  unfolding target_size_def word_size Let_def ..
huffman@46011
   976
huffman@46011
   977
lemma is_down:
huffman@46011
   978
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
   979
  shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)"
huffman@46011
   980
  unfolding is_down_def source_size target_size ..
huffman@46011
   981
huffman@46011
   982
lemma is_up:
huffman@46011
   983
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
   984
  shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)"
huffman@46011
   985
  unfolding is_up_def source_size target_size ..
haftmann@37660
   986
wenzelm@45604
   987
lemmas is_up_down = trans [OF is_up is_down [symmetric]]
haftmann@37660
   988
huffman@45811
   989
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
haftmann@37660
   990
  apply (unfold is_down)
haftmann@37660
   991
  apply safe
haftmann@37660
   992
  apply (rule ext)
haftmann@37660
   993
  apply (unfold ucast_def scast_def uint_sint)
haftmann@37660
   994
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
   995
  apply simp
haftmann@37660
   996
  done
haftmann@37660
   997
huffman@45811
   998
lemma word_rev_tf:
huffman@45811
   999
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
  1000
    rev (takefill False (len_of TYPE('a)) (rev bl))"
haftmann@37660
  1001
  unfolding of_bl_def uint_bl
haftmann@37660
  1002
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
haftmann@37660
  1003
huffman@45811
  1004
lemma word_rep_drop:
huffman@45811
  1005
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
  1006
    replicate (len_of TYPE('a) - length bl) False @
huffman@45811
  1007
    drop (length bl - len_of TYPE('a)) bl"
huffman@45811
  1008
  by (simp add: word_rev_tf takefill_alt rev_take)
haftmann@37660
  1009
haftmann@37660
  1010
lemma to_bl_ucast: 
haftmann@37660
  1011
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
haftmann@37660
  1012
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
haftmann@37660
  1013
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
haftmann@37660
  1014
  apply (unfold ucast_bl)
haftmann@37660
  1015
  apply (rule trans)
haftmann@37660
  1016
   apply (rule word_rep_drop)
haftmann@37660
  1017
  apply simp
haftmann@37660
  1018
  done
haftmann@37660
  1019
huffman@45811
  1020
lemma ucast_up_app [OF refl]:
haftmann@40827
  1021
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
haftmann@37660
  1022
    to_bl (uc w) = replicate n False @ (to_bl w)"
haftmann@37660
  1023
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
  1024
huffman@45811
  1025
lemma ucast_down_drop [OF refl]:
haftmann@40827
  1026
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
haftmann@37660
  1027
    to_bl (uc w) = drop n (to_bl w)"
haftmann@37660
  1028
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
  1029
huffman@45811
  1030
lemma scast_down_drop [OF refl]:
haftmann@40827
  1031
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
haftmann@37660
  1032
    to_bl (sc w) = drop n (to_bl w)"
haftmann@37660
  1033
  apply (subgoal_tac "sc = ucast")
haftmann@37660
  1034
   apply safe
haftmann@37660
  1035
   apply simp
huffman@45811
  1036
   apply (erule ucast_down_drop)
huffman@45811
  1037
  apply (rule down_cast_same [symmetric])
haftmann@37660
  1038
  apply (simp add : source_size target_size is_down)
haftmann@37660
  1039
  done
haftmann@37660
  1040
huffman@45811
  1041
lemma sint_up_scast [OF refl]:
haftmann@40827
  1042
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
haftmann@37660
  1043
  apply (unfold is_up)
haftmann@37660
  1044
  apply safe
haftmann@37660
  1045
  apply (simp add: scast_def word_sbin.eq_norm)
haftmann@37660
  1046
  apply (rule box_equals)
haftmann@37660
  1047
    prefer 3
haftmann@37660
  1048
    apply (rule word_sbin.norm_Rep)
haftmann@37660
  1049
   apply (rule sbintrunc_sbintrunc_l)
haftmann@37660
  1050
   defer
haftmann@37660
  1051
   apply (subst word_sbin.norm_Rep)
haftmann@37660
  1052
   apply (rule refl)
haftmann@37660
  1053
  apply simp
haftmann@37660
  1054
  done
haftmann@37660
  1055
huffman@45811
  1056
lemma uint_up_ucast [OF refl]:
haftmann@40827
  1057
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
haftmann@37660
  1058
  apply (unfold is_up)
haftmann@37660
  1059
  apply safe
haftmann@37660
  1060
  apply (rule bin_eqI)
haftmann@37660
  1061
  apply (fold word_test_bit_def)
haftmann@37660
  1062
  apply (auto simp add: nth_ucast)
haftmann@37660
  1063
  apply (auto simp add: test_bit_bin)
haftmann@37660
  1064
  done
huffman@45811
  1065
huffman@45811
  1066
lemma ucast_up_ucast [OF refl]:
huffman@45811
  1067
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
haftmann@37660
  1068
  apply (simp (no_asm) add: ucast_def)
haftmann@37660
  1069
  apply (clarsimp simp add: uint_up_ucast)
haftmann@37660
  1070
  done
haftmann@37660
  1071
    
huffman@45811
  1072
lemma scast_up_scast [OF refl]:
huffman@45811
  1073
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
haftmann@37660
  1074
  apply (simp (no_asm) add: scast_def)
haftmann@37660
  1075
  apply (clarsimp simp add: sint_up_scast)
haftmann@37660
  1076
  done
haftmann@37660
  1077
    
huffman@45811
  1078
lemma ucast_of_bl_up [OF refl]:
haftmann@40827
  1079
  "w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl"
haftmann@37660
  1080
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
haftmann@37660
  1081
haftmann@37660
  1082
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
haftmann@37660
  1083
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
haftmann@37660
  1084
haftmann@37660
  1085
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
haftmann@37660
  1086
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
haftmann@37660
  1087
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
haftmann@37660
  1088
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
haftmann@37660
  1089
haftmann@37660
  1090
lemma up_ucast_surj:
haftmann@40827
  1091
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1092
   surj (ucast :: 'a word => 'b word)"
haftmann@37660
  1093
  by (rule surjI, erule ucast_up_ucast_id)
haftmann@37660
  1094
haftmann@37660
  1095
lemma up_scast_surj:
haftmann@40827
  1096
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1097
   surj (scast :: 'a word => 'b word)"
haftmann@37660
  1098
  by (rule surjI, erule scast_up_scast_id)
haftmann@37660
  1099
haftmann@37660
  1100
lemma down_scast_inj:
haftmann@40827
  1101
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1102
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1103
  by (rule inj_on_inverseI, erule scast_down_scast_id)
haftmann@37660
  1104
haftmann@37660
  1105
lemma down_ucast_inj:
haftmann@40827
  1106
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1107
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1108
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)
haftmann@37660
  1109
haftmann@37660
  1110
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
haftmann@37660
  1111
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
huffman@45811
  1112
huffman@46646
  1113
lemma ucast_down_wi [OF refl]:
huffman@46646
  1114
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
huffman@46646
  1115
  apply (unfold is_down)
haftmann@37660
  1116
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
haftmann@37660
  1117
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1118
  apply (erule bintrunc_bintrunc_ge)
haftmann@37660
  1119
  done
huffman@45811
  1120
huffman@46646
  1121
lemma ucast_down_no [OF refl]:
huffman@47108
  1122
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin"
huffman@47108
  1123
  unfolding word_numeral_alt by clarify (rule ucast_down_wi)
huffman@46646
  1124
huffman@45811
  1125
lemma ucast_down_bl [OF refl]:
huffman@45811
  1126
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
huffman@46646
  1127
  unfolding of_bl_def by clarify (erule ucast_down_wi)
haftmann@37660
  1128
haftmann@37660
  1129
lemmas slice_def' = slice_def [unfolded word_size]
haftmann@37660
  1130
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
haftmann@37660
  1131
haftmann@37660
  1132
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
haftmann@37660
  1133
haftmann@37660
  1134
haftmann@37660
  1135
subsection {* Word Arithmetic *}
haftmann@37660
  1136
haftmann@37660
  1137
lemma word_less_alt: "(a < b) = (uint a < uint b)"
huffman@46012
  1138
  unfolding word_less_def word_le_def by (simp add: less_le)
haftmann@37660
  1139
haftmann@37660
  1140
lemma signed_linorder: "class.linorder word_sle word_sless"
wenzelm@46124
  1141
  by default (unfold word_sle_def word_sless_def, auto)
haftmann@37660
  1142
haftmann@37660
  1143
interpretation signed: linorder "word_sle" "word_sless"
haftmann@37660
  1144
  by (rule signed_linorder)
haftmann@37660
  1145
haftmann@37660
  1146
lemma udvdI: 
haftmann@40827
  1147
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
haftmann@37660
  1148
  by (auto simp: udvd_def)
haftmann@37660
  1149
huffman@47108
  1150
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1151
huffman@47108
  1152
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1153
huffman@47108
  1154
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1155
huffman@47108
  1156
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1157
huffman@47108
  1158
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1159
huffman@47108
  1160
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b
haftmann@37660
  1161
haftmann@54489
  1162
lemma word_m1_wi: "- 1 = word_of_int (- 1)" 
haftmann@54489
  1163
  using word_neg_numeral_alt [of Num.One] by simp
haftmann@37660
  1164
huffman@46648
  1165
lemma word_0_bl [simp]: "of_bl [] = 0"
huffman@46648
  1166
  unfolding of_bl_def by simp
haftmann@37660
  1167
haftmann@37660
  1168
lemma word_1_bl: "of_bl [True] = 1" 
huffman@46648
  1169
  unfolding of_bl_def by (simp add: bl_to_bin_def)
huffman@46648
  1170
huffman@46648
  1171
lemma uint_eq_0 [simp]: "uint 0 = 0"
huffman@46648
  1172
  unfolding word_0_wi word_ubin.eq_norm by simp
haftmann@37660
  1173
huffman@45995
  1174
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
huffman@46648
  1175
  by (simp add: of_bl_def bl_to_bin_rep_False)
haftmann@37660
  1176
huffman@45805
  1177
lemma to_bl_0 [simp]:
haftmann@37660
  1178
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
haftmann@37660
  1179
  unfolding uint_bl
huffman@46617
  1180
  by (simp add: word_size bin_to_bl_zero)
haftmann@37660
  1181
haftmann@37660
  1182
lemma uint_0_iff: "(uint x = 0) = (x = 0)"
haftmann@37660
  1183
  by (auto intro!: word_uint.Rep_eqD)
haftmann@37660
  1184
haftmann@37660
  1185
lemma unat_0_iff: "(unat x = 0) = (x = 0)"
haftmann@37660
  1186
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)
haftmann@37660
  1187
haftmann@37660
  1188
lemma unat_0 [simp]: "unat 0 = 0"
haftmann@37660
  1189
  unfolding unat_def by auto
haftmann@37660
  1190
haftmann@40827
  1191
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
haftmann@37660
  1192
  apply (unfold word_size)
haftmann@37660
  1193
  apply (rule box_equals)
haftmann@37660
  1194
    defer
haftmann@37660
  1195
    apply (rule word_uint.Rep_inverse)+
haftmann@37660
  1196
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1197
  apply simp
haftmann@37660
  1198
  done
haftmann@37660
  1199
huffman@45816
  1200
lemmas size_0_same = size_0_same' [unfolded word_size]
haftmann@37660
  1201
haftmann@37660
  1202
lemmas unat_eq_0 = unat_0_iff
haftmann@37660
  1203
lemmas unat_eq_zero = unat_0_iff
haftmann@37660
  1204
haftmann@37660
  1205
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
haftmann@37660
  1206
by (auto simp: unat_0_iff [symmetric])
haftmann@37660
  1207
huffman@45958
  1208
lemma ucast_0 [simp]: "ucast 0 = 0"
huffman@45995
  1209
  unfolding ucast_def by simp
huffman@45958
  1210
huffman@45958
  1211
lemma sint_0 [simp]: "sint 0 = 0"
huffman@45958
  1212
  unfolding sint_uint by simp
huffman@45958
  1213
huffman@45958
  1214
lemma scast_0 [simp]: "scast 0 = 0"
huffman@45995
  1215
  unfolding scast_def by simp
haftmann@37660
  1216
haftmann@37660
  1217
lemma sint_n1 [simp] : "sint -1 = -1"
haftmann@54489
  1218
  unfolding word_m1_wi word_sbin.eq_norm by simp
haftmann@54489
  1219
haftmann@54489
  1220
lemma scast_n1 [simp]: "scast (- 1) = - 1"
huffman@45958
  1221
  unfolding scast_def by simp
huffman@45958
  1222
huffman@45958
  1223
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
haftmann@37660
  1224
  unfolding word_1_wi
huffman@45995
  1225
  by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1)
huffman@45958
  1226
huffman@45958
  1227
lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
huffman@45958
  1228
  unfolding unat_def by simp
huffman@45958
  1229
huffman@45958
  1230
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
huffman@45995
  1231
  unfolding ucast_def by simp
haftmann@37660
  1232
haftmann@37660
  1233
(* now, to get the weaker results analogous to word_div/mod_def *)
haftmann@37660
  1234
haftmann@37660
  1235
lemmas word_arith_alts = 
huffman@46000
  1236
  word_sub_wi
huffman@46000
  1237
  word_arith_wis (* FIXME: duplicate *)
huffman@46000
  1238
haftmann@37660
  1239
subsection  "Transferring goals from words to ints"
haftmann@37660
  1240
haftmann@37660
  1241
lemma word_ths:  
haftmann@37660
  1242
  shows
haftmann@37660
  1243
  word_succ_p1:   "word_succ a = a + 1" and
haftmann@37660
  1244
  word_pred_m1:   "word_pred a = a - 1" and
haftmann@37660
  1245
  word_pred_succ: "word_pred (word_succ a) = a" and
haftmann@37660
  1246
  word_succ_pred: "word_succ (word_pred a) = a" and
haftmann@37660
  1247
  word_mult_succ: "word_succ a * b = b + a * b"
huffman@47374
  1248
  by (transfer, simp add: algebra_simps)+
haftmann@37660
  1249
huffman@45816
  1250
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
huffman@45816
  1251
  by simp
haftmann@37660
  1252
haftmann@37660
  1253
lemmas uint_word_ariths = 
wenzelm@45604
  1254
  word_arith_alts [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1255
haftmann@37660
  1256
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p]
haftmann@37660
  1257
haftmann@37660
  1258
(* similar expressions for sint (arith operations) *)
haftmann@37660
  1259
lemmas sint_word_ariths = uint_word_arith_bintrs
haftmann@37660
  1260
  [THEN uint_sint [symmetric, THEN trans],
haftmann@37660
  1261
  unfolded uint_sint bintr_arith1s bintr_ariths 
wenzelm@45604
  1262
    len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep]
wenzelm@45604
  1263
wenzelm@45604
  1264
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
wenzelm@45604
  1265
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1266
haftmann@37660
  1267
lemma word_pred_0_n1: "word_pred 0 = word_of_int -1"
huffman@47374
  1268
  unfolding word_pred_m1 by simp
haftmann@37660
  1269
haftmann@37660
  1270
lemma succ_pred_no [simp]:
huffman@47108
  1271
  "word_succ (numeral w) = numeral w + 1"
huffman@47108
  1272
  "word_pred (numeral w) = numeral w - 1"
haftmann@54489
  1273
  "word_succ (- numeral w) = - numeral w + 1"
haftmann@54489
  1274
  "word_pred (- numeral w) = - numeral w - 1"
huffman@47108
  1275
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1276
haftmann@37660
  1277
lemma word_sp_01 [simp] : 
haftmann@37660
  1278
  "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0"
huffman@47108
  1279
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1280
haftmann@37660
  1281
(* alternative approach to lifting arithmetic equalities *)
haftmann@37660
  1282
lemma word_of_int_Ex:
haftmann@37660
  1283
  "\<exists>y. x = word_of_int y"
haftmann@37660
  1284
  by (rule_tac x="uint x" in exI) simp
haftmann@37660
  1285
haftmann@37660
  1286
haftmann@37660
  1287
subsection "Order on fixed-length words"
haftmann@37660
  1288
haftmann@37660
  1289
lemma word_zero_le [simp] :
haftmann@37660
  1290
  "0 <= (y :: 'a :: len0 word)"
haftmann@37660
  1291
  unfolding word_le_def by auto
haftmann@37660
  1292
  
huffman@45816
  1293
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *)
haftmann@37660
  1294
  unfolding word_le_def
haftmann@37660
  1295
  by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1296
huffman@45816
  1297
lemma word_n1_ge [simp]: "y \<le> (-1::'a::len0 word)"
huffman@45816
  1298
  unfolding word_le_def
huffman@45816
  1299
  by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1300
haftmann@37660
  1301
lemmas word_not_simps [simp] = 
haftmann@37660
  1302
  word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD]
haftmann@37660
  1303
huffman@47108
  1304
lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> (y :: 'a :: len0 word)"
huffman@47108
  1305
  by (simp add: less_le)
huffman@47108
  1306
huffman@47108
  1307
lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y
haftmann@37660
  1308
haftmann@40827
  1309
lemma word_sless_alt: "(a <s b) = (sint a < sint b)"
haftmann@37660
  1310
  unfolding word_sle_def word_sless_def
haftmann@37660
  1311
  by (auto simp add: less_le)
haftmann@37660
  1312
haftmann@37660
  1313
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)"
haftmann@37660
  1314
  unfolding unat_def word_le_def
haftmann@37660
  1315
  by (rule nat_le_eq_zle [symmetric]) simp
haftmann@37660
  1316
haftmann@37660
  1317
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)"
haftmann@37660
  1318
  unfolding unat_def word_less_alt
haftmann@37660
  1319
  by (rule nat_less_eq_zless [symmetric]) simp
haftmann@37660
  1320
  
haftmann@37660
  1321
lemma wi_less: 
haftmann@37660
  1322
  "(word_of_int n < (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1323
    (n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1324
  unfolding word_less_alt by (simp add: word_uint.eq_norm)
haftmann@37660
  1325
haftmann@37660
  1326
lemma wi_le: 
haftmann@37660
  1327
  "(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1328
    (n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1329
  unfolding word_le_def by (simp add: word_uint.eq_norm)
haftmann@37660
  1330
haftmann@37660
  1331
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)"
haftmann@37660
  1332
  apply (unfold udvd_def)
haftmann@37660
  1333
  apply safe
haftmann@37660
  1334
   apply (simp add: unat_def nat_mult_distrib)
haftmann@37660
  1335
  apply (simp add: uint_nat int_mult)
haftmann@37660
  1336
  apply (rule exI)
haftmann@37660
  1337
  apply safe
haftmann@37660
  1338
   prefer 2
haftmann@37660
  1339
   apply (erule notE)
haftmann@37660
  1340
   apply (rule refl)
haftmann@37660
  1341
  apply force
haftmann@37660
  1342
  done
haftmann@37660
  1343
haftmann@37660
  1344
lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y"
haftmann@37660
  1345
  unfolding dvd_def udvd_nat_alt by force
haftmann@37660
  1346
wenzelm@45604
  1347
lemmas unat_mono = word_less_nat_alt [THEN iffD1]
haftmann@37660
  1348
haftmann@40827
  1349
lemma unat_minus_one: "x ~= 0 \<Longrightarrow> unat (x - 1) = unat x - 1"
haftmann@37660
  1350
  apply (unfold unat_def)
haftmann@37660
  1351
  apply (simp only: int_word_uint word_arith_alts rdmods)
haftmann@37660
  1352
  apply (subgoal_tac "uint x >= 1")
haftmann@37660
  1353
   prefer 2
haftmann@37660
  1354
   apply (drule contrapos_nn)
haftmann@37660
  1355
    apply (erule word_uint.Rep_inverse' [symmetric])
haftmann@37660
  1356
   apply (insert uint_ge_0 [of x])[1]
haftmann@37660
  1357
   apply arith
haftmann@37660
  1358
  apply (rule box_equals)
haftmann@37660
  1359
    apply (rule nat_diff_distrib)
haftmann@37660
  1360
     prefer 2
haftmann@37660
  1361
     apply assumption
haftmann@37660
  1362
    apply simp
haftmann@37660
  1363
   apply (subst mod_pos_pos_trivial)
haftmann@37660
  1364
     apply arith
haftmann@37660
  1365
    apply (insert uint_lt2p [of x])[1]
haftmann@37660
  1366
    apply arith
haftmann@37660
  1367
   apply (rule refl)
haftmann@37660
  1368
  apply simp
haftmann@37660
  1369
  done
haftmann@37660
  1370
    
haftmann@40827
  1371
lemma measure_unat: "p ~= 0 \<Longrightarrow> unat (p - 1) < unat p"
haftmann@37660
  1372
  by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric])
haftmann@37660
  1373
  
wenzelm@45604
  1374
lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
wenzelm@45604
  1375
lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
haftmann@37660
  1376
haftmann@37660
  1377
lemma uint_sub_lt2p [simp]: 
haftmann@37660
  1378
  "uint (x :: 'a :: len0 word) - uint (y :: 'b :: len0 word) < 
haftmann@37660
  1379
    2 ^ len_of TYPE('a)"
haftmann@37660
  1380
  using uint_ge_0 [of y] uint_lt2p [of x] by arith
haftmann@37660
  1381
haftmann@37660
  1382
haftmann@37660
  1383
subsection "Conditions for the addition (etc) of two words to overflow"
haftmann@37660
  1384
haftmann@37660
  1385
lemma uint_add_lem: 
haftmann@37660
  1386
  "(uint x + uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1387
    (uint (x + y :: 'a :: len0 word) = uint x + uint y)"
haftmann@37660
  1388
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1389
haftmann@37660
  1390
lemma uint_mult_lem: 
haftmann@37660
  1391
  "(uint x * uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1392
    (uint (x * y :: 'a :: len0 word) = uint x * uint y)"
haftmann@37660
  1393
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1394
haftmann@37660
  1395
lemma uint_sub_lem: 
haftmann@37660
  1396
  "(uint x >= uint y) = (uint (x - y) = uint x - uint y)"
haftmann@37660
  1397
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1398
haftmann@37660
  1399
lemma uint_add_le: "uint (x + y) <= uint x + uint y"
haftmann@37660
  1400
  unfolding uint_word_ariths by (auto simp: mod_add_if_z)
haftmann@37660
  1401
haftmann@37660
  1402
lemma uint_sub_ge: "uint (x - y) >= uint x - uint y"
haftmann@37660
  1403
  unfolding uint_word_ariths by (auto simp: mod_sub_if_z)
haftmann@37660
  1404
wenzelm@45604
  1405
lemmas uint_sub_if' = trans [OF uint_word_ariths(1) mod_sub_if_z, simplified]
wenzelm@45604
  1406
lemmas uint_plus_if' = trans [OF uint_word_ariths(2) mod_add_if_z, simplified]
haftmann@37660
  1407
haftmann@37660
  1408
haftmann@37660
  1409
subsection {* Definition of uint\_arith *}
haftmann@37660
  1410
haftmann@37660
  1411
lemma word_of_int_inverse:
haftmann@40827
  1412
  "word_of_int r = a \<Longrightarrow> 0 <= r \<Longrightarrow> r < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1413
   uint (a::'a::len0 word) = r"
haftmann@37660
  1414
  apply (erule word_uint.Abs_inverse' [rotated])
haftmann@37660
  1415
  apply (simp add: uints_num)
haftmann@37660
  1416
  done
haftmann@37660
  1417
haftmann@37660
  1418
lemma uint_split:
haftmann@37660
  1419
  fixes x::"'a::len0 word"
haftmann@37660
  1420
  shows "P (uint x) = 
haftmann@37660
  1421
         (ALL i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) --> P i)"
haftmann@37660
  1422
  apply (fold word_int_case_def)
haftmann@37660
  1423
  apply (auto dest!: word_of_int_inverse simp: int_word_uint int_mod_eq'
haftmann@37660
  1424
              split: word_int_split)
haftmann@37660
  1425
  done
haftmann@37660
  1426
haftmann@37660
  1427
lemma uint_split_asm:
haftmann@37660
  1428
  fixes x::"'a::len0 word"
haftmann@37660
  1429
  shows "P (uint x) = 
haftmann@37660
  1430
         (~(EX i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) & ~ P i))"
haftmann@37660
  1431
  by (auto dest!: word_of_int_inverse 
haftmann@37660
  1432
           simp: int_word_uint int_mod_eq'
haftmann@37660
  1433
           split: uint_split)
haftmann@37660
  1434
haftmann@37660
  1435
lemmas uint_splits = uint_split uint_split_asm
haftmann@37660
  1436
haftmann@37660
  1437
lemmas uint_arith_simps = 
haftmann@37660
  1438
  word_le_def word_less_alt
haftmann@37660
  1439
  word_uint.Rep_inject [symmetric] 
haftmann@37660
  1440
  uint_sub_if' uint_plus_if'
haftmann@37660
  1441
haftmann@37660
  1442
(* use this to stop, eg, 2 ^ len_of TYPE (32) being simplified *)
haftmann@40827
  1443
lemma power_False_cong: "False \<Longrightarrow> a ^ b = c ^ d" 
haftmann@37660
  1444
  by auto
haftmann@37660
  1445
haftmann@37660
  1446
(* uint_arith_tac: reduce to arithmetic on int, try to solve by arith *)
haftmann@37660
  1447
ML {*
wenzelm@51717
  1448
fun uint_arith_simpset ctxt = 
wenzelm@51717
  1449
  ctxt addsimps @{thms uint_arith_simps}
haftmann@37660
  1450
     delsimps @{thms word_uint.Rep_inject}
wenzelm@45620
  1451
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1452
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1453
haftmann@37660
  1454
fun uint_arith_tacs ctxt = 
haftmann@37660
  1455
  let
haftmann@37660
  1456
    fun arith_tac' n t =
haftmann@37660
  1457
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1458
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1459
  in 
wenzelm@42793
  1460
    [ clarify_tac ctxt 1,
wenzelm@51717
  1461
      full_simp_tac (uint_arith_simpset ctxt) 1,
wenzelm@51717
  1462
      ALLGOALS (full_simp_tac
wenzelm@51717
  1463
        (put_simpset HOL_ss ctxt
wenzelm@51717
  1464
          |> fold Splitter.add_split @{thms uint_splits}
wenzelm@51717
  1465
          |> fold Simplifier.add_cong @{thms power_False_cong})),
wenzelm@54742
  1466
      rewrite_goals_tac ctxt @{thms word_size}, 
haftmann@37660
  1467
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1468
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1469
                         REPEAT (dtac @{thm word_of_int_inverse} n 
haftmann@37660
  1470
                                 THEN atac n 
haftmann@37660
  1471
                                 THEN atac n)),
haftmann@37660
  1472
      TRYALL arith_tac' ]
haftmann@37660
  1473
  end
haftmann@37660
  1474
haftmann@37660
  1475
fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt))
haftmann@37660
  1476
*}
haftmann@37660
  1477
haftmann@37660
  1478
method_setup uint_arith = 
haftmann@37660
  1479
  {* Scan.succeed (SIMPLE_METHOD' o uint_arith_tac) *}
haftmann@37660
  1480
  "solving word arithmetic via integers and arith"
haftmann@37660
  1481
haftmann@37660
  1482
haftmann@37660
  1483
subsection "More on overflows and monotonicity"
haftmann@37660
  1484
haftmann@37660
  1485
lemma no_plus_overflow_uint_size: 
haftmann@37660
  1486
  "((x :: 'a :: len0 word) <= x + y) = (uint x + uint y < 2 ^ size x)"
haftmann@37660
  1487
  unfolding word_size by uint_arith
haftmann@37660
  1488
haftmann@37660
  1489
lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size]
haftmann@37660
  1490
haftmann@37660
  1491
lemma no_ulen_sub: "((x :: 'a :: len0 word) >= x - y) = (uint y <= uint x)"
haftmann@37660
  1492
  by uint_arith
haftmann@37660
  1493
haftmann@37660
  1494
lemma no_olen_add':
haftmann@37660
  1495
  fixes x :: "'a::len0 word"
haftmann@37660
  1496
  shows "(x \<le> y + x) = (uint y + uint x < 2 ^ len_of TYPE('a))"
huffman@45546
  1497
  by (simp add: add_ac no_olen_add)
haftmann@37660
  1498
wenzelm@45604
  1499
lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric]]
wenzelm@45604
  1500
wenzelm@45604
  1501
lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem]
wenzelm@45604
  1502
lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1]
wenzelm@45604
  1503
lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem]
haftmann@37660
  1504
lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def]
haftmann@37660
  1505
lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def]
wenzelm@45604
  1506
lemmas word_sub_le = word_sub_le_iff [THEN iffD2]
haftmann@37660
  1507
haftmann@37660
  1508
lemma word_less_sub1: 
haftmann@40827
  1509
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 < x) = (0 < x - 1)"
haftmann@37660
  1510
  by uint_arith
haftmann@37660
  1511
haftmann@37660
  1512
lemma word_le_sub1: 
haftmann@40827
  1513
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 <= x) = (0 <= x - 1)"
haftmann@37660
  1514
  by uint_arith
haftmann@37660
  1515
haftmann@37660
  1516
lemma sub_wrap_lt: 
haftmann@37660
  1517
  "((x :: 'a :: len0 word) < x - z) = (x < z)"
haftmann@37660
  1518
  by uint_arith
haftmann@37660
  1519
haftmann@37660
  1520
lemma sub_wrap: 
haftmann@37660
  1521
  "((x :: 'a :: len0 word) <= x - z) = (z = 0 | x < z)"
haftmann@37660
  1522
  by uint_arith
haftmann@37660
  1523
haftmann@37660
  1524
lemma plus_minus_not_NULL_ab: 
haftmann@40827
  1525
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> c ~= 0 \<Longrightarrow> x + c ~= 0"
haftmann@37660
  1526
  by uint_arith
haftmann@37660
  1527
haftmann@37660
  1528
lemma plus_minus_no_overflow_ab: 
haftmann@40827
  1529
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> x <= x + c" 
haftmann@37660
  1530
  by uint_arith
haftmann@37660
  1531
haftmann@37660
  1532
lemma le_minus': 
haftmann@40827
  1533
  "(a :: 'a :: len0 word) + c <= b \<Longrightarrow> a <= a + c \<Longrightarrow> c <= b - a"
haftmann@37660
  1534
  by uint_arith
haftmann@37660
  1535
haftmann@37660
  1536
lemma le_plus': 
haftmann@40827
  1537
  "(a :: 'a :: len0 word) <= b \<Longrightarrow> c <= b - a \<Longrightarrow> a + c <= b"
haftmann@37660
  1538
  by uint_arith
haftmann@37660
  1539
haftmann@37660
  1540
lemmas le_plus = le_plus' [rotated]
haftmann@37660
  1541
huffman@46011
  1542
lemmas le_minus = leD [THEN thin_rl, THEN le_minus'] (* FIXME *)
haftmann@37660
  1543
haftmann@37660
  1544
lemma word_plus_mono_right: 
haftmann@40827
  1545
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= x + z \<Longrightarrow> x + y <= x + z"
haftmann@37660
  1546
  by uint_arith
haftmann@37660
  1547
haftmann@37660
  1548
lemma word_less_minus_cancel: 
haftmann@40827
  1549
  "y - x < z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) < z"
haftmann@37660
  1550
  by uint_arith
haftmann@37660
  1551
haftmann@37660
  1552
lemma word_less_minus_mono_left: 
haftmann@40827
  1553
  "(y :: 'a :: len0 word) < z \<Longrightarrow> x <= y \<Longrightarrow> y - x < z - x"
haftmann@37660
  1554
  by uint_arith
haftmann@37660
  1555
haftmann@37660
  1556
lemma word_less_minus_mono:  
haftmann@40827
  1557
  "a < c \<Longrightarrow> d < b \<Longrightarrow> a - b < a \<Longrightarrow> c - d < c 
haftmann@40827
  1558
  \<Longrightarrow> a - b < c - (d::'a::len word)"
haftmann@37660
  1559
  by uint_arith
haftmann@37660
  1560
haftmann@37660
  1561
lemma word_le_minus_cancel: 
haftmann@40827
  1562
  "y - x <= z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) <= z"
haftmann@37660
  1563
  by uint_arith
haftmann@37660
  1564
haftmann@37660
  1565
lemma word_le_minus_mono_left: 
haftmann@40827
  1566
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= y \<Longrightarrow> y - x <= z - x"
haftmann@37660
  1567
  by uint_arith
haftmann@37660
  1568
haftmann@37660
  1569
lemma word_le_minus_mono:  
haftmann@40827
  1570
  "a <= c \<Longrightarrow> d <= b \<Longrightarrow> a - b <= a \<Longrightarrow> c - d <= c 
haftmann@40827
  1571
  \<Longrightarrow> a - b <= c - (d::'a::len word)"
haftmann@37660
  1572
  by uint_arith
haftmann@37660
  1573
haftmann@37660
  1574
lemma plus_le_left_cancel_wrap: 
haftmann@40827
  1575
  "(x :: 'a :: len0 word) + y' < x \<Longrightarrow> x + y < x \<Longrightarrow> (x + y' < x + y) = (y' < y)"
haftmann@37660
  1576
  by uint_arith
haftmann@37660
  1577
haftmann@37660
  1578
lemma plus_le_left_cancel_nowrap: 
haftmann@40827
  1579
  "(x :: 'a :: len0 word) <= x + y' \<Longrightarrow> x <= x + y \<Longrightarrow> 
haftmann@37660
  1580
    (x + y' < x + y) = (y' < y)" 
haftmann@37660
  1581
  by uint_arith
haftmann@37660
  1582
haftmann@37660
  1583
lemma word_plus_mono_right2: 
haftmann@40827
  1584
  "(a :: 'a :: len0 word) <= a + b \<Longrightarrow> c <= b \<Longrightarrow> a <= a + c"
haftmann@37660
  1585
  by uint_arith
haftmann@37660
  1586
haftmann@37660
  1587
lemma word_less_add_right: 
haftmann@40827
  1588
  "(x :: 'a :: len0 word) < y - z \<Longrightarrow> z <= y \<Longrightarrow> x + z < y"
haftmann@37660
  1589
  by uint_arith
haftmann@37660
  1590
haftmann@37660
  1591
lemma word_less_sub_right: 
haftmann@40827
  1592
  "(x :: 'a :: len0 word) < y + z \<Longrightarrow> y <= x \<Longrightarrow> x - y < z"
haftmann@37660
  1593
  by uint_arith
haftmann@37660
  1594
haftmann@37660
  1595
lemma word_le_plus_either: 
haftmann@40827
  1596
  "(x :: 'a :: len0 word) <= y | x <= z \<Longrightarrow> y <= y + z \<Longrightarrow> x <= y + z"
haftmann@37660
  1597
  by uint_arith
haftmann@37660
  1598
haftmann@37660
  1599
lemma word_less_nowrapI: 
haftmann@40827
  1600
  "(x :: 'a :: len0 word) < z - k \<Longrightarrow> k <= z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
haftmann@37660
  1601
  by uint_arith
haftmann@37660
  1602
haftmann@40827
  1603
lemma inc_le: "(i :: 'a :: len word) < m \<Longrightarrow> i + 1 <= m"
haftmann@37660
  1604
  by uint_arith
haftmann@37660
  1605
haftmann@37660
  1606
lemma inc_i: 
haftmann@40827
  1607
  "(1 :: 'a :: len word) <= i \<Longrightarrow> i < m \<Longrightarrow> 1 <= (i + 1) & i + 1 <= m"
haftmann@37660
  1608
  by uint_arith
haftmann@37660
  1609
haftmann@37660
  1610
lemma udvd_incr_lem:
haftmann@40827
  1611
  "up < uq \<Longrightarrow> up = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1612
    uq = ua + n' * uint K \<Longrightarrow> up + uint K <= uq"
haftmann@37660
  1613
  apply clarsimp
haftmann@37660
  1614
  apply (drule less_le_mult)
haftmann@37660
  1615
  apply safe
haftmann@37660
  1616
  done
haftmann@37660
  1617
haftmann@37660
  1618
lemma udvd_incr': 
haftmann@40827
  1619
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1620
    uint q = ua + n' * uint K \<Longrightarrow> p + K <= q" 
haftmann@37660
  1621
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1622
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1623
  apply (erule uint_add_le [THEN order_trans])
haftmann@37660
  1624
  done
haftmann@37660
  1625
haftmann@37660
  1626
lemma udvd_decr': 
haftmann@40827
  1627
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1628
    uint q = ua + n' * uint K \<Longrightarrow> p <= q - K"
haftmann@37660
  1629
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1630
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1631
  apply (drule le_diff_eq [THEN iffD2])
haftmann@37660
  1632
  apply (erule order_trans)
haftmann@37660
  1633
  apply (rule uint_sub_ge)
haftmann@37660
  1634
  done
haftmann@37660
  1635
huffman@45816
  1636
lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, unfolded add_0_left]
huffman@45816
  1637
lemmas udvd_incr0 = udvd_incr' [where ua=0, unfolded add_0_left]
huffman@45816
  1638
lemmas udvd_decr0 = udvd_decr' [where ua=0, unfolded add_0_left]
haftmann@37660
  1639
haftmann@37660
  1640
lemma udvd_minus_le': 
haftmann@40827
  1641
  "xy < k \<Longrightarrow> z udvd xy \<Longrightarrow> z udvd k \<Longrightarrow> xy <= k - z"
haftmann@37660
  1642
  apply (unfold udvd_def)
haftmann@37660
  1643
  apply clarify
haftmann@37660
  1644
  apply (erule (2) udvd_decr0)
haftmann@37660
  1645
  done
haftmann@37660
  1646
haftmann@37660
  1647
lemma udvd_incr2_K: 
haftmann@40827
  1648
  "p < a + s \<Longrightarrow> a <= a + s \<Longrightarrow> K udvd s \<Longrightarrow> K udvd p - a \<Longrightarrow> a <= p \<Longrightarrow> 
haftmann@40827
  1649
    0 < K \<Longrightarrow> p <= p + K & p + K <= a + s"
wenzelm@51286
  1650
  using [[simproc del: linordered_ring_less_cancel_factor]]
haftmann@37660
  1651
  apply (unfold udvd_def)
haftmann@37660
  1652
  apply clarify
haftmann@37660
  1653
  apply (simp add: uint_arith_simps split: split_if_asm)
haftmann@37660
  1654
   prefer 2 
haftmann@37660
  1655
   apply (insert uint_range' [of s])[1]
haftmann@37660
  1656
   apply arith
haftmann@37660
  1657
  apply (drule add_commute [THEN xtr1])
haftmann@37660
  1658
  apply (simp add: diff_less_eq [symmetric])
haftmann@37660
  1659
  apply (drule less_le_mult)
haftmann@37660
  1660
   apply arith
haftmann@37660
  1661
  apply simp
haftmann@37660
  1662
  done
haftmann@37660
  1663
haftmann@37660
  1664
(* links with rbl operations *)
haftmann@37660
  1665
lemma word_succ_rbl:
haftmann@40827
  1666
  "to_bl w = bl \<Longrightarrow> to_bl (word_succ w) = (rev (rbl_succ (rev bl)))"
haftmann@37660
  1667
  apply (unfold word_succ_def)
haftmann@37660
  1668
  apply clarify
haftmann@37660
  1669
  apply (simp add: to_bl_of_bin)
huffman@46654
  1670
  apply (simp add: to_bl_def rbl_succ)
haftmann@37660
  1671
  done
haftmann@37660
  1672
haftmann@37660
  1673
lemma word_pred_rbl:
haftmann@40827
  1674
  "to_bl w = bl \<Longrightarrow> to_bl (word_pred w) = (rev (rbl_pred (rev bl)))"
haftmann@37660
  1675
  apply (unfold word_pred_def)
haftmann@37660
  1676
  apply clarify
haftmann@37660
  1677
  apply (simp add: to_bl_of_bin)
huffman@46654
  1678
  apply (simp add: to_bl_def rbl_pred)
haftmann@37660
  1679
  done
haftmann@37660
  1680
haftmann@37660
  1681
lemma word_add_rbl:
haftmann@40827
  1682
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1683
    to_bl (v + w) = (rev (rbl_add (rev vbl) (rev wbl)))"
haftmann@37660
  1684
  apply (unfold word_add_def)
haftmann@37660
  1685
  apply clarify
haftmann@37660
  1686
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1687
  apply (simp add: to_bl_def rbl_add)
haftmann@37660
  1688
  done
haftmann@37660
  1689
haftmann@37660
  1690
lemma word_mult_rbl:
haftmann@40827
  1691
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1692
    to_bl (v * w) = (rev (rbl_mult (rev vbl) (rev wbl)))"
haftmann@37660
  1693
  apply (unfold word_mult_def)
haftmann@37660
  1694
  apply clarify
haftmann@37660
  1695
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1696
  apply (simp add: to_bl_def rbl_mult)
haftmann@37660
  1697
  done
haftmann@37660
  1698
haftmann@37660
  1699
lemma rtb_rbl_ariths:
haftmann@37660
  1700
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys"
haftmann@37660
  1701
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys"
haftmann@40827
  1702
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs"
haftmann@40827
  1703
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs"
haftmann@37660
  1704
  by (auto simp: rev_swap [symmetric] word_succ_rbl 
haftmann@37660
  1705
                 word_pred_rbl word_mult_rbl word_add_rbl)
haftmann@37660
  1706
haftmann@37660
  1707
haftmann@37660
  1708
subsection "Arithmetic type class instantiations"
haftmann@37660
  1709
haftmann@37660
  1710
lemmas word_le_0_iff [simp] =
haftmann@37660
  1711
  word_zero_le [THEN leD, THEN linorder_antisym_conv1]
haftmann@37660
  1712
haftmann@37660
  1713
lemma word_of_int_nat: 
haftmann@40827
  1714
  "0 <= x \<Longrightarrow> word_of_int x = of_nat (nat x)"
haftmann@37660
  1715
  by (simp add: of_nat_nat word_of_int)
haftmann@37660
  1716
huffman@46603
  1717
(* note that iszero_def is only for class comm_semiring_1_cancel,
huffman@46603
  1718
   which requires word length >= 1, ie 'a :: len word *) 
huffman@46603
  1719
lemma iszero_word_no [simp]:
huffman@47108
  1720
  "iszero (numeral bin :: 'a :: len word) = 
huffman@47108
  1721
    iszero (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
  1722
  using word_ubin.norm_eq_iff [where 'a='a, of "numeral bin" 0]
huffman@46603
  1723
  by (simp add: iszero_def [symmetric])
huffman@47108
  1724
    
huffman@47108
  1725
text {* Use @{text iszero} to simplify equalities between word numerals. *}
huffman@47108
  1726
huffman@47108
  1727
lemmas word_eq_numeral_iff_iszero [simp] =
huffman@47108
  1728
  eq_numeral_iff_iszero [where 'a="'a::len word"]
huffman@46603
  1729
haftmann@37660
  1730
haftmann@37660
  1731
subsection "Word and nat"
haftmann@37660
  1732
huffman@45811
  1733
lemma td_ext_unat [OF refl]:
haftmann@40827
  1734
  "n = len_of TYPE ('a :: len) \<Longrightarrow> 
haftmann@37660
  1735
    td_ext (unat :: 'a word => nat) of_nat 
haftmann@37660
  1736
    (unats n) (%i. i mod 2 ^ n)"
haftmann@37660
  1737
  apply (unfold td_ext_def' unat_def word_of_nat unats_uints)
haftmann@37660
  1738
  apply (auto intro!: imageI simp add : word_of_int_hom_syms)
haftmann@37660
  1739
  apply (erule word_uint.Abs_inverse [THEN arg_cong])
haftmann@37660
  1740
  apply (simp add: int_word_uint nat_mod_distrib nat_power_eq)
haftmann@37660
  1741
  done
haftmann@37660
  1742
wenzelm@45604
  1743
lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm]
haftmann@37660
  1744
haftmann@37660
  1745
interpretation word_unat:
haftmann@37660
  1746
  td_ext "unat::'a::len word => nat" 
haftmann@37660
  1747
         of_nat 
haftmann@37660
  1748
         "unats (len_of TYPE('a::len))"
haftmann@37660
  1749
         "%i. i mod 2 ^ len_of TYPE('a::len)"
haftmann@37660
  1750
  by (rule td_ext_unat)
haftmann@37660
  1751
haftmann@37660
  1752
lemmas td_unat = word_unat.td_thm
haftmann@37660
  1753
haftmann@37660
  1754
lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq]
haftmann@37660
  1755
haftmann@40827
  1756
lemma unat_le: "y <= unat (z :: 'a :: len word) \<Longrightarrow> y : unats (len_of TYPE ('a))"
haftmann@37660
  1757
  apply (unfold unats_def)
haftmann@37660
  1758
  apply clarsimp
haftmann@37660
  1759
  apply (rule xtrans, rule unat_lt2p, assumption) 
haftmann@37660
  1760
  done
haftmann@37660
  1761
haftmann@37660
  1762
lemma word_nchotomy:
haftmann@37660
  1763
  "ALL w. EX n. (w :: 'a :: len word) = of_nat n & n < 2 ^ len_of TYPE ('a)"
haftmann@37660
  1764
  apply (rule allI)
haftmann@37660
  1765
  apply (rule word_unat.Abs_cases)
haftmann@37660
  1766
  apply (unfold unats_def)
haftmann@37660
  1767
  apply auto
haftmann@37660
  1768
  done
haftmann@37660
  1769
haftmann@37660
  1770
lemma of_nat_eq:
haftmann@37660
  1771
  fixes w :: "'a::len word"
haftmann@37660
  1772
  shows "(of_nat n = w) = (\<exists>q. n = unat w + q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1773
  apply (rule trans)
haftmann@37660
  1774
   apply (rule word_unat.inverse_norm)
haftmann@37660
  1775
  apply (rule iffI)
haftmann@37660
  1776
   apply (rule mod_eqD)
haftmann@37660
  1777
   apply simp
haftmann@37660
  1778
  apply clarsimp
haftmann@37660
  1779
  done
haftmann@37660
  1780
haftmann@37660
  1781
lemma of_nat_eq_size: 
haftmann@37660
  1782
  "(of_nat n = w) = (EX q. n = unat w + q * 2 ^ size w)"
haftmann@37660
  1783
  unfolding word_size by (rule of_nat_eq)
haftmann@37660
  1784
haftmann@37660
  1785
lemma of_nat_0:
haftmann@37660
  1786
  "(of_nat m = (0::'a::len word)) = (\<exists>q. m = q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1787
  by (simp add: of_nat_eq)
haftmann@37660
  1788
huffman@45805
  1789
lemma of_nat_2p [simp]:
huffman@45805
  1790
  "of_nat (2 ^ len_of TYPE('a)) = (0::'a::len word)"
huffman@45805
  1791
  by (fact mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]])
haftmann@37660
  1792
haftmann@40827
  1793
lemma of_nat_gt_0: "of_nat k ~= 0 \<Longrightarrow> 0 < k"
haftmann@37660
  1794
  by (cases k) auto
haftmann@37660
  1795
haftmann@37660
  1796
lemma of_nat_neq_0: 
haftmann@40827
  1797
  "0 < k \<Longrightarrow> k < 2 ^ len_of TYPE ('a :: len) \<Longrightarrow> of_nat k ~= (0 :: 'a word)"
haftmann@37660
  1798
  by (clarsimp simp add : of_nat_0)
haftmann@37660
  1799
haftmann@37660
  1800
lemma Abs_fnat_hom_add:
haftmann@37660
  1801
  "of_nat a + of_nat b = of_nat (a + b)"
haftmann@37660
  1802
  by simp
haftmann@37660
  1803
haftmann@37660
  1804
lemma Abs_fnat_hom_mult:
haftmann@37660
  1805
  "of_nat a * of_nat b = (of_nat (a * b) :: 'a :: len word)"
huffman@46013
  1806
  by (simp add: word_of_nat wi_hom_mult zmult_int)
haftmann@37660
  1807
haftmann@37660
  1808
lemma Abs_fnat_hom_Suc:
haftmann@37660
  1809
  "word_succ (of_nat a) = of_nat (Suc a)"
huffman@46013
  1810
  by (simp add: word_of_nat wi_hom_succ add_ac)
haftmann@37660
  1811
haftmann@37660
  1812
lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0"
huffman@45995
  1813
  by simp
haftmann@37660
  1814
haftmann@37660
  1815
lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)"
huffman@45995
  1816
  by simp
haftmann@37660
  1817
haftmann@37660
  1818
lemmas Abs_fnat_homs = 
haftmann@37660
  1819
  Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc 
haftmann@37660
  1820
  Abs_fnat_hom_0 Abs_fnat_hom_1
haftmann@37660
  1821
haftmann@37660
  1822
lemma word_arith_nat_add:
haftmann@37660
  1823
  "a + b = of_nat (unat a + unat b)" 
haftmann@37660
  1824
  by simp
haftmann@37660
  1825
haftmann@37660
  1826
lemma word_arith_nat_mult:
haftmann@37660
  1827
  "a * b = of_nat (unat a * unat b)"
huffman@45995
  1828
  by (simp add: of_nat_mult)
haftmann@37660
  1829
    
haftmann@37660
  1830
lemma word_arith_nat_Suc:
haftmann@37660
  1831
  "word_succ a = of_nat (Suc (unat a))"
haftmann@37660
  1832
  by (subst Abs_fnat_hom_Suc [symmetric]) simp
haftmann@37660
  1833
haftmann@37660
  1834
lemma word_arith_nat_div:
haftmann@37660
  1835
  "a div b = of_nat (unat a div unat b)"
haftmann@37660
  1836
  by (simp add: word_div_def word_of_nat zdiv_int uint_nat)
haftmann@37660
  1837
haftmann@37660
  1838
lemma word_arith_nat_mod:
haftmann@37660
  1839
  "a mod b = of_nat (unat a mod unat b)"
haftmann@37660
  1840
  by (simp add: word_mod_def word_of_nat zmod_int uint_nat)
haftmann@37660
  1841
haftmann@37660
  1842
lemmas word_arith_nat_defs =
haftmann@37660
  1843
  word_arith_nat_add word_arith_nat_mult
haftmann@37660
  1844
  word_arith_nat_Suc Abs_fnat_hom_0
haftmann@37660
  1845
  Abs_fnat_hom_1 word_arith_nat_div
haftmann@37660
  1846
  word_arith_nat_mod 
haftmann@37660
  1847
huffman@45816
  1848
lemma unat_cong: "x = y \<Longrightarrow> unat x = unat y"
huffman@45816
  1849
  by simp
haftmann@37660
  1850
  
haftmann@37660
  1851
lemmas unat_word_ariths = word_arith_nat_defs
wenzelm@45604
  1852
  [THEN trans [OF unat_cong unat_of_nat]]
haftmann@37660
  1853
haftmann@37660
  1854
lemmas word_sub_less_iff = word_sub_le_iff
huffman@45816
  1855
  [unfolded linorder_not_less [symmetric] Not_eq_iff]
haftmann@37660
  1856
haftmann@37660
  1857
lemma unat_add_lem: 
haftmann@37660
  1858
  "(unat x + unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1859
    (unat (x + y :: 'a :: len word) = unat x + unat y)"
haftmann@37660
  1860
  unfolding unat_word_ariths
haftmann@37660
  1861
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1862
haftmann@37660
  1863
lemma unat_mult_lem: 
haftmann@37660
  1864
  "(unat x * unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1865
    (unat (x * y :: 'a :: len word) = unat x * unat y)"
haftmann@37660
  1866
  unfolding unat_word_ariths
haftmann@37660
  1867
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1868
wenzelm@45604
  1869
lemmas unat_plus_if' = trans [OF unat_word_ariths(1) mod_nat_add, simplified]
haftmann@37660
  1870
haftmann@37660
  1871
lemma le_no_overflow: 
haftmann@40827
  1872
  "x <= b \<Longrightarrow> a <= a + b \<Longrightarrow> x <= a + (b :: 'a :: len0 word)"
haftmann@37660
  1873
  apply (erule order_trans)
haftmann@37660
  1874
  apply (erule olen_add_eqv [THEN iffD1])
haftmann@37660
  1875
  done
haftmann@37660
  1876
wenzelm@45604
  1877
lemmas un_ui_le = trans [OF word_le_nat_alt [symmetric] word_le_def]
haftmann@37660
  1878
haftmann@37660
  1879
lemma unat_sub_if_size:
haftmann@37660
  1880
  "unat (x - y) = (if unat y <= unat x 
haftmann@37660
  1881
   then unat x - unat y 
haftmann@37660
  1882
   else unat x + 2 ^ size x - unat y)"
haftmann@37660
  1883
  apply (unfold word_size)
haftmann@37660
  1884
  apply (simp add: un_ui_le)
haftmann@37660
  1885
  apply (auto simp add: unat_def uint_sub_if')
haftmann@37660
  1886
   apply (rule nat_diff_distrib)
haftmann@37660
  1887
    prefer 3
haftmann@37660
  1888
    apply (simp add: algebra_simps)
haftmann@37660
  1889
    apply (rule nat_diff_distrib [THEN trans])
haftmann@37660
  1890
      prefer 3
haftmann@37660
  1891
      apply (subst nat_add_distrib)
haftmann@37660
  1892
        prefer 3
haftmann@37660
  1893
        apply (simp add: nat_power_eq)
haftmann@37660
  1894
       apply auto
haftmann@37660
  1895
  apply uint_arith
haftmann@37660
  1896
  done
haftmann@37660
  1897
haftmann@37660
  1898
lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size]
haftmann@37660
  1899
haftmann@37660
  1900
lemma unat_div: "unat ((x :: 'a :: len word) div y) = unat x div unat y"
haftmann@37660
  1901
  apply (simp add : unat_word_ariths)
haftmann@37660
  1902
  apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1903
  apply (rule div_le_dividend)
haftmann@37660
  1904
  done
haftmann@37660
  1905
haftmann@37660
  1906
lemma unat_mod: "unat ((x :: 'a :: len word) mod y) = unat x mod unat y"
haftmann@37660
  1907
  apply (clarsimp simp add : unat_word_ariths)
haftmann@37660
  1908
  apply (cases "unat y")
haftmann@37660
  1909
   prefer 2
haftmann@37660
  1910
   apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1911
   apply (rule mod_le_divisor)
haftmann@37660
  1912
   apply auto
haftmann@37660
  1913
  done
haftmann@37660
  1914
haftmann@37660
  1915
lemma uint_div: "uint ((x :: 'a :: len word) div y) = uint x div uint y"
haftmann@37660
  1916
  unfolding uint_nat by (simp add : unat_div zdiv_int)
haftmann@37660
  1917
haftmann@37660
  1918
lemma uint_mod: "uint ((x :: 'a :: len word) mod y) = uint x mod uint y"
haftmann@37660
  1919
  unfolding uint_nat by (simp add : unat_mod zmod_int)
haftmann@37660
  1920
haftmann@37660
  1921
haftmann@37660
  1922
subsection {* Definition of unat\_arith tactic *}
haftmann@37660
  1923
haftmann@37660
  1924
lemma unat_split:
haftmann@37660
  1925
  fixes x::"'a::len word"
haftmann@37660
  1926
  shows "P (unat x) = 
haftmann@37660
  1927
         (ALL n. of_nat n = x & n < 2^len_of TYPE('a) --> P n)"
haftmann@37660
  1928
  by (auto simp: unat_of_nat)
haftmann@37660
  1929
haftmann@37660
  1930
lemma unat_split_asm:
haftmann@37660
  1931
  fixes x::"'a::len word"
haftmann@37660
  1932
  shows "P (unat x) = 
haftmann@37660
  1933
         (~(EX n. of_nat n = x & n < 2^len_of TYPE('a) & ~ P n))"
haftmann@37660
  1934
  by (auto simp: unat_of_nat)
haftmann@37660
  1935
haftmann@37660
  1936
lemmas of_nat_inverse = 
haftmann@37660
  1937
  word_unat.Abs_inverse' [rotated, unfolded unats_def, simplified]
haftmann@37660
  1938
haftmann@37660
  1939
lemmas unat_splits = unat_split unat_split_asm
haftmann@37660
  1940
haftmann@37660
  1941
lemmas unat_arith_simps =
haftmann@37660
  1942
  word_le_nat_alt word_less_nat_alt
haftmann@37660
  1943
  word_unat.Rep_inject [symmetric]
haftmann@37660
  1944
  unat_sub_if' unat_plus_if' unat_div unat_mod
haftmann@37660
  1945
haftmann@37660
  1946
(* unat_arith_tac: tactic to reduce word arithmetic to nat, 
haftmann@37660
  1947
   try to solve via arith *)
haftmann@37660
  1948
ML {*
wenzelm@51717
  1949
fun unat_arith_simpset ctxt = 
wenzelm@51717
  1950
  ctxt addsimps @{thms unat_arith_simps}
haftmann@37660
  1951
     delsimps @{thms word_unat.Rep_inject}
wenzelm@45620
  1952
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1953
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1954
haftmann@37660
  1955
fun unat_arith_tacs ctxt =   
haftmann@37660
  1956
  let
haftmann@37660
  1957
    fun arith_tac' n t =
haftmann@37660
  1958
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1959
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1960
  in 
wenzelm@42793
  1961
    [ clarify_tac ctxt 1,
wenzelm@51717
  1962
      full_simp_tac (unat_arith_simpset ctxt) 1,
wenzelm@51717
  1963
      ALLGOALS (full_simp_tac
wenzelm@51717
  1964
        (put_simpset HOL_ss ctxt
wenzelm@51717
  1965
          |> fold Splitter.add_split @{thms unat_splits}
wenzelm@51717
  1966
          |> fold Simplifier.add_cong @{thms power_False_cong})),
wenzelm@54742
  1967
      rewrite_goals_tac ctxt @{thms word_size}, 
haftmann@37660
  1968
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1969
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1970
                         REPEAT (dtac @{thm of_nat_inverse} n THEN atac n)),
haftmann@37660
  1971
      TRYALL arith_tac' ] 
haftmann@37660
  1972
  end
haftmann@37660
  1973
haftmann@37660
  1974
fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt))
haftmann@37660
  1975
*}
haftmann@37660
  1976
haftmann@37660
  1977
method_setup unat_arith = 
haftmann@37660
  1978
  {* Scan.succeed (SIMPLE_METHOD' o unat_arith_tac) *}
haftmann@37660
  1979
  "solving word arithmetic via natural numbers and arith"
haftmann@37660
  1980
haftmann@37660
  1981
lemma no_plus_overflow_unat_size: 
haftmann@37660
  1982
  "((x :: 'a :: len word) <= x + y) = (unat x + unat y < 2 ^ size x)" 
haftmann@37660
  1983
  unfolding word_size by unat_arith
haftmann@37660
  1984
haftmann@37660
  1985
lemmas no_olen_add_nat = no_plus_overflow_unat_size [unfolded word_size]
haftmann@37660
  1986
wenzelm@45604
  1987
lemmas unat_plus_simple = trans [OF no_olen_add_nat unat_add_lem]
haftmann@37660
  1988
haftmann@37660
  1989
lemma word_div_mult: 
haftmann@40827
  1990
  "(0 :: 'a :: len word) < y \<Longrightarrow> unat x * unat y < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1991
    x * y div y = x"
haftmann@37660
  1992
  apply unat_arith
haftmann@37660
  1993
  apply clarsimp
haftmann@37660
  1994
  apply (subst unat_mult_lem [THEN iffD1])
haftmann@37660
  1995
  apply auto
haftmann@37660
  1996
  done
haftmann@37660
  1997
haftmann@40827
  1998
lemma div_lt': "(i :: 'a :: len word) <= k div x \<Longrightarrow> 
haftmann@37660
  1999
    unat i * unat x < 2 ^ len_of TYPE('a)"
haftmann@37660
  2000
  apply unat_arith
haftmann@37660
  2001
  apply clarsimp
haftmann@37660
  2002
  apply (drule mult_le_mono1)
haftmann@37660
  2003
  apply (erule order_le_less_trans)
haftmann@37660
  2004
  apply (rule xtr7 [OF unat_lt2p div_mult_le])
haftmann@37660
  2005
  done
haftmann@37660
  2006
haftmann@37660
  2007
lemmas div_lt'' = order_less_imp_le [THEN div_lt']
haftmann@37660
  2008
haftmann@40827
  2009
lemma div_lt_mult: "(i :: 'a :: len word) < k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x < k"
haftmann@37660
  2010
  apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  2011
  apply (simp add: unat_arith_simps)
haftmann@37660
  2012
  apply (drule (1) mult_less_mono1)
haftmann@37660
  2013
  apply (erule order_less_le_trans)
haftmann@37660
  2014
  apply (rule div_mult_le)
haftmann@37660
  2015
  done
haftmann@37660
  2016
haftmann@37660
  2017
lemma div_le_mult: 
haftmann@40827
  2018
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x <= k"
haftmann@37660
  2019
  apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  2020
  apply (simp add: unat_arith_simps)
haftmann@37660
  2021
  apply (drule mult_le_mono1)
haftmann@37660
  2022
  apply (erule order_trans)
haftmann@37660
  2023
  apply (rule div_mult_le)
haftmann@37660
  2024
  done
haftmann@37660
  2025
haftmann@37660
  2026
lemma div_lt_uint': 
haftmann@40827
  2027
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> uint i * uint x < 2 ^ len_of TYPE('a)"
haftmann@37660
  2028
  apply (unfold uint_nat)
haftmann@37660
  2029
  apply (drule div_lt')
haftmann@37660
  2030
  apply (simp add: zmult_int zless_nat_eq_int_zless [symmetric] 
haftmann@37660
  2031
                   nat_power_eq)
haftmann@37660
  2032
  done
haftmann@37660
  2033
haftmann@37660
  2034
lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint']
haftmann@37660
  2035
haftmann@37660
  2036
lemma word_le_exists': 
haftmann@40827
  2037
  "(x :: 'a :: len0 word) <= y \<Longrightarrow> 
haftmann@37660
  2038
    (EX z. y = x + z & uint x + uint z < 2 ^ len_of TYPE('a))"
haftmann@37660
  2039
  apply (rule exI)
haftmann@37660
  2040
  apply (rule conjI)
haftmann@37660
  2041
  apply (rule zadd_diff_inverse)
haftmann@37660
  2042
  apply uint_arith
haftmann@37660
  2043
  done
haftmann@37660
  2044
haftmann@37660
  2045
lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab]
haftmann@37660
  2046
haftmann@37660
  2047
lemmas plus_minus_no_overflow =
haftmann@37660
  2048
  order_less_imp_le [THEN plus_minus_no_overflow_ab]
haftmann@37660
  2049
  
haftmann@37660
  2050
lemmas mcs = word_less_minus_cancel word_less_minus_mono_left
haftmann@37660
  2051
  word_le_minus_cancel word_le_minus_mono_left
haftmann@37660
  2052
wenzelm@45604
  2053
lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2054
lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2055
lemmas word_plus_mcs = word_diff_ls [where y = "v + x", unfolded add_diff_cancel] for v x
haftmann@37660
  2056
haftmann@37660
  2057
lemmas le_unat_uoi = unat_le [THEN word_unat.Abs_inverse]
haftmann@37660
  2058
haftmann@37660
  2059
lemmas thd = refl [THEN [2] split_div_lemma [THEN iffD2], THEN conjunct1]
haftmann@37660
  2060
haftmann@37660
  2061
lemma thd1:
haftmann@37660
  2062
  "a div b * b \<le> (a::nat)"
haftmann@37660
  2063
  using gt_or_eq_0 [of b]
haftmann@37660
  2064
  apply (rule disjE)
haftmann@37660
  2065
   apply (erule xtr4 [OF thd mult_commute])
haftmann@37660
  2066
  apply clarsimp
haftmann@37660
  2067
  done
haftmann@37660
  2068
wenzelm@45604
  2069
lemmas uno_simps [THEN le_unat_uoi] = mod_le_divisor div_le_dividend thd1 
haftmann@37660
  2070
haftmann@37660
  2071
lemma word_mod_div_equality:
haftmann@37660
  2072
  "(n div b) * b + (n mod b) = (n :: 'a :: len word)"
haftmann@37660
  2073
  apply (unfold word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2074
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2075
  apply (erule disjE)
haftmann@37660
  2076
   apply (simp add: mod_div_equality uno_simps)
haftmann@37660
  2077
  apply simp
haftmann@37660
  2078
  done
haftmann@37660
  2079
haftmann@37660
  2080
lemma word_div_mult_le: "a div b * b <= (a::'a::len word)"
haftmann@37660
  2081
  apply (unfold word_le_nat_alt word_arith_nat_defs)
haftmann@37660
  2082
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2083
  apply (erule disjE)
haftmann@37660
  2084
   apply (simp add: div_mult_le uno_simps)
haftmann@37660
  2085
  apply simp
haftmann@37660
  2086
  done
haftmann@37660
  2087
haftmann@40827
  2088
lemma word_mod_less_divisor: "0 < n \<Longrightarrow> m mod n < (n :: 'a :: len word)"
haftmann@37660
  2089
  apply (simp only: word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2090
  apply (clarsimp simp add : uno_simps)
haftmann@37660
  2091
  done
haftmann@37660
  2092
haftmann@37660
  2093
lemma word_of_int_power_hom: 
haftmann@37660
  2094
  "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a :: len word)"
huffman@45995
  2095
  by (induct n) (simp_all add: wi_hom_mult [symmetric])
haftmann@37660
  2096
haftmann@37660
  2097
lemma word_arith_power_alt: 
haftmann@37660
  2098
  "a ^ n = (word_of_int (uint a ^ n) :: 'a :: len word)"
haftmann@37660
  2099
  by (simp add : word_of_int_power_hom [symmetric])
haftmann@37660
  2100
haftmann@37660
  2101
lemma of_bl_length_less: 
haftmann@40827
  2102
  "length x = k \<Longrightarrow> k < len_of TYPE('a) \<Longrightarrow> (of_bl x :: 'a :: len word) < 2 ^ k"
huffman@47108
  2103
  apply (unfold of_bl_def word_less_alt word_numeral_alt)
haftmann@37660
  2104
  apply safe
haftmann@37660
  2105
  apply (simp (no_asm) add: word_of_int_power_hom word_uint.eq_norm 
huffman@47108
  2106
                       del: word_of_int_numeral)
haftmann@37660
  2107
  apply (simp add: mod_pos_pos_trivial)
haftmann@37660
  2108
  apply (subst mod_pos_pos_trivial)
haftmann@37660
  2109
    apply (rule bl_to_bin_ge0)
haftmann@37660
  2110
   apply (rule order_less_trans)
haftmann@37660
  2111
    apply (rule bl_to_bin_lt2p)
haftmann@37660
  2112
   apply simp
huffman@46646
  2113
  apply (rule bl_to_bin_lt2p)
haftmann@37660
  2114
  done
haftmann@37660
  2115
haftmann@37660
  2116
haftmann@37660
  2117
subsection "Cardinality, finiteness of set of words"
haftmann@37660
  2118
huffman@45809
  2119
instance word :: (len0) finite
huffman@45809
  2120
  by (default, simp add: type_definition.univ [OF type_definition_word])
huffman@45809
  2121
huffman@45809
  2122
lemma card_word: "CARD('a::len0 word) = 2 ^ len_of TYPE('a)"
huffman@45809
  2123
  by (simp add: type_definition.card [OF type_definition_word] nat_power_eq)
haftmann@37660
  2124
haftmann@37660
  2125
lemma card_word_size: 
huffman@45809
  2126
  "card (UNIV :: 'a :: len0 word set) = (2 ^ size (x :: 'a word))"
haftmann@37660
  2127
unfolding word_size by (rule card_word)
haftmann@37660
  2128
haftmann@37660
  2129
haftmann@37660
  2130
subsection {* Bitwise Operations on Words *}
haftmann@37660
  2131
haftmann@37660
  2132
lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or
haftmann@37660
  2133
  
haftmann@37660
  2134
(* following definitions require both arithmetic and bit-wise word operations *)
haftmann@37660
  2135
haftmann@37660
  2136
(* to get word_no_log_defs from word_log_defs, using bin_log_bintrs *)
haftmann@37660
  2137
lemmas wils1 = bin_log_bintrs [THEN word_ubin.norm_eq_iff [THEN iffD1],
wenzelm@45604
  2138
  folded word_ubin.eq_norm, THEN eq_reflection]
haftmann@37660
  2139
haftmann@37660
  2140
(* the binary operations only *)
huffman@46013
  2141
(* BH: why is this needed? *)
haftmann@37660
  2142
lemmas word_log_binary_defs = 
haftmann@37660
  2143
  word_and_def word_or_def word_xor_def
haftmann@37660
  2144
huffman@46011
  2145
lemma word_wi_log_defs:
huffman@46011
  2146
  "NOT word_of_int a = word_of_int (NOT a)"
huffman@46011
  2147
  "word_of_int a AND word_of_int b = word_of_int (a AND b)"
huffman@46011
  2148
  "word_of_int a OR word_of_int b = word_of_int (a OR b)"
huffman@46011
  2149
  "word_of_int a XOR word_of_int b = word_of_int (a XOR b)"
huffman@47374
  2150
  by (transfer, rule refl)+
huffman@47372
  2151
huffman@46011
  2152
lemma word_no_log_defs [simp]:
huffman@47108
  2153
  "NOT (numeral a) = word_of_int (NOT (numeral a))"
haftmann@54489
  2154
  "NOT (- numeral a) = word_of_int (NOT (- numeral a))"
huffman@47108
  2155
  "numeral a AND numeral b = word_of_int (numeral a AND numeral b)"
haftmann@54489
  2156
  "numeral a AND - numeral b = word_of_int (numeral a AND - numeral b)"
haftmann@54489
  2157
  "- numeral a AND numeral b = word_of_int (- numeral a AND numeral b)"
haftmann@54489
  2158
  "- numeral a AND - numeral b = word_of_int (- numeral a AND - numeral b)"
huffman@47108
  2159
  "numeral a OR numeral b = word_of_int (numeral a OR numeral b)"
haftmann@54489
  2160
  "numeral a OR - numeral b = word_of_int (numeral a OR - numeral b)"
haftmann@54489
  2161
  "- numeral a OR numeral b = word_of_int (- numeral a OR numeral b)"
haftmann@54489
  2162
  "- numeral a OR - numeral b = word_of_int (- numeral a OR - numeral b)"
huffman@47108
  2163
  "numeral a XOR numeral b = word_of_int (numeral a XOR numeral b)"
haftmann@54489
  2164
  "numeral a XOR - numeral b = word_of_int (numeral a XOR - numeral b)"
haftmann@54489
  2165
  "- numeral a XOR numeral b = word_of_int (- numeral a XOR numeral b)"
haftmann@54489
  2166
  "- numeral a XOR - numeral b = word_of_int (- numeral a XOR - numeral b)"
huffman@47372
  2167
  by (transfer, rule refl)+
haftmann@37660
  2168
huffman@46064
  2169
text {* Special cases for when one of the arguments equals 1. *}
huffman@46064
  2170
huffman@46064
  2171
lemma word_bitwise_1_simps [simp]:
huffman@46064
  2172
  "NOT (1::'a::len0 word) = -2"
huffman@47108
  2173
  "1 AND numeral b = word_of_int (1 AND numeral b)"
haftmann@54489
  2174
  "1 AND - numeral b = word_of_int (1 AND - numeral b)"
huffman@47108
  2175
  "numeral a AND 1 = word_of_int (numeral a AND 1)"
haftmann@54489
  2176
  "- numeral a AND 1 = word_of_int (- numeral a AND 1)"
huffman@47108
  2177
  "1 OR numeral b = word_of_int (1 OR numeral b)"
haftmann@54489
  2178
  "1 OR - numeral b = word_of_int (1 OR - numeral b)"
huffman@47108
  2179
  "numeral a OR 1 = word_of_int (numeral a OR 1)"
haftmann@54489
  2180
  "- numeral a OR 1 = word_of_int (- numeral a OR 1)"
huffman@47108
  2181
  "1 XOR numeral b = word_of_int (1 XOR numeral b)"
haftmann@54489
  2182
  "1 XOR - numeral b = word_of_int (1 XOR - numeral b)"
huffman@47108
  2183
  "numeral a XOR 1 = word_of_int (numeral a XOR 1)"
haftmann@54489
  2184
  "- numeral a XOR 1 = word_of_int (- numeral a XOR 1)"
huffman@47372
  2185
  by (transfer, simp)+
huffman@46064
  2186
haftmann@37660
  2187
lemma uint_or: "uint (x OR y) = (uint x) OR (uint y)"
huffman@47372
  2188
  by (transfer, simp add: bin_trunc_ao)
haftmann@37660
  2189
haftmann@37660
  2190
lemma uint_and: "uint (x AND y) = (uint x) AND (uint y)"
huffman@47372
  2191
  by (transfer, simp add: bin_trunc_ao)
huffman@47372
  2192
huffman@47372
  2193
lemma test_bit_wi [simp]:
huffman@47372
  2194
  "(word_of_int x::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a) \<and> bin_nth x n"
huffman@47372
  2195
  unfolding word_test_bit_def
huffman@47372
  2196
  by (simp add: word_ubin.eq_norm nth_bintr)
huffman@47372
  2197
huffman@47372
  2198
lemma word_test_bit_transfer [transfer_rule]:
kuncar@51375
  2199
  "(fun_rel pcr_word (fun_rel op = op =))
huffman@47372
  2200
    (\<lambda>x n. n < len_of TYPE('a) \<and> bin_nth x n) (test_bit :: 'a::len0 word \<Rightarrow> _)"
kuncar@51375
  2201
  unfolding fun_rel_def word.pcr_cr_eq cr_word_def by simp
haftmann@37660
  2202
haftmann@37660
  2203
lemma word_ops_nth_size:
haftmann@40827
  2204
  "n < size (x::'a::len0 word) \<Longrightarrow> 
haftmann@37660
  2205
    (x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2206
    (x AND y) !! n = (x !! n & y !! n) & 
haftmann@37660
  2207
    (x XOR y) !! n = (x !! n ~= y !! n) & 
haftmann@37660
  2208
    (NOT x) !! n = (~ x !! n)"
huffman@47372
  2209
  unfolding word_size by transfer (simp add: bin_nth_ops)
haftmann@37660
  2210
haftmann@37660
  2211
lemma word_ao_nth:
haftmann@37660
  2212
  fixes x :: "'a::len0 word"
haftmann@37660
  2213
  shows "(x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2214
         (x AND y) !! n = (x !! n & y !! n)"
huffman@47372
  2215
  by transfer (auto simp add: bin_nth_ops)
huffman@46023
  2216
huffman@47108
  2217
lemma test_bit_numeral [simp]:
huffman@47108
  2218
  "(numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
huffman@47108
  2219
    n < len_of TYPE('a) \<and> bin_nth (numeral w) n"
huffman@47372
  2220
  by transfer (rule refl)
huffman@47108
  2221
huffman@47108
  2222
lemma test_bit_neg_numeral [simp]:
haftmann@54489
  2223
  "(- numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
haftmann@54489
  2224
    n < len_of TYPE('a) \<and> bin_nth (- numeral w) n"
huffman@47372
  2225
  by transfer (rule refl)
huffman@46023
  2226
huffman@46172
  2227
lemma test_bit_1 [simp]: "(1::'a::len word) !! n \<longleftrightarrow> n = 0"
huffman@47372
  2228
  by transfer auto
huffman@46172
  2229
  
huffman@46023
  2230
lemma nth_0 [simp]: "~ (0::'a::len0 word) !! n"
huffman@47372
  2231
  by transfer simp
huffman@46023
  2232
huffman@47108
  2233
lemma nth_minus1 [simp]: "(-1::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a)"
huffman@47372
  2234
  by transfer simp
huffman@47108
  2235
haftmann@37660
  2236
(* get from commutativity, associativity etc of int_and etc
haftmann@37660
  2237
  to same for word_and etc *)
haftmann@37660
  2238
haftmann@37660
  2239
lemmas bwsimps = 
huffman@46013
  2240
  wi_hom_add
haftmann@37660
  2241
  word_wi_log_defs
haftmann@37660
  2242
haftmann@37660
  2243
lemma word_bw_assocs:
haftmann@37660
  2244
  fixes x :: "'a::len0 word"
haftmann@37660
  2245
  shows
haftmann@37660
  2246
  "(x AND y) AND z = x AND y AND z"
haftmann@37660
  2247
  "(x OR y) OR z = x OR y OR z"
haftmann@37660
  2248
  "(x XOR y) XOR z = x XOR y XOR z"
huffman@46022
  2249
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2250
  
haftmann@37660
  2251
lemma word_bw_comms:
haftmann@37660
  2252
  fixes x :: "'a::len0 word"
haftmann@37660
  2253
  shows
haftmann@37660
  2254
  "x AND y = y AND x"
haftmann@37660
  2255
  "x OR y = y OR x"
haftmann@37660
  2256
  "x XOR y = y XOR x"
huffman@46022
  2257
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2258
  
haftmann@37660
  2259
lemma word_bw_lcs:
haftmann@37660
  2260
  fixes x :: "'a::len0 word"
haftmann@37660
  2261
  shows
haftmann@37660
  2262
  "y AND x AND z = x AND y AND z"
haftmann@37660
  2263
  "y OR x OR z = x OR y OR z"
haftmann@37660
  2264
  "y XOR x XOR z = x XOR y XOR z"
huffman@46022
  2265
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2266
haftmann@37660
  2267
lemma word_log_esimps [simp]:
haftmann@37660
  2268
  fixes x :: "'a::len0 word"
haftmann@37660
  2269
  shows
haftmann@37660
  2270
  "x AND 0 = 0"
haftmann@37660
  2271
  "x AND -1 = x"
haftmann@37660
  2272
  "x OR 0 = x"
haftmann@37660
  2273
  "x OR -1 = -1"
haftmann@37660
  2274
  "x XOR 0 = x"
haftmann@37660
  2275
  "x XOR -1 = NOT x"
haftmann@37660
  2276
  "0 AND x = 0"
haftmann@37660
  2277
  "-1 AND x = x"
haftmann@37660
  2278
  "0 OR x = x"
haftmann@37660
  2279
  "-1 OR x = -1"
haftmann@37660
  2280
  "0 XOR x = x"
haftmann@37660
  2281
  "-1 XOR x = NOT x"
huffman@46023
  2282
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2283
haftmann@37660
  2284
lemma word_not_dist:
haftmann@37660
  2285
  fixes x :: "'a::len0 word"
haftmann@37660
  2286
  shows
haftmann@37660
  2287
  "NOT (x OR y) = NOT x AND NOT y"
haftmann@37660
  2288
  "NOT (x AND y) = NOT x OR NOT y"
huffman@46022
  2289
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2290
haftmann@37660
  2291
lemma word_bw_same:
haftmann@37660
  2292
  fixes x :: "'a::len0 word"
haftmann@37660
  2293
  shows
haftmann@37660
  2294
  "x AND x = x"
haftmann@37660
  2295
  "x OR x = x"
haftmann@37660
  2296
  "x XOR x = 0"
huffman@46023
  2297
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2298
haftmann@37660
  2299
lemma word_ao_absorbs [simp]:
haftmann@37660
  2300
  fixes x :: "'a::len0 word"
haftmann@37660
  2301
  shows
haftmann@37660
  2302
  "x AND (y OR x) = x"
haftmann@37660
  2303
  "x OR y AND x = x"
haftmann@37660
  2304
  "x AND (x OR y) = x"
haftmann@37660
  2305
  "y AND x OR x = x"
haftmann@37660
  2306
  "(y OR x) AND x = x"
haftmann@37660
  2307
  "x OR x AND y = x"
haftmann@37660
  2308
  "(x OR y) AND x = x"
haftmann@37660
  2309
  "x AND y OR x = x"
huffman@46022
  2310
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2311
haftmann@37660
  2312
lemma word_not_not [simp]:
haftmann@37660
  2313
  "NOT NOT (x::'a::len0 word) = x"
huffman@46022
  2314
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2315
haftmann@37660
  2316
lemma word_ao_dist:
haftmann@37660
  2317
  fixes x :: "'a::len0 word"
haftmann@37660
  2318
  shows "(x OR y) AND z = x AND z OR y AND z"
huffman@46022
  2319
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2320
haftmann@37660
  2321
lemma word_oa_dist:
haftmann@37660
  2322
  fixes x :: "'a::len0 word"
haftmann@37660
  2323
  shows "x AND y OR z = (x OR z) AND (y OR z)"
huffman@46022
  2324
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2325
haftmann@37660
  2326
lemma word_add_not [simp]: 
haftmann@37660
  2327
  fixes x :: "'a::len0 word"
haftmann@37660
  2328
  shows "x + NOT x = -1"
huffman@47372
  2329
  by transfer (simp add: bin_add_not)
haftmann@37660
  2330
haftmann@37660
  2331
lemma word_plus_and_or [simp]:
haftmann@37660
  2332
  fixes x :: "'a::len0 word"
haftmann@37660
  2333
  shows "(x AND y) + (x OR y) = x + y"
huffman@47372
  2334
  by transfer (simp add: plus_and_or)
haftmann@37660
  2335
haftmann@37660
  2336
lemma leoa:   
haftmann@37660
  2337
  fixes x :: "'a::len0 word"
haftmann@40827
  2338
  shows "(w = (x OR y)) \<Longrightarrow> (y = (w AND y))" by auto
haftmann@37660
  2339
lemma leao: 
haftmann@37660
  2340
  fixes x' :: "'a::len0 word"
haftmann@40827
  2341
  shows "(w' = (x' AND y')) \<Longrightarrow> (x' = (x' OR w'))" by auto 
haftmann@37660
  2342
wenzelm@48196
  2343
lemma word_ao_equiv:
wenzelm@48196
  2344
  fixes w w' :: "'a::len0 word"
wenzelm@48196
  2345
  shows "(w = w OR w') = (w' = w AND w')"
wenzelm@48196
  2346
  by (auto intro: leoa leao)
haftmann@37660
  2347
haftmann@37660
  2348
lemma le_word_or2: "x <= x OR (y::'a::len0 word)"
haftmann@37660
  2349
  unfolding word_le_def uint_or
haftmann@37660
  2350
  by (auto intro: le_int_or) 
haftmann@37660
  2351
wenzelm@45604
  2352
lemmas le_word_or1 = xtr3 [OF word_bw_comms (2) le_word_or2]
wenzelm@45604
  2353
lemmas word_and_le1 = xtr3 [OF word_ao_absorbs (4) [symmetric] le_word_or2]
wenzelm@45604
  2354
lemmas word_and_le2 = xtr3 [OF word_ao_absorbs (8) [symmetric] le_word_or2]
haftmann@37660
  2355
haftmann@37660
  2356
lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" 
huffman@45550
  2357
  unfolding to_bl_def word_log_defs bl_not_bin
huffman@45550
  2358
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2359
haftmann@37660
  2360
lemma bl_word_xor: "to_bl (v XOR w) = map2 op ~= (to_bl v) (to_bl w)" 
haftmann@37660
  2361
  unfolding to_bl_def word_log_defs bl_xor_bin
huffman@45550
  2362
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2363
haftmann@37660
  2364
lemma bl_word_or: "to_bl (v OR w) = map2 op | (to_bl v) (to_bl w)" 
huffman@45550
  2365
  unfolding to_bl_def word_log_defs bl_or_bin
huffman@45550
  2366
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2367
haftmann@37660
  2368
lemma bl_word_and: "to_bl (v AND w) = map2 op & (to_bl v) (to_bl w)" 
huffman@45550
  2369
  unfolding to_bl_def word_log_defs bl_and_bin
huffman@45550
  2370
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2371
haftmann@37660
  2372
lemma word_lsb_alt: "lsb (w::'a::len0 word) = test_bit w 0"
haftmann@37660
  2373
  by (auto simp: word_test_bit_def word_lsb_def)
haftmann@37660
  2374
huffman@45805
  2375
lemma word_lsb_1_0 [simp]: "lsb (1::'a::len word) & ~ lsb (0::'b::len0 word)"
huffman@45550
  2376
  unfolding word_lsb_def uint_eq_0 uint_1 by simp
haftmann@37660
  2377
haftmann@37660
  2378
lemma word_lsb_last: "lsb (w::'a::len word) = last (to_bl w)"
haftmann@37660
  2379
  apply (unfold word_lsb_def uint_bl bin_to_bl_def) 
haftmann@37660
  2380
  apply (rule_tac bin="uint w" in bin_exhaust)
haftmann@37660
  2381
  apply (cases "size w")
haftmann@37660
  2382
   apply auto
haftmann@37660
  2383
   apply (auto simp add: bin_to_bl_aux_alt)
haftmann@37660
  2384
  done
haftmann@37660
  2385
haftmann@37660
  2386
lemma word_lsb_int: "lsb w = (uint w mod 2 = 1)"
huffman@45529
  2387
  unfolding word_lsb_def bin_last_def by auto
haftmann@37660
  2388
haftmann@37660
  2389
lemma word_msb_sint: "msb w = (sint w < 0)" 
huffman@46604
  2390
  unfolding word_msb_def sign_Min_lt_0 ..
haftmann@37660
  2391
huffman@46173
  2392
lemma msb_word_of_int:
huffman@46173
  2393
  "msb (word_of_int x::'a::len word) = bin_nth x (len_of TYPE('a) - 1)"
huffman@46173
  2394
  unfolding word_msb_def by (simp add: word_sbin.eq_norm bin_sign_lem)
huffman@46173
  2395
huffman@47108
  2396
lemma word_msb_numeral [simp]:
huffman@47108
  2397
  "msb (numeral w::'a::len word) = bin_nth (numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2398
  unfolding word_numeral_alt by (rule msb_word_of_int)
huffman@47108
  2399
huffman@47108
  2400
lemma word_msb_neg_numeral [simp]:
haftmann@54489
  2401
  "msb (- numeral w::'a::len word) = bin_nth (- numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2402
  unfolding word_neg_numeral_alt by (rule msb_word_of_int)
huffman@46173
  2403
huffman@46173
  2404
lemma word_msb_0 [simp]: "\<not> msb (0::'a::len word)"
huffman@46173
  2405
  unfolding word_msb_def by simp
huffman@46173
  2406
huffman@46173
  2407
lemma word_msb_1 [simp]: "msb (1::'a::len word) \<longleftrightarrow> len_of TYPE('a) = 1"
huffman@46173
  2408
  unfolding word_1_wi msb_word_of_int eq_iff [where 'a=nat]
huffman@46173
  2409
  by (simp add: Suc_le_eq)
huffman@45811
  2410
huffman@45811
  2411
lemma word_msb_nth:
huffman@45811
  2412
  "msb (w::'a::len word) = bin_nth (uint w) (len_of TYPE('a) - 1)"
huffman@46023
  2413
  unfolding word_msb_def sint_uint by (simp add: bin_sign_lem)
haftmann@37660
  2414
haftmann@37660
  2415
lemma word_msb_alt: "msb (w::'a::len word) = hd (to_bl w)"
haftmann@37660
  2416
  apply (unfold word_msb_nth uint_bl)
haftmann@37660
  2417
  apply (subst hd_conv_nth)
haftmann@37660
  2418
  apply (rule length_greater_0_conv [THEN iffD1])
haftmann@37660
  2419
   apply simp
haftmann@37660
  2420
  apply (simp add : nth_bin_to_bl word_size)
haftmann@37660
  2421
  done
haftmann@37660
  2422
huffman@45805
  2423
lemma word_set_nth [simp]:
haftmann@37660
  2424
  "set_bit w n (test_bit w n) = (w::'a::len0 word)"
haftmann@37660
  2425
  unfolding word_test_bit_def word_set_bit_def by auto
haftmann@37660
  2426
haftmann@37660
  2427
lemma bin_nth_uint':
haftmann@37660
  2428
  "bin_nth (uint w) n = (rev (bin_to_bl (size w) (uint w)) ! n & n < size w)"
haftmann@37660
  2429
  apply (unfold word_size)
haftmann@37660
  2430
  apply (safe elim!: bin_nth_uint_imp)
haftmann@37660
  2431
   apply (frule bin_nth_uint_imp)
haftmann@37660
  2432
   apply (fast dest!: bin_nth_bl)+
haftmann@37660
  2433
  done
haftmann@37660
  2434
haftmann@37660
  2435
lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size]
haftmann@37660
  2436
haftmann@37660
  2437
lemma test_bit_bl: "w !! n = (rev (to_bl w) ! n & n < size w)"
haftmann@37660
  2438
  unfolding to_bl_def word_test_bit_def word_size
haftmann@37660
  2439
  by (rule bin_nth_uint)
haftmann@37660
  2440
haftmann@40827
  2441
lemma to_bl_nth: "n < size w \<Longrightarrow> to_bl w ! n = w !! (size w - Suc n)"
haftmann@37660
  2442
  apply (unfold test_bit_bl)
haftmann@37660
  2443
  apply clarsimp
haftmann@37660
  2444
  apply (rule trans)
haftmann@37660
  2445
   apply (rule nth_rev_alt)
haftmann@37660
  2446
   apply (auto simp add: word_size)
haftmann@37660
  2447
  done
haftmann@37660
  2448
haftmann@37660
  2449
lemma test_bit_set: 
haftmann@37660
  2450
  fixes w :: "'a::len0 word"
haftmann@37660
  2451
  shows "(set_bit w n x) !! n = (n < size w & x)"
haftmann@37660
  2452
  unfolding word_size word_test_bit_def word_set_bit_def
haftmann@37660
  2453
  by (clarsimp simp add : word_ubin.eq_norm nth_bintr)
haftmann@37660
  2454
haftmann@37660
  2455
lemma test_bit_set_gen: 
haftmann@37660
  2456
  fixes w :: "'a::len0 word"
haftmann@37660
  2457
  shows "test_bit (set_bit w n x) m = 
haftmann@37660
  2458
         (if m = n then n < size w & x else test_bit w m)"
haftmann@37660
  2459
  apply (unfold word_size word_test_bit_def word_set_bit_def)
haftmann@37660
  2460
  apply (clarsimp simp add: word_ubin.eq_norm nth_bintr bin_nth_sc_gen)
haftmann@37660
  2461
  apply (auto elim!: test_bit_size [unfolded word_size]
haftmann@37660
  2462
              simp add: word_test_bit_def [symmetric])
haftmann@37660
  2463
  done
haftmann@37660
  2464
haftmann@37660
  2465
lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs"
haftmann@37660
  2466
  unfolding of_bl_def bl_to_bin_rep_F by auto
haftmann@37660
  2467
  
huffman@45811
  2468
lemma msb_nth:
haftmann@37660
  2469
  fixes w :: "'a::len word"
huffman@45811
  2470
  shows "msb w = w !! (len_of TYPE('a) - 1)"
huffman@45811
  2471
  unfolding word_msb_nth word_test_bit_def by simp
haftmann@37660
  2472
wenzelm@45604
  2473
lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2474
lemmas msb1 = msb0 [where i = 0]
haftmann@37660
  2475
lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]]
haftmann@37660
  2476
wenzelm@45604
  2477
lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2478
lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt]
haftmann@37660
  2479
huffman@45811
  2480
lemma td_ext_nth [OF refl refl refl, unfolded word_size]:
haftmann@40827
  2481
  "n = size (w::'a::len0 word) \<Longrightarrow> ofn = set_bits \<Longrightarrow> [w, ofn g] = l \<Longrightarrow> 
haftmann@37660
  2482
    td_ext test_bit ofn {f. ALL i. f i --> i < n} (%h i. h i & i < n)"
haftmann@37660
  2483
  apply (unfold word_size td_ext_def')
wenzelm@46008
  2484
  apply safe
haftmann@37660
  2485
     apply (rule_tac [3] ext)
haftmann@37660
  2486
     apply (rule_tac [4] ext)
haftmann@37660
  2487
     apply (unfold word_size of_nth_def test_bit_bl)
haftmann@37660
  2488
     apply safe
haftmann@37660
  2489
       defer
haftmann@37660
  2490
       apply (clarsimp simp: word_bl.Abs_inverse)+
haftmann@37660
  2491
  apply (rule word_bl.Rep_inverse')
haftmann@37660
  2492
  apply (rule sym [THEN trans])
haftmann@37660
  2493
  apply (rule bl_of_nth_nth)
haftmann@37660
  2494
  apply simp
haftmann@37660
  2495
  apply (rule bl_of_nth_inj)
haftmann@37660
  2496
  apply (clarsimp simp add : test_bit_bl word_size)
haftmann@37660
  2497
  done
haftmann@37660
  2498
haftmann@37660
  2499
interpretation test_bit:
haftmann@37660
  2500
  td_ext "op !! :: 'a::len0 word => nat => bool"
haftmann@37660
  2501
         set_bits
haftmann@37660
  2502
         "{f. \<forall>i. f i \<longrightarrow> i < len_of TYPE('a::len0)}"
haftmann@37660
  2503
         "(\<lambda>h i. h i \<and> i < len_of TYPE('a::len0))"
haftmann@37660
  2504
  by (rule td_ext_nth)
haftmann@37660
  2505
haftmann@37660
  2506
lemmas td_nth = test_bit.td_thm
haftmann@37660
  2507
huffman@45805
  2508
lemma word_set_set_same [simp]:
haftmann@37660
  2509
  fixes w :: "'a::len0 word"
haftmann@37660
  2510
  shows "set_bit (set_bit w n x) n y = set_bit w n y" 
haftmann@37660
  2511
  by (rule word_eqI) (simp add : test_bit_set_gen word_size)
haftmann@37660
  2512
    
haftmann@37660
  2513
lemma word_set_set_diff: 
haftmann@37660
  2514
  fixes w :: "'a::len0 word"
haftmann@37660
  2515
  assumes "m ~= n"
haftmann@37660
  2516
  shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x" 
wenzelm@41550
  2517
  by (rule word_eqI) (clarsimp simp add: test_bit_set_gen word_size assms)
huffman@46001
  2518
haftmann@37660
  2519
lemma nth_sint: 
haftmann@37660
  2520
  fixes w :: "'a::len word"
haftmann@37660
  2521
  defines "l \<equiv> len_of TYPE ('a)"
haftmann@37660
  2522
  shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
haftmann@37660
  2523
  unfolding sint_uint l_def
haftmann@37660
  2524
  by (clarsimp simp add: nth_sbintr word_test_bit_def [symmetric])
haftmann@37660
  2525
huffman@47108
  2526
lemma word_lsb_numeral [simp]:
huffman@47108
  2527
  "lsb (numeral bin :: 'a :: len word) = (bin_last (numeral bin) = 1)"
huffman@47108
  2528
  unfolding word_lsb_alt test_bit_numeral by simp
huffman@47108
  2529
huffman@47108
  2530
lemma word_lsb_neg_numeral [simp]:
haftmann@54489
  2531
  "lsb (- numeral bin :: 'a :: len word) = (bin_last (- numeral bin) = 1)"
huffman@47108
  2532
  unfolding word_lsb_alt test_bit_neg_numeral by simp
haftmann@37660
  2533
huffman@46173
  2534
lemma set_bit_word_of_int:
huffman@46173
  2535
  "set_bit (word_of_int x) n b = word_of_int (bin_sc n (if b then 1 else 0) x)"
huffman@46173
  2536
  unfolding word_set_bit_def
huffman@46173
  2537
  apply (rule word_eqI)
huffman@46173
  2538
  apply (simp add: word_size bin_nth_sc_gen word_ubin.eq_norm nth_bintr)
huffman@46173
  2539
  done
huffman@46173
  2540
huffman@47108
  2541
lemma word_set_numeral [simp]:
huffman@47108
  2542
  "set_bit (numeral bin::'a::len0 word) n b = 
huffman@47108
  2543
    word_of_int (bin_sc n (if b then 1 else 0) (numeral bin))"
huffman@47108
  2544
  unfolding word_numeral_alt by (rule set_bit_word_of_int)
huffman@47108
  2545
huffman@47108
  2546
lemma word_set_neg_numeral [simp]:
haftmann@54489
  2547
  "set_bit (- numeral bin::'a::len0 word) n b = 
haftmann@54489
  2548
    word_of_int (bin_sc n (if b then 1 else 0) (- numeral bin))"
huffman@47108
  2549
  unfolding word_neg_numeral_alt by (rule set_bit_word_of_int)
huffman@46173
  2550
huffman@46173
  2551
lemma word_set_bit_0 [simp]:
huffman@46173
  2552
  "set_bit 0 n b = word_of_int (bin_sc n (if b then 1 else 0) 0)"
huffman@46173
  2553
  unfolding word_0_wi by (rule set_bit_word_of_int)
huffman@46173
  2554
huffman@46173
  2555
lemma word_set_bit_1 [simp]:
huffman@46173
  2556
  "set_bit 1 n b = word_of_int (bin_sc n (if b then 1 else 0) 1)"
huffman@46173
  2557
  unfolding word_1_wi by (rule set_bit_word_of_int)
haftmann@37660
  2558
huffman@45805
  2559
lemma setBit_no [simp]:
huffman@47108
  2560
  "setBit (numeral bin) n = word_of_int (bin_sc n 1 (numeral bin))"
huffman@45805
  2561
  by (simp add: setBit_def)
huffman@45805
  2562
huffman@45805
  2563
lemma clearBit_no [simp]:
huffman@47108
  2564
  "clearBit (numeral bin) n = word_of_int (bin_sc n 0 (numeral bin))"
huffman@45805
  2565
  by (simp add: clearBit_def)
haftmann@37660
  2566
haftmann@37660
  2567
lemma to_bl_n1: 
haftmann@37660
  2568
  "to_bl (-1::'a::len0 word) = replicate (len_of TYPE ('a)) True"
haftmann@37660
  2569
  apply (rule word_bl.Abs_inverse')
haftmann@37660
  2570
   apply simp
haftmann@37660
  2571
  apply (rule word_eqI)
huffman@45805
  2572
  apply (clarsimp simp add: word_size)
haftmann@37660
  2573
  apply (auto simp add: word_bl.Abs_inverse test_bit_bl word_size)
haftmann@37660
  2574
  done
haftmann@37660
  2575
huffman@45805
  2576
lemma word_msb_n1 [simp]: "msb (-1::'a::len word)"
wenzelm@41550
  2577
  unfolding word_msb_alt to_bl_n1 by simp
haftmann@37660
  2578
haftmann@37660
  2579
lemma word_set_nth_iff: 
haftmann@37660
  2580
  "(set_bit w n b = w) = (w !! n = b | n >= size (w::'a::len0 word))"
haftmann@37660
  2581
  apply (rule iffI)
haftmann@37660
  2582
   apply (rule disjCI)
haftmann@37660
  2583
   apply (drule word_eqD)
haftmann@37660
  2584
   apply (erule sym [THEN trans])
haftmann@37660
  2585
   apply (simp add: test_bit_set)
haftmann@37660
  2586
  apply (erule disjE)
haftmann@37660
  2587
   apply clarsimp
haftmann@37660
  2588
  apply (rule word_eqI)
haftmann@37660
  2589
  apply (clarsimp simp add : test_bit_set_gen)
haftmann@37660
  2590
  apply (drule test_bit_size)
haftmann@37660
  2591
  apply force
haftmann@37660
  2592
  done
haftmann@37660
  2593
huffman@45811
  2594
lemma test_bit_2p:
huffman@45811
  2595
  "(word_of_int (2 ^ n)::'a::len word) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a)"
huffman@45811
  2596
  unfolding word_test_bit_def
haftmann@37660
  2597
  by (auto simp add: word_ubin.eq_norm nth_bintr nth_2p_bin)
haftmann@37660
  2598
haftmann@37660
  2599
lemma nth_w2p:
haftmann@37660
  2600
  "((2\<Colon>'a\<Colon>len word) ^ n) !! m \<longleftrightarrow> m = n \<and> m < len_of TYPE('a\<Colon>len)"
haftmann@37660
  2601
  unfolding test_bit_2p [symmetric] word_of_int [symmetric]
haftmann@37660
  2602
  by (simp add:  of_int_power)
haftmann@37660
  2603
haftmann@37660
  2604
lemma uint_2p: 
haftmann@40827
  2605
  "(0::'a::len word) < 2 ^ n \<Longrightarrow> uint (2 ^ n::'a::len word) = 2 ^ n"
haftmann@37660
  2606
  apply (unfold word_arith_power_alt)
haftmann@37660
  2607
  apply (case_tac "len_of TYPE ('a)")
haftmann@37660
  2608
   apply clarsimp
haftmann@37660
  2609
  apply (case_tac "nat")
haftmann@37660
  2610
   apply clarsimp
haftmann@37660
  2611
   apply (case_tac "n")
huffman@46001
  2612
    apply clarsimp
huffman@46001
  2613
   apply clarsimp
haftmann@37660
  2614
  apply (drule word_gt_0 [THEN iffD1])
haftmann@54489
  2615
  apply (safe intro!: word_eqI)
haftmann@54489
  2616
  apply (auto simp add: nth_2p_bin)
haftmann@54489
  2617
  apply (erule notE)
haftmann@54489
  2618
  apply (simp (no_asm_use) add: uint_word_of_int word_size)
haftmann@54489
  2619
  apply (subst mod_pos_pos_trivial)
haftmann@54489
  2620
  apply simp
haftmann@54489
  2621
  apply (rule power_strict_increasing)
haftmann@54489
  2622