src/HOL/Word/Word.thy
author haftmann
Sat Mar 01 08:21:46 2014 +0100 (2014-03-01)
changeset 55816 e8dd03241e86
parent 55415 05f5fdb8d093
child 55817 0bc0217387a5
permissions -rw-r--r--
cursory polishing: tuned proofs, tuned symbols, tuned headings
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(*  Title:      HOL/Word/Word.thy
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    Author:     Jeremy Dawson and Gerwin Klein, NICTA
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*)
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header {* A type of finite bit strings *}
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theory Word
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imports
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  Type_Length
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  "~~/src/HOL/Library/Boolean_Algebra"
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  Bits_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 @{file "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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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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definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word"
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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 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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  then have "PROP P (word_of_int (uint x))" .
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  find_theorems word_of_int uint
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  then show "PROP P x" by (simp add: word_of_int_uint)
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qed
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subsection {* Type conversions and casting *}
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definition sint :: "'a::len word \<Rightarrow> int"
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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 \<Rightarrow> nat"
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where
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  "unat w = nat (uint w)"
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definition uints :: "nat \<Rightarrow> int set"
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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 \<Rightarrow> int set"
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where
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  "sints n = range (sbintrunc (n - 1))"
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lemma uints_num:
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  "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:
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  "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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definition unats :: "nat \<Rightarrow> nat set"
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where
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  "unats n = {i. i < 2 ^ n}"
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definition norm_sint :: "nat \<Rightarrow> int \<Rightarrow> int"
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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 \<Rightarrow> 'b::len word"
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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 \<Rightarrow> 'b::len0 word"
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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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lemma word_size_gt_0 [iff]:
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  "0 < size (w::'a::len word)"
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  by (simp add: word_size)
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lemmas lens_gt_0 = word_size_gt_0 len_gt_0
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lemma lens_not_0 [iff]:
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  shows "size (w::'a::len word) \<noteq> 0"
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  and "len_of TYPE('a::len) \<noteq> 0"
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  by auto
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definition source_size :: "('a::len0 word \<Rightarrow> 'b) \<Rightarrow> nat"
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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 \<Rightarrow> 'b::len0 word) \<Rightarrow> nat"
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where
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  "target_size c = size (c undefined)"
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definition is_up :: "('a::len0 word \<Rightarrow> 'b::len0 word) \<Rightarrow> bool"
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where
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  "is_up c \<longleftrightarrow> source_size c \<le> target_size c"
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definition is_down :: "('a :: len0 word \<Rightarrow> 'b :: len0 word) \<Rightarrow> bool"
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where
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  "is_down c \<longleftrightarrow> target_size c \<le> source_size c"
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definition of_bl :: "bool list \<Rightarrow> 'a::len0 word"
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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 \<Rightarrow> bool list"
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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 \<Rightarrow> 'a word"
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where
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  "word_reverse w = of_bl (rev (to_bl w))"
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definition word_int_case :: "(int \<Rightarrow> 'b) \<Rightarrow> 'a::len0 word \<Rightarrow> 'b" 
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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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lemmas uint_0 = uint_nonnegative (* FIXME duplicate *)
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lemmas uint_lt = uint_bounded (* FIXME duplicate *)
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lemmas uint_mod_same = uint_idem (* FIXME duplicate *)
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lemma td_ext_uint: 
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  "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0))) 
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    (\<lambda>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 (fact 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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lemma td_ext_ubin:
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  "td_ext (uint :: 'a word \<Rightarrow> int) word_of_int (uints (len_of TYPE('a::len0)))
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    (bintrunc (len_of TYPE('a)))"
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  by (unfold no_bintr_alt1) (fact td_ext_uint)
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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 (fact td_ext_ubin)
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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
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  "cr_word = (\<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 {* 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)
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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"
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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)
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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 {* 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
huffman@45545
   342
  wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and
huffman@45545
   343
  wi_hom_neg: "- word_of_int a = word_of_int (- a)" and
huffman@46000
   344
  wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and
huffman@46000
   345
  wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
huffman@47374
   346
  by (transfer, simp)+
huffman@45545
   347
huffman@45545
   348
lemmas wi_hom_syms = wi_homs [symmetric]
huffman@45545
   349
huffman@46013
   350
lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi
huffman@46009
   351
huffman@46009
   352
lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]
huffman@45545
   353
huffman@45545
   354
instance word :: (len) comm_ring_1
huffman@45810
   355
proof
huffman@45810
   356
  have "0 < len_of TYPE('a)" by (rule len_gt_0)
huffman@45810
   357
  then show "(0::'a word) \<noteq> 1"
huffman@47372
   358
    by - (transfer, auto simp add: gr0_conv_Suc)
huffman@45810
   359
qed
huffman@45545
   360
huffman@45545
   361
lemma word_of_nat: "of_nat n = word_of_int (int n)"
huffman@45545
   362
  by (induct n) (auto simp add : word_of_int_hom_syms)
huffman@45545
   363
huffman@45545
   364
lemma word_of_int: "of_int = word_of_int"
huffman@45545
   365
  apply (rule ext)
huffman@45545
   366
  apply (case_tac x rule: int_diff_cases)
huffman@46013
   367
  apply (simp add: word_of_nat wi_hom_sub)
huffman@45545
   368
  done
huffman@45545
   369
haftmann@54848
   370
definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50)
haftmann@54848
   371
where
haftmann@40827
   372
  "a udvd b = (EX n>=0. uint b = n * uint a)"
haftmann@37660
   373
huffman@45547
   374
haftmann@55816
   375
subsection {* Ordering *}
huffman@45547
   376
huffman@45547
   377
instantiation word :: (len0) linorder
huffman@45547
   378
begin
huffman@45547
   379
haftmann@37660
   380
definition
haftmann@37660
   381
  word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"
haftmann@37660
   382
haftmann@37660
   383
definition
huffman@47108
   384
  word_less_def: "a < b \<longleftrightarrow> uint a < uint b"
haftmann@37660
   385
huffman@45547
   386
instance
huffman@45547
   387
  by default (auto simp: word_less_def word_le_def)
huffman@45547
   388
huffman@45547
   389
end
huffman@45547
   390
haftmann@54848
   391
definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50)
haftmann@54848
   392
where
haftmann@40827
   393
  "a <=s b = (sint a <= sint b)"
haftmann@37660
   394
haftmann@54848
   395
definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50)
haftmann@54848
   396
where
haftmann@40827
   397
  "(x <s y) = (x <=s y & x ~= y)"
haftmann@37660
   398
haftmann@37660
   399
haftmann@55816
   400
subsection {* Bit-wise operations *}
haftmann@37660
   401
haftmann@37660
   402
instantiation word :: (len0) bits
haftmann@37660
   403
begin
haftmann@37660
   404
huffman@47387
   405
lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT
huffman@47374
   406
  by (metis bin_trunc_not)
huffman@47374
   407
huffman@47387
   408
lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND
huffman@47374
   409
  by (metis bin_trunc_and)
huffman@47374
   410
huffman@47387
   411
lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR
huffman@47374
   412
  by (metis bin_trunc_or)
huffman@47374
   413
huffman@47387
   414
lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR
huffman@47374
   415
  by (metis bin_trunc_xor)
haftmann@37660
   416
haftmann@37660
   417
definition
haftmann@37660
   418
  word_test_bit_def: "test_bit a = bin_nth (uint a)"
haftmann@37660
   419
haftmann@37660
   420
definition
haftmann@37660
   421
  word_set_bit_def: "set_bit a n x =
haftmann@54847
   422
   word_of_int (bin_sc n x (uint a))"
haftmann@37660
   423
haftmann@37660
   424
definition
haftmann@37660
   425
  word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"
haftmann@37660
   426
haftmann@37660
   427
definition
haftmann@54847
   428
  word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a)"
haftmann@37660
   429
haftmann@54848
   430
definition shiftl1 :: "'a word \<Rightarrow> 'a word"
haftmann@54848
   431
where
haftmann@54847
   432
  "shiftl1 w = word_of_int (uint w BIT False)"
haftmann@37660
   433
haftmann@54848
   434
definition shiftr1 :: "'a word \<Rightarrow> 'a word"
haftmann@54848
   435
where
haftmann@37660
   436
  -- "shift right as unsigned or as signed, ie logical or arithmetic"
haftmann@37660
   437
  "shiftr1 w = word_of_int (bin_rest (uint w))"
haftmann@37660
   438
haftmann@37660
   439
definition
haftmann@37660
   440
  shiftl_def: "w << n = (shiftl1 ^^ n) w"
haftmann@37660
   441
haftmann@37660
   442
definition
haftmann@37660
   443
  shiftr_def: "w >> n = (shiftr1 ^^ n) w"
haftmann@37660
   444
haftmann@37660
   445
instance ..
haftmann@37660
   446
haftmann@37660
   447
end
haftmann@37660
   448
huffman@47611
   449
lemma [code]: shows
huffman@47374
   450
  word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" and
huffman@47374
   451
  word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" and
huffman@47374
   452
  word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" and
huffman@47374
   453
  word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
huffman@47374
   454
  unfolding bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def
huffman@47374
   455
  by simp_all
huffman@47374
   456
haftmann@37660
   457
instantiation word :: (len) bitss
haftmann@37660
   458
begin
haftmann@37660
   459
haftmann@37660
   460
definition
haftmann@37660
   461
  word_msb_def: 
huffman@46001
   462
  "msb a \<longleftrightarrow> bin_sign (sint a) = -1"
haftmann@37660
   463
haftmann@37660
   464
instance ..
haftmann@37660
   465
haftmann@37660
   466
end
haftmann@37660
   467
haftmann@54848
   468
definition setBit :: "'a :: len0 word => nat => 'a word"
haftmann@54848
   469
where 
haftmann@40827
   470
  "setBit w n = set_bit w n True"
haftmann@37660
   471
haftmann@54848
   472
definition clearBit :: "'a :: len0 word => nat => 'a word"
haftmann@54848
   473
where
haftmann@40827
   474
  "clearBit w n = set_bit w n False"
haftmann@37660
   475
haftmann@37660
   476
haftmann@55816
   477
subsection {* Shift operations *}
haftmann@37660
   478
haftmann@54848
   479
definition sshiftr1 :: "'a :: len word => 'a word"
haftmann@54848
   480
where 
haftmann@40827
   481
  "sshiftr1 w = word_of_int (bin_rest (sint w))"
haftmann@37660
   482
haftmann@54848
   483
definition bshiftr1 :: "bool => 'a :: len word => 'a word"
haftmann@54848
   484
where
haftmann@40827
   485
  "bshiftr1 b w = of_bl (b # butlast (to_bl w))"
haftmann@37660
   486
haftmann@54848
   487
definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55)
haftmann@54848
   488
where
haftmann@40827
   489
  "w >>> n = (sshiftr1 ^^ n) w"
haftmann@37660
   490
haftmann@54848
   491
definition mask :: "nat => 'a::len word"
haftmann@54848
   492
where
haftmann@40827
   493
  "mask n = (1 << n) - 1"
haftmann@37660
   494
haftmann@54848
   495
definition revcast :: "'a :: len0 word => 'b :: len0 word"
haftmann@54848
   496
where
haftmann@40827
   497
  "revcast w =  of_bl (takefill False (len_of TYPE('b)) (to_bl w))"
haftmann@37660
   498
haftmann@54848
   499
definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word"
haftmann@54848
   500
where
haftmann@40827
   501
  "slice1 n w = of_bl (takefill False n (to_bl w))"
haftmann@37660
   502
haftmann@54848
   503
definition slice :: "nat => 'a :: len0 word => 'b :: len0 word"
haftmann@54848
   504
where
haftmann@40827
   505
  "slice n w = slice1 (size w - n) w"
haftmann@37660
   506
haftmann@37660
   507
haftmann@55816
   508
subsection {* Rotation *}
haftmann@37660
   509
haftmann@54848
   510
definition rotater1 :: "'a list => 'a list"
haftmann@54848
   511
where
haftmann@40827
   512
  "rotater1 ys = 
haftmann@40827
   513
    (case ys of [] => [] | x # xs => last ys # butlast ys)"
haftmann@37660
   514
haftmann@54848
   515
definition rotater :: "nat => 'a list => 'a list"
haftmann@54848
   516
where
haftmann@40827
   517
  "rotater n = rotater1 ^^ n"
haftmann@37660
   518
haftmann@54848
   519
definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word"
haftmann@54848
   520
where
haftmann@40827
   521
  "word_rotr n w = of_bl (rotater n (to_bl w))"
haftmann@37660
   522
haftmann@54848
   523
definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word"
haftmann@54848
   524
where
haftmann@40827
   525
  "word_rotl n w = of_bl (rotate n (to_bl w))"
haftmann@37660
   526
haftmann@54848
   527
definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word"
haftmann@54848
   528
where
haftmann@40827
   529
  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
haftmann@40827
   530
                    else word_rotl (nat (- i)) w)"
haftmann@37660
   531
haftmann@37660
   532
haftmann@55816
   533
subsection {* Split and cat operations *}
haftmann@37660
   534
haftmann@54848
   535
definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word"
haftmann@54848
   536
where
haftmann@40827
   537
  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"
haftmann@37660
   538
haftmann@54848
   539
definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)"
haftmann@54848
   540
where
haftmann@40827
   541
  "word_split a = 
haftmann@40827
   542
   (case bin_split (len_of TYPE ('c)) (uint a) of 
haftmann@40827
   543
     (u, v) => (word_of_int u, word_of_int v))"
haftmann@37660
   544
haftmann@54848
   545
definition word_rcat :: "'a :: len0 word list => 'b :: len0 word"
haftmann@54848
   546
where
haftmann@40827
   547
  "word_rcat ws = 
haftmann@37660
   548
  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"
haftmann@37660
   549
haftmann@54848
   550
definition word_rsplit :: "'a :: len0 word => 'b :: len word list"
haftmann@54848
   551
where
haftmann@40827
   552
  "word_rsplit w = 
haftmann@37660
   553
  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"
haftmann@37660
   554
haftmann@54848
   555
definition max_word :: "'a::len word" -- "Largest representable machine integer."
haftmann@54848
   556
where
haftmann@40827
   557
  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"
haftmann@37660
   558
haftmann@55816
   559
lemmas of_nth_def = word_set_bits_def (* FIXME duplicate *)
haftmann@55816
   560
haftmann@37660
   561
huffman@46010
   562
subsection {* Theorems about typedefs *}
huffman@46010
   563
haftmann@37660
   564
lemma sint_sbintrunc': 
haftmann@37660
   565
  "sint (word_of_int bin :: 'a word) = 
haftmann@37660
   566
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
haftmann@37660
   567
  unfolding sint_uint 
haftmann@37660
   568
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)
haftmann@37660
   569
haftmann@37660
   570
lemma uint_sint: 
haftmann@37660
   571
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
haftmann@37660
   572
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)
haftmann@37660
   573
huffman@46057
   574
lemma bintr_uint:
huffman@46057
   575
  fixes w :: "'a::len0 word"
huffman@46057
   576
  shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w"
haftmann@37660
   577
  apply (subst word_ubin.norm_Rep [symmetric]) 
haftmann@37660
   578
  apply (simp only: bintrunc_bintrunc_min word_size)
haftmann@54863
   579
  apply (simp add: min.absorb2)
haftmann@37660
   580
  done
haftmann@37660
   581
huffman@46057
   582
lemma wi_bintr:
huffman@46057
   583
  "len_of TYPE('a::len0) \<le> n \<Longrightarrow>
huffman@46057
   584
    word_of_int (bintrunc n w) = (word_of_int w :: 'a word)"
haftmann@54863
   585
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min.absorb1)
haftmann@37660
   586
haftmann@37660
   587
lemma td_ext_sbin: 
haftmann@55816
   588
  "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len))) 
haftmann@37660
   589
    (sbintrunc (len_of TYPE('a) - 1))"
haftmann@37660
   590
  apply (unfold td_ext_def' sint_uint)
haftmann@37660
   591
  apply (simp add : word_ubin.eq_norm)
haftmann@37660
   592
  apply (cases "len_of TYPE('a)")
haftmann@37660
   593
   apply (auto simp add : sints_def)
haftmann@37660
   594
  apply (rule sym [THEN trans])
haftmann@37660
   595
  apply (rule word_ubin.Abs_norm)
haftmann@37660
   596
  apply (simp only: bintrunc_sbintrunc)
haftmann@37660
   597
  apply (drule sym)
haftmann@37660
   598
  apply simp
haftmann@37660
   599
  done
haftmann@37660
   600
haftmann@55816
   601
lemma td_ext_sint:
haftmann@55816
   602
  "td_ext (sint :: 'a word \<Rightarrow> int) word_of_int (sints (len_of TYPE('a::len)))
haftmann@55816
   603
     (\<lambda>w. (w + 2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
haftmann@55816
   604
         2 ^ (len_of TYPE('a) - 1))"
haftmann@55816
   605
  using td_ext_sbin [where ?'a = 'a] by (simp add: no_sbintr_alt2)
haftmann@37660
   606
haftmann@37660
   607
(* We do sint before sbin, before sint is the user version
haftmann@37660
   608
   and interpretations do not produce thm duplicates. I.e. 
haftmann@37660
   609
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
haftmann@37660
   610
   because the latter is the same thm as the former *)
haftmann@37660
   611
interpretation word_sint:
haftmann@37660
   612
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   613
          word_of_int 
haftmann@37660
   614
          "sints (len_of TYPE('a::len))"
haftmann@37660
   615
          "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) -
haftmann@37660
   616
               2 ^ (len_of TYPE('a::len) - 1)"
haftmann@37660
   617
  by (rule td_ext_sint)
haftmann@37660
   618
haftmann@37660
   619
interpretation word_sbin:
haftmann@37660
   620
  td_ext "sint ::'a::len word => int" 
haftmann@37660
   621
          word_of_int 
haftmann@37660
   622
          "sints (len_of TYPE('a::len))"
haftmann@37660
   623
          "sbintrunc (len_of TYPE('a::len) - 1)"
haftmann@37660
   624
  by (rule td_ext_sbin)
haftmann@37660
   625
wenzelm@45604
   626
lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]
haftmann@37660
   627
haftmann@37660
   628
lemmas td_sint = word_sint.td
haftmann@37660
   629
haftmann@37660
   630
lemma to_bl_def': 
haftmann@37660
   631
  "(to_bl :: 'a :: len0 word => bool list) =
haftmann@37660
   632
    bin_to_bl (len_of TYPE('a)) o uint"
wenzelm@44762
   633
  by (auto simp: to_bl_def)
haftmann@37660
   634
huffman@47108
   635
lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w
haftmann@37660
   636
huffman@45805
   637
lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
huffman@45805
   638
  by (fact uints_def [unfolded no_bintr_alt1])
huffman@45805
   639
huffman@47108
   640
lemma word_numeral_alt:
huffman@47108
   641
  "numeral b = word_of_int (numeral b)"
huffman@47108
   642
  by (induct b, simp_all only: numeral.simps word_of_int_homs)
huffman@47108
   643
huffman@47108
   644
declare word_numeral_alt [symmetric, code_abbrev]
huffman@47108
   645
huffman@47108
   646
lemma word_neg_numeral_alt:
haftmann@54489
   647
  "- numeral b = word_of_int (- numeral b)"
haftmann@54489
   648
  by (simp only: word_numeral_alt wi_hom_neg)
huffman@47108
   649
huffman@47108
   650
declare word_neg_numeral_alt [symmetric, code_abbrev]
huffman@47108
   651
huffman@47372
   652
lemma word_numeral_transfer [transfer_rule]:
kuncar@51375
   653
  "(fun_rel op = pcr_word) numeral numeral"
haftmann@54489
   654
  "(fun_rel op = pcr_word) (- numeral) (- numeral)"
haftmann@54489
   655
  apply (simp_all add: fun_rel_def word.pcr_cr_eq cr_word_def)
haftmann@54489
   656
  using word_numeral_alt [symmetric] word_neg_numeral_alt [symmetric] by blast+
huffman@47372
   657
huffman@45805
   658
lemma uint_bintrunc [simp]:
huffman@47108
   659
  "uint (numeral bin :: 'a word) = 
huffman@47108
   660
    bintrunc (len_of TYPE ('a :: len0)) (numeral bin)"
huffman@47108
   661
  unfolding word_numeral_alt by (rule word_ubin.eq_norm)
huffman@47108
   662
haftmann@54489
   663
lemma uint_bintrunc_neg [simp]: "uint (- numeral bin :: 'a word) = 
haftmann@54489
   664
    bintrunc (len_of TYPE ('a :: len0)) (- numeral bin)"
huffman@47108
   665
  by (simp only: word_neg_numeral_alt word_ubin.eq_norm)
haftmann@37660
   666
huffman@45805
   667
lemma sint_sbintrunc [simp]:
huffman@47108
   668
  "sint (numeral bin :: 'a word) = 
huffman@47108
   669
    sbintrunc (len_of TYPE ('a :: len) - 1) (numeral bin)"
huffman@47108
   670
  by (simp only: word_numeral_alt word_sbin.eq_norm)
huffman@47108
   671
haftmann@54489
   672
lemma sint_sbintrunc_neg [simp]: "sint (- numeral bin :: 'a word) = 
haftmann@54489
   673
    sbintrunc (len_of TYPE ('a :: len) - 1) (- numeral bin)"
huffman@47108
   674
  by (simp only: word_neg_numeral_alt word_sbin.eq_norm)
haftmann@37660
   675
huffman@45805
   676
lemma unat_bintrunc [simp]:
huffman@47108
   677
  "unat (numeral bin :: 'a :: len0 word) =
huffman@47108
   678
    nat (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
   679
  by (simp only: unat_def uint_bintrunc)
huffman@47108
   680
huffman@47108
   681
lemma unat_bintrunc_neg [simp]:
haftmann@54489
   682
  "unat (- numeral bin :: 'a :: len0 word) =
haftmann@54489
   683
    nat (bintrunc (len_of TYPE('a)) (- numeral bin))"
huffman@47108
   684
  by (simp only: unat_def uint_bintrunc_neg)
haftmann@37660
   685
haftmann@40827
   686
lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
haftmann@37660
   687
  apply (unfold word_size)
haftmann@37660
   688
  apply (rule word_uint.Rep_eqD)
haftmann@37660
   689
  apply (rule box_equals)
haftmann@37660
   690
    defer
haftmann@37660
   691
    apply (rule word_ubin.norm_Rep)+
haftmann@37660
   692
  apply simp
haftmann@37660
   693
  done
haftmann@37660
   694
huffman@45805
   695
lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)"
huffman@45805
   696
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   697
huffman@45805
   698
lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
huffman@45805
   699
  using word_uint.Rep [of x] by (simp add: uints_num)
huffman@45805
   700
huffman@45805
   701
lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)"
huffman@45805
   702
  using word_sint.Rep [of x] by (simp add: sints_num)
huffman@45805
   703
huffman@45805
   704
lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)"
huffman@45805
   705
  using word_sint.Rep [of x] by (simp add: sints_num)
haftmann@37660
   706
haftmann@37660
   707
lemma sign_uint_Pls [simp]: 
huffman@46604
   708
  "bin_sign (uint x) = 0"
huffman@47108
   709
  by (simp add: sign_Pls_ge_0)
haftmann@37660
   710
huffman@45805
   711
lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0"
huffman@45805
   712
  by (simp only: diff_less_0_iff_less uint_lt2p)
huffman@45805
   713
huffman@45805
   714
lemma uint_m2p_not_non_neg:
huffman@45805
   715
  "\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)"
huffman@45805
   716
  by (simp only: not_le uint_m2p_neg)
haftmann@37660
   717
haftmann@37660
   718
lemma lt2p_lem:
haftmann@55816
   719
  "len_of TYPE('a) \<le> n \<Longrightarrow> uint (w :: 'a::len0 word) < 2 ^ n"
haftmann@55816
   720
  by (metis bintr_uint bintrunc_mod2p int_mod_lem zless2p)
haftmann@37660
   721
huffman@45805
   722
lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
huffman@45805
   723
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])
haftmann@37660
   724
haftmann@40827
   725
lemma uint_nat: "uint w = int (unat w)"
haftmann@37660
   726
  unfolding unat_def by auto
haftmann@37660
   727
huffman@47108
   728
lemma uint_numeral:
huffman@47108
   729
  "uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   730
  unfolding word_numeral_alt
haftmann@37660
   731
  by (simp only: int_word_uint)
haftmann@37660
   732
huffman@47108
   733
lemma uint_neg_numeral:
haftmann@54489
   734
  "uint (- numeral b :: 'a :: len0 word) = - numeral b mod 2 ^ len_of TYPE('a)"
huffman@47108
   735
  unfolding word_neg_numeral_alt
huffman@47108
   736
  by (simp only: int_word_uint)
huffman@47108
   737
huffman@47108
   738
lemma unat_numeral: 
huffman@47108
   739
  "unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)"
haftmann@37660
   740
  apply (unfold unat_def)
huffman@47108
   741
  apply (clarsimp simp only: uint_numeral)
haftmann@37660
   742
  apply (rule nat_mod_distrib [THEN trans])
huffman@47108
   743
    apply (rule zero_le_numeral)
haftmann@37660
   744
   apply (simp_all add: nat_power_eq)
haftmann@37660
   745
  done
haftmann@37660
   746
huffman@47108
   747
lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + 
haftmann@37660
   748
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
haftmann@37660
   749
    2 ^ (len_of TYPE('a) - 1)"
huffman@47108
   750
  unfolding word_numeral_alt by (rule int_word_sint)
huffman@47108
   751
haftmann@55816
   752
lemma word_of_int_0 [simp, code_post]:
haftmann@55816
   753
  "word_of_int 0 = 0"
huffman@45958
   754
  unfolding word_0_wi ..
huffman@45958
   755
haftmann@55816
   756
lemma word_of_int_1 [simp, code_post]:
haftmann@55816
   757
  "word_of_int 1 = 1"
huffman@45958
   758
  unfolding word_1_wi ..
huffman@45958
   759
haftmann@54489
   760
lemma word_of_int_neg_1 [simp]: "word_of_int (- 1) = - 1"
haftmann@54489
   761
  by (simp add: wi_hom_syms)
haftmann@54489
   762
huffman@47108
   763
lemma word_of_int_numeral [simp] : 
huffman@47108
   764
  "(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)"
huffman@47108
   765
  unfolding word_numeral_alt ..
huffman@47108
   766
huffman@47108
   767
lemma word_of_int_neg_numeral [simp]:
haftmann@54489
   768
  "(word_of_int (- numeral bin) :: 'a :: len0 word) = (- numeral bin)"
haftmann@54489
   769
  unfolding word_numeral_alt wi_hom_syms ..
haftmann@37660
   770
haftmann@37660
   771
lemma word_int_case_wi: 
haftmann@37660
   772
  "word_int_case f (word_of_int i :: 'b word) = 
haftmann@37660
   773
    f (i mod 2 ^ len_of TYPE('b::len0))"
haftmann@37660
   774
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)
haftmann@37660
   775
haftmann@37660
   776
lemma word_int_split: 
haftmann@37660
   777
  "P (word_int_case f x) = 
haftmann@37660
   778
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
haftmann@37660
   779
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
haftmann@37660
   780
  unfolding word_int_case_def
haftmann@55816
   781
  by (auto simp: word_uint.eq_norm mod_pos_pos_trivial)
haftmann@37660
   782
haftmann@37660
   783
lemma word_int_split_asm: 
haftmann@37660
   784
  "P (word_int_case f x) = 
haftmann@37660
   785
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
haftmann@37660
   786
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
haftmann@37660
   787
  unfolding word_int_case_def
haftmann@55816
   788
  by (auto simp: word_uint.eq_norm mod_pos_pos_trivial)
huffman@45805
   789
wenzelm@45604
   790
lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
wenzelm@45604
   791
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]
haftmann@37660
   792
haftmann@37660
   793
lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
haftmann@37660
   794
  unfolding word_size by (rule uint_range')
haftmann@37660
   795
haftmann@37660
   796
lemma sint_range_size:
haftmann@37660
   797
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
haftmann@37660
   798
  unfolding word_size by (rule sint_range')
haftmann@37660
   799
huffman@45805
   800
lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x"
huffman@45805
   801
  unfolding word_size by (rule less_le_trans [OF sint_lt])
huffman@45805
   802
huffman@45805
   803
lemma sint_below_size:
huffman@45805
   804
  "x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w"
huffman@45805
   805
  unfolding word_size by (rule order_trans [OF _ sint_ge])
haftmann@37660
   806
haftmann@55816
   807
huffman@46010
   808
subsection {* Testing bits *}
huffman@46010
   809
haftmann@37660
   810
lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)"
haftmann@37660
   811
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)
haftmann@37660
   812
haftmann@37660
   813
lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
haftmann@37660
   814
  apply (unfold word_test_bit_def)
haftmann@37660
   815
  apply (subst word_ubin.norm_Rep [symmetric])
haftmann@37660
   816
  apply (simp only: nth_bintr word_size)
haftmann@37660
   817
  apply fast
haftmann@37660
   818
  done
haftmann@37660
   819
huffman@46021
   820
lemma word_eq_iff:
huffman@46021
   821
  fixes x y :: "'a::len0 word"
huffman@46021
   822
  shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)"
huffman@46021
   823
  unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric]
huffman@46021
   824
  by (metis test_bit_size [unfolded word_size])
huffman@46021
   825
huffman@46023
   826
lemma word_eqI [rule_format]:
haftmann@37660
   827
  fixes u :: "'a::len0 word"
haftmann@40827
   828
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
huffman@46021
   829
  by (simp add: word_size word_eq_iff)
haftmann@37660
   830
huffman@45805
   831
lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x"
huffman@45805
   832
  by simp
haftmann@37660
   833
haftmann@37660
   834
lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
haftmann@37660
   835
  unfolding word_test_bit_def word_size
haftmann@37660
   836
  by (simp add: nth_bintr [symmetric])
haftmann@37660
   837
haftmann@37660
   838
lemmas test_bit_bin = test_bit_bin' [unfolded word_size]
haftmann@37660
   839
huffman@46057
   840
lemma bin_nth_uint_imp:
huffman@46057
   841
  "bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)"
haftmann@37660
   842
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
haftmann@37660
   843
  apply (subst word_ubin.norm_Rep)
haftmann@37660
   844
  apply assumption
haftmann@37660
   845
  done
haftmann@37660
   846
huffman@46057
   847
lemma bin_nth_sint:
huffman@46057
   848
  fixes w :: "'a::len word"
huffman@46057
   849
  shows "len_of TYPE('a) \<le> n \<Longrightarrow>
huffman@46057
   850
    bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)"
haftmann@37660
   851
  apply (subst word_sbin.norm_Rep [symmetric])
huffman@46057
   852
  apply (auto simp add: nth_sbintr)
haftmann@37660
   853
  done
haftmann@37660
   854
haftmann@37660
   855
(* type definitions theorem for in terms of equivalent bool list *)
haftmann@37660
   856
lemma td_bl: 
haftmann@37660
   857
  "type_definition (to_bl :: 'a::len0 word => bool list) 
haftmann@37660
   858
                   of_bl  
haftmann@37660
   859
                   {bl. length bl = len_of TYPE('a)}"
haftmann@37660
   860
  apply (unfold type_definition_def of_bl_def to_bl_def)
haftmann@37660
   861
  apply (simp add: word_ubin.eq_norm)
haftmann@37660
   862
  apply safe
haftmann@37660
   863
  apply (drule sym)
haftmann@37660
   864
  apply simp
haftmann@37660
   865
  done
haftmann@37660
   866
haftmann@37660
   867
interpretation word_bl:
haftmann@37660
   868
  type_definition "to_bl :: 'a::len0 word => bool list"
haftmann@37660
   869
                  of_bl  
haftmann@37660
   870
                  "{bl. length bl = len_of TYPE('a::len0)}"
haftmann@55816
   871
  by (fact td_bl)
haftmann@37660
   872
huffman@45816
   873
lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]
wenzelm@45538
   874
haftmann@40827
   875
lemma word_size_bl: "size w = size (to_bl w)"
haftmann@37660
   876
  unfolding word_size by auto
haftmann@37660
   877
haftmann@37660
   878
lemma to_bl_use_of_bl:
haftmann@37660
   879
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
huffman@45816
   880
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])
haftmann@37660
   881
haftmann@37660
   882
lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
haftmann@37660
   883
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)
haftmann@37660
   884
haftmann@37660
   885
lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
haftmann@37660
   886
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)
haftmann@37660
   887
haftmann@40827
   888
lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
huffman@47108
   889
  by (metis word_rev_rev)
haftmann@37660
   890
huffman@45805
   891
lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
huffman@45805
   892
  by simp
huffman@45805
   893
huffman@45805
   894
lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))"
huffman@45805
   895
  unfolding word_bl_Rep' by (rule len_gt_0)
huffman@45805
   896
huffman@45805
   897
lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []"
huffman@45805
   898
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])
huffman@45805
   899
huffman@45805
   900
lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0"
huffman@45805
   901
  by (fact length_bl_gt_0 [THEN gr_implies_not0])
haftmann@37660
   902
huffman@46001
   903
lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
haftmann@37660
   904
  apply (unfold to_bl_def sint_uint)
haftmann@37660
   905
  apply (rule trans [OF _ bl_sbin_sign])
haftmann@37660
   906
  apply simp
haftmann@37660
   907
  done
haftmann@37660
   908
haftmann@37660
   909
lemma of_bl_drop': 
haftmann@40827
   910
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
haftmann@37660
   911
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
haftmann@37660
   912
  apply (unfold of_bl_def)
haftmann@37660
   913
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
haftmann@37660
   914
  done
haftmann@37660
   915
haftmann@37660
   916
lemma test_bit_of_bl:  
haftmann@37660
   917
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)"
haftmann@37660
   918
  apply (unfold of_bl_def word_test_bit_def)
haftmann@37660
   919
  apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl)
haftmann@37660
   920
  done
haftmann@37660
   921
haftmann@37660
   922
lemma no_of_bl: 
huffman@47108
   923
  "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) (numeral bin))"
huffman@47108
   924
  unfolding of_bl_def by simp
haftmann@37660
   925
haftmann@40827
   926
lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
haftmann@37660
   927
  unfolding word_size to_bl_def by auto
haftmann@37660
   928
haftmann@37660
   929
lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   930
  unfolding uint_bl by (simp add : word_size)
haftmann@37660
   931
haftmann@37660
   932
lemma to_bl_of_bin: 
haftmann@37660
   933
  "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
haftmann@37660
   934
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)
haftmann@37660
   935
huffman@47108
   936
lemma to_bl_numeral [simp]:
huffman@47108
   937
  "to_bl (numeral bin::'a::len0 word) =
huffman@47108
   938
    bin_to_bl (len_of TYPE('a)) (numeral bin)"
huffman@47108
   939
  unfolding word_numeral_alt by (rule to_bl_of_bin)
huffman@47108
   940
huffman@47108
   941
lemma to_bl_neg_numeral [simp]:
haftmann@54489
   942
  "to_bl (- numeral bin::'a::len0 word) =
haftmann@54489
   943
    bin_to_bl (len_of TYPE('a)) (- numeral bin)"
huffman@47108
   944
  unfolding word_neg_numeral_alt by (rule to_bl_of_bin)
haftmann@37660
   945
haftmann@37660
   946
lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
haftmann@37660
   947
  unfolding uint_bl by (simp add : word_size)
huffman@46011
   948
huffman@46011
   949
lemma uint_bl_bin:
huffman@46011
   950
  fixes x :: "'a::len0 word"
huffman@46011
   951
  shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x"
huffman@46011
   952
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])
wenzelm@45604
   953
haftmann@37660
   954
(* naturals *)
haftmann@37660
   955
lemma uints_unats: "uints n = int ` unats n"
haftmann@37660
   956
  apply (unfold unats_def uints_num)
haftmann@37660
   957
  apply safe
haftmann@37660
   958
  apply (rule_tac image_eqI)
haftmann@37660
   959
  apply (erule_tac nat_0_le [symmetric])
haftmann@37660
   960
  apply auto
haftmann@37660
   961
  apply (erule_tac nat_less_iff [THEN iffD2])
haftmann@37660
   962
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
haftmann@37660
   963
  apply (auto simp add : nat_power_eq int_power)
haftmann@37660
   964
  done
haftmann@37660
   965
haftmann@37660
   966
lemma unats_uints: "unats n = nat ` uints n"
haftmann@37660
   967
  by (auto simp add : uints_unats image_iff)
haftmann@37660
   968
huffman@46962
   969
lemmas bintr_num = word_ubin.norm_eq_iff
huffman@47108
   970
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
huffman@46962
   971
lemmas sbintr_num = word_sbin.norm_eq_iff
huffman@47108
   972
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
haftmann@37660
   973
haftmann@37660
   974
lemma num_of_bintr':
huffman@47108
   975
  "bintrunc (len_of TYPE('a :: len0)) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   976
    numeral a = (numeral b :: 'a word)"
huffman@46962
   977
  unfolding bintr_num by (erule subst, simp)
haftmann@37660
   978
haftmann@37660
   979
lemma num_of_sbintr':
huffman@47108
   980
  "sbintrunc (len_of TYPE('a :: len) - 1) (numeral a) = (numeral b) \<Longrightarrow> 
huffman@47108
   981
    numeral a = (numeral b :: 'a word)"
huffman@46962
   982
  unfolding sbintr_num by (erule subst, simp)
huffman@46962
   983
huffman@46962
   984
lemma num_abs_bintr:
huffman@47108
   985
  "(numeral x :: 'a word) =
huffman@47108
   986
    word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))"
huffman@47108
   987
  by (simp only: word_ubin.Abs_norm word_numeral_alt)
huffman@46962
   988
huffman@46962
   989
lemma num_abs_sbintr:
huffman@47108
   990
  "(numeral x :: 'a word) =
huffman@47108
   991
    word_of_int (sbintrunc (len_of TYPE('a::len) - 1) (numeral x))"
huffman@47108
   992
  by (simp only: word_sbin.Abs_norm word_numeral_alt)
huffman@46962
   993
haftmann@37660
   994
(** cast - note, no arg for new length, as it's determined by type of result,
haftmann@37660
   995
  thus in "cast w = w, the type means cast to length of w! **)
haftmann@37660
   996
haftmann@37660
   997
lemma ucast_id: "ucast w = w"
haftmann@37660
   998
  unfolding ucast_def by auto
haftmann@37660
   999
haftmann@37660
  1000
lemma scast_id: "scast w = w"
haftmann@37660
  1001
  unfolding scast_def by auto
haftmann@37660
  1002
haftmann@40827
  1003
lemma ucast_bl: "ucast w = of_bl (to_bl w)"
haftmann@37660
  1004
  unfolding ucast_def of_bl_def uint_bl
haftmann@37660
  1005
  by (auto simp add : word_size)
haftmann@37660
  1006
haftmann@37660
  1007
lemma nth_ucast: 
haftmann@37660
  1008
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
haftmann@37660
  1009
  apply (unfold ucast_def test_bit_bin)
haftmann@37660
  1010
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
haftmann@37660
  1011
  apply (fast elim!: bin_nth_uint_imp)
haftmann@37660
  1012
  done
haftmann@37660
  1013
haftmann@37660
  1014
(* for literal u(s)cast *)
haftmann@37660
  1015
huffman@46001
  1016
lemma ucast_bintr [simp]:
huffman@47108
  1017
  "ucast (numeral w ::'a::len0 word) = 
huffman@47108
  1018
   word_of_int (bintrunc (len_of TYPE('a)) (numeral w))"
haftmann@37660
  1019
  unfolding ucast_def by simp
huffman@47108
  1020
(* TODO: neg_numeral *)
haftmann@37660
  1021
huffman@46001
  1022
lemma scast_sbintr [simp]:
huffman@47108
  1023
  "scast (numeral w ::'a::len word) = 
huffman@47108
  1024
   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (numeral w))"
haftmann@37660
  1025
  unfolding scast_def by simp
haftmann@37660
  1026
huffman@46011
  1027
lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)"
huffman@46011
  1028
  unfolding source_size_def word_size Let_def ..
huffman@46011
  1029
huffman@46011
  1030
lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)"
huffman@46011
  1031
  unfolding target_size_def word_size Let_def ..
huffman@46011
  1032
huffman@46011
  1033
lemma is_down:
huffman@46011
  1034
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
  1035
  shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)"
huffman@46011
  1036
  unfolding is_down_def source_size target_size ..
huffman@46011
  1037
huffman@46011
  1038
lemma is_up:
huffman@46011
  1039
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
huffman@46011
  1040
  shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)"
huffman@46011
  1041
  unfolding is_up_def source_size target_size ..
haftmann@37660
  1042
wenzelm@45604
  1043
lemmas is_up_down = trans [OF is_up is_down [symmetric]]
haftmann@37660
  1044
huffman@45811
  1045
lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
haftmann@37660
  1046
  apply (unfold is_down)
haftmann@37660
  1047
  apply safe
haftmann@37660
  1048
  apply (rule ext)
haftmann@37660
  1049
  apply (unfold ucast_def scast_def uint_sint)
haftmann@37660
  1050
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1051
  apply simp
haftmann@37660
  1052
  done
haftmann@37660
  1053
huffman@45811
  1054
lemma word_rev_tf:
huffman@45811
  1055
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
  1056
    rev (takefill False (len_of TYPE('a)) (rev bl))"
haftmann@37660
  1057
  unfolding of_bl_def uint_bl
haftmann@37660
  1058
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
haftmann@37660
  1059
huffman@45811
  1060
lemma word_rep_drop:
huffman@45811
  1061
  "to_bl (of_bl bl::'a::len0 word) =
huffman@45811
  1062
    replicate (len_of TYPE('a) - length bl) False @
huffman@45811
  1063
    drop (length bl - len_of TYPE('a)) bl"
huffman@45811
  1064
  by (simp add: word_rev_tf takefill_alt rev_take)
haftmann@37660
  1065
haftmann@37660
  1066
lemma to_bl_ucast: 
haftmann@37660
  1067
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
haftmann@37660
  1068
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
haftmann@37660
  1069
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
haftmann@37660
  1070
  apply (unfold ucast_bl)
haftmann@37660
  1071
  apply (rule trans)
haftmann@37660
  1072
   apply (rule word_rep_drop)
haftmann@37660
  1073
  apply simp
haftmann@37660
  1074
  done
haftmann@37660
  1075
huffman@45811
  1076
lemma ucast_up_app [OF refl]:
haftmann@40827
  1077
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
haftmann@37660
  1078
    to_bl (uc w) = replicate n False @ (to_bl w)"
haftmann@37660
  1079
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
  1080
huffman@45811
  1081
lemma ucast_down_drop [OF refl]:
haftmann@40827
  1082
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
haftmann@37660
  1083
    to_bl (uc w) = drop n (to_bl w)"
haftmann@37660
  1084
  by (auto simp add : source_size target_size to_bl_ucast)
haftmann@37660
  1085
huffman@45811
  1086
lemma scast_down_drop [OF refl]:
haftmann@40827
  1087
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
haftmann@37660
  1088
    to_bl (sc w) = drop n (to_bl w)"
haftmann@37660
  1089
  apply (subgoal_tac "sc = ucast")
haftmann@37660
  1090
   apply safe
haftmann@37660
  1091
   apply simp
huffman@45811
  1092
   apply (erule ucast_down_drop)
huffman@45811
  1093
  apply (rule down_cast_same [symmetric])
haftmann@37660
  1094
  apply (simp add : source_size target_size is_down)
haftmann@37660
  1095
  done
haftmann@37660
  1096
huffman@45811
  1097
lemma sint_up_scast [OF refl]:
haftmann@40827
  1098
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
haftmann@37660
  1099
  apply (unfold is_up)
haftmann@37660
  1100
  apply safe
haftmann@37660
  1101
  apply (simp add: scast_def word_sbin.eq_norm)
haftmann@37660
  1102
  apply (rule box_equals)
haftmann@37660
  1103
    prefer 3
haftmann@37660
  1104
    apply (rule word_sbin.norm_Rep)
haftmann@37660
  1105
   apply (rule sbintrunc_sbintrunc_l)
haftmann@37660
  1106
   defer
haftmann@37660
  1107
   apply (subst word_sbin.norm_Rep)
haftmann@37660
  1108
   apply (rule refl)
haftmann@37660
  1109
  apply simp
haftmann@37660
  1110
  done
haftmann@37660
  1111
huffman@45811
  1112
lemma uint_up_ucast [OF refl]:
haftmann@40827
  1113
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
haftmann@37660
  1114
  apply (unfold is_up)
haftmann@37660
  1115
  apply safe
haftmann@37660
  1116
  apply (rule bin_eqI)
haftmann@37660
  1117
  apply (fold word_test_bit_def)
haftmann@37660
  1118
  apply (auto simp add: nth_ucast)
haftmann@37660
  1119
  apply (auto simp add: test_bit_bin)
haftmann@37660
  1120
  done
huffman@45811
  1121
huffman@45811
  1122
lemma ucast_up_ucast [OF refl]:
huffman@45811
  1123
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
haftmann@37660
  1124
  apply (simp (no_asm) add: ucast_def)
haftmann@37660
  1125
  apply (clarsimp simp add: uint_up_ucast)
haftmann@37660
  1126
  done
haftmann@37660
  1127
    
huffman@45811
  1128
lemma scast_up_scast [OF refl]:
huffman@45811
  1129
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
haftmann@37660
  1130
  apply (simp (no_asm) add: scast_def)
haftmann@37660
  1131
  apply (clarsimp simp add: sint_up_scast)
haftmann@37660
  1132
  done
haftmann@37660
  1133
    
huffman@45811
  1134
lemma ucast_of_bl_up [OF refl]:
haftmann@40827
  1135
  "w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl"
haftmann@37660
  1136
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
haftmann@37660
  1137
haftmann@37660
  1138
lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
haftmann@37660
  1139
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]
haftmann@37660
  1140
haftmann@37660
  1141
lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
haftmann@37660
  1142
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
haftmann@37660
  1143
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
haftmann@37660
  1144
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]
haftmann@37660
  1145
haftmann@37660
  1146
lemma up_ucast_surj:
haftmann@40827
  1147
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1148
   surj (ucast :: 'a word => 'b word)"
haftmann@37660
  1149
  by (rule surjI, erule ucast_up_ucast_id)
haftmann@37660
  1150
haftmann@37660
  1151
lemma up_scast_surj:
haftmann@40827
  1152
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1153
   surj (scast :: 'a word => 'b word)"
haftmann@37660
  1154
  by (rule surjI, erule scast_up_scast_id)
haftmann@37660
  1155
haftmann@37660
  1156
lemma down_scast_inj:
haftmann@40827
  1157
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
haftmann@37660
  1158
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1159
  by (rule inj_on_inverseI, erule scast_down_scast_id)
haftmann@37660
  1160
haftmann@37660
  1161
lemma down_ucast_inj:
haftmann@40827
  1162
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
haftmann@37660
  1163
   inj_on (ucast :: 'a word => 'b word) A"
haftmann@37660
  1164
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)
haftmann@37660
  1165
haftmann@37660
  1166
lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
haftmann@37660
  1167
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)
huffman@45811
  1168
huffman@46646
  1169
lemma ucast_down_wi [OF refl]:
huffman@46646
  1170
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
huffman@46646
  1171
  apply (unfold is_down)
haftmann@37660
  1172
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
haftmann@37660
  1173
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1174
  apply (erule bintrunc_bintrunc_ge)
haftmann@37660
  1175
  done
huffman@45811
  1176
huffman@46646
  1177
lemma ucast_down_no [OF refl]:
huffman@47108
  1178
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin"
huffman@47108
  1179
  unfolding word_numeral_alt by clarify (rule ucast_down_wi)
huffman@46646
  1180
huffman@45811
  1181
lemma ucast_down_bl [OF refl]:
huffman@45811
  1182
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
huffman@46646
  1183
  unfolding of_bl_def by clarify (erule ucast_down_wi)
haftmann@37660
  1184
haftmann@37660
  1185
lemmas slice_def' = slice_def [unfolded word_size]
haftmann@37660
  1186
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]
haftmann@37660
  1187
haftmann@37660
  1188
lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def
haftmann@37660
  1189
haftmann@37660
  1190
haftmann@37660
  1191
subsection {* Word Arithmetic *}
haftmann@37660
  1192
haftmann@37660
  1193
lemma word_less_alt: "(a < b) = (uint a < uint b)"
huffman@46012
  1194
  unfolding word_less_def word_le_def by (simp add: less_le)
haftmann@37660
  1195
haftmann@37660
  1196
lemma signed_linorder: "class.linorder word_sle word_sless"
wenzelm@46124
  1197
  by default (unfold word_sle_def word_sless_def, auto)
haftmann@37660
  1198
haftmann@37660
  1199
interpretation signed: linorder "word_sle" "word_sless"
haftmann@37660
  1200
  by (rule signed_linorder)
haftmann@37660
  1201
haftmann@37660
  1202
lemma udvdI: 
haftmann@40827
  1203
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
haftmann@37660
  1204
  by (auto simp: udvd_def)
haftmann@37660
  1205
huffman@47108
  1206
lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1207
huffman@47108
  1208
lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1209
huffman@47108
  1210
lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1211
huffman@47108
  1212
lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1213
huffman@47108
  1214
lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b
huffman@47108
  1215
huffman@47108
  1216
lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b
haftmann@37660
  1217
haftmann@54489
  1218
lemma word_m1_wi: "- 1 = word_of_int (- 1)" 
haftmann@54489
  1219
  using word_neg_numeral_alt [of Num.One] by simp
haftmann@37660
  1220
huffman@46648
  1221
lemma word_0_bl [simp]: "of_bl [] = 0"
huffman@46648
  1222
  unfolding of_bl_def by simp
haftmann@37660
  1223
haftmann@37660
  1224
lemma word_1_bl: "of_bl [True] = 1" 
huffman@46648
  1225
  unfolding of_bl_def by (simp add: bl_to_bin_def)
huffman@46648
  1226
huffman@46648
  1227
lemma uint_eq_0 [simp]: "uint 0 = 0"
huffman@46648
  1228
  unfolding word_0_wi word_ubin.eq_norm by simp
haftmann@37660
  1229
huffman@45995
  1230
lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
huffman@46648
  1231
  by (simp add: of_bl_def bl_to_bin_rep_False)
haftmann@37660
  1232
huffman@45805
  1233
lemma to_bl_0 [simp]:
haftmann@37660
  1234
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
haftmann@37660
  1235
  unfolding uint_bl
huffman@46617
  1236
  by (simp add: word_size bin_to_bl_zero)
haftmann@37660
  1237
haftmann@37660
  1238
lemma uint_0_iff: "(uint x = 0) = (x = 0)"
haftmann@37660
  1239
  by (auto intro!: word_uint.Rep_eqD)
haftmann@37660
  1240
haftmann@37660
  1241
lemma unat_0_iff: "(unat x = 0) = (x = 0)"
haftmann@37660
  1242
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)
haftmann@37660
  1243
haftmann@37660
  1244
lemma unat_0 [simp]: "unat 0 = 0"
haftmann@37660
  1245
  unfolding unat_def by auto
haftmann@37660
  1246
haftmann@40827
  1247
lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
haftmann@37660
  1248
  apply (unfold word_size)
haftmann@37660
  1249
  apply (rule box_equals)
haftmann@37660
  1250
    defer
haftmann@37660
  1251
    apply (rule word_uint.Rep_inverse)+
haftmann@37660
  1252
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
haftmann@37660
  1253
  apply simp
haftmann@37660
  1254
  done
haftmann@37660
  1255
huffman@45816
  1256
lemmas size_0_same = size_0_same' [unfolded word_size]
haftmann@37660
  1257
haftmann@37660
  1258
lemmas unat_eq_0 = unat_0_iff
haftmann@37660
  1259
lemmas unat_eq_zero = unat_0_iff
haftmann@37660
  1260
haftmann@37660
  1261
lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
haftmann@37660
  1262
by (auto simp: unat_0_iff [symmetric])
haftmann@37660
  1263
huffman@45958
  1264
lemma ucast_0 [simp]: "ucast 0 = 0"
huffman@45995
  1265
  unfolding ucast_def by simp
huffman@45958
  1266
huffman@45958
  1267
lemma sint_0 [simp]: "sint 0 = 0"
huffman@45958
  1268
  unfolding sint_uint by simp
huffman@45958
  1269
huffman@45958
  1270
lemma scast_0 [simp]: "scast 0 = 0"
huffman@45995
  1271
  unfolding scast_def by simp
haftmann@37660
  1272
haftmann@37660
  1273
lemma sint_n1 [simp] : "sint -1 = -1"
haftmann@54489
  1274
  unfolding word_m1_wi word_sbin.eq_norm by simp
haftmann@54489
  1275
haftmann@54489
  1276
lemma scast_n1 [simp]: "scast (- 1) = - 1"
huffman@45958
  1277
  unfolding scast_def by simp
huffman@45958
  1278
huffman@45958
  1279
lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
haftmann@37660
  1280
  unfolding word_1_wi
huffman@45995
  1281
  by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1)
huffman@45958
  1282
huffman@45958
  1283
lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
huffman@45958
  1284
  unfolding unat_def by simp
huffman@45958
  1285
huffman@45958
  1286
lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
huffman@45995
  1287
  unfolding ucast_def by simp
haftmann@37660
  1288
haftmann@37660
  1289
(* now, to get the weaker results analogous to word_div/mod_def *)
haftmann@37660
  1290
haftmann@37660
  1291
lemmas word_arith_alts = 
huffman@46000
  1292
  word_sub_wi
huffman@46000
  1293
  word_arith_wis (* FIXME: duplicate *)
huffman@46000
  1294
haftmann@55816
  1295
haftmann@55816
  1296
subsection {* Transferring goals from words to ints *}
haftmann@37660
  1297
haftmann@37660
  1298
lemma word_ths:  
haftmann@37660
  1299
  shows
haftmann@37660
  1300
  word_succ_p1:   "word_succ a = a + 1" and
haftmann@37660
  1301
  word_pred_m1:   "word_pred a = a - 1" and
haftmann@37660
  1302
  word_pred_succ: "word_pred (word_succ a) = a" and
haftmann@37660
  1303
  word_succ_pred: "word_succ (word_pred a) = a" and
haftmann@37660
  1304
  word_mult_succ: "word_succ a * b = b + a * b"
huffman@47374
  1305
  by (transfer, simp add: algebra_simps)+
haftmann@37660
  1306
huffman@45816
  1307
lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
huffman@45816
  1308
  by simp
haftmann@37660
  1309
haftmann@37660
  1310
lemmas uint_word_ariths = 
wenzelm@45604
  1311
  word_arith_alts [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1312
haftmann@37660
  1313
lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p]
haftmann@37660
  1314
haftmann@37660
  1315
(* similar expressions for sint (arith operations) *)
haftmann@37660
  1316
lemmas sint_word_ariths = uint_word_arith_bintrs
haftmann@37660
  1317
  [THEN uint_sint [symmetric, THEN trans],
haftmann@37660
  1318
  unfolded uint_sint bintr_arith1s bintr_ariths 
wenzelm@45604
  1319
    len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep]
wenzelm@45604
  1320
wenzelm@45604
  1321
lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
wenzelm@45604
  1322
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]
haftmann@37660
  1323
haftmann@37660
  1324
lemma word_pred_0_n1: "word_pred 0 = word_of_int -1"
huffman@47374
  1325
  unfolding word_pred_m1 by simp
haftmann@37660
  1326
haftmann@37660
  1327
lemma succ_pred_no [simp]:
huffman@47108
  1328
  "word_succ (numeral w) = numeral w + 1"
huffman@47108
  1329
  "word_pred (numeral w) = numeral w - 1"
haftmann@54489
  1330
  "word_succ (- numeral w) = - numeral w + 1"
haftmann@54489
  1331
  "word_pred (- numeral w) = - numeral w - 1"
huffman@47108
  1332
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1333
haftmann@37660
  1334
lemma word_sp_01 [simp] : 
haftmann@37660
  1335
  "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0"
huffman@47108
  1336
  unfolding word_succ_p1 word_pred_m1 by simp_all
haftmann@37660
  1337
haftmann@37660
  1338
(* alternative approach to lifting arithmetic equalities *)
haftmann@37660
  1339
lemma word_of_int_Ex:
haftmann@37660
  1340
  "\<exists>y. x = word_of_int y"
haftmann@37660
  1341
  by (rule_tac x="uint x" in exI) simp
haftmann@37660
  1342
haftmann@37660
  1343
haftmann@55816
  1344
subsection {* Order on fixed-length words *}
haftmann@37660
  1345
haftmann@37660
  1346
lemma word_zero_le [simp] :
haftmann@37660
  1347
  "0 <= (y :: 'a :: len0 word)"
haftmann@37660
  1348
  unfolding word_le_def by auto
haftmann@37660
  1349
  
huffman@45816
  1350
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *)
haftmann@37660
  1351
  unfolding word_le_def
haftmann@37660
  1352
  by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1353
huffman@45816
  1354
lemma word_n1_ge [simp]: "y \<le> (-1::'a::len0 word)"
huffman@45816
  1355
  unfolding word_le_def
huffman@45816
  1356
  by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto
haftmann@37660
  1357
haftmann@37660
  1358
lemmas word_not_simps [simp] = 
haftmann@37660
  1359
  word_zero_le [THEN leD] word_m1_ge [THEN leD] word_n1_ge [THEN leD]
haftmann@37660
  1360
huffman@47108
  1361
lemma word_gt_0: "0 < y \<longleftrightarrow> 0 \<noteq> (y :: 'a :: len0 word)"
huffman@47108
  1362
  by (simp add: less_le)
huffman@47108
  1363
huffman@47108
  1364
lemmas word_gt_0_no [simp] = word_gt_0 [of "numeral y"] for y
haftmann@37660
  1365
haftmann@40827
  1366
lemma word_sless_alt: "(a <s b) = (sint a < sint b)"
haftmann@37660
  1367
  unfolding word_sle_def word_sless_def
haftmann@37660
  1368
  by (auto simp add: less_le)
haftmann@37660
  1369
haftmann@37660
  1370
lemma word_le_nat_alt: "(a <= b) = (unat a <= unat b)"
haftmann@37660
  1371
  unfolding unat_def word_le_def
haftmann@37660
  1372
  by (rule nat_le_eq_zle [symmetric]) simp
haftmann@37660
  1373
haftmann@37660
  1374
lemma word_less_nat_alt: "(a < b) = (unat a < unat b)"
haftmann@37660
  1375
  unfolding unat_def word_less_alt
haftmann@37660
  1376
  by (rule nat_less_eq_zless [symmetric]) simp
haftmann@37660
  1377
  
haftmann@37660
  1378
lemma wi_less: 
haftmann@37660
  1379
  "(word_of_int n < (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1380
    (n mod 2 ^ len_of TYPE('a) < m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1381
  unfolding word_less_alt by (simp add: word_uint.eq_norm)
haftmann@37660
  1382
haftmann@37660
  1383
lemma wi_le: 
haftmann@37660
  1384
  "(word_of_int n <= (word_of_int m :: 'a :: len0 word)) = 
haftmann@37660
  1385
    (n mod 2 ^ len_of TYPE('a) <= m mod 2 ^ len_of TYPE('a))"
haftmann@37660
  1386
  unfolding word_le_def by (simp add: word_uint.eq_norm)
haftmann@37660
  1387
haftmann@37660
  1388
lemma udvd_nat_alt: "a udvd b = (EX n>=0. unat b = n * unat a)"
haftmann@37660
  1389
  apply (unfold udvd_def)
haftmann@37660
  1390
  apply safe
haftmann@37660
  1391
   apply (simp add: unat_def nat_mult_distrib)
haftmann@37660
  1392
  apply (simp add: uint_nat int_mult)
haftmann@37660
  1393
  apply (rule exI)
haftmann@37660
  1394
  apply safe
haftmann@37660
  1395
   prefer 2
haftmann@37660
  1396
   apply (erule notE)
haftmann@37660
  1397
   apply (rule refl)
haftmann@37660
  1398
  apply force
haftmann@37660
  1399
  done
haftmann@37660
  1400
haftmann@37660
  1401
lemma udvd_iff_dvd: "x udvd y <-> unat x dvd unat y"
haftmann@37660
  1402
  unfolding dvd_def udvd_nat_alt by force
haftmann@37660
  1403
wenzelm@45604
  1404
lemmas unat_mono = word_less_nat_alt [THEN iffD1]
haftmann@37660
  1405
haftmann@55816
  1406
lemma unat_minus_one:
haftmann@55816
  1407
  assumes "w \<noteq> 0"
haftmann@55816
  1408
  shows "unat (w - 1) = unat w - 1"
haftmann@55816
  1409
proof -
haftmann@55816
  1410
  have "0 \<le> uint w" by (fact uint_nonnegative)
haftmann@55816
  1411
  moreover from assms have "0 \<noteq> uint w" by (simp add: uint_0_iff)
haftmann@55816
  1412
  ultimately have "1 \<le> uint w" by arith
haftmann@55816
  1413
  from uint_lt2p [of w] have "uint w - 1 < 2 ^ len_of TYPE('a)" by arith
haftmann@55816
  1414
  with `1 \<le> uint w` have "(uint w - 1) mod 2 ^ len_of TYPE('a) = uint w - 1"
haftmann@55816
  1415
    by (auto intro: mod_pos_pos_trivial)
haftmann@55816
  1416
  with `1 \<le> uint w` have "nat ((uint w - 1) mod 2 ^ len_of TYPE('a)) = nat (uint w) - 1"
haftmann@55816
  1417
    by auto
haftmann@55816
  1418
  then show ?thesis
haftmann@55816
  1419
    by (simp only: unat_def int_word_uint word_arith_alts mod_diff_right_eq [symmetric])
haftmann@55816
  1420
qed
haftmann@55816
  1421
haftmann@40827
  1422
lemma measure_unat: "p ~= 0 \<Longrightarrow> unat (p - 1) < unat p"
haftmann@37660
  1423
  by (simp add: unat_minus_one) (simp add: unat_0_iff [symmetric])
haftmann@37660
  1424
  
wenzelm@45604
  1425
lemmas uint_add_ge0 [simp] = add_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
wenzelm@45604
  1426
lemmas uint_mult_ge0 [simp] = mult_nonneg_nonneg [OF uint_ge_0 uint_ge_0]
haftmann@37660
  1427
haftmann@37660
  1428
lemma uint_sub_lt2p [simp]: 
haftmann@37660
  1429
  "uint (x :: 'a :: len0 word) - uint (y :: 'b :: len0 word) < 
haftmann@37660
  1430
    2 ^ len_of TYPE('a)"
haftmann@37660
  1431
  using uint_ge_0 [of y] uint_lt2p [of x] by arith
haftmann@37660
  1432
haftmann@37660
  1433
haftmann@55816
  1434
subsection {* Conditions for the addition (etc) of two words to overflow *}
haftmann@37660
  1435
haftmann@37660
  1436
lemma uint_add_lem: 
haftmann@37660
  1437
  "(uint x + uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1438
    (uint (x + y :: 'a :: len0 word) = uint x + uint y)"
haftmann@37660
  1439
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1440
haftmann@37660
  1441
lemma uint_mult_lem: 
haftmann@37660
  1442
  "(uint x * uint y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1443
    (uint (x * y :: 'a :: len0 word) = uint x * uint y)"
haftmann@37660
  1444
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1445
haftmann@37660
  1446
lemma uint_sub_lem: 
haftmann@37660
  1447
  "(uint x >= uint y) = (uint (x - y) = uint x - uint y)"
haftmann@37660
  1448
  by (unfold uint_word_ariths) (auto intro!: trans [OF _ int_mod_lem])
haftmann@37660
  1449
haftmann@37660
  1450
lemma uint_add_le: "uint (x + y) <= uint x + uint y"
haftmann@55816
  1451
  unfolding uint_word_ariths by (metis uint_add_ge0 zmod_le_nonneg_dividend)
haftmann@37660
  1452
haftmann@37660
  1453
lemma uint_sub_ge: "uint (x - y) >= uint x - uint y"
haftmann@55816
  1454
  unfolding uint_word_ariths by (metis int_mod_ge uint_sub_lt2p zless2p)
haftmann@55816
  1455
haftmann@55816
  1456
lemma mod_add_if_z:
haftmann@55816
  1457
  "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==> 
haftmann@55816
  1458
   (x + y) mod z = (if x + y < z then x + y else x + y - z)"
haftmann@55816
  1459
  by (auto intro: int_mod_eq)
haftmann@55816
  1460
haftmann@55816
  1461
lemma uint_plus_if':
haftmann@55816
  1462
  "uint ((a::'a word) + b) =
haftmann@55816
  1463
  (if uint a + uint b < 2 ^ len_of TYPE('a::len0) then uint a + uint b
haftmann@55816
  1464
   else uint a + uint b - 2 ^ len_of TYPE('a))"
haftmann@55816
  1465
  using mod_add_if_z [of "uint a" _ "uint b"] by (simp add: uint_word_ariths)
haftmann@55816
  1466
haftmann@55816
  1467
lemma mod_sub_if_z:
haftmann@55816
  1468
  "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==> 
haftmann@55816
  1469
   (x - y) mod z = (if y <= x then x - y else x - y + z)"
haftmann@55816
  1470
  by (auto intro: int_mod_eq)
haftmann@55816
  1471
haftmann@55816
  1472
lemma uint_sub_if':
haftmann@55816
  1473
  "uint ((a::'a word) - b) =
haftmann@55816
  1474
  (if uint b \<le> uint a then uint a - uint b
haftmann@55816
  1475
   else uint a - uint b + 2 ^ len_of TYPE('a::len0))"
haftmann@55816
  1476
  using mod_sub_if_z [of "uint a" _ "uint b"] by (simp add: uint_word_ariths)
haftmann@55816
  1477
haftmann@55816
  1478
haftmann@55816
  1479
subsection {* Definition of @{text uint_arith} *}
haftmann@37660
  1480
haftmann@37660
  1481
lemma word_of_int_inverse:
haftmann@40827
  1482
  "word_of_int r = a \<Longrightarrow> 0 <= r \<Longrightarrow> r < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  1483
   uint (a::'a::len0 word) = r"
haftmann@37660
  1484
  apply (erule word_uint.Abs_inverse' [rotated])
haftmann@37660
  1485
  apply (simp add: uints_num)
haftmann@37660
  1486
  done
haftmann@37660
  1487
haftmann@37660
  1488
lemma uint_split:
haftmann@37660
  1489
  fixes x::"'a::len0 word"
haftmann@37660
  1490
  shows "P (uint x) = 
haftmann@37660
  1491
         (ALL i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) --> P i)"
haftmann@37660
  1492
  apply (fold word_int_case_def)
haftmann@55816
  1493
  apply (auto dest!: word_of_int_inverse simp: int_word_uint mod_pos_pos_trivial
haftmann@37660
  1494
              split: word_int_split)
haftmann@37660
  1495
  done
haftmann@37660
  1496
haftmann@37660
  1497
lemma uint_split_asm:
haftmann@37660
  1498
  fixes x::"'a::len0 word"
haftmann@37660
  1499
  shows "P (uint x) = 
haftmann@37660
  1500
         (~(EX i. word_of_int i = x & 0 <= i & i < 2^len_of TYPE('a) & ~ P i))"
haftmann@37660
  1501
  by (auto dest!: word_of_int_inverse 
haftmann@55816
  1502
           simp: int_word_uint mod_pos_pos_trivial
haftmann@37660
  1503
           split: uint_split)
haftmann@37660
  1504
haftmann@37660
  1505
lemmas uint_splits = uint_split uint_split_asm
haftmann@37660
  1506
haftmann@37660
  1507
lemmas uint_arith_simps = 
haftmann@37660
  1508
  word_le_def word_less_alt
haftmann@37660
  1509
  word_uint.Rep_inject [symmetric] 
haftmann@37660
  1510
  uint_sub_if' uint_plus_if'
haftmann@37660
  1511
haftmann@37660
  1512
(* use this to stop, eg, 2 ^ len_of TYPE (32) being simplified *)
haftmann@40827
  1513
lemma power_False_cong: "False \<Longrightarrow> a ^ b = c ^ d" 
haftmann@37660
  1514
  by auto
haftmann@37660
  1515
haftmann@37660
  1516
(* uint_arith_tac: reduce to arithmetic on int, try to solve by arith *)
haftmann@37660
  1517
ML {*
wenzelm@51717
  1518
fun uint_arith_simpset ctxt = 
wenzelm@51717
  1519
  ctxt addsimps @{thms uint_arith_simps}
haftmann@37660
  1520
     delsimps @{thms word_uint.Rep_inject}
wenzelm@45620
  1521
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  1522
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  1523
haftmann@37660
  1524
fun uint_arith_tacs ctxt = 
haftmann@37660
  1525
  let
haftmann@37660
  1526
    fun arith_tac' n t =
haftmann@37660
  1527
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  1528
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  1529
  in 
wenzelm@42793
  1530
    [ clarify_tac ctxt 1,
wenzelm@51717
  1531
      full_simp_tac (uint_arith_simpset ctxt) 1,
wenzelm@51717
  1532
      ALLGOALS (full_simp_tac
wenzelm@51717
  1533
        (put_simpset HOL_ss ctxt
wenzelm@51717
  1534
          |> fold Splitter.add_split @{thms uint_splits}
wenzelm@51717
  1535
          |> fold Simplifier.add_cong @{thms power_False_cong})),
wenzelm@54742
  1536
      rewrite_goals_tac ctxt @{thms word_size}, 
haftmann@37660
  1537
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  1538
                         REPEAT (etac conjE n) THEN
haftmann@37660
  1539
                         REPEAT (dtac @{thm word_of_int_inverse} n 
haftmann@37660
  1540
                                 THEN atac n 
haftmann@37660
  1541
                                 THEN atac n)),
haftmann@37660
  1542
      TRYALL arith_tac' ]
haftmann@37660
  1543
  end
haftmann@37660
  1544
haftmann@37660
  1545
fun uint_arith_tac ctxt = SELECT_GOAL (EVERY (uint_arith_tacs ctxt))
haftmann@37660
  1546
*}
haftmann@37660
  1547
haftmann@37660
  1548
method_setup uint_arith = 
haftmann@37660
  1549
  {* Scan.succeed (SIMPLE_METHOD' o uint_arith_tac) *}
haftmann@37660
  1550
  "solving word arithmetic via integers and arith"
haftmann@37660
  1551
haftmann@37660
  1552
haftmann@55816
  1553
subsection {* More on overflows and monotonicity *}
haftmann@37660
  1554
haftmann@37660
  1555
lemma no_plus_overflow_uint_size: 
haftmann@37660
  1556
  "((x :: 'a :: len0 word) <= x + y) = (uint x + uint y < 2 ^ size x)"
haftmann@37660
  1557
  unfolding word_size by uint_arith
haftmann@37660
  1558
haftmann@37660
  1559
lemmas no_olen_add = no_plus_overflow_uint_size [unfolded word_size]
haftmann@37660
  1560
haftmann@37660
  1561
lemma no_ulen_sub: "((x :: 'a :: len0 word) >= x - y) = (uint y <= uint x)"
haftmann@37660
  1562
  by uint_arith
haftmann@37660
  1563
haftmann@37660
  1564
lemma no_olen_add':
haftmann@37660
  1565
  fixes x :: "'a::len0 word"
haftmann@37660
  1566
  shows "(x \<le> y + x) = (uint y + uint x < 2 ^ len_of TYPE('a))"
huffman@45546
  1567
  by (simp add: add_ac no_olen_add)
haftmann@37660
  1568
wenzelm@45604
  1569
lemmas olen_add_eqv = trans [OF no_olen_add no_olen_add' [symmetric]]
wenzelm@45604
  1570
wenzelm@45604
  1571
lemmas uint_plus_simple_iff = trans [OF no_olen_add uint_add_lem]
wenzelm@45604
  1572
lemmas uint_plus_simple = uint_plus_simple_iff [THEN iffD1]
wenzelm@45604
  1573
lemmas uint_minus_simple_iff = trans [OF no_ulen_sub uint_sub_lem]
haftmann@37660
  1574
lemmas uint_minus_simple_alt = uint_sub_lem [folded word_le_def]
haftmann@37660
  1575
lemmas word_sub_le_iff = no_ulen_sub [folded word_le_def]
wenzelm@45604
  1576
lemmas word_sub_le = word_sub_le_iff [THEN iffD2]
haftmann@37660
  1577
haftmann@37660
  1578
lemma word_less_sub1: 
haftmann@40827
  1579
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 < x) = (0 < x - 1)"
haftmann@37660
  1580
  by uint_arith
haftmann@37660
  1581
haftmann@37660
  1582
lemma word_le_sub1: 
haftmann@40827
  1583
  "(x :: 'a :: len word) ~= 0 \<Longrightarrow> (1 <= x) = (0 <= x - 1)"
haftmann@37660
  1584
  by uint_arith
haftmann@37660
  1585
haftmann@37660
  1586
lemma sub_wrap_lt: 
haftmann@37660
  1587
  "((x :: 'a :: len0 word) < x - z) = (x < z)"
haftmann@37660
  1588
  by uint_arith
haftmann@37660
  1589
haftmann@37660
  1590
lemma sub_wrap: 
haftmann@37660
  1591
  "((x :: 'a :: len0 word) <= x - z) = (z = 0 | x < z)"
haftmann@37660
  1592
  by uint_arith
haftmann@37660
  1593
haftmann@37660
  1594
lemma plus_minus_not_NULL_ab: 
haftmann@40827
  1595
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> c ~= 0 \<Longrightarrow> x + c ~= 0"
haftmann@37660
  1596
  by uint_arith
haftmann@37660
  1597
haftmann@37660
  1598
lemma plus_minus_no_overflow_ab: 
haftmann@40827
  1599
  "(x :: 'a :: len0 word) <= ab - c \<Longrightarrow> c <= ab \<Longrightarrow> x <= x + c" 
haftmann@37660
  1600
  by uint_arith
haftmann@37660
  1601
haftmann@37660
  1602
lemma le_minus': 
haftmann@40827
  1603
  "(a :: 'a :: len0 word) + c <= b \<Longrightarrow> a <= a + c \<Longrightarrow> c <= b - a"
haftmann@37660
  1604
  by uint_arith
haftmann@37660
  1605
haftmann@37660
  1606
lemma le_plus': 
haftmann@40827
  1607
  "(a :: 'a :: len0 word) <= b \<Longrightarrow> c <= b - a \<Longrightarrow> a + c <= b"
haftmann@37660
  1608
  by uint_arith
haftmann@37660
  1609
haftmann@37660
  1610
lemmas le_plus = le_plus' [rotated]
haftmann@37660
  1611
huffman@46011
  1612
lemmas le_minus = leD [THEN thin_rl, THEN le_minus'] (* FIXME *)
haftmann@37660
  1613
haftmann@37660
  1614
lemma word_plus_mono_right: 
haftmann@40827
  1615
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= x + z \<Longrightarrow> x + y <= x + z"
haftmann@37660
  1616
  by uint_arith
haftmann@37660
  1617
haftmann@37660
  1618
lemma word_less_minus_cancel: 
haftmann@40827
  1619
  "y - x < z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) < z"
haftmann@37660
  1620
  by uint_arith
haftmann@37660
  1621
haftmann@37660
  1622
lemma word_less_minus_mono_left: 
haftmann@40827
  1623
  "(y :: 'a :: len0 word) < z \<Longrightarrow> x <= y \<Longrightarrow> y - x < z - x"
haftmann@37660
  1624
  by uint_arith
haftmann@37660
  1625
haftmann@37660
  1626
lemma word_less_minus_mono:  
haftmann@40827
  1627
  "a < c \<Longrightarrow> d < b \<Longrightarrow> a - b < a \<Longrightarrow> c - d < c 
haftmann@40827
  1628
  \<Longrightarrow> a - b < c - (d::'a::len word)"
haftmann@37660
  1629
  by uint_arith
haftmann@37660
  1630
haftmann@37660
  1631
lemma word_le_minus_cancel: 
haftmann@40827
  1632
  "y - x <= z - x \<Longrightarrow> x <= z \<Longrightarrow> (y :: 'a :: len0 word) <= z"
haftmann@37660
  1633
  by uint_arith
haftmann@37660
  1634
haftmann@37660
  1635
lemma word_le_minus_mono_left: 
haftmann@40827
  1636
  "(y :: 'a :: len0 word) <= z \<Longrightarrow> x <= y \<Longrightarrow> y - x <= z - x"
haftmann@37660
  1637
  by uint_arith
haftmann@37660
  1638
haftmann@37660
  1639
lemma word_le_minus_mono:  
haftmann@40827
  1640
  "a <= c \<Longrightarrow> d <= b \<Longrightarrow> a - b <= a \<Longrightarrow> c - d <= c 
haftmann@40827
  1641
  \<Longrightarrow> a - b <= c - (d::'a::len word)"
haftmann@37660
  1642
  by uint_arith
haftmann@37660
  1643
haftmann@37660
  1644
lemma plus_le_left_cancel_wrap: 
haftmann@40827
  1645
  "(x :: 'a :: len0 word) + y' < x \<Longrightarrow> x + y < x \<Longrightarrow> (x + y' < x + y) = (y' < y)"
haftmann@37660
  1646
  by uint_arith
haftmann@37660
  1647
haftmann@37660
  1648
lemma plus_le_left_cancel_nowrap: 
haftmann@40827
  1649
  "(x :: 'a :: len0 word) <= x + y' \<Longrightarrow> x <= x + y \<Longrightarrow> 
haftmann@37660
  1650
    (x + y' < x + y) = (y' < y)" 
haftmann@37660
  1651
  by uint_arith
haftmann@37660
  1652
haftmann@37660
  1653
lemma word_plus_mono_right2: 
haftmann@40827
  1654
  "(a :: 'a :: len0 word) <= a + b \<Longrightarrow> c <= b \<Longrightarrow> a <= a + c"
haftmann@37660
  1655
  by uint_arith
haftmann@37660
  1656
haftmann@37660
  1657
lemma word_less_add_right: 
haftmann@40827
  1658
  "(x :: 'a :: len0 word) < y - z \<Longrightarrow> z <= y \<Longrightarrow> x + z < y"
haftmann@37660
  1659
  by uint_arith
haftmann@37660
  1660
haftmann@37660
  1661
lemma word_less_sub_right: 
haftmann@40827
  1662
  "(x :: 'a :: len0 word) < y + z \<Longrightarrow> y <= x \<Longrightarrow> x - y < z"
haftmann@37660
  1663
  by uint_arith
haftmann@37660
  1664
haftmann@37660
  1665
lemma word_le_plus_either: 
haftmann@40827
  1666
  "(x :: 'a :: len0 word) <= y | x <= z \<Longrightarrow> y <= y + z \<Longrightarrow> x <= y + z"
haftmann@37660
  1667
  by uint_arith
haftmann@37660
  1668
haftmann@37660
  1669
lemma word_less_nowrapI: 
haftmann@40827
  1670
  "(x :: 'a :: len0 word) < z - k \<Longrightarrow> k <= z \<Longrightarrow> 0 < k \<Longrightarrow> x < x + k"
haftmann@37660
  1671
  by uint_arith
haftmann@37660
  1672
haftmann@40827
  1673
lemma inc_le: "(i :: 'a :: len word) < m \<Longrightarrow> i + 1 <= m"
haftmann@37660
  1674
  by uint_arith
haftmann@37660
  1675
haftmann@37660
  1676
lemma inc_i: 
haftmann@40827
  1677
  "(1 :: 'a :: len word) <= i \<Longrightarrow> i < m \<Longrightarrow> 1 <= (i + 1) & i + 1 <= m"
haftmann@37660
  1678
  by uint_arith
haftmann@37660
  1679
haftmann@37660
  1680
lemma udvd_incr_lem:
haftmann@40827
  1681
  "up < uq \<Longrightarrow> up = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1682
    uq = ua + n' * uint K \<Longrightarrow> up + uint K <= uq"
haftmann@37660
  1683
  apply clarsimp
haftmann@55816
  1684
  
haftmann@37660
  1685
  apply (drule less_le_mult)
haftmann@37660
  1686
  apply safe
haftmann@37660
  1687
  done
haftmann@37660
  1688
haftmann@37660
  1689
lemma udvd_incr': 
haftmann@40827
  1690
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1691
    uint q = ua + n' * uint K \<Longrightarrow> p + K <= q" 
haftmann@37660
  1692
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1693
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1694
  apply (erule uint_add_le [THEN order_trans])
haftmann@37660
  1695
  done
haftmann@37660
  1696
haftmann@37660
  1697
lemma udvd_decr': 
haftmann@40827
  1698
  "p < q \<Longrightarrow> uint p = ua + n * uint K \<Longrightarrow> 
haftmann@40827
  1699
    uint q = ua + n' * uint K \<Longrightarrow> p <= q - K"
haftmann@37660
  1700
  apply (unfold word_less_alt word_le_def)
haftmann@37660
  1701
  apply (drule (2) udvd_incr_lem)
haftmann@37660
  1702
  apply (drule le_diff_eq [THEN iffD2])
haftmann@37660
  1703
  apply (erule order_trans)
haftmann@37660
  1704
  apply (rule uint_sub_ge)
haftmann@37660
  1705
  done
haftmann@37660
  1706
huffman@45816
  1707
lemmas udvd_incr_lem0 = udvd_incr_lem [where ua=0, unfolded add_0_left]
huffman@45816
  1708
lemmas udvd_incr0 = udvd_incr' [where ua=0, unfolded add_0_left]
huffman@45816
  1709
lemmas udvd_decr0 = udvd_decr' [where ua=0, unfolded add_0_left]
haftmann@37660
  1710
haftmann@37660
  1711
lemma udvd_minus_le': 
haftmann@40827
  1712
  "xy < k \<Longrightarrow> z udvd xy \<Longrightarrow> z udvd k \<Longrightarrow> xy <= k - z"
haftmann@37660
  1713
  apply (unfold udvd_def)
haftmann@37660
  1714
  apply clarify
haftmann@37660
  1715
  apply (erule (2) udvd_decr0)
haftmann@37660
  1716
  done
haftmann@37660
  1717
haftmann@37660
  1718
lemma udvd_incr2_K: 
haftmann@40827
  1719
  "p < a + s \<Longrightarrow> a <= a + s \<Longrightarrow> K udvd s \<Longrightarrow> K udvd p - a \<Longrightarrow> a <= p \<Longrightarrow> 
haftmann@40827
  1720
    0 < K \<Longrightarrow> p <= p + K & p + K <= a + s"
wenzelm@51286
  1721
  using [[simproc del: linordered_ring_less_cancel_factor]]
haftmann@37660
  1722
  apply (unfold udvd_def)
haftmann@37660
  1723
  apply clarify
haftmann@37660
  1724
  apply (simp add: uint_arith_simps split: split_if_asm)
haftmann@37660
  1725
   prefer 2 
haftmann@37660
  1726
   apply (insert uint_range' [of s])[1]
haftmann@37660
  1727
   apply arith
haftmann@37660
  1728
  apply (drule add_commute [THEN xtr1])
haftmann@37660
  1729
  apply (simp add: diff_less_eq [symmetric])
haftmann@37660
  1730
  apply (drule less_le_mult)
haftmann@37660
  1731
   apply arith
haftmann@37660
  1732
  apply simp
haftmann@37660
  1733
  done
haftmann@37660
  1734
haftmann@37660
  1735
(* links with rbl operations *)
haftmann@37660
  1736
lemma word_succ_rbl:
haftmann@40827
  1737
  "to_bl w = bl \<Longrightarrow> to_bl (word_succ w) = (rev (rbl_succ (rev bl)))"
haftmann@37660
  1738
  apply (unfold word_succ_def)
haftmann@37660
  1739
  apply clarify
haftmann@37660
  1740
  apply (simp add: to_bl_of_bin)
huffman@46654
  1741
  apply (simp add: to_bl_def rbl_succ)
haftmann@37660
  1742
  done
haftmann@37660
  1743
haftmann@37660
  1744
lemma word_pred_rbl:
haftmann@40827
  1745
  "to_bl w = bl \<Longrightarrow> to_bl (word_pred w) = (rev (rbl_pred (rev bl)))"
haftmann@37660
  1746
  apply (unfold word_pred_def)
haftmann@37660
  1747
  apply clarify
haftmann@37660
  1748
  apply (simp add: to_bl_of_bin)
huffman@46654
  1749
  apply (simp add: to_bl_def rbl_pred)
haftmann@37660
  1750
  done
haftmann@37660
  1751
haftmann@37660
  1752
lemma word_add_rbl:
haftmann@40827
  1753
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1754
    to_bl (v + w) = (rev (rbl_add (rev vbl) (rev wbl)))"
haftmann@37660
  1755
  apply (unfold word_add_def)
haftmann@37660
  1756
  apply clarify
haftmann@37660
  1757
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1758
  apply (simp add: to_bl_def rbl_add)
haftmann@37660
  1759
  done
haftmann@37660
  1760
haftmann@37660
  1761
lemma word_mult_rbl:
haftmann@40827
  1762
  "to_bl v = vbl \<Longrightarrow> to_bl w = wbl \<Longrightarrow> 
haftmann@37660
  1763
    to_bl (v * w) = (rev (rbl_mult (rev vbl) (rev wbl)))"
haftmann@37660
  1764
  apply (unfold word_mult_def)
haftmann@37660
  1765
  apply clarify
haftmann@37660
  1766
  apply (simp add: to_bl_of_bin)
haftmann@37660
  1767
  apply (simp add: to_bl_def rbl_mult)
haftmann@37660
  1768
  done
haftmann@37660
  1769
haftmann@37660
  1770
lemma rtb_rbl_ariths:
haftmann@37660
  1771
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_succ w)) = rbl_succ ys"
haftmann@37660
  1772
  "rev (to_bl w) = ys \<Longrightarrow> rev (to_bl (word_pred w)) = rbl_pred ys"
haftmann@40827
  1773
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v * w)) = rbl_mult ys xs"
haftmann@40827
  1774
  "rev (to_bl v) = ys \<Longrightarrow> rev (to_bl w) = xs \<Longrightarrow> rev (to_bl (v + w)) = rbl_add ys xs"
haftmann@37660
  1775
  by (auto simp: rev_swap [symmetric] word_succ_rbl 
haftmann@37660
  1776
                 word_pred_rbl word_mult_rbl word_add_rbl)
haftmann@37660
  1777
haftmann@37660
  1778
haftmann@55816
  1779
subsection {* Arithmetic type class instantiations *}
haftmann@37660
  1780
haftmann@37660
  1781
lemmas word_le_0_iff [simp] =
haftmann@37660
  1782
  word_zero_le [THEN leD, THEN linorder_antisym_conv1]
haftmann@37660
  1783
haftmann@37660
  1784
lemma word_of_int_nat: 
haftmann@40827
  1785
  "0 <= x \<Longrightarrow> word_of_int x = of_nat (nat x)"
haftmann@37660
  1786
  by (simp add: of_nat_nat word_of_int)
haftmann@37660
  1787
huffman@46603
  1788
(* note that iszero_def is only for class comm_semiring_1_cancel,
huffman@46603
  1789
   which requires word length >= 1, ie 'a :: len word *) 
huffman@46603
  1790
lemma iszero_word_no [simp]:
huffman@47108
  1791
  "iszero (numeral bin :: 'a :: len word) = 
huffman@47108
  1792
    iszero (bintrunc (len_of TYPE('a)) (numeral bin))"
huffman@47108
  1793
  using word_ubin.norm_eq_iff [where 'a='a, of "numeral bin" 0]
huffman@46603
  1794
  by (simp add: iszero_def [symmetric])
huffman@47108
  1795
    
huffman@47108
  1796
text {* Use @{text iszero} to simplify equalities between word numerals. *}
huffman@47108
  1797
huffman@47108
  1798
lemmas word_eq_numeral_iff_iszero [simp] =
huffman@47108
  1799
  eq_numeral_iff_iszero [where 'a="'a::len word"]
huffman@46603
  1800
haftmann@37660
  1801
haftmann@55816
  1802
subsection {* Word and nat *}
haftmann@37660
  1803
huffman@45811
  1804
lemma td_ext_unat [OF refl]:
haftmann@40827
  1805
  "n = len_of TYPE ('a :: len) \<Longrightarrow> 
haftmann@37660
  1806
    td_ext (unat :: 'a word => nat) of_nat 
haftmann@37660
  1807
    (unats n) (%i. i mod 2 ^ n)"
haftmann@37660
  1808
  apply (unfold td_ext_def' unat_def word_of_nat unats_uints)
haftmann@37660
  1809
  apply (auto intro!: imageI simp add : word_of_int_hom_syms)
haftmann@37660
  1810
  apply (erule word_uint.Abs_inverse [THEN arg_cong])
haftmann@37660
  1811
  apply (simp add: int_word_uint nat_mod_distrib nat_power_eq)
haftmann@37660
  1812
  done
haftmann@37660
  1813
wenzelm@45604
  1814
lemmas unat_of_nat = td_ext_unat [THEN td_ext.eq_norm]
haftmann@37660
  1815
haftmann@37660
  1816
interpretation word_unat:
haftmann@37660
  1817
  td_ext "unat::'a::len word => nat" 
haftmann@37660
  1818
         of_nat 
haftmann@37660
  1819
         "unats (len_of TYPE('a::len))"
haftmann@37660
  1820
         "%i. i mod 2 ^ len_of TYPE('a::len)"
haftmann@37660
  1821
  by (rule td_ext_unat)
haftmann@37660
  1822
haftmann@37660
  1823
lemmas td_unat = word_unat.td_thm
haftmann@37660
  1824
haftmann@37660
  1825
lemmas unat_lt2p [iff] = word_unat.Rep [unfolded unats_def mem_Collect_eq]
haftmann@37660
  1826
haftmann@40827
  1827
lemma unat_le: "y <= unat (z :: 'a :: len word) \<Longrightarrow> y : unats (len_of TYPE ('a))"
haftmann@37660
  1828
  apply (unfold unats_def)
haftmann@37660
  1829
  apply clarsimp
haftmann@37660
  1830
  apply (rule xtrans, rule unat_lt2p, assumption) 
haftmann@37660
  1831
  done
haftmann@37660
  1832
haftmann@37660
  1833
lemma word_nchotomy:
haftmann@37660
  1834
  "ALL w. EX n. (w :: 'a :: len word) = of_nat n & n < 2 ^ len_of TYPE ('a)"
haftmann@37660
  1835
  apply (rule allI)
haftmann@37660
  1836
  apply (rule word_unat.Abs_cases)
haftmann@37660
  1837
  apply (unfold unats_def)
haftmann@37660
  1838
  apply auto
haftmann@37660
  1839
  done
haftmann@37660
  1840
haftmann@37660
  1841
lemma of_nat_eq:
haftmann@37660
  1842
  fixes w :: "'a::len word"
haftmann@37660
  1843
  shows "(of_nat n = w) = (\<exists>q. n = unat w + q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1844
  apply (rule trans)
haftmann@37660
  1845
   apply (rule word_unat.inverse_norm)
haftmann@37660
  1846
  apply (rule iffI)
haftmann@37660
  1847
   apply (rule mod_eqD)
haftmann@37660
  1848
   apply simp
haftmann@37660
  1849
  apply clarsimp
haftmann@37660
  1850
  done
haftmann@37660
  1851
haftmann@37660
  1852
lemma of_nat_eq_size: 
haftmann@37660
  1853
  "(of_nat n = w) = (EX q. n = unat w + q * 2 ^ size w)"
haftmann@37660
  1854
  unfolding word_size by (rule of_nat_eq)
haftmann@37660
  1855
haftmann@37660
  1856
lemma of_nat_0:
haftmann@37660
  1857
  "(of_nat m = (0::'a::len word)) = (\<exists>q. m = q * 2 ^ len_of TYPE('a))"
haftmann@37660
  1858
  by (simp add: of_nat_eq)
haftmann@37660
  1859
huffman@45805
  1860
lemma of_nat_2p [simp]:
huffman@45805
  1861
  "of_nat (2 ^ len_of TYPE('a)) = (0::'a::len word)"
huffman@45805
  1862
  by (fact mult_1 [symmetric, THEN iffD2 [OF of_nat_0 exI]])
haftmann@37660
  1863
haftmann@40827
  1864
lemma of_nat_gt_0: "of_nat k ~= 0 \<Longrightarrow> 0 < k"
haftmann@37660
  1865
  by (cases k) auto
haftmann@37660
  1866
haftmann@37660
  1867
lemma of_nat_neq_0: 
haftmann@40827
  1868
  "0 < k \<Longrightarrow> k < 2 ^ len_of TYPE ('a :: len) \<Longrightarrow> of_nat k ~= (0 :: 'a word)"
haftmann@37660
  1869
  by (clarsimp simp add : of_nat_0)
haftmann@37660
  1870
haftmann@37660
  1871
lemma Abs_fnat_hom_add:
haftmann@37660
  1872
  "of_nat a + of_nat b = of_nat (a + b)"
haftmann@37660
  1873
  by simp
haftmann@37660
  1874
haftmann@37660
  1875
lemma Abs_fnat_hom_mult:
haftmann@37660
  1876
  "of_nat a * of_nat b = (of_nat (a * b) :: 'a :: len word)"
huffman@46013
  1877
  by (simp add: word_of_nat wi_hom_mult zmult_int)
haftmann@37660
  1878
haftmann@37660
  1879
lemma Abs_fnat_hom_Suc:
haftmann@37660
  1880
  "word_succ (of_nat a) = of_nat (Suc a)"
huffman@46013
  1881
  by (simp add: word_of_nat wi_hom_succ add_ac)
haftmann@37660
  1882
haftmann@37660
  1883
lemma Abs_fnat_hom_0: "(0::'a::len word) = of_nat 0"
huffman@45995
  1884
  by simp
haftmann@37660
  1885
haftmann@37660
  1886
lemma Abs_fnat_hom_1: "(1::'a::len word) = of_nat (Suc 0)"
huffman@45995
  1887
  by simp
haftmann@37660
  1888
haftmann@37660
  1889
lemmas Abs_fnat_homs = 
haftmann@37660
  1890
  Abs_fnat_hom_add Abs_fnat_hom_mult Abs_fnat_hom_Suc 
haftmann@37660
  1891
  Abs_fnat_hom_0 Abs_fnat_hom_1
haftmann@37660
  1892
haftmann@37660
  1893
lemma word_arith_nat_add:
haftmann@37660
  1894
  "a + b = of_nat (unat a + unat b)" 
haftmann@37660
  1895
  by simp
haftmann@37660
  1896
haftmann@37660
  1897
lemma word_arith_nat_mult:
haftmann@37660
  1898
  "a * b = of_nat (unat a * unat b)"
huffman@45995
  1899
  by (simp add: of_nat_mult)
haftmann@37660
  1900
    
haftmann@37660
  1901
lemma word_arith_nat_Suc:
haftmann@37660
  1902
  "word_succ a = of_nat (Suc (unat a))"
haftmann@37660
  1903
  by (subst Abs_fnat_hom_Suc [symmetric]) simp
haftmann@37660
  1904
haftmann@37660
  1905
lemma word_arith_nat_div:
haftmann@37660
  1906
  "a div b = of_nat (unat a div unat b)"
haftmann@37660
  1907
  by (simp add: word_div_def word_of_nat zdiv_int uint_nat)
haftmann@37660
  1908
haftmann@37660
  1909
lemma word_arith_nat_mod:
haftmann@37660
  1910
  "a mod b = of_nat (unat a mod unat b)"
haftmann@37660
  1911
  by (simp add: word_mod_def word_of_nat zmod_int uint_nat)
haftmann@37660
  1912
haftmann@37660
  1913
lemmas word_arith_nat_defs =
haftmann@37660
  1914
  word_arith_nat_add word_arith_nat_mult
haftmann@37660
  1915
  word_arith_nat_Suc Abs_fnat_hom_0
haftmann@37660
  1916
  Abs_fnat_hom_1 word_arith_nat_div
haftmann@37660
  1917
  word_arith_nat_mod 
haftmann@37660
  1918
huffman@45816
  1919
lemma unat_cong: "x = y \<Longrightarrow> unat x = unat y"
huffman@45816
  1920
  by simp
haftmann@37660
  1921
  
haftmann@37660
  1922
lemmas unat_word_ariths = word_arith_nat_defs
wenzelm@45604
  1923
  [THEN trans [OF unat_cong unat_of_nat]]
haftmann@37660
  1924
haftmann@37660
  1925
lemmas word_sub_less_iff = word_sub_le_iff
huffman@45816
  1926
  [unfolded linorder_not_less [symmetric] Not_eq_iff]
haftmann@37660
  1927
haftmann@37660
  1928
lemma unat_add_lem: 
haftmann@37660
  1929
  "(unat x + unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1930
    (unat (x + y :: 'a :: len word) = unat x + unat y)"
haftmann@37660
  1931
  unfolding unat_word_ariths
haftmann@37660
  1932
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1933
haftmann@37660
  1934
lemma unat_mult_lem: 
haftmann@37660
  1935
  "(unat x * unat y < 2 ^ len_of TYPE('a)) = 
haftmann@37660
  1936
    (unat (x * y :: 'a :: len word) = unat x * unat y)"
haftmann@37660
  1937
  unfolding unat_word_ariths
haftmann@37660
  1938
  by (auto intro!: trans [OF _ nat_mod_lem])
haftmann@37660
  1939
wenzelm@45604
  1940
lemmas unat_plus_if' = trans [OF unat_word_ariths(1) mod_nat_add, simplified]
haftmann@37660
  1941
haftmann@37660
  1942
lemma le_no_overflow: 
haftmann@40827
  1943
  "x <= b \<Longrightarrow> a <= a + b \<Longrightarrow> x <= a + (b :: 'a :: len0 word)"
haftmann@37660
  1944
  apply (erule order_trans)
haftmann@37660
  1945
  apply (erule olen_add_eqv [THEN iffD1])
haftmann@37660
  1946
  done
haftmann@37660
  1947
wenzelm@45604
  1948
lemmas un_ui_le = trans [OF word_le_nat_alt [symmetric] word_le_def]
haftmann@37660
  1949
haftmann@37660
  1950
lemma unat_sub_if_size:
haftmann@37660
  1951
  "unat (x - y) = (if unat y <= unat x 
haftmann@37660
  1952
   then unat x - unat y 
haftmann@37660
  1953
   else unat x + 2 ^ size x - unat y)"
haftmann@37660
  1954
  apply (unfold word_size)
haftmann@37660
  1955
  apply (simp add: un_ui_le)
haftmann@37660
  1956
  apply (auto simp add: unat_def uint_sub_if')
haftmann@37660
  1957
   apply (rule nat_diff_distrib)
haftmann@37660
  1958
    prefer 3
haftmann@37660
  1959
    apply (simp add: algebra_simps)
haftmann@37660
  1960
    apply (rule nat_diff_distrib [THEN trans])
haftmann@37660
  1961
      prefer 3
haftmann@37660
  1962
      apply (subst nat_add_distrib)
haftmann@37660
  1963
        prefer 3
haftmann@37660
  1964
        apply (simp add: nat_power_eq)
haftmann@37660
  1965
       apply auto
haftmann@37660
  1966
  apply uint_arith
haftmann@37660
  1967
  done
haftmann@37660
  1968
haftmann@37660
  1969
lemmas unat_sub_if' = unat_sub_if_size [unfolded word_size]
haftmann@37660
  1970
haftmann@37660
  1971
lemma unat_div: "unat ((x :: 'a :: len word) div y) = unat x div unat y"
haftmann@37660
  1972
  apply (simp add : unat_word_ariths)
haftmann@37660
  1973
  apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1974
  apply (rule div_le_dividend)
haftmann@37660
  1975
  done
haftmann@37660
  1976
haftmann@37660
  1977
lemma unat_mod: "unat ((x :: 'a :: len word) mod y) = unat x mod unat y"
haftmann@37660
  1978
  apply (clarsimp simp add : unat_word_ariths)
haftmann@37660
  1979
  apply (cases "unat y")
haftmann@37660
  1980
   prefer 2
haftmann@37660
  1981
   apply (rule unat_lt2p [THEN xtr7, THEN nat_mod_eq'])
haftmann@37660
  1982
   apply (rule mod_le_divisor)
haftmann@37660
  1983
   apply auto
haftmann@37660
  1984
  done
haftmann@37660
  1985
haftmann@37660
  1986
lemma uint_div: "uint ((x :: 'a :: len word) div y) = uint x div uint y"
haftmann@37660
  1987
  unfolding uint_nat by (simp add : unat_div zdiv_int)
haftmann@37660
  1988
haftmann@37660
  1989
lemma uint_mod: "uint ((x :: 'a :: len word) mod y) = uint x mod uint y"
haftmann@37660
  1990
  unfolding uint_nat by (simp add : unat_mod zmod_int)
haftmann@37660
  1991
haftmann@37660
  1992
haftmann@55816
  1993
subsection {* Definition of @[text unat_arith} tactic *}
haftmann@37660
  1994
haftmann@37660
  1995
lemma unat_split:
haftmann@37660
  1996
  fixes x::"'a::len word"
haftmann@37660
  1997
  shows "P (unat x) = 
haftmann@37660
  1998
         (ALL n. of_nat n = x & n < 2^len_of TYPE('a) --> P n)"
haftmann@37660
  1999
  by (auto simp: unat_of_nat)
haftmann@37660
  2000
haftmann@37660
  2001
lemma unat_split_asm:
haftmann@37660
  2002
  fixes x::"'a::len word"
haftmann@37660
  2003
  shows "P (unat x) = 
haftmann@37660
  2004
         (~(EX n. of_nat n = x & n < 2^len_of TYPE('a) & ~ P n))"
haftmann@37660
  2005
  by (auto simp: unat_of_nat)
haftmann@37660
  2006
haftmann@37660
  2007
lemmas of_nat_inverse = 
haftmann@37660
  2008
  word_unat.Abs_inverse' [rotated, unfolded unats_def, simplified]
haftmann@37660
  2009
haftmann@37660
  2010
lemmas unat_splits = unat_split unat_split_asm
haftmann@37660
  2011
haftmann@37660
  2012
lemmas unat_arith_simps =
haftmann@37660
  2013
  word_le_nat_alt word_less_nat_alt
haftmann@37660
  2014
  word_unat.Rep_inject [symmetric]
haftmann@37660
  2015
  unat_sub_if' unat_plus_if' unat_div unat_mod
haftmann@37660
  2016
haftmann@37660
  2017
(* unat_arith_tac: tactic to reduce word arithmetic to nat, 
haftmann@37660
  2018
   try to solve via arith *)
haftmann@37660
  2019
ML {*
wenzelm@51717
  2020
fun unat_arith_simpset ctxt = 
wenzelm@51717
  2021
  ctxt addsimps @{thms unat_arith_simps}
haftmann@37660
  2022
     delsimps @{thms word_unat.Rep_inject}
wenzelm@45620
  2023
     |> fold Splitter.add_split @{thms split_if_asm}
wenzelm@45620
  2024
     |> fold Simplifier.add_cong @{thms power_False_cong}
haftmann@37660
  2025
haftmann@37660
  2026
fun unat_arith_tacs ctxt =   
haftmann@37660
  2027
  let
haftmann@37660
  2028
    fun arith_tac' n t =
haftmann@37660
  2029
      Arith_Data.verbose_arith_tac ctxt n t
haftmann@37660
  2030
        handle Cooper.COOPER _ => Seq.empty;
haftmann@37660
  2031
  in 
wenzelm@42793
  2032
    [ clarify_tac ctxt 1,
wenzelm@51717
  2033
      full_simp_tac (unat_arith_simpset ctxt) 1,
wenzelm@51717
  2034
      ALLGOALS (full_simp_tac
wenzelm@51717
  2035
        (put_simpset HOL_ss ctxt
wenzelm@51717
  2036
          |> fold Splitter.add_split @{thms unat_splits}
wenzelm@51717
  2037
          |> fold Simplifier.add_cong @{thms power_False_cong})),
wenzelm@54742
  2038
      rewrite_goals_tac ctxt @{thms word_size}, 
haftmann@37660
  2039
      ALLGOALS  (fn n => REPEAT (resolve_tac [allI, impI] n) THEN      
haftmann@37660
  2040
                         REPEAT (etac conjE n) THEN
haftmann@37660
  2041
                         REPEAT (dtac @{thm of_nat_inverse} n THEN atac n)),
haftmann@37660
  2042
      TRYALL arith_tac' ] 
haftmann@37660
  2043
  end
haftmann@37660
  2044
haftmann@37660
  2045
fun unat_arith_tac ctxt = SELECT_GOAL (EVERY (unat_arith_tacs ctxt))
haftmann@37660
  2046
*}
haftmann@37660
  2047
haftmann@37660
  2048
method_setup unat_arith = 
haftmann@37660
  2049
  {* Scan.succeed (SIMPLE_METHOD' o unat_arith_tac) *}
haftmann@37660
  2050
  "solving word arithmetic via natural numbers and arith"
haftmann@37660
  2051
haftmann@37660
  2052
lemma no_plus_overflow_unat_size: 
haftmann@37660
  2053
  "((x :: 'a :: len word) <= x + y) = (unat x + unat y < 2 ^ size x)" 
haftmann@37660
  2054
  unfolding word_size by unat_arith
haftmann@37660
  2055
haftmann@37660
  2056
lemmas no_olen_add_nat = no_plus_overflow_unat_size [unfolded word_size]
haftmann@37660
  2057
wenzelm@45604
  2058
lemmas unat_plus_simple = trans [OF no_olen_add_nat unat_add_lem]
haftmann@37660
  2059
haftmann@37660
  2060
lemma word_div_mult: 
haftmann@40827
  2061
  "(0 :: 'a :: len word) < y \<Longrightarrow> unat x * unat y < 2 ^ len_of TYPE('a) \<Longrightarrow> 
haftmann@37660
  2062
    x * y div y = x"
haftmann@37660
  2063
  apply unat_arith
haftmann@37660
  2064
  apply clarsimp
haftmann@37660
  2065
  apply (subst unat_mult_lem [THEN iffD1])
haftmann@37660
  2066
  apply auto
haftmann@37660
  2067
  done
haftmann@37660
  2068
haftmann@40827
  2069
lemma div_lt': "(i :: 'a :: len word) <= k div x \<Longrightarrow> 
haftmann@37660
  2070
    unat i * unat x < 2 ^ len_of TYPE('a)"
haftmann@37660
  2071
  apply unat_arith
haftmann@37660
  2072
  apply clarsimp
haftmann@37660
  2073
  apply (drule mult_le_mono1)
haftmann@37660
  2074
  apply (erule order_le_less_trans)
haftmann@37660
  2075
  apply (rule xtr7 [OF unat_lt2p div_mult_le])
haftmann@37660
  2076
  done
haftmann@37660
  2077
haftmann@37660
  2078
lemmas div_lt'' = order_less_imp_le [THEN div_lt']
haftmann@37660
  2079
haftmann@40827
  2080
lemma div_lt_mult: "(i :: 'a :: len word) < k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x < k"
haftmann@37660
  2081
  apply (frule div_lt'' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  2082
  apply (simp add: unat_arith_simps)
haftmann@37660
  2083
  apply (drule (1) mult_less_mono1)
haftmann@37660
  2084
  apply (erule order_less_le_trans)
haftmann@37660
  2085
  apply (rule div_mult_le)
haftmann@37660
  2086
  done
haftmann@37660
  2087
haftmann@37660
  2088
lemma div_le_mult: 
haftmann@40827
  2089
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> 0 < x \<Longrightarrow> i * x <= k"
haftmann@37660
  2090
  apply (frule div_lt' [THEN unat_mult_lem [THEN iffD1]])
haftmann@37660
  2091
  apply (simp add: unat_arith_simps)
haftmann@37660
  2092
  apply (drule mult_le_mono1)
haftmann@37660
  2093
  apply (erule order_trans)
haftmann@37660
  2094
  apply (rule div_mult_le)
haftmann@37660
  2095
  done
haftmann@37660
  2096
haftmann@37660
  2097
lemma div_lt_uint': 
haftmann@40827
  2098
  "(i :: 'a :: len word) <= k div x \<Longrightarrow> uint i * uint x < 2 ^ len_of TYPE('a)"
haftmann@37660
  2099
  apply (unfold uint_nat)
haftmann@37660
  2100
  apply (drule div_lt')
haftmann@37660
  2101
  apply (simp add: zmult_int zless_nat_eq_int_zless [symmetric] 
haftmann@37660
  2102
                   nat_power_eq)
haftmann@37660
  2103
  done
haftmann@37660
  2104
haftmann@37660
  2105
lemmas div_lt_uint'' = order_less_imp_le [THEN div_lt_uint']
haftmann@37660
  2106
haftmann@37660
  2107
lemma word_le_exists': 
haftmann@40827
  2108
  "(x :: 'a :: len0 word) <= y \<Longrightarrow> 
haftmann@37660
  2109
    (EX z. y = x + z & uint x + uint z < 2 ^ len_of TYPE('a))"
haftmann@37660
  2110
  apply (rule exI)
haftmann@37660
  2111
  apply (rule conjI)
haftmann@37660
  2112
  apply (rule zadd_diff_inverse)
haftmann@37660
  2113
  apply uint_arith
haftmann@37660
  2114
  done
haftmann@37660
  2115
haftmann@37660
  2116
lemmas plus_minus_not_NULL = order_less_imp_le [THEN plus_minus_not_NULL_ab]
haftmann@37660
  2117
haftmann@37660
  2118
lemmas plus_minus_no_overflow =
haftmann@37660
  2119
  order_less_imp_le [THEN plus_minus_no_overflow_ab]
haftmann@37660
  2120
  
haftmann@37660
  2121
lemmas mcs = word_less_minus_cancel word_less_minus_mono_left
haftmann@37660
  2122
  word_le_minus_cancel word_le_minus_mono_left
haftmann@37660
  2123
wenzelm@45604
  2124
lemmas word_l_diffs = mcs [where y = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2125
lemmas word_diff_ls = mcs [where z = "w + x", unfolded add_diff_cancel] for w x
wenzelm@45604
  2126
lemmas word_plus_mcs = word_diff_ls [where y = "v + x", unfolded add_diff_cancel] for v x
haftmann@37660
  2127
haftmann@37660
  2128
lemmas le_unat_uoi = unat_le [THEN word_unat.Abs_inverse]
haftmann@37660
  2129
haftmann@37660
  2130
lemmas thd = refl [THEN [2] split_div_lemma [THEN iffD2], THEN conjunct1]
haftmann@37660
  2131
haftmann@37660
  2132
lemma thd1:
haftmann@37660
  2133
  "a div b * b \<le> (a::nat)"
haftmann@37660
  2134
  using gt_or_eq_0 [of b]
haftmann@37660
  2135
  apply (rule disjE)
haftmann@37660
  2136
   apply (erule xtr4 [OF thd mult_commute])
haftmann@37660
  2137
  apply clarsimp
haftmann@37660
  2138
  done
haftmann@37660
  2139
wenzelm@45604
  2140
lemmas uno_simps [THEN le_unat_uoi] = mod_le_divisor div_le_dividend thd1 
haftmann@37660
  2141
haftmann@37660
  2142
lemma word_mod_div_equality:
haftmann@37660
  2143
  "(n div b) * b + (n mod b) = (n :: 'a :: len word)"
haftmann@37660
  2144
  apply (unfold word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2145
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2146
  apply (erule disjE)
haftmann@37660
  2147
   apply (simp add: mod_div_equality uno_simps)
haftmann@37660
  2148
  apply simp
haftmann@37660
  2149
  done
haftmann@37660
  2150
haftmann@37660
  2151
lemma word_div_mult_le: "a div b * b <= (a::'a::len word)"
haftmann@37660
  2152
  apply (unfold word_le_nat_alt word_arith_nat_defs)
haftmann@37660
  2153
  apply (cut_tac y="unat b" in gt_or_eq_0)
haftmann@37660
  2154
  apply (erule disjE)
haftmann@37660
  2155
   apply (simp add: div_mult_le uno_simps)
haftmann@37660
  2156
  apply simp
haftmann@37660
  2157
  done
haftmann@37660
  2158
haftmann@40827
  2159
lemma word_mod_less_divisor: "0 < n \<Longrightarrow> m mod n < (n :: 'a :: len word)"
haftmann@37660
  2160
  apply (simp only: word_less_nat_alt word_arith_nat_defs)
haftmann@37660
  2161
  apply (clarsimp simp add : uno_simps)
haftmann@37660
  2162
  done
haftmann@37660
  2163
haftmann@37660
  2164
lemma word_of_int_power_hom: 
haftmann@37660
  2165
  "word_of_int a ^ n = (word_of_int (a ^ n) :: 'a :: len word)"
huffman@45995
  2166
  by (induct n) (simp_all add: wi_hom_mult [symmetric])
haftmann@37660
  2167
haftmann@37660
  2168
lemma word_arith_power_alt: 
haftmann@37660
  2169
  "a ^ n = (word_of_int (uint a ^ n) :: 'a :: len word)"
haftmann@37660
  2170
  by (simp add : word_of_int_power_hom [symmetric])
haftmann@37660
  2171
haftmann@37660
  2172
lemma of_bl_length_less: 
haftmann@40827
  2173
  "length x = k \<Longrightarrow> k < len_of TYPE('a) \<Longrightarrow> (of_bl x :: 'a :: len word) < 2 ^ k"
huffman@47108
  2174
  apply (unfold of_bl_def word_less_alt word_numeral_alt)
haftmann@37660
  2175
  apply safe
haftmann@37660
  2176
  apply (simp (no_asm) add: word_of_int_power_hom word_uint.eq_norm 
huffman@47108
  2177
                       del: word_of_int_numeral)
haftmann@37660
  2178
  apply (simp add: mod_pos_pos_trivial)
haftmann@37660
  2179
  apply (subst mod_pos_pos_trivial)
haftmann@37660
  2180
    apply (rule bl_to_bin_ge0)
haftmann@37660
  2181
   apply (rule order_less_trans)
haftmann@37660
  2182
    apply (rule bl_to_bin_lt2p)
haftmann@37660
  2183
   apply simp
huffman@46646
  2184
  apply (rule bl_to_bin_lt2p)
haftmann@37660
  2185
  done
haftmann@37660
  2186
haftmann@37660
  2187
haftmann@55816
  2188
subsection {* Cardinality, finiteness of set of words *}
haftmann@37660
  2189
huffman@45809
  2190
instance word :: (len0) finite
huffman@45809
  2191
  by (default, simp add: type_definition.univ [OF type_definition_word])
huffman@45809
  2192
huffman@45809
  2193
lemma card_word: "CARD('a::len0 word) = 2 ^ len_of TYPE('a)"
huffman@45809
  2194
  by (simp add: type_definition.card [OF type_definition_word] nat_power_eq)
haftmann@37660
  2195
haftmann@37660
  2196
lemma card_word_size: 
huffman@45809
  2197
  "card (UNIV :: 'a :: len0 word set) = (2 ^ size (x :: 'a word))"
haftmann@37660
  2198
unfolding word_size by (rule card_word)
haftmann@37660
  2199
haftmann@37660
  2200
haftmann@37660
  2201
subsection {* Bitwise Operations on Words *}
haftmann@37660
  2202
haftmann@37660
  2203
lemmas bin_log_bintrs = bin_trunc_not bin_trunc_xor bin_trunc_and bin_trunc_or
haftmann@37660
  2204
  
haftmann@37660
  2205
(* following definitions require both arithmetic and bit-wise word operations *)
haftmann@37660
  2206
haftmann@37660
  2207
(* to get word_no_log_defs from word_log_defs, using bin_log_bintrs *)
haftmann@37660
  2208
lemmas wils1 = bin_log_bintrs [THEN word_ubin.norm_eq_iff [THEN iffD1],
wenzelm@45604
  2209
  folded word_ubin.eq_norm, THEN eq_reflection]
haftmann@37660
  2210
haftmann@37660
  2211
(* the binary operations only *)
huffman@46013
  2212
(* BH: why is this needed? *)
haftmann@37660
  2213
lemmas word_log_binary_defs = 
haftmann@37660
  2214
  word_and_def word_or_def word_xor_def
haftmann@37660
  2215
huffman@46011
  2216
lemma word_wi_log_defs:
huffman@46011
  2217
  "NOT word_of_int a = word_of_int (NOT a)"
huffman@46011
  2218
  "word_of_int a AND word_of_int b = word_of_int (a AND b)"
huffman@46011
  2219
  "word_of_int a OR word_of_int b = word_of_int (a OR b)"
huffman@46011
  2220
  "word_of_int a XOR word_of_int b = word_of_int (a XOR b)"
huffman@47374
  2221
  by (transfer, rule refl)+
huffman@47372
  2222
huffman@46011
  2223
lemma word_no_log_defs [simp]:
huffman@47108
  2224
  "NOT (numeral a) = word_of_int (NOT (numeral a))"
haftmann@54489
  2225
  "NOT (- numeral a) = word_of_int (NOT (- numeral a))"
huffman@47108
  2226
  "numeral a AND numeral b = word_of_int (numeral a AND numeral b)"
haftmann@54489
  2227
  "numeral a AND - numeral b = word_of_int (numeral a AND - numeral b)"
haftmann@54489
  2228
  "- numeral a AND numeral b = word_of_int (- numeral a AND numeral b)"
haftmann@54489
  2229
  "- numeral a AND - numeral b = word_of_int (- numeral a AND - numeral b)"
huffman@47108
  2230
  "numeral a OR numeral b = word_of_int (numeral a OR numeral b)"
haftmann@54489
  2231
  "numeral a OR - numeral b = word_of_int (numeral a OR - numeral b)"
haftmann@54489
  2232
  "- numeral a OR numeral b = word_of_int (- numeral a OR numeral b)"
haftmann@54489
  2233
  "- numeral a OR - numeral b = word_of_int (- numeral a OR - numeral b)"
huffman@47108
  2234
  "numeral a XOR numeral b = word_of_int (numeral a XOR numeral b)"
haftmann@54489
  2235
  "numeral a XOR - numeral b = word_of_int (numeral a XOR - numeral b)"
haftmann@54489
  2236
  "- numeral a XOR numeral b = word_of_int (- numeral a XOR numeral b)"
haftmann@54489
  2237
  "- numeral a XOR - numeral b = word_of_int (- numeral a XOR - numeral b)"
huffman@47372
  2238
  by (transfer, rule refl)+
haftmann@37660
  2239
huffman@46064
  2240
text {* Special cases for when one of the arguments equals 1. *}
huffman@46064
  2241
huffman@46064
  2242
lemma word_bitwise_1_simps [simp]:
huffman@46064
  2243
  "NOT (1::'a::len0 word) = -2"
huffman@47108
  2244
  "1 AND numeral b = word_of_int (1 AND numeral b)"
haftmann@54489
  2245
  "1 AND - numeral b = word_of_int (1 AND - numeral b)"
huffman@47108
  2246
  "numeral a AND 1 = word_of_int (numeral a AND 1)"
haftmann@54489
  2247
  "- numeral a AND 1 = word_of_int (- numeral a AND 1)"
huffman@47108
  2248
  "1 OR numeral b = word_of_int (1 OR numeral b)"
haftmann@54489
  2249
  "1 OR - numeral b = word_of_int (1 OR - numeral b)"
huffman@47108
  2250
  "numeral a OR 1 = word_of_int (numeral a OR 1)"
haftmann@54489
  2251
  "- numeral a OR 1 = word_of_int (- numeral a OR 1)"
huffman@47108
  2252
  "1 XOR numeral b = word_of_int (1 XOR numeral b)"
haftmann@54489
  2253
  "1 XOR - numeral b = word_of_int (1 XOR - numeral b)"
huffman@47108
  2254
  "numeral a XOR 1 = word_of_int (numeral a XOR 1)"
haftmann@54489
  2255
  "- numeral a XOR 1 = word_of_int (- numeral a XOR 1)"
huffman@47372
  2256
  by (transfer, simp)+
huffman@46064
  2257
haftmann@37660
  2258
lemma uint_or: "uint (x OR y) = (uint x) OR (uint y)"
huffman@47372
  2259
  by (transfer, simp add: bin_trunc_ao)
haftmann@37660
  2260
haftmann@37660
  2261
lemma uint_and: "uint (x AND y) = (uint x) AND (uint y)"
huffman@47372
  2262
  by (transfer, simp add: bin_trunc_ao)
huffman@47372
  2263
huffman@47372
  2264
lemma test_bit_wi [simp]:
huffman@47372
  2265
  "(word_of_int x::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a) \<and> bin_nth x n"
huffman@47372
  2266
  unfolding word_test_bit_def
huffman@47372
  2267
  by (simp add: word_ubin.eq_norm nth_bintr)
huffman@47372
  2268
huffman@47372
  2269
lemma word_test_bit_transfer [transfer_rule]:
kuncar@51375
  2270
  "(fun_rel pcr_word (fun_rel op = op =))
huffman@47372
  2271
    (\<lambda>x n. n < len_of TYPE('a) \<and> bin_nth x n) (test_bit :: 'a::len0 word \<Rightarrow> _)"
kuncar@51375
  2272
  unfolding fun_rel_def word.pcr_cr_eq cr_word_def by simp
haftmann@37660
  2273
haftmann@37660
  2274
lemma word_ops_nth_size:
haftmann@40827
  2275
  "n < size (x::'a::len0 word) \<Longrightarrow> 
haftmann@37660
  2276
    (x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2277
    (x AND y) !! n = (x !! n & y !! n) & 
haftmann@37660
  2278
    (x XOR y) !! n = (x !! n ~= y !! n) & 
haftmann@37660
  2279
    (NOT x) !! n = (~ x !! n)"
huffman@47372
  2280
  unfolding word_size by transfer (simp add: bin_nth_ops)
haftmann@37660
  2281
haftmann@37660
  2282
lemma word_ao_nth:
haftmann@37660
  2283
  fixes x :: "'a::len0 word"
haftmann@37660
  2284
  shows "(x OR y) !! n = (x !! n | y !! n) & 
haftmann@37660
  2285
         (x AND y) !! n = (x !! n & y !! n)"
huffman@47372
  2286
  by transfer (auto simp add: bin_nth_ops)
huffman@46023
  2287
huffman@47108
  2288
lemma test_bit_numeral [simp]:
huffman@47108
  2289
  "(numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
huffman@47108
  2290
    n < len_of TYPE('a) \<and> bin_nth (numeral w) n"
huffman@47372
  2291
  by transfer (rule refl)
huffman@47108
  2292
huffman@47108
  2293
lemma test_bit_neg_numeral [simp]:
haftmann@54489
  2294
  "(- numeral w :: 'a::len0 word) !! n \<longleftrightarrow>
haftmann@54489
  2295
    n < len_of TYPE('a) \<and> bin_nth (- numeral w) n"
huffman@47372
  2296
  by transfer (rule refl)
huffman@46023
  2297
huffman@46172
  2298
lemma test_bit_1 [simp]: "(1::'a::len word) !! n \<longleftrightarrow> n = 0"
huffman@47372
  2299
  by transfer auto
huffman@46172
  2300
  
huffman@46023
  2301
lemma nth_0 [simp]: "~ (0::'a::len0 word) !! n"
huffman@47372
  2302
  by transfer simp
huffman@46023
  2303
huffman@47108
  2304
lemma nth_minus1 [simp]: "(-1::'a::len0 word) !! n \<longleftrightarrow> n < len_of TYPE('a)"
huffman@47372
  2305
  by transfer simp
huffman@47108
  2306
haftmann@37660
  2307
(* get from commutativity, associativity etc of int_and etc
haftmann@37660
  2308
  to same for word_and etc *)
haftmann@37660
  2309
haftmann@37660
  2310
lemmas bwsimps = 
huffman@46013
  2311
  wi_hom_add
haftmann@37660
  2312
  word_wi_log_defs
haftmann@37660
  2313
haftmann@37660
  2314
lemma word_bw_assocs:
haftmann@37660
  2315
  fixes x :: "'a::len0 word"
haftmann@37660
  2316
  shows
haftmann@37660
  2317
  "(x AND y) AND z = x AND y AND z"
haftmann@37660
  2318
  "(x OR y) OR z = x OR y OR z"
haftmann@37660
  2319
  "(x XOR y) XOR z = x XOR y XOR z"
huffman@46022
  2320
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2321
  
haftmann@37660
  2322
lemma word_bw_comms:
haftmann@37660
  2323
  fixes x :: "'a::len0 word"
haftmann@37660
  2324
  shows
haftmann@37660
  2325
  "x AND y = y AND x"
haftmann@37660
  2326
  "x OR y = y OR x"
haftmann@37660
  2327
  "x XOR y = y XOR x"
huffman@46022
  2328
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2329
  
haftmann@37660
  2330
lemma word_bw_lcs:
haftmann@37660
  2331
  fixes x :: "'a::len0 word"
haftmann@37660
  2332
  shows
haftmann@37660
  2333
  "y AND x AND z = x AND y AND z"
haftmann@37660
  2334
  "y OR x OR z = x OR y OR z"
haftmann@37660
  2335
  "y XOR x XOR z = x XOR y XOR z"
huffman@46022
  2336
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2337
haftmann@37660
  2338
lemma word_log_esimps [simp]:
haftmann@37660
  2339
  fixes x :: "'a::len0 word"
haftmann@37660
  2340
  shows
haftmann@37660
  2341
  "x AND 0 = 0"
haftmann@37660
  2342
  "x AND -1 = x"
haftmann@37660
  2343
  "x OR 0 = x"
haftmann@37660
  2344
  "x OR -1 = -1"
haftmann@37660
  2345
  "x XOR 0 = x"
haftmann@37660
  2346
  "x XOR -1 = NOT x"
haftmann@37660
  2347
  "0 AND x = 0"
haftmann@37660
  2348
  "-1 AND x = x"
haftmann@37660
  2349
  "0 OR x = x"
haftmann@37660
  2350
  "-1 OR x = -1"
haftmann@37660
  2351
  "0 XOR x = x"
haftmann@37660
  2352
  "-1 XOR x = NOT x"
huffman@46023
  2353
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2354
haftmann@37660
  2355
lemma word_not_dist:
haftmann@37660
  2356
  fixes x :: "'a::len0 word"
haftmann@37660
  2357
  shows
haftmann@37660
  2358
  "NOT (x OR y) = NOT x AND NOT y"
haftmann@37660
  2359
  "NOT (x AND y) = NOT x OR NOT y"
huffman@46022
  2360
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2361
haftmann@37660
  2362
lemma word_bw_same:
haftmann@37660
  2363
  fixes x :: "'a::len0 word"
haftmann@37660
  2364
  shows
haftmann@37660
  2365
  "x AND x = x"
haftmann@37660
  2366
  "x OR x = x"
haftmann@37660
  2367
  "x XOR x = 0"
huffman@46023
  2368
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2369
haftmann@37660
  2370
lemma word_ao_absorbs [simp]:
haftmann@37660
  2371
  fixes x :: "'a::len0 word"
haftmann@37660
  2372
  shows
haftmann@37660
  2373
  "x AND (y OR x) = x"
haftmann@37660
  2374
  "x OR y AND x = x"
haftmann@37660
  2375
  "x AND (x OR y) = x"
haftmann@37660
  2376
  "y AND x OR x = x"
haftmann@37660
  2377
  "(y OR x) AND x = x"
haftmann@37660
  2378
  "x OR x AND y = x"
haftmann@37660
  2379
  "(x OR y) AND x = x"
haftmann@37660
  2380
  "x AND y OR x = x"
huffman@46022
  2381
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2382
haftmann@37660
  2383
lemma word_not_not [simp]:
haftmann@37660
  2384
  "NOT NOT (x::'a::len0 word) = x"
huffman@46022
  2385
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2386
haftmann@37660
  2387
lemma word_ao_dist:
haftmann@37660
  2388
  fixes x :: "'a::len0 word"
haftmann@37660
  2389
  shows "(x OR y) AND z = x AND z OR y AND z"
huffman@46022
  2390
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2391
haftmann@37660
  2392
lemma word_oa_dist:
haftmann@37660
  2393
  fixes x :: "'a::len0 word"
haftmann@37660
  2394
  shows "x AND y OR z = (x OR z) AND (y OR z)"
huffman@46022
  2395
  by (auto simp: word_eq_iff word_ops_nth_size [unfolded word_size])
haftmann@37660
  2396
haftmann@37660
  2397
lemma word_add_not [simp]: 
haftmann@37660
  2398
  fixes x :: "'a::len0 word"
haftmann@37660
  2399
  shows "x + NOT x = -1"
huffman@47372
  2400
  by transfer (simp add: bin_add_not)
haftmann@37660
  2401
haftmann@37660
  2402
lemma word_plus_and_or [simp]:
haftmann@37660
  2403
  fixes x :: "'a::len0 word"
haftmann@37660
  2404
  shows "(x AND y) + (x OR y) = x + y"
huffman@47372
  2405
  by transfer (simp add: plus_and_or)
haftmann@37660
  2406
haftmann@37660
  2407
lemma leoa:   
haftmann@37660
  2408
  fixes x :: "'a::len0 word"
haftmann@40827
  2409
  shows "(w = (x OR y)) \<Longrightarrow> (y = (w AND y))" by auto
haftmann@37660
  2410
lemma leao: 
haftmann@37660
  2411
  fixes x' :: "'a::len0 word"
haftmann@40827
  2412
  shows "(w' = (x' AND y')) \<Longrightarrow> (x' = (x' OR w'))" by auto 
haftmann@37660
  2413
wenzelm@48196
  2414
lemma word_ao_equiv:
wenzelm@48196
  2415
  fixes w w' :: "'a::len0 word"
wenzelm@48196
  2416
  shows "(w = w OR w') = (w' = w AND w')"
wenzelm@48196
  2417
  by (auto intro: leoa leao)
haftmann@37660
  2418
haftmann@37660
  2419
lemma le_word_or2: "x <= x OR (y::'a::len0 word)"
haftmann@37660
  2420
  unfolding word_le_def uint_or
haftmann@37660
  2421
  by (auto intro: le_int_or) 
haftmann@37660
  2422
wenzelm@45604
  2423
lemmas le_word_or1 = xtr3 [OF word_bw_comms (2) le_word_or2]
wenzelm@45604
  2424
lemmas word_and_le1 = xtr3 [OF word_ao_absorbs (4) [symmetric] le_word_or2]
wenzelm@45604
  2425
lemmas word_and_le2 = xtr3 [OF word_ao_absorbs (8) [symmetric] le_word_or2]
haftmann@37660
  2426
haftmann@37660
  2427
lemma bl_word_not: "to_bl (NOT w) = map Not (to_bl w)" 
huffman@45550
  2428
  unfolding to_bl_def word_log_defs bl_not_bin
huffman@45550
  2429
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2430
haftmann@37660
  2431
lemma bl_word_xor: "to_bl (v XOR w) = map2 op ~= (to_bl v) (to_bl w)" 
haftmann@37660
  2432
  unfolding to_bl_def word_log_defs bl_xor_bin
huffman@45550
  2433
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2434
haftmann@37660
  2435
lemma bl_word_or: "to_bl (v OR w) = map2 op | (to_bl v) (to_bl w)" 
huffman@45550
  2436
  unfolding to_bl_def word_log_defs bl_or_bin
huffman@45550
  2437
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2438
haftmann@37660
  2439
lemma bl_word_and: "to_bl (v AND w) = map2 op & (to_bl v) (to_bl w)" 
huffman@45550
  2440
  unfolding to_bl_def word_log_defs bl_and_bin
huffman@45550
  2441
  by (simp add: word_ubin.eq_norm)
haftmann@37660
  2442
haftmann@37660
  2443
lemma word_lsb_alt: "lsb (w::'a::len0 word) = test_bit w 0"
haftmann@37660
  2444
  by (auto simp: word_test_bit_def word_lsb_def)
haftmann@37660
  2445
huffman@45805
  2446
lemma word_lsb_1_0 [simp]: "lsb (1::'a::len word) & ~ lsb (0::'b::len0 word)"
huffman@45550
  2447
  unfolding word_lsb_def uint_eq_0 uint_1 by simp
haftmann@37660
  2448
haftmann@37660
  2449
lemma word_lsb_last: "lsb (w::'a::len word) = last (to_bl w)"
haftmann@37660
  2450
  apply (unfold word_lsb_def uint_bl bin_to_bl_def) 
haftmann@37660
  2451
  apply (rule_tac bin="uint w" in bin_exhaust)
haftmann@37660
  2452
  apply (cases "size w")
haftmann@37660
  2453
   apply auto
haftmann@37660
  2454
   apply (auto simp add: bin_to_bl_aux_alt)
haftmann@37660
  2455
  done
haftmann@37660
  2456
haftmann@37660
  2457
lemma word_lsb_int: "lsb w = (uint w mod 2 = 1)"
huffman@45529
  2458
  unfolding word_lsb_def bin_last_def by auto
haftmann@37660
  2459
haftmann@37660
  2460
lemma word_msb_sint: "msb w = (sint w < 0)" 
huffman@46604
  2461
  unfolding word_msb_def sign_Min_lt_0 ..
haftmann@37660
  2462
huffman@46173
  2463
lemma msb_word_of_int:
huffman@46173
  2464
  "msb (word_of_int x::'a::len word) = bin_nth x (len_of TYPE('a) - 1)"
huffman@46173
  2465
  unfolding word_msb_def by (simp add: word_sbin.eq_norm bin_sign_lem)
huffman@46173
  2466
huffman@47108
  2467
lemma word_msb_numeral [simp]:
huffman@47108
  2468
  "msb (numeral w::'a::len word) = bin_nth (numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2469
  unfolding word_numeral_alt by (rule msb_word_of_int)
huffman@47108
  2470
huffman@47108
  2471
lemma word_msb_neg_numeral [simp]:
haftmann@54489
  2472
  "msb (- numeral w::'a::len word) = bin_nth (- numeral w) (len_of TYPE('a) - 1)"
huffman@47108
  2473
  unfolding word_neg_numeral_alt by (rule msb_word_of_int)
huffman@46173
  2474
huffman@46173
  2475
lemma word_msb_0 [simp]: "\<not> msb (0::'a::len word)"
huffman@46173
  2476
  unfolding word_msb_def by simp
huffman@46173
  2477
huffman@46173
  2478
lemma word_msb_1 [simp]: "msb (1::'a::len word) \<longleftrightarrow> len_of TYPE('a) = 1"
huffman@46173
  2479
  unfolding word_1_wi msb_word_of_int eq_iff [where 'a=nat]
huffman@46173
  2480
  by (simp add: Suc_le_eq)
huffman@45811
  2481
huffman@45811
  2482
lemma word_msb_nth:
huffman@45811
  2483
  "msb (w::'a::len word) = bin_nth (uint w) (len_of TYPE('a) - 1)"
huffman@46023
  2484
  unfolding word_msb_def sint_uint by (simp add: bin_sign_lem)
haftmann@37660
  2485
haftmann@37660
  2486
lemma word_msb_alt: "msb (w::'a::len word) = hd (to_bl w)"
haftmann@37660
  2487
  apply (unfold word_msb_nth uint_bl)
haftmann@37660
  2488
  apply (subst hd_conv_nth)
haftmann@37660
  2489
  apply (rule length_greater_0_conv [THEN iffD1])
haftmann@37660
  2490
   apply simp
haftmann@37660
  2491
  apply (simp add : nth_bin_to_bl word_size)
haftmann@37660
  2492
  done
haftmann@37660
  2493
huffman@45805
  2494
lemma word_set_nth [simp]:
haftmann@37660
  2495
  "set_bit w n (test_bit w n) = (w::'a::len0 word)"
haftmann@37660
  2496
  unfolding word_test_bit_def word_set_bit_def by auto
haftmann@37660
  2497
haftmann@37660
  2498
lemma bin_nth_uint':
haftmann@37660
  2499
  "bin_nth (uint w) n = (rev (bin_to_bl (size w) (uint w)) ! n & n < size w)"
haftmann@37660
  2500
  apply (unfold word_size)
haftmann@37660
  2501
  apply (safe elim!: bin_nth_uint_imp)
haftmann@37660
  2502
   apply (frule bin_nth_uint_imp)
haftmann@37660
  2503
   apply (fast dest!: bin_nth_bl)+
haftmann@37660
  2504
  done
haftmann@37660
  2505
haftmann@37660
  2506
lemmas bin_nth_uint = bin_nth_uint' [unfolded word_size]
haftmann@37660
  2507
haftmann@37660
  2508
lemma test_bit_bl: "w !! n = (rev (to_bl w) ! n & n < size w)"
haftmann@37660
  2509
  unfolding to_bl_def word_test_bit_def word_size
haftmann@37660
  2510
  by (rule bin_nth_uint)
haftmann@37660
  2511
haftmann@40827
  2512
lemma to_bl_nth: "n < size w \<Longrightarrow> to_bl w ! n = w !! (size w - Suc n)"
haftmann@37660
  2513
  apply (unfold test_bit_bl)
haftmann@37660
  2514
  apply clarsimp
haftmann@37660
  2515
  apply (rule trans)
haftmann@37660
  2516
   apply (rule nth_rev_alt)
haftmann@37660
  2517
   apply (auto simp add: word_size)
haftmann@37660
  2518
  done
haftmann@37660
  2519
haftmann@37660
  2520
lemma test_bit_set: 
haftmann@37660
  2521
  fixes w :: "'a::len0 word"
haftmann@37660
  2522
  shows "(set_bit w n x) !! n = (n < size w & x)"
haftmann@37660
  2523
  unfolding word_size word_test_bit_def word_set_bit_def
haftmann@37660
  2524
  by (clarsimp simp add : word_ubin.eq_norm nth_bintr)
haftmann@37660
  2525
haftmann@37660
  2526
lemma test_bit_set_gen: 
haftmann@37660
  2527
  fixes w :: "'a::len0 word"
haftmann@37660
  2528
  shows "test_bit (set_bit w n x) m = 
haftmann@37660
  2529
         (if m = n then n < size w & x else test_bit w m)"
haftmann@37660
  2530
  apply (unfold word_size word_test_bit_def word_set_bit_def)
haftmann@37660
  2531
  apply (clarsimp simp add: word_ubin.eq_norm nth_bintr bin_nth_sc_gen)
haftmann@37660
  2532
  apply (auto elim!: test_bit_size [unfolded word_size]
haftmann@37660
  2533
              simp add: word_test_bit_def [symmetric])
haftmann@37660
  2534
  done
haftmann@37660
  2535
haftmann@37660
  2536
lemma of_bl_rep_False: "of_bl (replicate n False @ bs) = of_bl bs"
haftmann@37660
  2537
  unfolding of_bl_def bl_to_bin_rep_F by auto
haftmann@37660
  2538
  
huffman@45811
  2539
lemma msb_nth:
haftmann@37660
  2540
  fixes w :: "'a::len word"
huffman@45811
  2541
  shows "msb w = w !! (len_of TYPE('a) - 1)"
huffman@45811
  2542
  unfolding word_msb_nth word_test_bit_def by simp
haftmann@37660
  2543
wenzelm@45604
  2544
lemmas msb0 = len_gt_0 [THEN diff_Suc_less, THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2545
lemmas msb1 = msb0 [where i = 0]
haftmann@37660
  2546
lemmas word_ops_msb = msb1 [unfolded msb_nth [symmetric, unfolded One_nat_def]]
haftmann@37660
  2547
wenzelm@45604
  2548
lemmas lsb0 = len_gt_0 [THEN word_ops_nth_size [unfolded word_size]]
haftmann@37660
  2549
lemmas word_ops_lsb = lsb0 [unfolded word_lsb_alt]
haftmann@37660
  2550
huffman@45811
  2551
lemma td_ext_nth [OF refl refl refl, unfolded word_size]:
haftmann@40827
  2552
  "n = size (w::'a::len0 word) \<Longrightarrow> ofn = set_bits \<Longrightarrow> [w, ofn g] = l \<Longrightarrow> 
haftmann@37660
  2553
    td_ext test_bit ofn {f. ALL i. f i --> i < n} (%h i. h i & i < n)"
haftmann@37660
  2554
  apply (unfold word_size td_ext_def')
wenzelm@46008
  2555
  apply safe
haftmann@37660
  2556
     apply (rule_tac [3] ext)
haftmann@37660
  2557
     apply (rule_tac [4] ext)
haftmann@37660
  2558
     apply (unfold word_size of_nth_def test_bit_bl)
haftmann@37660
  2559
     apply safe
haftmann@37660
  2560
       defer
haftmann@37660
  2561
       apply (clarsimp simp: word_bl.Abs_inverse)+
haftmann@37660
  2562
  apply (rule word_bl.Rep_inverse')
haftmann@37660
  2563
  apply (rule sym [THEN trans])
haftmann@37660
  2564
  apply (rule bl_of_nth_nth)
haftmann@37660
  2565
  apply simp
haftmann@37660
  2566
  apply (rule bl_of_nth_inj)
haftmann@37660
  2567
  apply (clarsimp simp add : test_bit_bl word_size)
haftmann@37660
  2568
  done
haftmann@37660
  2569
haftmann@37660
  2570
interpretation test_bit:
haftmann@37660
  2571
  td_ext "op !! :: 'a::len0 word => nat => bool"
haftmann@37660
  2572
         set_bits
haftmann@37660
  2573
         "{f. \<forall>i. f i \<longrightarrow> i < len_of TYPE('a::len0)}"
haftmann@37660
  2574
         "(\<lambda>h i. h i \<and> i < len_of TYPE('a::len0))"
haftmann@37660
  2575
  by (rule td_ext_nth)
haftmann@37660
  2576
haftmann@37660
  2577
lemmas td_nth = test_bit.td_thm
haftmann@37660
  2578
huffman@45805
  2579
lemma word_set_set_same [simp]:
haftmann@37660
  2580
  fixes w :: "'a::len0 word"
haftmann@37660
  2581
  shows "set_bit (set_bit w n x) n y = set_bit w n y" 
haftmann@37660
  2582
  by (rule word_eqI) (simp add : test_bit_set_gen word_size)
haftmann@37660
  2583
    
haftmann@37660
  2584
lemma word_set_set_diff: 
haftmann@37660
  2585
  fixes w :: "'a::len0 word"
haftmann@37660
  2586
  assumes "m ~= n"
haftmann@37660
  2587
  shows "set_bit (set_bit w m x) n y = set_bit (set_bit w n y) m x" 
wenzelm@41550
  2588
  by (rule word_eqI) (clarsimp simp add: test_bit_set_gen word_size assms)
huffman@46001
  2589
haftmann@37660
  2590
lemma nth_sint: 
haftmann@37660
  2591
  fixes w :: "'a::len word"
haftmann@37660
  2592
  defines "l \<equiv> len_of TYPE ('a)"
haftmann@37660
  2593
  shows "bin_nth (sint w) n = (if n < l - 1 then w !! n else w !! (l - 1))"
haftmann@37660
  2594
  unfolding sint_uint l_def
haftmann@37660
  2595
  by (clarsimp simp add: nth_sbintr word_test_bit_def [symmetric])
haftmann@37660
  2596
huffman@47108
  2597
lemma word_lsb_numeral [simp]:
haftmann@54847
  2598
  "lsb (numeral bin :: 'a :: len word) \<longleftrightarrow> bin_last (numeral bin)"
huffman@47108
  2599
  unfolding word_lsb_alt test_bit_numeral by simp
huffman@47108
  2600
huffman@47108
  2601
lemma word_lsb_neg_numeral [simp]:
haftmann@54847
  2602
  "lsb (- numeral bin :: 'a :: len word) \<longleftrightarrow> bin_last (- numeral bin)"
huffman@47108
  2603
  unfolding word_lsb_alt test_bit_neg_numeral by simp
haftmann@37660
  2604
huffman@46173
  2605
lemma set_bit_word_of_int:
haftmann@54847
  2606
  "set_bit (word_of_int x) n b = word_of_int (bin_sc n b x)"
huffman@46173
  2607
  unfolding word_set_bit_def
huffman@46173
  2608
  apply (rule word_eqI)
huffman@46173
  2609
  apply (simp add: word_size bin_nth_sc_gen word_ubin.eq_norm nth_bintr)
huffman@46173
  2610
  done
huffman@46173
  2611
huffman@47108
  2612
lemma word_set_numeral [simp]:
huffman@47108
  2613
  "set_bit (numeral bin::'a::len0 word) n b = 
haftmann@54847
  2614
    word_of_int (bin_sc n b (numeral bin))"
huffman@47108
  2615
  unfolding word_numeral_alt by (rule set_bit_word_of_int)
huffman@47108
  2616