src/HOL/Word/Word.thy

remove now-unnecessary type annotations from lift_definition commands

(*  Title:      HOL/Word/Word.thy
    Author:     Jeremy Dawson and Gerwin Klein, NICTA
*)

header {* A type of finite bit strings *}

theory Word
imports
  Type_Length
  Misc_Typedef
  "~~/src/HOL/Library/Boolean_Algebra"
  Bool_List_Representation
uses ("~~/src/HOL/Word/Tools/smt_word.ML")
begin

text {* see @{text "Examples/WordExamples.thy"} for examples *}

subsection {* Type definition *}

typedef (open) 'a word = "{(0::int) ..< 2^len_of TYPE('a::len0)}"
  morphisms uint Abs_word by auto

lemma uint_nonnegative:
  "0 \<le> uint w"
  using word.uint [of w] by simp

lemma uint_bounded:
  fixes w :: "'a::len0 word"
  shows "uint w < 2 ^ len_of TYPE('a)"
  using word.uint [of w] by simp

lemma uint_idem:
  fixes w :: "'a::len0 word"
  shows "uint w mod 2 ^ len_of TYPE('a) = uint w"
  using uint_nonnegative uint_bounded by (rule mod_pos_pos_trivial)

definition word_of_int :: "int \<Rightarrow> 'a\<Colon>len0 word" where
  -- {* representation of words using unsigned or signed bins, 
        only difference in these is the type class *}
  "word_of_int k = Abs_word (k mod 2 ^ len_of TYPE('a))" 

lemma uint_word_of_int:
  "uint (word_of_int k :: 'a::len0 word) = k mod 2 ^ len_of TYPE('a)"
  by (auto simp add: word_of_int_def intro: Abs_word_inverse)

lemma word_of_int_uint:
  "word_of_int (uint w) = w"
  by (simp add: word_of_int_def uint_idem uint_inverse)

lemma word_uint_eq_iff:
  "a = b \<longleftrightarrow> uint a = uint b"
  by (simp add: uint_inject)

lemma word_uint_eqI:
  "uint a = uint b \<Longrightarrow> a = b"
  by (simp add: word_uint_eq_iff)


subsection {* Basic code generation setup *}

definition Word :: "int \<Rightarrow> 'a::len0 word"
where
  [code_post]: "Word = word_of_int"

lemma [code abstype]:
  "Word (uint w) = w"
  by (simp add: Word_def word_of_int_uint)

declare uint_word_of_int [code abstract]

instantiation word :: (len0) equal
begin

definition equal_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where
  "equal_word k l \<longleftrightarrow> HOL.equal (uint k) (uint l)"

instance proof
qed (simp add: equal equal_word_def word_uint_eq_iff)

end

notation fcomp (infixl "\<circ>>" 60)
notation scomp (infixl "\<circ>\<rightarrow>" 60)

instantiation word :: ("{len0, typerep}") random
begin

definition
  "random_word i = Random.range i \<circ>\<rightarrow> (\<lambda>k. Pair (
     let j = word_of_int (Code_Numeral.int_of k) :: 'a word
     in (j, \<lambda>_::unit. Code_Evaluation.term_of j)))"

instance ..

end

no_notation fcomp (infixl "\<circ>>" 60)
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)


subsection {* Type conversions and casting *}

definition sint :: "'a :: len word => int" where
  -- {* treats the most-significant-bit as a sign bit *}
  sint_uint: "sint w = sbintrunc (len_of TYPE ('a) - 1) (uint w)"

definition unat :: "'a :: len0 word => nat" where
  "unat w = nat (uint w)"

definition uints :: "nat => int set" where
  -- "the sets of integers representing the words"
  "uints n = range (bintrunc n)"

definition sints :: "nat => int set" where
  "sints n = range (sbintrunc (n - 1))"

definition unats :: "nat => nat set" where
  "unats n = {i. i < 2 ^ n}"

definition norm_sint :: "nat => int => int" where
  "norm_sint n w = (w + 2 ^ (n - 1)) mod 2 ^ n - 2 ^ (n - 1)"

definition scast :: "'a :: len word => 'b :: len word" where
  -- "cast a word to a different length"
  "scast w = word_of_int (sint w)"

definition ucast :: "'a :: len0 word => 'b :: len0 word" where
  "ucast w = word_of_int (uint w)"

instantiation word :: (len0) size
begin

definition
  word_size: "size (w :: 'a word) = len_of TYPE('a)"

instance ..

end

definition source_size :: "('a :: len0 word => 'b) => nat" where
  -- "whether a cast (or other) function is to a longer or shorter length"
  "source_size c = (let arb = undefined ; x = c arb in size arb)"  

definition target_size :: "('a => 'b :: len0 word) => nat" where
  "target_size c = size (c undefined)"

definition is_up :: "('a :: len0 word => 'b :: len0 word) => bool" where
  "is_up c \<longleftrightarrow> source_size c <= target_size c"

definition is_down :: "('a :: len0 word => 'b :: len0 word) => bool" where
  "is_down c \<longleftrightarrow> target_size c <= source_size c"

definition of_bl :: "bool list => 'a :: len0 word" where
  "of_bl bl = word_of_int (bl_to_bin bl)"

definition to_bl :: "'a :: len0 word => bool list" where
  "to_bl w = bin_to_bl (len_of TYPE ('a)) (uint w)"

definition word_reverse :: "'a :: len0 word => 'a word" where
  "word_reverse w = of_bl (rev (to_bl w))"

definition word_int_case :: "(int => 'b) => ('a :: len0 word) => 'b" where
  "word_int_case f w = f (uint w)"

translations
  "case x of XCONST of_int y => b" == "CONST word_int_case (%y. b) x"
  "case x of (XCONST of_int :: 'a) y => b" => "CONST word_int_case (%y. b) x"

subsection {* Type-definition locale instantiations *}

lemma word_size_gt_0 [iff]: "0 < size (w::'a::len word)"
  by (fact xtr1 [OF word_size len_gt_0])

lemmas lens_gt_0 = word_size_gt_0 len_gt_0
lemmas lens_not_0 [iff] = lens_gt_0 [THEN gr_implies_not0]

lemma uints_num: "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}"
  by (simp add: uints_def range_bintrunc)

lemma sints_num: "sints n = {i. - (2 ^ (n - 1)) \<le> i \<and> i < 2 ^ (n - 1)}"
  by (simp add: sints_def range_sbintrunc)

lemma 
  uint_0:"0 <= uint x" and 
  uint_lt: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
  by (auto simp: uint [unfolded atLeastLessThan_iff])

lemma uint_mod_same:
  "uint x mod 2 ^ len_of TYPE('a) = uint (x::'a::len0 word)"
  by (simp add: int_mod_eq uint_lt uint_0)

lemma td_ext_uint: 
  "td_ext (uint :: 'a word => int) word_of_int (uints (len_of TYPE('a::len0))) 
    (%w::int. w mod 2 ^ len_of TYPE('a))"
  apply (unfold td_ext_def')
  apply (simp add: uints_num word_of_int_def bintrunc_mod2p)
  apply (simp add: uint_mod_same uint_0 uint_lt
                   word.uint_inverse word.Abs_word_inverse int_mod_lem)
  done

interpretation word_uint:
  td_ext "uint::'a::len0 word \<Rightarrow> int" 
         word_of_int 
         "uints (len_of TYPE('a::len0))"
         "\<lambda>w. w mod 2 ^ len_of TYPE('a::len0)"
  by (rule td_ext_uint)

lemmas td_uint = word_uint.td_thm

lemmas int_word_uint = word_uint.eq_norm

lemmas td_ext_ubin = td_ext_uint 
  [unfolded len_gt_0 no_bintr_alt1 [symmetric]]

interpretation word_ubin:
  td_ext "uint::'a::len0 word \<Rightarrow> int" 
         word_of_int 
         "uints (len_of TYPE('a::len0))"
         "bintrunc (len_of TYPE('a::len0))"
  by (rule td_ext_ubin)

lemma split_word_all:
  "(\<And>x::'a::len0 word. PROP P x) \<equiv> (\<And>x. PROP P (word_of_int x))"
proof
  fix x :: "'a word"
  assume "\<And>x. PROP P (word_of_int x)"
  hence "PROP P (word_of_int (uint x))" .
  thus "PROP P x" by simp
qed

subsection {* Correspondence relation for theorem transfer *}

definition cr_word :: "int \<Rightarrow> 'a::len0 word \<Rightarrow> bool"
  where "cr_word \<equiv> (\<lambda>x y. word_of_int x = y)"

lemma Quotient_word:
  "Quotient (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y)
    word_of_int uint (cr_word :: _ \<Rightarrow> 'a::len0 word \<Rightarrow> bool)"
  unfolding Quotient_alt_def cr_word_def
  by (simp add: word_ubin.norm_eq_iff)

lemma reflp_word:
  "reflp (\<lambda>x y. bintrunc (len_of TYPE('a::len0)) x = bintrunc (len_of TYPE('a)) y)"
  by (simp add: reflp_def)

local_setup {*
  Lifting_Setup.setup_lifting_infr @{thm Quotient_word} @{thm reflp_word}
*}

text {* TODO: The next several lemmas could be generated automatically. *}

lemma bi_total_cr_word [transfer_rule]: "bi_total cr_word"
  using Quotient_word reflp_word by (rule Quotient_bi_total)

lemma right_unique_cr_word [transfer_rule]: "right_unique cr_word"
  using Quotient_word by (rule Quotient_right_unique)

lemma word_eq_transfer [transfer_rule]:
  "(fun_rel cr_word (fun_rel cr_word op =))
    (\<lambda>x y. bintrunc (len_of TYPE('a)) x = bintrunc (len_of TYPE('a)) y)
    (op = :: 'a::len0 word \<Rightarrow> 'a word \<Rightarrow> bool)"
  using Quotient_word by (rule Quotient_rel_eq_transfer)

lemma word_of_int_transfer [transfer_rule]:
  "(fun_rel op = cr_word) (\<lambda>x. x) word_of_int"
  using Quotient_word reflp_word by (rule Quotient_id_abs_transfer)

lemma uint_transfer [transfer_rule]:
  "(fun_rel cr_word op =) (bintrunc (len_of TYPE('a)))
    (uint :: 'a::len0 word \<Rightarrow> int)"
  unfolding fun_rel_def cr_word_def by (simp add: word_ubin.eq_norm)

subsection  "Arithmetic operations"

lift_definition word_succ :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x + 1"
  by (metis bintr_ariths(6))

lift_definition word_pred :: "'a::len0 word \<Rightarrow> 'a word" is "\<lambda>x. x - 1"
  by (metis bintr_ariths(7))

instantiation word :: (len0) "{neg_numeral, Divides.div, comm_monoid_mult, comm_ring}"
begin

lift_definition zero_word :: "'a word" is "0" .

lift_definition one_word :: "'a word" is "1" .

lift_definition plus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op +"
  by (metis bintr_ariths(2))

lift_definition minus_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op -"
  by (metis bintr_ariths(3))

lift_definition uminus_word :: "'a word \<Rightarrow> 'a word" is uminus
  by (metis bintr_ariths(5))

lift_definition times_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is "op *"
  by (metis bintr_ariths(4))

definition
  word_div_def: "a div b = word_of_int (uint a div uint b)"

definition
  word_mod_def: "a mod b = word_of_int (uint a mod uint b)"

instance
  by default (transfer, simp add: algebra_simps)+

end

text {* Legacy theorems: *}

lemma word_arith_wis: shows
  word_add_def: "a + b = word_of_int (uint a + uint b)" and
  word_sub_wi: "a - b = word_of_int (uint a - uint b)" and
  word_mult_def: "a * b = word_of_int (uint a * uint b)" and
  word_minus_def: "- a = word_of_int (- uint a)" and
  word_succ_alt: "word_succ a = word_of_int (uint a + 1)" and
  word_pred_alt: "word_pred a = word_of_int (uint a - 1)" and
  word_0_wi: "0 = word_of_int 0" and
  word_1_wi: "1 = word_of_int 1"
  unfolding plus_word_def minus_word_def times_word_def uminus_word_def
  unfolding word_succ_def word_pred_def zero_word_def one_word_def
  by simp_all

lemmas arths = 
  bintr_ariths [THEN word_ubin.norm_eq_iff [THEN iffD1], folded word_ubin.eq_norm]

lemma wi_homs: 
  shows
  wi_hom_add: "word_of_int a + word_of_int b = word_of_int (a + b)" and
  wi_hom_sub: "word_of_int a - word_of_int b = word_of_int (a - b)" and
  wi_hom_mult: "word_of_int a * word_of_int b = word_of_int (a * b)" and
  wi_hom_neg: "- word_of_int a = word_of_int (- a)" and
  wi_hom_succ: "word_succ (word_of_int a) = word_of_int (a + 1)" and
  wi_hom_pred: "word_pred (word_of_int a) = word_of_int (a - 1)"
  by (transfer, simp)+

lemmas wi_hom_syms = wi_homs [symmetric]

lemmas word_of_int_homs = wi_homs word_0_wi word_1_wi

lemmas word_of_int_hom_syms = word_of_int_homs [symmetric]

instance word :: (len) comm_ring_1
proof
  have "0 < len_of TYPE('a)" by (rule len_gt_0)
  then show "(0::'a word) \<noteq> 1"
    by - (transfer, auto simp add: gr0_conv_Suc)
qed

lemma word_of_nat: "of_nat n = word_of_int (int n)"
  by (induct n) (auto simp add : word_of_int_hom_syms)

lemma word_of_int: "of_int = word_of_int"
  apply (rule ext)
  apply (case_tac x rule: int_diff_cases)
  apply (simp add: word_of_nat wi_hom_sub)
  done

definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where
  "a udvd b = (EX n>=0. uint b = n * uint a)"


subsection "Ordering"

instantiation word :: (len0) linorder
begin

definition
  word_le_def: "a \<le> b \<longleftrightarrow> uint a \<le> uint b"

definition
  word_less_def: "a < b \<longleftrightarrow> uint a < uint b"

instance
  by default (auto simp: word_less_def word_le_def)

end

definition word_sle :: "'a :: len word => 'a word => bool" ("(_/ <=s _)" [50, 51] 50) where
  "a <=s b = (sint a <= sint b)"

definition word_sless :: "'a :: len word => 'a word => bool" ("(_/ <s _)" [50, 51] 50) where
  "(x <s y) = (x <=s y & x ~= y)"


subsection "Bit-wise operations"

instantiation word :: (len0) bits
begin

lift_definition bitNOT_word :: "'a word \<Rightarrow> 'a word" is bitNOT
  by (metis bin_trunc_not)

lift_definition bitAND_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitAND
  by (metis bin_trunc_and)

lift_definition bitOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitOR
  by (metis bin_trunc_or)

lift_definition bitXOR_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> 'a word" is bitXOR
  by (metis bin_trunc_xor)

definition
  word_test_bit_def: "test_bit a = bin_nth (uint a)"

definition
  word_set_bit_def: "set_bit a n x =
   word_of_int (bin_sc n (If x 1 0) (uint a))"

definition
  word_set_bits_def: "(BITS n. f n) = of_bl (bl_of_nth (len_of TYPE ('a)) f)"

definition
  word_lsb_def: "lsb a \<longleftrightarrow> bin_last (uint a) = 1"

definition shiftl1 :: "'a word \<Rightarrow> 'a word" where
  "shiftl1 w = word_of_int (uint w BIT 0)"

definition shiftr1 :: "'a word \<Rightarrow> 'a word" where
  -- "shift right as unsigned or as signed, ie logical or arithmetic"
  "shiftr1 w = word_of_int (bin_rest (uint w))"

definition
  shiftl_def: "w << n = (shiftl1 ^^ n) w"

definition
  shiftr_def: "w >> n = (shiftr1 ^^ n) w"

instance ..

end

lemma shows
  word_not_def: "NOT (a::'a::len0 word) = word_of_int (NOT (uint a))" and
  word_and_def: "(a::'a word) AND b = word_of_int (uint a AND uint b)" and
  word_or_def: "(a::'a word) OR b = word_of_int (uint a OR uint b)" and
  word_xor_def: "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"
  unfolding bitNOT_word_def bitAND_word_def bitOR_word_def bitXOR_word_def
  by simp_all

instantiation word :: (len) bitss
begin

definition
  word_msb_def: 
  "msb a \<longleftrightarrow> bin_sign (sint a) = -1"

instance ..

end

definition setBit :: "'a :: len0 word => nat => 'a word" where 
  "setBit w n = set_bit w n True"

definition clearBit :: "'a :: len0 word => nat => 'a word" where
  "clearBit w n = set_bit w n False"


subsection "Shift operations"

definition sshiftr1 :: "'a :: len word => 'a word" where 
  "sshiftr1 w = word_of_int (bin_rest (sint w))"

definition bshiftr1 :: "bool => 'a :: len word => 'a word" where
  "bshiftr1 b w = of_bl (b # butlast (to_bl w))"

definition sshiftr :: "'a :: len word => nat => 'a word" (infixl ">>>" 55) where
  "w >>> n = (sshiftr1 ^^ n) w"

definition mask :: "nat => 'a::len word" where
  "mask n = (1 << n) - 1"

definition revcast :: "'a :: len0 word => 'b :: len0 word" where
  "revcast w =  of_bl (takefill False (len_of TYPE('b)) (to_bl w))"

definition slice1 :: "nat => 'a :: len0 word => 'b :: len0 word" where
  "slice1 n w = of_bl (takefill False n (to_bl w))"

definition slice :: "nat => 'a :: len0 word => 'b :: len0 word" where
  "slice n w = slice1 (size w - n) w"


subsection "Rotation"

definition rotater1 :: "'a list => 'a list" where
  "rotater1 ys = 
    (case ys of [] => [] | x # xs => last ys # butlast ys)"

definition rotater :: "nat => 'a list => 'a list" where
  "rotater n = rotater1 ^^ n"

definition word_rotr :: "nat => 'a :: len0 word => 'a :: len0 word" where
  "word_rotr n w = of_bl (rotater n (to_bl w))"

definition word_rotl :: "nat => 'a :: len0 word => 'a :: len0 word" where
  "word_rotl n w = of_bl (rotate n (to_bl w))"

definition word_roti :: "int => 'a :: len0 word => 'a :: len0 word" where
  "word_roti i w = (if i >= 0 then word_rotr (nat i) w
                    else word_rotl (nat (- i)) w)"


subsection "Split and cat operations"

definition word_cat :: "'a :: len0 word => 'b :: len0 word => 'c :: len0 word" where
  "word_cat a b = word_of_int (bin_cat (uint a) (len_of TYPE ('b)) (uint b))"

definition word_split :: "'a :: len0 word => ('b :: len0 word) * ('c :: len0 word)" where
  "word_split a = 
   (case bin_split (len_of TYPE ('c)) (uint a) of 
     (u, v) => (word_of_int u, word_of_int v))"

definition word_rcat :: "'a :: len0 word list => 'b :: len0 word" where
  "word_rcat ws = 
  word_of_int (bin_rcat (len_of TYPE ('a)) (map uint ws))"

definition word_rsplit :: "'a :: len0 word => 'b :: len word list" where
  "word_rsplit w = 
  map word_of_int (bin_rsplit (len_of TYPE ('b)) (len_of TYPE ('a), uint w))"

definition max_word :: "'a::len word" -- "Largest representable machine integer." where
  "max_word = word_of_int (2 ^ len_of TYPE('a) - 1)"

primrec of_bool :: "bool \<Rightarrow> 'a::len word" where
  "of_bool False = 0"
| "of_bool True = 1"

(* FIXME: only provide one theorem name *)
lemmas of_nth_def = word_set_bits_def

subsection {* Theorems about typedefs *}

lemma sint_sbintrunc': 
  "sint (word_of_int bin :: 'a word) = 
    (sbintrunc (len_of TYPE ('a :: len) - 1) bin)"
  unfolding sint_uint 
  by (auto simp: word_ubin.eq_norm sbintrunc_bintrunc_lt)

lemma uint_sint: 
  "uint w = bintrunc (len_of TYPE('a)) (sint (w :: 'a :: len word))"
  unfolding sint_uint by (auto simp: bintrunc_sbintrunc_le)

lemma bintr_uint:
  fixes w :: "'a::len0 word"
  shows "len_of TYPE('a) \<le> n \<Longrightarrow> bintrunc n (uint w) = uint w"
  apply (subst word_ubin.norm_Rep [symmetric]) 
  apply (simp only: bintrunc_bintrunc_min word_size)
  apply (simp add: min_max.inf_absorb2)
  done

lemma wi_bintr:
  "len_of TYPE('a::len0) \<le> n \<Longrightarrow>
    word_of_int (bintrunc n w) = (word_of_int w :: 'a word)"
  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1)

lemma td_ext_sbin: 
  "td_ext (sint :: 'a word => int) word_of_int (sints (len_of TYPE('a::len))) 
    (sbintrunc (len_of TYPE('a) - 1))"
  apply (unfold td_ext_def' sint_uint)
  apply (simp add : word_ubin.eq_norm)
  apply (cases "len_of TYPE('a)")
   apply (auto simp add : sints_def)
  apply (rule sym [THEN trans])
  apply (rule word_ubin.Abs_norm)
  apply (simp only: bintrunc_sbintrunc)
  apply (drule sym)
  apply simp
  done

lemmas td_ext_sint = td_ext_sbin 
  [simplified len_gt_0 no_sbintr_alt2 Suc_pred' [symmetric]]

(* We do sint before sbin, before sint is the user version
   and interpretations do not produce thm duplicates. I.e. 
   we get the name word_sint.Rep_eqD, but not word_sbin.Req_eqD,
   because the latter is the same thm as the former *)
interpretation word_sint:
  td_ext "sint ::'a::len word => int" 
          word_of_int 
          "sints (len_of TYPE('a::len))"
          "%w. (w + 2^(len_of TYPE('a::len) - 1)) mod 2^len_of TYPE('a::len) -
               2 ^ (len_of TYPE('a::len) - 1)"
  by (rule td_ext_sint)

interpretation word_sbin:
  td_ext "sint ::'a::len word => int" 
          word_of_int 
          "sints (len_of TYPE('a::len))"
          "sbintrunc (len_of TYPE('a::len) - 1)"
  by (rule td_ext_sbin)

lemmas int_word_sint = td_ext_sint [THEN td_ext.eq_norm]

lemmas td_sint = word_sint.td

lemma to_bl_def': 
  "(to_bl :: 'a :: len0 word => bool list) =
    bin_to_bl (len_of TYPE('a)) o uint"
  by (auto simp: to_bl_def)

lemmas word_reverse_no_def [simp] = word_reverse_def [of "numeral w"] for w

lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"
  by (fact uints_def [unfolded no_bintr_alt1])

lemma word_numeral_alt:
  "numeral b = word_of_int (numeral b)"
  by (induct b, simp_all only: numeral.simps word_of_int_homs)

declare word_numeral_alt [symmetric, code_abbrev]

lemma word_neg_numeral_alt:
  "neg_numeral b = word_of_int (neg_numeral b)"
  by (simp only: neg_numeral_def word_numeral_alt wi_hom_neg)

declare word_neg_numeral_alt [symmetric, code_abbrev]

lemma word_numeral_transfer [transfer_rule]:
  "(fun_rel op = cr_word) numeral numeral"
  "(fun_rel op = cr_word) neg_numeral neg_numeral"
  unfolding fun_rel_def cr_word_def word_numeral_alt word_neg_numeral_alt
  by simp_all

lemma uint_bintrunc [simp]:
  "uint (numeral bin :: 'a word) = 
    bintrunc (len_of TYPE ('a :: len0)) (numeral bin)"
  unfolding word_numeral_alt by (rule word_ubin.eq_norm)

lemma uint_bintrunc_neg [simp]: "uint (neg_numeral bin :: 'a word) = 
    bintrunc (len_of TYPE ('a :: len0)) (neg_numeral bin)"
  by (simp only: word_neg_numeral_alt word_ubin.eq_norm)

lemma sint_sbintrunc [simp]:
  "sint (numeral bin :: 'a word) = 
    sbintrunc (len_of TYPE ('a :: len) - 1) (numeral bin)"
  by (simp only: word_numeral_alt word_sbin.eq_norm)

lemma sint_sbintrunc_neg [simp]: "sint (neg_numeral bin :: 'a word) = 
    sbintrunc (len_of TYPE ('a :: len) - 1) (neg_numeral bin)"
  by (simp only: word_neg_numeral_alt word_sbin.eq_norm)

lemma unat_bintrunc [simp]:
  "unat (numeral bin :: 'a :: len0 word) =
    nat (bintrunc (len_of TYPE('a)) (numeral bin))"
  by (simp only: unat_def uint_bintrunc)

lemma unat_bintrunc_neg [simp]:
  "unat (neg_numeral bin :: 'a :: len0 word) =
    nat (bintrunc (len_of TYPE('a)) (neg_numeral bin))"
  by (simp only: unat_def uint_bintrunc_neg)

lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 \<Longrightarrow> v = w"
  apply (unfold word_size)
  apply (rule word_uint.Rep_eqD)
  apply (rule box_equals)
    defer
    apply (rule word_ubin.norm_Rep)+
  apply simp
  done

lemma uint_ge_0 [iff]: "0 \<le> uint (x::'a::len0 word)"
  using word_uint.Rep [of x] by (simp add: uints_num)

lemma uint_lt2p [iff]: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
  using word_uint.Rep [of x] by (simp add: uints_num)

lemma sint_ge: "- (2 ^ (len_of TYPE('a) - 1)) \<le> sint (x::'a::len word)"
  using word_sint.Rep [of x] by (simp add: sints_num)

lemma sint_lt: "sint (x::'a::len word) < 2 ^ (len_of TYPE('a) - 1)"
  using word_sint.Rep [of x] by (simp add: sints_num)

lemma sign_uint_Pls [simp]: 
  "bin_sign (uint x) = 0"
  by (simp add: sign_Pls_ge_0)

lemma uint_m2p_neg: "uint (x::'a::len0 word) - 2 ^ len_of TYPE('a) < 0"
  by (simp only: diff_less_0_iff_less uint_lt2p)

lemma uint_m2p_not_non_neg:
  "\<not> 0 \<le> uint (x::'a::len0 word) - 2 ^ len_of TYPE('a)"
  by (simp only: not_le uint_m2p_neg)

lemma lt2p_lem:
  "len_of TYPE('a) <= n \<Longrightarrow> uint (w :: 'a :: len0 word) < 2 ^ n"
  by (rule xtr8 [OF _ uint_lt2p]) simp

lemma uint_le_0_iff [simp]: "uint x \<le> 0 \<longleftrightarrow> uint x = 0"
  by (fact uint_ge_0 [THEN leD, THEN linorder_antisym_conv1])

lemma uint_nat: "uint w = int (unat w)"
  unfolding unat_def by auto

lemma uint_numeral:
  "uint (numeral b :: 'a :: len0 word) = numeral b mod 2 ^ len_of TYPE('a)"
  unfolding word_numeral_alt
  by (simp only: int_word_uint)

lemma uint_neg_numeral:
  "uint (neg_numeral b :: 'a :: len0 word) = neg_numeral b mod 2 ^ len_of TYPE('a)"
  unfolding word_neg_numeral_alt
  by (simp only: int_word_uint)

lemma unat_numeral: 
  "unat (numeral b::'a::len0 word) = numeral b mod 2 ^ len_of TYPE ('a)"
  apply (unfold unat_def)
  apply (clarsimp simp only: uint_numeral)
  apply (rule nat_mod_distrib [THEN trans])
    apply (rule zero_le_numeral)
   apply (simp_all add: nat_power_eq)
  done

lemma sint_numeral: "sint (numeral b :: 'a :: len word) = (numeral b + 
    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -
    2 ^ (len_of TYPE('a) - 1)"
  unfolding word_numeral_alt by (rule int_word_sint)

lemma word_of_int_0 [simp, code_post]: "word_of_int 0 = 0"
  unfolding word_0_wi ..

lemma word_of_int_1 [simp, code_post]: "word_of_int 1 = 1"
  unfolding word_1_wi ..

lemma word_of_int_numeral [simp] : 
  "(word_of_int (numeral bin) :: 'a :: len0 word) = (numeral bin)"
  unfolding word_numeral_alt ..

lemma word_of_int_neg_numeral [simp]:
  "(word_of_int (neg_numeral bin) :: 'a :: len0 word) = (neg_numeral bin)"
  unfolding neg_numeral_def word_numeral_alt wi_hom_syms ..

lemma word_int_case_wi: 
  "word_int_case f (word_of_int i :: 'b word) = 
    f (i mod 2 ^ len_of TYPE('b::len0))"
  unfolding word_int_case_def by (simp add: word_uint.eq_norm)

lemma word_int_split: 
  "P (word_int_case f x) = 
    (ALL i. x = (word_of_int i :: 'b :: len0 word) & 
      0 <= i & i < 2 ^ len_of TYPE('b) --> P (f i))"
  unfolding word_int_case_def
  by (auto simp: word_uint.eq_norm int_mod_eq')

lemma word_int_split_asm: 
  "P (word_int_case f x) = 
    (~ (EX n. x = (word_of_int n :: 'b::len0 word) &
      0 <= n & n < 2 ^ len_of TYPE('b::len0) & ~ P (f n)))"
  unfolding word_int_case_def
  by (auto simp: word_uint.eq_norm int_mod_eq')

lemmas uint_range' = word_uint.Rep [unfolded uints_num mem_Collect_eq]
lemmas sint_range' = word_sint.Rep [unfolded One_nat_def sints_num mem_Collect_eq]

lemma uint_range_size: "0 <= uint w & uint w < 2 ^ size w"
  unfolding word_size by (rule uint_range')

lemma sint_range_size:
  "- (2 ^ (size w - Suc 0)) <= sint w & sint w < 2 ^ (size w - Suc 0)"
  unfolding word_size by (rule sint_range')

lemma sint_above_size: "2 ^ (size (w::'a::len word) - 1) \<le> x \<Longrightarrow> sint w < x"
  unfolding word_size by (rule less_le_trans [OF sint_lt])

lemma sint_below_size:
  "x \<le> - (2 ^ (size (w::'a::len word) - 1)) \<Longrightarrow> x \<le> sint w"
  unfolding word_size by (rule order_trans [OF _ sint_ge])

subsection {* Testing bits *}

lemma test_bit_eq_iff: "(test_bit (u::'a::len0 word) = test_bit v) = (u = v)"
  unfolding word_test_bit_def by (simp add: bin_nth_eq_iff)

lemma test_bit_size [rule_format] : "(w::'a::len0 word) !! n --> n < size w"
  apply (unfold word_test_bit_def)
  apply (subst word_ubin.norm_Rep [symmetric])
  apply (simp only: nth_bintr word_size)
  apply fast
  done

lemma word_eq_iff:
  fixes x y :: "'a::len0 word"
  shows "x = y \<longleftrightarrow> (\<forall>n<len_of TYPE('a). x !! n = y !! n)"
  unfolding uint_inject [symmetric] bin_eq_iff word_test_bit_def [symmetric]
  by (metis test_bit_size [unfolded word_size])

lemma word_eqI [rule_format]:
  fixes u :: "'a::len0 word"
  shows "(ALL n. n < size u --> u !! n = v !! n) \<Longrightarrow> u = v"
  by (simp add: word_size word_eq_iff)

lemma word_eqD: "(u::'a::len0 word) = v \<Longrightarrow> u !! x = v !! x"
  by simp

lemma test_bit_bin': "w !! n = (n < size w & bin_nth (uint w) n)"
  unfolding word_test_bit_def word_size
  by (simp add: nth_bintr [symmetric])

lemmas test_bit_bin = test_bit_bin' [unfolded word_size]

lemma bin_nth_uint_imp:
  "bin_nth (uint (w::'a::len0 word)) n \<Longrightarrow> n < len_of TYPE('a)"
  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
  apply (subst word_ubin.norm_Rep)
  apply assumption
  done

lemma bin_nth_sint:
  fixes w :: "'a::len word"
  shows "len_of TYPE('a) \<le> n \<Longrightarrow>
    bin_nth (sint w) n = bin_nth (sint w) (len_of TYPE('a) - 1)"
  apply (subst word_sbin.norm_Rep [symmetric])
  apply (auto simp add: nth_sbintr)
  done

(* type definitions theorem for in terms of equivalent bool list *)
lemma td_bl: 
  "type_definition (to_bl :: 'a::len0 word => bool list) 
                   of_bl  
                   {bl. length bl = len_of TYPE('a)}"
  apply (unfold type_definition_def of_bl_def to_bl_def)
  apply (simp add: word_ubin.eq_norm)
  apply safe
  apply (drule sym)
  apply simp
  done

interpretation word_bl:
  type_definition "to_bl :: 'a::len0 word => bool list"
                  of_bl  
                  "{bl. length bl = len_of TYPE('a::len0)}"
  by (rule td_bl)

lemmas word_bl_Rep' = word_bl.Rep [unfolded mem_Collect_eq, iff]

lemma word_size_bl: "size w = size (to_bl w)"
  unfolding word_size by auto

lemma to_bl_use_of_bl:
  "(to_bl w = bl) = (w = of_bl bl \<and> length bl = length (to_bl w))"
  by (fastforce elim!: word_bl.Abs_inverse [unfolded mem_Collect_eq])

lemma to_bl_word_rev: "to_bl (word_reverse w) = rev (to_bl w)"
  unfolding word_reverse_def by (simp add: word_bl.Abs_inverse)

lemma word_rev_rev [simp] : "word_reverse (word_reverse w) = w"
  unfolding word_reverse_def by (simp add : word_bl.Abs_inverse)

lemma word_rev_gal: "word_reverse w = u \<Longrightarrow> word_reverse u = w"
  by (metis word_rev_rev)

lemma word_rev_gal': "u = word_reverse w \<Longrightarrow> w = word_reverse u"
  by simp

lemma length_bl_gt_0 [iff]: "0 < length (to_bl (x::'a::len word))"
  unfolding word_bl_Rep' by (rule len_gt_0)

lemma bl_not_Nil [iff]: "to_bl (x::'a::len word) \<noteq> []"
  by (fact length_bl_gt_0 [unfolded length_greater_0_conv])

lemma length_bl_neq_0 [iff]: "length (to_bl (x::'a::len word)) \<noteq> 0"
  by (fact length_bl_gt_0 [THEN gr_implies_not0])

lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = -1)"
  apply (unfold to_bl_def sint_uint)
  apply (rule trans [OF _ bl_sbin_sign])
  apply simp
  done

lemma of_bl_drop': 
  "lend = length bl - len_of TYPE ('a :: len0) \<Longrightarrow> 
    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
  apply (unfold of_bl_def)
  apply (clarsimp simp add : trunc_bl2bin [symmetric])
  done

lemma test_bit_of_bl:  
  "(of_bl bl::'a::len0 word) !! n = (rev bl ! n \<and> n < len_of TYPE('a) \<and> n < length bl)"
  apply (unfold of_bl_def word_test_bit_def)
  apply (auto simp add: word_size word_ubin.eq_norm nth_bintr bin_nth_of_bl)
  done

lemma no_of_bl: 
  "(numeral bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) (numeral bin))"
  unfolding of_bl_def by simp

lemma uint_bl: "to_bl w = bin_to_bl (size w) (uint w)"
  unfolding word_size to_bl_def by auto

lemma to_bl_bin: "bl_to_bin (to_bl w) = uint w"
  unfolding uint_bl by (simp add : word_size)

lemma to_bl_of_bin: 
  "to_bl (word_of_int bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) bin"
  unfolding uint_bl by (clarsimp simp add: word_ubin.eq_norm word_size)

lemma to_bl_numeral [simp]:
  "to_bl (numeral bin::'a::len0 word) =
    bin_to_bl (len_of TYPE('a)) (numeral bin)"
  unfolding word_numeral_alt by (rule to_bl_of_bin)

lemma to_bl_neg_numeral [simp]:
  "to_bl (neg_numeral bin::'a::len0 word) =
    bin_to_bl (len_of TYPE('a)) (neg_numeral bin)"
  unfolding word_neg_numeral_alt by (rule to_bl_of_bin)

lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
  unfolding uint_bl by (simp add : word_size)

lemma uint_bl_bin:
  fixes x :: "'a::len0 word"
  shows "bl_to_bin (bin_to_bl (len_of TYPE('a)) (uint x)) = uint x"
  by (rule trans [OF bin_bl_bin word_ubin.norm_Rep])

(* naturals *)
lemma uints_unats: "uints n = int ` unats n"
  apply (unfold unats_def uints_num)
  apply safe
  apply (rule_tac image_eqI)
  apply (erule_tac nat_0_le [symmetric])
  apply auto
  apply (erule_tac nat_less_iff [THEN iffD2])
  apply (rule_tac [2] zless_nat_eq_int_zless [THEN iffD1])
  apply (auto simp add : nat_power_eq int_power)
  done

lemma unats_uints: "unats n = nat ` uints n"
  by (auto simp add : uints_unats image_iff)

lemmas bintr_num = word_ubin.norm_eq_iff
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b
lemmas sbintr_num = word_sbin.norm_eq_iff
  [of "numeral a" "numeral b", symmetric, folded word_numeral_alt] for a b

lemma num_of_bintr':
  "bintrunc (len_of TYPE('a :: len0)) (numeral a) = (numeral b) \<Longrightarrow> 
    numeral a = (numeral b :: 'a word)"
  unfolding bintr_num by (erule subst, simp)

lemma num_of_sbintr':
  "sbintrunc (len_of TYPE('a :: len) - 1) (numeral a) = (numeral b) \<Longrightarrow> 
    numeral a = (numeral b :: 'a word)"
  unfolding sbintr_num by (erule subst, simp)

lemma num_abs_bintr:
  "(numeral x :: 'a word) =
    word_of_int (bintrunc (len_of TYPE('a::len0)) (numeral x))"
  by (simp only: word_ubin.Abs_norm word_numeral_alt)

lemma num_abs_sbintr:
  "(numeral x :: 'a word) =
    word_of_int (sbintrunc (len_of TYPE('a::len) - 1) (numeral x))"
  by (simp only: word_sbin.Abs_norm word_numeral_alt)

(** cast - note, no arg for new length, as it's determined by type of result,
  thus in "cast w = w, the type means cast to length of w! **)

lemma ucast_id: "ucast w = w"
  unfolding ucast_def by auto

lemma scast_id: "scast w = w"
  unfolding scast_def by auto

lemma ucast_bl: "ucast w = of_bl (to_bl w)"
  unfolding ucast_def of_bl_def uint_bl
  by (auto simp add : word_size)

lemma nth_ucast: 
  "(ucast w::'a::len0 word) !! n = (w !! n & n < len_of TYPE('a))"
  apply (unfold ucast_def test_bit_bin)
  apply (simp add: word_ubin.eq_norm nth_bintr word_size) 
  apply (fast elim!: bin_nth_uint_imp)
  done

(* for literal u(s)cast *)

lemma ucast_bintr [simp]:
  "ucast (numeral w ::'a::len0 word) = 
   word_of_int (bintrunc (len_of TYPE('a)) (numeral w))"
  unfolding ucast_def by simp
(* TODO: neg_numeral *)

lemma scast_sbintr [simp]:
  "scast (numeral w ::'a::len word) = 
   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (numeral w))"
  unfolding scast_def by simp

lemma source_size: "source_size (c::'a::len0 word \<Rightarrow> _) = len_of TYPE('a)"
  unfolding source_size_def word_size Let_def ..

lemma target_size: "target_size (c::_ \<Rightarrow> 'b::len0 word) = len_of TYPE('b)"
  unfolding target_size_def word_size Let_def ..

lemma is_down:
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
  shows "is_down c \<longleftrightarrow> len_of TYPE('b) \<le> len_of TYPE('a)"
  unfolding is_down_def source_size target_size ..

lemma is_up:
  fixes c :: "'a::len0 word \<Rightarrow> 'b::len0 word"
  shows "is_up c \<longleftrightarrow> len_of TYPE('a) \<le> len_of TYPE('b)"
  unfolding is_up_def source_size target_size ..

lemmas is_up_down = trans [OF is_up is_down [symmetric]]

lemma down_cast_same [OF refl]: "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc = scast"
  apply (unfold is_down)
  apply safe
  apply (rule ext)
  apply (unfold ucast_def scast_def uint_sint)
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
  apply simp
  done

lemma word_rev_tf:
  "to_bl (of_bl bl::'a::len0 word) =
    rev (takefill False (len_of TYPE('a)) (rev bl))"
  unfolding of_bl_def uint_bl
  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)

lemma word_rep_drop:
  "to_bl (of_bl bl::'a::len0 word) =
    replicate (len_of TYPE('a) - length bl) False @
    drop (length bl - len_of TYPE('a)) bl"
  by (simp add: word_rev_tf takefill_alt rev_take)

lemma to_bl_ucast: 
  "to_bl (ucast (w::'b::len0 word) ::'a::len0 word) = 
   replicate (len_of TYPE('a) - len_of TYPE('b)) False @
   drop (len_of TYPE('b) - len_of TYPE('a)) (to_bl w)"
  apply (unfold ucast_bl)
  apply (rule trans)
   apply (rule word_rep_drop)
  apply simp
  done

lemma ucast_up_app [OF refl]:
  "uc = ucast \<Longrightarrow> source_size uc + n = target_size uc \<Longrightarrow> 
    to_bl (uc w) = replicate n False @ (to_bl w)"
  by (auto simp add : source_size target_size to_bl_ucast)

lemma ucast_down_drop [OF refl]:
  "uc = ucast \<Longrightarrow> source_size uc = target_size uc + n \<Longrightarrow> 
    to_bl (uc w) = drop n (to_bl w)"
  by (auto simp add : source_size target_size to_bl_ucast)

lemma scast_down_drop [OF refl]:
  "sc = scast \<Longrightarrow> source_size sc = target_size sc + n \<Longrightarrow> 
    to_bl (sc w) = drop n (to_bl w)"
  apply (subgoal_tac "sc = ucast")
   apply safe
   apply simp
   apply (erule ucast_down_drop)
  apply (rule down_cast_same [symmetric])
  apply (simp add : source_size target_size is_down)
  done

lemma sint_up_scast [OF refl]:
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> sint (sc w) = sint w"
  apply (unfold is_up)
  apply safe
  apply (simp add: scast_def word_sbin.eq_norm)
  apply (rule box_equals)
    prefer 3
    apply (rule word_sbin.norm_Rep)
   apply (rule sbintrunc_sbintrunc_l)
   defer
   apply (subst word_sbin.norm_Rep)
   apply (rule refl)
  apply simp
  done

lemma uint_up_ucast [OF refl]:
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> uint (uc w) = uint w"
  apply (unfold is_up)
  apply safe
  apply (rule bin_eqI)
  apply (fold word_test_bit_def)
  apply (auto simp add: nth_ucast)
  apply (auto simp add: test_bit_bin)
  done

lemma ucast_up_ucast [OF refl]:
  "uc = ucast \<Longrightarrow> is_up uc \<Longrightarrow> ucast (uc w) = ucast w"
  apply (simp (no_asm) add: ucast_def)
  apply (clarsimp simp add: uint_up_ucast)
  done
    
lemma scast_up_scast [OF refl]:
  "sc = scast \<Longrightarrow> is_up sc \<Longrightarrow> scast (sc w) = scast w"
  apply (simp (no_asm) add: scast_def)
  apply (clarsimp simp add: sint_up_scast)
  done
    
lemma ucast_of_bl_up [OF refl]:
  "w = of_bl bl \<Longrightarrow> size bl <= size w \<Longrightarrow> ucast w = of_bl bl"
  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)

lemmas ucast_up_ucast_id = trans [OF ucast_up_ucast ucast_id]
lemmas scast_up_scast_id = trans [OF scast_up_scast scast_id]

lemmas isduu = is_up_down [where c = "ucast", THEN iffD2]
lemmas isdus = is_up_down [where c = "scast", THEN iffD2]
lemmas ucast_down_ucast_id = isduu [THEN ucast_up_ucast_id]
lemmas scast_down_scast_id = isdus [THEN ucast_up_ucast_id]

lemma up_ucast_surj:
  "is_up (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
   surj (ucast :: 'a word => 'b word)"
  by (rule surjI, erule ucast_up_ucast_id)

lemma up_scast_surj:
  "is_up (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
   surj (scast :: 'a word => 'b word)"
  by (rule surjI, erule scast_up_scast_id)

lemma down_scast_inj:
  "is_down (scast :: 'b::len word => 'a::len word) \<Longrightarrow> 
   inj_on (ucast :: 'a word => 'b word) A"
  by (rule inj_on_inverseI, erule scast_down_scast_id)

lemma down_ucast_inj:
  "is_down (ucast :: 'b::len0 word => 'a::len0 word) \<Longrightarrow> 
   inj_on (ucast :: 'a word => 'b word) A"
  by (rule inj_on_inverseI, erule ucast_down_ucast_id)

lemma of_bl_append_same: "of_bl (X @ to_bl w) = w"
  by (rule word_bl.Rep_eqD) (simp add: word_rep_drop)

lemma ucast_down_wi [OF refl]:
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (word_of_int x) = word_of_int x"
  apply (unfold is_down)
  apply (clarsimp simp add: ucast_def word_ubin.eq_norm)
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
  apply (erule bintrunc_bintrunc_ge)
  done

lemma ucast_down_no [OF refl]:
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (numeral bin) = numeral bin"
  unfolding word_numeral_alt by clarify (rule ucast_down_wi)

lemma ucast_down_bl [OF refl]:
  "uc = ucast \<Longrightarrow> is_down uc \<Longrightarrow> uc (of_bl bl) = of_bl bl"
  unfolding of_bl_def by clarify (erule ucast_down_wi)

lemmas slice_def' = slice_def [unfolded word_size]
lemmas test_bit_def' = word_test_bit_def [THEN fun_cong]

lemmas word_log_defs = word_and_def word_or_def word_xor_def word_not_def


subsection {* Word Arithmetic *}

lemma word_less_alt: "(a < b) = (uint a < uint b)"
  unfolding word_less_def word_le_def by (simp add: less_le)

lemma signed_linorder: "class.linorder word_sle word_sless"
  by default (unfold word_sle_def word_sless_def, auto)

interpretation signed: linorder "word_sle" "word_sless"
  by (rule signed_linorder)

lemma udvdI: 
  "0 \<le> n \<Longrightarrow> uint b = n * uint a \<Longrightarrow> a udvd b"
  by (auto simp: udvd_def)

lemmas word_div_no [simp] = word_div_def [of "numeral a" "numeral b"] for a b

lemmas word_mod_no [simp] = word_mod_def [of "numeral a" "numeral b"] for a b

lemmas word_less_no [simp] = word_less_def [of "numeral a" "numeral b"] for a b

lemmas word_le_no [simp] = word_le_def [of "numeral a" "numeral b"] for a b

lemmas word_sless_no [simp] = word_sless_def [of "numeral a" "numeral b"] for a b

lemmas word_sle_no [simp] = word_sle_def [of "numeral a" "numeral b"] for a b

lemma word_1_no: "(1::'a::len0 word) = Numeral1"
  by (simp add: word_numeral_alt)

lemma word_m1_wi: "-1 = word_of_int -1" 
  by (rule word_neg_numeral_alt)

lemma word_0_bl [simp]: "of_bl [] = 0"
  unfolding of_bl_def by simp

lemma word_1_bl: "of_bl [True] = 1" 
  unfolding of_bl_def by (simp add: bl_to_bin_def)

lemma uint_eq_0 [simp]: "uint 0 = 0"
  unfolding word_0_wi word_ubin.eq_norm by simp

lemma of_bl_0 [simp]: "of_bl (replicate n False) = 0"
  by (simp add: of_bl_def bl_to_bin_rep_False)

lemma to_bl_0 [simp]:
  "to_bl (0::'a::len0 word) = replicate (len_of TYPE('a)) False"
  unfolding uint_bl
  by (simp add: word_size bin_to_bl_zero)

lemma uint_0_iff: "(uint x = 0) = (x = 0)"
  by (auto intro!: word_uint.Rep_eqD)

lemma unat_0_iff: "(unat x = 0) = (x = 0)"
  unfolding unat_def by (auto simp add : nat_eq_iff uint_0_iff)

lemma unat_0 [simp]: "unat 0 = 0"
  unfolding unat_def by auto

lemma size_0_same': "size w = 0 \<Longrightarrow> w = (v :: 'a :: len0 word)"
  apply (unfold word_size)
  apply (rule box_equals)
    defer
    apply (rule word_uint.Rep_inverse)+
  apply (rule word_ubin.norm_eq_iff [THEN iffD1])
  apply simp
  done

lemmas size_0_same = size_0_same' [unfolded word_size]

lemmas unat_eq_0 = unat_0_iff
lemmas unat_eq_zero = unat_0_iff

lemma unat_gt_0: "(0 < unat x) = (x ~= 0)"
by (auto simp: unat_0_iff [symmetric])

lemma ucast_0 [simp]: "ucast 0 = 0"
  unfolding ucast_def by simp

lemma sint_0 [simp]: "sint 0 = 0"
  unfolding sint_uint by simp

lemma scast_0 [simp]: "scast 0 = 0"
  unfolding scast_def by simp

lemma sint_n1 [simp] : "sint -1 = -1"
  unfolding word_m1_wi by (simp add: word_sbin.eq_norm)

lemma scast_n1 [simp]: "scast -1 = -1"
  unfolding scast_def by simp

lemma uint_1 [simp]: "uint (1::'a::len word) = 1"
  unfolding word_1_wi
  by (simp add: word_ubin.eq_norm bintrunc_minus_simps del: word_of_int_1)

lemma unat_1 [simp]: "unat (1::'a::len word) = 1"
  unfolding unat_def by simp

lemma ucast_1 [simp]: "ucast (1::'a::len word) = 1"
  unfolding ucast_def by simp

(* now, to get the weaker results analogous to word_div/mod_def *)

lemmas word_arith_alts = 
  word_sub_wi
  word_arith_wis (* FIXME: duplicate *)

subsection  "Transferring goals from words to ints"

lemma word_ths:  
  shows
  word_succ_p1:   "word_succ a = a + 1" and
  word_pred_m1:   "word_pred a = a - 1" and
  word_pred_succ: "word_pred (word_succ a) = a" and
  word_succ_pred: "word_succ (word_pred a) = a" and
  word_mult_succ: "word_succ a * b = b + a * b"
  by (transfer, simp add: algebra_simps)+

lemma uint_cong: "x = y \<Longrightarrow> uint x = uint y"
  by simp

lemmas uint_word_ariths = 
  word_arith_alts [THEN trans [OF uint_cong int_word_uint]]

lemmas uint_word_arith_bintrs = uint_word_ariths [folded bintrunc_mod2p]

(* similar expressions for sint (arith operations) *)
lemmas sint_word_ariths = uint_word_arith_bintrs
  [THEN uint_sint [symmetric, THEN trans],
  unfolded uint_sint bintr_arith1s bintr_ariths 
    len_gt_0 [THEN bin_sbin_eq_iff'] word_sbin.norm_Rep]

lemmas uint_div_alt = word_div_def [THEN trans [OF uint_cong int_word_uint]]
lemmas uint_mod_alt = word_mod_def [THEN trans [OF uint_cong int_word_uint]]

lemma word_pred_0_n1: "word_pred 0 = word_of_int -1"
  unfolding word_pred_m1 by simp

lemma succ_pred_no [simp]:
  "word_succ (numeral w) = numeral w + 1"
  "word_pred (numeral w) = numeral w - 1"
  "word_succ (neg_numeral w) = neg_numeral w + 1"
  "word_pred (neg_numeral w) = neg_numeral w - 1"
  unfolding word_succ_p1 word_pred_m1 by simp_all

lemma word_sp_01 [simp] : 
  "word_succ -1 = 0 & word_succ 0 = 1 & word_pred 0 = -1 & word_pred 1 = 0"
  unfolding word_succ_p1 word_pred_m1 by simp_all

(* alternative approach to lifting arithmetic equalities *)
lemma word_of_int_Ex:
  "\<exists>y. x = word_of_int y"
  by (rule_tac x="uint x" in exI) simp


subsection "Order on fixed-length words"

lemma word_zero_le [simp] :
  "0 <= (y :: 'a :: len0 word)"
  unfolding word_le_def by auto
  
lemma word_m1_ge [simp] : "word_pred 0 >= y" (* FIXME: delete *)
  unfolding word_le_def
  by (simp only : word_pred_0_n1 word_uint.eq_norm m1mod2k) auto

lemma word_n1_ge [simp]: "y \<le> (-1::'a::len0 word)"
  unfolding word_le_def
  by (simp only: word_m1_wi word_uint.eq_norm m1mod2k) auto