(*  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

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 w = Abs_word (bintrunc (len_of TYPE ('a)) w)" 

lemma uint_word_of_int [code]: "uint (word_of_int w \<Colon> 'a\<Colon>len0 word) = w mod 2 ^ len_of TYPE('a)"

  by (auto simp add: word_of_int_def bintrunc_mod2p intro: Abs_word_inverse)

code_datatype word_of_int

subsection {* Random instance *}

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

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

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

begin

definition

  "random_word i = Random.range (max i (2 ^ len_of TYPE('a))) \<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  "Arithmetic operations"

definition

  word_succ :: "'a :: len0 word => 'a word"

where

  "word_succ a = word_of_int (uint a + 1)"

definition

  word_pred :: "'a :: len0 word => 'a word"

where

  "word_pred a = word_of_int (uint a - 1)"

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

begin

definition

  word_0_wi: "0 = word_of_int 0"

definition

  word_1_wi: "1 = word_of_int 1"

definition

  word_add_def: "a + b = word_of_int (uint a + uint b)"

definition

  word_sub_wi: "a - b = word_of_int (uint a - uint b)"

definition

  word_minus_def: "- a = word_of_int (- uint a)"

definition

  word_mult_def: "a * b = word_of_int (uint a * uint b)"

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)"

definition

  word_number_of_def: "number_of w = word_of_int w"

lemmas word_arith_wis =

  word_add_def word_sub_wi word_mult_def word_minus_def 

  word_succ_def word_pred_def word_0_wi word_1_wi

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 (auto simp: word_arith_wis arths)

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

  by default (auto simp: split_word_all word_of_int_homs algebra_simps)

end

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"

    unfolding word_0_wi word_1_wi

    by (auto simp add: word_ubin.norm_eq_iff [symmetric] 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

instance word :: (len) number_ring

  by (default, simp add: word_number_of_def word_of_int)

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: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> (y \<Colon> 'a word)"

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

definition

  word_and_def: 

  "(a::'a word) AND b = word_of_int (uint a AND uint b)"

definition

  word_or_def:  

  "(a::'a word) OR b = word_of_int (uint a OR uint b)"

definition

  word_xor_def: 

  "(a::'a word) XOR b = word_of_int (uint a XOR uint b)"

definition

  word_not_def: 

  "NOT (a::'a word) = word_of_int (NOT (uint a))"

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

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 word_number_of_alt:

  "number_of b = word_of_int (number_of b)"

  by (simp add: number_of_eq word_number_of_def)

declare word_number_of_alt [symmetric, code_abbrev]

lemma word_no_wi: "number_of = word_of_int"

  by (auto simp: word_number_of_def)

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 "number_of w"] for w

lemma uints_mod: "uints n = range (\<lambda>w. w mod 2 ^ n)"

  by (fact uints_def [unfolded no_bintr_alt1])

lemma uint_bintrunc [simp]:

  "uint (number_of bin :: 'a word) =

    bintrunc (len_of TYPE ('a :: len0)) (number_of bin)"

  unfolding word_number_of_alt by (rule word_ubin.eq_norm)

lemma sint_sbintrunc [simp]:

  "sint (number_of bin :: 'a word) =

    sbintrunc (len_of TYPE ('a :: len) - 1) (number_of bin)"

  unfolding word_number_of_alt by (rule word_sbin.eq_norm)

lemma unat_bintrunc [simp]:

  "unat (number_of bin :: 'a :: len0 word) =

    nat (bintrunc (len_of TYPE('a)) (number_of bin))"

  unfolding unat_def nat_number_of_def 

  by (simp only: uint_bintrunc)

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 number_of_eq)

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_number_of:

  "uint (number_of b :: 'a :: len0 word) = number_of b mod 2 ^ len_of TYPE('a)"

  unfolding word_number_of_alt

  by (simp only: int_word_uint)

lemma unat_number_of: 

  "bin_sign (number_of b) = 0 \<Longrightarrow> 

  unat (number_of b::'a::len0 word) = number_of b mod 2 ^ len_of TYPE ('a)"

  apply (unfold unat_def)

  apply (clarsimp simp only: uint_number_of)

  apply (rule nat_mod_distrib [THEN trans])

    apply (erule sign_Pls_ge_0 [THEN iffD1])

   apply (simp_all add: nat_power_eq)

  done

lemma sint_number_of: "sint (number_of b :: 'a :: len word) = (number_of b + 

    2 ^ (len_of TYPE('a) - 1)) mod 2 ^ len_of TYPE('a) -

    2 ^ (len_of TYPE('a) - 1)"

  unfolding word_number_of_alt by (rule int_word_sint)

lemma word_of_int_0 [simp]: "word_of_int 0 = 0"

  unfolding word_0_wi ..

lemma word_of_int_1 [simp]: "word_of_int 1 = 1"

  unfolding word_1_wi ..

lemma word_of_int_bin [simp] : 

  "(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)"

  unfolding word_number_of_alt ..

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 auto

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: 

  "(number_of bin ::'a::len0 word) = of_bl (bin_to_bl (len_of TYPE ('a)) bin)"

  unfolding word_size of_bl_def by (simp add: word_number_of_def)

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_no_bin [simp]:

  "to_bl (number_of bin::'a::len0 word) = bin_to_bl (len_of TYPE('a)) (number_of bin)"

  unfolding word_number_of_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 "number_of a" "number_of b", symmetric, folded word_number_of_alt] for a b

lemmas sbintr_num = word_sbin.norm_eq_iff

  [of "number_of a" "number_of b", symmetric, folded word_number_of_alt] for a b

lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def]

lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def]

(* don't add these to simpset, since may want bintrunc n w to be simplified;

  may want these in reverse, but loop as simp rules, so use following *)

lemma num_of_bintr':

  "bintrunc (len_of TYPE('a :: len0)) (number_of a) = (number_of b) \<Longrightarrow> 

    number_of a = (number_of b :: 'a word)"

  unfolding bintr_num by (erule subst, simp)

lemma num_of_sbintr':

  "sbintrunc (len_of TYPE('a :: len) - 1) (number_of a) = (number_of b) \<Longrightarrow> 

    number_of a = (number_of b :: 'a word)"

  unfolding sbintr_num by (erule subst, simp)

lemma num_abs_bintr:

  "(number_of x :: 'a word) =

    word_of_int (bintrunc (len_of TYPE('a::len0)) (number_of x))"

  by (simp only: word_ubin.Abs_norm word_number_of_alt)

lemma num_abs_sbintr:

  "(number_of x :: 'a word) =

    word_of_int (sbintrunc (len_of TYPE('a::len) - 1) (number_of x))"

  by (simp only: word_sbin.Abs_norm word_number_of_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 (number_of w ::'a::len0 word) = 

   word_of_int (bintrunc (len_of TYPE('a)) (number_of w))"

  unfolding ucast_def by simp

lemma scast_sbintr [simp]:

  "scast (number_of w ::'a::len word) = 

   word_of_int (sbintrunc (len_of TYPE('a) - Suc 0) (number_of 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>