src/HOL/Word/WordDefinition.thy

-(* 
-  Author: Jeremy Dawson and Gerwin Klein, NICTA
-  
-  Basic definition of word type and basic theorems following from 
-  the definition of the word type 
-*) 
-
-header {* Definition of Word Type *}
-
-theory WordDefinition
-imports Type_Length Misc_Typedef Bool_List_Representation
-begin
-
-subsection {* Type definition *}
-
-typedef (open word) '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
-
-notation fcomp (infixl "o>" 60)
-notation scomp (infixl "o\<rightarrow>" 60)
-
-instantiation word :: ("{len0, typerep}") random
-begin
-
-definition
-  "random_word i = Random.range (max i (2 ^ len_of TYPE('a))) o\<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 "o>" 60)
-no_notation scomp (infixl "o\<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)"
-
-syntax
-  of_int :: "int => 'a"
-translations
-  "case x of CONST of_int y => b" == "CONST word_int_case (%y. b) x"
-
-
-subsection  "Arithmetic operations"
-
-instantiation word :: (len0) "{number, uminus, minus, plus, one, zero, times, Divides.div, ord, bit}"
-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"
-
-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)"
-
-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))"
-
-instance ..
-
-end
-
-definition
-  word_succ :: "'a :: len0 word => 'a word"
-where
-  "word_succ a = word_of_int (Int.succ (uint a))"
-
-definition
-  word_pred :: "'a :: len0 word => 'a word"
-where
-  "word_pred a = word_of_int (Int.pred (uint a))"
-
-definition udvd :: "'a::len word => 'a::len word => bool" (infixl "udvd" 50) where
-  "a udvd b == EX n>=0. uint b = n * uint a"
-
-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_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) = Int.Min"
-
-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 \<equiv> 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"
-
-
-lemmas of_nth_def = word_set_bits_def
-
-lemmas word_size_gt_0 [iff] = 
-  xtr1 [OF word_size len_gt_0, standard]
-lemmas lens_gt_0 = word_size_gt_0 len_gt_0
-lemmas lens_not_0 [iff] = lens_gt_0 [THEN gr_implies_not0, standard]
-
-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)
-
-lemmas atLeastLessThan_alt = atLeastLessThan_def [unfolded 
-  atLeast_def lessThan_def Collect_conj_eq [symmetric]]
-  
-lemma mod_in_reps: "m > 0 ==> y mod m : {0::int ..< m}"
-  unfolding atLeastLessThan_alt by auto
-
-lemma 
-  uint_0:"0 <= uint x" and 
-  uint_lt: "uint (x::'a::len0 word) < 2 ^ len_of TYPE('a)"
-  by (auto simp: uint [simplified])
-
-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
-
-lemmas int_word_uint = td_ext_uint [THEN td_ext.eq_norm, standard]
-
-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 td_ext_ubin = td_ext_uint 
-  [simplified 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 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': 
-  "n >= size w ==> bintrunc n (uint w) = uint w"
-  apply (unfold word_size)
-  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': 
-  "wb = word_of_int bin ==> n >= size wb ==> 
-    word_of_int (bintrunc n bin) = wb"
-  unfolding word_size
-  by (clarsimp simp add: word_ubin.norm_eq_iff [symmetric] min_max.inf_absorb1)
-
-lemmas bintr_uint = bintr_uint' [unfolded word_size]
-lemmas wi_bintr = wi_bintr' [unfolded word_size]
-
-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, standard]
-
-lemmas td_sint = word_sint.td
-
-lemma word_number_of_alt: "number_of b == word_of_int (number_of b)"
-  unfolding word_number_of_def by (simp add: number_of_eq)
-
-lemma word_no_wi: "number_of = word_of_int"
-  by (auto simp: word_number_of_def intro: ext)
-
-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 intro: ext)
-
-lemmas word_reverse_no_def [simp] = word_reverse_def [of "number_of w", standard]
-
-lemmas uints_mod = uints_def [unfolded no_bintr_alt1]
-
-lemma uint_bintrunc: "uint (number_of bin :: 'a word) = 
-    number_of (bintrunc (len_of TYPE ('a :: len0)) bin)"
-  unfolding word_number_of_def number_of_eq
-  by (auto intro: word_ubin.eq_norm) 
-
-lemma sint_sbintrunc: "sint (number_of bin :: 'a word) = 
-    number_of (sbintrunc (len_of TYPE ('a :: len) - 1) bin)" 
-  unfolding word_number_of_def number_of_eq
-  by (subst word_sbin.eq_norm) simp
-
-lemma unat_bintrunc: 
-  "unat (number_of bin :: 'a :: len0 word) =
-    number_of (bintrunc (len_of TYPE('a)) bin)"
-  unfolding unat_def nat_number_of_def 
-  by (simp only: uint_bintrunc)
-
-(* WARNING - these may not always be helpful *)
-declare 
-  uint_bintrunc [simp] 
-  sint_sbintrunc [simp] 
-  unat_bintrunc [simp]
-
-lemma size_0_eq: "size (w :: 'a :: len0 word) = 0 ==> 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
-
-lemmas uint_lem = word_uint.Rep [unfolded uints_num mem_Collect_eq]
-lemmas sint_lem = word_sint.Rep [unfolded sints_num mem_Collect_eq]
-lemmas uint_ge_0 [iff] = uint_lem [THEN conjunct1, standard]
-lemmas uint_lt2p [iff] = uint_lem [THEN conjunct2, standard]
-lemmas sint_ge = sint_lem [THEN conjunct1, standard]
-lemmas sint_lt = sint_lem [THEN conjunct2, standard]
-
-lemma sign_uint_Pls [simp]: 
-  "bin_sign (uint x) = Int.Pls"
-  by (simp add: sign_Pls_ge_0 number_of_eq)
-
-lemmas uint_m2p_neg = iffD2 [OF diff_less_0_iff_less uint_lt2p, standard]
-lemmas uint_m2p_not_non_neg = 
-  iffD2 [OF linorder_not_le uint_m2p_neg, standard]
-
-lemma lt2p_lem:
-  "len_of TYPE('a) <= n ==> uint (w :: 'a :: len0 word) < 2 ^ n"
-  by (rule xtr8 [OF _ uint_lt2p]) simp
-
-lemmas uint_le_0_iff [simp] = 
-  uint_ge_0 [THEN leD, THEN linorder_antisym_conv1, standard]
-
-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 b = Int.Pls ==> 
-  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_bin [simp] : 
-  "(word_of_int (number_of bin) :: 'a :: len0 word) = (number_of bin)"
-  unfolding word_number_of_alt by auto
-
-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, standard]
-lemmas sint_range' = word_sint.Rep [unfolded One_nat_def
-  sints_num mem_Collect_eq, standard]
-
-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')
-
-lemmas sint_above_size = sint_range_size
-  [THEN conjunct2, THEN [2] xtr8, folded One_nat_def, standard]
-
-lemmas sint_below_size = sint_range_size
-  [THEN conjunct1, THEN [2] order_trans, folded One_nat_def, standard]
-
-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_eqI [rule_format] : 
-  fixes u :: "'a::len0 word"
-  shows "(ALL n. n < size u --> u !! n = v !! n) ==> u = v"
-  apply (rule test_bit_eq_iff [THEN iffD1])
-  apply (rule ext)
-  apply (erule allE)
-  apply (erule impCE)
-   prefer 2
-   apply assumption
-  apply (auto dest!: test_bit_size simp add: word_size)
-  done
-
-lemmas word_eqD = test_bit_eq_iff [THEN iffD2, THEN fun_cong, standard]
-
-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) n --> n < size w"
-  apply (unfold word_size)
-  apply (rule impI)
-  apply (rule nth_bintr [THEN iffD1, THEN conjunct1])
-  apply (subst word_ubin.norm_Rep)
-  apply assumption
-  done
-
-lemma bin_nth_sint': 
-  "n >= size w --> bin_nth (sint w) n = bin_nth (sint w) (size w - 1)"
-  apply (rule impI)
-  apply (subst word_sbin.norm_Rep [symmetric])
-  apply (simp add : nth_sbintr word_size)
-  apply auto
-  done
-
-lemmas bin_nth_uint_imp = bin_nth_uint_imp' [rule_format, unfolded word_size]
-lemmas bin_nth_sint = bin_nth_sint' [rule_format, unfolded word_size]
-
-(* 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)
-
-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 (fastsimp elim!: word_bl.Abs_inverse [simplified])
-
-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 ==> word_reverse u = w"
-  by auto
-
-lemmas word_rev_gal' = sym [THEN word_rev_gal, symmetric, standard]
-
-lemmas length_bl_gt_0 [iff] = xtr1 [OF word_bl.Rep' len_gt_0, standard]
-lemmas bl_not_Nil [iff] = 
-  length_bl_gt_0 [THEN length_greater_0_conv [THEN iffD1], standard]
-lemmas length_bl_neq_0 [iff] = length_bl_gt_0 [THEN gr_implies_not0]
-
-lemma hd_bl_sign_sint: "hd (to_bl w) = (bin_sign (sint w) = Int.Min)"
-  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) ==> 
-    of_bl (drop lend bl) = (of_bl bl :: 'a word)"
-  apply (unfold of_bl_def)
-  apply (clarsimp simp add : trunc_bl2bin [symmetric])
-  done
-
-lemmas of_bl_no = of_bl_def [folded word_number_of_def]
-
-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_no 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)
-
-lemmas to_bl_no_bin [simp] = to_bl_of_bin [folded word_number_of_def]
-
-lemma to_bl_to_bin [simp] : "bl_to_bin (to_bl w) = uint w"
-  unfolding uint_bl by (simp add : word_size)
-  
-lemmas uint_bl_bin [simp] = trans [OF bin_bl_bin word_ubin.norm_Rep, standard]
-
-lemmas num_AB_u [simp] = word_uint.Rep_inverse 
-  [unfolded o_def word_number_of_def [symmetric], standard]
-lemmas num_AB_s [simp] = word_sint.Rep_inverse 
-  [unfolded o_def word_number_of_def [symmetric], standard]
-
-(* 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 
-  [symmetric, folded word_number_of_def, standard]
-lemmas sbintr_num = word_sbin.norm_eq_iff 
-  [symmetric, folded word_number_of_def, standard]
-
-lemmas num_of_bintr = word_ubin.Abs_norm [folded word_number_of_def, standard]
-lemmas num_of_sbintr = word_sbin.Abs_norm [folded word_number_of_def, standard];
-    
-(* 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)) a = b ==> 
-    number_of a = (number_of b :: 'a word)"
-  apply safe
-  apply (rule_tac num_of_bintr [symmetric])
-  done
-
-lemma num_of_sbintr':
-  "sbintrunc (len_of TYPE('a :: len) - 1) a = b ==> 
-    number_of a = (number_of b :: 'a word)"
-  apply safe
-  apply (rule_tac num_of_sbintr [symmetric])
-  done
-
-lemmas num_abs_bintr = sym [THEN trans,
-  OF num_of_bintr word_number_of_def, standard]
-lemmas num_abs_sbintr = sym [THEN trans,
-  OF num_of_sbintr word_number_of_def, standard]
-  
-(** 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) = 
-   number_of (bintrunc (len_of TYPE('a)) w)"
-  unfolding ucast_def by simp
-
-lemma scast_sbintr [simp]: 
-  "scast (number_of w ::'a::len word) = 
-   number_of (sbintrunc (len_of TYPE('a) - Suc 0) w)"
-  unfolding scast_def by simp
-
-lemmas source_size = source_size_def [unfolded Let_def word_size]
-lemmas target_size = target_size_def [unfolded Let_def word_size]
-lemmas is_down = is_down_def [unfolded source_size target_size]
-lemmas is_up = is_up_def [unfolded source_size target_size]
-
-lemmas is_up_down =  trans [OF is_up is_down [symmetric], standard]
-
-lemma down_cast_same': "uc = ucast ==> is_down uc ==> 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': 
-  "r = to_bl (of_bl bl) ==> r = rev (takefill False (length r) (rev bl))"
-  unfolding of_bl_def uint_bl
-  by (clarsimp simp add: bl_bin_bl_rtf word_ubin.eq_norm word_size)
-
-lemmas word_rev_tf = refl [THEN word_rev_tf', unfolded word_bl.Rep', standard]
-
-lemmas word_rep_drop = word_rev_tf [simplified takefill_alt,
-  simplified, simplified rev_take, simplified]
-
-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': 
-  "uc = ucast ==> source_size uc + n = target_size uc ==> 
-    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': 
-  "uc = ucast ==> source_size uc = target_size uc + n ==> 
-    to_bl (uc w) = drop n (to_bl w)"
-  by (auto simp add : source_size target_size to_bl_ucast)
-
-lemma scast_down_drop': 
-  "sc = scast ==> source_size sc = target_size sc + n ==> 
-    to_bl (sc w) = drop n (to_bl w)"
-  apply (subgoal_tac "sc = ucast")
-   apply safe
-   apply simp
-   apply (erule refl [THEN ucast_down_drop'])
-  apply (rule refl [THEN down_cast_same', symmetric])
-  apply (simp add : source_size target_size is_down)
-  done
-
-lemma sint_up_scast': 
-  "sc = scast ==> is_up sc ==> 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':
-  "uc = ucast ==> is_up uc ==> 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
-    
-lemmas down_cast_same = refl [THEN down_cast_same']
-lemmas ucast_up_app = refl [THEN ucast_up_app']
-lemmas ucast_down_drop = refl [THEN ucast_down_drop']
-lemmas scast_down_drop = refl [THEN scast_down_drop']
-lemmas uint_up_ucast = refl [THEN uint_up_ucast']
-lemmas sint_up_scast = refl [THEN sint_up_scast']
-
-lemma ucast_up_ucast': "uc = ucast ==> is_up uc ==> ucast (uc w) = ucast w"
-  apply (simp (no_asm) add: ucast_def)
-  apply (clarsimp simp add: uint_up_ucast)
-  done
-    
-lemma scast_up_scast': "sc = scast ==> is_up sc ==> 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': 
-  "w = of_bl bl ==> size bl <= size w ==> ucast w = of_bl bl"
-  by (auto simp add : nth_ucast word_size test_bit_of_bl intro!: word_eqI)
-
-lemmas ucast_up_ucast = refl [THEN ucast_up_ucast']
-lemmas scast_up_scast = refl [THEN scast_up_scast']
-lemmas ucast_of_bl_up = refl [THEN ucast_of_bl_up']
-
-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) ==> 
-   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) ==> 
-   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) ==> 
-   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) ==> 
-   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_no': 
-  "uc = ucast ==> is_down uc ==> uc (number_of bin) = number_of bin"
-  apply (unfold word_number_of_def 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
-    
-lemmas ucast_down_no = ucast_down_no' [OF refl]
-
-lemma ucast_down_bl': "uc = ucast ==> is_down uc ==> uc (of_bl bl) = of_bl bl"
-  unfolding of_bl_no by clarify (erule ucast_down_no)
-    
-lemmas ucast_down_bl = ucast_down_bl' [OF refl]
-
-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
-lemmas word_log_bin_defs = word_log_defs
-
-text {* Executable equality *}
-
-instantiation word :: ("{len0}") eq
-begin
-
-definition eq_word :: "'a word \<Rightarrow> 'a word \<Rightarrow> bool" where
-  "eq_word k l \<longleftrightarrow> HOL.eq (uint k) (uint l)"
-
-instance proof
-qed (simp add: eq eq_word_def)
-
-end
-
-end