src/HOL/Code_Numeral.thy
 author haftmann Thu Feb 23 21:25:59 2012 +0100 (2012-02-23) changeset 46635 cde737f9c911 parent 46547 d1dcb91a512e child 46638 fc315796794e permissions -rw-r--r--
moved predicate relations and conversion rules between set and predicate relations from Predicate.thy to Relation.thy; moved Predicate.thy upwards in theory hierarchy
```     1 (* Author: Florian Haftmann, TU Muenchen *)
```
```     2
```
```     3 header {* Type of target language numerals *}
```
```     4
```
```     5 theory Code_Numeral
```
```     6 imports Nat_Numeral Nat_Transfer Divides
```
```     7 begin
```
```     8
```
```     9 text {*
```
```    10   Code numerals are isomorphic to HOL @{typ nat} but
```
```    11   mapped to target-language builtin numerals.
```
```    12 *}
```
```    13
```
```    14 subsection {* Datatype of target language numerals *}
```
```    15
```
```    16 typedef (open) code_numeral = "UNIV \<Colon> nat set"
```
```    17   morphisms nat_of of_nat ..
```
```    18
```
```    19 lemma of_nat_nat_of [simp]:
```
```    20   "of_nat (nat_of k) = k"
```
```    21   by (rule nat_of_inverse)
```
```    22
```
```    23 lemma nat_of_of_nat [simp]:
```
```    24   "nat_of (of_nat n) = n"
```
```    25   by (rule of_nat_inverse) (rule UNIV_I)
```
```    26
```
```    27 lemma [measure_function]:
```
```    28   "is_measure nat_of" by (rule is_measure_trivial)
```
```    29
```
```    30 lemma code_numeral:
```
```    31   "(\<And>n\<Colon>code_numeral. PROP P n) \<equiv> (\<And>n\<Colon>nat. PROP P (of_nat n))"
```
```    32 proof
```
```    33   fix n :: nat
```
```    34   assume "\<And>n\<Colon>code_numeral. PROP P n"
```
```    35   then show "PROP P (of_nat n)" .
```
```    36 next
```
```    37   fix n :: code_numeral
```
```    38   assume "\<And>n\<Colon>nat. PROP P (of_nat n)"
```
```    39   then have "PROP P (of_nat (nat_of n))" .
```
```    40   then show "PROP P n" by simp
```
```    41 qed
```
```    42
```
```    43 lemma code_numeral_case:
```
```    44   assumes "\<And>n. k = of_nat n \<Longrightarrow> P"
```
```    45   shows P
```
```    46   by (rule assms [of "nat_of k"]) simp
```
```    47
```
```    48 lemma code_numeral_induct_raw:
```
```    49   assumes "\<And>n. P (of_nat n)"
```
```    50   shows "P k"
```
```    51 proof -
```
```    52   from assms have "P (of_nat (nat_of k))" .
```
```    53   then show ?thesis by simp
```
```    54 qed
```
```    55
```
```    56 lemma nat_of_inject [simp]:
```
```    57   "nat_of k = nat_of l \<longleftrightarrow> k = l"
```
```    58   by (rule nat_of_inject)
```
```    59
```
```    60 lemma of_nat_inject [simp]:
```
```    61   "of_nat n = of_nat m \<longleftrightarrow> n = m"
```
```    62   by (rule of_nat_inject) (rule UNIV_I)+
```
```    63
```
```    64 instantiation code_numeral :: zero
```
```    65 begin
```
```    66
```
```    67 definition [simp, code del]:
```
```    68   "0 = of_nat 0"
```
```    69
```
```    70 instance ..
```
```    71
```
```    72 end
```
```    73
```
```    74 definition Suc where [simp]:
```
```    75   "Suc k = of_nat (Nat.Suc (nat_of k))"
```
```    76
```
```    77 rep_datatype "0 \<Colon> code_numeral" Suc
```
```    78 proof -
```
```    79   fix P :: "code_numeral \<Rightarrow> bool"
```
```    80   fix k :: code_numeral
```
```    81   assume "P 0" then have init: "P (of_nat 0)" by simp
```
```    82   assume "\<And>k. P k \<Longrightarrow> P (Suc k)"
```
```    83     then have "\<And>n. P (of_nat n) \<Longrightarrow> P (Suc (of_nat n))" .
```
```    84     then have step: "\<And>n. P (of_nat n) \<Longrightarrow> P (of_nat (Nat.Suc n))" by simp
```
```    85   from init step have "P (of_nat (nat_of k))"
```
```    86     by (induct ("nat_of k")) simp_all
```
```    87   then show "P k" by simp
```
```    88 qed simp_all
```
```    89
```
```    90 declare code_numeral_case [case_names nat, cases type: code_numeral]
```
```    91 declare code_numeral.induct [case_names nat, induct type: code_numeral]
```
```    92
```
```    93 lemma code_numeral_decr [termination_simp]:
```
```    94   "k \<noteq> of_nat 0 \<Longrightarrow> nat_of k - Nat.Suc 0 < nat_of k"
```
```    95   by (cases k) simp
```
```    96
```
```    97 lemma [simp, code]:
```
```    98   "code_numeral_size = nat_of"
```
```    99 proof (rule ext)
```
```   100   fix k
```
```   101   have "code_numeral_size k = nat_size (nat_of k)"
```
```   102     by (induct k rule: code_numeral.induct) (simp_all del: zero_code_numeral_def Suc_def, simp_all)
```
```   103   also have "nat_size (nat_of k) = nat_of k" by (induct ("nat_of k")) simp_all
```
```   104   finally show "code_numeral_size k = nat_of k" .
```
```   105 qed
```
```   106
```
```   107 lemma [simp, code]:
```
```   108   "size = nat_of"
```
```   109 proof (rule ext)
```
```   110   fix k
```
```   111   show "size k = nat_of k"
```
```   112   by (induct k) (simp_all del: zero_code_numeral_def Suc_def, simp_all)
```
```   113 qed
```
```   114
```
```   115 lemmas [code del] = code_numeral.recs code_numeral.cases
```
```   116
```
```   117 lemma [code]:
```
```   118   "HOL.equal k l \<longleftrightarrow> HOL.equal (nat_of k) (nat_of l)"
```
```   119   by (cases k, cases l) (simp add: equal)
```
```   120
```
```   121 lemma [code nbe]:
```
```   122   "HOL.equal (k::code_numeral) k \<longleftrightarrow> True"
```
```   123   by (rule equal_refl)
```
```   124
```
```   125
```
```   126 subsection {* Code numerals as datatype of ints *}
```
```   127
```
```   128 instantiation code_numeral :: number
```
```   129 begin
```
```   130
```
```   131 definition
```
```   132   "number_of = of_nat o nat"
```
```   133
```
```   134 instance ..
```
```   135
```
```   136 end
```
```   137
```
```   138 lemma nat_of_number [simp]:
```
```   139   "nat_of (number_of k) = number_of k"
```
```   140   by (simp add: number_of_code_numeral_def nat_number_of_def number_of_is_id)
```
```   141
```
```   142 code_datatype "number_of \<Colon> int \<Rightarrow> code_numeral"
```
```   143
```
```   144
```
```   145 subsection {* Basic arithmetic *}
```
```   146
```
```   147 instantiation code_numeral :: "{minus, linordered_semidom, semiring_div, linorder}"
```
```   148 begin
```
```   149
```
```   150 definition [simp, code del]:
```
```   151   "(1\<Colon>code_numeral) = of_nat 1"
```
```   152
```
```   153 definition [simp, code del]:
```
```   154   "n + m = of_nat (nat_of n + nat_of m)"
```
```   155
```
```   156 definition [simp, code del]:
```
```   157   "n - m = of_nat (nat_of n - nat_of m)"
```
```   158
```
```   159 definition [simp, code del]:
```
```   160   "n * m = of_nat (nat_of n * nat_of m)"
```
```   161
```
```   162 definition [simp, code del]:
```
```   163   "n div m = of_nat (nat_of n div nat_of m)"
```
```   164
```
```   165 definition [simp, code del]:
```
```   166   "n mod m = of_nat (nat_of n mod nat_of m)"
```
```   167
```
```   168 definition [simp, code del]:
```
```   169   "n \<le> m \<longleftrightarrow> nat_of n \<le> nat_of m"
```
```   170
```
```   171 definition [simp, code del]:
```
```   172   "n < m \<longleftrightarrow> nat_of n < nat_of m"
```
```   173
```
```   174 instance proof
```
```   175 qed (auto simp add: code_numeral left_distrib intro: mult_commute)
```
```   176
```
```   177 end
```
```   178
```
```   179 lemma zero_code_numeral_code [code]:
```
```   180   "(0\<Colon>code_numeral) = Numeral0"
```
```   181   by (simp add: number_of_code_numeral_def Pls_def)
```
```   182
```
```   183 lemma [code_abbrev]: "Numeral0 = (0\<Colon>code_numeral)"
```
```   184   using zero_code_numeral_code ..
```
```   185
```
```   186 lemma one_code_numeral_code [code]:
```
```   187   "(1\<Colon>code_numeral) = Numeral1"
```
```   188   by (simp add: number_of_code_numeral_def Pls_def Bit1_def)
```
```   189
```
```   190 lemma [code_abbrev]: "Numeral1 = (1\<Colon>code_numeral)"
```
```   191   using one_code_numeral_code ..
```
```   192
```
```   193 lemma plus_code_numeral_code [code nbe]:
```
```   194   "of_nat n + of_nat m = of_nat (n + m)"
```
```   195   by simp
```
```   196
```
```   197 definition subtract :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
```
```   198   [simp]: "subtract = minus"
```
```   199
```
```   200 lemma subtract_code [code nbe]:
```
```   201   "subtract (of_nat n) (of_nat m) = of_nat (n - m)"
```
```   202   by simp
```
```   203
```
```   204 lemma minus_code_numeral_code [code]:
```
```   205   "minus = subtract"
```
```   206   by simp
```
```   207
```
```   208 lemma times_code_numeral_code [code nbe]:
```
```   209   "of_nat n * of_nat m = of_nat (n * m)"
```
```   210   by simp
```
```   211
```
```   212 lemma less_eq_code_numeral_code [code nbe]:
```
```   213   "of_nat n \<le> of_nat m \<longleftrightarrow> n \<le> m"
```
```   214   by simp
```
```   215
```
```   216 lemma less_code_numeral_code [code nbe]:
```
```   217   "of_nat n < of_nat m \<longleftrightarrow> n < m"
```
```   218   by simp
```
```   219
```
```   220 lemma code_numeral_zero_minus_one:
```
```   221   "(0::code_numeral) - 1 = 0"
```
```   222   by simp
```
```   223
```
```   224 lemma Suc_code_numeral_minus_one:
```
```   225   "Suc n - 1 = n"
```
```   226   by simp
```
```   227
```
```   228 lemma of_nat_code [code]:
```
```   229   "of_nat = Nat.of_nat"
```
```   230 proof
```
```   231   fix n :: nat
```
```   232   have "Nat.of_nat n = of_nat n"
```
```   233     by (induct n) simp_all
```
```   234   then show "of_nat n = Nat.of_nat n"
```
```   235     by (rule sym)
```
```   236 qed
```
```   237
```
```   238 lemma code_numeral_not_eq_zero: "i \<noteq> of_nat 0 \<longleftrightarrow> i \<ge> 1"
```
```   239   by (cases i) auto
```
```   240
```
```   241 definition nat_of_aux :: "code_numeral \<Rightarrow> nat \<Rightarrow> nat" where
```
```   242   "nat_of_aux i n = nat_of i + n"
```
```   243
```
```   244 lemma nat_of_aux_code [code]:
```
```   245   "nat_of_aux i n = (if i = 0 then n else nat_of_aux (i - 1) (Nat.Suc n))"
```
```   246   by (auto simp add: nat_of_aux_def code_numeral_not_eq_zero)
```
```   247
```
```   248 lemma nat_of_code [code]:
```
```   249   "nat_of i = nat_of_aux i 0"
```
```   250   by (simp add: nat_of_aux_def)
```
```   251
```
```   252 definition div_mod :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<times> code_numeral" where
```
```   253   [code del]: "div_mod n m = (n div m, n mod m)"
```
```   254
```
```   255 lemma [code]:
```
```   256   "div_mod n m = (if m = 0 then (0, n) else (n div m, n mod m))"
```
```   257   unfolding div_mod_def by auto
```
```   258
```
```   259 lemma [code]:
```
```   260   "n div m = fst (div_mod n m)"
```
```   261   unfolding div_mod_def by simp
```
```   262
```
```   263 lemma [code]:
```
```   264   "n mod m = snd (div_mod n m)"
```
```   265   unfolding div_mod_def by simp
```
```   266
```
```   267 definition int_of :: "code_numeral \<Rightarrow> int" where
```
```   268   "int_of = Nat.of_nat o nat_of"
```
```   269
```
```   270 lemma int_of_code [code]:
```
```   271   "int_of k = (if k = 0 then 0
```
```   272     else (if k mod 2 = 0 then 2 * int_of (k div 2) else 2 * int_of (k div 2) + 1))"
```
```   273 proof -
```
```   274   have "(nat_of k div 2) * 2 + nat_of k mod 2 = nat_of k"
```
```   275     by (rule mod_div_equality)
```
```   276   then have "int ((nat_of k div 2) * 2 + nat_of k mod 2) = int (nat_of k)"
```
```   277     by simp
```
```   278   then have "int (nat_of k) = int (nat_of k div 2) * 2 + int (nat_of k mod 2)"
```
```   279     unfolding of_nat_mult of_nat_add by simp
```
```   280   then show ?thesis by (auto simp add: int_of_def mult_ac)
```
```   281 qed
```
```   282
```
```   283
```
```   284 hide_const (open) of_nat nat_of Suc subtract int_of
```
```   285
```
```   286
```
```   287 subsection {* Code generator setup *}
```
```   288
```
```   289 text {* Implementation of code numerals by bounded integers *}
```
```   290
```
```   291 code_type code_numeral
```
```   292   (SML "int")
```
```   293   (OCaml "Big'_int.big'_int")
```
```   294   (Haskell "Integer")
```
```   295   (Scala "BigInt")
```
```   296
```
```   297 code_instance code_numeral :: equal
```
```   298   (Haskell -)
```
```   299
```
```   300 setup {*
```
```   301   Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}
```
```   302     false Code_Printer.literal_naive_numeral "SML"
```
```   303   #> fold (Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral}
```
```   304     false Code_Printer.literal_numeral) ["OCaml", "Haskell", "Scala"]
```
```   305 *}
```
```   306
```
```   307 code_reserved SML Int int
```
```   308 code_reserved Eval Integer
```
```   309
```
```   310 code_const "plus \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
```
```   311   (SML "Int.+/ ((_),/ (_))")
```
```   312   (OCaml "Big'_int.add'_big'_int")
```
```   313   (Haskell infixl 6 "+")
```
```   314   (Scala infixl 7 "+")
```
```   315   (Eval infixl 8 "+")
```
```   316
```
```   317 code_const "Code_Numeral.subtract \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
```
```   318   (SML "Int.max/ (_/ -/ _,/ 0 : int)")
```
```   319   (OCaml "Big'_int.max'_big'_int/ (Big'_int.sub'_big'_int/ _/ _)/ Big'_int.zero'_big'_int")
```
```   320   (Haskell "max/ (_/ -/ _)/ (0 :: Integer)")
```
```   321   (Scala "!(_/ -/ _).max(0)")
```
```   322   (Eval "Integer.max/ (_/ -/ _)/ 0")
```
```   323
```
```   324 code_const "times \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
```
```   325   (SML "Int.*/ ((_),/ (_))")
```
```   326   (OCaml "Big'_int.mult'_big'_int")
```
```   327   (Haskell infixl 7 "*")
```
```   328   (Scala infixl 8 "*")
```
```   329   (Eval infixl 8 "*")
```
```   330
```
```   331 code_const Code_Numeral.div_mod
```
```   332   (SML "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Int.div (n, m), Int.mod (n, m)))")
```
```   333   (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
```
```   334   (Haskell "divMod")
```
```   335   (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
```
```   336   (Eval "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Integer.div'_mod n m))")
```
```   337
```
```   338 code_const "HOL.equal \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
```
```   339   (SML "!((_ : Int.int) = _)")
```
```   340   (OCaml "Big'_int.eq'_big'_int")
```
```   341   (Haskell infix 4 "==")
```
```   342   (Scala infixl 5 "==")
```
```   343   (Eval "!((_ : int) = _)")
```
```   344
```
```   345 code_const "less_eq \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
```
```   346   (SML "Int.<=/ ((_),/ (_))")
```
```   347   (OCaml "Big'_int.le'_big'_int")
```
```   348   (Haskell infix 4 "<=")
```
```   349   (Scala infixl 4 "<=")
```
```   350   (Eval infixl 6 "<=")
```
```   351
```
```   352 code_const "less \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
```
```   353   (SML "Int.</ ((_),/ (_))")
```
```   354   (OCaml "Big'_int.lt'_big'_int")
```
```   355   (Haskell infix 4 "<")
```
```   356   (Scala infixl 4 "<")
```
```   357   (Eval infixl 6 "<")
```
```   358
```
```   359 code_modulename SML
```
```   360   Code_Numeral Arith
```
```   361
```
```   362 code_modulename OCaml
```
```   363   Code_Numeral Arith
```
```   364
```
```   365 code_modulename Haskell
```
```   366   Code_Numeral Arith
```
```   367
```
```   368 end
```
```   369
```