restrict unqualified imports from Haskell Prelude to a small set of fundamental operations
1 (* Author: Florian Haftmann, TU Muenchen *)
3 header {* Type of target language numerals *}
6 imports Nat_Transfer Divides
10 Code numerals are isomorphic to HOL @{typ nat} but
11 mapped to target-language builtin numerals.
14 subsection {* Datatype of target language numerals *}
16 typedef (open) code_numeral = "UNIV \<Colon> nat set"
17 morphisms nat_of of_nat ..
19 lemma of_nat_nat_of [simp]:
20 "of_nat (nat_of k) = k"
21 by (rule nat_of_inverse)
23 lemma nat_of_of_nat [simp]:
24 "nat_of (of_nat n) = n"
25 by (rule of_nat_inverse) (rule UNIV_I)
27 lemma [measure_function]:
28 "is_measure nat_of" by (rule is_measure_trivial)
31 "(\<And>n\<Colon>code_numeral. PROP P n) \<equiv> (\<And>n\<Colon>nat. PROP P (of_nat n))"
34 assume "\<And>n\<Colon>code_numeral. PROP P n"
35 then show "PROP P (of_nat n)" .
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
43 lemma code_numeral_case:
44 assumes "\<And>n. k = of_nat n \<Longrightarrow> P"
46 by (rule assms [of "nat_of k"]) simp
48 lemma code_numeral_induct_raw:
49 assumes "\<And>n. P (of_nat n)"
52 from assms have "P (of_nat (nat_of k))" .
53 then show ?thesis by simp
56 lemma nat_of_inject [simp]:
57 "nat_of k = nat_of l \<longleftrightarrow> k = l"
58 by (rule nat_of_inject)
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)+
64 instantiation code_numeral :: zero
67 definition [simp, code del]:
74 definition Suc where [simp]:
75 "Suc k = of_nat (Nat.Suc (nat_of k))"
77 rep_datatype "0 \<Colon> code_numeral" Suc
79 fix P :: "code_numeral \<Rightarrow> bool"
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
90 declare code_numeral_case [case_names nat, cases type: code_numeral]
91 declare code_numeral.induct [case_names nat, induct type: code_numeral]
93 lemma code_numeral_decr [termination_simp]:
94 "k \<noteq> of_nat 0 \<Longrightarrow> nat_of k - Nat.Suc 0 < nat_of k"
98 "code_numeral_size = nat_of"
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" .
111 show "size k = nat_of k"
112 by (induct k) (simp_all del: zero_code_numeral_def Suc_def, simp_all)
115 lemmas [code del] = code_numeral.recs code_numeral.cases
118 "HOL.equal k l \<longleftrightarrow> HOL.equal (nat_of k) (nat_of l)"
119 by (cases k, cases l) (simp add: equal)
122 "HOL.equal (k::code_numeral) k \<longleftrightarrow> True"
126 subsection {* Basic arithmetic *}
128 instantiation code_numeral :: "{minus, linordered_semidom, semiring_div, linorder}"
131 definition [simp, code del]:
132 "(1\<Colon>code_numeral) = of_nat 1"
134 definition [simp, code del]:
135 "n + m = of_nat (nat_of n + nat_of m)"
137 definition [simp, code del]:
138 "n - m = of_nat (nat_of n - nat_of m)"
140 definition [simp, code del]:
141 "n * m = of_nat (nat_of n * nat_of m)"
143 definition [simp, code del]:
144 "n div m = of_nat (nat_of n div nat_of m)"
146 definition [simp, code del]:
147 "n mod m = of_nat (nat_of n mod nat_of m)"
149 definition [simp, code del]:
150 "n \<le> m \<longleftrightarrow> nat_of n \<le> nat_of m"
152 definition [simp, code del]:
153 "n < m \<longleftrightarrow> nat_of n < nat_of m"
156 qed (auto simp add: code_numeral left_distrib intro: mult_commute)
160 lemma nat_of_numeral [simp]: "nat_of (numeral k) = numeral k"
161 by (induct k rule: num_induct) (simp_all add: numeral_inc)
163 definition Num :: "num \<Rightarrow> code_numeral"
164 where [simp, code_abbrev]: "Num = numeral"
166 code_datatype "0::code_numeral" Num
168 lemma one_code_numeral_code [code]:
169 "(1\<Colon>code_numeral) = Numeral1"
172 lemma [code_abbrev]: "Numeral1 = (1\<Colon>code_numeral)"
173 using one_code_numeral_code ..
175 lemma plus_code_numeral_code [code nbe]:
176 "of_nat n + of_nat m = of_nat (n + m)"
179 lemma minus_code_numeral_code [code nbe]:
180 "of_nat n - of_nat m = of_nat (n - m)"
183 lemma times_code_numeral_code [code nbe]:
184 "of_nat n * of_nat m = of_nat (n * m)"
187 lemma less_eq_code_numeral_code [code nbe]:
188 "of_nat n \<le> of_nat m \<longleftrightarrow> n \<le> m"
191 lemma less_code_numeral_code [code nbe]:
192 "of_nat n < of_nat m \<longleftrightarrow> n < m"
195 lemma code_numeral_zero_minus_one:
196 "(0::code_numeral) - 1 = 0"
199 lemma Suc_code_numeral_minus_one:
203 lemma of_nat_code [code]:
204 "of_nat = Nat.of_nat"
207 have "Nat.of_nat n = of_nat n"
208 by (induct n) simp_all
209 then show "of_nat n = Nat.of_nat n"
213 lemma code_numeral_not_eq_zero: "i \<noteq> of_nat 0 \<longleftrightarrow> i \<ge> 1"
216 definition nat_of_aux :: "code_numeral \<Rightarrow> nat \<Rightarrow> nat" where
217 "nat_of_aux i n = nat_of i + n"
219 lemma nat_of_aux_code [code]:
220 "nat_of_aux i n = (if i = 0 then n else nat_of_aux (i - 1) (Nat.Suc n))"
221 by (auto simp add: nat_of_aux_def code_numeral_not_eq_zero)
223 lemma nat_of_code [code]:
224 "nat_of i = nat_of_aux i 0"
225 by (simp add: nat_of_aux_def)
227 definition div_mod :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral \<times> code_numeral" where
228 [code del]: "div_mod n m = (n div m, n mod m)"
231 "div_mod n m = (if m = 0 then (0, n) else (n div m, n mod m))"
232 unfolding div_mod_def by auto
235 "n div m = fst (div_mod n m)"
236 unfolding div_mod_def by simp
239 "n mod m = snd (div_mod n m)"
240 unfolding div_mod_def by simp
242 definition int_of :: "code_numeral \<Rightarrow> int" where
243 "int_of = Nat.of_nat o nat_of"
245 lemma int_of_code [code]:
246 "int_of k = (if k = 0 then 0
247 else (if k mod 2 = 0 then 2 * int_of (k div 2) else 2 * int_of (k div 2) + 1))"
249 have "(nat_of k div 2) * 2 + nat_of k mod 2 = nat_of k"
250 by (rule mod_div_equality)
251 then have "int ((nat_of k div 2) * 2 + nat_of k mod 2) = int (nat_of k)"
253 then have "int (nat_of k) = int (nat_of k div 2) * 2 + int (nat_of k mod 2)"
254 unfolding of_nat_mult of_nat_add by simp
255 then show ?thesis by (auto simp add: int_of_def mult_ac)
259 hide_const (open) of_nat nat_of Suc int_of
262 subsection {* Code generator setup *}
264 text {* Implementation of code numerals by bounded integers *}
266 code_type code_numeral
268 (OCaml "Big'_int.big'_int")
272 code_instance code_numeral :: equal
276 Numeral.add_code @{const_name Num}
277 false Code_Printer.literal_naive_numeral "SML"
278 #> fold (Numeral.add_code @{const_name Num}
279 false Code_Printer.literal_numeral) ["OCaml", "Haskell", "Scala"]
282 code_reserved SML Int int
283 code_reserved Eval Integer
285 code_const "0::code_numeral"
287 (OCaml "Big'_int.zero'_big'_int")
291 code_const "plus \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
292 (SML "Int.+/ ((_),/ (_))")
293 (OCaml "Big'_int.add'_big'_int")
294 (Haskell infixl 6 "+")
298 code_const "minus \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
299 (SML "Int.max/ (0 : int,/ Int.-/ ((_),/ (_)))")
300 (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int/ (Big'_int.sub'_big'_int/ _/ _)")
301 (Haskell "Prelude.max/ (0 :: Integer)/ (_/ -/ _)")
302 (Scala "!(_/ -/ _).max(0)")
303 (Eval "Integer.max/ 0/ (_/ -/ _)")
305 code_const "times \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral"
306 (SML "Int.*/ ((_),/ (_))")
307 (OCaml "Big'_int.mult'_big'_int")
308 (Haskell infixl 7 "*")
312 code_const Code_Numeral.div_mod
313 (SML "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Int.div (n, m), Int.mod (n, m)))")
314 (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
316 (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
317 (Eval "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (Integer.div'_mod n m))")
319 code_const "HOL.equal \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
320 (SML "!((_ : Int.int) = _)")
321 (OCaml "Big'_int.eq'_big'_int")
322 (Haskell infix 4 "==")
323 (Scala infixl 5 "==")
324 (Eval "!((_ : int) = _)")
326 code_const "less_eq \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
327 (SML "Int.<=/ ((_),/ (_))")
328 (OCaml "Big'_int.le'_big'_int")
329 (Haskell infix 4 "<=")
330 (Scala infixl 4 "<=")
333 code_const "less \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
334 (SML "Int.</ ((_),/ (_))")
335 (OCaml "Big'_int.lt'_big'_int")
336 (Haskell infix 4 "<")
343 code_modulename OCaml
346 code_modulename Haskell