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 *)
3 header {* Type of target language numerals *}
6 imports Nat_Numeral 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 {* Code numerals as datatype of ints *}
128 instantiation code_numeral :: number
132 "number_of = of_nat o nat"
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)
142 code_datatype "number_of \<Colon> int \<Rightarrow> code_numeral"
145 subsection {* Basic arithmetic *}
147 instantiation code_numeral :: "{minus, linordered_semidom, semiring_div, linorder}"
150 definition [simp, code del]:
151 "(1\<Colon>code_numeral) = of_nat 1"
153 definition [simp, code del]:
154 "n + m = of_nat (nat_of n + nat_of m)"
156 definition [simp, code del]:
157 "n - m = of_nat (nat_of n - nat_of m)"
159 definition [simp, code del]:
160 "n * m = of_nat (nat_of n * nat_of m)"
162 definition [simp, code del]:
163 "n div m = of_nat (nat_of n div nat_of m)"
165 definition [simp, code del]:
166 "n mod m = of_nat (nat_of n mod nat_of m)"
168 definition [simp, code del]:
169 "n \<le> m \<longleftrightarrow> nat_of n \<le> nat_of m"
171 definition [simp, code del]:
172 "n < m \<longleftrightarrow> nat_of n < nat_of m"
175 qed (auto simp add: code_numeral left_distrib intro: mult_commute)
179 lemma zero_code_numeral_code [code]:
180 "(0\<Colon>code_numeral) = Numeral0"
181 by (simp add: number_of_code_numeral_def Pls_def)
183 lemma [code_abbrev]: "Numeral0 = (0\<Colon>code_numeral)"
184 using zero_code_numeral_code ..
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)
190 lemma [code_abbrev]: "Numeral1 = (1\<Colon>code_numeral)"
191 using one_code_numeral_code ..
193 lemma plus_code_numeral_code [code nbe]:
194 "of_nat n + of_nat m = of_nat (n + m)"
197 definition subtract :: "code_numeral \<Rightarrow> code_numeral \<Rightarrow> code_numeral" where
198 [simp]: "subtract = minus"
200 lemma subtract_code [code nbe]:
201 "subtract (of_nat n) (of_nat m) = of_nat (n - m)"
204 lemma minus_code_numeral_code [code]:
208 lemma times_code_numeral_code [code nbe]:
209 "of_nat n * of_nat m = of_nat (n * m)"
212 lemma less_eq_code_numeral_code [code nbe]:
213 "of_nat n \<le> of_nat m \<longleftrightarrow> n \<le> m"
216 lemma less_code_numeral_code [code nbe]:
217 "of_nat n < of_nat m \<longleftrightarrow> n < m"
220 lemma code_numeral_zero_minus_one:
221 "(0::code_numeral) - 1 = 0"
224 lemma Suc_code_numeral_minus_one:
228 lemma of_nat_code [code]:
229 "of_nat = Nat.of_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"
238 lemma code_numeral_not_eq_zero: "i \<noteq> of_nat 0 \<longleftrightarrow> i \<ge> 1"
241 definition nat_of_aux :: "code_numeral \<Rightarrow> nat \<Rightarrow> nat" where
242 "nat_of_aux i n = nat_of i + n"
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)
248 lemma nat_of_code [code]:
249 "nat_of i = nat_of_aux i 0"
250 by (simp add: nat_of_aux_def)
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)"
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
260 "n div m = fst (div_mod n m)"
261 unfolding div_mod_def by simp
264 "n mod m = snd (div_mod n m)"
265 unfolding div_mod_def by simp
267 definition int_of :: "code_numeral \<Rightarrow> int" where
268 "int_of = Nat.of_nat o nat_of"
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))"
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)"
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)
284 hide_const (open) of_nat nat_of Suc subtract int_of
287 subsection {* Code generator setup *}
289 text {* Implementation of code numerals by bounded integers *}
291 code_type code_numeral
293 (OCaml "Big'_int.big'_int")
297 code_instance code_numeral :: equal
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"]
307 code_reserved SML Int int
308 code_reserved Eval Integer
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 "+")
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")
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 "*")
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 _)")
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))")
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) = _)")
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 "<=")
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 "<")
362 code_modulename OCaml
365 code_modulename Haskell