src/HOL/Library/Numeral_Type.thy
 author huffman Thu Feb 19 16:51:46 2009 -0800 (2009-02-19) changeset 29997 f6756c097c2d parent 29629 5111ce425e7a child 29998 19e1ef628b25 permissions -rw-r--r--
number_ring instances for numeral types
1 (*  Title:      HOL/Library/Numeral_Type.thy
2     Author:     Brian Huffman
3 *)
5 header {* Numeral Syntax for Types *}
7 theory Numeral_Type
8 imports Plain "~~/src/HOL/Presburger"
9 begin
11 subsection {* Preliminary lemmas *}
12 (* These should be moved elsewhere *)
14 lemma (in type_definition) univ:
15   "UNIV = Abs ` A"
16 proof
17   show "Abs ` A \<subseteq> UNIV" by (rule subset_UNIV)
18   show "UNIV \<subseteq> Abs ` A"
19   proof
20     fix x :: 'b
21     have "x = Abs (Rep x)" by (rule Rep_inverse [symmetric])
22     moreover have "Rep x \<in> A" by (rule Rep)
23     ultimately show "x \<in> Abs ` A" by (rule image_eqI)
24   qed
25 qed
27 lemma (in type_definition) card: "card (UNIV :: 'b set) = card A"
28   by (simp add: univ card_image inj_on_def Abs_inject)
31 subsection {* Cardinalities of types *}
33 syntax "_type_card" :: "type => nat" ("(1CARD/(1'(_')))")
35 translations "CARD(t)" => "CONST card (CONST UNIV \<Colon> t set)"
37 typed_print_translation {*
38 let
39   fun card_univ_tr' show_sorts _ [Const (@{const_name UNIV}, Type(_,[T,_]))] =
40     Syntax.const "_type_card" \$ Syntax.term_of_typ show_sorts T;
41 in [(@{const_syntax card}, card_univ_tr')]
42 end
43 *}
45 lemma card_unit: "CARD(unit) = 1"
46   unfolding UNIV_unit by simp
48 lemma card_bool: "CARD(bool) = 2"
49   unfolding UNIV_bool by simp
51 lemma card_prod: "CARD('a::finite \<times> 'b::finite) = CARD('a) * CARD('b)"
52   unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product)
54 lemma card_sum: "CARD('a::finite + 'b::finite) = CARD('a) + CARD('b)"
55   unfolding UNIV_Plus_UNIV [symmetric] by (simp only: finite card_Plus)
57 lemma card_option: "CARD('a::finite option) = Suc CARD('a)"
58   unfolding insert_None_conv_UNIV [symmetric]
59   apply (subgoal_tac "(None::'a option) \<notin> range Some")
61   apply fast
62   done
64 lemma card_set: "CARD('a::finite set) = 2 ^ CARD('a)"
65   unfolding Pow_UNIV [symmetric]
66   by (simp only: card_Pow finite numeral_2_eq_2)
68 lemma card_finite_pos [simp]: "0 < CARD('a::finite)"
69   unfolding neq0_conv [symmetric] by simp
72 subsection {* Numeral Types *}
74 typedef (open) num0 = "UNIV :: nat set" ..
75 typedef (open) num1 = "UNIV :: unit set" ..
77 typedef (open) 'a bit0 = "{0 ..< 2 * int CARD('a::finite)}"
78 proof
79   show "0 \<in> {0 ..< 2 * int CARD('a)}"
80     by simp
81 qed
83 typedef (open) 'a bit1 = "{0 ..< 1 + 2 * int CARD('a::finite)}"
84 proof
85   show "0 \<in> {0 ..< 1 + 2 * int CARD('a)}"
86     by simp
87 qed
89 instance num1 :: finite
90 proof
91   show "finite (UNIV::num1 set)"
92     unfolding type_definition.univ [OF type_definition_num1]
93     using finite by (rule finite_imageI)
94 qed
96 instance bit0 :: (finite) finite
97 proof
98   show "finite (UNIV::'a bit0 set)"
99     unfolding type_definition.univ [OF type_definition_bit0]
100     by simp
101 qed
103 instance bit1 :: (finite) finite
104 proof
105   show "finite (UNIV::'a bit1 set)"
106     unfolding type_definition.univ [OF type_definition_bit1]
107     by simp
108 qed
110 lemma card_num1: "CARD(num1) = 1"
111   unfolding type_definition.card [OF type_definition_num1]
112   by (simp only: card_unit)
114 lemma card_bit0: "CARD('a::finite bit0) = 2 * CARD('a)"
115   unfolding type_definition.card [OF type_definition_bit0]
116   by simp
118 lemma card_bit1: "CARD('a::finite bit1) = Suc (2 * CARD('a))"
119   unfolding type_definition.card [OF type_definition_bit1]
120   by simp
122 lemma card_num0: "CARD (num0) = 0"
123   by (simp add: infinite_UNIV_nat card_eq_0_iff type_definition.card [OF type_definition_num0])
125 lemmas card_univ_simps [simp] =
126   card_unit
127   card_bool
128   card_prod
129   card_sum
130   card_option
131   card_set
132   card_num1
133   card_bit0
134   card_bit1
135   card_num0
138 subsection {* Locale for modular arithmetic subtypes *}
140 locale mod_type =
141   fixes n :: int
142   and Rep :: "'a::{zero,one,plus,times,uminus,minus,power} \<Rightarrow> int"
143   and Abs :: "int \<Rightarrow> 'a::{zero,one,plus,times,uminus,minus,power}"
144   assumes type: "type_definition Rep Abs {0..<n}"
145   and size1: "1 < n"
146   and zero_def: "0 = Abs 0"
147   and one_def:  "1 = Abs 1"
148   and add_def:  "x + y = Abs ((Rep x + Rep y) mod n)"
149   and mult_def: "x * y = Abs ((Rep x * Rep y) mod n)"
150   and diff_def: "x - y = Abs ((Rep x - Rep y) mod n)"
151   and minus_def: "- x = Abs ((- Rep x) mod n)"
152   and power_def: "x ^ k = Abs (Rep x ^ k mod n)"
153 begin
155 lemma size0: "0 < n"
156 by (cut_tac size1, simp)
158 lemmas definitions =
159   zero_def one_def add_def mult_def minus_def diff_def power_def
161 lemma Rep_less_n: "Rep x < n"
162 by (rule type_definition.Rep [OF type, simplified, THEN conjunct2])
164 lemma Rep_le_n: "Rep x \<le> n"
165 by (rule Rep_less_n [THEN order_less_imp_le])
167 lemma Rep_inject_sym: "x = y \<longleftrightarrow> Rep x = Rep y"
168 by (rule type_definition.Rep_inject [OF type, symmetric])
170 lemma Rep_inverse: "Abs (Rep x) = x"
171 by (rule type_definition.Rep_inverse [OF type])
173 lemma Abs_inverse: "m \<in> {0..<n} \<Longrightarrow> Rep (Abs m) = m"
174 by (rule type_definition.Abs_inverse [OF type])
176 lemma Rep_Abs_mod: "Rep (Abs (m mod n)) = m mod n"
177 by (simp add: Abs_inverse IntDiv.pos_mod_conj [OF size0])
179 lemma Rep_Abs_0: "Rep (Abs 0) = 0"
180 by (simp add: Abs_inverse size0)
182 lemma Rep_0: "Rep 0 = 0"
183 by (simp add: zero_def Rep_Abs_0)
185 lemma Rep_Abs_1: "Rep (Abs 1) = 1"
186 by (simp add: Abs_inverse size1)
188 lemma Rep_1: "Rep 1 = 1"
189 by (simp add: one_def Rep_Abs_1)
191 lemma Rep_mod: "Rep x mod n = Rep x"
192 apply (rule_tac x=x in type_definition.Abs_cases [OF type])
193 apply (simp add: type_definition.Abs_inverse [OF type])
195 done
197 lemmas Rep_simps =
198   Rep_inject_sym Rep_inverse Rep_Abs_mod Rep_mod Rep_Abs_0 Rep_Abs_1
200 lemma comm_ring_1: "OFCLASS('a, comm_ring_1_class)"
201 apply (intro_classes, unfold definitions)
202 apply (simp_all add: Rep_simps zmod_simps ring_simps)
203 done
205 lemma recpower: "OFCLASS('a, recpower_class)"
206 apply (intro_classes, unfold definitions)
208                      mod_pos_pos_trivial size1)
209 done
211 end
213 locale mod_ring = mod_type +
214   constrains n :: int
215   and Rep :: "'a::{number_ring,power} \<Rightarrow> int"
216   and Abs :: "int \<Rightarrow> 'a::{number_ring,power}"
217 begin
219 lemma of_nat_eq: "of_nat k = Abs (int k mod n)"
220 apply (induct k)
223 done
225 lemma of_int_eq: "of_int z = Abs (z mod n)"
226 apply (cases z rule: int_diff_cases)
227 apply (simp add: Rep_simps of_nat_eq diff_def zmod_simps)
228 done
230 lemma Rep_number_of:
231   "Rep (number_of w) = number_of w mod n"
232 by (simp add: number_of_eq of_int_eq Rep_Abs_mod)
234 lemma iszero_number_of:
235   "iszero (number_of w::'a) \<longleftrightarrow> number_of w mod n = 0"
236 by (simp add: Rep_simps number_of_eq of_int_eq iszero_def zero_def)
238 lemma cases:
239   assumes 1: "\<And>z. \<lbrakk>(x::'a) = of_int z; 0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P"
240   shows "P"
241 apply (cases x rule: type_definition.Abs_cases [OF type])
242 apply (rule_tac z="y" in 1)
243 apply (simp_all add: of_int_eq mod_pos_pos_trivial)
244 done
246 lemma induct:
247   "(\<And>z. \<lbrakk>0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P (of_int z)) \<Longrightarrow> P (x::'a)"
248 by (cases x rule: cases) simp
250 end
253 subsection {* Number ring instances *}
255 instantiation
256   bit0 and bit1 :: (finite) "{zero,one,plus,times,uminus,minus,power}"
257 begin
259 definition Abs_bit0' :: "int \<Rightarrow> 'a bit0" where
260   "Abs_bit0' x = Abs_bit0 (x mod (2 * int CARD('a)))"
262 definition Abs_bit1' :: "int \<Rightarrow> 'a bit1" where
263   "Abs_bit1' x = Abs_bit1 (x mod (1 + 2 * int CARD('a)))"
265 definition "0 = Abs_bit0 0"
266 definition "1 = Abs_bit0 1"
267 definition "x + y = Abs_bit0' (Rep_bit0 x + Rep_bit0 y)"
268 definition "x * y = Abs_bit0' (Rep_bit0 x * Rep_bit0 y)"
269 definition "x - y = Abs_bit0' (Rep_bit0 x - Rep_bit0 y)"
270 definition "- x = Abs_bit0' (- Rep_bit0 x)"
271 definition "x ^ k = Abs_bit0' (Rep_bit0 x ^ k)"
273 definition "0 = Abs_bit1 0"
274 definition "1 = Abs_bit1 1"
275 definition "x + y = Abs_bit1' (Rep_bit1 x + Rep_bit1 y)"
276 definition "x * y = Abs_bit1' (Rep_bit1 x * Rep_bit1 y)"
277 definition "x - y = Abs_bit1' (Rep_bit1 x - Rep_bit1 y)"
278 definition "- x = Abs_bit1' (- Rep_bit1 x)"
279 definition "x ^ k = Abs_bit1' (Rep_bit1 x ^ k)"
281 instance ..
283 end
285 interpretation bit0!:
286   mod_type "2 * int CARD('a::finite)"
287            "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
288            "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
289 apply (rule mod_type.intro)
290 apply (rule type_definition_bit0)
291 using card_finite_pos [where ?'a='a] apply arith
292 apply (rule zero_bit0_def)
293 apply (rule one_bit0_def)
294 apply (rule plus_bit0_def [unfolded Abs_bit0'_def])
295 apply (rule times_bit0_def [unfolded Abs_bit0'_def])
296 apply (rule minus_bit0_def [unfolded Abs_bit0'_def])
297 apply (rule uminus_bit0_def [unfolded Abs_bit0'_def])
298 apply (rule power_bit0_def [unfolded Abs_bit0'_def])
299 done
301 interpretation bit1!:
302   mod_type "1 + 2 * int CARD('a::finite)"
303            "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
304            "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
305 apply (rule mod_type.intro)
306 apply (rule type_definition_bit1)
307 apply simp
308 apply (rule zero_bit1_def)
309 apply (rule one_bit1_def)
310 apply (rule plus_bit1_def [unfolded Abs_bit1'_def])
311 apply (rule times_bit1_def [unfolded Abs_bit1'_def])
312 apply (rule minus_bit1_def [unfolded Abs_bit1'_def])
313 apply (rule uminus_bit1_def [unfolded Abs_bit1'_def])
314 apply (rule power_bit1_def [unfolded Abs_bit1'_def])
315 done
317 instance bit0 :: (finite) "{comm_ring_1,recpower}"
318   by (rule bit0.comm_ring_1 bit0.recpower)+
320 instance bit1 :: (finite) "{comm_ring_1,recpower}"
321   by (rule bit1.comm_ring_1 bit1.recpower)+
323 instantiation bit0 and bit1 :: (finite) number_ring
324 begin
326 definition "(number_of w :: _ bit0) = of_int w"
328 definition "(number_of w :: _ bit1) = of_int w"
330 instance proof
331 qed (rule number_of_bit0_def number_of_bit1_def)+
333 end
335 interpretation bit0!:
336   mod_ring "2 * int CARD('a::finite)"
337            "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
338            "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
339   ..
341 interpretation bit1!:
342   mod_ring "1 + 2 * int CARD('a::finite)"
343            "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
344            "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
345   ..
347 text {* Set up cases, induction, and arithmetic *}
349 lemmas bit0_cases [cases type: bit0, case_names of_int] = bit0.cases
350 lemmas bit1_cases [cases type: bit1, case_names of_int] = bit1.cases
352 lemmas bit0_induct [induct type: bit0, case_names of_int] = bit0.induct
353 lemmas bit1_induct [induct type: bit1, case_names of_int] = bit1.induct
355 lemmas bit0_iszero_number_of [simp] = bit0.iszero_number_of
356 lemmas bit1_iszero_number_of [simp] = bit1.iszero_number_of
358 declare power_Suc [where ?'a="'a::finite bit0", standard, simp]
359 declare power_Suc [where ?'a="'a::finite bit1", standard, simp]
362 subsection {* Syntax *}
364 syntax
365   "_NumeralType" :: "num_const => type"  ("_")
366   "_NumeralType0" :: type ("0")
367   "_NumeralType1" :: type ("1")
369 translations
370   "_NumeralType1" == (type) "num1"
371   "_NumeralType0" == (type) "num0"
373 parse_translation {*
374 let
376 val num1_const = Syntax.const "Numeral_Type.num1";
377 val num0_const = Syntax.const "Numeral_Type.num0";
378 val B0_const = Syntax.const "Numeral_Type.bit0";
379 val B1_const = Syntax.const "Numeral_Type.bit1";
381 fun mk_bintype n =
382   let
383     fun mk_bit n = if n = 0 then B0_const else B1_const;
384     fun bin_of n =
385       if n = 1 then num1_const
386       else if n = 0 then num0_const
387       else if n = ~1 then raise TERM ("negative type numeral", [])
388       else
389         let val (q, r) = Integer.div_mod n 2;
390         in mk_bit r \$ bin_of q end;
391   in bin_of n end;
393 fun numeral_tr (*"_NumeralType"*) [Const (str, _)] =
394       mk_bintype (valOf (Int.fromString str))
395   | numeral_tr (*"_NumeralType"*) ts = raise TERM ("numeral_tr", ts);
397 in [("_NumeralType", numeral_tr)] end;
398 *}
400 print_translation {*
401 let
402 fun int_of [] = 0
403   | int_of (b :: bs) = b + 2 * int_of bs;
405 fun bin_of (Const ("num0", _)) = []
406   | bin_of (Const ("num1", _)) = [1]
407   | bin_of (Const ("bit0", _) \$ bs) = 0 :: bin_of bs
408   | bin_of (Const ("bit1", _) \$ bs) = 1 :: bin_of bs
409   | bin_of t = raise TERM("bin_of", [t]);
411 fun bit_tr' b [t] =
412   let
413     val rev_digs = b :: bin_of t handle TERM _ => raise Match
414     val i = int_of rev_digs;
415     val num = string_of_int (abs i);
416   in
417     Syntax.const "_NumeralType" \$ Syntax.free num
418   end
419   | bit_tr' b _ = raise Match;
421 in [("bit0", bit_tr' 0), ("bit1", bit_tr' 1)] end;
422 *}
425 subsection {* Classes with at least 1 and 2  *}
427 text {* Class finite already captures "at least 1" *}
429 lemma zero_less_card_finite [simp]:
430   "0 < CARD('a::finite)"
431 proof (cases "CARD('a::finite) = 0")
432   case False thus ?thesis by (simp del: card_0_eq)
433 next
434   case True
435   thus ?thesis by (simp add: finite)
436 qed
438 lemma one_le_card_finite [simp]:
439   "Suc 0 <= CARD('a::finite)"
440   by (simp add: less_Suc_eq_le [symmetric] zero_less_card_finite)
443 text {* Class for cardinality "at least 2" *}
445 class card2 = finite +
446   assumes two_le_card: "2 <= CARD('a)"
448 lemma one_less_card: "Suc 0 < CARD('a::card2)"
449   using two_le_card [where 'a='a] by simp
451 instance bit0 :: (finite) card2
452   by intro_classes (simp add: one_le_card_finite)
454 instance bit1 :: (finite) card2
455   by intro_classes (simp add: one_le_card_finite)
457 subsection {* Examples *}
459 lemma "CARD(0) = 0" by simp
460 lemma "CARD(17) = 17" by simp
461 lemma "8 * 11 ^ 3 - 6 = (2::5)" by simp
463 end