src/HOL/Library/Numeral_Type.thy
 author huffman Thu Jun 11 09:03:24 2009 -0700 (2009-06-11) changeset 31563 ded2364d14d4 parent 31080 21ffc770ebc0 child 33035 15eab423e573 permissions -rw-r--r--
cleaned up some proofs
1 (*  Title:      HOL/Library/Numeral_Type.thy
2     Author:     Brian Huffman
3 *)
5 header {* Numeral Syntax for Types *}
7 theory Numeral_Type
8 imports Main
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_syntax 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 [simp]: "CARD(unit) = 1"
46   unfolding UNIV_unit by simp
48 lemma card_bool [simp]: "CARD(bool) = 2"
49   unfolding UNIV_bool by simp
51 lemma card_prod [simp]: "CARD('a \<times> 'b) = CARD('a::finite) * CARD('b::finite)"
52   unfolding UNIV_Times_UNIV [symmetric] by (simp only: card_cartesian_product)
54 lemma card_sum [simp]: "CARD('a + 'b) = CARD('a::finite) + CARD('b::finite)"
55   unfolding UNIV_Plus_UNIV [symmetric] by (simp only: finite card_Plus)
57 lemma card_option [simp]: "CARD('a option) = Suc CARD('a::finite)"
58   unfolding UNIV_option_conv
59   apply (subgoal_tac "(None::'a option) \<notin> range Some")
61   apply fast
62   done
64 lemma card_set [simp]: "CARD('a set) = 2 ^ CARD('a::finite)"
65   unfolding Pow_UNIV [symmetric]
66   by (simp only: card_Pow finite numeral_2_eq_2)
68 lemma card_nat [simp]: "CARD(nat) = 0"
69   by (simp add: infinite_UNIV_nat card_eq_0_iff)
72 subsection {* Classes with at least 1 and 2  *}
74 text {* Class finite already captures "at least 1" *}
76 lemma zero_less_card_finite [simp]: "0 < CARD('a::finite)"
77   unfolding neq0_conv [symmetric] by simp
79 lemma one_le_card_finite [simp]: "Suc 0 \<le> CARD('a::finite)"
80   by (simp add: less_Suc_eq_le [symmetric])
82 text {* Class for cardinality "at least 2" *}
84 class card2 = finite +
85   assumes two_le_card: "2 \<le> CARD('a)"
87 lemma one_less_card: "Suc 0 < CARD('a::card2)"
88   using two_le_card [where 'a='a] by simp
90 lemma one_less_int_card: "1 < int CARD('a::card2)"
91   using one_less_card [where 'a='a] by simp
94 subsection {* Numeral Types *}
96 typedef (open) num0 = "UNIV :: nat set" ..
97 typedef (open) num1 = "UNIV :: unit set" ..
99 typedef (open) 'a bit0 = "{0 ..< 2 * int CARD('a::finite)}"
100 proof
101   show "0 \<in> {0 ..< 2 * int CARD('a)}"
102     by simp
103 qed
105 typedef (open) 'a bit1 = "{0 ..< 1 + 2 * int CARD('a::finite)}"
106 proof
107   show "0 \<in> {0 ..< 1 + 2 * int CARD('a)}"
108     by simp
109 qed
111 lemma card_num0 [simp]: "CARD (num0) = 0"
112   unfolding type_definition.card [OF type_definition_num0]
113   by simp
115 lemma card_num1 [simp]: "CARD(num1) = 1"
116   unfolding type_definition.card [OF type_definition_num1]
117   by (simp only: card_unit)
119 lemma card_bit0 [simp]: "CARD('a bit0) = 2 * CARD('a::finite)"
120   unfolding type_definition.card [OF type_definition_bit0]
121   by simp
123 lemma card_bit1 [simp]: "CARD('a bit1) = Suc (2 * CARD('a::finite))"
124   unfolding type_definition.card [OF type_definition_bit1]
125   by simp
127 instance num1 :: finite
128 proof
129   show "finite (UNIV::num1 set)"
130     unfolding type_definition.univ [OF type_definition_num1]
131     using finite by (rule finite_imageI)
132 qed
134 instance bit0 :: (finite) card2
135 proof
136   show "finite (UNIV::'a bit0 set)"
137     unfolding type_definition.univ [OF type_definition_bit0]
138     by simp
139   show "2 \<le> CARD('a bit0)"
140     by simp
141 qed
143 instance bit1 :: (finite) card2
144 proof
145   show "finite (UNIV::'a bit1 set)"
146     unfolding type_definition.univ [OF type_definition_bit1]
147     by simp
148   show "2 \<le> CARD('a bit1)"
149     by simp
150 qed
153 subsection {* Locale for modular arithmetic subtypes *}
155 locale mod_type =
156   fixes n :: int
157   and Rep :: "'a::{zero,one,plus,times,uminus,minus} \<Rightarrow> int"
158   and Abs :: "int \<Rightarrow> 'a::{zero,one,plus,times,uminus,minus}"
159   assumes type: "type_definition Rep Abs {0..<n}"
160   and size1: "1 < n"
161   and zero_def: "0 = Abs 0"
162   and one_def:  "1 = Abs 1"
163   and add_def:  "x + y = Abs ((Rep x + Rep y) mod n)"
164   and mult_def: "x * y = Abs ((Rep x * Rep y) mod n)"
165   and diff_def: "x - y = Abs ((Rep x - Rep y) mod n)"
166   and minus_def: "- x = Abs ((- Rep x) mod n)"
167 begin
169 lemma size0: "0 < n"
170 by (cut_tac size1, simp)
172 lemmas definitions =
173   zero_def one_def add_def mult_def minus_def diff_def
175 lemma Rep_less_n: "Rep x < n"
176 by (rule type_definition.Rep [OF type, simplified, THEN conjunct2])
178 lemma Rep_le_n: "Rep x \<le> n"
179 by (rule Rep_less_n [THEN order_less_imp_le])
181 lemma Rep_inject_sym: "x = y \<longleftrightarrow> Rep x = Rep y"
182 by (rule type_definition.Rep_inject [OF type, symmetric])
184 lemma Rep_inverse: "Abs (Rep x) = x"
185 by (rule type_definition.Rep_inverse [OF type])
187 lemma Abs_inverse: "m \<in> {0..<n} \<Longrightarrow> Rep (Abs m) = m"
188 by (rule type_definition.Abs_inverse [OF type])
190 lemma Rep_Abs_mod: "Rep (Abs (m mod n)) = m mod n"
191 by (simp add: Abs_inverse IntDiv.pos_mod_conj [OF size0])
193 lemma Rep_Abs_0: "Rep (Abs 0) = 0"
194 by (simp add: Abs_inverse size0)
196 lemma Rep_0: "Rep 0 = 0"
197 by (simp add: zero_def Rep_Abs_0)
199 lemma Rep_Abs_1: "Rep (Abs 1) = 1"
200 by (simp add: Abs_inverse size1)
202 lemma Rep_1: "Rep 1 = 1"
203 by (simp add: one_def Rep_Abs_1)
205 lemma Rep_mod: "Rep x mod n = Rep x"
206 apply (rule_tac x=x in type_definition.Abs_cases [OF type])
207 apply (simp add: type_definition.Abs_inverse [OF type])
209 done
211 lemmas Rep_simps =
212   Rep_inject_sym Rep_inverse Rep_Abs_mod Rep_mod Rep_Abs_0 Rep_Abs_1
214 lemma comm_ring_1: "OFCLASS('a, comm_ring_1_class)"
215 apply (intro_classes, unfold definitions)
216 apply (simp_all add: Rep_simps zmod_simps ring_simps)
217 done
219 end
221 locale mod_ring = mod_type +
222   constrains n :: int
223   and Rep :: "'a::{number_ring} \<Rightarrow> int"
224   and Abs :: "int \<Rightarrow> 'a::{number_ring}"
225 begin
227 lemma of_nat_eq: "of_nat k = Abs (int k mod n)"
228 apply (induct k)
231 done
233 lemma of_int_eq: "of_int z = Abs (z mod n)"
234 apply (cases z rule: int_diff_cases)
235 apply (simp add: Rep_simps of_nat_eq diff_def zmod_simps)
236 done
238 lemma Rep_number_of:
239   "Rep (number_of w) = number_of w mod n"
240 by (simp add: number_of_eq of_int_eq Rep_Abs_mod)
242 lemma iszero_number_of:
243   "iszero (number_of w::'a) \<longleftrightarrow> number_of w mod n = 0"
244 by (simp add: Rep_simps number_of_eq of_int_eq iszero_def zero_def)
246 lemma cases:
247   assumes 1: "\<And>z. \<lbrakk>(x::'a) = of_int z; 0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P"
248   shows "P"
249 apply (cases x rule: type_definition.Abs_cases [OF type])
250 apply (rule_tac z="y" in 1)
251 apply (simp_all add: of_int_eq mod_pos_pos_trivial)
252 done
254 lemma induct:
255   "(\<And>z. \<lbrakk>0 \<le> z; z < n\<rbrakk> \<Longrightarrow> P (of_int z)) \<Longrightarrow> P (x::'a)"
256 by (cases x rule: cases) simp
258 end
261 subsection {* Number ring instances *}
263 text {*
264   Unfortunately a number ring instance is not possible for
265   @{typ num1}, since 0 and 1 are not distinct.
266 *}
268 instantiation num1 :: "{comm_ring,comm_monoid_mult,number}"
269 begin
271 lemma num1_eq_iff: "(x::num1) = (y::num1) \<longleftrightarrow> True"
272   by (induct x, induct y) simp
274 instance proof
277 end
279 instantiation
280   bit0 and bit1 :: (finite) "{zero,one,plus,times,uminus,minus}"
281 begin
283 definition Abs_bit0' :: "int \<Rightarrow> 'a bit0" where
284   "Abs_bit0' x = Abs_bit0 (x mod int CARD('a bit0))"
286 definition Abs_bit1' :: "int \<Rightarrow> 'a bit1" where
287   "Abs_bit1' x = Abs_bit1 (x mod int CARD('a bit1))"
289 definition "0 = Abs_bit0 0"
290 definition "1 = Abs_bit0 1"
291 definition "x + y = Abs_bit0' (Rep_bit0 x + Rep_bit0 y)"
292 definition "x * y = Abs_bit0' (Rep_bit0 x * Rep_bit0 y)"
293 definition "x - y = Abs_bit0' (Rep_bit0 x - Rep_bit0 y)"
294 definition "- x = Abs_bit0' (- Rep_bit0 x)"
296 definition "0 = Abs_bit1 0"
297 definition "1 = Abs_bit1 1"
298 definition "x + y = Abs_bit1' (Rep_bit1 x + Rep_bit1 y)"
299 definition "x * y = Abs_bit1' (Rep_bit1 x * Rep_bit1 y)"
300 definition "x - y = Abs_bit1' (Rep_bit1 x - Rep_bit1 y)"
301 definition "- x = Abs_bit1' (- Rep_bit1 x)"
303 instance ..
305 end
307 interpretation bit0:
308   mod_type "int CARD('a::finite bit0)"
309            "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
310            "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
311 apply (rule mod_type.intro)
312 apply (simp add: int_mult type_definition_bit0)
313 apply (rule one_less_int_card)
314 apply (rule zero_bit0_def)
315 apply (rule one_bit0_def)
316 apply (rule plus_bit0_def [unfolded Abs_bit0'_def])
317 apply (rule times_bit0_def [unfolded Abs_bit0'_def])
318 apply (rule minus_bit0_def [unfolded Abs_bit0'_def])
319 apply (rule uminus_bit0_def [unfolded Abs_bit0'_def])
320 done
322 interpretation bit1:
323   mod_type "int CARD('a::finite bit1)"
324            "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
325            "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
326 apply (rule mod_type.intro)
327 apply (simp add: int_mult type_definition_bit1)
328 apply (rule one_less_int_card)
329 apply (rule zero_bit1_def)
330 apply (rule one_bit1_def)
331 apply (rule plus_bit1_def [unfolded Abs_bit1'_def])
332 apply (rule times_bit1_def [unfolded Abs_bit1'_def])
333 apply (rule minus_bit1_def [unfolded Abs_bit1'_def])
334 apply (rule uminus_bit1_def [unfolded Abs_bit1'_def])
335 done
337 instance bit0 :: (finite) comm_ring_1
338   by (rule bit0.comm_ring_1)+
340 instance bit1 :: (finite) comm_ring_1
341   by (rule bit1.comm_ring_1)+
343 instantiation bit0 and bit1 :: (finite) number_ring
344 begin
346 definition "(number_of w :: _ bit0) = of_int w"
348 definition "(number_of w :: _ bit1) = of_int w"
350 instance proof
351 qed (rule number_of_bit0_def number_of_bit1_def)+
353 end
355 interpretation bit0:
356   mod_ring "int CARD('a::finite bit0)"
357            "Rep_bit0 :: 'a::finite bit0 \<Rightarrow> int"
358            "Abs_bit0 :: int \<Rightarrow> 'a::finite bit0"
359   ..
361 interpretation bit1:
362   mod_ring "int CARD('a::finite bit1)"
363            "Rep_bit1 :: 'a::finite bit1 \<Rightarrow> int"
364            "Abs_bit1 :: int \<Rightarrow> 'a::finite bit1"
365   ..
367 text {* Set up cases, induction, and arithmetic *}
369 lemmas bit0_cases [case_names of_int, cases type: bit0] = bit0.cases
370 lemmas bit1_cases [case_names of_int, cases type: bit1] = bit1.cases
372 lemmas bit0_induct [case_names of_int, induct type: bit0] = bit0.induct
373 lemmas bit1_induct [case_names of_int, induct type: bit1] = bit1.induct
375 lemmas bit0_iszero_number_of [simp] = bit0.iszero_number_of
376 lemmas bit1_iszero_number_of [simp] = bit1.iszero_number_of
379 subsection {* Syntax *}
381 syntax
382   "_NumeralType" :: "num_const => type"  ("_")
383   "_NumeralType0" :: type ("0")
384   "_NumeralType1" :: type ("1")
386 translations
387   "_NumeralType1" == (type) "num1"
388   "_NumeralType0" == (type) "num0"
390 parse_translation {*
391 let
393 val num1_const = Syntax.const "Numeral_Type.num1";
394 val num0_const = Syntax.const "Numeral_Type.num0";
395 val B0_const = Syntax.const "Numeral_Type.bit0";
396 val B1_const = Syntax.const "Numeral_Type.bit1";
398 fun mk_bintype n =
399   let
400     fun mk_bit n = if n = 0 then B0_const else B1_const;
401     fun bin_of n =
402       if n = 1 then num1_const
403       else if n = 0 then num0_const
404       else if n = ~1 then raise TERM ("negative type numeral", [])
405       else
406         let val (q, r) = Integer.div_mod n 2;
407         in mk_bit r \$ bin_of q end;
408   in bin_of n end;
410 fun numeral_tr (*"_NumeralType"*) [Const (str, _)] =
411       mk_bintype (valOf (Int.fromString str))
412   | numeral_tr (*"_NumeralType"*) ts = raise TERM ("numeral_tr", ts);
414 in [("_NumeralType", numeral_tr)] end;
415 *}
417 print_translation {*
418 let
419 fun int_of [] = 0
420   | int_of (b :: bs) = b + 2 * int_of bs;
422 fun bin_of (Const ("num0", _)) = []
423   | bin_of (Const ("num1", _)) = [1]
424   | bin_of (Const ("bit0", _) \$ bs) = 0 :: bin_of bs
425   | bin_of (Const ("bit1", _) \$ bs) = 1 :: bin_of bs
426   | bin_of t = raise TERM("bin_of", [t]);
428 fun bit_tr' b [t] =
429   let
430     val rev_digs = b :: bin_of t handle TERM _ => raise Match
431     val i = int_of rev_digs;
432     val num = string_of_int (abs i);
433   in
434     Syntax.const "_NumeralType" \$ Syntax.free num
435   end
436   | bit_tr' b _ = raise Match;
438 in [("bit0", bit_tr' 0), ("bit1", bit_tr' 1)] end;
439 *}
441 subsection {* Examples *}
443 lemma "CARD(0) = 0" by simp
444 lemma "CARD(17) = 17" by simp
445 lemma "8 * 11 ^ 3 - 6 = (2::5)" by simp
447 end