author | krauss |
Mon, 31 Aug 2009 20:34:48 +0200 | |
changeset 32463 | 3a0a65ca2261 |
parent 32264 | 0be31453f698 |
child 32554 | 4ccd84fb19d3 |
permissions | -rw-r--r-- |
31708 | 1 |
(* Title: HOL/Library/NatTransfer.thy |
2 |
Authors: Jeremy Avigad and Amine Chaieb |
|
3 |
||
4 |
Sets up transfer from nats to ints and |
|
5 |
back. |
|
6 |
*) |
|
7 |
||
8 |
||
9 |
header {* NatTransfer *} |
|
10 |
||
11 |
theory NatTransfer |
|
12 |
imports Main Parity |
|
13 |
uses ("Tools/transfer_data.ML") |
|
14 |
begin |
|
15 |
||
16 |
subsection {* A transfer Method between isomorphic domains*} |
|
17 |
||
18 |
definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool" |
|
19 |
where "TransferMorphism a B = True" |
|
20 |
||
21 |
use "Tools/transfer_data.ML" |
|
22 |
||
23 |
setup TransferData.setup |
|
24 |
||
25 |
||
26 |
subsection {* Set up transfer from nat to int *} |
|
27 |
||
28 |
(* set up transfer direction *) |
|
29 |
||
30 |
lemma TransferMorphism_nat_int: "TransferMorphism nat (op <= (0::int))" |
|
31 |
by (simp add: TransferMorphism_def) |
|
32 |
||
33 |
declare TransferMorphism_nat_int[transfer |
|
34 |
add mode: manual |
|
35 |
return: nat_0_le |
|
36 |
labels: natint |
|
37 |
] |
|
38 |
||
39 |
(* basic functions and relations *) |
|
40 |
||
41 |
lemma transfer_nat_int_numerals: |
|
42 |
"(0::nat) = nat 0" |
|
43 |
"(1::nat) = nat 1" |
|
44 |
"(2::nat) = nat 2" |
|
45 |
"(3::nat) = nat 3" |
|
46 |
by auto |
|
47 |
||
48 |
definition |
|
49 |
tsub :: "int \<Rightarrow> int \<Rightarrow> int" |
|
50 |
where |
|
51 |
"tsub x y = (if x >= y then x - y else 0)" |
|
52 |
||
53 |
lemma tsub_eq: "x >= y \<Longrightarrow> tsub x y = x - y" |
|
54 |
by (simp add: tsub_def) |
|
55 |
||
56 |
||
57 |
lemma transfer_nat_int_functions: |
|
58 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) + (nat y) = nat (x + y)" |
|
59 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) * (nat y) = nat (x * y)" |
|
60 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) - (nat y) = nat (tsub x y)" |
|
61 |
"(x::int) >= 0 \<Longrightarrow> (nat x)^n = nat (x^n)" |
|
62 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)" |
|
63 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)" |
|
64 |
by (auto simp add: eq_nat_nat_iff nat_mult_distrib |
|
65 |
nat_power_eq nat_div_distrib nat_mod_distrib tsub_def) |
|
66 |
||
67 |
lemma transfer_nat_int_function_closures: |
|
68 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x + y >= 0" |
|
69 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x * y >= 0" |
|
70 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> tsub x y >= 0" |
|
71 |
"(x::int) >= 0 \<Longrightarrow> x^n >= 0" |
|
72 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0" |
|
73 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0" |
|
74 |
"(0::int) >= 0" |
|
75 |
"(1::int) >= 0" |
|
76 |
"(2::int) >= 0" |
|
77 |
"(3::int) >= 0" |
|
78 |
"int z >= 0" |
|
79 |
apply (auto simp add: zero_le_mult_iff tsub_def) |
|
80 |
apply (case_tac "y = 0") |
|
81 |
apply auto |
|
82 |
apply (subst pos_imp_zdiv_nonneg_iff, auto) |
|
83 |
apply (case_tac "y = 0") |
|
84 |
apply force |
|
85 |
apply (rule pos_mod_sign) |
|
86 |
apply arith |
|
87 |
done |
|
88 |
||
89 |
lemma transfer_nat_int_relations: |
|
90 |
"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
|
91 |
(nat (x::int) = nat y) = (x = y)" |
|
92 |
"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
|
93 |
(nat (x::int) < nat y) = (x < y)" |
|
94 |
"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
|
95 |
(nat (x::int) <= nat y) = (x <= y)" |
|
96 |
"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
|
97 |
(nat (x::int) dvd nat y) = (x dvd y)" |
|
98 |
by (auto simp add: zdvd_int even_nat_def) |
|
99 |
||
100 |
declare TransferMorphism_nat_int[transfer add return: |
|
101 |
transfer_nat_int_numerals |
|
102 |
transfer_nat_int_functions |
|
103 |
transfer_nat_int_function_closures |
|
104 |
transfer_nat_int_relations |
|
105 |
] |
|
106 |
||
107 |
||
108 |
(* first-order quantifiers *) |
|
109 |
||
110 |
lemma transfer_nat_int_quantifiers: |
|
111 |
"(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))" |
|
112 |
"(EX (x::nat). P x) = (EX (x::int). x >= 0 & P (nat x))" |
|
113 |
by (rule all_nat, rule ex_nat) |
|
114 |
||
115 |
(* should we restrict these? *) |
|
116 |
lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
|
117 |
(ALL x. Q x \<longrightarrow> P x) = (ALL x. Q x \<longrightarrow> P' x)" |
|
118 |
by auto |
|
119 |
||
120 |
lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
|
121 |
(EX x. Q x \<and> P x) = (EX x. Q x \<and> P' x)" |
|
122 |
by auto |
|
123 |
||
124 |
declare TransferMorphism_nat_int[transfer add |
|
125 |
return: transfer_nat_int_quantifiers |
|
126 |
cong: all_cong ex_cong] |
|
127 |
||
128 |
||
129 |
(* if *) |
|
130 |
||
131 |
lemma nat_if_cong: "(if P then (nat x) else (nat y)) = |
|
132 |
nat (if P then x else y)" |
|
133 |
by auto |
|
134 |
||
135 |
declare TransferMorphism_nat_int [transfer add return: nat_if_cong] |
|
136 |
||
137 |
||
138 |
(* operations with sets *) |
|
139 |
||
140 |
definition |
|
141 |
nat_set :: "int set \<Rightarrow> bool" |
|
142 |
where |
|
143 |
"nat_set S = (ALL x:S. x >= 0)" |
|
144 |
||
145 |
lemma transfer_nat_int_set_functions: |
|
146 |
"card A = card (int ` A)" |
|
147 |
"{} = nat ` ({}::int set)" |
|
148 |
"A Un B = nat ` (int ` A Un int ` B)" |
|
149 |
"A Int B = nat ` (int ` A Int int ` B)" |
|
150 |
"{x. P x} = nat ` {x. x >= 0 & P(nat x)}" |
|
151 |
"{..n} = nat ` {0..int n}" |
|
152 |
"{m..n} = nat ` {int m..int n}" (* need all variants of these! *) |
|
153 |
apply (rule card_image [symmetric]) |
|
154 |
apply (auto simp add: inj_on_def image_def) |
|
155 |
apply (rule_tac x = "int x" in bexI) |
|
156 |
apply auto |
|
157 |
apply (rule_tac x = "int x" in bexI) |
|
158 |
apply auto |
|
159 |
apply (rule_tac x = "int x" in bexI) |
|
160 |
apply auto |
|
161 |
apply (rule_tac x = "int x" in exI) |
|
162 |
apply auto |
|
163 |
apply (rule_tac x = "int x" in bexI) |
|
164 |
apply auto |
|
165 |
apply (rule_tac x = "int x" in bexI) |
|
166 |
apply auto |
|
167 |
done |
|
168 |
||
169 |
lemma transfer_nat_int_set_function_closures: |
|
170 |
"nat_set {}" |
|
171 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
|
172 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
|
173 |
"x >= 0 \<Longrightarrow> nat_set {x..y}" |
|
174 |
"nat_set {x. x >= 0 & P x}" |
|
175 |
"nat_set (int ` C)" |
|
176 |
"nat_set A \<Longrightarrow> x : A \<Longrightarrow> x >= 0" (* does it hurt to turn this on? *) |
|
177 |
unfolding nat_set_def apply auto |
|
178 |
done |
|
179 |
||
180 |
lemma transfer_nat_int_set_relations: |
|
181 |
"(finite A) = (finite (int ` A))" |
|
182 |
"(x : A) = (int x : int ` A)" |
|
183 |
"(A = B) = (int ` A = int ` B)" |
|
184 |
"(A < B) = (int ` A < int ` B)" |
|
185 |
"(A <= B) = (int ` A <= int ` B)" |
|
186 |
||
187 |
apply (rule iffI) |
|
188 |
apply (erule finite_imageI) |
|
189 |
apply (erule finite_imageD) |
|
190 |
apply (auto simp add: image_def expand_set_eq inj_on_def) |
|
191 |
apply (drule_tac x = "int x" in spec, auto) |
|
192 |
apply (drule_tac x = "int x" in spec, auto) |
|
193 |
apply (drule_tac x = "int x" in spec, auto) |
|
194 |
done |
|
195 |
||
196 |
lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow> |
|
197 |
(int ` nat ` A = A)" |
|
198 |
by (auto simp add: nat_set_def image_def) |
|
199 |
||
200 |
lemma transfer_nat_int_set_cong: "(!!x. x >= 0 \<Longrightarrow> P x = P' x) \<Longrightarrow> |
|
201 |
{(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}" |
|
202 |
by auto |
|
203 |
||
204 |
declare TransferMorphism_nat_int[transfer add |
|
205 |
return: transfer_nat_int_set_functions |
|
206 |
transfer_nat_int_set_function_closures |
|
207 |
transfer_nat_int_set_relations |
|
208 |
transfer_nat_int_set_return_embed |
|
209 |
cong: transfer_nat_int_set_cong |
|
210 |
] |
|
211 |
||
212 |
||
213 |
(* setsum and setprod *) |
|
214 |
||
215 |
(* this handles the case where the *domain* of f is nat *) |
|
216 |
lemma transfer_nat_int_sum_prod: |
|
217 |
"setsum f A = setsum (%x. f (nat x)) (int ` A)" |
|
218 |
"setprod f A = setprod (%x. f (nat x)) (int ` A)" |
|
219 |
apply (subst setsum_reindex) |
|
220 |
apply (unfold inj_on_def, auto) |
|
221 |
apply (subst setprod_reindex) |
|
222 |
apply (unfold inj_on_def o_def, auto) |
|
223 |
done |
|
224 |
||
225 |
(* this handles the case where the *range* of f is nat *) |
|
226 |
lemma transfer_nat_int_sum_prod2: |
|
227 |
"setsum f A = nat(setsum (%x. int (f x)) A)" |
|
228 |
"setprod f A = nat(setprod (%x. int (f x)) A)" |
|
229 |
apply (subst int_setsum [symmetric]) |
|
230 |
apply auto |
|
231 |
apply (subst int_setprod [symmetric]) |
|
232 |
apply auto |
|
233 |
done |
|
234 |
||
235 |
lemma transfer_nat_int_sum_prod_closure: |
|
236 |
"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
|
237 |
"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
|
238 |
unfolding nat_set_def |
|
239 |
apply (rule setsum_nonneg) |
|
240 |
apply auto |
|
241 |
apply (rule setprod_nonneg) |
|
242 |
apply auto |
|
243 |
done |
|
244 |
||
245 |
(* this version doesn't work, even with nat_set A \<Longrightarrow> |
|
246 |
x : A \<Longrightarrow> x >= 0 turned on. Why not? |
|
247 |
||
248 |
also: what does =simp=> do? |
|
249 |
||
250 |
lemma transfer_nat_int_sum_prod_closure: |
|
251 |
"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
|
252 |
"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
|
253 |
unfolding nat_set_def simp_implies_def |
|
254 |
apply (rule setsum_nonneg) |
|
255 |
apply auto |
|
256 |
apply (rule setprod_nonneg) |
|
257 |
apply auto |
|
258 |
done |
|
259 |
*) |
|
260 |
||
261 |
(* Making A = B in this lemma doesn't work. Why not? |
|
262 |
Also, why aren't setsum_cong and setprod_cong enough, |
|
263 |
with the previously mentioned rule turned on? *) |
|
264 |
||
265 |
lemma transfer_nat_int_sum_prod_cong: |
|
266 |
"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
|
267 |
setsum f A = setsum g B" |
|
268 |
"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
|
269 |
setprod f A = setprod g B" |
|
270 |
unfolding nat_set_def |
|
271 |
apply (subst setsum_cong, assumption) |
|
272 |
apply auto [2] |
|
273 |
apply (subst setprod_cong, assumption, auto) |
|
274 |
done |
|
275 |
||
276 |
declare TransferMorphism_nat_int[transfer add |
|
277 |
return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2 |
|
278 |
transfer_nat_int_sum_prod_closure |
|
279 |
cong: transfer_nat_int_sum_prod_cong] |
|
280 |
||
281 |
(* lists *) |
|
282 |
||
283 |
definition |
|
284 |
embed_list :: "nat list \<Rightarrow> int list" |
|
285 |
where |
|
286 |
"embed_list l = map int l"; |
|
287 |
||
288 |
definition |
|
289 |
nat_list :: "int list \<Rightarrow> bool" |
|
290 |
where |
|
291 |
"nat_list l = nat_set (set l)"; |
|
292 |
||
293 |
definition |
|
294 |
return_list :: "int list \<Rightarrow> nat list" |
|
295 |
where |
|
296 |
"return_list l = map nat l"; |
|
297 |
||
298 |
thm nat_0_le; |
|
299 |
||
300 |
lemma transfer_nat_int_list_return_embed: "nat_list l \<longrightarrow> |
|
301 |
embed_list (return_list l) = l"; |
|
302 |
unfolding embed_list_def return_list_def nat_list_def nat_set_def |
|
303 |
apply (induct l); |
|
304 |
apply auto; |
|
305 |
done; |
|
306 |
||
307 |
lemma transfer_nat_int_list_functions: |
|
308 |
"l @ m = return_list (embed_list l @ embed_list m)" |
|
309 |
"[] = return_list []"; |
|
310 |
unfolding return_list_def embed_list_def; |
|
311 |
apply auto; |
|
312 |
apply (induct l, auto); |
|
313 |
apply (induct m, auto); |
|
314 |
done; |
|
315 |
||
316 |
(* |
|
317 |
lemma transfer_nat_int_fold1: "fold f l x = |
|
318 |
fold (%x. f (nat x)) (embed_list l) x"; |
|
319 |
*) |
|
320 |
||
321 |
||
322 |
||
323 |
||
324 |
subsection {* Set up transfer from int to nat *} |
|
325 |
||
326 |
(* set up transfer direction *) |
|
327 |
||
328 |
lemma TransferMorphism_int_nat: "TransferMorphism int (UNIV :: nat set)" |
|
329 |
by (simp add: TransferMorphism_def) |
|
330 |
||
331 |
declare TransferMorphism_int_nat[transfer add |
|
332 |
mode: manual |
|
333 |
(* labels: int-nat *) |
|
334 |
return: nat_int |
|
335 |
] |
|
336 |
||
337 |
||
338 |
(* basic functions and relations *) |
|
339 |
||
340 |
definition |
|
341 |
is_nat :: "int \<Rightarrow> bool" |
|
342 |
where |
|
343 |
"is_nat x = (x >= 0)" |
|
344 |
||
345 |
lemma transfer_int_nat_numerals: |
|
346 |
"0 = int 0" |
|
347 |
"1 = int 1" |
|
348 |
"2 = int 2" |
|
349 |
"3 = int 3" |
|
350 |
by auto |
|
351 |
||
352 |
lemma transfer_int_nat_functions: |
|
353 |
"(int x) + (int y) = int (x + y)" |
|
354 |
"(int x) * (int y) = int (x * y)" |
|
355 |
"tsub (int x) (int y) = int (x - y)" |
|
356 |
"(int x)^n = int (x^n)" |
|
357 |
"(int x) div (int y) = int (x div y)" |
|
358 |
"(int x) mod (int y) = int (x mod y)" |
|
359 |
by (auto simp add: int_mult tsub_def int_power zdiv_int zmod_int) |
|
360 |
||
361 |
lemma transfer_int_nat_function_closures: |
|
362 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)" |
|
363 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)" |
|
364 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)" |
|
365 |
"is_nat x \<Longrightarrow> is_nat (x^n)" |
|
366 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)" |
|
367 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)" |
|
368 |
"is_nat 0" |
|
369 |
"is_nat 1" |
|
370 |
"is_nat 2" |
|
371 |
"is_nat 3" |
|
372 |
"is_nat (int z)" |
|
373 |
by (simp_all only: is_nat_def transfer_nat_int_function_closures) |
|
374 |
||
375 |
lemma transfer_int_nat_relations: |
|
376 |
"(int x = int y) = (x = y)" |
|
377 |
"(int x < int y) = (x < y)" |
|
378 |
"(int x <= int y) = (x <= y)" |
|
379 |
"(int x dvd int y) = (x dvd y)" |
|
380 |
"(even (int x)) = (even x)" |
|
381 |
by (auto simp add: zdvd_int even_nat_def) |
|
382 |
||
32121 | 383 |
lemma UNIV_apply: |
384 |
"UNIV x = True" |
|
32264
0be31453f698
Set.UNIV and Set.empty are mere abbreviations for top and bot
haftmann
parents:
32121
diff
changeset
|
385 |
by (simp add: top_fun_eq top_bool_eq) |
32121 | 386 |
|
31708 | 387 |
declare TransferMorphism_int_nat[transfer add return: |
388 |
transfer_int_nat_numerals |
|
389 |
transfer_int_nat_functions |
|
390 |
transfer_int_nat_function_closures |
|
391 |
transfer_int_nat_relations |
|
32121 | 392 |
UNIV_apply |
31708 | 393 |
] |
394 |
||
395 |
||
396 |
(* first-order quantifiers *) |
|
397 |
||
398 |
lemma transfer_int_nat_quantifiers: |
|
399 |
"(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))" |
|
400 |
"(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))" |
|
401 |
apply (subst all_nat) |
|
402 |
apply auto [1] |
|
403 |
apply (subst ex_nat) |
|
404 |
apply auto |
|
405 |
done |
|
406 |
||
407 |
declare TransferMorphism_int_nat[transfer add |
|
408 |
return: transfer_int_nat_quantifiers] |
|
409 |
||
410 |
||
411 |
(* if *) |
|
412 |
||
413 |
lemma int_if_cong: "(if P then (int x) else (int y)) = |
|
414 |
int (if P then x else y)" |
|
415 |
by auto |
|
416 |
||
417 |
declare TransferMorphism_int_nat [transfer add return: int_if_cong] |
|
418 |
||
419 |
||
420 |
||
421 |
(* operations with sets *) |
|
422 |
||
423 |
lemma transfer_int_nat_set_functions: |
|
424 |
"nat_set A \<Longrightarrow> card A = card (nat ` A)" |
|
425 |
"{} = int ` ({}::nat set)" |
|
426 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)" |
|
427 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)" |
|
428 |
"{x. x >= 0 & P x} = int ` {x. P(int x)}" |
|
429 |
"is_nat m \<Longrightarrow> is_nat n \<Longrightarrow> {m..n} = int ` {nat m..nat n}" |
|
430 |
(* need all variants of these! *) |
|
431 |
by (simp_all only: is_nat_def transfer_nat_int_set_functions |
|
432 |
transfer_nat_int_set_function_closures |
|
433 |
transfer_nat_int_set_return_embed nat_0_le |
|
434 |
cong: transfer_nat_int_set_cong) |
|
435 |
||
436 |
lemma transfer_int_nat_set_function_closures: |
|
437 |
"nat_set {}" |
|
438 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
|
439 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
|
440 |
"is_nat x \<Longrightarrow> nat_set {x..y}" |
|
441 |
"nat_set {x. x >= 0 & P x}" |
|
442 |
"nat_set (int ` C)" |
|
443 |
"nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x" |
|
444 |
by (simp_all only: transfer_nat_int_set_function_closures is_nat_def) |
|
445 |
||
446 |
lemma transfer_int_nat_set_relations: |
|
447 |
"nat_set A \<Longrightarrow> finite A = finite (nat ` A)" |
|
448 |
"is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)" |
|
449 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)" |
|
450 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)" |
|
451 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)" |
|
452 |
by (simp_all only: is_nat_def transfer_nat_int_set_relations |
|
453 |
transfer_nat_int_set_return_embed nat_0_le) |
|
454 |
||
455 |
lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A" |
|
456 |
by (simp only: transfer_nat_int_set_relations |
|
457 |
transfer_nat_int_set_function_closures |
|
458 |
transfer_nat_int_set_return_embed nat_0_le) |
|
459 |
||
460 |
lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow> |
|
461 |
{(x::nat). P x} = {x. P' x}" |
|
462 |
by auto |
|
463 |
||
464 |
declare TransferMorphism_int_nat[transfer add |
|
465 |
return: transfer_int_nat_set_functions |
|
466 |
transfer_int_nat_set_function_closures |
|
467 |
transfer_int_nat_set_relations |
|
468 |
transfer_int_nat_set_return_embed |
|
469 |
cong: transfer_int_nat_set_cong |
|
470 |
] |
|
471 |
||
472 |
||
473 |
(* setsum and setprod *) |
|
474 |
||
475 |
(* this handles the case where the *domain* of f is int *) |
|
476 |
lemma transfer_int_nat_sum_prod: |
|
477 |
"nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)" |
|
478 |
"nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)" |
|
479 |
apply (subst setsum_reindex) |
|
480 |
apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff) |
|
481 |
apply (subst setprod_reindex) |
|
482 |
apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff |
|
483 |
cong: setprod_cong) |
|
484 |
done |
|
485 |
||
486 |
(* this handles the case where the *range* of f is int *) |
|
487 |
lemma transfer_int_nat_sum_prod2: |
|
488 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)" |
|
489 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> |
|
490 |
setprod f A = int(setprod (%x. nat (f x)) A)" |
|
491 |
unfolding is_nat_def |
|
492 |
apply (subst int_setsum, auto) |
|
493 |
apply (subst int_setprod, auto simp add: cong: setprod_cong) |
|
494 |
done |
|
495 |
||
496 |
declare TransferMorphism_int_nat[transfer add |
|
497 |
return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2 |
|
498 |
cong: setsum_cong setprod_cong] |
|
499 |
||
500 |
||
501 |
subsection {* Test it out *} |
|
502 |
||
503 |
(* nat to int *) |
|
504 |
||
505 |
lemma ex1: "(x::nat) + y = y + x" |
|
506 |
by auto |
|
507 |
||
508 |
thm ex1 [transferred] |
|
509 |
||
510 |
lemma ex2: "(a::nat) div b * b + a mod b = a" |
|
511 |
by (rule mod_div_equality) |
|
512 |
||
513 |
thm ex2 [transferred] |
|
514 |
||
515 |
lemma ex3: "ALL (x::nat). ALL y. EX z. z >= x + y" |
|
516 |
by auto |
|
517 |
||
518 |
thm ex3 [transferred natint] |
|
519 |
||
520 |
lemma ex4: "(x::nat) >= y \<Longrightarrow> (x - y) + y = x" |
|
521 |
by auto |
|
522 |
||
523 |
thm ex4 [transferred] |
|
524 |
||
525 |
lemma ex5: "(2::nat) * (SUM i <= n. i) = n * (n + 1)" |
|
526 |
by (induct n rule: nat_induct, auto) |
|
527 |
||
528 |
thm ex5 [transferred] |
|
529 |
||
530 |
theorem ex6: "0 <= (n::int) \<Longrightarrow> 2 * \<Sum>{0..n} = n * (n + 1)" |
|
531 |
by (rule ex5 [transferred]) |
|
532 |
||
533 |
thm ex6 [transferred] |
|
534 |
||
535 |
thm ex5 [transferred, transferred] |
|
536 |
||
537 |
end |