author | blanchet |
Mon, 23 Jan 2012 17:40:32 +0100 | |
changeset 46320 | 0b8b73b49848 |
parent 42870 | 36abaf4cce1f |
child 47255 | 30a1692557b0 |
permissions | -rw-r--r-- |
31708 | 1 |
|
32554 | 2 |
(* Authors: Jeremy Avigad and Amine Chaieb *) |
31708 | 3 |
|
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
4 |
header {* Generic transfer machinery; specific transfer from nats to ints and back. *} |
31708 | 5 |
|
32558 | 6 |
theory Nat_Transfer |
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
7 |
imports Nat_Numeral |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
8 |
uses ("Tools/transfer.ML") |
31708 | 9 |
begin |
10 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
11 |
subsection {* Generic transfer machinery *} |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
12 |
|
35821 | 13 |
definition transfer_morphism:: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> bool" |
42870
36abaf4cce1f
clarified vacuous nature of predicate "transfer_morphism" -- equivalent to previous definiton
krauss
parents:
39302
diff
changeset
|
14 |
where "transfer_morphism f A \<longleftrightarrow> True" |
35644 | 15 |
|
42870
36abaf4cce1f
clarified vacuous nature of predicate "transfer_morphism" -- equivalent to previous definiton
krauss
parents:
39302
diff
changeset
|
16 |
lemma transfer_morphismI[intro]: "transfer_morphism f A" |
36abaf4cce1f
clarified vacuous nature of predicate "transfer_morphism" -- equivalent to previous definiton
krauss
parents:
39302
diff
changeset
|
17 |
by (simp add: transfer_morphism_def) |
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
18 |
|
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
19 |
use "Tools/transfer.ML" |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
20 |
|
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
21 |
setup Transfer.setup |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
22 |
|
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
23 |
|
31708 | 24 |
subsection {* Set up transfer from nat to int *} |
25 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
26 |
text {* set up transfer direction *} |
31708 | 27 |
|
42870
36abaf4cce1f
clarified vacuous nature of predicate "transfer_morphism" -- equivalent to previous definiton
krauss
parents:
39302
diff
changeset
|
28 |
lemma transfer_morphism_nat_int: "transfer_morphism nat (op <= (0::int))" .. |
31708 | 29 |
|
35683 | 30 |
declare transfer_morphism_nat_int [transfer add |
31 |
mode: manual |
|
31708 | 32 |
return: nat_0_le |
35683 | 33 |
labels: nat_int |
31708 | 34 |
] |
35 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
36 |
text {* basic functions and relations *} |
31708 | 37 |
|
35683 | 38 |
lemma transfer_nat_int_numerals [transfer key: transfer_morphism_nat_int]: |
31708 | 39 |
"(0::nat) = nat 0" |
40 |
"(1::nat) = nat 1" |
|
41 |
"(2::nat) = nat 2" |
|
42 |
"(3::nat) = nat 3" |
|
43 |
by auto |
|
44 |
||
45 |
definition |
|
46 |
tsub :: "int \<Rightarrow> int \<Rightarrow> int" |
|
47 |
where |
|
48 |
"tsub x y = (if x >= y then x - y else 0)" |
|
49 |
||
50 |
lemma tsub_eq: "x >= y \<Longrightarrow> tsub x y = x - y" |
|
51 |
by (simp add: tsub_def) |
|
52 |
||
35683 | 53 |
lemma transfer_nat_int_functions [transfer key: transfer_morphism_nat_int]: |
31708 | 54 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) + (nat y) = nat (x + y)" |
55 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) * (nat y) = nat (x * y)" |
|
56 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) - (nat y) = nat (tsub x y)" |
|
57 |
"(x::int) >= 0 \<Longrightarrow> (nat x)^n = nat (x^n)" |
|
58 |
by (auto simp add: eq_nat_nat_iff nat_mult_distrib |
|
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
59 |
nat_power_eq tsub_def) |
31708 | 60 |
|
35683 | 61 |
lemma transfer_nat_int_function_closures [transfer key: transfer_morphism_nat_int]: |
31708 | 62 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x + y >= 0" |
63 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x * y >= 0" |
|
64 |
"(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> tsub x y >= 0" |
|
65 |
"(x::int) >= 0 \<Longrightarrow> x^n >= 0" |
|
66 |
"(0::int) >= 0" |
|
67 |
"(1::int) >= 0" |
|
68 |
"(2::int) >= 0" |
|
69 |
"(3::int) >= 0" |
|
70 |
"int z >= 0" |
|
33340
a165b97f3658
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
33318
diff
changeset
|
71 |
by (auto simp add: zero_le_mult_iff tsub_def) |
31708 | 72 |
|
35683 | 73 |
lemma transfer_nat_int_relations [transfer key: transfer_morphism_nat_int]: |
31708 | 74 |
"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
75 |
(nat (x::int) = nat y) = (x = y)" |
|
76 |
"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
|
77 |
(nat (x::int) < nat y) = (x < y)" |
|
78 |
"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
|
79 |
(nat (x::int) <= nat y) = (x <= y)" |
|
80 |
"x >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> |
|
81 |
(nat (x::int) dvd nat y) = (x dvd y)" |
|
32558 | 82 |
by (auto simp add: zdvd_int) |
31708 | 83 |
|
84 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
85 |
text {* first-order quantifiers *} |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
86 |
|
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
87 |
lemma all_nat: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x\<ge>0. P (nat x))" |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
88 |
by (simp split add: split_nat) |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
89 |
|
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
90 |
lemma ex_nat: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. 0 \<le> x \<and> P (nat x))" |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
91 |
proof |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
92 |
assume "\<exists>x. P x" |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
93 |
then obtain x where "P x" .. |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
94 |
then have "int x \<ge> 0 \<and> P (nat (int x))" by simp |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
95 |
then show "\<exists>x\<ge>0. P (nat x)" .. |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
96 |
next |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
97 |
assume "\<exists>x\<ge>0. P (nat x)" |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
98 |
then show "\<exists>x. P x" by auto |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
99 |
qed |
31708 | 100 |
|
35683 | 101 |
lemma transfer_nat_int_quantifiers [transfer key: transfer_morphism_nat_int]: |
31708 | 102 |
"(ALL (x::nat). P x) = (ALL (x::int). x >= 0 \<longrightarrow> P (nat x))" |
103 |
"(EX (x::nat). P x) = (EX (x::int). x >= 0 & P (nat x))" |
|
104 |
by (rule all_nat, rule ex_nat) |
|
105 |
||
106 |
(* should we restrict these? *) |
|
107 |
lemma all_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
|
108 |
(ALL x. Q x \<longrightarrow> P x) = (ALL x. Q x \<longrightarrow> P' x)" |
|
109 |
by auto |
|
110 |
||
111 |
lemma ex_cong: "(\<And>x. Q x \<Longrightarrow> P x = P' x) \<Longrightarrow> |
|
112 |
(EX x. Q x \<and> P x) = (EX x. Q x \<and> P' x)" |
|
113 |
by auto |
|
114 |
||
35644 | 115 |
declare transfer_morphism_nat_int [transfer add |
31708 | 116 |
cong: all_cong ex_cong] |
117 |
||
118 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
119 |
text {* if *} |
31708 | 120 |
|
35683 | 121 |
lemma nat_if_cong [transfer key: transfer_morphism_nat_int]: |
122 |
"(if P then (nat x) else (nat y)) = nat (if P then x else y)" |
|
31708 | 123 |
by auto |
124 |
||
125 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
126 |
text {* operations with sets *} |
31708 | 127 |
|
128 |
definition |
|
129 |
nat_set :: "int set \<Rightarrow> bool" |
|
130 |
where |
|
131 |
"nat_set S = (ALL x:S. x >= 0)" |
|
132 |
||
133 |
lemma transfer_nat_int_set_functions: |
|
134 |
"card A = card (int ` A)" |
|
135 |
"{} = nat ` ({}::int set)" |
|
136 |
"A Un B = nat ` (int ` A Un int ` B)" |
|
137 |
"A Int B = nat ` (int ` A Int int ` B)" |
|
138 |
"{x. P x} = nat ` {x. x >= 0 & P(nat x)}" |
|
139 |
apply (rule card_image [symmetric]) |
|
140 |
apply (auto simp add: inj_on_def image_def) |
|
141 |
apply (rule_tac x = "int x" in bexI) |
|
142 |
apply auto |
|
143 |
apply (rule_tac x = "int x" in bexI) |
|
144 |
apply auto |
|
145 |
apply (rule_tac x = "int x" in bexI) |
|
146 |
apply auto |
|
147 |
apply (rule_tac x = "int x" in exI) |
|
148 |
apply auto |
|
149 |
done |
|
150 |
||
151 |
lemma transfer_nat_int_set_function_closures: |
|
152 |
"nat_set {}" |
|
153 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
|
154 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
|
155 |
"nat_set {x. x >= 0 & P x}" |
|
156 |
"nat_set (int ` C)" |
|
157 |
"nat_set A \<Longrightarrow> x : A \<Longrightarrow> x >= 0" (* does it hurt to turn this on? *) |
|
158 |
unfolding nat_set_def apply auto |
|
159 |
done |
|
160 |
||
161 |
lemma transfer_nat_int_set_relations: |
|
162 |
"(finite A) = (finite (int ` A))" |
|
163 |
"(x : A) = (int x : int ` A)" |
|
164 |
"(A = B) = (int ` A = int ` B)" |
|
165 |
"(A < B) = (int ` A < int ` B)" |
|
166 |
"(A <= B) = (int ` A <= int ` B)" |
|
167 |
apply (rule iffI) |
|
168 |
apply (erule finite_imageI) |
|
169 |
apply (erule finite_imageD) |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
170 |
apply (auto simp add: image_def set_eq_iff inj_on_def) |
31708 | 171 |
apply (drule_tac x = "int x" in spec, auto) |
172 |
apply (drule_tac x = "int x" in spec, auto) |
|
173 |
apply (drule_tac x = "int x" in spec, auto) |
|
174 |
done |
|
175 |
||
176 |
lemma transfer_nat_int_set_return_embed: "nat_set A \<Longrightarrow> |
|
177 |
(int ` nat ` A = A)" |
|
178 |
by (auto simp add: nat_set_def image_def) |
|
179 |
||
180 |
lemma transfer_nat_int_set_cong: "(!!x. x >= 0 \<Longrightarrow> P x = P' x) \<Longrightarrow> |
|
181 |
{(x::int). x >= 0 & P x} = {x. x >= 0 & P' x}" |
|
182 |
by auto |
|
183 |
||
35644 | 184 |
declare transfer_morphism_nat_int [transfer add |
31708 | 185 |
return: transfer_nat_int_set_functions |
186 |
transfer_nat_int_set_function_closures |
|
187 |
transfer_nat_int_set_relations |
|
188 |
transfer_nat_int_set_return_embed |
|
189 |
cong: transfer_nat_int_set_cong |
|
190 |
] |
|
191 |
||
192 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
193 |
text {* setsum and setprod *} |
31708 | 194 |
|
195 |
(* this handles the case where the *domain* of f is nat *) |
|
196 |
lemma transfer_nat_int_sum_prod: |
|
197 |
"setsum f A = setsum (%x. f (nat x)) (int ` A)" |
|
198 |
"setprod f A = setprod (%x. f (nat x)) (int ` A)" |
|
199 |
apply (subst setsum_reindex) |
|
200 |
apply (unfold inj_on_def, auto) |
|
201 |
apply (subst setprod_reindex) |
|
202 |
apply (unfold inj_on_def o_def, auto) |
|
203 |
done |
|
204 |
||
205 |
(* this handles the case where the *range* of f is nat *) |
|
206 |
lemma transfer_nat_int_sum_prod2: |
|
207 |
"setsum f A = nat(setsum (%x. int (f x)) A)" |
|
208 |
"setprod f A = nat(setprod (%x. int (f x)) A)" |
|
209 |
apply (subst int_setsum [symmetric]) |
|
210 |
apply auto |
|
211 |
apply (subst int_setprod [symmetric]) |
|
212 |
apply auto |
|
213 |
done |
|
214 |
||
215 |
lemma transfer_nat_int_sum_prod_closure: |
|
216 |
"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
|
217 |
"nat_set A \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
|
218 |
unfolding nat_set_def |
|
219 |
apply (rule setsum_nonneg) |
|
220 |
apply auto |
|
221 |
apply (rule setprod_nonneg) |
|
222 |
apply auto |
|
223 |
done |
|
224 |
||
225 |
(* this version doesn't work, even with nat_set A \<Longrightarrow> |
|
226 |
x : A \<Longrightarrow> x >= 0 turned on. Why not? |
|
227 |
||
228 |
also: what does =simp=> do? |
|
229 |
||
230 |
lemma transfer_nat_int_sum_prod_closure: |
|
231 |
"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setsum f A >= 0" |
|
232 |
"(!!x. x : A ==> f x >= (0::int)) \<Longrightarrow> setprod f A >= 0" |
|
233 |
unfolding nat_set_def simp_implies_def |
|
234 |
apply (rule setsum_nonneg) |
|
235 |
apply auto |
|
236 |
apply (rule setprod_nonneg) |
|
237 |
apply auto |
|
238 |
done |
|
239 |
*) |
|
240 |
||
241 |
(* Making A = B in this lemma doesn't work. Why not? |
|
242 |
Also, why aren't setsum_cong and setprod_cong enough, |
|
243 |
with the previously mentioned rule turned on? *) |
|
244 |
||
245 |
lemma transfer_nat_int_sum_prod_cong: |
|
246 |
"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
|
247 |
setsum f A = setsum g B" |
|
248 |
"A = B \<Longrightarrow> nat_set B \<Longrightarrow> (!!x. x >= 0 \<Longrightarrow> f x = g x) \<Longrightarrow> |
|
249 |
setprod f A = setprod g B" |
|
250 |
unfolding nat_set_def |
|
251 |
apply (subst setsum_cong, assumption) |
|
252 |
apply auto [2] |
|
253 |
apply (subst setprod_cong, assumption, auto) |
|
254 |
done |
|
255 |
||
35644 | 256 |
declare transfer_morphism_nat_int [transfer add |
31708 | 257 |
return: transfer_nat_int_sum_prod transfer_nat_int_sum_prod2 |
258 |
transfer_nat_int_sum_prod_closure |
|
259 |
cong: transfer_nat_int_sum_prod_cong] |
|
260 |
||
261 |
||
262 |
subsection {* Set up transfer from int to nat *} |
|
263 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
264 |
text {* set up transfer direction *} |
31708 | 265 |
|
42870
36abaf4cce1f
clarified vacuous nature of predicate "transfer_morphism" -- equivalent to previous definiton
krauss
parents:
39302
diff
changeset
|
266 |
lemma transfer_morphism_int_nat: "transfer_morphism int (\<lambda>n. True)" .. |
31708 | 267 |
|
35644 | 268 |
declare transfer_morphism_int_nat [transfer add |
31708 | 269 |
mode: manual |
270 |
return: nat_int |
|
35683 | 271 |
labels: int_nat |
31708 | 272 |
] |
273 |
||
274 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
275 |
text {* basic functions and relations *} |
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
276 |
|
31708 | 277 |
definition |
278 |
is_nat :: "int \<Rightarrow> bool" |
|
279 |
where |
|
280 |
"is_nat x = (x >= 0)" |
|
281 |
||
282 |
lemma transfer_int_nat_numerals: |
|
283 |
"0 = int 0" |
|
284 |
"1 = int 1" |
|
285 |
"2 = int 2" |
|
286 |
"3 = int 3" |
|
287 |
by auto |
|
288 |
||
289 |
lemma transfer_int_nat_functions: |
|
290 |
"(int x) + (int y) = int (x + y)" |
|
291 |
"(int x) * (int y) = int (x * y)" |
|
292 |
"tsub (int x) (int y) = int (x - y)" |
|
293 |
"(int x)^n = int (x^n)" |
|
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
294 |
by (auto simp add: int_mult tsub_def int_power) |
31708 | 295 |
|
296 |
lemma transfer_int_nat_function_closures: |
|
297 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x + y)" |
|
298 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x * y)" |
|
299 |
"is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (tsub x y)" |
|
300 |
"is_nat x \<Longrightarrow> is_nat (x^n)" |
|
301 |
"is_nat 0" |
|
302 |
"is_nat 1" |
|
303 |
"is_nat 2" |
|
304 |
"is_nat 3" |
|
305 |
"is_nat (int z)" |
|
306 |
by (simp_all only: is_nat_def transfer_nat_int_function_closures) |
|
307 |
||
308 |
lemma transfer_int_nat_relations: |
|
309 |
"(int x = int y) = (x = y)" |
|
310 |
"(int x < int y) = (x < y)" |
|
311 |
"(int x <= int y) = (x <= y)" |
|
312 |
"(int x dvd int y) = (x dvd y)" |
|
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
313 |
by (auto simp add: zdvd_int) |
32121 | 314 |
|
35644 | 315 |
declare transfer_morphism_int_nat [transfer add return: |
31708 | 316 |
transfer_int_nat_numerals |
317 |
transfer_int_nat_functions |
|
318 |
transfer_int_nat_function_closures |
|
319 |
transfer_int_nat_relations |
|
320 |
] |
|
321 |
||
322 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
323 |
text {* first-order quantifiers *} |
31708 | 324 |
|
325 |
lemma transfer_int_nat_quantifiers: |
|
326 |
"(ALL (x::int) >= 0. P x) = (ALL (x::nat). P (int x))" |
|
327 |
"(EX (x::int) >= 0. P x) = (EX (x::nat). P (int x))" |
|
328 |
apply (subst all_nat) |
|
329 |
apply auto [1] |
|
330 |
apply (subst ex_nat) |
|
331 |
apply auto |
|
332 |
done |
|
333 |
||
35644 | 334 |
declare transfer_morphism_int_nat [transfer add |
31708 | 335 |
return: transfer_int_nat_quantifiers] |
336 |
||
337 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
338 |
text {* if *} |
31708 | 339 |
|
340 |
lemma int_if_cong: "(if P then (int x) else (int y)) = |
|
341 |
int (if P then x else y)" |
|
342 |
by auto |
|
343 |
||
35644 | 344 |
declare transfer_morphism_int_nat [transfer add return: int_if_cong] |
31708 | 345 |
|
346 |
||
347 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
348 |
text {* operations with sets *} |
31708 | 349 |
|
350 |
lemma transfer_int_nat_set_functions: |
|
351 |
"nat_set A \<Longrightarrow> card A = card (nat ` A)" |
|
352 |
"{} = int ` ({}::nat set)" |
|
353 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Un B = int ` (nat ` A Un nat ` B)" |
|
354 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> A Int B = int ` (nat ` A Int nat ` B)" |
|
355 |
"{x. x >= 0 & P x} = int ` {x. P(int x)}" |
|
356 |
(* need all variants of these! *) |
|
357 |
by (simp_all only: is_nat_def transfer_nat_int_set_functions |
|
358 |
transfer_nat_int_set_function_closures |
|
359 |
transfer_nat_int_set_return_embed nat_0_le |
|
360 |
cong: transfer_nat_int_set_cong) |
|
361 |
||
362 |
lemma transfer_int_nat_set_function_closures: |
|
363 |
"nat_set {}" |
|
364 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Un B)" |
|
365 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> nat_set (A Int B)" |
|
366 |
"nat_set {x. x >= 0 & P x}" |
|
367 |
"nat_set (int ` C)" |
|
368 |
"nat_set A \<Longrightarrow> x : A \<Longrightarrow> is_nat x" |
|
369 |
by (simp_all only: transfer_nat_int_set_function_closures is_nat_def) |
|
370 |
||
371 |
lemma transfer_int_nat_set_relations: |
|
372 |
"nat_set A \<Longrightarrow> finite A = finite (nat ` A)" |
|
373 |
"is_nat x \<Longrightarrow> nat_set A \<Longrightarrow> (x : A) = (nat x : nat ` A)" |
|
374 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A = B) = (nat ` A = nat ` B)" |
|
375 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A < B) = (nat ` A < nat ` B)" |
|
376 |
"nat_set A \<Longrightarrow> nat_set B \<Longrightarrow> (A <= B) = (nat ` A <= nat ` B)" |
|
377 |
by (simp_all only: is_nat_def transfer_nat_int_set_relations |
|
378 |
transfer_nat_int_set_return_embed nat_0_le) |
|
379 |
||
380 |
lemma transfer_int_nat_set_return_embed: "nat ` int ` A = A" |
|
381 |
by (simp only: transfer_nat_int_set_relations |
|
382 |
transfer_nat_int_set_function_closures |
|
383 |
transfer_nat_int_set_return_embed nat_0_le) |
|
384 |
||
385 |
lemma transfer_int_nat_set_cong: "(!!x. P x = P' x) \<Longrightarrow> |
|
386 |
{(x::nat). P x} = {x. P' x}" |
|
387 |
by auto |
|
388 |
||
35644 | 389 |
declare transfer_morphism_int_nat [transfer add |
31708 | 390 |
return: transfer_int_nat_set_functions |
391 |
transfer_int_nat_set_function_closures |
|
392 |
transfer_int_nat_set_relations |
|
393 |
transfer_int_nat_set_return_embed |
|
394 |
cong: transfer_int_nat_set_cong |
|
395 |
] |
|
396 |
||
397 |
||
33318
ddd97d9dfbfb
moved Nat_Transfer before Divides; distributed Nat_Transfer setup accordingly
haftmann
parents:
32558
diff
changeset
|
398 |
text {* setsum and setprod *} |
31708 | 399 |
|
400 |
(* this handles the case where the *domain* of f is int *) |
|
401 |
lemma transfer_int_nat_sum_prod: |
|
402 |
"nat_set A \<Longrightarrow> setsum f A = setsum (%x. f (int x)) (nat ` A)" |
|
403 |
"nat_set A \<Longrightarrow> setprod f A = setprod (%x. f (int x)) (nat ` A)" |
|
404 |
apply (subst setsum_reindex) |
|
405 |
apply (unfold inj_on_def nat_set_def, auto simp add: eq_nat_nat_iff) |
|
406 |
apply (subst setprod_reindex) |
|
407 |
apply (unfold inj_on_def nat_set_def o_def, auto simp add: eq_nat_nat_iff |
|
408 |
cong: setprod_cong) |
|
409 |
done |
|
410 |
||
411 |
(* this handles the case where the *range* of f is int *) |
|
412 |
lemma transfer_int_nat_sum_prod2: |
|
413 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> setsum f A = int(setsum (%x. nat (f x)) A)" |
|
414 |
"(!!x. x:A \<Longrightarrow> is_nat (f x)) \<Longrightarrow> |
|
415 |
setprod f A = int(setprod (%x. nat (f x)) A)" |
|
416 |
unfolding is_nat_def |
|
417 |
apply (subst int_setsum, auto) |
|
418 |
apply (subst int_setprod, auto simp add: cong: setprod_cong) |
|
419 |
done |
|
420 |
||
35644 | 421 |
declare transfer_morphism_int_nat [transfer add |
31708 | 422 |
return: transfer_int_nat_sum_prod transfer_int_nat_sum_prod2 |
423 |
cong: setsum_cong setprod_cong] |
|
424 |
||
425 |
end |