src/HOL/Library/FuncSet.thy
 author Christian Sternagel Thu Aug 30 15:44:03 2012 +0900 (2012-08-30) changeset 49093 fdc301f592c4 parent 47761 dfe747e72fa8 child 50104 de19856feb54 permissions -rw-r--r--
1 (*  Title:      HOL/Library/FuncSet.thy
2     Author:     Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
3 *)
5 header {* Pi and Function Sets *}
7 theory FuncSet
8 imports Hilbert_Choice Main
9 begin
11 definition
12   Pi :: "['a set, 'a => 'b set] => ('a => 'b) set" where
13   "Pi A B = {f. \<forall>x. x \<in> A --> f x \<in> B x}"
15 definition
16   extensional :: "'a set => ('a => 'b) set" where
17   "extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
19 definition
20   "restrict" :: "['a => 'b, 'a set] => ('a => 'b)" where
21   "restrict f A = (%x. if x \<in> A then f x else undefined)"
23 abbreviation
24   funcset :: "['a set, 'b set] => ('a => 'b) set"
25     (infixr "->" 60) where
26   "A -> B == Pi A (%_. B)"
28 notation (xsymbols)
29   funcset  (infixr "\<rightarrow>" 60)
31 syntax
32   "_Pi"  :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3PI _:_./ _)" 10)
33   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3%_:_./ _)" [0,0,3] 3)
35 syntax (xsymbols)
36   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
37   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
39 syntax (HTML output)
40   "_Pi" :: "[pttrn, 'a set, 'b set] => ('a => 'b) set"  ("(3\<Pi> _\<in>_./ _)"   10)
41   "_lam" :: "[pttrn, 'a set, 'a => 'b] => ('a=>'b)"  ("(3\<lambda>_\<in>_./ _)" [0,0,3] 3)
43 translations
44   "PI x:A. B" == "CONST Pi A (%x. B)"
45   "%x:A. f" == "CONST restrict (%x. f) A"
47 definition
48   "compose" :: "['a set, 'b => 'c, 'a => 'b] => ('a => 'c)" where
49   "compose A g f = (\<lambda>x\<in>A. g (f x))"
52 subsection{*Basic Properties of @{term Pi}*}
54 lemma Pi_I[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
57 lemma Pi_I'[simp]: "(!!x. x : A --> f x : B x) ==> f : Pi A B"
60 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
63 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
66 lemma Pi_iff: "f \<in> Pi I X \<longleftrightarrow> (\<forall>i\<in>I. f i \<in> X i)"
67   unfolding Pi_def by auto
69 lemma PiE [elim]:
70   "f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
71 by(auto simp: Pi_def)
73 lemma Pi_cong:
74   "(\<And> w. w \<in> A \<Longrightarrow> f w = g w) \<Longrightarrow> f \<in> Pi A B \<longleftrightarrow> g \<in> Pi A B"
75   by (auto simp: Pi_def)
77 lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
78   by auto
80 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
83 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
84 by auto
86 lemma Pi_eq_empty[simp]: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
87 apply (simp add: Pi_def, auto)
88 txt{*Converse direction requires Axiom of Choice to exhibit a function
89 picking an element from each non-empty @{term "B x"}*}
90 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
91 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
92 done
94 lemma Pi_empty [simp]: "Pi {} B = UNIV"
97 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
99 (*
100 lemma funcset_id [simp]: "(%x. x): A -> A"
102 *)
103 text{*Covariance of Pi-sets in their second argument*}
104 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
105 by auto
107 text{*Contravariance of Pi-sets in their first argument*}
108 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
109 by auto
111 lemma prod_final:
112   assumes 1: "fst \<circ> f \<in> Pi A B" and 2: "snd \<circ> f \<in> Pi A C"
113   shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
114 proof (rule Pi_I)
115   fix z
116   assume z: "z \<in> A"
117   have "f z = (fst (f z), snd (f z))"
118     by simp
119   also have "...  \<in> B z \<times> C z"
120     by (metis SigmaI PiE o_apply 1 2 z)
121   finally show "f z \<in> B z \<times> C z" .
122 qed
125 subsection{*Composition With a Restricted Domain: @{term compose}*}
127 lemma funcset_compose:
128   "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
129 by (simp add: Pi_def compose_def restrict_def)
131 lemma compose_assoc:
132     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
133       ==> compose A h (compose A g f) = compose A (compose B h g) f"
134 by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
136 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
137 by (simp add: compose_def restrict_def)
139 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
140   by (auto simp add: image_def compose_eq)
143 subsection{*Bounded Abstraction: @{term restrict}*}
145 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
146   by (simp add: Pi_def restrict_def)
148 lemma restrictI[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
149   by (simp add: Pi_def restrict_def)
151 lemma restrict_apply [simp]:
152     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
155 lemma restrict_ext:
156     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
157   by (simp add: fun_eq_iff Pi_def restrict_def)
159 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
160   by (simp add: inj_on_def restrict_def)
162 lemma Id_compose:
163     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
164   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
166 lemma compose_Id:
167     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
168   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
170 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
171   by (auto simp add: restrict_def)
174 subsection{*Bijections Between Sets*}
176 text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
177 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
179 lemma bij_betwI:
180 assumes "f \<in> A \<rightarrow> B" and "g \<in> B \<rightarrow> A"
181     and g_f: "\<And>x. x\<in>A \<Longrightarrow> g (f x) = x" and f_g: "\<And>y. y\<in>B \<Longrightarrow> f (g y) = y"
182 shows "bij_betw f A B"
183 unfolding bij_betw_def
184 proof
185   show "inj_on f A" by (metis g_f inj_on_def)
186 next
187   have "f ` A \<subseteq> B" using `f \<in> A \<rightarrow> B` by auto
188   moreover
189   have "B \<subseteq> f ` A" by auto (metis Pi_mem `g \<in> B \<rightarrow> A` f_g image_iff)
190   ultimately show "f ` A = B" by blast
191 qed
193 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
194 by (auto simp add: bij_betw_def)
196 lemma inj_on_compose:
197   "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
198 by (auto simp add: bij_betw_def inj_on_def compose_eq)
200 lemma bij_betw_compose:
201   "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
202 apply (simp add: bij_betw_def compose_eq inj_on_compose)
203 apply (auto simp add: compose_def image_def)
204 done
206 lemma bij_betw_restrict_eq [simp]:
207   "bij_betw (restrict f A) A B = bij_betw f A B"
211 subsection{*Extensionality*}
213 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
216 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
217 by (simp add: restrict_def extensional_def)
219 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
222 lemma extensionalityI:
223   "[| f \<in> extensional A; g \<in> extensional A;
224       !!x. x\<in>A ==> f x = g x |] ==> f = g"
225 by (force simp add: fun_eq_iff extensional_def)
227 lemma extensional_restrict:  "f \<in> extensional A \<Longrightarrow> restrict f A = f"
228 by(rule extensionalityI[OF restrict_extensional]) auto
230 lemma inv_into_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_into A f x) : B -> A"
231 by (unfold inv_into_def) (fast intro: someI2)
233 lemma compose_inv_into_id:
234   "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
235 apply (simp add: bij_betw_def compose_def)
236 apply (rule restrict_ext, auto)
237 done
239 lemma compose_id_inv_into:
240   "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
242 apply (rule restrict_ext)
244 done
247 subsection{*Cardinality*}
249 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
250 by (rule card_inj_on_le) auto
252 lemma card_bij:
253   "[|f \<in> A\<rightarrow>B; inj_on f A;
254      g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
255 by (blast intro: card_inj order_antisym)
257 subsection {* Extensional Function Spaces *}
259 definition extensional_funcset
260 where "extensional_funcset S T = (S -> T) \<inter> (extensional S)"
262 lemma extensional_empty[simp]: "extensional {} = {%x. undefined}"
263 unfolding extensional_def by auto
265 lemma extensional_funcset_empty_domain: "extensional_funcset {} T = {%x. undefined}"
266 unfolding extensional_funcset_def by simp
268 lemma extensional_funcset_empty_range:
269   assumes "S \<noteq> {}"
270   shows "extensional_funcset S {} = {}"
271 using assms unfolding extensional_funcset_def by auto
273 lemma extensional_funcset_arb:
274   assumes "f \<in> extensional_funcset S T" "x \<notin> S"
275   shows "f x = undefined"
276 using assms
277 unfolding extensional_funcset_def by auto (auto dest!: extensional_arb)
279 lemma extensional_funcset_mem: "f \<in> extensional_funcset S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T"
280 unfolding extensional_funcset_def by auto
282 lemma extensional_subset: "f : extensional A ==> A <= B ==> f : extensional B"
283 unfolding extensional_def by auto
285 lemma extensional_funcset_extend_domainI: "\<lbrakk> y \<in> T; f \<in> extensional_funcset S T\<rbrakk> \<Longrightarrow> f(x := y) \<in> extensional_funcset (insert x S) T"
286 unfolding extensional_funcset_def extensional_def by auto
288 lemma extensional_funcset_restrict_domain:
289   "x \<notin> S \<Longrightarrow> f \<in> extensional_funcset (insert x S) T \<Longrightarrow> f(x := undefined) \<in> extensional_funcset S T"
290 unfolding extensional_funcset_def extensional_def by auto
292 lemma extensional_funcset_extend_domain_eq:
293   assumes "x \<notin> S"
294   shows
295     "extensional_funcset (insert x S) T = (\<lambda>(y, g). g(x := y)) ` {(y, g). y \<in> T \<and> g \<in> extensional_funcset S T}"
296   using assms
297 proof -
298   {
299     fix f
300     assume "f : extensional_funcset (insert x S) T"
301     from this assms have "f : (%(y, g). g(x := y)) ` (T <*> extensional_funcset S T)"
302       unfolding image_iff
303       apply (rule_tac x="(f x, f(x := undefined))" in bexI)
304     apply (auto intro: extensional_funcset_extend_domainI extensional_funcset_restrict_domain extensional_funcset_mem) done
305   }
306   moreover
307   {
308     fix f
309     assume "f : (%(y, g). g(x := y)) ` (T <*> extensional_funcset S T)"
310     from this assms have "f : extensional_funcset (insert x S) T"
311       by (auto intro: extensional_funcset_extend_domainI)
312   }
313   ultimately show ?thesis by auto
314 qed
316 lemma extensional_funcset_fun_upd_restricts_rangeI:  "\<forall> y \<in> S. f x \<noteq> f y ==> f : extensional_funcset (insert x S) T ==> f(x := undefined) : extensional_funcset S (T - {f x})"
317 unfolding extensional_funcset_def extensional_def
318 apply auto
319 apply (case_tac "x = xa")
320 apply auto done
322 lemma extensional_funcset_fun_upd_extends_rangeI:
323   assumes "a \<in> T" "f : extensional_funcset S (T - {a})"
324   shows "f(x := a) : extensional_funcset (insert x S) T"
325   using assms unfolding extensional_funcset_def extensional_def by auto
327 subsubsection {* Injective Extensional Function Spaces *}
329 lemma extensional_funcset_fun_upd_inj_onI:
330   assumes "f \<in> extensional_funcset S (T - {a})" "inj_on f S"
331   shows "inj_on (f(x := a)) S"
332   using assms unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
334 lemma extensional_funcset_extend_domain_inj_on_eq:
335   assumes "x \<notin> S"
336   shows"{f. f \<in> extensional_funcset (insert x S) T \<and> inj_on f (insert x S)} =
337     (%(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S}"
338 proof -
339   from assms show ?thesis
340     apply auto
341     apply (auto intro: extensional_funcset_fun_upd_inj_onI extensional_funcset_fun_upd_extends_rangeI)
342     apply (auto simp add: image_iff inj_on_def)
343     apply (rule_tac x="xa x" in exI)
344     apply (auto intro: extensional_funcset_mem)
345     apply (rule_tac x="xa(x := undefined)" in exI)
346     apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
347     apply (auto dest!: extensional_funcset_mem split: split_if_asm)
348     done
349 qed
351 lemma extensional_funcset_extend_domain_inj_onI:
352   assumes "x \<notin> S"
353   shows "inj_on (\<lambda>(y, g). g(x := y)) {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S}"
354 proof -
355   from assms show ?thesis
356     apply (auto intro!: inj_onI)
357     apply (metis fun_upd_same)
358     by (metis assms extensional_funcset_arb fun_upd_triv fun_upd_upd)
359 qed
362 subsubsection {* Cardinality *}
364 lemma card_extensional_funcset:
365   assumes "finite S"
366   shows "card (extensional_funcset S T) = (card T) ^ (card S)"
367 using assms
368 proof (induct rule: finite_induct)
369   case empty
370   show ?case
371     by (auto simp add: extensional_funcset_empty_domain)
372 next
373   case (insert x S)
374   {
375     fix g g' y y'
376     assume assms: "g \<in> extensional_funcset S T"
377       "g' \<in> extensional_funcset S T"
378       "y \<in> T" "y' \<in> T"
379       "g(x := y) = g'(x := y')"
380     from this have "y = y'"
381       by (metis fun_upd_same)
382     have "g = g'"
383       by (metis assms(1) assms(2) assms(5) extensional_funcset_arb fun_upd_triv fun_upd_upd insert(2))
384   from `y = y'` `g = g'` have "y = y' & g = g'" by simp
385   }
386   from this have "inj_on (\<lambda>(y, g). g (x := y)) (T \<times> extensional_funcset S T)"
387     by (auto intro: inj_onI)
388   from this insert.hyps show ?case
389     by (simp add: extensional_funcset_extend_domain_eq card_image card_cartesian_product)
390 qed
392 lemma finite_extensional_funcset:
393   assumes "finite S" "finite T"
394   shows "finite (extensional_funcset S T)"
395 proof -
396   from card_extensional_funcset[OF assms(1), of T] assms(2)
397   have "(card (extensional_funcset S T) \<noteq> 0) \<or> (S \<noteq> {} \<and> T = {})"
398     by auto
399   from this show ?thesis
400   proof
401     assume "card (extensional_funcset S T) \<noteq> 0"
402     from this show ?thesis
403       by (auto intro: card_ge_0_finite)
404   next
405     assume "S \<noteq> {} \<and> T = {}"
406     from this show ?thesis
407       by (auto simp add: extensional_funcset_empty_range)
408   qed
409 qed
411 end