src/HOL/Library/FuncSet.thy
 author hoelzl Fri Nov 16 18:45:57 2012 +0100 (2012-11-16) changeset 50104 de19856feb54 parent 47761 dfe747e72fa8 child 50123 69b35a75caf3 permissions -rw-r--r--
move theorems to be more generally useable
```     1 (*  Title:      HOL/Library/FuncSet.thy
```
```     2     Author:     Florian Kammueller and Lawrence C Paulson, Lukas Bulwahn
```
```     3 *)
```
```     4
```
```     5 header {* Pi and Function Sets *}
```
```     6
```
```     7 theory FuncSet
```
```     8 imports Hilbert_Choice Main
```
```     9 begin
```
```    10
```
```    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}"
```
```    14
```
```    15 definition
```
```    16   extensional :: "'a set => ('a => 'b) set" where
```
```    17   "extensional A = {f. \<forall>x. x~:A --> f x = undefined}"
```
```    18
```
```    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)"
```
```    22
```
```    23 abbreviation
```
```    24   funcset :: "['a set, 'b set] => ('a => 'b) set"
```
```    25     (infixr "->" 60) where
```
```    26   "A -> B == Pi A (%_. B)"
```
```    27
```
```    28 notation (xsymbols)
```
```    29   funcset  (infixr "\<rightarrow>" 60)
```
```    30
```
```    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)
```
```    34
```
```    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)
```
```    38
```
```    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)
```
```    42
```
```    43 translations
```
```    44   "PI x:A. B" == "CONST Pi A (%x. B)"
```
```    45   "%x:A. f" == "CONST restrict (%x. f) A"
```
```    46
```
```    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))"
```
```    50
```
```    51
```
```    52 subsection{*Basic Properties of @{term Pi}*}
```
```    53
```
```    54 lemma Pi_I[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> f \<in> Pi A B"
```
```    55   by (simp add: Pi_def)
```
```    56
```
```    57 lemma Pi_I'[simp]: "(!!x. x : A --> f x : B x) ==> f : Pi A B"
```
```    58 by(simp add:Pi_def)
```
```    59
```
```    60 lemma funcsetI: "(!!x. x \<in> A ==> f x \<in> B) ==> f \<in> A -> B"
```
```    61   by (simp add: Pi_def)
```
```    62
```
```    63 lemma Pi_mem: "[|f: Pi A B; x \<in> A|] ==> f x \<in> B x"
```
```    64   by (simp add: Pi_def)
```
```    65
```
```    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
```
```    68
```
```    69 lemma PiE [elim]:
```
```    70   "f : Pi A B ==> (f x : B x ==> Q) ==> (x ~: A ==> Q) ==> Q"
```
```    71 by(auto simp: Pi_def)
```
```    72
```
```    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)
```
```    76
```
```    77 lemma funcset_id [simp]: "(\<lambda>x. x) \<in> A \<rightarrow> A"
```
```    78   by auto
```
```    79
```
```    80 lemma funcset_mem: "[|f \<in> A -> B; x \<in> A|] ==> f x \<in> B"
```
```    81   by (simp add: Pi_def)
```
```    82
```
```    83 lemma funcset_image: "f \<in> A\<rightarrow>B ==> f ` A \<subseteq> B"
```
```    84   by auto
```
```    85
```
```    86 lemma image_subset_iff_funcset: "F ` A \<subseteq> B \<longleftrightarrow> F \<in> A \<rightarrow> B"
```
```    87   by auto
```
```    88
```
```    89 lemma Pi_eq_empty[simp]: "((PI x: A. B x) = {}) = (\<exists>x\<in>A. B(x) = {})"
```
```    90 apply (simp add: Pi_def, auto)
```
```    91 txt{*Converse direction requires Axiom of Choice to exhibit a function
```
```    92 picking an element from each non-empty @{term "B x"}*}
```
```    93 apply (drule_tac x = "%u. SOME y. y \<in> B u" in spec, auto)
```
```    94 apply (cut_tac P= "%y. y \<in> B x" in some_eq_ex, auto)
```
```    95 done
```
```    96
```
```    97 lemma Pi_empty [simp]: "Pi {} B = UNIV"
```
```    98 by (simp add: Pi_def)
```
```    99
```
```   100 lemma Pi_UNIV [simp]: "A -> UNIV = UNIV"
```
```   101 by (simp add: Pi_def)
```
```   102 (*
```
```   103 lemma funcset_id [simp]: "(%x. x): A -> A"
```
```   104   by (simp add: Pi_def)
```
```   105 *)
```
```   106 text{*Covariance of Pi-sets in their second argument*}
```
```   107 lemma Pi_mono: "(!!x. x \<in> A ==> B x <= C x) ==> Pi A B <= Pi A C"
```
```   108 by auto
```
```   109
```
```   110 text{*Contravariance of Pi-sets in their first argument*}
```
```   111 lemma Pi_anti_mono: "A' <= A ==> Pi A B <= Pi A' B"
```
```   112 by auto
```
```   113
```
```   114 lemma prod_final:
```
```   115   assumes 1: "fst \<circ> f \<in> Pi A B" and 2: "snd \<circ> f \<in> Pi A C"
```
```   116   shows "f \<in> (\<Pi> z \<in> A. B z \<times> C z)"
```
```   117 proof (rule Pi_I)
```
```   118   fix z
```
```   119   assume z: "z \<in> A"
```
```   120   have "f z = (fst (f z), snd (f z))"
```
```   121     by simp
```
```   122   also have "...  \<in> B z \<times> C z"
```
```   123     by (metis SigmaI PiE o_apply 1 2 z)
```
```   124   finally show "f z \<in> B z \<times> C z" .
```
```   125 qed
```
```   126
```
```   127
```
```   128 subsection{*Composition With a Restricted Domain: @{term compose}*}
```
```   129
```
```   130 lemma funcset_compose:
```
```   131   "[| f \<in> A -> B; g \<in> B -> C |]==> compose A g f \<in> A -> C"
```
```   132 by (simp add: Pi_def compose_def restrict_def)
```
```   133
```
```   134 lemma compose_assoc:
```
```   135     "[| f \<in> A -> B; g \<in> B -> C; h \<in> C -> D |]
```
```   136       ==> compose A h (compose A g f) = compose A (compose B h g) f"
```
```   137 by (simp add: fun_eq_iff Pi_def compose_def restrict_def)
```
```   138
```
```   139 lemma compose_eq: "x \<in> A ==> compose A g f x = g(f(x))"
```
```   140 by (simp add: compose_def restrict_def)
```
```   141
```
```   142 lemma surj_compose: "[| f ` A = B; g ` B = C |] ==> compose A g f ` A = C"
```
```   143   by (auto simp add: image_def compose_eq)
```
```   144
```
```   145
```
```   146 subsection{*Bounded Abstraction: @{term restrict}*}
```
```   147
```
```   148 lemma restrict_in_funcset: "(!!x. x \<in> A ==> f x \<in> B) ==> (\<lambda>x\<in>A. f x) \<in> A -> B"
```
```   149   by (simp add: Pi_def restrict_def)
```
```   150
```
```   151 lemma restrictI[intro!]: "(!!x. x \<in> A ==> f x \<in> B x) ==> (\<lambda>x\<in>A. f x) \<in> Pi A B"
```
```   152   by (simp add: Pi_def restrict_def)
```
```   153
```
```   154 lemma restrict_apply [simp]:
```
```   155     "(\<lambda>y\<in>A. f y) x = (if x \<in> A then f x else undefined)"
```
```   156   by (simp add: restrict_def)
```
```   157
```
```   158 lemma restrict_ext:
```
```   159     "(!!x. x \<in> A ==> f x = g x) ==> (\<lambda>x\<in>A. f x) = (\<lambda>x\<in>A. g x)"
```
```   160   by (simp add: fun_eq_iff Pi_def restrict_def)
```
```   161
```
```   162 lemma inj_on_restrict_eq [simp]: "inj_on (restrict f A) A = inj_on f A"
```
```   163   by (simp add: inj_on_def restrict_def)
```
```   164
```
```   165 lemma Id_compose:
```
```   166     "[|f \<in> A -> B;  f \<in> extensional A|] ==> compose A (\<lambda>y\<in>B. y) f = f"
```
```   167   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   168
```
```   169 lemma compose_Id:
```
```   170     "[|g \<in> A -> B;  g \<in> extensional A|] ==> compose A g (\<lambda>x\<in>A. x) = g"
```
```   171   by (auto simp add: fun_eq_iff compose_def extensional_def Pi_def)
```
```   172
```
```   173 lemma image_restrict_eq [simp]: "(restrict f A) ` A = f ` A"
```
```   174   by (auto simp add: restrict_def)
```
```   175
```
```   176
```
```   177 subsection{*Bijections Between Sets*}
```
```   178
```
```   179 text{*The definition of @{const bij_betw} is in @{text "Fun.thy"}, but most of
```
```   180 the theorems belong here, or need at least @{term Hilbert_Choice}.*}
```
```   181
```
```   182 lemma bij_betwI:
```
```   183 assumes "f \<in> A \<rightarrow> B" and "g \<in> B \<rightarrow> A"
```
```   184     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"
```
```   185 shows "bij_betw f A B"
```
```   186 unfolding bij_betw_def
```
```   187 proof
```
```   188   show "inj_on f A" by (metis g_f inj_on_def)
```
```   189 next
```
```   190   have "f ` A \<subseteq> B" using `f \<in> A \<rightarrow> B` by auto
```
```   191   moreover
```
```   192   have "B \<subseteq> f ` A" by auto (metis Pi_mem `g \<in> B \<rightarrow> A` f_g image_iff)
```
```   193   ultimately show "f ` A = B" by blast
```
```   194 qed
```
```   195
```
```   196 lemma bij_betw_imp_funcset: "bij_betw f A B \<Longrightarrow> f \<in> A \<rightarrow> B"
```
```   197 by (auto simp add: bij_betw_def)
```
```   198
```
```   199 lemma inj_on_compose:
```
```   200   "[| bij_betw f A B; inj_on g B |] ==> inj_on (compose A g f) A"
```
```   201 by (auto simp add: bij_betw_def inj_on_def compose_eq)
```
```   202
```
```   203 lemma bij_betw_compose:
```
```   204   "[| bij_betw f A B; bij_betw g B C |] ==> bij_betw (compose A g f) A C"
```
```   205 apply (simp add: bij_betw_def compose_eq inj_on_compose)
```
```   206 apply (auto simp add: compose_def image_def)
```
```   207 done
```
```   208
```
```   209 lemma bij_betw_restrict_eq [simp]:
```
```   210   "bij_betw (restrict f A) A B = bij_betw f A B"
```
```   211 by (simp add: bij_betw_def)
```
```   212
```
```   213
```
```   214 subsection{*Extensionality*}
```
```   215
```
```   216 lemma extensional_arb: "[|f \<in> extensional A; x\<notin> A|] ==> f x = undefined"
```
```   217 by (simp add: extensional_def)
```
```   218
```
```   219 lemma restrict_extensional [simp]: "restrict f A \<in> extensional A"
```
```   220 by (simp add: restrict_def extensional_def)
```
```   221
```
```   222 lemma compose_extensional [simp]: "compose A f g \<in> extensional A"
```
```   223 by (simp add: compose_def)
```
```   224
```
```   225 lemma extensionalityI:
```
```   226   "[| f \<in> extensional A; g \<in> extensional A;
```
```   227       !!x. x\<in>A ==> f x = g x |] ==> f = g"
```
```   228 by (force simp add: fun_eq_iff extensional_def)
```
```   229
```
```   230 lemma extensional_restrict:  "f \<in> extensional A \<Longrightarrow> restrict f A = f"
```
```   231 by(rule extensionalityI[OF restrict_extensional]) auto
```
```   232
```
```   233 lemma inv_into_funcset: "f ` A = B ==> (\<lambda>x\<in>B. inv_into A f x) : B -> A"
```
```   234 by (unfold inv_into_def) (fast intro: someI2)
```
```   235
```
```   236 lemma compose_inv_into_id:
```
```   237   "bij_betw f A B ==> compose A (\<lambda>y\<in>B. inv_into A f y) f = (\<lambda>x\<in>A. x)"
```
```   238 apply (simp add: bij_betw_def compose_def)
```
```   239 apply (rule restrict_ext, auto)
```
```   240 done
```
```   241
```
```   242 lemma compose_id_inv_into:
```
```   243   "f ` A = B ==> compose B f (\<lambda>y\<in>B. inv_into A f y) = (\<lambda>x\<in>B. x)"
```
```   244 apply (simp add: compose_def)
```
```   245 apply (rule restrict_ext)
```
```   246 apply (simp add: f_inv_into_f)
```
```   247 done
```
```   248
```
```   249
```
```   250 subsection{*Cardinality*}
```
```   251
```
```   252 lemma card_inj: "[|f \<in> A\<rightarrow>B; inj_on f A; finite B|] ==> card(A) \<le> card(B)"
```
```   253 by (rule card_inj_on_le) auto
```
```   254
```
```   255 lemma card_bij:
```
```   256   "[|f \<in> A\<rightarrow>B; inj_on f A;
```
```   257      g \<in> B\<rightarrow>A; inj_on g B; finite A; finite B|] ==> card(A) = card(B)"
```
```   258 by (blast intro: card_inj order_antisym)
```
```   259
```
```   260 subsection {* Extensional Function Spaces *}
```
```   261
```
```   262 definition extensional_funcset
```
```   263 where "extensional_funcset S T = (S -> T) \<inter> (extensional S)"
```
```   264
```
```   265 lemma extensional_empty[simp]: "extensional {} = {%x. undefined}"
```
```   266 unfolding extensional_def by auto
```
```   267
```
```   268 lemma extensional_funcset_empty_domain: "extensional_funcset {} T = {%x. undefined}"
```
```   269 unfolding extensional_funcset_def by simp
```
```   270
```
```   271 lemma extensional_funcset_empty_range:
```
```   272   assumes "S \<noteq> {}"
```
```   273   shows "extensional_funcset S {} = {}"
```
```   274 using assms unfolding extensional_funcset_def by auto
```
```   275
```
```   276 lemma extensional_funcset_arb:
```
```   277   assumes "f \<in> extensional_funcset S T" "x \<notin> S"
```
```   278   shows "f x = undefined"
```
```   279 using assms
```
```   280 unfolding extensional_funcset_def by auto (auto dest!: extensional_arb)
```
```   281
```
```   282 lemma extensional_funcset_mem: "f \<in> extensional_funcset S T \<Longrightarrow> x \<in> S \<Longrightarrow> f x \<in> T"
```
```   283 unfolding extensional_funcset_def by auto
```
```   284
```
```   285 lemma extensional_subset: "f : extensional A ==> A <= B ==> f : extensional B"
```
```   286 unfolding extensional_def by auto
```
```   287
```
```   288 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"
```
```   289 unfolding extensional_funcset_def extensional_def by auto
```
```   290
```
```   291 lemma extensional_funcset_restrict_domain:
```
```   292   "x \<notin> S \<Longrightarrow> f \<in> extensional_funcset (insert x S) T \<Longrightarrow> f(x := undefined) \<in> extensional_funcset S T"
```
```   293 unfolding extensional_funcset_def extensional_def by auto
```
```   294
```
```   295 lemma extensional_funcset_extend_domain_eq:
```
```   296   assumes "x \<notin> S"
```
```   297   shows
```
```   298     "extensional_funcset (insert x S) T = (\<lambda>(y, g). g(x := y)) ` {(y, g). y \<in> T \<and> g \<in> extensional_funcset S T}"
```
```   299   using assms
```
```   300 proof -
```
```   301   {
```
```   302     fix f
```
```   303     assume "f : extensional_funcset (insert x S) T"
```
```   304     from this assms have "f : (%(y, g). g(x := y)) ` (T <*> extensional_funcset S T)"
```
```   305       unfolding image_iff
```
```   306       apply (rule_tac x="(f x, f(x := undefined))" in bexI)
```
```   307     apply (auto intro: extensional_funcset_extend_domainI extensional_funcset_restrict_domain extensional_funcset_mem) done
```
```   308   }
```
```   309   moreover
```
```   310   {
```
```   311     fix f
```
```   312     assume "f : (%(y, g). g(x := y)) ` (T <*> extensional_funcset S T)"
```
```   313     from this assms have "f : extensional_funcset (insert x S) T"
```
```   314       by (auto intro: extensional_funcset_extend_domainI)
```
```   315   }
```
```   316   ultimately show ?thesis by auto
```
```   317 qed
```
```   318
```
```   319 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})"
```
```   320 unfolding extensional_funcset_def extensional_def
```
```   321 apply auto
```
```   322 apply (case_tac "x = xa")
```
```   323 apply auto done
```
```   324
```
```   325 lemma extensional_funcset_fun_upd_extends_rangeI:
```
```   326   assumes "a \<in> T" "f : extensional_funcset S (T - {a})"
```
```   327   shows "f(x := a) : extensional_funcset (insert x S) T"
```
```   328   using assms unfolding extensional_funcset_def extensional_def by auto
```
```   329
```
```   330 subsubsection {* Injective Extensional Function Spaces *}
```
```   331
```
```   332 lemma extensional_funcset_fun_upd_inj_onI:
```
```   333   assumes "f \<in> extensional_funcset S (T - {a})" "inj_on f S"
```
```   334   shows "inj_on (f(x := a)) S"
```
```   335   using assms unfolding extensional_funcset_def by (auto intro!: inj_on_fun_updI)
```
```   336
```
```   337 lemma extensional_funcset_extend_domain_inj_on_eq:
```
```   338   assumes "x \<notin> S"
```
```   339   shows"{f. f \<in> extensional_funcset (insert x S) T \<and> inj_on f (insert x S)} =
```
```   340     (%(y, g). g(x:=y)) ` {(y, g). y \<in> T \<and> g \<in> extensional_funcset S (T - {y}) \<and> inj_on g S}"
```
```   341 proof -
```
```   342   from assms show ?thesis
```
```   343     apply auto
```
```   344     apply (auto intro: extensional_funcset_fun_upd_inj_onI extensional_funcset_fun_upd_extends_rangeI)
```
```   345     apply (auto simp add: image_iff inj_on_def)
```
```   346     apply (rule_tac x="xa x" in exI)
```
```   347     apply (auto intro: extensional_funcset_mem)
```
```   348     apply (rule_tac x="xa(x := undefined)" in exI)
```
```   349     apply (auto intro!: extensional_funcset_fun_upd_restricts_rangeI)
```
```   350     apply (auto dest!: extensional_funcset_mem split: split_if_asm)
```
```   351     done
```
```   352 qed
```
```   353
```
```   354 lemma extensional_funcset_extend_domain_inj_onI:
```
```   355   assumes "x \<notin> S"
```
```   356   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}"
```
```   357 proof -
```
```   358   from assms show ?thesis
```
```   359     apply (auto intro!: inj_onI)
```
```   360     apply (metis fun_upd_same)
```
```   361     by (metis assms extensional_funcset_arb fun_upd_triv fun_upd_upd)
```
```   362 qed
```
```   363
```
```   364
```
```   365 subsubsection {* Cardinality *}
```
```   366
```
```   367 lemma card_extensional_funcset:
```
```   368   assumes "finite S"
```
```   369   shows "card (extensional_funcset S T) = (card T) ^ (card S)"
```
```   370 using assms
```
```   371 proof (induct rule: finite_induct)
```
```   372   case empty
```
```   373   show ?case
```
```   374     by (auto simp add: extensional_funcset_empty_domain)
```
```   375 next
```
```   376   case (insert x S)
```
```   377   {
```
```   378     fix g g' y y'
```
```   379     assume assms: "g \<in> extensional_funcset S T"
```
```   380       "g' \<in> extensional_funcset S T"
```
```   381       "y \<in> T" "y' \<in> T"
```
```   382       "g(x := y) = g'(x := y')"
```
```   383     from this have "y = y'"
```
```   384       by (metis fun_upd_same)
```
```   385     have "g = g'"
```
```   386       by (metis assms(1) assms(2) assms(5) extensional_funcset_arb fun_upd_triv fun_upd_upd insert(2))
```
```   387   from `y = y'` `g = g'` have "y = y' & g = g'" by simp
```
```   388   }
```
```   389   from this have "inj_on (\<lambda>(y, g). g (x := y)) (T \<times> extensional_funcset S T)"
```
```   390     by (auto intro: inj_onI)
```
```   391   from this insert.hyps show ?case
```
```   392     by (simp add: extensional_funcset_extend_domain_eq card_image card_cartesian_product)
```
```   393 qed
```
```   394
```
```   395 lemma finite_extensional_funcset:
```
```   396   assumes "finite S" "finite T"
```
```   397   shows "finite (extensional_funcset S T)"
```
```   398 proof -
```
```   399   from card_extensional_funcset[OF assms(1), of T] assms(2)
```
```   400   have "(card (extensional_funcset S T) \<noteq> 0) \<or> (S \<noteq> {} \<and> T = {})"
```
```   401     by auto
```
```   402   from this show ?thesis
```
```   403   proof
```
```   404     assume "card (extensional_funcset S T) \<noteq> 0"
```
```   405     from this show ?thesis
```
```   406       by (auto intro: card_ge_0_finite)
```
```   407   next
```
```   408     assume "S \<noteq> {} \<and> T = {}"
```
```   409     from this show ?thesis
```
```   410       by (auto simp add: extensional_funcset_empty_range)
```
```   411   qed
```
```   412 qed
```
```   413
```
```   414 end
```