src/HOL/Fun.thy
author huffman
Tue Sep 13 17:07:33 2011 -0700 (2011-09-13)
changeset 44921 58eef4843641
parent 44890 22f665a2e91c
child 44928 7ef6505bde7f
permissions -rw-r--r--
tuned proofs
     1 (*  Title:      HOL/Fun.thy
     2     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
     3     Copyright   1994  University of Cambridge
     4 *)
     5 
     6 header {* Notions about functions *}
     7 
     8 theory Fun
     9 imports Complete_Lattices
    10 uses ("Tools/enriched_type.ML")
    11 begin
    12 
    13 lemma apply_inverse:
    14   "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
    15   by auto
    16 
    17 
    18 subsection {* The Identity Function @{text id} *}
    19 
    20 definition id :: "'a \<Rightarrow> 'a" where
    21   "id = (\<lambda>x. x)"
    22 
    23 lemma id_apply [simp]: "id x = x"
    24   by (simp add: id_def)
    25 
    26 lemma image_id [simp]: "id ` Y = Y"
    27   by (simp add: id_def)
    28 
    29 lemma vimage_id [simp]: "id -` A = A"
    30   by (simp add: id_def)
    31 
    32 
    33 subsection {* The Composition Operator @{text "f \<circ> g"} *}
    34 
    35 definition comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55) where
    36   "f o g = (\<lambda>x. f (g x))"
    37 
    38 notation (xsymbols)
    39   comp  (infixl "\<circ>" 55)
    40 
    41 notation (HTML output)
    42   comp  (infixl "\<circ>" 55)
    43 
    44 lemma o_apply [simp]: "(f o g) x = f (g x)"
    45 by (simp add: comp_def)
    46 
    47 lemma o_assoc: "f o (g o h) = f o g o h"
    48 by (simp add: comp_def)
    49 
    50 lemma id_o [simp]: "id o g = g"
    51 by (simp add: comp_def)
    52 
    53 lemma o_id [simp]: "f o id = f"
    54 by (simp add: comp_def)
    55 
    56 lemma o_eq_dest:
    57   "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
    58   by (simp only: comp_def) (fact fun_cong)
    59 
    60 lemma o_eq_elim:
    61   "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
    62   by (erule meta_mp) (fact o_eq_dest) 
    63 
    64 lemma image_compose: "(f o g) ` r = f`(g`r)"
    65 by (simp add: comp_def, blast)
    66 
    67 lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
    68   by auto
    69 
    70 lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
    71 by (unfold comp_def, blast)
    72 
    73 
    74 subsection {* The Forward Composition Operator @{text fcomp} *}
    75 
    76 definition fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60) where
    77   "f \<circ>> g = (\<lambda>x. g (f x))"
    78 
    79 lemma fcomp_apply [simp]:  "(f \<circ>> g) x = g (f x)"
    80   by (simp add: fcomp_def)
    81 
    82 lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"
    83   by (simp add: fcomp_def)
    84 
    85 lemma id_fcomp [simp]: "id \<circ>> g = g"
    86   by (simp add: fcomp_def)
    87 
    88 lemma fcomp_id [simp]: "f \<circ>> id = f"
    89   by (simp add: fcomp_def)
    90 
    91 code_const fcomp
    92   (Eval infixl 1 "#>")
    93 
    94 no_notation fcomp (infixl "\<circ>>" 60)
    95 
    96 
    97 subsection {* Mapping functions *}
    98 
    99 definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where
   100   "map_fun f g h = g \<circ> h \<circ> f"
   101 
   102 lemma map_fun_apply [simp]:
   103   "map_fun f g h x = g (h (f x))"
   104   by (simp add: map_fun_def)
   105 
   106 
   107 subsection {* Injectivity and Bijectivity *}
   108 
   109 definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where -- "injective"
   110   "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
   111 
   112 definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where -- "bijective"
   113   "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
   114 
   115 text{*A common special case: functions injective, surjective or bijective over
   116 the entire domain type.*}
   117 
   118 abbreviation
   119   "inj f \<equiv> inj_on f UNIV"
   120 
   121 abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where -- "surjective"
   122   "surj f \<equiv> (range f = UNIV)"
   123 
   124 abbreviation
   125   "bij f \<equiv> bij_betw f UNIV UNIV"
   126 
   127 text{* The negated case: *}
   128 translations
   129 "\<not> CONST surj f" <= "CONST range f \<noteq> CONST UNIV"
   130 
   131 lemma injI:
   132   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
   133   shows "inj f"
   134   using assms unfolding inj_on_def by auto
   135 
   136 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
   137   by (unfold inj_on_def, blast)
   138 
   139 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
   140 by (simp add: inj_on_def)
   141 
   142 lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
   143 by (force simp add: inj_on_def)
   144 
   145 lemma inj_on_cong:
   146   "(\<And> a. a : A \<Longrightarrow> f a = g a) \<Longrightarrow> inj_on f A = inj_on g A"
   147 unfolding inj_on_def by auto
   148 
   149 lemma inj_on_strict_subset:
   150   "\<lbrakk> inj_on f B; A < B \<rbrakk> \<Longrightarrow> f`A < f`B"
   151 unfolding inj_on_def unfolding image_def by blast
   152 
   153 lemma inj_comp:
   154   "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
   155   by (simp add: inj_on_def)
   156 
   157 lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
   158   by (simp add: inj_on_def fun_eq_iff)
   159 
   160 lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
   161 by (simp add: inj_on_eq_iff)
   162 
   163 lemma inj_on_id[simp]: "inj_on id A"
   164   by (simp add: inj_on_def)
   165 
   166 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
   167 by (simp add: inj_on_def)
   168 
   169 lemma inj_on_Int: "\<lbrakk>inj_on f A; inj_on f B\<rbrakk> \<Longrightarrow> inj_on f (A \<inter> B)"
   170 unfolding inj_on_def by blast
   171 
   172 lemma inj_on_INTER:
   173   "\<lbrakk>I \<noteq> {}; \<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)\<rbrakk> \<Longrightarrow> inj_on f (\<Inter> i \<in> I. A i)"
   174 unfolding inj_on_def by blast
   175 
   176 lemma inj_on_Inter:
   177   "\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> inj_on f A\<rbrakk> \<Longrightarrow> inj_on f (Inter S)"
   178 unfolding inj_on_def by blast
   179 
   180 lemma inj_on_UNION_chain:
   181   assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
   182          INJ: "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
   183   shows "inj_on f (\<Union> i \<in> I. A i)"
   184 proof(unfold inj_on_def UNION_def, auto)
   185   fix i j x y
   186   assume *: "i \<in> I" "j \<in> I" and **: "x \<in> A i" "y \<in> A j"
   187          and ***: "f x = f y"
   188   show "x = y"
   189   proof-
   190     {assume "A i \<le> A j"
   191      with ** have "x \<in> A j" by auto
   192      with INJ * ** *** have ?thesis
   193      by(auto simp add: inj_on_def)
   194     }
   195     moreover
   196     {assume "A j \<le> A i"
   197      with ** have "y \<in> A i" by auto
   198      with INJ * ** *** have ?thesis
   199      by(auto simp add: inj_on_def)
   200     }
   201     ultimately show ?thesis using  CH * by blast
   202   qed
   203 qed
   204 
   205 lemma surj_id: "surj id"
   206 by simp
   207 
   208 lemma bij_id[simp]: "bij id"
   209 by (simp add: bij_betw_def)
   210 
   211 lemma inj_onI:
   212     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
   213 by (simp add: inj_on_def)
   214 
   215 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
   216 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
   217 
   218 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
   219 by (unfold inj_on_def, blast)
   220 
   221 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
   222 by (blast dest!: inj_onD)
   223 
   224 lemma comp_inj_on:
   225      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
   226 by (simp add: comp_def inj_on_def)
   227 
   228 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
   229 apply(simp add:inj_on_def image_def)
   230 apply blast
   231 done
   232 
   233 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
   234   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
   235 apply(unfold inj_on_def)
   236 apply blast
   237 done
   238 
   239 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
   240 by (unfold inj_on_def, blast)
   241 
   242 lemma inj_singleton: "inj (%s. {s})"
   243 by (simp add: inj_on_def)
   244 
   245 lemma inj_on_empty[iff]: "inj_on f {}"
   246 by(simp add: inj_on_def)
   247 
   248 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
   249 by (unfold inj_on_def, blast)
   250 
   251 lemma inj_on_Un:
   252  "inj_on f (A Un B) =
   253   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
   254 apply(unfold inj_on_def)
   255 apply (blast intro:sym)
   256 done
   257 
   258 lemma inj_on_insert[iff]:
   259   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
   260 apply(unfold inj_on_def)
   261 apply (blast intro:sym)
   262 done
   263 
   264 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
   265 apply(unfold inj_on_def)
   266 apply (blast)
   267 done
   268 
   269 lemma comp_inj_on_iff:
   270   "inj_on f A \<Longrightarrow> inj_on f' (f ` A) \<longleftrightarrow> inj_on (f' o f) A"
   271 by(auto simp add: comp_inj_on inj_on_def)
   272 
   273 lemma inj_on_imageI2:
   274   "inj_on (f' o f) A \<Longrightarrow> inj_on f A"
   275 by(auto simp add: comp_inj_on inj_on_def)
   276 
   277 lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
   278   by auto
   279 
   280 lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"
   281   using *[symmetric] by auto
   282 
   283 lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
   284   by (simp add: surj_def)
   285 
   286 lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
   287   by (simp add: surj_def, blast)
   288 
   289 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
   290 apply (simp add: comp_def surj_def, clarify)
   291 apply (drule_tac x = y in spec, clarify)
   292 apply (drule_tac x = x in spec, blast)
   293 done
   294 
   295 lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
   296   unfolding bij_betw_def by auto
   297 
   298 lemma bij_betw_empty1:
   299   assumes "bij_betw f {} A"
   300   shows "A = {}"
   301 using assms unfolding bij_betw_def by blast
   302 
   303 lemma bij_betw_empty2:
   304   assumes "bij_betw f A {}"
   305   shows "A = {}"
   306 using assms unfolding bij_betw_def by blast
   307 
   308 lemma inj_on_imp_bij_betw:
   309   "inj_on f A \<Longrightarrow> bij_betw f A (f ` A)"
   310 unfolding bij_betw_def by simp
   311 
   312 lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
   313   unfolding bij_betw_def ..
   314 
   315 lemma bijI: "[| inj f; surj f |] ==> bij f"
   316 by (simp add: bij_def)
   317 
   318 lemma bij_is_inj: "bij f ==> inj f"
   319 by (simp add: bij_def)
   320 
   321 lemma bij_is_surj: "bij f ==> surj f"
   322 by (simp add: bij_def)
   323 
   324 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
   325 by (simp add: bij_betw_def)
   326 
   327 lemma bij_betw_trans:
   328   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
   329 by(auto simp add:bij_betw_def comp_inj_on)
   330 
   331 lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
   332   by (rule bij_betw_trans)
   333 
   334 lemma bij_betw_comp_iff:
   335   "bij_betw f A A' \<Longrightarrow> bij_betw f' A' A'' \<longleftrightarrow> bij_betw (f' o f) A A''"
   336 by(auto simp add: bij_betw_def inj_on_def)
   337 
   338 lemma bij_betw_comp_iff2:
   339   assumes BIJ: "bij_betw f' A' A''" and IM: "f ` A \<le> A'"
   340   shows "bij_betw f A A' \<longleftrightarrow> bij_betw (f' o f) A A''"
   341 using assms
   342 proof(auto simp add: bij_betw_comp_iff)
   343   assume *: "bij_betw (f' \<circ> f) A A''"
   344   thus "bij_betw f A A'"
   345   using IM
   346   proof(auto simp add: bij_betw_def)
   347     assume "inj_on (f' \<circ> f) A"
   348     thus "inj_on f A" using inj_on_imageI2 by blast
   349   next
   350     fix a' assume **: "a' \<in> A'"
   351     hence "f' a' \<in> A''" using BIJ unfolding bij_betw_def by auto
   352     then obtain a where 1: "a \<in> A \<and> f'(f a) = f' a'" using *
   353     unfolding bij_betw_def by force
   354     hence "f a \<in> A'" using IM by auto
   355     hence "f a = a'" using BIJ ** 1 unfolding bij_betw_def inj_on_def by auto
   356     thus "a' \<in> f ` A" using 1 by auto
   357   qed
   358 qed
   359 
   360 lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
   361 proof -
   362   have i: "inj_on f A" and s: "f ` A = B"
   363     using assms by(auto simp:bij_betw_def)
   364   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
   365   { fix a b assume P: "?P b a"
   366     hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
   367     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
   368     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
   369   } note g = this
   370   have "inj_on ?g B"
   371   proof(rule inj_onI)
   372     fix x y assume "x:B" "y:B" "?g x = ?g y"
   373     from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
   374     from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
   375     from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
   376   qed
   377   moreover have "?g ` B = A"
   378   proof(auto simp:image_def)
   379     fix b assume "b:B"
   380     with s obtain a where P: "?P b a" unfolding image_def by blast
   381     thus "?g b \<in> A" using g[OF P] by auto
   382   next
   383     fix a assume "a:A"
   384     then obtain b where P: "?P b a" using s unfolding image_def by blast
   385     then have "b:B" using s unfolding image_def by blast
   386     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
   387   qed
   388   ultimately show ?thesis by(auto simp:bij_betw_def)
   389 qed
   390 
   391 lemma bij_betw_cong:
   392   "(\<And> a. a \<in> A \<Longrightarrow> f a = g a) \<Longrightarrow> bij_betw f A A' = bij_betw g A A'"
   393 unfolding bij_betw_def inj_on_def by force
   394 
   395 lemma bij_betw_id[intro, simp]:
   396   "bij_betw id A A"
   397 unfolding bij_betw_def id_def by auto
   398 
   399 lemma bij_betw_id_iff:
   400   "bij_betw id A B \<longleftrightarrow> A = B"
   401 by(auto simp add: bij_betw_def)
   402 
   403 lemma bij_betw_combine:
   404   assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
   405   shows "bij_betw f (A \<union> C) (B \<union> D)"
   406   using assms unfolding bij_betw_def inj_on_Un image_Un by auto
   407 
   408 lemma bij_betw_UNION_chain:
   409   assumes CH: "\<And> i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<le> A j \<or> A j \<le> A i" and
   410          BIJ: "\<And> i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)"
   411   shows "bij_betw f (\<Union> i \<in> I. A i) (\<Union> i \<in> I. A' i)"
   412 proof(unfold bij_betw_def, auto simp add: image_def)
   413   have "\<And> i. i \<in> I \<Longrightarrow> inj_on f (A i)"
   414   using BIJ bij_betw_def[of f] by auto
   415   thus "inj_on f (\<Union> i \<in> I. A i)"
   416   using CH inj_on_UNION_chain[of I A f] by auto
   417 next
   418   fix i x
   419   assume *: "i \<in> I" "x \<in> A i"
   420   hence "f x \<in> A' i" using BIJ bij_betw_def[of f] by auto
   421   thus "\<exists>j \<in> I. f x \<in> A' j" using * by blast
   422 next
   423   fix i x'
   424   assume *: "i \<in> I" "x' \<in> A' i"
   425   hence "\<exists>x \<in> A i. x' = f x" using BIJ bij_betw_def[of f] by blast
   426   thus "\<exists>j \<in> I. \<exists>x \<in> A j. x' = f x"
   427   using * by blast
   428 qed
   429 
   430 lemma bij_betw_Disj_Un:
   431   assumes DISJ: "A \<inter> B = {}" and DISJ': "A' \<inter> B' = {}" and
   432           B1: "bij_betw f A A'" and B2: "bij_betw f B B'"
   433   shows "bij_betw f (A \<union> B) (A' \<union> B')"
   434 proof-
   435   have 1: "inj_on f A \<and> inj_on f B"
   436   using B1 B2 by (auto simp add: bij_betw_def)
   437   have 2: "f`A = A' \<and> f`B = B'"
   438   using B1 B2 by (auto simp add: bij_betw_def)
   439   hence "f`(A - B) \<inter> f`(B - A) = {}"
   440   using DISJ DISJ' by blast
   441   hence "inj_on f (A \<union> B)"
   442   using 1 by (auto simp add: inj_on_Un)
   443   (*  *)
   444   moreover
   445   have "f`(A \<union> B) = A' \<union> B'"
   446   using 2 by auto
   447   ultimately show ?thesis
   448   unfolding bij_betw_def by auto
   449 qed
   450 
   451 lemma bij_betw_subset:
   452   assumes BIJ: "bij_betw f A A'" and
   453           SUB: "B \<le> A" and IM: "f ` B = B'"
   454   shows "bij_betw f B B'"
   455 using assms
   456 by(unfold bij_betw_def inj_on_def, auto simp add: inj_on_def)
   457 
   458 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
   459 by simp
   460 
   461 lemma surj_vimage_empty:
   462   assumes "surj f" shows "f -` A = {} \<longleftrightarrow> A = {}"
   463   using surj_image_vimage_eq[OF `surj f`, of A]
   464   by (intro iffI) fastforce+
   465 
   466 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
   467 by (simp add: inj_on_def, blast)
   468 
   469 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
   470 by (blast intro: sym)
   471 
   472 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
   473 by (unfold inj_on_def, blast)
   474 
   475 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
   476 apply (unfold bij_def)
   477 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
   478 done
   479 
   480 lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
   481 by(blast dest: inj_onD)
   482 
   483 lemma inj_on_image_Int:
   484    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
   485 apply (simp add: inj_on_def, blast)
   486 done
   487 
   488 lemma inj_on_image_set_diff:
   489    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
   490 apply (simp add: inj_on_def, blast)
   491 done
   492 
   493 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
   494 by (simp add: inj_on_def, blast)
   495 
   496 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
   497 by (simp add: inj_on_def, blast)
   498 
   499 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
   500 by (blast dest: injD)
   501 
   502 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
   503 by (simp add: inj_on_def, blast)
   504 
   505 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
   506 by (blast dest: injD)
   507 
   508 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
   509 lemma image_INT:
   510    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
   511     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
   512 apply (simp add: inj_on_def, blast)
   513 done
   514 
   515 (*Compare with image_INT: no use of inj_on, and if f is surjective then
   516   it doesn't matter whether A is empty*)
   517 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
   518 apply (simp add: bij_def)
   519 apply (simp add: inj_on_def surj_def, blast)
   520 done
   521 
   522 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
   523 by auto
   524 
   525 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
   526 by (auto simp add: inj_on_def)
   527 
   528 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
   529 apply (simp add: bij_def)
   530 apply (rule equalityI)
   531 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
   532 done
   533 
   534 lemma inj_vimage_singleton: "inj f \<Longrightarrow> f -` {a} \<subseteq> {THE x. f x = a}"
   535   -- {* The inverse image of a singleton under an injective function
   536          is included in a singleton. *}
   537   apply (auto simp add: inj_on_def)
   538   apply (blast intro: the_equality [symmetric])
   539   done
   540 
   541 lemma inj_on_vimage_singleton:
   542   "inj_on f A \<Longrightarrow> f -` {a} \<inter> A \<subseteq> {THE x. x \<in> A \<and> f x = a}"
   543   by (auto simp add: inj_on_def intro: the_equality [symmetric])
   544 
   545 lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
   546   by (auto intro!: inj_onI)
   547 
   548 lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
   549   by (auto intro!: inj_onI dest: strict_mono_eq)
   550 
   551 
   552 subsection{*Function Updating*}
   553 
   554 definition fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
   555   "fun_upd f a b == % x. if x=a then b else f x"
   556 
   557 nonterminal updbinds and updbind
   558 
   559 syntax
   560   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
   561   ""         :: "updbind => updbinds"             ("_")
   562   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
   563   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
   564 
   565 translations
   566   "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
   567   "f(x:=y)" == "CONST fun_upd f x y"
   568 
   569 (* Hint: to define the sum of two functions (or maps), use sum_case.
   570          A nice infix syntax could be defined (in Datatype.thy or below) by
   571 notation
   572   sum_case  (infixr "'(+')"80)
   573 *)
   574 
   575 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
   576 apply (simp add: fun_upd_def, safe)
   577 apply (erule subst)
   578 apply (rule_tac [2] ext, auto)
   579 done
   580 
   581 (* f x = y ==> f(x:=y) = f *)
   582 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
   583 
   584 (* f(x := f x) = f *)
   585 lemmas fun_upd_triv = refl [THEN fun_upd_idem]
   586 declare fun_upd_triv [iff]
   587 
   588 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
   589 by (simp add: fun_upd_def)
   590 
   591 (* fun_upd_apply supersedes these two,   but they are useful
   592    if fun_upd_apply is intentionally removed from the simpset *)
   593 lemma fun_upd_same: "(f(x:=y)) x = y"
   594 by simp
   595 
   596 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
   597 by simp
   598 
   599 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
   600 by (simp add: fun_eq_iff)
   601 
   602 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
   603 by (rule ext, auto)
   604 
   605 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
   606 by (fastforce simp:inj_on_def image_def)
   607 
   608 lemma fun_upd_image:
   609      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
   610 by auto
   611 
   612 lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
   613   by auto
   614 
   615 lemma UNION_fun_upd:
   616   "UNION J (A(i:=B)) = (UNION (J-{i}) A \<union> (if i\<in>J then B else {}))"
   617 by (auto split: if_splits)
   618 
   619 
   620 subsection {* @{text override_on} *}
   621 
   622 definition override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b" where
   623   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
   624 
   625 lemma override_on_emptyset[simp]: "override_on f g {} = f"
   626 by(simp add:override_on_def)
   627 
   628 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
   629 by(simp add:override_on_def)
   630 
   631 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
   632 by(simp add:override_on_def)
   633 
   634 
   635 subsection {* @{text swap} *}
   636 
   637 definition swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" where
   638   "swap a b f = f (a := f b, b:= f a)"
   639 
   640 lemma swap_self [simp]: "swap a a f = f"
   641 by (simp add: swap_def)
   642 
   643 lemma swap_commute: "swap a b f = swap b a f"
   644 by (rule ext, simp add: fun_upd_def swap_def)
   645 
   646 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
   647 by (rule ext, simp add: fun_upd_def swap_def)
   648 
   649 lemma swap_triple:
   650   assumes "a \<noteq> c" and "b \<noteq> c"
   651   shows "swap a b (swap b c (swap a b f)) = swap a c f"
   652   using assms by (simp add: fun_eq_iff swap_def)
   653 
   654 lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
   655 by (rule ext, simp add: fun_upd_def swap_def)
   656 
   657 lemma swap_image_eq [simp]:
   658   assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
   659 proof -
   660   have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
   661     using assms by (auto simp: image_iff swap_def)
   662   then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
   663   with subset[of f] show ?thesis by auto
   664 qed
   665 
   666 lemma inj_on_imp_inj_on_swap:
   667   "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
   668   by (simp add: inj_on_def swap_def, blast)
   669 
   670 lemma inj_on_swap_iff [simp]:
   671   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
   672 proof
   673   assume "inj_on (swap a b f) A"
   674   with A have "inj_on (swap a b (swap a b f)) A"
   675     by (iprover intro: inj_on_imp_inj_on_swap)
   676   thus "inj_on f A" by simp
   677 next
   678   assume "inj_on f A"
   679   with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
   680 qed
   681 
   682 lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
   683   by simp
   684 
   685 lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
   686   by simp
   687 
   688 lemma bij_betw_swap_iff [simp]:
   689   "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
   690   by (auto simp: bij_betw_def)
   691 
   692 lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
   693   by simp
   694 
   695 hide_const (open) swap
   696 
   697 subsection {* Inversion of injective functions *}
   698 
   699 definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
   700   "the_inv_into A f == %x. THE y. y : A & f y = x"
   701 
   702 lemma the_inv_into_f_f:
   703   "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
   704 apply (simp add: the_inv_into_def inj_on_def)
   705 apply blast
   706 done
   707 
   708 lemma f_the_inv_into_f:
   709   "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
   710 apply (simp add: the_inv_into_def)
   711 apply (rule the1I2)
   712  apply(blast dest: inj_onD)
   713 apply blast
   714 done
   715 
   716 lemma the_inv_into_into:
   717   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
   718 apply (simp add: the_inv_into_def)
   719 apply (rule the1I2)
   720  apply(blast dest: inj_onD)
   721 apply blast
   722 done
   723 
   724 lemma the_inv_into_onto[simp]:
   725   "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
   726 by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
   727 
   728 lemma the_inv_into_f_eq:
   729   "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
   730   apply (erule subst)
   731   apply (erule the_inv_into_f_f, assumption)
   732   done
   733 
   734 lemma the_inv_into_comp:
   735   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
   736   the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
   737 apply (rule the_inv_into_f_eq)
   738   apply (fast intro: comp_inj_on)
   739  apply (simp add: f_the_inv_into_f the_inv_into_into)
   740 apply (simp add: the_inv_into_into)
   741 done
   742 
   743 lemma inj_on_the_inv_into:
   744   "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
   745 by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
   746 
   747 lemma bij_betw_the_inv_into:
   748   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
   749 by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
   750 
   751 abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
   752   "the_inv f \<equiv> the_inv_into UNIV f"
   753 
   754 lemma the_inv_f_f:
   755   assumes "inj f"
   756   shows "the_inv f (f x) = x" using assms UNIV_I
   757   by (rule the_inv_into_f_f)
   758 
   759 
   760 text{*compatibility*}
   761 lemmas o_def = comp_def
   762 
   763 
   764 subsection {* Cantor's Paradox *}
   765 
   766 lemma Cantors_paradox [no_atp]:
   767   "\<not>(\<exists>f. f ` A = Pow A)"
   768 proof clarify
   769   fix f assume "f ` A = Pow A" hence *: "Pow A \<le> f ` A" by blast
   770   let ?X = "{a \<in> A. a \<notin> f a}"
   771   have "?X \<in> Pow A" unfolding Pow_def by auto
   772   with * obtain x where "x \<in> A \<and> f x = ?X" by blast
   773   thus False by best
   774 qed
   775 
   776 subsection {* Setup *} 
   777 
   778 subsubsection {* Proof tools *}
   779 
   780 text {* simplifies terms of the form
   781   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
   782 
   783 simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
   784 let
   785   fun gen_fun_upd NONE T _ _ = NONE
   786     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
   787   fun dest_fun_T1 (Type (_, T :: Ts)) = T
   788   fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
   789     let
   790       fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
   791             if v aconv x then SOME g else gen_fun_upd (find g) T v w
   792         | find t = NONE
   793     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
   794 
   795   fun proc ss ct =
   796     let
   797       val ctxt = Simplifier.the_context ss
   798       val t = Thm.term_of ct
   799     in
   800       case find_double t of
   801         (T, NONE) => NONE
   802       | (T, SOME rhs) =>
   803           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
   804             (fn _ =>
   805               rtac eq_reflection 1 THEN
   806               rtac ext 1 THEN
   807               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
   808     end
   809 in proc end
   810 *}
   811 
   812 
   813 subsubsection {* Code generator *}
   814 
   815 types_code
   816   "fun"  ("(_ ->/ _)")
   817 attach (term_of) {*
   818 fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
   819 *}
   820 attach (test) {*
   821 fun gen_fun_type aF aT bG bT i =
   822   let
   823     val tab = Unsynchronized.ref [];
   824     fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
   825       (aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()
   826   in
   827     (fn x =>
   828        case AList.lookup op = (!tab) x of
   829          NONE =>
   830            let val p as (y, _) = bG i
   831            in (tab := (x, p) :: !tab; y) end
   832        | SOME (y, _) => y,
   833      fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
   834   end;
   835 *}
   836 
   837 code_const "op \<circ>"
   838   (SML infixl 5 "o")
   839   (Haskell infixr 9 ".")
   840 
   841 code_const "id"
   842   (Haskell "id")
   843 
   844 
   845 subsubsection {* Functorial structure of types *}
   846 
   847 use "Tools/enriched_type.ML"
   848 
   849 end