src/HOL/Fun.thy
author paulson
Tue Oct 20 16:32:51 2009 +0100 (2009-10-20)
changeset 33044 fd0a9c794ec1
parent 32998 31b19fa0de0b
child 33057 764547b68538
permissions -rw-r--r--
Some new lemmas concerning sets
     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_Lattice
    10 uses ("Tools/transfer.ML")
    11 begin
    12 
    13 text{*As a simplification rule, it replaces all function equalities by
    14   first-order equalities.*}
    15 lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
    16 apply (rule iffI)
    17 apply (simp (no_asm_simp))
    18 apply (rule ext)
    19 apply (simp (no_asm_simp))
    20 done
    21 
    22 lemma apply_inverse:
    23   "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
    24   by auto
    25 
    26 
    27 subsection {* The Identity Function @{text id} *}
    28 
    29 definition
    30   id :: "'a \<Rightarrow> 'a"
    31 where
    32   "id = (\<lambda>x. x)"
    33 
    34 lemma id_apply [simp]: "id x = x"
    35   by (simp add: id_def)
    36 
    37 lemma image_ident [simp]: "(%x. x) ` Y = Y"
    38 by blast
    39 
    40 lemma image_id [simp]: "id ` Y = Y"
    41 by (simp add: id_def)
    42 
    43 lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
    44 by blast
    45 
    46 lemma vimage_id [simp]: "id -` A = A"
    47 by (simp add: id_def)
    48 
    49 
    50 subsection {* The Composition Operator @{text "f \<circ> g"} *}
    51 
    52 definition
    53   comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
    54 where
    55   "f o g = (\<lambda>x. f (g x))"
    56 
    57 notation (xsymbols)
    58   comp  (infixl "\<circ>" 55)
    59 
    60 notation (HTML output)
    61   comp  (infixl "\<circ>" 55)
    62 
    63 text{*compatibility*}
    64 lemmas o_def = comp_def
    65 
    66 lemma o_apply [simp]: "(f o g) x = f (g x)"
    67 by (simp add: comp_def)
    68 
    69 lemma o_assoc: "f o (g o h) = f o g o h"
    70 by (simp add: comp_def)
    71 
    72 lemma id_o [simp]: "id o g = g"
    73 by (simp add: comp_def)
    74 
    75 lemma o_id [simp]: "f o id = f"
    76 by (simp add: comp_def)
    77 
    78 lemma image_compose: "(f o g) ` r = f`(g`r)"
    79 by (simp add: comp_def, blast)
    80 
    81 lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
    82   by auto
    83 
    84 lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
    85 by (unfold comp_def, blast)
    86 
    87 
    88 subsection {* The Forward Composition Operator @{text fcomp} *}
    89 
    90 definition
    91   fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60)
    92 where
    93   "f o> g = (\<lambda>x. g (f x))"
    94 
    95 lemma fcomp_apply:  "(f o> g) x = g (f x)"
    96   by (simp add: fcomp_def)
    97 
    98 lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)"
    99   by (simp add: fcomp_def)
   100 
   101 lemma id_fcomp [simp]: "id o> g = g"
   102   by (simp add: fcomp_def)
   103 
   104 lemma fcomp_id [simp]: "f o> id = f"
   105   by (simp add: fcomp_def)
   106 
   107 code_const fcomp
   108   (Eval infixl 1 "#>")
   109 
   110 no_notation fcomp (infixl "o>" 60)
   111 
   112 
   113 subsection {* Injectivity and Surjectivity *}
   114 
   115 constdefs
   116   inj_on :: "['a => 'b, 'a set] => bool"  -- "injective"
   117   "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
   118 
   119 text{*A common special case: functions injective over the entire domain type.*}
   120 
   121 abbreviation
   122   "inj f == inj_on f UNIV"
   123 
   124 definition
   125   bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
   126   [code del]: "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"
   127 
   128 constdefs
   129   surj :: "('a => 'b) => bool"                   (*surjective*)
   130   "surj f == ! y. ? x. y=f(x)"
   131 
   132   bij :: "('a => 'b) => bool"                    (*bijective*)
   133   "bij f == inj f & surj f"
   134 
   135 lemma injI:
   136   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
   137   shows "inj f"
   138   using assms unfolding inj_on_def by auto
   139 
   140 text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
   141 lemma datatype_injI:
   142     "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
   143 by (simp add: inj_on_def)
   144 
   145 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
   146   by (unfold inj_on_def, blast)
   147 
   148 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
   149 by (simp add: inj_on_def)
   150 
   151 lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
   152 by (force simp add: inj_on_def)
   153 
   154 lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
   155 by (simp add: inj_on_eq_iff)
   156 
   157 lemma inj_on_id[simp]: "inj_on id A"
   158   by (simp add: inj_on_def) 
   159 
   160 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
   161 by (simp add: inj_on_def) 
   162 
   163 lemma surj_id[simp]: "surj id"
   164 by (simp add: surj_def) 
   165 
   166 lemma bij_id[simp]: "bij id"
   167 by (simp add: bij_def inj_on_id surj_id) 
   168 
   169 lemma inj_onI:
   170     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
   171 by (simp add: inj_on_def)
   172 
   173 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
   174 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
   175 
   176 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
   177 by (unfold inj_on_def, blast)
   178 
   179 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
   180 by (blast dest!: inj_onD)
   181 
   182 lemma comp_inj_on:
   183      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
   184 by (simp add: comp_def inj_on_def)
   185 
   186 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
   187 apply(simp add:inj_on_def image_def)
   188 apply blast
   189 done
   190 
   191 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
   192   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
   193 apply(unfold inj_on_def)
   194 apply blast
   195 done
   196 
   197 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
   198 by (unfold inj_on_def, blast)
   199 
   200 lemma inj_singleton: "inj (%s. {s})"
   201 by (simp add: inj_on_def)
   202 
   203 lemma inj_on_empty[iff]: "inj_on f {}"
   204 by(simp add: inj_on_def)
   205 
   206 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
   207 by (unfold inj_on_def, blast)
   208 
   209 lemma inj_on_Un:
   210  "inj_on f (A Un B) =
   211   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
   212 apply(unfold inj_on_def)
   213 apply (blast intro:sym)
   214 done
   215 
   216 lemma inj_on_insert[iff]:
   217   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
   218 apply(unfold inj_on_def)
   219 apply (blast intro:sym)
   220 done
   221 
   222 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
   223 apply(unfold inj_on_def)
   224 apply (blast)
   225 done
   226 
   227 lemma surjI: "(!! x. g(f x) = x) ==> surj g"
   228 apply (simp add: surj_def)
   229 apply (blast intro: sym)
   230 done
   231 
   232 lemma surj_range: "surj f ==> range f = UNIV"
   233 by (auto simp add: surj_def)
   234 
   235 lemma surjD: "surj f ==> EX x. y = f x"
   236 by (simp add: surj_def)
   237 
   238 lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
   239 by (simp add: surj_def, blast)
   240 
   241 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
   242 apply (simp add: comp_def surj_def, clarify)
   243 apply (drule_tac x = y in spec, clarify)
   244 apply (drule_tac x = x in spec, blast)
   245 done
   246 
   247 lemma bijI: "[| inj f; surj f |] ==> bij f"
   248 by (simp add: bij_def)
   249 
   250 lemma bij_is_inj: "bij f ==> inj f"
   251 by (simp add: bij_def)
   252 
   253 lemma bij_is_surj: "bij f ==> surj f"
   254 by (simp add: bij_def)
   255 
   256 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
   257 by (simp add: bij_betw_def)
   258 
   259 lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
   260 by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
   261 
   262 lemma bij_betw_trans:
   263   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
   264 by(auto simp add:bij_betw_def comp_inj_on)
   265 
   266 lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
   267 proof -
   268   have i: "inj_on f A" and s: "f ` A = B"
   269     using assms by(auto simp:bij_betw_def)
   270   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
   271   { fix a b assume P: "?P b a"
   272     hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
   273     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
   274     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
   275   } note g = this
   276   have "inj_on ?g B"
   277   proof(rule inj_onI)
   278     fix x y assume "x:B" "y:B" "?g x = ?g y"
   279     from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
   280     from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
   281     from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
   282   qed
   283   moreover have "?g ` B = A"
   284   proof(auto simp:image_def)
   285     fix b assume "b:B"
   286     with s obtain a where P: "?P b a" unfolding image_def by blast
   287     thus "?g b \<in> A" using g[OF P] by auto
   288   next
   289     fix a assume "a:A"
   290     then obtain b where P: "?P b a" using s unfolding image_def by blast
   291     then have "b:B" using s unfolding image_def by blast
   292     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
   293   qed
   294   ultimately show ?thesis by(auto simp:bij_betw_def)
   295 qed
   296 
   297 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
   298 by (simp add: surj_range)
   299 
   300 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
   301 by (simp add: inj_on_def, blast)
   302 
   303 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
   304 apply (unfold surj_def)
   305 apply (blast intro: sym)
   306 done
   307 
   308 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
   309 by (unfold inj_on_def, blast)
   310 
   311 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
   312 apply (unfold bij_def)
   313 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
   314 done
   315 
   316 lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
   317 by(blast dest: inj_onD)
   318 
   319 lemma inj_on_image_Int:
   320    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
   321 apply (simp add: inj_on_def, blast)
   322 done
   323 
   324 lemma inj_on_image_set_diff:
   325    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
   326 apply (simp add: inj_on_def, blast)
   327 done
   328 
   329 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
   330 by (simp add: inj_on_def, blast)
   331 
   332 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
   333 by (simp add: inj_on_def, blast)
   334 
   335 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
   336 by (blast dest: injD)
   337 
   338 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
   339 by (simp add: inj_on_def, blast)
   340 
   341 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
   342 by (blast dest: injD)
   343 
   344 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
   345 lemma image_INT:
   346    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
   347     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
   348 apply (simp add: inj_on_def, blast)
   349 done
   350 
   351 (*Compare with image_INT: no use of inj_on, and if f is surjective then
   352   it doesn't matter whether A is empty*)
   353 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
   354 apply (simp add: bij_def)
   355 apply (simp add: inj_on_def surj_def, blast)
   356 done
   357 
   358 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
   359 by (auto simp add: surj_def)
   360 
   361 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
   362 by (auto simp add: inj_on_def)
   363 
   364 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
   365 apply (simp add: bij_def)
   366 apply (rule equalityI)
   367 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
   368 done
   369 
   370 
   371 subsection{*Function Updating*}
   372 
   373 constdefs
   374   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
   375   "fun_upd f a b == % x. if x=a then b else f x"
   376 
   377 nonterminals
   378   updbinds updbind
   379 syntax
   380   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
   381   ""         :: "updbind => updbinds"             ("_")
   382   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
   383   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
   384 
   385 translations
   386   "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
   387   "f(x:=y)"                     == "fun_upd f x y"
   388 
   389 (* Hint: to define the sum of two functions (or maps), use sum_case.
   390          A nice infix syntax could be defined (in Datatype.thy or below) by
   391 consts
   392   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
   393 translations
   394  "fun_sum" == sum_case
   395 *)
   396 
   397 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
   398 apply (simp add: fun_upd_def, safe)
   399 apply (erule subst)
   400 apply (rule_tac [2] ext, auto)
   401 done
   402 
   403 (* f x = y ==> f(x:=y) = f *)
   404 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
   405 
   406 (* f(x := f x) = f *)
   407 lemmas fun_upd_triv = refl [THEN fun_upd_idem]
   408 declare fun_upd_triv [iff]
   409 
   410 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
   411 by (simp add: fun_upd_def)
   412 
   413 (* fun_upd_apply supersedes these two,   but they are useful
   414    if fun_upd_apply is intentionally removed from the simpset *)
   415 lemma fun_upd_same: "(f(x:=y)) x = y"
   416 by simp
   417 
   418 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
   419 by simp
   420 
   421 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
   422 by (simp add: expand_fun_eq)
   423 
   424 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
   425 by (rule ext, auto)
   426 
   427 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
   428 by(fastsimp simp:inj_on_def image_def)
   429 
   430 lemma fun_upd_image:
   431      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
   432 by auto
   433 
   434 lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
   435 by(auto intro: ext)
   436 
   437 
   438 subsection {* @{text override_on} *}
   439 
   440 definition
   441   override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
   442 where
   443   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
   444 
   445 lemma override_on_emptyset[simp]: "override_on f g {} = f"
   446 by(simp add:override_on_def)
   447 
   448 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
   449 by(simp add:override_on_def)
   450 
   451 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
   452 by(simp add:override_on_def)
   453 
   454 
   455 subsection {* @{text swap} *}
   456 
   457 definition
   458   swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
   459 where
   460   "swap a b f = f (a := f b, b:= f a)"
   461 
   462 lemma swap_self: "swap a a f = f"
   463 by (simp add: swap_def)
   464 
   465 lemma swap_commute: "swap a b f = swap b a f"
   466 by (rule ext, simp add: fun_upd_def swap_def)
   467 
   468 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
   469 by (rule ext, simp add: fun_upd_def swap_def)
   470 
   471 lemma inj_on_imp_inj_on_swap:
   472   "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
   473 by (simp add: inj_on_def swap_def, blast)
   474 
   475 lemma inj_on_swap_iff [simp]:
   476   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
   477 proof 
   478   assume "inj_on (swap a b f) A"
   479   with A have "inj_on (swap a b (swap a b f)) A" 
   480     by (iprover intro: inj_on_imp_inj_on_swap) 
   481   thus "inj_on f A" by simp 
   482 next
   483   assume "inj_on f A"
   484   with A show "inj_on (swap a b f) A" by(iprover intro: inj_on_imp_inj_on_swap)
   485 qed
   486 
   487 lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
   488 apply (simp add: surj_def swap_def, clarify)
   489 apply (case_tac "y = f b", blast)
   490 apply (case_tac "y = f a", auto)
   491 done
   492 
   493 lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
   494 proof 
   495   assume "surj (swap a b f)"
   496   hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
   497   thus "surj f" by simp 
   498 next
   499   assume "surj f"
   500   thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
   501 qed
   502 
   503 lemma bij_swap_iff: "bij (swap a b f) = bij f"
   504 by (simp add: bij_def)
   505 
   506 hide (open) const swap
   507 
   508 
   509 subsection {* Inversion of injective functions *}
   510 
   511 definition the_inv_onto :: "'a set => ('a => 'b) => ('b => 'a)" where
   512 "the_inv_onto A f == %x. THE y. y : A & f y = x"
   513 
   514 lemma the_inv_onto_f_f:
   515   "[| inj_on f A;  x : A |] ==> the_inv_onto A f (f x) = x"
   516 apply (simp add: the_inv_onto_def inj_on_def)
   517 apply (blast intro: the_equality)
   518 done
   519 
   520 lemma f_the_inv_onto_f:
   521   "inj_on f A ==> y : f`A  ==> f (the_inv_onto A f y) = y"
   522 apply (simp add: the_inv_onto_def)
   523 apply (rule the1I2)
   524  apply(blast dest: inj_onD)
   525 apply blast
   526 done
   527 
   528 lemma the_inv_onto_into:
   529   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_onto A f x : B"
   530 apply (simp add: the_inv_onto_def)
   531 apply (rule the1I2)
   532  apply(blast dest: inj_onD)
   533 apply blast
   534 done
   535 
   536 lemma the_inv_onto_onto[simp]:
   537   "inj_on f A ==> the_inv_onto A f ` (f ` A) = A"
   538 by (fast intro:the_inv_onto_into the_inv_onto_f_f[symmetric])
   539 
   540 lemma the_inv_onto_f_eq:
   541   "[| inj_on f A; f x = y; x : A |] ==> the_inv_onto A f y = x"
   542   apply (erule subst)
   543   apply (erule the_inv_onto_f_f, assumption)
   544   done
   545 
   546 lemma the_inv_onto_comp:
   547   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
   548   the_inv_onto A (f o g) x = (the_inv_onto A g o the_inv_onto (g ` A) f) x"
   549 apply (rule the_inv_onto_f_eq)
   550   apply (fast intro: comp_inj_on)
   551  apply (simp add: f_the_inv_onto_f the_inv_onto_into)
   552 apply (simp add: the_inv_onto_into)
   553 done
   554 
   555 lemma inj_on_the_inv_onto:
   556   "inj_on f A \<Longrightarrow> inj_on (the_inv_onto A f) (f ` A)"
   557 by (auto intro: inj_onI simp: image_def the_inv_onto_f_f)
   558 
   559 lemma bij_betw_the_inv_onto:
   560   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_onto A f) B A"
   561 by (auto simp add: bij_betw_def inj_on_the_inv_onto the_inv_onto_into)
   562 
   563 abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
   564   "the_inv f \<equiv> the_inv_onto UNIV f"
   565 
   566 lemma the_inv_f_f:
   567   assumes "inj f"
   568   shows "the_inv f (f x) = x" using assms UNIV_I
   569   by (rule the_inv_onto_f_f)
   570 
   571 
   572 subsection {* Proof tool setup *} 
   573 
   574 text {* simplifies terms of the form
   575   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
   576 
   577 simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
   578 let
   579   fun gen_fun_upd NONE T _ _ = NONE
   580     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
   581   fun dest_fun_T1 (Type (_, T :: Ts)) = T
   582   fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
   583     let
   584       fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
   585             if v aconv x then SOME g else gen_fun_upd (find g) T v w
   586         | find t = NONE
   587     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
   588 
   589   fun proc ss ct =
   590     let
   591       val ctxt = Simplifier.the_context ss
   592       val t = Thm.term_of ct
   593     in
   594       case find_double t of
   595         (T, NONE) => NONE
   596       | (T, SOME rhs) =>
   597           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
   598             (fn _ =>
   599               rtac eq_reflection 1 THEN
   600               rtac ext 1 THEN
   601               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
   602     end
   603 in proc end
   604 *}
   605 
   606 
   607 subsection {* Generic transfer procedure *}
   608 
   609 definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool"
   610   where "TransferMorphism a B \<longleftrightarrow> True"
   611 
   612 use "Tools/transfer.ML"
   613 
   614 setup Transfer.setup
   615 
   616 
   617 subsection {* Code generator setup *}
   618 
   619 types_code
   620   "fun"  ("(_ ->/ _)")
   621 attach (term_of) {*
   622 fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
   623 *}
   624 attach (test) {*
   625 fun gen_fun_type aF aT bG bT i =
   626   let
   627     val tab = Unsynchronized.ref [];
   628     fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
   629       (aT --> bT) --> aT --> bT --> aT --> bT) $ t $ aF x $ y ()
   630   in
   631     (fn x =>
   632        case AList.lookup op = (!tab) x of
   633          NONE =>
   634            let val p as (y, _) = bG i
   635            in (tab := (x, p) :: !tab; y) end
   636        | SOME (y, _) => y,
   637      fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
   638   end;
   639 *}
   640 
   641 code_const "op \<circ>"
   642   (SML infixl 5 "o")
   643   (Haskell infixr 9 ".")
   644 
   645 code_const "id"
   646   (Haskell "id")
   647 
   648 end