src/HOL/Fun.thy
author haftmann
Wed Sep 26 20:27:55 2007 +0200 (2007-09-26)
changeset 24728 e2b3a1065676
parent 24286 7619080e49f0
child 25886 7753e0d81b7a
permissions -rw-r--r--
moved Finite_Set before Datatype
     1 (*  Title:      HOL/Fun.thy
     2     ID:         $Id$
     3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
     4     Copyright   1994  University of Cambridge
     5 *)
     6 
     7 header {* Notions about functions *}
     8 
     9 theory Fun
    10 imports Set
    11 begin
    12 
    13 constdefs
    14   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
    15   "fun_upd f a b == % x. if x=a then b else f x"
    16 
    17 nonterminals
    18   updbinds updbind
    19 syntax
    20   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
    21   ""         :: "updbind => updbinds"             ("_")
    22   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
    23   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
    24 
    25 translations
    26   "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
    27   "f(x:=y)"                     == "fun_upd f x y"
    28 
    29 (* Hint: to define the sum of two functions (or maps), use sum_case.
    30          A nice infix syntax could be defined (in Datatype.thy or below) by
    31 consts
    32   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
    33 translations
    34  "fun_sum" == sum_case
    35 *)
    36 
    37 definition
    38   override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
    39 where
    40   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
    41 
    42 definition
    43   id :: "'a \<Rightarrow> 'a"
    44 where
    45   "id = (\<lambda>x. x)"
    46 
    47 definition
    48   comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
    49 where
    50   "f o g = (\<lambda>x. f (g x))"
    51 
    52 notation (xsymbols)
    53   comp  (infixl "\<circ>" 55)
    54 
    55 notation (HTML output)
    56   comp  (infixl "\<circ>" 55)
    57 
    58 text{*compatibility*}
    59 lemmas o_def = comp_def
    60 
    61 constdefs
    62   inj_on :: "['a => 'b, 'a set] => bool"         (*injective*)
    63   "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
    64 
    65 text{*A common special case: functions injective over the entire domain type.*}
    66 
    67 abbreviation
    68   "inj f == inj_on f UNIV"
    69 
    70 constdefs
    71   surj :: "('a => 'b) => bool"                   (*surjective*)
    72   "surj f == ! y. ? x. y=f(x)"
    73 
    74   bij :: "('a => 'b) => bool"                    (*bijective*)
    75   "bij f == inj f & surj f"
    76 
    77 
    78 
    79 text{*As a simplification rule, it replaces all function equalities by
    80   first-order equalities.*}
    81 lemma expand_fun_eq: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
    82 apply (rule iffI)
    83 apply (simp (no_asm_simp))
    84 apply (rule ext)
    85 apply (simp (no_asm_simp))
    86 done
    87 
    88 lemma apply_inverse:
    89     "[| f(x)=u;  !!x. P(x) ==> g(f(x)) = x;  P(x) |] ==> x=g(u)"
    90 by auto
    91 
    92 
    93 text{*The Identity Function: @{term id}*}
    94 lemma id_apply [simp]: "id x = x"
    95 by (simp add: id_def)
    96 
    97 lemma inj_on_id[simp]: "inj_on id A"
    98 by (simp add: inj_on_def) 
    99 
   100 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
   101 by (simp add: inj_on_def) 
   102 
   103 lemma surj_id[simp]: "surj id"
   104 by (simp add: surj_def) 
   105 
   106 lemma bij_id[simp]: "bij id"
   107 by (simp add: bij_def inj_on_id surj_id) 
   108 
   109 
   110 
   111 subsection{*The Composition Operator: @{term "f \<circ> g"}*}
   112 
   113 lemma o_apply [simp]: "(f o g) x = f (g x)"
   114 by (simp add: comp_def)
   115 
   116 lemma o_assoc: "f o (g o h) = f o g o h"
   117 by (simp add: comp_def)
   118 
   119 lemma id_o [simp]: "id o g = g"
   120 by (simp add: comp_def)
   121 
   122 lemma o_id [simp]: "f o id = f"
   123 by (simp add: comp_def)
   124 
   125 lemma image_compose: "(f o g) ` r = f`(g`r)"
   126 by (simp add: comp_def, blast)
   127 
   128 lemma image_eq_UN: "f`A = (UN x:A. {f x})"
   129 by blast
   130 
   131 lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
   132 by (unfold comp_def, blast)
   133 
   134 
   135 subsection{*The Injectivity Predicate, @{term inj}*}
   136 
   137 text{*NB: @{term inj} now just translates to @{term inj_on}*}
   138 
   139 
   140 text{*For Proofs in @{text "Tools/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 (*Useful with the simplifier*)
   152 lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)"
   153 by (force simp add: inj_on_def)
   154 
   155 
   156 subsection{*The Predicate @{term inj_on}: Injectivity On A Restricted Domain*}
   157 
   158 lemma inj_onI:
   159     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
   160 by (simp add: inj_on_def)
   161 
   162 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
   163 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
   164 
   165 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
   166 by (unfold inj_on_def, blast)
   167 
   168 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
   169 by (blast dest!: inj_onD)
   170 
   171 lemma comp_inj_on:
   172      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
   173 by (simp add: comp_def inj_on_def)
   174 
   175 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
   176 apply(simp add:inj_on_def image_def)
   177 apply blast
   178 done
   179 
   180 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
   181   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
   182 apply(unfold inj_on_def)
   183 apply blast
   184 done
   185 
   186 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
   187 by (unfold inj_on_def, blast)
   188 
   189 lemma inj_singleton: "inj (%s. {s})"
   190 by (simp add: inj_on_def)
   191 
   192 lemma inj_on_empty[iff]: "inj_on f {}"
   193 by(simp add: inj_on_def)
   194 
   195 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
   196 by (unfold inj_on_def, blast)
   197 
   198 lemma inj_on_Un:
   199  "inj_on f (A Un B) =
   200   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
   201 apply(unfold inj_on_def)
   202 apply (blast intro:sym)
   203 done
   204 
   205 lemma inj_on_insert[iff]:
   206   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
   207 apply(unfold inj_on_def)
   208 apply (blast intro:sym)
   209 done
   210 
   211 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
   212 apply(unfold inj_on_def)
   213 apply (blast)
   214 done
   215 
   216 
   217 subsection{*The Predicate @{term surj}: Surjectivity*}
   218 
   219 lemma surjI: "(!! x. g(f x) = x) ==> surj g"
   220 apply (simp add: surj_def)
   221 apply (blast intro: sym)
   222 done
   223 
   224 lemma surj_range: "surj f ==> range f = UNIV"
   225 by (auto simp add: surj_def)
   226 
   227 lemma surjD: "surj f ==> EX x. y = f x"
   228 by (simp add: surj_def)
   229 
   230 lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
   231 by (simp add: surj_def, blast)
   232 
   233 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
   234 apply (simp add: comp_def surj_def, clarify)
   235 apply (drule_tac x = y in spec, clarify)
   236 apply (drule_tac x = x in spec, blast)
   237 done
   238 
   239 
   240 
   241 subsection{*The Predicate @{term bij}: Bijectivity*}
   242 
   243 lemma bijI: "[| inj f; surj f |] ==> bij f"
   244 by (simp add: bij_def)
   245 
   246 lemma bij_is_inj: "bij f ==> inj f"
   247 by (simp add: bij_def)
   248 
   249 lemma bij_is_surj: "bij f ==> surj f"
   250 by (simp add: bij_def)
   251 
   252 
   253 subsection{*Facts About the Identity Function*}
   254 
   255 text{*We seem to need both the @{term id} forms and the @{term "\<lambda>x. x"}
   256 forms. The latter can arise by rewriting, while @{term id} may be used
   257 explicitly.*}
   258 
   259 lemma image_ident [simp]: "(%x. x) ` Y = Y"
   260 by blast
   261 
   262 lemma image_id [simp]: "id ` Y = Y"
   263 by (simp add: id_def)
   264 
   265 lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
   266 by blast
   267 
   268 lemma vimage_id [simp]: "id -` A = A"
   269 by (simp add: id_def)
   270 
   271 lemma vimage_image_eq [noatp]: "f -` (f ` A) = {y. EX x:A. f x = f y}"
   272 by (blast intro: sym)
   273 
   274 lemma image_vimage_subset: "f ` (f -` A) <= A"
   275 by blast
   276 
   277 lemma image_vimage_eq [simp]: "f ` (f -` A) = A Int range f"
   278 by blast
   279 
   280 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
   281 by (simp add: surj_range)
   282 
   283 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
   284 by (simp add: inj_on_def, blast)
   285 
   286 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
   287 apply (unfold surj_def)
   288 apply (blast intro: sym)
   289 done
   290 
   291 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
   292 by (unfold inj_on_def, blast)
   293 
   294 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
   295 apply (unfold bij_def)
   296 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
   297 done
   298 
   299 lemma image_Int_subset: "f`(A Int B) <= f`A Int f`B"
   300 by blast
   301 
   302 lemma image_diff_subset: "f`A - f`B <= f`(A - B)"
   303 by blast
   304 
   305 lemma inj_on_image_Int:
   306    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
   307 apply (simp add: inj_on_def, blast)
   308 done
   309 
   310 lemma inj_on_image_set_diff:
   311    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
   312 apply (simp add: inj_on_def, blast)
   313 done
   314 
   315 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
   316 by (simp add: inj_on_def, blast)
   317 
   318 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
   319 by (simp add: inj_on_def, blast)
   320 
   321 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
   322 by (blast dest: injD)
   323 
   324 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
   325 by (simp add: inj_on_def, blast)
   326 
   327 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
   328 by (blast dest: injD)
   329 
   330 lemma image_UN: "(f ` (UNION A B)) = (UN x:A.(f ` (B x)))"
   331 by blast
   332 
   333 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
   334 lemma image_INT:
   335    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
   336     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
   337 apply (simp add: inj_on_def, blast)
   338 done
   339 
   340 (*Compare with image_INT: no use of inj_on, and if f is surjective then
   341   it doesn't matter whether A is empty*)
   342 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
   343 apply (simp add: bij_def)
   344 apply (simp add: inj_on_def surj_def, blast)
   345 done
   346 
   347 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
   348 by (auto simp add: surj_def)
   349 
   350 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
   351 by (auto simp add: inj_on_def)
   352 
   353 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
   354 apply (simp add: bij_def)
   355 apply (rule equalityI)
   356 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
   357 done
   358 
   359 
   360 subsection{*Function Updating*}
   361 
   362 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
   363 apply (simp add: fun_upd_def, safe)
   364 apply (erule subst)
   365 apply (rule_tac [2] ext, auto)
   366 done
   367 
   368 (* f x = y ==> f(x:=y) = f *)
   369 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
   370 
   371 (* f(x := f x) = f *)
   372 lemmas fun_upd_triv = refl [THEN fun_upd_idem]
   373 declare fun_upd_triv [iff]
   374 
   375 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
   376 by (simp add: fun_upd_def)
   377 
   378 (* fun_upd_apply supersedes these two,   but they are useful
   379    if fun_upd_apply is intentionally removed from the simpset *)
   380 lemma fun_upd_same: "(f(x:=y)) x = y"
   381 by simp
   382 
   383 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
   384 by simp
   385 
   386 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
   387 by (simp add: expand_fun_eq)
   388 
   389 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
   390 by (rule ext, auto)
   391 
   392 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
   393 by(fastsimp simp:inj_on_def image_def)
   394 
   395 lemma fun_upd_image:
   396      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
   397 by auto
   398 
   399 subsection{* @{text override_on} *}
   400 
   401 lemma override_on_emptyset[simp]: "override_on f g {} = f"
   402 by(simp add:override_on_def)
   403 
   404 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
   405 by(simp add:override_on_def)
   406 
   407 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
   408 by(simp add:override_on_def)
   409 
   410 subsection{* swap *}
   411 
   412 definition
   413   swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
   414 where
   415   "swap a b f = f (a := f b, b:= f a)"
   416 
   417 lemma swap_self: "swap a a f = f"
   418 by (simp add: swap_def)
   419 
   420 lemma swap_commute: "swap a b f = swap b a f"
   421 by (rule ext, simp add: fun_upd_def swap_def)
   422 
   423 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
   424 by (rule ext, simp add: fun_upd_def swap_def)
   425 
   426 lemma inj_on_imp_inj_on_swap:
   427   "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
   428 by (simp add: inj_on_def swap_def, blast)
   429 
   430 lemma inj_on_swap_iff [simp]:
   431   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
   432 proof 
   433   assume "inj_on (swap a b f) A"
   434   with A have "inj_on (swap a b (swap a b f)) A" 
   435     by (iprover intro: inj_on_imp_inj_on_swap) 
   436   thus "inj_on f A" by simp 
   437 next
   438   assume "inj_on f A"
   439   with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
   440 qed
   441 
   442 lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
   443 apply (simp add: surj_def swap_def, clarify)
   444 apply (rule_tac P = "y = f b" in case_split_thm, blast)
   445 apply (rule_tac P = "y = f a" in case_split_thm, auto)
   446   --{*We don't yet have @{text case_tac}*}
   447 done
   448 
   449 lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
   450 proof 
   451   assume "surj (swap a b f)"
   452   hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) 
   453   thus "surj f" by simp 
   454 next
   455   assume "surj f"
   456   thus "surj (swap a b f)" by (rule surj_imp_surj_swap) 
   457 qed
   458 
   459 lemma bij_swap_iff: "bij (swap a b f) = bij f"
   460 by (simp add: bij_def)
   461 
   462 
   463 subsection {* Proof tool setup *} 
   464 
   465 text {* simplifies terms of the form
   466   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
   467 
   468 simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
   469 let
   470   fun gen_fun_upd NONE T _ _ = NONE
   471     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) $ f $ x $ y)
   472   fun dest_fun_T1 (Type (_, T :: Ts)) = T
   473   fun find_double (t as Const (@{const_name fun_upd},T) $ f $ x $ y) =
   474     let
   475       fun find (Const (@{const_name fun_upd},T) $ g $ v $ w) =
   476             if v aconv x then SOME g else gen_fun_upd (find g) T v w
   477         | find t = NONE
   478     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
   479 
   480   fun proc ss ct =
   481     let
   482       val ctxt = Simplifier.the_context ss
   483       val t = Thm.term_of ct
   484     in
   485       case find_double t of
   486         (T, NONE) => NONE
   487       | (T, SOME rhs) =>
   488           SOME (Goal.prove ctxt [] [] (Term.equals T $ t $ rhs)
   489             (fn _ =>
   490               rtac eq_reflection 1 THEN
   491               rtac ext 1 THEN
   492               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
   493     end
   494 in proc end
   495 *}
   496 
   497 
   498 subsection {* Code generator setup *}
   499 
   500 code_const "op \<circ>"
   501   (SML infixl 5 "o")
   502   (Haskell infixr 9 ".")
   503 
   504 code_const "id"
   505   (Haskell "id")
   506 
   507 
   508 subsection {* ML legacy bindings *} 
   509 
   510 ML {*
   511 val set_cs = claset() delrules [equalityI]
   512 *}
   513 
   514 ML {*
   515 val id_apply = @{thm id_apply}
   516 val id_def = @{thm id_def}
   517 val o_apply = @{thm o_apply}
   518 val o_assoc = @{thm o_assoc}
   519 val o_def = @{thm o_def}
   520 val injD = @{thm injD}
   521 val datatype_injI = @{thm datatype_injI}
   522 val range_ex1_eq = @{thm range_ex1_eq}
   523 val expand_fun_eq = @{thm expand_fun_eq}
   524 *}
   525 
   526 end