src/HOL/Fun.thy
 author nipkow Sat Oct 17 13:46:39 2009 +0200 (2009-10-17) changeset 32961 61431a41ddd5 parent 32740 9dd0a2f83429 child 32988 d1d4d7a08a66 permissions -rw-r--r--
```     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 UN_o: "UNION A (g o f) = UNION (f`A) g"
```
```    82 by (unfold comp_def, blast)
```
```    83
```
```    84
```
```    85 subsection {* The Forward Composition Operator @{text fcomp} *}
```
```    86
```
```    87 definition
```
```    88   fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o>" 60)
```
```    89 where
```
```    90   "f o> g = (\<lambda>x. g (f x))"
```
```    91
```
```    92 lemma fcomp_apply:  "(f o> g) x = g (f x)"
```
```    93   by (simp add: fcomp_def)
```
```    94
```
```    95 lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)"
```
```    96   by (simp add: fcomp_def)
```
```    97
```
```    98 lemma id_fcomp [simp]: "id o> g = g"
```
```    99   by (simp add: fcomp_def)
```
```   100
```
```   101 lemma fcomp_id [simp]: "f o> id = f"
```
```   102   by (simp add: fcomp_def)
```
```   103
```
```   104 code_const fcomp
```
```   105   (Eval infixl 1 "#>")
```
```   106
```
```   107 no_notation fcomp (infixl "o>" 60)
```
```   108
```
```   109
```
```   110 subsection {* Injectivity and Surjectivity *}
```
```   111
```
```   112 constdefs
```
```   113   inj_on :: "['a => 'b, 'a set] => bool"  -- "injective"
```
```   114   "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
```
```   115
```
```   116 text{*A common special case: functions injective over the entire domain type.*}
```
```   117
```
```   118 abbreviation
```
```   119   "inj f == inj_on f UNIV"
```
```   120
```
```   121 definition
```
```   122   bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
```
```   123   [code del]: "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"
```
```   124
```
```   125 constdefs
```
```   126   surj :: "('a => 'b) => bool"                   (*surjective*)
```
```   127   "surj f == ! y. ? x. y=f(x)"
```
```   128
```
```   129   bij :: "('a => 'b) => bool"                    (*bijective*)
```
```   130   "bij f == inj f & surj f"
```
```   131
```
```   132 lemma injI:
```
```   133   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
```
```   134   shows "inj f"
```
```   135   using assms unfolding inj_on_def by auto
```
```   136
```
```   137 text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
```
```   138 lemma datatype_injI:
```
```   139     "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
```
```   140 by (simp add: inj_on_def)
```
```   141
```
```   142 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
```
```   143   by (unfold inj_on_def, blast)
```
```   144
```
```   145 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
```
```   146 by (simp add: inj_on_def)
```
```   147
```
```   148 (*Useful with the simplifier*)
```
```   149 lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)"
```
```   150 by (force simp add: inj_on_def)
```
```   151
```
```   152 lemma inj_on_id[simp]: "inj_on id A"
```
```   153   by (simp add: inj_on_def)
```
```   154
```
```   155 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
```
```   156 by (simp add: inj_on_def)
```
```   157
```
```   158 lemma surj_id[simp]: "surj id"
```
```   159 by (simp add: surj_def)
```
```   160
```
```   161 lemma bij_id[simp]: "bij id"
```
```   162 by (simp add: bij_def inj_on_id surj_id)
```
```   163
```
```   164 lemma inj_onI:
```
```   165     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
```
```   166 by (simp add: inj_on_def)
```
```   167
```
```   168 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
```
```   169 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
```
```   170
```
```   171 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
```
```   172 by (unfold inj_on_def, blast)
```
```   173
```
```   174 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
```
```   175 by (blast dest!: inj_onD)
```
```   176
```
```   177 lemma comp_inj_on:
```
```   178      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
```
```   179 by (simp add: comp_def inj_on_def)
```
```   180
```
```   181 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
```
```   182 apply(simp add:inj_on_def image_def)
```
```   183 apply blast
```
```   184 done
```
```   185
```
```   186 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
```
```   187   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
```
```   188 apply(unfold inj_on_def)
```
```   189 apply blast
```
```   190 done
```
```   191
```
```   192 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
```
```   193 by (unfold inj_on_def, blast)
```
```   194
```
```   195 lemma inj_singleton: "inj (%s. {s})"
```
```   196 by (simp add: inj_on_def)
```
```   197
```
```   198 lemma inj_on_empty[iff]: "inj_on f {}"
```
```   199 by(simp add: inj_on_def)
```
```   200
```
```   201 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
```
```   202 by (unfold inj_on_def, blast)
```
```   203
```
```   204 lemma inj_on_Un:
```
```   205  "inj_on f (A Un B) =
```
```   206   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
```
```   207 apply(unfold inj_on_def)
```
```   208 apply (blast intro:sym)
```
```   209 done
```
```   210
```
```   211 lemma inj_on_insert[iff]:
```
```   212   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
```
```   213 apply(unfold inj_on_def)
```
```   214 apply (blast intro:sym)
```
```   215 done
```
```   216
```
```   217 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
```
```   218 apply(unfold inj_on_def)
```
```   219 apply (blast)
```
```   220 done
```
```   221
```
```   222 lemma surjI: "(!! x. g(f x) = x) ==> surj g"
```
```   223 apply (simp add: surj_def)
```
```   224 apply (blast intro: sym)
```
```   225 done
```
```   226
```
```   227 lemma surj_range: "surj f ==> range f = UNIV"
```
```   228 by (auto simp add: surj_def)
```
```   229
```
```   230 lemma surjD: "surj f ==> EX x. y = f x"
```
```   231 by (simp add: surj_def)
```
```   232
```
```   233 lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
```
```   234 by (simp add: surj_def, blast)
```
```   235
```
```   236 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
```
```   237 apply (simp add: comp_def surj_def, clarify)
```
```   238 apply (drule_tac x = y in spec, clarify)
```
```   239 apply (drule_tac x = x in spec, blast)
```
```   240 done
```
```   241
```
```   242 lemma bijI: "[| inj f; surj f |] ==> bij f"
```
```   243 by (simp add: bij_def)
```
```   244
```
```   245 lemma bij_is_inj: "bij f ==> inj f"
```
```   246 by (simp add: bij_def)
```
```   247
```
```   248 lemma bij_is_surj: "bij f ==> surj f"
```
```   249 by (simp add: bij_def)
```
```   250
```
```   251 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
```
```   252 by (simp add: bij_betw_def)
```
```   253
```
```   254 lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
```
```   255 by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
```
```   256
```
```   257 lemma bij_betw_trans:
```
```   258   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
```
```   259 by(auto simp add:bij_betw_def comp_inj_on)
```
```   260
```
```   261 lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
```
```   262 proof -
```
```   263   have i: "inj_on f A" and s: "f ` A = B"
```
```   264     using assms by(auto simp:bij_betw_def)
```
```   265   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
```
```   266   { fix a b assume P: "?P b a"
```
```   267     hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
```
```   268     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
```
```   269     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
```
```   270   } note g = this
```
```   271   have "inj_on ?g B"
```
```   272   proof(rule inj_onI)
```
```   273     fix x y assume "x:B" "y:B" "?g x = ?g y"
```
```   274     from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
```
```   275     from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
```
```   276     from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
```
```   277   qed
```
```   278   moreover have "?g ` B = A"
```
```   279   proof(auto simp:image_def)
```
```   280     fix b assume "b:B"
```
```   281     with s obtain a where P: "?P b a" unfolding image_def by blast
```
```   282     thus "?g b \<in> A" using g[OF P] by auto
```
```   283   next
```
```   284     fix a assume "a:A"
```
```   285     then obtain b where P: "?P b a" using s unfolding image_def by blast
```
```   286     then have "b:B" using s unfolding image_def by blast
```
```   287     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
```
```   288   qed
```
```   289   ultimately show ?thesis by(auto simp:bij_betw_def)
```
```   290 qed
```
```   291
```
```   292 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
```
```   293 by (simp add: surj_range)
```
```   294
```
```   295 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
```
```   296 by (simp add: inj_on_def, blast)
```
```   297
```
```   298 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
```
```   299 apply (unfold surj_def)
```
```   300 apply (blast intro: sym)
```
```   301 done
```
```   302
```
```   303 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
```
```   304 by (unfold inj_on_def, blast)
```
```   305
```
```   306 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
```
```   307 apply (unfold bij_def)
```
```   308 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
```
```   309 done
```
```   310
```
```   311 lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
```
```   312 by(blast dest: inj_onD)
```
```   313
```
```   314 lemma inj_on_image_Int:
```
```   315    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
```
```   316 apply (simp add: inj_on_def, blast)
```
```   317 done
```
```   318
```
```   319 lemma inj_on_image_set_diff:
```
```   320    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
```
```   321 apply (simp add: inj_on_def, blast)
```
```   322 done
```
```   323
```
```   324 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
```
```   325 by (simp add: inj_on_def, blast)
```
```   326
```
```   327 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
```
```   328 by (simp add: inj_on_def, blast)
```
```   329
```
```   330 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
```
```   331 by (blast dest: injD)
```
```   332
```
```   333 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
```
```   334 by (simp add: inj_on_def, blast)
```
```   335
```
```   336 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
```
```   337 by (blast dest: injD)
```
```   338
```
```   339 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
```
```   340 lemma image_INT:
```
```   341    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
```
```   342     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   343 apply (simp add: inj_on_def, blast)
```
```   344 done
```
```   345
```
```   346 (*Compare with image_INT: no use of inj_on, and if f is surjective then
```
```   347   it doesn't matter whether A is empty*)
```
```   348 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   349 apply (simp add: bij_def)
```
```   350 apply (simp add: inj_on_def surj_def, blast)
```
```   351 done
```
```   352
```
```   353 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
```
```   354 by (auto simp add: surj_def)
```
```   355
```
```   356 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
```
```   357 by (auto simp add: inj_on_def)
```
```   358
```
```   359 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
```
```   360 apply (simp add: bij_def)
```
```   361 apply (rule equalityI)
```
```   362 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
```
```   363 done
```
```   364
```
```   365
```
```   366 subsection{*Function Updating*}
```
```   367
```
```   368 constdefs
```
```   369   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
```
```   370   "fun_upd f a b == % x. if x=a then b else f x"
```
```   371
```
```   372 nonterminals
```
```   373   updbinds updbind
```
```   374 syntax
```
```   375   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
```
```   376   ""         :: "updbind => updbinds"             ("_")
```
```   377   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
```
```   378   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
```
```   379
```
```   380 translations
```
```   381   "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
```
```   382   "f(x:=y)"                     == "fun_upd f x y"
```
```   383
```
```   384 (* Hint: to define the sum of two functions (or maps), use sum_case.
```
```   385          A nice infix syntax could be defined (in Datatype.thy or below) by
```
```   386 consts
```
```   387   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
```
```   388 translations
```
```   389  "fun_sum" == sum_case
```
```   390 *)
```
```   391
```
```   392 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
```
```   393 apply (simp add: fun_upd_def, safe)
```
```   394 apply (erule subst)
```
```   395 apply (rule_tac [2] ext, auto)
```
```   396 done
```
```   397
```
```   398 (* f x = y ==> f(x:=y) = f *)
```
```   399 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
```
```   400
```
```   401 (* f(x := f x) = f *)
```
```   402 lemmas fun_upd_triv = refl [THEN fun_upd_idem]
```
```   403 declare fun_upd_triv [iff]
```
```   404
```
```   405 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
```
```   406 by (simp add: fun_upd_def)
```
```   407
```
```   408 (* fun_upd_apply supersedes these two,   but they are useful
```
```   409    if fun_upd_apply is intentionally removed from the simpset *)
```
```   410 lemma fun_upd_same: "(f(x:=y)) x = y"
```
```   411 by simp
```
```   412
```
```   413 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
```
```   414 by simp
```
```   415
```
```   416 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
```
```   417 by (simp add: expand_fun_eq)
```
```   418
```
```   419 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
```
```   420 by (rule ext, auto)
```
```   421
```
```   422 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
```
```   423 by(fastsimp simp:inj_on_def image_def)
```
```   424
```
```   425 lemma fun_upd_image:
```
```   426      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
```
```   427 by auto
```
```   428
```
```   429 lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
```
```   430 by(auto intro: ext)
```
```   431
```
```   432
```
```   433 subsection {* @{text override_on} *}
```
```   434
```
```   435 definition
```
```   436   override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   437 where
```
```   438   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
```
```   439
```
```   440 lemma override_on_emptyset[simp]: "override_on f g {} = f"
```
```   441 by(simp add:override_on_def)
```
```   442
```
```   443 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
```
```   444 by(simp add:override_on_def)
```
```   445
```
```   446 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
```
```   447 by(simp add:override_on_def)
```
```   448
```
```   449
```
```   450 subsection {* @{text swap} *}
```
```   451
```
```   452 definition
```
```   453   swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
```
```   454 where
```
```   455   "swap a b f = f (a := f b, b:= f a)"
```
```   456
```
```   457 lemma swap_self: "swap a a f = f"
```
```   458 by (simp add: swap_def)
```
```   459
```
```   460 lemma swap_commute: "swap a b f = swap b a f"
```
```   461 by (rule ext, simp add: fun_upd_def swap_def)
```
```   462
```
```   463 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
```
```   464 by (rule ext, simp add: fun_upd_def swap_def)
```
```   465
```
```   466 lemma inj_on_imp_inj_on_swap:
```
```   467   "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
```
```   468 by (simp add: inj_on_def swap_def, blast)
```
```   469
```
```   470 lemma inj_on_swap_iff [simp]:
```
```   471   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
```
```   472 proof
```
```   473   assume "inj_on (swap a b f) A"
```
```   474   with A have "inj_on (swap a b (swap a b f)) A"
```
```   475     by (iprover intro: inj_on_imp_inj_on_swap)
```
```   476   thus "inj_on f A" by simp
```
```   477 next
```
```   478   assume "inj_on f A"
```
```   479   with A show "inj_on (swap a b f) A" by(iprover intro: inj_on_imp_inj_on_swap)
```
```   480 qed
```
```   481
```
```   482 lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
```
```   483 apply (simp add: surj_def swap_def, clarify)
```
```   484 apply (case_tac "y = f b", blast)
```
```   485 apply (case_tac "y = f a", auto)
```
```   486 done
```
```   487
```
```   488 lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
```
```   489 proof
```
```   490   assume "surj (swap a b f)"
```
```   491   hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap)
```
```   492   thus "surj f" by simp
```
```   493 next
```
```   494   assume "surj f"
```
```   495   thus "surj (swap a b f)" by (rule surj_imp_surj_swap)
```
```   496 qed
```
```   497
```
```   498 lemma bij_swap_iff: "bij (swap a b f) = bij f"
```
```   499 by (simp add: bij_def)
```
```   500
```
```   501 hide (open) const swap
```
```   502
```
```   503
```
```   504 subsection {* Inversion of injective functions *}
```
```   505
```
```   506 definition inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
```
```   507   "inv f y = (THE x. f x = y)"
```
```   508
```
```   509 lemma inv_f_f:
```
```   510   assumes "inj f"
```
```   511   shows "inv f (f x) = x"
```
```   512 proof -
```
```   513   from assms have "(THE x'. f x' = f x) = (THE x'. x' = x)"
```
```   514     by (simp only: inj_eq)
```
```   515   also have "... = x" by (rule the_eq_trivial)
```
```   516   finally show ?thesis by (unfold inv_def)
```
```   517 qed
```
```   518
```
```   519 lemma f_inv_f:
```
```   520   assumes "inj f"
```
```   521   and "y \<in> range f"
```
```   522   shows "f (inv f y) = y"
```
```   523 proof (unfold inv_def)
```
```   524   from `y \<in> range f` obtain x where "y = f x" ..
```
```   525   then have "f x = y" ..
```
```   526   then show "f (THE x. f x = y) = y"
```
```   527   proof (rule theI)
```
```   528     fix x' assume "f x' = y"
```
```   529     with `f x = y` have "f x' = f x" by simp
```
```   530     with `inj f` show "x' = x" by (rule injD)
```
```   531   qed
```
```   532 qed
```
```   533
```
```   534 hide (open) const inv
```
```   535
```
```   536 definition the_inv_onto :: "'a set => ('a => 'b) => ('b => 'a)" where
```
```   537 "the_inv_onto A f == %x. THE y. y : A & f y = x"
```
```   538
```
```   539 lemma the_inv_onto_f_f:
```
```   540   "[| inj_on f A;  x : A |] ==> the_inv_onto A f (f x) = x"
```
```   541 apply (simp add: the_inv_onto_def inj_on_def)
```
```   542 apply (blast intro: the_equality)
```
```   543 done
```
```   544
```
```   545 lemma f_the_inv_onto_f:
```
```   546   "inj_on f A ==> y : f`A  ==> f (the_inv_onto A f y) = y"
```
```   547 apply (simp add: the_inv_onto_def)
```
```   548 apply (rule the1I2)
```
```   549  apply(blast dest: inj_onD)
```
```   550 apply blast
```
```   551 done
```
```   552
```
```   553 lemma the_inv_onto_into:
```
```   554   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_onto A f x : B"
```
```   555 apply (simp add: the_inv_onto_def)
```
```   556 apply (rule the1I2)
```
```   557  apply(blast dest: inj_onD)
```
```   558 apply blast
```
```   559 done
```
```   560
```
```   561 lemma the_inv_onto_onto[simp]:
```
```   562   "inj_on f A ==> the_inv_onto A f ` (f ` A) = A"
```
```   563 by (fast intro:the_inv_onto_into the_inv_onto_f_f[symmetric])
```
```   564
```
```   565 lemma the_inv_onto_f_eq:
```
```   566   "[| inj_on f A; f x = y; x : A |] ==> the_inv_onto A f y = x"
```
```   567   apply (erule subst)
```
```   568   apply (erule the_inv_onto_f_f, assumption)
```
```   569   done
```
```   570
```
```   571 lemma the_inv_onto_comp:
```
```   572   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
```
```   573   the_inv_onto A (f o g) x = (the_inv_onto A g o the_inv_onto (g ` A) f) x"
```
```   574 apply (rule the_inv_onto_f_eq)
```
```   575   apply (fast intro: comp_inj_on)
```
```   576  apply (simp add: f_the_inv_onto_f the_inv_onto_into)
```
```   577 apply (simp add: the_inv_onto_into)
```
```   578 done
```
```   579
```
```   580 lemma inj_on_the_inv_onto:
```
```   581   "inj_on f A \<Longrightarrow> inj_on (the_inv_onto A f) (f ` A)"
```
```   582 by (auto intro: inj_onI simp: image_def the_inv_onto_f_f)
```
```   583
```
```   584 lemma bij_betw_the_inv_onto:
```
```   585   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_onto A f) B A"
```
```   586 by (auto simp add: bij_betw_def inj_on_the_inv_onto the_inv_onto_into)
```
```   587
```
```   588
```
```   589 subsection {* Proof tool setup *}
```
```   590
```
```   591 text {* simplifies terms of the form
```
```   592   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
```
```   593
```
```   594 simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
```
```   595 let
```
```   596   fun gen_fun_upd NONE T _ _ = NONE
```
```   597     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y)
```
```   598   fun dest_fun_T1 (Type (_, T :: Ts)) = T
```
```   599   fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) =
```
```   600     let
```
```   601       fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) =
```
```   602             if v aconv x then SOME g else gen_fun_upd (find g) T v w
```
```   603         | find t = NONE
```
```   604     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
```
```   605
```
```   606   fun proc ss ct =
```
```   607     let
```
```   608       val ctxt = Simplifier.the_context ss
```
```   609       val t = Thm.term_of ct
```
```   610     in
```
```   611       case find_double t of
```
```   612         (T, NONE) => NONE
```
```   613       | (T, SOME rhs) =>
```
```   614           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
```
```   615             (fn _ =>
```
```   616               rtac eq_reflection 1 THEN
```
```   617               rtac ext 1 THEN
```
```   618               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
```
```   619     end
```
```   620 in proc end
```
```   621 *}
```
```   622
```
```   623
```
```   624 subsection {* Generic transfer procedure *}
```
```   625
```
```   626 definition TransferMorphism:: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> bool"
```
```   627   where "TransferMorphism a B \<longleftrightarrow> True"
```
```   628
```
```   629 use "Tools/transfer.ML"
```
```   630
```
```   631 setup Transfer.setup
```
```   632
```
```   633
```
```   634 subsection {* Code generator setup *}
```
```   635
```
```   636 types_code
```
```   637   "fun"  ("(_ ->/ _)")
```
```   638 attach (term_of) {*
```
```   639 fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
```
```   640 *}
```
```   641 attach (test) {*
```
```   642 fun gen_fun_type aF aT bG bT i =
```
```   643   let
```
```   644     val tab = Unsynchronized.ref [];
```
```   645     fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
```
```   646       (aT --> bT) --> aT --> bT --> aT --> bT) \$ t \$ aF x \$ y ()
```
```   647   in
```
```   648     (fn x =>
```
```   649        case AList.lookup op = (!tab) x of
```
```   650          NONE =>
```
```   651            let val p as (y, _) = bG i
```
```   652            in (tab := (x, p) :: !tab; y) end
```
```   653        | SOME (y, _) => y,
```
```   654      fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
```
```   655   end;
```
```   656 *}
```
```   657
```
```   658 code_const "op \<circ>"
```
```   659   (SML infixl 5 "o")
```
```   660   (Haskell infixr 9 ".")
```
```   661
```
```   662 code_const "id"
```
```   663   (Haskell "id")
```
```   664
```
```   665 end
```