src/HOL/Fun.thy
 author hoelzl Mon Nov 22 10:34:33 2010 +0100 (2010-11-22) changeset 40702 cf26dd7395e4 parent 40602 91e583511113 child 40703 d1fc454d6735 permissions -rw-r--r--
Replace surj by abbreviation; remove surj_on.
```     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 begin
```
```    11
```
```    12 text{*As a simplification rule, it replaces all function equalities by
```
```    13   first-order equalities.*}
```
```    14 lemma fun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f x = g x)"
```
```    15 apply (rule iffI)
```
```    16 apply (simp (no_asm_simp))
```
```    17 apply (rule ext)
```
```    18 apply (simp (no_asm_simp))
```
```    19 done
```
```    20
```
```    21 lemma apply_inverse:
```
```    22   "f x = u \<Longrightarrow> (\<And>x. P x \<Longrightarrow> g (f x) = x) \<Longrightarrow> P x \<Longrightarrow> x = g u"
```
```    23   by auto
```
```    24
```
```    25
```
```    26 subsection {* The Identity Function @{text id} *}
```
```    27
```
```    28 definition
```
```    29   id :: "'a \<Rightarrow> 'a"
```
```    30 where
```
```    31   "id = (\<lambda>x. x)"
```
```    32
```
```    33 lemma id_apply [simp]: "id x = x"
```
```    34   by (simp add: id_def)
```
```    35
```
```    36 lemma image_ident [simp]: "(%x. x) ` Y = Y"
```
```    37 by blast
```
```    38
```
```    39 lemma image_id [simp]: "id ` Y = Y"
```
```    40 by (simp add: id_def)
```
```    41
```
```    42 lemma vimage_ident [simp]: "(%x. x) -` Y = Y"
```
```    43 by blast
```
```    44
```
```    45 lemma vimage_id [simp]: "id -` A = A"
```
```    46 by (simp add: id_def)
```
```    47
```
```    48
```
```    49 subsection {* The Composition Operator @{text "f \<circ> g"} *}
```
```    50
```
```    51 definition
```
```    52   comp :: "('b \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "o" 55)
```
```    53 where
```
```    54   "f o g = (\<lambda>x. f (g x))"
```
```    55
```
```    56 notation (xsymbols)
```
```    57   comp  (infixl "\<circ>" 55)
```
```    58
```
```    59 notation (HTML output)
```
```    60   comp  (infixl "\<circ>" 55)
```
```    61
```
```    62 text{*compatibility*}
```
```    63 lemmas o_def = comp_def
```
```    64
```
```    65 lemma o_apply [simp]: "(f o g) x = f (g x)"
```
```    66 by (simp add: comp_def)
```
```    67
```
```    68 lemma o_assoc: "f o (g o h) = f o g o h"
```
```    69 by (simp add: comp_def)
```
```    70
```
```    71 lemma id_o [simp]: "id o g = g"
```
```    72 by (simp add: comp_def)
```
```    73
```
```    74 lemma o_id [simp]: "f o id = f"
```
```    75 by (simp add: comp_def)
```
```    76
```
```    77 lemma o_eq_dest:
```
```    78   "a o b = c o d \<Longrightarrow> a (b v) = c (d v)"
```
```    79   by (simp only: o_def) (fact fun_cong)
```
```    80
```
```    81 lemma o_eq_elim:
```
```    82   "a o b = c o d \<Longrightarrow> ((\<And>v. a (b v) = c (d v)) \<Longrightarrow> R) \<Longrightarrow> R"
```
```    83   by (erule meta_mp) (fact o_eq_dest)
```
```    84
```
```    85 lemma image_compose: "(f o g) ` r = f`(g`r)"
```
```    86 by (simp add: comp_def, blast)
```
```    87
```
```    88 lemma vimage_compose: "(g \<circ> f) -` x = f -` (g -` x)"
```
```    89   by auto
```
```    90
```
```    91 lemma UN_o: "UNION A (g o f) = UNION (f`A) g"
```
```    92 by (unfold comp_def, blast)
```
```    93
```
```    94
```
```    95 subsection {* The Forward Composition Operator @{text fcomp} *}
```
```    96
```
```    97 definition
```
```    98   fcomp :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'a \<Rightarrow> 'c" (infixl "\<circ>>" 60)
```
```    99 where
```
```   100   "f \<circ>> g = (\<lambda>x. g (f x))"
```
```   101
```
```   102 lemma fcomp_apply [simp]:  "(f \<circ>> g) x = g (f x)"
```
```   103   by (simp add: fcomp_def)
```
```   104
```
```   105 lemma fcomp_assoc: "(f \<circ>> g) \<circ>> h = f \<circ>> (g \<circ>> h)"
```
```   106   by (simp add: fcomp_def)
```
```   107
```
```   108 lemma id_fcomp [simp]: "id \<circ>> g = g"
```
```   109   by (simp add: fcomp_def)
```
```   110
```
```   111 lemma fcomp_id [simp]: "f \<circ>> id = f"
```
```   112   by (simp add: fcomp_def)
```
```   113
```
```   114 code_const fcomp
```
```   115   (Eval infixl 1 "#>")
```
```   116
```
```   117 no_notation fcomp (infixl "\<circ>>" 60)
```
```   118
```
```   119
```
```   120 subsection {* Mapping functions *}
```
```   121
```
```   122 definition map_fun :: "('c \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'd" where
```
```   123   "map_fun f g h = g \<circ> h \<circ> f"
```
```   124
```
```   125 lemma map_fun_apply [simp]:
```
```   126   "map_fun f g h x = g (h (f x))"
```
```   127   by (simp add: map_fun_def)
```
```   128
```
```   129 type_mapper map_fun
```
```   130   by (simp_all add: fun_eq_iff)
```
```   131
```
```   132
```
```   133 subsection {* Injectivity and Bijectivity *}
```
```   134
```
```   135 definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where -- "injective"
```
```   136   "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
```
```   137
```
```   138 definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where -- "bijective"
```
```   139   "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
```
```   140
```
```   141 text{*A common special case: functions injective, surjective or bijective over
```
```   142 the entire domain type.*}
```
```   143
```
```   144 abbreviation
```
```   145   "inj f \<equiv> inj_on f UNIV"
```
```   146
```
```   147 abbreviation surj :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" where -- "surjective"
```
```   148   "surj f \<equiv> (range f = UNIV)"
```
```   149
```
```   150 abbreviation
```
```   151   "bij f \<equiv> bij_betw f UNIV UNIV"
```
```   152
```
```   153 lemma injI:
```
```   154   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
```
```   155   shows "inj f"
```
```   156   using assms unfolding inj_on_def by auto
```
```   157
```
```   158 text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
```
```   159 lemma datatype_injI:
```
```   160     "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
```
```   161 by (simp add: inj_on_def)
```
```   162
```
```   163 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
```
```   164   by (unfold inj_on_def, blast)
```
```   165
```
```   166 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
```
```   167 by (simp add: inj_on_def)
```
```   168
```
```   169 lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
```
```   170 by (force simp add: inj_on_def)
```
```   171
```
```   172 lemma inj_comp:
```
```   173   "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
```
```   174   by (simp add: inj_on_def)
```
```   175
```
```   176 lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
```
```   177   by (simp add: inj_on_def fun_eq_iff)
```
```   178
```
```   179 lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
```
```   180 by (simp add: inj_on_eq_iff)
```
```   181
```
```   182 lemma inj_on_id[simp]: "inj_on id A"
```
```   183   by (simp add: inj_on_def)
```
```   184
```
```   185 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
```
```   186 by (simp add: inj_on_def)
```
```   187
```
```   188 lemma surj_id: "surj id"
```
```   189 by simp
```
```   190
```
```   191 lemma bij_id[simp]: "bij id"
```
```   192 by (simp add: bij_betw_def)
```
```   193
```
```   194 lemma inj_onI:
```
```   195     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
```
```   196 by (simp add: inj_on_def)
```
```   197
```
```   198 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
```
```   199 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
```
```   200
```
```   201 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
```
```   202 by (unfold inj_on_def, blast)
```
```   203
```
```   204 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
```
```   205 by (blast dest!: inj_onD)
```
```   206
```
```   207 lemma comp_inj_on:
```
```   208      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
```
```   209 by (simp add: comp_def inj_on_def)
```
```   210
```
```   211 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
```
```   212 apply(simp add:inj_on_def image_def)
```
```   213 apply blast
```
```   214 done
```
```   215
```
```   216 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
```
```   217   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
```
```   218 apply(unfold inj_on_def)
```
```   219 apply blast
```
```   220 done
```
```   221
```
```   222 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
```
```   223 by (unfold inj_on_def, blast)
```
```   224
```
```   225 lemma inj_singleton: "inj (%s. {s})"
```
```   226 by (simp add: inj_on_def)
```
```   227
```
```   228 lemma inj_on_empty[iff]: "inj_on f {}"
```
```   229 by(simp add: inj_on_def)
```
```   230
```
```   231 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
```
```   232 by (unfold inj_on_def, blast)
```
```   233
```
```   234 lemma inj_on_Un:
```
```   235  "inj_on f (A Un B) =
```
```   236   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
```
```   237 apply(unfold inj_on_def)
```
```   238 apply (blast intro:sym)
```
```   239 done
```
```   240
```
```   241 lemma inj_on_insert[iff]:
```
```   242   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
```
```   243 apply(unfold inj_on_def)
```
```   244 apply (blast intro:sym)
```
```   245 done
```
```   246
```
```   247 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
```
```   248 apply(unfold inj_on_def)
```
```   249 apply (blast)
```
```   250 done
```
```   251
```
```   252 lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
```
```   253   by auto
```
```   254
```
```   255 lemma surjI: assumes *: "\<And> x. g (f x) = x" shows "surj g"
```
```   256   using *[symmetric] by auto
```
```   257
```
```   258 lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
```
```   259   by (simp add: surj_def)
```
```   260
```
```   261 lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
```
```   262   by (simp add: surj_def, blast)
```
```   263
```
```   264 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
```
```   265 apply (simp add: comp_def surj_def, clarify)
```
```   266 apply (drule_tac x = y in spec, clarify)
```
```   267 apply (drule_tac x = x in spec, blast)
```
```   268 done
```
```   269
```
```   270 lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
```
```   271   unfolding bij_betw_def by auto
```
```   272
```
```   273 lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
```
```   274   unfolding bij_betw_def ..
```
```   275
```
```   276 lemma bijI: "[| inj f; surj f |] ==> bij f"
```
```   277 by (simp add: bij_def)
```
```   278
```
```   279 lemma bij_is_inj: "bij f ==> inj f"
```
```   280 by (simp add: bij_def)
```
```   281
```
```   282 lemma bij_is_surj: "bij f ==> surj f"
```
```   283 by (simp add: bij_def)
```
```   284
```
```   285 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
```
```   286 by (simp add: bij_betw_def)
```
```   287
```
```   288 lemma bij_betw_trans:
```
```   289   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
```
```   290 by(auto simp add:bij_betw_def comp_inj_on)
```
```   291
```
```   292 lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
```
```   293   by (rule bij_betw_trans)
```
```   294
```
```   295 lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
```
```   296 proof -
```
```   297   have i: "inj_on f A" and s: "f ` A = B"
```
```   298     using assms by(auto simp:bij_betw_def)
```
```   299   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
```
```   300   { fix a b assume P: "?P b a"
```
```   301     hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
```
```   302     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
```
```   303     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
```
```   304   } note g = this
```
```   305   have "inj_on ?g B"
```
```   306   proof(rule inj_onI)
```
```   307     fix x y assume "x:B" "y:B" "?g x = ?g y"
```
```   308     from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
```
```   309     from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
```
```   310     from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
```
```   311   qed
```
```   312   moreover have "?g ` B = A"
```
```   313   proof(auto simp:image_def)
```
```   314     fix b assume "b:B"
```
```   315     with s obtain a where P: "?P b a" unfolding image_def by blast
```
```   316     thus "?g b \<in> A" using g[OF P] by auto
```
```   317   next
```
```   318     fix a assume "a:A"
```
```   319     then obtain b where P: "?P b a" using s unfolding image_def by blast
```
```   320     then have "b:B" using s unfolding image_def by blast
```
```   321     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
```
```   322   qed
```
```   323   ultimately show ?thesis by(auto simp:bij_betw_def)
```
```   324 qed
```
```   325
```
```   326 lemma bij_betw_combine:
```
```   327   assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
```
```   328   shows "bij_betw f (A \<union> C) (B \<union> D)"
```
```   329   using assms unfolding bij_betw_def inj_on_Un image_Un by auto
```
```   330
```
```   331 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
```
```   332 by simp
```
```   333
```
```   334 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
```
```   335 by (simp add: inj_on_def, blast)
```
```   336
```
```   337 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
```
```   338 by (blast intro: sym)
```
```   339
```
```   340 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
```
```   341 by (unfold inj_on_def, blast)
```
```   342
```
```   343 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
```
```   344 apply (unfold bij_def)
```
```   345 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
```
```   346 done
```
```   347
```
```   348 lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
```
```   349 by(blast dest: inj_onD)
```
```   350
```
```   351 lemma inj_on_image_Int:
```
```   352    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
```
```   353 apply (simp add: inj_on_def, blast)
```
```   354 done
```
```   355
```
```   356 lemma inj_on_image_set_diff:
```
```   357    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
```
```   358 apply (simp add: inj_on_def, blast)
```
```   359 done
```
```   360
```
```   361 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
```
```   362 by (simp add: inj_on_def, blast)
```
```   363
```
```   364 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
```
```   365 by (simp add: inj_on_def, blast)
```
```   366
```
```   367 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
```
```   368 by (blast dest: injD)
```
```   369
```
```   370 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
```
```   371 by (simp add: inj_on_def, blast)
```
```   372
```
```   373 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
```
```   374 by (blast dest: injD)
```
```   375
```
```   376 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
```
```   377 lemma image_INT:
```
```   378    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
```
```   379     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   380 apply (simp add: inj_on_def, blast)
```
```   381 done
```
```   382
```
```   383 (*Compare with image_INT: no use of inj_on, and if f is surjective then
```
```   384   it doesn't matter whether A is empty*)
```
```   385 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   386 apply (simp add: bij_def)
```
```   387 apply (simp add: inj_on_def surj_def, blast)
```
```   388 done
```
```   389
```
```   390 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
```
```   391 by auto
```
```   392
```
```   393 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
```
```   394 by (auto simp add: inj_on_def)
```
```   395
```
```   396 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
```
```   397 apply (simp add: bij_def)
```
```   398 apply (rule equalityI)
```
```   399 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
```
```   400 done
```
```   401
```
```   402 lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
```
```   403   by (auto intro!: inj_onI)
```
```   404
```
```   405 lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
```
```   406   by (auto intro!: inj_onI dest: strict_mono_eq)
```
```   407
```
```   408 subsection{*Function Updating*}
```
```   409
```
```   410 definition
```
```   411   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
```
```   412   "fun_upd f a b == % x. if x=a then b else f x"
```
```   413
```
```   414 nonterminals
```
```   415   updbinds updbind
```
```   416 syntax
```
```   417   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
```
```   418   ""         :: "updbind => updbinds"             ("_")
```
```   419   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
```
```   420   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
```
```   421
```
```   422 translations
```
```   423   "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
```
```   424   "f(x:=y)" == "CONST fun_upd f x y"
```
```   425
```
```   426 (* Hint: to define the sum of two functions (or maps), use sum_case.
```
```   427          A nice infix syntax could be defined (in Datatype.thy or below) by
```
```   428 notation
```
```   429   sum_case  (infixr "'(+')"80)
```
```   430 *)
```
```   431
```
```   432 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
```
```   433 apply (simp add: fun_upd_def, safe)
```
```   434 apply (erule subst)
```
```   435 apply (rule_tac [2] ext, auto)
```
```   436 done
```
```   437
```
```   438 (* f x = y ==> f(x:=y) = f *)
```
```   439 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
```
```   440
```
```   441 (* f(x := f x) = f *)
```
```   442 lemmas fun_upd_triv = refl [THEN fun_upd_idem]
```
```   443 declare fun_upd_triv [iff]
```
```   444
```
```   445 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
```
```   446 by (simp add: fun_upd_def)
```
```   447
```
```   448 (* fun_upd_apply supersedes these two,   but they are useful
```
```   449    if fun_upd_apply is intentionally removed from the simpset *)
```
```   450 lemma fun_upd_same: "(f(x:=y)) x = y"
```
```   451 by simp
```
```   452
```
```   453 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
```
```   454 by simp
```
```   455
```
```   456 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
```
```   457 by (simp add: fun_eq_iff)
```
```   458
```
```   459 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
```
```   460 by (rule ext, auto)
```
```   461
```
```   462 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
```
```   463 by (fastsimp simp:inj_on_def image_def)
```
```   464
```
```   465 lemma fun_upd_image:
```
```   466      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
```
```   467 by auto
```
```   468
```
```   469 lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
```
```   470 by (auto intro: ext)
```
```   471
```
```   472
```
```   473 subsection {* @{text override_on} *}
```
```   474
```
```   475 definition
```
```   476   override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   477 where
```
```   478   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
```
```   479
```
```   480 lemma override_on_emptyset[simp]: "override_on f g {} = f"
```
```   481 by(simp add:override_on_def)
```
```   482
```
```   483 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
```
```   484 by(simp add:override_on_def)
```
```   485
```
```   486 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
```
```   487 by(simp add:override_on_def)
```
```   488
```
```   489
```
```   490 subsection {* @{text swap} *}
```
```   491
```
```   492 definition
```
```   493   swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
```
```   494 where
```
```   495   "swap a b f = f (a := f b, b:= f a)"
```
```   496
```
```   497 lemma swap_self [simp]: "swap a a f = f"
```
```   498 by (simp add: swap_def)
```
```   499
```
```   500 lemma swap_commute: "swap a b f = swap b a f"
```
```   501 by (rule ext, simp add: fun_upd_def swap_def)
```
```   502
```
```   503 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
```
```   504 by (rule ext, simp add: fun_upd_def swap_def)
```
```   505
```
```   506 lemma swap_triple:
```
```   507   assumes "a \<noteq> c" and "b \<noteq> c"
```
```   508   shows "swap a b (swap b c (swap a b f)) = swap a c f"
```
```   509   using assms by (simp add: fun_eq_iff swap_def)
```
```   510
```
```   511 lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
```
```   512 by (rule ext, simp add: fun_upd_def swap_def)
```
```   513
```
```   514 lemma swap_image_eq [simp]:
```
```   515   assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
```
```   516 proof -
```
```   517   have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
```
```   518     using assms by (auto simp: image_iff swap_def)
```
```   519   then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
```
```   520   with subset[of f] show ?thesis by auto
```
```   521 qed
```
```   522
```
```   523 lemma inj_on_imp_inj_on_swap:
```
```   524   "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
```
```   525   by (simp add: inj_on_def swap_def, blast)
```
```   526
```
```   527 lemma inj_on_swap_iff [simp]:
```
```   528   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
```
```   529 proof
```
```   530   assume "inj_on (swap a b f) A"
```
```   531   with A have "inj_on (swap a b (swap a b f)) A"
```
```   532     by (iprover intro: inj_on_imp_inj_on_swap)
```
```   533   thus "inj_on f A" by simp
```
```   534 next
```
```   535   assume "inj_on f A"
```
```   536   with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
```
```   537 qed
```
```   538
```
```   539 lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
```
```   540   by simp
```
```   541
```
```   542 lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
```
```   543   by simp
```
```   544
```
```   545 lemma bij_betw_swap_iff [simp]:
```
```   546   "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
```
```   547   by (auto simp: bij_betw_def)
```
```   548
```
```   549 lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
```
```   550   by simp
```
```   551
```
```   552 hide_const (open) swap
```
```   553
```
```   554 subsection {* Inversion of injective functions *}
```
```   555
```
```   556 definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
```
```   557 "the_inv_into A f == %x. THE y. y : A & f y = x"
```
```   558
```
```   559 lemma the_inv_into_f_f:
```
```   560   "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
```
```   561 apply (simp add: the_inv_into_def inj_on_def)
```
```   562 apply blast
```
```   563 done
```
```   564
```
```   565 lemma f_the_inv_into_f:
```
```   566   "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
```
```   567 apply (simp add: the_inv_into_def)
```
```   568 apply (rule the1I2)
```
```   569  apply(blast dest: inj_onD)
```
```   570 apply blast
```
```   571 done
```
```   572
```
```   573 lemma the_inv_into_into:
```
```   574   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
```
```   575 apply (simp add: the_inv_into_def)
```
```   576 apply (rule the1I2)
```
```   577  apply(blast dest: inj_onD)
```
```   578 apply blast
```
```   579 done
```
```   580
```
```   581 lemma the_inv_into_onto[simp]:
```
```   582   "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
```
```   583 by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
```
```   584
```
```   585 lemma the_inv_into_f_eq:
```
```   586   "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
```
```   587   apply (erule subst)
```
```   588   apply (erule the_inv_into_f_f, assumption)
```
```   589   done
```
```   590
```
```   591 lemma the_inv_into_comp:
```
```   592   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
```
```   593   the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
```
```   594 apply (rule the_inv_into_f_eq)
```
```   595   apply (fast intro: comp_inj_on)
```
```   596  apply (simp add: f_the_inv_into_f the_inv_into_into)
```
```   597 apply (simp add: the_inv_into_into)
```
```   598 done
```
```   599
```
```   600 lemma inj_on_the_inv_into:
```
```   601   "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
```
```   602 by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
```
```   603
```
```   604 lemma bij_betw_the_inv_into:
```
```   605   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
```
```   606 by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
```
```   607
```
```   608 abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
```
```   609   "the_inv f \<equiv> the_inv_into UNIV f"
```
```   610
```
```   611 lemma the_inv_f_f:
```
```   612   assumes "inj f"
```
```   613   shows "the_inv f (f x) = x" using assms UNIV_I
```
```   614   by (rule the_inv_into_f_f)
```
```   615
```
```   616
```
```   617 subsection {* Proof tool setup *}
```
```   618
```
```   619 text {* simplifies terms of the form
```
```   620   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
```
```   621
```
```   622 simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
```
```   623 let
```
```   624   fun gen_fun_upd NONE T _ _ = NONE
```
```   625     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y)
```
```   626   fun dest_fun_T1 (Type (_, T :: Ts)) = T
```
```   627   fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) =
```
```   628     let
```
```   629       fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) =
```
```   630             if v aconv x then SOME g else gen_fun_upd (find g) T v w
```
```   631         | find t = NONE
```
```   632     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
```
```   633
```
```   634   fun proc ss ct =
```
```   635     let
```
```   636       val ctxt = Simplifier.the_context ss
```
```   637       val t = Thm.term_of ct
```
```   638     in
```
```   639       case find_double t of
```
```   640         (T, NONE) => NONE
```
```   641       | (T, SOME rhs) =>
```
```   642           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
```
```   643             (fn _ =>
```
```   644               rtac eq_reflection 1 THEN
```
```   645               rtac ext 1 THEN
```
```   646               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
```
```   647     end
```
```   648 in proc end
```
```   649 *}
```
```   650
```
```   651
```
```   652 subsection {* Code generator setup *}
```
```   653
```
```   654 types_code
```
```   655   "fun"  ("(_ ->/ _)")
```
```   656 attach (term_of) {*
```
```   657 fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
```
```   658 *}
```
```   659 attach (test) {*
```
```   660 fun gen_fun_type aF aT bG bT i =
```
```   661   let
```
```   662     val tab = Unsynchronized.ref [];
```
```   663     fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
```
```   664       (aT --> bT) --> aT --> bT --> aT --> bT) \$ t \$ aF x \$ y ()
```
```   665   in
```
```   666     (fn x =>
```
```   667        case AList.lookup op = (!tab) x of
```
```   668          NONE =>
```
```   669            let val p as (y, _) = bG i
```
```   670            in (tab := (x, p) :: !tab; y) end
```
```   671        | SOME (y, _) => y,
```
```   672      fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
```
```   673   end;
```
```   674 *}
```
```   675
```
```   676 code_const "op \<circ>"
```
```   677   (SML infixl 5 "o")
```
```   678   (Haskell infixr 9 ".")
```
```   679
```
```   680 code_const "id"
```
```   681   (Haskell "id")
```
```   682
```
```   683 end
```