src/HOL/Fun.thy
 author haftmann Thu Nov 18 17:01:15 2010 +0100 (2010-11-18) changeset 40602 91e583511113 parent 39302 d7728f65b353 child 40702 cf26dd7395e4 permissions -rw-r--r--
map_fun combinator in theory Fun
```     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, Surjectivity 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 surj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> bool" where -- "surjective"
```
```   139   "surj_on f B \<longleftrightarrow> B \<subseteq> range f"
```
```   140
```
```   141 definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where -- "bijective"
```
```   142   "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
```
```   143
```
```   144 text{*A common special case: functions injective over the entire domain type.*}
```
```   145
```
```   146 abbreviation
```
```   147   "inj f \<equiv> inj_on f UNIV"
```
```   148
```
```   149 abbreviation
```
```   150   "surj f \<equiv> surj_on f UNIV"
```
```   151
```
```   152 abbreviation
```
```   153   "bij f \<equiv> bij_betw f UNIV UNIV"
```
```   154
```
```   155 lemma injI:
```
```   156   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
```
```   157   shows "inj f"
```
```   158   using assms unfolding inj_on_def by auto
```
```   159
```
```   160 text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
```
```   161 lemma datatype_injI:
```
```   162     "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
```
```   163 by (simp add: inj_on_def)
```
```   164
```
```   165 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
```
```   166   by (unfold inj_on_def, blast)
```
```   167
```
```   168 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
```
```   169 by (simp add: inj_on_def)
```
```   170
```
```   171 lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
```
```   172 by (force simp add: inj_on_def)
```
```   173
```
```   174 lemma inj_comp:
```
```   175   "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
```
```   176   by (simp add: inj_on_def)
```
```   177
```
```   178 lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
```
```   179   by (simp add: inj_on_def fun_eq_iff)
```
```   180
```
```   181 lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
```
```   182 by (simp add: inj_on_eq_iff)
```
```   183
```
```   184 lemma inj_on_id[simp]: "inj_on id A"
```
```   185   by (simp add: inj_on_def)
```
```   186
```
```   187 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
```
```   188 by (simp add: inj_on_def)
```
```   189
```
```   190 lemma surj_id[simp]: "surj_on id A"
```
```   191 by (simp add: surj_on_def)
```
```   192
```
```   193 lemma bij_id[simp]: "bij id"
```
```   194 by (simp add: bij_betw_def)
```
```   195
```
```   196 lemma inj_onI:
```
```   197     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
```
```   198 by (simp add: inj_on_def)
```
```   199
```
```   200 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
```
```   201 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
```
```   202
```
```   203 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
```
```   204 by (unfold inj_on_def, blast)
```
```   205
```
```   206 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
```
```   207 by (blast dest!: inj_onD)
```
```   208
```
```   209 lemma comp_inj_on:
```
```   210      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
```
```   211 by (simp add: comp_def inj_on_def)
```
```   212
```
```   213 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
```
```   214 apply(simp add:inj_on_def image_def)
```
```   215 apply blast
```
```   216 done
```
```   217
```
```   218 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
```
```   219   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
```
```   220 apply(unfold inj_on_def)
```
```   221 apply blast
```
```   222 done
```
```   223
```
```   224 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
```
```   225 by (unfold inj_on_def, blast)
```
```   226
```
```   227 lemma inj_singleton: "inj (%s. {s})"
```
```   228 by (simp add: inj_on_def)
```
```   229
```
```   230 lemma inj_on_empty[iff]: "inj_on f {}"
```
```   231 by(simp add: inj_on_def)
```
```   232
```
```   233 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
```
```   234 by (unfold inj_on_def, blast)
```
```   235
```
```   236 lemma inj_on_Un:
```
```   237  "inj_on f (A Un B) =
```
```   238   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
```
```   239 apply(unfold inj_on_def)
```
```   240 apply (blast intro:sym)
```
```   241 done
```
```   242
```
```   243 lemma inj_on_insert[iff]:
```
```   244   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
```
```   245 apply(unfold inj_on_def)
```
```   246 apply (blast intro:sym)
```
```   247 done
```
```   248
```
```   249 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
```
```   250 apply(unfold inj_on_def)
```
```   251 apply (blast)
```
```   252 done
```
```   253
```
```   254 lemma surj_onI: "(\<And>x. x \<in> B \<Longrightarrow> g (f x) = x) \<Longrightarrow> surj_on g B"
```
```   255   by (simp add: surj_on_def) (blast intro: sym)
```
```   256
```
```   257 lemma surj_onD: "surj_on f B \<Longrightarrow> y \<in> B \<Longrightarrow> \<exists>x. y = f x"
```
```   258   by (auto simp: surj_on_def)
```
```   259
```
```   260 lemma surj_on_range_iff: "surj_on f B \<longleftrightarrow> (\<exists>A. f ` A = B)"
```
```   261   unfolding surj_on_def by (auto intro!: exI[of _ "f -` B"])
```
```   262
```
```   263 lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
```
```   264   by (simp add: surj_on_def subset_eq image_iff)
```
```   265
```
```   266 lemma surjI: "(\<And> x. g (f x) = x) \<Longrightarrow> surj g"
```
```   267   by (blast intro: surj_onI)
```
```   268
```
```   269 lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
```
```   270   by (simp add: surj_def)
```
```   271
```
```   272 lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
```
```   273   by (simp add: surj_def, blast)
```
```   274
```
```   275 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
```
```   276 apply (simp add: comp_def surj_def, clarify)
```
```   277 apply (drule_tac x = y in spec, clarify)
```
```   278 apply (drule_tac x = x in spec, blast)
```
```   279 done
```
```   280
```
```   281 lemma surj_range: "surj f \<Longrightarrow> range f = UNIV"
```
```   282   by (auto simp add: surj_on_def)
```
```   283
```
```   284 lemma surj_range_iff: "surj f \<longleftrightarrow> range f = UNIV"
```
```   285   unfolding surj_on_def by auto
```
```   286
```
```   287 lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
```
```   288   unfolding bij_betw_def surj_range_iff by auto
```
```   289
```
```   290 lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
```
```   291   unfolding surj_range_iff bij_betw_def ..
```
```   292
```
```   293 lemma bijI: "[| inj f; surj f |] ==> bij f"
```
```   294 by (simp add: bij_def)
```
```   295
```
```   296 lemma bij_is_inj: "bij f ==> inj f"
```
```   297 by (simp add: bij_def)
```
```   298
```
```   299 lemma bij_is_surj: "bij f ==> surj f"
```
```   300 by (simp add: bij_def)
```
```   301
```
```   302 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
```
```   303 by (simp add: bij_betw_def)
```
```   304
```
```   305 lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> surj_on f B"
```
```   306 by (auto simp: bij_betw_def surj_on_range_iff)
```
```   307
```
```   308 lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
```
```   309 by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
```
```   310
```
```   311 lemma bij_betw_trans:
```
```   312   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
```
```   313 by(auto simp add:bij_betw_def comp_inj_on)
```
```   314
```
```   315 lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
```
```   316 proof -
```
```   317   have i: "inj_on f A" and s: "f ` A = B"
```
```   318     using assms by(auto simp:bij_betw_def)
```
```   319   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
```
```   320   { fix a b assume P: "?P b a"
```
```   321     hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
```
```   322     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
```
```   323     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
```
```   324   } note g = this
```
```   325   have "inj_on ?g B"
```
```   326   proof(rule inj_onI)
```
```   327     fix x y assume "x:B" "y:B" "?g x = ?g y"
```
```   328     from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
```
```   329     from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
```
```   330     from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
```
```   331   qed
```
```   332   moreover have "?g ` B = A"
```
```   333   proof(auto simp:image_def)
```
```   334     fix b assume "b:B"
```
```   335     with s obtain a where P: "?P b a" unfolding image_def by blast
```
```   336     thus "?g b \<in> A" using g[OF P] by auto
```
```   337   next
```
```   338     fix a assume "a:A"
```
```   339     then obtain b where P: "?P b a" using s unfolding image_def by blast
```
```   340     then have "b:B" using s unfolding image_def by blast
```
```   341     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
```
```   342   qed
```
```   343   ultimately show ?thesis by(auto simp:bij_betw_def)
```
```   344 qed
```
```   345
```
```   346 lemma bij_betw_combine:
```
```   347   assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
```
```   348   shows "bij_betw f (A \<union> C) (B \<union> D)"
```
```   349   using assms unfolding bij_betw_def inj_on_Un image_Un by auto
```
```   350
```
```   351 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
```
```   352 by (simp add: surj_range)
```
```   353
```
```   354 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
```
```   355 by (simp add: inj_on_def, blast)
```
```   356
```
```   357 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
```
```   358 apply (unfold surj_def)
```
```   359 apply (blast intro: sym)
```
```   360 done
```
```   361
```
```   362 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
```
```   363 by (unfold inj_on_def, blast)
```
```   364
```
```   365 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
```
```   366 apply (unfold bij_def)
```
```   367 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
```
```   368 done
```
```   369
```
```   370 lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
```
```   371 by(blast dest: inj_onD)
```
```   372
```
```   373 lemma inj_on_image_Int:
```
```   374    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
```
```   375 apply (simp add: inj_on_def, blast)
```
```   376 done
```
```   377
```
```   378 lemma inj_on_image_set_diff:
```
```   379    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
```
```   380 apply (simp add: inj_on_def, blast)
```
```   381 done
```
```   382
```
```   383 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
```
```   384 by (simp add: inj_on_def, blast)
```
```   385
```
```   386 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
```
```   387 by (simp add: inj_on_def, blast)
```
```   388
```
```   389 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
```
```   390 by (blast dest: injD)
```
```   391
```
```   392 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
```
```   393 by (simp add: inj_on_def, blast)
```
```   394
```
```   395 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
```
```   396 by (blast dest: injD)
```
```   397
```
```   398 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
```
```   399 lemma image_INT:
```
```   400    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
```
```   401     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   402 apply (simp add: inj_on_def, blast)
```
```   403 done
```
```   404
```
```   405 (*Compare with image_INT: no use of inj_on, and if f is surjective then
```
```   406   it doesn't matter whether A is empty*)
```
```   407 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   408 apply (simp add: bij_def)
```
```   409 apply (simp add: inj_on_def surj_def, blast)
```
```   410 done
```
```   411
```
```   412 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
```
```   413 by (auto simp add: surj_def)
```
```   414
```
```   415 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
```
```   416 by (auto simp add: inj_on_def)
```
```   417
```
```   418 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
```
```   419 apply (simp add: bij_def)
```
```   420 apply (rule equalityI)
```
```   421 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
```
```   422 done
```
```   423
```
```   424 lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
```
```   425   by (auto intro!: inj_onI)
```
```   426
```
```   427 lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
```
```   428   by (auto intro!: inj_onI dest: strict_mono_eq)
```
```   429
```
```   430 subsection{*Function Updating*}
```
```   431
```
```   432 definition
```
```   433   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
```
```   434   "fun_upd f a b == % x. if x=a then b else f x"
```
```   435
```
```   436 nonterminals
```
```   437   updbinds updbind
```
```   438 syntax
```
```   439   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
```
```   440   ""         :: "updbind => updbinds"             ("_")
```
```   441   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
```
```   442   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
```
```   443
```
```   444 translations
```
```   445   "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
```
```   446   "f(x:=y)" == "CONST fun_upd f x y"
```
```   447
```
```   448 (* Hint: to define the sum of two functions (or maps), use sum_case.
```
```   449          A nice infix syntax could be defined (in Datatype.thy or below) by
```
```   450 notation
```
```   451   sum_case  (infixr "'(+')"80)
```
```   452 *)
```
```   453
```
```   454 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
```
```   455 apply (simp add: fun_upd_def, safe)
```
```   456 apply (erule subst)
```
```   457 apply (rule_tac [2] ext, auto)
```
```   458 done
```
```   459
```
```   460 (* f x = y ==> f(x:=y) = f *)
```
```   461 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
```
```   462
```
```   463 (* f(x := f x) = f *)
```
```   464 lemmas fun_upd_triv = refl [THEN fun_upd_idem]
```
```   465 declare fun_upd_triv [iff]
```
```   466
```
```   467 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
```
```   468 by (simp add: fun_upd_def)
```
```   469
```
```   470 (* fun_upd_apply supersedes these two,   but they are useful
```
```   471    if fun_upd_apply is intentionally removed from the simpset *)
```
```   472 lemma fun_upd_same: "(f(x:=y)) x = y"
```
```   473 by simp
```
```   474
```
```   475 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
```
```   476 by simp
```
```   477
```
```   478 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
```
```   479 by (simp add: fun_eq_iff)
```
```   480
```
```   481 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
```
```   482 by (rule ext, auto)
```
```   483
```
```   484 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
```
```   485 by (fastsimp simp:inj_on_def image_def)
```
```   486
```
```   487 lemma fun_upd_image:
```
```   488      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
```
```   489 by auto
```
```   490
```
```   491 lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
```
```   492 by (auto intro: ext)
```
```   493
```
```   494
```
```   495 subsection {* @{text override_on} *}
```
```   496
```
```   497 definition
```
```   498   override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   499 where
```
```   500   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
```
```   501
```
```   502 lemma override_on_emptyset[simp]: "override_on f g {} = f"
```
```   503 by(simp add:override_on_def)
```
```   504
```
```   505 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
```
```   506 by(simp add:override_on_def)
```
```   507
```
```   508 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
```
```   509 by(simp add:override_on_def)
```
```   510
```
```   511
```
```   512 subsection {* @{text swap} *}
```
```   513
```
```   514 definition
```
```   515   swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
```
```   516 where
```
```   517   "swap a b f = f (a := f b, b:= f a)"
```
```   518
```
```   519 lemma swap_self [simp]: "swap a a f = f"
```
```   520 by (simp add: swap_def)
```
```   521
```
```   522 lemma swap_commute: "swap a b f = swap b a f"
```
```   523 by (rule ext, simp add: fun_upd_def swap_def)
```
```   524
```
```   525 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
```
```   526 by (rule ext, simp add: fun_upd_def swap_def)
```
```   527
```
```   528 lemma swap_triple:
```
```   529   assumes "a \<noteq> c" and "b \<noteq> c"
```
```   530   shows "swap a b (swap b c (swap a b f)) = swap a c f"
```
```   531   using assms by (simp add: fun_eq_iff swap_def)
```
```   532
```
```   533 lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
```
```   534 by (rule ext, simp add: fun_upd_def swap_def)
```
```   535
```
```   536 lemma swap_image_eq [simp]:
```
```   537   assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
```
```   538 proof -
```
```   539   have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
```
```   540     using assms by (auto simp: image_iff swap_def)
```
```   541   then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
```
```   542   with subset[of f] show ?thesis by auto
```
```   543 qed
```
```   544
```
```   545 lemma inj_on_imp_inj_on_swap:
```
```   546   "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
```
```   547   by (simp add: inj_on_def swap_def, blast)
```
```   548
```
```   549 lemma inj_on_swap_iff [simp]:
```
```   550   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
```
```   551 proof
```
```   552   assume "inj_on (swap a b f) A"
```
```   553   with A have "inj_on (swap a b (swap a b f)) A"
```
```   554     by (iprover intro: inj_on_imp_inj_on_swap)
```
```   555   thus "inj_on f A" by simp
```
```   556 next
```
```   557   assume "inj_on f A"
```
```   558   with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
```
```   559 qed
```
```   560
```
```   561 lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
```
```   562   unfolding surj_range_iff by simp
```
```   563
```
```   564 lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
```
```   565   unfolding surj_range_iff by simp
```
```   566
```
```   567 lemma bij_betw_swap_iff [simp]:
```
```   568   "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
```
```   569   by (auto simp: bij_betw_def)
```
```   570
```
```   571 lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
```
```   572   by simp
```
```   573
```
```   574 hide_const (open) swap
```
```   575
```
```   576 subsection {* Inversion of injective functions *}
```
```   577
```
```   578 definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
```
```   579 "the_inv_into A f == %x. THE y. y : A & f y = x"
```
```   580
```
```   581 lemma the_inv_into_f_f:
```
```   582   "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
```
```   583 apply (simp add: the_inv_into_def inj_on_def)
```
```   584 apply blast
```
```   585 done
```
```   586
```
```   587 lemma f_the_inv_into_f:
```
```   588   "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
```
```   589 apply (simp add: the_inv_into_def)
```
```   590 apply (rule the1I2)
```
```   591  apply(blast dest: inj_onD)
```
```   592 apply blast
```
```   593 done
```
```   594
```
```   595 lemma the_inv_into_into:
```
```   596   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
```
```   597 apply (simp add: the_inv_into_def)
```
```   598 apply (rule the1I2)
```
```   599  apply(blast dest: inj_onD)
```
```   600 apply blast
```
```   601 done
```
```   602
```
```   603 lemma the_inv_into_onto[simp]:
```
```   604   "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
```
```   605 by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
```
```   606
```
```   607 lemma the_inv_into_f_eq:
```
```   608   "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
```
```   609   apply (erule subst)
```
```   610   apply (erule the_inv_into_f_f, assumption)
```
```   611   done
```
```   612
```
```   613 lemma the_inv_into_comp:
```
```   614   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
```
```   615   the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
```
```   616 apply (rule the_inv_into_f_eq)
```
```   617   apply (fast intro: comp_inj_on)
```
```   618  apply (simp add: f_the_inv_into_f the_inv_into_into)
```
```   619 apply (simp add: the_inv_into_into)
```
```   620 done
```
```   621
```
```   622 lemma inj_on_the_inv_into:
```
```   623   "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
```
```   624 by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
```
```   625
```
```   626 lemma bij_betw_the_inv_into:
```
```   627   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
```
```   628 by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
```
```   629
```
```   630 abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
```
```   631   "the_inv f \<equiv> the_inv_into UNIV f"
```
```   632
```
```   633 lemma the_inv_f_f:
```
```   634   assumes "inj f"
```
```   635   shows "the_inv f (f x) = x" using assms UNIV_I
```
```   636   by (rule the_inv_into_f_f)
```
```   637
```
```   638
```
```   639 subsection {* Proof tool setup *}
```
```   640
```
```   641 text {* simplifies terms of the form
```
```   642   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
```
```   643
```
```   644 simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
```
```   645 let
```
```   646   fun gen_fun_upd NONE T _ _ = NONE
```
```   647     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y)
```
```   648   fun dest_fun_T1 (Type (_, T :: Ts)) = T
```
```   649   fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) =
```
```   650     let
```
```   651       fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) =
```
```   652             if v aconv x then SOME g else gen_fun_upd (find g) T v w
```
```   653         | find t = NONE
```
```   654     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
```
```   655
```
```   656   fun proc ss ct =
```
```   657     let
```
```   658       val ctxt = Simplifier.the_context ss
```
```   659       val t = Thm.term_of ct
```
```   660     in
```
```   661       case find_double t of
```
```   662         (T, NONE) => NONE
```
```   663       | (T, SOME rhs) =>
```
```   664           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
```
```   665             (fn _ =>
```
```   666               rtac eq_reflection 1 THEN
```
```   667               rtac ext 1 THEN
```
```   668               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
```
```   669     end
```
```   670 in proc end
```
```   671 *}
```
```   672
```
```   673
```
```   674 subsection {* Code generator setup *}
```
```   675
```
```   676 types_code
```
```   677   "fun"  ("(_ ->/ _)")
```
```   678 attach (term_of) {*
```
```   679 fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
```
```   680 *}
```
```   681 attach (test) {*
```
```   682 fun gen_fun_type aF aT bG bT i =
```
```   683   let
```
```   684     val tab = Unsynchronized.ref [];
```
```   685     fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
```
```   686       (aT --> bT) --> aT --> bT --> aT --> bT) \$ t \$ aF x \$ y ()
```
```   687   in
```
```   688     (fn x =>
```
```   689        case AList.lookup op = (!tab) x of
```
```   690          NONE =>
```
```   691            let val p as (y, _) = bG i
```
```   692            in (tab := (x, p) :: !tab; y) end
```
```   693        | SOME (y, _) => y,
```
```   694      fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
```
```   695   end;
```
```   696 *}
```
```   697
```
```   698 code_const "op \<circ>"
```
```   699   (SML infixl 5 "o")
```
```   700   (Haskell infixr 9 ".")
```
```   701
```
```   702 code_const "id"
```
```   703   (Haskell "id")
```
```   704
```
```   705 end
```