src/HOL/Fun.thy
 author nipkow Mon Sep 13 11:13:15 2010 +0200 (2010-09-13) changeset 39302 d7728f65b353 parent 39213 297cd703f1f0 child 40602 91e583511113 permissions -rw-r--r--
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
```     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 {* Injectivity, Surjectivity and Bijectivity *}
```
```   121
```
```   122 definition inj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> bool" where -- "injective"
```
```   123   "inj_on f A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. f x = f y \<longrightarrow> x = y)"
```
```   124
```
```   125 definition surj_on :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> bool" where -- "surjective"
```
```   126   "surj_on f B \<longleftrightarrow> B \<subseteq> range f"
```
```   127
```
```   128 definition bij_betw :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" where -- "bijective"
```
```   129   "bij_betw f A B \<longleftrightarrow> inj_on f A \<and> f ` A = B"
```
```   130
```
```   131 text{*A common special case: functions injective over the entire domain type.*}
```
```   132
```
```   133 abbreviation
```
```   134   "inj f \<equiv> inj_on f UNIV"
```
```   135
```
```   136 abbreviation
```
```   137   "surj f \<equiv> surj_on f UNIV"
```
```   138
```
```   139 abbreviation
```
```   140   "bij f \<equiv> bij_betw f UNIV UNIV"
```
```   141
```
```   142 lemma injI:
```
```   143   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
```
```   144   shows "inj f"
```
```   145   using assms unfolding inj_on_def by auto
```
```   146
```
```   147 text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*}
```
```   148 lemma datatype_injI:
```
```   149     "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
```
```   150 by (simp add: inj_on_def)
```
```   151
```
```   152 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
```
```   153   by (unfold inj_on_def, blast)
```
```   154
```
```   155 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
```
```   156 by (simp add: inj_on_def)
```
```   157
```
```   158 lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)"
```
```   159 by (force simp add: inj_on_def)
```
```   160
```
```   161 lemma inj_comp:
```
```   162   "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (f \<circ> g)"
```
```   163   by (simp add: inj_on_def)
```
```   164
```
```   165 lemma inj_fun: "inj f \<Longrightarrow> inj (\<lambda>x y. f x)"
```
```   166   by (simp add: inj_on_def fun_eq_iff)
```
```   167
```
```   168 lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)"
```
```   169 by (simp add: inj_on_eq_iff)
```
```   170
```
```   171 lemma inj_on_id[simp]: "inj_on id A"
```
```   172   by (simp add: inj_on_def)
```
```   173
```
```   174 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
```
```   175 by (simp add: inj_on_def)
```
```   176
```
```   177 lemma surj_id[simp]: "surj_on id A"
```
```   178 by (simp add: surj_on_def)
```
```   179
```
```   180 lemma bij_id[simp]: "bij id"
```
```   181 by (simp add: bij_betw_def)
```
```   182
```
```   183 lemma inj_onI:
```
```   184     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
```
```   185 by (simp add: inj_on_def)
```
```   186
```
```   187 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
```
```   188 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
```
```   189
```
```   190 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
```
```   191 by (unfold inj_on_def, blast)
```
```   192
```
```   193 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
```
```   194 by (blast dest!: inj_onD)
```
```   195
```
```   196 lemma comp_inj_on:
```
```   197      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
```
```   198 by (simp add: comp_def inj_on_def)
```
```   199
```
```   200 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
```
```   201 apply(simp add:inj_on_def image_def)
```
```   202 apply blast
```
```   203 done
```
```   204
```
```   205 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
```
```   206   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
```
```   207 apply(unfold inj_on_def)
```
```   208 apply blast
```
```   209 done
```
```   210
```
```   211 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
```
```   212 by (unfold inj_on_def, blast)
```
```   213
```
```   214 lemma inj_singleton: "inj (%s. {s})"
```
```   215 by (simp add: inj_on_def)
```
```   216
```
```   217 lemma inj_on_empty[iff]: "inj_on f {}"
```
```   218 by(simp add: inj_on_def)
```
```   219
```
```   220 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
```
```   221 by (unfold inj_on_def, blast)
```
```   222
```
```   223 lemma inj_on_Un:
```
```   224  "inj_on f (A Un B) =
```
```   225   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
```
```   226 apply(unfold inj_on_def)
```
```   227 apply (blast intro:sym)
```
```   228 done
```
```   229
```
```   230 lemma inj_on_insert[iff]:
```
```   231   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
```
```   232 apply(unfold inj_on_def)
```
```   233 apply (blast intro:sym)
```
```   234 done
```
```   235
```
```   236 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
```
```   237 apply(unfold inj_on_def)
```
```   238 apply (blast)
```
```   239 done
```
```   240
```
```   241 lemma surj_onI: "(\<And>x. x \<in> B \<Longrightarrow> g (f x) = x) \<Longrightarrow> surj_on g B"
```
```   242   by (simp add: surj_on_def) (blast intro: sym)
```
```   243
```
```   244 lemma surj_onD: "surj_on f B \<Longrightarrow> y \<in> B \<Longrightarrow> \<exists>x. y = f x"
```
```   245   by (auto simp: surj_on_def)
```
```   246
```
```   247 lemma surj_on_range_iff: "surj_on f B \<longleftrightarrow> (\<exists>A. f ` A = B)"
```
```   248   unfolding surj_on_def by (auto intro!: exI[of _ "f -` B"])
```
```   249
```
```   250 lemma surj_def: "surj f \<longleftrightarrow> (\<forall>y. \<exists>x. y = f x)"
```
```   251   by (simp add: surj_on_def subset_eq image_iff)
```
```   252
```
```   253 lemma surjI: "(\<And> x. g (f x) = x) \<Longrightarrow> surj g"
```
```   254   by (blast intro: surj_onI)
```
```   255
```
```   256 lemma surjD: "surj f \<Longrightarrow> \<exists>x. y = f x"
```
```   257   by (simp add: surj_def)
```
```   258
```
```   259 lemma surjE: "surj f \<Longrightarrow> (\<And>x. y = f x \<Longrightarrow> C) \<Longrightarrow> C"
```
```   260   by (simp add: surj_def, blast)
```
```   261
```
```   262 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
```
```   263 apply (simp add: comp_def surj_def, clarify)
```
```   264 apply (drule_tac x = y in spec, clarify)
```
```   265 apply (drule_tac x = x in spec, blast)
```
```   266 done
```
```   267
```
```   268 lemma surj_range: "surj f \<Longrightarrow> range f = UNIV"
```
```   269   by (auto simp add: surj_on_def)
```
```   270
```
```   271 lemma surj_range_iff: "surj f \<longleftrightarrow> range f = UNIV"
```
```   272   unfolding surj_on_def by auto
```
```   273
```
```   274 lemma bij_betw_imp_surj: "bij_betw f A UNIV \<Longrightarrow> surj f"
```
```   275   unfolding bij_betw_def surj_range_iff by auto
```
```   276
```
```   277 lemma bij_def: "bij f \<longleftrightarrow> inj f \<and> surj f"
```
```   278   unfolding surj_range_iff bij_betw_def ..
```
```   279
```
```   280 lemma bijI: "[| inj f; surj f |] ==> bij f"
```
```   281 by (simp add: bij_def)
```
```   282
```
```   283 lemma bij_is_inj: "bij f ==> inj f"
```
```   284 by (simp add: bij_def)
```
```   285
```
```   286 lemma bij_is_surj: "bij f ==> surj f"
```
```   287 by (simp add: bij_def)
```
```   288
```
```   289 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
```
```   290 by (simp add: bij_betw_def)
```
```   291
```
```   292 lemma bij_betw_imp_surj_on: "bij_betw f A B \<Longrightarrow> surj_on f B"
```
```   293 by (auto simp: bij_betw_def surj_on_range_iff)
```
```   294
```
```   295 lemma bij_comp: "bij f \<Longrightarrow> bij g \<Longrightarrow> bij (g o f)"
```
```   296 by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range)
```
```   297
```
```   298 lemma bij_betw_trans:
```
```   299   "bij_betw f A B \<Longrightarrow> bij_betw g B C \<Longrightarrow> bij_betw (g o f) A C"
```
```   300 by(auto simp add:bij_betw_def comp_inj_on)
```
```   301
```
```   302 lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
```
```   303 proof -
```
```   304   have i: "inj_on f A" and s: "f ` A = B"
```
```   305     using assms by(auto simp:bij_betw_def)
```
```   306   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
```
```   307   { fix a b assume P: "?P b a"
```
```   308     hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
```
```   309     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
```
```   310     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
```
```   311   } note g = this
```
```   312   have "inj_on ?g B"
```
```   313   proof(rule inj_onI)
```
```   314     fix x y assume "x:B" "y:B" "?g x = ?g y"
```
```   315     from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
```
```   316     from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
```
```   317     from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
```
```   318   qed
```
```   319   moreover have "?g ` B = A"
```
```   320   proof(auto simp:image_def)
```
```   321     fix b assume "b:B"
```
```   322     with s obtain a where P: "?P b a" unfolding image_def by blast
```
```   323     thus "?g b \<in> A" using g[OF P] by auto
```
```   324   next
```
```   325     fix a assume "a:A"
```
```   326     then obtain b where P: "?P b a" using s unfolding image_def by blast
```
```   327     then have "b:B" using s unfolding image_def by blast
```
```   328     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
```
```   329   qed
```
```   330   ultimately show ?thesis by(auto simp:bij_betw_def)
```
```   331 qed
```
```   332
```
```   333 lemma bij_betw_combine:
```
```   334   assumes "bij_betw f A B" "bij_betw f C D" "B \<inter> D = {}"
```
```   335   shows "bij_betw f (A \<union> C) (B \<union> D)"
```
```   336   using assms unfolding bij_betw_def inj_on_Un image_Un by auto
```
```   337
```
```   338 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
```
```   339 by (simp add: surj_range)
```
```   340
```
```   341 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
```
```   342 by (simp add: inj_on_def, blast)
```
```   343
```
```   344 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
```
```   345 apply (unfold surj_def)
```
```   346 apply (blast intro: sym)
```
```   347 done
```
```   348
```
```   349 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
```
```   350 by (unfold inj_on_def, blast)
```
```   351
```
```   352 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
```
```   353 apply (unfold bij_def)
```
```   354 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
```
```   355 done
```
```   356
```
```   357 lemma inj_on_Un_image_eq_iff: "inj_on f (A \<union> B) \<Longrightarrow> f ` A = f ` B \<longleftrightarrow> A = B"
```
```   358 by(blast dest: inj_onD)
```
```   359
```
```   360 lemma inj_on_image_Int:
```
```   361    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
```
```   362 apply (simp add: inj_on_def, blast)
```
```   363 done
```
```   364
```
```   365 lemma inj_on_image_set_diff:
```
```   366    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
```
```   367 apply (simp add: inj_on_def, blast)
```
```   368 done
```
```   369
```
```   370 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
```
```   371 by (simp add: inj_on_def, blast)
```
```   372
```
```   373 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
```
```   374 by (simp add: inj_on_def, blast)
```
```   375
```
```   376 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
```
```   377 by (blast dest: injD)
```
```   378
```
```   379 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
```
```   380 by (simp add: inj_on_def, blast)
```
```   381
```
```   382 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
```
```   383 by (blast dest: injD)
```
```   384
```
```   385 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
```
```   386 lemma image_INT:
```
```   387    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
```
```   388     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   389 apply (simp add: inj_on_def, blast)
```
```   390 done
```
```   391
```
```   392 (*Compare with image_INT: no use of inj_on, and if f is surjective then
```
```   393   it doesn't matter whether A is empty*)
```
```   394 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   395 apply (simp add: bij_def)
```
```   396 apply (simp add: inj_on_def surj_def, blast)
```
```   397 done
```
```   398
```
```   399 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
```
```   400 by (auto simp add: surj_def)
```
```   401
```
```   402 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
```
```   403 by (auto simp add: inj_on_def)
```
```   404
```
```   405 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
```
```   406 apply (simp add: bij_def)
```
```   407 apply (rule equalityI)
```
```   408 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
```
```   409 done
```
```   410
```
```   411 lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A"
```
```   412   by (auto intro!: inj_onI)
```
```   413
```
```   414 lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \<Longrightarrow> inj_on f A"
```
```   415   by (auto intro!: inj_onI dest: strict_mono_eq)
```
```   416
```
```   417 subsection{*Function Updating*}
```
```   418
```
```   419 definition
```
```   420   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where
```
```   421   "fun_upd f a b == % x. if x=a then b else f x"
```
```   422
```
```   423 nonterminals
```
```   424   updbinds updbind
```
```   425 syntax
```
```   426   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
```
```   427   ""         :: "updbind => updbinds"             ("_")
```
```   428   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
```
```   429   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000, 0] 900)
```
```   430
```
```   431 translations
```
```   432   "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs"
```
```   433   "f(x:=y)" == "CONST fun_upd f x y"
```
```   434
```
```   435 (* Hint: to define the sum of two functions (or maps), use sum_case.
```
```   436          A nice infix syntax could be defined (in Datatype.thy or below) by
```
```   437 notation
```
```   438   sum_case  (infixr "'(+')"80)
```
```   439 *)
```
```   440
```
```   441 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
```
```   442 apply (simp add: fun_upd_def, safe)
```
```   443 apply (erule subst)
```
```   444 apply (rule_tac [2] ext, auto)
```
```   445 done
```
```   446
```
```   447 (* f x = y ==> f(x:=y) = f *)
```
```   448 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
```
```   449
```
```   450 (* f(x := f x) = f *)
```
```   451 lemmas fun_upd_triv = refl [THEN fun_upd_idem]
```
```   452 declare fun_upd_triv [iff]
```
```   453
```
```   454 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
```
```   455 by (simp add: fun_upd_def)
```
```   456
```
```   457 (* fun_upd_apply supersedes these two,   but they are useful
```
```   458    if fun_upd_apply is intentionally removed from the simpset *)
```
```   459 lemma fun_upd_same: "(f(x:=y)) x = y"
```
```   460 by simp
```
```   461
```
```   462 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
```
```   463 by simp
```
```   464
```
```   465 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
```
```   466 by (simp add: fun_eq_iff)
```
```   467
```
```   468 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
```
```   469 by (rule ext, auto)
```
```   470
```
```   471 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
```
```   472 by (fastsimp simp:inj_on_def image_def)
```
```   473
```
```   474 lemma fun_upd_image:
```
```   475      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
```
```   476 by auto
```
```   477
```
```   478 lemma fun_upd_comp: "f \<circ> (g(x := y)) = (f \<circ> g)(x := f y)"
```
```   479 by (auto intro: ext)
```
```   480
```
```   481
```
```   482 subsection {* @{text override_on} *}
```
```   483
```
```   484 definition
```
```   485   override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   486 where
```
```   487   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
```
```   488
```
```   489 lemma override_on_emptyset[simp]: "override_on f g {} = f"
```
```   490 by(simp add:override_on_def)
```
```   491
```
```   492 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
```
```   493 by(simp add:override_on_def)
```
```   494
```
```   495 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
```
```   496 by(simp add:override_on_def)
```
```   497
```
```   498
```
```   499 subsection {* @{text swap} *}
```
```   500
```
```   501 definition
```
```   502   swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
```
```   503 where
```
```   504   "swap a b f = f (a := f b, b:= f a)"
```
```   505
```
```   506 lemma swap_self [simp]: "swap a a f = f"
```
```   507 by (simp add: swap_def)
```
```   508
```
```   509 lemma swap_commute: "swap a b f = swap b a f"
```
```   510 by (rule ext, simp add: fun_upd_def swap_def)
```
```   511
```
```   512 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
```
```   513 by (rule ext, simp add: fun_upd_def swap_def)
```
```   514
```
```   515 lemma swap_triple:
```
```   516   assumes "a \<noteq> c" and "b \<noteq> c"
```
```   517   shows "swap a b (swap b c (swap a b f)) = swap a c f"
```
```   518   using assms by (simp add: fun_eq_iff swap_def)
```
```   519
```
```   520 lemma comp_swap: "f \<circ> swap a b g = swap a b (f \<circ> g)"
```
```   521 by (rule ext, simp add: fun_upd_def swap_def)
```
```   522
```
```   523 lemma swap_image_eq [simp]:
```
```   524   assumes "a \<in> A" "b \<in> A" shows "swap a b f ` A = f ` A"
```
```   525 proof -
```
```   526   have subset: "\<And>f. swap a b f ` A \<subseteq> f ` A"
```
```   527     using assms by (auto simp: image_iff swap_def)
```
```   528   then have "swap a b (swap a b f) ` A \<subseteq> (swap a b f) ` A" .
```
```   529   with subset[of f] show ?thesis by auto
```
```   530 qed
```
```   531
```
```   532 lemma inj_on_imp_inj_on_swap:
```
```   533   "\<lbrakk>inj_on f A; a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> inj_on (swap a b f) A"
```
```   534   by (simp add: inj_on_def swap_def, blast)
```
```   535
```
```   536 lemma inj_on_swap_iff [simp]:
```
```   537   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A \<longleftrightarrow> inj_on f A"
```
```   538 proof
```
```   539   assume "inj_on (swap a b f) A"
```
```   540   with A have "inj_on (swap a b (swap a b f)) A"
```
```   541     by (iprover intro: inj_on_imp_inj_on_swap)
```
```   542   thus "inj_on f A" by simp
```
```   543 next
```
```   544   assume "inj_on f A"
```
```   545   with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
```
```   546 qed
```
```   547
```
```   548 lemma surj_imp_surj_swap: "surj f \<Longrightarrow> surj (swap a b f)"
```
```   549   unfolding surj_range_iff by simp
```
```   550
```
```   551 lemma surj_swap_iff [simp]: "surj (swap a b f) \<longleftrightarrow> surj f"
```
```   552   unfolding surj_range_iff by simp
```
```   553
```
```   554 lemma bij_betw_swap_iff [simp]:
```
```   555   "\<lbrakk> x \<in> A; y \<in> A \<rbrakk> \<Longrightarrow> bij_betw (swap x y f) A B \<longleftrightarrow> bij_betw f A B"
```
```   556   by (auto simp: bij_betw_def)
```
```   557
```
```   558 lemma bij_swap_iff [simp]: "bij (swap a b f) \<longleftrightarrow> bij f"
```
```   559   by simp
```
```   560
```
```   561 hide_const (open) swap
```
```   562
```
```   563 subsection {* Inversion of injective functions *}
```
```   564
```
```   565 definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where
```
```   566 "the_inv_into A f == %x. THE y. y : A & f y = x"
```
```   567
```
```   568 lemma the_inv_into_f_f:
```
```   569   "[| inj_on f A;  x : A |] ==> the_inv_into A f (f x) = x"
```
```   570 apply (simp add: the_inv_into_def inj_on_def)
```
```   571 apply blast
```
```   572 done
```
```   573
```
```   574 lemma f_the_inv_into_f:
```
```   575   "inj_on f A ==> y : f`A  ==> f (the_inv_into A f y) = y"
```
```   576 apply (simp add: the_inv_into_def)
```
```   577 apply (rule the1I2)
```
```   578  apply(blast dest: inj_onD)
```
```   579 apply blast
```
```   580 done
```
```   581
```
```   582 lemma the_inv_into_into:
```
```   583   "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B"
```
```   584 apply (simp add: the_inv_into_def)
```
```   585 apply (rule the1I2)
```
```   586  apply(blast dest: inj_onD)
```
```   587 apply blast
```
```   588 done
```
```   589
```
```   590 lemma the_inv_into_onto[simp]:
```
```   591   "inj_on f A ==> the_inv_into A f ` (f ` A) = A"
```
```   592 by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric])
```
```   593
```
```   594 lemma the_inv_into_f_eq:
```
```   595   "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x"
```
```   596   apply (erule subst)
```
```   597   apply (erule the_inv_into_f_f, assumption)
```
```   598   done
```
```   599
```
```   600 lemma the_inv_into_comp:
```
```   601   "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==>
```
```   602   the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x"
```
```   603 apply (rule the_inv_into_f_eq)
```
```   604   apply (fast intro: comp_inj_on)
```
```   605  apply (simp add: f_the_inv_into_f the_inv_into_into)
```
```   606 apply (simp add: the_inv_into_into)
```
```   607 done
```
```   608
```
```   609 lemma inj_on_the_inv_into:
```
```   610   "inj_on f A \<Longrightarrow> inj_on (the_inv_into A f) (f ` A)"
```
```   611 by (auto intro: inj_onI simp: image_def the_inv_into_f_f)
```
```   612
```
```   613 lemma bij_betw_the_inv_into:
```
```   614   "bij_betw f A B \<Longrightarrow> bij_betw (the_inv_into A f) B A"
```
```   615 by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into)
```
```   616
```
```   617 abbreviation the_inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
```
```   618   "the_inv f \<equiv> the_inv_into UNIV f"
```
```   619
```
```   620 lemma the_inv_f_f:
```
```   621   assumes "inj f"
```
```   622   shows "the_inv f (f x) = x" using assms UNIV_I
```
```   623   by (rule the_inv_into_f_f)
```
```   624
```
```   625
```
```   626 subsection {* Proof tool setup *}
```
```   627
```
```   628 text {* simplifies terms of the form
```
```   629   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
```
```   630
```
```   631 simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
```
```   632 let
```
```   633   fun gen_fun_upd NONE T _ _ = NONE
```
```   634     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y)
```
```   635   fun dest_fun_T1 (Type (_, T :: Ts)) = T
```
```   636   fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) =
```
```   637     let
```
```   638       fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) =
```
```   639             if v aconv x then SOME g else gen_fun_upd (find g) T v w
```
```   640         | find t = NONE
```
```   641     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
```
```   642
```
```   643   fun proc ss ct =
```
```   644     let
```
```   645       val ctxt = Simplifier.the_context ss
```
```   646       val t = Thm.term_of ct
```
```   647     in
```
```   648       case find_double t of
```
```   649         (T, NONE) => NONE
```
```   650       | (T, SOME rhs) =>
```
```   651           SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs))
```
```   652             (fn _ =>
```
```   653               rtac eq_reflection 1 THEN
```
```   654               rtac ext 1 THEN
```
```   655               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
```
```   656     end
```
```   657 in proc end
```
```   658 *}
```
```   659
```
```   660
```
```   661 subsection {* Code generator setup *}
```
```   662
```
```   663 types_code
```
```   664   "fun"  ("(_ ->/ _)")
```
```   665 attach (term_of) {*
```
```   666 fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
```
```   667 *}
```
```   668 attach (test) {*
```
```   669 fun gen_fun_type aF aT bG bT i =
```
```   670   let
```
```   671     val tab = Unsynchronized.ref [];
```
```   672     fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
```
```   673       (aT --> bT) --> aT --> bT --> aT --> bT) \$ t \$ aF x \$ y ()
```
```   674   in
```
```   675     (fn x =>
```
```   676        case AList.lookup op = (!tab) x of
```
```   677          NONE =>
```
```   678            let val p as (y, _) = bG i
```
```   679            in (tab := (x, p) :: !tab; y) end
```
```   680        | SOME (y, _) => y,
```
```   681      fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT)))
```
```   682   end;
```
```   683 *}
```
```   684
```
```   685 code_const "op \<circ>"
```
```   686   (SML infixl 5 "o")
```
```   687   (Haskell infixr 9 ".")
```
```   688
```
```   689 code_const "id"
```
```   690   (Haskell "id")
```
```   691
```
```   692 end
```