src/HOL/Fun.thy
 author haftmann Tue Feb 26 20:38:12 2008 +0100 (2008-02-26) changeset 26147 ae2bf929e33c parent 26105 ae06618225ec child 26342 0f65fa163304 permissions -rw-r--r--
moved some set lemmas to Set.thy
```     1 (*  Title:      HOL/Fun.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tobias Nipkow, Cambridge University Computer Laboratory
```
```     4     Copyright   1994  University of Cambridge
```
```     5 *)
```
```     6
```
```     7 header {* Notions about functions *}
```
```     8
```
```     9 theory Fun
```
```    10 imports Set
```
```    11 begin
```
```    12
```
```    13 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 {* Injectivity and Surjectivity *}
```
```    86
```
```    87 constdefs
```
```    88   inj_on :: "['a => 'b, 'a set] => bool"  -- "injective"
```
```    89   "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y"
```
```    90
```
```    91 text{*A common special case: functions injective over the entire domain type.*}
```
```    92
```
```    93 abbreviation
```
```    94   "inj f == inj_on f UNIV"
```
```    95
```
```    96 definition
```
```    97   bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective"
```
```    98   "bij_betw f A B \<longleftrightarrow> inj_on f A & f ` A = B"
```
```    99
```
```   100 constdefs
```
```   101   surj :: "('a => 'b) => bool"                   (*surjective*)
```
```   102   "surj f == ! y. ? x. y=f(x)"
```
```   103
```
```   104   bij :: "('a => 'b) => bool"                    (*bijective*)
```
```   105   "bij f == inj f & surj f"
```
```   106
```
```   107 lemma injI:
```
```   108   assumes "\<And>x y. f x = f y \<Longrightarrow> x = y"
```
```   109   shows "inj f"
```
```   110   using assms unfolding inj_on_def by auto
```
```   111
```
```   112 text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*}
```
```   113 lemma datatype_injI:
```
```   114     "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)"
```
```   115 by (simp add: inj_on_def)
```
```   116
```
```   117 theorem range_ex1_eq: "inj f \<Longrightarrow> b : range f = (EX! x. b = f x)"
```
```   118   by (unfold inj_on_def, blast)
```
```   119
```
```   120 lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y"
```
```   121 by (simp add: inj_on_def)
```
```   122
```
```   123 (*Useful with the simplifier*)
```
```   124 lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)"
```
```   125 by (force simp add: inj_on_def)
```
```   126
```
```   127 lemma inj_on_id[simp]: "inj_on id A"
```
```   128   by (simp add: inj_on_def)
```
```   129
```
```   130 lemma inj_on_id2[simp]: "inj_on (%x. x) A"
```
```   131 by (simp add: inj_on_def)
```
```   132
```
```   133 lemma surj_id[simp]: "surj id"
```
```   134 by (simp add: surj_def)
```
```   135
```
```   136 lemma bij_id[simp]: "bij id"
```
```   137 by (simp add: bij_def inj_on_id surj_id)
```
```   138
```
```   139 lemma inj_onI:
```
```   140     "(!! x y. [|  x:A;  y:A;  f(x) = f(y) |] ==> x=y) ==> inj_on f A"
```
```   141 by (simp add: inj_on_def)
```
```   142
```
```   143 lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A"
```
```   144 by (auto dest:  arg_cong [of concl: g] simp add: inj_on_def)
```
```   145
```
```   146 lemma inj_onD: "[| inj_on f A;  f(x)=f(y);  x:A;  y:A |] ==> x=y"
```
```   147 by (unfold inj_on_def, blast)
```
```   148
```
```   149 lemma inj_on_iff: "[| inj_on f A;  x:A;  y:A |] ==> (f(x)=f(y)) = (x=y)"
```
```   150 by (blast dest!: inj_onD)
```
```   151
```
```   152 lemma comp_inj_on:
```
```   153      "[| inj_on f A;  inj_on g (f`A) |] ==> inj_on (g o f) A"
```
```   154 by (simp add: comp_def inj_on_def)
```
```   155
```
```   156 lemma inj_on_imageI: "inj_on (g o f) A \<Longrightarrow> inj_on g (f ` A)"
```
```   157 apply(simp add:inj_on_def image_def)
```
```   158 apply blast
```
```   159 done
```
```   160
```
```   161 lemma inj_on_image_iff: "\<lbrakk> ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y);
```
```   162   inj_on f A \<rbrakk> \<Longrightarrow> inj_on g (f ` A) = inj_on g A"
```
```   163 apply(unfold inj_on_def)
```
```   164 apply blast
```
```   165 done
```
```   166
```
```   167 lemma inj_on_contraD: "[| inj_on f A;  ~x=y;  x:A;  y:A |] ==> ~ f(x)=f(y)"
```
```   168 by (unfold inj_on_def, blast)
```
```   169
```
```   170 lemma inj_singleton: "inj (%s. {s})"
```
```   171 by (simp add: inj_on_def)
```
```   172
```
```   173 lemma inj_on_empty[iff]: "inj_on f {}"
```
```   174 by(simp add: inj_on_def)
```
```   175
```
```   176 lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A"
```
```   177 by (unfold inj_on_def, blast)
```
```   178
```
```   179 lemma inj_on_Un:
```
```   180  "inj_on f (A Un B) =
```
```   181   (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})"
```
```   182 apply(unfold inj_on_def)
```
```   183 apply (blast intro:sym)
```
```   184 done
```
```   185
```
```   186 lemma inj_on_insert[iff]:
```
```   187   "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))"
```
```   188 apply(unfold inj_on_def)
```
```   189 apply (blast intro:sym)
```
```   190 done
```
```   191
```
```   192 lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)"
```
```   193 apply(unfold inj_on_def)
```
```   194 apply (blast)
```
```   195 done
```
```   196
```
```   197 lemma surjI: "(!! x. g(f x) = x) ==> surj g"
```
```   198 apply (simp add: surj_def)
```
```   199 apply (blast intro: sym)
```
```   200 done
```
```   201
```
```   202 lemma surj_range: "surj f ==> range f = UNIV"
```
```   203 by (auto simp add: surj_def)
```
```   204
```
```   205 lemma surjD: "surj f ==> EX x. y = f x"
```
```   206 by (simp add: surj_def)
```
```   207
```
```   208 lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C"
```
```   209 by (simp add: surj_def, blast)
```
```   210
```
```   211 lemma comp_surj: "[| surj f;  surj g |] ==> surj (g o f)"
```
```   212 apply (simp add: comp_def surj_def, clarify)
```
```   213 apply (drule_tac x = y in spec, clarify)
```
```   214 apply (drule_tac x = x in spec, blast)
```
```   215 done
```
```   216
```
```   217 lemma bijI: "[| inj f; surj f |] ==> bij f"
```
```   218 by (simp add: bij_def)
```
```   219
```
```   220 lemma bij_is_inj: "bij f ==> inj f"
```
```   221 by (simp add: bij_def)
```
```   222
```
```   223 lemma bij_is_surj: "bij f ==> surj f"
```
```   224 by (simp add: bij_def)
```
```   225
```
```   226 lemma bij_betw_imp_inj_on: "bij_betw f A B \<Longrightarrow> inj_on f A"
```
```   227 by (simp add: bij_betw_def)
```
```   228
```
```   229 lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A"
```
```   230 proof -
```
```   231   have i: "inj_on f A" and s: "f ` A = B"
```
```   232     using assms by(auto simp:bij_betw_def)
```
```   233   let ?P = "%b a. a:A \<and> f a = b" let ?g = "%b. The (?P b)"
```
```   234   { fix a b assume P: "?P b a"
```
```   235     hence ex1: "\<exists>a. ?P b a" using s unfolding image_def by blast
```
```   236     hence uex1: "\<exists>!a. ?P b a" by(blast dest:inj_onD[OF i])
```
```   237     hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp
```
```   238   } note g = this
```
```   239   have "inj_on ?g B"
```
```   240   proof(rule inj_onI)
```
```   241     fix x y assume "x:B" "y:B" "?g x = ?g y"
```
```   242     from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast
```
```   243     from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast
```
```   244     from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp
```
```   245   qed
```
```   246   moreover have "?g ` B = A"
```
```   247   proof(auto simp:image_def)
```
```   248     fix b assume "b:B"
```
```   249     with s obtain a where P: "?P b a" unfolding image_def by blast
```
```   250     thus "?g b \<in> A" using g[OF P] by auto
```
```   251   next
```
```   252     fix a assume "a:A"
```
```   253     then obtain b where P: "?P b a" using s unfolding image_def by blast
```
```   254     then have "b:B" using s unfolding image_def by blast
```
```   255     with g[OF P] show "\<exists>b\<in>B. a = ?g b" by blast
```
```   256   qed
```
```   257   ultimately show ?thesis by(auto simp:bij_betw_def)
```
```   258 qed
```
```   259
```
```   260 lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A"
```
```   261 by (simp add: surj_range)
```
```   262
```
```   263 lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A"
```
```   264 by (simp add: inj_on_def, blast)
```
```   265
```
```   266 lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A"
```
```   267 apply (unfold surj_def)
```
```   268 apply (blast intro: sym)
```
```   269 done
```
```   270
```
```   271 lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A"
```
```   272 by (unfold inj_on_def, blast)
```
```   273
```
```   274 lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)"
```
```   275 apply (unfold bij_def)
```
```   276 apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD)
```
```   277 done
```
```   278
```
```   279 lemma inj_on_image_Int:
```
```   280    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A Int B) = f`A Int f`B"
```
```   281 apply (simp add: inj_on_def, blast)
```
```   282 done
```
```   283
```
```   284 lemma inj_on_image_set_diff:
```
```   285    "[| inj_on f C;  A<=C;  B<=C |] ==> f`(A-B) = f`A - f`B"
```
```   286 apply (simp add: inj_on_def, blast)
```
```   287 done
```
```   288
```
```   289 lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B"
```
```   290 by (simp add: inj_on_def, blast)
```
```   291
```
```   292 lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B"
```
```   293 by (simp add: inj_on_def, blast)
```
```   294
```
```   295 lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)"
```
```   296 by (blast dest: injD)
```
```   297
```
```   298 lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)"
```
```   299 by (simp add: inj_on_def, blast)
```
```   300
```
```   301 lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)"
```
```   302 by (blast dest: injD)
```
```   303
```
```   304 (*injectivity's required.  Left-to-right inclusion holds even if A is empty*)
```
```   305 lemma image_INT:
```
```   306    "[| inj_on f C;  ALL x:A. B x <= C;  j:A |]
```
```   307     ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   308 apply (simp add: inj_on_def, blast)
```
```   309 done
```
```   310
```
```   311 (*Compare with image_INT: no use of inj_on, and if f is surjective then
```
```   312   it doesn't matter whether A is empty*)
```
```   313 lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)"
```
```   314 apply (simp add: bij_def)
```
```   315 apply (simp add: inj_on_def surj_def, blast)
```
```   316 done
```
```   317
```
```   318 lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)"
```
```   319 by (auto simp add: surj_def)
```
```   320
```
```   321 lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)"
```
```   322 by (auto simp add: inj_on_def)
```
```   323
```
```   324 lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)"
```
```   325 apply (simp add: bij_def)
```
```   326 apply (rule equalityI)
```
```   327 apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset)
```
```   328 done
```
```   329
```
```   330
```
```   331 subsection{*Function Updating*}
```
```   332
```
```   333 constdefs
```
```   334   fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)"
```
```   335   "fun_upd f a b == % x. if x=a then b else f x"
```
```   336
```
```   337 nonterminals
```
```   338   updbinds updbind
```
```   339 syntax
```
```   340   "_updbind" :: "['a, 'a] => updbind"             ("(2_ :=/ _)")
```
```   341   ""         :: "updbind => updbinds"             ("_")
```
```   342   "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _")
```
```   343   "_Update"  :: "['a, updbinds] => 'a"            ("_/'((_)')" [1000,0] 900)
```
```   344
```
```   345 translations
```
```   346   "_Update f (_updbinds b bs)"  == "_Update (_Update f b) bs"
```
```   347   "f(x:=y)"                     == "fun_upd f x y"
```
```   348
```
```   349 (* Hint: to define the sum of two functions (or maps), use sum_case.
```
```   350          A nice infix syntax could be defined (in Datatype.thy or below) by
```
```   351 consts
```
```   352   fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80)
```
```   353 translations
```
```   354  "fun_sum" == sum_case
```
```   355 *)
```
```   356
```
```   357 lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)"
```
```   358 apply (simp add: fun_upd_def, safe)
```
```   359 apply (erule subst)
```
```   360 apply (rule_tac [2] ext, auto)
```
```   361 done
```
```   362
```
```   363 (* f x = y ==> f(x:=y) = f *)
```
```   364 lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard]
```
```   365
```
```   366 (* f(x := f x) = f *)
```
```   367 lemmas fun_upd_triv = refl [THEN fun_upd_idem]
```
```   368 declare fun_upd_triv [iff]
```
```   369
```
```   370 lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)"
```
```   371 by (simp add: fun_upd_def)
```
```   372
```
```   373 (* fun_upd_apply supersedes these two,   but they are useful
```
```   374    if fun_upd_apply is intentionally removed from the simpset *)
```
```   375 lemma fun_upd_same: "(f(x:=y)) x = y"
```
```   376 by simp
```
```   377
```
```   378 lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z"
```
```   379 by simp
```
```   380
```
```   381 lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)"
```
```   382 by (simp add: expand_fun_eq)
```
```   383
```
```   384 lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)"
```
```   385 by (rule ext, auto)
```
```   386
```
```   387 lemma inj_on_fun_updI: "\<lbrakk> inj_on f A; y \<notin> f`A \<rbrakk> \<Longrightarrow> inj_on (f(x:=y)) A"
```
```   388 by(fastsimp simp:inj_on_def image_def)
```
```   389
```
```   390 lemma fun_upd_image:
```
```   391      "f(x:=y) ` A = (if x \<in> A then insert y (f ` (A-{x})) else f ` A)"
```
```   392 by auto
```
```   393
```
```   394
```
```   395 subsection {* @{text override_on} *}
```
```   396
```
```   397 definition
```
```   398   override_on :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> 'b"
```
```   399 where
```
```   400   "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
```
```   401
```
```   402 lemma override_on_emptyset[simp]: "override_on f g {} = f"
```
```   403 by(simp add:override_on_def)
```
```   404
```
```   405 lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a"
```
```   406 by(simp add:override_on_def)
```
```   407
```
```   408 lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a"
```
```   409 by(simp add:override_on_def)
```
```   410
```
```   411
```
```   412 subsection {* @{text swap} *}
```
```   413
```
```   414 definition
```
```   415   swap :: "'a \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
```
```   416 where
```
```   417   "swap a b f = f (a := f b, b:= f a)"
```
```   418
```
```   419 lemma swap_self: "swap a a f = f"
```
```   420 by (simp add: swap_def)
```
```   421
```
```   422 lemma swap_commute: "swap a b f = swap b a f"
```
```   423 by (rule ext, simp add: fun_upd_def swap_def)
```
```   424
```
```   425 lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f"
```
```   426 by (rule ext, simp add: fun_upd_def swap_def)
```
```   427
```
```   428 lemma inj_on_imp_inj_on_swap:
```
```   429   "[|inj_on f A; a \<in> A; b \<in> A|] ==> inj_on (swap a b f) A"
```
```   430 by (simp add: inj_on_def swap_def, blast)
```
```   431
```
```   432 lemma inj_on_swap_iff [simp]:
```
```   433   assumes A: "a \<in> A" "b \<in> A" shows "inj_on (swap a b f) A = inj_on f A"
```
```   434 proof
```
```   435   assume "inj_on (swap a b f) A"
```
```   436   with A have "inj_on (swap a b (swap a b f)) A"
```
```   437     by (iprover intro: inj_on_imp_inj_on_swap)
```
```   438   thus "inj_on f A" by simp
```
```   439 next
```
```   440   assume "inj_on f A"
```
```   441   with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap)
```
```   442 qed
```
```   443
```
```   444 lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)"
```
```   445 apply (simp add: surj_def swap_def, clarify)
```
```   446 apply (rule_tac P = "y = f b" in case_split_thm, blast)
```
```   447 apply (rule_tac P = "y = f a" in case_split_thm, auto)
```
```   448   --{*We don't yet have @{text case_tac}*}
```
```   449 done
```
```   450
```
```   451 lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f"
```
```   452 proof
```
```   453   assume "surj (swap a b f)"
```
```   454   hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap)
```
```   455   thus "surj f" by simp
```
```   456 next
```
```   457   assume "surj f"
```
```   458   thus "surj (swap a b f)" by (rule surj_imp_surj_swap)
```
```   459 qed
```
```   460
```
```   461 lemma bij_swap_iff: "bij (swap a b f) = bij f"
```
```   462 by (simp add: bij_def)
```
```   463
```
```   464
```
```   465 subsection {* Proof tool setup *}
```
```   466
```
```   467 text {* simplifies terms of the form
```
```   468   f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *}
```
```   469
```
```   470 simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ =>
```
```   471 let
```
```   472   fun gen_fun_upd NONE T _ _ = NONE
```
```   473     | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y)
```
```   474   fun dest_fun_T1 (Type (_, T :: Ts)) = T
```
```   475   fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) =
```
```   476     let
```
```   477       fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) =
```
```   478             if v aconv x then SOME g else gen_fun_upd (find g) T v w
```
```   479         | find t = NONE
```
```   480     in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end
```
```   481
```
```   482   fun proc ss ct =
```
```   483     let
```
```   484       val ctxt = Simplifier.the_context ss
```
```   485       val t = Thm.term_of ct
```
```   486     in
```
```   487       case find_double t of
```
```   488         (T, NONE) => NONE
```
```   489       | (T, SOME rhs) =>
```
```   490           SOME (Goal.prove ctxt [] [] (Term.equals T \$ t \$ rhs)
```
```   491             (fn _ =>
```
```   492               rtac eq_reflection 1 THEN
```
```   493               rtac ext 1 THEN
```
```   494               simp_tac (Simplifier.inherit_context ss @{simpset}) 1))
```
```   495     end
```
```   496 in proc end
```
```   497 *}
```
```   498
```
```   499
```
```   500 subsection {* Code generator setup *}
```
```   501
```
```   502 types_code
```
```   503   "fun"  ("(_ ->/ _)")
```
```   504 attach (term_of) {*
```
```   505 fun term_of_fun_type _ aT _ bT _ = Free ("<function>", aT --> bT);
```
```   506 *}
```
```   507 attach (test) {*
```
```   508 fun gen_fun_type aF aT bG bT i =
```
```   509   let
```
```   510     val tab = ref [];
```
```   511     fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd",
```
```   512       (aT --> bT) --> aT --> bT --> aT --> bT) \$ t \$ aF x \$ y ()
```
```   513   in
```
```   514     (fn x =>
```
```   515        case AList.lookup op = (!tab) x of
```
```   516          NONE =>
```
```   517            let val p as (y, _) = bG i
```
```   518            in (tab := (x, p) :: !tab; y) end
```
```   519        | SOME (y, _) => y,
```
```   520      fn () => Basics.fold mk_upd (!tab) (Const ("arbitrary", aT --> bT)))
```
```   521   end;
```
```   522 *}
```
```   523
```
```   524 code_const "op \<circ>"
```
```   525   (SML infixl 5 "o")
```
```   526   (Haskell infixr 9 ".")
```
```   527
```
```   528 code_const "id"
```
```   529   (Haskell "id")
```
```   530
```
```   531
```
```   532 subsection {* ML legacy bindings *}
```
```   533
```
```   534 ML {*
```
```   535 val set_cs = claset() delrules [equalityI]
```
```   536 *}
```
```   537
```
```   538 ML {*
```
```   539 val id_apply = @{thm id_apply}
```
```   540 val id_def = @{thm id_def}
```
```   541 val o_apply = @{thm o_apply}
```
```   542 val o_assoc = @{thm o_assoc}
```
```   543 val o_def = @{thm o_def}
```
```   544 val injD = @{thm injD}
```
```   545 val datatype_injI = @{thm datatype_injI}
```
```   546 val range_ex1_eq = @{thm range_ex1_eq}
```
```   547 val expand_fun_eq = @{thm expand_fun_eq}
```
```   548 *}
```
```   549
```
```   550 end
```