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.
 clasohm@1475 ` 1` ```(* Title: HOL/Fun.thy ``` clasohm@1475 ` 2` ``` Author: Tobias Nipkow, Cambridge University Computer Laboratory ``` clasohm@923 ` 3` ``` Copyright 1994 University of Cambridge ``` huffman@18154 ` 4` ```*) ``` clasohm@923 ` 5` huffman@18154 ` 6` ```header {* Notions about functions *} ``` clasohm@923 ` 7` paulson@15510 ` 8` ```theory Fun ``` haftmann@32139 ` 9` ```imports Complete_Lattice ``` nipkow@15131 ` 10` ```begin ``` nipkow@2912 ` 11` haftmann@26147 ` 12` ```text{*As a simplification rule, it replaces all function equalities by ``` haftmann@26147 ` 13` ``` first-order equalities.*} ``` nipkow@39302 ` 14` ```lemma fun_eq_iff: "f = g \ (\x. f x = g x)" ``` haftmann@26147 ` 15` ```apply (rule iffI) ``` haftmann@26147 ` 16` ```apply (simp (no_asm_simp)) ``` haftmann@26147 ` 17` ```apply (rule ext) ``` haftmann@26147 ` 18` ```apply (simp (no_asm_simp)) ``` haftmann@26147 ` 19` ```done ``` oheimb@5305 ` 20` haftmann@26147 ` 21` ```lemma apply_inverse: ``` haftmann@26357 ` 22` ``` "f x = u \ (\x. P x \ g (f x) = x) \ P x \ x = g u" ``` haftmann@26147 ` 23` ``` by auto ``` nipkow@2912 ` 24` wenzelm@12258 ` 25` haftmann@26147 ` 26` ```subsection {* The Identity Function @{text id} *} ``` paulson@6171 ` 27` haftmann@22744 ` 28` ```definition ``` haftmann@22744 ` 29` ``` id :: "'a \ 'a" ``` haftmann@22744 ` 30` ```where ``` haftmann@22744 ` 31` ``` "id = (\x. x)" ``` nipkow@13910 ` 32` haftmann@26147 ` 33` ```lemma id_apply [simp]: "id x = x" ``` haftmann@26147 ` 34` ``` by (simp add: id_def) ``` haftmann@26147 ` 35` haftmann@26147 ` 36` ```lemma image_ident [simp]: "(%x. x) ` Y = Y" ``` haftmann@26147 ` 37` ```by blast ``` haftmann@26147 ` 38` haftmann@26147 ` 39` ```lemma image_id [simp]: "id ` Y = Y" ``` haftmann@26147 ` 40` ```by (simp add: id_def) ``` haftmann@26147 ` 41` haftmann@26147 ` 42` ```lemma vimage_ident [simp]: "(%x. x) -` Y = Y" ``` haftmann@26147 ` 43` ```by blast ``` haftmann@26147 ` 44` haftmann@26147 ` 45` ```lemma vimage_id [simp]: "id -` A = A" ``` haftmann@26147 ` 46` ```by (simp add: id_def) ``` haftmann@26147 ` 47` haftmann@26147 ` 48` haftmann@26147 ` 49` ```subsection {* The Composition Operator @{text "f \ g"} *} ``` haftmann@26147 ` 50` haftmann@22744 ` 51` ```definition ``` haftmann@22744 ` 52` ``` comp :: "('b \ 'c) \ ('a \ 'b) \ 'a \ 'c" (infixl "o" 55) ``` haftmann@22744 ` 53` ```where ``` haftmann@22744 ` 54` ``` "f o g = (\x. f (g x))" ``` oheimb@11123 ` 55` wenzelm@21210 ` 56` ```notation (xsymbols) ``` wenzelm@19656 ` 57` ``` comp (infixl "\" 55) ``` wenzelm@19656 ` 58` wenzelm@21210 ` 59` ```notation (HTML output) ``` wenzelm@19656 ` 60` ``` comp (infixl "\" 55) ``` wenzelm@19656 ` 61` paulson@13585 ` 62` ```text{*compatibility*} ``` paulson@13585 ` 63` ```lemmas o_def = comp_def ``` nipkow@2912 ` 64` paulson@13585 ` 65` ```lemma o_apply [simp]: "(f o g) x = f (g x)" ``` paulson@13585 ` 66` ```by (simp add: comp_def) ``` paulson@13585 ` 67` paulson@13585 ` 68` ```lemma o_assoc: "f o (g o h) = f o g o h" ``` paulson@13585 ` 69` ```by (simp add: comp_def) ``` paulson@13585 ` 70` paulson@13585 ` 71` ```lemma id_o [simp]: "id o g = g" ``` paulson@13585 ` 72` ```by (simp add: comp_def) ``` paulson@13585 ` 73` paulson@13585 ` 74` ```lemma o_id [simp]: "f o id = f" ``` paulson@13585 ` 75` ```by (simp add: comp_def) ``` paulson@13585 ` 76` haftmann@34150 ` 77` ```lemma o_eq_dest: ``` haftmann@34150 ` 78` ``` "a o b = c o d \ a (b v) = c (d v)" ``` haftmann@34150 ` 79` ``` by (simp only: o_def) (fact fun_cong) ``` haftmann@34150 ` 80` haftmann@34150 ` 81` ```lemma o_eq_elim: ``` haftmann@34150 ` 82` ``` "a o b = c o d \ ((\v. a (b v) = c (d v)) \ R) \ R" ``` haftmann@34150 ` 83` ``` by (erule meta_mp) (fact o_eq_dest) ``` haftmann@34150 ` 84` paulson@13585 ` 85` ```lemma image_compose: "(f o g) ` r = f`(g`r)" ``` paulson@13585 ` 86` ```by (simp add: comp_def, blast) ``` paulson@13585 ` 87` paulson@33044 ` 88` ```lemma vimage_compose: "(g \ f) -` x = f -` (g -` x)" ``` paulson@33044 ` 89` ``` by auto ``` paulson@33044 ` 90` paulson@13585 ` 91` ```lemma UN_o: "UNION A (g o f) = UNION (f`A) g" ``` paulson@13585 ` 92` ```by (unfold comp_def, blast) ``` paulson@13585 ` 93` paulson@13585 ` 94` haftmann@26588 ` 95` ```subsection {* The Forward Composition Operator @{text fcomp} *} ``` haftmann@26357 ` 96` haftmann@26357 ` 97` ```definition ``` haftmann@37751 ` 98` ``` fcomp :: "('a \ 'b) \ ('b \ 'c) \ 'a \ 'c" (infixl "\>" 60) ``` haftmann@26357 ` 99` ```where ``` haftmann@37751 ` 100` ``` "f \> g = (\x. g (f x))" ``` haftmann@26357 ` 101` haftmann@37751 ` 102` ```lemma fcomp_apply [simp]: "(f \> g) x = g (f x)" ``` haftmann@26357 ` 103` ``` by (simp add: fcomp_def) ``` haftmann@26357 ` 104` haftmann@37751 ` 105` ```lemma fcomp_assoc: "(f \> g) \> h = f \> (g \> h)" ``` haftmann@26357 ` 106` ``` by (simp add: fcomp_def) ``` haftmann@26357 ` 107` haftmann@37751 ` 108` ```lemma id_fcomp [simp]: "id \> g = g" ``` haftmann@26357 ` 109` ``` by (simp add: fcomp_def) ``` haftmann@26357 ` 110` haftmann@37751 ` 111` ```lemma fcomp_id [simp]: "f \> id = f" ``` haftmann@26357 ` 112` ``` by (simp add: fcomp_def) ``` haftmann@26357 ` 113` haftmann@31202 ` 114` ```code_const fcomp ``` haftmann@31202 ` 115` ``` (Eval infixl 1 "#>") ``` haftmann@31202 ` 116` haftmann@37751 ` 117` ```no_notation fcomp (infixl "\>" 60) ``` haftmann@26588 ` 118` haftmann@26357 ` 119` haftmann@40602 ` 120` ```subsection {* Mapping functions *} ``` haftmann@40602 ` 121` haftmann@40602 ` 122` ```definition map_fun :: "('c \ 'a) \ ('b \ 'd) \ ('a \ 'b) \ 'c \ 'd" where ``` haftmann@40602 ` 123` ``` "map_fun f g h = g \ h \ f" ``` haftmann@40602 ` 124` haftmann@40602 ` 125` ```lemma map_fun_apply [simp]: ``` haftmann@40602 ` 126` ``` "map_fun f g h x = g (h (f x))" ``` haftmann@40602 ` 127` ``` by (simp add: map_fun_def) ``` haftmann@40602 ` 128` haftmann@40602 ` 129` ```type_mapper map_fun ``` haftmann@40602 ` 130` ``` by (simp_all add: fun_eq_iff) ``` haftmann@40602 ` 131` haftmann@40602 ` 132` hoelzl@40702 ` 133` ```subsection {* Injectivity and Bijectivity *} ``` hoelzl@39076 ` 134` hoelzl@39076 ` 135` ```definition inj_on :: "('a \ 'b) \ 'a set \ bool" where -- "injective" ``` hoelzl@39076 ` 136` ``` "inj_on f A \ (\x\A. \y\A. f x = f y \ x = y)" ``` haftmann@26147 ` 137` hoelzl@39076 ` 138` ```definition bij_betw :: "('a \ 'b) \ 'a set \ 'b set \ bool" where -- "bijective" ``` hoelzl@39076 ` 139` ``` "bij_betw f A B \ inj_on f A \ f ` A = B" ``` haftmann@26147 ` 140` hoelzl@40702 ` 141` ```text{*A common special case: functions injective, surjective or bijective over ``` hoelzl@40702 ` 142` ```the entire domain type.*} ``` haftmann@26147 ` 143` haftmann@26147 ` 144` ```abbreviation ``` hoelzl@39076 ` 145` ``` "inj f \ inj_on f UNIV" ``` haftmann@26147 ` 146` hoelzl@40702 ` 147` ```abbreviation surj :: "('a \ 'b) \ bool" where -- "surjective" ``` hoelzl@40702 ` 148` ``` "surj f \ (range f = UNIV)" ``` paulson@13585 ` 149` hoelzl@39076 ` 150` ```abbreviation ``` hoelzl@39076 ` 151` ``` "bij f \ bij_betw f UNIV UNIV" ``` haftmann@26147 ` 152` haftmann@26147 ` 153` ```lemma injI: ``` haftmann@26147 ` 154` ``` assumes "\x y. f x = f y \ x = y" ``` haftmann@26147 ` 155` ``` shows "inj f" ``` haftmann@26147 ` 156` ``` using assms unfolding inj_on_def by auto ``` paulson@13585 ` 157` haftmann@31775 ` 158` ```text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*} ``` paulson@13585 ` 159` ```lemma datatype_injI: ``` paulson@13585 ` 160` ``` "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)" ``` paulson@13585 ` 161` ```by (simp add: inj_on_def) ``` paulson@13585 ` 162` berghofe@13637 ` 163` ```theorem range_ex1_eq: "inj f \ b : range f = (EX! x. b = f x)" ``` berghofe@13637 ` 164` ``` by (unfold inj_on_def, blast) ``` berghofe@13637 ` 165` paulson@13585 ` 166` ```lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y" ``` paulson@13585 ` 167` ```by (simp add: inj_on_def) ``` paulson@13585 ` 168` nipkow@32988 ` 169` ```lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)" ``` paulson@13585 ` 170` ```by (force simp add: inj_on_def) ``` paulson@13585 ` 171` haftmann@38620 ` 172` ```lemma inj_comp: ``` haftmann@38620 ` 173` ``` "inj f \ inj g \ inj (f \ g)" ``` haftmann@38620 ` 174` ``` by (simp add: inj_on_def) ``` haftmann@38620 ` 175` haftmann@38620 ` 176` ```lemma inj_fun: "inj f \ inj (\x y. f x)" ``` nipkow@39302 ` 177` ``` by (simp add: inj_on_def fun_eq_iff) ``` haftmann@38620 ` 178` nipkow@32988 ` 179` ```lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)" ``` nipkow@32988 ` 180` ```by (simp add: inj_on_eq_iff) ``` nipkow@32988 ` 181` haftmann@26147 ` 182` ```lemma inj_on_id[simp]: "inj_on id A" ``` hoelzl@39076 ` 183` ``` by (simp add: inj_on_def) ``` paulson@13585 ` 184` haftmann@26147 ` 185` ```lemma inj_on_id2[simp]: "inj_on (%x. x) A" ``` hoelzl@39076 ` 186` ```by (simp add: inj_on_def) ``` haftmann@26147 ` 187` hoelzl@40702 ` 188` ```lemma surj_id: "surj id" ``` hoelzl@40702 ` 189` ```by simp ``` haftmann@26147 ` 190` hoelzl@39101 ` 191` ```lemma bij_id[simp]: "bij id" ``` hoelzl@39076 ` 192` ```by (simp add: bij_betw_def) ``` paulson@13585 ` 193` paulson@13585 ` 194` ```lemma inj_onI: ``` paulson@13585 ` 195` ``` "(!! x y. [| x:A; y:A; f(x) = f(y) |] ==> x=y) ==> inj_on f A" ``` paulson@13585 ` 196` ```by (simp add: inj_on_def) ``` paulson@13585 ` 197` paulson@13585 ` 198` ```lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" ``` paulson@13585 ` 199` ```by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) ``` paulson@13585 ` 200` paulson@13585 ` 201` ```lemma inj_onD: "[| inj_on f A; f(x)=f(y); x:A; y:A |] ==> x=y" ``` paulson@13585 ` 202` ```by (unfold inj_on_def, blast) ``` paulson@13585 ` 203` paulson@13585 ` 204` ```lemma inj_on_iff: "[| inj_on f A; x:A; y:A |] ==> (f(x)=f(y)) = (x=y)" ``` paulson@13585 ` 205` ```by (blast dest!: inj_onD) ``` paulson@13585 ` 206` paulson@13585 ` 207` ```lemma comp_inj_on: ``` paulson@13585 ` 208` ``` "[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A" ``` paulson@13585 ` 209` ```by (simp add: comp_def inj_on_def) ``` paulson@13585 ` 210` nipkow@15303 ` 211` ```lemma inj_on_imageI: "inj_on (g o f) A \ inj_on g (f ` A)" ``` nipkow@15303 ` 212` ```apply(simp add:inj_on_def image_def) ``` nipkow@15303 ` 213` ```apply blast ``` nipkow@15303 ` 214` ```done ``` nipkow@15303 ` 215` nipkow@15439 ` 216` ```lemma inj_on_image_iff: "\ ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); ``` nipkow@15439 ` 217` ``` inj_on f A \ \ inj_on g (f ` A) = inj_on g A" ``` nipkow@15439 ` 218` ```apply(unfold inj_on_def) ``` nipkow@15439 ` 219` ```apply blast ``` nipkow@15439 ` 220` ```done ``` nipkow@15439 ` 221` paulson@13585 ` 222` ```lemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)" ``` paulson@13585 ` 223` ```by (unfold inj_on_def, blast) ``` wenzelm@12258 ` 224` paulson@13585 ` 225` ```lemma inj_singleton: "inj (%s. {s})" ``` paulson@13585 ` 226` ```by (simp add: inj_on_def) ``` paulson@13585 ` 227` nipkow@15111 ` 228` ```lemma inj_on_empty[iff]: "inj_on f {}" ``` nipkow@15111 ` 229` ```by(simp add: inj_on_def) ``` nipkow@15111 ` 230` nipkow@15303 ` 231` ```lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A" ``` paulson@13585 ` 232` ```by (unfold inj_on_def, blast) ``` paulson@13585 ` 233` nipkow@15111 ` 234` ```lemma inj_on_Un: ``` nipkow@15111 ` 235` ``` "inj_on f (A Un B) = ``` nipkow@15111 ` 236` ``` (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})" ``` nipkow@15111 ` 237` ```apply(unfold inj_on_def) ``` nipkow@15111 ` 238` ```apply (blast intro:sym) ``` nipkow@15111 ` 239` ```done ``` nipkow@15111 ` 240` nipkow@15111 ` 241` ```lemma inj_on_insert[iff]: ``` nipkow@15111 ` 242` ``` "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))" ``` nipkow@15111 ` 243` ```apply(unfold inj_on_def) ``` nipkow@15111 ` 244` ```apply (blast intro:sym) ``` nipkow@15111 ` 245` ```done ``` nipkow@15111 ` 246` nipkow@15111 ` 247` ```lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)" ``` nipkow@15111 ` 248` ```apply(unfold inj_on_def) ``` nipkow@15111 ` 249` ```apply (blast) ``` nipkow@15111 ` 250` ```done ``` nipkow@15111 ` 251` hoelzl@40702 ` 252` ```lemma surj_def: "surj f \ (\y. \x. y = f x)" ``` hoelzl@40702 ` 253` ``` by auto ``` hoelzl@39076 ` 254` hoelzl@40702 ` 255` ```lemma surjI: assumes *: "\ x. g (f x) = x" shows "surj g" ``` hoelzl@40702 ` 256` ``` using *[symmetric] by auto ``` paulson@13585 ` 257` hoelzl@39076 ` 258` ```lemma surjD: "surj f \ \x. y = f x" ``` hoelzl@39076 ` 259` ``` by (simp add: surj_def) ``` paulson@13585 ` 260` hoelzl@39076 ` 261` ```lemma surjE: "surj f \ (\x. y = f x \ C) \ C" ``` hoelzl@39076 ` 262` ``` by (simp add: surj_def, blast) ``` paulson@13585 ` 263` paulson@13585 ` 264` ```lemma comp_surj: "[| surj f; surj g |] ==> surj (g o f)" ``` paulson@13585 ` 265` ```apply (simp add: comp_def surj_def, clarify) ``` paulson@13585 ` 266` ```apply (drule_tac x = y in spec, clarify) ``` paulson@13585 ` 267` ```apply (drule_tac x = x in spec, blast) ``` paulson@13585 ` 268` ```done ``` paulson@13585 ` 269` hoelzl@39074 ` 270` ```lemma bij_betw_imp_surj: "bij_betw f A UNIV \ surj f" ``` hoelzl@40702 ` 271` ``` unfolding bij_betw_def by auto ``` hoelzl@39074 ` 272` hoelzl@39076 ` 273` ```lemma bij_def: "bij f \ inj f \ surj f" ``` hoelzl@40702 ` 274` ``` unfolding bij_betw_def .. ``` hoelzl@39074 ` 275` paulson@13585 ` 276` ```lemma bijI: "[| inj f; surj f |] ==> bij f" ``` paulson@13585 ` 277` ```by (simp add: bij_def) ``` paulson@13585 ` 278` paulson@13585 ` 279` ```lemma bij_is_inj: "bij f ==> inj f" ``` paulson@13585 ` 280` ```by (simp add: bij_def) ``` paulson@13585 ` 281` paulson@13585 ` 282` ```lemma bij_is_surj: "bij f ==> surj f" ``` paulson@13585 ` 283` ```by (simp add: bij_def) ``` paulson@13585 ` 284` nipkow@26105 ` 285` ```lemma bij_betw_imp_inj_on: "bij_betw f A B \ inj_on f A" ``` nipkow@26105 ` 286` ```by (simp add: bij_betw_def) ``` nipkow@26105 ` 287` nipkow@31438 ` 288` ```lemma bij_betw_trans: ``` nipkow@31438 ` 289` ``` "bij_betw f A B \ bij_betw g B C \ bij_betw (g o f) A C" ``` nipkow@31438 ` 290` ```by(auto simp add:bij_betw_def comp_inj_on) ``` nipkow@31438 ` 291` hoelzl@40702 ` 292` ```lemma bij_comp: "bij f \ bij g \ bij (g o f)" ``` hoelzl@40702 ` 293` ``` by (rule bij_betw_trans) ``` hoelzl@40702 ` 294` nipkow@26105 ` 295` ```lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A" ``` nipkow@26105 ` 296` ```proof - ``` nipkow@26105 ` 297` ``` have i: "inj_on f A" and s: "f ` A = B" ``` nipkow@26105 ` 298` ``` using assms by(auto simp:bij_betw_def) ``` nipkow@26105 ` 299` ``` let ?P = "%b a. a:A \ f a = b" let ?g = "%b. The (?P b)" ``` nipkow@26105 ` 300` ``` { fix a b assume P: "?P b a" ``` nipkow@26105 ` 301` ``` hence ex1: "\a. ?P b a" using s unfolding image_def by blast ``` nipkow@26105 ` 302` ``` hence uex1: "\!a. ?P b a" by(blast dest:inj_onD[OF i]) ``` nipkow@26105 ` 303` ``` hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp ``` nipkow@26105 ` 304` ``` } note g = this ``` nipkow@26105 ` 305` ``` have "inj_on ?g B" ``` nipkow@26105 ` 306` ``` proof(rule inj_onI) ``` nipkow@26105 ` 307` ``` fix x y assume "x:B" "y:B" "?g x = ?g y" ``` nipkow@26105 ` 308` ``` from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast ``` nipkow@26105 ` 309` ``` from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast ``` nipkow@26105 ` 310` ``` from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp ``` nipkow@26105 ` 311` ``` qed ``` nipkow@26105 ` 312` ``` moreover have "?g ` B = A" ``` nipkow@26105 ` 313` ``` proof(auto simp:image_def) ``` nipkow@26105 ` 314` ``` fix b assume "b:B" ``` nipkow@26105 ` 315` ``` with s obtain a where P: "?P b a" unfolding image_def by blast ``` nipkow@26105 ` 316` ``` thus "?g b \ A" using g[OF P] by auto ``` nipkow@26105 ` 317` ``` next ``` nipkow@26105 ` 318` ``` fix a assume "a:A" ``` nipkow@26105 ` 319` ``` then obtain b where P: "?P b a" using s unfolding image_def by blast ``` nipkow@26105 ` 320` ``` then have "b:B" using s unfolding image_def by blast ``` nipkow@26105 ` 321` ``` with g[OF P] show "\b\B. a = ?g b" by blast ``` nipkow@26105 ` 322` ``` qed ``` nipkow@26105 ` 323` ``` ultimately show ?thesis by(auto simp:bij_betw_def) ``` nipkow@26105 ` 324` ```qed ``` nipkow@26105 ` 325` hoelzl@39075 ` 326` ```lemma bij_betw_combine: ``` hoelzl@39075 ` 327` ``` assumes "bij_betw f A B" "bij_betw f C D" "B \ D = {}" ``` hoelzl@39075 ` 328` ``` shows "bij_betw f (A \ C) (B \ D)" ``` hoelzl@39075 ` 329` ``` using assms unfolding bij_betw_def inj_on_Un image_Un by auto ``` hoelzl@39075 ` 330` paulson@13585 ` 331` ```lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A" ``` hoelzl@40702 ` 332` ```by simp ``` paulson@13585 ` 333` paulson@13585 ` 334` ```lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A" ``` paulson@13585 ` 335` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 336` paulson@13585 ` 337` ```lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A" ``` hoelzl@40702 ` 338` ```by (blast intro: sym) ``` paulson@13585 ` 339` paulson@13585 ` 340` ```lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A" ``` paulson@13585 ` 341` ```by (unfold inj_on_def, blast) ``` paulson@13585 ` 342` paulson@13585 ` 343` ```lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)" ``` paulson@13585 ` 344` ```apply (unfold bij_def) ``` paulson@13585 ` 345` ```apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) ``` paulson@13585 ` 346` ```done ``` paulson@13585 ` 347` nipkow@31438 ` 348` ```lemma inj_on_Un_image_eq_iff: "inj_on f (A \ B) \ f ` A = f ` B \ A = B" ``` nipkow@31438 ` 349` ```by(blast dest: inj_onD) ``` nipkow@31438 ` 350` paulson@13585 ` 351` ```lemma inj_on_image_Int: ``` paulson@13585 ` 352` ``` "[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B" ``` paulson@13585 ` 353` ```apply (simp add: inj_on_def, blast) ``` paulson@13585 ` 354` ```done ``` paulson@13585 ` 355` paulson@13585 ` 356` ```lemma inj_on_image_set_diff: ``` paulson@13585 ` 357` ``` "[| inj_on f C; A<=C; B<=C |] ==> f`(A-B) = f`A - f`B" ``` paulson@13585 ` 358` ```apply (simp add: inj_on_def, blast) ``` paulson@13585 ` 359` ```done ``` paulson@13585 ` 360` paulson@13585 ` 361` ```lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" ``` paulson@13585 ` 362` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 363` paulson@13585 ` 364` ```lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B" ``` paulson@13585 ` 365` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 366` paulson@13585 ` 367` ```lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)" ``` paulson@13585 ` 368` ```by (blast dest: injD) ``` paulson@13585 ` 369` paulson@13585 ` 370` ```lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" ``` paulson@13585 ` 371` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 372` paulson@13585 ` 373` ```lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" ``` paulson@13585 ` 374` ```by (blast dest: injD) ``` paulson@13585 ` 375` paulson@13585 ` 376` ```(*injectivity's required. Left-to-right inclusion holds even if A is empty*) ``` paulson@13585 ` 377` ```lemma image_INT: ``` paulson@13585 ` 378` ``` "[| inj_on f C; ALL x:A. B x <= C; j:A |] ``` paulson@13585 ` 379` ``` ==> f ` (INTER A B) = (INT x:A. f ` B x)" ``` paulson@13585 ` 380` ```apply (simp add: inj_on_def, blast) ``` paulson@13585 ` 381` ```done ``` paulson@13585 ` 382` paulson@13585 ` 383` ```(*Compare with image_INT: no use of inj_on, and if f is surjective then ``` paulson@13585 ` 384` ``` it doesn't matter whether A is empty*) ``` paulson@13585 ` 385` ```lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)" ``` paulson@13585 ` 386` ```apply (simp add: bij_def) ``` paulson@13585 ` 387` ```apply (simp add: inj_on_def surj_def, blast) ``` paulson@13585 ` 388` ```done ``` paulson@13585 ` 389` paulson@13585 ` 390` ```lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)" ``` hoelzl@40702 ` 391` ```by auto ``` paulson@13585 ` 392` paulson@13585 ` 393` ```lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)" ``` paulson@13585 ` 394` ```by (auto simp add: inj_on_def) ``` paulson@5852 ` 395` paulson@13585 ` 396` ```lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)" ``` paulson@13585 ` 397` ```apply (simp add: bij_def) ``` paulson@13585 ` 398` ```apply (rule equalityI) ``` paulson@13585 ` 399` ```apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) ``` paulson@13585 ` 400` ```done ``` paulson@13585 ` 401` hoelzl@35584 ` 402` ```lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A" ``` hoelzl@35580 ` 403` ``` by (auto intro!: inj_onI) ``` paulson@13585 ` 404` hoelzl@35584 ` 405` ```lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \ inj_on f A" ``` hoelzl@35584 ` 406` ``` by (auto intro!: inj_onI dest: strict_mono_eq) ``` hoelzl@35584 ` 407` paulson@13585 ` 408` ```subsection{*Function Updating*} ``` paulson@13585 ` 409` haftmann@35416 ` 410` ```definition ``` haftmann@35416 ` 411` ``` fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where ``` haftmann@26147 ` 412` ``` "fun_upd f a b == % x. if x=a then b else f x" ``` haftmann@26147 ` 413` haftmann@26147 ` 414` ```nonterminals ``` haftmann@26147 ` 415` ``` updbinds updbind ``` haftmann@26147 ` 416` ```syntax ``` haftmann@26147 ` 417` ``` "_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") ``` haftmann@26147 ` 418` ``` "" :: "updbind => updbinds" ("_") ``` haftmann@26147 ` 419` ``` "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") ``` wenzelm@35115 ` 420` ``` "_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000, 0] 900) ``` haftmann@26147 ` 421` haftmann@26147 ` 422` ```translations ``` wenzelm@35115 ` 423` ``` "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" ``` wenzelm@35115 ` 424` ``` "f(x:=y)" == "CONST fun_upd f x y" ``` haftmann@26147 ` 425` haftmann@26147 ` 426` ```(* Hint: to define the sum of two functions (or maps), use sum_case. ``` haftmann@26147 ` 427` ``` A nice infix syntax could be defined (in Datatype.thy or below) by ``` wenzelm@35115 ` 428` ```notation ``` wenzelm@35115 ` 429` ``` sum_case (infixr "'(+')"80) ``` haftmann@26147 ` 430` ```*) ``` haftmann@26147 ` 431` paulson@13585 ` 432` ```lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" ``` paulson@13585 ` 433` ```apply (simp add: fun_upd_def, safe) ``` paulson@13585 ` 434` ```apply (erule subst) ``` paulson@13585 ` 435` ```apply (rule_tac [2] ext, auto) ``` paulson@13585 ` 436` ```done ``` paulson@13585 ` 437` paulson@13585 ` 438` ```(* f x = y ==> f(x:=y) = f *) ``` paulson@13585 ` 439` ```lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard] ``` paulson@13585 ` 440` paulson@13585 ` 441` ```(* f(x := f x) = f *) ``` paulson@17084 ` 442` ```lemmas fun_upd_triv = refl [THEN fun_upd_idem] ``` paulson@17084 ` 443` ```declare fun_upd_triv [iff] ``` paulson@13585 ` 444` paulson@13585 ` 445` ```lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)" ``` paulson@17084 ` 446` ```by (simp add: fun_upd_def) ``` paulson@13585 ` 447` paulson@13585 ` 448` ```(* fun_upd_apply supersedes these two, but they are useful ``` paulson@13585 ` 449` ``` if fun_upd_apply is intentionally removed from the simpset *) ``` paulson@13585 ` 450` ```lemma fun_upd_same: "(f(x:=y)) x = y" ``` paulson@13585 ` 451` ```by simp ``` paulson@13585 ` 452` paulson@13585 ` 453` ```lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" ``` paulson@13585 ` 454` ```by simp ``` paulson@13585 ` 455` paulson@13585 ` 456` ```lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)" ``` nipkow@39302 ` 457` ```by (simp add: fun_eq_iff) ``` paulson@13585 ` 458` paulson@13585 ` 459` ```lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" ``` paulson@13585 ` 460` ```by (rule ext, auto) ``` paulson@13585 ` 461` nipkow@15303 ` 462` ```lemma inj_on_fun_updI: "\ inj_on f A; y \ f`A \ \ inj_on (f(x:=y)) A" ``` krauss@34209 ` 463` ```by (fastsimp simp:inj_on_def image_def) ``` nipkow@15303 ` 464` paulson@15510 ` 465` ```lemma fun_upd_image: ``` paulson@15510 ` 466` ``` "f(x:=y) ` A = (if x \ A then insert y (f ` (A-{x})) else f ` A)" ``` paulson@15510 ` 467` ```by auto ``` paulson@15510 ` 468` nipkow@31080 ` 469` ```lemma fun_upd_comp: "f \ (g(x := y)) = (f \ g)(x := f y)" ``` krauss@34209 ` 470` ```by (auto intro: ext) ``` nipkow@31080 ` 471` haftmann@26147 ` 472` haftmann@26147 ` 473` ```subsection {* @{text override_on} *} ``` haftmann@26147 ` 474` haftmann@26147 ` 475` ```definition ``` haftmann@26147 ` 476` ``` override_on :: "('a \ 'b) \ ('a \ 'b) \ 'a set \ 'a \ 'b" ``` haftmann@26147 ` 477` ```where ``` haftmann@26147 ` 478` ``` "override_on f g A = (\a. if a \ A then g a else f a)" ``` nipkow@13910 ` 479` nipkow@15691 ` 480` ```lemma override_on_emptyset[simp]: "override_on f g {} = f" ``` nipkow@15691 ` 481` ```by(simp add:override_on_def) ``` nipkow@13910 ` 482` nipkow@15691 ` 483` ```lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" ``` nipkow@15691 ` 484` ```by(simp add:override_on_def) ``` nipkow@13910 ` 485` nipkow@15691 ` 486` ```lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" ``` nipkow@15691 ` 487` ```by(simp add:override_on_def) ``` nipkow@13910 ` 488` haftmann@26147 ` 489` haftmann@26147 ` 490` ```subsection {* @{text swap} *} ``` paulson@15510 ` 491` haftmann@22744 ` 492` ```definition ``` haftmann@22744 ` 493` ``` swap :: "'a \ 'a \ ('a \ 'b) \ ('a \ 'b)" ``` haftmann@22744 ` 494` ```where ``` haftmann@22744 ` 495` ``` "swap a b f = f (a := f b, b:= f a)" ``` paulson@15510 ` 496` huffman@34101 ` 497` ```lemma swap_self [simp]: "swap a a f = f" ``` nipkow@15691 ` 498` ```by (simp add: swap_def) ``` paulson@15510 ` 499` paulson@15510 ` 500` ```lemma swap_commute: "swap a b f = swap b a f" ``` paulson@15510 ` 501` ```by (rule ext, simp add: fun_upd_def swap_def) ``` paulson@15510 ` 502` paulson@15510 ` 503` ```lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" ``` paulson@15510 ` 504` ```by (rule ext, simp add: fun_upd_def swap_def) ``` paulson@15510 ` 505` huffman@34145 ` 506` ```lemma swap_triple: ``` huffman@34145 ` 507` ``` assumes "a \ c" and "b \ c" ``` huffman@34145 ` 508` ``` shows "swap a b (swap b c (swap a b f)) = swap a c f" ``` nipkow@39302 ` 509` ``` using assms by (simp add: fun_eq_iff swap_def) ``` huffman@34145 ` 510` huffman@34101 ` 511` ```lemma comp_swap: "f \ swap a b g = swap a b (f \ g)" ``` huffman@34101 ` 512` ```by (rule ext, simp add: fun_upd_def swap_def) ``` huffman@34101 ` 513` hoelzl@39076 ` 514` ```lemma swap_image_eq [simp]: ``` hoelzl@39076 ` 515` ``` assumes "a \ A" "b \ A" shows "swap a b f ` A = f ` A" ``` hoelzl@39076 ` 516` ```proof - ``` hoelzl@39076 ` 517` ``` have subset: "\f. swap a b f ` A \ f ` A" ``` hoelzl@39076 ` 518` ``` using assms by (auto simp: image_iff swap_def) ``` hoelzl@39076 ` 519` ``` then have "swap a b (swap a b f) ` A \ (swap a b f) ` A" . ``` hoelzl@39076 ` 520` ``` with subset[of f] show ?thesis by auto ``` hoelzl@39076 ` 521` ```qed ``` hoelzl@39076 ` 522` paulson@15510 ` 523` ```lemma inj_on_imp_inj_on_swap: ``` hoelzl@39076 ` 524` ``` "\inj_on f A; a \ A; b \ A\ \ inj_on (swap a b f) A" ``` hoelzl@39076 ` 525` ``` by (simp add: inj_on_def swap_def, blast) ``` paulson@15510 ` 526` paulson@15510 ` 527` ```lemma inj_on_swap_iff [simp]: ``` hoelzl@39076 ` 528` ``` assumes A: "a \ A" "b \ A" shows "inj_on (swap a b f) A \ inj_on f A" ``` hoelzl@39075 ` 529` ```proof ``` paulson@15510 ` 530` ``` assume "inj_on (swap a b f) A" ``` hoelzl@39075 ` 531` ``` with A have "inj_on (swap a b (swap a b f)) A" ``` hoelzl@39075 ` 532` ``` by (iprover intro: inj_on_imp_inj_on_swap) ``` hoelzl@39075 ` 533` ``` thus "inj_on f A" by simp ``` paulson@15510 ` 534` ```next ``` paulson@15510 ` 535` ``` assume "inj_on f A" ``` krauss@34209 ` 536` ``` with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap) ``` paulson@15510 ` 537` ```qed ``` paulson@15510 ` 538` hoelzl@39076 ` 539` ```lemma surj_imp_surj_swap: "surj f \ surj (swap a b f)" ``` hoelzl@40702 ` 540` ``` by simp ``` paulson@15510 ` 541` hoelzl@39076 ` 542` ```lemma surj_swap_iff [simp]: "surj (swap a b f) \ surj f" ``` hoelzl@40702 ` 543` ``` by simp ``` haftmann@21547 ` 544` hoelzl@39076 ` 545` ```lemma bij_betw_swap_iff [simp]: ``` hoelzl@39076 ` 546` ``` "\ x \ A; y \ A \ \ bij_betw (swap x y f) A B \ bij_betw f A B" ``` hoelzl@39076 ` 547` ``` by (auto simp: bij_betw_def) ``` hoelzl@39076 ` 548` hoelzl@39076 ` 549` ```lemma bij_swap_iff [simp]: "bij (swap a b f) \ bij f" ``` hoelzl@39076 ` 550` ``` by simp ``` hoelzl@39075 ` 551` wenzelm@36176 ` 552` ```hide_const (open) swap ``` haftmann@21547 ` 553` haftmann@31949 ` 554` ```subsection {* Inversion of injective functions *} ``` haftmann@31949 ` 555` nipkow@33057 ` 556` ```definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where ``` nipkow@33057 ` 557` ```"the_inv_into A f == %x. THE y. y : A & f y = x" ``` nipkow@32961 ` 558` nipkow@33057 ` 559` ```lemma the_inv_into_f_f: ``` nipkow@33057 ` 560` ``` "[| inj_on f A; x : A |] ==> the_inv_into A f (f x) = x" ``` nipkow@33057 ` 561` ```apply (simp add: the_inv_into_def inj_on_def) ``` krauss@34209 ` 562` ```apply blast ``` nipkow@32961 ` 563` ```done ``` nipkow@32961 ` 564` nipkow@33057 ` 565` ```lemma f_the_inv_into_f: ``` nipkow@33057 ` 566` ``` "inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y" ``` nipkow@33057 ` 567` ```apply (simp add: the_inv_into_def) ``` nipkow@32961 ` 568` ```apply (rule the1I2) ``` nipkow@32961 ` 569` ``` apply(blast dest: inj_onD) ``` nipkow@32961 ` 570` ```apply blast ``` nipkow@32961 ` 571` ```done ``` nipkow@32961 ` 572` nipkow@33057 ` 573` ```lemma the_inv_into_into: ``` nipkow@33057 ` 574` ``` "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B" ``` nipkow@33057 ` 575` ```apply (simp add: the_inv_into_def) ``` nipkow@32961 ` 576` ```apply (rule the1I2) ``` nipkow@32961 ` 577` ``` apply(blast dest: inj_onD) ``` nipkow@32961 ` 578` ```apply blast ``` nipkow@32961 ` 579` ```done ``` nipkow@32961 ` 580` nipkow@33057 ` 581` ```lemma the_inv_into_onto[simp]: ``` nipkow@33057 ` 582` ``` "inj_on f A ==> the_inv_into A f ` (f ` A) = A" ``` nipkow@33057 ` 583` ```by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric]) ``` nipkow@32961 ` 584` nipkow@33057 ` 585` ```lemma the_inv_into_f_eq: ``` nipkow@33057 ` 586` ``` "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x" ``` nipkow@32961 ` 587` ``` apply (erule subst) ``` nipkow@33057 ` 588` ``` apply (erule the_inv_into_f_f, assumption) ``` nipkow@32961 ` 589` ``` done ``` nipkow@32961 ` 590` nipkow@33057 ` 591` ```lemma the_inv_into_comp: ``` nipkow@32961 ` 592` ``` "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==> ``` nipkow@33057 ` 593` ``` the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x" ``` nipkow@33057 ` 594` ```apply (rule the_inv_into_f_eq) ``` nipkow@32961 ` 595` ``` apply (fast intro: comp_inj_on) ``` nipkow@33057 ` 596` ``` apply (simp add: f_the_inv_into_f the_inv_into_into) ``` nipkow@33057 ` 597` ```apply (simp add: the_inv_into_into) ``` nipkow@32961 ` 598` ```done ``` nipkow@32961 ` 599` nipkow@33057 ` 600` ```lemma inj_on_the_inv_into: ``` nipkow@33057 ` 601` ``` "inj_on f A \ inj_on (the_inv_into A f) (f ` A)" ``` nipkow@33057 ` 602` ```by (auto intro: inj_onI simp: image_def the_inv_into_f_f) ``` nipkow@32961 ` 603` nipkow@33057 ` 604` ```lemma bij_betw_the_inv_into: ``` nipkow@33057 ` 605` ``` "bij_betw f A B \ bij_betw (the_inv_into A f) B A" ``` nipkow@33057 ` 606` ```by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into) ``` nipkow@32961 ` 607` berghofe@32998 ` 608` ```abbreviation the_inv :: "('a \ 'b) \ ('b \ 'a)" where ``` nipkow@33057 ` 609` ``` "the_inv f \ the_inv_into UNIV f" ``` berghofe@32998 ` 610` berghofe@32998 ` 611` ```lemma the_inv_f_f: ``` berghofe@32998 ` 612` ``` assumes "inj f" ``` berghofe@32998 ` 613` ``` shows "the_inv f (f x) = x" using assms UNIV_I ``` nipkow@33057 ` 614` ``` by (rule the_inv_into_f_f) ``` berghofe@32998 ` 615` haftmann@31949 ` 616` haftmann@22845 ` 617` ```subsection {* Proof tool setup *} ``` haftmann@22845 ` 618` haftmann@22845 ` 619` ```text {* simplifies terms of the form ``` haftmann@22845 ` 620` ``` f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *} ``` haftmann@22845 ` 621` wenzelm@24017 ` 622` ```simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ => ``` haftmann@22845 ` 623` ```let ``` haftmann@22845 ` 624` ``` fun gen_fun_upd NONE T _ _ = NONE ``` wenzelm@24017 ` 625` ``` | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y) ``` haftmann@22845 ` 626` ``` fun dest_fun_T1 (Type (_, T :: Ts)) = T ``` haftmann@22845 ` 627` ``` fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) = ``` haftmann@22845 ` 628` ``` let ``` haftmann@22845 ` 629` ``` fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) = ``` haftmann@22845 ` 630` ``` if v aconv x then SOME g else gen_fun_upd (find g) T v w ``` haftmann@22845 ` 631` ``` | find t = NONE ``` haftmann@22845 ` 632` ``` in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end ``` wenzelm@24017 ` 633` wenzelm@24017 ` 634` ``` fun proc ss ct = ``` wenzelm@24017 ` 635` ``` let ``` wenzelm@24017 ` 636` ``` val ctxt = Simplifier.the_context ss ``` wenzelm@24017 ` 637` ``` val t = Thm.term_of ct ``` wenzelm@24017 ` 638` ``` in ``` wenzelm@24017 ` 639` ``` case find_double t of ``` wenzelm@24017 ` 640` ``` (T, NONE) => NONE ``` wenzelm@24017 ` 641` ``` | (T, SOME rhs) => ``` wenzelm@27330 ` 642` ``` SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) ``` wenzelm@24017 ` 643` ``` (fn _ => ``` wenzelm@24017 ` 644` ``` rtac eq_reflection 1 THEN ``` wenzelm@24017 ` 645` ``` rtac ext 1 THEN ``` wenzelm@24017 ` 646` ``` simp_tac (Simplifier.inherit_context ss @{simpset}) 1)) ``` wenzelm@24017 ` 647` ``` end ``` wenzelm@24017 ` 648` ```in proc end ``` haftmann@22845 ` 649` ```*} ``` haftmann@22845 ` 650` haftmann@22845 ` 651` haftmann@21870 ` 652` ```subsection {* Code generator setup *} ``` haftmann@21870 ` 653` berghofe@25886 ` 654` ```types_code ``` berghofe@25886 ` 655` ``` "fun" ("(_ ->/ _)") ``` berghofe@25886 ` 656` ```attach (term_of) {* ``` berghofe@25886 ` 657` ```fun term_of_fun_type _ aT _ bT _ = Free ("", aT --> bT); ``` berghofe@25886 ` 658` ```*} ``` berghofe@25886 ` 659` ```attach (test) {* ``` berghofe@25886 ` 660` ```fun gen_fun_type aF aT bG bT i = ``` berghofe@25886 ` 661` ``` let ``` wenzelm@32740 ` 662` ``` val tab = Unsynchronized.ref []; ``` berghofe@25886 ` 663` ``` fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd", ``` berghofe@25886 ` 664` ``` (aT --> bT) --> aT --> bT --> aT --> bT) \$ t \$ aF x \$ y () ``` berghofe@25886 ` 665` ``` in ``` berghofe@25886 ` 666` ``` (fn x => ``` berghofe@25886 ` 667` ``` case AList.lookup op = (!tab) x of ``` berghofe@25886 ` 668` ``` NONE => ``` berghofe@25886 ` 669` ``` let val p as (y, _) = bG i ``` berghofe@25886 ` 670` ``` in (tab := (x, p) :: !tab; y) end ``` berghofe@25886 ` 671` ``` | SOME (y, _) => y, ``` berghofe@28711 ` 672` ``` fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT))) ``` berghofe@25886 ` 673` ``` end; ``` berghofe@25886 ` 674` ```*} ``` berghofe@25886 ` 675` haftmann@21870 ` 676` ```code_const "op \" ``` haftmann@21870 ` 677` ``` (SML infixl 5 "o") ``` haftmann@21870 ` 678` ``` (Haskell infixr 9 ".") ``` haftmann@21870 ` 679` haftmann@21906 ` 680` ```code_const "id" ``` haftmann@21906 ` 681` ``` (Haskell "id") ``` haftmann@21906 ` 682` nipkow@2912 ` 683` ```end ```