src/HOL/Fun.thy
 author nipkow Tue Sep 07 10:05:19 2010 +0200 (2010-09-07) changeset 39198 f967a16dfcdd parent 39101 606432dd1896 child 39213 297cd703f1f0 permissions -rw-r--r--
expand_fun_eq -> ext_iff
expand_set_eq -> set_ext_iff
Naming in line now with multisets
 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@39198 ` 14` ```lemma ext_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` hoelzl@39076 ` 120` ```subsection {* Injectivity, Surjectivity and Bijectivity *} ``` hoelzl@39076 ` 121` hoelzl@39076 ` 122` ```definition inj_on :: "('a \ 'b) \ 'a set \ bool" where -- "injective" ``` hoelzl@39076 ` 123` ``` "inj_on f A \ (\x\A. \y\A. f x = f y \ x = y)" ``` haftmann@26147 ` 124` hoelzl@39076 ` 125` ```definition surj_on :: "('a \ 'b) \ 'b set \ bool" where -- "surjective" ``` hoelzl@39076 ` 126` ``` "surj_on f B \ B \ range f" ``` hoelzl@39076 ` 127` hoelzl@39076 ` 128` ```definition bij_betw :: "('a \ 'b) \ 'a set \ 'b set \ bool" where -- "bijective" ``` hoelzl@39076 ` 129` ``` "bij_betw f A B \ inj_on f A \ f ` A = B" ``` haftmann@26147 ` 130` haftmann@26147 ` 131` ```text{*A common special case: functions injective over the entire domain type.*} ``` haftmann@26147 ` 132` haftmann@26147 ` 133` ```abbreviation ``` hoelzl@39076 ` 134` ``` "inj f \ inj_on f UNIV" ``` haftmann@26147 ` 135` hoelzl@39076 ` 136` ```abbreviation ``` hoelzl@39076 ` 137` ``` "surj f \ surj_on f UNIV" ``` paulson@13585 ` 138` hoelzl@39076 ` 139` ```abbreviation ``` hoelzl@39076 ` 140` ``` "bij f \ bij_betw f UNIV UNIV" ``` haftmann@26147 ` 141` haftmann@26147 ` 142` ```lemma injI: ``` haftmann@26147 ` 143` ``` assumes "\x y. f x = f y \ x = y" ``` haftmann@26147 ` 144` ``` shows "inj f" ``` haftmann@26147 ` 145` ``` using assms unfolding inj_on_def by auto ``` paulson@13585 ` 146` haftmann@31775 ` 147` ```text{*For Proofs in @{text "Tools/Datatype/datatype_rep_proofs"}*} ``` paulson@13585 ` 148` ```lemma datatype_injI: ``` paulson@13585 ` 149` ``` "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)" ``` paulson@13585 ` 150` ```by (simp add: inj_on_def) ``` paulson@13585 ` 151` berghofe@13637 ` 152` ```theorem range_ex1_eq: "inj f \ b : range f = (EX! x. b = f x)" ``` berghofe@13637 ` 153` ``` by (unfold inj_on_def, blast) ``` berghofe@13637 ` 154` paulson@13585 ` 155` ```lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y" ``` paulson@13585 ` 156` ```by (simp add: inj_on_def) ``` paulson@13585 ` 157` nipkow@32988 ` 158` ```lemma inj_on_eq_iff: "inj_on f A ==> x:A ==> y:A ==> (f(x) = f(y)) = (x=y)" ``` paulson@13585 ` 159` ```by (force simp add: inj_on_def) ``` paulson@13585 ` 160` haftmann@38620 ` 161` ```lemma inj_comp: ``` haftmann@38620 ` 162` ``` "inj f \ inj g \ inj (f \ g)" ``` haftmann@38620 ` 163` ``` by (simp add: inj_on_def) ``` haftmann@38620 ` 164` haftmann@38620 ` 165` ```lemma inj_fun: "inj f \ inj (\x y. f x)" ``` nipkow@39198 ` 166` ``` by (simp add: inj_on_def ext_iff) ``` haftmann@38620 ` 167` nipkow@32988 ` 168` ```lemma inj_eq: "inj f ==> (f(x) = f(y)) = (x=y)" ``` nipkow@32988 ` 169` ```by (simp add: inj_on_eq_iff) ``` nipkow@32988 ` 170` haftmann@26147 ` 171` ```lemma inj_on_id[simp]: "inj_on id A" ``` hoelzl@39076 ` 172` ``` by (simp add: inj_on_def) ``` paulson@13585 ` 173` haftmann@26147 ` 174` ```lemma inj_on_id2[simp]: "inj_on (%x. x) A" ``` hoelzl@39076 ` 175` ```by (simp add: inj_on_def) ``` haftmann@26147 ` 176` hoelzl@39076 ` 177` ```lemma surj_id[simp]: "surj_on id A" ``` hoelzl@39076 ` 178` ```by (simp add: surj_on_def) ``` haftmann@26147 ` 179` hoelzl@39101 ` 180` ```lemma bij_id[simp]: "bij id" ``` hoelzl@39076 ` 181` ```by (simp add: bij_betw_def) ``` paulson@13585 ` 182` paulson@13585 ` 183` ```lemma inj_onI: ``` paulson@13585 ` 184` ``` "(!! x y. [| x:A; y:A; f(x) = f(y) |] ==> x=y) ==> inj_on f A" ``` paulson@13585 ` 185` ```by (simp add: inj_on_def) ``` paulson@13585 ` 186` paulson@13585 ` 187` ```lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" ``` paulson@13585 ` 188` ```by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) ``` paulson@13585 ` 189` paulson@13585 ` 190` ```lemma inj_onD: "[| inj_on f A; f(x)=f(y); x:A; y:A |] ==> x=y" ``` paulson@13585 ` 191` ```by (unfold inj_on_def, blast) ``` paulson@13585 ` 192` paulson@13585 ` 193` ```lemma inj_on_iff: "[| inj_on f A; x:A; y:A |] ==> (f(x)=f(y)) = (x=y)" ``` paulson@13585 ` 194` ```by (blast dest!: inj_onD) ``` paulson@13585 ` 195` paulson@13585 ` 196` ```lemma comp_inj_on: ``` paulson@13585 ` 197` ``` "[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A" ``` paulson@13585 ` 198` ```by (simp add: comp_def inj_on_def) ``` paulson@13585 ` 199` nipkow@15303 ` 200` ```lemma inj_on_imageI: "inj_on (g o f) A \ inj_on g (f ` A)" ``` nipkow@15303 ` 201` ```apply(simp add:inj_on_def image_def) ``` nipkow@15303 ` 202` ```apply blast ``` nipkow@15303 ` 203` ```done ``` nipkow@15303 ` 204` nipkow@15439 ` 205` ```lemma inj_on_image_iff: "\ ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); ``` nipkow@15439 ` 206` ``` inj_on f A \ \ inj_on g (f ` A) = inj_on g A" ``` nipkow@15439 ` 207` ```apply(unfold inj_on_def) ``` nipkow@15439 ` 208` ```apply blast ``` nipkow@15439 ` 209` ```done ``` nipkow@15439 ` 210` paulson@13585 ` 211` ```lemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)" ``` paulson@13585 ` 212` ```by (unfold inj_on_def, blast) ``` wenzelm@12258 ` 213` paulson@13585 ` 214` ```lemma inj_singleton: "inj (%s. {s})" ``` paulson@13585 ` 215` ```by (simp add: inj_on_def) ``` paulson@13585 ` 216` nipkow@15111 ` 217` ```lemma inj_on_empty[iff]: "inj_on f {}" ``` nipkow@15111 ` 218` ```by(simp add: inj_on_def) ``` nipkow@15111 ` 219` nipkow@15303 ` 220` ```lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A" ``` paulson@13585 ` 221` ```by (unfold inj_on_def, blast) ``` paulson@13585 ` 222` nipkow@15111 ` 223` ```lemma inj_on_Un: ``` nipkow@15111 ` 224` ``` "inj_on f (A Un B) = ``` nipkow@15111 ` 225` ``` (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})" ``` nipkow@15111 ` 226` ```apply(unfold inj_on_def) ``` nipkow@15111 ` 227` ```apply (blast intro:sym) ``` nipkow@15111 ` 228` ```done ``` nipkow@15111 ` 229` nipkow@15111 ` 230` ```lemma inj_on_insert[iff]: ``` nipkow@15111 ` 231` ``` "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))" ``` nipkow@15111 ` 232` ```apply(unfold inj_on_def) ``` nipkow@15111 ` 233` ```apply (blast intro:sym) ``` nipkow@15111 ` 234` ```done ``` nipkow@15111 ` 235` nipkow@15111 ` 236` ```lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)" ``` nipkow@15111 ` 237` ```apply(unfold inj_on_def) ``` nipkow@15111 ` 238` ```apply (blast) ``` nipkow@15111 ` 239` ```done ``` nipkow@15111 ` 240` hoelzl@39076 ` 241` ```lemma surj_onI: "(\x. x \ B \ g (f x) = x) \ surj_on g B" ``` hoelzl@39076 ` 242` ``` by (simp add: surj_on_def) (blast intro: sym) ``` hoelzl@39076 ` 243` hoelzl@39076 ` 244` ```lemma surj_onD: "surj_on f B \ y \ B \ \x. y = f x" ``` hoelzl@39076 ` 245` ``` by (auto simp: surj_on_def) ``` hoelzl@39076 ` 246` hoelzl@39076 ` 247` ```lemma surj_on_range_iff: "surj_on f B \ (\A. f ` A = B)" ``` hoelzl@39076 ` 248` ``` unfolding surj_on_def by (auto intro!: exI[of _ "f -` B"]) ``` paulson@13585 ` 249` hoelzl@39076 ` 250` ```lemma surj_def: "surj f \ (\y. \x. y = f x)" ``` hoelzl@39076 ` 251` ``` by (simp add: surj_on_def subset_eq image_iff) ``` hoelzl@39076 ` 252` hoelzl@39076 ` 253` ```lemma surjI: "(\ x. g (f x) = x) \ surj g" ``` hoelzl@39076 ` 254` ``` by (blast intro: surj_onI) ``` paulson@13585 ` 255` hoelzl@39076 ` 256` ```lemma surjD: "surj f \ \x. y = f x" ``` hoelzl@39076 ` 257` ``` by (simp add: surj_def) ``` paulson@13585 ` 258` hoelzl@39076 ` 259` ```lemma surjE: "surj f \ (\x. y = f x \ C) \ C" ``` hoelzl@39076 ` 260` ``` by (simp add: surj_def, blast) ``` paulson@13585 ` 261` paulson@13585 ` 262` ```lemma comp_surj: "[| surj f; surj g |] ==> surj (g o f)" ``` paulson@13585 ` 263` ```apply (simp add: comp_def surj_def, clarify) ``` paulson@13585 ` 264` ```apply (drule_tac x = y in spec, clarify) ``` paulson@13585 ` 265` ```apply (drule_tac x = x in spec, blast) ``` paulson@13585 ` 266` ```done ``` paulson@13585 ` 267` hoelzl@39076 ` 268` ```lemma surj_range: "surj f \ range f = UNIV" ``` hoelzl@39076 ` 269` ``` by (auto simp add: surj_on_def) ``` hoelzl@39076 ` 270` hoelzl@39074 ` 271` ```lemma surj_range_iff: "surj f \ range f = UNIV" ``` hoelzl@39076 ` 272` ``` unfolding surj_on_def by auto ``` hoelzl@39074 ` 273` hoelzl@39074 ` 274` ```lemma bij_betw_imp_surj: "bij_betw f A UNIV \ surj f" ``` hoelzl@39074 ` 275` ``` unfolding bij_betw_def surj_range_iff by auto ``` hoelzl@39074 ` 276` hoelzl@39076 ` 277` ```lemma bij_def: "bij f \ inj f \ surj f" ``` hoelzl@39076 ` 278` ``` unfolding surj_range_iff bij_betw_def .. ``` hoelzl@39074 ` 279` paulson@13585 ` 280` ```lemma bijI: "[| inj f; surj f |] ==> bij f" ``` paulson@13585 ` 281` ```by (simp add: bij_def) ``` paulson@13585 ` 282` paulson@13585 ` 283` ```lemma bij_is_inj: "bij f ==> inj f" ``` paulson@13585 ` 284` ```by (simp add: bij_def) ``` paulson@13585 ` 285` paulson@13585 ` 286` ```lemma bij_is_surj: "bij f ==> surj f" ``` paulson@13585 ` 287` ```by (simp add: bij_def) ``` paulson@13585 ` 288` nipkow@26105 ` 289` ```lemma bij_betw_imp_inj_on: "bij_betw f A B \ inj_on f A" ``` nipkow@26105 ` 290` ```by (simp add: bij_betw_def) ``` nipkow@26105 ` 291` hoelzl@39076 ` 292` ```lemma bij_betw_imp_surj_on: "bij_betw f A B \ surj_on f B" ``` hoelzl@39076 ` 293` ```by (auto simp: bij_betw_def surj_on_range_iff) ``` hoelzl@39076 ` 294` nipkow@32337 ` 295` ```lemma bij_comp: "bij f \ bij g \ bij (g o f)" ``` nipkow@32337 ` 296` ```by(fastsimp intro: comp_inj_on comp_surj simp: bij_def surj_range) ``` nipkow@32337 ` 297` nipkow@31438 ` 298` ```lemma bij_betw_trans: ``` nipkow@31438 ` 299` ``` "bij_betw f A B \ bij_betw g B C \ bij_betw (g o f) A C" ``` nipkow@31438 ` 300` ```by(auto simp add:bij_betw_def comp_inj_on) ``` nipkow@31438 ` 301` nipkow@26105 ` 302` ```lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A" ``` nipkow@26105 ` 303` ```proof - ``` nipkow@26105 ` 304` ``` have i: "inj_on f A" and s: "f ` A = B" ``` nipkow@26105 ` 305` ``` using assms by(auto simp:bij_betw_def) ``` nipkow@26105 ` 306` ``` let ?P = "%b a. a:A \ f a = b" let ?g = "%b. The (?P b)" ``` nipkow@26105 ` 307` ``` { fix a b assume P: "?P b a" ``` nipkow@26105 ` 308` ``` hence ex1: "\a. ?P b a" using s unfolding image_def by blast ``` nipkow@26105 ` 309` ``` hence uex1: "\!a. ?P b a" by(blast dest:inj_onD[OF i]) ``` nipkow@26105 ` 310` ``` hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp ``` nipkow@26105 ` 311` ``` } note g = this ``` nipkow@26105 ` 312` ``` have "inj_on ?g B" ``` nipkow@26105 ` 313` ``` proof(rule inj_onI) ``` nipkow@26105 ` 314` ``` fix x y assume "x:B" "y:B" "?g x = ?g y" ``` nipkow@26105 ` 315` ``` from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast ``` nipkow@26105 ` 316` ``` from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast ``` nipkow@26105 ` 317` ``` from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp ``` nipkow@26105 ` 318` ``` qed ``` nipkow@26105 ` 319` ``` moreover have "?g ` B = A" ``` nipkow@26105 ` 320` ``` proof(auto simp:image_def) ``` nipkow@26105 ` 321` ``` fix b assume "b:B" ``` nipkow@26105 ` 322` ``` with s obtain a where P: "?P b a" unfolding image_def by blast ``` nipkow@26105 ` 323` ``` thus "?g b \ A" using g[OF P] by auto ``` nipkow@26105 ` 324` ``` next ``` nipkow@26105 ` 325` ``` fix a assume "a:A" ``` nipkow@26105 ` 326` ``` then obtain b where P: "?P b a" using s unfolding image_def by blast ``` nipkow@26105 ` 327` ``` then have "b:B" using s unfolding image_def by blast ``` nipkow@26105 ` 328` ``` with g[OF P] show "\b\B. a = ?g b" by blast ``` nipkow@26105 ` 329` ``` qed ``` nipkow@26105 ` 330` ``` ultimately show ?thesis by(auto simp:bij_betw_def) ``` nipkow@26105 ` 331` ```qed ``` nipkow@26105 ` 332` hoelzl@39075 ` 333` ```lemma bij_betw_combine: ``` hoelzl@39075 ` 334` ``` assumes "bij_betw f A B" "bij_betw f C D" "B \ D = {}" ``` hoelzl@39075 ` 335` ``` shows "bij_betw f (A \ C) (B \ D)" ``` hoelzl@39075 ` 336` ``` using assms unfolding bij_betw_def inj_on_Un image_Un by auto ``` hoelzl@39075 ` 337` paulson@13585 ` 338` ```lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A" ``` paulson@13585 ` 339` ```by (simp add: surj_range) ``` paulson@13585 ` 340` paulson@13585 ` 341` ```lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A" ``` paulson@13585 ` 342` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 343` paulson@13585 ` 344` ```lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A" ``` paulson@13585 ` 345` ```apply (unfold surj_def) ``` paulson@13585 ` 346` ```apply (blast intro: sym) ``` paulson@13585 ` 347` ```done ``` paulson@13585 ` 348` paulson@13585 ` 349` ```lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A" ``` paulson@13585 ` 350` ```by (unfold inj_on_def, blast) ``` paulson@13585 ` 351` paulson@13585 ` 352` ```lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)" ``` paulson@13585 ` 353` ```apply (unfold bij_def) ``` paulson@13585 ` 354` ```apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) ``` paulson@13585 ` 355` ```done ``` paulson@13585 ` 356` nipkow@31438 ` 357` ```lemma inj_on_Un_image_eq_iff: "inj_on f (A \ B) \ f ` A = f ` B \ A = B" ``` nipkow@31438 ` 358` ```by(blast dest: inj_onD) ``` nipkow@31438 ` 359` paulson@13585 ` 360` ```lemma inj_on_image_Int: ``` paulson@13585 ` 361` ``` "[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B" ``` paulson@13585 ` 362` ```apply (simp add: inj_on_def, blast) ``` paulson@13585 ` 363` ```done ``` paulson@13585 ` 364` paulson@13585 ` 365` ```lemma inj_on_image_set_diff: ``` paulson@13585 ` 366` ``` "[| inj_on f C; A<=C; B<=C |] ==> f`(A-B) = f`A - f`B" ``` paulson@13585 ` 367` ```apply (simp add: inj_on_def, blast) ``` paulson@13585 ` 368` ```done ``` paulson@13585 ` 369` paulson@13585 ` 370` ```lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" ``` paulson@13585 ` 371` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 372` paulson@13585 ` 373` ```lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B" ``` paulson@13585 ` 374` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 375` paulson@13585 ` 376` ```lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)" ``` paulson@13585 ` 377` ```by (blast dest: injD) ``` paulson@13585 ` 378` paulson@13585 ` 379` ```lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" ``` paulson@13585 ` 380` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 381` paulson@13585 ` 382` ```lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" ``` paulson@13585 ` 383` ```by (blast dest: injD) ``` paulson@13585 ` 384` paulson@13585 ` 385` ```(*injectivity's required. Left-to-right inclusion holds even if A is empty*) ``` paulson@13585 ` 386` ```lemma image_INT: ``` paulson@13585 ` 387` ``` "[| inj_on f C; ALL x:A. B x <= C; j:A |] ``` paulson@13585 ` 388` ``` ==> f ` (INTER A B) = (INT x:A. f ` B x)" ``` paulson@13585 ` 389` ```apply (simp add: inj_on_def, blast) ``` paulson@13585 ` 390` ```done ``` paulson@13585 ` 391` paulson@13585 ` 392` ```(*Compare with image_INT: no use of inj_on, and if f is surjective then ``` paulson@13585 ` 393` ``` it doesn't matter whether A is empty*) ``` paulson@13585 ` 394` ```lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)" ``` paulson@13585 ` 395` ```apply (simp add: bij_def) ``` paulson@13585 ` 396` ```apply (simp add: inj_on_def surj_def, blast) ``` paulson@13585 ` 397` ```done ``` paulson@13585 ` 398` paulson@13585 ` 399` ```lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)" ``` paulson@13585 ` 400` ```by (auto simp add: surj_def) ``` paulson@13585 ` 401` paulson@13585 ` 402` ```lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)" ``` paulson@13585 ` 403` ```by (auto simp add: inj_on_def) ``` paulson@5852 ` 404` paulson@13585 ` 405` ```lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)" ``` paulson@13585 ` 406` ```apply (simp add: bij_def) ``` paulson@13585 ` 407` ```apply (rule equalityI) ``` paulson@13585 ` 408` ```apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) ``` paulson@13585 ` 409` ```done ``` paulson@13585 ` 410` hoelzl@35584 ` 411` ```lemma (in ordered_ab_group_add) inj_uminus[simp, intro]: "inj_on uminus A" ``` hoelzl@35580 ` 412` ``` by (auto intro!: inj_onI) ``` paulson@13585 ` 413` hoelzl@35584 ` 414` ```lemma (in linorder) strict_mono_imp_inj_on: "strict_mono f \ inj_on f A" ``` hoelzl@35584 ` 415` ``` by (auto intro!: inj_onI dest: strict_mono_eq) ``` hoelzl@35584 ` 416` paulson@13585 ` 417` ```subsection{*Function Updating*} ``` paulson@13585 ` 418` haftmann@35416 ` 419` ```definition ``` haftmann@35416 ` 420` ``` fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" where ``` haftmann@26147 ` 421` ``` "fun_upd f a b == % x. if x=a then b else f x" ``` haftmann@26147 ` 422` haftmann@26147 ` 423` ```nonterminals ``` haftmann@26147 ` 424` ``` updbinds updbind ``` haftmann@26147 ` 425` ```syntax ``` haftmann@26147 ` 426` ``` "_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") ``` haftmann@26147 ` 427` ``` "" :: "updbind => updbinds" ("_") ``` haftmann@26147 ` 428` ``` "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") ``` wenzelm@35115 ` 429` ``` "_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000, 0] 900) ``` haftmann@26147 ` 430` haftmann@26147 ` 431` ```translations ``` wenzelm@35115 ` 432` ``` "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" ``` wenzelm@35115 ` 433` ``` "f(x:=y)" == "CONST fun_upd f x y" ``` haftmann@26147 ` 434` haftmann@26147 ` 435` ```(* Hint: to define the sum of two functions (or maps), use sum_case. ``` haftmann@26147 ` 436` ``` A nice infix syntax could be defined (in Datatype.thy or below) by ``` wenzelm@35115 ` 437` ```notation ``` wenzelm@35115 ` 438` ``` sum_case (infixr "'(+')"80) ``` haftmann@26147 ` 439` ```*) ``` haftmann@26147 ` 440` paulson@13585 ` 441` ```lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" ``` paulson@13585 ` 442` ```apply (simp add: fun_upd_def, safe) ``` paulson@13585 ` 443` ```apply (erule subst) ``` paulson@13585 ` 444` ```apply (rule_tac [2] ext, auto) ``` paulson@13585 ` 445` ```done ``` paulson@13585 ` 446` paulson@13585 ` 447` ```(* f x = y ==> f(x:=y) = f *) ``` paulson@13585 ` 448` ```lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard] ``` paulson@13585 ` 449` paulson@13585 ` 450` ```(* f(x := f x) = f *) ``` paulson@17084 ` 451` ```lemmas fun_upd_triv = refl [THEN fun_upd_idem] ``` paulson@17084 ` 452` ```declare fun_upd_triv [iff] ``` paulson@13585 ` 453` paulson@13585 ` 454` ```lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)" ``` paulson@17084 ` 455` ```by (simp add: fun_upd_def) ``` paulson@13585 ` 456` paulson@13585 ` 457` ```(* fun_upd_apply supersedes these two, but they are useful ``` paulson@13585 ` 458` ``` if fun_upd_apply is intentionally removed from the simpset *) ``` paulson@13585 ` 459` ```lemma fun_upd_same: "(f(x:=y)) x = y" ``` paulson@13585 ` 460` ```by simp ``` paulson@13585 ` 461` paulson@13585 ` 462` ```lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" ``` paulson@13585 ` 463` ```by simp ``` paulson@13585 ` 464` paulson@13585 ` 465` ```lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)" ``` nipkow@39198 ` 466` ```by (simp add: ext_iff) ``` paulson@13585 ` 467` paulson@13585 ` 468` ```lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" ``` paulson@13585 ` 469` ```by (rule ext, auto) ``` paulson@13585 ` 470` nipkow@15303 ` 471` ```lemma inj_on_fun_updI: "\ inj_on f A; y \ f`A \ \ inj_on (f(x:=y)) A" ``` krauss@34209 ` 472` ```by (fastsimp simp:inj_on_def image_def) ``` nipkow@15303 ` 473` paulson@15510 ` 474` ```lemma fun_upd_image: ``` paulson@15510 ` 475` ``` "f(x:=y) ` A = (if x \ A then insert y (f ` (A-{x})) else f ` A)" ``` paulson@15510 ` 476` ```by auto ``` paulson@15510 ` 477` nipkow@31080 ` 478` ```lemma fun_upd_comp: "f \ (g(x := y)) = (f \ g)(x := f y)" ``` krauss@34209 ` 479` ```by (auto intro: ext) ``` nipkow@31080 ` 480` haftmann@26147 ` 481` haftmann@26147 ` 482` ```subsection {* @{text override_on} *} ``` haftmann@26147 ` 483` haftmann@26147 ` 484` ```definition ``` haftmann@26147 ` 485` ``` override_on :: "('a \ 'b) \ ('a \ 'b) \ 'a set \ 'a \ 'b" ``` haftmann@26147 ` 486` ```where ``` haftmann@26147 ` 487` ``` "override_on f g A = (\a. if a \ A then g a else f a)" ``` nipkow@13910 ` 488` nipkow@15691 ` 489` ```lemma override_on_emptyset[simp]: "override_on f g {} = f" ``` nipkow@15691 ` 490` ```by(simp add:override_on_def) ``` nipkow@13910 ` 491` nipkow@15691 ` 492` ```lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" ``` nipkow@15691 ` 493` ```by(simp add:override_on_def) ``` nipkow@13910 ` 494` nipkow@15691 ` 495` ```lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" ``` nipkow@15691 ` 496` ```by(simp add:override_on_def) ``` nipkow@13910 ` 497` haftmann@26147 ` 498` haftmann@26147 ` 499` ```subsection {* @{text swap} *} ``` paulson@15510 ` 500` haftmann@22744 ` 501` ```definition ``` haftmann@22744 ` 502` ``` swap :: "'a \ 'a \ ('a \ 'b) \ ('a \ 'b)" ``` haftmann@22744 ` 503` ```where ``` haftmann@22744 ` 504` ``` "swap a b f = f (a := f b, b:= f a)" ``` paulson@15510 ` 505` huffman@34101 ` 506` ```lemma swap_self [simp]: "swap a a f = f" ``` nipkow@15691 ` 507` ```by (simp add: swap_def) ``` paulson@15510 ` 508` paulson@15510 ` 509` ```lemma swap_commute: "swap a b f = swap b a f" ``` paulson@15510 ` 510` ```by (rule ext, simp add: fun_upd_def swap_def) ``` paulson@15510 ` 511` paulson@15510 ` 512` ```lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" ``` paulson@15510 ` 513` ```by (rule ext, simp add: fun_upd_def swap_def) ``` paulson@15510 ` 514` huffman@34145 ` 515` ```lemma swap_triple: ``` huffman@34145 ` 516` ``` assumes "a \ c" and "b \ c" ``` huffman@34145 ` 517` ``` shows "swap a b (swap b c (swap a b f)) = swap a c f" ``` nipkow@39198 ` 518` ``` using assms by (simp add: ext_iff swap_def) ``` huffman@34145 ` 519` huffman@34101 ` 520` ```lemma comp_swap: "f \ swap a b g = swap a b (f \ g)" ``` huffman@34101 ` 521` ```by (rule ext, simp add: fun_upd_def swap_def) ``` huffman@34101 ` 522` hoelzl@39076 ` 523` ```lemma swap_image_eq [simp]: ``` hoelzl@39076 ` 524` ``` assumes "a \ A" "b \ A" shows "swap a b f ` A = f ` A" ``` hoelzl@39076 ` 525` ```proof - ``` hoelzl@39076 ` 526` ``` have subset: "\f. swap a b f ` A \ f ` A" ``` hoelzl@39076 ` 527` ``` using assms by (auto simp: image_iff swap_def) ``` hoelzl@39076 ` 528` ``` then have "swap a b (swap a b f) ` A \ (swap a b f) ` A" . ``` hoelzl@39076 ` 529` ``` with subset[of f] show ?thesis by auto ``` hoelzl@39076 ` 530` ```qed ``` hoelzl@39076 ` 531` paulson@15510 ` 532` ```lemma inj_on_imp_inj_on_swap: ``` hoelzl@39076 ` 533` ``` "\inj_on f A; a \ A; b \ A\ \ inj_on (swap a b f) A" ``` hoelzl@39076 ` 534` ``` by (simp add: inj_on_def swap_def, blast) ``` paulson@15510 ` 535` paulson@15510 ` 536` ```lemma inj_on_swap_iff [simp]: ``` hoelzl@39076 ` 537` ``` assumes A: "a \ A" "b \ A" shows "inj_on (swap a b f) A \ inj_on f A" ``` hoelzl@39075 ` 538` ```proof ``` paulson@15510 ` 539` ``` assume "inj_on (swap a b f) A" ``` hoelzl@39075 ` 540` ``` with A have "inj_on (swap a b (swap a b f)) A" ``` hoelzl@39075 ` 541` ``` by (iprover intro: inj_on_imp_inj_on_swap) ``` hoelzl@39075 ` 542` ``` thus "inj_on f A" by simp ``` paulson@15510 ` 543` ```next ``` paulson@15510 ` 544` ``` assume "inj_on f A" ``` krauss@34209 ` 545` ``` with A show "inj_on (swap a b f) A" by (iprover intro: inj_on_imp_inj_on_swap) ``` paulson@15510 ` 546` ```qed ``` paulson@15510 ` 547` hoelzl@39076 ` 548` ```lemma surj_imp_surj_swap: "surj f \ surj (swap a b f)" ``` hoelzl@39076 ` 549` ``` unfolding surj_range_iff by simp ``` paulson@15510 ` 550` hoelzl@39076 ` 551` ```lemma surj_swap_iff [simp]: "surj (swap a b f) \ surj f" ``` hoelzl@39076 ` 552` ``` unfolding surj_range_iff by simp ``` haftmann@21547 ` 553` hoelzl@39076 ` 554` ```lemma bij_betw_swap_iff [simp]: ``` hoelzl@39076 ` 555` ``` "\ x \ A; y \ A \ \ bij_betw (swap x y f) A B \ bij_betw f A B" ``` hoelzl@39076 ` 556` ``` by (auto simp: bij_betw_def) ``` hoelzl@39076 ` 557` hoelzl@39076 ` 558` ```lemma bij_swap_iff [simp]: "bij (swap a b f) \ bij f" ``` hoelzl@39076 ` 559` ``` by simp ``` hoelzl@39075 ` 560` wenzelm@36176 ` 561` ```hide_const (open) swap ``` haftmann@21547 ` 562` haftmann@31949 ` 563` ```subsection {* Inversion of injective functions *} ``` haftmann@31949 ` 564` nipkow@33057 ` 565` ```definition the_inv_into :: "'a set => ('a => 'b) => ('b => 'a)" where ``` nipkow@33057 ` 566` ```"the_inv_into A f == %x. THE y. y : A & f y = x" ``` nipkow@32961 ` 567` nipkow@33057 ` 568` ```lemma the_inv_into_f_f: ``` nipkow@33057 ` 569` ``` "[| inj_on f A; x : A |] ==> the_inv_into A f (f x) = x" ``` nipkow@33057 ` 570` ```apply (simp add: the_inv_into_def inj_on_def) ``` krauss@34209 ` 571` ```apply blast ``` nipkow@32961 ` 572` ```done ``` nipkow@32961 ` 573` nipkow@33057 ` 574` ```lemma f_the_inv_into_f: ``` nipkow@33057 ` 575` ``` "inj_on f A ==> y : f`A ==> f (the_inv_into A f y) = y" ``` nipkow@33057 ` 576` ```apply (simp add: the_inv_into_def) ``` nipkow@32961 ` 577` ```apply (rule the1I2) ``` nipkow@32961 ` 578` ``` apply(blast dest: inj_onD) ``` nipkow@32961 ` 579` ```apply blast ``` nipkow@32961 ` 580` ```done ``` nipkow@32961 ` 581` nipkow@33057 ` 582` ```lemma the_inv_into_into: ``` nipkow@33057 ` 583` ``` "[| inj_on f A; x : f ` A; A <= B |] ==> the_inv_into A f x : B" ``` nipkow@33057 ` 584` ```apply (simp add: the_inv_into_def) ``` nipkow@32961 ` 585` ```apply (rule the1I2) ``` nipkow@32961 ` 586` ``` apply(blast dest: inj_onD) ``` nipkow@32961 ` 587` ```apply blast ``` nipkow@32961 ` 588` ```done ``` nipkow@32961 ` 589` nipkow@33057 ` 590` ```lemma the_inv_into_onto[simp]: ``` nipkow@33057 ` 591` ``` "inj_on f A ==> the_inv_into A f ` (f ` A) = A" ``` nipkow@33057 ` 592` ```by (fast intro:the_inv_into_into the_inv_into_f_f[symmetric]) ``` nipkow@32961 ` 593` nipkow@33057 ` 594` ```lemma the_inv_into_f_eq: ``` nipkow@33057 ` 595` ``` "[| inj_on f A; f x = y; x : A |] ==> the_inv_into A f y = x" ``` nipkow@32961 ` 596` ``` apply (erule subst) ``` nipkow@33057 ` 597` ``` apply (erule the_inv_into_f_f, assumption) ``` nipkow@32961 ` 598` ``` done ``` nipkow@32961 ` 599` nipkow@33057 ` 600` ```lemma the_inv_into_comp: ``` nipkow@32961 ` 601` ``` "[| inj_on f (g ` A); inj_on g A; x : f ` g ` A |] ==> ``` nipkow@33057 ` 602` ``` the_inv_into A (f o g) x = (the_inv_into A g o the_inv_into (g ` A) f) x" ``` nipkow@33057 ` 603` ```apply (rule the_inv_into_f_eq) ``` nipkow@32961 ` 604` ``` apply (fast intro: comp_inj_on) ``` nipkow@33057 ` 605` ``` apply (simp add: f_the_inv_into_f the_inv_into_into) ``` nipkow@33057 ` 606` ```apply (simp add: the_inv_into_into) ``` nipkow@32961 ` 607` ```done ``` nipkow@32961 ` 608` nipkow@33057 ` 609` ```lemma inj_on_the_inv_into: ``` nipkow@33057 ` 610` ``` "inj_on f A \ inj_on (the_inv_into A f) (f ` A)" ``` nipkow@33057 ` 611` ```by (auto intro: inj_onI simp: image_def the_inv_into_f_f) ``` nipkow@32961 ` 612` nipkow@33057 ` 613` ```lemma bij_betw_the_inv_into: ``` nipkow@33057 ` 614` ``` "bij_betw f A B \ bij_betw (the_inv_into A f) B A" ``` nipkow@33057 ` 615` ```by (auto simp add: bij_betw_def inj_on_the_inv_into the_inv_into_into) ``` nipkow@32961 ` 616` berghofe@32998 ` 617` ```abbreviation the_inv :: "('a \ 'b) \ ('b \ 'a)" where ``` nipkow@33057 ` 618` ``` "the_inv f \ the_inv_into UNIV f" ``` berghofe@32998 ` 619` berghofe@32998 ` 620` ```lemma the_inv_f_f: ``` berghofe@32998 ` 621` ``` assumes "inj f" ``` berghofe@32998 ` 622` ``` shows "the_inv f (f x) = x" using assms UNIV_I ``` nipkow@33057 ` 623` ``` by (rule the_inv_into_f_f) ``` berghofe@32998 ` 624` haftmann@31949 ` 625` haftmann@22845 ` 626` ```subsection {* Proof tool setup *} ``` haftmann@22845 ` 627` haftmann@22845 ` 628` ```text {* simplifies terms of the form ``` haftmann@22845 ` 629` ``` f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *} ``` haftmann@22845 ` 630` wenzelm@24017 ` 631` ```simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ => ``` haftmann@22845 ` 632` ```let ``` haftmann@22845 ` 633` ``` fun gen_fun_upd NONE T _ _ = NONE ``` wenzelm@24017 ` 634` ``` | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y) ``` haftmann@22845 ` 635` ``` fun dest_fun_T1 (Type (_, T :: Ts)) = T ``` haftmann@22845 ` 636` ``` fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) = ``` haftmann@22845 ` 637` ``` let ``` haftmann@22845 ` 638` ``` fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) = ``` haftmann@22845 ` 639` ``` if v aconv x then SOME g else gen_fun_upd (find g) T v w ``` haftmann@22845 ` 640` ``` | find t = NONE ``` haftmann@22845 ` 641` ``` in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end ``` wenzelm@24017 ` 642` wenzelm@24017 ` 643` ``` fun proc ss ct = ``` wenzelm@24017 ` 644` ``` let ``` wenzelm@24017 ` 645` ``` val ctxt = Simplifier.the_context ss ``` wenzelm@24017 ` 646` ``` val t = Thm.term_of ct ``` wenzelm@24017 ` 647` ``` in ``` wenzelm@24017 ` 648` ``` case find_double t of ``` wenzelm@24017 ` 649` ``` (T, NONE) => NONE ``` wenzelm@24017 ` 650` ``` | (T, SOME rhs) => ``` wenzelm@27330 ` 651` ``` SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) ``` wenzelm@24017 ` 652` ``` (fn _ => ``` wenzelm@24017 ` 653` ``` rtac eq_reflection 1 THEN ``` wenzelm@24017 ` 654` ``` rtac ext 1 THEN ``` wenzelm@24017 ` 655` ``` simp_tac (Simplifier.inherit_context ss @{simpset}) 1)) ``` wenzelm@24017 ` 656` ``` end ``` wenzelm@24017 ` 657` ```in proc end ``` haftmann@22845 ` 658` ```*} ``` haftmann@22845 ` 659` haftmann@22845 ` 660` haftmann@21870 ` 661` ```subsection {* Code generator setup *} ``` haftmann@21870 ` 662` berghofe@25886 ` 663` ```types_code ``` berghofe@25886 ` 664` ``` "fun" ("(_ ->/ _)") ``` berghofe@25886 ` 665` ```attach (term_of) {* ``` berghofe@25886 ` 666` ```fun term_of_fun_type _ aT _ bT _ = Free ("", aT --> bT); ``` berghofe@25886 ` 667` ```*} ``` berghofe@25886 ` 668` ```attach (test) {* ``` berghofe@25886 ` 669` ```fun gen_fun_type aF aT bG bT i = ``` berghofe@25886 ` 670` ``` let ``` wenzelm@32740 ` 671` ``` val tab = Unsynchronized.ref []; ``` berghofe@25886 ` 672` ``` fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd", ``` berghofe@25886 ` 673` ``` (aT --> bT) --> aT --> bT --> aT --> bT) \$ t \$ aF x \$ y () ``` berghofe@25886 ` 674` ``` in ``` berghofe@25886 ` 675` ``` (fn x => ``` berghofe@25886 ` 676` ``` case AList.lookup op = (!tab) x of ``` berghofe@25886 ` 677` ``` NONE => ``` berghofe@25886 ` 678` ``` let val p as (y, _) = bG i ``` berghofe@25886 ` 679` ``` in (tab := (x, p) :: !tab; y) end ``` berghofe@25886 ` 680` ``` | SOME (y, _) => y, ``` berghofe@28711 ` 681` ``` fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT))) ``` berghofe@25886 ` 682` ``` end; ``` berghofe@25886 ` 683` ```*} ``` berghofe@25886 ` 684` haftmann@21870 ` 685` ```code_const "op \" ``` haftmann@21870 ` 686` ``` (SML infixl 5 "o") ``` haftmann@21870 ` 687` ``` (Haskell infixr 9 ".") ``` haftmann@21870 ` 688` haftmann@21906 ` 689` ```code_const "id" ``` haftmann@21906 ` 690` ``` (Haskell "id") ``` haftmann@21906 ` 691` nipkow@2912 ` 692` ```end ```