src/HOL/Fun.thy
 author nipkow Thu Jun 04 13:26:32 2009 +0200 (2009-06-04) changeset 31438 a1c4c1500abe parent 31202 52d332f8f909 child 31604 eb2f9d709296 permissions -rw-r--r--
A few finite lemmas
 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@22886 ` 9` ```imports Set ``` 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.*} ``` haftmann@26147 ` 14` ```lemma expand_fun_eq: "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` paulson@13585 ` 77` ```lemma image_compose: "(f o g) ` r = f`(g`r)" ``` paulson@13585 ` 78` ```by (simp add: comp_def, blast) ``` paulson@13585 ` 79` paulson@13585 ` 80` ```lemma UN_o: "UNION A (g o f) = UNION (f`A) g" ``` paulson@13585 ` 81` ```by (unfold comp_def, blast) ``` paulson@13585 ` 82` paulson@13585 ` 83` haftmann@26588 ` 84` ```subsection {* The Forward Composition Operator @{text fcomp} *} ``` haftmann@26357 ` 85` haftmann@26357 ` 86` ```definition ``` haftmann@26357 ` 87` ``` fcomp :: "('a \ 'b) \ ('b \ 'c) \ 'a \ 'c" (infixl "o>" 60) ``` haftmann@26357 ` 88` ```where ``` haftmann@26357 ` 89` ``` "f o> g = (\x. g (f x))" ``` haftmann@26357 ` 90` haftmann@26357 ` 91` ```lemma fcomp_apply: "(f o> g) x = g (f x)" ``` haftmann@26357 ` 92` ``` by (simp add: fcomp_def) ``` haftmann@26357 ` 93` haftmann@26357 ` 94` ```lemma fcomp_assoc: "(f o> g) o> h = f o> (g o> h)" ``` haftmann@26357 ` 95` ``` by (simp add: fcomp_def) ``` haftmann@26357 ` 96` haftmann@26357 ` 97` ```lemma id_fcomp [simp]: "id o> g = g" ``` haftmann@26357 ` 98` ``` by (simp add: fcomp_def) ``` haftmann@26357 ` 99` haftmann@26357 ` 100` ```lemma fcomp_id [simp]: "f o> id = f" ``` haftmann@26357 ` 101` ``` by (simp add: fcomp_def) ``` haftmann@26357 ` 102` haftmann@31202 ` 103` ```code_const fcomp ``` haftmann@31202 ` 104` ``` (Eval infixl 1 "#>") ``` haftmann@31202 ` 105` haftmann@26588 ` 106` ```no_notation fcomp (infixl "o>" 60) ``` haftmann@26588 ` 107` haftmann@26357 ` 108` haftmann@26147 ` 109` ```subsection {* Injectivity and Surjectivity *} ``` haftmann@26147 ` 110` haftmann@26147 ` 111` ```constdefs ``` haftmann@26147 ` 112` ``` inj_on :: "['a => 'b, 'a set] => bool" -- "injective" ``` haftmann@26147 ` 113` ``` "inj_on f A == ! x:A. ! y:A. f(x)=f(y) --> x=y" ``` haftmann@26147 ` 114` haftmann@26147 ` 115` ```text{*A common special case: functions injective over the entire domain type.*} ``` haftmann@26147 ` 116` haftmann@26147 ` 117` ```abbreviation ``` haftmann@26147 ` 118` ``` "inj f == inj_on f UNIV" ``` paulson@13585 ` 119` haftmann@26147 ` 120` ```definition ``` haftmann@26147 ` 121` ``` bij_betw :: "('a => 'b) => 'a set => 'b set => bool" where -- "bijective" ``` haftmann@28562 ` 122` ``` [code del]: "bij_betw f A B \ inj_on f A & f ` A = B" ``` haftmann@26147 ` 123` haftmann@26147 ` 124` ```constdefs ``` haftmann@26147 ` 125` ``` surj :: "('a => 'b) => bool" (*surjective*) ``` haftmann@26147 ` 126` ``` "surj f == ! y. ? x. y=f(x)" ``` paulson@13585 ` 127` haftmann@26147 ` 128` ``` bij :: "('a => 'b) => bool" (*bijective*) ``` haftmann@26147 ` 129` ``` "bij f == inj f & surj f" ``` haftmann@26147 ` 130` haftmann@26147 ` 131` ```lemma injI: ``` haftmann@26147 ` 132` ``` assumes "\x y. f x = f y \ x = y" ``` haftmann@26147 ` 133` ``` shows "inj f" ``` haftmann@26147 ` 134` ``` using assms unfolding inj_on_def by auto ``` paulson@13585 ` 135` paulson@13585 ` 136` ```text{*For Proofs in @{text "Tools/datatype_rep_proofs"}*} ``` paulson@13585 ` 137` ```lemma datatype_injI: ``` paulson@13585 ` 138` ``` "(!! x. ALL y. f(x) = f(y) --> x=y) ==> inj(f)" ``` paulson@13585 ` 139` ```by (simp add: inj_on_def) ``` paulson@13585 ` 140` berghofe@13637 ` 141` ```theorem range_ex1_eq: "inj f \ b : range f = (EX! x. b = f x)" ``` berghofe@13637 ` 142` ``` by (unfold inj_on_def, blast) ``` berghofe@13637 ` 143` paulson@13585 ` 144` ```lemma injD: "[| inj(f); f(x) = f(y) |] ==> x=y" ``` paulson@13585 ` 145` ```by (simp add: inj_on_def) ``` paulson@13585 ` 146` paulson@13585 ` 147` ```(*Useful with the simplifier*) ``` paulson@13585 ` 148` ```lemma inj_eq: "inj(f) ==> (f(x) = f(y)) = (x=y)" ``` paulson@13585 ` 149` ```by (force simp add: inj_on_def) ``` paulson@13585 ` 150` haftmann@26147 ` 151` ```lemma inj_on_id[simp]: "inj_on id A" ``` haftmann@26147 ` 152` ``` by (simp add: inj_on_def) ``` paulson@13585 ` 153` haftmann@26147 ` 154` ```lemma inj_on_id2[simp]: "inj_on (%x. x) A" ``` haftmann@26147 ` 155` ```by (simp add: inj_on_def) ``` haftmann@26147 ` 156` haftmann@26147 ` 157` ```lemma surj_id[simp]: "surj id" ``` haftmann@26147 ` 158` ```by (simp add: surj_def) ``` haftmann@26147 ` 159` haftmann@26147 ` 160` ```lemma bij_id[simp]: "bij id" ``` haftmann@26147 ` 161` ```by (simp add: bij_def inj_on_id surj_id) ``` paulson@13585 ` 162` paulson@13585 ` 163` ```lemma inj_onI: ``` paulson@13585 ` 164` ``` "(!! x y. [| x:A; y:A; f(x) = f(y) |] ==> x=y) ==> inj_on f A" ``` paulson@13585 ` 165` ```by (simp add: inj_on_def) ``` paulson@13585 ` 166` paulson@13585 ` 167` ```lemma inj_on_inverseI: "(!!x. x:A ==> g(f(x)) = x) ==> inj_on f A" ``` paulson@13585 ` 168` ```by (auto dest: arg_cong [of concl: g] simp add: inj_on_def) ``` paulson@13585 ` 169` paulson@13585 ` 170` ```lemma inj_onD: "[| inj_on f A; f(x)=f(y); x:A; y:A |] ==> x=y" ``` paulson@13585 ` 171` ```by (unfold inj_on_def, blast) ``` paulson@13585 ` 172` paulson@13585 ` 173` ```lemma inj_on_iff: "[| inj_on f A; x:A; y:A |] ==> (f(x)=f(y)) = (x=y)" ``` paulson@13585 ` 174` ```by (blast dest!: inj_onD) ``` paulson@13585 ` 175` paulson@13585 ` 176` ```lemma comp_inj_on: ``` paulson@13585 ` 177` ``` "[| inj_on f A; inj_on g (f`A) |] ==> inj_on (g o f) A" ``` paulson@13585 ` 178` ```by (simp add: comp_def inj_on_def) ``` paulson@13585 ` 179` nipkow@15303 ` 180` ```lemma inj_on_imageI: "inj_on (g o f) A \ inj_on g (f ` A)" ``` nipkow@15303 ` 181` ```apply(simp add:inj_on_def image_def) ``` nipkow@15303 ` 182` ```apply blast ``` nipkow@15303 ` 183` ```done ``` nipkow@15303 ` 184` nipkow@15439 ` 185` ```lemma inj_on_image_iff: "\ ALL x:A. ALL y:A. (g(f x) = g(f y)) = (g x = g y); ``` nipkow@15439 ` 186` ``` inj_on f A \ \ inj_on g (f ` A) = inj_on g A" ``` nipkow@15439 ` 187` ```apply(unfold inj_on_def) ``` nipkow@15439 ` 188` ```apply blast ``` nipkow@15439 ` 189` ```done ``` nipkow@15439 ` 190` paulson@13585 ` 191` ```lemma inj_on_contraD: "[| inj_on f A; ~x=y; x:A; y:A |] ==> ~ f(x)=f(y)" ``` paulson@13585 ` 192` ```by (unfold inj_on_def, blast) ``` wenzelm@12258 ` 193` paulson@13585 ` 194` ```lemma inj_singleton: "inj (%s. {s})" ``` paulson@13585 ` 195` ```by (simp add: inj_on_def) ``` paulson@13585 ` 196` nipkow@15111 ` 197` ```lemma inj_on_empty[iff]: "inj_on f {}" ``` nipkow@15111 ` 198` ```by(simp add: inj_on_def) ``` nipkow@15111 ` 199` nipkow@15303 ` 200` ```lemma subset_inj_on: "[| inj_on f B; A <= B |] ==> inj_on f A" ``` paulson@13585 ` 201` ```by (unfold inj_on_def, blast) ``` paulson@13585 ` 202` nipkow@15111 ` 203` ```lemma inj_on_Un: ``` nipkow@15111 ` 204` ``` "inj_on f (A Un B) = ``` nipkow@15111 ` 205` ``` (inj_on f A & inj_on f B & f`(A-B) Int f`(B-A) = {})" ``` nipkow@15111 ` 206` ```apply(unfold inj_on_def) ``` nipkow@15111 ` 207` ```apply (blast intro:sym) ``` nipkow@15111 ` 208` ```done ``` nipkow@15111 ` 209` nipkow@15111 ` 210` ```lemma inj_on_insert[iff]: ``` nipkow@15111 ` 211` ``` "inj_on f (insert a A) = (inj_on f A & f a ~: f`(A-{a}))" ``` nipkow@15111 ` 212` ```apply(unfold inj_on_def) ``` nipkow@15111 ` 213` ```apply (blast intro:sym) ``` nipkow@15111 ` 214` ```done ``` nipkow@15111 ` 215` nipkow@15111 ` 216` ```lemma inj_on_diff: "inj_on f A ==> inj_on f (A-B)" ``` nipkow@15111 ` 217` ```apply(unfold inj_on_def) ``` nipkow@15111 ` 218` ```apply (blast) ``` nipkow@15111 ` 219` ```done ``` nipkow@15111 ` 220` paulson@13585 ` 221` ```lemma surjI: "(!! x. g(f x) = x) ==> surj g" ``` paulson@13585 ` 222` ```apply (simp add: surj_def) ``` paulson@13585 ` 223` ```apply (blast intro: sym) ``` paulson@13585 ` 224` ```done ``` paulson@13585 ` 225` paulson@13585 ` 226` ```lemma surj_range: "surj f ==> range f = UNIV" ``` paulson@13585 ` 227` ```by (auto simp add: surj_def) ``` paulson@13585 ` 228` paulson@13585 ` 229` ```lemma surjD: "surj f ==> EX x. y = f x" ``` paulson@13585 ` 230` ```by (simp add: surj_def) ``` paulson@13585 ` 231` paulson@13585 ` 232` ```lemma surjE: "surj f ==> (!!x. y = f x ==> C) ==> C" ``` paulson@13585 ` 233` ```by (simp add: surj_def, blast) ``` paulson@13585 ` 234` paulson@13585 ` 235` ```lemma comp_surj: "[| surj f; surj g |] ==> surj (g o f)" ``` paulson@13585 ` 236` ```apply (simp add: comp_def surj_def, clarify) ``` paulson@13585 ` 237` ```apply (drule_tac x = y in spec, clarify) ``` paulson@13585 ` 238` ```apply (drule_tac x = x in spec, blast) ``` paulson@13585 ` 239` ```done ``` paulson@13585 ` 240` paulson@13585 ` 241` ```lemma bijI: "[| inj f; surj f |] ==> bij f" ``` paulson@13585 ` 242` ```by (simp add: bij_def) ``` paulson@13585 ` 243` paulson@13585 ` 244` ```lemma bij_is_inj: "bij f ==> inj f" ``` paulson@13585 ` 245` ```by (simp add: bij_def) ``` paulson@13585 ` 246` paulson@13585 ` 247` ```lemma bij_is_surj: "bij f ==> surj f" ``` paulson@13585 ` 248` ```by (simp add: bij_def) ``` paulson@13585 ` 249` nipkow@26105 ` 250` ```lemma bij_betw_imp_inj_on: "bij_betw f A B \ inj_on f A" ``` nipkow@26105 ` 251` ```by (simp add: bij_betw_def) ``` nipkow@26105 ` 252` nipkow@31438 ` 253` ```lemma bij_betw_trans: ``` nipkow@31438 ` 254` ``` "bij_betw f A B \ bij_betw g B C \ bij_betw (g o f) A C" ``` nipkow@31438 ` 255` ```by(auto simp add:bij_betw_def comp_inj_on) ``` nipkow@31438 ` 256` nipkow@26105 ` 257` ```lemma bij_betw_inv: assumes "bij_betw f A B" shows "EX g. bij_betw g B A" ``` nipkow@26105 ` 258` ```proof - ``` nipkow@26105 ` 259` ``` have i: "inj_on f A" and s: "f ` A = B" ``` nipkow@26105 ` 260` ``` using assms by(auto simp:bij_betw_def) ``` nipkow@26105 ` 261` ``` let ?P = "%b a. a:A \ f a = b" let ?g = "%b. The (?P b)" ``` nipkow@26105 ` 262` ``` { fix a b assume P: "?P b a" ``` nipkow@26105 ` 263` ``` hence ex1: "\a. ?P b a" using s unfolding image_def by blast ``` nipkow@26105 ` 264` ``` hence uex1: "\!a. ?P b a" by(blast dest:inj_onD[OF i]) ``` nipkow@26105 ` 265` ``` hence " ?g b = a" using the1_equality[OF uex1, OF P] P by simp ``` nipkow@26105 ` 266` ``` } note g = this ``` nipkow@26105 ` 267` ``` have "inj_on ?g B" ``` nipkow@26105 ` 268` ``` proof(rule inj_onI) ``` nipkow@26105 ` 269` ``` fix x y assume "x:B" "y:B" "?g x = ?g y" ``` nipkow@26105 ` 270` ``` from s `x:B` obtain a1 where a1: "?P x a1" unfolding image_def by blast ``` nipkow@26105 ` 271` ``` from s `y:B` obtain a2 where a2: "?P y a2" unfolding image_def by blast ``` nipkow@26105 ` 272` ``` from g[OF a1] a1 g[OF a2] a2 `?g x = ?g y` show "x=y" by simp ``` nipkow@26105 ` 273` ``` qed ``` nipkow@26105 ` 274` ``` moreover have "?g ` B = A" ``` nipkow@26105 ` 275` ``` proof(auto simp:image_def) ``` nipkow@26105 ` 276` ``` fix b assume "b:B" ``` nipkow@26105 ` 277` ``` with s obtain a where P: "?P b a" unfolding image_def by blast ``` nipkow@26105 ` 278` ``` thus "?g b \ A" using g[OF P] by auto ``` nipkow@26105 ` 279` ``` next ``` nipkow@26105 ` 280` ``` fix a assume "a:A" ``` nipkow@26105 ` 281` ``` then obtain b where P: "?P b a" using s unfolding image_def by blast ``` nipkow@26105 ` 282` ``` then have "b:B" using s unfolding image_def by blast ``` nipkow@26105 ` 283` ``` with g[OF P] show "\b\B. a = ?g b" by blast ``` nipkow@26105 ` 284` ``` qed ``` nipkow@26105 ` 285` ``` ultimately show ?thesis by(auto simp:bij_betw_def) ``` nipkow@26105 ` 286` ```qed ``` nipkow@26105 ` 287` paulson@13585 ` 288` ```lemma surj_image_vimage_eq: "surj f ==> f ` (f -` A) = A" ``` paulson@13585 ` 289` ```by (simp add: surj_range) ``` paulson@13585 ` 290` paulson@13585 ` 291` ```lemma inj_vimage_image_eq: "inj f ==> f -` (f ` A) = A" ``` paulson@13585 ` 292` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 293` paulson@13585 ` 294` ```lemma vimage_subsetD: "surj f ==> f -` B <= A ==> B <= f ` A" ``` paulson@13585 ` 295` ```apply (unfold surj_def) ``` paulson@13585 ` 296` ```apply (blast intro: sym) ``` paulson@13585 ` 297` ```done ``` paulson@13585 ` 298` paulson@13585 ` 299` ```lemma vimage_subsetI: "inj f ==> B <= f ` A ==> f -` B <= A" ``` paulson@13585 ` 300` ```by (unfold inj_on_def, blast) ``` paulson@13585 ` 301` paulson@13585 ` 302` ```lemma vimage_subset_eq: "bij f ==> (f -` B <= A) = (B <= f ` A)" ``` paulson@13585 ` 303` ```apply (unfold bij_def) ``` paulson@13585 ` 304` ```apply (blast del: subsetI intro: vimage_subsetI vimage_subsetD) ``` paulson@13585 ` 305` ```done ``` paulson@13585 ` 306` nipkow@31438 ` 307` ```lemma inj_on_Un_image_eq_iff: "inj_on f (A \ B) \ f ` A = f ` B \ A = B" ``` nipkow@31438 ` 308` ```by(blast dest: inj_onD) ``` nipkow@31438 ` 309` paulson@13585 ` 310` ```lemma inj_on_image_Int: ``` paulson@13585 ` 311` ``` "[| inj_on f C; A<=C; B<=C |] ==> f`(A Int B) = f`A Int f`B" ``` paulson@13585 ` 312` ```apply (simp add: inj_on_def, blast) ``` paulson@13585 ` 313` ```done ``` paulson@13585 ` 314` paulson@13585 ` 315` ```lemma inj_on_image_set_diff: ``` paulson@13585 ` 316` ``` "[| inj_on f C; A<=C; B<=C |] ==> f`(A-B) = f`A - f`B" ``` paulson@13585 ` 317` ```apply (simp add: inj_on_def, blast) ``` paulson@13585 ` 318` ```done ``` paulson@13585 ` 319` paulson@13585 ` 320` ```lemma image_Int: "inj f ==> f`(A Int B) = f`A Int f`B" ``` paulson@13585 ` 321` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 322` paulson@13585 ` 323` ```lemma image_set_diff: "inj f ==> f`(A-B) = f`A - f`B" ``` paulson@13585 ` 324` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 325` paulson@13585 ` 326` ```lemma inj_image_mem_iff: "inj f ==> (f a : f`A) = (a : A)" ``` paulson@13585 ` 327` ```by (blast dest: injD) ``` paulson@13585 ` 328` paulson@13585 ` 329` ```lemma inj_image_subset_iff: "inj f ==> (f`A <= f`B) = (A<=B)" ``` paulson@13585 ` 330` ```by (simp add: inj_on_def, blast) ``` paulson@13585 ` 331` paulson@13585 ` 332` ```lemma inj_image_eq_iff: "inj f ==> (f`A = f`B) = (A = B)" ``` paulson@13585 ` 333` ```by (blast dest: injD) ``` paulson@13585 ` 334` paulson@13585 ` 335` ```(*injectivity's required. Left-to-right inclusion holds even if A is empty*) ``` paulson@13585 ` 336` ```lemma image_INT: ``` paulson@13585 ` 337` ``` "[| inj_on f C; ALL x:A. B x <= C; j:A |] ``` paulson@13585 ` 338` ``` ==> f ` (INTER A B) = (INT x:A. f ` B x)" ``` paulson@13585 ` 339` ```apply (simp add: inj_on_def, blast) ``` paulson@13585 ` 340` ```done ``` paulson@13585 ` 341` paulson@13585 ` 342` ```(*Compare with image_INT: no use of inj_on, and if f is surjective then ``` paulson@13585 ` 343` ``` it doesn't matter whether A is empty*) ``` paulson@13585 ` 344` ```lemma bij_image_INT: "bij f ==> f ` (INTER A B) = (INT x:A. f ` B x)" ``` paulson@13585 ` 345` ```apply (simp add: bij_def) ``` paulson@13585 ` 346` ```apply (simp add: inj_on_def surj_def, blast) ``` paulson@13585 ` 347` ```done ``` paulson@13585 ` 348` paulson@13585 ` 349` ```lemma surj_Compl_image_subset: "surj f ==> -(f`A) <= f`(-A)" ``` paulson@13585 ` 350` ```by (auto simp add: surj_def) ``` paulson@13585 ` 351` paulson@13585 ` 352` ```lemma inj_image_Compl_subset: "inj f ==> f`(-A) <= -(f`A)" ``` paulson@13585 ` 353` ```by (auto simp add: inj_on_def) ``` paulson@5852 ` 354` paulson@13585 ` 355` ```lemma bij_image_Compl_eq: "bij f ==> f`(-A) = -(f`A)" ``` paulson@13585 ` 356` ```apply (simp add: bij_def) ``` paulson@13585 ` 357` ```apply (rule equalityI) ``` paulson@13585 ` 358` ```apply (simp_all (no_asm_simp) add: inj_image_Compl_subset surj_Compl_image_subset) ``` paulson@13585 ` 359` ```done ``` paulson@13585 ` 360` paulson@13585 ` 361` paulson@13585 ` 362` ```subsection{*Function Updating*} ``` paulson@13585 ` 363` haftmann@26147 ` 364` ```constdefs ``` haftmann@26147 ` 365` ``` fun_upd :: "('a => 'b) => 'a => 'b => ('a => 'b)" ``` haftmann@26147 ` 366` ``` "fun_upd f a b == % x. if x=a then b else f x" ``` haftmann@26147 ` 367` haftmann@26147 ` 368` ```nonterminals ``` haftmann@26147 ` 369` ``` updbinds updbind ``` haftmann@26147 ` 370` ```syntax ``` haftmann@26147 ` 371` ``` "_updbind" :: "['a, 'a] => updbind" ("(2_ :=/ _)") ``` haftmann@26147 ` 372` ``` "" :: "updbind => updbinds" ("_") ``` haftmann@26147 ` 373` ``` "_updbinds":: "[updbind, updbinds] => updbinds" ("_,/ _") ``` haftmann@26147 ` 374` ``` "_Update" :: "['a, updbinds] => 'a" ("_/'((_)')" [1000,0] 900) ``` haftmann@26147 ` 375` haftmann@26147 ` 376` ```translations ``` haftmann@26147 ` 377` ``` "_Update f (_updbinds b bs)" == "_Update (_Update f b) bs" ``` haftmann@26147 ` 378` ``` "f(x:=y)" == "fun_upd f x y" ``` haftmann@26147 ` 379` haftmann@26147 ` 380` ```(* Hint: to define the sum of two functions (or maps), use sum_case. ``` haftmann@26147 ` 381` ``` A nice infix syntax could be defined (in Datatype.thy or below) by ``` haftmann@26147 ` 382` ```consts ``` haftmann@26147 ` 383` ``` fun_sum :: "('a => 'c) => ('b => 'c) => (('a+'b) => 'c)" (infixr "'(+')"80) ``` haftmann@26147 ` 384` ```translations ``` haftmann@26147 ` 385` ``` "fun_sum" == sum_case ``` haftmann@26147 ` 386` ```*) ``` haftmann@26147 ` 387` paulson@13585 ` 388` ```lemma fun_upd_idem_iff: "(f(x:=y) = f) = (f x = y)" ``` paulson@13585 ` 389` ```apply (simp add: fun_upd_def, safe) ``` paulson@13585 ` 390` ```apply (erule subst) ``` paulson@13585 ` 391` ```apply (rule_tac [2] ext, auto) ``` paulson@13585 ` 392` ```done ``` paulson@13585 ` 393` paulson@13585 ` 394` ```(* f x = y ==> f(x:=y) = f *) ``` paulson@13585 ` 395` ```lemmas fun_upd_idem = fun_upd_idem_iff [THEN iffD2, standard] ``` paulson@13585 ` 396` paulson@13585 ` 397` ```(* f(x := f x) = f *) ``` paulson@17084 ` 398` ```lemmas fun_upd_triv = refl [THEN fun_upd_idem] ``` paulson@17084 ` 399` ```declare fun_upd_triv [iff] ``` paulson@13585 ` 400` paulson@13585 ` 401` ```lemma fun_upd_apply [simp]: "(f(x:=y))z = (if z=x then y else f z)" ``` paulson@17084 ` 402` ```by (simp add: fun_upd_def) ``` paulson@13585 ` 403` paulson@13585 ` 404` ```(* fun_upd_apply supersedes these two, but they are useful ``` paulson@13585 ` 405` ``` if fun_upd_apply is intentionally removed from the simpset *) ``` paulson@13585 ` 406` ```lemma fun_upd_same: "(f(x:=y)) x = y" ``` paulson@13585 ` 407` ```by simp ``` paulson@13585 ` 408` paulson@13585 ` 409` ```lemma fun_upd_other: "z~=x ==> (f(x:=y)) z = f z" ``` paulson@13585 ` 410` ```by simp ``` paulson@13585 ` 411` paulson@13585 ` 412` ```lemma fun_upd_upd [simp]: "f(x:=y,x:=z) = f(x:=z)" ``` paulson@13585 ` 413` ```by (simp add: expand_fun_eq) ``` paulson@13585 ` 414` paulson@13585 ` 415` ```lemma fun_upd_twist: "a ~= c ==> (m(a:=b))(c:=d) = (m(c:=d))(a:=b)" ``` paulson@13585 ` 416` ```by (rule ext, auto) ``` paulson@13585 ` 417` nipkow@15303 ` 418` ```lemma inj_on_fun_updI: "\ inj_on f A; y \ f`A \ \ inj_on (f(x:=y)) A" ``` nipkow@15303 ` 419` ```by(fastsimp simp:inj_on_def image_def) ``` nipkow@15303 ` 420` paulson@15510 ` 421` ```lemma fun_upd_image: ``` paulson@15510 ` 422` ``` "f(x:=y) ` A = (if x \ A then insert y (f ` (A-{x})) else f ` A)" ``` paulson@15510 ` 423` ```by auto ``` paulson@15510 ` 424` nipkow@31080 ` 425` ```lemma fun_upd_comp: "f \ (g(x := y)) = (f \ g)(x := f y)" ``` nipkow@31080 ` 426` ```by(auto intro: ext) ``` nipkow@31080 ` 427` haftmann@26147 ` 428` haftmann@26147 ` 429` ```subsection {* @{text override_on} *} ``` haftmann@26147 ` 430` haftmann@26147 ` 431` ```definition ``` haftmann@26147 ` 432` ``` override_on :: "('a \ 'b) \ ('a \ 'b) \ 'a set \ 'a \ 'b" ``` haftmann@26147 ` 433` ```where ``` haftmann@26147 ` 434` ``` "override_on f g A = (\a. if a \ A then g a else f a)" ``` nipkow@13910 ` 435` nipkow@15691 ` 436` ```lemma override_on_emptyset[simp]: "override_on f g {} = f" ``` nipkow@15691 ` 437` ```by(simp add:override_on_def) ``` nipkow@13910 ` 438` nipkow@15691 ` 439` ```lemma override_on_apply_notin[simp]: "a ~: A ==> (override_on f g A) a = f a" ``` nipkow@15691 ` 440` ```by(simp add:override_on_def) ``` nipkow@13910 ` 441` nipkow@15691 ` 442` ```lemma override_on_apply_in[simp]: "a : A ==> (override_on f g A) a = g a" ``` nipkow@15691 ` 443` ```by(simp add:override_on_def) ``` nipkow@13910 ` 444` haftmann@26147 ` 445` haftmann@26147 ` 446` ```subsection {* @{text swap} *} ``` paulson@15510 ` 447` haftmann@22744 ` 448` ```definition ``` haftmann@22744 ` 449` ``` swap :: "'a \ 'a \ ('a \ 'b) \ ('a \ 'b)" ``` haftmann@22744 ` 450` ```where ``` haftmann@22744 ` 451` ``` "swap a b f = f (a := f b, b:= f a)" ``` paulson@15510 ` 452` paulson@15510 ` 453` ```lemma swap_self: "swap a a f = f" ``` nipkow@15691 ` 454` ```by (simp add: swap_def) ``` paulson@15510 ` 455` paulson@15510 ` 456` ```lemma swap_commute: "swap a b f = swap b a f" ``` paulson@15510 ` 457` ```by (rule ext, simp add: fun_upd_def swap_def) ``` paulson@15510 ` 458` paulson@15510 ` 459` ```lemma swap_nilpotent [simp]: "swap a b (swap a b f) = f" ``` paulson@15510 ` 460` ```by (rule ext, simp add: fun_upd_def swap_def) ``` paulson@15510 ` 461` paulson@15510 ` 462` ```lemma inj_on_imp_inj_on_swap: ``` haftmann@22744 ` 463` ``` "[|inj_on f A; a \ A; b \ A|] ==> inj_on (swap a b f) A" ``` paulson@15510 ` 464` ```by (simp add: inj_on_def swap_def, blast) ``` paulson@15510 ` 465` paulson@15510 ` 466` ```lemma inj_on_swap_iff [simp]: ``` paulson@15510 ` 467` ``` assumes A: "a \ A" "b \ A" shows "inj_on (swap a b f) A = inj_on f A" ``` paulson@15510 ` 468` ```proof ``` paulson@15510 ` 469` ``` assume "inj_on (swap a b f) A" ``` paulson@15510 ` 470` ``` with A have "inj_on (swap a b (swap a b f)) A" ``` nipkow@17589 ` 471` ``` by (iprover intro: inj_on_imp_inj_on_swap) ``` paulson@15510 ` 472` ``` thus "inj_on f A" by simp ``` paulson@15510 ` 473` ```next ``` paulson@15510 ` 474` ``` assume "inj_on f A" ``` nipkow@27165 ` 475` ``` with A show "inj_on (swap a b f) A" by(iprover intro: inj_on_imp_inj_on_swap) ``` paulson@15510 ` 476` ```qed ``` paulson@15510 ` 477` paulson@15510 ` 478` ```lemma surj_imp_surj_swap: "surj f ==> surj (swap a b f)" ``` paulson@15510 ` 479` ```apply (simp add: surj_def swap_def, clarify) ``` wenzelm@27125 ` 480` ```apply (case_tac "y = f b", blast) ``` wenzelm@27125 ` 481` ```apply (case_tac "y = f a", auto) ``` paulson@15510 ` 482` ```done ``` paulson@15510 ` 483` paulson@15510 ` 484` ```lemma surj_swap_iff [simp]: "surj (swap a b f) = surj f" ``` paulson@15510 ` 485` ```proof ``` paulson@15510 ` 486` ``` assume "surj (swap a b f)" ``` paulson@15510 ` 487` ``` hence "surj (swap a b (swap a b f))" by (rule surj_imp_surj_swap) ``` paulson@15510 ` 488` ``` thus "surj f" by simp ``` paulson@15510 ` 489` ```next ``` paulson@15510 ` 490` ``` assume "surj f" ``` paulson@15510 ` 491` ``` thus "surj (swap a b f)" by (rule surj_imp_surj_swap) ``` paulson@15510 ` 492` ```qed ``` paulson@15510 ` 493` paulson@15510 ` 494` ```lemma bij_swap_iff: "bij (swap a b f) = bij f" ``` paulson@15510 ` 495` ```by (simp add: bij_def) ``` haftmann@21547 ` 496` nipkow@27188 ` 497` ```hide (open) const swap ``` haftmann@21547 ` 498` haftmann@22845 ` 499` ```subsection {* Proof tool setup *} ``` haftmann@22845 ` 500` haftmann@22845 ` 501` ```text {* simplifies terms of the form ``` haftmann@22845 ` 502` ``` f(...,x:=y,...,x:=z,...) to f(...,x:=z,...) *} ``` haftmann@22845 ` 503` wenzelm@24017 ` 504` ```simproc_setup fun_upd2 ("f(v := w, x := y)") = {* fn _ => ``` haftmann@22845 ` 505` ```let ``` haftmann@22845 ` 506` ``` fun gen_fun_upd NONE T _ _ = NONE ``` wenzelm@24017 ` 507` ``` | gen_fun_upd (SOME f) T x y = SOME (Const (@{const_name fun_upd}, T) \$ f \$ x \$ y) ``` haftmann@22845 ` 508` ``` fun dest_fun_T1 (Type (_, T :: Ts)) = T ``` haftmann@22845 ` 509` ``` fun find_double (t as Const (@{const_name fun_upd},T) \$ f \$ x \$ y) = ``` haftmann@22845 ` 510` ``` let ``` haftmann@22845 ` 511` ``` fun find (Const (@{const_name fun_upd},T) \$ g \$ v \$ w) = ``` haftmann@22845 ` 512` ``` if v aconv x then SOME g else gen_fun_upd (find g) T v w ``` haftmann@22845 ` 513` ``` | find t = NONE ``` haftmann@22845 ` 514` ``` in (dest_fun_T1 T, gen_fun_upd (find f) T x y) end ``` wenzelm@24017 ` 515` wenzelm@24017 ` 516` ``` fun proc ss ct = ``` wenzelm@24017 ` 517` ``` let ``` wenzelm@24017 ` 518` ``` val ctxt = Simplifier.the_context ss ``` wenzelm@24017 ` 519` ``` val t = Thm.term_of ct ``` wenzelm@24017 ` 520` ``` in ``` wenzelm@24017 ` 521` ``` case find_double t of ``` wenzelm@24017 ` 522` ``` (T, NONE) => NONE ``` wenzelm@24017 ` 523` ``` | (T, SOME rhs) => ``` wenzelm@27330 ` 524` ``` SOME (Goal.prove ctxt [] [] (Logic.mk_equals (t, rhs)) ``` wenzelm@24017 ` 525` ``` (fn _ => ``` wenzelm@24017 ` 526` ``` rtac eq_reflection 1 THEN ``` wenzelm@24017 ` 527` ``` rtac ext 1 THEN ``` wenzelm@24017 ` 528` ``` simp_tac (Simplifier.inherit_context ss @{simpset}) 1)) ``` wenzelm@24017 ` 529` ``` end ``` wenzelm@24017 ` 530` ```in proc end ``` haftmann@22845 ` 531` ```*} ``` haftmann@22845 ` 532` haftmann@22845 ` 533` haftmann@21870 ` 534` ```subsection {* Code generator setup *} ``` haftmann@21870 ` 535` berghofe@25886 ` 536` ```types_code ``` berghofe@25886 ` 537` ``` "fun" ("(_ ->/ _)") ``` berghofe@25886 ` 538` ```attach (term_of) {* ``` berghofe@25886 ` 539` ```fun term_of_fun_type _ aT _ bT _ = Free ("", aT --> bT); ``` berghofe@25886 ` 540` ```*} ``` berghofe@25886 ` 541` ```attach (test) {* ``` berghofe@25886 ` 542` ```fun gen_fun_type aF aT bG bT i = ``` berghofe@25886 ` 543` ``` let ``` berghofe@25886 ` 544` ``` val tab = ref []; ``` berghofe@25886 ` 545` ``` fun mk_upd (x, (_, y)) t = Const ("Fun.fun_upd", ``` berghofe@25886 ` 546` ``` (aT --> bT) --> aT --> bT --> aT --> bT) \$ t \$ aF x \$ y () ``` berghofe@25886 ` 547` ``` in ``` berghofe@25886 ` 548` ``` (fn x => ``` berghofe@25886 ` 549` ``` case AList.lookup op = (!tab) x of ``` berghofe@25886 ` 550` ``` NONE => ``` berghofe@25886 ` 551` ``` let val p as (y, _) = bG i ``` berghofe@25886 ` 552` ``` in (tab := (x, p) :: !tab; y) end ``` berghofe@25886 ` 553` ``` | SOME (y, _) => y, ``` berghofe@28711 ` 554` ``` fn () => Basics.fold mk_upd (!tab) (Const ("HOL.undefined", aT --> bT))) ``` berghofe@25886 ` 555` ``` end; ``` berghofe@25886 ` 556` ```*} ``` berghofe@25886 ` 557` haftmann@21870 ` 558` ```code_const "op \" ``` haftmann@21870 ` 559` ``` (SML infixl 5 "o") ``` haftmann@21870 ` 560` ``` (Haskell infixr 9 ".") ``` haftmann@21870 ` 561` haftmann@21906 ` 562` ```code_const "id" ``` haftmann@21906 ` 563` ``` (Haskell "id") ``` haftmann@21906 ` 564` nipkow@2912 ` 565` ```end ```