5250

1 
(* Title: HOL/ex/PiSets.thy


2 
ID: $Id$


3 
Author: Florian Kammueller, University of Cambridge


4 


5 
Pi sets and their application.


6 
*)


7 


8 
(* One abbreviation for my major simp application *)


9 
fun afs thms = (asm_full_simp_tac (simpset() addsimps thms));


10 
(* strip outer quantifiers and lift implication *)


11 
fun strip i = (REPEAT ((rtac ballI i)


12 
ORELSE (rtac allI i)


13 
ORELSE (rtac impI i)));


14 
(* eresolve but leave the eliminated assumptions (improves unification) *)


15 
goal HOL.thy "!! P. [ P ] ==> P & P";


16 
by (Fast_tac 1);


17 
val double = result();


18 


19 
fun re_tac rule r i = ((rotate_tac (r  1) i)


20 
THEN (dtac double i)


21 
THEN (rotate_tac ~1 i)


22 
THEN (etac conjE i)


23 
THEN (rotate_tac ~1 i)


24 
THEN (etac rule i));


25 


26 
(* individual theorems for convenient use *)


27 
val [p1,p2] = goal HOL.thy "[P == Q; P] ==> Q";


28 
by (fold_goals_tac [p1]);


29 
br p2 1;


30 
val apply_def = result();


31 


32 
goal HOL.thy "!! P x y. x = y ==> P(x) = P(y)";


33 
be ssubst 1;


34 
br refl 1;


35 
val extend = result();


36 


37 
val [p1] = goal HOL.thy "P ==> ~~P";


38 
br notI 1;


39 
br (p1 RSN(2, notE)) 1;


40 
ba 1;


41 
val notnotI = result();


42 


43 
val [p1] = goal Set.thy "? x. x: S ==> S ~= {}";


44 
br contrapos 1;


45 
br (p1 RS notnotI) 1;


46 
be ssubst 1;


47 
br notI 1;


48 
be exE 1;


49 
be emptyE 1;


50 
val ExEl_NotEmpty = result();


51 


52 


53 
val [p1] = goal HOL.thy "~x ==> x = False";


54 
val l1 = (p1 RS (not_def RS apply_def)) RS mp;


55 
val l2 = read_instantiate [("P","x")] FalseE;


56 
br iffI 1;


57 
br l1 1;


58 
br l2 2;


59 
ba 1;


60 
ba 1;


61 
val NoteqFalseEq = result();


62 


63 
val [p1] = goal HOL.thy "~ (! x. ~P(x)) ==> ? x. P(x)";


64 
br exCI 1;


65 
(* 1. ! x. ~ P x ==> P ?a *)


66 
val l1 = p1 RS NoteqFalseEq;


67 
(* l1 = (! x. ~ P x) = False *)


68 
val l2 = l1 RS iffD1;


69 
val l3 = l1 RS iffD2;


70 
val l4 = read_instantiate [("P", "P ?a")] FalseE;


71 
br (l2 RS l4) 1;


72 
ba 1;


73 
val NotAllNot_Ex = result();


74 


75 
val [p1] = goal HOL.thy "~(? x. P(x)) ==> ! x. ~ P(x)";


76 
br notnotD 1;


77 
br (p1 RS contrapos) 1;


78 
br NotAllNot_Ex 1;


79 
ba 1;


80 
val NotEx_AllNot = result();


81 


82 
goal Set.thy "!!S. ~ (? x. x : S) ==> S = {}";


83 
by (Fast_tac 1);


84 
val NoEl_Empty = result();


85 


86 
goal Set.thy "!!S. S ~= {} ==> ? x. x : S";


87 
by (Fast_tac 1);


88 
val NotEmpty_ExEl = result();


89 


90 
goal PiSets.thy "!!S. S = {} ==> ! x. x ~: S";


91 
by (Fast_tac 1);


92 
val Empty_NoElem = result();


93 


94 


95 
val [q1,q2] = goal HOL.thy "[ b = a ; (P a) ] ==> (P b)";


96 
br (q1 RS ssubst) 1;


97 
br q2 1;


98 
val forw_subst = result();


99 


100 
val [q1,q2] = goal HOL.thy "[ a = b ; (P a) ] ==> (P b)";


101 
br (q1 RS subst) 1;


102 
br q2 1;


103 
val forw_ssubst = result();


104 


105 
goal Prod.thy "((fst A),(fst(snd A)),(fst (snd (snd A))),(snd(snd(snd A)))) = A";


106 
br (surjective_pairing RS subst) 1;


107 
br (surjective_pairing RS subst) 1;


108 
br (surjective_pairing RS subst) 1;


109 
br refl 1;


110 
val blow4 = result();


111 


112 
goal Prod.thy "!! P a b. (%(a,b). P a b) A ==> P (fst A)(snd A)";


113 
by (Step_tac 1);


114 
by (afs [fst_conv,snd_conv] 1);


115 
val apply_pair = result();


116 


117 
goal Prod.thy "!! P a b c d. (%(a,b,c,d). P a b c d) A \


118 
\ ==> P (fst A)(fst(snd A))(fst (snd (snd A)))(snd(snd(snd A)))";


119 
bd (blow4 RS forw_subst) 1;


120 
by (afs [split_def] 1);


121 
val apply_quadr = result();


122 


123 
goal Prod.thy "!! A B x. x: A Times B ==> x = (fst x, snd x)";


124 
br (surjective_pairing RS subst) 1;


125 
br refl 1;


126 
val surj_pair_forw = result();


127 


128 
goal Prod.thy "!! A B x. x: A Times B ==> fst x: A";


129 
by (forward_tac [surj_pair_forw] 1);


130 
bd forw_ssubst 1;


131 
ba 1;


132 
be SigmaD1 1;


133 
val TimesE1 = result();


134 


135 
goal Prod.thy "!! A B x. x: A Times B ==> snd x: B";


136 
by (forward_tac [surj_pair_forw] 1);


137 
bd forw_ssubst 1;


138 
ba 1;


139 
be SigmaD2 1;


140 
val TimesE2 = result();


141 


142 
(* > and Pi *)


143 


144 
goal PiSets.thy "!! A B. A > B == {f. ! x. if x: A then f(x) : B else f(x) = (@ y. True)}";


145 
by (simp_tac (simpset() addsimps [Pi_def]) 1);


146 
val funcset_def = result();


147 


148 


149 
val [q1,q2] = goal PiSets.thy


150 
"[ !!x. x: A ==> f x: B x; !!x. x ~: A ==> f(x) = (@ y. True)] ==> f: Pi A B";


151 
by (rewrite_goals_tac [Pi_def]);


152 
br CollectI 1;


153 
br allI 1;


154 
by (case_tac "x : A" 1);


155 
br (if_P RS ssubst) 1;


156 
ba 1;


157 
be q1 1;


158 
br (if_not_P RS ssubst) 1;


159 
ba 1;


160 
be q2 1;


161 
val Pi_I = result();


162 


163 
goal PiSets.thy


164 
"!! A f. [ !!x. x: A ==> f x: B; !!x. x ~: A ==> f(x) = (@ y. True)] ==> f: A > B";


165 
by (afs [Pi_I] 1);


166 
val funcsetI = result();


167 


168 
val [q1,q2,q3] = goal PiSets.thy


169 
"[ !! x y. [ x: A; y: B ] ==> f x y: C; \


170 
\ !! x. [ x ~: A ] ==> f x = (@ y. True);\


171 
\ !! x y. [ x : A; y ~: B ] ==> f x y = (@ y. True) ] ==> f: A > B > C";


172 
by (simp_tac (simpset() addsimps [q1,q2,q3,funcsetI]) 1);


173 
val funcsetI2 = result();


174 


175 
goal PiSets.thy "!! f A B. [f: A > B; x: A] ==> f x: B";


176 
by (afs [funcset_def] 1);


177 
val funcsetE1 = result();


178 


179 
goal PiSets.thy "!! f A B. [f: Pi A B; x: A] ==> f x: B x";


180 
by (afs [Pi_def] 1);


181 
val PiE1 = result();


182 


183 
goal PiSets.thy "!! f A B. [f: A > B; x~: A] ==> f x = (@ y. True)";


184 
by (afs [funcset_def] 1);


185 
val funcsetE2 = result();


186 


187 
goal PiSets.thy "!! f A B. [f: Pi A B; x~: A] ==> f x = (@ y. True)";


188 
by (afs [Pi_def] 1);


189 
val PiE2 = result();


190 


191 
goal PiSets.thy "!! f A B. [f: A > B > C; x : A; y ~: B] ==> f x y = (@ y. True)";


192 
by (afs [funcset_def] 1);


193 
val funcset2E2 = result();


194 


195 


196 
goal PiSets.thy "!! f A B C. [ f: A > B > C; x: A; y: B ] ==> f x y: C";


197 
by (afs [funcset_def] 1);


198 
val funcset2E1 = result();


199 


200 
goal PiSets.thy "!! f g A B. [ f: A > B; g: A > B; ! x: A. f x = g x ]\


201 
\ ==> f = g";


202 
br ext 1;


203 
by (case_tac "x : A" 1);


204 
by (Fast_tac 1);


205 
by (fast_tac (claset() addSDs [funcsetE2] addEs [ssubst]) 1);


206 
val function_extensionality = result();


207 


208 
goal PiSets.thy "!! f g A B. [ f: Pi A B; g: Pi A B; ! x: A. f x = g x ]\


209 
\ ==> f = g";


210 
br ext 1;


211 
by (case_tac "x : A" 1);


212 
by (Fast_tac 1);


213 
by (fast_tac (claset() addSDs [PiE2] addEs [ssubst]) 1);


214 
val Pi_extensionality = result();


215 


216 
(* compose *)


217 
goal PiSets.thy "!! A B C f g. [ f: A > B; g: B > C ]==> compose A g f: A > C";


218 
br funcsetI 1;


219 
by (rewrite_goals_tac [compose_def,restrict_def]);


220 
by (afs [funcsetE1] 1);


221 
br (if_not_P RS ssubst) 1;


222 
ba 1;


223 
br refl 1;


224 
val funcset_compose = result();


225 


226 
goal PiSets.thy "!! A B C f g h. [ f: A > B; g: B > C; h: C > D ]\


227 
\ ==> compose A h (compose A g f) = compose A (compose B h g) f";


228 
by (res_inst_tac [("A","A")] function_extensionality 1);


229 
br funcset_compose 1;


230 
br funcset_compose 1;


231 
ba 1;


232 
ba 1;


233 
ba 1;


234 
br funcset_compose 1;


235 
ba 1;


236 
br funcset_compose 1;


237 
ba 1;


238 
ba 1;


239 
br ballI 1;


240 
by (rewrite_goals_tac [compose_def,restrict_def]);


241 
by (afs [funcsetE1,if_P RS ssubst] 1);


242 
val compose_assoc = result();


243 


244 
goal PiSets.thy "!! A B C f g x. [ f: A > B; g: B > C; x: A ]==> compose A g f x = g(f(x))";


245 
by (afs [compose_def, restrict_def, if_P RS ssubst] 1);


246 
val composeE1 = result();


247 


248 
goal PiSets.thy "!! A B C g f.[ f : A > B; f `` A = B; g: B > C; g `` B = C ]\


249 
\ ==> compose A g f `` A = C";


250 
br equalityI 1;


251 
br subsetI 1;


252 
be imageE 1;


253 
by (rotate_tac 4 1);


254 
be ssubst 1;


255 
br (funcset_compose RS funcsetE1) 1;


256 
ba 1;


257 
ba 1;


258 
ba 1;


259 
br subsetI 1;


260 
by (hyp_subst_tac 1);


261 
be imageE 1;


262 
by (rotate_tac 3 1);


263 
be ssubst 1;


264 
be imageE 1;


265 
by (rotate_tac 3 1);


266 
be ssubst 1;


267 
be (composeE1 RS subst) 1;


268 
ba 1;


269 
ba 1;


270 
br imageI 1;


271 
ba 1;


272 
val surj_compose = result();


273 


274 


275 
goal PiSets.thy "!! A B C g f.[ f : A > B; g: B > C; f `` A = B; inj_on f A; inj_on g B ]\


276 
\ ==> inj_on (compose A g f) A";


277 
br inj_onI 1;


278 
by (forward_tac [composeE1] 1);


279 
ba 1;


280 
ba 1;


281 
by (forward_tac [composeE1] 1);


282 
ba 1;


283 
by (rotate_tac 7 1);


284 
ba 1;


285 
by (step_tac (claset() addSEs [inj_onD]) 1);


286 
by (rotate_tac 5 1);


287 
be subst 1;


288 
be subst 1;


289 
ba 1;


290 
be imageI 1;


291 
be imageI 1;


292 
val inj_on_compose = result();


293 


294 


295 
(* restrict / lam *)


296 
goal PiSets.thy "!! f A B. [ f `` A <= B ] ==> (lam x: A. f x) : A > B";


297 
by (rewrite_goals_tac [restrict_def]);


298 
br funcsetI 1;


299 
by (afs [if_P RS ssubst] 1);


300 
be subsetD 1;


301 
be imageI 1;


302 
by (afs [if_not_P RS ssubst] 1);


303 
val restrict_in_funcset = result();


304 


305 
goal PiSets.thy "!! f A B. [ ! x: A. f x: B ] ==> (lam x: A. f x) : A > B";


306 
br restrict_in_funcset 1;


307 
by (afs [image_def] 1);


308 
by (Step_tac 1);


309 
by (Fast_tac 1);


310 
val restrictI = result();


311 


312 
goal PiSets.thy "!! f A B. [ ! x: A. f x: B x ] ==> (lam x: A. f x) : Pi A B";


313 
by (rewrite_goals_tac [restrict_def]);


314 
br Pi_I 1;


315 
by (afs [if_P RS ssubst] 1);


316 
by (Asm_full_simp_tac 1);


317 
val restrictI_Pi = result();


318 


319 
(* The following proof has to be redone *)


320 
goal PiSets.thy "!! f A B C.[ f `` A <= B > C ] ==> (lam x: A. lam y: B. f x y) : A > B > C";


321 
br restrict_in_funcset 1;


322 
by (afs [image_def] 1);


323 
by (afs [Pi_def,subset_def] 1);


324 
by (afs [restrict_def] 1);


325 
by (Step_tac 1);


326 
by (Asm_full_simp_tac 1);


327 
by (dres_inst_tac [("x","f xa")] spec 1);


328 
bd mp 1;


329 
br bexI 1;


330 
br refl 1;


331 
ba 1;


332 
by (dres_inst_tac [("x","xb")] spec 1);


333 
by (Asm_full_simp_tac 1);


334 
(* fini 1 *)


335 
by (Asm_full_simp_tac 1);


336 
val restrict_in_funcset2 = result();


337 


338 
goal PiSets.thy "!! f A B C.[ !x: A. ! y: B. f x y: C ] ==> (lam x: A. lam y: B. f x y) : A > B > C";


339 
br restrict_in_funcset 1;


340 
by (afs [image_def] 1);


341 
by (afs [Pi_def,subset_def] 1);


342 
by (afs [restrict_def] 1);


343 
by (Step_tac 1);


344 
by (Asm_full_simp_tac 1);


345 
by (Asm_full_simp_tac 1);


346 
val restrictI2 = result();


347 


348 


349 
(* goal PiSets.thy "!! f A B. [ f `` A <= UNION A B ] ==> (lam x: A. f x) : Pi A B"; *)


350 


351 
goal PiSets.thy "!! f A B. [ x: A ] ==> (lam y: A. f y) x = f x";


352 
by (afs [restrict_def] 1);


353 
val restrictE1 = result();


354 


355 
goal PiSets.thy "!! f A B. [ x: A; f : A > B ] ==> (lam y: A. f y) x : B";


356 
by (afs [restrictE1,funcsetE1] 1);


357 
val restrictE1a = result();


358 


359 
goal PiSets.thy "!! f A B. [ x ~: A ] ==> (lam y: A. f y) x = (@ y. True)";


360 
by (afs [restrict_def] 1);


361 
val restrictE2 = result();


362 


363 
(* It would be nice to have this, but this doesn't work unfortunately


364 
see PiSets.ML: Pi_subset1


365 
goal PiSets.thy "!! A B. [ A <= B ; ! x: A. f x : C] ==> (lam x: B. f(x)): A > C"; *)


366 


367 
goal PiSets.thy "!! f A B x y. [ x: A; y: B ] ==> \


368 
\ (lam a: A. lam b: B. f a b) x y = f x y";


369 
by (afs [restrictE1] 1);


370 
val restrict2E1 = result();


371 


372 
(* New restrict2E1: *)


373 
goal PiSets.thy "!! A B. [ x : A; y : B x] ==> (lam a:A. lam b: (B a). f a b) x y = f x y" ;


374 
by (afs [restrictE1] 1);


375 
val restrict2E1a = result();


376 


377 
goal PiSets.thy "!! f A B x y. [ x: A; y: B; z: C ] ==> \


378 
\ (lam a: A. lam b: B. lam c: C. f a b c) x y z = f x y z";


379 
by (afs [restrictE1] 1);


380 
val restrict3E1 = result();


381 


382 
goal PiSets.thy "!! f A B x y. [ x: A; y ~: B ] ==> \


383 
\ (lam a: A. lam b: B. f a b) x y = (@ y. True)";


384 
by (afs [restrictE1,restrictE2] 1);


385 
val restrict2E2 = result();


386 


387 


388 
goal PiSets.thy "!! f g A B. [ ! x: A. f x = g x ]\


389 
\ ==> (lam x: A. f x) = (lam x: A. g x)";


390 
br ext 1;


391 
by (case_tac "x: A" 1);


392 
by (afs [restrictE1] 1);


393 
by (afs [restrictE2] 1);


394 
val restrict_ext = result();


395 


396 
(* Invers *)


397 


398 
goal PiSets.thy "!! f A B.[f `` A = B; x: B ] ==> ? y: A. f y = x";


399 
by (rewrite_goals_tac [image_def]);


400 
bd equalityD2 1;


401 
bd subsetD 1;


402 
ba 1;


403 
bd CollectD 1;


404 
be bexE 1;


405 
bd sym 1;


406 
be bexI 1;


407 
ba 1;


408 
val surj_image = result();


409 


410 
val [q1,q2] = goal PiSets.thy "[ f `` A = B; f : A > B ] \


411 
\ ==> (lam x: B. (Inv A f) x) : B > A";


412 
br restrict_in_funcset 1;


413 
by (rewrite_goals_tac [image_def]);


414 
br subsetI 1;


415 
bd CollectD 1;


416 
be bexE 1;


417 
be ssubst 1;


418 
bd (q1 RS surj_image) 1;


419 
be bexE 1;


420 
be subst 1;


421 
by (rewrite_goals_tac [Inv_def]);


422 
by (res_inst_tac [("Q","f(@ ya. ya : A & f ya = f y) = f y")] conjunct1 1);


423 
br (q1 RS surj_image RS (Bex_def RS apply_def) RS (Ex_def RS apply_def)) 1;


424 
be (q2 RS funcsetE1) 1;


425 
val Inv_funcset = result();


426 


427 


428 
val [q1,q2,q3] = goal PiSets.thy "[ f: A > B; inj_on f A; f `` A = B ]\


429 
\ ==> ! x: A. (lam y: B. (Inv A f) y)(f x) = x";


430 
br ballI 1;


431 
br (restrictE1 RS ssubst) 1;


432 
be (q1 RS funcsetE1) 1;


433 
by (rewrite_goals_tac [Inv_def]);


434 
br (q2 RS inj_onD) 1;


435 
ba 3;


436 
by (res_inst_tac [("P","(@ y. y : A & f y = f x) : A")] conjunct2 1);


437 
br (q3 RS surj_image RS (Bex_def RS apply_def) RS (Ex_def RS apply_def)) 1;


438 
be (q1 RS funcsetE1) 1;


439 
by (res_inst_tac [("Q","f (@ y. y : A & f y = f x) = f x")] conjunct1 1);


440 
br (q3 RS surj_image RS (Bex_def RS apply_def) RS (Ex_def RS apply_def)) 1;


441 
be (q1 RS funcsetE1) 1;


442 
val Inv_f_f = result();


443 


444 
val [q1,q2] = goal PiSets.thy "[ f: A > B; f `` A = B ]\


445 
\ ==> ! x: B. f ((lam y: B. (Inv A f y)) x) = x";


446 
br ballI 1;


447 
br (restrictE1 RS ssubst) 1;


448 
ba 1;


449 
by (rewrite_goals_tac [Inv_def]);


450 
by (res_inst_tac [("P","(@ y. y : A & f y = x) : A")] conjunct2 1);


451 
br (q2 RS surj_image RS (Bex_def RS apply_def) RS (Ex_def RS apply_def)) 1;


452 
ba 1;


453 
val f_Inv_f = result();


454 


455 
val [q1,q2,q3] = goal PiSets.thy "[ f: A > B; inj_on f A; f `` A = B ]\


456 
\ ==> compose A (lam y:B. (Inv A f) y) f = (lam x: A. x)";


457 
br function_extensionality 1;


458 
br funcset_compose 1;


459 
br q1 1;


460 
br (q1 RS (q3 RS Inv_funcset)) 1;


461 
br restrict_in_funcset 1;


462 
by (Fast_tac 1);


463 
br ballI 1;


464 
by (afs [compose_def] 1);


465 
br (restrictE1 RS ssubst) 1;


466 
ba 1;


467 
br (restrictE1 RS ssubst) 1;


468 
ba 1;


469 
be (q3 RS (q2 RS (q1 RS Inv_f_f)) RS bspec) 1;


470 
val comp_Inv_id = result();


471 


472 


473 
(* Pi and its application @@ *)


474 


475 
goal PiSets.thy "!! A B. (PI x: A. B x) ~= {} ==> ! x: A. B(x) ~= {}";


476 
bd NotEmpty_ExEl 1;


477 
be exE 1;


478 
by (rewrite_goals_tac [Pi_def]);


479 
bd CollectD 1;


480 
br ballI 1;


481 
br ExEl_NotEmpty 1;


482 
by (res_inst_tac [("x","x xa")] exI 1);


483 
by (afs [if_P RS subst] 1);


484 
val Pi_total1 = result();


485 


486 
goal Set.thy "!! M P. ? x: M . P x ==> (~ (! x: M. ~ P x))";


487 
by (Fast_tac 1);


488 
val SetEx_NotNotAll = result();


489 


490 
goal PiSets.thy "!! A B. ? x: A. B(x) = {} ==> (PI x: A. B x) = {}";


491 
br notnotD 1;


492 
br (Pi_total1 RSN(2,contrapos)) 1;


493 
ba 2;


494 
be SetEx_NotNotAll 1;


495 
val Pi_total2 = result();


496 


497 
val [q1,q2] = goal PiSets.thy "[a : A; Pi A B ~= {} ] ==> (Pi A B) @@ a = B(a)";


498 
by (rewrite_goals_tac [Fset_apply_def]);


499 
br equalityI 1;


500 
br subsetI 1;


501 
be imageE 1;


502 
be ssubst 1;


503 
by (rewrite_goals_tac [Pi_def]);


504 
bd CollectD 1;


505 
bd spec 1;


506 
br (q1 RS if_P RS subst) 1;


507 
ba 1;


508 
br subsetI 1;


509 
by (rewrite_goals_tac [image_def]);


510 
br CollectI 1;


511 
br exE 1;


512 
br (q2 RS NotEmpty_ExEl) 1;


513 
by (res_inst_tac [("x","%y. if (y = a) then x else xa y")] bexI 1);


514 
by (Simp_tac 1);


515 
by (Simp_tac 1);


516 
br allI 1;


517 
by (case_tac "xb: A" 1);


518 
by (afs [if_P RS ssubst] 1);


519 
by (case_tac "xb = a" 1);


520 
by (afs [if_P RS ssubst] 1);


521 
by (afs [if_not_P RS ssubst] 1);


522 
by (rewrite_goals_tac [Pi_def]);


523 
by (afs [if_P RS ssubst] 1);


524 
by (afs [if_not_P RS ssubst] 1);


525 
by (case_tac "xb = a" 1);


526 
by (afs [if_P RS ssubst] 1);


527 
by (hyp_subst_tac 1);


528 
by (afs [q1] 1);


529 
by (afs [if_not_P RS ssubst] 1);


530 
val Pi_app_def = result();


531 


532 
goal PiSets.thy "!! a A B C. [ a: A; (PI x: A. PI y: B x. C x y) ~= {} ] ==> (PI y: B a. C a y) ~= {}";


533 
bd NotEmpty_ExEl 1;


534 
be exE 1;


535 
by (rewrite_goals_tac [Pi_def]);


536 
bd CollectD 1;


537 
bd spec 1;


538 
br ExEl_NotEmpty 1;


539 
br exI 1;


540 
be (if_P RS eq_reflection RS apply_def) 1;


541 
ba 1;


542 
val NotEmptyPiStep = result();


543 


544 
val [q1,q2,q3] = goal PiSets.thy


545 
"[a : A; b: B a; (PI x: A. PI y: B x. C x y) ~= {} ] ==> (PI x: A. PI y: B x. C x y) @@ a @@ b = C a b";


546 
by (fold_goals_tac [q3 RS (q1 RS NotEmptyPiStep) RS (q2 RS Pi_app_def) RS eq_reflection]);


547 
by (fold_goals_tac [q3 RS (q1 RS Pi_app_def) RS eq_reflection]);


548 
br refl 1;


549 
val Pi_app2_def = result();


550 


551 
(* Sigma does a better job ( the following is from PiSig.ML) *)


552 
goal PiSets.thy "!! A b a. [ a: A; Pi A B ~= {} ]\


553 
\ ==> Sigma A B ^^ {a} = Pi A B @@ a";


554 
br (Pi_app_def RS ssubst) 1;


555 
ba 1;


556 
ba 1;


557 
by (afs [Sigma_def,Domain_def,converse_def,Range_def,Image_def] 1);


558 
by (rewrite_goals_tac [Bex_def]);


559 
by (Fast_tac 1);


560 
val PiSig_image_eq = result();


561 


562 
goal PiSets.thy "!! A b a. [ a: A ]\


563 
\ ==> Sigma A B ^^ {a} = B a";


564 
by (Fast_tac 1);


565 
val Sigma_app_def = result();


566 


567 
(* Bijection between Pi in terms of => and Pi in terms of Sigma *)


568 
goal PiSets.thy "!! f. f: Pi A B ==> PiBij A B f <= Sigma A B";


569 
by (afs [PiBij_def,Pi_def,restrictE1] 1);


570 
br subsetI 1;


571 
by (split_all_tac 1);


572 
bd CollectD 1;


573 
by (Asm_full_simp_tac 1);


574 
val PiBij_subset_Sigma = result();


575 


576 
goal PiSets.thy "!! f. f: Pi A B ==> (! x: A. (?! y. (x, y): (PiBij A B f)))";


577 
by (afs [PiBij_def,restrictE1] 1);


578 
br ballI 1;


579 
br ex1I 1;


580 
ba 2;


581 
br refl 1;


582 
val PiBij_unique = result();


583 


584 
goal PiSets.thy "!! f. f: Pi A B ==> (! x: A. (?! y. y: B x & (x, y): (PiBij A B f)))";


585 
by (afs [PiBij_def,restrictE1] 1);


586 
br ballI 1;


587 
br ex1I 1;


588 
be conjunct2 2;


589 
by (afs [PiE1] 1);


590 
val PiBij_unique2 = result();


591 


592 
goal PiSets.thy "!! f. f: Pi A B ==> PiBij A B f : Graph A B";


593 
by (afs [Graph_def,PiBij_unique,PiBij_subset_Sigma] 1);


594 
val PiBij_in_Graph = result();


595 


596 
goal PiSets.thy "!! A B. PiBij A B: Pi A B > Graph A B";


597 
by (afs [PiBij_def] 1);


598 
br restrictI 1;


599 
by (strip 1);


600 
by (afs [Graph_def] 1);


601 
br conjI 1;


602 
br subsetI 1;


603 
by (strip 2);


604 
br ex1I 2;


605 
br refl 2;


606 
ba 2;


607 
by (split_all_tac 1);


608 
by (afs [Pi_def] 1);


609 
val PiBij_func = result();


610 


611 
goal PiSets.thy "!! A f g x. [ f: Pi A B; g: Pi A B; \


612 
\ {(x, y). x: A & y = f x} = {(x, y). x: A & y = g x}; x: A ]\


613 
\ ==> f x = g x";


614 
be equalityE 1;


615 
by (rewrite_goals_tac [subset_def]);


616 
by (dres_inst_tac [("x","(x, f x)")] bspec 1);


617 
by (Fast_tac 1);


618 
by (Fast_tac 1);


619 
val set_eq_lemma = result();


620 


621 
goal PiSets.thy "!! A B. inj_on (PiBij A B) (Pi A B)";


622 
br inj_onI 1;


623 
br Pi_extensionality 1;


624 
ba 1;


625 
ba 1;


626 
by (strip 1);


627 
by (afs [PiBij_def,restrictE1] 1);


628 
by (re_tac set_eq_lemma 2 1);


629 
ba 1;


630 
ba 2;


631 
by (afs [restrictE1] 1);


632 
be subst 1;


633 
by (afs [restrictE1] 1);


634 
val inj_PiBij = result();


635 


636 
goal HOL.thy "!! P . ?! x. P x ==> ? x. P x";


637 
by (Blast_tac 1);


638 
val Ex1_Ex = result();


639 


640 
goal PiSets.thy "!!A B. PiBij A B `` (Pi A B) = Graph A B";


641 
br equalityI 1;


642 
by (afs [image_def] 1);


643 
br subsetI 1;


644 
by (Asm_full_simp_tac 1);


645 
be bexE 1;


646 
be ssubst 1;


647 
by (afs [PiBij_in_Graph] 1);


648 
br subsetI 1;


649 
by (afs [image_def] 1);


650 
by (res_inst_tac [("x","lam a: A. @ y. (a, y): x")] bexI 1);


651 
br restrictI_Pi 2;


652 
by (strip 2);


653 
br ex1E 2;


654 
by (afs [Graph_def] 2);


655 
be conjE 2;


656 
bd bspec 2;


657 
ba 2;


658 
ba 2;


659 
br (select_equality RS ssubst) 2;


660 
ba 2;


661 
by (Blast_tac 2);


662 
(* gets hung up on by (afs [Graph_def] 2); *)


663 
by (SELECT_GOAL (rewrite_goals_tac [Graph_def]) 2);


664 
by (Blast_tac 2);


665 
(* x = PiBij A B (lam a:A. @ y. (a, y) : x) *)


666 
by (afs [PiBij_def,Graph_def] 1);


667 
br (restrictE1 RS ssubst) 1;


668 
br restrictI_Pi 1;


669 
(* again like the old 2. subgoal *)


670 
by (strip 1);


671 
br ex1E 1;


672 
be conjE 1;


673 
bd bspec 1;


674 
ba 1;


675 
ba 1;


676 
br (select_equality RS ssubst) 1;


677 
ba 1;


678 
by (Blast_tac 1);


679 
by (Blast_tac 1);


680 
(* *)


681 
br equalityI 1;


682 
br subsetI 1;


683 
br CollectI 1;


684 
by (split_all_tac 1);


685 
by (Simp_tac 1);


686 
br conjI 1;


687 
by (Blast_tac 1);


688 
be conjE 1;


689 
bd subsetD 1;


690 
ba 1;


691 
bd SigmaD1 1;


692 
bd bspec 1;


693 
ba 1;


694 
br (restrictE1 RS ssubst) 1;


695 
ba 1;


696 
br sym 1;


697 
br select_equality 1;


698 
ba 1;


699 
by (Blast_tac 1);


700 
(* {(xa,y). xa : A & y = (lam a:A. @ y. (a, y) : x) xa} <= x *)


701 
br subsetI 1;


702 
by (Asm_full_simp_tac 1);


703 
by (split_all_tac 1);


704 
by (Asm_full_simp_tac 1);


705 
be conjE 1;


706 
be conjE 1;


707 
by (afs [restrictE1] 1);


708 
bd bspec 1;


709 
ba 1;


710 
bd Ex1_Ex 1;


711 
by (rewrite_goals_tac [Ex_def]);


712 
ba 1;


713 
val surj_PiBij = result();


714 


715 


716 
goal PiSets.thy "!! A B. [ f: Pi A B ] ==> \


717 
\ (lam y: Graph A B. (Inv (Pi A B)(PiBij A B)) y)(PiBij A B f) = f";


718 
br (Inv_f_f RS bspec) 1;


719 
ba 4;


720 
by (afs [PiBij_func] 1);


721 
by (afs [inj_PiBij] 1);


722 
by (afs [surj_PiBij] 1);


723 
val PiBij_bij1 = result();


724 


725 
goal PiSets.thy "!! A B. [ f: Graph A B ] ==> \


726 
\ (PiBij A B) ((lam y: (Graph A B). (Inv (Pi A B)(PiBij A B)) y) f) = f";


727 
br (PiBij_func RS (f_Inv_f RS bspec)) 1;


728 
by (afs [surj_PiBij] 1);


729 
ba 1;


730 
val PiBij_bij2 = result();


731 


732 
goal PiSets.thy "!! g f. [ ! x. g( f x) = x ] ==> inj f";


733 
br injI 1;


734 
by (dres_inst_tac [("f","g")] arg_cong 1);


735 
by (forw_inst_tac [("x","x")] spec 1);


736 
by (rotate_tac 2 1);


737 
be subst 1;


738 
by (forw_inst_tac [("x","y")] spec 1);


739 
by (rotate_tac 2 1);


740 
be subst 1;


741 
ba 1;


742 
val inj_lemma = result();


743 


744 
goal PiSets.thy "!! g f. [ ! x. g( f x) = x ] ==> surj g";


745 
by (afs [surj_def] 1);


746 
br allI 1;


747 
by (res_inst_tac [("x","f y")] exI 1);


748 
bd spec 1;


749 
be sym 1;


750 
val surj_lemma = result();


751 


752 
goal PiSets.thy "Pi {} B == {f. !x. f x = (@ y. True)}";


753 
by (afs [Pi_def] 1);


754 
val empty_Pi = result();


755 


756 
goal PiSets.thy "(% x. (@ y. True)) : Pi {} B";


757 
by (afs [empty_Pi] 1);


758 
val empty_Pi_func = result();


759 


760 
goal Set.thy "!! A B. [ A <= B; x ~: B ] ==> x ~: A";


761 
by (Blast_tac 1);


762 
val subsetND = result();


763 


764 


765 
goal PiSets.thy "!! A B C . [ ! x: A. B x <= C x ] ==> Pi A B <= Pi A C";


766 
br subsetI 1;


767 
br Pi_I 1;


768 
by (afs [Pi_def] 2);


769 
bd bspec 1;


770 
ba 1;


771 
be subsetD 1;


772 
by (afs [PiE1] 1);


773 
val Pi_subset1 = result();
