author  paulson 
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child 15422  cbdddc0efadf 
permissions  rwrr 
10213  1 
(* Title: HOL/Product_Type.thy 
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ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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Copyright 1992 University of Cambridge 

11777  5 
*) 
10213  6 

11838  7 
header {* Cartesian products *} 
10213  8 

15131  9 
theory Product_Type 
15140  10 
imports Fun 
15131  11 
files ("Tools/split_rule.ML") 
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begin 

11838  13 

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subsection {* Unit *} 

15 

16 
typedef unit = "{True}" 

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proof 

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show "True : ?unit" by blast 

19 
qed 

20 

21 
constdefs 

22 
Unity :: unit ("'(')") 

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"() == Abs_unit True" 

24 

25 
lemma unit_eq: "u = ()" 

26 
by (induct u) (simp add: unit_def Unity_def) 

27 

28 
text {* 

29 
Simplification procedure for @{thm [source] unit_eq}. Cannot use 

30 
this rule directly  it loops! 

31 
*} 

32 

33 
ML_setup {* 

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val unit_eq_proc = 
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let val unit_meta_eq = mk_meta_eq (thm "unit_eq") in 

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Simplifier.simproc (Theory.sign_of (the_context ())) "unit_eq" ["x::unit"] 

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(fn _ => fn _ => fn t => if HOLogic.is_unit t then None else Some unit_meta_eq) 

38 
end; 

11838  39 

40 
Addsimprocs [unit_eq_proc]; 

41 
*} 

42 

43 
lemma unit_all_eq1: "(!!x::unit. PROP P x) == PROP P ()" 

44 
by simp 

45 

46 
lemma unit_all_eq2: "(!!x::unit. PROP P) == PROP P" 

47 
by (rule triv_forall_equality) 

48 

49 
lemma unit_induct [induct type: unit]: "P () ==> P x" 

50 
by simp 

51 

52 
text {* 

53 
This rewrite counters the effect of @{text unit_eq_proc} on @{term 

54 
[source] "%u::unit. f u"}, replacing it by @{term [source] 

55 
f} rather than by @{term [source] "%u. f ()"}. 

56 
*} 

57 

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lemma unit_abs_eta_conv [simp]: "(%u::unit. f ()) = f" 

59 
by (rule ext) simp 

10213  60 

61 

11838  62 
subsection {* Pairs *} 
10213  63 

11777  64 
subsubsection {* Type definition *} 
10213  65 

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constdefs 

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Pair_Rep :: "['a, 'b] => ['a, 'b] => bool" 
11032  68 
"Pair_Rep == (%a b. %x y. x=a & y=b)" 
10213  69 

70 
global 

71 

72 
typedef (Prod) 

11838  73 
('a, 'b) "*" (infixr 20) 
11032  74 
= "{f. EX a b. f = Pair_Rep (a::'a) (b::'b)}" 
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proof 
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fix a b show "Pair_Rep a b : ?Prod" 
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by blast 
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qed 
10213  79 

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syntax (xsymbols) 
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"*" :: "[type, type] => type" ("(_ \\<times>/ _)" [21, 20] 20) 
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syntax (HTML output) 
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"*" :: "[type, type] => type" ("(_ \\<times>/ _)" [21, 20] 20) 
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local 
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subsubsection {* Abstract constants and syntax *} 

89 

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global 

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consts 

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fst :: "'a * 'b => 'a" 
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snd :: "'a * 'b => 'b" 
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split :: "[['a, 'b] => 'c, 'a * 'b] => 'c" 
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curry :: "['a * 'b => 'c, 'a, 'b] => 'c" 
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prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd" 
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Pair :: "['a, 'b] => 'a * 'b" 
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Sigma :: "['a set, 'a => 'b set] => ('a * 'b) set" 
10213  100 

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local 
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text {* 
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Patterns  extends predefined type @{typ pttrn} used in 

105 
abstractions. 

106 
*} 

10213  107 

108 
nonterminals 

109 
tuple_args patterns 

110 

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syntax 

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"_tuple" :: "'a => tuple_args => 'a * 'b" ("(1'(_,/ _'))") 

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"_tuple_arg" :: "'a => tuple_args" ("_") 

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"_tuple_args" :: "'a => tuple_args => tuple_args" ("_,/ _") 

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"_pattern" :: "[pttrn, patterns] => pttrn" ("'(_,/ _')") 
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"" :: "pttrn => patterns" ("_") 
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"_patterns" :: "[pttrn, patterns] => patterns" ("_,/ _") 
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"@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10) 
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"@Times" ::"['a set, 'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80) 
10213  120 

121 
translations 

122 
"(x, y)" == "Pair x y" 

123 
"_tuple x (_tuple_args y z)" == "_tuple x (_tuple_arg (_tuple y z))" 

124 
"%(x,y,zs).b" == "split(%x (y,zs).b)" 

125 
"%(x,y).b" == "split(%x y. b)" 

126 
"_abs (Pair x y) t" => "%(x,y).t" 

127 
(* The last rule accommodates tuples in `case C ... (x,y) ... => ...' 

128 
The (x,y) is parsed as `Pair x y' because it is logic, not pttrn *) 

129 

130 
"SIGMA x:A. B" => "Sigma A (%x. B)" 

131 
"A <*> B" => "Sigma A (_K B)" 

132 

14359  133 
(* reconstructs pattern from (nested) splits, avoiding etacontraction of body*) 
134 
(* works best with enclosing "let", if "let" does not avoid etacontraction *) 

135 
print_translation {* 

136 
let fun split_tr' [Abs (x,T,t as (Abs abs))] = 

137 
(* split (%x y. t) => %(x,y) t *) 

138 
let val (y,t') = atomic_abs_tr' abs; 

139 
val (x',t'') = atomic_abs_tr' (x,T,t'); 

140 

141 
in Syntax.const "_abs" $ (Syntax.const "_pattern" $x'$y) $ t'' end 

142 
 split_tr' [Abs (x,T,(s as Const ("split",_)$t))] = 

143 
(* split (%x. (split (%y z. t))) => %(x,y,z). t *) 

144 
let val (Const ("_abs",_)$(Const ("_pattern",_)$y$z)$t') = split_tr' [t]; 

145 
val (x',t'') = atomic_abs_tr' (x,T,t'); 

146 
in Syntax.const "_abs"$ 

147 
(Syntax.const "_pattern"$x'$(Syntax.const "_patterns"$y$z))$t'' end 

148 
 split_tr' [Const ("split",_)$t] = 

149 
(* split (split (%x y z. t)) => %((x,y),z). t *) 

150 
split_tr' [(split_tr' [t])] (* inner split_tr' creates next pattern *) 

151 
 split_tr' [Const ("_abs",_)$x_y$(Abs abs)] = 

152 
(* split (%pttrn z. t) => %(pttrn,z). t *) 

153 
let val (z,t) = atomic_abs_tr' abs; 

154 
in Syntax.const "_abs" $ (Syntax.const "_pattern" $x_y$z) $ t end 

155 
 split_tr' _ = raise Match; 

156 
in [("split", split_tr')] 

157 
end 

158 
*} 

159 

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text{*Deleted xsymbol and html support using @{text"\\<Sigma>"} (Sigma) because of the danger of confusion with Sum.*} 
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syntax (xsymbols) 
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"@Times" :: "['a set, 'a => 'b set] => ('a * 'b) set" ("_ \\<times> _" [81, 80] 80) 
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syntax (HTML output) 
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"@Times" :: "['a set, 'a => 'b set] => ('a * 'b) set" ("_ \\<times> _" [81, 80] 80) 
14565  166 

11032  167 
print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))] *} 
10213  168 

169 

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subsubsection {* Definitions *} 
10213  171 

172 
defs 

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Pair_def: "Pair a b == Abs_Prod(Pair_Rep a b)" 
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fst_def: "fst p == THE a. EX b. p = (a, b)" 
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snd_def: "snd p == THE b. EX a. p = (a, b)" 
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split_def: "split == (%c p. c (fst p) (snd p))" 
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curry_def: "curry == (%c x y. c (x,y))" 
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prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))" 
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Sigma_def: "Sigma A B == UN x:A. UN y:B(x). {(x, y)}" 
10213  180 

181 

11966  182 
subsubsection {* Lemmas and proof tool setup *} 
11838  183 

184 
lemma ProdI: "Pair_Rep a b : Prod" 

185 
by (unfold Prod_def) blast 

186 

187 
lemma Pair_Rep_inject: "Pair_Rep a b = Pair_Rep a' b' ==> a = a' & b = b'" 

188 
apply (unfold Pair_Rep_def) 

14208  189 
apply (drule fun_cong [THEN fun_cong], blast) 
11838  190 
done 
10213  191 

11838  192 
lemma inj_on_Abs_Prod: "inj_on Abs_Prod Prod" 
193 
apply (rule inj_on_inverseI) 

194 
apply (erule Abs_Prod_inverse) 

195 
done 

196 

197 
lemma Pair_inject: 

198 
"(a, b) = (a', b') ==> (a = a' ==> b = b' ==> R) ==> R" 

199 
proof  

200 
case rule_context [unfolded Pair_def] 

201 
show ?thesis 

202 
apply (rule inj_on_Abs_Prod [THEN inj_onD, THEN Pair_Rep_inject, THEN conjE]) 

203 
apply (rule rule_context ProdI)+ 

204 
. 

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qed 
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11838  207 
lemma Pair_eq [iff]: "((a, b) = (a', b')) = (a = a' & b = b')" 
208 
by (blast elim!: Pair_inject) 

209 

210 
lemma fst_conv [simp]: "fst (a, b) = a" 

211 
by (unfold fst_def) blast 

212 

213 
lemma snd_conv [simp]: "snd (a, b) = b" 

214 
by (unfold snd_def) blast 

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lemma fst_eqD: "fst (x, y) = a ==> x = a" 
217 
by simp 

218 

219 
lemma snd_eqD: "snd (x, y) = a ==> y = a" 

220 
by simp 

221 

222 
lemma PairE_lemma: "EX x y. p = (x, y)" 

223 
apply (unfold Pair_def) 

224 
apply (rule Rep_Prod [unfolded Prod_def, THEN CollectE]) 

225 
apply (erule exE, erule exE, rule exI, rule exI) 

226 
apply (rule Rep_Prod_inverse [symmetric, THEN trans]) 

227 
apply (erule arg_cong) 

228 
done 

11032  229 

11838  230 
lemma PairE [cases type: *]: "(!!x y. p = (x, y) ==> Q) ==> Q" 
231 
by (insert PairE_lemma [of p]) blast 

232 

233 
ML_setup {* 

234 
local val PairE = thm "PairE" in 

235 
fun pair_tac s = 

236 
EVERY' [res_inst_tac [("p", s)] PairE, hyp_subst_tac, K prune_params_tac]; 

237 
end; 

238 
*} 

11032  239 

11838  240 
lemma surjective_pairing: "p = (fst p, snd p)" 
241 
 {* Do not add as rewrite rule: invalidates some proofs in IMP *} 

242 
by (cases p) simp 

243 

244 
declare surjective_pairing [symmetric, simp] 

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lemma surj_pair [simp]: "EX x y. z = (x, y)" 
247 
apply (rule exI) 

248 
apply (rule exI) 

249 
apply (rule surjective_pairing) 

250 
done 

251 

252 
lemma split_paired_all: "(!!x. PROP P x) == (!!a b. PROP P (a, b))" 

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proof 
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assume "!!x. PROP P x" 
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thus "PROP P (a, b)" . 
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next 
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assume "!!a b. PROP P (a, b)" 
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hence "PROP P (fst x, snd x)" . 
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thus "PROP P x" by simp 
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qed 
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11838  264 
lemmas split_tupled_all = split_paired_all unit_all_eq2 
265 

266 
text {* 

267 
The rule @{thm [source] split_paired_all} does not work with the 

268 
Simplifier because it also affects premises in congrence rules, 

269 
where this can lead to premises of the form @{text "!!a b. ... = 

270 
?P(a, b)"} which cannot be solved by reflexivity. 

271 
*} 

272 

273 
ML_setup " 

274 
(* replace parameters of product type by individual component parameters *) 

275 
val safe_full_simp_tac = generic_simp_tac true (true, false, false); 

276 
local (* filtering with exists_paired_all is an essential optimization *) 

277 
fun exists_paired_all (Const (\"all\", _) $ Abs (_, T, t)) = 

278 
can HOLogic.dest_prodT T orelse exists_paired_all t 

279 
 exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u 

280 
 exists_paired_all (Abs (_, _, t)) = exists_paired_all t 

281 
 exists_paired_all _ = false; 

282 
val ss = HOL_basic_ss 

283 
addsimps [thm \"split_paired_all\", thm \"unit_all_eq2\", thm \"unit_abs_eta_conv\"] 

284 
addsimprocs [unit_eq_proc]; 

285 
in 

286 
val split_all_tac = SUBGOAL (fn (t, i) => 

287 
if exists_paired_all t then safe_full_simp_tac ss i else no_tac); 

288 
val unsafe_split_all_tac = SUBGOAL (fn (t, i) => 

289 
if exists_paired_all t then full_simp_tac ss i else no_tac); 

290 
fun split_all th = 

291 
if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th; 

292 
end; 

293 

294 
claset_ref() := claset() addSbefore (\"split_all_tac\", split_all_tac); 

295 
" 

296 

297 
lemma split_paired_All [simp]: "(ALL x. P x) = (ALL a b. P (a, b))" 

298 
 {* @{text "[iff]"} is not a good idea because it makes @{text blast} loop *} 

299 
by fast 

300 

14189  301 
lemma curry_split [simp]: "curry (split f) = f" 
302 
by (simp add: curry_def split_def) 

303 

304 
lemma split_curry [simp]: "split (curry f) = f" 

305 
by (simp add: curry_def split_def) 

306 

307 
lemma curryI [intro!]: "f (a,b) ==> curry f a b" 

308 
by (simp add: curry_def) 

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lemma curryD [dest!]: "curry f a b ==> f (a,b)" 
14189  311 
by (simp add: curry_def) 
312 

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lemma curryE: "[ curry f a b ; f (a,b) ==> Q ] ==> Q" 
14189  314 
by (simp add: curry_def) 
315 

316 
lemma curry_conv [simp]: "curry f a b = f (a,b)" 

317 
by (simp add: curry_def) 

318 

11838  319 
lemma prod_induct [induct type: *]: "!!x. (!!a b. P (a, b)) ==> P x" 
320 
by fast 

321 

322 
lemma split_paired_Ex [simp]: "(EX x. P x) = (EX a b. P (a, b))" 

323 
by fast 

324 

325 
lemma split_conv [simp]: "split c (a, b) = c a b" 

326 
by (simp add: split_def) 

327 

328 
lemmas split = split_conv  {* for backwards compatibility *} 

329 

330 
lemmas splitI = split_conv [THEN iffD2, standard] 

331 
lemmas splitD = split_conv [THEN iffD1, standard] 

332 

333 
lemma split_Pair_apply: "split (%x y. f (x, y)) = f" 

334 
 {* Subsumes the old @{text split_Pair} when @{term f} is the identity function. *} 

335 
apply (rule ext) 

14208  336 
apply (tactic {* pair_tac "x" 1 *}, simp) 
11838  337 
done 
338 

339 
lemma split_paired_The: "(THE x. P x) = (THE (a, b). P (a, b))" 

340 
 {* Can't be added to simpset: loops! *} 

341 
by (simp add: split_Pair_apply) 

342 

343 
lemma The_split: "The (split P) = (THE xy. P (fst xy) (snd xy))" 

344 
by (simp add: split_def) 

345 

346 
lemma Pair_fst_snd_eq: "!!s t. (s = t) = (fst s = fst t & snd s = snd t)" 

14208  347 
by (simp only: split_tupled_all, simp) 
11838  348 

349 
lemma prod_eqI [intro?]: "fst p = fst q ==> snd p = snd q ==> p = q" 

350 
by (simp add: Pair_fst_snd_eq) 

351 

352 
lemma split_weak_cong: "p = q ==> split c p = split c q" 

353 
 {* Prevents simplification of @{term c}: much faster *} 

354 
by (erule arg_cong) 

355 

356 
lemma split_eta: "(%(x, y). f (x, y)) = f" 

357 
apply (rule ext) 

358 
apply (simp only: split_tupled_all) 

359 
apply (rule split_conv) 

360 
done 

361 

362 
lemma cond_split_eta: "(!!x y. f x y = g (x, y)) ==> (%(x, y). f x y) = g" 

363 
by (simp add: split_eta) 

364 

365 
text {* 

366 
Simplification procedure for @{thm [source] cond_split_eta}. Using 

367 
@{thm [source] split_eta} as a rewrite rule is not general enough, 

368 
and using @{thm [source] cond_split_eta} directly would render some 

369 
existing proofs very inefficient; similarly for @{text 

370 
split_beta}. *} 

371 

372 
ML_setup {* 

373 

374 
local 

375 
val cond_split_eta = thm "cond_split_eta"; 

376 
fun Pair_pat k 0 (Bound m) = (m = k) 

377 
 Pair_pat k i (Const ("Pair", _) $ Bound m $ t) = i > 0 andalso 

378 
m = k+i andalso Pair_pat k (i1) t 

379 
 Pair_pat _ _ _ = false; 

380 
fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t 

381 
 no_args k i (t $ u) = no_args k i t andalso no_args k i u 

382 
 no_args k i (Bound m) = m < k orelse m > k+i 

383 
 no_args _ _ _ = true; 

384 
fun split_pat tp i (Abs (_,_,t)) = if tp 0 i t then Some (i,t) else None 

385 
 split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t 

386 
 split_pat tp i _ = None; 

13480
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset

387 
fun metaeq sg lhs rhs = mk_meta_eq (Tactic.prove sg [] [] 
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset

388 
(HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs))) 
bb72bd43c6c3
use Tactic.prove instead of prove_goalw_cterm in internal proofs!
wenzelm
parents:
13462
diff
changeset

389 
(K (simp_tac (HOL_basic_ss addsimps [cond_split_eta]) 1))); 
11838  390 

391 
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t 

392 
 beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse 

393 
(beta_term_pat k i t andalso beta_term_pat k i u) 

394 
 beta_term_pat k i t = no_args k i t; 

395 
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg 

396 
 eta_term_pat _ _ _ = false; 

397 
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t) 

398 
 subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg 

399 
else (subst arg k i t $ subst arg k i u) 

400 
 subst arg k i t = t; 

401 
fun beta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t) $ arg) = 

402 
(case split_pat beta_term_pat 1 t of 

403 
Some (i,f) => Some (metaeq sg s (subst arg 0 i f)) 

404 
 None => None) 

405 
 beta_proc _ _ _ = None; 

406 
fun eta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t)) = 

407 
(case split_pat eta_term_pat 1 t of 

408 
Some (_,ft) => Some (metaeq sg s (let val (f $ arg) = ft in f end)) 

409 
 None => None) 

410 
 eta_proc _ _ _ = None; 

411 
in 

13462  412 
val split_beta_proc = Simplifier.simproc (Theory.sign_of (the_context ())) 
413 
"split_beta" ["split f z"] beta_proc; 

414 
val split_eta_proc = Simplifier.simproc (Theory.sign_of (the_context ())) 

415 
"split_eta" ["split f"] eta_proc; 

11838  416 
end; 
417 

418 
Addsimprocs [split_beta_proc, split_eta_proc]; 

419 
*} 

420 

421 
lemma split_beta: "(%(x, y). P x y) z = P (fst z) (snd z)" 

422 
by (subst surjective_pairing, rule split_conv) 

423 

424 
lemma split_split: "R (split c p) = (ALL x y. p = (x, y) > R (c x y))" 

425 
 {* For use with @{text split} and the Simplifier. *} 

426 
apply (subst surjective_pairing) 

14208  427 
apply (subst split_conv, blast) 
11838  428 
done 
429 

430 
text {* 

431 
@{thm [source] split_split} could be declared as @{text "[split]"} 

432 
done after the Splitter has been speeded up significantly; 

433 
precompute the constants involved and don't do anything unless the 

434 
current goal contains one of those constants. 

435 
*} 

436 

437 
lemma split_split_asm: "R (split c p) = (~(EX x y. p = (x, y) & (~R (c x y))))" 

14208  438 
by (subst split_split, simp) 
11838  439 

440 

441 
text {* 

442 
\medskip @{term split} used as a logical connective or set former. 

443 

444 
\medskip These rules are for use with @{text blast}; could instead 

445 
call @{text simp} using @{thm [source] split} as rewrite. *} 

446 

447 
lemma splitI2: "!!p. [ !!a b. p = (a, b) ==> c a b ] ==> split c p" 

448 
apply (simp only: split_tupled_all) 

449 
apply (simp (no_asm_simp)) 

450 
done 

451 

452 
lemma splitI2': "!!p. [ !!a b. (a, b) = p ==> c a b x ] ==> split c p x" 

453 
apply (simp only: split_tupled_all) 

454 
apply (simp (no_asm_simp)) 

455 
done 

456 

457 
lemma splitE: "split c p ==> (!!x y. p = (x, y) ==> c x y ==> Q) ==> Q" 

458 
by (induct p) (auto simp add: split_def) 

459 

460 
lemma splitE': "split c p z ==> (!!x y. p = (x, y) ==> c x y z ==> Q) ==> Q" 

461 
by (induct p) (auto simp add: split_def) 

462 

463 
lemma splitE2: 

464 
"[ Q (split P z); !!x y. [z = (x, y); Q (P x y)] ==> R ] ==> R" 

465 
proof  

466 
assume q: "Q (split P z)" 

467 
assume r: "!!x y. [z = (x, y); Q (P x y)] ==> R" 

468 
show R 

469 
apply (rule r surjective_pairing)+ 

470 
apply (rule split_beta [THEN subst], rule q) 

471 
done 

472 
qed 

473 

474 
lemma splitD': "split R (a,b) c ==> R a b c" 

475 
by simp 

476 

477 
lemma mem_splitI: "z: c a b ==> z: split c (a, b)" 

478 
by simp 

479 

480 
lemma mem_splitI2: "!!p. [ !!a b. p = (a, b) ==> z: c a b ] ==> z: split c p" 

14208  481 
by (simp only: split_tupled_all, simp) 
11838  482 

483 
lemma mem_splitE: "[ z: split c p; !!x y. [ p = (x,y); z: c x y ] ==> Q ] ==> Q" 

484 
proof  

485 
case rule_context [unfolded split_def] 

486 
show ?thesis by (rule rule_context surjective_pairing)+ 

487 
qed 

488 

489 
declare mem_splitI2 [intro!] mem_splitI [intro!] splitI2' [intro!] splitI2 [intro!] splitI [intro!] 

490 
declare mem_splitE [elim!] splitE' [elim!] splitE [elim!] 

491 

492 
ML_setup " 

493 
local (* filtering with exists_p_split is an essential optimization *) 

494 
fun exists_p_split (Const (\"split\",_) $ _ $ (Const (\"Pair\",_)$_$_)) = true 

495 
 exists_p_split (t $ u) = exists_p_split t orelse exists_p_split u 

496 
 exists_p_split (Abs (_, _, t)) = exists_p_split t 

497 
 exists_p_split _ = false; 

498 
val ss = HOL_basic_ss addsimps [thm \"split_conv\"]; 

499 
in 

500 
val split_conv_tac = SUBGOAL (fn (t, i) => 

501 
if exists_p_split t then safe_full_simp_tac ss i else no_tac); 

502 
end; 

503 
(* This prevents applications of splitE for already splitted arguments leading 

504 
to quite timeconsuming computations (in particular for nested tuples) *) 

505 
claset_ref() := claset() addSbefore (\"split_conv_tac\", split_conv_tac); 

506 
" 

507 

508 
lemma split_eta_SetCompr [simp]: "(%u. EX x y. u = (x, y) & P (x, y)) = P" 

14208  509 
by (rule ext, fast) 
11838  510 

511 
lemma split_eta_SetCompr2 [simp]: "(%u. EX x y. u = (x, y) & P x y) = split P" 

14208  512 
by (rule ext, fast) 
11838  513 

514 
lemma split_part [simp]: "(%(a,b). P & Q a b) = (%ab. P & split Q ab)" 

515 
 {* Allows simplifications of nested splits in case of independent predicates. *} 

14208  516 
apply (rule ext, blast) 
11838  517 
done 
518 

14337
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

519 
(* Do NOT make this a simp rule as it 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

520 
a) only helps in special situations 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

521 
b) can lead to nontermination in the presence of split_def 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

522 
*) 
e13731554e50
undid split_comp_eq[simp] because it leads to nontermination together with split_def!
nipkow
parents:
14208
diff
changeset

523 
lemma split_comp_eq: 
14101  524 
"(%u. f (g (fst u)) (snd u)) = (split (%x. f (g x)))" 
525 
by (rule ext, auto) 

526 

11838  527 
lemma The_split_eq [simp]: "(THE (x',y'). x = x' & y = y') = (x, y)" 
528 
by blast 

529 

530 
(* 

531 
the following would be slightly more general, 

532 
but cannot be used as rewrite rule: 

533 
### Cannot add premise as rewrite rule because it contains (type) unknowns: 

534 
### ?y = .x 

535 
Goal "[ P y; !!x. P x ==> x = y ] ==> (@(x',y). x = x' & P y) = (x,y)" 

14208  536 
by (rtac some_equality 1) 
537 
by ( Simp_tac 1) 

538 
by (split_all_tac 1) 

539 
by (Asm_full_simp_tac 1) 

11838  540 
qed "The_split_eq"; 
541 
*) 

542 

543 
lemma injective_fst_snd: "!!x y. [fst x = fst y; snd x = snd y] ==> x = y" 

544 
by auto 

545 

546 

547 
text {* 

548 
\bigskip @{term prod_fun}  action of the product functor upon 

549 
functions. 

550 
*} 

551 

552 
lemma prod_fun [simp]: "prod_fun f g (a, b) = (f a, g b)" 

553 
by (simp add: prod_fun_def) 

554 

555 
lemma prod_fun_compose: "prod_fun (f1 o f2) (g1 o g2) = (prod_fun f1 g1 o prod_fun f2 g2)" 

556 
apply (rule ext) 

14208  557 
apply (tactic {* pair_tac "x" 1 *}, simp) 
11838  558 
done 
559 

560 
lemma prod_fun_ident [simp]: "prod_fun (%x. x) (%y. y) = (%z. z)" 

561 
apply (rule ext) 

14208  562 
apply (tactic {* pair_tac "z" 1 *}, simp) 
11838  563 
done 
564 

565 
lemma prod_fun_imageI [intro]: "(a, b) : r ==> (f a, g b) : prod_fun f g ` r" 

566 
apply (rule image_eqI) 

14208  567 
apply (rule prod_fun [symmetric], assumption) 
11838  568 
done 
569 

570 
lemma prod_fun_imageE [elim!]: 

571 
"[ c: (prod_fun f g)`r; !!x y. [ c=(f(x),g(y)); (x,y):r ] ==> P 

572 
] ==> P" 

573 
proof  

574 
case rule_context 

575 
assume major: "c: (prod_fun f g)`r" 

576 
show ?thesis 

577 
apply (rule major [THEN imageE]) 

578 
apply (rule_tac p = x in PairE) 

579 
apply (rule rule_context) 

580 
prefer 2 

581 
apply blast 

582 
apply (blast intro: prod_fun) 

583 
done 

584 
qed 

585 

586 

14101  587 
constdefs 
588 
upd_fst :: "('a => 'c) => 'a * 'b => 'c * 'b" 

589 
"upd_fst f == prod_fun f id" 

590 

591 
upd_snd :: "('b => 'c) => 'a * 'b => 'a * 'c" 

592 
"upd_snd f == prod_fun id f" 

593 

594 
lemma upd_fst_conv [simp]: "upd_fst f (x,y) = (f x,y)" 

595 
by (simp add: upd_fst_def) 

596 

597 
lemma upd_snd_conv [simp]: "upd_snd f (x,y) = (x,f y)" 

598 
by (simp add: upd_snd_def) 

599 

11838  600 
text {* 
601 
\bigskip Disjoint union of a family of sets  Sigma. 

602 
*} 

603 

604 
lemma SigmaI [intro!]: "[ a:A; b:B(a) ] ==> (a,b) : Sigma A B" 

605 
by (unfold Sigma_def) blast 

606 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

607 
lemma SigmaE [elim!]: 
11838  608 
"[ c: Sigma A B; 
609 
!!x y.[ x:A; y:B(x); c=(x,y) ] ==> P 

610 
] ==> P" 

611 
 {* The general elimination rule. *} 

612 
by (unfold Sigma_def) blast 

613 

614 
text {* 

15404  615 
Elimination of @{term "(a, b) : A \\<times> B"}  introduces no 
11838  616 
eigenvariables. 
617 
*} 

618 

619 
lemma SigmaD1: "(a, b) : Sigma A B ==> a : A" 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

620 
by blast 
11838  621 

622 
lemma SigmaD2: "(a, b) : Sigma A B ==> b : B a" 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

623 
by blast 
11838  624 

625 
lemma SigmaE2: 

626 
"[ (a, b) : Sigma A B; 

627 
[ a:A; b:B(a) ] ==> P 

628 
] ==> P" 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

629 
by blast 
11838  630 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

631 
lemma Sigma_cong: 
15404  632 
"\\<lbrakk>A = B; !!x. x \\<in> B \\<Longrightarrow> C x = D x\\<rbrakk> 
633 
\\<Longrightarrow> (SIGMA x: A. C x) = (SIGMA x: B. D x)" 

14952
47455995693d
removal of xsymbol syntax <Sigma> for dependent products
paulson
parents:
14565
diff
changeset

634 
by auto 
11838  635 

636 
lemma Sigma_mono: "[ A <= C; !!x. x:A ==> B x <= D x ] ==> Sigma A B <= Sigma C D" 

637 
by blast 

638 

639 
lemma Sigma_empty1 [simp]: "Sigma {} B = {}" 

640 
by blast 

641 

642 
lemma Sigma_empty2 [simp]: "A <*> {} = {}" 

643 
by blast 

644 

645 
lemma UNIV_Times_UNIV [simp]: "UNIV <*> UNIV = UNIV" 

646 
by auto 

647 

648 
lemma Compl_Times_UNIV1 [simp]: " (UNIV <*> A) = UNIV <*> (A)" 

649 
by auto 

650 

651 
lemma Compl_Times_UNIV2 [simp]: " (A <*> UNIV) = (A) <*> UNIV" 

652 
by auto 

653 

654 
lemma mem_Sigma_iff [iff]: "((a,b): Sigma A B) = (a:A & b:B(a))" 

655 
by blast 

656 

657 
lemma Times_subset_cancel2: "x:C ==> (A <*> C <= B <*> C) = (A <= B)" 

658 
by blast 

659 

660 
lemma Times_eq_cancel2: "x:C ==> (A <*> C = B <*> C) = (A = B)" 

661 
by (blast elim: equalityE) 

662 

663 
lemma SetCompr_Sigma_eq: 

664 
"Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))" 

665 
by blast 

666 

667 
text {* 

668 
\bigskip Complex rules for Sigma. 

669 
*} 

670 

671 
lemma Collect_split [simp]: "{(a,b). P a & Q b} = Collect P <*> Collect Q" 

672 
by blast 

673 

674 
lemma UN_Times_distrib: 

675 
"(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)" 

676 
 {* Suggested by Pierre Chartier *} 

677 
by blast 

678 

679 
lemma split_paired_Ball_Sigma [simp]: 

680 
"(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))" 

681 
by blast 

682 

683 
lemma split_paired_Bex_Sigma [simp]: 

684 
"(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))" 

685 
by blast 

686 

687 
lemma Sigma_Un_distrib1: "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))" 

688 
by blast 

689 

690 
lemma Sigma_Un_distrib2: "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))" 

691 
by blast 

692 

693 
lemma Sigma_Int_distrib1: "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))" 

694 
by blast 

695 

696 
lemma Sigma_Int_distrib2: "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))" 

697 
by blast 

698 

699 
lemma Sigma_Diff_distrib1: "(SIGMA i:I  J. C(i)) = (SIGMA i:I. C(i))  (SIGMA j:J. C(j))" 

700 
by blast 

701 

702 
lemma Sigma_Diff_distrib2: "(SIGMA i:I. A(i)  B(i)) = (SIGMA i:I. A(i))  (SIGMA i:I. B(i))" 

703 
by blast 

704 

705 
lemma Sigma_Union: "Sigma (Union X) B = (UN A:X. Sigma A B)" 

706 
by blast 

707 

708 
text {* 

709 
Nondependent versions are needed to avoid the need for higherorder 

710 
matching, especially when the rules are reoriented. 

711 
*} 

712 

713 
lemma Times_Un_distrib1: "(A Un B) <*> C = (A <*> C) Un (B <*> C)" 

714 
by blast 

715 

716 
lemma Times_Int_distrib1: "(A Int B) <*> C = (A <*> C) Int (B <*> C)" 

717 
by blast 

718 

719 
lemma Times_Diff_distrib1: "(A  B) <*> C = (A <*> C)  (B <*> C)" 

720 
by blast 

721 

722 

11493  723 
lemma pair_imageI [intro]: "(a, b) : A ==> f a b : (%(a, b). f a b) ` A" 
11777  724 
apply (rule_tac x = "(a, b)" in image_eqI) 
725 
apply auto 

726 
done 

727 

11493  728 

11838  729 
text {* 
730 
Setup of internal @{text split_rule}. 

731 
*} 

732 

11032  733 
constdefs 
11425  734 
internal_split :: "('a => 'b => 'c) => 'a * 'b => 'c" 
11032  735 
"internal_split == split" 
736 

737 
lemma internal_split_conv: "internal_split c (a, b) = c a b" 

738 
by (simp only: internal_split_def split_conv) 

739 

740 
hide const internal_split 

741 

11025
a70b796d9af8
converted to Isar therory, adding attributes complete_split and split_format
oheimb
parents:
10289
diff
changeset

742 
use "Tools/split_rule.ML" 
11032  743 
setup SplitRule.setup 
10213  744 

15394  745 

746 
subsection {* Code generator setup *} 

747 

748 
types_code 

749 
"*" ("(_ */ _)") 

750 

751 
consts_code 

752 
"Pair" ("(_,/ _)") 

753 
"fst" ("fst") 

754 
"snd" ("snd") 

755 

756 
ML {* 

757 
fun term_of_id_42 f T g U (x, y) = HOLogic.pair_const T U $ f x $ g y; 

758 

759 
fun gen_id_42 aG bG i = (aG i, bG i); 

760 

761 
local 

762 

763 
fun strip_abs 0 t = ([], t) 

764 
 strip_abs i (Abs (s, T, t)) = 

765 
let 

766 
val s' = Codegen.new_name t s; 

767 
val v = Free (s', T) 

768 
in apfst (cons v) (strip_abs (i1) (subst_bound (v, t))) end 

769 
 strip_abs i (u as Const ("split", _) $ t) = (case strip_abs (i+1) t of 

770 
(v :: v' :: vs, u) => (HOLogic.mk_prod (v, v') :: vs, u) 

771 
 _ => ([], u)) 

772 
 strip_abs i t = ([], t); 

773 

774 
fun let_codegen thy gr dep brack (t as Const ("Let", _) $ _ $ _) = 

775 
let 

776 
fun dest_let (l as Const ("Let", _) $ t $ u) = 

777 
(case strip_abs 1 u of 

778 
([p], u') => apfst (cons (p, t)) (dest_let u') 

779 
 _ => ([], l)) 

780 
 dest_let t = ([], t); 

781 
fun mk_code (gr, (l, r)) = 

782 
let 

783 
val (gr1, pl) = Codegen.invoke_codegen thy dep false (gr, l); 

784 
val (gr2, pr) = Codegen.invoke_codegen thy dep false (gr1, r); 

785 
in (gr2, (pl, pr)) end 

786 
in case dest_let t of 

787 
([], _) => None 

788 
 (ps, u) => 

789 
let 

790 
val (gr1, qs) = foldl_map mk_code (gr, ps); 

791 
val (gr2, pu) = Codegen.invoke_codegen thy dep false (gr1, u) 

792 
in 

793 
Some (gr2, Pretty.blk (0, [Pretty.str "let ", Pretty.blk (0, flat 

794 
(separate [Pretty.str ";", Pretty.brk 1] (map (fn (pl, pr) => 

795 
[Pretty.block [Pretty.str "val ", pl, Pretty.str " =", 

796 
Pretty.brk 1, pr]]) qs))), 

797 
Pretty.brk 1, Pretty.str "in ", pu, 

798 
Pretty.brk 1, Pretty.str "end"])) 

799 
end 

800 
end 

801 
 let_codegen thy gr dep brack t = None; 

802 

803 
fun split_codegen thy gr dep brack (t as Const ("split", _) $ _) = 

804 
(case strip_abs 1 t of 

805 
([p], u) => 

806 
let 

807 
val (gr1, q) = Codegen.invoke_codegen thy dep false (gr, p); 

808 
val (gr2, pu) = Codegen.invoke_codegen thy dep false (gr1, u) 

809 
in 

810 
Some (gr2, Pretty.block [Pretty.str "(fn ", q, Pretty.str " =>", 

811 
Pretty.brk 1, pu, Pretty.str ")"]) 

812 
end 

813 
 _ => None) 

814 
 split_codegen thy gr dep brack t = None; 

815 

816 
in 

817 

818 
val prod_codegen_setup = 

819 
[Codegen.add_codegen "let_codegen" let_codegen, 

820 
Codegen.add_codegen "split_codegen" split_codegen]; 

821 

822 
end; 

823 
*} 

824 

825 
setup prod_codegen_setup 

826 

15404  827 
ML 
828 
{* 

829 
val Collect_split = thm "Collect_split"; 

830 
val Compl_Times_UNIV1 = thm "Compl_Times_UNIV1"; 

831 
val Compl_Times_UNIV2 = thm "Compl_Times_UNIV2"; 

832 
val PairE = thm "PairE"; 

833 
val PairE_lemma = thm "PairE_lemma"; 

834 
val Pair_Rep_inject = thm "Pair_Rep_inject"; 

835 
val Pair_def = thm "Pair_def"; 

836 
val Pair_eq = thm "Pair_eq"; 

837 
val Pair_fst_snd_eq = thm "Pair_fst_snd_eq"; 

838 
val Pair_inject = thm "Pair_inject"; 

839 
val ProdI = thm "ProdI"; 

840 
val SetCompr_Sigma_eq = thm "SetCompr_Sigma_eq"; 

841 
val SigmaD1 = thm "SigmaD1"; 

842 
val SigmaD2 = thm "SigmaD2"; 

843 
val SigmaE = thm "SigmaE"; 

844 
val SigmaE2 = thm "SigmaE2"; 

845 
val SigmaI = thm "SigmaI"; 

846 
val Sigma_Diff_distrib1 = thm "Sigma_Diff_distrib1"; 

847 
val Sigma_Diff_distrib2 = thm "Sigma_Diff_distrib2"; 

848 
val Sigma_Int_distrib1 = thm "Sigma_Int_distrib1"; 

849 
val Sigma_Int_distrib2 = thm "Sigma_Int_distrib2"; 

850 
val Sigma_Un_distrib1 = thm "Sigma_Un_distrib1"; 

851 
val Sigma_Un_distrib2 = thm "Sigma_Un_distrib2"; 

852 
val Sigma_Union = thm "Sigma_Union"; 

853 
val Sigma_def = thm "Sigma_def"; 

854 
val Sigma_empty1 = thm "Sigma_empty1"; 

855 
val Sigma_empty2 = thm "Sigma_empty2"; 

856 
val Sigma_mono = thm "Sigma_mono"; 

857 
val The_split = thm "The_split"; 

858 
val The_split_eq = thm "The_split_eq"; 

859 
val The_split_eq = thm "The_split_eq"; 

860 
val Times_Diff_distrib1 = thm "Times_Diff_distrib1"; 

861 
val Times_Int_distrib1 = thm "Times_Int_distrib1"; 

862 
val Times_Un_distrib1 = thm "Times_Un_distrib1"; 

863 
val Times_eq_cancel2 = thm "Times_eq_cancel2"; 

864 
val Times_subset_cancel2 = thm "Times_subset_cancel2"; 

865 
val UNIV_Times_UNIV = thm "UNIV_Times_UNIV"; 

866 
val UN_Times_distrib = thm "UN_Times_distrib"; 

867 
val Unity_def = thm "Unity_def"; 

868 
val cond_split_eta = thm "cond_split_eta"; 

869 
val fst_conv = thm "fst_conv"; 

870 
val fst_def = thm "fst_def"; 

871 
val fst_eqD = thm "fst_eqD"; 

872 
val inj_on_Abs_Prod = thm "inj_on_Abs_Prod"; 

873 
val injective_fst_snd = thm "injective_fst_snd"; 

874 
val mem_Sigma_iff = thm "mem_Sigma_iff"; 

875 
val mem_splitE = thm "mem_splitE"; 

876 
val mem_splitI = thm "mem_splitI"; 

877 
val mem_splitI2 = thm "mem_splitI2"; 

878 
val prod_eqI = thm "prod_eqI"; 

879 
val prod_fun = thm "prod_fun"; 

880 
val prod_fun_compose = thm "prod_fun_compose"; 

881 
val prod_fun_def = thm "prod_fun_def"; 

882 
val prod_fun_ident = thm "prod_fun_ident"; 

883 
val prod_fun_imageE = thm "prod_fun_imageE"; 

884 
val prod_fun_imageI = thm "prod_fun_imageI"; 

885 
val prod_induct = thm "prod_induct"; 

886 
val snd_conv = thm "snd_conv"; 

887 
val snd_def = thm "snd_def"; 

888 
val snd_eqD = thm "snd_eqD"; 

889 
val split = thm "split"; 

890 
val splitD = thm "splitD"; 

891 
val splitD' = thm "splitD'"; 

892 
val splitE = thm "splitE"; 

893 
val splitE' = thm "splitE'"; 

894 
val splitE2 = thm "splitE2"; 

895 
val splitI = thm "splitI"; 

896 
val splitI2 = thm "splitI2"; 

897 
val splitI2' = thm "splitI2'"; 

898 
val split_Pair_apply = thm "split_Pair_apply"; 

899 
val split_beta = thm "split_beta"; 

900 
val split_conv = thm "split_conv"; 

901 
val split_def = thm "split_def"; 

902 
val split_eta = thm "split_eta"; 

903 
val split_eta_SetCompr = thm "split_eta_SetCompr"; 

904 
val split_eta_SetCompr2 = thm "split_eta_SetCompr2"; 

905 
val split_paired_All = thm "split_paired_All"; 

906 
val split_paired_Ball_Sigma = thm "split_paired_Ball_Sigma"; 

907 
val split_paired_Bex_Sigma = thm "split_paired_Bex_Sigma"; 

908 
val split_paired_Ex = thm "split_paired_Ex"; 

909 
val split_paired_The = thm "split_paired_The"; 

910 
val split_paired_all = thm "split_paired_all"; 

911 
val split_part = thm "split_part"; 

912 
val split_split = thm "split_split"; 

913 
val split_split_asm = thm "split_split_asm"; 

914 
val split_tupled_all = thms "split_tupled_all"; 

915 
val split_weak_cong = thm "split_weak_cong"; 

916 
val surj_pair = thm "surj_pair"; 

917 
val surjective_pairing = thm "surjective_pairing"; 

918 
val unit_abs_eta_conv = thm "unit_abs_eta_conv"; 

919 
val unit_all_eq1 = thm "unit_all_eq1"; 

920 
val unit_all_eq2 = thm "unit_all_eq2"; 

921 
val unit_eq = thm "unit_eq"; 

922 
val unit_induct = thm "unit_induct"; 

923 
*} 

924 

10213  925 
end 