author  wenzelm 
Wed, 25 Jul 2012 18:05:07 +0200  
changeset 48502  fd03877ad5bc 
parent 41526  54b4686704af 
child 48891  c0eafbd55de3 
permissions  rwrr 
17441  1 
(* Title: CTT/CTT.thy 
0  2 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
3 
Copyright 1993 University of Cambridge 

4 
*) 

5 

17441  6 
header {* Constructive Type Theory *} 
0  7 

17441  8 
theory CTT 
9 
imports Pure 

19761  10 
uses "~~/src/Provers/typedsimp.ML" ("rew.ML") 
17441  11 
begin 
12 

39557
fe5722fce758
renamed structure PureThy to Pure_Thy and moved most content to Global_Theory, to emphasize that this is globalonly;
wenzelm
parents:
35762
diff
changeset

13 
setup Pure_Thy.old_appl_syntax_setup 
26956
1309a6a0a29f
setup PureThy.old_appl_syntax_setup  theory Pure provides regular application syntax by default;
wenzelm
parents:
26391
diff
changeset

14 

17441  15 
typedecl i 
16 
typedecl t 

17 
typedecl o 

0  18 

19 
consts 

20 
(*Types*) 

17441  21 
F :: "t" 
22 
T :: "t" (*F is empty, T contains one element*) 

0  23 
contr :: "i=>i" 
24 
tt :: "i" 

25 
(*Natural numbers*) 

26 
N :: "t" 

27 
succ :: "i=>i" 

28 
rec :: "[i, i, [i,i]=>i] => i" 

29 
(*Unions*) 

17441  30 
inl :: "i=>i" 
31 
inr :: "i=>i" 

0  32 
when :: "[i, i=>i, i=>i]=>i" 
33 
(*General Sum and Binary Product*) 

34 
Sum :: "[t, i=>t]=>t" 

17441  35 
fst :: "i=>i" 
36 
snd :: "i=>i" 

0  37 
split :: "[i, [i,i]=>i] =>i" 
38 
(*General Product and Function Space*) 

39 
Prod :: "[t, i=>t]=>t" 

14765  40 
(*Types*) 
22808  41 
Plus :: "[t,t]=>t" (infixr "+" 40) 
0  42 
(*Equality type*) 
43 
Eq :: "[t,i,i]=>t" 

44 
eq :: "i" 

45 
(*Judgements*) 

46 
Type :: "t => prop" ("(_ type)" [10] 5) 

10467
e6e7205e9e91
xsymbol support for Pi, Sigma, >, : (membership)
paulson
parents:
3837
diff
changeset

47 
Eqtype :: "[t,t]=>prop" ("(_ =/ _)" [10,10] 5) 
0  48 
Elem :: "[i, t]=>prop" ("(_ /: _)" [10,10] 5) 
10467
e6e7205e9e91
xsymbol support for Pi, Sigma, >, : (membership)
paulson
parents:
3837
diff
changeset

49 
Eqelem :: "[i,i,t]=>prop" ("(_ =/ _ :/ _)" [10,10,10] 5) 
0  50 
Reduce :: "[i,i]=>prop" ("Reduce[_,_]") 
51 
(*Types*) 

14765  52 

0  53 
(*Functions*) 
54 
lambda :: "(i => i) => i" (binder "lam " 10) 

22808  55 
app :: "[i,i]=>i" (infixl "`" 60) 
0  56 
(*Natural numbers*) 
41310  57 
Zero :: "i" ("0") 
0  58 
(*Pairing*) 
59 
pair :: "[i,i]=>i" ("(1<_,/_>)") 

60 

14765  61 
syntax 
19761  62 
"_PROD" :: "[idt,t,t]=>t" ("(3PROD _:_./ _)" 10) 
63 
"_SUM" :: "[idt,t,t]=>t" ("(3SUM _:_./ _)" 10) 

0  64 
translations 
35054  65 
"PROD x:A. B" == "CONST Prod(A, %x. B)" 
66 
"SUM x:A. B" == "CONST Sum(A, %x. B)" 

19761  67 

68 
abbreviation 

21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset

69 
Arrow :: "[t,t]=>t" (infixr ">" 30) where 
19761  70 
"A > B == PROD _:A. B" 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset

71 
abbreviation 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset

72 
Times :: "[t,t]=>t" (infixr "*" 50) where 
19761  73 
"A * B == SUM _:A. B" 
0  74 

21210  75 
notation (xsymbols) 
21524  76 
lambda (binder "\<lambda>\<lambda>" 10) and 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset

77 
Elem ("(_ /\<in> _)" [10,10] 5) and 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset

78 
Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset

79 
Arrow (infixr "\<longrightarrow>" 30) and 
19761  80 
Times (infixr "\<times>" 50) 
17441  81 

21210  82 
notation (HTML output) 
21524  83 
lambda (binder "\<lambda>\<lambda>" 10) and 
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset

84 
Elem ("(_ /\<in> _)" [10,10] 5) and 
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21210
diff
changeset

85 
Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and 
19761  86 
Times (infixr "\<times>" 50) 
17441  87 

10467
e6e7205e9e91
xsymbol support for Pi, Sigma, >, : (membership)
paulson
parents:
3837
diff
changeset

88 
syntax (xsymbols) 
21524  89 
"_PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10) 
90 
"_SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10) 

10467
e6e7205e9e91
xsymbol support for Pi, Sigma, >, : (membership)
paulson
parents:
3837
diff
changeset

91 

14565  92 
syntax (HTML output) 
21524  93 
"_PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10) 
94 
"_SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10) 

14565  95 

17441  96 
axioms 
0  97 

98 
(*Reduction: a weaker notion than equality; a hack for simplification. 

99 
Reduce[a,b] means either that a=b:A for some A or else that "a" and "b" 

100 
are textually identical.*) 

101 

102 
(*does not verify a:A! Sound because only trans_red uses a Reduce premise 

103 
No new theorems can be proved about the standard judgements.*) 

17441  104 
refl_red: "Reduce[a,a]" 
105 
red_if_equal: "a = b : A ==> Reduce[a,b]" 

106 
trans_red: "[ a = b : A; Reduce[b,c] ] ==> a = c : A" 

0  107 

108 
(*Reflexivity*) 

109 

17441  110 
refl_type: "A type ==> A = A" 
111 
refl_elem: "a : A ==> a = a : A" 

0  112 

113 
(*Symmetry*) 

114 

17441  115 
sym_type: "A = B ==> B = A" 
116 
sym_elem: "a = b : A ==> b = a : A" 

0  117 

118 
(*Transitivity*) 

119 

17441  120 
trans_type: "[ A = B; B = C ] ==> A = C" 
121 
trans_elem: "[ a = b : A; b = c : A ] ==> a = c : A" 

0  122 

17441  123 
equal_types: "[ a : A; A = B ] ==> a : B" 
124 
equal_typesL: "[ a = b : A; A = B ] ==> a = b : B" 

0  125 

126 
(*Substitution*) 

127 

17441  128 
subst_type: "[ a : A; !!z. z:A ==> B(z) type ] ==> B(a) type" 
129 
subst_typeL: "[ a = c : A; !!z. z:A ==> B(z) = D(z) ] ==> B(a) = D(c)" 

0  130 

17441  131 
subst_elem: "[ a : A; !!z. z:A ==> b(z):B(z) ] ==> b(a):B(a)" 
132 
subst_elemL: 

0  133 
"[ a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) ] ==> b(a)=d(c) : B(a)" 
134 

135 

136 
(*The type N  natural numbers*) 

137 

17441  138 
NF: "N type" 
139 
NI0: "0 : N" 

140 
NI_succ: "a : N ==> succ(a) : N" 

141 
NI_succL: "a = b : N ==> succ(a) = succ(b) : N" 

0  142 

17441  143 
NE: 
144 
"[ p: N; a: C(0); !!u v. [ u: N; v: C(u) ] ==> b(u,v): C(succ(u)) ] 

3837  145 
==> rec(p, a, %u v. b(u,v)) : C(p)" 
0  146 

17441  147 
NEL: 
148 
"[ p = q : N; a = c : C(0); 

149 
!!u v. [ u: N; v: C(u) ] ==> b(u,v) = d(u,v): C(succ(u)) ] 

3837  150 
==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)" 
0  151 

17441  152 
NC0: 
153 
"[ a: C(0); !!u v. [ u: N; v: C(u) ] ==> b(u,v): C(succ(u)) ] 

3837  154 
==> rec(0, a, %u v. b(u,v)) = a : C(0)" 
0  155 

17441  156 
NC_succ: 
157 
"[ p: N; a: C(0); 

158 
!!u v. [ u: N; v: C(u) ] ==> b(u,v): C(succ(u)) ] ==> 

3837  159 
rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))" 
0  160 

161 
(*The fourth Peano axiom. See page 91 of MartinLof's book*) 

17441  162 
zero_ne_succ: 
0  163 
"[ a: N; 0 = succ(a) : N ] ==> 0: F" 
164 

165 

166 
(*The Product of a family of types*) 

167 

17441  168 
ProdF: "[ A type; !!x. x:A ==> B(x) type ] ==> PROD x:A. B(x) type" 
0  169 

17441  170 
ProdFL: 
171 
"[ A = C; !!x. x:A ==> B(x) = D(x) ] ==> 

3837  172 
PROD x:A. B(x) = PROD x:C. D(x)" 
0  173 

17441  174 
ProdI: 
3837  175 
"[ A type; !!x. x:A ==> b(x):B(x)] ==> lam x. b(x) : PROD x:A. B(x)" 
0  176 

17441  177 
ProdIL: 
178 
"[ A type; !!x. x:A ==> b(x) = c(x) : B(x)] ==> 

3837  179 
lam x. b(x) = lam x. c(x) : PROD x:A. B(x)" 
0  180 

17441  181 
ProdE: "[ p : PROD x:A. B(x); a : A ] ==> p`a : B(a)" 
182 
ProdEL: "[ p=q: PROD x:A. B(x); a=b : A ] ==> p`a = q`b : B(a)" 

0  183 

17441  184 
ProdC: 
185 
"[ a : A; !!x. x:A ==> b(x) : B(x)] ==> 

3837  186 
(lam x. b(x)) ` a = b(a) : B(a)" 
0  187 

17441  188 
ProdC2: 
3837  189 
"p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)" 
0  190 

191 

192 
(*The Sum of a family of types*) 

193 

17441  194 
SumF: "[ A type; !!x. x:A ==> B(x) type ] ==> SUM x:A. B(x) type" 
195 
SumFL: 

3837  196 
"[ A = C; !!x. x:A ==> B(x) = D(x) ] ==> SUM x:A. B(x) = SUM x:C. D(x)" 
0  197 

17441  198 
SumI: "[ a : A; b : B(a) ] ==> <a,b> : SUM x:A. B(x)" 
199 
SumIL: "[ a=c:A; b=d:B(a) ] ==> <a,b> = <c,d> : SUM x:A. B(x)" 

0  200 

17441  201 
SumE: 
202 
"[ p: SUM x:A. B(x); !!x y. [ x:A; y:B(x) ] ==> c(x,y): C(<x,y>) ] 

3837  203 
==> split(p, %x y. c(x,y)) : C(p)" 
0  204 

17441  205 
SumEL: 
206 
"[ p=q : SUM x:A. B(x); 

207 
!!x y. [ x:A; y:B(x) ] ==> c(x,y)=d(x,y): C(<x,y>)] 

3837  208 
==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)" 
0  209 

17441  210 
SumC: 
211 
"[ a: A; b: B(a); !!x y. [ x:A; y:B(x) ] ==> c(x,y): C(<x,y>) ] 

3837  212 
==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)" 
0  213 

17441  214 
fst_def: "fst(a) == split(a, %x y. x)" 
215 
snd_def: "snd(a) == split(a, %x y. y)" 

0  216 

217 

218 
(*The sum of two types*) 

219 

17441  220 
PlusF: "[ A type; B type ] ==> A+B type" 
221 
PlusFL: "[ A = C; B = D ] ==> A+B = C+D" 

0  222 

17441  223 
PlusI_inl: "[ a : A; B type ] ==> inl(a) : A+B" 
224 
PlusI_inlL: "[ a = c : A; B type ] ==> inl(a) = inl(c) : A+B" 

0  225 

17441  226 
PlusI_inr: "[ A type; b : B ] ==> inr(b) : A+B" 
227 
PlusI_inrL: "[ A type; b = d : B ] ==> inr(b) = inr(d) : A+B" 

0  228 

17441  229 
PlusE: 
230 
"[ p: A+B; !!x. x:A ==> c(x): C(inl(x)); 

231 
!!y. y:B ==> d(y): C(inr(y)) ] 

3837  232 
==> when(p, %x. c(x), %y. d(y)) : C(p)" 
0  233 

17441  234 
PlusEL: 
235 
"[ p = q : A+B; !!x. x: A ==> c(x) = e(x) : C(inl(x)); 

236 
!!y. y: B ==> d(y) = f(y) : C(inr(y)) ] 

3837  237 
==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)" 
0  238 

17441  239 
PlusC_inl: 
240 
"[ a: A; !!x. x:A ==> c(x): C(inl(x)); 

241 
!!y. y:B ==> d(y): C(inr(y)) ] 

3837  242 
==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))" 
0  243 

17441  244 
PlusC_inr: 
245 
"[ b: B; !!x. x:A ==> c(x): C(inl(x)); 

246 
!!y. y:B ==> d(y): C(inr(y)) ] 

3837  247 
==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))" 
0  248 

249 

250 
(*The type Eq*) 

251 

17441  252 
EqF: "[ A type; a : A; b : A ] ==> Eq(A,a,b) type" 
253 
EqFL: "[ A=B; a=c: A; b=d : A ] ==> Eq(A,a,b) = Eq(B,c,d)" 

254 
EqI: "a = b : A ==> eq : Eq(A,a,b)" 

255 
EqE: "p : Eq(A,a,b) ==> a = b : A" 

0  256 

257 
(*By equality of types, can prove C(p) from C(eq), an elimination rule*) 

17441  258 
EqC: "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)" 
0  259 

260 
(*The type F*) 

261 

17441  262 
FF: "F type" 
263 
FE: "[ p: F; C type ] ==> contr(p) : C" 

264 
FEL: "[ p = q : F; C type ] ==> contr(p) = contr(q) : C" 

0  265 

266 
(*The type T 

267 
MartinLof's book (page 68) discusses elimination and computation. 

268 
Elimination can be derived by computation and equality of types, 

269 
but with an extra premise C(x) type x:T. 

270 
Also computation can be derived from elimination. *) 

271 

17441  272 
TF: "T type" 
273 
TI: "tt : T" 

274 
TE: "[ p : T; c : C(tt) ] ==> c : C(p)" 

275 
TEL: "[ p = q : T; c = d : C(tt) ] ==> c = d : C(p)" 

276 
TC: "p : T ==> p = tt : T" 

0  277 

19761  278 

279 
subsection "Tactics and derived rules for Constructive Type Theory" 

280 

281 
(*Formation rules*) 

282 
lemmas form_rls = NF ProdF SumF PlusF EqF FF TF 

283 
and formL_rls = ProdFL SumFL PlusFL EqFL 

284 

285 
(*Introduction rules 

286 
OMITTED: EqI, because its premise is an eqelem, not an elem*) 

287 
lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI 

288 
and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL 

289 

290 
(*Elimination rules 

291 
OMITTED: EqE, because its conclusion is an eqelem, not an elem 

292 
TE, because it does not involve a constructor *) 

293 
lemmas elim_rls = NE ProdE SumE PlusE FE 

294 
and elimL_rls = NEL ProdEL SumEL PlusEL FEL 

295 

296 
(*OMITTED: eqC are TC because they make rewriting loop: p = un = un = ... *) 

297 
lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr 

298 

299 
(*rules with conclusion a:A, an elem judgement*) 

300 
lemmas element_rls = intr_rls elim_rls 

301 

302 
(*Definitions are (meta)equality axioms*) 

303 
lemmas basic_defs = fst_def snd_def 

304 

305 
(*Compare with standard version: B is applied to UNSIMPLIFIED expression! *) 

306 
lemma SumIL2: "[ c=a : A; d=b : B(a) ] ==> <c,d> = <a,b> : Sum(A,B)" 

307 
apply (rule sym_elem) 

308 
apply (rule SumIL) 

309 
apply (rule_tac [!] sym_elem) 

310 
apply assumption+ 

311 
done 

312 

313 
lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL 

314 

315 
(*Exploit p:Prod(A,B) to create the assumption z:B(a). 

316 
A more natural form of product elimination. *) 

317 
lemma subst_prodE: 

318 
assumes "p: Prod(A,B)" 

319 
and "a: A" 

320 
and "!!z. z: B(a) ==> c(z): C(z)" 

321 
shows "c(p`a): C(p`a)" 

41526  322 
apply (rule assms ProdE)+ 
19761  323 
done 
324 

325 

326 
subsection {* Tactics for type checking *} 

327 

328 
ML {* 

329 

330 
local 

331 

332 
fun is_rigid_elem (Const("CTT.Elem",_) $ a $ _) = not(is_Var (head_of a)) 

333 
 is_rigid_elem (Const("CTT.Eqelem",_) $ a $ _ $ _) = not(is_Var (head_of a)) 

334 
 is_rigid_elem (Const("CTT.Type",_) $ a) = not(is_Var (head_of a)) 

335 
 is_rigid_elem _ = false 

336 

337 
in 

338 

339 
(*Try solving a:A or a=b:A by assumption provided a is rigid!*) 

340 
val test_assume_tac = SUBGOAL(fn (prem,i) => 

341 
if is_rigid_elem (Logic.strip_assums_concl prem) 

342 
then assume_tac i else no_tac) 

343 

344 
fun ASSUME tf i = test_assume_tac i ORELSE tf i 

345 

346 
end; 

347 

348 
*} 

349 

350 
(*For simplification: type formation and checking, 

351 
but no equalities between terms*) 

352 
lemmas routine_rls = form_rls formL_rls refl_type element_rls 

353 

354 
ML {* 

355 
local 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

356 
val equal_rls = @{thms form_rls} @ @{thms element_rls} @ @{thms intrL_rls} @ 
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

357 
@{thms elimL_rls} @ @{thms refl_elem} 
19761  358 
in 
359 

360 
fun routine_tac rls prems = ASSUME (filt_resolve_tac (prems @ rls) 4); 

361 

362 
(*Solve all subgoals "A type" using formation rules. *) 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

363 
val form_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac @{thms form_rls} 1)); 
19761  364 

365 
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) 

366 
fun typechk_tac thms = 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

367 
let val tac = filt_resolve_tac (thms @ @{thms form_rls} @ @{thms element_rls}) 3 
19761  368 
in REPEAT_FIRST (ASSUME tac) end 
369 

370 
(*Solve a:A (a flexible, A rigid) by introduction rules. 

371 
Cannot use stringtrees (filt_resolve_tac) since 

372 
goals like ?a:SUM(A,B) have a trivial headstring *) 

373 
fun intr_tac thms = 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

374 
let val tac = filt_resolve_tac(thms @ @{thms form_rls} @ @{thms intr_rls}) 1 
19761  375 
in REPEAT_FIRST (ASSUME tac) end 
376 

377 
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) 

378 
fun equal_tac thms = 

379 
REPEAT_FIRST (ASSUME (filt_resolve_tac (thms @ equal_rls) 3)) 

0  380 

17441  381 
end 
19761  382 

383 
*} 

384 

385 

386 
subsection {* Simplification *} 

387 

388 
(*To simplify the type in a goal*) 

389 
lemma replace_type: "[ B = A; a : A ] ==> a : B" 

390 
apply (rule equal_types) 

391 
apply (rule_tac [2] sym_type) 

392 
apply assumption+ 

393 
done 

394 

395 
(*Simplify the parameter of a unary type operator.*) 

396 
lemma subst_eqtyparg: 

23467  397 
assumes 1: "a=c : A" 
398 
and 2: "!!z. z:A ==> B(z) type" 

19761  399 
shows "B(a)=B(c)" 
400 
apply (rule subst_typeL) 

401 
apply (rule_tac [2] refl_type) 

23467  402 
apply (rule 1) 
403 
apply (erule 2) 

19761  404 
done 
405 

406 
(*Simplification rules for Constructive Type Theory*) 

407 
lemmas reduction_rls = comp_rls [THEN trans_elem] 

408 

409 
ML {* 

410 
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. 

411 
Uses other intro rules to avoid changing flexible goals.*) 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

412 
val eqintr_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac (@{thm EqI} :: @{thms intr_rls}) 1)) 
19761  413 

414 
(** Tactics that instantiate CTTrules. 

415 
Vars in the given terms will be incremented! 

416 
The (rtac EqE i) lets them apply to equality judgements. **) 

417 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

418 
fun NE_tac ctxt sp i = 
27239  419 
TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm NE} i 
19761  420 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

421 
fun SumE_tac ctxt sp i = 
27239  422 
TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm SumE} i 
19761  423 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

424 
fun PlusE_tac ctxt sp i = 
27239  425 
TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm PlusE} i 
19761  426 

427 
(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) 

428 

429 
(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *) 

430 
fun add_mp_tac i = 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

431 
rtac @{thm subst_prodE} i THEN assume_tac i THEN assume_tac i 
19761  432 

433 
(*Finds P>Q and P in the assumptions, replaces implication by Q *) 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

434 
fun mp_tac i = etac @{thm subst_prodE} i THEN assume_tac i 
19761  435 

436 
(*"safe" when regarded as predicate calculus rules*) 

437 
val safe_brls = sort (make_ord lessb) 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

438 
[ (true, @{thm FE}), (true,asm_rl), 
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

439 
(false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ] 
19761  440 

441 
val unsafe_brls = 

27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

442 
[ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}), 
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

443 
(true, @{thm subst_prodE}) ] 
19761  444 

445 
(*0 subgoals vs 1 or more*) 

446 
val (safe0_brls, safep_brls) = 

447 
List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls 

448 

449 
fun safestep_tac thms i = 

450 
form_tac ORELSE 

451 
resolve_tac thms i ORELSE 

452 
biresolve_tac safe0_brls i ORELSE mp_tac i ORELSE 

453 
DETERM (biresolve_tac safep_brls i) 

454 

455 
fun safe_tac thms i = DEPTH_SOLVE_1 (safestep_tac thms i) 

456 

457 
fun step_tac thms = safestep_tac thms ORELSE' biresolve_tac unsafe_brls 

458 

459 
(*Fails unless it solves the goal!*) 

460 
fun pc_tac thms = DEPTH_SOLVE_1 o (step_tac thms) 

461 
*} 

462 

463 
use "rew.ML" 

464 

465 

466 
subsection {* The elimination rules for fst/snd *} 

467 

468 
lemma SumE_fst: "p : Sum(A,B) ==> fst(p) : A" 

469 
apply (unfold basic_defs) 

470 
apply (erule SumE) 

471 
apply assumption 

472 
done 

473 

474 
(*The first premise must be p:Sum(A,B) !!*) 

475 
lemma SumE_snd: 

476 
assumes major: "p: Sum(A,B)" 

477 
and "A type" 

478 
and "!!x. x:A ==> B(x) type" 

479 
shows "snd(p) : B(fst(p))" 

480 
apply (unfold basic_defs) 

481 
apply (rule major [THEN SumE]) 

482 
apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) 

26391  483 
apply (tactic {* typechk_tac @{thms assms} *}) 
19761  484 
done 
485 

486 
end 