author  wenzelm 
Mon, 10 Nov 2014 21:49:48 +0100  
changeset 58963  26bf09b95dda 
parent 58889  5b7a9633cfa8 
child 58972  5b026cfc5f04 
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 

58889  6 
section {* Constructive Type Theory *} 
0  7 

17441  8 
theory CTT 
9 
imports Pure 

10 
begin 

11 

48891  12 
ML_file "~~/src/Provers/typedsimp.ML" 
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 

0  96 
(*Reduction: a weaker notion than equality; a hack for simplification. 
97 
Reduce[a,b] means either that a=b:A for some A or else that "a" and "b" 

98 
are textually identical.*) 

99 

100 
(*does not verify a:A! Sound because only trans_red uses a Reduce premise 

101 
No new theorems can be proved about the standard judgements.*) 

51308  102 
axiomatization where 
103 
refl_red: "\<And>a. Reduce[a,a]" and 

104 
red_if_equal: "\<And>a b A. a = b : A ==> Reduce[a,b]" and 

105 
trans_red: "\<And>a b c A. [ a = b : A; Reduce[b,c] ] ==> a = c : A" and 

0  106 

107 
(*Reflexivity*) 

108 

51308  109 
refl_type: "\<And>A. A type ==> A = A" and 
110 
refl_elem: "\<And>a A. a : A ==> a = a : A" and 

0  111 

112 
(*Symmetry*) 

113 

51308  114 
sym_type: "\<And>A B. A = B ==> B = A" and 
115 
sym_elem: "\<And>a b A. a = b : A ==> b = a : A" and 

0  116 

117 
(*Transitivity*) 

118 

51308  119 
trans_type: "\<And>A B C. [ A = B; B = C ] ==> A = C" and 
120 
trans_elem: "\<And>a b c A. [ a = b : A; b = c : A ] ==> a = c : A" and 

0  121 

51308  122 
equal_types: "\<And>a A B. [ a : A; A = B ] ==> a : B" and 
123 
equal_typesL: "\<And>a b A B. [ a = b : A; A = B ] ==> a = b : B" and 

0  124 

125 
(*Substitution*) 

126 

51308  127 
subst_type: "\<And>a A B. [ a : A; !!z. z:A ==> B(z) type ] ==> B(a) type" and 
128 
subst_typeL: "\<And>a c A B D. [ a = c : A; !!z. z:A ==> B(z) = D(z) ] ==> B(a) = D(c)" and 

0  129 

51308  130 
subst_elem: "\<And>a b A B. [ a : A; !!z. z:A ==> b(z):B(z) ] ==> b(a):B(a)" and 
17441  131 
subst_elemL: 
51308  132 
"\<And>a b c d A B. [ a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) ] ==> b(a)=d(c) : B(a)" and 
0  133 

134 

135 
(*The type N  natural numbers*) 

136 

51308  137 
NF: "N type" and 
138 
NI0: "0 : N" and 

139 
NI_succ: "\<And>a. a : N ==> succ(a) : N" and 

140 
NI_succL: "\<And>a b. a = b : N ==> succ(a) = succ(b) : N" and 

0  141 

17441  142 
NE: 
51308  143 
"\<And>p a b C. [ p: N; a: C(0); !!u v. [ u: N; v: C(u) ] ==> b(u,v): C(succ(u)) ] 
144 
==> rec(p, a, %u v. b(u,v)) : C(p)" and 

0  145 

17441  146 
NEL: 
51308  147 
"\<And>p q a b c d C. [ p = q : N; a = c : C(0); 
17441  148 
!!u v. [ u: N; v: C(u) ] ==> b(u,v) = d(u,v): C(succ(u)) ] 
51308  149 
==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)" and 
0  150 

17441  151 
NC0: 
51308  152 
"\<And>a b C. [ a: C(0); !!u v. [ u: N; v: C(u) ] ==> b(u,v): C(succ(u)) ] 
153 
==> rec(0, a, %u v. b(u,v)) = a : C(0)" and 

0  154 

17441  155 
NC_succ: 
51308  156 
"\<And>p a b C. [ p: N; a: C(0); 
17441  157 
!!u v. [ u: N; v: C(u) ] ==> b(u,v): C(succ(u)) ] ==> 
51308  158 
rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))" and 
0  159 

160 
(*The fourth Peano axiom. See page 91 of MartinLof's book*) 

17441  161 
zero_ne_succ: 
51308  162 
"\<And>a. [ a: N; 0 = succ(a) : N ] ==> 0: F" and 
0  163 

164 

165 
(*The Product of a family of types*) 

166 

51308  167 
ProdF: "\<And>A B. [ A type; !!x. x:A ==> B(x) type ] ==> PROD x:A. B(x) type" and 
0  168 

17441  169 
ProdFL: 
51308  170 
"\<And>A B C D. [ A = C; !!x. x:A ==> B(x) = D(x) ] ==> 
171 
PROD x:A. B(x) = PROD x:C. D(x)" and 

0  172 

17441  173 
ProdI: 
51308  174 
"\<And>b A B. [ A type; !!x. x:A ==> b(x):B(x)] ==> lam x. b(x) : PROD x:A. B(x)" and 
0  175 

17441  176 
ProdIL: 
51308  177 
"\<And>b c A B. [ A type; !!x. x:A ==> b(x) = c(x) : B(x)] ==> 
178 
lam x. b(x) = lam x. c(x) : PROD x:A. B(x)" and 

0  179 

51308  180 
ProdE: "\<And>p a A B. [ p : PROD x:A. B(x); a : A ] ==> p`a : B(a)" and 
181 
ProdEL: "\<And>p q a b A B. [ p=q: PROD x:A. B(x); a=b : A ] ==> p`a = q`b : B(a)" and 

0  182 

17441  183 
ProdC: 
51308  184 
"\<And>a b A B. [ a : A; !!x. x:A ==> b(x) : B(x)] ==> 
185 
(lam x. b(x)) ` a = b(a) : B(a)" and 

0  186 

17441  187 
ProdC2: 
51308  188 
"\<And>p A B. p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)" and 
0  189 

190 

191 
(*The Sum of a family of types*) 

192 

51308  193 
SumF: "\<And>A B. [ A type; !!x. x:A ==> B(x) type ] ==> SUM x:A. B(x) type" and 
17441  194 
SumFL: 
51308  195 
"\<And>A B C D. [ A = C; !!x. x:A ==> B(x) = D(x) ] ==> SUM x:A. B(x) = SUM x:C. D(x)" and 
0  196 

51308  197 
SumI: "\<And>a b A B. [ a : A; b : B(a) ] ==> <a,b> : SUM x:A. B(x)" and 
198 
SumIL: "\<And>a b c d A B. [ a=c:A; b=d:B(a) ] ==> <a,b> = <c,d> : SUM x:A. B(x)" and 

0  199 

17441  200 
SumE: 
51308  201 
"\<And>p c A B C. [ p: SUM x:A. B(x); !!x y. [ x:A; y:B(x) ] ==> c(x,y): C(<x,y>) ] 
202 
==> split(p, %x y. c(x,y)) : C(p)" and 

0  203 

17441  204 
SumEL: 
51308  205 
"\<And>p q c d A B C. [ p=q : SUM x:A. B(x); 
17441  206 
!!x y. [ x:A; y:B(x) ] ==> c(x,y)=d(x,y): C(<x,y>)] 
51308  207 
==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)" and 
0  208 

17441  209 
SumC: 
51308  210 
"\<And>a b c A B C. [ a: A; b: B(a); !!x y. [ x:A; y:B(x) ] ==> c(x,y): C(<x,y>) ] 
211 
==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)" and 

0  212 

51308  213 
fst_def: "\<And>a. fst(a) == split(a, %x y. x)" and 
214 
snd_def: "\<And>a. snd(a) == split(a, %x y. y)" and 

0  215 

216 

217 
(*The sum of two types*) 

218 

51308  219 
PlusF: "\<And>A B. [ A type; B type ] ==> A+B type" and 
220 
PlusFL: "\<And>A B C D. [ A = C; B = D ] ==> A+B = C+D" and 

0  221 

51308  222 
PlusI_inl: "\<And>a A B. [ a : A; B type ] ==> inl(a) : A+B" and 
223 
PlusI_inlL: "\<And>a c A B. [ a = c : A; B type ] ==> inl(a) = inl(c) : A+B" and 

0  224 

51308  225 
PlusI_inr: "\<And>b A B. [ A type; b : B ] ==> inr(b) : A+B" and 
226 
PlusI_inrL: "\<And>b d A B. [ A type; b = d : B ] ==> inr(b) = inr(d) : A+B" and 

0  227 

17441  228 
PlusE: 
51308  229 
"\<And>p c d A B C. [ p: A+B; !!x. x:A ==> c(x): C(inl(x)); 
17441  230 
!!y. y:B ==> d(y): C(inr(y)) ] 
51308  231 
==> when(p, %x. c(x), %y. d(y)) : C(p)" and 
0  232 

17441  233 
PlusEL: 
51308  234 
"\<And>p q c d e f A B C. [ p = q : A+B; !!x. x: A ==> c(x) = e(x) : C(inl(x)); 
17441  235 
!!y. y: B ==> d(y) = f(y) : C(inr(y)) ] 
51308  236 
==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)" and 
0  237 

17441  238 
PlusC_inl: 
51308  239 
"\<And>a c d A C. [ a: A; !!x. x:A ==> c(x): C(inl(x)); 
17441  240 
!!y. y:B ==> d(y): C(inr(y)) ] 
51308  241 
==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))" and 
0  242 

17441  243 
PlusC_inr: 
51308  244 
"\<And>b c d A B C. [ b: B; !!x. x:A ==> c(x): C(inl(x)); 
17441  245 
!!y. y:B ==> d(y): C(inr(y)) ] 
51308  246 
==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))" and 
0  247 

248 

249 
(*The type Eq*) 

250 

51308  251 
EqF: "\<And>a b A. [ A type; a : A; b : A ] ==> Eq(A,a,b) type" and 
252 
EqFL: "\<And>a b c d A B. [ A=B; a=c: A; b=d : A ] ==> Eq(A,a,b) = Eq(B,c,d)" and 

253 
EqI: "\<And>a b A. a = b : A ==> eq : Eq(A,a,b)" and 

254 
EqE: "\<And>p a b A. p : Eq(A,a,b) ==> a = b : A" and 

0  255 

256 
(*By equality of types, can prove C(p) from C(eq), an elimination rule*) 

51308  257 
EqC: "\<And>p a b A. p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)" and 
0  258 

259 
(*The type F*) 

260 

51308  261 
FF: "F type" and 
262 
FE: "\<And>p C. [ p: F; C type ] ==> contr(p) : C" and 

263 
FEL: "\<And>p q C. [ p = q : F; C type ] ==> contr(p) = contr(q) : C" and 

0  264 

265 
(*The type T 

266 
MartinLof's book (page 68) discusses elimination and computation. 

267 
Elimination can be derived by computation and equality of types, 

268 
but with an extra premise C(x) type x:T. 

269 
Also computation can be derived from elimination. *) 

270 

51308  271 
TF: "T type" and 
272 
TI: "tt : T" and 

273 
TE: "\<And>p c C. [ p : T; c : C(tt) ] ==> c : C(p)" and 

274 
TEL: "\<And>p q c d C. [ p = q : T; c = d : C(tt) ] ==> c = d : C(p)" and 

275 
TC: "\<And>p. p : T ==> p = tt : T" 

0  276 

19761  277 

278 
subsection "Tactics and derived rules for Constructive Type Theory" 

279 

280 
(*Formation rules*) 

281 
lemmas form_rls = NF ProdF SumF PlusF EqF FF TF 

282 
and formL_rls = ProdFL SumFL PlusFL EqFL 

283 

284 
(*Introduction rules 

285 
OMITTED: EqI, because its premise is an eqelem, not an elem*) 

286 
lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI 

287 
and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL 

288 

289 
(*Elimination rules 

290 
OMITTED: EqE, because its conclusion is an eqelem, not an elem 

291 
TE, because it does not involve a constructor *) 

292 
lemmas elim_rls = NE ProdE SumE PlusE FE 

293 
and elimL_rls = NEL ProdEL SumEL PlusEL FEL 

294 

295 
(*OMITTED: eqC are TC because they make rewriting loop: p = un = un = ... *) 

296 
lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr 

297 

298 
(*rules with conclusion a:A, an elem judgement*) 

299 
lemmas element_rls = intr_rls elim_rls 

300 

301 
(*Definitions are (meta)equality axioms*) 

302 
lemmas basic_defs = fst_def snd_def 

303 

304 
(*Compare with standard version: B is applied to UNSIMPLIFIED expression! *) 

305 
lemma SumIL2: "[ c=a : A; d=b : B(a) ] ==> <c,d> = <a,b> : Sum(A,B)" 

306 
apply (rule sym_elem) 

307 
apply (rule SumIL) 

308 
apply (rule_tac [!] sym_elem) 

309 
apply assumption+ 

310 
done 

311 

312 
lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL 

313 

314 
(*Exploit p:Prod(A,B) to create the assumption z:B(a). 

315 
A more natural form of product elimination. *) 

316 
lemma subst_prodE: 

317 
assumes "p: Prod(A,B)" 

318 
and "a: A" 

319 
and "!!z. z: B(a) ==> c(z): C(z)" 

320 
shows "c(p`a): C(p`a)" 

41526  321 
apply (rule assms ProdE)+ 
19761  322 
done 
323 

324 

325 
subsection {* Tactics for type checking *} 

326 

327 
ML {* 

328 

329 
local 

330 

56250  331 
fun is_rigid_elem (Const(@{const_name Elem},_) $ a $ _) = not(is_Var (head_of a)) 
332 
 is_rigid_elem (Const(@{const_name Eqelem},_) $ a $ _ $ _) = not(is_Var (head_of a)) 

333 
 is_rigid_elem (Const(@{const_name Type},_) $ a) = not(is_Var (head_of a)) 

19761  334 
 is_rigid_elem _ = false 
335 

336 
in 

337 

338 
(*Try solving a:A or a=b:A by assumption provided a is rigid!*) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

339 
fun test_assume_tac ctxt = SUBGOAL(fn (prem,i) => 
19761  340 
if is_rigid_elem (Logic.strip_assums_concl prem) 
58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

341 
then assume_tac ctxt i else no_tac) 
19761  342 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

343 
fun ASSUME ctxt tf i = test_assume_tac ctxt i ORELSE tf i 
19761  344 

345 
end; 

346 

347 
*} 

348 

349 
(*For simplification: type formation and checking, 

350 
but no equalities between terms*) 

351 
lemmas routine_rls = form_rls formL_rls refl_type element_rls 

352 

353 
ML {* 

354 
local 

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

355 
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

356 
@{thms elimL_rls} @ @{thms refl_elem} 
19761  357 
in 
358 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

359 
fun routine_tac rls ctxt prems = ASSUME ctxt (filt_resolve_tac (prems @ rls) 4); 
19761  360 

361 
(*Solve all subgoals "A type" using formation rules. *) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

362 
fun form_tac ctxt = REPEAT_FIRST (ASSUME ctxt (filt_resolve_tac @{thms form_rls} 1)); 
19761  363 

364 
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

365 
fun typechk_tac ctxt thms = 
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

366 
let val tac = filt_resolve_tac (thms @ @{thms form_rls} @ @{thms element_rls}) 3 
58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

367 
in REPEAT_FIRST (ASSUME ctxt tac) end 
19761  368 

369 
(*Solve a:A (a flexible, A rigid) by introduction rules. 

370 
Cannot use stringtrees (filt_resolve_tac) since 

371 
goals like ?a:SUM(A,B) have a trivial headstring *) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

372 
fun intr_tac ctxt thms = 
27208
5fe899199f85
proper context for tactics derived from res_inst_tac;
wenzelm
parents:
26956
diff
changeset

373 
let val tac = filt_resolve_tac(thms @ @{thms form_rls} @ @{thms intr_rls}) 1 
58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

374 
in REPEAT_FIRST (ASSUME ctxt tac) end 
19761  375 

376 
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

377 
fun equal_tac ctxt thms = 
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

378 
REPEAT_FIRST (ASSUME ctxt (filt_resolve_tac (thms @ equal_rls) 3)) 
0  379 

17441  380 
end 
19761  381 

382 
*} 

383 

384 

385 
subsection {* Simplification *} 

386 

387 
(*To simplify the type in a goal*) 

388 
lemma replace_type: "[ B = A; a : A ] ==> a : B" 

389 
apply (rule equal_types) 

390 
apply (rule_tac [2] sym_type) 

391 
apply assumption+ 

392 
done 

393 

394 
(*Simplify the parameter of a unary type operator.*) 

395 
lemma subst_eqtyparg: 

23467  396 
assumes 1: "a=c : A" 
397 
and 2: "!!z. z:A ==> B(z) type" 

19761  398 
shows "B(a)=B(c)" 
399 
apply (rule subst_typeL) 

400 
apply (rule_tac [2] refl_type) 

23467  401 
apply (rule 1) 
402 
apply (erule 2) 

19761  403 
done 
404 

405 
(*Simplification rules for Constructive Type Theory*) 

406 
lemmas reduction_rls = comp_rls [THEN trans_elem] 

407 

408 
ML {* 

409 
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. 

410 
Uses other intro rules to avoid changing flexible goals.*) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

411 
fun eqintr_tac ctxt = 
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

412 
REPEAT_FIRST (ASSUME ctxt (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) *) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

430 
fun add_mp_tac ctxt i = 
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

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

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

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

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

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

449 
fun safestep_tac ctxt thms i = 
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

450 
form_tac ctxt ORELSE 
19761  451 
resolve_tac thms i ORELSE 
58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

452 
biresolve_tac safe0_brls i ORELSE mp_tac ctxt i ORELSE 
19761  453 
DETERM (biresolve_tac safep_brls i) 
454 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

455 
fun safe_tac ctxt thms i = DEPTH_SOLVE_1 (safestep_tac ctxt thms i) 
19761  456 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

457 
fun step_tac ctxt thms = safestep_tac ctxt thms ORELSE' biresolve_tac unsafe_brls 
19761  458 

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

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

460 
fun pc_tac ctxt thms = DEPTH_SOLVE_1 o (step_tac ctxt thms) 
19761  461 
*} 
462 

48891  463 
ML_file "rew.ML" 
19761  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]) 

58963
26bf09b95dda
proper context for assume_tac (atac remains as fallback without context);
wenzelm
parents:
58889
diff
changeset

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

486 
end 