author  paulson 
Thu, 26 Sep 1996 12:47:47 +0200  
changeset 2031  03a843f0f447 
parent 1950  97f1c6bf3ace 
child 2058  ff04984186e9 
permissions  rwrr 
1300  1 
(* Title: HOL/MiniML/W.ML 
2 
ID: $Id$ 

3 
Author: Dieter Nazareth and Tobias Nipkow 

4 
Copyright 1995 TU Muenchen 

5 

6 
Correctness and completeness of type inference algorithm W 

7 
*) 

8 

9 
open W; 

10 

11 
Addsimps [Suc_le_lessD]; 

1950
97f1c6bf3ace
Miniscoping rules are deleted, as these brittle proofs
paulson
parents:
1925
diff
changeset

12 
Delsimps (ex_simps @ all_simps); 
1300  13 

14 
(* correctness of W with respect to has_type *) 

1525  15 
goal W.thy 
16 
"!a s t m n . Ok (s,t,m) = W e a n > $s a  e :: t"; 

1300  17 
by (expr.induct_tac "e" 1); 
18 
(* case Var n *) 

19 
by (asm_simp_tac (!simpset setloop (split_tac [expand_if])) 1); 

20 
(* case Abs e *) 

21 
by (asm_full_simp_tac (!simpset addsimps [app_subst_list] 

22 
setloop (split_tac [expand_bind])) 1); 

23 
by (strip_tac 1); 

24 
by (eres_inst_tac [("x","TVar(n) # a")] allE 1); 

2031  25 
by ( fast_tac (HOL_cs addss (!simpset addsimps [eq_sym_conv])) 1); 
1300  26 
(* case App e1 e2 *) 
27 
by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1); 

28 
by (strip_tac 1); 

2031  29 
by ( rename_tac "sa ta na sb tb nb sc" 1); 
1300  30 
by (res_inst_tac [("t2.0","$ sc tb")] has_type.AppI 1); 
31 
by (res_inst_tac [("s1","sc")] (app_subst_TVar RS subst) 1); 

32 
by (rtac (app_subst_Fun RS subst) 1); 

1525  33 
by (res_inst_tac [("t","$sc(tb > (TVar nb))"),("s","$sc($sb ta)")] subst 1); 
1300  34 
by (Asm_full_simp_tac 1); 
35 
by (simp_tac (HOL_ss addsimps [subst_comp_tel RS sym]) 1); 

36 
by ( (rtac has_type_cl_sub 1) THEN (rtac has_type_cl_sub 1)); 

37 
by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1); 

1669  38 
by (asm_full_simp_tac (!simpset addsimps [subst_comp_tel RS sym,o_def,has_type_cl_sub,eq_sym_conv]) 1); 
1486  39 
qed_spec_mp "W_correct"; 
1300  40 

41 
val has_type_casesE = map(has_type.mk_cases expr.simps) 

1465  42 
[" s  Var n :: t"," s  Abs e :: t","s  App e1 e2 ::t"]; 
1300  43 

44 

45 
(* the resulting type variable is always greater or equal than the given one *) 

46 
goal thy 

1465  47 
"!a n s t m. W e a n = Ok (s,t,m) > n<=m"; 
1300  48 
by (expr.induct_tac "e" 1); 
49 
(* case Var(n) *) 

50 
by (fast_tac (HOL_cs addss (!simpset setloop (split_tac [expand_if]))) 1); 

51 
(* case Abs e *) 

52 
by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1); 

53 
by (fast_tac (HOL_cs addDs [Suc_leD]) 1); 

54 
(* case App e1 e2 *) 

55 
by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1); 

56 
by (strip_tac 1); 

57 
by (rename_tac "s t na sa ta nb sb sc tb m" 1); 

58 
by (eres_inst_tac [("x","a")] allE 1); 

59 
by (eres_inst_tac [("x","n")] allE 1); 

60 
by (eres_inst_tac [("x","$ s a")] allE 1); 

61 
by (eres_inst_tac [("x","s")] allE 1); 

62 
by (eres_inst_tac [("x","t")] allE 1); 

63 
by (eres_inst_tac [("x","na")] allE 1); 

64 
by (eres_inst_tac [("x","na")] allE 1); 

65 
by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1); 

66 
by (etac conjE 1); 

67 
by (eres_inst_tac [("x","sa")] allE 1); 

68 
by (eres_inst_tac [("x","ta")] allE 1); 

69 
by (eres_inst_tac [("x","nb")] allE 1); 

70 
by (etac conjE 1); 

71 
by (res_inst_tac [("j","na")] le_trans 1); 

72 
by (Asm_simp_tac 1); 

1669  73 
by (Asm_simp_tac 1); 
1486  74 
qed_spec_mp "W_var_ge"; 
1300  75 

76 
Addsimps [W_var_ge]; 

77 

78 
goal thy 

1465  79 
"!! s. Ok (s,t,m) = W e a n ==> n<=m"; 
1300  80 
by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1); 
81 
qed "W_var_geD"; 

82 

83 

84 
(* auxiliary lemma *) 

85 
goal Maybe.thy "(y = Ok x) = (Ok x = y)"; 

2031  86 
by ( simp_tac (!simpset addsimps [eq_sym_conv]) 1); 
1300  87 
qed "rotate_Ok"; 
88 

89 

90 
(* resulting type variable is new *) 

91 
goal thy 

92 
"!n a s t m. new_tv n a > W e a n = Ok (s,t,m) > \ 

1525  93 
\ new_tv m s & new_tv m t"; 
1300  94 
by (expr.induct_tac "e" 1); 
95 
(* case Var n *) 

96 
by (fast_tac (HOL_cs addss (!simpset 

1465  97 
addsimps [id_subst_def,list_all_nth,new_tv_list,new_tv_subst] 
1300  98 
setloop (split_tac [expand_if]))) 1); 
99 

100 
(* case Abs e *) 

101 
by (simp_tac (!simpset addsimps [new_tv_subst,new_tv_Suc_list] 

102 
setloop (split_tac [expand_bind])) 1); 

103 
by (strip_tac 1); 

104 
by (eres_inst_tac [("x","Suc n")] allE 1); 

105 
by (eres_inst_tac [("x","(TVar n)#a")] allE 1); 

106 
by (fast_tac (HOL_cs addss (!simpset 

1465  107 
addsimps [new_tv_subst,new_tv_Suc_list])) 1); 
1300  108 

109 
(* case App e1 e2 *) 

110 
by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1); 

111 
by (strip_tac 1); 

112 
by (rename_tac "s t na sa ta nb sb sc tb m" 1); 

113 
by (eres_inst_tac [("x","n")] allE 1); 

114 
by (eres_inst_tac [("x","a")] allE 1); 

115 
by (eres_inst_tac [("x","s")] allE 1); 

116 
by (eres_inst_tac [("x","t")] allE 1); 

117 
by (eres_inst_tac [("x","na")] allE 1); 

118 
by (eres_inst_tac [("x","na")] allE 1); 

119 
by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 1); 

120 
by (eres_inst_tac [("x","$ s a")] allE 1); 

121 
by (eres_inst_tac [("x","sa")] allE 1); 

122 
by (eres_inst_tac [("x","ta")] allE 1); 

123 
by (eres_inst_tac [("x","nb")] allE 1); 

2031  124 
by ( asm_full_simp_tac (!simpset addsimps [o_def,rotate_Ok]) 1); 
1300  125 
by (rtac conjI 1); 
126 
by (rtac new_tv_subst_comp_2 1); 

127 
by (rtac new_tv_subst_comp_2 1); 

128 
by (rtac (lessI RS less_imp_le RS new_tv_subst_le) 1); 

129 
by (res_inst_tac [("n","na")] new_tv_subst_le 1); 

130 
by (asm_full_simp_tac (!simpset addsimps [rotate_Ok]) 1); 

131 
by (Asm_simp_tac 1); 

132 
by (fast_tac (HOL_cs addDs [W_var_geD] addIs 

133 
[new_tv_list_le,new_tv_subst_tel,lessI RS less_imp_le RS new_tv_subst_le]) 

134 
1); 

1465  135 
by (etac (sym RS mgu_new) 1); 
1925  136 
by (best_tac (HOL_cs addDs [W_var_geD] 
2031  137 
addIs [new_tv_subst_te,new_tv_list_le, 
138 
new_tv_subst_tel, 

139 
lessI RS less_imp_le RS new_tv_le, 

140 
lessI RS less_imp_le RS new_tv_subst_le, 

141 
new_tv_le]) 1); 

1925  142 
by (fast_tac (HOL_cs addDs [W_var_geD] 
2031  143 
addIs [new_tv_list_le,new_tv_subst_tel,new_tv_le] 
144 
addss (!simpset)) 1); 

1465  145 
by (rtac (lessI RS new_tv_subst_var) 1); 
146 
by (etac (sym RS mgu_new) 1); 

1925  147 
by (best_tac (HOL_cs addSIs [lessI RS less_imp_le RS new_tv_le,new_tv_subst_te] 
148 
addDs [W_var_geD] 

2031  149 
addIs [new_tv_list_le, 
150 
new_tv_subst_tel, 

151 
lessI RS less_imp_le RS new_tv_subst_le, 

152 
new_tv_le] 

153 
addss !simpset) 1); 

1925  154 
by (fast_tac (HOL_cs addDs [W_var_geD] 
2031  155 
addIs [new_tv_list_le,new_tv_subst_tel,new_tv_le] 
1925  156 
addss (!simpset)) 1); 
1486  157 
qed_spec_mp "new_tv_W"; 
1300  158 

159 

160 
goal thy 

1465  161 
"!n a s t m v. W e a n = Ok (s,t,m) > \ 
1300  162 
\ (v:free_tv s  v:free_tv t) > v<n > v:free_tv a"; 
163 
by (expr.induct_tac "e" 1); 

164 
(* case Var n *) 

165 
by (fast_tac (HOL_cs addIs [nth_mem,subsetD,ftv_mem_sub_ftv_list] 

166 
addss (!simpset setloop (split_tac [expand_if]))) 1); 

167 

168 
(* case Abs e *) 

169 
by (asm_full_simp_tac (!simpset addsimps 

170 
[free_tv_subst] setloop (split_tac [expand_bind])) 1); 

171 
by (strip_tac 1); 

172 
by (rename_tac "s t na sa ta m v" 1); 

173 
by (eres_inst_tac [("x","Suc n")] allE 1); 

174 
by (eres_inst_tac [("x","TVar n # a")] allE 1); 

175 
by (eres_inst_tac [("x","s")] allE 1); 

176 
by (eres_inst_tac [("x","t")] allE 1); 

177 
by (eres_inst_tac [("x","na")] allE 1); 

178 
by (eres_inst_tac [("x","v")] allE 1); 

1669  179 
by (fast_tac (HOL_cs addIs [cod_app_subst] 
180 
addss (!simpset addsimps [less_Suc_eq])) 1); 

1300  181 

182 
(* case App e1 e2 *) 

183 
by (simp_tac (!simpset setloop (split_tac [expand_bind])) 1); 

184 
by (strip_tac 1); 

185 
by (rename_tac "s t na sa ta nb sb sc tb m v" 1); 

186 
by (eres_inst_tac [("x","n")] allE 1); 

187 
by (eres_inst_tac [("x","a")] allE 1); 

188 
by (eres_inst_tac [("x","s")] allE 1); 

189 
by (eres_inst_tac [("x","t")] allE 1); 

190 
by (eres_inst_tac [("x","na")] allE 1); 

191 
by (eres_inst_tac [("x","na")] allE 1); 

192 
by (eres_inst_tac [("x","v")] allE 1); 

193 
(* second case *) 

194 
by (eres_inst_tac [("x","$ s a")] allE 1); 

195 
by (eres_inst_tac [("x","sa")] allE 1); 

196 
by (eres_inst_tac [("x","ta")] allE 1); 

197 
by (eres_inst_tac [("x","nb")] allE 1); 

198 
by (eres_inst_tac [("x","v")] allE 1); 

199 
by (safe_tac (empty_cs addSIs [conjI,impI] addSEs [conjE]) ); 

1669  200 
by (asm_full_simp_tac (!simpset addsimps [rotate_Ok,o_def]) 1); 
1465  201 
by (dtac W_var_geD 1); 
202 
by (dtac W_var_geD 1); 

1300  203 
by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) ); 
204 
by (fast_tac (HOL_cs addDs [free_tv_comp_subst RS subsetD,sym RS mgu_free, 

205 
codD,free_tv_app_subst_te RS subsetD,free_tv_app_subst_tel RS subsetD, 

206 
less_le_trans,less_not_refl2,subsetD] 

207 
addEs [UnE] 

208 
addss !simpset) 1); 

209 
by (Asm_full_simp_tac 1); 

1465  210 
by (dtac (sym RS W_var_geD) 1); 
211 
by (dtac (sym RS W_var_geD) 1); 

1300  212 
by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) ); 
213 
by (fast_tac (HOL_cs addDs [mgu_free, codD,free_tv_subst_var RS subsetD, 

214 
free_tv_app_subst_te RS subsetD,free_tv_app_subst_tel RS subsetD, 

215 
less_le_trans,subsetD] 

216 
addEs [UnE] 

217 
addss !simpset) 1); 

1486  218 
qed_spec_mp "free_tv_W"; 
1300  219 

220 
(* Completeness of W w.r.t. has_type *) 

221 
goal thy 

1525  222 
"!s' a t' n. $s' a  e :: t' > new_tv n a > \ 
223 
\ (? s t. (? m. W e a n = Ok (s,t,m)) & \ 

224 
\ (? r. $s' a = $r ($s a) & t' = $r t))"; 

1300  225 
by (expr.induct_tac "e" 1); 
226 
(* case Var n *) 

227 
by (strip_tac 1); 

228 
by (simp_tac (!simpset addcongs [conj_cong] 

2031  229 
setloop (split_tac [expand_if])) 1); 
1300  230 
by (eresolve_tac has_type_casesE 1); 
231 
by (asm_full_simp_tac (!simpset addsimps [eq_sym_conv,app_subst_list]) 1); 

232 
by (res_inst_tac [("x","id_subst")] exI 1); 

233 
by (res_inst_tac [("x","nth nat a")] exI 1); 

234 
by (Simp_tac 1); 

235 
by (res_inst_tac [("x","s'")] exI 1); 

236 
by (Asm_simp_tac 1); 

237 

238 
(* case Abs e *) 

239 
by (strip_tac 1); 

240 
by (eresolve_tac has_type_casesE 1); 

241 
by (eres_inst_tac [("x","%x.if x=n then t1 else (s' x)")] allE 1); 

242 
by (eres_inst_tac [("x","(TVar n)#a")] allE 1); 

243 
by (eres_inst_tac [("x","t2")] allE 1); 

244 
by (eres_inst_tac [("x","Suc n")] allE 1); 

245 
by (fast_tac (HOL_cs addss (!simpset addcongs [conj_cong] 

2031  246 
setloop (split_tac [expand_bind]))) 1); 
1300  247 

248 
(* case App e1 e2 *) 

249 
by (strip_tac 1); 

250 
by (eresolve_tac has_type_casesE 1); 

251 
by (eres_inst_tac [("x","s'")] allE 1); 

252 
by (eres_inst_tac [("x","a")] allE 1); 

1400  253 
by (eres_inst_tac [("x","t2 > t'")] allE 1); 
1300  254 
by (eres_inst_tac [("x","n")] allE 1); 
255 
by (safe_tac HOL_cs); 

256 
by (eres_inst_tac [("x","r")] allE 1); 

257 
by (eres_inst_tac [("x","$ s a")] allE 1); 

258 
by (eres_inst_tac [("x","t2")] allE 1); 

259 
by (eres_inst_tac [("x","m")] allE 1); 

1465  260 
by (dtac asm_rl 1); 
261 
by (dtac asm_rl 1); 

262 
by (dtac asm_rl 1); 

1300  263 
by (Asm_full_simp_tac 1); 
264 
by (safe_tac HOL_cs); 

265 
by (fast_tac HOL_cs 1); 

266 
by (fast_tac (HOL_cs addIs [sym RS W_var_geD,new_tv_W RS 

2031  267 
conjunct1,new_tv_list_le,new_tv_subst_tel]) 1); 
1300  268 

269 
by (subgoal_tac 

270 
"$ (%x.if x=ma then t' else (if x:(free_tv t  free_tv sa) then r x \ 

271 
\ else ra x)) ($ sa t) = \ 

272 
\ $ (%x.if x=ma then t' else (if x:(free_tv t  free_tv sa) then r x \ 

1400  273 
\ else ra x)) (ta > (TVar ma))" 1); 
1300  274 
by (res_inst_tac [("t","$ (%x. if x = ma then t' else \ 
275 
\ (if x:(free_tv t  free_tv sa) then r x else ra x)) ($ sa t)"), 

1400  276 
("s","($ ra ta) > t'")] ssubst 2); 
1300  277 
by (asm_simp_tac (!simpset addsimps [subst_comp_te]) 2); 
1465  278 
by (rtac eq_free_eq_subst_te 2); 
1300  279 
by (strip_tac 2); 
280 
by (subgoal_tac "na ~=ma" 2); 

281 
by (fast_tac (HOL_cs addDs [new_tv_W,sym RS W_var_geD, 

2031  282 
new_tv_not_free_tv,new_tv_le]) 3); 
1300  283 
by (case_tac "na:free_tv sa" 2); 
284 
(* na ~: free_tv sa *) 

285 
by (asm_simp_tac (!simpset addsimps [not_free_impl_id] 

2031  286 
setloop (split_tac [expand_if])) 3); 
1300  287 
(* na : free_tv sa *) 
1400  288 
by (dres_inst_tac [("ts1","$ s a")] (subst_comp_tel RSN (2,trans)) 2); 
1465  289 
by (dtac eq_subst_tel_eq_free 2); 
1300  290 
by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2); 
291 
by (Asm_simp_tac 2); 

292 
by (case_tac "na:dom sa" 2); 

293 
(* na ~: dom sa *) 

294 
by (asm_full_simp_tac (!simpset addsimps [dom_def] 

2031  295 
setloop (split_tac [expand_if])) 3); 
1300  296 
(* na : dom sa *) 
1465  297 
by (rtac eq_free_eq_subst_te 2); 
1300  298 
by (strip_tac 2); 
299 
by (subgoal_tac "nb ~= ma" 2); 

300 
by ((forward_tac [new_tv_W] 3) THEN (atac 3)); 

1465  301 
by (etac conjE 3); 
302 
by (dtac new_tv_subst_tel 3); 

1300  303 
by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 3); 
304 
by (fast_tac (set_cs addDs [new_tv_W,new_tv_not_free_tv] addss 

2031  305 
(!simpset addsimps [cod_def,free_tv_subst])) 3); 
1300  306 
by (fast_tac (set_cs addss (!simpset addsimps [cod_def,free_tv_subst] 
2031  307 
setloop (split_tac [expand_if]))) 2); 
1300  308 

309 
by (Simp_tac 2); 

1465  310 
by (rtac eq_free_eq_subst_te 2); 
1300  311 
by (strip_tac 2 ); 
312 
by (subgoal_tac "na ~= ma" 2); 

313 
by ((forward_tac [new_tv_W] 3) THEN (atac 3)); 

1465  314 
by (etac conjE 3); 
315 
by (dtac (sym RS W_var_geD) 3); 

1300  316 
by (fast_tac (HOL_cs addDs [new_tv_list_le,new_tv_subst_tel,new_tv_W,new_tv_not_free_tv]) 3); 
317 
by (case_tac "na: free_tv t  free_tv sa" 2); 

318 
(* case na ~: free_tv t  free_tv sa *) 

2031  319 
by ( asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 3); 
1300  320 
(* case na : free_tv t  free_tv sa *) 
2031  321 
by ( asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 2); 
1400  322 
by (dres_inst_tac [("ts1","$ s a")] (subst_comp_tel RSN (2,trans)) 2); 
1465  323 
by (dtac eq_subst_tel_eq_free 2); 
1300  324 
by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2); 
1486  325 
by (asm_full_simp_tac (!simpset addsimps [free_tv_subst,dom_def,de_Morgan_disj]) 2); 
1300  326 

327 
by (asm_simp_tac (!simpset setloop (split_tac [expand_bind])) 1); 

328 
by (safe_tac HOL_cs ); 

1465  329 
by (dtac mgu_Ok 1); 
2031  330 
by ( fast_tac (HOL_cs addss !simpset) 1); 
1300  331 
by (REPEAT (resolve_tac [exI,conjI] 1)); 
332 
by (fast_tac HOL_cs 1); 

333 
by (fast_tac HOL_cs 1); 

334 
by ((dtac mgu_mg 1) THEN (atac 1)); 

1465  335 
by (etac exE 1); 
1300  336 
by (res_inst_tac [("x","rb")] exI 1); 
1465  337 
by (rtac conjI 1); 
1300  338 
by (dres_inst_tac [("x","ma")] fun_cong 2); 
2031  339 
by ( asm_full_simp_tac (!simpset addsimps [eq_sym_conv]) 2); 
1300  340 
by (simp_tac (!simpset addsimps [subst_comp_tel RS sym]) 1); 
1400  341 
by (res_inst_tac [("ts2","($ sa ($ s a))")] ((subst_comp_tel RS sym) RSN 
1300  342 
(2,trans)) 1); 
2031  343 
by ( asm_full_simp_tac (!simpset addsimps [o_def,eq_sym_conv]) 1); 
1465  344 
by (rtac eq_free_eq_subst_tel 1); 
2031  345 
by ( safe_tac HOL_cs ); 
1300  346 
by (subgoal_tac "ma ~= na" 1); 
347 
by ((forward_tac [new_tv_W] 2) THEN (atac 2)); 

1465  348 
by (etac conjE 2); 
349 
by (dtac new_tv_subst_tel 2); 

1300  350 
by (fast_tac (HOL_cs addIs [new_tv_list_le] addDs [sym RS W_var_geD]) 2); 
1486  351 
by (( forw_inst_tac [("n","m")] (sym RSN (2,new_tv_W)) 2) THEN (atac 2)); 
1465  352 
by (etac conjE 2); 
353 
by (dtac (free_tv_app_subst_tel RS subsetD) 2); 

1300  354 
by (fast_tac (set_cs addDs [W_var_geD,new_tv_list_le,codD, 
355 
new_tv_not_free_tv]) 2); 

356 
by (case_tac "na: free_tv t  free_tv sa" 1); 

357 
(* case na ~: free_tv t  free_tv sa *) 

358 
by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 2); 

359 
(* case na : free_tv t  free_tv sa *) 

360 
by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1); 

1465  361 
by (dtac (free_tv_app_subst_tel RS subsetD) 1); 
1300  362 
by (fast_tac (set_cs addDs [codD,subst_comp_tel RSN (2,trans), 
363 
eq_subst_tel_eq_free] addss ((!simpset addsimps 

1486  364 
[de_Morgan_disj,free_tv_subst,dom_def]))) 1); 
1525  365 
qed_spec_mp "W_complete_lemma"; 
366 

367 
goal W.thy 

368 
"!!e. []  e :: t' ==> (? s t. (? m. W e [] n = Ok(s,t,m)) & \ 

369 
\ (? r. t' = $r t))"; 

2031  370 
by (cut_inst_tac [("a","[]"),("s'","id_subst"),("e","e"),("t'","t'")] 
1525  371 
W_complete_lemma 1); 
2031  372 
by (ALLGOALS Asm_full_simp_tac); 
1525  373 
qed "W_complete"; 