author | nipkow |
Fri, 16 Oct 1998 17:32:29 +0200 | |
changeset 5655 | afd75136b236 |
parent 5348 | 5f6416d64a94 |
child 6073 | fba734ba6894 |
permissions | -rw-r--r-- |
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 |
*) |
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8 |
||
9 |
open W; |
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10 |
||
5348 | 11 |
Addsimps [Suc_le_lessD]; Delsimps [less_imp_le]; (*the combination loops*) |
1300 | 12 |
|
13 |
val has_type_casesE = map(has_type.mk_cases expr.simps) |
|
2525 | 14 |
[" A |- Var n :: t"," A |- Abs e :: t","A |- App e1 e2 ::t","A |- LET e1 e2 ::t" ]; |
1300 | 15 |
|
16 |
(* the resulting type variable is always greater or equal than the given one *) |
|
5069 | 17 |
Goal |
2525 | 18 |
"!A n S t m. W e A n = Some (S,t,m) --> n<=m"; |
5184 | 19 |
by (induct_tac "e" 1); |
1300 | 20 |
(* case Var(n) *) |
4686 | 21 |
by (simp_tac (simpset() addsplits [split_option_bind]) 1); |
1300 | 22 |
(* case Abs e *) |
4089 | 23 |
by (simp_tac (simpset() addsplits [split_option_bind]) 1); |
1300 | 24 |
by (fast_tac (HOL_cs addDs [Suc_leD]) 1); |
25 |
(* case App e1 e2 *) |
|
4089 | 26 |
by (simp_tac (simpset() addsplits [split_option_bind]) 1); |
4423 | 27 |
by (blast_tac (claset() addIs [le_SucI,le_trans]) 1); |
2525 | 28 |
(* case LET e1 e2 *) |
4089 | 29 |
by (simp_tac (simpset() addsplits [split_option_bind]) 1); |
4423 | 30 |
by (blast_tac (claset() addIs [le_trans]) 1); |
1486 | 31 |
qed_spec_mp "W_var_ge"; |
1300 | 32 |
|
33 |
Addsimps [W_var_ge]; |
|
34 |
||
5069 | 35 |
Goal |
5118 | 36 |
"Some (S,t,m) = W e A n ==> n<=m"; |
4089 | 37 |
by (asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 1); |
1300 | 38 |
qed "W_var_geD"; |
39 |
||
5118 | 40 |
Goal "new_tv n A ==> Some (S,t,m) = W e A n ==> new_tv m A"; |
2525 | 41 |
by (dtac W_var_geD 1); |
42 |
by (rtac new_scheme_list_le 1); |
|
3018 | 43 |
by (assume_tac 1); |
44 |
by (assume_tac 1); |
|
2525 | 45 |
qed "new_tv_compatible_W"; |
1300 | 46 |
|
5118 | 47 |
Goal "new_tv n sch --> new_tv (n + (min_new_bound_tv sch)) (bound_typ_inst (%b. TVar (b + n)) sch)"; |
5184 | 48 |
by (induct_tac "sch" 1); |
4727 | 49 |
by (Asm_full_simp_tac 1); |
50 |
by (asm_full_simp_tac (simpset() addsimps [add_commute]) 1); |
|
2525 | 51 |
by (strip_tac 1); |
52 |
by (Asm_full_simp_tac 1); |
|
53 |
by (etac conjE 1); |
|
54 |
by (rtac conjI 1); |
|
4727 | 55 |
by (rtac new_tv_le 1); |
56 |
by (assume_tac 2); |
|
57 |
by (rtac add_le_mono 1); |
|
58 |
by (Simp_tac 1); |
|
59 |
by (simp_tac (simpset() addsimps [max_def]) 1); |
|
2525 | 60 |
by (rtac new_tv_le 1); |
4727 | 61 |
by (assume_tac 2); |
2525 | 62 |
by (rtac add_le_mono 1); |
4727 | 63 |
by (Simp_tac 1); |
4686 | 64 |
by (simp_tac (simpset() addsimps [max_def]) 1); |
2525 | 65 |
by (strip_tac 1); |
66 |
by (dtac not_leE 1); |
|
67 |
by (rtac less_or_eq_imp_le 1); |
|
68 |
by (Fast_tac 1); |
|
69 |
qed_spec_mp "new_tv_bound_typ_inst_sch"; |
|
70 |
||
71 |
Addsimps [new_tv_bound_typ_inst_sch]; |
|
1300 | 72 |
|
73 |
(* resulting type variable is new *) |
|
5069 | 74 |
Goal |
2525 | 75 |
"!n A S t m. new_tv n A --> W e A n = Some (S,t,m) --> \ |
76 |
\ new_tv m S & new_tv m t"; |
|
5184 | 77 |
by (induct_tac "e" 1); |
1300 | 78 |
(* case Var n *) |
4686 | 79 |
by (simp_tac (simpset() addsplits [split_option_bind]) 1); |
2525 | 80 |
by (strip_tac 1); |
81 |
by (dtac new_tv_nth_nat_A 1); |
|
3018 | 82 |
by (assume_tac 1); |
4033 | 83 |
by (Asm_simp_tac 1); |
1300 | 84 |
(* case Abs e *) |
4089 | 85 |
by (simp_tac (simpset() addsimps [new_tv_subst,new_tv_Suc_list] |
4072 | 86 |
addsplits [split_option_bind]) 1); |
1300 | 87 |
by (strip_tac 1); |
88 |
by (eres_inst_tac [("x","Suc n")] allE 1); |
|
2525 | 89 |
by (eres_inst_tac [("x","(FVar n)#A")] allE 1); |
4089 | 90 |
by (fast_tac (HOL_cs addss (simpset() |
1465 | 91 |
addsimps [new_tv_subst,new_tv_Suc_list])) 1); |
1300 | 92 |
(* case App e1 e2 *) |
4089 | 93 |
by (simp_tac (simpset() addsplits [split_option_bind]) 1); |
1300 | 94 |
by (strip_tac 1); |
4033 | 95 |
by (rename_tac "S1 t1 n1 S2 t2 n2 S3" 1); |
1300 | 96 |
by (eres_inst_tac [("x","n")] allE 1); |
2525 | 97 |
by (eres_inst_tac [("x","A")] allE 1); |
4033 | 98 |
by (eres_inst_tac [("x","S1")] allE 1); |
99 |
by (eres_inst_tac [("x","t1")] allE 1); |
|
2525 | 100 |
by (eres_inst_tac [("x","n1")] allE 1); |
101 |
by (eres_inst_tac [("x","n1")] allE 1); |
|
4089 | 102 |
by (asm_full_simp_tac (simpset() addsimps [eq_sym_conv] delsimps all_simps) 1); |
4033 | 103 |
by (eres_inst_tac [("x","$S1 A")] allE 1); |
104 |
by (eres_inst_tac [("x","S2")] allE 1); |
|
105 |
by (eres_inst_tac [("x","t2")] allE 1); |
|
2525 | 106 |
by (eres_inst_tac [("x","n2")] allE 1); |
4089 | 107 |
by ( asm_full_simp_tac (simpset() addsimps [o_def,rotate_Some]) 1); |
1300 | 108 |
by (rtac conjI 1); |
109 |
by (rtac new_tv_subst_comp_2 1); |
|
110 |
by (rtac new_tv_subst_comp_2 1); |
|
2525 | 111 |
by (rtac (lessI RS less_imp_le RS new_tv_le) 1); |
112 |
by (res_inst_tac [("n","n1")] new_tv_subst_le 1); |
|
4089 | 113 |
by (asm_full_simp_tac (simpset() addsimps [rotate_Some]) 1); |
1300 | 114 |
by (Asm_simp_tac 1); |
115 |
by (fast_tac (HOL_cs addDs [W_var_geD] addIs |
|
2525 | 116 |
[new_scheme_list_le,new_tv_subst_scheme_list,lessI RS less_imp_le RS new_tv_subst_le]) |
1300 | 117 |
1); |
1465 | 118 |
by (etac (sym RS mgu_new) 1); |
2525 | 119 |
by (best_tac (HOL_cs addDs [W_var_geD] addIs [new_tv_subst_te,new_scheme_list_le, |
120 |
new_tv_subst_scheme_list,lessI RS less_imp_le RS new_tv_le,lessI RS less_imp_le RS |
|
121 |
new_tv_subst_le,new_tv_le]) 1); |
|
122 |
by (fast_tac (HOL_cs addDs [W_var_geD] addIs |
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123 |
[new_scheme_list_le,new_tv_subst_scheme_list,new_tv_le] |
|
4089 | 124 |
addss (simpset())) 1); |
1465 | 125 |
by (rtac (lessI RS new_tv_subst_var) 1); |
126 |
by (etac (sym RS mgu_new) 1); |
|
1925 | 127 |
by (best_tac (HOL_cs addSIs [lessI RS less_imp_le RS new_tv_le,new_tv_subst_te] |
2525 | 128 |
addDs [W_var_geD] addIs |
129 |
[new_scheme_list_le,new_tv_subst_scheme_list,lessI RS less_imp_le RS |
|
4089 | 130 |
new_tv_subst_le,new_tv_le] addss simpset()) 1); |
2525 | 131 |
by (fast_tac (HOL_cs addDs [W_var_geD] addIs |
132 |
[new_scheme_list_le,new_tv_subst_scheme_list,new_tv_le] |
|
4089 | 133 |
addss (simpset())) 1); |
4033 | 134 |
(* 41: case LET e1 e2 *) |
4089 | 135 |
by (simp_tac (simpset() addsplits [split_option_bind]) 1); |
2525 | 136 |
by (strip_tac 1); |
4423 | 137 |
by (EVERY1[etac allE,etac allE,etac allE,etac allE,etac allE,mp_tac,mp_tac]); |
2525 | 138 |
by (etac conjE 1); |
4423 | 139 |
by (EVERY[etac allE 1,etac allE 1,etac allE 1,etac allE 1,etac allE 1, |
4033 | 140 |
etac impE 1, mp_tac 2]); |
141 |
by (SELECT_GOAL(rewtac new_tv_def)1); |
|
142 |
by (Asm_simp_tac 1); |
|
143 |
by (REPEAT(dtac W_var_ge 1)); |
|
144 |
by (rtac allI 1); |
|
145 |
by (strip_tac 1); |
|
146 |
by (SELECT_GOAL(rewtac free_tv_subst) 1); |
|
147 |
by (dtac (free_tv_app_subst_scheme_list RS subsetD) 1); |
|
4089 | 148 |
by (best_tac (claset() addEs [less_le_trans]) 1); |
2525 | 149 |
by (etac conjE 1); |
150 |
by (rtac conjI 1); |
|
151 |
by (rtac new_tv_subst_comp_2 1); |
|
4033 | 152 |
by (etac (W_var_ge RS new_tv_subst_le) 1); |
3018 | 153 |
by (assume_tac 1); |
154 |
by (assume_tac 1); |
|
155 |
by (assume_tac 1); |
|
1486 | 156 |
qed_spec_mp "new_tv_W"; |
1300 | 157 |
|
5118 | 158 |
Goal "(v ~: free_tv sch) --> (v : free_tv (bound_typ_inst (TVar o S) sch)) --> (? x. v = S x)"; |
5184 | 159 |
by (induct_tac "sch" 1); |
2525 | 160 |
by (Asm_full_simp_tac 1); |
161 |
by (Asm_full_simp_tac 1); |
|
162 |
by (strip_tac 1); |
|
163 |
by (rtac exI 1); |
|
164 |
by (rtac refl 1); |
|
165 |
by (Asm_full_simp_tac 1); |
|
166 |
qed_spec_mp "free_tv_bound_typ_inst1"; |
|
167 |
||
168 |
Addsimps [free_tv_bound_typ_inst1]; |
|
1300 | 169 |
|
5069 | 170 |
Goal |
2525 | 171 |
"!n A S t m v. W e A n = Some (S,t,m) --> \ |
172 |
\ (v:free_tv S | v:free_tv t) --> v<n --> v:free_tv A"; |
|
5184 | 173 |
by (induct_tac "e" 1); |
1300 | 174 |
(* case Var n *) |
4089 | 175 |
by (simp_tac (simpset() addsimps |
4686 | 176 |
[free_tv_subst] addsplits [split_option_bind]) 1); |
2525 | 177 |
by (strip_tac 1); |
4502 | 178 |
by (case_tac "v : free_tv (A!nat)" 1); |
2525 | 179 |
by (Asm_full_simp_tac 1); |
180 |
by (dtac free_tv_bound_typ_inst1 1); |
|
4089 | 181 |
by (simp_tac (simpset() addsimps [o_def]) 1); |
2525 | 182 |
by (etac exE 1); |
183 |
by (rotate_tac 3 1); |
|
184 |
by (Asm_full_simp_tac 1); |
|
1300 | 185 |
(* case Abs e *) |
4089 | 186 |
by (asm_full_simp_tac (simpset() addsimps |
4072 | 187 |
[free_tv_subst] addsplits [split_option_bind] delsimps all_simps) 1); |
1300 | 188 |
by (strip_tac 1); |
2525 | 189 |
by (rename_tac "S t n1 S1 t1 m v" 1); |
1300 | 190 |
by (eres_inst_tac [("x","Suc n")] allE 1); |
2525 | 191 |
by (eres_inst_tac [("x","FVar n # A")] allE 1); |
192 |
by (eres_inst_tac [("x","S")] allE 1); |
|
1300 | 193 |
by (eres_inst_tac [("x","t")] allE 1); |
2525 | 194 |
by (eres_inst_tac [("x","n1")] allE 1); |
1300 | 195 |
by (eres_inst_tac [("x","v")] allE 1); |
2525 | 196 |
by (best_tac (HOL_cs addIs [cod_app_subst] |
4089 | 197 |
addss (simpset() addsimps [less_Suc_eq])) 1); |
1300 | 198 |
(* case App e1 e2 *) |
4089 | 199 |
by (simp_tac (simpset() addsplits [split_option_bind] delsimps all_simps) 1); |
1300 | 200 |
by (strip_tac 1); |
2525 | 201 |
by (rename_tac "S t n1 S1 t1 n2 S2 S3 t2 m v" 1); |
1300 | 202 |
by (eres_inst_tac [("x","n")] allE 1); |
2525 | 203 |
by (eres_inst_tac [("x","A")] allE 1); |
204 |
by (eres_inst_tac [("x","S")] allE 1); |
|
1300 | 205 |
by (eres_inst_tac [("x","t")] allE 1); |
2525 | 206 |
by (eres_inst_tac [("x","n1")] allE 1); |
207 |
by (eres_inst_tac [("x","n1")] allE 1); |
|
1300 | 208 |
by (eres_inst_tac [("x","v")] allE 1); |
209 |
(* second case *) |
|
2525 | 210 |
by (eres_inst_tac [("x","$ S A")] allE 1); |
211 |
by (eres_inst_tac [("x","S1")] allE 1); |
|
212 |
by (eres_inst_tac [("x","t1")] allE 1); |
|
213 |
by (eres_inst_tac [("x","n2")] allE 1); |
|
1300 | 214 |
by (eres_inst_tac [("x","v")] allE 1); |
215 |
by (safe_tac (empty_cs addSIs [conjI,impI] addSEs [conjE]) ); |
|
4089 | 216 |
by (asm_full_simp_tac (simpset() addsimps [rotate_Some,o_def]) 1); |
1465 | 217 |
by (dtac W_var_geD 1); |
218 |
by (dtac W_var_geD 1); |
|
1300 | 219 |
by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) ); |
220 |
by (fast_tac (HOL_cs addDs [free_tv_comp_subst RS subsetD,sym RS mgu_free, |
|
2525 | 221 |
codD,free_tv_app_subst_te RS subsetD,free_tv_app_subst_scheme_list RS subsetD, |
1300 | 222 |
less_le_trans,less_not_refl2,subsetD] |
223 |
addEs [UnE] |
|
4089 | 224 |
addss simpset()) 1); |
1300 | 225 |
by (Asm_full_simp_tac 1); |
1465 | 226 |
by (dtac (sym RS W_var_geD) 1); |
227 |
by (dtac (sym RS W_var_geD) 1); |
|
1300 | 228 |
by ( (forward_tac [less_le_trans] 1) THEN (assume_tac 1) ); |
229 |
by (fast_tac (HOL_cs addDs [mgu_free, codD,free_tv_subst_var RS subsetD, |
|
2525 | 230 |
free_tv_app_subst_te RS subsetD,free_tv_app_subst_scheme_list RS subsetD, |
231 |
less_le_trans,subsetD] |
|
232 |
addEs [UnE] |
|
5655
afd75136b236
Mods because of: Installed trans_tac in solver of simpset().
nipkow
parents:
5348
diff
changeset
|
233 |
addss (simpset() setSolver unsafe_solver)) 1); |
2525 | 234 |
(* LET e1 e2 *) |
4089 | 235 |
by (simp_tac (simpset() addsplits [split_option_bind] delsimps all_simps) 1); |
2525 | 236 |
by (strip_tac 1); |
237 |
by (rename_tac "nat A S1 t1 n2 S2 t2 m2 S t m v" 1); |
|
238 |
by (eres_inst_tac [("x","nat")] allE 1); |
|
239 |
by (eres_inst_tac [("x","A")] allE 1); |
|
240 |
by (eres_inst_tac [("x","S1")] allE 1); |
|
241 |
by (eres_inst_tac [("x","t1")] allE 1); |
|
242 |
by (rotate_tac 1 1); |
|
243 |
by (eres_inst_tac [("x","n2")] allE 1); |
|
244 |
by (rotate_tac 4 1); |
|
245 |
by (eres_inst_tac [("x","v")] allE 1); |
|
246 |
by (mp_tac 1); |
|
247 |
by (EVERY1 [etac allE,etac allE,etac allE,etac allE,etac allE,eres_inst_tac [("x","v")] allE]); |
|
248 |
by (mp_tac 1); |
|
249 |
by (Asm_full_simp_tac 1); |
|
250 |
by (rtac conjI 1); |
|
4089 | 251 |
by (fast_tac (claset() addSDs [codD,free_tv_app_subst_scheme_list RS subsetD,free_tv_o_subst RS subsetD,W_var_ge] |
2525 | 252 |
addDs [less_le_trans]) 1); |
4089 | 253 |
by (fast_tac (claset() addSDs [codD,free_tv_app_subst_scheme_list RS subsetD,W_var_ge] |
2525 | 254 |
addDs [less_le_trans]) 1); |
1486 | 255 |
qed_spec_mp "free_tv_W"; |
1300 | 256 |
|
5118 | 257 |
Goal "(!x. x : A --> x ~: B) ==> A Int B = {}"; |
2525 | 258 |
by (Fast_tac 1); |
2625 | 259 |
val weaken_A_Int_B_eq_empty = result(); |
2525 | 260 |
|
5118 | 261 |
Goal "x ~: A | x : B ==> x ~: A - B"; |
2525 | 262 |
by (Fast_tac 1); |
2625 | 263 |
val weaken_not_elem_A_minus_B = result(); |
2525 | 264 |
|
265 |
(* correctness of W with respect to has_type *) |
|
5069 | 266 |
Goal |
2525 | 267 |
"!A S t m n . new_tv n A --> Some (S,t,m) = W e A n --> $S A |- e :: t"; |
5184 | 268 |
by (induct_tac "e" 1); |
2525 | 269 |
(* case Var n *) |
4686 | 270 |
by (Asm_full_simp_tac 1); |
2525 | 271 |
by (strip_tac 1); |
272 |
by (rtac has_type.VarI 1); |
|
273 |
by (Simp_tac 1); |
|
4089 | 274 |
by (simp_tac (simpset() addsimps [is_bound_typ_instance]) 1); |
2525 | 275 |
by (rtac exI 1); |
276 |
by (rtac refl 1); |
|
277 |
(* case Abs e *) |
|
4089 | 278 |
by (asm_full_simp_tac (simpset() addsimps [app_subst_list] |
4072 | 279 |
addsplits [split_option_bind]) 1); |
2525 | 280 |
by (strip_tac 1); |
281 |
by (eres_inst_tac [("x","(mk_scheme (TVar n)) # A")] allE 1); |
|
282 |
by (Asm_full_simp_tac 1); |
|
283 |
by (rtac has_type.AbsI 1); |
|
284 |
by (dtac (le_refl RS le_SucI RS new_scheme_list_le) 1); |
|
3018 | 285 |
by (dtac sym 1); |
2525 | 286 |
by (REPEAT (etac allE 1)); |
287 |
by (etac impE 1); |
|
288 |
by (mp_tac 2); |
|
289 |
by (Asm_simp_tac 1); |
|
3018 | 290 |
by (assume_tac 1); |
2525 | 291 |
(* case App e1 e2 *) |
4089 | 292 |
by (simp_tac (simpset() addsplits [split_option_bind]) 1); |
2525 | 293 |
by (strip_tac 1); |
294 |
by (rename_tac "S1 t1 n1 S2 t2 n2 S3" 1); |
|
295 |
by (res_inst_tac [("t2.0","$ S3 t2")] has_type.AppI 1); |
|
296 |
by (res_inst_tac [("S1","S3")] (app_subst_TVar RS subst) 1); |
|
297 |
by (rtac (app_subst_Fun RS subst) 1); |
|
298 |
by (res_inst_tac [("t","$S3 (t2 -> (TVar n2))"),("s","$S3 ($S2 t1)")] subst 1); |
|
299 |
by (Asm_full_simp_tac 1); |
|
300 |
by (simp_tac (HOL_ss addsimps [subst_comp_scheme_list RS sym]) 1); |
|
301 |
by ((rtac (has_type_cl_sub RS spec) 1) THEN (rtac (has_type_cl_sub RS spec) 1)); |
|
4089 | 302 |
by (asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 1); |
303 |
by (asm_full_simp_tac (simpset() addsimps [subst_comp_scheme_list RS sym,o_def,has_type_cl_sub,eq_sym_conv]) 1); |
|
2525 | 304 |
by (rtac (has_type_cl_sub RS spec) 1); |
305 |
by (forward_tac [new_tv_W] 1); |
|
3018 | 306 |
by (assume_tac 1); |
2525 | 307 |
by (dtac conjunct1 1); |
308 |
by (dtac conjunct1 1); |
|
309 |
by (forward_tac [new_tv_subst_scheme_list] 1); |
|
310 |
by (rtac new_scheme_list_le 1); |
|
311 |
by (rtac W_var_ge 1); |
|
3018 | 312 |
by (assume_tac 1); |
313 |
by (assume_tac 1); |
|
2525 | 314 |
by (etac thin_rl 1); |
315 |
by (REPEAT (etac allE 1)); |
|
3018 | 316 |
by (dtac sym 1); |
317 |
by (dtac sym 1); |
|
2525 | 318 |
by (etac thin_rl 1); |
319 |
by (etac thin_rl 1); |
|
320 |
by (mp_tac 1); |
|
321 |
by (mp_tac 1); |
|
3018 | 322 |
by (assume_tac 1); |
2525 | 323 |
(* case Let e1 e2 *) |
4089 | 324 |
by (simp_tac (simpset() addsplits [split_option_bind]) 1); |
2525 | 325 |
by (strip_tac 1); |
326 |
by (rename_tac "w q m1 S1 t1 m2 S2 t n2" 1); |
|
327 |
by (res_inst_tac [("t1.0","$ S2 t1")] has_type.LETI 1); |
|
4089 | 328 |
by (simp_tac (simpset() addsimps [o_def]) 1); |
2525 | 329 |
by (simp_tac (HOL_ss addsimps [subst_comp_scheme_list RS sym]) 1); |
330 |
by (rtac (has_type_cl_sub RS spec) 1); |
|
331 |
by (dres_inst_tac [("x","A")] spec 1); |
|
332 |
by (dres_inst_tac [("x","S1")] spec 1); |
|
333 |
by (dres_inst_tac [("x","t1")] spec 1); |
|
334 |
by (dres_inst_tac [("x","m2")] spec 1); |
|
335 |
by (rotate_tac 4 1); |
|
336 |
by (dres_inst_tac [("x","m1")] spec 1); |
|
337 |
by (mp_tac 1); |
|
4089 | 338 |
by (asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 1); |
339 |
by (simp_tac (simpset() addsimps [o_def]) 1); |
|
2525 | 340 |
by (simp_tac (HOL_ss addsimps [subst_comp_scheme_list RS sym]) 1); |
341 |
by (rtac (gen_subst_commutes RS sym RS subst) 1); |
|
342 |
by (rtac (app_subst_Cons RS subst) 2); |
|
343 |
by (etac thin_rl 2); |
|
344 |
by (dres_inst_tac [("x","gen ($S1 A) t1 # $ S1 A")] spec 2); |
|
345 |
by (dres_inst_tac [("x","S2")] spec 2); |
|
346 |
by (dres_inst_tac [("x","t")] spec 2); |
|
347 |
by (dres_inst_tac [("x","n2")] spec 2); |
|
348 |
by (dres_inst_tac [("x","m2")] spec 2); |
|
349 |
by (forward_tac [new_tv_W] 2); |
|
3018 | 350 |
by (assume_tac 2); |
2525 | 351 |
by (dtac conjunct1 2); |
352 |
by (dtac conjunct1 2); |
|
353 |
by (forward_tac [new_tv_subst_scheme_list] 2); |
|
354 |
by (rtac new_scheme_list_le 2); |
|
355 |
by (rtac W_var_ge 2); |
|
3018 | 356 |
by (assume_tac 2); |
357 |
by (assume_tac 2); |
|
2525 | 358 |
by (etac impE 2); |
359 |
by (res_inst_tac [("A","$ S1 A")] new_tv_only_depends_on_free_tv_scheme_list 2); |
|
360 |
by (Simp_tac 2); |
|
361 |
by (Fast_tac 2); |
|
3018 | 362 |
by (assume_tac 2); |
2525 | 363 |
by (Asm_full_simp_tac 2); |
364 |
by (rtac weaken_A_Int_B_eq_empty 1); |
|
365 |
by (rtac allI 1); |
|
366 |
by (strip_tac 1); |
|
367 |
by (rtac weaken_not_elem_A_minus_B 1); |
|
368 |
by (case_tac "x < m2" 1); |
|
369 |
by (rtac disjI2 1); |
|
370 |
by (rtac (free_tv_gen_cons RS subst) 1); |
|
371 |
by (rtac free_tv_W 1); |
|
3018 | 372 |
by (assume_tac 1); |
2525 | 373 |
by (Asm_full_simp_tac 1); |
3018 | 374 |
by (assume_tac 1); |
2525 | 375 |
by (rtac disjI1 1); |
376 |
by (dtac new_tv_W 1); |
|
3018 | 377 |
by (assume_tac 1); |
2525 | 378 |
by (dtac conjunct2 1); |
379 |
by (dtac conjunct2 1); |
|
380 |
by (rtac new_tv_not_free_tv 1); |
|
381 |
by (rtac new_tv_le 1); |
|
3018 | 382 |
by (assume_tac 2); |
4089 | 383 |
by (asm_full_simp_tac (simpset() addsimps [not_less_iff_le]) 1); |
2525 | 384 |
qed_spec_mp "W_correct_lemma"; |
385 |
||
1300 | 386 |
(* Completeness of W w.r.t. has_type *) |
5069 | 387 |
Goal |
2525 | 388 |
"!S' A t' n. $S' A |- e :: t' --> new_tv n A --> \ |
389 |
\ (? S t. (? m. W e A n = Some (S,t,m)) & \ |
|
390 |
\ (? R. $S' A = $R ($S A) & t' = $R t))"; |
|
5184 | 391 |
by (induct_tac "e" 1); |
1300 | 392 |
(* case Var n *) |
393 |
by (strip_tac 1); |
|
4686 | 394 |
by (simp_tac (simpset() addcongs [conj_cong]) 1); |
1300 | 395 |
by (eresolve_tac has_type_casesE 1); |
4089 | 396 |
by (asm_full_simp_tac (simpset() addsimps [is_bound_typ_instance]) 1); |
2525 | 397 |
by (etac exE 1); |
398 |
by (hyp_subst_tac 1); |
|
399 |
by (rename_tac "S" 1); |
|
400 |
by (res_inst_tac [("x","%x. (if x < n then S' x else S (x - n))")] exI 1); |
|
401 |
by (rtac conjI 1); |
|
1300 | 402 |
by (Asm_simp_tac 1); |
4089 | 403 |
by (asm_simp_tac (simpset() addsimps [(bound_typ_inst_composed_subst RS sym),new_tv_nth_nat_A,o_def,nth_subst] |
2525 | 404 |
delsimps [bound_typ_inst_composed_subst]) 1); |
2749
2f477a0e690d
Deleted steps made redundant by the stronger eq_assume_tac
paulson
parents:
2637
diff
changeset
|
405 |
(** LEVEL 12 **) |
1300 | 406 |
(* case Abs e *) |
407 |
by (strip_tac 1); |
|
408 |
by (eresolve_tac has_type_casesE 1); |
|
3842 | 409 |
by (eres_inst_tac [("x","%x. if x=n then t1 else (S' x)")] allE 1); |
2525 | 410 |
by (eres_inst_tac [("x","(FVar n)#A")] allE 1); |
1300 | 411 |
by (eres_inst_tac [("x","t2")] allE 1); |
412 |
by (eres_inst_tac [("x","Suc n")] allE 1); |
|
2749
2f477a0e690d
Deleted steps made redundant by the stronger eq_assume_tac
paulson
parents:
2637
diff
changeset
|
413 |
by (best_tac (HOL_cs addSDs [mk_scheme_injective] |
4089 | 414 |
addss (simpset() addcongs [conj_cong] |
4072 | 415 |
addsplits [split_option_bind])) 1); |
2749
2f477a0e690d
Deleted steps made redundant by the stronger eq_assume_tac
paulson
parents:
2637
diff
changeset
|
416 |
(** LEVEL 19 **) |
1300 | 417 |
|
418 |
(* case App e1 e2 *) |
|
419 |
by (strip_tac 1); |
|
420 |
by (eresolve_tac has_type_casesE 1); |
|
2525 | 421 |
by (eres_inst_tac [("x","S'")] allE 1); |
422 |
by (eres_inst_tac [("x","A")] allE 1); |
|
1400 | 423 |
by (eres_inst_tac [("x","t2 -> t'")] allE 1); |
1300 | 424 |
by (eres_inst_tac [("x","n")] allE 1); |
425 |
by (safe_tac HOL_cs); |
|
2749
2f477a0e690d
Deleted steps made redundant by the stronger eq_assume_tac
paulson
parents:
2637
diff
changeset
|
426 |
(** LEVEL 26 **) |
2525 | 427 |
by (eres_inst_tac [("x","R")] allE 1); |
428 |
by (eres_inst_tac [("x","$ S A")] allE 1); |
|
1300 | 429 |
by (eres_inst_tac [("x","t2")] allE 1); |
430 |
by (eres_inst_tac [("x","m")] allE 1); |
|
431 |
by (Asm_full_simp_tac 1); |
|
432 |
by (safe_tac HOL_cs); |
|
433 |
by (fast_tac (HOL_cs addIs [sym RS W_var_geD,new_tv_W RS |
|
2525 | 434 |
conjunct1,new_scheme_list_le,new_tv_subst_scheme_list]) 1); |
2779
9c42ae57b5f4
The contr_tac, which replaces a fast_tac, is needed only because eq_assume_tac
paulson
parents:
2749
diff
changeset
|
435 |
(** LEVEL 35 **) |
1300 | 436 |
by (subgoal_tac |
3842 | 437 |
"$ (%x. if x=ma then t' else (if x:(free_tv t - free_tv Sa) then R x \ |
2525 | 438 |
\ else Ra x)) ($ Sa t) = \ |
3842 | 439 |
\ $ (%x. if x=ma then t' else (if x:(free_tv t - free_tv Sa) then R x \ |
2525 | 440 |
\ else Ra x)) (ta -> (TVar ma))" 1); |
1300 | 441 |
by (res_inst_tac [("t","$ (%x. if x = ma then t' else \ |
2525 | 442 |
\ (if x:(free_tv t - free_tv Sa) then R x else Ra x)) ($ Sa t)"), |
443 |
("s","($ Ra ta) -> t'")] ssubst 2); |
|
4089 | 444 |
by (asm_simp_tac (simpset() addsimps [subst_comp_te]) 2); |
1465 | 445 |
by (rtac eq_free_eq_subst_te 2); |
1300 | 446 |
by (strip_tac 2); |
447 |
by (subgoal_tac "na ~=ma" 2); |
|
2749
2f477a0e690d
Deleted steps made redundant by the stronger eq_assume_tac
paulson
parents:
2637
diff
changeset
|
448 |
by (best_tac (HOL_cs addDs [new_tv_W,sym RS W_var_geD, |
2525 | 449 |
new_tv_not_free_tv,new_tv_le]) 3); |
450 |
by (case_tac "na:free_tv Sa" 2); |
|
451 |
(* n1 ~: free_tv S1 *) |
|
452 |
by (forward_tac [not_free_impl_id] 3); |
|
4686 | 453 |
by (Asm_simp_tac 3); |
2525 | 454 |
(* na : free_tv Sa *) |
455 |
by (dres_inst_tac [("A1","$ S A")] (subst_comp_scheme_list RSN (2,trans)) 2); |
|
456 |
by (dtac eq_subst_scheme_list_eq_free 2); |
|
1300 | 457 |
by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2); |
458 |
by (Asm_simp_tac 2); |
|
2525 | 459 |
by (case_tac "na:dom Sa" 2); |
460 |
(* na ~: dom Sa *) |
|
4686 | 461 |
by (asm_full_simp_tac (simpset() addsimps [dom_def]) 3); |
2525 | 462 |
(* na : dom Sa *) |
1465 | 463 |
by (rtac eq_free_eq_subst_te 2); |
1300 | 464 |
by (strip_tac 2); |
465 |
by (subgoal_tac "nb ~= ma" 2); |
|
466 |
by ((forward_tac [new_tv_W] 3) THEN (atac 3)); |
|
1465 | 467 |
by (etac conjE 3); |
2525 | 468 |
by (dtac new_tv_subst_scheme_list 3); |
469 |
by (fast_tac (HOL_cs addIs [new_scheme_list_le] addDs [sym RS W_var_geD]) 3); |
|
1300 | 470 |
by (fast_tac (set_cs addDs [new_tv_W,new_tv_not_free_tv] addss |
4089 | 471 |
(simpset() addsimps [cod_def,free_tv_subst])) 3); |
4686 | 472 |
by (fast_tac (set_cs addss (simpset() addsimps [cod_def,free_tv_subst])) 2); |
1300 | 473 |
by (Simp_tac 2); |
1465 | 474 |
by (rtac eq_free_eq_subst_te 2); |
1300 | 475 |
by (strip_tac 2 ); |
476 |
by (subgoal_tac "na ~= ma" 2); |
|
477 |
by ((forward_tac [new_tv_W] 3) THEN (atac 3)); |
|
1465 | 478 |
by (etac conjE 3); |
479 |
by (dtac (sym RS W_var_geD) 3); |
|
2525 | 480 |
by (fast_tac (HOL_cs addDs [new_scheme_list_le,new_tv_subst_scheme_list,new_tv_W,new_tv_not_free_tv]) 3); |
481 |
by (case_tac "na: free_tv t - free_tv Sa" 2); |
|
482 |
(* case na ~: free_tv t - free_tv Sa *) |
|
4686 | 483 |
by (Asm_full_simp_tac 3); |
2793
b30c41754c86
Modified proofs because simplifier does not eta-contract any longer.
nipkow
parents:
2779
diff
changeset
|
484 |
by (Fast_tac 3); |
2525 | 485 |
(* case na : free_tv t - free_tv Sa *) |
4686 | 486 |
by (Asm_full_simp_tac 2); |
2525 | 487 |
by (dres_inst_tac [("A1","$ S A")] (subst_comp_scheme_list RSN (2,trans)) 2); |
488 |
by (dtac eq_subst_scheme_list_eq_free 2); |
|
1300 | 489 |
by (fast_tac (HOL_cs addIs [free_tv_W,free_tv_le_new_tv] addDs [new_tv_W]) 2); |
2793
b30c41754c86
Modified proofs because simplifier does not eta-contract any longer.
nipkow
parents:
2779
diff
changeset
|
490 |
(** LEVEL 75 **) |
4089 | 491 |
by (asm_full_simp_tac (simpset() addsimps [free_tv_subst,dom_def]) 2); |
492 |
by (asm_simp_tac (simpset() addsplits [split_option_bind]) 1); |
|
1300 | 493 |
by (safe_tac HOL_cs ); |
2525 | 494 |
by (dtac mgu_Some 1); |
4089 | 495 |
by ( fast_tac (HOL_cs addss simpset()) 1); |
2793
b30c41754c86
Modified proofs because simplifier does not eta-contract any longer.
nipkow
parents:
2779
diff
changeset
|
496 |
(** LEVEL 80 *) |
1300 | 497 |
by ((dtac mgu_mg 1) THEN (atac 1)); |
1465 | 498 |
by (etac exE 1); |
2525 | 499 |
by (res_inst_tac [("x","Rb")] exI 1); |
1465 | 500 |
by (rtac conjI 1); |
1300 | 501 |
by (dres_inst_tac [("x","ma")] fun_cong 2); |
4089 | 502 |
by ( asm_full_simp_tac (simpset() addsimps [eq_sym_conv]) 2); |
503 |
by (simp_tac (simpset() addsimps [subst_comp_scheme_list]) 1); |
|
504 |
by (simp_tac (simpset() addsimps [subst_comp_scheme_list RS sym]) 1); |
|
2793
b30c41754c86
Modified proofs because simplifier does not eta-contract any longer.
nipkow
parents:
2779
diff
changeset
|
505 |
by (res_inst_tac [("A2","($ Sa ($ S A))")] |
b30c41754c86
Modified proofs because simplifier does not eta-contract any longer.
nipkow
parents:
2779
diff
changeset
|
506 |
((subst_comp_scheme_list RS sym) RSN (2,trans)) 1); |
4089 | 507 |
by ( asm_full_simp_tac (simpset() addsimps [o_def,eq_sym_conv]) 1); |
2525 | 508 |
by (rtac eq_free_eq_subst_scheme_list 1); |
3018 | 509 |
by ( safe_tac HOL_cs ); |
1300 | 510 |
by (subgoal_tac "ma ~= na" 1); |
511 |
by ((forward_tac [new_tv_W] 2) THEN (atac 2)); |
|
1465 | 512 |
by (etac conjE 2); |
2525 | 513 |
by (dtac new_tv_subst_scheme_list 2); |
514 |
by (fast_tac (HOL_cs addIs [new_scheme_list_le] addDs [sym RS W_var_geD]) 2); |
|
2793
b30c41754c86
Modified proofs because simplifier does not eta-contract any longer.
nipkow
parents:
2779
diff
changeset
|
515 |
by (forw_inst_tac [("n","m")] new_tv_W 2 THEN atac 2); |
1465 | 516 |
by (etac conjE 2); |
2525 | 517 |
by (dtac (free_tv_app_subst_scheme_list RS subsetD) 2); |
2793
b30c41754c86
Modified proofs because simplifier does not eta-contract any longer.
nipkow
parents:
2779
diff
changeset
|
518 |
by (fast_tac (set_cs addDs [sym RS W_var_geD,new_scheme_list_le,codD, |
1300 | 519 |
new_tv_not_free_tv]) 2); |
2525 | 520 |
by (case_tac "na: free_tv t - free_tv Sa" 1); |
521 |
(* case na ~: free_tv t - free_tv Sa *) |
|
4686 | 522 |
by (Asm_full_simp_tac 2); |
2525 | 523 |
(* case na : free_tv t - free_tv Sa *) |
4686 | 524 |
by (Asm_full_simp_tac 1); |
2525 | 525 |
by (dtac (free_tv_app_subst_scheme_list RS subsetD) 1); |
526 |
by (fast_tac (set_cs addDs [codD,subst_comp_scheme_list RSN (2,trans), |
|
4089 | 527 |
eq_subst_scheme_list_eq_free] addss ((simpset() addsimps |
2525 | 528 |
[free_tv_subst,dom_def]))) 1); |
2083
b56425a385b9
Tidied some proofs: changed needed for de Morgan laws
paulson
parents:
2058
diff
changeset
|
529 |
by (Fast_tac 1); |
2525 | 530 |
(* case Let e1 e2 *) |
531 |
by (eresolve_tac has_type_casesE 1); |
|
532 |
by (eres_inst_tac [("x","S'")] allE 1); |
|
533 |
by (eres_inst_tac [("x","A")] allE 1); |
|
534 |
by (eres_inst_tac [("x","t1")] allE 1); |
|
535 |
by (eres_inst_tac [("x","n")] allE 1); |
|
536 |
by (mp_tac 1); |
|
537 |
by (mp_tac 1); |
|
538 |
by (safe_tac HOL_cs); |
|
539 |
by (Asm_simp_tac 1); |
|
540 |
by (eres_inst_tac [("x","R")] allE 1); |
|
541 |
by (eres_inst_tac [("x","gen ($ S A) t # $ S A")] allE 1); |
|
542 |
by (eres_inst_tac [("x","t'")] allE 1); |
|
543 |
by (eres_inst_tac [("x","m")] allE 1); |
|
544 |
by (Asm_full_simp_tac 1); |
|
545 |
by (dtac mp 1); |
|
546 |
by (rtac has_type_le_env 1); |
|
3018 | 547 |
by (assume_tac 1); |
2525 | 548 |
by (Simp_tac 1); |
549 |
by (rtac gen_bound_typ_instance 1); |
|
550 |
by (dtac mp 1); |
|
551 |
by (forward_tac [new_tv_compatible_W] 1); |
|
552 |
by (rtac sym 1); |
|
3018 | 553 |
by (assume_tac 1); |
4089 | 554 |
by (fast_tac (claset() addDs [new_tv_compatible_gen,new_tv_subst_scheme_list,new_tv_W]) 1); |
2525 | 555 |
by (safe_tac HOL_cs); |
556 |
by (Asm_full_simp_tac 1); |
|
557 |
by (res_inst_tac [("x","Ra")] exI 1); |
|
4089 | 558 |
by (simp_tac (simpset() addsimps [o_def,subst_comp_scheme_list RS sym]) 1); |
1525 | 559 |
qed_spec_mp "W_complete_lemma"; |
560 |
||
5069 | 561 |
Goal |
5118 | 562 |
"[] |- e :: t' ==> (? S t. (? m. W e [] n = Some(S,t,m)) & \ |
2525 | 563 |
\ (? R. t' = $ R t))"; |
3018 | 564 |
by (cut_inst_tac [("A","[]"),("S'","id_subst"),("e","e"),("t'","t'")] |
1525 | 565 |
W_complete_lemma 1); |
3018 | 566 |
by (ALLGOALS Asm_full_simp_tac); |
1525 | 567 |
qed "W_complete"; |