1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/Fsub/defn".
16 include "logic/equality.ma".
18 include "datatypes/bool.ma".
19 include "nat/compare.ma".
20 include "list/list.ma".
21 include "Fsub/util.ma".
23 (*** representation of Fsub types ***)
24 inductive Typ : Set \def
25 | TVar : nat \to Typ (* type var *)
26 | TFree: nat \to Typ (* free type name *)
27 | Top : Typ (* maximum type *)
28 | Arrow : Typ \to Typ \to Typ (* functions *)
29 | Forall : Typ \to Typ \to Typ. (* universal type *)
31 (*** representation of Fsub terms ***)
32 inductive Term : Set \def
33 | Var : nat \to Term (* variable *)
34 | Free : nat \to Term (* free name *)
35 | Abs : Typ \to Term \to Term (* abstraction *)
36 | App : Term \to Term \to Term (* function application *)
37 | TAbs : Typ \to Term \to Term (* type abstraction *)
38 | TApp : Term \to Typ \to Term. (* type application *)
40 (* representation of bounds *)
42 record bound : Set \def {
43 istype : bool; (* is subtyping bound? *)
44 name : nat ; (* name *)
45 btype : Typ (* type to which the name is bound *)
48 (* representation of Fsub typing environments *)
49 (*definition Env \def (list bound).
50 definition Empty \def (nil bound).
51 definition Cons \def \lambda G,X,T.((mk_bound false X T) :: G).
52 definition TCons \def \lambda G,X,T.((mk_bound true X T) :: G).
54 definition env_append : Env \to Env \to Env \def \lambda G,H.(H @ G). *)
56 (*** Various kinds of substitution, not all will be used probably ***)
58 (* substitutes i-th dangling index in type T with type U *)
59 let rec subst_type_nat T U i \def
61 [ (TVar n) \Rightarrow match (eqb n i) with
63 | false \Rightarrow T]
64 | (TFree X) \Rightarrow T
66 | (Arrow T1 T2) \Rightarrow (Arrow (subst_type_nat T1 U i) (subst_type_nat T2 U i))
67 | (Forall T1 T2) \Rightarrow (Forall (subst_type_nat T1 U i) (subst_type_nat T2 U (S i))) ].
69 (* substitutes 0-th dangling index in type T with type U *)
70 (*let rec subst_type_O T U \def subst_type_nat T U O.*)
72 (* substitutes 0-th dangling index in term t with term u *)
73 (*let rec subst_term_O t u \def
76 [ (Var n) \Rightarrow match (eqb n i) with
78 | false \Rightarrow t0]
79 | (Free X) \Rightarrow t0
80 | (Abs T1 t1) \Rightarrow (Abs T1 (aux t1 (S i)))
81 | (App t1 t2) \Rightarrow (App (aux t1 i) (aux t2 i))
82 | (TAbs T1 t1) \Rightarrow (TAbs T1 (aux t1 (S i)))
83 | (TApp t1 T1) \Rightarrow (TApp (aux t1 i) T1) ]
86 (* substitutes 0-th dangling index in term T, which shall be a TVar,
88 let rec subst_term_tO t T \def
91 [ (Var n) \Rightarrow t0
92 | (Free X) \Rightarrow t0
93 | (Abs T1 t1) \Rightarrow (Abs (subst_type_nat T1 T i) (aux t1 (S i)))
94 | (App t1 t2) \Rightarrow (App (aux t1 i) (aux t2 i))
95 | (TAbs T1 t1) \Rightarrow (TAbs (subst_type_nat T1 T i) (aux t1 (S i)))
96 | (TApp t1 T1) \Rightarrow (TApp (aux t1 i) (subst_type_nat T1 T i)) ]
99 (* substitutes (TFree X) in type T with type U *)
100 let rec subst_type_tfree_type T X U on T \def
102 [ (TVar n) \Rightarrow T
103 | (TFree Y) \Rightarrow match (eqb X Y) with
105 | false \Rightarrow T ]
107 | (Arrow T1 T2) \Rightarrow (Arrow (subst_type_tfree_type T1 X U)
108 (subst_type_tfree_type T2 X U))
109 | (Forall T1 T2) \Rightarrow (Forall (subst_type_tfree_type T1 X U)
110 (subst_type_tfree_type T2 X U)) ].*)
112 (*** height of T's syntactic tree ***)
116 [(TVar n) \Rightarrow (S O)
117 |(TFree X) \Rightarrow (S O)
118 |Top \Rightarrow (S O)
119 |(Arrow T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2)))
120 |(Forall T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2)))].
123 \lambda G:(list bound).match G with
124 [ nil \Rightarrow (mk_bound false O Top)
125 | (cons b H) \Rightarrow b].
127 definition head_nat \def
128 \lambda G:(list nat).match G with
130 | (cons n H) \Rightarrow n].
132 (*** definitions about lists ***)
134 (*(* var binding is in env judgement *)
135 definition var_bind_in_env : bound \to (list bound) \to Prop \def
136 \lambda b,G.(in_list bound b G).*)
138 definition fv_env : (list bound) \to (list nat) \def
139 \lambda G.(map ? ? (\lambda b.match b with
140 [(mk_bound B X T) \Rightarrow X]) G).
142 (*(* variable is in env judgement *)
143 definition var_in_env : nat \to (list bound) \to Prop \def
144 \lambda x,G.(in_list nat x (fv_env G)).
146 definition var_type_in_env : nat \to (list bound) \to Prop \def
147 \lambda x,G.\exists T.(var_bind_in_env (mk_bound true x T) G).*)
149 let rec fv_type T \def
151 [(TVar n) \Rightarrow []
152 |(TFree x) \Rightarrow [x]
154 |(Arrow U V) \Rightarrow ((fv_type U) @ (fv_type V))
155 |(Forall U V) \Rightarrow ((fv_type U) @ (fv_type V))].
157 (*** Type Well-Formedness judgement ***)
159 inductive WFType : (list bound) \to Typ \to Prop \def
160 | WFT_TFree : \forall X,G.(in_list ? X (fv_env G))
161 \to (WFType G (TFree X))
162 | WFT_Top : \forall G.(WFType G Top)
163 | WFT_Arrow : \forall G,T,U.(WFType G T) \to (WFType G U) \to
164 (WFType G (Arrow T U))
165 | WFT_Forall : \forall G,T,U.(WFType G T) \to
167 (\lnot (in_list ? X (fv_env G))) \to
168 (\lnot (in_list ? X (fv_type U))) \to
169 (WFType ((mk_bound true X T) :: G)
170 (subst_type_nat U (TFree X) O))) \to
171 (WFType G (Forall T U)).
173 (*** Environment Well-Formedness judgement ***)
175 inductive WFEnv : (list bound) \to Prop \def
176 | WFE_Empty : (WFEnv (nil ?))
177 | WFE_cons : \forall B,X,T,G.(WFEnv G) \to
178 \lnot (in_list ? X (fv_env G)) \to
179 (WFType G T) \to (WFEnv ((mk_bound B X T) :: G)).
181 (*** Subtyping judgement ***)
182 inductive JSubtype : (list bound) \to Typ \to Typ \to Prop \def
183 | SA_Top : \forall G.\forall T:Typ.(WFEnv G) \to
184 (WFType G T) \to (JSubtype G T Top)
185 | SA_Refl_TVar : \forall G.\forall X:nat.(WFEnv G)
186 \to (in_list ? X (fv_env G))
187 \to (JSubtype G (TFree X) (TFree X))
188 | SA_Trans_TVar : \forall G.\forall X:nat.\forall T:Typ.
190 (in_list ? (mk_bound true X U) G) \to
191 (JSubtype G U T) \to (JSubtype G (TFree X) T)
192 | SA_Arrow : \forall G.\forall S1,S2,T1,T2:Typ.
193 (JSubtype G T1 S1) \to (JSubtype G S2 T2) \to
194 (JSubtype G (Arrow S1 S2) (Arrow T1 T2))
195 | SA_All : \forall G.\forall S1,S2,T1,T2:Typ.
196 (JSubtype G T1 S1) \to
197 (\forall X:nat.\lnot (in_list ? X (fv_env G)) \to
198 (JSubtype ((mk_bound true X T1) :: G)
199 (subst_type_nat S2 (TFree X) O) (subst_type_nat T2 (TFree X) O))) \to
200 (JSubtype G (Forall S1 S2) (Forall T1 T2)).
202 notation "hvbox(e ⊢ break ta ⊴ break tb)"
203 non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
204 interpretation "Fsub subtype judgement" 'subjudg e ta tb =
205 (cic:/matita/Fsub/defn/JSubtype.ind#xpointer(1/1) e ta tb).
207 notation > "hvbox(\Forall S.T)"
208 non associative with precedence 60 for @{ 'forall $S $T}.
209 notation < "hvbox('All' \sub S. break T)"
210 non associative with precedence 60 for @{ 'forall $S $T}.
211 interpretation "universal type" 'forall S T =
212 (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/5) S T).
214 notation "#x" with precedence 79 for @{'tvar $x}.
215 interpretation "bound tvar" 'tvar x =
216 (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/1) x).
218 notation "!x" with precedence 79 for @{'tname $x}.
219 interpretation "bound tname" 'tname x =
220 (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/2) x).
222 notation "⊤" with precedence 90 for @{'toptype}.
223 interpretation "toptype" 'toptype =
224 (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/3)).
226 notation "hvbox(s break ⇛ t)"
227 right associative with precedence 55 for @{ 'arrow $s $t }.
228 interpretation "arrow type" 'arrow S T =
229 (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/4) S T).
231 notation "hvbox(S [# n ↦ T])"
232 non associative with precedence 80 for @{ 'substvar $S $T $n }.
233 interpretation "subst bound var" 'substvar S T n =
234 (cic:/matita/Fsub/defn/subst_type_nat.con S T n).
236 notation "hvbox(|T|)"
237 non associative with precedence 30 for @{ 'tlen $T }.
238 interpretation "type length" 'tlen T =
239 (cic:/matita/Fsub/defn/t_len.con T).
241 notation > "hvbox(x ∈ l)"
242 non associative with precedence 30 for @{ 'inlist $x $l }.
243 notation < "hvbox(x \nbsp ∈ \nbsp l)"
244 non associative with precedence 30 for @{ 'inlist $x $l }.
245 interpretation "item in list" 'inlist x l =
246 (cic:/matita/Fsub/util/in_list.ind#xpointer(1/1) _ x l).
248 notation "hvbox(!X ⊴ T)"
249 non associative with precedence 60 for @{ 'subtypebound $X $T }.
250 interpretation "subtyping bound" 'subtypebound X T =
251 (cic:/matita/Fsub/defn/bound.ind#xpointer(1/1/1) true X T).
253 (*notation < "hvbox(e break ⊢ ta \nbsp 'V' \nbsp tb (= \above \alpha))"
254 non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
255 notation > "hvbox(e break ⊢ ta 'Fall' break tb)"
256 non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.
257 notation "hvbox(e break ⊢ ta \lessdot break tb)"
258 non associative with precedence 30 for @{ 'subjudg $e $ta $tb }.*)
260 (****** PROOFS ********)
262 (*lemma subst_O_nat : \forall T,U.((subst_type_O T U) = T[#O↦U]).
263 intros;elim T;simplify;reflexivity;
266 (*** theorems about lists ***)
268 (* FIXME: these definitions shouldn't be part of the poplmark challenge
269 - use destruct instead, when hopefully it will get fixed... *)
271 lemma inj_head : \forall h1,h2:bound.\forall t1,t2:(list bound).
272 (h1::t1 = h2::t2) \to h1 = h2.
274 lapply (eq_f ? ? head ? ? H).simplify in Hletin.assumption.
277 lemma inj_head_nat : \forall h1,h2:nat.\forall t1,t2:(list nat).
278 (h1::t1 = h2::t2) \to (h1 = h2).
280 lapply (eq_f ? ? head_nat ? ? H).simplify in Hletin.assumption.
283 lemma inj_tail : \forall A.\forall h1,h2:A.\forall t1,t2:(list A).
284 ((h1::t1) = (h2::t2)) \to (t1 = t2).
285 intros.lapply (eq_f ? ? (tail ?) ? ? H).simplify in Hletin.assumption.
290 lemma boundinenv_natinfv : \forall x,G.
291 (\exists B,T.(in_list ? (mk_bound B x T) G)) \to
292 (in_list ? x (fv_env G)).
294 [elim H;elim H1;lapply (in_list_nil ? ? H2);elim Hletin
295 |elim H1;elim H2;inversion H3
296 [intros;rewrite < H4;simplify;apply in_Base
297 |intros;elim a3;simplify;apply in_Skip;
298 lapply (inj_tail ? ? ? ? ? H7);rewrite > Hletin in H;apply H;
303 |rewrite > H6;assumption]]]]
306 lemma nat_in_list_case : \forall G,H,n.(in_list nat n (H @ G)) \to
307 (in_list nat n G) \lor (in_list nat n H).
309 [simplify in H1;left;assumption
310 |simplify in H2;inversion H2
311 [intros;lapply (inj_head_nat ? ? ? ? H4);rewrite > Hletin;
313 |intros;lapply (inj_tail ? ? ? ? ? H6);rewrite < Hletin in H3;
314 rewrite > H5 in H1;lapply (H1 H3);elim Hletin1
315 [left;assumption|right;apply in_Skip;assumption]]]
318 lemma natinG_or_inH_to_natinGH : \forall G,H,n.
319 (in_list nat n G) \lor (in_list nat n H) \to
320 (in_list nat n (H @ G)).
324 |simplify;apply in_Skip;assumption]
325 |generalize in match H2;elim H2
326 [simplify;apply in_Base
327 |lapply (H4 H3);simplify;apply in_Skip;assumption]]
330 lemma natinfv_boundinenv : \forall x,G.(in_list ? x (fv_env G)) \to
331 \exists B,T.(in_list ? (mk_bound B x T) G).
333 [simplify;intro;lapply (in_list_nil ? ? H);elim Hletin
334 |intros 3;elim t;simplify in H1;inversion H1
335 [intros;rewrite < H2;simplify;apply ex_intro
339 |lapply (inj_head_nat ? ? ? ? H3);rewrite > H2;rewrite < Hletin;
341 |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2;
342 rewrite < H4 in H2;lapply (H H2);elim Hletin1;elim H6;apply ex_intro
346 |apply in_Skip;rewrite < H4;assumption]]]]
349 theorem varinT_varinT_subst : \forall X,Y,T.
350 (in_list ? X (fv_type T)) \to \forall n.
351 (in_list ? X (fv_type (subst_type_nat T (TFree Y) n))).
353 [simplify in H;elim (in_list_nil ? ? H)
354 |simplify in H;simplify;assumption
355 |simplify in H;elim (in_list_nil ? ? H)
356 |simplify in H2;simplify;elim (nat_in_list_case ? ? ? H2);
357 apply natinG_or_inH_to_natinGH;
360 |simplify in H2;simplify;elim (nat_in_list_case ? ? ? H2);
361 apply natinG_or_inH_to_natinGH;
363 |right;apply (H H3)]]
366 lemma incl_bound_fv : \forall l1,l2.(incl ? l1 l2) \to
367 (incl ? (fv_env l1) (fv_env l2)).
368 intros.unfold in H.unfold.intros.apply boundinenv_natinfv.
369 lapply (natinfv_boundinenv ? ? H1).elim Hletin.elim H2.apply ex_intro
376 lemma incl_nat_cons : \forall x,l1,l2.
377 (incl nat l1 l2) \to (incl nat (x :: l1) (x :: l2)).
378 intros.unfold in H.unfold.intros.inversion H1
379 [intros;lapply (inj_head_nat ? ? ? ? H3);rewrite > Hletin;apply in_Base
380 |intros;apply in_Skip;apply H;lapply (inj_tail ? ? ? ? ? H5);rewrite > Hletin;
384 lemma WFT_env_incl : \forall G,T.(WFType G T) \to
385 \forall H.(incl ? (fv_env G) (fv_env H)) \to (WFType H T).
387 [apply WFT_TFree;unfold in H3;apply (H3 ? H1)
389 |apply WFT_Arrow [apply (H2 ? H6)|apply (H4 ? H6)]
393 [unfold;intro;apply H7;apply(H6 ? H9)
395 |simplify;apply (incl_nat_cons ? ? ? H6)]]]
398 lemma fv_env_extends : \forall H,x,B,C,T,U,G.
399 (fv_env (H @ ((mk_bound B x T) :: G))) =
400 (fv_env (H @ ((mk_bound C x U) :: G))).
402 [simplify;reflexivity
403 |elim t;simplify;rewrite > H1;reflexivity]
406 lemma lookup_env_extends : \forall G,H,B,C,D,T,U,V,x,y.
407 (in_list ? (mk_bound D y V) (H @ ((mk_bound C x U) :: G))) \to
409 (in_list ? (mk_bound D y V) (H @ ((mk_bound B x T) :: G))).
411 [simplify in H1;(*FIXME*)generalize in match H1;intro;inversion H1
412 [intros;lapply (inj_head ? ? ? ? H5);rewrite < H4 in Hletin;
413 destruct Hletin;absurd (y = x) [symmetry;assumption|assumption]
414 |intros;simplify;lapply (inj_tail ? ? ? ? ? H7);rewrite > Hletin;
415 apply in_Skip;assumption]
416 |(*FIXME*)generalize in match H2;intro;inversion H2
417 [intros;simplify in H6;lapply (inj_head ? ? ? ? H6);rewrite > Hletin;
418 simplify;apply in_Base
419 |simplify;intros;lapply (inj_tail ? ? ? ? ? H8);rewrite > Hletin in H1;
420 rewrite > H7 in H1;apply in_Skip;apply (H1 H5 H3)]]
423 lemma in_FV_subst : \forall x,T,U,n.(in_list ? x (fv_type T)) \to
424 (in_list ? x (fv_type (subst_type_nat T U n))).
426 [simplify in H;inversion H
427 [intros;lapply (sym_eq ? ? ? H2);absurd (a::l = [])
428 [assumption|apply nil_cons]
429 |intros;lapply (sym_eq ? ? ? H4);absurd (a1::l = [])
430 [assumption|apply nil_cons]]
431 |2,3:simplify;simplify in H;assumption
432 |*:simplify in H2;simplify;apply natinG_or_inH_to_natinGH;
433 lapply (nat_in_list_case ? ? ? H2);elim Hletin
434 [1,3:left;apply (H1 ? H3)
435 |*:right;apply (H ? H3)]]
438 (*** lemma on fresh names ***)
440 lemma fresh_name : \forall l:(list nat).\exists n.\lnot (in_list ? n l).
441 cut (\forall l:(list nat).\exists n.\forall m.
442 (n \leq m) \to \lnot (in_list ? m l))
443 [intros;lapply (Hcut l);elim Hletin;apply ex_intro
445 |apply H;constructor 1]
449 |intros;unfold;intro;inversion H1
450 [intros;lapply (sym_eq ? ? ? H3);absurd (a::l1 = [])
451 [assumption|apply nil_cons]
452 |intros;lapply (sym_eq ? ? ? H5);absurd (a1::l1 = [])
453 [assumption|apply nil_cons]]]
454 |elim H;lapply (decidable_eq_nat a t);elim Hletin
457 |intros;unfold;intro;inversion H4
458 [intros;lapply (inj_head_nat ? ? ? ? H6);rewrite < Hletin1 in H5;
459 rewrite < H2 in H5;rewrite > H5 in H3;
460 apply (not_le_Sn_n ? H3)
461 |intros;lapply (inj_tail ? ? ? ? ? H8);rewrite < Hletin1 in H5;
463 apply (H1 m ? H5);lapply (le_S ? ? H3);
464 apply (le_S_S_to_le ? ? Hletin2)]]
465 |cut ((leb a t) = true \lor (leb a t) = false)
469 |intros;unfold;intro;inversion H5
470 [intros;lapply (inj_head_nat ? ? ? ? H7);rewrite > H6 in H4;
471 rewrite < Hletin1 in H4;apply (not_le_Sn_n ? H4)
472 |intros;lapply (inj_tail ? ? ? ? ? H9);
473 rewrite < Hletin1 in H6;lapply (H1 a1)
475 |lapply (leb_to_Prop a t);rewrite > H3 in Hletin2;
476 simplify in Hletin2;rewrite < H8;
477 apply (trans_le ? ? ? Hletin2);
478 apply (trans_le ? ? ? ? H4);constructor 2;constructor 1]]]
481 |intros;lapply (leb_to_Prop a t);rewrite > H3 in Hletin1;
482 simplify in Hletin1;lapply (not_le_to_lt ? ? Hletin1);
483 unfold in Hletin2;unfold;intro;inversion H5
484 [intros;lapply (inj_head_nat ? ? ? ? H7);
485 rewrite < Hletin3 in H6;rewrite > H6 in H4;
487 |intros;lapply (inj_tail ? ? ? ? ? H9);
488 rewrite < Hletin3 in H6;rewrite < H8 in H6;
489 apply (H1 ? H4 H6)]]]
490 |elim (leb a t);autobatch]]]]
493 (*** lemmata on well-formedness ***)
495 lemma fv_WFT : \forall T,x,G.(WFType G T) \to (in_list ? x (fv_type T)) \to
496 (in_list ? x (fv_env G)).
498 [simplify in H2;inversion H2
499 [intros;lapply (inj_head_nat ? ? ? ? H4);rewrite < Hletin;assumption
500 |intros;lapply (inj_tail ? ? ? ? ? H6);rewrite < Hletin in H3;
502 [intros;lapply (sym_eq ? ? ? H8);absurd (a2 :: l2 = [])
503 [assumption|apply nil_cons]
504 |intros;lapply (sym_eq ? ? ? H10);
505 absurd (a3 :: l2 = []) [assumption|apply nil_cons]]]
506 |simplify in H1;lapply (in_list_nil ? x H1);elim Hletin
507 |simplify in H5;lapply (nat_in_list_case ? ? ? H5);elim Hletin
510 |simplify in H5;lapply (nat_in_list_case ? ? ? H5);elim Hletin
511 [lapply (fresh_name ((fv_type t1) @ (fv_env l)));elim Hletin1;
512 cut ((\lnot (in_list ? a (fv_type t1))) \land
513 (\lnot (in_list ? a (fv_env l))))
514 [elim Hcut;lapply (H4 ? H9 H8)
516 [simplify in Hletin2;
517 (* FIXME trick *);generalize in match Hletin2;intro;
519 [intros;lapply (inj_head_nat ? ? ? ? H12);
520 rewrite < Hletin3 in H11;lapply (Hcut1 H11);elim Hletin4
521 |intros;lapply (inj_tail ? ? ? ? ? H14);rewrite > Hletin3;
523 |unfold;intro;apply H8;rewrite < H10;assumption]
524 |apply in_FV_subst;assumption]
526 [unfold;intro;apply H7;apply natinG_or_inH_to_natinGH;right;
528 |unfold;intro;apply H7;apply natinG_or_inH_to_natinGH;left;
533 (*** some exotic inductions and related lemmas ***)
535 lemma not_t_len_lt_SO : \forall T.\lnot (t_len T) < (S O).
537 [1,2,3:simplify;unfold;intro;unfold in H;elim (not_le_Sn_n ? H)
538 |*:simplify;unfold;rewrite > max_case;elim (leb (t_len t) (t_len t1))
539 [1,3:simplify in H2;apply H1;apply (trans_lt ? ? ? ? H2);unfold;constructor 1
540 |*:simplify in H2;apply H;apply (trans_lt ? ? ? ? H2);unfold;constructor 1]]
543 lemma Typ_len_ind : \forall P:Typ \to Prop.
544 (\forall U.(\forall V.((t_len V) < (t_len U)) \to (P V))
547 cut (\forall P:Typ \to Prop.
548 (\forall U.(\forall V.((t_len V) < (t_len U)) \to (P V))
550 \to \forall T,n.(n = (t_len T)) \to (P T))
551 [intros;apply (Hcut ? H ? (t_len T));reflexivity
552 |intros 4;generalize in match T;apply (nat_elim1 n);intros;
553 generalize in match H2;elim t
554 [1,2,3:apply H;intros;simplify in H4;elim (not_t_len_lt_SO ? H4)
555 |*:apply H;intros;apply (H1 (t_len V))
556 [1,3:rewrite > H5;assumption
560 lemma t_len_arrow1 : \forall T1,T2.(t_len T1) < (t_len (Arrow T1 T2)).
562 (* FIXME!!! BUG?!?! *)
563 cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with
565 | true ⇒ (t_len T1) ])
566 [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor
567 (leb (t_len T1) (t_len T2)) = true)
568 [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1
569 [rewrite > H;rewrite > H in Hletin;simplify;constructor 1
570 |rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin;
571 unfold;apply le_S_S;assumption]
572 |elim (leb (t_len T1) (t_len T2));autobatch]
573 |elim T1;simplify;reflexivity]
576 lemma t_len_arrow2 : \forall T1,T2.(t_len T2) < (t_len (Arrow T1 T2)).
578 (* FIXME!!! BUG?!?! *)
579 cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with
580 [ false \Rightarrow (t_len T2)
581 | true \Rightarrow (t_len T1) ])
582 [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor
583 (leb (t_len T1) (t_len T2)) = true)
584 [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1
585 [rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin;
586 lapply (not_le_to_lt ? ? Hletin);unfold in Hletin1;unfold;
587 constructor 2;assumption
588 |rewrite > H;simplify;unfold;constructor 1]
589 |elim (leb (t_len T1) (t_len T2));autobatch]
590 |elim T1;simplify;reflexivity]
593 lemma t_len_forall1 : \forall T1,T2.(t_len T1) < (t_len (Forall T1 T2)).
595 (* FIXME!!! BUG?!?! *)
596 cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with
597 [ false \Rightarrow (t_len T2)
598 | true \Rightarrow (t_len T1) ])
599 [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor
600 (leb (t_len T1) (t_len T2)) = true)
601 [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1
602 [rewrite > H;rewrite > H in Hletin;simplify;constructor 1
603 |rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin;
604 unfold;apply le_S_S;assumption]
605 |elim (leb (t_len T1) (t_len T2));autobatch]
606 |elim T1;simplify;reflexivity]
609 lemma t_len_forall2 : \forall T1,T2.(t_len T2) < (t_len (Forall T1 T2)).
611 (* FIXME!!! BUG?!?! *)
612 cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with
613 [ false \Rightarrow (t_len T2)
614 | true \Rightarrow (t_len T1) ])
615 [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor
616 (leb (t_len T1) (t_len T2)) = true)
617 [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1
618 [rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin;
619 lapply (not_le_to_lt ? ? Hletin);unfold in Hletin1;unfold;
620 constructor 2;assumption
621 |rewrite > H;simplify;unfold;constructor 1]
622 |elim (leb (t_len T1) (t_len T2));autobatch]
623 |elim T1;simplify;reflexivity]
626 lemma eq_t_len_TFree_subst : \forall T,n,X.(t_len T) =
627 (t_len (subst_type_nat T (TFree X) n)).
629 [simplify;elim (eqb n n1);simplify;reflexivity
630 |2,3:simplify;reflexivity
631 |simplify;lapply (H n X);lapply (H1 n X);rewrite < Hletin;rewrite < Hletin1;
633 |simplify;lapply (H n X);lapply (H1 (S n) X);rewrite < Hletin;
634 rewrite < Hletin1;reflexivity]
637 (*** lemmata relating subtyping and well-formedness ***)
639 lemma JS_to_WFE : \forall G,T,U.(G \vdash T ⊴ U) \to (WFEnv G).
640 intros;elim H;assumption.
643 lemma JS_to_WFT : \forall G,T,U.(JSubtype G T U) \to ((WFType G T) \land
646 [split [assumption|apply WFT_Top]
647 |split;apply WFT_TFree;assumption
649 [apply WFT_TFree;apply boundinenv_natinfv;apply ex_intro
650 [apply true | apply ex_intro [apply t1 |assumption]]
652 |elim H2;elim H4;split;apply WFT_Arrow;assumption
654 [apply (WFT_Forall ? ? ? H6);intros;elim (H4 X H7);
655 apply (WFT_env_incl ? ? H9);simplify;unfold;intros;assumption
656 |apply (WFT_Forall ? ? ? H5);intros;elim (H4 X H7);
657 apply (WFT_env_incl ? ? H10);simplify;unfold;intros;assumption]]
660 lemma JS_to_WFT1 : \forall G,T,U.(JSubtype G T U) \to (WFType G T).
661 intros;lapply (JS_to_WFT ? ? ? H);elim Hletin;assumption.
664 lemma JS_to_WFT2 : \forall G,T,U.(JSubtype G T U) \to (WFType G U).
665 intros;lapply (JS_to_WFT ? ? ? H);elim Hletin;assumption.
668 lemma WFE_Typ_subst : \forall H,x,B,C,T,U,G.
669 (WFEnv (H @ ((mk_bound B x T) :: G))) \to (WFType G U) \to
670 (WFEnv (H @ ((mk_bound C x U) :: G))).
672 [simplify;intros;(*FIXME*)generalize in match H1;intro;inversion H1
673 [intros;lapply (nil_cons ? G (mk_bound B x T));lapply (Hletin H4);
675 |intros;lapply (inj_tail ? ? ? ? ? H8);lapply (inj_head ? ? ? ? H8);
676 destruct Hletin1;rewrite < Hletin in H6;rewrite < Hletin in H4;
677 rewrite < Hcut1 in H6;apply (WFE_cons ? ? ? ? H4 H6 H2)]
678 |intros;simplify;generalize in match H2;elim t;simplify in H4;
680 [intros;absurd (mk_bound b n t1::l@(mk_bound B x T::G)=[])
683 |intros;lapply (inj_tail ? ? ? ? ? H9);lapply (inj_head ? ? ? ? H9);
684 destruct Hletin1;apply WFE_cons
686 [rewrite > Hletin;assumption
688 |rewrite > Hcut1;generalize in match H7;rewrite < Hletin;
689 rewrite > (fv_env_extends ? x B C T U);intro;assumption
690 |rewrite < Hletin in H8;rewrite > Hcut2;
691 apply (WFT_env_incl ? ? H8);rewrite > (fv_env_extends ? x B C T U);
692 unfold;intros;assumption]]]
695 lemma WFE_bound_bound : \forall B,x,T,U,G. (WFEnv G) \to
696 (in_list ? (mk_bound B x T) G) \to
697 (in_list ? (mk_bound B x U) G) \to T = U.
699 [lapply (in_list_nil ? ? H1);elim Hletin
701 [intros;rewrite < H7 in H8;lapply (inj_head ? ? ? ? H8);
702 rewrite > Hletin in H5;inversion H5
703 [intros;rewrite < H9 in H10;lapply (inj_head ? ? ? ? H10);
704 destruct Hletin1;symmetry;assumption
705 |intros;lapply (inj_tail ? ? ? ? ? H12);rewrite < Hletin1 in H9;
706 rewrite < H11 in H9;lapply (boundinenv_natinfv x l)
707 [destruct Hletin;rewrite < Hcut1 in Hletin2;lapply (H3 Hletin2);
710 [apply B|apply ex_intro [apply T|assumption]]]]
711 |intros;lapply (inj_tail ? ? ? ? ? H10);rewrite < H9 in H7;
712 rewrite < Hletin in H7;(*FIXME*)generalize in match H5;intro;inversion H5
713 [intros;rewrite < H12 in H13;lapply (inj_head ? ? ? ? H13);
714 destruct Hletin1;rewrite < Hcut1 in H7;
715 lapply (boundinenv_natinfv n l)
716 [lapply (H3 Hletin2);elim Hletin3
718 [apply B|apply ex_intro [apply U|assumption]]]
719 |intros;apply (H2 ? H7);rewrite > H14;lapply (inj_tail ? ? ? ? ? H15);
720 rewrite > Hletin1;assumption]]]