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 : Type \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 : Type \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 : Type \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 (* notation "hvbox(\Forall S. break T)"
57 non associative with precedence 90
58 for @{ 'forall $S $T}.
76 notation "hvbox(s break \mapsto t)"
77 right associative with precedence 55
78 for @{ 'arrow $s $t }.
80 interpretation "universal type" 'forall S T = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/5) S T).
82 interpretation "bound var" 'var x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/1) x).
84 interpretation "bound tvar" 'tvar x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/3) x).
86 interpretation "bound tname" 'tname x = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/2) x).
88 interpretation "arrow type" 'arrow S T = (cic:/matita/Fsub/defn/Typ.ind#xpointer(1/1/4) S T). *)
90 (*** Various kinds of substitution, not all will be used probably ***)
92 (* substitutes i-th dangling index in type T with type U *)
93 let rec subst_type_nat T U i \def
95 [ (TVar n) \Rightarrow match (eqb n i) with
97 | false \Rightarrow T]
98 | (TFree X) \Rightarrow T
100 | (Arrow T1 T2) \Rightarrow (Arrow (subst_type_nat T1 U i) (subst_type_nat T2 U i))
101 | (Forall T1 T2) \Rightarrow (Forall (subst_type_nat T1 U i) (subst_type_nat T2 U (S i))) ].
103 (* substitutes 0-th dangling index in type T with type U *)
104 let rec subst_type_O T U \def subst_type_nat T U O.
106 (* substitutes 0-th dangling index in term t with term u *)
107 let rec subst_term_O t u \def
108 let rec aux t0 i \def
110 [ (Var n) \Rightarrow match (eqb n i) with
112 | false \Rightarrow t0]
113 | (Free X) \Rightarrow t0
114 | (Abs T1 t1) \Rightarrow (Abs T1 (aux t1 (S i)))
115 | (App t1 t2) \Rightarrow (App (aux t1 i) (aux t2 i))
116 | (TAbs T1 t1) \Rightarrow (TAbs T1 (aux t1 (S i)))
117 | (TApp t1 T1) \Rightarrow (TApp (aux t1 i) T1) ]
120 (* substitutes 0-th dangling index in term T, which shall be a TVar,
122 let rec subst_term_tO t T \def
123 let rec aux t0 i \def
125 [ (Var n) \Rightarrow t0
126 | (Free X) \Rightarrow t0
127 | (Abs T1 t1) \Rightarrow (Abs (subst_type_nat T1 T i) (aux t1 (S i)))
128 | (App t1 t2) \Rightarrow (App (aux t1 i) (aux t2 i))
129 | (TAbs T1 t1) \Rightarrow (TAbs (subst_type_nat T1 T i) (aux t1 (S i)))
130 | (TApp t1 T1) \Rightarrow (TApp (aux t1 i) (subst_type_nat T1 T i)) ]
133 (* substitutes (TFree X) in type T with type U *)
134 let rec subst_type_tfree_type T X U on T \def
136 [ (TVar n) \Rightarrow T
137 | (TFree Y) \Rightarrow match (eqb X Y) with
139 | false \Rightarrow T ]
141 | (Arrow T1 T2) \Rightarrow (Arrow (subst_type_tfree_type T1 X U)
142 (subst_type_tfree_type T2 X U))
143 | (Forall T1 T2) \Rightarrow (Forall (subst_type_tfree_type T1 X U)
144 (subst_type_tfree_type T2 X U)) ].
146 (*** height of T's syntactic tree ***)
150 [(TVar n) \Rightarrow (S O)
151 |(TFree X) \Rightarrow (S O)
152 |Top \Rightarrow (S O)
153 |(Arrow T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2)))
154 |(Forall T1 T2) \Rightarrow (S (max (t_len T1) (t_len T2)))].
157 \lambda G:(list bound).match G with
158 [ nil \Rightarrow (mk_bound false O Top)
159 | (cons b H) \Rightarrow b].
161 definition head_nat \def
162 \lambda G:(list nat).match G with
164 | (cons n H) \Rightarrow n].
166 (*** definitions about lists ***)
168 (* var binding is in env judgement *)
169 definition var_bind_in_env : bound \to Env \to Prop \def
170 \lambda b,G.(in_list bound b G).
172 definition fv_env : (list bound) \to (list nat) \def
173 \lambda G.(map ? ? (\lambda b.match b with
174 [(mk_bound B X T) \Rightarrow X]) G).
176 (* variable is in env judgement *)
177 definition var_in_env : nat \to Env \to Prop \def
178 \lambda x,G.(in_list nat x (fv_env G)).
180 definition var_type_in_env : nat \to Env \to Prop \def
181 \lambda x,G.\exists T.(var_bind_in_env (mk_bound true x T) G).
183 let rec fv_type T \def
185 [(TVar n) \Rightarrow []
186 |(TFree x) \Rightarrow [x]
188 |(Arrow U V) \Rightarrow ((fv_type U) @ (fv_type V))
189 |(Forall U V) \Rightarrow ((fv_type U) @ (fv_type V))].
191 (*** Type Well-Formedness judgement ***)
193 inductive WFType : Env \to Typ \to Prop \def
194 | WFT_TFree : \forall X,G.(in_list ? X (fv_env G))
195 \to (WFType G (TFree X))
196 | WFT_Top : \forall G.(WFType G Top)
197 | WFT_Arrow : \forall G,T,U.(WFType G T) \to (WFType G U) \to
198 (WFType G (Arrow T U))
199 | WFT_Forall : \forall G,T,U.(WFType G T) \to
201 (\lnot (in_list ? X (fv_env G))) \to
202 (\lnot (in_list ? X (fv_type U))) \to
203 (WFType ((mk_bound true X T) :: G)
204 (subst_type_O U (TFree X)))) \to
205 (WFType G (Forall T U)).
207 (*** Environment Well-Formedness judgement ***)
209 inductive WFEnv : Env \to Prop \def
210 | WFE_Empty : (WFEnv Empty)
211 | WFE_cons : \forall B,X,T,G.(WFEnv G) \to
212 \lnot (in_list ? X (fv_env G)) \to
213 (WFType G T) \to (WFEnv ((mk_bound B X T) :: G)).
215 (*** Subtyping judgement ***)
216 inductive JSubtype : Env \to Typ \to Typ \to Prop \def
217 | SA_Top : \forall G:Env.\forall T:Typ.(WFEnv G) \to
218 (WFType G T) \to (JSubtype G T Top)
219 | SA_Refl_TVar : \forall G:Env.\forall X:nat.(WFEnv G) \to (var_in_env X G)
220 \to (JSubtype G (TFree X) (TFree X))
221 | SA_Trans_TVar : \forall G:Env.\forall X:nat.\forall T:Typ.
223 (var_bind_in_env (mk_bound true X U) G) \to
224 (JSubtype G U T) \to (JSubtype G (TFree X) T)
225 | SA_Arrow : \forall G:Env.\forall S1,S2,T1,T2:Typ.
226 (JSubtype G T1 S1) \to (JSubtype G S2 T2) \to
227 (JSubtype G (Arrow S1 S2) (Arrow T1 T2))
228 | SA_All : \forall G:Env.\forall S1,S2,T1,T2:Typ.
229 (JSubtype G T1 S1) \to
230 (\forall X:nat.\lnot (var_in_env X G) \to
231 (JSubtype ((mk_bound true X T1) :: G)
232 (subst_type_O S2 (TFree X)) (subst_type_O T2 (TFree X)))) \to
233 (JSubtype G (Forall S1 S2) (Forall T1 T2)).
235 (*** Typing judgement ***)
236 inductive JType : Env \to Term \to Typ \to Prop \def
237 | T_Var : \forall G:Env.\forall x:nat.\forall T:Typ.
238 (WFEnv G) \to (var_bind_in_env (mk_bound false x T) G) \to
240 | T_Abs : \forall G.\forall T1,T2:Typ.\forall t2:Term.
242 (JType ((mk_bound false x T1)::G) (subst_term_O t2 (Free x)) T2) \to
243 (JType G (Abs T1 t2) (Arrow T1 T2))
244 | T_App : \forall G.\forall t1,t2:Term.\forall T2:Typ.
245 \forall T1:Typ.(JType G t1 (Arrow T1 T2)) \to (JType G t2 T1) \to
246 (JType G (App t1 t2) T2)
247 | T_TAbs : \forall G:Env.\forall T1,T2:Typ.\forall t2:Term.
249 (JType ((mk_bound true X T1)::G)
250 (subst_term_tO t2 (TFree X)) (subst_type_O T2 (TFree X)))
251 \to (JType G (TAbs T1 t2) (Forall T1 T2))
252 | T_TApp : \forall G:Env.\forall t1:Term.\forall T2,T12:Typ.
253 \forall X:nat.\forall T11:Typ.
254 (JType G t1 (Forall T11 (subst_type_tfree_type T12 X (TVar O)))) \to
256 \to (JType G (TApp t1 T2) (subst_type_tfree_type T12 X T2))
257 | T_Sub : \forall G:Env.\forall t:Term.\forall T:Typ.
258 \forall S:Typ.(JType G t S) \to (JSubtype G S T) \to (JType G t T).
260 (*** definitions about swaps ***)
262 let rec swap_Typ u v T on T \def
264 [(TVar n) \Rightarrow (TVar n)
265 |(TFree X) \Rightarrow (TFree (swap u v X))
267 |(Arrow T1 T2) \Rightarrow (Arrow (swap_Typ u v T1) (swap_Typ u v T2))
268 |(Forall T1 T2) \Rightarrow (Forall (swap_Typ u v T1) (swap_Typ u v T2))].
270 definition swap_bound : nat \to nat \to bound \to bound \def
271 \lambda u,v,b.match b with
272 [(mk_bound B X T) \Rightarrow (mk_bound B (swap u v X) (swap_Typ u v T))].
274 definition swap_Env : nat \to nat \to Env \to Env \def
275 \lambda u,v,G.(map ? ? (\lambda b.(swap_bound u v b)) G).
277 (****** PROOFS ********)
279 lemma subst_O_nat : \forall T,U.((subst_type_O T U) = (subst_type_nat T U O)).
280 intros;elim T;simplify;reflexivity;
283 (*** theorems about lists ***)
285 (* FIXME: these definitions shouldn't be part of the poplmark challenge
286 - use destruct instead, when hopefully it will get fixed... *)
288 lemma inj_head : \forall h1,h2:bound.\forall t1,t2:Env.
289 ((h1::t1) = (h2::t2)) \to (h1 = h2).
291 lapply (eq_f ? ? head ? ? H).simplify in Hletin.assumption.
294 lemma inj_head_nat : \forall h1,h2:nat.\forall t1,t2:(list nat).
295 ((h1::t1) = (h2::t2)) \to (h1 = h2).
297 lapply (eq_f ? ? head_nat ? ? H).simplify in Hletin.assumption.
300 lemma inj_tail : \forall A.\forall h1,h2:A.\forall t1,t2:(list A).
301 ((h1::t1) = (h2::t2)) \to (t1 = t2).
302 intros.lapply (eq_f ? ? (tail ?) ? ? H).simplify in Hletin.assumption.
307 lemma var_notinbG_notinG : \forall G,x,b.
308 (\lnot (var_in_env x (b::G)))
309 \to \lnot (var_in_env x G).
310 intros 3.elim b.unfold.intro.elim H.unfold.simplify.constructor 2.exact H1.
313 lemma boundinenv_natinfv : \forall x,G.
314 (\exists B,T.(in_list ? (mk_bound B x T) G)) \to
315 (in_list ? x (fv_env G)).
317 [elim H;elim H1;lapply (in_list_nil ? ? H2);elim Hletin
318 |elim H1;elim H2;inversion H3
319 [intros;rewrite < H4;simplify;apply in_Base
320 |intros;elim a3;simplify;apply in_Skip;
321 lapply (inj_tail ? ? ? ? ? H7);rewrite > Hletin in H;apply H;
326 |rewrite > H6;assumption]]]]
329 lemma nat_in_list_case : \forall G,H,n.(in_list nat n (H @ G)) \to
330 (in_list nat n G) \lor (in_list nat n H).
332 [simplify in H1;left;assumption
333 |simplify in H2;inversion H2
334 [intros;lapply (inj_head_nat ? ? ? ? H4);rewrite > Hletin;
336 |intros;lapply (inj_tail ? ? ? ? ? H6);rewrite < Hletin in H3;
337 rewrite > H5 in H1;lapply (H1 H3);elim Hletin1
338 [left;assumption|right;apply in_Skip;assumption]]]
341 lemma natinG_or_inH_to_natinGH : \forall G,H,n.
342 (in_list nat n G) \lor (in_list nat n H) \to
343 (in_list nat n (H @ G)).
347 |simplify;apply in_Skip;assumption]
348 |generalize in match H2;elim H2
349 [simplify;apply in_Base
350 |lapply (H4 H3);simplify;apply in_Skip;assumption]]
353 lemma natinfv_boundinenv : \forall x,G.(in_list ? x (fv_env G)) \to
354 \exists B,T.(in_list ? (mk_bound B x T) G).
356 [simplify;intro;lapply (in_list_nil ? ? H);elim Hletin
357 |intros 3;elim t;simplify in H1;inversion H1
358 [intros;rewrite < H2;simplify;apply ex_intro
362 |lapply (inj_head_nat ? ? ? ? H3);rewrite > H2;rewrite < Hletin;
364 |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2;
365 rewrite < H4 in H2;lapply (H H2);elim Hletin1;elim H6;apply ex_intro
369 |apply in_Skip;rewrite < H4;assumption]]]]
372 lemma incl_bound_fv : \forall l1,l2.(incl ? l1 l2) \to
373 (incl ? (fv_env l1) (fv_env l2)).
374 intros.unfold in H.unfold.intros.apply boundinenv_natinfv.
375 lapply (natinfv_boundinenv ? ? H1).elim Hletin.elim H2.apply ex_intro
382 (* lemma incl_cons : \forall x,l1,l2.
383 (incl bound l1 l2) \to (incl bound (x :: l1) (x :: l2)).
384 intros.unfold in H.unfold.intros.inversion H1
385 [intros;lapply (inj_head ? ? ? ? H3);rewrite > Hletin;apply in_Base
386 |intros;apply in_Skip;apply H;lapply (inj_tail ? ? ? ? ? H5);rewrite > Hletin;
390 lemma incl_nat_cons : \forall x,l1,l2.
391 (incl nat l1 l2) \to (incl nat (x :: l1) (x :: l2)).
392 intros.unfold in H.unfold.intros.inversion H1
393 [intros;lapply (inj_head_nat ? ? ? ? H3);rewrite > Hletin;apply in_Base
394 |intros;apply in_Skip;apply H;lapply (inj_tail ? ? ? ? ? H5);rewrite > Hletin;
398 lemma boundin_envappend_case : \forall G,H,b.(var_bind_in_env b (H @ G)) \to
399 (var_bind_in_env b G) \lor (var_bind_in_env b H).
401 [simplify in H1;left;assumption
402 |unfold in H2;inversion H2
403 [intros;simplify in H4;lapply (inj_head ? ? ? ? H4);rewrite > Hletin;
405 |intros;simplify in H6;lapply (inj_tail ? ? ? ? ? H6);rewrite < Hletin in H3;
406 rewrite > H5 in H1;lapply (H1 H3);elim Hletin1
407 [left;assumption|right;apply in_Skip;assumption]]]
410 lemma varin_envappend_case: \forall G,H,x.(var_in_env x (H @ G)) \to
411 (var_in_env x G) \lor (var_in_env x H).
413 [simplify;intro;left;assumption
414 |intros 2;elim t;simplify in H2;inversion H2
415 [intros;lapply (inj_head_nat ? ? ? ? H4);rewrite > Hletin;right;
416 simplify;constructor 1
417 |intros;lapply (inj_tail ? ? ? ? ? H6);
419 [rewrite < H5;elim Hletin1
420 [left;assumption|right;simplify;constructor 2;assumption]
421 |unfold var_in_env;unfold fv_env;rewrite > Hletin;rewrite > H5;
425 lemma boundinG_or_boundinH_to_boundinGH : \forall G,H,b.
426 (var_bind_in_env b G) \lor (var_bind_in_env b H) \to
427 (var_bind_in_env b (H @ G)).
431 |simplify;apply in_Skip;assumption]
432 |generalize in match H2;elim H2
433 [simplify;apply in_Base
434 |lapply (H4 H3);simplify;apply in_Skip;assumption]]
438 lemma varinG_or_varinH_to_varinGH : \forall G,H,x.
439 (var_in_env x G) \lor (var_in_env x H) \to
440 (var_in_env x (H @ G)).
444 |elim t;simplify;constructor 2;apply (H2 H3)]
446 [simplify;intro;lapply (in_list_nil nat x H2);elim Hletin
447 |intros 2;elim t;simplify in H3;inversion H3
448 [intros;lapply (inj_head_nat ? ? ? ? H5);rewrite > Hletin;simplify;
450 |intros;simplify;constructor 2;rewrite < H6;apply H2;
451 lapply (inj_tail ? ? ? ? ? H7);rewrite > H6;unfold;unfold fv_env;
452 rewrite > Hletin;assumption]]]
455 lemma varbind_to_append : \forall G,b.(var_bind_in_env b G) \to
456 \exists G1,G2.(G = (G2 @ (b :: G1))).
457 intros.generalize in match H.elim H
458 [apply ex_intro [apply l|apply ex_intro [apply Empty|reflexivity]]
459 |lapply (H2 H1);elim Hletin;elim H4;rewrite > H5;
461 [apply a2|apply ex_intro [apply (a1 :: a3)|simplify;reflexivity]]]
465 lemma WFT_env_incl : \forall G,T.(WFType G T) \to
466 \forall H.(incl ? (fv_env G) (fv_env H)) \to (WFType H T).
467 intros 4.generalize in match H1.elim H
468 [apply WFT_TFree;unfold in H3;apply (H3 ? H2)
470 |apply WFT_Arrow [apply (H3 ? H6)|apply (H5 ? H6)]
474 [unfold;intro;unfold in H7;apply H7;unfold in H6;apply(H6 ? H9)
476 |simplify;apply (incl_nat_cons ? ? ? H6)]]]
479 lemma fv_env_extends : \forall H,x,B,C,T,U,G.
480 (fv_env (H @ ((mk_bound B x T) :: G))) =
481 (fv_env (H @ ((mk_bound C x U) :: G))).
483 [simplify;reflexivity
484 |elim t;simplify;rewrite > H1;reflexivity]
487 lemma lookup_env_extends : \forall G,H,B,C,D,T,U,V,x,y.
488 (in_list ? (mk_bound D y V) (H @ ((mk_bound C x U) :: G))) \to
490 (in_list ? (mk_bound D y V) (H @ ((mk_bound B x T) :: G))).
492 [simplify in H1;(*FIXME*)generalize in match H1;intro;inversion H1
493 [intros;lapply (inj_head ? ? ? ? H5);rewrite < H4 in Hletin;
494 destruct Hletin;absurd (y = x) [symmetry;assumption|assumption]
495 |intros;simplify;lapply (inj_tail ? ? ? ? ? H7);rewrite > Hletin;
496 apply in_Skip;assumption]
497 |(*FIXME*)generalize in match H2;intro;inversion H2
498 [intros;simplify in H6;lapply (inj_head ? ? ? ? H6);rewrite > Hletin;
499 simplify;apply in_Base
500 |simplify;intros;lapply (inj_tail ? ? ? ? ? H8);rewrite > Hletin in H1;
501 rewrite > H7 in H1;apply in_Skip;apply (H1 H5 H3)]]
505 (*** theorems about swaps ***)
507 lemma fv_subst_type_nat : \forall x,T,y,n.(in_list ? x (fv_type T)) \to
508 (in_list ? x (fv_type (subst_type_nat T (TFree y) n))).
510 [intros;simplify in H;elim (in_list_nil ? ? H)
511 |2,3:simplify;intros;assumption
512 |*:intros;simplify in H2;elim (nat_in_list_case ? ? ? H2)
513 [1,3:simplify;apply natinG_or_inH_to_natinGH;left;apply (H1 ? H3)
514 |*:simplify;apply natinG_or_inH_to_natinGH;right;apply (H ? H3)]]
517 lemma fv_subst_type_O : \forall x,T,y.(in_list ? x (fv_type T)) \to
518 (in_list ? x (fv_type (subst_type_O T (TFree y)))).
519 intros;rewrite > subst_O_nat;apply (fv_subst_type_nat ? ? ? ? H);
522 lemma swap_Typ_inv : \forall u,v,T.(swap_Typ u v (swap_Typ u v T)) = T.
524 [1,3:simplify;reflexivity
525 |simplify;rewrite > swap_inv;reflexivity
526 |*:simplify;rewrite > H;rewrite > H1;reflexivity]
529 lemma swap_Typ_not_free : \forall u,v,T.\lnot (in_list ? u (fv_type T)) \to
530 \lnot (in_list ? v (fv_type T)) \to (swap_Typ u v T) = T.
532 [1,3:intros;simplify;reflexivity
533 |simplify;intros;cut (n \neq u \land n \neq v)
534 [elim Hcut;rewrite > (swap_other ? ? ? H2 H3);reflexivity
536 [unfold;intro;apply H;rewrite > H2;apply in_Base
537 |unfold;intro;apply H1;rewrite > H2;apply in_Base]]
538 |*:simplify;intros;cut ((\lnot (in_list ? u (fv_type t)) \land
539 \lnot (in_list ? u (fv_type t1))) \land
540 (\lnot (in_list ? v (fv_type t)) \land
541 \lnot (in_list ? v (fv_type t1))))
542 [1,3:elim Hcut;elim H4;elim H5;clear Hcut H4 H5;rewrite > (H H6 H8);
543 rewrite > (H1 H7 H9);reflexivity
545 [1,3:split;unfold;intro;apply H2;apply natinG_or_inH_to_natinGH;auto
546 |*:split;unfold;intro;apply H3;apply natinG_or_inH_to_natinGH;auto]]]
549 lemma subst_type_nat_swap : \forall u,v,T,X,m.
550 (swap_Typ u v (subst_type_nat T (TFree X) m)) =
551 (subst_type_nat (swap_Typ u v T) (TFree (swap u v X)) m).
553 [simplify;elim (eqb_case n m);rewrite > H;simplify;reflexivity
554 |2,3:simplify;reflexivity
555 |*:simplify;rewrite > H;rewrite > H1;reflexivity]
558 lemma subst_type_O_swap : \forall u,v,T,X.
559 (swap_Typ u v (subst_type_O T (TFree X))) =
560 (subst_type_O (swap_Typ u v T) (TFree (swap u v X))).
561 intros 4;rewrite > (subst_O_nat (swap_Typ u v T));rewrite > (subst_O_nat T);
562 apply subst_type_nat_swap;
565 lemma in_fv_type_swap : \forall u,v,x,T.((in_list ? x (fv_type T)) \to
566 (in_list ? (swap u v x) (fv_type (swap_Typ u v T)))) \land
567 ((in_list ? (swap u v x) (fv_type (swap_Typ u v T))) \to
568 (in_list ? x (fv_type T))).
571 [1,3:simplify;intros;elim (in_list_nil ? ? H)
572 |simplify;intros;cut (x = n)
573 [rewrite > Hcut;apply in_Base
575 [intros;lapply (inj_head_nat ? ? ? ? H2);rewrite > Hletin;
577 |intros;lapply (inj_tail ? ? ? ? ? H4);rewrite < Hletin in H1;
578 elim (in_list_nil ? ? H1)]]
579 |*:simplify;intros;elim (nat_in_list_case ? ? ? H2)
580 [1,3:apply natinG_or_inH_to_natinGH;left;apply (H1 H3)
581 |*:apply natinG_or_inH_to_natinGH;right;apply (H H3)]]
583 [1,3:simplify;intros;elim (in_list_nil ? ? H)
584 |simplify;intros;cut ((swap u v x) = (swap u v n))
585 [lapply (swap_inj ? ? ? ? Hcut);rewrite > Hletin;apply in_Base
587 [intros;lapply (inj_head_nat ? ? ? ? H2);rewrite > Hletin;
589 |intros;lapply (inj_tail ? ? ? ? ? H4);rewrite < Hletin in H1;
590 elim (in_list_nil ? ? H1)]]
591 |*:simplify;intros;elim (nat_in_list_case ? ? ? H2)
592 [1,3:apply natinG_or_inH_to_natinGH;left;apply (H1 H3)
593 |*:apply natinG_or_inH_to_natinGH;right;apply (H H3)]]]
596 lemma lookup_swap : \forall x,u,v,T,B,G.(in_list ? (mk_bound B x T) G) \to
597 (in_list ? (mk_bound B (swap u v x) (swap_Typ u v T)) (swap_Env u v G)).
599 [intros;elim (in_list_nil ? ? H)
600 |intro;elim t;simplify;inversion H1
601 [intros;lapply (inj_head ? ? ? ? H3);rewrite < H2 in Hletin;
602 destruct Hletin;rewrite > Hcut;rewrite > Hcut1;rewrite > Hcut2;
604 |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2;
605 rewrite < H4 in H2;apply in_Skip;apply (H H2)]]
608 lemma in_FV_subst : \forall x,T,U,n.(in_list ? x (fv_type T)) \to
609 (in_list ? x (fv_type (subst_type_nat T U n))).
611 [simplify in H;inversion H
612 [intros;lapply (sym_eq ? ? ? H2);absurd (a::l = [])
613 [assumption|apply nil_cons]
614 |intros;lapply (sym_eq ? ? ? H4);absurd (a1::l = [])
615 [assumption|apply nil_cons]]
616 |2,3:simplify;simplify in H;assumption
617 |*:simplify in H2;simplify;apply natinG_or_inH_to_natinGH;
618 lapply (nat_in_list_case ? ? ? H2);elim Hletin
619 [1,3:left;apply (H1 ? H3)
620 |*:right;apply (H ? H3)]]
623 lemma in_dom_swap : \forall u,v,x,G.
624 ((in_list ? x (fv_env G)) \to
625 (in_list ? (swap u v x) (fv_env (swap_Env u v G)))) \land
626 ((in_list ? (swap u v x) (fv_env (swap_Env u v G))) \to
627 (in_list ? x (fv_env G))).
630 [simplify;intro;elim (in_list_nil ? ? H)
631 |intro;elim t 0;simplify;intros;inversion H1
632 [intros;lapply (inj_head_nat ? ? ? ? H3);rewrite > Hletin;apply in_Base
633 |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2;
634 rewrite > H4 in H;apply in_Skip;apply (H H2)]]
636 [simplify;intro;elim (in_list_nil ? ? H)
637 |intro;elim t 0;simplify;intros;inversion H1
638 [intros;lapply (inj_head_nat ? ? ? ? H3);rewrite < H2 in Hletin;
639 lapply (swap_inj ? ? ? ? Hletin);rewrite > Hletin1;apply in_Base
640 |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite < Hletin in H2;
641 rewrite > H4 in H;apply in_Skip;apply (H H2)]]]
644 (*** lemma on fresh names ***)
646 lemma fresh_name : \forall l:(list nat).\exists n.\lnot (in_list ? n l).
647 cut (\forall l:(list nat).\exists n.\forall m.
648 (n \leq m) \to \lnot (in_list ? m l))
649 [intros;lapply (Hcut l);elim Hletin;apply ex_intro
651 |apply H;constructor 1]
655 |intros;unfold;intro;inversion H1
656 [intros;lapply (sym_eq ? ? ? H3);absurd (a::l1 = [])
657 [assumption|apply nil_cons]
658 |intros;lapply (sym_eq ? ? ? H5);absurd (a1::l1 = [])
659 [assumption|apply nil_cons]]]
660 |elim H;lapply (decidable_eq_nat a t);elim Hletin
663 |intros;unfold;intro;inversion H4
664 [intros;lapply (inj_head_nat ? ? ? ? H6);rewrite < Hletin1 in H5;
665 rewrite < H2 in H5;rewrite > H5 in H3;
666 apply (not_le_Sn_n ? H3)
667 |intros;lapply (inj_tail ? ? ? ? ? H8);rewrite < Hletin1 in H5;
669 apply (H1 m ? H5);lapply (le_S ? ? H3);
670 apply (le_S_S_to_le ? ? Hletin2)]]
671 |cut ((leb a t) = true \lor (leb a t) = false)
675 |intros;unfold;intro;inversion H5
676 [intros;lapply (inj_head_nat ? ? ? ? H7);rewrite > H6 in H4;
677 rewrite < Hletin1 in H4;apply (not_le_Sn_n ? H4)
678 |intros;lapply (inj_tail ? ? ? ? ? H9);
679 rewrite < Hletin1 in H6;lapply (H1 a1)
681 |lapply (leb_to_Prop a t);rewrite > H3 in Hletin2;
682 simplify in Hletin2;rewrite < H8;
683 apply (trans_le ? ? ? Hletin2);
684 apply (trans_le ? ? ? ? H4);constructor 2;constructor 1]]]
687 |intros;lapply (leb_to_Prop a t);rewrite > H3 in Hletin1;
688 simplify in Hletin1;lapply (not_le_to_lt ? ? Hletin1);
689 unfold in Hletin2;unfold;intro;inversion H5
690 [intros;lapply (inj_head_nat ? ? ? ? H7);
691 rewrite < Hletin3 in H6;rewrite > H6 in H4;
693 |intros;lapply (inj_tail ? ? ? ? ? H9);
694 rewrite < Hletin3 in H6;rewrite < H8 in H6;
695 apply (H1 ? H4 H6)]]]
696 |elim (leb a t);auto]]]]
699 (*** lemmas on well-formedness ***)
701 lemma fv_WFT : \forall T,x,G.(WFType G T) \to (in_list ? x (fv_type T)) \to
702 (in_list ? x (fv_env G)).
704 [simplify in H2;inversion H2
705 [intros;lapply (inj_head_nat ? ? ? ? H4);rewrite < Hletin;assumption
706 |intros;lapply (inj_tail ? ? ? ? ? H6);rewrite < Hletin in H3;
708 [intros;lapply (sym_eq ? ? ? H8);absurd (a2 :: l2 = [])
709 [assumption|apply nil_cons]
710 |intros;lapply (sym_eq ? ? ? H10);
711 absurd (a3 :: l2 = []) [assumption|apply nil_cons]]]
712 |simplify in H1;lapply (in_list_nil ? x H1);elim Hletin
713 |simplify in H5;lapply (nat_in_list_case ? ? ? H5);elim Hletin
716 |simplify in H5;lapply (nat_in_list_case ? ? ? H5);elim Hletin
717 [lapply (fresh_name ((fv_type t1) @ (fv_env e)));elim Hletin1;
718 cut ((\lnot (in_list ? a (fv_type t1))) \land
719 (\lnot (in_list ? a (fv_env e))))
720 [elim Hcut;lapply (H4 ? H9 H8)
722 [simplify in Hletin2;
723 (* FIXME trick *);generalize in match Hletin2;intro;
725 [intros;lapply (inj_head_nat ? ? ? ? H12);
726 rewrite < Hletin3 in H11;lapply (Hcut1 H11);elim Hletin4
727 |intros;lapply (inj_tail ? ? ? ? ? H14);rewrite > Hletin3;
729 |unfold;intro;apply H8;rewrite < H10;assumption]
730 |rewrite > subst_O_nat;apply in_FV_subst;assumption]
732 [unfold;intro;apply H7;apply natinG_or_inH_to_natinGH;right;
734 |unfold;intro;apply H7;apply natinG_or_inH_to_natinGH;left;
739 lemma WFE_consG_to_WFT : \forall G.\forall b,X,T.
740 (WFEnv ((mk_bound b X T)::G)) \to (WFType G T).
743 [intro;reduce in H1;destruct H1
744 |intros;lapply (inj_head ? ? ? ? H5);lapply (inj_tail ? ? ? ? ? H5);
745 destruct Hletin;rewrite > Hletin1;rewrite > Hcut2;assumption]
748 lemma WFE_consG_WFE_G : \forall G.\forall b.
749 (WFEnv (b::G)) \to (WFEnv G).
752 [intro;reduce in H1;destruct H1
753 |intros;lapply (inj_tail ? ? ? ? ? H5);rewrite > Hletin;assumption]
756 (* silly, but later useful *)
758 lemma env_append_weaken : \forall G,H.(WFEnv (H @ G)) \to
761 [simplify;unfold;intros;assumption
762 |simplify in H2;simplify;unfold;intros;apply in_Skip;apply H1
763 [apply (WFE_consG_WFE_G ? ? H2)
767 lemma WFT_swap : \forall u,v,G,T.(WFType G T) \to
768 (WFType (swap_Env u v G) (swap_Typ u v T)).
770 [simplify;apply WFT_TFree;lapply (natinfv_boundinenv ? ? H1);elim Hletin;
771 elim H2;apply boundinenv_natinfv;apply ex_intro
774 [apply (swap_Typ u v a1)
775 |apply lookup_swap;assumption]]
776 |simplify;apply WFT_Top
777 |simplify;apply WFT_Arrow
778 [assumption|assumption]
779 |simplify;apply WFT_Forall
781 |intros;rewrite < (swap_inv u v);
782 cut (\lnot (in_list ? (swap u v X) (fv_type t1)))
783 [cut (\lnot (in_list ? (swap u v X) (fv_env e)))
784 [generalize in match (H4 ? Hcut1 Hcut);simplify;
785 rewrite > subst_type_O_swap;intro;assumption
786 |lapply (in_dom_swap u v (swap u v X) e);elim Hletin;unfold;
787 intros;lapply (H7 H9);rewrite > (swap_inv u v) in Hletin1;
789 |generalize in match (in_fv_type_swap u v (swap u v X) t1);intros;
790 elim H7;unfold;intro;lapply (H8 H10);
791 rewrite > (swap_inv u v) in Hletin;apply (H6 Hletin)]]]
794 lemma WFE_swap : \forall u,v,G.(WFEnv G) \to (WFEnv (swap_Env u v G)).
796 [intro;simplify;assumption
797 |intros 2;elim t;simplify;constructor 2
798 [apply H;apply (WFE_consG_WFE_G ? ? H1)
799 |unfold;intro;lapply (in_dom_swap u v n l);elim Hletin;lapply (H4 H2);
800 (* FIXME trick *)generalize in match H1;intro;inversion H1
801 [intros;absurd ((mk_bound b n t1)::l = [])
802 [assumption|apply nil_cons]
803 |intros;lapply (inj_head ? ? ? ? H10);lapply (inj_tail ? ? ? ? ? H10);
804 destruct Hletin2;rewrite < Hcut1 in H8;rewrite < Hletin3 in H8;
806 |apply (WFT_swap u v l t1);inversion H1
807 [intro;absurd ((mk_bound b n t1)::l = [])
808 [assumption|apply nil_cons]
809 |intros;lapply (inj_head ? ? ? ? H6);lapply (inj_tail ? ? ? ? ? H6);
810 destruct Hletin;rewrite > Hletin1;rewrite > Hcut2;assumption]]]
813 (*** some exotic inductions and related lemmas ***)
815 lemma not_t_len_lt_SO : \forall T.\lnot (t_len T) < (S O).
817 [1,2,3:simplify;unfold;intro;unfold in H;elim (not_le_Sn_n ? H)
818 |*:simplify;unfold;rewrite > max_case;elim (leb (t_len t) (t_len t1))
819 [1,3:simplify in H2;apply H1;apply (trans_lt ? ? ? ? H2);unfold;constructor 1
820 |*:simplify in H2;apply H;apply (trans_lt ? ? ? ? H2);unfold;constructor 1]]
823 lemma t_len_gt_O : \forall T.(t_len T) > O.
825 [1,2,3:simplify;unfold;unfold;constructor 1
826 |*:simplify;lapply (max_case (t_len t) (t_len t1));rewrite > Hletin;
827 elim (leb (t_len t) (t_len t1))
828 [1,3:simplify;unfold;unfold;constructor 2;unfold in H1;unfold in H1;assumption
829 |*:simplify;unfold;unfold;constructor 2;unfold in H;unfold in H;assumption]]
832 lemma Typ_len_ind : \forall P:Typ \to Prop.
833 (\forall U.(\forall V.((t_len V) < (t_len U)) \to (P V))
836 cut (\forall P:Typ \to Prop.
837 (\forall U.(\forall V.((t_len V) < (t_len U)) \to (P V))
839 \to \forall T,n.(n = (t_len T)) \to (P T))
840 [intros;apply (Hcut ? H ? (t_len T));reflexivity
841 |intros 4;generalize in match T;apply (nat_elim1 n);intros;
842 generalize in match H2;elim t
843 [1,2,3:apply H;intros;simplify in H4;elim (not_t_len_lt_SO ? H4)
844 |*:apply H;intros;apply (H1 (t_len V))
845 [1,3:rewrite > H5;assumption
849 lemma t_len_arrow1 : \forall T1,T2.(t_len T1) < (t_len (Arrow T1 T2)).
851 (* FIXME!!! BUG?!?! *)
852 cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with
853 [ false \Rightarrow (t_len T2)
854 | true \Rightarrow (t_len T1) ])
855 [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor
856 (leb (t_len T1) (t_len T2)) = true)
857 [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1
858 [rewrite > H;rewrite > H in Hletin;simplify;constructor 1
859 |rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin;
860 unfold;apply le_S_S;assumption]
861 |elim (leb (t_len T1) (t_len T2));auto]
862 |elim T1;simplify;reflexivity]
865 lemma t_len_arrow2 : \forall T1,T2.(t_len T2) < (t_len (Arrow T1 T2)).
867 (* FIXME!!! BUG?!?! *)
868 cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with
869 [ false \Rightarrow (t_len T2)
870 | true \Rightarrow (t_len T1) ])
871 [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor
872 (leb (t_len T1) (t_len T2)) = true)
873 [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1
874 [rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin;
875 lapply (not_le_to_lt ? ? Hletin);unfold in Hletin1;unfold;
876 constructor 2;assumption
877 |rewrite > H;simplify;unfold;constructor 1]
878 |elim (leb (t_len T1) (t_len T2));auto]
879 |elim T1;simplify;reflexivity]
882 lemma t_len_forall1 : \forall T1,T2.(t_len T1) < (t_len (Forall T1 T2)).
884 (* FIXME!!! BUG?!?! *)
885 cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with
886 [ false \Rightarrow (t_len T2)
887 | true \Rightarrow (t_len T1) ])
888 [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor
889 (leb (t_len T1) (t_len T2)) = true)
890 [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1
891 [rewrite > H;rewrite > H in Hletin;simplify;constructor 1
892 |rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin;
893 unfold;apply le_S_S;assumption]
894 |elim (leb (t_len T1) (t_len T2));auto]
895 |elim T1;simplify;reflexivity]
898 lemma t_len_forall2 : \forall T1,T2.(t_len T2) < (t_len (Forall T1 T2)).
900 (* FIXME!!! BUG?!?! *)
901 cut ((max (t_len T1) (t_len T2)) = match (leb (t_len T1) (t_len T2)) with
902 [ false \Rightarrow (t_len T2)
903 | true \Rightarrow (t_len T1) ])
904 [rewrite > Hcut;cut ((leb (t_len T1) (t_len T2)) = false \lor
905 (leb (t_len T1) (t_len T2)) = true)
906 [lapply (leb_to_Prop (t_len T1) (t_len T2));elim Hcut1
907 [rewrite > H;rewrite > H in Hletin;simplify;simplify in Hletin;
908 lapply (not_le_to_lt ? ? Hletin);unfold in Hletin1;unfold;
909 constructor 2;assumption
910 |rewrite > H;simplify;unfold;constructor 1]
911 |elim (leb (t_len T1) (t_len T2));auto]
912 |elim T1;simplify;reflexivity]
915 lemma eq_t_len_TFree_subst : \forall T,n,X.(t_len T) =
916 (t_len (subst_type_nat T (TFree X) n)).
918 [simplify;elim (eqb n n1);simplify;reflexivity
919 |2,3:simplify;reflexivity
920 |simplify;lapply (H n X);lapply (H1 n X);rewrite < Hletin;rewrite < Hletin1;
922 |simplify;lapply (H n X);lapply (H1 (S n) X);rewrite < Hletin;
923 rewrite < Hletin1;reflexivity]
926 lemma swap_env_not_free : \forall u,v,G.(WFEnv G) \to
927 \lnot (in_list ? u (fv_env G)) \to
928 \lnot (in_list ? v (fv_env G)) \to
929 (swap_Env u v G) = G.
931 [simplify;intros;reflexivity
932 |intros 2;elim t 0;simplify;intros;lapply (notin_cons ? ? ? ? H2);
933 lapply (notin_cons ? ? ? ? H3);elim Hletin;elim Hletin1;
934 lapply (swap_other ? ? ? H4 H6);lapply (WFE_consG_to_WFT ? ? ? ? H1);
935 cut (\lnot (in_list ? u (fv_type t1)))
936 [cut (\lnot (in_list ? v (fv_type t1)))
937 [lapply (swap_Typ_not_free ? ? ? Hcut Hcut1);
938 lapply (WFE_consG_WFE_G ? ? H1);
939 lapply (H Hletin5 H5 H7);
940 rewrite > Hletin2;rewrite > Hletin4;rewrite > Hletin6;reflexivity
941 |unfold;intro;apply H7;
942 apply (fv_WFT ? ? ? Hletin3 H8)]
943 |unfold;intro;apply H5;apply (fv_WFT ? ? ? Hletin3 H8)]]
946 (*** alternate "constructor" for universal types' well-formedness ***)
948 lemma WFT_Forall2 : \forall G,X,T,T1,T2.
951 \lnot (in_list ? X (fv_type T2)) \to
952 \lnot (in_list ? X (fv_env G)) \to
953 (WFType ((mk_bound true X T)::G)
954 (subst_type_O T2 (TFree X))) \to
955 (WFType G (Forall T1 T2)).
956 intros.apply WFT_Forall
958 |intros;generalize in match (WFT_swap X X1 ? ? H4);simplify;
960 rewrite > (swap_env_not_free X X1 G H H3 H5);
961 rewrite > subst_type_O_swap;rewrite > swap_left;
962 rewrite > (swap_Typ_not_free ? ? T2 H2 H6);
963 intro;apply (WFT_env_incl ? ? H7);unfold;simplify;intros;assumption]
966 (*** lemmas relating subtyping and well-formedness ***)
968 lemma JS_to_WFE : \forall G,T,U.(JSubtype G T U) \to (WFEnv G).
969 intros;elim H;assumption.
972 lemma JS_to_WFT : \forall G,T,U.(JSubtype G T U) \to ((WFType G T) \land
975 [split [assumption|apply WFT_Top]
976 |split;apply WFT_TFree;assumption
978 [apply WFT_TFree;apply boundinenv_natinfv;apply ex_intro
979 [apply true | apply ex_intro [apply t1 |assumption]]
981 |elim H2;elim H4;split;apply WFT_Arrow;assumption
983 [lapply (fresh_name ((fv_env e) @ (fv_type t1)));
984 elim Hletin;cut ((\lnot (in_list ? a (fv_env e))) \land
985 (\lnot (in_list ? a (fv_type t1))))
986 [elim Hcut;apply (WFT_Forall2 ? a t2 ? ? (JS_to_WFE ? ? ? H1) H6 H9 H8);
987 lapply (H4 ? H8);elim Hletin1;assumption
988 |split;unfold;intro;apply H7;apply natinG_or_inH_to_natinGH
991 |lapply (fresh_name ((fv_env e) @ (fv_type t3)));
992 elim Hletin;cut ((\lnot (in_list ? a (fv_env e))) \land
993 (\lnot (in_list ? a (fv_type t3))))
994 [elim Hcut;apply (WFT_Forall2 ? a t2 ? ? (JS_to_WFE ? ? ? H1) H5 H9 H8);
995 lapply (H4 ? H8);elim Hletin1;assumption
996 |split;unfold;intro;apply H7;apply natinG_or_inH_to_natinGH
1001 lemma JS_to_WFT1 : \forall G,T,U.(JSubtype G T U) \to (WFType G T).
1002 intros;lapply (JS_to_WFT ? ? ? H);elim Hletin;assumption.
1005 lemma JS_to_WFT2 : \forall G,T,U.(JSubtype G T U) \to (WFType G U).
1006 intros;lapply (JS_to_WFT ? ? ? H);elim Hletin;assumption.
1009 (*** lemma relating subtyping and swaps ***)
1011 lemma JS_swap : \forall u,v,G,T,U.(JSubtype G T U) \to
1012 (JSubtype (swap_Env u v G) (swap_Typ u v T) (swap_Typ u v U)).
1014 [simplify;apply SA_Top
1015 [apply WFE_swap;assumption
1016 |apply WFT_swap;assumption]
1017 |simplify;apply SA_Refl_TVar
1018 [apply WFE_swap;assumption
1019 |unfold in H2;unfold;lapply (in_dom_swap u v n e);elim Hletin;
1021 |simplify;apply SA_Trans_TVar
1022 [apply (swap_Typ u v t1)
1023 |apply lookup_swap;assumption
1025 |simplify;apply SA_Arrow;assumption
1026 |simplify;apply SA_All
1028 |intros;lapply (H4 (swap u v X))
1029 [simplify in Hletin;rewrite > subst_type_O_swap in Hletin;
1030 rewrite > subst_type_O_swap in Hletin;rewrite > swap_inv in Hletin;
1032 |unfold;intro;apply H5;unfold;
1033 lapply (in_dom_swap u v (swap u v X) e);
1034 elim Hletin;rewrite > swap_inv in H7;apply H7;assumption]]]
1037 lemma fresh_WFT : \forall x,G,T.(WFType G T) \to \lnot (in_list ? x (fv_env G))
1038 \to \lnot (in_list ? x (fv_type T)).
1039 intros;unfold;intro;apply H1;apply (fv_WFT ? ? ? H H2);
1042 lemma fresh_subst_type_O : \forall x,T1,B,G,T,y.
1043 (WFType ((mk_bound B x T1)::G) (subst_type_O T (TFree x))) \to
1044 \lnot (in_list ? y (fv_env G)) \to (x \neq y) \to
1045 \lnot (in_list ? y (fv_type T)).
1046 intros;unfold;intro;
1047 cut (in_list ? y (fv_env ((mk_bound B x T1) :: G)))
1048 [simplify in Hcut;inversion Hcut
1049 [intros;apply H2;lapply (inj_head_nat ? ? ? ? H5);rewrite < H4 in Hletin;
1051 |intros;apply H1;rewrite > H6;lapply (inj_tail ? ? ? ? ? H7);
1052 rewrite > Hletin;assumption]
1053 |apply (fv_WFT (subst_type_O T (TFree x)) ? ? H);
1054 apply fv_subst_type_O;assumption]
1057 (*** alternate "constructor" for subtyping between universal types ***)
1059 lemma SA_All2 : \forall G,S1,S2,T1,T2,X.(JSubtype G T1 S1) \to
1060 \lnot (in_list ? X (fv_env G)) \to
1061 \lnot (in_list ? X (fv_type S2)) \to
1062 \lnot (in_list ? X (fv_type T2)) \to
1063 (JSubtype ((mk_bound true X T1) :: G)
1064 (subst_type_O S2 (TFree X))
1065 (subst_type_O T2 (TFree X))) \to
1066 (JSubtype G (Forall S1 S2) (Forall T1 T2)).
1067 intros;apply (SA_All ? ? ? ? ? H);intros;
1068 lapply (decidable_eq_nat X X1);elim Hletin
1069 [rewrite < H6;assumption
1070 |elim (JS_to_WFT ? ? ? H);elim (JS_to_WFT ? ? ? H4);
1071 cut (\lnot (in_list ? X1 (fv_type S2)))
1072 [cut (\lnot (in_list ? X1 (fv_type T2)))
1073 [cut (((mk_bound true X1 T1)::G) =
1074 (swap_Env X X1 ((mk_bound true X T1)::G)))
1076 cut (((subst_type_O S2 (TFree X1)) =
1077 (swap_Typ X X1 (subst_type_O S2 (TFree X)))) \land
1078 ((subst_type_O T2 (TFree X1)) =
1079 (swap_Typ X X1 (subst_type_O T2 (TFree X)))))
1080 [elim Hcut3;rewrite > H11;rewrite > H12;apply JS_swap;
1083 [rewrite > (subst_type_O_swap X X1 S2 X);
1084 rewrite > (swap_Typ_not_free X X1 S2 H2 Hcut);
1085 rewrite > swap_left;reflexivity
1086 |rewrite > (subst_type_O_swap X X1 T2 X);
1087 rewrite > (swap_Typ_not_free X X1 T2 H3 Hcut1);
1088 rewrite > swap_left;reflexivity]]
1089 |simplify;lapply (JS_to_WFE ? ? ? H);
1090 rewrite > (swap_env_not_free X X1 G Hletin1 H1 H5);
1091 cut ((\lnot (in_list ? X (fv_type T1))) \land
1092 (\lnot (in_list ? X1 (fv_type T1))))
1093 [elim Hcut2;rewrite > (swap_Typ_not_free X X1 T1 H11 H12);
1094 rewrite > swap_left;reflexivity
1096 [unfold;intro;apply H1;apply (fv_WFT T1 X G H7 H11)
1097 |unfold;intro;apply H5;apply (fv_WFT T1 X1 G H7 H11)]]]
1098 |unfold;intro;apply H5;lapply (fv_WFT ? X1 ? H10)
1100 [intros;simplify in H13;lapply (inj_head_nat ? ? ? ? H13);
1101 rewrite < H12 in Hletin2;lapply (H6 Hletin2);elim Hletin3
1102 |intros;simplify in H15;lapply (inj_tail ? ? ? ? ? H15);
1103 rewrite < Hletin2 in H12;rewrite < H14 in H12;lapply (H5 H12);
1105 |rewrite > subst_O_nat;apply in_FV_subst;assumption]]
1106 |unfold;intro;apply H5;lapply (fv_WFT ? X1 ? H9)
1108 [intros;simplify in H13;lapply (inj_head_nat ? ? ? ? H13);
1109 rewrite < H12 in Hletin2;lapply (H6 Hletin2);elim Hletin3
1110 |intros;simplify in H15;lapply (inj_tail ? ? ? ? ? H15);
1111 rewrite < Hletin2 in H12;rewrite < H14 in H12;lapply (H5 H12);
1113 |rewrite > subst_O_nat;apply in_FV_subst;assumption]]]
1116 lemma WFE_Typ_subst : \forall H,x,B,C,T,U,G.
1117 (WFEnv (H @ ((mk_bound B x T) :: G))) \to (WFType G U) \to
1118 (WFEnv (H @ ((mk_bound C x U) :: G))).
1120 [simplify;intros;(*FIXME*)generalize in match H1;intro;inversion H1
1121 [intros;lapply (nil_cons ? G (mk_bound B x T));lapply (Hletin H4);
1123 |intros;lapply (inj_tail ? ? ? ? ? H8);lapply (inj_head ? ? ? ? H8);
1124 destruct Hletin1;rewrite < Hletin in H6;rewrite < Hletin in H4;
1125 rewrite < Hcut1 in H6;apply (WFE_cons ? ? ? ? H4 H6 H2)]
1126 |intros;simplify;generalize in match H2;elim t;simplify in H4;
1128 [intros;absurd (mk_bound b n t1::l@(mk_bound B x T::G)=Empty)
1131 |intros;lapply (inj_tail ? ? ? ? ? H9);lapply (inj_head ? ? ? ? H9);
1132 destruct Hletin1;apply WFE_cons
1134 [rewrite > Hletin;assumption
1136 |rewrite > Hcut1;generalize in match H7;rewrite < Hletin;
1137 rewrite > (fv_env_extends ? x B C T U);intro;assumption
1138 |rewrite < Hletin in H8;rewrite > Hcut2;
1139 apply (WFT_env_incl ? ? H8);rewrite > (fv_env_extends ? x B C T U);
1140 unfold;intros;assumption]]]
1143 lemma t_len_pred: \forall T,m.(S (t_len T)) \leq m \to (t_len T) \leq (pred m).
1145 [elim (not_le_Sn_O ? H)
1146 |simplify;apply (le_S_S_to_le ? ? H1)]
1149 lemma pred_m_lt_m : \forall m,T.(t_len T) \leq m \to (pred m) < m.
1152 [4,5:simplify in H2;elim (not_le_Sn_O ? H2)
1153 |*:simplify in H;elim (not_le_Sn_n ? H)]
1154 |intros;simplify;unfold;constructor 1]
1157 lemma WFE_bound_bound : \forall B,x,T,U,G. (WFEnv G) \to
1158 (in_list ? (mk_bound B x T) G) \to
1159 (in_list ? (mk_bound B x U) G) \to T = U.
1161 [lapply (in_list_nil ? ? H1);elim Hletin
1163 [intros;rewrite < H7 in H8;lapply (inj_head ? ? ? ? H8);
1164 rewrite > Hletin in H5;inversion H5
1165 [intros;rewrite < H9 in H10;lapply (inj_head ? ? ? ? H10);
1166 destruct Hletin1;symmetry;assumption
1167 |intros;lapply (inj_tail ? ? ? ? ? H12);rewrite < Hletin1 in H9;
1168 rewrite < H11 in H9;lapply (boundinenv_natinfv x e)
1169 [destruct Hletin;rewrite < Hcut1 in Hletin2;lapply (H3 Hletin2);
1172 [apply B|apply ex_intro [apply T|assumption]]]]
1173 |intros;lapply (inj_tail ? ? ? ? ? H10);rewrite < H9 in H7;
1174 rewrite < Hletin in H7;(*FIXME*)generalize in match H5;intro;inversion H5
1175 [intros;rewrite < H12 in H13;lapply (inj_head ? ? ? ? H13);
1176 destruct Hletin1;rewrite < Hcut1 in H7;
1177 lapply (boundinenv_natinfv n e)
1178 [lapply (H3 Hletin2);elim Hletin3
1180 [apply B|apply ex_intro [apply U|assumption]]]
1181 |intros;apply (H2 ? H7);rewrite > H14;lapply (inj_tail ? ? ? ? ? H15);
1182 rewrite > Hletin1;assumption]]]