(**************************************************************************)
set "baseuri" "cic:/matita/Fsub/part1a/".
-include "logic/equality.ma".
-include "nat/nat.ma".
-include "datatypes/bool.ma".
-include "nat/compare.ma".
-include "Fsub/util.ma".
include "Fsub/defn.ma".
(*** Lemma A.1 (Reflexivity) ***)
-theorem JS_Refl : \forall T,G.(WFType G T) \to (WFEnv G) \to (JSubtype G T T).
-apply Typ_len_ind;intro;elim U
- [(* FIXME *) generalize in match H1;intro;inversion H1
- [intros;destruct H6
- |intros;destruct H5
- |*:intros;destruct H9]
- |apply (SA_Refl_TVar ? ? H2);(*FIXME*)generalize in match H1;intro;
- inversion H1
- [intros;destruct H6;rewrite > Hcut;assumption
- |intros;destruct H5
- |*:intros;destruct H9]
- |apply (SA_Top ? ? H2 H1)
- |cut ((WFType G t) \land (WFType G t1))
- [elim Hcut;apply SA_Arrow
- [apply H2
- [apply t_len_arrow1
- |*:assumption]
- |apply H2
- [apply t_len_arrow2
- |*:assumption]]
- |(* no shortcut? *)
- (*FIXME*)generalize in match H3;intro;inversion H3
- [intros;destruct H8
- |intros;destruct H7
- |intros;destruct H11;rewrite > Hcut;rewrite > Hcut1;split;assumption
- |intros;destruct H11]]
- |cut (WFType G t)
- [lapply (H2 t ? ? Hcut H4)
- [apply t_len_forall1
- |apply (SA_All ? ? ? ? ? Hletin);intros;apply H2
- [rewrite < eq_t_len_TFree_subst;
- apply t_len_forall2
- |generalize in match H3;intro;inversion H3
- [intros;destruct H9
- |intros;destruct H8
- |intros;destruct H12
- |intros;destruct H12;subst;apply H9
- [assumption
- |unfold;intro;apply H5;
- elim (fresh_name ((fv_env l)@(fv_type t3)));
- cut ((\lnot (in_list ? a (fv_env l))) \land
- (\lnot (in_list ? a (fv_type t3))))
- [elim Hcut1;lapply (H9 ? H13 H14);
- lapply (fv_WFT ? X ? Hletin1)
- [simplify in Hletin2;inversion Hletin2
- [intros;lapply (inj_head_nat ? ? ? ? H16);subst;
- elim (H14 H11)
- |intros;lapply (inj_tail ? ? ? ? ? H18);
- rewrite < Hletin3 in H15;assumption]
- |apply varinT_varinT_subst;
- assumption]
- |split;unfold;intro;apply H12;apply natinG_or_inH_to_natinGH
- [right;assumption
- |left;assumption]]]]
- |apply WFE_cons;assumption]]
- |(*FIXME*)generalize in match H3;intro;inversion H3
- [intros;destruct H8
- |intros;destruct H7
- |intros;destruct H11
- |intros;destruct H11;subst;assumption]]]
+theorem JS_Refl : ∀T,G.WFType G T → WFEnv G → G ⊢ T ⊴ T.
+intros 3;elim H
+ [apply (SA_Refl_TVar l n H2 H1);
+ |apply (SA_Top l Top H1 (WFT_Top l));
+ |apply (SA_Arrow l t t1 t t1 (H2 H5) (H4 H5))
+ |apply (SA_All ? ? ? ? ? (H2 H5));intros;apply (H4 X H6)
+ [intro;apply H6;apply (fv_WFT (Forall t t1) X l)
+ [apply (WFT_Forall ? ? ? H1 H3)
+ |simplify;autobatch]
+ |autobatch]]
qed.
(*
|apply WFE_cons
[1,2:assumption
|apply (JS_to_WFT1 ? ? ? Hletin)]
- |unfold;intros;inversion H9
- [intros;lapply (inj_head ? ? ? ? H11);rewrite > Hletin1;apply in_Base
- |intros;lapply (inj_tail ? ? ? ? ? H13);rewrite < Hletin1 in H10;
- apply in_Skip;apply (H7 ? H10)]]]
+ |unfold;intros;elim (in_cons_case ? ? ? ? H9)
+ [rewrite > H10;apply in_Base
+ |elim H10;apply (in_Skip ? ? ? ? ? H11);apply (H7 ? H12)]]]
+qed.
+
+lemma decidable_eq_Typ : \forall S,T:Typ.(S = T) ∨ (S ≠ T).
+intro;elim S
+ [elim T
+ [elim (decidable_eq_nat n n1)
+ [rewrite > H;left;reflexivity
+ |right;intro;destruct H1;apply (H Hcut)]
+ |2,3:right;intro;destruct H
+ |*:right;intro;destruct H2]
+ |elim T
+ [2:elim (decidable_eq_nat n n1)
+ [rewrite > H;left;reflexivity
+ |right;intro;destruct H1;apply (H Hcut)]
+ |1,3:right;intro;destruct H
+ |*:right;intro;destruct H2]
+ |elim T
+ [3:left;reflexivity
+ |1,2:right;intro;destruct H
+ |*:right;intro;destruct H2]
+ |elim T
+ [1,2,3:right;intro;destruct H2
+ |elim (H t2)
+ [rewrite > H4;elim (H1 t3)
+ [rewrite > H5;left;reflexivity
+ |right;intro;apply H5;destruct H6;assumption]
+ |right;intro;apply H4;destruct H5;assumption]
+ |right;intro;destruct H4]
+ |elim T
+ [1,2,3:right;intro;destruct H2
+ |right;intro;destruct H4
+ |elim (H t2)
+ [rewrite > H4;elim (H1 t3)
+ [rewrite > H5;left;reflexivity
+ |right;intro;apply H5;destruct H6;assumption]
+ |right;intro;apply H4;destruct H5;assumption]]]
qed.
+lemma decidable_eq_bound: ∀b1,b2:bound.(b1 = b2) ∨ (b1 ≠ b2).
+intros;elim b1;elim b2;elim (decidable_eq_nat n n1)
+ [rewrite < H;elim (decidable_eq_Typ t t1)
+ [rewrite < H1;elim (bool_to_decidable_eq b b3)
+ [rewrite > H2;left;reflexivity
+ |right;intro;destruct H3;apply (H2 Hcut)]
+ |right;intro;destruct H2;apply (H1 Hcut1)]
+ |right;intro;destruct H1;apply (H Hcut1)]
+qed.
+
(* Lemma A.3 (Transitivity and Narrowing) *)
lemma JS_trans_narrow : \forall Q.
[apply (SA_Trans_TVar ? ? ? P)
[rewrite < H10;elim l1
[simplify;constructor 1
- |simplify;constructor 2;assumption]
+ |simplify;elim (decidable_eq_bound (mk_bound true X P) t2)
+ [rewrite < H12;apply in_Base
+ |apply (in_Skip ? ? ? ? ? H12);assumption]]
|apply H7
[lapply (H6 ? H7 H8 H9);lapply (JS_to_WFE ? ? ? Hletin);
apply (JS_weakening ? ? ? H3 ? Hletin1);unfold;intros;
- elim l1;simplify;constructor 2;assumption
+ elim l1
+ [simplify;
+ elim (decidable_eq_bound x (mk_bound true X P))
+ [rewrite < H12;apply in_Base
+ |apply (in_Skip ? ? ? ? ? H12);assumption]
+ |simplify;elim (decidable_eq_bound x t2)
+ [rewrite < H13;apply in_Base
+ |apply (in_Skip ? ? ? ? ? H13);assumption]]
|lapply (WFE_bound_bound true n t1 U ? ? H4)
[apply (JS_to_WFE ? ? ? H5)
|rewrite < Hletin;apply (H6 ? H7 H8 H9)
|rewrite > H9;rewrite > H10;elim l1;simplify
[constructor 1
- |constructor 2;assumption]]]]
+ |elim (decidable_eq_bound (mk_bound true n U) t2)
+ [rewrite > H12;apply in_Base
+ |apply (in_Skip ? ? ? ? ? H12);assumption]]]]]
|apply (SA_Trans_TVar ? ? ? t1)
[rewrite > H9 in H4;
apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H4);
projects / helm.git / blobdiff