include "Fsub/defn.ma".
-axiom daemon : False.
-
(*** Lemma A.1 (Reflexivity) ***)
-theorem JS_Refl : ∀T,G.WFType G T → WFEnv G → G ⊢ T ⊴ T.
-intros 3; elim H;
- [1,2,3: autobatch
- | apply SA_All; [ autobatch | intros;autobatch depth=4 size=10]]
+theorem JS_Refl : ∀T,G.(G ⊢ T) → G ⊢ ♦ → G ⊢ T ⊴ T.
+intros 3; elim H;try autobatch;
+apply SA_All; [ autobatch | intros;autobatch depth=4 size=10]
qed.
(*
* set inclusion.
*)
-lemma JS_weakening : ∀G,T,U.G ⊢ T ⊴ U → ∀H.WFEnv H → incl ? G H → H ⊢ T ⊴ U.
-intros 4; elim H;
- [1,2,3,4: autobatch depth=4 size=7
- | apply (SA_All ? ? ? ? ? (H2 ? H6 H7));
- intros; apply H4;autobatch depth=4 size=7]
+lemma JS_weakening : ∀G,T,U.G ⊢ T ⊴ U → ∀H.H ⊢ ♦ → G ⊆ H → H ⊢ T ⊴ U.
+intros 4; elim H;try autobatch depth=4 size=7;
+apply (SA_All ? ? ? ? ? (H2 ? H6 H7));
+intros; autobatch depth=6 width=4 size=13;
qed.
inverter JS_indinv for JSubtype (%?%).
theorem narrowing:∀X,G,G1,U,P,M,N.
G1 ⊢ P ⊴ U → (∀G2,T.G2@G1 ⊢ U ⊴ T → G2@G1 ⊢ P ⊴ T) → G ⊢ M ⊴ N →
- ∀l.G=l@(mk_bound true X U::G1) → l@(mk_bound true X P::G1) ⊢ M ⊴ N.
+ ∀l.G=l@ !X ⊴ U::G1 → l@ !X ⊴ P::G1 ⊢ M ⊴ N.
intros 10.elim H2; destruct;
[letin x \def fv_env. letin y ≝incl. autobatch depth=4 size=8.
| autobatch depth=4 size=7;
| elim (decidable_eq_nat X n)
[apply (SA_Trans_TVar ? ? ? P); destruct;
[ autobatch
- | lapply (WFE_bound_bound true X t1 U ? ? H3); autobatch]
+ | lapply (WFE_bound_bound X t1 U ? ? H3); autobatch]
| apply (SA_Trans_TVar ? ? ? t1); autobatch]
| autobatch
| apply SA_All;
| intros; apply (H6 ? ? (mk_bound true X1 t2::l1)); autobatch]]
qed.
-lemma JS_trans_prova: ∀T,G1.WFType G1 T →
-∀G,R,U.incl ? (fv_env G1) (fv_env G) → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
+lemma JS_trans_prova: ∀T,G1.(G1 ⊢ T) →
+ ∀G,R,U.fv_env G1 ⊆ fv_env G → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
intros 3;elim H;clear H;
[elim H3 using JS_indinv;destruct;autobatch
|inversion H3; intros; destruct; assumption
- |*: elim H6 using JS_indinv;destruct;
+ |*:elim H6 using JS_indinv;destruct;
[1,3: autobatch
|*: inversion H7; intros; destruct;
[1,2: autobatch depth=4 width=4 size=9
[4: apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H11 ? ? [])
[intros;apply H2;try unfold;intros;autobatch;
|*:autobatch]
+ |3:apply incl_cons;apply H5
|*:autobatch]]]]]
qed.
qed.
theorem JS_narrow : ∀G1,G2,X,P,Q,T,U.
- (G2 @ (mk_bound true X Q :: G1)) ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
- (G2 @ (mk_bound true X P :: G1)) ⊢ T ⊴ U.
+ G2 @ !X ⊴ Q :: G1 ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
+ G2 @ !X ⊴ P :: G1 ⊢ T ⊴ U.
intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);autobatch.
-qed.
+qed.
\ No newline at end of file