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; apply (H4 ? H6);
- [ intro; apply H6; apply (fv_WFT ? ? ? (WFT_Forall ? ? ? H1 H3));
- simplify; autobatch
- | autobatch]]]
+ | apply SA_All; [ autobatch | intros; autobatch depth=4 size=10]
+ ]
qed.
(*
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 width=4 size=7
- | apply (SA_All ? ? ? ? ? (H2 ? H6 H7));intros;
- apply H4
- [ intro; autobatch
- | apply WFE_cons; autobatch
- | unfold;intros; elim (in_list_cons_case ? ? ? ? H9); destruct; autobatch]]
+ [1,2,3,4: autobatch depth=4 size=7
+ | apply (SA_All ? ? ? ? ? (H2 ? H6 H7));
+ intros; apply H4; autobatch depth=4 size=7]
qed.
lemma JSubtype_inv:
intros;
generalize in match (refl_eq ? T);
generalize in match (refl_eq ? G);
- elim H5 in ⊢ (? ? ? % → ? ? ? % → %); destruct;
- [1,2,3,4: autobatch depth=10 width=10 size=8
- | apply H4; first [assumption | autobatch]]
+ elim H5 in ⊢ (? ? ? % → ? ? ? % → %); destruct; autobatch width=4 size=7;
qed.
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.
intros 10.elim H2; destruct;
- [1,2,4: autobatch width=10 depth=10 size=8
+ [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
- | rewrite > append_cons; apply H1;
- lapply (WFE_bound_bound true X t1 U ? ? H3); destruct;
- [1,3: autobatch
- | rewrite < append_cons; autobatch
- ]]
- | apply (SA_Trans_TVar ? ? ? t1)
- [ apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H3);
- intro; autobatch
- | autobatch]]
+ | lapply (WFE_bound_bound true X t1 U ? ? H3); autobatch]
+ | apply (SA_Trans_TVar ? ? ? t1); autobatch]
+ | autobatch
| apply SA_All;
[ autobatch
- | intros;
- apply (H6 ? ? (mk_bound true X1 t2::l1))
- [ rewrite > fv_env_extends; autobatch
- | autobatch]]]
+ | intros; apply (H6 ? ? (mk_bound true X1 t2::l1));autobatch]]
qed.
lemma JS_trans_prova: ∀T,G1.WFType G1 T →
[1,2: autobatch depth=4 width=4 size=9
| apply SA_Top
[ assumption
- | apply WFT_Forall;
- [ autobatch
- | intros;lapply (H8 ? H11);
- autobatch]]
+ | apply WFT_Forall;intros;autobatch depth=4]
| apply SA_All
[ autobatch
- | intros;apply (H4 X);
- [intro; autobatch;
- |intro; apply H13;apply H5; apply (WFT_to_incl ? ? ? H3);
- assumption
- |simplify;autobatch
- |apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H9 ? ? [])
- [intros;apply H2
- [unfold;intros;lapply (H5 ? H15);rewrite > fv_append;
- autobatch
- |apply (JS_weakening ? ? ? H9)
- [autobatch
- |unfold;intros;autobatch]
- |assumption]
+ | intros;apply (H4 X);simplify;
+ [4: apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H9 ? ? [])
+ [intros;apply H2;try unfold;intros;autobatch;
|*:autobatch]
- |autobatch]]]]]
+ |*: autobatch]]]]]
qed.
theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
-intros 5; autobatch.
+intros 5; apply (JS_trans_prova ? G); autobatch depth=2.
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.
-intros; apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
-intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
- [autobatch|unfold;intros;autobatch]
+intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
+intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);autobatch.
qed.