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 include "Fsub/defn.ma".
17 (*** Lemma A.1 (Reflexivity) ***)
18 theorem JS_Refl : ∀T,G.WFType G T → WFEnv G → G ⊢ T ⊴ T.
21 | apply SA_All; [ autobatch | intros;autobatch depth=4 size=10]]
25 * A slightly more general variant to lemma A.2.2, where weakening isn't
26 * defined as concatenation of any two disjoint environments, but as
30 lemma JS_weakening : ∀G,T,U.G ⊢ T ⊴ U → ∀H.WFEnv H → incl ? G H → H ⊢ T ⊴ U.
32 [1,2,3,4: autobatch depth=4 size=7
33 | apply (SA_All ? ? ? ? ? (H2 ? H6 H7));
34 intros; apply H4;autobatch depth=4 size=7]
37 inverter JS_indinv for JSubtype (%?%).
39 theorem narrowing:∀X,G,G1,U,P,M,N.
40 G1 ⊢ P ⊴ U → (∀G2,T.G2@G1 ⊢ U ⊴ T → G2@G1 ⊢ P ⊴ T) → G ⊢ M ⊴ N →
41 ∀l.G=l@(mk_bound true X U::G1) → l@(mk_bound true X P::G1) ⊢ M ⊴ N.
42 intros 10.elim H2; destruct;
43 [letin x \def fv_env. letin y ≝incl. autobatch depth=4 size=8.
44 | autobatch depth=4 size=7;
45 | elim (decidable_eq_nat X n)
46 [apply (SA_Trans_TVar ? ? ? P); destruct;
48 | lapply (WFE_bound_bound true X t1 U ? ? H3); autobatch]
49 | apply (SA_Trans_TVar ? ? ? t1); autobatch]
53 | intros; apply (H6 ? ? (mk_bound true X1 t2::l1)); autobatch]]
56 lemma JS_trans_prova: ∀T,G1.WFType G1 T →
57 ∀G,R,U.incl ? (fv_env G1) (fv_env G) → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
58 intros 3;elim H;clear H;
59 [elim H3 using JS_indinv;destruct;autobatch
60 |inversion H3; intros; destruct; assumption
61 |*: elim H6 using JS_indinv;destruct;
63 |*: inversion H7; intros; destruct;
64 [1,2: autobatch depth=4 width=4 size=9
67 | apply WFT_Forall;intros;autobatch depth=4]
70 | intros;apply (H4 X);simplify;
71 [4: apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H11 ? ? [])
72 [intros;apply H2;try unfold;intros;autobatch;
77 theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
78 intros 5; apply (JS_trans_prova ? G); autobatch depth=2.
81 theorem JS_narrow : ∀G1,G2,X,P,Q,T,U.
82 (G2 @ (mk_bound true X Q :: G1)) ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
83 (G2 @ (mk_bound true X P :: G1)) ⊢ T ⊴ U.
84 intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
85 intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);autobatch.