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/defn2.ma".
17 (*** Lemma A.1 (Reflexivity) ***)
18 theorem JS_Refl : ∀T,G.WFType G T → WFEnv G → G ⊢ T ⊴ T.
23 | intros; apply (H4 ? H6);
24 [ intro; apply H6; apply (fv_WFT ? ? ? (WFT_Forall ? ? ? H1 H3));
30 * A slightly more general variant to lemma A.2.2, where weakening isn't
31 * defined as concatenation of any two disjoint environments, but as
35 lemma JS_weakening : ∀G,T,U.G ⊢ T ⊴ U → ∀H.WFEnv H → incl ? G H → H ⊢ T ⊴ U.
37 [1,2,3,4: autobatch depth=4 width=4 size=7
38 | apply (SA_All ? ? ? ? ? (H2 ? H6 H7));intros;
41 | apply WFE_cons; autobatch
42 | unfold;intros; elim (in_list_cons_case ? ? ? ? H9); destruct; autobatch]]
46 ∀G:list bound.∀T1,T:Typ.
47 ∀P:list bound → Typ → Typ → Prop.
48 (∀t. WFEnv G → WFType G t → T=Top → P G t Top) →
49 (∀n. WFEnv G → n ∈ fv_env G → T=TFree n → P G (TFree n) (TFree n)) →
51 (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ t → P G t1 t → T=t → P G (TFree n) T) →
52 (∀s1,s2,t1,t2. G ⊢ t1 ⊴ s1 → G ⊢ s2 ⊴ t2 → T=Arrow t1 t2 → P G (Arrow s1 s2) (Arrow t1 t2)) →
53 (∀s1,s2,t1,t2. G ⊢ t1 ⊴ s1 →
54 (∀X. ¬(X ∈ fv_env G) → (mk_bound true X t1)::G ⊢ subst_type_nat s2 (TFree X) O ⊴ subst_type_nat t2 (TFree X) O)
55 → T=Forall t1 t2 → P G (Forall s1 s2) (Forall t1 t2)) →
56 G ⊢ T1 ⊴ T → P G T1 T.
58 generalize in match (refl_eq ? T);
59 generalize in match (refl_eq ? G);
60 elim H5 in ⊢ (? ? ? % → ? ? ? % → %); destruct;
61 [1,2,3,4: autobatch depth=10 width=10 size=8
62 | apply H4; first [assumption | autobatch]]
65 theorem narrowing:∀X,G,G1,U,P,M,N.
66 G1 ⊢ P ⊴ U → (∀G2,T.G2@G1 ⊢ U ⊴ T → G2@G1 ⊢ P ⊴ T) → G ⊢ M ⊴ N →
67 ∀l.G=l@(mk_bound true X U::G1) → l@(mk_bound true X P::G1) ⊢ M ⊴ N.
68 intros 10.elim H2; destruct;
69 [1,2,4: autobatch width=10 depth=10 size=8
70 | elim (decidable_eq_nat X n)
71 [apply (SA_Trans_TVar ? ? ? P); destruct;
73 | rewrite > append_cons; apply H1;
74 lapply (WFE_bound_bound true X t1 U ? ? H3); destruct;
76 | rewrite < append_cons; autobatch
78 | apply (SA_Trans_TVar ? ? ? t1)
79 [ apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H3);
85 apply (H6 ? ? (mk_bound true X1 t2::l1))
86 [ rewrite > fv_env_extends; autobatch
90 lemma JS_trans_prova: ∀T,G1.WFType G1 T →
91 ∀G,R,U.incl ? (fv_env G1) (fv_env G) → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
92 intros 3;elim H;clear H; try autobatch;
93 [ apply (JSubtype_inv ? ? ? ? ? ? ? ? ? H3); intros; destruct; autobatch
94 | inversion H3; intros; destruct; assumption
95 |*: apply (JSubtype_inv ? ? ? ? ? ? ? ? ? H6); intros; destruct;
97 |*: inversion H7; intros; destruct;
98 [1,2: autobatch depth=4 width=4 size=9
103 | intros;lapply (H8 ? H11);
107 | intros;apply (H4 X);
109 |intro; apply H13;apply H5; apply (WFT_to_incl ? ? ? H3);
112 |apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H9 ? ? [])
114 [unfold;intros;lapply (H5 ? H15);rewrite > fv_append;
116 |apply (JS_weakening ? ? ? H9)
118 |unfold;intros;autobatch]
124 theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
128 theorem JS_narrow : ∀G1,G2,X,P,Q,T,U.
129 (G2 @ (mk_bound true X Q :: G1)) ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
130 (G2 @ (mk_bound true X P :: G1)) ⊢ T ⊴ U.
131 intros; apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
132 intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
133 [autobatch|unfold;intros;autobatch]