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.
20 [apply SA_Refl_TVar [apply H2|assumption]
21 |apply SA_Top [assumption|apply WFT_Top]
22 |apply (SA_Arrow ? ? ? ? ? (H2 H5) (H4 H5))
23 |apply (SA_All ? ? ? ? ? (H2 H5));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 [apply (SA_Top ? ? H4);apply (WFT_env_incl ? ? H2 ? (incl_bound_fv ? ? H5))
38 |apply (SA_Refl_TVar ? ? H4);apply (incl_bound_fv ? ? H5 ? H2)
39 |apply (SA_Trans_TVar ? ? ? ? ? (H3 ? H5 H6));apply H6;assumption
40 |apply (SA_Arrow ? ? ? ? ? (H2 ? H6 H7) (H4 ? H6 H7))
41 |apply (SA_All ? ? ? ? ? (H2 ? H6 H7));intros;apply H4
42 [unfold;intro;apply H8;apply (incl_bound_fv ? ? H7 ? H9)
43 |apply (WFE_cons ? ? ? ? H6 H8);autobatch
44 |unfold;intros;inversion H9;intros
45 [destruct H11;apply in_list_head
46 |destruct H13;apply in_list_cons;apply (H7 ? H10)]]]
49 theorem narrowing:∀X,G,G1,U,P,M,N.
50 G1 ⊢ P ⊴ U → (∀G2,T.G2@G1 ⊢ U ⊴ T → G2@G1 ⊢ P ⊴ T) → G ⊢ M ⊴ N →
51 ∀l.G=l@(mk_bound true X U::G1) → l@(mk_bound true X P::G1) ⊢ M ⊴ N.
55 apply (WFE_Typ_subst ? ? ? ? ? ? ? H3 (JS_to_WFT1 ? ? ? H))
56 |rewrite > H5 in H4;apply (WFT_env_incl ? ? H4);apply incl_fv_env]
58 [rewrite > H5 in H3;apply (WFE_Typ_subst ? ? ? ? ? ? ? H3);
59 apply (JS_to_WFT1 ? ? ? H)
60 |rewrite > H5 in H4;rewrite < fv_env_extends;apply H4]
61 |elim (decidable_eq_nat X n)
62 [apply (SA_Trans_TVar ? ? ? P)
63 [rewrite < H7;elim l1;simplify
64 [constructor 1|constructor 2;assumption]
65 |rewrite > append_cons;apply H1;
66 lapply (WFE_bound_bound true n t1 U ? ? H3)
67 [apply (JS_to_WFE ? ? ? H4)
68 |rewrite < Hletin;rewrite < append_cons;apply (H5 ? H6)
69 |rewrite < H7;rewrite > H6;elim l1;simplify
70 [constructor 1|constructor 2;assumption]]]
71 |apply (SA_Trans_TVar ? ? ? t1)
72 [rewrite > H6 in H3;apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H3);
73 unfold;intro;apply H7;symmetry;assumption
75 |apply (SA_Arrow ? ? ? ? ? (H4 ? H7) (H6 ? H7))
76 |apply (SA_All ? ? ? ? ? (H4 ? H7));intros;
77 apply (H6 ? ? (mk_bound true X1 t2::l1))
78 [rewrite > H7;rewrite > fv_env_extends;apply H8
79 |simplify;rewrite < H7;reflexivity]]
82 lemma JSubtype_Arrow_inv:
83 ∀G:list bound.∀T1,T2,T3:Typ.
84 ∀P:list bound → Typ → Prop.
86 (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ (Arrow T2 T3) → P G t1 → P G (TFree n)) →
87 (∀t,t1. G ⊢ T2 ⊴ t → G ⊢ t1 ⊴ T3 → P G (Arrow t t1)) →
88 G ⊢ T1 ⊴ (Arrow T2 T3) → P G T1.
90 generalize in match (refl_eq ? (Arrow T2 T3));
91 generalize in match (refl_eq ? G);
92 elim H2 in ⊢ (? ? ? % → ? ? ? % → %);
95 | lapply (H5 H6 H7); destruct; clear H5;
105 lemma JSubtype_Forall_inv:
106 ∀G:list bound.∀T1,T2,T3:Typ.
107 ∀P:list bound → Typ → Prop.
109 (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ (Forall T2 T3) → P G t1 → P G (TFree n)) →
110 (∀t,t1. G ⊢ T2 ⊴ t → (∀X. ¬(X ∈ fv_env G) → (mk_bound true X T2)::G ⊢ subst_type_nat t1 (TFree X) O ⊴ subst_type_nat T3 (TFree X) O)
111 → P G (Forall t t1)) →
112 G ⊢ T1 ⊴ (Forall T2 T3) → P G T1.
114 generalize in match (refl_eq ? (Forall T2 T3));
115 generalize in match (refl_eq ? G);
116 elim H2 in ⊢ (? ? ? % → ? ? ? % → %);
119 | lapply (H5 H6 H7); destruct; clear H5;
130 lemma JS_trans_prova: ∀T,G1.WFType G1 T →
131 ∀G,R,U.incl ? (fv_env G1) (fv_env G) → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
132 intros 3;elim H;clear H; try autobatch;
133 [rewrite > (JSubtype_Top ? ? H3);autobatch
134 |apply (JSubtype_Arrow_inv ? ? ? ? ? ? ? H6); intros;
136 | inversion H7;intros; destruct; autobatch depth=4 width=4 size=9]
137 |apply (JSubtype_Forall_inv ? ? ? ? ? ? ? H6); intros;
139 | inversion H7;intros; destruct;
144 | intros;lapply (H8 ? H11);
148 | intros;apply (H4 X);
149 [intro;apply H13;apply H5;assumption
150 |intro;apply H13;apply H5;apply (WFT_to_incl ? ? ? H3);
153 |apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H9 ? ? [])
155 [unfold;intros;lapply (H5 ? H15);rewrite > fv_append;
157 |apply (JS_weakening ? ? ? H9)
159 |unfold;intros;autobatch]
165 theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
166 intros 5;apply (JS_trans_prova ? G);autobatch;
169 theorem JS_narrow : ∀G1,G2,X,P,Q,T,U.
170 (G2 @ (mk_bound true X Q :: G1)) ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
171 (G2 @ (mk_bound true X P :: G1)) ⊢ T ⊴ U.
172 intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
173 intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
174 [autobatch|unfold;intros;autobatch]