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.
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]]
83 ∀G:list bound.∀T1,T:Typ.
84 ∀P:list bound → Typ → Prop.
85 (∀t. WFEnv G → WFType G t → T=Top → P G t) →
86 (∀n. T=TFree n → P G (TFree n)) →
88 (mk_bound true n t1) ∈ G → G ⊢ t1 ⊴ T → P G t1 → P G (TFree n)) →
89 (∀s1,s2,t1,t2. G ⊢ t1 ⊴ s1 → G ⊢ s2 ⊴ t2 → T=Arrow t1 t2 → P G (Arrow s1 s2)) →
90 (∀s1,s2,t1,t2. G ⊢ t1 ⊴ s1 →
91 (∀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)
92 → T=Forall t1 t2 → P G (Forall s1 s2)) →
95 generalize in match (refl_eq ? T);
96 generalize in match (refl_eq ? G);
97 elim H5 in ⊢ (? ? ? % → ? ? ? % → %);
98 [1,2: destruct; autobatch
99 | rewrite < H9 in H6 H7 H8 ⊢ %;
100 rewrite < H10 in H7 H8;
102 | rewrite < H10 in H6 H8 ⊢ %;
104 | rewrite < H10 in H6 H8 ⊢ %;
105 apply (H4 t t1 t2 t3); assumption
110 lemma JS_trans_prova: ∀T,G1.WFType G1 T →
111 ∀G,R,U.incl ? (fv_env G1) (fv_env G) → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
112 intros 3;elim H;clear H; try autobatch;
113 [ apply (JSubtype_inv ? ? ? ? ? ? ? ? ? H3); intros; destruct; autobatch
114 | inversion H3; intros; destruct; assumption
115 |*: apply (JSubtype_inv ? ? ? ? ? ? ? ? ? H6); intros; destruct;
117 |*: inversion H7; intros; destruct;
118 [1,2: autobatch depth=4 width=4 size=9
123 | intros;lapply (H8 ? H11);
127 | intros;apply (H4 X);
129 |intro; apply H13;apply H5; apply (WFT_to_incl ? ? ? H3);
132 |apply (narrowing X (mk_bound true X t::G) ? ? ? ? ? H9 ? ? [])
134 [unfold;intros;lapply (H5 ? H15);rewrite > fv_append;
136 |apply (JS_weakening ? ? ? H9)
138 |unfold;intros;autobatch]
144 theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
145 intros 5;apply (JS_trans_prova ? G);autobatch;
148 theorem JS_narrow : ∀G1,G2,X,P,Q,T,U.
149 (G2 @ (mk_bound true X Q :: G1)) ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
150 (G2 @ (mk_bound true X P :: G1)) ⊢ T ⊴ U.
151 intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
152 intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
153 [autobatch|unfold;intros;autobatch]