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.
53 [letin x \def fv_env. letin y ≝incl.
54 (* autobatch depth=4 size=8 by SA_Top, WFE_Typ_subst, H3, JS_to_WFT1, H, H4, WFT_env_incl, incl_fv_env]*)
56 [autobatch by WFE_Typ_subst, H3, JS_to_WFT1, H.
59 apply (WFE_Typ_subst ? ? ? ? ? ? ? H3 (JS_to_WFT1 ? ? ? H)) *)
60 |autobatch by H4, WFT_env_incl, incl_fv_env]
61 (* rewrite > H5 in H4;apply (WFT_env_incl ? ? H4);apply incl_fv_env] *)
62 |autobatch depth=4 by SA_Refl_TVar, WFE_Typ_subst, H3, JS_to_WFT1, H, H4.
65 [autobatch by WFE_Typ_subst, H3, JS_to_WFT1, H.
67 rewrite > H5 in H3;apply (WFE_Typ_subst ? ? ? ? ? ? ? H3);
68 apply (JS_to_WFT1 ? ? ? H) *)
69 |autobatch by H4. (* rewrite > H5 in H4;rewrite < fv_env_extends;apply H4*)] *)
70 |elim (decidable_eq_nat X n)
71 [apply (SA_Trans_TVar ? ? ? P)
72 [rewrite < H7;elim l1;simplify
73 [constructor 1|constructor 2;assumption]
75 lapply (WFE_bound_bound true n t1 U ? ? H3);
76 [autobatch. (* apply (JS_to_WFE ? ? ? H4) *)
77 |autobatch. (* rewrite < Hletin;rewrite < append_cons;apply (H5 ? H6) *)
78 |destruct.elim l1;autobatch.
80 |(* autobatch depth=4 size=7 by SA_Trans_TVar, lookup_env_extends, H3, sym_neq, H5, H6, H7. *)
81 apply (SA_Trans_TVar ? ? ? t1);
82 [autobatch by lookup_env_extends, H3, sym_neq, H7.
83 (* rewrite > H6 in H3; apply (lookup_env_extends ? ? ? ? ? ? ? ? ? ? H3);
84 unfold;intro;apply H7;symmetry;assumption *)
86 |autobatch; (* apply (SA_Arrow ? ? ? ? ? (H4 ? H7) (H6 ? H7)) *)
87 |apply (SA_All ? ? ? ? ? (H4 ? H7));intros;autobatch]
89 apply (H6 ? ? (mk_bound true X1 t2::l1))
90 [rewrite > H7;rewrite > fv_env_extends;apply H8
91 |simplify;rewrite < H7;reflexivity]] *)
94 lemma JS_trans_prova: ∀T,G1.WFType G1 T →
95 ∀G,R,U.incl ? (fv_env G1) (fv_env G) → G ⊢ R ⊴ T → G ⊢ T ⊴ U → G ⊢ R ⊴ U.
96 intros 3;elim H;clear H; try autobatch;
98 rewrite > (JSubtype_Top ? ? H3);autobatch
99 |generalize in match H7;generalize in match H4;generalize in match H2;
100 generalize in match H5;clear H7 H4 H2 H5;
101 generalize in match (refl_eq ? (Arrow t t1));
102 elim H6 in ⊢ (? ? ? %→%); clear H6; intros; destruct;
103 [apply (SA_Trans_TVar ? ? ? ? H);apply (H4 ? ? H8 H9);autobatch
104 |inversion H11;intros; destruct; autobatch depth=4 width=4 size=9;
106 |generalize in match H7;generalize in match H4;generalize in match H2;
107 generalize in match H5;clear H7 H4 H2 H5;
108 generalize in match (refl_eq ? (Forall t t1));elim H6 in ⊢ (? ? ? %→%);destruct;
109 [apply (SA_Trans_TVar ? ? ? ? H);apply (H4 ? H7 H8 H9 H10);reflexivity
110 |inversion H11;intros;destruct;
115 |intros;lapply (H4 ? H13);autobatch]]
117 [autobatch paramodulation
118 |intros;apply (H10 X)
119 [intro;apply H15;apply H8;assumption
120 |intro;apply H15;apply H8;apply (WFT_to_incl ? ? ? H3);
123 |apply (narrowing X (mk_bound true X t::l1)
126 [unfold;intros;lapply (H8 ? H17);rewrite > fv_append;
128 |apply (JS_weakening ? ? ? H7)
130 |unfold;intros;autobatch]
136 theorem JS_trans : ∀G,T,U,V.G ⊢ T ⊴ U → G ⊢ U ⊴ V → G ⊢ T ⊴ V.
137 intros 5;apply (JS_trans_prova ? G);autobatch;
140 theorem JS_narrow : ∀G1,G2,X,P,Q,T,U.
141 (G2 @ (mk_bound true X Q :: G1)) ⊢ T ⊴ U → G1 ⊢ P ⊴ Q →
142 (G2 @ (mk_bound true X P :: G1)) ⊢ T ⊴ U.
143 intros;apply (narrowing ? ? ? ? ? ? ? H1 ? H) [|autobatch]
144 intros;apply (JS_trans ? ? ? ? ? H2);apply (JS_weakening ? ? ? H1);
145 [autobatch|unfold;intros;autobatch]