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 "basic_2/notation/relations/lazyeq_3.ma".
16 include "basic_2/static/lfxs.ma".
18 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
20 definition lfeq: relation3 term lenv lenv ≝ lfxs ceq.
23 "equivalence on referred entries (local environment)"
24 'LazyEq T L1 L2 = (lfeq T L1 L2).
26 definition lfeq_transitive: predicate (relation3 lenv term term) ≝
27 λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2.
29 (* Basic properties ***********************************************************)
31 lemma lfeq_atom: ∀I. ⋆ ≡[⓪{I}] ⋆.
32 /2 width=1 by lfxs_atom/ qed.
34 lemma lfeq_sort: ∀I,L1,L2,V1,V2,s.
35 L1 ≡[⋆s] L2 → L1.ⓑ{I}V1 ≡[⋆s] L2.ⓑ{I}V2.
36 /2 width=1 by lfxs_sort/ qed.
38 lemma lfeq_zero: ∀I,L1,L2,V.
39 L1 ≡[V] L2 → L1.ⓑ{I}V ≡[#0] L2.ⓑ{I}V.
40 /2 width=1 by lfxs_zero/ qed.
42 lemma lfeq_lref: ∀I,L1,L2,V1,V2,i.
43 L1 ≡[#i] L2 → L1.ⓑ{I}V1 ≡[#⫯i] L2.ⓑ{I}V2.
44 /2 width=1 by lfxs_lref/ qed.
46 lemma lfeq_gref: ∀I,L1,L2,V1,V2,l.
47 L1 ≡[§l] L2 → L1.ⓑ{I}V1 ≡[§l] L2.ⓑ{I}V2.
48 /2 width=1 by lfxs_gref/ qed.
50 (* Basic inversion lemmas ***************************************************)
52 lemma lfeq_inv_atom_sn: ∀I,Y2. ⋆ ≡[⓪{I}] Y2 → Y2 = ⋆.
53 /2 width=3 by lfxs_inv_atom_sn/ qed-.
55 lemma lfeq_inv_atom_dx: ∀I,Y1. Y1 ≡[⓪{I}] ⋆ → Y1 = ⋆.
56 /2 width=3 by lfxs_inv_atom_dx/ qed-.
58 lemma lfeq_inv_zero: ∀Y1,Y2. Y1 ≡[#0] Y2 →
60 ∃∃I,L1,L2,V. L1 ≡[V] L2 &
61 Y1 = L1.ⓑ{I}V & Y2 = L2.ⓑ{I}V.
62 #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
63 /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
66 lemma lfeq_inv_lref: ∀Y1,Y2,i. Y1 ≡[#⫯i] Y2 →
68 ∃∃I,L1,L2,V1,V2. L1 ≡[#i] L2 &
69 Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
70 /2 width=1 by lfxs_inv_lref/ qed-.
72 lemma lfeq_inv_bind: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 →
73 L1 ≡[V] L2 ∧ L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
74 /2 width=2 by lfxs_inv_bind/ qed-.
76 lemma lfeq_inv_flat: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 →
77 L1 ≡[V] L2 ∧ L1 ≡[T] L2.
78 /2 width=2 by lfxs_inv_flat/ qed-.
80 (* Advanced inversion lemmas ************************************************)
82 lemma lfeq_inv_zero_pair_sn: ∀I,Y2,L1,V. L1.ⓑ{I}V ≡[#0] Y2 →
83 ∃∃L2. L1 ≡[V] L2 & Y2 = L2.ⓑ{I}V.
84 #I #Y2 #L1 #V #H elim (lfxs_inv_zero_pair_sn … H) -H /2 width=3 by ex2_intro/
87 lemma lfeq_inv_zero_pair_dx: ∀I,Y1,L2,V. Y1 ≡[#0] L2.ⓑ{I}V →
88 ∃∃L1. L1 ≡[V] L2 & Y1 = L1.ⓑ{I}V.
89 #I #Y1 #L2 #V #H elim (lfxs_inv_zero_pair_dx … H) -H
90 #L1 #X #HL12 #HX #H destruct /2 width=3 by ex2_intro/
93 lemma lfeq_inv_lref_pair_sn: ∀I,Y2,L1,V1,i. L1.ⓑ{I}V1 ≡[#⫯i] Y2 →
94 ∃∃L2,V2. L1 ≡[#i] L2 & Y2 = L2.ⓑ{I}V2.
95 /2 width=2 by lfxs_inv_lref_pair_sn/ qed-.
97 lemma lfeq_inv_lref_pair_dx: ∀I,Y1,L2,V2,i. Y1 ≡[#⫯i] L2.ⓑ{I}V2 →
98 ∃∃L1,V1. L1 ≡[#i] L2 & Y1 = L1.ⓑ{I}V1.
99 /2 width=2 by lfxs_inv_lref_pair_dx/ qed-.
101 (* Basic forward lemmas *****************************************************)
103 lemma lfeq_fwd_bind_sn: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 → L1 ≡[V] L2.
104 /2 width=4 by lfxs_fwd_bind_sn/ qed-.
106 lemma lfeq_fwd_bind_dx: ∀p,I,L1,L2,V,T.
107 L1 ≡[ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
108 /2 width=2 by lfxs_fwd_bind_dx/ qed-.
110 lemma lfeq_fwd_flat_sn: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[V] L2.
111 /2 width=3 by lfxs_fwd_flat_sn/ qed-.
113 lemma lfeq_fwd_flat_dx: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 → L1 ≡[T] L2.
114 /2 width=3 by lfxs_fwd_flat_dx/ qed-.
116 lemma lfeq_fwd_pair_sn: ∀I,L1,L2,V,T. L1 ≡[②{I}V.T] L2 → L1 ≡[V] L2.
117 /2 width=3 by lfxs_fwd_pair_sn/ qed-.
119 (* Advanceded forward lemmas with generic extension on referred entries *****)
121 lemma lfex_fwd_lfxs_refl: ∀R. (∀L. reflexive … (R L)) →
122 ∀L1,L2,T. L1 ≡[T] L2 → L1 ⦻*[R, T] L2.
123 /2 width=3 by lfxs_co/ qed-.
125 (* Basic_2A1: removed theorems 30:
126 lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref
127 lleq_fwd_drop_sn lleq_fwd_drop_dx
128 lleq_fwd_bind_sn lleq_fwd_bind_dx lleq_fwd_flat_sn lleq_fwd_flat_dx
129 lleq_sort lleq_skip lleq_lref lleq_free lleq_gref lleq_bind lleq_flat
130 lleq_refl lleq_Y lleq_sym lleq_ge_up lleq_ge lleq_bind_O llpx_sn_lrefl
131 lleq_trans lleq_canc_sn lleq_canc_dx lleq_nlleq_trans nlleq_lleq_div