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2 (*       ___                                                              *)
3 (*      ||M||                                                             *)
4 (*      ||A||       A project by Andrea Asperti                           *)
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14
15 include "static_2/static/reqg_reqg.ma".
16 include "static_2/static/feqg.ma".
17
18 (* GENERIC EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES *********************)
19
20 (* Advanced properties ******************************************************)
21
22 lemma feqg_sym (S):
23       reflexive … S → symmetric … S →
24       tri_symmetric … (feqg S).
25 #S #H1S #H2S #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1
26 /3 width=1 by feqg_intro_dx, reqg_sym, teqg_sym/
27 qed-.
28
29 lemma feqg_dec (S):
30       (∀s1,s2. Decidable … (S s1 s2)) →
31       ∀G1,G2,L1,L2,T1,T2. Decidable (❨G1,L1,T1❩ ≛[S] ❨G2,L2,T2❩).
32 #S #HS #G1 #G2 #L1 #L2 #T1 #T2
33 elim (eq_genv_dec G1 G2) #HnG destruct
34 [ elim (reqg_dec … HS L1 L2 T1) #HnL
35   [ elim (teqg_dec … HS T1 T2) #HnT
36     [ /3 width=1 by feqg_intro_sn, or_introl/ ]
37   ]
38 ]
39 @or_intror #H
40 elim (feqg_inv_gen_sn … H) -H #H #HL #HT destruct
41 /2 width=1 by/
42 qed-. 
43
44 (* Main properties **********************************************************)
45
46 theorem feqg_trans (S):
47         reflexive … S → symmetric … S → Transitive … S →
48         tri_transitive … (feqg S).
49 #S #H1S #H2S #H3S #G1 #G #L1 #L #T1 #T * -G -L -T
50 #L #T #HL1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
51 /4 width=8 by feqg_intro_sn, reqg_trans, teqg_reqg_div, teqg_trans/
52 qed-.
53
54 theorem feqg_canc_sn (S):
55         reflexive … S → symmetric … S → Transitive … S →
56         ∀G,G1,L,L1,T,T1. ❨G,L,T❩ ≛[S] ❨G1,L1,T1❩ →
57         ∀G2,L2,T2. ❨G,L,T❩ ≛[S] ❨G2,L2,T2❩ → ❨G1,L1,T1❩ ≛[S] ❨G2,L2,T2❩.
58 /3 width=5 by feqg_trans, feqg_sym/ qed-.
59
60 theorem feqg_canc_dx (S):
61         reflexive … S → symmetric … S → Transitive … S →
62         ∀G1,G,L1,L,T1,T. ❨G1,L1,T1❩ ≛[S] ❨G,L,T❩ →
63         ∀G2,L2,T2. ❨G2,L2,T2❩ ≛[S] ❨G,L,T❩ → ❨G1,L1,T1❩ ≛[S] ❨G2,L2,T2❩.
64 /3 width=5 by feqg_trans, feqg_sym/ qed-.
65
66 lemma feqg_reqg_trans (S) (G2) (L) (T2):
67       reflexive … S → symmetric … S → Transitive … S →
68       ∀G1,L1,T1. ❨G1,L1,T1❩ ≛[S] ❨G2,L,T2❩ →
69       ∀L2. L ≛[S,T2] L2 → ❨G1,L1,T1❩ ≛[S] ❨G2,L2,T2❩.
70 #S #G2 #L #T2 #H1S #H2S #H3S #G1 #L1 #T1 #H1 #L2 #HL2
71 /4 width=5 by feqg_trans, feqg_intro_sn, teqg_refl/
72 qed-.
73
74 (* Inversion lemmas with generic equivalence on terms ***********************)
75
76 (* Basic_2A1: uses: feqg_tneqg_repl_dx *)
77 lemma feqg_tneqg_trans (S) (G1) (G2) (L1) (L2) (T):
78       reflexive … S → symmetric … S → Transitive … S →
79       ∀T1. ❨G1,L1,T1❩ ≛[S] ❨G2,L2,T❩ →
80       ∀T2. (T ≛[S] T2 → ⊥) → (T1 ≛[S] T2 → ⊥).
81 #S #G1 #G2 #L1 #L2 #T #H1S #H2S #H3S #T1 #H1 #T2 #HnT2 #HT12
82 elim (feqg_inv_gen_sn … H1) -H1 #_ #_ #HnT1 -G1 -G2 -L1 -L2
83 /3 width=3 by teqg_canc_sn/
84 qed-.