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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/static/lfxs_fqup.ma".
16 include "basic_2/i_static/tc_lfxs.ma".
18 (* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
20 (* Advanced properties ******************************************************)
22 lemma tc_lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀T. reflexive … (tc_lfxs R T).
23 /3 width=1 by lfxs_refl, inj/ qed.
25 (* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
26 lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
27 ∀L,V1,V2. LTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤**[R, T] L.ⓑ{I}V2.
28 #R #HR #L #V1 #V2 #H elim H -V2
29 /3 width=3 by tc_lfxs_step_dx, lfxs_pair, inj/
32 (* Advanced eliminators *****************************************************)
34 lemma tc_lfxs_ind_sn: ∀R. (∀L. reflexive … (R L)) →
35 ∀L1,T. ∀R0:predicate …. R0 L1 →
36 (∀L,L2. L1 ⪤**[R, T] L → L ⪤*[R, T] L2 → R0 L → R0 L2) →
37 ∀L2. L1 ⪤**[R, T] L2 → R0 L2.
38 #R #HR #L1 #T #R0 #HL1 #IHL1 #L2 #HL12
39 @(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by lfxs_refl/
42 lemma tc_lfxs_ind_dx: ∀R. (∀L. reflexive … (R L)) →
43 ∀L2,T. ∀R0:predicate …. R0 L2 →
44 (∀L1,L. L1 ⪤*[R, T] L → L ⪤**[R, T] L2 → R0 L → R0 L1) →
45 ∀L1. L1 ⪤**[R, T] L2 → R0 L1.
46 #R #HR #L2 #R0 #HL2 #IHL2 #L1 #HL12
47 @(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by lfxs_refl/
50 (* Advanced inversion lemmas ************************************************)
52 lemma tc_lfxs_inv_bind_void: ∀R. (∀L. reflexive … (R L)) →
53 ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
54 L1 ⪤**[R, V] L2 ∧ L1.ⓧ ⪤**[R, T] L2.ⓧ.
55 #R #HR #p #I #L1 #L2 #V #T #H @(tc_lfxs_ind_sn … HR … H) -L2
56 [ /3 width=1 by tc_lfxs_refl, conj/
57 | #L #L2 #_ #H * elim (lfxs_inv_bind_void … H) -H /3 width=3 by tc_lfxs_step_dx, conj/
61 (* Advanced forward lemmas **************************************************)
63 lemma tc_lfxs_fwd_bind_dx_void: ∀R. (∀L. reflexive … (R L)) →
64 ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
66 #R #HR #p #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_bind_void … H) -H //