--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/relationstarstar_4.ma".
+include "basic_2/static/lfxs.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+definition tc_lfxs (R) (T): relation lenv ≝ TC … (lfxs R T).
+
+interpretation "iterated extension on referred entries (local environment)"
+ 'RelationStarStar R T L1 L2 = (tc_lfxs R T L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma tc_lfxs_atom: ∀R,I. ⋆ ⦻**[R, ⓪{I}] ⋆.
+/2 width=1 by inj/ qed.
+
+lemma tc_lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
+ L1 ⦻**[R, ⋆s] L2 → L1.ⓑ{I}V1 ⦻**[R, ⋆s] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
+/3 width=4 by lfxs_sort, step, inj/
+qed.
+
+lemma tc_lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
+ L1 ⦻**[R, #i] L2 → L1.ⓑ{I}V1 ⦻**[R, #⫯i] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
+/3 width=4 by lfxs_lref, step, inj/
+qed.
+
+lemma tc_lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
+ L1 ⦻**[R, §l] L2 → L1.ⓑ{I}V1 ⦻**[R, §l] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
+/3 width=4 by lfxs_gref, step, inj/
+qed.
+
+lemma tc_lfxs_sym: ∀R. lexs_frees_confluent R cfull →
+ (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀T. symmetric … (tc_lfxs R T).
+#R #H1R #H2R #T #L1 #L2 #H elim H -L2
+/4 width=3 by lfxs_sym, TC_strap, inj/
+qed-.
+
+lemma tc_lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+ ∀L1,L2,T. L1 ⦻**[R1, T] L2 → L1 ⦻**[R2, T] L2.
+#R1 #R2 #HR #L1 #L2 #T #H elim H -L2
+/4 width=5 by lfxs_co, step, inj/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma tc_lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻**[R, ⓪{I}] Y2 → Y2 = ⋆.
+#R #I #Y2 #H elim H -Y2 /3 width=3 by inj, lfxs_inv_atom_sn/
+qed-.
+
+lemma tc_lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻**[R, ⓪{I}] ⋆ → Y1 = ⋆.
+#R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
+/3 width=3 by inj, lfxs_inv_atom_dx/
+qed-.
+
+lemma tc_lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻**[R, ⋆s] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, ⋆s] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #Y1 #Y2 #s #H elim H -Y2
+[ #Y2 #H elim (lfxs_inv_sort … H) -H *
+ /4 width=8 by ex3_5_intro, inj, or_introl, or_intror, conj/
+| #Y #Y2 #_ #H elim (lfxs_inv_sort … H) -H *
+ [ #H #H2 * * /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
+ | #I #L #L2 #V #V2 #HL2 #H #H2 * *
+ [ #H1 #H0 destruct
+ | #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct
+ /4 width=8 by ex3_5_intro, step, or_intror/
+ ]
+ ]
+]
+qed-.
+(*
+lemma tc_lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻**[R, #0] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #Y1 #Y2 #H elim H -Y2
+[ #Y2 #H elim (lfxs_inv_zero … H) -H *
+ /4 width=9 by ex4_5_intro, inj, or_introl, or_intror, conj/
+| #Y #Y2 #_ #H elim (lfxs_inv_zero … H) -H *
+ [ #H #H2 * * /3 width=9 by ex4_5_intro, or_introl, or_intror, conj/
+ | #I #L #L2 #V #V2 #HL2 #HV2 #H #H2 * *
+ [ #H1 #H0 destruct
+ | #I0 #L1 #L0 #V1 #V0 #HL10 #HV10 #H1 #H0 destruct
+ @or_intror @ex4_5_intro [6,7: |*: /width=7/ ]
+
+ /4 width=8 by ex3_5_intro, step, or_intror/
+ ]
+ ]
+]
+qed-.
+
+
+
+
+
+#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
+[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
+| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
+ /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
+[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
+| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
+ /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2
+[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
+| lapply (frees_inv_gref … H1) -H1 #Hf
+ elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
+ elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
+ /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
+ L1 ⦻*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
+/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+lemma lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 →
+ L1 ⦻*[R, V] L2 ∧ L1 ⦻*[R, T] L2.
+#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
+/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
+/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
+qed-.
+
+lemma lfxs_fwd_bind_dx: ∀R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
+qed-.
+
+lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
+qed-.
+
+lemma lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, T] L2.
+#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
+qed-.
+
+lemma lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
+qed-.
+
+(* Basic_2A1: removed theorems 24:
+ llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
+ llpx_sn_bind llpx_sn_flat
+ llpx_sn_inv_bind llpx_sn_inv_flat
+ llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
+ llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
+ llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
+ llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx
+*)
+*)
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