+(**************************************************************************)
+(* ___ *)
+(* ||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 "ground_2/lib/star.ma".
+include "basic_2/relocation/lexs.ma".
+
+(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
+
+(* Properties with transitive closure ***************************************)
+
+lemma lexs_tc_refl: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+ ∀f. reflexive … (TC … (lexs RN RP f)).
+/3 width=1 by lexs_refl, TC_reflexive/ qed.
+
+lemma lexs_tc_next_sn: ∀RN,RP. (∀L. reflexive … (RN L)) →
+ ∀f,I2,L1,L2. TC … (lexs RN RP f) L1 L2 → ∀I1. RN L1 I1 I2 →
+ TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
+/3 width=3 by lexs_next, TC_strap, inj/
+qed.
+
+lemma lexs_tc_next_dx: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+ ∀f,I1,I2,L1. (LTC … RN) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
+ TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
+/4 width=5 by lexs_refl, lexs_next, step, inj/
+qed.
+
+lemma lexs_tc_push_sn: ∀RN,RP. (∀L. reflexive … (RP L)) →
+ ∀f,I2,L1,L2. TC … (lexs RN RP f) L1 L2 → ∀I1. RP L1 I1 I2 →
+ TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRP #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
+/3 width=3 by lexs_push, TC_strap, inj/
+qed.
+
+lemma lexs_tc_push_dx: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+ ∀f,I1,I2,L1. (LTC … RP) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
+ TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
+/4 width=5 by lexs_refl, lexs_push, step, inj/
+qed.
+
+lemma lexs_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → L1 ⪤*[LTC … RN, RP, f] L2.
+#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by lexs_push, lexs_next, inj/
+qed.
+
+lemma lexs_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → L1 ⪤*[RN, LTC … RP, f] L2.
+#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by lexs_push, lexs_next, inj/
+qed.
+
+(* Main properties with transitive closure **********************************)
+
+theorem lexs_tc_next: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+ ∀f,I1,I2,L1. (LTC … RN) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
+ TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
+/4 width=5 by lexs_tc_next_sn, lexs_tc_refl, trans_TC/
+qed.
+
+theorem lexs_tc_push: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+ ∀f,I1,I2,L1. (LTC … RP) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
+ TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
+/4 width=5 by lexs_tc_push_sn, lexs_tc_refl, trans_TC/
+qed.
+
+(* Basic_2A1: uses: TC_lpx_sn_ind *)
+theorem lexs_tc_step_dx: ∀RN,RP. (∀f. 𝐈⦃f⦄ → s_rs_transitive … RP (λ_.lexs RN RP f)) →
+ ∀f,L1,L. L1 ⪤*[RN, RP, f] L → 𝐈⦃f⦄ →
+ ∀L2. L ⪤*[RN, LTC … RP, f] L2 → L1⪤* [RN, LTC … RP, f] L2.
+#RN #RP #HRP #f #L1 #L #H elim H -f -L1 -L
+[ #f #_ #Y #H -HRP >(lexs_inv_atom1 … H) -Y // ]
+#f #I1 #I #L1 #L #HL1 #HI1 #IH #Hf #Y #H
+[ elim (isid_inv_next … Hf) -Hf //
+| lapply (isid_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
+ elim (lexs_inv_push1 … H) -H #I2 #L2 #HL2 #HI2 #H destruct
+ @lexs_push [ /2 width=1 by/ ] -L2 -IH
+ @(TC_strap … HI1) -HI1
+ @(HRP … HL1) // (**) (* auto fails *)
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: TC_lpx_sn_inv_lpx_sn_LTC *)
+lemma lexs_tc_dx: ∀RN,RP. (∀f. 𝐈⦃f⦄ → s_rs_transitive … RP (λ_.lexs RN RP f)) →
+ ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (lexs RN RP f) L1 L2 → L1 ⪤*[RN, LTC … RP, f] L2.
+#RN #RP #HRP #f #Hf #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
+/3 width=3 by lexs_tc_step_dx, lexs_tc_inj_dx/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lexs_inv_tc_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+ ∀f,L1,L2. L1 ⪤*[LTC … RN, RP, f] L2 → TC … (lexs RN RP f) L1 L2.
+#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
+/2 width=1 by lexs_tc_next, lexs_tc_push_sn, lexs_atom, inj/
+qed-.
+
+(* Basic_2A1: uses: lpx_sn_LTC_TC_lpx_sn *)
+lemma lexs_inv_tc_dx: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+ ∀f,L1,L2. L1 ⪤*[RN, LTC … RP, f] L2 → TC … (lexs RN RP f) L1 L2.
+#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
+/2 width=1 by lexs_tc_push, lexs_tc_next_sn, lexs_atom, inj/
+qed-.