(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
lemma tc_lfxs_pair_refl: ∀R. c_reflexive … R →
- ∀L,V1,V2. LTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤**[R, T] L.ⓑ{I}V2.
+ ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤**[R, T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #H elim H -V2
/3 width=3 by tc_lfxs_step_dx, lfxs_pair_refl, inj/
qed.
+lemma tc_lfxs_tc: ∀R,L1,L2,T,f. 𝐈⦃f⦄ → TC … (lexs cfull (cext2 R) f) L1 L2 →
+ L1 ⪤**[R, T] L2.
+#R #L1 #L2 #T #f #Hf #H elim H -L2
+[ elim (frees_total L1 T) | #L elim (frees_total L T) ]
+/5 width=7 by lexs_sdj, tc_lfxs_step_dx, sdj_isid_sn, inj, ex2_intro/
+qed.
+
(* Advanced eliminators *****************************************************)
lemma tc_lfxs_ind_sn: ∀R. c_reflexive … R →
- ∀L1,T. ∀R0:predicate …. R0 L1 →
- (∀L,L2. L1 ⪤**[R, T] L → L ⪤*[R, T] L2 → R0 L → R0 L2) →
- ∀L2. L1 ⪤**[R, T] L2 → R0 L2.
-#R #HR #L1 #T #R0 #HL1 #IHL1 #L2 #HL12
+ ∀L1,T. ∀Q:predicate …. Q L1 →
+ (∀L,L2. L1 ⪤**[R, T] L → L ⪤*[R, T] L2 → Q L → Q L2) →
+ ∀L2. L1 ⪤**[R, T] L2 → Q L2.
+#R #HR #L1 #T #Q #HL1 #IHL1 #L2 #HL12
@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by lfxs_refl/
qed-.
lemma tc_lfxs_ind_dx: ∀R. c_reflexive … R →
- ∀L2,T. ∀R0:predicate …. R0 L2 →
- (∀L1,L. L1 ⪤*[R, T] L → L ⪤**[R, T] L2 → R0 L → R0 L1) →
- ∀L1. L1 ⪤**[R, T] L2 → R0 L1.
-#R #HR #L2 #R0 #HL2 #IHL2 #L1 #HL12
+ ∀L2,T. ∀Q:predicate …. Q L2 →
+ (∀L1,L. L1 ⪤*[R, T] L → L ⪤**[R, T] L2 → Q L → Q L1) →
+ ∀L1. L1 ⪤**[R, T] L2 → Q L1.
+#R #HR #L2 #Q #HL2 #IHL2 #L1 #HL12
@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by lfxs_refl/
qed-.