-(* Basic properties *********************************************************)
-
-lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆.
-/3 width=3 by lexs_atom, frees_atom, ex2_intro/
-qed.
-
-lemma 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 * /3 width=3 by lexs_push, frees_sort, ex2_intro/
-qed.
-
-lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/
-qed.
-
-lemma 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 * /3 width=3 by lexs_push, frees_lref, ex2_intro/
-qed.
-
-lemma 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 * /3 width=3 by lexs_push, frees_gref, ex2_intro/
-qed.
-
-lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1.
- L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V1 →
- ∀V2. R L1 V V2 →
- L1.ⓑ{I}V ⦻*[R, T] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR
-/3 width=5 by lexs_pair_repl, ex2_intro/
-qed-.
-
-lemma lfxs_sym: ∀R. lexs_frees_confluent R cfull →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (lfxs R T).
-#R #H1R #H2R #T #L1 #L2 * #f1 #Hf1 #HL12 elim (H1R … Hf1 … HL12) -Hf1
-/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/
-qed-.
-
-lemma 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 * /4 width=7 by lexs_co, ex2_intro/
-qed-.
-
-lemma lfxs_isid: ∀R1,R2,L1,L2,T1,T2.
- (∀f. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → 𝐈⦃f⦄) →
- (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≡ f) →
- L1 ⦻*[R1, T1] L2 → L1 ⦻*[R2, T2] L2.
-#R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
-/4 width=7 by lexs_co_isid, ex2_intro/
-qed-.
-