/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_inv_sort: ∀R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
#R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2
qed-.
lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
- (Y1 = ⋆ ∧ 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 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
qed-.
lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
- (Y1 = ⋆ ∧ 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 *