+
+(* Inversions with inner condition for path *********************************)
+
+lemma lift_inv_append_inner_sn (q1) (q2) (p) (f):
+ q1 Ļµ š ā q1āq2 = ā[f]p ā
+ āāp1,p2. āp1 = q1 & ā[ā[p1]f]p2 = q2 & p1āp2 = p.
+#q1 @(list_ind_rcons ā¦ q1) -q1
+[ #q2 #p #f #Hq1 <list_append_empty_sn #H destruct
+ /2 width=5 by ex3_2_intro/
+| #q1 * [ #n1 ] #_ #q2 #p #f #Hq2
+ [ elim (pic_inv_d_dx ā¦ Hq2)
+ | <list_append_rcons_sn #H0
+ elim (lift_inv_append_proper_dx ā¦ H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (lift_path_inv_m_sn ā¦ (sym_eq ā¦ H2))
+ | <list_append_rcons_sn #H0
+ elim (lift_inv_append_proper_dx ā¦ H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (lift_path_inv_L_sn ā¦ (sym_eq ā¦ H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+ @(ex3_2_intro ā¦ (p1ār2āš)) [1,3: // ]
+ [ <structure_append <structure_L_dx <Hr2 -Hr2 //
+ | <list_append_assoc <list_append_rcons_sn //
+ ]
+ | <list_append_rcons_sn #H0
+ elim (lift_inv_append_proper_dx ā¦ H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (lift_path_inv_A_sn ā¦ (sym_eq ā¦ H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+ @(ex3_2_intro ā¦ (p1ār2āš)) [1,3: // ]
+ [ <structure_append <structure_A_dx <Hr2 -Hr2 //
+ | <list_append_assoc <list_append_rcons_sn //
+ ]
+ | <list_append_rcons_sn #H0
+ elim (lift_inv_append_proper_dx ā¦ H0) -H0 // #p1 #p2 #H1 #H2 #H3 destruct
+ elim (lift_path_inv_S_sn ā¦ (sym_eq ā¦ H2)) -H2 #r2 #s2 #Hr2 #Hs2 #H0 destruct
+ @(ex3_2_intro ā¦ (p1ār2āš¦)) [1,3: // ]
+ [ <structure_append <structure_S_dx <Hr2 -Hr2 //
+ | <list_append_assoc <list_append_rcons_sn //
+ ]
+ ]
+]
+qed-.