+
+lemma structure_S_sn (p):
+ (π¦ββp) = β(π¦βp).
+#p <structure_append //
+qed.
+
+(* Basic inversions *********************************************************)
+
+lemma eq_inv_d_dx_structure (h) (q) (p):
+ qβπ±h = βp β β₯.
+#h #q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0 /2 width=1 by/
+| <structure_m_dx #H0 /2 width=1 by/
+| <structure_L_dx #H0 destruct
+| <structure_A_dx #H0 destruct
+| <structure_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_m_dx_structure (q) (p):
+ qβπΊ = βp β β₯.
+#q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0 /2 width=1 by/
+| <structure_m_dx #H0 /2 width=1 by/
+| <structure_L_dx #H0 destruct
+| <structure_A_dx #H0 destruct
+| <structure_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_L_dx_structure (q) (p):
+ qβπ = βp β
+ ββr1,r2. q = βr1 & π = βr2 & r1βπβr2 = p.
+#q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_m_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_L_dx #H0 destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <structure_A_dx #H0 destruct
+| <structure_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_A_dx_structure (q) (p):
+ qβπ = βp β
+ ββr1,r2. q = βr1 & π = βr2 & r1βπβr2 = p.
+#q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_m_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_L_dx #H0 destruct
+| <structure_A_dx #H0 destruct -IH
+ /2 width=5 by ex3_2_intro/
+| <structure_S_dx #H0 destruct
+]
+qed-.
+
+lemma eq_inv_S_dx_structure (q) (p):
+ qβπ¦ = βp β
+ ββr1,r2. q = βr1 & π = βr2 & r1βπ¦βr2 = p.
+#q #p elim p -p [| * [ #k ] #p #IH ]
+[ <structure_empty #H0 destruct
+| <structure_d_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_m_dx #H0
+ elim IH -IH // -H0 #r1 #r2 #H1 #H0 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| <structure_L_dx #H0 destruct
+| <structure_A_dx #H0 destruct
+| <structure_S_dx #H0 destruct -IH
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Main inversions **********************************************************)
+
+theorem eq_inv_append_structure (p) (q) (r):
+ pβq = βr β
+ ββr1,r2.p = βr1 & q = βr2 & r1βr2 = r.
+#p #q elim q -q [| * [ #k ] #q #IH ] #r
+[ <list_append_empty_sn #H0 destruct
+ /2 width=5 by ex3_2_intro/
+| #H0 elim (eq_inv_d_dx_structure β¦ H0)
+| #H0 elim (eq_inv_m_dx_structure β¦ H0)
+| #H0 elim (eq_inv_L_dx_structure β¦ H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (IH β¦ Hr1) -IH -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro β¦ s1 (s2βπβr2)) //
+ <structure_append <structure_L_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_A_dx_structure β¦ H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (IH β¦ Hr1) -IH -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro β¦ s1 (s2βπβr2)) //
+ <structure_append <structure_A_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_S_dx_structure β¦ H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (IH β¦ Hr1) -IH -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro β¦ s1 (s2βπ¦βr2)) //
+ <structure_append <structure_S_sn <Hr2 -Hr2 //
+]
+qed-.
+
+(* Inversions with path_lcons ***********************************************)
+
+lemma eq_inv_d_sn_structure (h) (q) (p):
+ (π±hβq) = βp β β₯.
+#h #q #p >list_cons_comm #H0
+elim (eq_inv_append_structure β¦ H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_d_dx_structure β¦ H0)
+qed-.
+
+lemma eq_inv_m_sn_structure (q) (p):
+ (πΊ βq) = βp β β₯.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure β¦ H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_m_dx_structure β¦ H0)
+qed-.
+
+lemma eq_inv_L_sn_structure (q) (p):
+ (πβq) = βp β
+ ββr1,r2. π = βr1 & q = βr2 & r1βπβr2 = p.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure β¦ H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_L_dx_structure β¦ H0) -H0 #s1 #s2 #H1 #H2 #H3 destruct
+@(ex3_2_intro β¦ s1 (s2βr2)) // -s1
+<structure_append <H2 -s2 //
+qed-.
+
+lemma eq_inv_A_sn_structure (q) (p):
+ (πβq) = βp β
+ ββr1,r2. π = βr1 & q = βr2 & r1βπβr2 = p.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure β¦ H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_A_dx_structure β¦ H0) -H0 #s1 #s2 #H1 #H2 #H3 destruct
+@(ex3_2_intro β¦ s1 (s2βr2)) // -s1
+<structure_append <H2 -s2 //
+qed-.
+
+lemma eq_inv_S_sn_structure (q) (p):
+ (π¦βq) = βp β
+ ββr1,r2. π = βr1 & q = βr2 & r1βπ¦βr2 = p.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure β¦ H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_S_dx_structure β¦ H0) -H0 #s1 #s2 #H1 #H2 #H3 destruct
+@(ex3_2_intro β¦ s1 (s2βr2)) // -s1
+<structure_append <H2 -s2 //
+qed-.