include "delayed_updating/syntax/path.ma".
include "delayed_updating/notation/functions/circled_times_1.ma".
+include "ground/xoa/ex_3_2.ma".
(* STRUCTURE FOR PATH *******************************************************)
theorem structure_idem (p):
⊗p = ⊗⊗p.
-#p elim p -p [| * [ #k ] #p #IH ] //
+#p elim p -p //
+* [ #k ] #p #IH //
qed.
theorem structure_append (p) (q):
⊗p●⊗q = ⊗(p●q).
-#p #q elim q -q [| * [ #k ] #q #IH ]
-[||*: <list_append_lcons_sn ] //
+#p #q elim q -q //
+* [ #k ] #q #IH //
+<list_append_lcons_sn //
qed.
(* Constructions with path_lcons ********************************************)
lemma structure_d_sn (p) (k):
⊗p = ⊗(𝗱k◗p).
-#p #n <structure_append //
+#p #k <structure_append //
qed.
lemma structure_m_sn (p):
(𝗦◗⊗p) = ⊗(𝗦◗p).
#p <structure_append //
qed.
+
+(* Basic inversions *********************************************************)
+
+lemma eq_inv_d_dx_structure (h) (q) (p):
+ q◖𝗱h = ⊗p → ⊥.
+#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):
+ q◖𝗺 = ⊗p → ⊥.
+#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):
+ q◖𝗟 = ⊗p →
+ ∃∃r1,r2. q = ⊗r1 & 𝐞 = ⊗r2 & r1●𝗟◗r2 = 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):
+ q◖𝗔 = ⊗p →
+ ∃∃r1,r2. q = ⊗r1 & 𝐞 = ⊗r2 & r1●𝗔◗r2 = 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):
+ q◖𝗦 = ⊗p →
+ ∃∃r1,r2. q = ⊗r1 & 𝐞 = ⊗r2 & r1●𝗦◗r2 = 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):
+ p●q = ⊗r →
+ ∃∃r1,r2.p = ⊗r1 & q = ⊗r2 & r1●r2 = 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 … H0)
+| #H0 elim (eq_inv_m_dx_structure … H0)
+| #H0 elim (eq_inv_L_dx_structure … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (IH … Hr1) -IH -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗟◗r2)) //
+ <structure_append <structure_L_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_A_dx_structure … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (IH … Hr1) -IH -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗔◗r2)) //
+ <structure_append <structure_A_sn <Hr2 -Hr2 //
+| #H0 elim (eq_inv_S_dx_structure … H0) -H0 #r1 #r2 #Hr1 #Hr2 #H0 destruct
+ elim (IH … Hr1) -IH -Hr1 #s1 #s2 #H1 #H2 #H3 destruct
+ @(ex3_2_intro … s1 (s2●𝗦◗r2)) //
+ <structure_append <structure_S_sn <Hr2 -Hr2 //
+]
+qed-.
+
+(* Inversions with path_lcons ***********************************************)
+
+lemma eq_inv_d_sn_structure (h) (q) (p):
+ (𝗱h◗q) = ⊗p → ⊥.
+#h #q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_d_dx_structure … H0)
+qed-.
+
+lemma eq_inv_m_sn_structure (q) (p):
+ (𝗺 ◗q) = ⊗p → ⊥.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_m_dx_structure … H0)
+qed-.
+
+lemma eq_inv_L_sn_structure (q) (p):
+ (𝗟◗q) = ⊗p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ⊗r2 & r1●𝗟◗r2 = p.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_L_dx_structure … H0) -H0 #s1 #s2 #H1 #H2 #H3 destruct
+@(ex3_2_intro … s1 (s2●r2)) // -s1
+<structure_append <H2 -s2 //
+qed-.
+
+lemma eq_inv_A_sn_structure (q) (p):
+ (𝗔◗q) = ⊗p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ⊗r2 & r1●𝗔◗r2 = p.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_A_dx_structure … H0) -H0 #s1 #s2 #H1 #H2 #H3 destruct
+@(ex3_2_intro … s1 (s2●r2)) // -s1
+<structure_append <H2 -s2 //
+qed-.
+
+lemma eq_inv_S_sn_structure (q) (p):
+ (𝗦◗q) = ⊗p →
+ ∃∃r1,r2. 𝐞 = ⊗r1 & q = ⊗r2 & r1●𝗦◗r2 = p.
+#q #p >list_cons_comm #H0
+elim (eq_inv_append_structure … H0) -H0 #r1 #r2
+<list_cons_comm #H0 #H1 #H2 destruct
+elim (eq_inv_S_dx_structure … H0) -H0 #s1 #s2 #H1 #H2 #H3 destruct
+@(ex3_2_intro … s1 (s2●r2)) // -s1
+<structure_append <H2 -s2 //
+qed-.
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