X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fdelayed_updating%2Fsyntax%2Fpath_structure.ma;h=3b91954fd83ca4a90115038959bdd2dc718f778d;hb=41a54a797af98d2867d4bf979d424283fb44a1fc;hp=a9d9dd18f362a187272cd5b7474fd2e261a2296a;hpb=306205b6853874cf485152222593b57249c6e7fa;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure.ma
index a9d9dd18f..3b91954fd 100644
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_structure.ma
@@ -14,6 +14,7 @@
 
 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 *******************************************************)
 
@@ -99,3 +100,159 @@ lemma structure_S_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-.