update in delayd_updating

[helm.git] / matita / matita / contribs / lambdadelta / delayed_updating / syntax / path_closed.ma

diff --git a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
index d8b00cabc8e8ec67d1a721c0580de19d2c8fab08..620db955c30943a906f75ca63045380069e576e0 100644 (file)
--- a/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
+++ b/matita/matita/contribs/lambdadelta/delayed_updating/syntax/path_closed.ma
@@ -14,35 +14,188 @@
 
 include "delayed_updating/syntax/path.ma".
 include "delayed_updating/notation/functions/class_c_1.ma".
+include "ground/arith/nat_plus.ma".
+include "ground/arith/nat_pred_succ.ma".
 include "ground/lib/subset.ma".
-
-include "delayed_updating/syntax/path_depth.ma".
-include "delayed_updating/syntax/path_height.ma".
+include "ground/generated/insert_eq_1.ma".
 
 (* CLOSED CONDITION FOR PATH ************************************************)
 
-axiom pcc: relation2 nat path.
+inductive pcc: relation2 nat path ≝
+| pcc_empty:
+  pcc (𝟎) (𝐞)
+| pcc_d_dx (p) (n) (k):
+  pcc (n+ninj k) p → pcc n (p◖𝗱k)
+| pcc_m_dx (p) (n):
+  pcc n p → pcc n (p◖𝗺)
+| pcc_L_dx (p) (n):
+  pcc n p → pcc (↑n) (p◖𝗟)
+| pcc_A_dx (p) (n):
+  pcc n p → pcc n (p◖𝗔)
+| pcc_S_dx (p) (n):
+  pcc n p → pcc n (p◖𝗦)
+.
 
 interpretation
   "closed condition (path)"
   'ClassC n = (pcc n).
 
-(* Basic destructions *******************************************************)
+(* Basic inversions ********************************************************)
+
+lemma pcc_inv_empty (n):
+      (𝐞) ϵ 𝐂❨n❩ → 𝟎 = n.
+#n @(insert_eq_1 … (𝐞))
+#x * -n //
+#p #n [ #k ] #_ #H0 destruct
+qed-.
+
+lemma pcc_inv_d_dx (p) (n) (k):
+      p◖𝗱k ϵ 𝐂❨n❩ → p ϵ 𝐂❨n+k❩.
+#p #n #h @(insert_eq_1 … (p◖𝗱h))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_m_dx (p) (n):
+      p◖𝗺 ϵ 𝐂❨n❩ → p ϵ 𝐂❨n❩.
+#p #n @(insert_eq_1 … (p◖𝗺))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_L_dx (p) (n):
+      p◖𝗟 ϵ 𝐂❨n❩ →
+      ∧∧ p ϵ 𝐂❨↓n❩ & n = ↑↓n.
+#p #n @(insert_eq_1 … (p◖𝗟))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct
+<npred_succ /2 width=1 by conj/
+qed-.
+
+lemma pcc_inv_A_dx (p) (n):
+      p◖𝗔 ϵ 𝐂❨n❩ → p ϵ 𝐂❨n❩.
+#p #n @(insert_eq_1 … (p◖𝗔))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+lemma pcc_inv_S_dx (p) (n):
+      p◖𝗦 ϵ 𝐂❨n❩ → p ϵ 𝐂❨n❩.
+#p #n @(insert_eq_1 … (p◖𝗦))
+#x * -x -n
+[|*: #x #n [ #k ] #Hx ] #H0 destruct //
+qed-.
+
+(* Advanced inversions ******************************************************)
+
+lemma pcc_inv_empty_succ (n):
+      (𝐞) ϵ 𝐂❨↑n❩ → ⊥.
+#n #H0
+lapply (pcc_inv_empty … H0) -H0 #H0
+/2 width=7 by eq_inv_zero_nsucc/
+qed-.
+
+lemma pcc_inv_L_dx_zero (p):
+      p◖𝗟 ϵ 𝐂❨𝟎❩ → ⊥.
+#p #H0
+elim (pcc_inv_L_dx … H0) -H0 #_ #H0
+/2 width=7 by eq_inv_zero_nsucc/
+qed-.
+
+lemma pcc_inv_L_dx_succ (p) (n):
+      p◖𝗟 ϵ 𝐂❨↑n❩ → p ϵ 𝐂❨n❩.
+#p #n #H0
+elim (pcc_inv_L_dx … H0) -H0 //
+qed-.
+
+(* Main constructions with path_append **************************************)
+
+theorem pcc_append_bi (p) (q) (m) (n):
+        p ϵ 𝐂❨m❩ → q ϵ 𝐂❨n❩ → p●q ϵ 𝐂❨m+n❩.
+#p #q #m #n #Hm #Hm elim Hm -Hm // -Hm
+#p #n [ #k ] #_ #IH [3: <nplus_succ_dx ]
+/2 width=1 by pcc_d_dx, pcc_m_dx, pcc_L_dx, pcc_A_dx, pcc_S_dx/
+qed.
+
+(* Constructions with path_lcons ********************************************)
+
+lemma pcc_m_sn (q) (n):
+      q ϵ 𝐂❨n❩ → (𝗺◗q) ϵ 𝐂❨n❩.
+#q #n #Hq
+lapply (pcc_append_bi (𝐞◖𝗺) … Hq) -Hq
+/2 width=3 by pcc_m_dx/
+qed.
+
+lemma pcc_L_sn (q) (n):
+      q ϵ 𝐂❨n❩ → (𝗟◗q) ϵ 𝐂❨↑n❩.
+#q #n #Hq
+lapply (pcc_append_bi (𝐞◖𝗟) … Hq) -Hq
+/2 width=3 by pcc_L_dx/
+qed.
 
-axiom pcc_empty:
-      (𝐞) ϵ 𝐂❨𝟎❩.
+lemma pcc_A_sn (q) (n):
+      q ϵ 𝐂❨n❩ → (𝗔◗q) ϵ 𝐂❨n❩.
+#q #n #Hq
+lapply (pcc_append_bi (𝐞◖𝗔) … Hq) -Hq
+/2 width=3 by pcc_A_dx/
+qed.
 
-axiom pcc_d (p) (d) (n:pnat):
-      p ϵ 𝐂❨d+n❩ → p◖𝗱n ϵ 𝐂❨d❩.
+lemma pcc_S_sn (q) (n):
+      q ϵ 𝐂❨n❩ → (𝗦◗q) ϵ 𝐂❨n❩.
+#q #n #Hq
+lapply (pcc_append_bi (𝐞◖𝗦) … Hq) -Hq
+/2 width=3 by pcc_S_dx/
+qed.
 
-axiom pcc_L (p) (d):
-      p ϵ 𝐂❨d❩ → p◖𝗟 ϵ 𝐂❨↑d❩.
+(* Main inversions **********************************************************)
 
-axiom pcc_A (p) (d):
-      p ϵ 𝐂❨d❩ → p◖𝗔 ϵ 𝐂❨d❩.
+theorem pcc_mono (q) (n1):
+        q ϵ 𝐂❨n1❩ → ∀n2. q ϵ 𝐂❨n2❩ → n1 = n2.
+#q1 #n1 #Hn1 elim Hn1 -q1 -n1
+[|*: #q1 #n1 [ #k1 ] #_ #IH ] #n2 #Hn2
+[ <(pcc_inv_empty … Hn2) -n2 //
+| lapply (pcc_inv_d_dx … Hn2) -Hn2 #Hn2
+  lapply (IH … Hn2) -q1 #H0
+  /2 width=2 by eq_inv_nplus_bi_dx/
+| lapply (pcc_inv_m_dx … Hn2) -Hn2 #Hn2
+  <(IH … Hn2) -q1 -n2 //
+| elim (pcc_inv_L_dx … Hn2) -Hn2 #Hn2 #H0
+  >(IH … Hn2) -q1 //
+| lapply (pcc_inv_A_dx … Hn2) -Hn2 #Hn2
+  <(IH … Hn2) -q1 -n2 //
+| lapply (pcc_inv_S_dx … Hn2) -Hn2 #Hn2
+  <(IH … Hn2) -q1 -n2 //
+]
+qed-.
 
-axiom pcc_S (p) (d):
-      p ϵ 𝐂❨d❩ → p◖𝗦 ϵ 𝐂❨d❩.
+theorem pcc_inj_L_sn (p1) (p2) (q1) (n):
+        q1 ϵ 𝐂❨n❩ → ∀q2. q2 ϵ 𝐂❨n❩ →
+        p1●𝗟◗q1 = p2●𝗟◗q2 → q1 = q2.
+#p1 #p2 #q1 #n #Hq1 elim Hq1 -q1 -n
+[|*: #q1 #n1 [ #k1 ] #_ #IH ] * //
+[1,3,5,7,9,11: #l2 #q2 ] #Hq2
+<list_append_lcons_sn <list_append_lcons_sn #H0
+elim (eq_inv_list_lcons_bi ????? H0) -H0 #H0 #H1 destruct
+[ elim (pcc_inv_L_dx_zero … Hq2)
+| lapply (pcc_inv_d_dx … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| lapply (pcc_inv_m_dx … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| lapply (pcc_inv_L_dx_succ … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| lapply (pcc_inv_A_dx … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| lapply (pcc_inv_S_dx … Hq2) -Hq2 #Hq2
+  <(IH … Hq2) //
+| elim (pcc_inv_empty_succ … Hq2)
+]
+qed-.
 
-axiom pcc_des_gen (p) (d):
-      p ϵ 𝐂❨d❩ → d + ♯p = ❘p❘.
+theorem pcc_inv_L_sn (q) (n) (m):
+        (𝗟◗q) ϵ 𝐂❨n❩ → q ϵ 𝐂❨m❩ →
+        ∧∧ ↓n = m & n = ↑↓n.
+#q #n #m #H1q #H2q
+lapply (pcc_L_sn … H2q) -H2q #H2q
+<(pcc_mono … H2q … H1q) -q -n
+/2 width=1 by conj/
+qed-.