milestone update in ground_2 and basic_2A

[helm.git] / matita / matita / contribs / lambdadelta / basic_2A / etc / cpr / tpss_tpss.etc

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_tpss.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/tpss_tpss.etc
new file mode 100644 (file)
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/substitution/tps_tps.ma".
+include "basic_2/unfold/tpss_lift.ma".
+
+(* PARTIAL UNFOLD ON TERMS **************************************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma tpss_inv_SO2: ∀L,T1,T2,d. L ⊢ T1 ▶* [d, 1] T2 → L ⊢ T1 ▶ [d, 1] T2.
+#L #T1 #T2 #d #H @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1
+lapply (tps_trans_ge … IHT1 … HT2 ?) //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma tpss_strip_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
+                     ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 →
+                     ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+lemma tpss_strip_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
+                      ∀L2,T2,d2,e2. L2 ⊢ T0 ▶ [d2, e2] T2 →
+                      (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+                      ∃∃T. L2 ⊢ T1 ▶ [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+lemma tpss_strap1_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
+                        ∀T2,d2,e2. L ⊢ T0 ▶ [d2, e2] T2 → d2 + e2 ≤ d1 →
+                        ∃∃T. L ⊢ T1 ▶ [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
+/3 width=3/ qed.
+
+lemma tpss_strap2_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶ [d1, e1] T0 →
+                        ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
+                        ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶ [d1, e1] T2.
+/3 width=3/ qed.
+
+lemma tpss_split_up: ∀L,T1,T2,d,e. L ⊢ T1 ▶* [d, e] T2 →
+                     ∀i. d ≤ i → i ≤ d + e →
+                     ∃∃T. L ⊢ T1 ▶* [d, i - d] T & L ⊢ T ▶* [i, d + e - i] T2.
+#L #T1 #T2 #d #e #H #i #Hdi #Hide @(tpss_ind … H) -T2
+[ /2 width=3/
+| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
+  elim (tps_split_up … HT12 … Hdi Hide) -HT12 -Hide #T0 #HT0 #HT02
+  elim (tpss_strap1_down … HT3 … HT0 ?) -T [2: >commutative_plus /2 width=1/ ]
+  /3 width=7 by ex2_intro, step/ (**) (* just /3 width=7/ is too slow *)
+]
+qed.
+
+lemma tpss_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 ▶* [dt, et] U2 →
+                         ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+                         d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+                         ∃∃T2. K ⊢ T1 ▶* [d, dt + et - (d + e)] T2 &
+                               ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tpss_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tpss_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tpss_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tpss_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 -HLK -HTU1 // <minus_plus_m_m /2 width=3/
+qed.
+
+(* Main properties **********************************************************)
+
+theorem tpss_conf_eq: ∀L,T0,T1,d1,e1. L ⊢ T0 ▶* [d1, e1] T1 →
+                      ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 →
+                      ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+theorem tpss_conf_neq: ∀L1,T0,T1,d1,e1. L1 ⊢ T0 ▶* [d1, e1] T1 →
+                       ∀L2,T2,d2,e2. L2 ⊢ T0 ▶* [d2, e2] T2 →
+                       (d1 + e1 ≤ d2 ∨ d2 + e2 ≤ d1) →
+                       ∃∃T. L2 ⊢ T1 ▶* [d2, e2] T & L1 ⊢ T2 ▶* [d1, e1] T.
+/3 width=3/ qed.
+
+theorem tpss_trans_eq: ∀L,T1,T,T2,d,e.
+                       L ⊢ T1 ▶* [d, e] T → L ⊢ T ▶* [d, e] T2 →
+                       L ⊢ T1 ▶* [d, e] T2.
+/2 width=3/ qed.
+
+theorem tpss_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 ▶* [d1, e1] T0 →
+                         ∀T2,d2,e2. L ⊢ T0 ▶* [d2, e2] T2 → d2 + e2 ≤ d1 →
+                         ∃∃T. L ⊢ T1 ▶* [d2, e2] T & L ⊢ T ▶* [d1, e1] T2.
+/3 width=3/ qed.