- degree-based equivalene for terms

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / etc_2A1 / delift / thin_delift.etc

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/thin_delift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc_2A1/delift/thin_delift.etc
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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/unfold/delift_tpss.ma".
+include "basic_2/unfold/delift_ltpss.ma".
+include "basic_2/unfold/thin.ma".
+
+(* BASIC DELIFT ON LOCAL ENVIRONMENTS ***************************************)
+
+(* Inversion lemmas on inverse basic term relocation ************************)
+
+lemma thin_inv_delift1: ∀I,K1,V1,L2,d,e. ▼*[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
+                        ∃∃K2,V2. ▼*[d - 1, e] K1 ≡ K2 &
+                                 K1 ⊢ ▼*[d - 1, e] V1 ≡ V2 &
+                                 L2 = K2. ⓑ{I} V2.
+#I #K1 #V1 #L2 #d #e * #X #HK1 #HL2 #e
+elim (ltpss_sn_inv_tpss11 … HK1 ?) -HK1 // #K #V #HK1 #HV1 #H destruct
+elim (ldrop_inv_skip1 … HL2 ?) -HL2 // #K2 #V2 #HK2 #HV2 #H destruct /3 width=5/
+qed-.
+
+(* Properties on inverse basic term relocation ******************************)
+
+lemma thin_delift: ∀L1,L2,d,e. ▼*[d, e] L1 ≡ L2 → ∀V1,V2. L1 ⊢ ▼*[d, e] V1 ≡ V2 →
+                   ∀I. ▼*[d + 1, e] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
+#L1 #L2 #d #e * #L #HL1 #HL2 #V1 #V2 * #V #HV1 #HV2 #I
+elim (ltpss_sn_tpss_conf … HV1 … HL1) -HV1 #V0 #HV10 #HV0
+lapply (tpss_inv_lift1_eq … HV0 … HV2) -HV0 #H destruct
+lapply (ltpss_sn_tpss_trans_eq … HV10 … HL1) -HV10 /3 width=5/
+qed.
+
+lemma thin_delift_tpss_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+                                ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+                                ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
+                                ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
+                                      L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdedd
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_le … HU1 … HUT1 … HYK ?) -HU1 -HUT1 -HYK // -Hdedd #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_le: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+                               ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+                               ∀K. ▼*[dd, ee] L ≡ K → d + e ≤ dd →
+                               ∃∃T2. K ⊢ T1 ▶* [d, e] T2 &
+                                     L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.
+
+lemma thin_delift_tpss_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+                                   ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+                                   ∀K. ▼*[dd, ee] L ≡ K →
+                                   d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+                                   ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+                                         L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hdde #Hddee
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_le_up … HU1 … HUT1 … HYK ? ? ?) -HU1 -HUT1 -HYK // -Hdd -Hdde -Hddee #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_le_up: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+                                  ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+                                  ∀K. ▼*[dd, ee] L ≡ K →
+                                  d ≤ dd → dd ≤ d + e → d + e ≤ dd + ee →
+                                  ∃∃T2. K ⊢ T1 ▶* [d, dd - d] T2 &
+                                        L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=6 by thin_delift_tpss_conf_le_up, tpss_strap2/ qed. (**) (* too slow without trace *)
+
+lemma thin_delift_tpss_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶* [d, e] U2 →
+                                ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+                                ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+                                ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+                                      L ⊢ ▼*[dd, ee] U2 ≡ T2.
+#L #U1 #U2 #d #e #HU12 #T1 #dd #ee #HUT1 #K * #Y #HLY #HYK #Hdd #Hddee
+lapply (delift_ltpss_sn_conf_eq … HUT1 … HLY) -HUT1 #HUT1
+elim (ltpss_sn_tpss_conf … HU12 … HLY) -HU12 #U #HU1 #HU2
+elim (delift_tpss_conf_be … HU1 … HUT1 … HYK ? ?) -HU1 -HUT1 -HYK // -Hdd -Hddee #T #HT1 #HUT
+lapply (ltpss_sn_delift_trans_eq … HLY … HUT) -HLY -HUT #HUT
+lapply (tpss_delift_trans_eq … HU2 … HUT) -U /2 width=3/
+qed.
+
+lemma thin_delift_tps_conf_be: ∀L,U1,U2,d,e. L ⊢ U1 ▶ [d, e] U2 →
+                               ∀T1,dd,ee. L ⊢ ▼*[dd, ee] U1 ≡ T1 →
+                               ∀K. ▼*[dd, ee] L ≡ K → d ≤ dd → dd + ee ≤ d + e →
+                               ∃∃T2. K ⊢ T1 ▶* [d, e - ee] T2 &
+                                     L ⊢ ▼*[dd, ee] U2 ≡ T2.
+/3 width=3/ qed.