- partial commit (unfold component only)

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / unfold / ltpss_dx_ldrop.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss_dx_ldrop.ma b/matita/matita/contribs/lambda_delta/basic_2/unfold/ltpss_dx_ldrop.ma
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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/ltpss_dx.ma".
+
+(* DX PARALLEL UNFOLD ON LOCAL ENVIRONMENTS *********************************)
+
+lemma ltpss_dx_ldrop_conf_ge: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+                              ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+                              d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K0 #K1 #I #V0 #V1 #e1 #_ #_ #IHK01 #L2 #e2 #H #He12
+  elim (le_inv_plus_l … He12) #_ #He2
+  lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+  lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2
+  elim (le_inv_plus_l … Hd1e2) #_ #He2
+  lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+  lapply (IHK01 … HK0L2 ?) -K0 /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_ldrop_trans_ge: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
+                               ∀L2,e2. ⇩[0, e2] L0 ≡ L2 →
+                               d1 + e1 ≤ e2 → ⇩[0, e2] L1 ≡ L2.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H //
+| //
+| normalize #K1 #K0 #I #V1 #V0 #e1 #_ #_ #IHK10 #L2 #e2 #H #He12
+  elim (le_inv_plus_l … He12) #_ #He2
+  lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+  lapply (IHK10 … HK0L2 ?) -K0 /2 width=1/
+| #K0 #K1 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2
+  elim (le_inv_plus_l … Hd1e2) #_ #He2
+  lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+  lapply (IHK10 … HK0L2 ?) -IHK10 -HK0L2 /2 width=1/
+]
+qed.
+
+lemma ltpss_dx_ldrop_conf_be: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+                              ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+                              ∃∃L. L2 ▶* [0, d1 + e1 - e2] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+  lapply (le_n_O_to_eq … He2) -He2 #H destruct
+  lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #_ #He21
+  lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+  [ -IHK01 -He21 destruct <minus_n_O /3 width=3/
+  | -HK01 -HV01 <minus_le_minus_minus_comm //
+    elim (IHK01 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+  ]
+| #K0 #K1 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK01 #L2 #e2 #H #Hd1e2 #He2de1
+  elim (le_inv_plus_l … Hd1e2) #_ #He2
+  <minus_le_minus_minus_comm //
+  lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+  elim (IHK01 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_dx_ldrop_trans_be: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
+                               ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
+                               ∃∃L. L ▶* [0, d1 + e1 - e2] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| normalize #L #I #V #L2 #e2 #HL2 #_ #He2
+  lapply (le_n_O_to_eq … He2) -He2 #H destruct
+  lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #_ #He21
+  lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+  [ -IHK10 -He21 destruct <minus_n_O /3 width=3/
+  | -HK10 -HV10 <minus_le_minus_minus_comm //
+    elim (IHK10 … HK0L2 ? ?) -K0 // /2 width=1/ /3 width=3/
+  ]
+| #K1 #K0 #I #V1 #V0 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHK10 #L2 #e2 #H #Hd1e2 #He2de1
+  elim (le_inv_plus_l … Hd1e2) #_ #He2
+  <minus_le_minus_minus_comm //
+  lapply (ldrop_inv_ldrop1 … H ?) -H // #HK0L2
+  elim (IHK10 … HK0L2 ? ?) -K0 /2 width=1/ /3 width=3/
+]
+qed.
+
+lemma ltpss_dx_ldrop_conf_le: ∀L0,L1,d1,e1. L0 ▶* [d1, e1] L1 →
+                              ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+                              ∃∃L. L2 ▶* [d1 - e2, e1] L & ⇩[0, e2] L1 ≡ L.
+#L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K0 #K1 #I #V0 #V1 #e1 #HK01 #HV01 #_ #L2 #e2 #H #He2
+  lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+  lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV01 #IHK01 #L2 #e2 #H #He2d1
+  lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+  [ -IHK01 -He2d1 destruct <minus_n_O /3 width=3/
+  | -HK01 -HV01 <minus_le_minus_minus_comm //
+    elim (IHK01 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+  ]
+]
+qed.
+
+lemma ltpss_dx_ldrop_trans_le: ∀L1,L0,d1,e1. L1 ▶* [d1, e1] L0 →
+                               ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
+                               ∃∃L. L ▶* [d1 - e2, e1] L2 & ⇩[0, e2] L1 ≡ L.
+#L1 #L0 #d1 #e1 #H elim H -L1 -L0 -d1 -e1
+[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+| /2 width=3/
+| normalize #K1 #K0 #I #V1 #V0 #e1 #HK10 #HV10 #_ #L2 #e2 #H #He2
+  lapply (le_n_O_to_eq … He2) -He2 #He2 destruct
+  lapply (ldrop_inv_refl … H) -H #H destruct /3 width=3/
+| #K1 #K0 #I #V1 #V0 #d1 #e1 #HK10 #HV10 #IHK10 #L2 #e2 #H #He2d1
+  lapply (ldrop_inv_O1 … H) -H * * #He2 #HK0L2
+  [ -IHK10 -He2d1 destruct <minus_n_O /3 width=3/
+  | -HK10 -HV10 <minus_le_minus_minus_comm //
+    elim (IHK10 … HK0L2 ?) -K0 /2 width=1/ /3 width=3/
+  ]
+]
+qed.