milestone update in ground_2 and basic_2A

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ldrop_lbotr.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/ldrop_lbotr.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/relocation/lsubr_lbotr.ma".
+include "basic_2/relocation/ldrop_ldrop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Inversion lemmas about local env. full refinement for substitution *******)
+
+(* Note: ldrop_ldrop not needed *)
+lemma lbotr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ⊒[d, e] L →
+                       d ≤ i → i < d + e → I = Abbr.
+#I #L elim L -L
+[ #K #V #i #H
+  lapply (ldrop_inv_atom1 … H) -H #H destruct
+| #L #J #W #IHL #K #V #i #H
+  elim (ldrop_inv_O1 … H) -H *
+  [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct
+    lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct
+    lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H
+    elim (lsubr_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct //
+  | #Hi #HLK #d @(nat_ind_plus … d) -d
+    [ #e #H #_ #Hide
+      elim (lbotr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct
+      @(IHL … HLK … HL) -IHL -HLK -HL // /2 width=1/
+    | #d #_ #e #H #Hdi #Hide
+      lapply (lbotr_inv_skip … H ?) -H // #HL
+      @(IHL … HLK … HL) -IHL -HLK -HL /2 width=1/
+    ]
+  ]
+]
+qed-.
+
+(* Properties about local env. full refinement for substitution *************)
+
+(* Note: ldrop_ldrop not needed *)
+lemma lbotr_ldrop: ∀L,d,e.
+                   (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) →
+                   ⊒[d, e] L.
+#L elim L -L //
+#L #I #V #IHL #d @(nat_ind_plus … d) -d
+[ #e @(nat_ind_plus … e) -e //
+  #e #_ #H0
+  >(H0 I L V 0 ? ? ?) //
+  /5 width=6 by lbotr_abbr, ldrop_ldrop, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
+| #d #_ #e #H0
+  /5 width=6 by lbotr_skip, ldrop_ldrop, le_S_S, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
+]
+qed.
+
+lemma lbotr_ldrop_trans_le: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ⊒[dd, ee] L1 →
+                            dd + ee ≤ d → ⊒[dd, ee] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
+@lbotr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
+lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid
+elim (ldrop_trans_le … HL12 … HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H
+elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct
+@(lbotr_inv_ldrop … HLK1 … HL1) -L1 -K1 -V1 //
+qed.
+
+lemma lbotr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
+                               ∀dd,ee. ⊒[dd, ee] L1 →
+                               dd ≤ d + e → d + e ≤ dd + ee →
+                               ⊒[d, dd + ee - d - e] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee
+@lbotr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2
+lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie
+>commutative_plus in Hiddee; >minus_minus_comm <plus_minus_m_m /2 width=1/ -Hddee #Hiddee
+lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hdi  #HL1K2
+@(lbotr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus // -Hddie /2 width=1/
+qed.
+
+lemma lbotr_ldrop_trans_ge: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ⊒[dd, ee] L1 →
+                            d + e ≤ dd → ⊒[dd - e, ee] L2.
+#L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
+@lbotr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
+elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd
+>plus_minus in Hiddee; // #Hiddee
+lapply (transitive_le … Hdde Hddi) -Hdde #Hid
+lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hid #HL1K2
+@(lbotr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus /2 width=1/
+qed.