milestone update in ground_2 and basic_2A

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

diff --git a/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cnf_cif.etc b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cnf_cif.etc
new file mode 100644 (file)
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+++ b/matita/matita/contribs/lambdadelta/basic_2A/etc/cpr/cnf_cif.etc
@@ -0,0 +1,106 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/reducibility/cif.ma".
+include "basic_2/reducibility/cnf_lift.ma".
+
+(* CONTEXT-SENSITIVE NORMAL TERMS *******************************************)
+
+(* Main properties **********************************************************)
+
+lemma tps_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
+[ //
+| #L #K #V #W #i #d #e #_ #_ #HLK #_ #H -d -e
+  elim (cif_inv_delta … HLK ?) //
+| #L #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
+  elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
+  lapply (IHV12 … HV1) -IHV12 -HV1 #H destruct /3 width=1/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H
+  elim (cif_inv_flat … H) -H #HV1 #HT1 #_ #_ /3 width=1/
+]
+qed.
+
+lemma tpss_cif_eq: ∀L,T1,T2,d,e. L ⊢ T1 ▶*[d, e] T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#L #T1 #T2 #d #e #H @(tpss_ind … H) -T2 //
+#T #T2 #_ #HT2 #IHT1 #HT1
+lapply (IHT1 HT1) -IHT1 #H destruct /2 width=5/
+qed.
+
+lemma tpr_cif_eq: ∀T1,T2. T1 ➡ T2 → ∀L. L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#T1 #T2 #H elim H -T1 -T2
+[ //
+| * #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #L #H
+  [ elim (cif_inv_appl … H) -H #HV1 #HT1 #_
+    >IHV1 -IHV1 // -HV1 >IHT1 -IHT1 //
+  | elim (cif_inv_ri2 … H) /2 width=1/
+  ]
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #_ #_ #L #H
+  elim (cif_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+| #a * #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV1 #IHT1 #L #H
+  [ lapply (tps_lsubr_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+    elim (cif_inv_bind … H) -H #HV1 #HT1 * #H destruct
+    lapply (IHV1 … HV1) -IHV1 -HV1 #H destruct
+    lapply (IHT1 … HT1) -IHT1 #H destruct
+    lapply (tps_cif_eq … HT2 ?) -HT2 //
+  | <(tps_inv_refl_SO2 … HT2 ?) -HT2 //
+    elim (cif_inv_ib2 … H) -H /2 width=1/ /3 width=2/
+  ]
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #_ #L #H
+  elim (cif_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+| #V1 #T1 #T #T2 #_ #_ #_ #L #H
+  elim (cif_inv_ri2 … H) /2 width=1/
+| #V1 #T1 #T2 #_ #_ #L #H
+  elim (cif_inv_ri2 … H) /2 width=1/
+]
+qed.
+
+lemma cpr_cif_eq: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ 𝐈⦃T1⦄ → T1 = T2.
+#L #T1 #T2 * #T0 #HT10 #HT02 #HT1
+lapply (tpr_cif_eq … HT10 … HT1) -HT10 #H destruct /2 width=5/
+qed.
+
+theorem cif_cnf: ∀L,T. L ⊢ 𝐈⦃T⦄ → L ⊢ 𝐍⦃T⦄.
+/3 width=3/ qed.
+
+(* Note: this property is unusual *)
+lemma cnf_crf_false: ∀L,T. L ⊢ 𝐑⦃T⦄ → L ⊢ 𝐍⦃T⦄ → ⊥.
+#L #T #H elim H -L -T
+[ #L #K #V #i #HLK #H
+  elim (cnf_inv_delta … HLK H)
+| #L #V #T #_ #IHV #H
+  elim (cnf_inv_appl … H) -H /2 width=1/
+| #L #V #T #_ #IHT #H
+  elim (cnf_inv_appl … H) -H /2 width=1/
+| #I #L #V #T * #H1 #H2 destruct
+  [ elim (cnf_inv_zeta … H2)
+  | elim (cnf_inv_tau … H2)
+  ]
+|5,6: #a * [ elim a ] #L #V #T * #H1 #_ #IH #H2 destruct
+  [1,3: elim (cnf_inv_abbr … H2) -H2 /2 width=1/
+  |*: elim (cnf_inv_abst … H2) -H2 /2 width=1/
+  ]
+| #a #L #V #W #T #H
+  elim (cnf_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+| #a #L #V #W #T #H
+  elim (cnf_inv_appl … H) -H #_ #_ #H
+  elim (simple_inv_bind … H)
+]
+qed.
+
+theorem cnf_cif: ∀L,T. L ⊢ 𝐍⦃T⦄ → L ⊢ 𝐈⦃T⦄.
+/2 width=4/ qed.