- main lemmas about abstract reducibility candidates closed

[helm.git] / matita / matita / contribs / lambda_delta / Basic_2 / unfold / delift.ma

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/unfold/delift.ma b/matita/matita/contribs/lambda_delta/Basic_2/unfold/delift.ma
index 5d19b650e70155c34c82def311d8ba3771dcea8c..ec2d6c373e69b977e83cb96310caa44d46860700 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/unfold/delift.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/unfold/delift.ma
@@ -17,7 +17,68 @@ include "Basic_2/unfold/tpss.ma".
 (* DELIFT ON TERMS **********************************************************)
 
 definition delift: nat → nat → lenv → relation term ≝
-                   λd,e,L,T1,T2. â\88\83â\88\83T. L â\8a¢ T1 [d, e] â\89«* T & â\86\91[d, e] T2 ≡ T.
+                   λd,e,L,T1,T2. â\88\83â\88\83T. L â\8a¢ T1 [d, e] â\96¶* T & â\87§[d, e] T2 ≡ T.
 
 interpretation "delift (term)"
    'TSubst L T1 d e T2 = (delift d e L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma delift_lsubs_conf: ∀L1,T1,T2,d,e. L1 ⊢ T1 [d, e] ≡ T2 →
+                         ∀L2. L1 [d, e] ≼ L2 → L2 ⊢ T1 [d, e] ≡ T2.
+#L1 #T1 #T2 #d #e * /3 width=3/
+qed.
+
+lemma delift_bind: ∀I,L,V1,V2,T1,T2,d,e.
+                   L ⊢ V1 [d, e] ≡ V2 → L. ⓑ{I} V2 ⊢ T1 [d+1, e] ≡ T2 →
+                   L ⊢ ⓑ{I} V1. T1 [d, e] ≡ ⓑ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * #T #HT1 #HT2
+lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V) ?) -HT1 /2 width=1/ /3 width=5/
+qed.
+
+lemma delift_flat: ∀I,L,V1,V2,T1,T2,d,e.
+                   L ⊢ V1 [d, e] ≡ V2 → L ⊢ T1 [d, e] ≡ T2 →
+                   L ⊢ ⓕ{I} V1. T1 [d, e] ≡ ⓕ{I} V2. T2.
+#I #L #V1 #V2 #T1 #T2 #d #e * #V #HV1 #HV2 * /3 width=5/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma delift_fwd_sort1: ∀L,U2,d,e,k. L ⊢ ⋆k [d, e] ≡ U2 → U2 = ⋆k.
+#L #U2 #d #e #k * #U #HU
+>(tpss_inv_sort1 … HU) -HU #HU2
+>(lift_inv_sort2 … HU2) -HU2 //
+qed-.
+
+lemma delift_fwd_gref1: ∀L,U2,d,e,p. L ⊢ §p [d, e] ≡ U2 → U2 = §p.
+#L #U #d #e #p * #U #HU
+>(tpss_inv_gref1 … HU) -HU #HU2
+>(lift_inv_gref2 … HU2) -HU2 //
+qed-.
+
+lemma delift_fwd_bind1: ∀I,L,V1,T1,U2,d,e. L ⊢ ⓑ{I} V1. T1 [d, e] ≡ U2 →
+                        ∃∃V2,T2. L ⊢ V1 [d, e] ≡ V2 &
+                                 L. ⓑ{I} V2 ⊢ T1 [d+1, e] ≡ T2 &
+                                 U2 = ⓑ{I} V2. T2.
+#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_bind1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_bind2 … HU2) -HU2 #V2 #T2 #HV2 #HT2
+lapply (tpss_lsubs_conf … HT1 (L. ⓑ{I} V2) ?) -HT1 /2 width=1/ /3 width=5/
+qed-.
+
+lemma delift_fwd_flat1: ∀I,L,V1,T1,U2,d,e. L ⊢ ⓕ{I} V1. T1 [d, e] ≡ U2 →
+                        ∃∃V2,T2. L ⊢ V1 [d, e] ≡ V2 &
+                                 L ⊢ T1 [d, e] ≡ T2 &
+                                 U2 = ⓕ{I} V2. T2.
+#I #L #V1 #T1 #U2 #d #e * #U #HU #HU2
+elim (tpss_inv_flat1 … HU) -HU #V #T #HV1 #HT1 #X destruct
+elim (lift_inv_flat2 … HU2) -HU2 /3 width=5/
+qed-.
+
+(* Basic Inversion lemmas ***************************************************)
+
+lemma delift_inv_refl_O2: ∀L,T1,T2,d. L ⊢ T1 [d, 0] ≡ T2 → T1 = T2.
+#L #T1 #T2 #d * #T #HT1
+>(tpss_inv_refl_O2 … HT1) -HT1 #HT2
+>(lift_inv_refl_O2 … HT2) -HT2 //
+qed-.