- we introduced the pointer_step rc in the perspective of proving

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / substitution / lift_lift.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift.ma
index e70389973b3ee3a9d97991d90ea52096e7e02855..3e18bff3275543a3d1daf190235cd529f928ec72 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/substitution/lift_lift.ma
@@ -29,7 +29,7 @@ theorem lift_inj: ∀d,e,T1,U. ⇧[d,e] T1 ≡ U → ∀T2. ⇧[d,e] T2 ≡ U 
   lapply (lift_inv_lref2_ge … HX ?) -HX // /2 width=1/
 | #p #d #e #X #HX
   lapply (lift_inv_gref2 … HX) -HX //
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
   elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
   elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
@@ -58,7 +58,7 @@ theorem lift_div_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
   ]
 | #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
   lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3/
-| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+| #a #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
   lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
   elim (IHW … HW2 ?) // -IHW -HW2 #W0 #HW2 #HW1
   >plus_plus_comm_23 in HU2; #HU2 elim (IHU … HU2 ?) /2 width=1/ /3 width=5/
@@ -88,7 +88,7 @@ theorem lift_div_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
     @lift_lref_ge_minus_eq [ >plus_minus_commutative // | /2 width=1/ ]
   ]
 | #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3/
-| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+| #a #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
   elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
   elim (IHV1 … HV2 ? ?) -V // >plus_plus_comm_23 in HT2; #HT2
   elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/
@@ -109,7 +109,7 @@ theorem lift_mono: ∀d,e,T,U1. ⇧[d,e] T ≡ U1 → ∀U2. ⇧[d,e] T ≡ U2 
   lapply (lift_inv_lref1_ge … HX ?) -HX //
 | #p #d #e #X #HX
   lapply (lift_inv_gref1 … HX) -HX //
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
   elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
   elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1/
@@ -133,7 +133,7 @@ theorem lift_trans_be: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
   ]
 | #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
   >(lift_inv_gref1 … HT2) -HT2 //
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
+| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
   elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
   lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
   lapply (IHT12 … HT20 ? ?) /2 width=1/
@@ -160,7 +160,7 @@ theorem lift_trans_le: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
   >plus_plus_comm_23 /4 width=3/
 | #p #d1 #e1 #d2 #e2 #X #HX #_
   >(lift_inv_gref1 … HX) -HX /2 width=3/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
+| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
   elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
   elim (IHV12 … HV20 ?) -IHV12 -HV20 //
   elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ /3 width=5/
@@ -187,7 +187,7 @@ theorem lift_trans_ge: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
   elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct /4 width=3/
 | #p #d1 #e1 #d2 #e2 #X #HX #_
   >(lift_inv_gref1 … HX) -HX /2 width=3/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
   elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
   elim (IHV12 … HV20 ?) -IHV12 -HV20 //
   elim (IHT12 … HT20 ?) -IHT12 -HT20 /2 width=1/ #T
@@ -198,3 +198,20 @@ theorem lift_trans_ge: ∀d1,e1,T1,T. ⇧[d1, e1] T1 ≡ T →
   elim (IHT12 … HT20 ?) -IHT12 -HT20 // /3 width=5/
 ]
 qed.
+
+(* Advanced properties ******************************************************)
+
+lemma lift_conf_O1: ∀T,T1,d1,e1. ⇧[d1, e1] T ≡ T1 → ∀T2,e2. ⇧[0, e2] T ≡ T2 →
+                    ∃∃T0. ⇧[0, e2] T1 ≡ T0 & ⇧[d1 + e2, e1] T2 ≡ T0.
+#T #T1 #d1 #e1 #HT1 #T2 #e2 #HT2
+elim (lift_total T1 0 e2) #T0 #HT10
+elim (lift_trans_le … HT1 … HT10 ?) -HT1 // #X #HTX #HT20
+lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3/
+qed.
+
+lemma lift_conf_be: ∀T,T1,d,e1. ⇧[d, e1] T ≡ T1 → ∀T2,e2. ⇧[d, e2] T ≡ T2 →
+                    e1 ≤ e2 → ⇧[d + e1, e2 - e1] T1 ≡ T2.
+#T #T1 #d #e1 #HT1 #T2 #e2 #HT2 #He12
+elim (lift_split … HT2 (d+e1) e1 ? ? ?) -HT2 // #X #H
+>(lift_mono … H … HT1) -T //
+qed.