- the definition of the framework for strong normalization continues ...

[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 cfcf976e00bbfd07dc73e8d9d2e1ac59c3e56699..3666799d455ddbe7ffa3a4fff82ad67ba3df51f1 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
@@ -50,7 +50,7 @@ theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
   elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct
   [ -Hd12 lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3 width=3/
-  | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_inv_plus_plus_r … H) -H #H
+  | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r … H) -H #H
     elim (le_inv_plus_l … H) -H #Hide2 #He2i
     lapply (transitive_le … Hd12 Hide2) -Hd12 #Hd12
     >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⊢ (? ? ? %); // -He2i
@@ -69,6 +69,36 @@ theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
 ]
 qed.
 
+(* Note: apparently this was missing in Basic_1 *)
+theorem lift_div_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+                     ∀e,e2,T2. ↑[d1 + e, e2] T2 ≡ T →
+                     e ≤ e1 → e1 ≤ e + e2 →
+                     ∃∃T0. ↑[d1, e] T0 ≡ T2 & ↑[d1, e + e2 - e1] T0 ≡ T1.
+#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
+[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3/
+| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+  >(lift_inv_lref2_lt … H) -H [ /3 width=3/ | /2 width=3/ ]
+| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+  elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2
+  [ elim (lift_inv_lref2_be … H ? ?) -H // /2 width=1/
+  | >(lift_inv_lref2_ge … H ?) -H //
+    lapply (le_plus_to_minus … Hie1d1e2) #Hd1e21i
+    elim (le_inv_plus_l … Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1
+    @ex2_1_intro [2: /2 width=1/ | skip ] -Hd1e12
+    @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
+  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/
+| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+  elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
+  elim (IHV1 … HV2 ? ?) -V //
+  elim (IHT1 … HT2 ? ?) -T // -He1 -He1e2 /3 width=5/
+]
+qed.
+
 theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
 #d #e #T #U1 #H elim H -d -e -T -U1
 [ #k #d #e #X #HX