- xoa: bug fix and improvement

[helm.git] / matita / matita / lib / lambda-delta / substitution / lift_fun.ma

diff --git a/matita/matita/lib/lambda-delta/substitution/lift_fun.ma b/matita/matita/lib/lambda-delta/substitution/lift_fun.ma
index ee54dede2e11dce091eecdc18dd3d3579ed3e1ef..58d718fb7f6bfc9562e18bec49992ce252fe4321 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/lift_fun.ma
+++ b/matita/matita/lib/lambda-delta/substitution/lift_fun.ma
@@ -16,10 +16,46 @@ include "lambda-delta/substitution/lift_defs.ma".
 
 (* RELOCATION ***************************************************************)
 
-(* the functional properties **************************************************)
+(* Functional properties ****************************************************)
 
-axiom lift_total: ∀d,e,T1. ∃T2. ↑[d,e] T1 ≡ T2.
+lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
+#T1 elim T1 -T1
+[ /2/
+| #i #d #e elim (lt_or_ge i d) /3/
+| * #I #V1 #T1 #IHV1 #IHT1 #d #e
+  elim (IHV1 d e) -IHV1 #V2 #HV12
+  [ elim (IHT1 (d+1) e) -IHT1 /3/
+  | elim (IHT1 d e) -IHT1 /3/
+  ]
+]
+qed.
 
-axiom lift_mono:  ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
+lemma lift_mono:  ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
+#d #e #T #U1 #H elim H -H d e T U1
+[ #k #d #e #X #HX
+  lapply (lift_inv_sort1 … HX) -HX //
+| #i #d #e #Hid #X #HX 
+  lapply (lift_inv_lref1_lt … HX ?) -HX //
+| #i #d #e #Hdi #X #HX 
+  lapply (lift_inv_lref1_ge … HX ?) -HX //
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+]
+qed.
 
-axiom lift_inj:  ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2.
+lemma lift_inj:  ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2.
+#d #e #T1 #U #H elim H -H d e T1 U
+[ #k #d #e #X #HX
+  lapply (lift_inv_sort2 … HX) -HX //
+| #i #d #e #Hid #X #HX 
+  lapply (lift_inv_lref2_lt … HX ?) -HX //
+| #i #d #e #Hdi #X #HX 
+  lapply (lift_inv_lref2_ge … HX ?) -HX /2/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+  elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct -X /3/
+]
+qed.