- transitivity of parallel telescopic substitution closed!

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

diff --git a/matita/matita/lib/lambda-delta/substitution/lift_main.ma b/matita/matita/lib/lambda-delta/substitution/lift_main.ma
index e847db483f6bb3645276538ecad95ed54a49c83a..26680f364cd3a680a8b5d15b4d1d774b813efcb1 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/lift_main.ma
+++ b/matita/matita/lib/lambda-delta/substitution/lift_main.ma
@@ -16,7 +16,22 @@ include "lambda-delta/substitution/lift_defs.ma".
 
 (* RELOCATION ***************************************************************)
 
-(* the main properies *******************************************************)
+(* Main properies ***********************************************************)
+
+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.
 
 lemma lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
                    ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
@@ -57,7 +72,7 @@ lemma lift_free: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
   lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
 | #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
   lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
-  <(plus_plus_minus_m_m e1 e2 i) /3/
+  <(arith_d1 i e2 e1) // /3/
 | #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
   elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
   elim (IHT (d2+1) … ? ? He12) /3 width = 5/
@@ -67,6 +82,21 @@ lemma lift_free: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
 ]
 qed.
 
+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.
+
 lemma lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
                      ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
                      d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
@@ -116,6 +146,28 @@ lemma lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
 ]
 qed.
 
-axiom lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+lemma lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
                      ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
                      ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
+#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
+[ #k #d1 #e1 #d2 #e2 #X #HX #_
+  >(lift_inv_sort1 … HX) -HX /2/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
+  lapply (lt_to_le_to_lt … (d1+e1) Hid1 ?) // #Hid1e
+  lapply (lt_to_le_to_lt … (d2-e1) Hid1 ?) /2/ #Hid2e
+  lapply (lt_to_le_to_lt … Hid1e Hded) -Hid1e Hded #Hid2
+  lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct -X /3/
+| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
+  elim (lift_inv_lref1 … HX) -HX * #Hied #HX destruct -X;
+  [2: >plus_plus_comm_23] /4/
+| #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 -X;
+  elim (IHV12 … HV20 ?) -IHV12 HV20 //
+  elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ #T
+  <plus_minus /3 width=5/
+| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+  elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
+  elim (IHV12 … HV20 ?) -IHV12 HV20 //
+  elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+]
+qed.