- contex-free normal forms started

[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 f9c99457db747c0e0c0b1983e4873c2aa52235fe..6feb72e036f9f8813e46d0eb6b9d561d588c7448 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
@@ -12,13 +12,13 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Basic-2/substitution/lift.ma".
+include "Basic_2/substitution/lift.ma".
 
 (* RELOCATION ***************************************************************)
 
 (* Main properies ***********************************************************)
 
-(* Basic-1: was: lift_inj *)
+(* Basic_1: was: lift_inj *)
 theorem 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
@@ -27,6 +27,8 @@ theorem lift_inj:  ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U 
   lapply (lift_inv_lref2_lt … HX ?) -HX //
 | #i #d #e #Hdi #X #HX 
   lapply (lift_inv_lref2_ge … HX ?) -HX /2/
+| #p #d #e #X #HX
+  lapply (lift_inv_gref2 … HX) -HX //
 | #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
@@ -34,7 +36,7 @@ theorem lift_inj:  ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U 
 ]
 qed.
 
-(* Basic-1: was: lift_gen_lift *)
+(* Basic_1: was: lift_gen_lift *)
 theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
                      ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
                      d1 ≤ d2 →
@@ -44,28 +46,30 @@ theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
   lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
   lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2
-  lapply (lift_inv_lref2_lt … Hi ?) -Hi /3/
+  lapply (lift_inv_lref2_lt … Hi ?) -Hi /2/ /3/
 | #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
   elim (lift_inv_lref2 … Hi) -Hi * #Hid2 #H destruct -T2
   [ -Hd12; lapply (lt_plus_to_lt_l … Hid2) -Hid2 #Hid2 /3/
   | -Hid1; lapply (arith1 … Hid2) -Hid2 #Hid2
     @(ex2_1_intro … #(i - e2))
     [ >le_plus_minus_comm [ @lift_lref_ge @(transitive_le … Hd12) -Hd12 /2/ | -Hd12 /2/ ]
-    | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /3/
+    | -Hd12 >(plus_minus_m_m i e2) in ⊢ (? ? ? ? %) /2/ /3/
     ]
   ]
+| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+  lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct -T2 /3/
 | #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 -T2;
   elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
-  >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /3 width = 5/
+  >plus_plus_comm_23 in HU2 #HU2 elim (IHU … HU2 ?) /2/ /3 width = 5/
 | #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
   lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct -T2;
   elim (IHW … HW2 ?) // -IHW HW2 #W0 #HW2 #HW1
-  elim (IHU … HU2 ?) /3 width = 5/
+  elim (IHU … HU2 ?) // /3 width = 5/
 ]
 qed.
 
-theorem lift_mono:  ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
+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 -H d e T U1
 [ #k #d #e #X #HX
   lapply (lift_inv_sort1 … HX) -HX //
@@ -73,6 +77,8 @@ theorem lift_mono:  ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2
   lapply (lift_inv_lref1_lt … HX ?) -HX //
 | #i #d #e #Hdi #X #HX 
   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
   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
@@ -80,7 +86,7 @@ theorem lift_mono:  ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2
 ]
 qed.
 
-(* Basic-1: was: lift_free (left to right) *)
+(* Basic_1: was: lift_free (left to right) *)
 theorem 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.
@@ -95,6 +101,8 @@ theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
   [ @(transitive_le … Hd21 ?) -Hd21 /2/
   | -Hd21 /2/
   ]
+| #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
   elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
   lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
@@ -102,11 +110,11 @@ theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
   elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct -X;
   lapply (IHV12 … HV20 ? ?) // -IHV12 HV20 #HV10
-  lapply (IHT12 … HT20 ? ?) /2/
+  lapply (IHT12 … HT20 ? ?) // /2/
 ]
 qed.
 
-(* Basic-1: was: lift_d (right to left) *)
+(* Basic_1: was: lift_d (right to left) *)
 theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
                        ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
                        ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
@@ -115,23 +123,25 @@ theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
   >(lift_inv_sort1 … HX) -HX /2/
 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
   lapply (lt_to_le_to_lt … (d1+e2) Hid1 ?) // #Hie2
-  elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /4/
+  elim (lift_inv_lref1 … HX) -HX * #Hid2 #HX destruct -X /3/ /4/
 | #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
   lapply (transitive_le … Hd21 Hid1) -Hd21 #Hid2
   lapply (lift_inv_lref1_ge … HX ?) -HX /2/ #HX destruct -X;
   >plus_plus_comm_23 /4/
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+  >(lift_inv_gref1 … HX) -HX /2/
 | #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 -X;
   elim (IHV12 … HV20 ?) -IHV12 HV20 //
-  elim (IHT12 … HT20 ?) -IHT12 HT20 /3 width=5/
+  elim (IHT12 … HT20 ?) -IHT12 HT20 /2/ /3 width=5/
 | #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
   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/
+  elim (IHT12 … HT20 ?) -IHT12 HT20 // /3 width=5/
 ]
 qed.
 
-(* Basic-1: was: lift_d (left to right) *)
+(* Basic_1: was: lift_d (left to right) *)
 theorem 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.
@@ -145,15 +155,17 @@ theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
   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/
+  [ /4/ | >plus_plus_comm_23 /4/ ]
+| #p #d1 #e1 #d2 #e2 #X #HX #_
+  >(lift_inv_gref1 … HX) -HX /2/
 | #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/
+  <plus_minus /2/ /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/
+  elim (IHT12 … HT20 ?) -IHT12 HT20 // /3 width=5/
 ]
 qed.