(* *)
(**************************************************************************)
-include "basic_2/substitution/ldrop_append.ma".
+include "basic_2/substitution/drop_append.ma".
include "basic_2/multiple/frees.ma".
(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
(* Properties on append for local environments ******************************)
-lemma frees_append: āL2,U,d,i. L2 ā¢ i Ļµ š
*[d]ā¦Uā¦ ā i ā¤ |L2| ā
- āL1. L1 @@ L2 ā¢ i Ļµ š
*[d]ā¦Uā¦.
-#L2 #U #d #i #H elim H -L2 -U -d -i /3 width=2 by frees_eq/
-#I #L2 #K2 #U #W #d #i #j #Hdj #Hji #HnU #HLK2 #_ #IHW #Hi #L1
-lapply (ldrop_fwd_length_minus2 ā¦ HLK2) normalize #H0
-lapply (ldrop_O1_append_sn_le ā¦ HLK2 ā¦ L1) -HLK2
-[ -I -L1 -K2 -U -W -d /3 width=3 by lt_to_le, lt_to_le_to_lt/
+lemma frees_append: āL2,U,l,i. L2 ā¢ i Ļµ š
*[l]ā¦Uā¦ ā i ā¤ |L2| ā
+ āL1. L1 @@ L2 ā¢ i Ļµ š
*[l]ā¦Uā¦.
+#L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
+#I #L2 #K2 #U #W #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW #Hi #L1
+lapply (drop_fwd_length_minus2 ā¦ HLK2) normalize #H0
+lapply (drop_O1_append_sn_le ā¦ HLK2 ā¦ L1) -HLK2
+[ -I -L1 -K2 -U -W -l /4 width=3 by ylt_yle_trans, ylt_inv_inj, lt_to_le/
| #HLK2 @(frees_be ā¦ HnU HLK2) // -HnU -HLK2 @IHW -IHW
- >(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
+ >(minus_plus_m_m (|K2|) 1) >H0 -H0 <yminus_inj >yminus_SO2
+ /3 width=1 by monotonic_yle_minus_dx, yle_pred/
]
qed.
(* Inversion lemmas on append for local environments ************************)
-fact frees_inv_append_aux: āL,U,d,i. L ā¢ i Ļµ š
*[d]ā¦Uā¦ ā āL1,L2. L = L1 @@ L2 ā
- i ā¤ |L2| ā L2 ā¢ i Ļµ š
*[d]ā¦Uā¦.
-#L #U #d #i #H elim H -L -U -d -i /3 width=2 by frees_eq/
-#Z #L #Y #U #X #d #i #j #Hdj #Hji #HnU #HLY #_ #IHW #L1 #L2 #H #Hi destruct
-elim (ldrop_O1_lt (ā») L2 j) [2: -Z -Y -L1 -X -U -d /2 width=3 by lt_to_le_to_lt/ ]
-#I #K2 #W #HLK2 lapply (ldrop_fwd_length_minus2 ā¦ HLK2) normalize #H0
-lapply (ldrop_O1_inv_append1_le ā¦ HLY ā¦ HLK2) -HLY
-[ -Z -I -Y -K2 -L1 -X -U -W -d /3 width=3 by lt_to_le, lt_to_le_to_lt/
+fact frees_inv_append_aux: āL,U,l,i. L ā¢ i Ļµ š
*[l]ā¦Uā¦ ā āL1,L2. L = L1 @@ L2 ā
+ i ā¤ |L2| ā L2 ā¢ i Ļµ š
*[l]ā¦Uā¦.
+#L #U #l #i #H elim H -L -U -l -i /3 width=2 by frees_eq/
+#Z #L #Y #U #X #l #i #j #Hlj #Hji #HnU #HLY #_ #IHW #L1 #L2 #H #Hi destruct
+elim (drop_O1_lt (ā») L2 j) [2: -Z -Y -L1 -X -U -l /3 width=3 by ylt_yle_trans, ylt_inv_inj/ ]
+#I #K2 #W #HLK2 lapply (drop_fwd_length_minus2 ā¦ HLK2) normalize #H0
+lapply (drop_O1_inv_append1_le ā¦ HLY ā¦ HLK2) -HLY
+[ -Z -I -Y -K2 -L1 -X -U -W -l /4 width=3 by ylt_yle_trans, ylt_inv_inj, lt_to_le/
| normalize #H destruct
@(frees_be ā¦ HnU HLK2) -HnU -HLK2 // @IHW -IHW //
- >(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
+ >(minus_plus_m_m (|K2|) 1) >H0 -H0 <yminus_inj >yminus_SO2
+ /3 width=1 by monotonic_yle_minus_dx, yle_pred/
]
qed-.
-lemma frees_inv_append: āL1,L2,U,d,i. L1 @@ L2 ā¢ i Ļµ š
*[d]ā¦Uā¦ ā
- i ā¤ |L2| ā L2 ā¢ i Ļµ š
*[d]ā¦Uā¦.
+lemma frees_inv_append: āL1,L2,U,l,i. L1 @@ L2 ā¢ i Ļµ š
*[l]ā¦Uā¦ ā
+ i ā¤ |L2| ā L2 ā¢ i Ļµ š
*[l]ā¦Uā¦.
/2 width=4 by frees_inv_append_aux/ qed-.