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15 include "basic_2A/substitution/drop_append.ma".
16 include "basic_2A/multiple/frees.ma".
18 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
20 (* Properties on append for local environments ******************************)
22 lemma frees_append: āL2,U,l,i. L2 ā¢ i Ļµ š
*[l]ā¦Uā¦ ā i ā¤ |L2| ā
23 āL1. L1 ā L2 ā¢ i Ļµ š
*[l]ā¦Uā¦.
24 #L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
25 #I #L2 #K2 #U #W #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW #Hi #L1
26 lapply (drop_fwd_length_minus2 ā¦ HLK2) normalize #H0
27 lapply (drop_O1_append_sn_le ā¦ HLK2 ā¦ L1) -HLK2
28 [ -I -L1 -K2 -U -W -l /3 width=3 by lt_to_le, lt_to_le_to_lt/
29 | #HLK2 @(frees_be ā¦ HnU HLK2) // -HnU -HLK2 @IHW -IHW
30 >(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
34 (* Inversion lemmas on append for local environments ************************)
36 fact frees_inv_append_aux: āL,U,l,i. L ā¢ i Ļµ š
*[l]ā¦Uā¦ ā āL1,L2. L = L1 ā L2 ā
37 i ā¤ |L2| ā L2 ā¢ i Ļµ š
*[l]ā¦Uā¦.
38 #L #U #l #i #H elim H -L -U -l -i /3 width=2 by frees_eq/
39 #Z #L #Y #U #X #l #i #j #Hlj #Hji #HnU #HLY #_ #IHW #L1 #L2 #H #Hi destruct
40 elim (drop_O1_lt (ā») L2 j) [2: -Z -Y -L1 -X -U -l /2 width=3 by lt_to_le_to_lt/ ]
41 #I #K2 #W #HLK2 lapply (drop_fwd_length_minus2 ā¦ HLK2) normalize #H0
42 lapply (drop_O1_inv_append1_le ā¦ HLY ā¦ HLK2) -HLY
43 [ -Z -I -Y -K2 -L1 -X -U -W -l /3 width=3 by lt_to_le, lt_to_le_to_lt/
44 | normalize #H destruct
45 @(frees_be ā¦ HnU HLK2) -HnU -HLK2 // @IHW -IHW //
46 >(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
50 lemma frees_inv_append: āL1,L2,U,l,i. L1 ā L2 ā¢ i Ļµ š
*[l]ā¦Uā¦ ā
51 i ā¤ |L2| ā L2 ā¢ i Ļµ š
*[l]ā¦Uā¦.
52 /2 width=4 by frees_inv_append_aux/ qed-.