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15 include "basic_2A/substitution/lift_neg.ma".
16 include "basic_2A/substitution/lift_lift.ma".
17 include "basic_2A/substitution/cpy.ma".
19 (* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
21 (* Inversion lemmas on negated relocation ***********************************)
23 lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
24 ∀i. l ≤ yinj i → (∀T2. ⬆[i, 1] T2 ≡ U2 → ⊥) →
25 (∀T1. ⬆[i, 1] T1 ≡ U1 → ⊥) ∨
26 ∃∃I,K,W,j. l ≤ yinj j & j < i & ⬇[j]L ≡ K.ⓑ{I}W &
27 (∀V. ⬆[i-j-1, 1] V ≡ W → ⊥) & (∀T1. ⬆[j, 1] T1 ≡ U1 → ⊥).
28 #G #L #U1 #U2 #l #m #H elim H -G -L -U1 -U2 -l -m
29 [ /3 width=2 by or_introl/
30 | #I #G #L #K #V #W #j #l #m #Hlj #Hjlm #HLK #HVW #i #Hli #HnW
31 elim (lt_or_ge j i) #Hij
32 [ @or_intror @(ex5_4_intro … HLK) // -HLK
33 [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V //
34 #Y #HXY >minus_plus <plus_minus_m_m /2 width=2 by/
35 | -HnW /2 width=7 by lift_inv_lref2_be/
37 | elim (lift_split … HVW i j) -HVW //
38 #X #_ #H elim HnW -HnW //
40 | #a #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_bind … H) -H
41 [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
42 [ /4 width=9 by nlift_bind_sn, or_introl/
43 | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/
45 | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
46 [ /4 width=9 by nlift_bind_dx, or_introl/
47 | * #J #K #W #j #Hlj #Hji #HLK #HnW
48 elim (yle_inv_succ1 … Hlj) -Hlj #Hlj #Hj
49 lapply (ylt_O … Hj) -Hj #Hj
50 lapply (drop_inv_drop1_lt … HLK ?) // -HLK #HLK
51 >(plus_minus_m_m j 1) in ⊢ (%→?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ]
52 #HnU1 <minus_le_minus_minus_comm in HnW;
53 /5 width=9 by nlift_bind_dx, monotonic_lt_pred, lt_plus_to_minus_r, ex5_4_intro, or_intror/
56 | #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_flat … H) -H
57 [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
58 [ /4 width=9 by nlift_flat_sn, or_introl/
59 | * /5 width=9 by nlift_flat_sn, ex5_4_intro, or_intror/
61 | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 //
62 [ /4 width=9 by nlift_flat_dx, or_introl/
63 | * /5 width=9 by nlift_flat_dx, ex5_4_intro, or_intror/