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4 (* ||A|| A project by Andrea Asperti *)
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15 include "basic_2/substitution/cpys_lift.ma".
16 include "basic_2/substitution/cofrees.ma".
18 (* CONTEXT-SENSITIVE EXCLUSION FROM FREE VARIABLES **************************)
20 (* Advanced properties ******************************************************)
22 lemma cofrees_lref_skip: ∀L,d,i,j. j < i → yinj j < d → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
23 #L #d #i #j #Hji #Hjd #X #H elim (cpys_inv_lref1_Y2 … H) -H
24 [ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
25 | * #I #K #W1 #W2 #Hdj elim (ylt_yle_false … Hdj) -i -I -L -K -W1 -W2 -X //
29 lemma cofrees_lref_lt: ∀L,d,i,j. i < j → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
30 #L #d #i #j #Hij #X #H elim (cpys_inv_lref1_Y2 … H) -H
31 [ #H destruct /3 width=2 by lift_lref_ge_minus, ex_intro/
32 | * #I #K #V1 #V2 #_ #_ #_ #H -I -L -K -V1 -d
33 elim (lift_split … H i j) /2 width=2 by lt_to_le, ex_intro/
37 lemma cofrees_lref_gt: ∀I,L,K,W,d,i,j. j < i → ⇩[j] L ≡ K.ⓑ{I}W →
38 K ⊢ (i-j-1) ~ϵ 𝐅*[O]⦃W⦄ → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
39 #I #L #K #W1 #d #i #j #Hji #HLK #HW1 #X #H elim (cpys_inv_lref1_Y2 … H) -H
40 [ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
41 | * #I0 #K0 #W0 #W2 #Hdj #HLK0 #HW12 #HW2 lapply (ldrop_mono … HLK0 … HLK) -L
42 #H destruct elim (HW1 … HW12) -I -K -W1 -d
43 #V2 #HVW2 elim (lift_trans_le … HVW2 … HW2) -W2 //
44 >minus_plus <plus_minus_m_m /2 width=2 by ex_intro/
48 lemma cofrees_lref_free: ∀L,d,i,j. |L| ≤ j → j < i → L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄.
49 #L #d #i #j #Hj #Hji #X #H elim (cpys_inv_lref1_Y2 … H) -H
50 [ #H destruct /3 width=2 by lift_lref_lt, ex_intro/
51 | * #I #K #W1 #W2 #_ #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
52 #H elim (lt_refl_false j) -d -i -K -W1 -W2 -X /2 width=3 by lt_to_le_to_lt/
56 (* Advanced negated inversion lemmas ****************************************)
58 lemma frees_inv_lref_gt: ∀L,d,i,j. j < i → (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) →
59 ∃∃I,K,W. ⇩[j] L ≡ K.ⓑ{I}W & (K ⊢ (i-j-1) ~ϵ 𝐅*[0]⦃W⦄ → ⊥) & d ≤ yinj j.
60 #L #d #i #j #Hji #H elim (ylt_split j d) #Hjd
61 [ elim H -H /2 width=6 by cofrees_lref_skip/
62 | elim (lt_or_ge j (|L|)) #Hj
63 [ elim (ldrop_O1_lt … Hj) -Hj /4 width=10 by cofrees_lref_gt, ex3_3_intro/
64 | elim H -H /2 width=6 by cofrees_lref_free/
69 lemma frees_inv_lref_free: ∀L,d,i,j. (L ⊢ i ~ϵ 𝐅*[d]⦃#j⦄ → ⊥) → |L| ≤ j → j = i.
70 #L #d #i #j #H #Hj elim (lt_or_eq_or_gt i j) //
71 #Hij elim H -H /2 width=6 by cofrees_lref_lt, cofrees_lref_free/
74 lemma frees_inv_gen: ∀L,U,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
75 ∃∃U0. ⦃⋆, L⦄ ⊢ U ▶*[d, ∞] U0 & (∀T. ⇧[i, 1] T ≡ U0 → ⊥).
76 #L #U @(f2_ind … rfw … L U) -L -U
78 [ -IH #k #_ #d #i #H elim H -H //
79 | #j #Hn #d #i #H elim (lt_or_eq_or_gt i j)
80 [ -n #Hij elim H -H /2 width=5 by cofrees_lref_lt/
81 | -H -n #H destruct /3 width=7 by lift_inv_lref2_be, ex2_intro/
82 | #Hji elim (frees_inv_lref_gt … H) // -H
83 #I #K #W1 #HLK #H #Hdj elim (IH … H) /2 width=2 by ldrop_fwd_rfw/ -H -n
84 #W2 #HW12 #HnW2 elim (lift_total W2 0 (j+1))
85 #U2 #HWU2 @(ex2_intro … U2) /2 width=7 by cpys_subst_Y2/ -I -L -K -W1 -d
86 #T2 #HTU2 elim (lift_div_le … HWU2 (i-j-1) 1 T2) /2 width=2 by/ -W2
87 >minus_plus <plus_minus_m_m //
89 | -IH #p #_ #d #i #H elim H -H //
90 | #a #I #W #U #Hn #d #i #H elim (frees_inv_bind … H) -H
91 #H elim (IH … H) // -H -n
92 /4 width=9 by cpys_bind, nlift_bind_dx, nlift_bind_sn, ex2_intro/
93 | #I #W #U #Hn #d #i #H elim (frees_inv_flat … H) -H
94 #H elim (IH … H) // -H -n
95 /4 width=9 by cpys_flat, nlift_flat_dx, nlift_flat_sn, ex2_intro/