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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "ground_2/ynat/ynat_plus.ma".
16 include "basic_2/notation/relations/freestar_4.ma".
17 include "basic_2/relocation/lift_neg.ma".
18 include "basic_2/relocation/ldrop.ma".
20 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
22 inductive frees: relation4 ynat lenv term nat ≝
23 | frees_eq: ∀L,U,d,i. (∀T. ⇧[i, 1] T ≡ U → ⊥) → frees d L U i
24 | frees_be: ∀I,L,K,U,W,d,i,j. d ≤ yinj j → j < i →
25 (∀T. ⇧[j, 1] T ≡ U → ⊥) → ⇩[j]L ≡ K.ⓑ{I}W →
26 frees 0 K W (i-j-1) → frees d L U i.
29 "context-sensitive free variables (term)"
30 'FreeStar L i d U = (frees d L U i).
32 (* Basic inversion lemmas ***************************************************)
34 lemma frees_inv: ∀L,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
35 (∀T. ⇧[i, 1] T ≡ U → ⊥) ∨
36 ∃∃I,K,W,j. d ≤ yinj j & j < i & (∀T. ⇧[j, 1] T ≡ U → ⊥) &
37 ⇩[j]L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
38 #L #U #d #i * -L -U -d -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
41 lemma frees_inv_sort: ∀L,d,i,k. L ⊢ i ϵ 𝐅*[d]⦃⋆k⦄ → ⊥.
42 #L #d #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
45 lemma frees_inv_gref: ∀L,d,i,p. L ⊢ i ϵ 𝐅*[d]⦃§p⦄ → ⊥.
46 #L #d #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
49 lemma frees_inv_lref: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ →
51 ∃∃I,K,W. d ≤ yinj j & j < i & ⇩[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
52 #L #d #x #i #H elim (frees_inv … H) -H
53 [ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
54 | * #I #K #W #j #Hdj #Hji #Hnx #HLK #HW
55 >(nlift_inv_lref_be_SO … Hnx) -x /3 width=5 by ex4_3_intro, or_intror/
59 lemma frees_inv_lref_free: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → |L| ≤ j → j = i.
60 #L #d #j #i #H #Hj elim (frees_inv_lref … H) -H //
61 * #I #K #W #_ #_ #HLK lapply (ldrop_fwd_length_lt2 … HLK) -I
62 #H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
65 lemma frees_inv_lref_skip: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → yinj j < d → j = i.
66 #L #d #j #i #H #Hjd elim (frees_inv_lref … H) -H //
67 * #I #K #W #Hdj elim (ylt_yle_false … Hdj) -Hdj //
70 lemma frees_inv_lref_ge: ∀L,d,j,i. L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → i ≤ j → j = i.
71 #L #d #j #i #H #Hij elim (frees_inv_lref … H) -H //
72 * #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -d /2 width=3 by lt_to_le_to_lt/
75 lemma frees_inv_lref_lt: ∀L,d,j,i.L ⊢ i ϵ 𝐅*[d]⦃#j⦄ → j < i →
76 ∃∃I,K,W. d ≤ yinj j & ⇩[j] L ≡ K.ⓑ{I}W & K ⊢ (i-j-1) ϵ 𝐅*[yinj 0]⦃W⦄.
77 #L #d #j #i #H #Hji elim (frees_inv_lref … H) -H
78 [ #H elim (lt_refl_false j) //
79 | * /2 width=5 by ex3_3_intro/
83 lemma frees_inv_bind: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄ →
84 L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ .
85 #a #J #L #V #U #d #i #H elim (frees_inv … H) -H
86 [ #HnX elim (nlift_inv_bind … HnX) -HnX
87 /4 width=2 by frees_eq, or_intror, or_introl/
88 | * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
89 [ /4 width=9 by frees_be, or_introl/
90 | #HnT @or_intror @(frees_be … HnT) -HnT
91 [4,5,6: /2 width=1 by ldrop_drop, yle_succ, lt_minus_to_plus/
92 |7: >minus_plus_plus_l //
99 lemma frees_inv_flat: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄ →
100 L ⊢ i ϵ 𝐅*[d]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[d]⦃U⦄ .
101 #J #L #V #U #d #i #H elim (frees_inv … H) -H
102 [ #HnX elim (nlift_inv_flat … HnX) -HnX
103 /4 width=2 by frees_eq, or_intror, or_introl/
104 | * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
105 /4 width=9 by frees_be, or_intror, or_introl/
109 (* Basic properties *********************************************************)
111 lemma frees_lref_eq: ∀L,d,i. L ⊢ i ϵ 𝐅*[d]⦃#i⦄.
112 /3 width=7 by frees_eq, lift_inv_lref2_be/ qed.
114 lemma frees_lref_be: ∀I,L,K,W,d,i,j. d ≤ yinj j → j < i → ⇩[j]L ≡ K.ⓑ{I}W →
115 K ⊢ i-j-1 ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[d]⦃#j⦄.
116 /3 width=9 by frees_be, lift_inv_lref2_be/ qed.
118 lemma frees_bind_sn: ∀a,I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
119 L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
120 #a #I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
121 /4 width=9 by frees_be, frees_eq, nlift_bind_sn/
124 lemma frees_bind_dx: ∀a,I,L,W,U,d,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[⫯d]⦃U⦄ →
125 L ⊢ i ϵ 𝐅*[d]⦃ⓑ{a,I}W.U⦄.
126 #a #J #L #V #U #d #i #H elim (frees_inv … H) -H
127 [ /4 width=9 by frees_eq, nlift_bind_dx/
128 | * #I #K #W #j #Hdj #Hji #HnU #HLK #HW
129 elim (yle_inv_succ1 … Hdj) -Hdj <yminus_SO2 #Hyj #H
130 lapply (ylt_O … H) -H #Hj
131 >(plus_minus_m_m j 1) in HnU; // <minus_le_minus_minus_comm in HW;
132 /4 width=9 by frees_be, nlift_bind_dx, ldrop_inv_drop1_lt, lt_plus_to_minus/
136 lemma frees_flat_sn: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃W⦄ →
137 L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
138 #I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
139 /4 width=9 by frees_be, frees_eq, nlift_flat_sn/
142 lemma frees_flat_dx: ∀I,L,W,U,d,i. L ⊢ i ϵ 𝐅*[d]⦃U⦄ →
143 L ⊢ i ϵ 𝐅*[d]⦃ⓕ{I}W.U⦄.
144 #I #L #W #U #d #i #H elim (frees_inv … H) -H [|*]
145 /4 width=9 by frees_be, frees_eq, nlift_flat_dx/
148 lemma frees_weak: ∀L,U,d1,i. L ⊢ i ϵ 𝐅*[d1]⦃U⦄ →
149 ∀d2. d2 ≤ d1 → L ⊢ i ϵ 𝐅*[d2]⦃U⦄.
150 #L #U #d1 #i #H elim H -L -U -d1 -i
151 /3 width=9 by frees_be, frees_eq, yle_trans/
154 (* Advanced inversion lemmas ************************************************)
156 lemma frees_inv_bind_O: ∀a,I,L,W,U,i. L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄ →
157 L ⊢ i ϵ 𝐅*[0]⦃W⦄ ∨ L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ .
158 #a #I #L #W #U #i #H elim (frees_inv_bind … H) -H
159 /3 width=3 by frees_weak, or_intror, or_introl/