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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 *)
10 (* \ / This file is distributed under the terms of the *)
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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/substitution/lift_neg.ma".
18 include "basic_2/substitution/drop.ma".
20 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
22 inductive frees: relation4 ynat lenv term ynat ≝
23 | frees_eq: ∀L,U,l,i. (∀T. ⬆[i, 1] T ≡ U → ⊥) → frees l L U i
24 | frees_be: ∀I,L,K,U,W,l,i,j. l ≤ yinj j → yinj j < i →
25 (∀T. ⬆[j, 1] T ≡ U → ⊥) → ⬇[j]L ≡ K.ⓑ{I}W →
26 frees 0 K W (⫰(i-j)) → frees l L U i.
29 "context-sensitive free variables (term)"
30 'FreeStar L i l U = (frees l L U i).
32 definition frees_trans: predicate (relation3 lenv term term) ≝
33 λR. ∀L,U1,U2,i. R L U1 U2 → L ⊢ i ϵ 𝐅*[0]⦃U2⦄ → L ⊢ i ϵ 𝐅*[0]⦃U1⦄.
35 (* Basic inversion lemmas ***************************************************)
37 lemma frees_inv: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
38 (∀T. ⬆[i, 1] T ≡ U → ⊥) ∨
39 ∃∃I,K,W,j. l ≤ yinj j & j < i & (∀T. ⬆[j, 1] T ≡ U → ⊥) &
40 ⬇[j]L ≡ K.ⓑ{I}W & K ⊢ ⫰(i-j) ϵ 𝐅*[yinj 0]⦃W⦄.
41 #L #U #l #i * -L -U -l -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
44 lemma frees_inv_sort: ∀L,l,i,k. L ⊢ i ϵ 𝐅*[l]⦃⋆k⦄ → ⊥.
45 #L #l #i #k #H elim (frees_inv … H) -H [|*] /2 width=2 by/
48 lemma frees_inv_gref: ∀L,l,i,p. L ⊢ i ϵ 𝐅*[l]⦃§p⦄ → ⊥.
49 #L #l #i #p #H elim (frees_inv … H) -H [|*] /2 width=2 by/
52 lemma frees_inv_lref: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ →
54 ∃∃I,K,W. l ≤ yinj j & yinj j < i & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ ⫰(i-j) ϵ 𝐅*[yinj 0]⦃W⦄.
55 #L #l #x #i #H elim (frees_inv … H) -H
56 [ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
57 | * #I #K #W #j #Hlj #Hji #Hnx #HLK #HW
58 lapply (nlift_inv_lref_be_SO … Hnx) -Hnx #H
59 lapply (yinj_inj … H) -H #H destruct
60 /3 width=5 by ex4_3_intro, or_intror/
64 lemma frees_inv_lref_free: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → |L| ≤ j → yinj j = i.
65 #L #l #j #i #H #Hj elim (frees_inv_lref … H) -H //
66 * #I #K #W #_ #_ #HLK lapply (drop_fwd_length_lt2 … HLK) -I
67 #H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
70 lemma frees_inv_lref_skip: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → yinj j < l → yinj j = i.
71 #L #l #j #i #H #Hjl elim (frees_inv_lref … H) -H //
72 * #I #K #W #Hlj elim (ylt_yle_false … Hlj) -Hlj //
75 lemma frees_inv_lref_ge: ∀L,l,j,i. L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → i ≤ j → yinj j = i.
76 #L #l #j #i #H #Hij elim (frees_inv_lref … H) -H //
77 * #I #K #W #_ #Hji elim (ylt_yle_false … Hji Hij)
80 lemma frees_inv_lref_lt: ∀L,l,j,i.L ⊢ i ϵ 𝐅*[l]⦃#j⦄ → j < i →
81 ∃∃I,K,W. l ≤ yinj j & ⬇[j] L ≡ K.ⓑ{I}W & K ⊢ ⫰(i-j) ϵ 𝐅*[yinj 0]⦃W⦄.
82 #L #l #j #i #H #Hji elim (frees_inv_lref … H) -H
83 [ #H elim (ylt_yle_false … Hji) //
84 | * /2 width=5 by ex3_3_intro/
88 lemma frees_inv_bind: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄ →
89 L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L.ⓑ{I}W ⊢ ⫯i ϵ 𝐅*[⫯l]⦃U⦄ .
90 #a #J #L #V #U #l #i #H elim (frees_inv … H) -H
91 [ #HnX elim (nlift_inv_bind … HnX) -HnX
92 /4 width=2 by frees_eq, or_intror, or_introl/
93 | * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_bind … HnX) -HnX
94 [ /4 width=9 by frees_be, or_introl/
95 | #HnT @or_intror @(frees_be … HnT) -HnT
96 [4: lapply (yle_succ … Hlj) // (**)
97 |5: lapply (ylt_succ … Hji) // (**)
98 |6: /2 width=4 by drop_drop/
99 |7: <yminus_succ in HW; // (**)
106 lemma frees_inv_flat: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄ →
107 L ⊢ i ϵ 𝐅*[l]⦃W⦄ ∨ L ⊢ i ϵ 𝐅*[l]⦃U⦄ .
108 #J #L #V #U #l #i #H elim (frees_inv … H) -H
109 [ #HnX elim (nlift_inv_flat … HnX) -HnX
110 /4 width=2 by frees_eq, or_intror, or_introl/
111 | * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_flat … HnX) -HnX
112 /4 width=9 by frees_be, or_intror, or_introl/
116 (* Basic properties *********************************************************)
118 lemma frees_lref_eq: ∀L,l,i. L ⊢ i ϵ 𝐅*[l]⦃#i⦄.
119 /4 width=7 by frees_eq, lift_inv_lref2_be, ylt_inj/ qed.
121 lemma frees_lref_be: ∀I,L,K,W,l,i,j. l ≤ yinj j → j < i → ⬇[j]L ≡ K.ⓑ{I}W →
122 K ⊢ ⫰(i-j) ϵ 𝐅*[0]⦃W⦄ → L ⊢ i ϵ 𝐅*[l]⦃#j⦄.
123 /4 width=9 by frees_be, lift_inv_lref2_be, ylt_inj/ qed.
125 lemma frees_bind_sn: ∀a,I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
126 L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
127 #a #I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
128 /4 width=9 by frees_be, frees_eq, nlift_bind_sn/
131 lemma frees_bind_dx: ∀a,I,L,W,U,l,i. L.ⓑ{I}W ⊢ ⫯i ϵ 𝐅*[⫯l]⦃U⦄ →
132 L ⊢ i ϵ 𝐅*[l]⦃ⓑ{a,I}W.U⦄.
133 #a #J #L #V #U #l #i #H elim (frees_inv … H) -H
134 [ /4 width=9 by frees_eq, nlift_bind_dx/
135 | * #I #K #W #j #Hlj elim (yle_inv_succ1 … Hlj) -Hlj #Hlj
137 lapply (ylt_O … Hj) -Hj #Hj #H
138 lapply (ylt_inv_succ … H) -H #Hji #HnU #HLK #HW
139 @(frees_be … Hlj Hji … HW) -HW -Hlj -Hji (**) (* explicit constructor *)
140 [2: #X #H lapply (nlift_bind_dx … H) /2 width=2 by/ (**)
141 |3: lapply (drop_inv_drop1_lt … HLK ?) -HLK //
146 lemma frees_flat_sn: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃W⦄ →
147 L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
148 #I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
149 /4 width=9 by frees_be, frees_eq, nlift_flat_sn/
152 lemma frees_flat_dx: ∀I,L,W,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
153 L ⊢ i ϵ 𝐅*[l]⦃ⓕ{I}W.U⦄.
154 #I #L #W #U #l #i #H elim (frees_inv … H) -H [|*]
155 /4 width=9 by frees_be, frees_eq, nlift_flat_dx/
158 lemma frees_weak: ∀L,U,l1,i. L ⊢ i ϵ 𝐅*[l1]⦃U⦄ →
159 ∀l2. l2 ≤ l1 → L ⊢ i ϵ 𝐅*[l2]⦃U⦄.
160 #L #U #l1 #i #H elim H -L -U -l1 -i
161 /3 width=9 by frees_be, frees_eq, yle_trans/
164 (* Advanced inversion lemmas ************************************************)
166 lemma frees_inv_bind_O: ∀a,I,L,W,U,i. L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄ →
167 L ⊢ i ϵ 𝐅*[0]⦃W⦄ ∨ L.ⓑ{I}W ⊢ ⫯i ϵ 𝐅*[0]⦃U⦄ .
168 #a #I #L #W #U #i #H elim (frees_inv_bind … H) -H
169 /3 width=3 by frees_weak, or_intror, or_introl/