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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 *)
11 (* v GNU General Public License Version 2 *)
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15 include "ground/lib/arith_2a.ma".
16 include "ground/ynat/ynat_minus_sn.ma".
17 include "basic_2A/substitution/drop_drop.ma".
18 include "basic_2A/multiple/frees.ma".
20 (* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
22 (* Advanced properties ******************************************************)
24 lemma frees_dec: ∀L,U,l,i. Decidable (frees l L U i).
25 #L #U @(f2_ind … rfw … L U) -L -U
27 [ -IH /3 width=5 by frees_inv_sort, or_intror/
28 | #j #Hx #l #i elim (lt_or_eq_or_gt i j) #Hji
29 [ -x @or_intror #H elim (lt_refl_false i)
30 lapply (frees_inv_lref_ge … H ?) -L -l /2 width=1 by lt_to_le/
31 | -x /2 width=1 by or_introl/
32 | elim (ylt_split j l) #Hli
33 [ -x @or_intror #H elim (lt_refl_false i)
34 lapply (frees_inv_lref_skip … H ?) -L //
35 | elim (lt_or_ge j (|L|)) #Hj
36 [ elim (drop_O1_lt (Ⓕ) L j) // -Hj #I #K #W #HLK destruct
37 elim (IH K W … 0 (i-j-1)) -IH [1,3: /3 width=5 by frees_lref_be, drop_fwd_rfw, or_introl/ ] #HnW
38 @or_intror #H elim (frees_inv_lref_lt … H) // #Z #Y #X #_ #HLY -l
39 lapply (drop_mono … HLY … HLK) -L #H destruct /2 width=1 by/
40 | -x @or_intror #H elim (lt_refl_false i)
41 lapply (frees_inv_lref_free … H ?) -l //
45 | -IH /3 width=5 by frees_inv_gref, or_intror/
46 | #a #I #W #U #Hx #l #i destruct
47 elim (IH L W … l i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
48 elim (IH (L.ⓑ{I}W) U … (↑l) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
49 @or_intror #H elim (frees_inv_bind … H) -H /2 width=1 by/
50 | #I #W #U #Hx #l #i destruct
51 elim (IH L W … l i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
52 elim (IH L U … l i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
53 @or_intror #H elim (frees_inv_flat … H) -H /2 width=1 by/
57 alias symbol "UpArrow" (instance 2) = "ynat successor".
58 alias symbol "minus" (instance 6) = "natural minus".
59 alias symbol "minus" (instance 5) = "natural minus".
60 alias symbol "RDrop" (instance 9) = "basic slicing (local environment) lget".
61 lemma frees_S: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[yinj l]⦃U⦄ → ∀I,K,W. ⬇[l] L ≡ K.ⓑ{I}W →
62 (K ⊢ i-l-1 ϵ 𝐅*[0]⦃W⦄ → ⊥) → L ⊢ i ϵ 𝐅*[↑l]⦃U⦄.
63 #L #U #l #i #H elim (frees_inv … H) -H /3 width=2 by frees_eq/
64 * #I #K #W #j #Hlj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
65 lapply (yle_inv_inj … Hlj) -Hlj #Hlj
66 elim (le_to_or_lt_eq … Hlj) -Hlj
67 [ -I0 -K0 -W0 /3 width=9 by frees_be, yle_inj/
68 | -Hji -HnU #H destruct
69 lapply (drop_mono … HLK0 … HLK) #H destruct -I
70 elim HnW0 -L -U -HnW0 //
74 (* Note: lemma 1250 *)
75 lemma frees_bind_dx_O: ∀a,I,L,W,U,i. L.ⓑ{I}W ⊢ i+1 ϵ 𝐅*[0]⦃U⦄ →
76 L ⊢ i ϵ 𝐅*[0]⦃ⓑ{a,I}W.U⦄.
77 #a #I #L #W #U #i #HU elim (frees_dec L W 0 i)
78 /4 width=5 by frees_S, frees_bind_dx, frees_bind_sn/
81 (* Properties on relocation *************************************************)
83 lemma frees_lift_ge: ∀K,T,l,i. K ⊢ i ϵ𝐅*[l]⦃T⦄ →
84 ∀L,s,l0,m0. ⬇[s, l0, m0] L ≡ K →
85 ∀U. ⬆[l0, m0] T ≡ U → l0 ≤ i →
87 #K #T #l #i #H elim H -K -T -l -i
88 [ #K #T #l #i #HnT #L #s #l0 #m0 #_ #U #HTU #Hl0i -K -s
89 @frees_eq #X #HXU elim (lift_div_le … HTU … HXU) -U /2 width=2 by/
90 | #I #K #K0 #T #V #l #i #j #Hlj #Hji #HnT #HK0 #HV #IHV #L #s #l0 #m0 #HLK #U #HTU #Hl0i
91 elim (lt_or_ge j l0) #H1
92 [ elim (drop_trans_lt … HLK … HK0) // -K #L0 #W #HL0 #HLK0 #HVW
93 @(frees_be … HL0) -HL0 -HV
94 /3 width=3 by lt_plus_to_minus_r, lt_to_le_to_lt/
95 [ #X #HXU >(plus_minus_m_m l0 1) in HTU; /2 width=2 by ltn_to_ltO/ #HTU
96 elim (lift_div_le … HXU … HTU ?) -U /2 width=2 by monotonic_pred/
97 | >minus_plus <plus_minus // <minus_plus
98 /3 width=5 by monotonic_le_minus_l2/
100 | lapply (drop_trans_ge … HLK … HK0 ?) // -K #HLK0
101 lapply (drop_inv_gen … HLK0) >commutative_plus -HLK0 #HLK0
102 @(frees_be … HLK0) -HLK0 -IHV
103 /2 width=1 by yle_plus_dx1_trans, lt_minus_to_plus/
104 #X #HXU elim (lift_div_le … HTU … HXU) -U /2 width=2 by/
109 (* Inversion lemmas on relocation *******************************************)
111 lemma frees_inv_lift_be: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
112 ∀K,s,l0,m0. ⬇[s, l0, m0+1] L ≡ K →
113 ∀T. ⬆[l0, m0+1] T ≡ U → l0 ≤ i → i ≤ l0 + m0 → ⊥.
114 #L #U #l #i #H elim H -L -U -l -i
115 [ #L #U #l #i #HnU #K #s #l0 #m0 #_ #T #HTU #Hl0i #Hilm0
116 elim (lift_split … HTU i m0) -HTU /2 width=2 by/
117 | #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hl0i #Hilm0
118 elim (lt_or_ge j l0) #H1
119 [ elim (drop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
121 [ /2 width=1 by monotonic_le_minus_l2/
122 | >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
124 | elim (lift_split … HTU j m0) -HTU /3 width=3 by lt_to_le_to_lt, lt_to_le/
129 lemma frees_inv_lift_ge: ∀L,U,l,i. L ⊢ i ϵ 𝐅*[l]⦃U⦄ →
130 ∀K,s,l0,m0. ⬇[s, l0, m0] L ≡ K →
131 ∀T. ⬆[l0, m0] T ≡ U → l0 + m0 ≤ i →
132 K ⊢ i-m0 ϵ𝐅*[l-m0]⦃T⦄.
133 #L #U #l #i #H elim H -L -U -l -i
134 [ #L #U #l #i #HnU #K #s #l0 #m0 #HLK #T #HTU #Hlm0i -L -s
135 elim (le_inv_plus_l … Hlm0i) -Hlm0i #Hl0im0 #Hm0i @frees_eq #X #HXT -K
136 elim (lift_trans_le … HXT … HTU) -T // <plus_minus_m_m /2 width=2 by/
137 | #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hlm0i
138 elim (lt_or_ge j l0) #H1
139 [ elim (drop_conf_lt … HLK … HLK0) -L // #L0 #V #H #HKL0 #HVW
140 elim (le_inv_plus_l … Hlm0i) #H0 #Hm0i
142 [ /3 width=1 by yle_plus_dx1_trans, monotonic_yle_minus_dx/
143 | /2 width=3 by lt_to_le_to_lt/
144 | #X #HXT elim (lift_trans_ge … HXT … HTU) -T /2 width=2 by/
145 | lapply (IHW … HKL0 … HVW ?) // -I -K -K0 -L0 -V -W -T -U -s
146 >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
148 | elim (lt_or_ge j (l0+m0)) [ >commutative_plus |] #H2
149 [ -L -I -W lapply (lt_plus_to_minus ???? H2) // -H2 #H2
150 elim (lift_split … HTU j (m0-1)) -HTU //
151 [ >minus_minus_associative /2 width=2 by ltn_to_ltO/ <minus_n_n
152 #X #_ #H elim (HnU … H)
153 | >commutative_plus /3 width=1 by le_minus_to_plus, monotonic_pred/
155 | lapply (drop_conf_ge … HLK … HLK0 ?) // -L #HK0
156 elim (le_inv_plus_l … H2) -H2 #H2 #Hm0j
158 [ /2 width=1 by monotonic_yle_minus_dx/
159 | /2 width=1 by monotonic_lt_minus_l/
160 | #X #HXT elim (lift_trans_le … HXT … HTU) -T // <plus_minus_m_m /2 width=2 by/
161 | >arith_b1 /2 width=5 by/