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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 "basic_2/substitution/lsubs_sfr.ma".
16 include "basic_2/substitution/ldrop_ldrop.ma".
18 (* DROPPING *****************************************************************)
20 (* Inversion lemmas about local env. full refinement for substitution *******)
22 (* Note: ldrop_ldrop not needed *)
23 lemma sfr_inv_ldrop: ∀I,L,K,V,i. ⇩[0, i] L ≡ K. ⓑ{I}V → ∀d,e. ≽ [d, e] L →
24 d ≤ i → i < d + e → I = Abbr.
27 lapply (ldrop_inv_atom1 … H) -H #H destruct
28 | #L #J #W #IHL #K #V #i #H
29 elim (ldrop_inv_O1 … H) -H *
30 [ -IHL #H1 #H2 #d #e #HL #Hdi #Hide destruct
31 lapply (le_n_O_to_eq … Hdi) -Hdi #H destruct
32 lapply (HL … (L.ⓓW) ?) -HL /2 width=1/ #H
33 elim (lsubs_inv_abbr2 … H ?) -H // -Hide #K #_ #H destruct //
34 | #Hi #HLK #d @(nat_ind_plus … d) -d
36 elim (sfr_inv_bind … H ?) -H [2: /2 width=2/ ] #HL #H destruct
37 @(IHL … HLK … HL) -IHL -HLK -HL // /2 width=1/
38 | #d #_ #e #H #Hdi #Hide
39 lapply (sfr_inv_skip … H ?) -H // #HL
40 @(IHL … HLK … HL) -IHL -HLK -HL /2 width=1/
46 (* Properties about local env. full refinement for substitution *************)
48 (* Note: ldrop_ldrop not needed *)
49 lemma sfr_ldrop: ∀L,d,e.
50 (∀I,K,V,i. d ≤ i → i < d + e → ⇩[0, i] L ≡ K. ⓑ{I}V → I = Abbr) →
53 #L #I #V #IHL #d @(nat_ind_plus … d) -d
54 [ #e @(nat_ind_plus … e) -e //
56 >(H0 I L V 0 ? ? ?) //
57 /5 width=6 by sfr_abbr, ldrop_ldrop, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
59 /5 width=6 by sfr_skip, ldrop_ldrop, le_S_S, lt_minus_to_plus_r/ (**) (* auto now too slow without trace *)
63 lemma sfr_ldrop_trans_le: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 →
64 dd + ee ≤ d → ≽ [dd, ee] L2.
65 #L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
66 @sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
67 lapply (lt_to_le_to_lt … Hiddee Hddee) -Hddee #Hid
68 elim (ldrop_trans_le … HL12 … HLK2 ?) -L2 /2 width=2/ #X #HLK1 #H
69 elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K1 #V1 #HK12 #HV21 #H destruct
70 @(sfr_inv_ldrop … HLK1 … HL1) -L1 -K1 -V1 //
73 lemma sfr_ldrop_trans_be_up: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 →
74 ∀dd,ee. ≽ [dd, ee] L1 →
75 dd ≤ d + e → d + e ≤ dd + ee →
76 ≽ [d, dd + ee - d - e] L2.
77 #L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hdde #Hddee
78 @sfr_ldrop #I #K2 #V2 #i #Hdi #Hiddee #HLK2
79 lapply (transitive_le ? ? (i+e)… Hdde ?) -Hdde /2 width=1/ #Hddie
80 >commutative_plus in Hiddee; >minus_minus_comm <plus_minus_m_m /2 width=1/ -Hddee #Hiddee
81 lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hdi #HL1K2
82 @(sfr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus // -Hddie /2 width=1/
85 lemma sfr_ldrop_trans_ge: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ∀dd,ee. ≽ [dd, ee] L1 →
86 d + e ≤ dd → ≽ [dd - e, ee] L2.
87 #L1 #L2 #d #e #HL12 #dd #ee #HL1 #Hddee
88 @sfr_ldrop #I #K2 #V2 #i #Hddi #Hiddee #HLK2
89 elim (le_inv_plus_l … Hddee) -Hddee #Hdde #Hedd
90 >plus_minus in Hiddee; // #Hiddee
91 lapply (transitive_le … Hdde Hddi) -Hdde #Hid
92 lapply (ldrop_trans_ge … HL12 … HLK2 ?) -L2 // -Hid #HL1K2
93 @(sfr_inv_ldrop … HL1K2 … HL1) -L1 >commutative_plus /2 width=1/