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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/lift_lift.ma".
16 include "Basic_2/substitution/ldrop.ma".
18 (* DROPPING *****************************************************************)
20 (* Main properties **********************************************************)
22 (* Basic_1: was: ldrop_mono *)
23 theorem ldrop_mono: ∀d,e,L,L1. ⇩[d, e] L ≡ L1 →
24 ∀L2. ⇩[d, e] L ≡ L2 → L1 = L2.
25 #d #e #L #L1 #H elim H -d -e -L -L1
27 >(ldrop_inv_atom1 … H) -L2 //
29 <(ldrop_inv_refl … HL12) -L2 //
30 | #L #K #I #V #e #_ #IHLK #L2 #H
31 lapply (ldrop_inv_ldrop1 … H ?) -H // /2 width=1/
32 | #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
33 elim (ldrop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
34 >(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
35 >(IHLK1 … HLK2) -IHLK1 -HLK2 //
39 (* Basic_1: was: ldrop_conf_ge *)
40 theorem ldrop_conf_ge: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
41 ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
42 ⇩[0, e2 - e1] L1 ≡ L2.
43 #d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1
45 >(ldrop_inv_atom1 … H) -L2 //
47 | #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
48 lapply (ldrop_inv_ldrop1 … H ?) -H /2 width=2/ #HL2
49 <minus_plus >minus_minus_comm /3 width=1/
50 | #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
51 lapply (transitive_le 1 … Hdee2) // #He2
52 lapply (ldrop_inv_ldrop1 … H ?) -H // -He2 #HL2
53 lapply (transitive_le (1 + e) … Hdee2) // #Hee2
54 @ldrop_ldrop_lt >minus_minus_comm /3 width=1/ (**) (* explicit constructor *)
58 (* Basic_1: was: ldrop_conf_lt *)
59 theorem ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
60 ∀e2,K2,I,V2. ⇩[0, e2] L ≡ K2. 𝕓{I} V2 →
61 e2 < d1 → let d ≝ d1 - e2 - 1 in
62 ∃∃K1,V1. ⇩[0, e2] L1 ≡ K1. 𝕓{I} V1 &
63 ⇩[d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2.
64 #d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1
65 [ #d #e #e2 #K2 #I #V2 #H
66 lapply (ldrop_inv_atom1 … H) -H #H destruct
67 | #L #I #V #e2 #K2 #J #V2 #_ #H
68 elim (lt_zero_false … H)
69 | #L1 #L2 #I #V #e #_ #_ #e2 #K2 #J #V2 #_ #H
70 elim (lt_zero_false … H)
71 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #K2 #J #V #H #He2d
72 elim (ldrop_inv_O1 … H) -H *
73 [ -IHL12 -He2d #H1 #H2 destruct /2 width=5/
74 | -HL12 -HV12 #He #HLK
75 elim (IHL12 … HLK ?) -IHL12 -HLK [ <minus_minus /3 width=5/ | /2 width=1/ ] (**) (* a bit slow *)
80 (* Basic_1: was: ldrop_trans_le *)
81 theorem ldrop_trans_le: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
82 ∀e2,L2. ⇩[0, e2] L ≡ L2 → e2 ≤ d1 →
83 ∃∃L0. ⇩[0, e2] L1 ≡ L0 & ⇩[d1 - e2, e1] L0 ≡ L2.
84 #d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L
86 >(ldrop_inv_atom1 … H) -L2 /2 width=3/
87 | #K #I #V #e2 #L2 #HL2 #H
88 lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
89 | #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
90 lapply (le_n_O_to_eq … H) -H #H destruct
91 elim (IHL12 … HL2 ?) -IHL12 -HL2 // #L0 #H #HL0
92 lapply (ldrop_inv_refl … H) -H #H destruct /3 width=5/
93 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
94 elim (ldrop_inv_O1 … H) -H *
95 [ -He2d -IHL12 #H1 #H2 destruct /3 width=5/
96 | -HL12 -HV12 #He2 #HL2
97 elim (IHL12 … HL2 ?) -L2 [ >minus_le_minus_minus_comm // /3 width=3/ | /2 width=1/ ]
102 (* Basic_1: was: ldrop_trans_ge *)
103 theorem ldrop_trans_ge: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
104 ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2.
105 #d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L
107 >(ldrop_inv_atom1 … H) -H -L2 //
110 | #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2
111 lapply (lt_to_le_to_lt 0 … Hde2) // #He2
112 lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
113 lapply (ldrop_inv_ldrop1 … H ?) -H // #HL2
114 @ldrop_ldrop_lt // >le_plus_minus // @IHL12 /2 width=1/ (**) (* explicit constructor *)
118 theorem ldrop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L.
119 ⇩[d1, e1] L1 ≡ L → ⇩[0, e2] L ≡ L2 → d1 ≤ e2 →
120 ⇩[0, e2 + e1] L1 ≡ L2.
121 #e1 #e1 #e2 >commutative_plus /2 width=5/
124 (* Basic_1: was: ldrop_conf_rev *)
125 axiom ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L →
126 ∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1.