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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 "Basic-2/substitution/lift_lift.ma".
16 include "Basic-2/substitution/drop.ma".
18 (* DROPPING *****************************************************************)
20 (* Main properties **********************************************************)
22 theorem drop_mono: ∀d,e,L,L1. ↓[d, e] L ≡ L1 →
23 ∀L2. ↓[d, e] L ≡ L2 → L1 = L2.
24 #d #e #L #L1 #H elim H -H d e L L1
26 >(drop_inv_sort1 … H) -H L2 //
27 | #K1 #K2 #I #V #HK12 #_ #L2 #HL12
28 <(drop_inv_refl … HK12) -HK12 K2
29 <(drop_inv_refl … HL12) -HL12 L2 //
30 | #L #K #I #V #e #_ #IHLK #L2 #H
31 lapply (drop_inv_drop1 … H ?) -H /2/
32 | #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
33 elim (drop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct -X
34 >(lift_inj … HVT1 … HVT2) -HVT1 HVT2
35 >(IHLK1 … HLK2) -IHLK1 HLK2 //
39 theorem drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
40 ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
41 ↓[0, e2 - e1] L1 ≡ L2.
42 #d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
44 >(drop_inv_sort1 … H) -H L2 //
45 | #K1 #K2 #I #V #HK12 #_ #e2 #L2 #H #_ <minus_n_O
46 <(drop_inv_refl … HK12) -HK12 K2 //
47 | #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
48 lapply (drop_inv_drop1 … H ?) -H /2/ #HL2
50 | #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
51 lapply (transitive_le 1 … Hdee2) // #He2
52 lapply (drop_inv_drop1 … H ?) -H // -He2 #HL2
53 lapply (transitive_le (1 + e) … Hdee2) // #Hee2
54 @drop_drop_lt >minus_minus_comm /3/ (**) (* explicit constructor *)
58 theorem drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 →
59 ∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 →
60 e2 < d1 → let d ≝ d1 - e2 - 1 in
61 ∃∃K1,V1. ↓[0, e2] L1 ≡ K1. 𝕓{I} V1 &
62 ↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2.
63 #d1 #e1 #L #L1 #H elim H -H d1 e1 L L1
64 [ #d #e #e2 #K2 #I #V2 #H
65 lapply (drop_inv_sort1 … H) -H #H destruct
66 | #L1 #L2 #I #V #_ #_ #e2 #K2 #J #V2 #_ #H
67 elim (lt_zero_false … H)
68 | #L1 #L2 #I #V #e #_ #_ #e2 #K2 #J #V2 #_ #H
69 elim (lt_zero_false … H)
70 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #K2 #J #V #H #He2d
71 elim (drop_inv_O1 … H) -H *
72 [ -IHL12 He2d #H1 #H2 destruct -e2 K2 J V /2 width=5/
73 | -HL12 -HV12 #He #HLK
74 elim (IHL12 … HLK ?) -IHL12 HLK [ <minus_minus /3 width=5/ | /2/ ] (**) (* a bit slow *)
79 theorem drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
80 ∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 →
81 ∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2.
82 #d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
84 >(drop_inv_sort1 … H) -H L2 /2/
85 | #K1 #K2 #I #V #HK12 #_ #e2 #L2 #HL2 #H
86 >(drop_inv_refl … HK12) -HK12 K1;
87 lapply (le_O_to_eq_O … H) -H #H destruct -e2 /2/
88 | #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
89 lapply (le_O_to_eq_O … H) -H #H destruct -e2;
90 elim (IHL12 … HL2 ?) -IHL12 HL2 // #L0 #H #HL0
91 lapply (drop_inv_refl … H) -H #H destruct -L1 /3 width=5/
92 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
93 elim (drop_inv_O1 … H) -H *
94 [ -He2d IHL12 #H1 #H2 destruct -e2 L /3 width=5/
95 | -HL12 HV12 #He2 #HL2
96 elim (IHL12 … HL2 ?) -IHL12 HL2 L2
97 [ >minus_le_minus_minus_comm // /3/ | /2/ ]
102 theorem drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
103 ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2.
104 #d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L
106 >(drop_inv_sort1 … H) -H L2 //
107 | #K1 #K2 #I #V #HK12 #_ #e2 #L2 #H #_ normalize
108 >(drop_inv_refl … HK12) -HK12 K1 //
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 (drop_inv_drop1 … H ?) -H // #HL2
114 @drop_drop_lt // >le_plus_minus // @IHL12 /2/ (**) (* explicit constructor *)
118 theorem drop_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 axiom drop_div: ∀e1,L1,L. ↓[0, e1] L1 ≡ L → ∀e2,L2. ↓[0, e2] L2 ≡ L →
125 ∃∃L0. ↓[0, e1] L0 ≡ L2 & ↓[e1, e2] L0 ≡ L1.