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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/leq.ma".
16 include "Basic-2/substitution/lift.ma".
18 (* DROPPING *****************************************************************)
20 inductive drop: lenv → nat → nat → lenv → Prop ≝
21 | drop_sort: ∀d,e. drop (⋆) d e (⋆)
22 | drop_comp: ∀L1,L2,I,V. drop L1 0 0 L2 → drop (L1. 𝕓{I} V) 0 0 (L2. 𝕓{I} V)
23 | drop_drop: ∀L1,L2,I,V,e. drop L1 0 e L2 → drop (L1. 𝕓{I} V) 0 (e + 1) L2
24 | drop_skip: ∀L1,L2,I,V1,V2,d,e.
25 drop L1 d e L2 → ↑[d,e] V2 ≡ V1 →
26 drop (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
29 interpretation "dropping" 'RDrop L1 d e L2 = (drop L1 d e L2).
31 (* Basic inversion lemmas ***************************************************)
33 lemma drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
34 #d #e #L1 #L2 #H elim H -H d e L1 L2
36 | #L1 #L2 #I #V #_ #IHL12 #H1 #H2
37 >(IHL12 H1 H2) -IHL12 H1 H2 L1 //
38 | #L1 #L2 #I #V #e #_ #_ #_ #H
39 elim (plus_S_eq_O_false … H)
40 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H
41 elim (plus_S_eq_O_false … H)
45 lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
48 lemma drop_inv_sort1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
50 #d #e #L1 #L2 * -d e L1 L2
52 | #L1 #L2 #I #V #_ #H destruct
53 | #L1 #L2 #I #V #e #_ #H destruct
54 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
58 lemma drop_inv_sort1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
61 lemma drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
62 ∀K,I,V. L1 = K. 𝕓{I} V →
63 (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
64 (0 < e ∧ ↓[d, e - 1] K ≡ L2).
65 #d #e #L1 #L2 * -d e L1 L2
66 [ #d #e #_ #K #I #V #H destruct
67 | #L1 #L2 #I #V #HL12 #H #K #J #W #HX destruct -L1 I V
68 >(drop_inv_refl … HL12) -HL12 K /3/
69 | #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/
70 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
74 lemma drop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
75 (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
76 (0 < e ∧ ↓[0, e - 1] K ≡ L2).
79 lemma drop_inv_drop1: ∀e,K,I,V,L2.
80 ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
81 #e #K #I #V #L2 #H #He
82 elim (drop_inv_O1 … H) -H * // #H destruct -e;
83 elim (lt_refl_false … He)
86 lemma drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
87 ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
88 ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
91 #d #e #L1 #L2 * -d e L1 L2
92 [ #d #e #_ #I #K #V #H destruct
93 | #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
94 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
95 | #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z
100 lemma drop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
101 ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
102 ↑[d - 1, e] V2 ≡ V1 &
106 lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
107 ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
108 ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
109 ↑[d - 1, e] V2 ≡ V1 &
111 #d #e #L1 #L2 * -d e L1 L2
112 [ #d #e #_ #I #K #V #H destruct
113 | #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
114 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
115 | #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z
120 lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
121 ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
125 (* Basic properties *********************************************************)
127 lemma drop_refl: ∀L. ↓[0, 0] L ≡ L.
131 lemma drop_drop_lt: ∀L1,L2,I,V,e.
132 ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
133 #L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
136 lemma drop_leq_drop1: ∀L1,L2,d,e. L1 [d, e] ≈ L2 →
137 ∀I,K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{I} V →
139 ∃∃K2. K1 [0, d + e - i - 1] ≈ K2 &
140 ↓[0, i] L2 ≡ K2. 𝕓{I} V.
141 #L1 #L2 #d #e #H elim H -H L1 L2 d e
142 [ #d #e #I #K1 #V #i #H
143 lapply (drop_inv_sort1 … H) -H #H destruct
144 | #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #I #K1 #V #i #_ #_ #H
145 elim (lt_zero_false … H)
146 | #L1 #L2 #I #V #e #HL12 #IHL12 #J #K1 #W #i #H #_ #Hie
147 elim (drop_inv_O1 … H) -H * #Hi #HLK1
148 [ -IHL12 Hie; destruct -i K1 J W;
149 <minus_n_O <minus_plus_m_m /2/
151 elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
153 | #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #I #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
154 lapply (plus_S_le_to_pos … Hdi) #Hi
155 lapply (drop_inv_drop1 … H ?) -H // #HLK1
156 elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
160 (* Basic forvard lemmas *****************************************************)
162 lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
165 [ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
166 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
167 elim (drop_inv_O1 … H) -H * #He #H
168 [ -IHL1; destruct -e K2 I2 V2 /2/
169 | @drop_drop >(plus_minus_m_m e 1) /2/
174 lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e.
175 ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
177 [ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
178 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
179 elim (drop_inv_O1 … H) -H * #He #H
180 [ -IHL1; destruct -e K2 I2 V2 //
181 | lapply (IHL1 … H) -IHL1 H #HeK1 whd in ⊢ (? ? %) /2/
186 lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
188 [ #L2 #e #H >(drop_inv_sort1 … H) -H //
189 | #K1 #I1 #V1 #IHL1 #L2 #e #H
190 elim (drop_inv_O1 … H) -H * #He #H
191 [ -IHL1; destruct -e L2 //
192 | lapply (IHL1 … H) -IHL1 H #H >H -H; normalize
193 >minus_le_minus_minus_comm //