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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/grammar/lenv_weight.ma".
16 include "Basic_2/grammar/lsubs.ma".
17 include "Basic_2/substitution/lift.ma".
19 (* LOCAL ENVIRONMENT SLICING ************************************************)
21 (* Basic_1: includes: ldrop_skip_bind *)
22 inductive ldrop: nat → nat → relation lenv ≝
23 | ldrop_atom: ∀d,e. ldrop d e (⋆) (⋆)
24 | ldrop_pair: ∀L,I,V. ldrop 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V)
25 | ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. 𝕓{I} V) L2
26 | ldrop_skip: ∀L1,L2,I,V1,V2,d,e.
27 ldrop d e L1 L2 → ↑[d,e] V2 ≡ V1 →
28 ldrop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
31 interpretation "ldropping" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
33 (* Basic inversion lemmas ***************************************************)
35 fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
36 #d #e #L1 #L2 * -d -e -L1 -L2
39 | #L1 #L2 #I #V #e #_ #_ #H
40 elim (plus_S_eq_O_false … H)
41 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H
42 elim (plus_S_eq_O_false … H)
46 (* Basic_1: was: ldrop_gen_refl *)
47 lemma ldrop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
50 fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
52 #d #e #L1 #L2 * -d -e -L1 -L2
54 | #L #I #V #H destruct
55 | #L1 #L2 #I #V #e #_ #H destruct
56 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
60 (* Basic_1: was: ldrop_gen_sort *)
61 lemma ldrop_inv_atom1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
64 fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
65 ∀K,I,V. L1 = K. 𝕓{I} V →
66 (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
67 (0 < e ∧ ↓[d, e - 1] K ≡ L2).
68 #d #e #L1 #L2 * -d -e -L1 -L2
69 [ #d #e #_ #K #I #V #H destruct
70 | #L #I #V #_ #K #J #W #HX destruct /3 width=1/
71 | #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct /3 width=1/
72 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
76 lemma ldrop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
77 (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
78 (0 < e ∧ ↓[0, e - 1] K ≡ L2).
81 (* Basic_1: was: ldrop_gen_ldrop *)
82 lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
83 ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
84 #e #K #I #V #L2 #H #He
85 elim (ldrop_inv_O1 … H) -H * // #H destruct
86 elim (lt_refl_false … He)
89 fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
90 ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
91 ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
94 #d #e #L1 #L2 * -d -e -L1 -L2
95 [ #d #e #_ #I #K #V #H destruct
96 | #L #I #V #H elim (lt_refl_false … H)
97 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
98 | #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct /2 width=5/
102 (* Basic_1: was: ldrop_gen_skip_l *)
103 lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
104 ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
105 ↑[d - 1, e] V2 ≡ V1 &
109 fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
110 ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
111 ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
112 ↑[d - 1, e] V2 ≡ V1 &
114 #d #e #L1 #L2 * -d -e -L1 -L2
115 [ #d #e #_ #I #K #V #H destruct
116 | #L #I #V #H elim (lt_refl_false … H)
117 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
118 | #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct /2 width=5/
122 (* Basic_1: was: ldrop_gen_skip_r *)
123 lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
124 ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
128 (* Basic properties *********************************************************)
130 (* Basic_1: was by definition: ldrop_refl *)
131 lemma ldrop_refl: ∀L. ↓[0, 0] L ≡ L.
135 lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
136 ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
137 #L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
140 lemma ldrop_lsubs_ldrop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
141 ∀K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{Abbr} V →
143 ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
144 ↓[0, i] L2 ≡ K2. 𝕓{Abbr} V.
145 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
147 lapply (ldrop_inv_atom1 … H) -H #H destruct
148 | #L1 #L2 #K1 #V #i #_ #_ #H
149 elim (lt_zero_false … H)
150 | #L1 #L2 #V #e #HL12 #IHL12 #K1 #W #i #H #_ #Hie
151 elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
152 [ -IHL12 -Hie destruct
153 <minus_n_O <minus_plus_m_m // /2 width=3/
155 elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /4 width=3/
157 | #L1 #L2 #I #V1 #V2 #e #_ #IHL12 #K1 #W #i #H #_ #Hie
158 elim (ldrop_inv_O1 … H) -H * #Hi #HLK1
159 [ -IHL12 -Hie -Hi destruct
160 | elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hie >minus_minus_comm >arith_b1 // /3 width=3/
162 | #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
163 elim (le_inv_plus_l … Hdi) #Hdim #Hi
164 lapply (ldrop_inv_ldrop1 … H ?) -H // #HLK1
165 elim (IHL12 … HLK1 ? ?) -IHL12 -HLK1 // /2 width=1/ -Hdi -Hide >minus_minus_comm >arith_b1 // /3 width=3/
169 (* Basic forvard lemmas *****************************************************)
171 (* Basic_1: was: ldrop_S *)
172 lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
175 [ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
176 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
177 elim (ldrop_inv_O1 … H) -H * #He #H
178 [ -IHL1 destruct /2 width=1/
179 | @ldrop_ldrop >(plus_minus_m_m e 1) // /2 width=3/
184 lemma ldrop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
185 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize
187 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
188 >(tw_lift … HV21) -HV21 /2 width=1/
192 lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
193 ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
195 [ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
196 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
197 elim (ldrop_inv_O1 … H) -H * #He #H
199 | lapply (IHL1 … H) -IHL1 -H #HeK1 whd in ⊢ (? ? %); /2 width=1/
204 lemma ldrop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
206 [ #L2 #e #H >(ldrop_inv_atom1 … H) -H //
207 | #K1 #I1 #V1 #IHL1 #L2 #e #H
208 elim (ldrop_inv_O1 … H) -H * #He #H
210 | lapply (IHL1 … H) -IHL1 -H #H >H -H normalize
211 >minus_le_minus_minus_comm //
216 (* Basic_1: removed theorems 49:
218 cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
219 ldrop_clear ldrop_clear_O ldrop_clear_S
220 clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
221 clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
222 getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
223 getl_clear_bind getl_clear_conf getl_dec getl_ldrop getl_ldrop_conf_lt
224 getl_ldrop_conf_ge getl_conf_ge_ldrop getl_ldrop_conf_rev
225 ldrop_getl_trans_lt ldrop_getl_trans_le ldrop_getl_trans_ge
226 getl_ldrop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
227 getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
228 getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono