1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "ground_2/lib/lstar.ma".
16 include "basic_2/notation/relations/rdrop_4.ma".
17 include "basic_2/grammar/lenv_length.ma".
18 include "basic_2/grammar/cl_restricted_weight.ma".
19 include "basic_2/relocation/lift.ma".
21 (* LOCAL ENVIRONMENT SLICING ************************************************)
23 (* Basic_1: includes: drop_skip_bind *)
24 inductive ldrop: relation4 nat nat lenv lenv ≝
25 | ldrop_atom : ∀d. ldrop d 0 (⋆) (⋆)
26 | ldrop_pair : ∀L,I,V. ldrop 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
27 | ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. ⓑ{I} V) L2
28 | ldrop_skip : ∀L1,L2,I,V1,V2,d,e.
29 ldrop d e L1 L2 → ⇧[d,e] V2 ≡ V1 →
30 ldrop (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
33 interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
35 definition l_liftable: predicate (lenv → relation term) ≝
36 λR. ∀K,T1,T2. R K T1 T2 → ∀L,d,e. ⇩[d, e] L ≡ K →
37 ∀U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → R L U1 U2.
39 definition l_deliftable_sn: predicate (lenv → relation term) ≝
40 λR. ∀L,U1,U2. R L U1 U2 → ∀K,d,e. ⇩[d, e] L ≡ K →
41 ∀T1. ⇧[d, e] T1 ≡ U1 →
42 ∃∃T2. ⇧[d, e] T2 ≡ U2 & R K T1 T2.
44 definition dropable_sn: predicate (relation lenv) ≝
45 λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀L2. R L1 L2 →
46 ∃∃K2. R K1 K2 & ⇩[d, e] L2 ≡ K2.
48 definition dedropable_sn: predicate (relation lenv) ≝
49 λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. R K1 K2 →
50 ∃∃L2. R L1 L2 & ⇩[d, e] L2 ≡ K2.
52 definition dropable_dx: predicate (relation lenv) ≝
53 λR. ∀L1,L2. R L1 L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
54 ∃∃K1. ⇩[0, e] L1 ≡ K1 & R K1 K2.
56 (* Basic inversion lemmas ***************************************************)
58 fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → L1 = ⋆ →
60 #d #e #L1 #L2 * -d -e -L1 -L2
62 | #L #I #V #H destruct
63 | #L1 #L2 #I #V #e #_ #H destruct
64 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
68 (* Basic_1: was: drop_gen_sort *)
69 lemma ldrop_inv_atom1: ∀d,e,L2. ⇩[d, e] ⋆ ≡ L2 → L2 = ⋆ ∧ e = 0.
70 /2 width=4 by ldrop_inv_atom1_aux/ qed-.
72 fact ldrop_inv_O1_pair1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 →
73 ∀K,I,V. L1 = K. ⓑ{I} V →
74 (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
75 (0 < e ∧ ⇩[d, e - 1] K ≡ L2).
76 #d #e #L1 #L2 * -d -e -L1 -L2
77 [ #d #_ #K #I #V #H destruct
78 | #L #I #V #_ #K #J #W #HX destruct /3 width=1/
79 | #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct /3 width=1/
80 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
84 lemma ldrop_inv_O1_pair1: ∀e,K,I,V,L2. ⇩[0, e] K. ⓑ{I} V ≡ L2 →
85 (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
86 (0 < e ∧ ⇩[0, e - 1] K ≡ L2).
87 /2 width=3 by ldrop_inv_O1_pair1_aux/ qed-.
89 lemma ldrop_inv_pair1: ∀K,I,V,L2. ⇩[0, 0] K. ⓑ{I} V ≡ L2 → L2 = K. ⓑ{I} V.
91 elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct
92 elim (lt_refl_false … H)
95 (* Basic_1: was: drop_gen_drop *)
96 lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
97 ⇩[0, e] K. ⓑ{I} V ≡ L2 → 0 < e → ⇩[0, e - 1] K ≡ L2.
98 #e #K #I #V #L2 #H #He
99 elim (ldrop_inv_O1_pair1 … H) -H * // #H destruct
100 elim (lt_refl_false … He)
103 fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
104 ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
105 ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
106 ⇧[d - 1, e] V2 ≡ V1 &
108 #d #e #L1 #L2 * -d -e -L1 -L2
109 [ #d #_ #I #K #V #H destruct
110 | #L #I #V #H elim (lt_refl_false … H)
111 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
112 | #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct /2 width=5/
116 (* Basic_1: was: drop_gen_skip_l *)
117 lemma ldrop_inv_skip1: ∀d,e,I,K1,V1,L2. ⇩[d, e] K1. ⓑ{I} V1 ≡ L2 → 0 < d →
118 ∃∃K2,V2. ⇩[d - 1, e] K1 ≡ K2 &
119 ⇧[d - 1, e] V2 ≡ V1 &
121 /2 width=3 by ldrop_inv_skip1_aux/ qed-.
123 lemma ldrop_inv_O1_pair2: ∀I,K,V,e,L1. ⇩[0, e] L1 ≡ K. ⓑ{I} V →
124 (e = 0 ∧ L1 = K. ⓑ{I} V) ∨
125 ∃∃I1,K1,V1. ⇩[0, e - 1] K1 ≡ K. ⓑ{I} V & L1 = K1.ⓑ{I1}V1 & 0 < e.
127 [ #H elim (ldrop_inv_atom1 … H) -H #H destruct
129 elim (ldrop_inv_O1_pair1 … H) -H *
130 [ #H1 #H2 destruct /3 width=1/
136 fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → 0 < d →
137 ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
138 ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 &
139 ⇧[d - 1, e] V2 ≡ V1 &
141 #d #e #L1 #L2 * -d -e -L1 -L2
142 [ #d #_ #I #K #V #H destruct
143 | #L #I #V #H elim (lt_refl_false … H)
144 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
145 | #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct /2 width=5/
149 (* Basic_1: was: drop_gen_skip_r *)
150 lemma ldrop_inv_skip2: ∀d,e,I,L1,K2,V2. ⇩[d, e] L1 ≡ K2. ⓑ{I} V2 → 0 < d →
151 ∃∃K1,V1. ⇩[d - 1, e] K1 ≡ K2 & ⇧[d - 1, e] V2 ≡ V1 &
153 /2 width=3 by ldrop_inv_skip2_aux/ qed-.
155 (* Basic properties *********************************************************)
157 (* Basic_1: was by definition: drop_refl *)
158 lemma ldrop_refl: ∀L,d. ⇩[d, 0] L ≡ L.
160 #L #I #V #IHL #d @(nat_ind_plus … d) -d // /2 width=1/
163 lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
164 ⇩[0, e - 1] L1 ≡ L2 → 0 < e → ⇩[0, e] L1. ⓑ{I} V ≡ L2.
165 #L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
168 lemma ldrop_skip_lt: ∀L1,L2,I,V1,V2,d,e.
169 ⇩[d - 1, e] L1 ≡ L2 → ⇧[d - 1, e] V2 ≡ V1 → 0 < d →
170 ⇩[d, e] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2.
171 #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) // /2 width=1/
174 lemma ldrop_O1_le: ∀e,L. e ≤ |L| → ∃K. ⇩[0, e] L ≡ K.
175 #e @(nat_ind_plus … e) -e /2 width=2/
177 [ #H lapply (le_n_O_to_eq … H) -H >commutative_plus normalize #H destruct
178 | #L #I #V normalize #H
179 elim (IHe L) -IHe /2 width=1/ -H /3 width=2/
183 lemma ldrop_O1_lt: ∀L,e. e < |L| → ∃∃I,K,V. ⇩[0, e] L ≡ K.ⓑ{I}V.
185 [ #e #H elim (lt_zero_false … H)
186 | #L #I #V #IHL #e @(nat_ind_plus … e) -e /2 width=4/
188 elim (IHL e) -IHL /2 width=1/ -H /3 width=4/
192 lemma l_liftable_LTC: ∀R. l_liftable R → l_liftable (LTC … R).
193 #R #HR #K #T1 #T2 #H elim H -T2
195 | #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #HTU1 #U2 #HTU2
196 elim (lift_total T d e) /4 width=11 by step/ (**) (* auto too slow without trace *)
200 lemma l_deliftable_sn_LTC: ∀R. l_deliftable_sn R → l_deliftable_sn (LTC … R).
201 #R #HR #L #U1 #U2 #H elim H -U2
202 [ #U2 #HU12 #K #d #e #HLK #T1 #HTU1
203 elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3/
204 | #U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #T1 #HTU1
205 elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
206 elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5/
210 lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
211 #R #HR #L1 #K1 #d #e #HLK1 #L2 #H elim H -L2
213 elim (HR … HLK1 … HL12) -HR -L1 /3 width=3/
214 | #L #L2 #_ #HL2 * #K #HK1 #HLK
215 elim (HR … HLK … HL2) -HR -L /3 width=3/
219 lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
220 #R #HR #L1 #K1 #d #e #HLK1 #K2 #H elim H -K2
222 elim (HR … HLK1 … HK12) -HR -K1 /3 width=3/
223 | #K #K2 #_ #HK2 * #L #HL1 #HLK
224 elim (HR … HLK … HK2) -HR -K /3 width=3/
228 lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
229 #R #HR #L1 #L2 #H elim H -L2
230 [ #L2 #HL12 #K2 #e #HLK2
231 elim (HR … HL12 … HLK2) -HR -L2 /3 width=3/
232 | #L #L2 #_ #HL2 #IHL1 #K2 #e #HLK2
233 elim (HR … HL2 … HLK2) -HR -L2 #K #HLK #HK2
234 elim (IHL1 … HLK) -L /3 width=5/
238 lemma l_deliftable_sn_llstar: ∀R. l_deliftable_sn R →
239 ∀l. l_deliftable_sn (llstar … R l).
240 #R #HR #l #L #U1 #U2 #H @(lstar_ind_r … l U2 H) -l -U2
242 | #l #U #U2 #_ #HU2 #IHU1 #K #d #e #HLK #T1 #HTU1
243 elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
244 elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5/
248 (* Basic forvard lemmas *****************************************************)
250 (* Basic_1: was: drop_S *)
251 lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ⇩[O, e] L1 ≡ K2. ⓑ{I2} V2 →
254 [ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H * #H destruct
255 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
256 elim (ldrop_inv_O1_pair1 … H) -H * #He #H
257 [ -IHL1 destruct /2 width=1/
258 | @ldrop_ldrop >(plus_minus_m_m e 1) // /2 width=3/
263 lemma ldrop_fwd_length: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L1| = |L2| + e.
264 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize /2 width=1/
267 lemma ldrop_fwd_length_minus2: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L2| = |L1| - e.
268 #L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1/
271 lemma ldrop_fwd_length_minus4: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → e = |L1| - |L2|.
272 #L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
275 lemma ldrop_fwd_length_le2: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → e ≤ |L1|.
276 #L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
279 lemma ldrop_fwd_length_le4: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L2| ≤ |L1|.
280 #L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H //
283 lemma ldrop_fwd_length_lt2: ∀L1,I2,K2,V2,d,e.
284 ⇩[d, e] L1 ≡ K2. ⓑ{I2} V2 → e < |L1|.
285 #L1 #I2 #K2 #V2 #d #e #H
286 lapply (ldrop_fwd_length … H) normalize in ⊢ (%→?); -I2 -V2 //
289 lemma ldrop_fwd_length_lt4: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → 0 < e → |L2| < |L1|.
290 #L1 #L2 #d #e #H lapply (ldrop_fwd_length … H) -H /2 width=1/
293 lemma ldrop_fwd_length_eq: ∀L1,L2,K1,K2,d,e. ⇩[d, e] L1 ≡ K1 → ⇩[d, e] L2 ≡ K2 →
294 |L1| = |L2| → |K1| = |K2|.
295 #L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
296 lapply (ldrop_fwd_length … HLK1) -HLK1
297 lapply (ldrop_fwd_length … HLK2) -HLK2
298 /2 width=2 by injective_plus_r/ (**) (* full auto fails *)
301 lemma ldrop_fwd_lw: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
302 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize
304 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
305 >(lift_fwd_tw … HV21) -HV21 /2 width=1/
309 lemma ldrop_fwd_lw_lt: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → 0 < e → ♯{L2} < ♯{L1}.
310 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e //
311 [ #L #I #V #H elim (lt_refl_false … H)
312 | #L1 #L2 #I #V #e #HL12 #_ #_
313 lapply (ldrop_fwd_lw … HL12) -HL12 #HL12
314 @(le_to_lt_to_lt … HL12) -HL12 //
315 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I
316 >(lift_fwd_tw … HV21) -V2 /3 by lt_minus_to_plus/ (**) (* auto too slow without trace *)
320 lemma ldrop_fwd_rfw: ∀I,L,K,V,i. ⇩[O, i] L ≡ K.ⓑ{I}V → ♯{K, V} < ♯{L, #i}.
321 #I #L #K #V #i #HLK lapply (ldrop_fwd_lw … HLK) -HLK
322 normalize in ⊢ (%→?%%); /2 width=1 by le_S_S/
325 (* Advanced inversion lemmas ************************************************)
327 fact ldrop_inv_O2_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → e = 0 → L1 = L2.
328 #d #e #L1 #L2 #H elim H -d -e -L1 -L2
331 | #L1 #L2 #I #V #e #_ #_ >commutative_plus normalize #H destruct
332 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12 #H
333 >(IHL12 H) -L1 >(lift_inv_O2_aux … HV21 … H) -V2 -d -e //
337 (* Basic_1: was: drop_gen_refl *)
338 lemma ldrop_inv_O2: ∀L1,L2,d. ⇩[d, 0] L1 ≡ L2 → L1 = L2.
339 /2 width=4 by ldrop_inv_O2_aux/ qed-.
341 lemma ldrop_inv_length_eq: ∀L1,L2,d,e. ⇩[d, e] L1 ≡ L2 → |L1| = |L2| → e = 0.
342 #L1 #L2 #d #e #H #HL12 lapply (ldrop_fwd_length_minus4 … H) //
345 lemma ldrop_inv_refl: ∀L,d,e. ⇩[d, e] L ≡ L → e = 0.
346 /2 width=5 by ldrop_inv_length_eq/ qed-.
348 (* Basic_1: removed theorems 50:
349 drop_ctail drop_skip_flat
350 cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
351 drop_clear drop_clear_O drop_clear_S
352 clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
353 clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
354 getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
355 getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
356 getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
357 drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
358 getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
359 getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
360 getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono