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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 "ground/xoa/ex_4_2.ma".
16 include "basic_2A/substitution/drop_drop.ma".
17 include "basic_2A/multiple/llpx_sn_lreq.ma".
19 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
21 (* Advanced forward lemmas **************************************************)
23 lemma llpx_sn_fwd_lref_dx: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 →
24 ∀I,K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
26 ∃∃K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 &
28 #R #L1 #L2 #l #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
29 [ #_ #H elim (lt_refl_false i)
30 lapply (drop_fwd_length_lt2 … HLK2) -HLK2
31 /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
32 | /2 width=1 by or_introl/
33 | #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hli
34 lapply (drop_mono … HLK22 … HLK2) -L2 #H destruct
35 /3 width=5 by ex4_2_intro, or_intror/
39 lemma llpx_sn_fwd_lref_sn: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 →
40 ∀I,K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 →
42 ∃∃K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 &
44 #R #L1 #L2 #l #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
45 [ #H #_ elim (lt_refl_false i)
46 lapply (drop_fwd_length_lt2 … HLK1) -HLK1
47 /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
48 | /2 width=1 by or_introl/
49 | #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hli
50 lapply (drop_mono … HLK11 … HLK1) -L1 #H destruct
51 /3 width=5 by ex4_2_intro, or_intror/
55 (* Advanced inversion lemmas ************************************************)
57 lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → l ≤ i →
58 ∀I,K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
59 ∃∃K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 &
60 llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
61 #R #L1 #L2 #l #i #H #Hli #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
62 [ #H elim (ylt_yle_false … H Hli)
63 | * /2 width=5 by ex3_2_intro/
67 lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → l ≤ i →
68 ∀I,K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 →
69 ∃∃K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 &
70 llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
71 #R #L1 #L2 #l #i #H #Hli #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
72 [ #H elim (ylt_yle_false … H Hli)
73 | * /2 width=5 by ex3_2_intro/
77 lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,l,i. llpx_sn R l (#i) L1 L2 → l ≤ i →
79 ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
80 ∧∧ I1 = I2 & llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
81 #R #L1 #L2 #l #i #HL12 #Hli #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
82 elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -l
83 #J #Y #HY lapply (drop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/
86 fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,l0. llpx_sn R l0 T L1 L2 → ∀l. l0 = l + 1 →
87 ∀K1,K2,I,V1,V2. ⬇[l] L1 ≡ K1.ⓑ{I}V1 → ⬇[l] L2 ≡ K2.ⓑ{I}V2 →
88 llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R l T L1 L2.
89 #R #L1 #L2 #T #l0 #H elim H -L1 -L2 -T -l0
90 /2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/
91 [ #L1 #L2 #l0 #i #HL12 #Hil #l #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
92 elim (yle_split_eq i l) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hil
93 #H destruct /2 width=9 by llpx_sn_lref/
94 | #I #L1 #L2 #K11 #K22 #V1 #V2 #l0 #i #Hl0i #HLK11 #HLK22 #HK12 #HV12 #_ #l #H #K1 #K2 #J #W1 #W2 #_ #_ #_ #_ destruct
95 /3 width=9 by llpx_sn_lref, yle_pred_sn/
96 | #a #I #L1 #L2 #V #T #l0 #_ #_ #IHV #IHT #l #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
97 /4 width=9 by llpx_sn_bind, drop_drop/
98 | #I #L1 #L2 #V #T #l0 #_ #_ #IHV #IHT #l #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
99 /3 width=9 by llpx_sn_flat/
103 lemma llpx_sn_inv_S: ∀R,L1,L2,T,l. llpx_sn R (l + 1) T L1 L2 →
104 ∀K1,K2,I,V1,V2. ⬇[l] L1 ≡ K1.ⓑ{I}V1 → ⬇[l] L2 ≡ K2.ⓑ{I}V2 →
105 llpx_sn R 0 V1 K1 K2 → R K1 V1 V2 → llpx_sn R l T L1 L2.
106 /2 width=9 by llpx_sn_inv_S_aux/ qed-.
108 lemma llpx_sn_inv_bind_O: ∀R. (∀L. reflexive … (R L)) →
109 ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
110 llpx_sn R 0 V L1 L2 ∧ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
111 #R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind … H) -H
112 /3 width=9 by drop_pair, conj, llpx_sn_inv_S/
115 (* More advanced forward lemmas *********************************************)
117 lemma llpx_sn_fwd_bind_O_dx: ∀R. (∀L. reflexive … (R L)) →
118 ∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
119 llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
120 #R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O … H) -H //
123 (* Advanced properties ******************************************************)
125 lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) →
126 ∀J,W1,W2. llpx_sn R 0 W1 L1 L2 → R L1 W1 W2 → llpx_sn R 0 T (L1.ⓑ{J}W1) (L2.ⓑ{J}W2).
127 /3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-.
129 lemma llpx_sn_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
130 ∀T,L1,L2,l. Decidable (llpx_sn R l T L1 L2).
131 #R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
133 [ #k #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_sort/
134 | #i #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|))
135 [ #HL12 #l elim (ylt_split i l) /3 width=1 by llpx_sn_skip, or_introl/
136 #Hli elim (lt_or_ge i (|L1|)) #HiL1
137 elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/
138 elim (drop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2
139 elim (drop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1
140 elim (eq_bind2_dec I2 I1)
141 [ #H2 elim (HR K1 V1 V2) -HR
142 [ #H3 elim (IH K1 V1 … K2 0) destruct
143 /3 width=9 by llpx_sn_lref, drop_fwd_rfw, or_introl/
147 #H elim (llpx_sn_fwd_lref … H) -H [1,3,4,6,7,9: * ]
148 [1,3,5: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
149 |7,8,9: /2 width=4 by ylt_yle_false/
151 #Z #Y1 #Y2 #X1 #X2 #HLY1 #HLY2 #HY12 #HX12
152 lapply (drop_mono … HLY1 … HLK1) -HLY1 -HLK1
153 lapply (drop_mono … HLY2 … HLK2) -HLY2 -HLK2
154 #H #H0 destruct /2 width=1 by/
156 | #p #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/
157 | #a #I #V #T #Hx #L2 #l destruct
158 elim (IH L1 V … L2 l) /2 width=1 by/
159 elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (↑l)) -IH /3 width=1 by or_introl, llpx_sn_bind/
161 #H elim (llpx_sn_inv_bind … H) -H /2 width=1 by/
162 | #I #V #T #Hx #L2 #l destruct
163 elim (IH L1 V … L2 l) /2 width=1 by/
164 elim (IH L1 T … L2 l) -IH /3 width=1 by or_introl, llpx_sn_flat/
166 #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/
168 -x /4 width=4 by llpx_sn_fwd_length, or_intror/
171 (* Properties on relocation *************************************************)
173 lemma llpx_sn_lift_le: ∀R. d_liftable R →
174 ∀K1,K2,T,l0. llpx_sn R l0 T K1 K2 →
175 ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
176 ∀U. ⬆[l, m] T ≡ U → l0 ≤ l → llpx_sn R l0 U L1 L2.
177 #R #HR #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0
178 [ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
179 lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
180 /2 width=1 by llpx_sn_sort/
181 | #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
183 [ lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
184 /2 width=1 by llpx_sn_skip/
185 | elim (ylt_yle_false … Hil0) -L1 -L2 -K1 -K2 -m -Hil0
186 /3 width=3 by yle_trans, yle_inj/
188 | #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
189 * #Hli #H destruct [ -HK12 | -IHK12 ]
190 [ elim (drop_trans_lt … HLK1 … HK11) // -K1
191 elim (drop_trans_lt … HLK2 … HK22) // -Hli -K2
192 /3 width=18 by llpx_sn_lref/
193 | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1
194 lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hli -Hl0 -K2
195 /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/
197 | #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
199 lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
200 [ /3 width=7 by llpx_sn_free, drop_fwd_be/
201 | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1
202 lapply (drop_fwd_length … HLK2) -HLK2 #HLK2
203 @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
205 | #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
206 lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l -m
207 /2 width=1 by llpx_sn_gref/
208 | #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H
209 #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/
210 | #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H
211 #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
215 lemma llpx_sn_lift_ge: ∀R,K1,K2,T,l0. llpx_sn R l0 T K1 K2 →
216 ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
217 ∀U. ⬆[l, m] T ≡ U → l ≤ l0 → llpx_sn R (l0+m) U L1 L2.
218 #R #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0
219 [ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 … H) -X
220 lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
221 /2 width=1 by llpx_sn_sort/
222 | #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 … H) -H
224 lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2
225 [ /3 width=3 by llpx_sn_skip, ylt_plus_dx1_trans/
226 | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/
228 | #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
230 [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -m -Hil0
231 /3 width=3 by ylt_yle_trans, ylt_inj/
232 | lapply (drop_trans_ge_comm … HLK1 … HK11 ?) // -K1
233 lapply (drop_trans_ge_comm … HLK2 … HK22 ?) // -Hil -Hl0 -K2
234 /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/
236 | #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 … H) -H
238 lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -HK12
239 [ /3 width=7 by llpx_sn_free, drop_fwd_be/
240 | lapply (drop_fwd_length … HLK1) -HLK1 #HLK1
241 lapply (drop_fwd_length … HLK2) -HLK2 #HLK2
242 @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
244 | #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 … H) -X
245 lapply (drop_fwd_length_eq2 … HLK1 HLK2 HK12) -K1 -K2 -l
246 /2 width=1 by llpx_sn_gref/
247 | #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 … H) -H
248 #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, drop_skip, yle_succ/
249 | #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat1 … H) -H
250 #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
254 (* Inversion lemmas on relocation *******************************************)
256 lemma llpx_sn_inv_lift_le: ∀R. d_deliftable_sn R →
257 ∀L1,L2,U,l0. llpx_sn R l0 U L1 L2 →
258 ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
259 ∀T. ⬆[l, m] T ≡ U → l0 ≤ l → llpx_sn R l0 T K1 K2.
260 #R #HR #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
261 [ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
262 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m
263 /2 width=1 by llpx_sn_sort/
264 | #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 … H) -H
266 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
267 [ /2 width=1 by llpx_sn_skip/
268 | /3 width=3 by llpx_sn_skip, yle_ylt_trans/
270 | #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H
271 * #Hil #H destruct [ -HK12 | -IHK12 ]
272 [ elim (drop_conf_lt … HLK1 … HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1
273 elim (drop_conf_lt … HLK2 … HLK22) // -Hil -L2 #L2 #V2 #HKL2 #HKL22 #HVW2
274 elim (HR … HW12 … HKL11 … HVW1) -HR #V0 #HV0 #HV12
275 lapply (lift_inj … HV0 … HVW2) -HV0 -HVW2 #H destruct
276 /3 width=10 by llpx_sn_lref/
277 | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1
278 lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0
279 elim (le_inv_plus_l … Hil) -Hil /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *)
281 | #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 … H) -H
283 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12)
284 [ lapply (drop_fwd_length_le4 … HLK1) -HLK1
285 lapply (drop_fwd_length_le4 … HLK2) -HLK2
286 #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
287 | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
288 lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
289 /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
291 | #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
292 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l -m
293 /2 width=1 by llpx_sn_gref/
294 | #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind2 … H) -H
295 #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_skip, yle_succ/
296 | #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_flat2 … H) -H
297 #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
301 lemma llpx_sn_inv_lift_be: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 →
302 ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
303 ∀T. ⬆[l, m] T ≡ U → l ≤ l0 → l0 ≤ yinj l + m → llpx_sn R l T K1 K2.
304 #R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
305 [ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 … H) -X
306 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m
307 /2 width=1 by llpx_sn_sort/
308 | #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H
310 [ lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
311 -Hil0 /3 width=1 by llpx_sn_skip, ylt_inj/
312 | elim (ylt_yle_false … Hil0) -L1 -L2 -Hl0 -Hil0
313 /3 width=3 by yle_trans, yle_inj/ (**) (* slow *)
315 | #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H
317 [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hl0m -Hil0
318 /3 width=3 by ylt_yle_trans, ylt_inj/
319 | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1
320 lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hil0 -Hl0 -Hl0m
321 elim (le_inv_plus_l … Hil) -Hil /3 width=9 by llpx_sn_lref, yle_inj/
323 | #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 … H) -H
325 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12)
326 [ lapply (drop_fwd_length_le4 … HLK1) -HLK1
327 lapply (drop_fwd_length_le4 … HLK2) -HLK2
328 #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
329 | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
330 lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
331 /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
333 | #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 … H) -X
334 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l0 -m
335 /2 width=1 by llpx_sn_gref/
336 | #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_bind2 … H) -H
337 >commutative_plus #V #T #HVW #HTU #H destruct
338 @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
339 @(IHU … HTU) -IHU -HTU /2 width=1 by drop_skip, yle_succ/
340 | #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_flat2 … H) -H
341 #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
345 lemma llpx_sn_inv_lift_ge: ∀R,L1,L2,U,l0. llpx_sn R l0 U L1 L2 →
346 ∀K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
347 ∀T. ⬆[l, m] T ≡ U → yinj l + m ≤ l0 → llpx_sn R (l0-m) T K1 K2.
348 #R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
349 [ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 … H) -X
350 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l
351 /2 width=1 by llpx_sn_sort/
352 | #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H
353 * #Hil #H destruct [ -Hil0 | -Hlml0 ]
354 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2
355 [ /4 width=3 by llpx_sn_skip, yle_plus1_to_minus_inj2, ylt_yle_trans, ylt_inj/
356 | elim (le_inv_plus_l … Hil) -Hil #_
357 /4 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx, yle_inj/
359 | #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H
361 [ elim (ylt_yle_false … Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hil0
362 /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/
363 | lapply (drop_conf_ge … HLK1 … HLK11 ?) // -L1
364 lapply (drop_conf_ge … HLK2 … HLK22 ?) // -L2 -Hlml0 -Hil
365 /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/
367 | #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 … H) -H
369 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12)
370 [ lapply (drop_fwd_length_le4 … HLK1) -HLK1
371 lapply (drop_fwd_length_le4 … HLK2) -HLK2
372 #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
373 | lapply (drop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
374 lapply (drop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
375 /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
377 | #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 … H) -X
378 lapply (drop_fwd_length_eq1 … HLK1 HLK2 HL12) -L1 -L2 -l
379 /2 width=1 by llpx_sn_gref/
380 | #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_bind2 … H) -H
381 #V #T #HVW #HTU #H destruct
382 @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
383 <yminus_succ1_inj /2 width=2 by yle_fwd_plus_sn2/
384 @(IHU … HTU) -IHU -HTU /2 width=1 by drop_skip, yle_succ/
385 | #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_flat2 … H) -H
386 #V #T #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
390 (* Advanced inversion lemmas on relocation **********************************)
392 lemma llpx_sn_inv_lift_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
393 ∀K1,K2,m. ⬇[m] L1 ≡ K1 → ⬇[m] L2 ≡ K2 →
394 ∀T. ⬆[0, m] T ≡ U → llpx_sn R 0 T K1 K2.
395 /2 width=11 by llpx_sn_inv_lift_be/ qed-.
397 lemma llpx_sn_drop_conf_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
398 ∀K1,m. ⬇[m] L1 ≡ K1 → ∀T. ⬆[0, m] T ≡ U →
399 ∃∃K2. ⬇[m] L2 ≡ K2 & llpx_sn R 0 T K1 K2.
400 #R #L1 #L2 #U #HU #K1 #m #HLK1 #T #HTU elim (llpx_sn_fwd_drop_sn … HU … HLK1)
401 /3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
404 lemma llpx_sn_drop_trans_O: ∀R,L1,L2,U. llpx_sn R 0 U L1 L2 →
405 ∀K2,m. ⬇[m] L2 ≡ K2 → ∀T. ⬆[0, m] T ≡ U →
406 ∃∃K1. ⬇[m] L1 ≡ K1 & llpx_sn R 0 T K1 K2.
407 #R #L1 #L2 #U #HU #K2 #m #HLK2 #T #HTU elim (llpx_sn_fwd_drop_dx … HU … HLK2)
408 /3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
411 (* Inversion lemmas on negated lazy pointwise extension *********************)
413 lemma nllpx_sn_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
414 ∀a,I,L1,L2,V,T,l. (llpx_sn R l (ⓑ{a,I}V.T) L1 L2 → ⊥) →
415 (llpx_sn R l V L1 L2 → ⊥) ∨ (llpx_sn R (↑l) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
416 #R #HR #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_dec … HR V L1 L2 l)
417 /4 width=1 by llpx_sn_bind, or_intror, or_introl/
420 lemma nllpx_sn_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
421 ∀I,L1,L2,V,T,l. (llpx_sn R l (ⓕ{I}V.T) L1 L2 → ⊥) →
422 (llpx_sn R l V L1 L2 → ⊥) ∨ (llpx_sn R l T L1 L2 → ⊥).
423 #R #HR #I #L1 #L2 #V #T #l #H elim (llpx_sn_dec … HR V L1 L2 l)
424 /4 width=1 by llpx_sn_flat, or_intror, or_introl/
427 lemma nllpx_sn_inv_bind_O: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
428 ∀a,I,L1,L2,V,T. (llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 → ⊥) →
429 (llpx_sn R 0 V L1 L2 → ⊥) ∨ (llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
430 #R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec … HR V L1 L2 0)
431 /4 width=1 by llpx_sn_bind_O, or_intror, or_introl/