(* *)
(**************************************************************************)
-include "basic_2/computation/dxprs_lift.ma".
+include "basic_2/computation/cpds_lift.ma".
include "basic_2/equivalence/cpcs_cpcs.ma".
include "basic_2/dynamic/snv.ma".
(* Relocation properties ****************************************************)
-lemma snv_lift: ∀h,g,K,T. ⦃h, K⦄ ⊢ T ¡[g] → ∀L,d,e. ⇩[d, e] L ≡ K →
- ∀U. ⇧[d, e] T ≡ U → ⦃h, L⦄ ⊢ U ¡[g].
-#h #g #K #T #H elim H -K -T
-[ #K #k #L #d #e #_ #X #H
+lemma snv_lift: ∀h,g,G,K,T. ⦃G, K⦄ ⊢ T ¡[h, g] → ∀L,s,d,e. ⇩[s, d, e] L ≡ K →
+ ∀U. ⇧[d, e] T ≡ U → ⦃G, L⦄ ⊢ U ¡[h, g].
+#h #g #G #K #T #H elim H -G -K -T
+[ #G #K #k #L #s #d #e #_ #X #H
>(lift_inv_sort1 … H) -X -K -d -e //
-| #I #K #K0 #V #i #HK0 #_ #IHV #L #d #e #HLK #X #H
+| #I #G #K #K0 #V #i #HK0 #_ #IHV #L #s #d #e #HLK #X #H
elim (lift_inv_lref1 … H) * #Hid #H destruct
- [ elim (ldrop_trans_le … HLK … HK0 ?) -K /2 width=2/ #X #HL0 #H
- elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #L0 #W #HLK0 #HVW #H destruct
- /3 width=8/
- | lapply (ldrop_trans_ge … HLK … HK0 ?) -K // -Hid /3 width=8/
+ [ elim (ldrop_trans_le … HLK … HK0) -K /2 width=2 by lt_to_le/ #X #HL0 #H
+ elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #L0 #W #HLK0 #HVW #H destruct
+ /3 width=9 by snv_lref/
+ | lapply (ldrop_trans_ge … HLK … HK0 ?) -K
+ /3 width=9 by snv_lref, ldrop_inv_gen/
]
-| #a #I #K #V #T #_ #_ #IHV #IHT #L #d #e #HLK #X #H
+| #a #I #G #K #V #T #_ #_ #IHV #IHT #L #s #d #e #HLK #X #H
elim (lift_inv_bind1 … H) -H #W #U #HVW #HTU #H destruct
- /4 width=4/
-| #a #K #V #V0 #V1 #T #T1 #l #_ #_ #HV0 #HV01 #HT1 #IHV #IHT #L #d #e #HLK #X #H
+ /4 width=5 by snv_bind, ldrop_skip/
+| #a #G #K #V #V0 #V1 #T #T1 #l #_ #_ #Hl #HV0 #HV01 #HT1 #IHV #IHT #L #s #d #e #HLK #X #H
elim (lift_inv_flat1 … H) -H #W #U #HVW #HTU #H destruct
elim (lift_total V0 d e) #W0 #HVW0
elim (lift_total V1 d e) #W1 #HVW1
elim (lift_total T1 (d+1) e) #U1 #HTU1
@(snv_appl … a … W0 … W1 … U1 l)
- [ /2 width=4/ | /2 width=4/ | /2 width=9/ | /2 width=9/ ]
- @(dxprs_lift … HLK … HTU … HT1) /2 width=1/
-| #K #V0 #T #V #l #_ #_ #HTV #HV0 #IHV0 #IHT #L #d #e #HLK #X #H
+ [1,2,3,4,5: /2 width=10 by cprs_lift, sta_lift, da_lift/ ]
+ @(cpds_lift … HT1 … HLK … HTU) /2 width=1 by lift_bind/ (**) (* full auto raises typecjhecker failure *)
+| #G #K #V0 #T #V #l #_ #_ #Hl #HTV #HV0 #IHV0 #IHT #L #s #d #e #HLK #X #H
elim (lift_inv_flat1 … H) -H #W0 #U #HVW0 #HTU #H destruct
- elim (lift_total V d e) #W #HVW
- @(snv_cast … W l) [ /2 width=4/ | /2 width=4/ | /2 width=9/ | /2 width=9/ ]
+ elim (lift_total V d e)
+ /3 width=12 by snv_cast, cpcs_lift, sta_lift, da_lift/
]
qed.
-lemma snv_inv_lift: ∀h,g,L,U. ⦃h, L⦄ ⊢ U ¡[g] → ∀K,d,e. ⇩[d, e] L ≡ K →
- ∀T. ⇧[d, e] T ≡ U → ⦃h, K⦄ ⊢ T ¡[g].
-#h #g #L #U #H elim H -L -U
-[ #L #k #K #d #e #_ #X #H
+lemma snv_inv_lift: ∀h,g,G,L,U. ⦃G, L⦄ ⊢ U ¡[h, g] → ∀K,s,d,e. ⇩[s, d, e] L ≡ K →
+ ∀T. ⇧[d, e] T ≡ U → ⦃G, K⦄ ⊢ T ¡[h, g].
+#h #g #G #L #U #H elim H -G -L -U
+[ #G #L #k #K #s #d #e #_ #X #H
>(lift_inv_sort2 … H) -X -L -d -e //
-| #I #L #L0 #W #i #HL0 #_ #IHW #K #d #e #HLK #X #H
+| #I #G #L #L0 #W #i #HL0 #_ #IHW #K #s #d #e #HLK #X #H
elim (lift_inv_lref2 … H) * #Hid #H destruct
- [ elim (ldrop_conf_le … HLK … HL0 ?) -L /2 width=2/ #X #HK0 #H
- elim (ldrop_inv_skip1 … H ?) -H /2 width=1/ -Hid #K0 #V #HLK0 #HVW #H destruct
- /3 width=8/
- | lapply (ldrop_conf_ge … HLK … HL0 ?) -L // -Hid /3 width=8/
+ [ elim (ldrop_conf_le … HLK … HL0) -L /2 width=2 by lt_to_le/ #X #HK0 #H
+ elim (ldrop_inv_skip1 … H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K0 #V #HLK0 #HVW #H destruct
+ /3 width=12 by snv_lref/
+ | lapply (ldrop_conf_ge … HLK … HL0 ?) -L /3 width=9 by snv_lref/
]
-| #a #I #L #W #U #_ #_ #IHW #IHU #K #d #e #HLK #X #H
- elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct /4 width=4/
-| #a #L #W #W0 #W1 #U #U1 #l #_ #_ #HW0 #HW01 #HU1 #IHW #IHU #K #d #e #HLK #X #H
+| #a #I #G #L #W #U #_ #_ #IHW #IHU #K #s #d #e #HLK #X #H
+ elim (lift_inv_bind2 … H) -H #V #T #HVW #HTU #H destruct
+ /4 width=5 by snv_bind, ldrop_skip/
+| #a #G #L #W #W0 #W1 #U #U1 #l #_ #_ #Hl #HW0 #HW01 #HU1 #IHW #IHU #K #s #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #V #T #HVW #HTU #H destruct
- elim (ssta_inv_lift1 … HW0 … HLK … HVW) -HW0 #V0 #HV0 #HVW0
- elim (cprs_inv_lift1 … HLK … HVW0 … HW01) -W0 #V1 #HVW1 #HV01
- elim (dxprs_inv_lift1 … HLK … HTU … HU1) -HU1 #X #H #HTU
+ lapply (da_inv_lift … Hl … HLK … HVW) -Hl #Hl
+ elim (sta_inv_lift1 … HW0 … HLK … HVW) -HW0 #V0 #HVW0 #HV0
+ elim (cprs_inv_lift1 … HW01 … HLK … HVW0) -W0 #V1 #HVW1 #HV01
+ elim (cpds_inv_lift1 … HU1 … HLK … HTU) -HU1 #X #H #HTU
elim (lift_inv_bind2 … H) -H #Y #T1 #HY #HTU1 #H destruct
- lapply (lift_inj … HY … HVW1) -HY #H destruct /3 width=8/
-| #L #W0 #U #W #l #_ #_ #HUW #HW0 #IHW0 #IHU #K #d #e #HLK #X #H
+ lapply (lift_inj … HY … HVW1) -HY #H destruct
+ /3 width=8 by snv_appl/
+| #G #L #W0 #U #W #l #_ #_ #Hl #HUW #HW0 #IHW0 #IHU #K #s #d #e #HLK #X #H
elim (lift_inv_flat2 … H) -H #V0 #T #HVW0 #HTU #H destruct
- elim (ssta_inv_lift1 … HUW … HLK … HTU) -HUW #V #HTV #HVW
- lapply (cpcs_inv_lift … HLK … HVW … HVW0 ?) // -W /3 width=4/
+ lapply (da_inv_lift … Hl … HLK … HTU) -Hl #Hl
+ elim (sta_inv_lift1 … HUW … HLK … HTU) -HUW #V #HVW #HTV
+ lapply (cpcs_inv_lift G … HLK … HVW … HVW0 ?) // -W
+ /3 width=8 by snv_cast/
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma snv_fqu_conf: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1⦄ ⊢ T1 ¡[h, g] → ⦃G2, L2⦄ ⊢ T2 ¡[h, g].
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I1 #G1 #L1 #V1 #H
+ elim (snv_inv_lref … H) -H #I2 #L2 #V2 #H #HV2
+ lapply (ldrop_inv_O2 … H) -H #H destruct //
+|2: *
+|5,6: /3 width=8 by snv_inv_lift/
+]
+[1,3: #a #I #G1 #L1 #V1 #T1 #H elim (snv_inv_bind … H) -H //
+|2,4: * #G1 #L1 #V1 #T1 #H
+ [1,3: elim (snv_inv_appl … H) -H //
+ |2,4: elim (snv_inv_cast … H) -H //
+ ]
]
qed-.
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