(* *)
(**************************************************************************)
-include "basic_2/computation/cpds_cpds.ma".
-include "basic_2/dynamic/lsubsv_cpds.ma".
-include "basic_2/dynamic/lsubsv_cpcs.ma".
+include "basic_2/dynamic/lsubsv_scpds.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
(* Properties concerning stratified native validity *************************)
-fact snv_lsubsv_aux: ∀h,g,L0,T0.
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
- (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_lsubsv h g L1 T1) →
- ∀L1,T1. L0 = L1 → T0 = T1 → IH_snv_lsubsv h g L1 T1.
-#h #g #L0 #T0 #IH4 #IH3 #IH2 #IH1 #L2 * * [||||*] //
-[ #i #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
- elim (snv_inv_lref … H) -H #I2 #K2 #W2 #HLK2 #HW2
- elim (lsubsv_ldrop_O1_trans … HL12 … HLK2) -HL12 #X #H #HLK1
+lemma lsubsv_snv_trans: ∀h,o,G,L2,T. ⦃G, L2⦄ ⊢ T ¡[h, o] →
+ ∀L1. G ⊢ L1 ⫃¡[h, o] L2 → ⦃G, L1⦄ ⊢ T ¡[h, o].
+#h #o #G #L2 #T #H elim H -G -L2 -T //
+[ #I #G #L2 #K2 #V #i #HLK2 #_ #IHV #L1 #HL12
+ elim (lsubsv_drop_O1_trans … HL12 … HLK2) -L2 #X #H #HLK1
elim (lsubsv_inv_pair2 … H) -H * #K1
- [ #HK12 #H destruct
- /5 width=8 by snv_lref, fsupp_ygt, fsupp_lref/ (**) (* auto too slow without trace *)
- | #V #W1 #V2 #l #HV #_ #_ #_ #_ #_ #H destruct /2 width=5/
+ [ #HK12 #H destruct /3 width=5 by snv_lref/
+ | #W #d #HVW #_ #_ #_ #_ #H1 #H2 destruct -IHV
+ /3 width=6 by shnv_inv_snv, snv_lref/
]
-| #p #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2 -IH1
- elim (snv_inv_gref … H)
-| #a #I #V #T #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
- elim (snv_inv_bind … H) -H /4 width=4/
-| #V #T #HL0 #HT0 #H #L1 #HL12 destruct
- elim (snv_inv_appl … H) -H #a #W #W0 #U #l #HV #HT #HVW #HW0 #HTU
- lapply (lsubsv_cprs_trans … HL12 … HW0) -HW0 #HW0
- elim (lsubsv_ssta_trans … HVW … HL12) -HVW #W1 #HVW1 #HW1
- lapply (cpcs_cprs_strap1 … HW1 … HW0) -W #HW10
- elim (lsubsv_cpds_aux … IH4 IH3 IH2 IH1 … HL12 … HTU) -IH4 -IH3 -IH2 -HTU // /2 width=1/ #X #HTU #H
- elim (cprs_inv_abst1 … H) -H #W #U2 #HW0 #HU2 #H destruct
- lapply (cpcs_cprs_strap1 … HW10 … HW0) -W0 #H
- elim (cpcs_inv_cprs … H) -H #W0 #HW10 #HW0
- lapply (cpds_cprs_trans … (ⓛ{a}W0.U2) HTU ?) [ /2 width=1/ ] -HTU -HW0
- /4 width=8 by snv_appl, fsupp_ygt/ (**) (* auto too slow without trace *)
-| #W #T #HL0 #HT0 #H #L1 #HL12 destruct -IH4 -IH3 -IH2
- elim (snv_inv_cast … H) -H #U #l #HW #HT #HTU #HUW
- lapply (lsubsv_cpcs_trans … HL12 … HUW) -HUW #HUW
- elim (lsubsv_ssta_trans … HTU … HL12) -HTU #U0 #HTU0 #HU0
- lapply (cpcs_trans … HU0 … HUW) -U /4 width=4 by snv_cast, fsupp_ygt/ (**) (* auto too slow without trace *)
+| #a #I #G #L2 #V #T #_ #_ #IHV #IHT #L1 #HL12 destruct
+ /4 width=1 by snv_bind, lsubsv_pair/
+| #a #G #L2 #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 #IHV #IHT #L1 #HL12
+ elim (lsubsv_scpds_trans … HVW0 … HL12) -HVW0 #V0 #HV0 #HWV0
+ elim (lsubsv_scpds_trans … HTU0 … HL12) -HTU0 #X #HT0 #H
+ elim (cprs_inv_abst1 … H) -H #W #T0 #HW0 #_ #H destruct
+ elim (cprs_conf … HWV0 … HW0) -W0
+ /4 width=10 by snv_appl, scpds_cprs_trans, cprs_bind/
+| #G #L2 #U #T #U0 #_ #_ #HU0 #HTU0 #IHU #IHT #L1 #HL12
+ elim (lsubsv_scpds_trans … HTU0 … HL12) -HTU0 #X0 #HTX0 #H1
+ elim (lsubsv_scpds_trans … HU0 … HL12) -HU0 #X #HUX #H2
+ elim (cprs_conf … H1 … H2) -U0 /3 width=5 by snv_cast, scpds_cprs_trans/
]
qed-.