(* Properties on nat-iterated stratified static type assignment for terms ***)
fact snv_lsstas_aux: ∀h,g,G0,L0,T0.
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_lsstas_cpr_lpr h g G1 L1 T1) →
- (∀G1,L1,T1. ⦃G0, L0, T0⦄ >[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsstas h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ ⊃≥[h, g] ⦃G1, L1, T1⦄ → IH_snv_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ ⊃≥[h, g] ⦃G1, L1, T1⦄ → IH_da_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ ⊃≥[h, g] ⦃G1, L1, T1⦄ → IH_lsstas_cpr_lpr h g G1 L1 T1) →
+ (∀G1,L1,T1. ⦃G0, L0, T0⦄ ⊃≥[h, g] ⦃G1, L1, T1⦄ → IH_snv_lsstas h g G1 L1 T1) →
∀G1,L1,T1. G0 = G1 → L0 = L1 → T0 = T1 → IH_snv_lsstas h g G1 L1 T1.
#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
[ #k #HG0 #HL0 #HT0 #_ #l1 #l2 #Hl21 #Hl1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
elim (lsstas_inv_lref1 … H2) -H2 * #K0 #Y0 #X0 [2,4: #l0 ] #HLK0 [1,2: #HYl0 ] #HYX0 #HX0
lapply (ldrop_mono … HLK0 … HLK1) -HLK0 #H destruct
[ lapply (le_plus_to_le_r … Hl21) -Hl21 #Hl21 ]
- lapply (fsupp_lref … G1 … HLK1) #H
- lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /4 width=8 by fsupp_ygt, snv_lift/
+ lapply (fqup_lref … G1 … HLK1) #H
+ lapply (ldrop_fwd_ldrop2 … HLK1) -HLK1 /4 width=8 by fqup_fpbr, snv_lift/
| #p #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
elim (snv_inv_gref … H1)
| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct -IH4 -IH3 -IH2
elim (snv_inv_bind … H1) -H1 #HV1 #HT1
lapply (da_inv_bind … Hl1) -Hl1 #Hl1
- elim (lsstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct /4 width=8 by fsupp_ygt, snv_bind/
+ elim (lsstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct /4 width=8 by fqup_fpbr, snv_bind/
| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct
elim (snv_inv_appl … H1) -H1 #a #W1 #W0 #T0 #l0 #HV1 #HT1 #Hl0 #HVW1 #HW10 #HT10
lapply (da_inv_flat … Hl1) -Hl1 #Hl1
elim (lsstas_inv_appl1 … H2) -H2 #U1 #HTU1 #H destruct
- lapply (IH1 … HT1 … Hl1 … HTU1) /2 width=1 by fsupp_ygt/ #HU1
- elim (lsstas_cpds_aux … IH1 IH4 IH3 IH2 … Hl1 … HTU1 … HT10) -IH4 -IH3 -IH2 -IH1 /2 width=1 by fsupp_ygt/ -T1 -l1 #X #l #_ #H #HU10 -l2
+ lapply (IH1 … HT1 … Hl1 … HTU1) /2 width=1 by fqup_fpbr/ #HU1
+ elim (lsstas_cpds_aux … IH1 IH4 IH3 IH2 … Hl1 … HTU1 … HT10) -IH4 -IH3 -IH2 -IH1 /2 width=1 by fqup_fpbr/ -T1 -l1 #X #l #_ #H #HU10 -l2
elim (lsstas_inv_bind1 … H) -H #U0 #_ #H destruct -T0 -l
elim (cpes_inv_abst2 … HU10) -HU10 #W2 #U2 #HU12 #HU02
elim (cprs_inv_abst … HU02) -HU02 #HW02 #_
[ lapply (lsstas_inv_O … H2) -H2 #H destruct // ]
elim (snv_inv_cast … H1) -H1
lapply (da_inv_flat … Hl1) -Hl1
- lapply (lsstas_inv_cast1 … H2) -H2 /3 width=8 by fsupp_ygt/
+ lapply (lsstas_inv_cast1 … H2) -H2 /3 width=8 by fqup_fpbr/
]
qed-.