(* Properties on sn parallel reduction for local environments ***************)
fact lsstas_cpr_lpr_aux: ∀h,g,G0,L0,T0.
- (∀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, 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 = G1 → L0 = L1 → T0 = T1 → IH_lsstas_cpr_lpr h g G1 L1 T1.
#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
[ #k #_ #_ #_ #_ #l1 #l2 #_ #_ #X2 #H2 #X3 #H3 #L2 #_ -IH4 -IH3 -IH2 -IH1
[ lapply (da_mono … Hl … HWl1) -Hl #H destruct
lapply (le_plus_to_le_r … Hl21) -Hl21 #Hl21
]
- lapply (fsupp_lref … G1 … HLK1) #HKV1
+ lapply (fqup_lref … G1 … HLK1) #HKV1
elim (lpr_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
elim (lpr_inv_pair1 … H) -H #K2 [ #W2 | #V2 ] #HK12 [ #HW12 | #HV12 ] #H destruct
lapply (ldrop_fwd_ldrop2 … HLK2) #H2
|2,4: * #K0 #V0 #X0 #H #HVX0 #HX0
lapply (ldrop_mono … H … HLK1) -H -HLK1 #H destruct
]
- [ elim (IH1 … HWl1 … HW0 … HW12 … HK12) -IH1 -HW0 /2 width=1 by fsupp_ygt/ #V2 #HWV2 #HV2
+ [ elim (IH1 … HWl1 … HW0 … HW12 … HK12) -IH1 -HW0 /2 width=1 by fqup_fpbr/ #V2 #HWV2 #HV2
elim (lift_total V2 0 (i+1))
- /6 width=11 by fsupp_ygt, cpcs_lift, lsstas_ldec, ex2_intro/
- | elim (IH1 … HVl1 … HW0 … HV12 … HK12) -IH1 -HVl1 -HW0 -HV12 -HK12 -IH2 /2 width=1 by fsupp_ygt/ #W2 #HVW2 #HW02
+ /6 width=11 by fqup_fpbr, cpcs_lift, lsstas_ldec, ex2_intro/
+ | elim (IH1 … HVl1 … HW0 … HV12 … HK12) -IH1 -HVl1 -HW0 -HV12 -HK12 -IH2 /2 width=1 by fqup_fpbr/ #W2 #HVW2 #HW02
elim (lift_total W2 0 (i+1))
/4 width=11 by cpcs_lift, lsstas_ldef, ex2_intro/
- | elim (IH1 … HVl1 … HW0 … HVX0 … HK12) -IH1 -HVl1 -HW0 -HVX0 -HK12 -IH2 -V2 /2 width=1 by fsupp_ygt/ -l1 #W2 #HXW2 #HW02
+ | elim (IH1 … HVl1 … HW0 … HVX0 … HK12) -IH1 -HVl1 -HW0 -HVX0 -HK12 -IH2 -V2 /2 width=1 by fqup_fpbr/ -l1 #W2 #HXW2 #HW02
elim (lift_total W2 0 (i+1))
/3 width=11 by cpcs_lift, lsstas_lift, ex2_intro/
]
elim (lsstas_inv_bind1 … H2) -H2 #U1 #HTU1 #H destruct
elim (cpr_inv_bind1 … H3) -H3 *
[ #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH1 … Hl1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hl1 -HTU1 -HT12 /2 width=1 by fsupp_ygt, lpr_pair/ -T1
+ elim (IH1 … Hl1 … HTU1 … HT12 (L2.ⓑ{I}V2)) -IH1 -Hl1 -HTU1 -HT12 /2 width=1 by fqup_fpbr, lpr_pair/ -T1
/4 width=5 by cpcs_bind2, lpr_cpr_conf, lsstas_bind, ex2_intro/
| #T3 #HT13 #HXT3 #H1 #H2 destruct
- elim (IH1 … Hl1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fsupp_ygt, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
+ elim (IH1 … Hl1 … HTU1 … HT13 (L2.ⓓV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fqup_fpbr, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
elim (lsstas_inv_lift1 … HTU3 L2 … HXT3) -T3
/5 width=8 by cpcs_cpr_strap1, cpcs_bind1, cpr_zeta, ldrop_ldrop, ex2_intro/
]
elim (cpr_inv_appl1 … H3) -H3 *
[ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -W10 -U10 -HV1 -IH4 -IH3 -IH2
elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1
- /4 width=5 by fsupp_ygt, cpcs_flat, lpr_cpr_conf, lsstas_appl, ex2_intro/
+ /4 width=5 by fqup_fpbr, cpcs_flat, lpr_cpr_conf, lsstas_appl, ex2_intro/
| #b #V2 #W2 #W3 #T2 #T3 #HV12 #HW23 #HT23 #H1 #H2 destruct
elim (snv_inv_bind … HT1) -HT1 #HW2 #HT2
lapply (da_inv_bind … Hl1) -Hl1 #Hl1
lapply (cprs_div … HW10 … HW20) -W0 #HW12
lapply (ssta_da_conf … HVW1 … Hl0) <minus_plus_m_m #H
elim (snv_fwd_da … HW2) #l #Hl
- lapply (IH4 … HV1 … 1 … Hl0 W1 ?) /2 width=1 by fsupp_ygt, ssta_lsstas/ #HW1
+ lapply (IH4 … HV1 … 1 … Hl0 W1 ?) /2 width=1 by fqup_fpbr, ssta_lsstas/ #HW1
lapply (da_cpcs_aux … IH3 IH2 … H … Hl … HW12) // -H
- /3 width=5 by ygt_fpbs_trans, fsupp_ygt, ssta_fpbs/ #H destruct
- lapply (IH3 … HV12 … HL12) /2 width=1 by fsupp_ygt/ #HV2
- lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fsupp_ygt/ #HV2l
- elim (IH1 … 1 … Hl0 … W1 … HV12 … HL12) /2 width=1 by fsupp_ygt, ssta_lsstas/ -HVW1 #W4 #H #HW14
+ /3 width=5 by fpbr_fpbs_trans, fqup_fpbr, ssta_fpbs/ #H destruct
+ lapply (IH3 … HV12 … HL12) /2 width=1 by fqup_fpbr/ #HV2
+ lapply (IH2 … Hl0 … HV12 … HL12) /2 width=1 by fqup_fpbr/ #HV2l
+ elim (IH1 … 1 … Hl0 … W1 … HV12 … HL12) /2 width=1 by fqup_fpbr, ssta_lsstas/ -HVW1 #W4 #H #HW14
lapply (lsstas_inv_SO … H) #HV2W4
lapply (ssta_da_conf … HV2W4 … HV2l) <minus_plus_m_m #HW4l
- lapply (IH4 … HV2 … HV2l … H) -H /3 width=5 by ygt_fpbs_trans, fsupp_ygt, cpr_lpr_fpbs/ #HW4
- lapply (IH3 … HW23 … HL12) /2 width=1 by fsupp_ygt/ #HW3
- lapply (IH2 … Hl … HW23 … HL12) /2 width=1 by fsupp_ygt/ #HW3l
- elim (IH1 … Hl1 … HTU2 … HT23 (L2.ⓛW3)) -HTU2 /2 width=1 by fsupp_ygt, lpr_pair/ #U3 #HTU3 #HU23
+ lapply (IH4 … HV2 … HV2l … H) -H /3 width=5 by fpbr_fpbs_trans, fqup_fpbr, cpr_lpr_fpbs/ #HW4
+ lapply (IH3 … HW23 … HL12) /2 width=1 by fqup_fpbr/ #HW3
+ lapply (IH2 … Hl … HW23 … HL12) /2 width=1 by fqup_fpbr/ #HW3l
+ elim (IH1 … Hl1 … HTU2 … HT23 (L2.ⓛW3)) -HTU2 /2 width=1 by fqup_fpbr, lpr_pair/ #U3 #HTU3 #HU23
lapply (cpcs_cpr_strap1 … HW12 … HW23) #H
lapply (lpr_cpcs_conf … HL12 … H) -H #H
lapply (cpcs_canc_sn … HW14 H) -H #HW43
@(cpcs_cpr_strap1 … (ⓐV2.ⓛ{b}W3.U3)) /2 width=1 by cpr_beta/
/4 width=3 by cpcs_flat, cpcs_bind2, lpr_cpr_conf/
| -U3
- @(lsubsv_abbr … l) /3 width=7 by fsupp_ygt/
+ @(lsubsv_abbr … l) /3 width=7 by fqup_fpbr/
#W #W0 #l0 #Hl0 #HV2W #HW30
lapply (lsstas_ssta_conf_pos … HV2W4 … HV2W) -HV2W #HW4W
@(lsstas_cpcs_lpr_aux … IH3 IH2 IH1 … Hl0 … HW4W … Hl0 … HW30 … HW43) //
- [ /3 width=9 by ygt_fpbs_trans, fsupp_ygt, cpr_lpr_ssta_fpbs/
- | /3 width=5 by ygt_fpbs_trans, fsupp_ygt, cpr_lpr_fpbs/
+ [ /3 width=9 by fpbr_fpbs_trans, fqup_fpbr, cpr_lpr_ssta_fpbs/
+ | /3 width=5 by fpbr_fpbs_trans, fqup_fpbr, cpr_lpr_fpbs/
]
- | -IH1 -IH3 -IH4 /3 width=9 by fsupp_ygt, lpr_pair/
+ | -IH1 -IH3 -IH4 /3 width=9 by fqup_fpbr, lpr_pair/
]
| #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -l0 -W1 -W10 -HV1 -IH4 -IH3 -IH2
elim (snv_inv_bind … HT1) -HT1 #_ #HT0
lapply (da_inv_bind … Hl1) -Hl1 #Hl1
elim (lsstas_inv_bind1 … HTU1) -HTU1 #U0 #HTU0 #H destruct
- elim (IH1 … Hl1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hl1 -HTU0 /2 width=1 by fsupp_ygt, lpr_pair/ -T0 #U2 #HTU2 #HU02
+ elim (IH1 … Hl1 … HTU0 … HT02 (L2.ⓓW2)) -IH1 -Hl1 -HTU0 /2 width=1 by fqup_fpbr, lpr_pair/ -T0 #U2 #HTU2 #HU02
lapply (lpr_cpr_conf … HL12 … HV10) -HV10 #HV10
lapply (lpr_cpr_conf … HL12 … HW02) -L1 #HW02
lapply (cpcs_bind2 b … HW02 … HU02) -HW02 -HU02 #HU02
elim (cpr_inv_cast1 … H3) -H3
[ * #U2 #T2 #_ #HT12 #H destruct
elim (IH1 … Hl1 … HTU1 … HT12 … HL12) -IH1 -Hl1 -HTU1 -HL12
- /3 width=3 by fsupp_ygt, lsstas_cast, ex2_intro/
+ /3 width=3 by fqup_fpbr, lsstas_cast, ex2_intro/
| #HT1X3
elim (IH1 … Hl1 … HTU1 … HT1X3 … HL12) -IH1 -Hl1 -HTU1 -HL12
- /2 width=3 by fsupp_ygt, ex2_intro/
+ /2 width=3 by fqup_fpbr, ex2_intro/
]
]
qed-.