(* Basic_1: was: sc3_arity_csubc *)
theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
gcp RR RS RP → gcr RR RS RP RP →
- â\88\80G,L1,T,A. â¦\83G, L1â¦\84 â\8a¢ T â\81\9d A â\86\92 â\88\80b,f,L0. â¬\87*[b, f] L0 â\89¡ L1 →
- â\88\80T0. â¬\86*[f] T â\89¡ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
+ â\88\80G,L1,T,A. â¦\83G, L1â¦\84 â\8a¢ T â\81\9d A â\86\92 â\88\80b,f,L0. â¬\87*[b, f] L0 â\89\98 L1 →
+ â\88\80T0. â¬\86*[f] T â\89\98 T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
#RR #RS #RP #H1RP #H2RP #G #L1 #T @(fqup_wf_ind_eq (Ⓣ) … G L1 T) -G -L1 -T
#Z #Y #X #IH #G #L1 * [ * | * [ #p ] * ]
lapply (drops_tls_at … Hf … HY) -Hf -HY #HY
elim (drops_inv_skip2 … HY) -HY #Z #K0 #HK01 #HZ #H destruct
elim (liftsb_inv_pair_sn … HZ) -HZ #V0 #HV10 #H destruct
- elim (lifts_total V0 (ð\9d\90\94â\9d´â«¯j❵)) #V #HV0
+ elim (lifts_total V0 (ð\9d\90\94â\9d´â\86\91j❵)) #V #HV0
elim (lsubc_drops_trans_isuni … HL20 … HLK0) -HL20 -HLK0 // #Y #HLK2 #H
elim (lsubc_inv_bind2 … H) -H *
[ #K2 #HK20 #H destruct
lapply (drops_isuni_fwd_drop2 … HLK2) // #HLK2b
lapply (aaa_lifts … HKV1 … HK01 … HV10) -HKV1 -HK01 -HV10 #HKV0A
lapply (aaa_mono … HKV0B … HKV0A) #H destruct -HKV0B -HKV0A
- elim (lifts_total V2 (ð\9d\90\94â\9d´â«¯j❵)) #V3 #HV23
+ elim (lifts_total V2 (ð\9d\90\94â\9d´â\86\91j❵)) #V3 #HV23
lapply (s5 … HA … G … (◊) … (ⓝW2.V2) (ⓝV.V3) ????)
[3: |*: /2 width=9 by drops_inv_gen, lifts_flat/ ] -HLK2
lapply (s7 … HA G L2 (◊)) -HA /3 width=7 by acr_lifts/
elim (drops_lsubc_trans … H1RP … HL32 … HL20) -L2 #L2 #HL32 #HL20
lapply (aaa_lifts … HW … (f3∘f) L2 … W3 ?) -HW
[4: |*: /2 width=8 by drops_trans, lifts_trans/ ] #HW3
- @(IH â\80¦ ((â\86\91f3)â\88\98â\86\91f) … (L2. ⓛW3)) -IH
+ @(IH â\80¦ ((⫯f3)â\88\98⫯f) … (L2. ⓛW3)) -IH
/4 width=12 by lsubc_beta, drops_trans, drops_skip, lifts_trans, ext2_pair/
| #V #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
elim (aaa_inv_appl … HA) -HA #B #HV #HT