(* Basic_1: was: sc3_arity_csubc *)
theorem aacr_aaa_csubc_lifts: ∀RR,RS,RP.
acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
- ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L0,des. ⇩*[des] L0 ≡ L1 →
+ ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L0,des. ⇩*[Ⓕ, des] L0 ≡ L1 →
∀T0. ⇧*[des] T ≡ T0 → ∀L2. G ⊢ L2 ⊑[RP] L0 →
⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
#RR #RS #RP #H1RP #H2RP #G #L1 #T #A #H elim H -G -L1 -T -A
@(aacr_abst … H1RP H2RP) [ /2 width=5 by/ ]
#L3 #V3 #W3 #T3 #des3 #HL32 #HW03 #HT03 #H1B #H2B
elim (ldrops_lsubc_trans … H1RP H2RP … HL32 … HL02) -L2 #L2 #HL32 #HL20
- lapply (aaa_lifts … L2 W3 … (des @@ des3) … HLWB) -HLWB /2 width=3 by ldrops_trans, lifts_trans/ #HLW2B
- @(IHA (L2. ⓛW3) … (des + 1 @@ des3 + 1)) -IHA /2 width=3/ /3 width=5 by lsubc_abbr, ldrops_trans, ldrops_skip/
+ lapply (aaa_lifts … L2 W3 … (des @@ des3) … HLWB) -HLWB /2 width=4 by ldrops_trans, lifts_trans/ #HLW2B
+ @(IHA (L2. ⓛW3) … (des + 1 @@ des3 + 1)) -IHA
+ /3 width=5 by lsubc_abbr, ldrops_trans, ldrops_skip, lifts_trans/
| #G #L #V #T #B #A #_ #_ #IHB #IHA #L0 #des #HL0 #X #H #L2 #HL20
elim (lifts_inv_flat1 … H) -H #V0 #T0 #HV0 #HT0 #H destruct
/3 width=10 by ldrops_nil, lifts_nil/