∀G,L,Vs. all … (RP G L) Vs →
∀T. 𝐒❪T❫ → nf RR RS G L T → C G L (ⒶVs.T).
-(* Note: this generalizes Tait's ii *)
+(* Note: this generalizes Tait's ii, or Girard's CR3 *)
definition S3 ≝ λC:candidate.
∀a,G,L,Vs,V,T,W.
C G L (ⒶVs.ⓓ[a]ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ[a]W.T).
definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
- C G L (ⒶVs.V2) → ⇧*[↑i] V1 ≘ V2 →
- ⇩*[i] L ≘ K.ⓑ[I]V1 → C G L (ⒶVs.#i).
+ C G L (ⒶVs.V2) → ⇧[↑i] V1 ≘ V2 →
+ ⇩[i] L ≘ K.ⓑ[I]V1 → C G L (ⒶVs.#i).
definition S6 ≝ λRP,C:candidate.
- ∀G,L,V1b,V2b. ⇧*[1] V1b ≘ V2b →
+ ∀G,L,V1b,V2b. ⇧[1] V1b ≘ V2b →
∀a,V,T. C G (L.ⓓV) (ⒶV2b.T) → RP G L V → C G L (ⒶV1b.ⓓ[a]V.T).
definition S7 ≝ λC:candidate.
letin s ≝ 0 (* one sort must exist *)
lapply (cp1 … H1RP G L s) #HK
lapply (s2 … IHB G L (Ⓔ) … HK) // #HB
- lapply (H (ð\9d\90\88ð\9d\90\9d) L (⋆s) T ? ? ?) -H
+ lapply (H (ð\9d\90¢) L (⋆s) T ? ? ?) -H
/3 width=6 by s1, cp3, drops_refl, lifts_refl/
| #G #L #Vs #HVs #T #H1T #H2T #f #L0 #V0 #X #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct