∀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).
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
elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
lapply (drops_trans … HL0 … HLK ??) [3: |*: // ] -HLK #H
- elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK0 #HY
+ elim (drops_split_trans … H) -H [ |*: /2 width=6 by pr_after_nat_uni/ ] #Y #HLK0 #HY
lapply (drops_tls_at … Hf … HY) -HY #HY
elim (drops_inv_skip2 … HY) -HY #Z #K0 #HK0 #HZ #H destruct
elim (liftsb_inv_pair_sn … HZ) -HZ #W1 #HVW1 #H destruct
elim (lifts_total W1 (𝐔❨↑j❩)) #W2 #HW12
lapply (lifts_trans … HVW1 … HW12 ??) -HVW1 [3: |*: // ] #H
- lapply (lifts_conf … HV12 … H f ?) -V1 [ /2 width=3 by after_uni_succ_sn/ ] #HVW2
+ lapply (lifts_conf … HV12 … H f ?) -V1 [ /2 width=3 by pr_pat_after_uni_tls/ ] #HVW2
@(s5 … IHA … (V0⨮V0s) … HW12) /3 width=4 by drops_inv_gen, lifts_applv/
| #G #L #V1s #V2s #HV12s #p #V #T #HA #HV #f #L0 #V10 #X #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V10s #X0 #HV10s #H0 #H destruct
@(HA … (⫯f)) /3 width=2 by drops_skip, ext2_pair/
[ @lifts_applv //
lapply (liftsv_trans … HV10s … HV120s ??) -V10s [3: |*: // ] #H
- elim (liftsv_split_trans … H (𝐔❨1❩) (⫯f)) /2 width=1 by after_uni_one_sn/ #V10s #HV10s #HV120s
+ elim (liftsv_split_trans … H (𝐔❨1❩) (⫯f)) /2 width=1 by pr_after_unit_sn/ #V10s #HV10s #HV120s
>(liftsv_mono … HV12s … HV10s) -V1s //
| @(acr_lifts … H1RP … HB … HV120) /3 width=2 by drops_refl, drops_drop/
]