(* Note: this is Tait's iii, or Girard's CR4 *)
definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
∀G,L,Vs. all … (RP G L) Vs →
- â\88\80T. ð\9d\90\92â\9dªTâ\9d« → nf RR RS G L T → C G L (ⒶVs.T).
+ â\88\80T. ð\9d\90\92â\9d¨Tâ\9d© → nf RR RS G L T → C G L (ⒶVs.T).
(* Note: this generalizes Tait's ii, or Girard's CR3 *)
definition S3 ≝ λC:candidate.
qed.
lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- â\88\80p,G,L,W,T,A,B. â\9dªG,L,Wâ\9d« ϵ ⟦B⟧[RP] → (
+ â\88\80p,G,L,W,T,A,B. â\9d¨G,L,Wâ\9d© ϵ ⟦B⟧[RP] → (
∀b,f,L0,V0,W0,T0. ⇩*[b,f] L0 ≘ L → ⇧*[f] W ≘ W0 → ⇧*[⫯f] T ≘ T0 →
- â\9dªG,L0,V0â\9d« ϵ â\9f¦Bâ\9f§[RP] â\86\92 â\9dªG,L0,W0â\9d« ϵ â\9f¦Bâ\9f§[RP] â\86\92 â\9dªG,L0.â\93\93â\93\9dW0.V0,T0â\9d« ϵ ⟦A⟧[RP]
+ â\9d¨G,L0,V0â\9d© ϵ â\9f¦Bâ\9f§[RP] â\86\92 â\9d¨G,L0,W0â\9d© ϵ â\9f¦Bâ\9f§[RP] â\86\92 â\9d¨G,L0.â\93\93â\93\9dW0.V0,T0â\9d© ϵ ⟦A⟧[RP]
) →
- â\9dªG,L,â\93\9b[p]W.Tâ\9d« ϵ ⟦②B.A⟧[RP].
+ â\9d¨G,L,â\93\9b[p]W.Tâ\9d© ϵ ⟦②B.A⟧[RP].
#RR #RS #RP #H1RP #H2RP #p #G #L #W #T #A #B #HW #HA #f #L0 #V0 #X #HL0 #H #HB
lapply (acr_gcr … H1RP H2RP A) #HCA
lapply (acr_gcr … H1RP H2RP B) #HCB