(* 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 →
- ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
+ ∀T. 𝐒⦃T⦄ → nf RR RS G L T → C G L (ⒶVs.T).
(* Note: this generalizes Tait's ii *)
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 S4 ≝ λRP,C:candidate.
- ∀G,L,Vs. all … (RP G L) Vs → ∀s. C G L (ⒶVs.⋆s).
-
definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
- C G L (â\92¶Vs.V2) â\86\92 â¬\86*[↑i] V1 ≘ V2 →
- â¬\87*[i] L ≘ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
+ C G L (â\92¶Vs.V2) â\86\92 â\87§*[↑i] V1 ≘ V2 →
+ â\87©*[i] L ≘ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
definition S6 ≝ λRP,C:candidate.
- â\88\80G,L,V1b,V2b. â¬\86*[1] V1b ≘ V2b →
+ â\88\80G,L,V1b,V2b. â\87§*[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.
{ s1: S1 RP C;
s2: S2 RR RS RP C;
s3: S3 C;
- s4: S4 RP C;
s5: S5 C;
s6: S6 RP C;
s7: S7 C
(* the functional construction for candidates *)
definition cfun: candidate → candidate → candidate ≝
λC1,C2,G,K,T. ∀f,L,W,U.
- â¬\87*[â\92»,f] L â\89\98 K â\86\92 â¬\86*[f] T ≘ U → C1 G L W → C2 G L (ⓐW.U).
+ â\87©*[â\92»,f] L â\89\98 K â\86\92 â\87§*[f] T ≘ U → C1 G L W → C2 G L (ⓐW.U).
(* the reducibility candidate associated to an atomic arity *)
rec definition acr (RP:candidate) (A:aarity) on A: candidate ≝
(* Basic_1: was:
sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast
*)
+(* Note: one sort must exist *)
lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
∀A. gcr RR RS RP (acr RP A).
#RR #RS #RP #H1RP #H2RP #A elim A -A //
#B #A #IHB #IHA @mk_gcr
[ #G #L #T #H
- elim (cp1 … H1RP G L) #s #HK
+ letin s ≝ 0 (* one sort must exist *)
+ lapply (cp1 … H1RP G L s) #HK
lapply (s2 … IHB G L (Ⓔ) … HK) // #HB
lapply (H (𝐈𝐝) L (⋆s) T ? ? ?) -H
/3 width=6 by s1, cp3, drops_refl, lifts_refl/
elim (lifts_inv_flat1 … H0) -H0 #U0 #X #HU0 #HX #H destruct
elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
@(s3 … IHA … (V0⨮V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
-| #G #L #Vs #HVs #s #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- >(lifts_inv_sort1 … H0) -X0
- lapply (s1 … IHB … HB) #HV0
- @(s4 … IHA … (V0⨮V0s)) /3 width=7 by gcp2_all, conj/
| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #f #L0 #V0 #X #HL0 #H #HB
elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
∀p,G,L,W,T,A,B. ⦃G,L,W⦄ ϵ[RP] 〚B〛 → (
- â\88\80b,f,L0,V0,W0,T0. â¬\87*[b,f] L0 â\89\98 L â\86\92 â¬\86*[f] W â\89\98 W0 â\86\92 â¬\86*[⫯f] T ≘ T0 →
+ â\88\80b,f,L0,V0,W0,T0. â\87©*[b,f] L0 â\89\98 L â\86\92 â\87§*[f] W â\89\98 W0 â\86\92 â\87§*[⫯f] T ≘ T0 →
⦃G,L0,V0⦄ ϵ[RP] 〚B〛 → ⦃G,L0,W0⦄ ϵ[RP] 〚B〛 → ⦃G,L0.ⓓⓝW0.V0,T0⦄ ϵ[RP] 〚A〛
) →
⦃G,L,ⓛ{p}W.T⦄ ϵ[RP] 〚②B.A〛.