(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
+definition S0 ≝ λC:relation3 genv lenv term. ∀G,L2,L1,T1,d,e.
+ C G L1 T1 → ∀T2. ⇩[Ⓕ, d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C G L2 T2.
+
+definition S0s ≝ λC:relation3 genv lenv term.
+ ∀G,L1,L2,des. ⇩*[Ⓕ, des] L2 ≡ L1 →
+ ∀T1,T2. ⇧*[des] T1 ≡ T2 → C G L1 T1 → C G L2 T2.
+
(* Note: this is Girard's CR1 *)
definition S1 ≝ λRP,C:relation3 genv lenv term.
∀G,L,T. C G L T → RP G L T.
definition S7 ≝ λC:relation3 genv lenv term.
∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
-definition S8 ≝ λC:relation3 genv lenv term. ∀G,L2,L1,T1,d,e.
- C G L1 T1 → ∀T2. ⇩[Ⓕ, d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C G L2 T2.
-
-definition S8s ≝ λC:relation3 genv lenv term.
- ∀G,L1,L2,des. ⇩*[Ⓕ, des] L2 ≡ L1 →
- ∀T1,T2. ⇧*[des] T1 ≡ T2 → C G L1 T1 → C G L2 T2.
-
(* properties of the abstract candidate of reducibility *)
record acr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:relation3 genv lenv term) : Prop ≝
-{ s1: S1 RP C;
+{ s0: S0 C;
+ 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;
- s8: S8 C
+ s7: S7 C
}.
(* the abstract candidate of reducibility associated to an atomic arity *)
(* Basic properties *********************************************************)
(* Basic_1: was: sc3_lift1 *)
-lemma acr_lifts: ∀C. S8 C → S8s C.
+lemma acr_lifts: ∀C. S0 C → S0s C.
#C #HC #G #L1 #L2 #des #H elim H -L1 -L2 -des
[ #L #T1 #T2 #H #HT1 <(lifts_inv_nil … H) -H //
| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
RP G L V → RP G L0 V0.
#RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV
@acr_lifts /width=7 by/
-@(s8 … HRP)
+@(s0 … HRP)
qed.
(* Basic_1: was only: sns3_lifts1 *)
∀A. acr RR RS RP (aacr RP A).
#RR #RS #RP #H1RP #H2RP #A elim A -A normalize //
#B #A #IHB #IHA @mk_acr normalize
-[ #G #L #T #H
+[ /3 width=7 by ldrops_cons, lifts_cons/
+| #G #L #T #H
elim (cp1 … H1RP G L) #k #HK
lapply (H ? (⋆k) ? (⟠) ? ? ?) -H
[1,3: // |2,4: skip
| @(s2 … IHB … (◊)) //
- | #H @(cp3 … H1RP … k) @(s1 … IHA) //
+ | #H @(cp2 … H1RP … k) @(s1 … IHA) //
]
| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H
elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
lapply (s1 … IHB … HB) #HV0
@(s2 … IHA … (V0 @ V0s))
- /3 width=14 by rp_liftsv_all, acp_lifts, cp2, lifts_simple_dx, conj/
+ /3 width=14 by rp_liftsv_all, acp_lifts, cp0, lifts_simple_dx, conj/
| #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #des #HB #HL0 #H
elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
elim (liftv_total 0 1 V10s) #V20s #HV120s
@(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=7 by rp_lifts, liftv_cons/
@(HA … (des + 1)) /2 width=2 by ldrops_skip/
- [ @(s8 … IHB … HB … HV120) /2 width=2 by ldrop_drop/
+ [ @(s0 … IHB … HB … HV120) /2 width=2 by ldrop_drop/
| @lifts_applv //
elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
>(liftv_mono … HV12s … HV10s) -V1s //
elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
@(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
-| /3 width=7 by ldrops_cons, lifts_cons/
]
qed.
lapply (aacr_acr … H1RP H2RP A) #HCA
lapply (aacr_acr … H1RP H2RP B) #HCB
elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
-lapply (acr_lifts … HL0 … HW0 HW) -HW [ @(s8 … HCB) ] #HW0
+lapply (acr_lifts … HL0 … HW0 HW) -HW [ @(s0 … HCB) ] #HW0
@(s3 … HCA … (◊))
@(s6 … HCA … (◊) (◊)) //
[ @(HA … HL0) //
| lapply (s1 … HCB) -HCB #HCB
- @(cp4 … H1RP) /2 width=1 by/
+ @(cp3 … H1RP) /2 width=1 by/
]
qed.