⬇[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
definition S6 ≝ λRP,C:candidate.
- ∀G,L,V1c,V2c. ⬆[0, 1] V1c ≡ V2c →
- ∀a,V,T. C G (L.ⓓV) (ⒶV2c.T) → RP G L V → C G L (ⒶV1c.ⓓ{a}V.T).
+ ∀G,L,V1b,V2b. ⬆[0, 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.
∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
(* requirements for the generic reducibility candidate *)
record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
-{ c1: S1 RP C;
- c2: S2 RR RS RP C;
- c3: S3 C;
- c4: S4 RP C;
- c5: S5 C;
- c6: S6 RP C;
- c7: S7 C
+{ b1: S1 RP C;
+ b2: S2 RR RS RP C;
+ b3: S3 C;
+ b4: S4 RP C;
+ b5: S5 C;
+ b6: S6 RP C;
+ b7: S7 C
}.
(* the functional construction for candidates *)
[ #G #L #T #H
elim (cp1 … H1RP G L) #s #HK
lapply (H L (⋆s) T (◊) ? ? ?) -H //
- [ lapply (c2 … IHB G L (◊) … HK) //
- | /3 width=6 by c1, cp3/
+ [ lapply (b2 … IHB G L (◊) … HK) //
+ | /3 width=6 by b1, cp3/
]
| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #cs #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0c #T0 #HV0c #HT0 #H destruct
- lapply (c1 … IHB … HB) #HV0
- @(c2 … IHA … (V0 @ V0c))
+ elim (lifts_inv_applv1 … H) -H #V0b #T0 #HV0b #HT0 #H destruct
+ lapply (b1 … IHB … HB) #HV0
+ @(b2 … IHA … (V0 @ V0b))
/3 width=14 by gcp2_lifts_all, gcp2_lifts, gcp0_lifts, lifts_simple_dx, conj/
| #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #cs #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct
+ elim (lifts_inv_applv1 … H) -H #V0b #Y #HV0b #HY #H destruct
elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct
elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
- @(c3 … IHA … (V0 @ V0c)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
+ @(b3 … IHA … (V0 @ V0b)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
| #G #L #Vs #HVs #s #L0 #V0 #X #cs #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct
+ elim (lifts_inv_applv1 … H) -H #V0b #Y #HV0b #HY #H destruct
>(lifts_inv_sort1 … HY) -Y
- lapply (c1 … IHB … HB) #HV0
- @(c4 … IHA … (V0 @ V0c)) /3 width=7 by gcp2_lifts_all, conj/
+ lapply (b1 … IHB … HB) #HV0
+ @(b4 … IHA … (V0 @ V0b)) /3 width=7 by gcp2_lifts_all, conj/
| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #cs #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct
+ elim (lifts_inv_applv1 … H) -H #V0b #Y #HV0b #HY #H destruct
elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
elim (drops_drop_trans … HL0 … HLK) #X #cs0 #i1 #HL02 #H #Hi1 #Hcs0
>(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
elim (lifts_lift_trans … Hcs0 … HVW1 … HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2
>(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
- @(c5 … IHA … (V0 @ V0c) … HW12 HL02) /3 width=5 by lifts_applv/
-| #G #L #V1c #V2c #HV12c #a #V #T #HA #HV #L0 #V10 #X #cs #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V10c #Y #HV10c #HY #H destruct
+ @(b5 … IHA … (V0 @ V0b) … HW12 HL02) /3 width=5 by lifts_applv/
+| #G #L #V1b #V2b #HV12b #a #V #T #HA #HV #L0 #V10 #X #cs #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V10b #Y #HV10b #HY #H destruct
elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct
elim (lift_total V10 0 1) #V20 #HV120
- elim (liftv_total 0 1 V10c) #V20c #HV120c
- @(c6 … IHA … (V10 @ V10c) (V20 @ V20c)) /3 width=7 by gcp2_lifts, liftv_cons/
+ elim (liftv_total 0 1 V10b) #V20b #HV120b
+ @(b6 … IHA … (V10 @ V10b) (V20 @ V20b)) /3 width=7 by gcp2_lifts, liftv_cons/
@(HA … (cs + 1)) /2 width=2 by drops_skip/
[ @lifts_applv //
- elim (liftsv_liftv_trans_le … HV10c … HV120c) -V10c #V10c #HV10c #HV120c
- >(liftv_mono … HV12c … HV10c) -V1c //
+ elim (liftsv_liftv_trans_le … HV10b … HV120b) -V10b #V10b #HV10b #HV120b
+ >(liftv_mono … HV12b … HV10b) -V1b //
| @(gcr_lift … H1RP … HB … HV120) /2 width=2 by drop_drop/
]
| #G #L #Vs #T #W #HA #HW #L0 #V0 #X #cs #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct
+ elim (lifts_inv_applv1 … H) -H #V0b #Y #HV0b #HY #H destruct
elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
- @(c7 … IHA … (V0 @ V0c)) /3 width=5 by lifts_applv/
+ @(b7 … IHA … (V0 @ V0b)) /3 width=5 by lifts_applv/
]
qed.
lapply (acr_gcr … H1RP H2RP B) #HCB
elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
lapply (gcr_lifts … H1RP … HL0 … HW0 HW) -HW #HW0
-lapply (c3 … HCA … a G L0 (◊)) #H @H -H
-lapply (c6 … HCA G L0 (◊) (◊) ?) // #H @H -H
+lapply (b3 … HCA … a G L0 (◊)) #H @H -H
+lapply (b6 … HCA G L0 (◊) (◊) ?) // #H @H -H
[ @(HA … HL0) //
-| lapply (c1 … HCB) -HCB #HCB
- lapply (c7 … H2RP G L0 (◊)) /3 width=1 by/
+| lapply (b1 … HCB) -HCB #HCB
+ lapply (b7 … H2RP G L0 (◊)) /3 width=1 by/
]
qed.