∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
definition S5 ≝ λC:relation3 genv lenv term. ∀I,G,L,K,Vs,V1,V2,i.
- C G L (ⒶVs.V2) → ⇧[0, i + 1] V1 ≡ V2 →
- ⇩[0, i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
+ C G L (ⒶVs.V2) → ⇧[0, i+1] V1 ≡ V2 →
+ ⇩[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
definition S6 ≝ λRP,C:relation3 genv lenv term.
∀G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
∀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.
+ 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 →
+ ∀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 *)
λT. match A with
[ AAtom ⇒ RP G L T
| APair B A ⇒ ∀L0,V0,T0,des.
- aacr RP B G L0 V0 → ⇩*[des] L0 ≡ L → ⇧*[des] T ≡ T0 →
+ aacr RP B G L0 V0 → ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] T ≡ T0 →
aacr RP A G L0 (ⓐV0.T0)
].
(* Basic_1: was: sc3_lift1 *)
lemma acr_lifts: ∀C. S8 C → S8s C.
#C #HC #G #L1 #L2 #des #H elim H -L1 -L2 -des
-[ #L #T1 #T2 #H #HT1
- <(lifts_inv_nil … H) -H //
+[ #L #T1 #T2 #H #HT1 <(lifts_inv_nil … H) -H //
| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
- elim (lifts_inv_cons … H) -H /3 width=9 by/
+ elim (lifts_inv_cons … H) -H /3 width=10 by/
]
qed.
lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λG,L,T. RP G L T) →
- ∀des,G,L0,L,V,V0. ⇩*[des] L0 ≡ L → ⇧*[des] V ≡ V0 →
+ ∀des,G,L0,L,V,V0. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] V ≡ V0 →
RP G L V → RP G L0 V0.
#RR #RS #RP #HRP #des #G #L0 #L #V #V0 #HL0 #HV0 #HV
-@acr_lifts /width=6 by/
+@acr_lifts /width=7 by/
@(s8 … HRP)
qed.
(* Basic_1: was only: sns3_lifts1 *)
lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λG,L,T. RP G L T) →
- â\88\80des,G,L0,L,Vs,V0s. â\87§*[des] Vs â\89¡ V0s â\86\92 â\87©*[des] L0 â\89¡ L →
+ â\88\80des,G,L0,L,Vs,V0s. â\87©*[â\92», des] L0 â\89¡ L â\86\92 â\87§*[des] Vs â\89¡ V0s →
all … (RP G L) Vs → all … (RP G L0) V0s.
-#RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #H elim H -Vs -V0s normalize //
-#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s #HL0 * /3 width=6 by rp_lifts, conj/
+#RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #HL0 #H elim H -Vs -V0s normalize //
+#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s * /3 width=7 by rp_lifts, conj/
qed.
(* Basic_1: was:
elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
lapply (s1 … IHB … HB) #HV0
@(s2 … IHA … (V0 @ V0s))
- /3 width=13 by rp_liftsv_all, acp_lifts, cp2, lifts_simple_dx, conj/
+ /3 width=14 by rp_liftsv_all, acp_lifts, cp2, 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 (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
- @(s3 … IHA … (V0 @ V0s)) /5 width=5 by lifts_applv, lifts_flat, lifts_bind/
-| #G #L #Vs #HVs #k #L0 #V0 #X #hdes #HB #HL0 #H
+ @(s3 … IHA … (V0 @ V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
+| #G #L #Vs #HVs #k #L0 #V0 #X #des #HB #HL0 #H
elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
>(lifts_inv_sort1 … HY) -Y
lapply (s1 … IHB … HB) #HV0
- @(s4 … IHA … (V0 @ V0s)) /3 width=6 by rp_liftsv_all, conj/
+ @(s4 … IHA … (V0 @ V0s)) /3 width=7 by rp_liftsv_all, conj/
| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H
elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
>(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
- @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=4 by lifts_applv/
+ @(s5 … IHA … (V0 @ V0s) … HW12 HL02) /3 width=5 by lifts_applv/
| #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #des #HB #HL0 #H
elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #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 V10s) #V20s #HV120s
- @(s6 … IHA … (V10 @ V10s) (V20 @ V20s)) /3 width=6 by rp_lifts, liftv_cons/
- @(HA … (des + 1)) /2 width=1 by ldrops_skip/
- [ @(s8 … IHB … HB … HV120) /2 width=1 by ldrop_ldrop/
+ @(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/
| @lifts_applv //
elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
>(liftv_mono … HV12s … HV10s) -V1s //
| #G #L #Vs #T #W #HA #HW #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 #W0 #T0 #HW0 #HT0 #H destruct
- @(s7 … IHA … (V0 @ V0s)) /3 width=4 by lifts_applv/
+ @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
| /3 width=7 by ldrops_cons, lifts_cons/
]
qed.
lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
∀a,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
- ∀L0,V0,W0,T0,des. ⇩*[des] L0 ≡ L → ⇧*[des ] W ≡ W0 → ⇧*[des + 1] T ≡ T0 →
+ ∀L0,V0,W0,T0,des. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des ] W ≡ W0 → ⇧*[des + 1] T ≡ T0 →
⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
) →
⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
projects / helm.git / blobdiff