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 → ∀k. C G L (ⒶVs.⋆k).
+ ∀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\87§[0, i+1] 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¬\86[0, i+1] V1 ≡ V2 →
+ â¬\87[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
definition S6 ≝ λRP,C:candidate.
- ∀G,L,V1s,V2s. ⇧[0, 1] V1s ≡ V2s →
- ∀a,V,T. C G (L.ⓓV) (ⒶV2s.T) → RP G L V → C G L (ⒶV1s.ⓓ{a}V.T).
+ ∀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).
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 ≝
-{ 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
+{ 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
}.
(* the functional construction for candidates *)
definition cfun: candidate → candidate → candidate ≝
- λC1,C2,G,K,T. ∀L,W,U,des.
- â\87©*[â\92», des] L â\89¡ K â\86\92 â\87§*[des] T ≡ U → C1 G L W → C2 G L (ⓐW.U).
+ λC1,C2,G,K,T. ∀L,W,U,cs.
+ â¬\87*[â\92», cs] L â\89¡ K â\86\92 â¬\86*[cs] T ≡ U → C1 G L W → C2 G L (ⓐW.U).
(* the reducibility candidate associated to an atomic arity *)
-let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
+rec definition acr (RP:candidate) (A:aarity) on A: candidate ≝
match A with
[ AAtom ⇒ RP
| APair B A ⇒ cfun (acr RP B) (acr RP A)
(* Basic properties *********************************************************)
(* Basic 1: was: sc3_lift *)
-lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. l_liftable1 (acr RP A G) (Ⓕ).
+lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftable1 (acr RP A G) (Ⓕ).
#RR #RS #RP #H #A elim A -A
/3 width=8 by cp2, drops_cons, lifts_cons/
qed.
(* Basic_1: was: sc3_lift1 *)
-lemma gcr_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. l_liftables1 (acr RP A G) (Ⓕ).
-#RR #RS #RP #H #A #G @l1_liftable_liftables /2 width=7 by gcr_lift/
+lemma gcr_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftables1 (acr RP A G) (Ⓕ).
+#RR #RS #RP #H #A #G @d1_liftable_liftables /2 width=7 by gcr_lift/
qed.
(* Basic_1: was:
#RR #RS #RP #H1RP #H2RP #A elim A -A //
#B #A #IHB #IHA @mk_gcr
[ #G #L #T #H
- elim (cp1 … H1RP G L) #k #HK
- lapply (H L (⋆k) T (◊) ? ? ?) -H //
- [ lapply (s2 … IHB G L (◊) … HK) //
- | /3 width=6 by s1, cp3/
+ 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/
]
-| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
- lapply (s1 … IHB … HB) #HV0
- @(s2 … IHA … (V0 @ V0s))
+| #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))
/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 #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+| #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_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=6 by lifts_applv, lifts_flat, lifts_bind/
-| #G #L #Vs #HVs #k #L0 #V0 #X #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+ @(c3 … IHA … (V0 @ V0c)) /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
>(lifts_inv_sort1 … HY) -Y
- lapply (s1 … IHB … HB) #HV0
- @(s4 … IHA … (V0 @ V0s)) /3 width=7 by gcp2_lifts_all, conj/
-| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+ lapply (c1 … IHB … HB) #HV0
+ @(c4 … IHA … (V0 @ V0c)) /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_lref1 … HY) -HY #i0 #Hi0 #H destruct
- elim (drops_drop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
+ elim (drops_drop_trans … HL0 … HLK) #X #cs0 #i1 #HL02 #H #Hi1 #Hcs0
>(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
- elim (drops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct
+ elim (drops_inv_skip2 … Hcs0 … H) -H -cs0 #L2 #W1 #cs0 #Hcs0 #HLK #HVW1 #H destruct
elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
- elim (lifts_lift_trans … Hdes0 … HVW1 … HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
+ elim (lifts_lift_trans … Hcs0 … HVW1 … HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2
>(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
- @(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 #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V10s #Y #HV10s #HY #H destruct
+ @(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
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=7 by gcp2_lifts, liftv_cons/
- @(HA … (des + 1)) /2 width=2 by drops_skip/
+ elim (liftv_total 0 1 V10c) #V20c #HV120c
+ @(c6 … IHA … (V10 @ V10c) (V20 @ V20c)) /3 width=7 by gcp2_lifts, liftv_cons/
+ @(HA … (cs + 1)) /2 width=2 by drops_skip/
[ @lifts_applv //
- elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
- >(liftv_mono … HV12s … HV10s) -V1s //
+ elim (liftsv_liftv_trans_le … HV10c … HV120c) -V10c #V10c #HV10c #HV120c
+ >(liftv_mono … HV12c … HV10c) -V1c //
| @(gcr_lift … H1RP … HB … HV120) /2 width=2 by drop_drop/
]
-| #G #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #Y #HV0s #HY #H destruct
+| #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_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
- @(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
+ @(c7 … IHA … (V0 @ V0c)) /3 width=5 by lifts_applv/
]
qed.
lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
∀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,cs. ⬇*[Ⓕ, cs] L0 ≡ L → ⬆*[cs] W ≡ W0 → ⬆*[cs + 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〛.
-#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HL0 #H #HB
+#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #cs #HL0 #H #HB
lapply (acr_gcr … H1RP H2RP A) #HCA
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 (s3 … HCA … a G L0 (◊)) #H @H -H
-lapply (s6 … HCA G L0 (◊) (◊) ?) // #H @H -H
+lapply (c3 … HCA … a G L0 (◊)) #H @H -H
+lapply (c6 … HCA G L0 (◊) (◊) ?) // #H @H -H
[ @(HA … HL0) //
-| lapply (s1 … HCB) -HCB #HCB
- lapply (s7 … H2RP G L0 (◊)) /3 width=1 by/
+| lapply (c1 … HCB) -HCB #HCB
+ lapply (c7 … H2RP G L0 (◊)) /3 width=1 by/
]
qed.
projects / helm.git / blobdiff