(**************************************************************************)
include "Basic_2/grammar/aarity.ma".
-include "Basic_2/unfold/lifts_vector.ma".
+include "Basic_2/unfold/gr2_gr2.ma".
+include "Basic_2/unfold/lifts_lift_vector.ma".
+include "Basic_2/unfold/ldrops_ldrop.ma".
include "Basic_2/computation/acp.ma".
(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
(* Note: this is Tait's ii *)
definition S3 ≝ λRP,C:lenv→predicate term.
- ∀L,Vs,V,T,W. C L (ⒶVs. 𝕔{Abbr}V. T) → RP L W → C L (ⒶVs. 𝕔{Appl}V. 𝕔{Abst}W. T).
+ ∀L,Vs,V,T,W. C L (ⒶVs. ⓓV. T) → RP L W → C L (ⒶVs. ⓐV. ⓛW. T).
+
+definition S4 ≝ λRP,C:lenv→predicate term. ∀L,K,Vs,V1,V2,i.
+ C L (ⒶVs. V2) → ⇧[0, i + 1] V1 ≡ V2 →
+ ⇩[0, i] L ≡ K. ⓓV1 → C L (Ⓐ Vs. #i).
definition S5 ≝ λRP,C:lenv→predicate term.
- â\88\80L,V1s,V2s. â\87\91[0, 1] V1s ≡ V2s →
- ∀V,T. C (L. 𝕓{Abbr}V) (ⒶV2s. T) → RP L V → C L (ⒶV1s. 𝕔{Abbr}V. T).
+ â\88\80L,V1s,V2s. â\87§[0, 1] V1s ≡ V2s →
+ ∀V,T. C (L. ⓓV) (ⒶV2s. T) → RP L V → C L (ⒶV1s. ⓓV. T).
definition S6 ≝ λRP,C:lenv→predicate term.
- ∀L,Vs,T,W. C L (ⒶVs. T) → RP L W → C L (ⒶVs. 𝕔{Cast}W. T).
+ ∀L,Vs,T,W. C L (ⒶVs. T) → RP L W → C L (ⒶVs. ⓣW. T).
definition S7 ≝ λC:lenv→predicate term. ∀L1,L2,T1,T2,d,e.
- C L1 T1 â\86\92 â\87\93[d, e] L2 â\89¡ L1 â\86\92 â\87\91[d, e] T1 ≡ T2 → C L2 T2.
+ C L1 T1 â\86\92 â\87©[d, e] L2 â\89¡ L1 â\86\92 â\87§[d, e] T1 ≡ T2 → C L2 T2.
definition S7s ≝ λC:lenv→predicate term.
- â\88\80L1,L2,des. â\87\93[des] L2 ≡ L1 →
- â\88\80T1,T2. â\87\91[des] T1 ≡ T2 → C L1 T1 → C L2 T2.
+ â\88\80L1,L2,des. â\87©*[des] L2 ≡ L1 →
+ â\88\80T1,T2. â\87§*[des] T1 ≡ T2 → C L1 T1 → C L2 T2.
(* properties of the abstract candidate of reducibility *)
record acr (RR:lenv->relation term) (RS:relation term) (RP,C:lenv→predicate term) : Prop ≝
{ s1: S1 RP C;
s2: S2 RR RS RP C;
s3: S3 RP C;
+ s4: S4 RP C;
s5: S5 RP C;
s6: S6 RP C;
s7: S7 C
let rec aacr (RP:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝
λT. match A with
[ AAtom ⇒ RP L T
-| APair B A â\87\92 â\88\80L0,V0,T0,des. aacr RP B L0 V0 â\86\92 â\87\93[des] L0 â\89¡ L â\86\92 â\87\91[des] T ≡ T0 →
- aacr RP A L0 (𝕔{Appl} V0. T0)
+| APair B A â\87\92 â\88\80L0,V0,T0,des. aacr RP B L0 V0 â\86\92 â\87©*[des] L0 â\89¡ L â\86\92 â\87§*[des] T ≡ T0 →
+ aacr RP A L0 (ⓐV0. T0)
].
interpretation
qed.
lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
- â\88\80des,L0,L,V,V0. â\87\93[des] L0 â\89¡ L â\86\92 â\87\91[des] V ≡ V0 →
+ â\88\80des,L0,L,V,V0. â\87©*[des] L0 â\89¡ L â\86\92 â\87§*[des] V ≡ V0 →
RP L V → RP L0 V0.
#RR #RS #RP #HRP #des #L0 #L #V #V0 #HL0 #HV0 #HV
@acr_lifts /width=6/
qed.
lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
- â\88\80des,L0,L,Vs,V0s. â\87\91[des] Vs â\89¡ V0s â\86\92 â\87\93[des] L0 ≡ L →
+ â\88\80des,L0,L,Vs,V0s. â\87§*[des] Vs â\89¡ V0s â\86\92 â\87©*[des] L0 ≡ L →
all … (RP L) Vs → all … (RP L0) V0s.
#RR #RS #RP #HRP #des #L0 #L #Vs #V0s #H elim H -Vs -V0s normalize //
#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s #HL0 * #HT1 #HT1s
@conj /2 width=1/ /2 width=6 by rp_lifts/
qed.
-axiom aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
+lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
∀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
[ #L #T #H
- lapply (H ? (⋆0) ? ⟠ ? ? ?) -H
+ lapply (H ? (⋆0) ? ⟠ ? ? ?) -H
[1,3: // |2,4: skip
| @(s2 … IHB … ◊) // /2 width=2/
| #H @(cp3 … H1RP … 0) @(s1 … IHA) //
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)) /2 width=6 by rp_lifts/ /4 width=5/
+| #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 (ldrops_ldrop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
+ >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02
+ elim (ldrops_inv_skip2 … Hdes0 … H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct
+ elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
+ elim (lifts_lift_trans … HVW1 … HW12 … Hdes0) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
+ >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2
+ @(s4 … IHA … (V0 :: V0s) … HW12 HL02) /3 width=4/
| #L #V1s #V2s #HV12s #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
@(s5 … IHA … (V10 :: V10s) (V20 :: V20s)) /2 width=1/ /2 width=6 by rp_lifts/
- @(HA … (ss des)) /2 width=1/
+ @(HA … (des + 1)) /2 width=1/
[ @(s7 … IHB … HB … HV120) /2 width=1/
- | @liftsv_applv //
+ | @lifts_applv //
+ elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
+ >(liftv_mono … HV12s … HV10s) -V1s //
]
| #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
- @(s6 … IHA … (V0 :: V0s)) /2 width=6 by rp_lifts/ /3 width=4/
+ @(s6 … IHA … (V0 :: V0s)) /2 width=6 by rp_lifts/ /3 width=4/
| /3 width=7/
]
qed.
-*)
+
lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λL,T. RP L T) →
∀L,W,T,A,B. RP L W → (
- â\88\80L0,V0,T0,des. â\87\93[des] L0 â\89¡ L â\86\92 â\87\91[ss des] T ≡ T0 →
- ⦃L0, V0⦄ [RP] ϵ 〚B〛→ ⦃L0. 𝕓{Abbr} V0, T0⦄ [RP] ϵ 〚A〛
+ â\88\80L0,V0,T0,des. â\87©*[des] L0 â\89¡ L â\86\92 â\87§*[des + 1] T ≡ T0 →
+ ⦃L0, V0⦄ [RP] ϵ 〚B〛 → ⦃L0. ⓓV0, T0⦄ [RP] ϵ 〚A〛
) →
- ⦃L, 𝕓{Abst} W. T⦄ [RP] ϵ 〚𝕔 B. A〛.
+ ⦃L, ⓛW. T⦄ [RP] ϵ 〚②B. A〛.
#RR #RS #RP #H1RP #H2RP #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H
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 (s1 … HCB) -HCB #HCB
+lapply (s1 … HCB) -HCB #HCB
@(s3 … HCA … ◊) /2 width=6 by rp_lifts/
@(s5 … HCA … ◊ ◊) // /2 width=1/ /2 width=3/
qed.
projects / helm.git / blobdiff