(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
-definition S0 ≝ λC:relation3 genv lenv term. ∀G,L2,L1,T1,d,e.
+definition S0 ≝ λC:candidate. ∀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.
+definition S0s ≝ λC:candidate.
∀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.
+definition S1 ≝ λRP,C:candidate.
∀G,L,T. C G L T → RP G L T.
(* Note: this is Tait's iii, or Girard's CR4 *)
-definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:relation3 genv lenv term.
+definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
∀G,L,Vs. all … (RP G L) Vs →
∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
(* Note: this generalizes Tait's ii *)
-definition S3 ≝ λC:relation3 genv lenv term.
+definition S3 ≝ λC:candidate.
∀a,G,L,Vs,V,T,W.
C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
-definition S4 ≝ λRP,C:relation3 genv lenv term.
+definition S4 ≝ λRP,C:candidate.
∀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.
+definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,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.
+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).
-definition S7 ≝ λC:relation3 genv lenv term.
+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).
(* 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 ≝
+record acr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
{ s0: S0 C;
s1: S1 RP C;
s2: S2 RR RS RP C;
s7: S7 C
}.
-(* the abstract candidate of reducibility associated to an atomic arity *)
-let rec aacr (RP:relation3 genv lenv term) (A:aarity) (G:genv) (L:lenv) on A: predicate term ≝
-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 A G L0 (ⓐV0.T0)
+(* the functional construction for candidates *)
+definition cfun: candidate → candidate → candidate ≝
+ λC1,C2,G,K,T. ∀L,V,U,des.
+ ⇩*[Ⓕ, des] L ≡ K → ⇧*[des] T ≡ U → C1 G L V → C2 G L (ⓐV.U).
+
+(* the candidate associated to an atomic arity *)
+let rec aacr (RP:candidate) (A:aarity) on A: candidate ≝
+match A with
+[ AAtom ⇒ RP
+| APair B A ⇒ cfun (aacr RP B) (aacr RP A)
].
interpretation
]
qed.
-lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λG,L,T. RP G L T) →
+lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP RP →
∀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
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) →
+lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP RP →
∀des,G,L0,L,Vs,V0s. ⇩*[Ⓕ, des] L0 ≡ L → ⇧*[des] Vs ≡ V0s →
all … (RP G L) Vs → all … (RP G L0) V0s.
#RR #RS #RP #HRP #des #G #L0 #L #Vs #V0s #HL0 #H elim H -Vs -V0s normalize //
(* Basic_1: was:
sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
*)
-lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
+lemma aacr_acr: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP RP →
∀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
elim (cp1 … H1RP G L) #k #HK
lapply (H ? (⋆k) ? (⟠) ? ? ?) -H
- [1,3: // |2,4: skip
+ [3,5: // |2,4: skip
| @(s2 … IHB … (◊)) //
| #H @(cp2 … H1RP … k) @(s1 … IHA) //
]
-| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H
+| #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))
/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
+| #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
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 #HB #HL0 #H
+| #G #L #Vs #HVs #k #L0 #V0 #X #des #HL0 #H #HB
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=7 by rp_liftsv_all, conj/
-| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HB #HL0 #H
+| #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
elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct
elim (drops_drop_trans … HL0 … HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
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=5 by lifts_applv/
-| #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #des #HB #HL0 #H
+| #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
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 rp_lifts, liftv_cons/
@(HA … (des + 1)) /2 width=2 by drops_skip/
- [ @(s0 … IHB … HB … HV120) /2 width=2 by drop_drop/
- | @lifts_applv //
+ [ @lifts_applv //
elim (liftsv_liftv_trans_le … HV10s … HV120s) -V10s #V10s #HV10s #HV120s
>(liftv_mono … HV12s … HV10s) -V1s //
+ | @(s0 … IHB … HB … HV120) /2 width=2 by drop_drop/
]
-| #G #L #Vs #T #W #HA #HW #L0 #V0 #X #des #HB #HL0 #H
+| #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
elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct
@(s7 … IHA … (V0 @ V0s)) /3 width=5 by lifts_applv/
]
qed.
-lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr RR RS RP (λG,L,T. RP G L T) →
+lemma aacr_abst: ∀RR,RS,RP. acp RR RS RP → acr 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,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〛.
-#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HB #HL0 #H
+#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #des #HL0 #H #HB
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
projects / helm.git / blobdiff