(**************************************************************************)
include "Basic_2/grammar/aarity.ma".
-include "Basic_2/grammar/lenv.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 CANDIDATES OF REDUCIBILITY **************************************)
+(* ABSTRACT COMPUTATION PROPERTIES ******************************************)
+
+(* Note: this is Girard's CR1 *)
+definition S1 ≝ λRP,C:lenv→predicate term.
+ ∀L,T. C L T → RP L T.
+
+(* Note: this is Tait's iii, or Girard's CR4 *)
+definition S2 ≝ λRR:lenv→relation term. λRS:relation term. λRP,C:lenv→predicate term.
+ ∀L,Vs. all … (RP L) Vs →
+ ∀T. 𝕊[T] → NF … (RR L) RS T → C L (ⒶVs.T).
+
+(* Note: this is Tait's ii *)
+definition S3 ≝ λRP,C:lenv→predicate term.
+ ∀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.
+ ∀L,V1s,V2s. ⇧[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. ⓣW. T).
+
+definition S7 ≝ λC:lenv→predicate term. ∀L1,L2,T1,T2,d,e.
+ C L1 T1 → ⇩[d, e] L2 ≡ L1 → ⇧[d, e] T1 ≡ T2 → C L2 T2.
+
+definition S7s ≝ λC:lenv→predicate term.
+ ∀L1,L2,des. ⇩*[des] L2 ≡ L1 →
+ ∀T1,T2. ⇧*[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
+}.
(* the abstract candidate of reducibility associated to an atomic arity *)
-let rec acr (R:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝
+let rec aacr (RP:lenv→predicate term) (A:aarity) (L:lenv) on A: predicate term ≝
λT. match A with
-[ AAtom ⇒ R L T
-| APair B A ⇒ ∀V. acr R B L V → acr R A L (𝕔{Appl} V. T)
+[ AAtom ⇒ RP L T
+| APair B A ⇒ ∀L0,V0,T0,des. aacr RP B L0 V0 → ⇩*[des] L0 ≡ L → ⇧*[des] T ≡ T0 →
+ aacr RP A L0 (ⓐV0. T0)
].
interpretation
- "candidate of reducibility (abstract)"
- 'InEInt R L T A = (acr R A L T).
+ "candidate of reducibility of an atomic arity (abstract)"
+ 'InEInt RP L T A = (aacr RP A L T).
+
+(* Basic properties *********************************************************)
+
+lemma acr_lifts: ∀C. S7 C → S7s C.
+#C #HC #L1 #L2 #des #H elim H -L1 -L2 -des
+[ #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/
+]
+qed.
+
+lemma rp_lifts: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
+ ∀des,L0,L,V,V0. ⇩*[des] L0 ≡ L → ⇧*[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/
+@(s7 … HRP)
+qed.
+
+lemma rp_liftsv_all: ∀RR,RS,RP. acr RR RS RP (λL,T. RP L T) →
+ ∀des,L0,L,Vs,V0s. ⇧*[des] Vs ≡ V0s → ⇩*[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.
+
+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
+ [1,3: // |2,4: skip
+ | @(s2 … IHB … ◊) // /2 width=2/
+ | #H @(cp3 … H1RP … 0) @(s1 … IHA) //
+ ]
+| #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #des #HB #HL0 #H
+ elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
+ lapply (s1 … IHB … HB) #HV0
+ @(s2 … IHA … (V0 :: V0s)) /2 width=4 by lifts_simple_dx/ /3 width=6/
+| #L #Vs #U #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 #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 … Hdes0 … HVW1 … HW12) // -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 … (des + 1)) /2 width=1/
+ [ @(s7 … IHB … HB … HV120) /2 width=1/
+ | @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/
+| /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 → (
+ ∀L0,V0,T0,des. ⇩*[des] L0 ≡ L → ⇧*[des + 1] T ≡ T0 →
+ ⦃L0, V0⦄ [RP] ϵ 〚B〛 → ⦃L0. ⓓV0, T0⦄ [RP] ϵ 〚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
+@(s3 … HCA … ◊) /2 width=6 by rp_lifts/
+@(s5 … HCA … ◊ ◊) // /2 width=1/ /2 width=3/
+qed.
projects / helm.git / blobdiff