--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/ineint_5.ma".
+include "basic_2/grammar/aarity.ma".
+include "basic_2/multiple/mr2_mr2.ma".
+include "basic_2/multiple/lifts_lift_vector.ma".
+include "basic_2/multiple/drops_drop.ma".
+include "basic_2/computation/gcp.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+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: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: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: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: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:candidate.
+ ∀G,L,Vs. all … (RP G L) Vs → ∀k. C G L (ⒶVs.⋆k).
+
+definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
+ ⇩[i] L ≡ K.ⓑ{I}V1 → ⇧[0, i+1] V1 ≡ V2 →
+ C G L (ⒶVs.V2) → 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).
+
+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 ≝
+{ (* s0: S0 C; *)
+ 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
+}.
+
+(* the functional construction for candidates *)
+definition cfun: candidate → candidate → candidate ≝
+ λC1,C2,G,K,T. ∀V. C1 G K V → C2 G K (ⓐV.T).
+
+(* the reducibility candidate associated to an atomic arity *)
+let rec acr (RP:candidate) (A:aarity) on A: candidate ≝
+match A with
+[ AAtom ⇒ RP
+| APair B A ⇒ cfun (acr RP B) (acr RP A)
+].
+
+interpretation
+ "candidate of reducibility of an atomic arity (abstract)"
+ 'InEInt RP G L T A = (acr RP A G L T).
+
+(* Basic properties *********************************************************)
+(*
+(* Basic_1: was: sc3_lift1 *)
+lemma gcr_lifts: ∀C. S0 C → S0s C.
+#C #HC #G #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=10 by/
+]
+qed.
+*)
+axiom rp_lift: ∀RP. S0 RP.
+
+
+axiom rp_lifts: ∀RR,RS,RP. gcr 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
+@gcr_lifts /width=7 by/
+@(s0 … HRP)
+qed.
+*)
+(* Basic_1: was only: sns3_lifts1 *)
+axiom rp_liftsv_all: ∀RR,RS,RP. gcr 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 //
+#T1s #T2s #T1 #T2 #HT12 #_ #IHT2s * /3 width=7 by rp_lifts, conj/
+qed.
+*)
+
+lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀A. S0 (acr RP A).
+#RR #RS #RP #H1RP #H2RP #A elim A -A /2 width=7 by rp_lift/
+#B #A #IHB #IHA #G #L2 #L1 #T1 #d #e #IH #T2 #HL21 #HT12 #V #HB
+@(IHA … HL21) [3: @(lift_flat … HT12) |1: skip |
+
+(* Basic_1: was:
+ sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast sc3_lift
+*)
+lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀A. gcr RR RS RP (acr RP A).
+#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 (⋆k) ?) -H
+ [ lapply (s2 … IHB G L (◊) … HK) //
+ | #H @(cp2 … H1RP … k) @(s1 … IHA) //
+ ]
+| #G #L #Vs #HVs #T #H1T #H2T #V #HB
+ lapply (s1 … IHB … HB) #HV
+ @(s2 … IHA … (V @ Vs))
+ /3 width=14 by rp_liftsv_all, gcp_lifts, cp0, lifts_simple_dx, conj/
+| #a #G #L #Vs #U #T #W #HA #V #HB
+ @(s3 … IHA … (V @ Vs)) /2 width=1 by/
+| #G #L #Vs #HVs #k #V #HB
+ lapply (s1 … IHB … HB) #HV
+ @(s4 … IHA … (V @ Vs)) /3 width=7 by rp_liftsv_all, conj/
+| #I #G #L #K #Vs #V1 #V2 #i #HLK #HV12 #HA #V #HB
+ @(s5 … IHA … (V @ Vs) … HLK HV12) /2 width=1 by/
+| #G #L #V1s #V2s #HV12s #a #W #T #HA #HW #V1 #HB
+ elim (lift_total V1 0 1) #V2 #HV12
+ @(s6 … IHA … (V1 @ V1s) (V2 @ V2s)) /2 width=1 by liftv_cons/
+ @HA @(gcr_lift … H1RP H2RP … HB … HV12) /2 width=2 by drop_drop/
+| #G #L #Vs #T #W #HA #HW #V #HB
+ @(s7 … IHA … (V @ Vs)) /2 width=1 by/
+]
+qed.
+
+lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀a,G,L,W,T,B,A. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
+ ∀V. ⦃G, L, V⦄ ϵ[RP] 〚B〛 → ⦃G, L.ⓓⓝW.V, T⦄ ϵ[RP] 〚A〛
+ ) →
+ ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛.
+#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #B #A #HW #HA #L0 #V0 #X #des #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 … HL0 … HW0 HW) -HW [ @(s0 … HCB) ] #HW0
+lapply (s3 … HCA … a G L0 (◊)) #H @H -H
+lapply (s6 … HCA G L0 (◊) (◊) ?) // #H @H -H
+[ @(HA … HL0) //
+| lapply (s1 … HCB) -HCB #HCB
+ lapply (s7 … H2RP G L0 (◊)) /3 width=1 by/
+]
+qed.
+
+(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
+(* Basic_1: removed local theorems 1: sc3_sn3_abst *)
projects / helm.git / blobdiff