+++ /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/syntax/aarity.ma".
-include "basic_2/relocation/lifts_simple.ma".
-include "basic_2/relocation/lifts_lifts_vector.ma".
-include "basic_2/relocation/drops_drops.ma".
-include "basic_2/static/gcp.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-(* 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 → ∀s. C G L (ⒶVs.⋆s).
-
-definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
- C G L (ⒶVs.V2) → ⬆*[↑i] V1 ≘ V2 →
- ⬇*[i] L ≘ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
-
-definition S6 ≝ λRP,C:candidate.
- ∀G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
- ∀a,V,T. C G (L.ⓓV) (ⒶV2b.T) → RP G L V → C G L (ⒶV1b.ⓓ{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
-}.
-
-(* the functional construction for candidates *)
-definition cfun: candidate → candidate → candidate ≝
- λC1,C2,G,K,T. ∀f,L,W,U.
- ⬇*[Ⓕ, f] L ≘ K → ⬆*[f] T ≘ U → C1 G L W → C2 G L (ⓐW.U).
-
-(* the reducibility candidate associated to an atomic arity *)
-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)
-].
-
-interpretation
- "reducibility candidate of an atomic arity (abstract)"
- 'InEInt RP G L T A = (acr RP A G L T).
-
-(* Basic properties *********************************************************)
-
-(* Note: this requires Ⓕ-slicing in cfun since b is unknown in d_liftable_1 *)
-(* Note: this requires multiple relocation *)
-(* Basic 1: includes: sc3_lift *)
-(* Basic 2A1: includes: gcr_lift *)
-(* Basic 2A1: note: gcr_lift should be acr_lift *)
-(* Basic_1: was: sc3_lift1 *)
-(* Basic 2A1: was: gcr_lifts *)
-(* Basic 2A1: note: gcr_lifts should be acr_lifts *)
-lemma acr_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftable1 (acr RP A G).
-#RR #RS #RP #H #A #G elim A -A
-[ /2 width=7 by cp2/
-| #B #A #HB #HA #K #T #HKT #b #f #L #HLK #U #HTU #f0 #L0 #W #U0 #HL0 #HU0 #HW
- lapply (drops_trans … HL0 … HLK ??) [3:|*: // ] -L #HL0K
- lapply (lifts_trans … HTU … HU0 ??) [3:|*: // ] -U #HTU0
- /2 width=3 by/ (**) (* full auto fails *)
-]
-qed-.
-
-(* Basic_1: was:
- sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast
-*)
-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) #s #HK
- lapply (s2 … IHB G L (Ⓔ) … HK) // #HB
- lapply (H (𝐈𝐝) L (⋆s) T ? ? ?) -H
- /3 width=6 by s1, cp3, drops_refl, lifts_refl/
-| #G #L #Vs #HVs #T #H1T #H2T #f #L0 #V0 #X #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=13 by cp0, gcp2_all, lifts_simple_dx, conj/
-| #p #G #L #Vs #U #T #W #HA #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- elim (lifts_inv_flat1 … H0) -H0 #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 #s #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- >(lifts_inv_sort1 … H0) -X0
- lapply (s1 … IHB … HB) #HV0
- @(s4 … IHA … (V0⨮V0s)) /3 width=7 by gcp2_all, conj/
-| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
- lapply (drops_trans … HL0 … HLK ??) [3: |*: // ] -HLK #H
- elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK0 #HY
- lapply (drops_tls_at … Hf … HY) -HY #HY
- elim (drops_inv_skip2 … HY) -HY #Z #K0 #HK0 #HZ #H destruct
- elim (liftsb_inv_pair_sn … HZ) -HZ #W1 #HVW1 #H destruct
- elim (lifts_total W1 (𝐔❴↑j❵)) #W2 #HW12
- lapply (lifts_trans … HVW1 … HW12 ??) -HVW1 [3: |*: // ] #H
- lapply (lifts_conf … HV12 … H f ?) -V1 [ /2 width=3 by after_uni_succ_sn/ ] #HVW2
- @(s5 … IHA … (V0⨮V0s) … HW12) /3 width=4 by drops_inv_gen, lifts_applv/
-| #G #L #V1s #V2s #HV12s #p #V #T #HA #HV #f #L0 #V10 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V10s #X0 #HV10s #H0 #H destruct
- elim (lifts_inv_bind1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
- elim (lifts_total V10 (𝐔❴1❵)) #V20 #HV120
- elim (liftsv_total (𝐔❴1❵) V10s) #V20s #HV120s
- @(s6 … IHA … (V10⨮V10s) (V20⨮V20s)) /3 width=7 by cp2, liftsv_cons/
- @(HA … (⫯f)) /3 width=2 by drops_skip, ext2_pair/
- [ @lifts_applv //
- lapply (liftsv_trans … HV10s … HV120s ??) -V10s [3: |*: // ] #H
- elim (liftsv_split_trans … H (𝐔❴1❵) (⫯f)) /2 width=1 by after_uni_one_sn/ #V10s #HV10s #HV120s
- >(liftsv_mono … HV12s … HV10s) -V1s //
- | @(acr_lifts … H1RP … HB … HV120) /3 width=2 by drops_refl, drops_drop/
- ]
-| #G #L #Vs #T #W #HA #HW #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- elim (lifts_inv_flat1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
- @(s7 … IHA … (V0⨮V0s)) /3 width=5 by lifts_applv/
-]
-qed.
-
-lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀p,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
- ∀b,f,L0,V0,W0,T0. ⬇*[b, f] L0 ≘ L → ⬆*[f] W ≘ W0 → ⬆*[⫯f] T ≘ T0 →
- ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
- ) →
- ⦃G, L, ⓛ{p}W.T⦄ ϵ[RP] 〚②B.A〛.
-#RR #RS #RP #H1RP #H2RP #p #G #L #W #T #A #B #HW #HA #f #L0 #V0 #X #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 (acr_lifts … H1RP … HW … HL0 … HW0) -HW #HW0
-lapply (s3 … HCA … p 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