λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≡ K →
∀U. ⬆*[f] T ≡ U → R L U.
+definition d_deliftable1: predicate (relation2 lenv term) ≝
+ λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≡ K →
+ ∀T. ⬆*[f] T ≡ U → R K T.
+
definition d_liftable2: predicate (lenv → relation term) ≝
λR. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≡ K →
∀U1. ⬆*[f] T1 ≡ U1 →
+++ /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/syntax/genv.ma".
-include "basic_2/relocation/drops_vector.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-definition nf ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- λG,L,T. NF … (RR G L) RS T.
-
-definition candidate: Type[0] ≝ relation3 genv lenv term.
-
-definition CP0 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- ∀G. d_liftable1 (nf RR RS G).
-
-definition CP1 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- ∀G,L. ∃s. NF … (RR G L) RS (⋆s).
-
-definition CP2 ≝ λRP:candidate. ∀G. d_liftable1 (RP G).
-
-definition CP3 ≝ λRP:candidate.
- ∀G,L,T,s. RP G L (ⓐ⋆s.T) → RP G L T.
-
-(* requirements for generic computation properties *)
-(* Basic_1: includes: nf2_lift1 *)
-(* Basic_2A1: includes: gcp0_lifts *)
-(* Basic_2A1: includes: gcp2_lifts *)
-record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop ≝
-{ cp0: CP0 RR RS;
- cp1: CP1 RR RS;
- cp2: CP2 RP;
- cp3: CP3 RP
-}.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was only: sns3_lifts1 *)
-(* Basic_2A1: was: gcp2_lifts_all *)
-lemma gcp2_all: ∀RR,RS,RP. gcp RR RS RP → ∀G. d_liftable1_all (RP G).
-/3 width=7 by cp2, d1_liftable_liftable_all/ qed.
+++ /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/static/aaa_aaa.ma".
-include "basic_2/rt_computation/lsubc_drops.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: sc3_arity_csubc *)
-theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
- gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀b,f,L0. ⬇*[b, f] L0 ≡ L1 →
- ∀T0. ⬆*[f] T ≡ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
- ⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
-#RR #RS #RP #H1RP #H2RP #G #L1 #T @(fqup_wf_ind_eq … G L1 T) -G -L1 -T
-#Z #Y #X #IH #G #L1 * [ * | * [ #p ] * ]
-[ #s #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
- lapply (aaa_inv_sort … HA) -HA #H destruct
- >(lifts_inv_sort1 … H0) -H0
- lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
- lapply (s4 … HAtom G L2 (◊)) /2 width=1 by/
-| #i #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_lref_drops … HA) -HA #I #K1 #V1 #HLK1 #HKV1
- elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
- lapply (acr_gcr … H1RP H2RP A) #HA
- lapply (drops_trans … HL01 … HLK1 ??) -HL01 [3: |*: // ] #H
- elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK0 #HY
- lapply (drops_tls_at … Hf … HY) -Hf -HY #HY
- elim (drops_inv_skip2 … HY) -HY #K0 #V0 #HK01 #HV10 #H destruct
- elim (lifts_total V0 (𝐔❴⫯j❵)) #V #HV0
- elim (lsubc_drops_trans_isuni … HL20 … HLK0) -HL20 -HLK0 // #Y #HLK2 #H
- elim (lsubc_inv_pair2 … H) -H *
- [ #K2 #HK20 #H destruct
- lapply (drops_isuni_fwd_drop2 … HLK2) // #HLK2b
- lapply (s5 … HA ? G ? ? (◊) … HV0 ?) -HA
- /4 width=11 by acr_lifts, fqup_lref, drops_inv_gen/
- | #K2 #V2 #B #HKV2 #HK2V0 #HKV0B #_ #H1 #H2 destruct -IH -HLK1
- lapply (drops_isuni_fwd_drop2 … HLK2) // #HLK2b
- lapply (aaa_lifts … HKV1 … HK01 … HV10) -HKV1 -HK01 -HV10 #HKV0A
- lapply (aaa_mono … HKV0B … HKV0A) #H destruct -HKV0B -HKV0A
- elim (lifts_total V2 (𝐔❴⫯j❵)) #V3 #HV23
- lapply (s5 … HA … G … (◊) … (ⓝV0.V2) (ⓝV.V3) ????)
- [3: |*: /2 width=9 by drops_inv_gen, lifts_flat/ ] -HLK2
- lapply (s7 … HA G L2 (◊)) -HA /3 width=7 by acr_lifts/
- ]
-| #l #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
- elim (aaa_inv_gref … HA)
-| #V #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_abbr … HA) -HA #B #HV #HT
- elim (lifts_inv_bind1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
- lapply (acr_gcr … H1RP H2RP A) #HA
- lapply (acr_gcr … H1RP H2RP B) #HB
- lapply (s1 … HB) -HB #HB
- lapply (s6 … HA G L2 (◊) (◊)) /4 width=10 by lsubc_pair, liftsv_nil, drops_skip/
-| #W #T #HG #HL #HT #Z0 #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_abst … HA) -HA #B #A #HW #HT #H destruct
- elim (lifts_inv_bind1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
- @(acr_abst … H1RP H2RP) /2 width=10 by/
- #b3 #f3 #L3 #V3 #W3 #T3 #HL32 #HW03 #HT03 #H1B #H2B
- elim (drops_lsubc_trans … H1RP … HL32 … HL20) -L2 #L2 #HL32 #HL20
- lapply (aaa_lifts … HW … (f3∘f) L2 … W3 ?) -HW
- [4: |*: /2 width=8 by drops_trans, lifts_trans/ ] #HW3
- @(IH … ((↑f3)∘↑f) … (L2. ⓛW3)) -IH
- /3 width=12 by lsubc_beta, drops_trans, drops_skip, lifts_trans/
-| #V #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_appl … HA) -HA #B #HV #HT
- elim (lifts_inv_flat1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
- lapply (IH … HT … HL01 … HT0 … HL20) -HT -HT0
- /3 width=10 by drops_refl, lifts_refl/
-| #W #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_cast … HA) -HA #HW #HT
- elim (lifts_inv_flat1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
- lapply (acr_gcr … H1RP H2RP A) #HA
- lapply (s7 … HA G L2 (◊)) /3 width=10 by/
-]
-qed.
-
-(* Basic_1: was: sc3_arity *)
-lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
-/3 width=9 by drops_refl, lifts_refl, acr_aaa_csubc_lifts/ qed.
-
-lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
-#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
-lapply (acr_gcr … H1RP H2RP A) #HA
-@(s1 … HA) /2 width=4 by acr_aaa/
-qed.
+++ /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/rt_computation/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 #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 #s #f #L0 #V0 #X #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 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 #Y #HV0s #HY #H destruct
- elim (lifts_inv_lref1 … HY) -HY #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 #K0 #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 #Y #HV10s #HY #H destruct
- elim (lifts_inv_bind1 … HY) -HY #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)) /2 width=2 by drops_skip/
- [ @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 #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 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 *)
+++ /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/lrsubeqc_4.ma".
-include "basic_2/static/aaa.ma".
-include "basic_2/rt_computation/gcp_cr.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-
-inductive lsubc (RP) (G): relation lenv ≝
-| lsubc_atom: lsubc RP G (⋆) (⋆)
-| lsubc_pair: ∀I,L1,L2,V. lsubc RP G L1 L2 → lsubc RP G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lsubc_beta: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
- lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
-.
-
-interpretation
- "local environment refinement (generic reducibility)"
- 'LRSubEqC RP G L1 L2 = (lsubc RP G L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubc_inv_atom1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 = ⋆ → L2 = ⋆.
-#RP #G #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: was just: csubc_gen_sort_r *)
-lemma lsubc_inv_atom1: ∀RP,G,L2. G ⊢ ⋆ ⫃[RP] L2 → L2 = ⋆.
-/2 width=5 by lsubc_inv_atom1_aux/ qed-.
-
-fact lsubc_inv_pair1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
- (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃[RP] K2 &
- L2 = K2. ⓛW & X = ⓝW.V & I = Abbr.
-#RP #G #L1 #L2 * -L1 -L2
-[ #I #K1 #V #H destruct
-| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #V #H destruct /3 width=10 by ex7_4_intro, or_intror/
-]
-qed-.
-
-(* Basic_1: was: csubc_gen_head_r *)
-lemma lsubc_inv_pair1: ∀RP,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃[RP] L2 →
- (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓑ{I}X) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃[RP] K2 &
- L2 = K2.ⓛW & X = ⓝW.V & I = Abbr.
-/2 width=3 by lsubc_inv_pair1_aux/ qed-.
-
-fact lsubc_inv_atom2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L2 = ⋆ → L1 = ⋆.
-#RP #G #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: was just: csubc_gen_sort_l *)
-lemma lsubc_inv_atom2: ∀RP,G,L1. G ⊢ L1 ⫃[RP] ⋆ → L1 = ⋆.
-/2 width=5 by lsubc_inv_atom2_aux/ qed-.
-
-fact lsubc_inv_pair2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2,W. L2 = K2.ⓑ{I} W →
- (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1. ⓑ{I} W) ∨
- ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃[RP] K2 &
- L1 = K1.ⓓⓝW.V & I = Abst.
-#RP #G #L1 #L2 * -L1 -L2
-[ #I #K2 #W #H destruct
-| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #W #H destruct /3 width=8 by ex6_3_intro, or_intror/
-]
-qed-.
-
-(* Basic_1: was just: csubc_gen_head_l *)
-lemma lsubc_inv_pair2: ∀RP,I,G,L1,K2,W. G ⊢ L1 ⫃[RP] K2.ⓑ{I} W →
- (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1.ⓑ{I} W) ∨
- ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃[RP] K2 &
- L1 = K1.ⓓⓝW.V & I = Abst.
-/2 width=3 by lsubc_inv_pair2_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was just: csubc_refl *)
-lemma lsubc_refl: ∀RP,G,L. G ⊢ L ⫃[RP] L.
-#RP #G #L elim L -L /2 width=1 by lsubc_pair/
-qed.
-
-(* Basic_1: removed theorems 3:
- csubc_clear_conf csubc_getl_conf csubc_csuba
-*)
+++ /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/static/aaa_drops.ma".
-include "basic_2/rt_computation/lsubc.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-
-(* Properties with generic slicing ******************************************)
-
-(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
-(* Basic_1: includes: csubc_drop_conf_O *)
-(* Basic_2A1: includes: lsubc_drop_O1_trans *)
-lemma lsubc_drops_trans_isuni: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 →
- ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≡ K2 →
- ∃∃K1. ⬇*[b, f] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
-#RP #G #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #HL12 #IH #b #f #K2 #Hf #H
- elim (drops_inv_pair1_isuni … Hf H) -Hf -H *
- [ #Hf #H destruct -IH
- /3 width=3 by lsubc_pair, drops_refl, ex2_intro/
- | #g #Hg #HLK2 #H destruct -HL12
- elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
- ]
-| #L1 #L2 #V #W #A #HV #H1W #H2W #HL12 #IH #b #f #K2 #Hf #H
- elim (drops_inv_pair1_isuni … Hf H) -Hf -H *
- [ #Hf #H destruct -IH
- /3 width=8 by drops_refl, lsubc_beta, ex2_intro/
- | #g #Hg #HLK2 #H destruct -HL12
- elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
- ]
-]
-qed-.
-
-(* Basic_1: was: csubc_drop1_conf_rev *)
-(* Basic_1: includes: csubc_drop_conf_rev *)
-(* Basic_2A1: includes: drop_lsubc_trans *)
-lemma drops_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
- ∀b,f,G,L1,K1. ⬇*[b, f] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
- ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[b, f] L2 ≡ K2.
-#RR #RS #RP #HR #b #f #G #L1 #K1 #H elim H -f -L1 -K1
-[ #f #Hf #Y #H lapply (lsubc_inv_atom1 … H) -H
- #H destruct /4 width=3 by lsubc_atom, drops_atom, ex2_intro/
-| #f #I #L1 #K1 #V1 #_ #IH #K2 #HK12 elim (IH … HK12) -K1
- /3 width=5 by lsubc_pair, drops_drop, ex2_intro/
-| #f #I #L1 #K1 #V1 #V2 #HLK1 #HV21 #IH #X #H elim (lsubc_inv_pair1 … H) -H *
- [ #K2 #HK12 #H destruct -HLK1
- elim (IH … HK12) -K1 /3 width=5 by lsubc_pair, drops_skip, ex2_intro/
- | #K2 #V #W2 #A #HV #H1W2 #H2W2 #HK12 #H1 #H2 #H3 destruct
- elim (lifts_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
- elim (IH … HK12) -IH -HK12 #K #HL1K #HK2
- lapply (acr_lifts … HR … HV … HLK1 … HV3) -HV
- lapply (acr_lifts … HR … H1W2 … HLK1 … HW23) -H1W2
- /4 width=10 by lsubc_beta, aaa_lifts, drops_skip, ex2_intro/
- ]
-]
-qed-.
+++ /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/static/lsuba.ma".
-include "basic_2/rt_computation/gcp_aaa.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-
-(* Properties with lenv refinement for atomic arity assignment **************)
-
-lemma lsuba_lsubc: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → G ⊢ L1 ⫃[RP] L2.
-#RR #RS #RP #H1RP #H2RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubc_pair/
-#L1 #L2 #W #V #A #H elim (aaa_inv_cast … H) -H /3 width=4 by acr_aaa, lsubc_beta/
-qed.
+++ /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/static/lsubr.ma".
-include "basic_2/rt_computation/lsubc.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-
-(* Forward lemmas with restricted refinement for local environments *********)
-
-lemma lsubc_fwd_lsubr: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 ⫃ L2.
-#RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
-qed-.
-lsubc.ma lsubc_drops.ma lsubc_lsubr.ma lsubc_lsuba.ma
-gcp.ma gcp_cr.ma gcp_aaa.ma
cpxs.ma
(* *)
(**************************************************************************)
+include "basic_2/relocation/lifts_tdeq.ma".
include "basic_2/rt_transition/cpx_drops.ma".
include "basic_2/rt_transition/cnx.ma".
#I #K #V1 #V2 #HLK lapply (drops_mono … Hi … HLK) -L #H destruct
qed.
+(* Basic_2A1: includes: cnx_lift *)
+lemma cnx_lifts: ∀h,o,G. d_liftable1 … (cnx h o G).
+#h #o #G #K #T #HT #b #f #L #HLK #U #HTU #U0 #H
+elim (cpx_inv_lifts … H … HLK … HTU) -b -L #T0 #HTU0 #HT0
+lapply (HT … HT0) -G -K #HT0
+elim (tdeq_lifts … HT0 … HTU) -T #X #HX #HU
+<(lifts_mono … HX … HTU0) -T0 //
+qed-.
+
(* Inversion lemmas with generic slicing ************************************)
(* Basic_2A1: was: cnx_inv_delta *)
/2 width=5 by lifts_inv_lref2_uni_lt/
qed-.
-(*
-(* Relocation properties ****************************************************)
-
-lemma cnx_lift: ∀h,o,G,L0,L,T,T0,c,l,k. ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄ → ⬇[c, l, k] L0 ≡ L →
- ⬆[l, k] T ≡ T0 → ⦃G, L0⦄ ⊢ ➡[h, o] 𝐍⦃T0⦄.
-#h #o #G #L0 #L #T #T0 #c #l #k #HLT #HL0 #HT0 #X #H
-elim (cpx_inv_lift1 … H … HL0 … HT0) -L0 #T1 #HT10 #HT1
-<(HLT … HT1) in HT0; -L #HT0
->(lift_mono … HT10 … HT0) -T1 -X //
-qed.
-
-lemma cnx_inv_lift: ∀h,o,G,L0,L,T,T0,c,l,k. ⦃G, L0⦄ ⊢ ➡[h, o] 𝐍⦃T0⦄ → ⬇[c, l, k] L0 ≡ L →
- ⬆[l, k] T ≡ T0 → ⦃G, L⦄ ⊢ ➡[h, o] 𝐍⦃T⦄.
-#h #o #G #L0 #L #T #T0 #c #l #k #HLT0 #HL0 #HT0 #X #H
-elim (lift_total X l k) #X0 #HX0
-lapply (cpx_lift … H … HL0 … HT0 … HX0) -L #HTX0
->(HLT0 … HTX0) in HX0; -L0 -X0 #H
->(lift_inj … H … HT0) -T0 -X -l -k //
+(* Basic_2A1: includes: cnx_inv_lift *)
+lemma cnx_inv_lifts: ∀h,o,G. d_deliftable1 … (cnx h o G).
+#h #o #G #L #U #HU #b #f #K #HLK #T #HTU #T0 #H
+elim (cpx_lifts … H … HLK … HTU) -b -K #U0 #HTU0 #HU0
+lapply (HU … HU0) -G -L #HU0
+elim (tdeq_inv_lifts … HU0 … HTU) -U #X #HX #HT
+<(lifts_inj … HX … HTU0) -U0 //
qed-.
-*)
cpg.ma cpg_simple.ma cpg_drops.ma cpg_lsubr.ma
cpx.ma cpx_simple.ma cpx_drops.ma cpx_fqus.ma cpx_lsubr.ma cpx_lfxs.ma cpx_lfdeq.ma
lfpx.ma lfpx_length.ma lfpx_drops.ma lfpx_fqup.ma lfpx_frees.ma lfpx_aaa.ma
-cnx.ma cnx_simple.ma
+cnx.ma cnx_simple.ma cnx_drops.ma
cpm.ma cpm_simple.ma cpm_drops.ma cpm_lsubr.ma cpm_lfxs.ma cpm_cpx.ma
cpr.ma cpr_drops.ma
lfpr.ma lfpr_length.ma lfpr_drops.ma lfpr_fqup.ma lfpr_frees.ma lfpr_aaa.ma lfpr_lfpx.ma lfpr_lfpr.ma
--- /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/syntax/genv.ma".
+include "basic_2/relocation/drops_vector.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+definition nf ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+ λG,L,T. NF … (RR G L) RS T.
+
+definition candidate: Type[0] ≝ relation3 genv lenv term.
+
+definition CP0 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+ ∀G. d_liftable1 (nf RR RS G).
+
+definition CP1 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+ ∀G,L. ∃s. NF … (RR G L) RS (⋆s).
+
+definition CP2 ≝ λRP:candidate. ∀G. d_liftable1 (RP G).
+
+definition CP3 ≝ λRP:candidate.
+ ∀G,L,T,s. RP G L (ⓐ⋆s.T) → RP G L T.
+
+(* requirements for generic computation properties *)
+(* Basic_1: includes: nf2_lift1 *)
+(* Basic_2A1: includes: gcp0_lifts *)
+(* Basic_2A1: includes: gcp2_lifts *)
+record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop ≝
+{ cp0: CP0 RR RS;
+ cp1: CP1 RR RS;
+ cp2: CP2 RP;
+ cp3: CP3 RP
+}.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was only: sns3_lifts1 *)
+(* Basic_2A1: was: gcp2_lifts_all *)
+lemma gcp2_all: ∀RR,RS,RP. gcp RR RS RP → ∀G. d_liftable1_all (RP G).
+/3 width=7 by cp2, d1_liftable_liftable_all/ qed.
--- /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/static/aaa_aaa.ma".
+include "basic_2/static/lsubc_drops.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: sc3_arity_csubc *)
+theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
+ gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀b,f,L0. ⬇*[b, f] L0 ≡ L1 →
+ ∀T0. ⬆*[f] T ≡ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
+ ⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
+#RR #RS #RP #H1RP #H2RP #G #L1 #T @(fqup_wf_ind_eq … G L1 T) -G -L1 -T
+#Z #Y #X #IH #G #L1 * [ * | * [ #p ] * ]
+[ #s #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
+ lapply (aaa_inv_sort … HA) -HA #H destruct
+ >(lifts_inv_sort1 … H0) -H0
+ lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
+ lapply (s4 … HAtom G L2 (◊)) /2 width=1 by/
+| #i #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_lref_drops … HA) -HA #I #K1 #V1 #HLK1 #HKV1
+ elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (drops_trans … HL01 … HLK1 ??) -HL01 [3: |*: // ] #H
+ elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK0 #HY
+ lapply (drops_tls_at … Hf … HY) -Hf -HY #HY
+ elim (drops_inv_skip2 … HY) -HY #K0 #V0 #HK01 #HV10 #H destruct
+ elim (lifts_total V0 (𝐔❴⫯j❵)) #V #HV0
+ elim (lsubc_drops_trans_isuni … HL20 … HLK0) -HL20 -HLK0 // #Y #HLK2 #H
+ elim (lsubc_inv_pair2 … H) -H *
+ [ #K2 #HK20 #H destruct
+ lapply (drops_isuni_fwd_drop2 … HLK2) // #HLK2b
+ lapply (s5 … HA ? G ? ? (◊) … HV0 ?) -HA
+ /4 width=11 by acr_lifts, fqup_lref, drops_inv_gen/
+ | #K2 #V2 #B #HKV2 #HK2V0 #HKV0B #_ #H1 #H2 destruct -IH -HLK1
+ lapply (drops_isuni_fwd_drop2 … HLK2) // #HLK2b
+ lapply (aaa_lifts … HKV1 … HK01 … HV10) -HKV1 -HK01 -HV10 #HKV0A
+ lapply (aaa_mono … HKV0B … HKV0A) #H destruct -HKV0B -HKV0A
+ elim (lifts_total V2 (𝐔❴⫯j❵)) #V3 #HV23
+ lapply (s5 … HA … G … (◊) … (ⓝV0.V2) (ⓝV.V3) ????)
+ [3: |*: /2 width=9 by drops_inv_gen, lifts_flat/ ] -HLK2
+ lapply (s7 … HA G L2 (◊)) -HA /3 width=7 by acr_lifts/
+ ]
+| #l #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
+ elim (aaa_inv_gref … HA)
+| #V #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_abbr … HA) -HA #B #HV #HT
+ elim (lifts_inv_bind1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (acr_gcr … H1RP H2RP B) #HB
+ lapply (s1 … HB) -HB #HB
+ lapply (s6 … HA G L2 (◊) (◊)) /4 width=10 by lsubc_pair, liftsv_nil, drops_skip/
+| #W #T #HG #HL #HT #Z0 #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_abst … HA) -HA #B #A #HW #HT #H destruct
+ elim (lifts_inv_bind1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
+ @(acr_abst … H1RP H2RP) /2 width=10 by/
+ #b3 #f3 #L3 #V3 #W3 #T3 #HL32 #HW03 #HT03 #H1B #H2B
+ elim (drops_lsubc_trans … H1RP … HL32 … HL20) -L2 #L2 #HL32 #HL20
+ lapply (aaa_lifts … HW … (f3∘f) L2 … W3 ?) -HW
+ [4: |*: /2 width=8 by drops_trans, lifts_trans/ ] #HW3
+ @(IH … ((↑f3)∘↑f) … (L2. ⓛW3)) -IH
+ /3 width=12 by lsubc_beta, drops_trans, drops_skip, lifts_trans/
+| #V #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_appl … HA) -HA #B #HV #HT
+ elim (lifts_inv_flat1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
+ lapply (IH … HT … HL01 … HT0 … HL20) -HT -HT0
+ /3 width=10 by drops_refl, lifts_refl/
+| #W #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_cast … HA) -HA #HW #HT
+ elim (lifts_inv_flat1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (s7 … HA G L2 (◊)) /3 width=10 by/
+]
+qed.
+
+(* Basic_1: was: sc3_arity *)
+lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
+/3 width=9 by drops_refl, lifts_refl, acr_aaa_csubc_lifts/ qed.
+
+lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
+#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
+lapply (acr_gcr … H1RP H2RP A) #HA
+@(s1 … HA) /2 width=4 by acr_aaa/
+qed.
--- /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 #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 #s #f #L0 #V0 #X #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 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 #Y #HV0s #HY #H destruct
+ elim (lifts_inv_lref1 … HY) -HY #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 #K0 #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 #Y #HV10s #HY #H destruct
+ elim (lifts_inv_bind1 … HY) -HY #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)) /2 width=2 by drops_skip/
+ [ @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 #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 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 *)
--- /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/lrsubeqc_4.ma".
+include "basic_2/static/aaa.ma".
+include "basic_2/static/gcp_cr.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+inductive lsubc (RP) (G): relation lenv ≝
+| lsubc_atom: lsubc RP G (⋆) (⋆)
+| lsubc_pair: ∀I,L1,L2,V. lsubc RP G L1 L2 → lsubc RP G (L1.ⓑ{I}V) (L2.ⓑ{I}V)
+| lsubc_beta: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
+ lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+ "local environment refinement (generic reducibility)"
+ 'LRSubEqC RP G L1 L2 = (lsubc RP G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubc_inv_atom1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 = ⋆ → L2 = ⋆.
+#RP #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was just: csubc_gen_sort_r *)
+lemma lsubc_inv_atom1: ∀RP,G,L2. G ⊢ ⋆ ⫃[RP] L2 → L2 = ⋆.
+/2 width=5 by lsubc_inv_atom1_aux/ qed-.
+
+fact lsubc_inv_pair1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1,X. L1 = K1.ⓑ{I}X →
+ (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
+ L2 = K2. ⓛW & X = ⓝW.V & I = Abbr.
+#RP #G #L1 #L2 * -L1 -L2
+[ #I #K1 #V #H destruct
+| #J #L1 #L2 #V #HL12 #I #K1 #W #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #V #H destruct /3 width=10 by ex7_4_intro, or_intror/
+]
+qed-.
+
+(* Basic_1: was: csubc_gen_head_r *)
+lemma lsubc_inv_pair1: ∀RP,I,G,K1,L2,X. G ⊢ K1.ⓑ{I}X ⫃[RP] L2 →
+ (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓑ{I}X) ∨
+ ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
+ L2 = K2.ⓛW & X = ⓝW.V & I = Abbr.
+/2 width=3 by lsubc_inv_pair1_aux/ qed-.
+
+fact lsubc_inv_atom2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L2 = ⋆ → L1 = ⋆.
+#RP #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #V #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was just: csubc_gen_sort_l *)
+lemma lsubc_inv_atom2: ∀RP,G,L1. G ⊢ L1 ⫃[RP] ⋆ → L1 = ⋆.
+/2 width=5 by lsubc_inv_atom2_aux/ qed-.
+
+fact lsubc_inv_pair2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2,W. L2 = K2.ⓑ{I} W →
+ (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1. ⓑ{I} W) ∨
+ ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
+ L1 = K1.ⓓⓝW.V & I = Abst.
+#RP #G #L1 #L2 * -L1 -L2
+[ #I #K2 #W #H destruct
+| #J #L1 #L2 #V #HL12 #I #K2 #W #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #W #H destruct /3 width=8 by ex6_3_intro, or_intror/
+]
+qed-.
+
+(* Basic_1: was just: csubc_gen_head_l *)
+lemma lsubc_inv_pair2: ∀RP,I,G,L1,K2,W. G ⊢ L1 ⫃[RP] K2.ⓑ{I} W →
+ (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1.ⓑ{I} W) ∨
+ ∃∃K1,V,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
+ L1 = K1.ⓓⓝW.V & I = Abst.
+/2 width=3 by lsubc_inv_pair2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: csubc_refl *)
+lemma lsubc_refl: ∀RP,G,L. G ⊢ L ⫃[RP] L.
+#RP #G #L elim L -L /2 width=1 by lsubc_pair/
+qed.
+
+(* Basic_1: removed theorems 3:
+ csubc_clear_conf csubc_getl_conf csubc_csuba
+*)
--- /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/static/aaa_drops.ma".
+include "basic_2/static/lsubc.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+(* Properties with generic slicing ******************************************)
+
+(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
+(* Basic_1: includes: csubc_drop_conf_O *)
+(* Basic_2A1: includes: lsubc_drop_O1_trans *)
+lemma lsubc_drops_trans_isuni: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 →
+ ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≡ K2 →
+ ∃∃K1. ⬇*[b, f] L1 ≡ K1 & G ⊢ K1 ⫃[RP] K2.
+#RP #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #HL12 #IH #b #f #K2 #Hf #H
+ elim (drops_inv_pair1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=3 by lsubc_pair, drops_refl, ex2_intro/
+ | #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
+ ]
+| #L1 #L2 #V #W #A #HV #H1W #H2W #HL12 #IH #b #f #K2 #Hf #H
+ elim (drops_inv_pair1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=8 by drops_refl, lsubc_beta, ex2_intro/
+ | #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
+ ]
+]
+qed-.
+
+(* Basic_1: was: csubc_drop1_conf_rev *)
+(* Basic_1: includes: csubc_drop_conf_rev *)
+(* Basic_2A1: includes: drop_lsubc_trans *)
+lemma drops_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
+ ∀b,f,G,L1,K1. ⬇*[b, f] L1 ≡ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
+ ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[b, f] L2 ≡ K2.
+#RR #RS #RP #HR #b #f #G #L1 #K1 #H elim H -f -L1 -K1
+[ #f #Hf #Y #H lapply (lsubc_inv_atom1 … H) -H
+ #H destruct /4 width=3 by lsubc_atom, drops_atom, ex2_intro/
+| #f #I #L1 #K1 #V1 #_ #IH #K2 #HK12 elim (IH … HK12) -K1
+ /3 width=5 by lsubc_pair, drops_drop, ex2_intro/
+| #f #I #L1 #K1 #V1 #V2 #HLK1 #HV21 #IH #X #H elim (lsubc_inv_pair1 … H) -H *
+ [ #K2 #HK12 #H destruct -HLK1
+ elim (IH … HK12) -K1 /3 width=5 by lsubc_pair, drops_skip, ex2_intro/
+ | #K2 #V #W2 #A #HV #H1W2 #H2W2 #HK12 #H1 #H2 #H3 destruct
+ elim (lifts_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
+ elim (IH … HK12) -IH -HK12 #K #HL1K #HK2
+ lapply (acr_lifts … HR … HV … HLK1 … HV3) -HV
+ lapply (acr_lifts … HR … H1W2 … HLK1 … HW23) -H1W2
+ /4 width=10 by lsubc_beta, aaa_lifts, drops_skip, ex2_intro/
+ ]
+]
+qed-.
--- /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/static/lsuba.ma".
+include "basic_2/static/gcp_aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+(* Properties with lenv refinement for atomic arity assignment **************)
+
+lemma lsuba_lsubc: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → G ⊢ L1 ⫃[RP] L2.
+#RR #RS #RP #H1RP #H2RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubc_pair/
+#L1 #L2 #W #V #A #H elim (aaa_inv_cast … H) -H /3 width=4 by acr_aaa, lsubc_beta/
+qed.
--- /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/static/lsubr.ma".
+include "basic_2/static/lsubc.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+(* Forward lemmas with restricted refinement for local environments *********)
+
+lemma lsubc_fwd_lsubr: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 ⫃ L2.
+#RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_pair, lsubr_beta/
+qed-.
[ "cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" * ]
}
]
- [ { "generic reducibility" * } {
- [ "lsubc ( ? ⊢ ? ⫃[?] ? )" "lsubc_drop" + "lsubc_drops" + "lsubc_lsubr" + "lsubc_lsuba" * ]
- [ "gcp_cr ( ⦃?,?,?⦄ ϵ[?] 〚?〛 )" "gcp_aaa" * ]
- [ "gcp" *]
- }
- ]
}
]
class "water"
]
class "green"
[ { "static typing" * } {
+ [ { "generic reducibility" * } {
+ [ "lsubc ( ? ⊢ ? ⫃[?] ? )" "lsubc_drop" + "lsubc_drops" + "lsubc_lsubr" + "lsubc_lsuba" * ]
+ [ "gcp_cr ( ⦃?,?,?⦄ ϵ[?] 〚?〛 )" "gcp_aaa" * ]
+ [ "gcp" *]
+ }
+ ]
[ { "atomic arity assignment" * } {
[ "lsuba ( ? ⊢ ? ⫃⁝ ? )" "lsuba_drops" + "lsuba_lsubr" + "lsuba_aaa" + "lsuba_lsuba" * ]
[ "aaa ( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_lfdeq" + "aaa_aaa" * ]
projects / helm.git / commitdiff