(* *)
(**************************************************************************)
-include "basic_2/static/aaa_lift.ma".
-include "basic_2/static/lsuba_aaa.ma".
-include "basic_2/reduction/lpx_drop.ma".
+include "static_2/static/aaa_drops.ma".
+include "static_2/static/lsuba_aaa.ma".
+include "basic_2/rt_transition/lpx_drops.ma".
-(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
-(* Properties on atomic arity assignment for terms **************************)
+(* Properties with atomic arity assignment for terms ************************)
(* Note: lemma 500 *)
-lemma cpx_lpx_aaa_conf: ∀h,o,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A →
- ∀T2. ⦃G, L1⦄ ⊢ T1 ➡[h, o] T2 →
- ∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#h #o #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #o #L1 #s #X #H
- elim (cpx_inv_sort1 … H) -H // * //
-| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHV1 #X #H #L2 #HL12
- elim (cpx_inv_lref1 … H) -H
- [ #H destruct
- elim (lpx_drop_conf … HLK1 … HL12) -L1 #X #H #HLK2
- elim (lpx_inv_pair1 … H) -H
- #K2 #V2 #HK12 #HV12 #H destruct /3 width=6 by aaa_lref/
- | * #J #Y #Z #V2 #H #HV12 #HV2
- lapply (drop_mono … H … HLK1) -H #H destruct
- elim (lpx_drop_conf … HLK1 … HL12) -L1 #Z #H #HLK2
- elim (lpx_inv_pair1 … H) -H #K2 #V0 #HK12 #_ #H destruct
- /3 width=8 by aaa_lift, drop_fwd_drop2/
+(* Basic_2A1: was: cpx_lpx_aaa_conf *)
+lemma cpx_aaa_conf_lpx (h): ∀G,L1,T1,A. ⦃G,L1⦄ ⊢ T1 ⁝ A →
+ ∀T2. ⦃G,L1⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L2. ⦃G,L1⦄ ⊢ ⬈[h] L2 → ⦃G,L2⦄ ⊢ T2 ⁝ A.
+#h #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #G #L1 #s #X #H
+ elim (cpx_inv_sort1 … H) -H #H destruct //
+| #I #G #K1 #V1 #B #_ #IH #X #HX #Y #HY
+ elim (lpx_inv_pair_sn … HY) -HY #K2 #V2 #HK12 #HV12 #H destruct
+ elim (cpx_inv_zero1_pair … HX) -HX
+ [ #H destruct /3 width=1 by aaa_zero/
+ | * #V #HV1 #HVX -HV12
+ /4 width=7 by aaa_lifts, drops_refl, drops_drop, true/
]
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+| #I1 #G #K1 #A #i #_ #IH #X #HX #Y #HY
+ elim (lpx_inv_bind_sn … HY) -HY #I2 #K2 #HK12 #_ #H destruct
+ elim (cpx_inv_lref1_bind … HX) -HX
+ [ #H destruct /3 width=1 by aaa_lref/
+ | * #T #HT #HTX
+ /4 width=7 by aaa_lifts, drops_refl, drops_drop, true/
+ ]
+| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
elim (cpx_inv_abbr1 … H) -H *
[ #V2 #T2 #HV12 #HT12 #H destruct /4 width=2 by lpx_pair, aaa_abbr/
- | #T2 #HT12 #HT2 #H destruct -IHV1
- /4 width=8 by lpx_pair, aaa_inv_lift, drop_drop/
+ | #X1 #HXT1 #HX1 #H destruct -IHV1
+ elim (cpx_lifts_sn … HX1 (Ⓣ) … (L1.ⓓV1) … HXT1) -X1
+ /4 width=7 by lpx_pair, aaa_inv_lifts, drops_refl, drops_drop/
]
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
+| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
elim (cpx_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
/4 width=1 by lpx_pair, aaa_abst/
| #G #L1 #V1 #T1 #B #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
elim (cpx_inv_appl1 … H) -H *
[ #V2 #T2 #HV12 #HT12 #H destruct /3 width=3 by aaa_appl/
- | #b #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
+ | #q #V2 #W1 #W2 #U1 #U2 #HV12 #HW12 #HU12 #H1 #H2 destruct
lapply (IHV1 … HV12 … HL12) -IHV1 -HV12 #HV2
- lapply (IHT1 (ⓛ{b}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+ lapply (IHT1 (ⓛ{q}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
elim (aaa_inv_abst … H) -H #B0 #A0 #HW1 #HU2 #H destruct
/5 width=6 by lsuba_aaa_trans, lsuba_beta, aaa_abbr, aaa_cast/
- | #b #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
- lapply (aaa_lift G L2 … B … (L2.ⓓW2) … HV2) -HV2 /2 width=2 by drop_drop/ #HV2
- lapply (IHT1 (ⓓ{b}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
+ | #q #V #V2 #W1 #W2 #U1 #U2 #HV1 #HV2 #HW12 #HU12 #H1 #H2 destruct
+ lapply (aaa_lifts G L2 … B … (L2.ⓓW2) … HV2) -HV2 /3 width=2 by drops_refl, drops_drop/ #HV2
+ lapply (IHT1 (ⓓ{q}W2.U2) … HL12) -IHT1 /2 width=1 by cpx_bind/ -L1 #H
elim (aaa_inv_abbr … H) -H /3 width=3 by aaa_abbr, aaa_appl/
]
| #G #L1 #V1 #T1 #A #_ #_ #IHV1 #IHT1 #X #H #L2 #HL12
]
qed-.
-lemma cpx_aaa_conf: ∀h,o,G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
-/2 width=7 by cpx_lpx_aaa_conf/ qed-.
-
-lemma lpx_aaa_conf: ∀h,o,G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
-/2 width=7 by cpx_lpx_aaa_conf/ qed-.
-
-lemma cpr_aaa_conf: ∀G,L,T1,A. ⦃G, L⦄ ⊢ T1 ⁝ A → ∀T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T2 ⁝ A.
-/3 width=5 by cpx_aaa_conf, cpr_cpx/ qed-.
+lemma cpx_aaa_conf (h): ∀G,L. Conf3 … (aaa G L) (cpx h G L).
+/2 width=7 by cpx_aaa_conf_lpx/ qed-.
-lemma lpr_aaa_conf: ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L2⦄ ⊢ T ⁝ A.
-/3 width=5 by lpx_aaa_conf, lpr_lpx/ qed-.
+lemma lpx_aaa_conf (h): ∀G,T. Conf3 … (λL. aaa G L T) (lpx h G).
+/2 width=7 by cpx_aaa_conf_lpx/ qed-.