(* *)
(**************************************************************************)
-include "basic_2/syntax/tsts_tsts.ma".
-include "basic_2/computation/cpxs_cpxs.ma".
-include "basic_2/computation/csx_alt.ma".
-include "basic_2/computation/csx_lift.ma".
+include "basic_2/rt_computation/cpxs_lpx.ma".
+include "basic_2/rt_computation/csx_cpxs.ma".
(* CONTEXT-SENSITIVE EXTENDED STRONGLY NORMALIZING TERMS ********************)
-(* Advanced properties ******************************************************)
+(* Properties with unbound parallel rt-transition on all entries ************)
-lemma csx_lpx_conf: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 →
- ∀T. ⦃G, L1⦄ ⊢ ⬊*[h, o] T → ⦃G, L2⦄ ⊢ ⬊*[h, o] T.
-#h #o #G #L1 #L2 #HL12 #T #H @(csx_ind_alt … H) -T
+lemma csx_lpx_conf: ∀h,o,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ →
+ ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+#h #o #G #L1 #T #H @(csx_ind_cpxs … H) -T
/4 width=3 by csx_intro, lpx_cpx_trans/
qed-.
-lemma csx_abst: ∀h,o,a,G,L,W. ⦃G, L⦄ ⊢ ⬊*[h, o] W →
- ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓛ{a}W.T.
-#h #o #a #G #L #W #HW @(csx_ind … HW) -W #W #_ #IHW #T #HT @(csx_ind … HT) -T #T #HT #IHT
+(* Advanced properties ******************************************************)
+
+lemma csx_abst: ∀h,o,p,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃W⦄ →
+ ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓛ{p}W.T⦄.
+#h #o #p #G #L #W #HW
+@(csx_ind … HW) -W #W #_ #IHW #T #HT
+@(csx_ind … HT) -T #T #HT #IHT
@csx_intro #X #H1 #H2
-elim (cpx_inv_abst1 … H1) -H1
-#W0 #T0 #HLW0 #HLT0 #H destruct
-elim (eq_false_inv_tpair_sn … H2) -H2
-[ -IHT #H lapply (csx_cpx_trans … HLT0) // -HT
- #HT0 lapply (csx_lpx_conf … (L.ⓛW0) … HT0) -HT0 /3 width=1 by lpx_pair/
-| -IHW -HLW0 -HT * #H destruct /3 width=1 by/
+elim (cpx_inv_abst1 … H1) -H1 #W0 #T0 #HLW0 #HLT0 #H destruct
+elim (tdneq_inv_pair … H2) -H2
+[ #H elim H -H //
+| -IHT #H lapply (csx_cpx_trans … o … HLT0) // -HT #HT0
+ /4 width=5 by csx_lpx_conf, lpx_pair/
+| -IHW -HT /4 width=3 by csx_cpx_trans, cpx_pair_sn/
]
qed.
-lemma csx_abbr: ∀h,o,a,G,L,V. ⦃G, L⦄ ⊢ ⬊*[h, o] V →
- ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬊*[h, o] T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}V. T.
-#h #o #a #G #L #V #HV elim HV -V #V #_ #IHV #T #HT @(csx_ind_alt … HT) -T #T #HT #IHT
+lemma csx_abbr: ∀h,o,p,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ →
+ ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.T⦄.
+#h #o #p #G #L #V #HV
+@(csx_ind … HV) -V #V #_ #IHV #T #HT
+@(csx_ind_cpxs … HT) -T #T #HT #IHT
@csx_intro #X #H1 #H2
elim (cpx_inv_abbr1 … H1) -H1 *
[ #V1 #T1 #HLV1 #HLT1 #H destruct
- elim (eq_false_inv_tpair_sn … H2) -H2
- [ /4 width=5 by csx_cpx_trans, csx_lpx_conf, lpx_pair/
- | -IHV -HLV1 * #H destruct /3 width=1 by cpx_cpxs/
+ elim (tdneq_inv_pair … H2) -H2
+ [ #H elim H -H //
+ | /4 width=5 by csx_cpx_trans, csx_lpx_conf, lpx_pair/
+ | -IHV /4 width=3 by csx_cpx_trans, cpx_cpxs, cpx_pair_sn/
]
-| -IHV -IHT -H2
- /3 width=8 by csx_cpx_trans, csx_inv_lift, drop_drop/
+| #U #HUT #HUX #_ -p
+ /5 width=7 by csx_cpx_trans, csx_inv_lifts, drops_drop, drops_refl/
]
qed.
-fact csx_appl_beta_aux: ∀h,o,a,G,L,U1. ⦃G, L⦄ ⊢ ⬊*[h, o] U1 →
- ∀V,W,T1. U1 = ⓓ{a}ⓝW.V.T1 → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.ⓛ{a}W.T1.
-#h #o #a #G #L #X #H @(csx_ind … H) -X
-#X #HT1 #IHT1 #V #W #T1 #H1 destruct
-@csx_intro #X #H1 #H2
-elim (cpx_inv_appl1 … H1) -H1 *
-[ -HT1 #V0 #Y #HLV0 #H #H0 destruct
- elim (cpx_inv_abst1 … H) -H #W0 #T0 #HLW0 #HLT0 #H destruct
- @IHT1 -IHT1 [4: // | skip |3: #H destruct /2 width=1 by/ ] -H2
- lapply (lsubr_cpx_trans … HLT0 (L.ⓓⓝW.V) ?) -HLT0 /3 width=1 by cpx_bind, cpx_flat, lsubr_beta/
-| -IHT1 -H2 #b #V0 #W0 #W2 #T0 #T2 #HLV0 #HLW02 #HLT02 #H1 #H3 destruct
- lapply (lsubr_cpx_trans … HLT02 (L.ⓓⓝW0.V) ?) -HLT02
- /4 width=5 by csx_cpx_trans, cpx_bind, cpx_flat, lsubr_beta/
-| -HT1 -IHT1 -H2 #b #V0 #V1 #W0 #W1 #T0 #T3 #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: was just: sn3_beta *)
-lemma csx_appl_beta: ∀h,o,a,G,L,V,W,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}ⓝW.V.T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV.ⓛ{a}W.T.
-/2 width=3 by csx_appl_beta_aux/ qed.
-
-fact csx_appl_theta_aux: ∀h,o,a,G,L,U. ⦃G, L⦄ ⊢ ⬊*[h, o] U → ∀V1,V2. ⬆[0, 1] V1 ≡ V2 →
- ∀V,T. U = ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV1.ⓓ{a}V.T.
-#h #o #a #G #L #X #H @(csx_ind_alt … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
+fact csx_appl_theta_aux: ∀h,o,p,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≘ V2 →
+ ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
+#h #o #p #G #L #X #H
+@(csx_ind_cpxs … H) -X #X #HVT #IHVT #V1 #V2 #HV12 #V #T #H destruct
lapply (csx_fwd_pair_sn … HVT) #HV
lapply (csx_fwd_bind_dx … HVT) -HVT #HVT
@csx_intro #X #HL #H
[ -HV #V0 #Y #HLV10 #HL #H0 destruct
elim (cpx_inv_abbr1 … HL) -HL *
[ #V3 #T3 #HV3 #HLT3 #H0 destruct
- elim (lift_total V0 0 1) #V4 #HV04
- elim (eq_term_dec (ⓓ{a}V.ⓐV2.T) (ⓓ{a}V3.ⓐV4.T3))
- [ -IHVT #H0 destruct
- elim (eq_false_inv_tpair_sn … H) -H
- [ -HLV10 -HV3 -HLT3 -HVT
- >(lift_inj … HV12 … HV04) -V4
- #H elim H //
- | * #_ #H elim H //
- ]
- | -H -HVT #H
- lapply (cpx_lift … HLV10 (L.ⓓV) (Ⓕ) … HV12 … HV04) -HLV10 -HV12 /2 width=1 by drop_drop/ #HV24
- @(IHVT … H … HV04) -IHVT /4 width=1 by cpx_cpxs, cpx_bind, cpx_flat/
+ elim (cpx_lifts_sn … HLV10 (Ⓣ) … (L.ⓓV) … HV12) -HLV10 /3 width=1 by drops_refl, drops_drop/ #V4 #HV04 #HV24
+ elim (tdeq_dec h o (ⓓ{p}V.ⓐV2.T) (ⓓ{p}V3.ⓐV4.T3)) #H0
+ [ -IHVT -HV3 -HV24 -HLT3
+ elim (tdeq_inv_pair … H0) -H0 #_ #HV3 #H0
+ elim (tdeq_inv_pair … H0) -H0 #_ #HV24 #HT3
+ elim (tdneq_inv_pair … H) -H #H elim H -H -G -L
+ /3 width=6 by tdeq_inv_lifts_bi, tdeq_pair/
+ | -V1 @(IHVT … H0 … HV04) -o -V0 /4 width=1 by cpx_cpxs, cpx_flat, cpx_bind/
]
- | -H -IHVT #T0 #HLT0 #HT0 #H0 destruct
- lapply (csx_cpx_trans … HVT (ⓐV2.T0) ?) /2 width=1 by cpx_flat/ -T #HVT0
- lapply (csx_inv_lift … L … (Ⓕ) … 1 HVT0 ? ? ?) -HVT0
- /3 width=5 by csx_cpx_trans, cpx_pair_sn, drop_drop, lift_flat/
+ | #T0 #HT0 #HLT0 #H0 destruct -H -IHVT
+ lapply (csx_inv_lifts … HVT (Ⓣ) … L ???) -HVT
+ [5:|*: /3 width=4 by drops_refl, drops_drop, lifts_flat/ ] -V2 -T #HVT
+ /3 width=5 by csx_cpx_trans, cpx_flat/
]
| -HV -HV12 -HVT -IHVT -H #b #V0 #W0 #W1 #T0 #T1 #_ #_ #_ #H destruct
| -IHVT -H #b #V0 #V3 #W0 #W1 #T0 #T1 #HLV10 #HV03 #HLW01 #HLT01 #H1 #H2 destruct
- lapply (cpx_lift … HLV10 (L. ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /2 width=2 by drop_drop/ #HLV23
+ lapply (cpx_lifts_bi … HLV10 (Ⓣ) … (L.ⓓW0) … HV12 … HV03) -HLV10 -HV12 -HV03 /3 width=1 by drops_refl, drops_drop/ #HLV23
@csx_abbr /2 width=3 by csx_cpx_trans/ -HV
@(csx_lpx_conf … (L.ⓓW0)) /2 width=1 by lpx_pair/ -W1
/4 width=5 by csx_cpxs_trans, cpx_cpxs, cpx_flat/
]
qed-.
-lemma csx_appl_theta: ∀h,o,a,V1,V2. ⬆[0, 1] V1 ≡ V2 →
- ∀G,L,V,T. ⦃G, L⦄ ⊢ ⬊*[h, o] ⓓ{a}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬊*[h, o] ⓐV1.ⓓ{a}V.T.
+lemma csx_appl_theta: ∀h,o,p,G,L,V,V2,T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ →
+ ∀V1. ⬆*[1] V1 ≘ V2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
/2 width=5 by csx_appl_theta_aux/ qed.
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