(* Properties with unbound parallel rt-transition on all entries ************)
-lemma csx_lpx_conf: ∀h,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ →
- ∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L2⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma csx_lpx_conf (h) (G) (L1):
+ ∀T. ❪G,L1❫ ⊢ ⬈*𝐒[h] T →
+ ∀L2. ❪G,L1❫ ⊢ ⬈[h] L2 → ❪G,L2❫ ⊢ ⬈*𝐒[h] T.
#h #G #L1 #T #H @(csx_ind_cpxs … H) -T
/4 width=3 by csx_intro, lpx_cpx_trans/
qed-.
(* Advanced properties ******************************************************)
-lemma csx_abst: ∀h,p,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ →
- ∀T. ⦃G, L.ⓛW⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓛ{p}W.T⦄.
-#h #p #G #L #W #HW
+lemma csx_abst (h) (G) (L):
+ ∀p,W. ❪G,L❫ ⊢ ⬈*𝐒[h] W →
+ ∀T. ❪G,L.ⓛW❫ ⊢ ⬈*𝐒[h] T → ❪G,L❫ ⊢ ⬈*𝐒[h] ⓛ[p]W.T.
+#h #G #L #p #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 (tdneq_inv_pair … H2) -H2
+elim (tneqx_inv_pair … H2) -H2
[ #H elim H -H //
| -IHT #H lapply (csx_cpx_trans … HLT0) // -HT #HT0
/4 width=5 by csx_lpx_conf, lpx_pair/
]
qed.
-lemma csx_abbr: ∀h,p,G,L,V. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ →
- ∀T. ⦃G, L.ⓓV⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.T⦄.
-#h #p #G #L #V #HV
+lemma csx_abbr (h) (G) (L):
+ ∀p,V. ❪G,L❫ ⊢ ⬈*𝐒[h] V →
+ ∀T. ❪G,L.ⓓV❫ ⊢ ⬈*𝐒[h] T → ❪G,L❫ ⊢ ⬈*𝐒[h] ⓓ[p]V.T.
+#h #G #L #p #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 (tdneq_inv_pair … H2) -H2
+ elim (tneqx_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/
]
qed.
-fact csx_appl_theta_aux: ∀h,p,G,L,U. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃U⦄ → ∀V1,V2. ⬆*[1] V1 ≘ V2 →
- ∀V,T. U = ⓓ{p}V.ⓐV2.T → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
-#h #p #G #L #X #H
+lemma csx_bind (h) (G) (L):
+ ∀p,I,V. ❪G,L❫ ⊢ ⬈*𝐒[h] V →
+ ∀T. ❪G,L.ⓑ[I]V❫ ⊢ ⬈*𝐒[h] T → ❪G,L❫ ⊢ ⬈*𝐒[h] ⓑ[p,I]V.T.
+#h #G #L #p * #V #HV #T #HT
+/2 width=1 by csx_abbr, csx_abst/
+qed.
+
+fact csx_appl_theta_aux (h) (G) (L):
+ ∀p,U. ❪G,L❫ ⊢ ⬈*𝐒[h] U → ∀V1,V2. ⇧[1] V1 ≘ V2 →
+ ∀V,T. U = ⓓ[p]V.ⓐV2.T → ❪G,L❫ ⊢ ⬈*𝐒[h] ⓐV1.ⓓ[p]V.T.
+#h #G #L #p #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
elim (cpx_inv_abbr1 … HL) -HL *
[ #V3 #T3 #HV3 #HLT3 #H0 destruct
elim (cpx_lifts_sn … HLV10 (Ⓣ) … (L.ⓓV) … HV12) -HLV10 /3 width=1 by drops_refl, drops_drop/ #V4 #HV04 #HV24
- elim (tdeq_dec (ⓓ{p}V.ⓐV2.T) (ⓓ{p}V3.ⓐV4.T3)) #H0
+ elim (teqx_dec (ⓓ[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/
+ elim (teqx_inv_pair … H0) -H0 #_ #HV3 #H0
+ elim (teqx_inv_pair … H0) -H0 #_ #HV24 #HT3
+ elim (tneqx_inv_pair … H) -H #H elim H -H -G -L
+ /3 width=6 by teqx_inv_lifts_bi, teqx_pair/
| -V1 @(IHVT … H0 … HV04) -V0 /4 width=1 by cpx_cpxs, cpx_flat, cpx_bind/
]
| #T0 #HT0 #HLT0 #H0 destruct -H -IHVT
]
qed-.
-lemma csx_appl_theta: ∀h,p,G,L,V,V2,T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓓ{p}V.ⓐV2.T⦄ →
- ∀V1. ⬆*[1] V1 ≘ V2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓐV1.ⓓ{p}V.T⦄.
+lemma csx_appl_theta (h) (G) (L):
+ ∀p,V,V2,T. ❪G,L❫ ⊢ ⬈*𝐒[h] ⓓ[p]V.ⓐV2.T →
+ ∀V1. ⇧[1] V1 ≘ V2 → ❪G,L❫ ⊢ ⬈*𝐒[h] ⓐV1.ⓓ[p]V.T.
/2 width=5 by csx_appl_theta_aux/ qed.