include "basic_2/rt_computation/lpxs_lpx.ma".
-(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
+(* EXTENDED PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS *************)
(* Properties with context-sensitive extended rt-computation for terms ******)
(* Basic_2A1: was: cpxs_bind2 *)
-lemma cpxs_bind_dx (h) (G): ∀L,V1,V2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I,T1,T2. ⦃G,L.ⓑ{I}V2⦄ ⊢ T1 ⬈*[h] T2 →
- ∀p. ⦃G,L⦄ ⊢ ⓑ{p,I}V1.T1 ⬈*[h] ⓑ{p,I}V2.T2.
+lemma cpxs_bind_alt (G):
+ ∀L,V1,V2. ❨G,L❩ ⊢ V1 ⬈* V2 →
+ ∀I,T1,T2. ❨G,L.ⓑ[I]V2❩ ⊢ T1 ⬈* T2 →
+ ∀p. ❨G,L❩ ⊢ ⓑ[p,I]V1.T1 ⬈* ⓑ[p,I]V2.T2.
/4 width=5 by lpxs_cpxs_trans, lpxs_pair, cpxs_bind/ qed.
(* Inversion lemmas with context-sensitive ext rt-computation for terms *****)
-lemma cpxs_inv_abst1 (h) (G): ∀p,L,V1,T1,U2. ⦃G,L⦄ ⊢ ⓛ{p}V1.T1 ⬈*[h] U2 →
- ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L.ⓛV1⦄ ⊢ T1 ⬈*[h] T2 &
- U2 = ⓛ{p}V2.T2.
-#h #G #p #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
+lemma cpxs_inv_abst1 (G):
+ ∀p,L,V1,T1,U2. ❨G,L❩ ⊢ ⓛ[p]V1.T1 ⬈* U2 →
+ ∃∃V2,T2. ❨G,L❩ ⊢ V1 ⬈* V2 & ❨G,L.ⓛV1❩ ⊢ T1 ⬈* T2 & U2 = ⓛ[p]V2.T2.
+#G #p #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5 by ex3_2_intro/
#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
lapply (lpxs_cpx_trans … HT02 (L.ⓛV1) ?)
qed-.
(* Basic_2A1: was: cpxs_inv_abbr1 *)
-lemma cpxs_inv_abbr1_dx (h) (p) (G) (L):
- ∀V1,T1,U2. ⦃G,L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 →
- ∨∨ ∃∃V2,T2. ⦃G,L⦄ ⊢ V1 ⬈*[h] V2 & ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 &
- U2 = ⓓ{p}V2.T2
- | ∃∃T2. ⦃G,L.ⓓV1⦄ ⊢ T1 ⬈*[h] T2 & ⬆*[1] U2 ≘ T2 & p = Ⓣ.
-#h #p #G #L #V1 #T1 #U2 #H
+lemma cpxs_inv_abbr1_dx (p) (G) (L):
+ ∀V1,T1,U2. ❨G,L❩ ⊢ ⓓ[p]V1.T1 ⬈* U2 →
+ ∨∨ ∃∃V2,T2. ❨G,L❩ ⊢ V1 ⬈* V2 & ❨G,L.ⓓV1❩ ⊢ T1 ⬈* T2 & U2 = ⓓ[p]V2.T2
+ | ∃∃T2. ❨G,L.ⓓV1❩ ⊢ T1 ⬈* T2 & ⇧[1] U2 ≘ T2 & p = Ⓣ.
+#p #G #L #V1 #T1 #U2 #H
@(cpxs_ind … H) -U2 /3 width=5 by ex3_2_intro, or_introl/
#U0 #U2 #_ #HU02 * *
[ #V0 #T0 #HV10 #HT10 #H destruct