(* Advanced properties ******************************************************)
-lemma lpxs_pair: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 → ∀V1,V2. ⦃h, L1⦄ ⊢ V1 ➡*[g] V2 →
- ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[g] L2.ⓑ{I}V2.
+lemma lpxs_pair: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 → ∀V1,V2. ⦃h, L1⦄ ⊢ V1 ➡*[h, g] V2 →
+ ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
/2 width=1 by TC_lpx_sn_pair/ qed.
(* Advanced inversion lemmas ************************************************)
-lemma lpxs_inv_pair1: ∀h,g,I,K1,L2,V1. ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡*[g] L2 →
- ∃∃K2,V2. ⦃h, K1⦄ ⊢ ➡*[g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[g] V2 & L2 = K2.ⓑ{I}V2.
+lemma lpxs_inv_pair1: ∀h,g,I,K1,L2,V1. ⦃h, K1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2 →
+ ∃∃K2,V2. ⦃h, K1⦄ ⊢ ➡*[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[h, g] V2 & L2 = K2.ⓑ{I}V2.
/3 width=3 by TC_lpx_sn_inv_pair1, lpx_cpxs_trans/ qed-.
-lemma lpxs_inv_pair2: ∀h,g,I,L1,K2,V2. ⦃h, L1⦄ ⊢ ➡*[g] K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⦃h, K1⦄ ⊢ ➡*[g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[g] V2 & L1 = K1.ⓑ{I}V1.
+lemma lpxs_inv_pair2: ∀h,g,I,L1,K2,V2. ⦃h, L1⦄ ⊢ ➡*[h, g] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⦃h, K1⦄ ⊢ ➡*[h, g] K2 & ⦃h, K1⦄ ⊢ V1 ➡*[h, g] V2 & L1 = K1.ⓑ{I}V1.
/3 width=3 by TC_lpx_sn_inv_pair2, lpx_cpxs_trans/ qed-.
(* Properties on context-sensitive extended parallel computation for terms **)
lemma lpxs_cpxs_trans: ∀h,g. s_rs_trans … (cpx h g) (lpxs h g).
/3 width=5 by s_r_trans_TC1, lpxs_cpx_trans/ qed-.
-lemma cpxs_bind2: ∀h,g,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 →
- ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[g] T2 →
- ∀a. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[g] ⓑ{a,I}V2.T2.
+lemma cpxs_bind2: ∀h,g,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+ ∀I,T1,T2. ⦃h, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
#h #g #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
lapply (lpxs_cpxs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
qed.
(* Inversion lemmas on context-sensitive ext parallel computation for terms *)
-lemma cpxs_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[g] U2 →
- ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 ➡*[g] T2 &
+lemma cpxs_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡*[h, g] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 ➡*[h, g] T2 &
U2 = ⓛ{a}V2.T2.
#h #g #a #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /2 width=5/
#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
lapply (cpxs_trans … HT10 … HT02) -T0 /2 width=5/
qed-.
-lemma cpxs_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[g] U2 → (
- ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡*[g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[g] T2 &
+lemma cpxs_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡*[h, g] U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 &
U2 = ⓓ{a}V2.T2
) ∨
- ∃∃T2. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
+ ∃∃T2. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡*[h, g] T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
#h #g #a #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
#U0 #U2 #_ #HU02 * *
[ #V0 #T0 #HV10 #HT10 #H destruct
(* More advanced properties *************************************************)
-lemma lpxs_pair2: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[g] L2 →
- ∀V1,V2. ⦃h, L2⦄ ⊢ V1 ➡*[g] V2 → ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[g] L2.ⓑ{I}V2.
+lemma lpxs_pair2: ∀h,g,I,L1,L2. ⦃h, L1⦄ ⊢ ➡*[h, g] L2 →
+ ∀V1,V2. ⦃h, L2⦄ ⊢ V1 ➡*[h, g] V2 → ⦃h, L1.ⓑ{I}V1⦄ ⊢ ➡*[h, g] L2.ⓑ{I}V2.
/3 width=3 by lpxs_pair, lpxs_cpxs_trans/ qed.