(* Properties with uncounted context-sensitive rt-computation for terms *****)
-lemma lfpxs_pair: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
-#h #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
-/3 width=3 by lfpxs_strap1, lfpx_pair/
-qed.
+(* Basic_2A1: uses: lpxs_pair lpxs_pair_refl *)
+lemma lfpxs_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈*[h, T] L.ⓑ{I}V2.
+/2 width=1 by tc_lfxs_pair_refl/ qed.
-(* Basic_2A1: was just: lpxs_cpx_trans *)
+(* Basic_2A1: uses: lpxs_cpx_trans *)
lemma lfpxs_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (lfpxs h G).
-/3 width=5 by s_r_trans_LTC2, lfpx_cpxs_trans/ qed-.
+#h #G @s_r_trans_LTC2 @lfpx_cpxs_trans (**) (* auto fails *)
+qed-.
(* Note: lfpxs_cpx_conf does not hold, thus we cannot invoke s_r_trans_LTC1 *)
-(* Basic_2A1: was just: lpxs_cpxs_trans *)
+(* Basic_2A1: uses: lpxs_cpxs_trans *)
lemma lfpxs_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (lfpxs h G).
#h #G @s_r_to_s_rs_trans @s_r_trans_LTC2
@s_rs_trans_TC1 /2 width=3 by lfpx_cpxs_trans/ (**) (* full auto too slow *)
lemma cpxs_bind2: ∀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.
-/4 width=3 by lfpxs_cpxs_trans, lfpxs_pair, cpxs_bind/ qed.
+/4 width=3 by lfpxs_cpxs_trans, lfpxs_pair_refl, cpxs_bind/ qed.
(* Advanced inversion lemmas on uncounted rt-computation for terms **********)
#U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
elim (cpx_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
lapply (lfpxs_cpx_trans … HT02 (L.ⓛV1) ?)
-/3 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro/
+/3 width=5 by lfpxs_pair_refl, cpxs_trans, cpxs_strap1, ex3_2_intro/
qed-.
lemma cpxs_inv_abbr1: ∀h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈*[h] U2 → (
elim (cpx_inv_abbr1 … HU02) -HU02 *
[ #V2 #T2 #HV02 #HT02 #H destruct
lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?)
- /4 width=5 by lfpxs_pair, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
+ /4 width=5 by lfpxs_pair_refl, cpxs_trans, cpxs_strap1, ex3_2_intro, or_introl/
| #T2 #HT02 #HUT2
lapply (lfpxs_cpx_trans … HT02 (L.ⓓV1) ?) -HT02
- /4 width=3 by lfpxs_pair, cpxs_trans, ex3_intro, or_intror/
+ /4 width=3 by lfpxs_pair_refl, cpxs_trans, ex3_intro, or_intror/
]
| #U1 #HTU1 #HU01
- elim (cpx_lifts … HU02 (Ⓣ) … (L.ⓓV1) … HU01)
+ elim (cpx_lifts_sn … HU02 (Ⓣ) … (L.ⓓV1) … HU01)
/4 width=3 by cpxs_strap1, drops_refl, drops_drop, ex3_intro, or_intror/
]
qed-.