(* *)
(**************************************************************************)
-include "basic_2/rt_transition/lfpx_lfdeq.ma".
+include "basic_2/rt_transition/lpx_rdeq.ma".
include "basic_2/rt_computation/csx_drops.ma".
-(* STRONGLY NORMALIZING TERMS FOR UNCOUNTED PARALLEL RT-TRANSITION **********)
+(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
(* Advanced properties ******************************************************)
-lemma csx_tdeq_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀T2. T1 ≛[h, o] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
+lemma csx_tdeq_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T2. T1 ≛ T2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #G #L #T1 #H @(csx_ind … H) -T1 #T #_ #IH #T2 #HT2
@csx_intro #T1 #HT21 #HnT21 elim (tdeq_cpx_trans … HT2 … HT21) -HT21
/4 width=5 by tdeq_repl/
qed-.
-lemma csx_cpx_trans: ∀h,o,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
-elim (tdeq_dec h o T1 T2) /3 width=4 by csx_tdeq_trans/
+lemma csx_cpx_trans: ∀h,G,L,T1. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
+ ∀T2. ⦃G, L⦄ ⊢ T1 ⬈[h] T2 → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T2⦄.
+#h #G #L #T1 #H @(csx_ind … H) -T1 #T1 #HT1 #IHT1 #T2 #HLT12
+elim (tdeq_dec T1 T2) /3 width=4 by csx_tdeq_trans/
qed-.
(* Basic_1: was just: sn3_cast *)
-lemma csx_cast: ∀h,o,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃W⦄ →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃ⓝW.T⦄.
-#h #o #G #L #W #HW @(csx_ind … HW) -W
+lemma csx_cast: ∀h,G,L,W. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃W⦄ →
+ ∀T. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃ⓝW.T⦄.
+#h #G #L #W #HW @(csx_ind … HW) -W
#W #HW #IHW #T #HT @(csx_ind … HT) -T
#T #HT #IHT @csx_intro
#X #H1 #H2 elim (cpx_inv_cast1 … H1) -H1
(* Basic_1: was just: sn3_abbr *)
(* Basic_2A1: was: csx_lref_bind *)
-lemma csx_lref_pair: ∀h,o,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
- ⦃G, K⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄ → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃#i⦄.
-#h #o #I #G #L #K #V #i #HLK #HV
+lemma csx_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
+ ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄ → ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄.
+#h #I #G #L #K #V #i #HLK #HV
@csx_intro #X #H #Hi elim (cpx_inv_lref1_drops … H) -H
[ #H destruct elim Hi //
| -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
(* Basic_1: was: sn3_gen_def *)
(* Basic_2A1: was: csx_inv_lref_bind *)
-lemma csx_inv_lref_pair: ∀h,o,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
- ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃#i⦄ → ⦃G, K⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
-#h #o #I #G #L #K #V #i #HLK #Hi
+lemma csx_inv_lref_pair: ∀h,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V →
+ ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ → ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+#h #I #G #L #K #V #i #HLK #Hi
elim (lifts_total V (𝐔❴↑i❵))
/4 width=9 by csx_inv_lifts, csx_cpx_trans, cpx_delta_drops, drops_isuni_fwd_drop2/
qed-.
-lemma csx_inv_lref: ∀h,o,G,L,i. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃#i⦄ →
+lemma csx_inv_lref: ∀h,G,L,i. ⦃G, L⦄ ⊢ ⬈*[h] 𝐒⦃#i⦄ →
∨∨ ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆
| ∃∃I,K. ⬇*[i] L ≘ K.ⓤ{I}
- | ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ ⬈*[h, o] 𝐒⦃V⦄.
-#h #o #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/
+ | ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ ⬈*[h] 𝐒⦃V⦄.
+#h #G #L #i #H elim (drops_F_uni L i) /2 width=1 by or3_intro0/
* * /4 width=9 by csx_inv_lref_pair, ex2_3_intro, ex1_2_intro, or3_intro2, or3_intro1/
qed-.