include "basic_2/syntax/tdeq_tdeq.ma".
include "basic_2/rt_transition/lfpx_lfdeq.ma".
-include "basic_2/rt_computation/csx.ma".
+include "basic_2/rt_computation/csx_drops.ma".
(* STRONGLY NORMALIZING TERMS FOR UNCOUNTED 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⦄.
+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
@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⦄.
+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/
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⦄.
+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
#W #HW #IHW #T #HT @(csx_ind … HT) -T
#T #HT #IHT @csx_intro
#X #H1 #H2 elim (cpx_inv_cast1 … H1) -H1
[ * #W0 #T0 #HLW0 #HLT0 #H destruct
- elim (tdeq_false_inv_pair … H2) -H2
+ elim (tdneq_inv_pair … H2) -H2
[ -W -T #H elim H -H //
| -HW -IHT /3 width=3 by csx_cpx_trans/
| -HW -HT -IHW /4 width=3 by csx_cpx_trans, cpx_pair_sn/
|*: /3 width=3 by csx_cpx_trans/
]
qed.
+
+(* Basic_1: was just: sn3_abbr *)
+(* Basic_2A1: was: csx_lref_bind *)
+lemma csx_lref_drops: ∀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
+@csx_intro #X #H #Hi elim (cpx_inv_lref1_drops … H) -H
+[ #H destruct elim Hi //
+| -Hi * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HV1
+ lapply (drops_mono … HLK0 … HLK) -HLK #H destruct
+ /3 width=8 by csx_lifts, csx_cpx_trans, drops_isuni_fwd_drop2/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_1: was: sn3_gen_def *)
+(* Basic_2A1: was: csx_inv_lref_bind *)
+lemma csx_inv_lref_drops: ∀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
+elim (lifts_total V (𝐔❴⫯i❵))
+/4 width=9 by csx_inv_lifts, csx_cpx_trans, cpx_delta_drops, drops_isuni_fwd_drop2/
+qed-.