include "basic_2/rt_computation/cpxs_cpxs.ma".
include "basic_2/rt_computation/csx_csx.ma".
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-(* Properties with uncounted context-sensitive rt-computation for terms *****)
+(* Properties with unbound context-sensitive rt-computation for terms *******)
(* Basic_1: was just: sn3_intro *)
lemma csx_intro_cpxs: ∀h,o,G,L,T1.
- (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T2 â\86\92 (T1 â\89¡[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
+ (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T2 â\86\92 (T1 â\89\9b[h, o] T2 → ⊥) → ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄) →
⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄.
/4 width=1 by cpx_cpxs, csx_intro/ qed-.
/2 width=3 by csx_cpx_trans/
qed-.
-(* Eliminators with uncounted context-sensitive rt-computation for terms ****)
+(* Eliminators with unbound context-sensitive rt-computation for terms ******)
-lemma csx_ind_cpxs_tdeq: ∀h,o,G,L. ∀R:predicate term.
+lemma csx_ind_cpxs_tdeq: ∀h,o,G,L. ∀Q:predicate term.
(∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T2 â\86\92 (T1 â\89¡[h, o] T2 â\86\92 â\8a¥) â\86\92 R T2) â\86\92 R T1
+ (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T2 â\86\92 (T1 â\89\9b[h, o] T2 â\86\92 â\8a¥) â\86\92 Q T2) â\86\92 Q T1
) →
∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- â\88\80T0. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T0 â\86\92 â\88\80T2. T0 â\89¡[h, o] T2 â\86\92 R T2.
-#h #o #G #L #R #IH #T1 #H @(csx_ind … H) -T1
+ â\88\80T0. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T0 â\86\92 â\88\80T2. T0 â\89\9b[h, o] T2 â\86\92 Q T2.
+#h #o #G #L #Q #IH #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IH1 #T0 #HT10 #T2 #HT02
@IH -IH /3 width=3 by csx_cpxs_trans, csx_tdeq_trans/ -HT1 #V2 #HTV2 #HnTV2
lapply (tdeq_tdneq_trans … HT02 … HnTV2) -HnTV2 #H
elim (tdeq_dec h o T1 T0) #H
[ lapply (tdeq_tdneq_trans … H … HnTV0) -H -HnTV0 #Hn10
lapply (cpxs_trans … HT10 … HTV0) -T0 #H10
- elim (cpxs_tdneq_inv_step_sn … H10 … Hn10) -H10 -Hn10
+ elim (cpxs_tdneq_fwd_step_sn … H10 … Hn10) -H10 -Hn10
/3 width=8 by tdeq_trans/
-| elim (cpxs_tdneq_inv_step_sn … HT10 … H) -HT10 -H #T #V #HT1 #HnT1 #HTV #HVT0
+| elim (cpxs_tdneq_fwd_step_sn … HT10 … H) -HT10 -H #T #V #HT1 #HnT1 #HTV #HVT0
elim (tdeq_cpxs_trans … HVT0 … HTV0) -T0
/3 width=8 by cpxs_trans, tdeq_trans/
]
qed-.
(* Basic_2A1: was: csx_ind_alt *)
-lemma csx_ind_cpxs: ∀h,o,G,L. ∀R:predicate term.
+lemma csx_ind_cpxs: ∀h,o,G,L. ∀Q:predicate term.
(∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T2 â\86\92 (T1 â\89¡[h, o] T2 â\86\92 â\8a¥) â\86\92 R T2) â\86\92 R T1
+ (â\88\80T2. â¦\83G, Lâ¦\84 â\8a¢ T1 â¬\88*[h] T2 â\86\92 (T1 â\89\9b[h, o] T2 â\86\92 â\8a¥) â\86\92 Q T2) â\86\92 Q T1
) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → R T.
-#h #o #G #L #R #IH #T #HT
+ ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → Q T.
+#h #o #G #L #Q #IH #T #HT
@(csx_ind_cpxs_tdeq … IH … HT) -IH -HT // (**) (* full auto fails *)
qed-.