(* 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-.
lemma csx_ind_cpxs_tdeq: ∀h,o,G,L. ∀R: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 → ⊥) → R T2) → 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 → ⊥) → R T2) → R 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 → R T2.
+ 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 → R T2.
#h #o #G #L #R #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_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
-lapply (tndeq_tdeq_canc_dx … H … HV02) -H #HnTV0
+lapply (tdneq_tdeq_canc_dx … H … HV02) -H #HnTV0
elim (tdeq_dec h o T1 T0) #H
[ lapply (tdeq_tdneq_trans … H … HnTV0) -H -HnTV0 #Hn10
lapply (cpxs_trans … HT10 … HTV0) -T0 #H10
]
qed-.
-lemma csx_ind_alt: ∀h,o,G,L. ∀R:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≡[h, o] T2 → ⊥) → R T2) → R T1
- ) →
- ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → R T.
+(* Basic_2A1: was: csx_ind_alt *)
+lemma csx_ind_cpxs: ∀h,o,G,L. ∀R:predicate term.
+ (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+ (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → R T2) → R T1
+ ) →
+ ∀T. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → R T.
#h #o #G #L #R #IH #T #HT
@(csx_ind_cpxs_tdeq … IH … HT) -IH -HT // (**) (* full auto fails *)
qed-.