-lemma csx_ind_cpxs_tdeq: ∀h,o,G,L. ∀Q:predicate term.
- (∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- (∀T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) → Q T2) → Q T1
- ) →
- ∀T1. ⦃G, L⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀T0. ⦃G, L⦄ ⊢ T1 ⬈*[h] T0 → ∀T2. T0 ≛[h, o] T2 → 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_cpxs_trans … HT02 … HTV2) -T2 #V0 #HTV0 #HV02
-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
- elim (cpxs_tdneq_fwd_step_sn … H10 … Hn10) -H10 -Hn10
- /3 width=8 by tdeq_trans/
-| 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/
+fact csx_ind_cpxs_aux (G) (L):
+ ∀Q:predicate term.
+ (∀T1. ❪G,L❫ ⊢ ⬈*𝐒 T1 →
+ (∀T2. ❪G,L❫ ⊢ T1 ⬈* T2 → (T1 ≅ T2 → ⊥) → Q T2) → Q T1
+ ) →
+ ∀T1. ❪G,L❫ ⊢ ⬈*𝐒 T1 →
+ ∀T2. ❪G,L❫ ⊢ T1 ⬈* T2 → Q T2.
+#G #L #Q #IH #T1 #H @(csx_ind … H) -T1
+#T1 #HT1 #IH1 #T2 #HT12
+@IH -IH /2 width=3 by csx_cpxs_trans/ -HT1 #V2 #HTV2 #HnTV2
+elim (teqx_dec T1 T2) #H
+[ lapply (teqg_tneqg_trans … H … HnTV2) // -H -HnTV2 #Hn12
+ lapply (cpxs_trans … HT12 … HTV2) -T2 #H12
+ elim (cpxs_tneqg_fwd_step_sn … H12 … Hn12) /2 width=1 by sfull_dec/ -H12 -Hn12
+ /3 width=4 by/
+| elim (cpxs_tneqg_fwd_step_sn … HT12 … H) -HT12 -H
+ /3 width=6 by cpxs_trans, sfull_dec/