@csn_intro #X #H1 #H2
elim (cpr_inv_abst1 … H1 I V) -H1
#W0 #T0 #HLW0 #HLT0 #H destruct
-elim (eq_false_inv_tpair … H2) -H2
+elim (eq_false_inv_tpair_sn … H2) -H2
[ /3 width=5/
| -HLW0 * #H destruct /3 width=1/
]
qed.
-(*
-axiom eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2.
- (②{I} V1. T1 = ②{I} V2. T2 → False) →
- (T1 = T2 → False) ∨ (T1 = T2 ∧ (V1 = V2 → False)).
-
-#I #V1 #T1 #V2 #T2 #H
-elim (term_eq_dec V1 V2) /3 width=1/ #HV12 destruct
-@or_intror @conj // #HT12 destruct /2 width=1/
-qed-.
-lemma csn_appl_simple: ∀L,T. L ⊢ ⬇* T → 𝐒[T] → ∀V. L ⊢ ⬇* V → L ⊢ ⬇* ⓐV. T.
-#L #T #H elim H -T #T #_ #IHT #HT #V #H @(csn_ind … H) -V #V #HV #IHV
+(* Basic_1: was only: sn3_appl_appl *)
+lemma csn_appl_simple: ∀L,V. L ⊢ ⬇* V → ∀T1.
+ (∀T2. L ⊢ T1 ➡ T2 → (T1 = T2 → False) → L ⊢ ⬇* ⓐV. T2) →
+ 𝐒[T1] → L ⊢ ⬇* ⓐV. T1.
+#L #V #H @(csn_ind … H) -V #V #_ #IHV #T1 #IHT1 #HT1
@csn_intro #X #H1 #H2
elim (cpr_inv_appl1_simple … H1 ?) // -H1
-#V0 #T0 #HLV0 #HLT0 #H destruct
+#V0 #T0 #HLV0 #HLT10 #H destruct
elim (eq_false_inv_tpair_dx … H2) -H2
-[ -IHV #HT0 @IHT -IHT // -HLT0 /2 width=1/ -HT0 /2 width=3/
-| -HV -HT -IHT -HLT0 * #H #HV0 destruct /3 width=1/
+[ -IHV -HT1 #HT10
+ @(csn_cpr_trans … (ⓐV.T0)) /2 width=1/ -HLV0
+ @IHT1 -IHT1 // /2 width=1/
+| -HLT10 * #H #HV0 destruct
+ @IHV -IHV // -HT1 /2 width=1/ -HV0
+ #T2 #HLT02 #HT02
+ @(csn_cpr_trans … (ⓐV.T2)) /2 width=1/ -HLV0
+ @IHT1 -IHT1 // -HLT02 /2 width=1/
+]
+qed.
+
+(* Basic_1: was only: sn3_appl_appls *)
+lemma csn_appl_appls_simple: ∀L,V. L ⊢ ⬇* V → ∀Vs,T1.
+ (∀T2. L ⊢ ⒶVs.T1 ➡ T2 → (ⒶVs.T1 = T2 → False) → L ⊢ ⬇* ⓐV. T2) →
+ 𝐒[T1] → L ⊢ ⬇* ⓐV. ⒶVs. T1.
+#L #V #HV #Vs elim Vs -Vs
+[ @csn_appl_simple //
+| #V0 #Vs #_ #T1 #HT1 #_
+ @csn_appl_simple // -HV @HT1
]
qed.
-*)
\ No newline at end of file