| TPair: item2 → term → term → term (* binary item construction *)
.
-interpretation "sort (term)" 'Star k = (TAtom (Sort k)).
+interpretation "term construction (atomic)"
+ 'Item0 I = (TAtom I).
-interpretation "local reference (term)" 'LRef i = (TAtom (LRef i)).
+interpretation "term construction (binary)"
+ 'SnItem2 I T1 T2 = (TPair I T1 T2).
-interpretation "global reference (term)" 'GRef p = (TAtom (GRef p)).
+interpretation "term binding construction (binary)"
+ 'SnBind2 I T1 T2 = (TPair (Bind2 I) T1 T2).
-interpretation "term construction (atomic)" 'SItem I = (TAtom I).
+interpretation "term flat construction (binary)"
+ 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
-interpretation "term construction (binary)" 'SItem I T1 T2 = (TPair I T1 T2).
+interpretation "sort (term)"
+ 'Star k = (TAtom (Sort k)).
-interpretation "term binding construction (binary)" 'SBind I T1 T2 = (TPair (Bind I) T1 T2).
+interpretation "local reference (term)"
+ 'LRef i = (TAtom (LRef i)).
-interpretation "term flat construction (binary)" 'SFlat I T1 T2 = (TPair (Flat I) T1 T2).
+interpretation "global reference (term)"
+ 'GRef p = (TAtom (GRef p)).
+
+interpretation "abbreviation (term)"
+ 'SnAbbr T1 T2 = (TPair (Bind2 Abbr) T1 T2).
+
+interpretation "abstraction (term)"
+ 'SnAbst T1 T2 = (TPair (Bind2 Abst) T1 T2).
+
+interpretation "application (term)"
+ 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
+
+interpretation "native type annotation (term)"
+ 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: term_dec *)
+axiom term_eq_dec: ∀T1,T2:term. Decidable (T1 = T2).
(* Basic inversion lemmas ***************************************************)
-lemma discr_tpair_xy_x: ∀I,T,V. 𝕔{I} V. T = V → False.
+lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → False.
#I #T #V elim V -V
[ #J #H destruct
| #J #W #U #IHW #_ #H destruct
qed-.
(* Basic_1: was: thead_x_y_y *)
-lemma discr_tpair_xy_y: ∀I,V,T. 𝕔{I} V. T = T → False.
+lemma discr_tpair_xy_y: ∀I,V,T. ②{I} V. T = T → False.
#I #V #T elim T -T
[ #J #H destruct
| #J #W #U #_ #IHU #H destruct
]
qed-.
-(* Basic properties *********************************************************)
+lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2.
+ (②{I} V1. T1 = ②{I} V2. T2 → False) →
+ (V1 = V2 → False) ∨ (V1 = V2 ∧ (T1 = T2 → 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-.
-(* Basic_1: was: term_dec *)
-axiom term_eq_dec: ∀T1,T2:term. Decidable (T1 = T2).
+lemma 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 T1 T2) /3 width=1/ #HT12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1/
+qed-.
+
+lemma eq_false_inv_beta: ∀V1,V2,W1,W2,T1,T2.
+ (ⓐV1. ⓛW1. T1 = ⓐV2. ⓛW2 .T2 →False) →
+ (W1 = W2 → False) ∨
+ (W1 = W2 ∧ (ⓓV1. T1 = ⓓV2. T2 → False)).
+#V1 #V2 #W1 #W2 #T1 #T2 #H
+elim (eq_false_inv_tpair_sn … H) -H
+[ #HV12 elim (term_eq_dec W1 W2) /3 width=1/
+ #H destruct @or_intror @conj // #H destruct /2 width=1/
+| * #H1 #H2 destruct
+ elim (eq_false_inv_tpair_sn … H2) -H2 /3 width=1/
+ * #H #HT12 destruct
+ @or_intror @conj // #H destruct /2 width=1/
+]
+qed.
(* Basic_1: removed theorems 3:
not_void_abst not_abbr_void not_abst_void