'SnItem2 I T1 T2 = (TPair I T1 T2).
interpretation "term binding construction (binary)"
- 'SnBind2 I T1 T2 = (TPair (Bind2 I) T1 T2).
+ 'SnBind2 a I T1 T2 = (TPair (Bind2 a I) T1 T2).
+
+interpretation "term positive binding construction (binary)"
+ 'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2).
+
+interpretation "term negative binding construction (binary)"
+ 'SnBind2Neg I T1 T2 = (TPair (Bind2 false I) T1 T2).
interpretation "term flat construction (binary)"
'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
'GRef p = (TAtom (GRef p)).
interpretation "abbreviation (term)"
- 'SnAbbr T1 T2 = (TPair (Bind2 Abbr) T1 T2).
+ 'SnAbbr a T1 T2 = (TPair (Bind2 a Abbr) T1 T2).
+
+interpretation "positive abbreviation (term)"
+ 'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2).
+
+interpretation "negative abbreviation (term)"
+ 'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2).
interpretation "abstraction (term)"
- 'SnAbst T1 T2 = (TPair (Bind2 Abst) T1 T2).
+ 'SnAbst a T1 T2 = (TPair (Bind2 a Abst) T1 T2).
+
+interpretation "positive abstraction (term)"
+ 'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2).
+
+interpretation "negative abstraction (term)"
+ 'SnAbstNeg T1 T2 = (TPair (Bind2 false Abst) T1 T2).
interpretation "application (term)"
'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
(* Basic inversion lemmas ***************************************************)
-lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → False.
+lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → ⊥.
#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. ②{I} V. T = T → False.
+lemma discr_tpair_xy_y: ∀I,V,T. ②{I} V. T = T → ⊥.
#I #V #T elim T -T
[ #J #H destruct
| #J #W #U #_ #IHU #H destruct
qed-.
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)).
+ (②{I} V1. T1 = ②{I} V2. T2 → ⊥) →
+ (V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)).
#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 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)).
+ (②{I} V1. T1 = ②{I} V2. T2 → ⊥) →
+ (T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)).
#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
+lemma eq_false_inv_beta: ∀a,V1,V2,W1,W2,T1,T2.
+ (ⓐV1. ⓛ{a}W1. T1 = ⓐV2. ⓛ{a}W2 .T2 → ⊥) →
+ (W1 = W2 → ⊥) ∨
+ (W1 = W2 ∧ (ⓓ{a}V1. T1 = ⓓ{a}V2. T2 → ⊥)).
+#a #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/