'SnItem2 I T1 T2 = (TPair I T1 T2).
interpretation "term binding construction (binary)"
- 'SnBind2 a I T1 T2 = (TPair (Bind2 a I) T1 T2).
+ 'SnBind2 p I T1 T2 = (TPair (Bind2 p I) T1 T2).
interpretation "term positive binding construction (binary)"
'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2).
'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
interpretation "sort (term)"
- 'Star k = (TAtom (Sort k)).
+ 'Star s = (TAtom (Sort s)).
interpretation "local reference (term)"
'LRef i = (TAtom (LRef i)).
interpretation "global reference (term)"
- 'GRef p = (TAtom (GRef p)).
+ 'GRef l = (TAtom (GRef l)).
interpretation "abbreviation (term)"
- 'SnAbbr a T1 T2 = (TPair (Bind2 a Abbr) T1 T2).
+ 'SnAbbr p T1 T2 = (TPair (Bind2 p Abbr) T1 T2).
interpretation "positive abbreviation (term)"
'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2).
'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2).
interpretation "abstraction (term)"
- 'SnAbst a T1 T2 = (TPair (Bind2 a Abst) T1 T2).
+ 'SnAbst p T1 T2 = (TPair (Bind2 p Abst) T1 T2).
interpretation "positive abstraction (term)"
'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2).
(* Basic properties *********************************************************)
(* Basic_1: was: term_dec *)
-axiom eq_term_dec: ∀T1,T2:term. Decidable (T1 = T2).
+lemma eq_term_dec: ∀T1,T2:term. Decidable (T1 = T2).
+#T1 elim T1 -T1 #I1 [| #V1 #T1 #IHV1 #IHT1 ] * #I2 [2,4: #V2 #T2 ]
+[1,4: @or_intror #H destruct
+| elim (eq_item2_dec I1 I2) #HI
+ [ elim (IHV1 V2) -IHV1 #HV
+ [ elim (IHT1 T2) -IHT1 /2 width=1 by or_introl/ #HT ]
+ ]
+ @or_intror #H destruct /2 width=1 by/
+| elim (eq_item0_dec I1 I2) /2 width=1 by or_introl/ #HI
+ @or_intror #H destruct /2 width=1 by/
+]
+qed-.
(* Basic inversion lemmas ***************************************************)
-lemma destruct_tpair_tpair: ∀I1,I2,T1,T2,V1,V2. ②{I1}T1.V1 = ②{I2}T2.V2 →
- ∧∧T1 = T2 & I1 = I2 & V1 = V2.
+fact destruct_tatom_tatom_aux: ∀I1,I2. ⓪{I1} = ⓪{I2} → I1 = I2.
+#I1 #I2 #H destruct //
+qed-.
+
+fact destruct_tpair_tpair_aux: ∀I1,I2,T1,T2,V1,V2. ②{I1}T1.V1 = ②{I2}T2.V2 →
+ ∧∧T1 = T2 & I1 = I2 & V1 = V2.
#I1 #I2 #T1 #T2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
qed-.
-lemma discr_tpair_xy_x: ∀I,T,V. ②{I} V. T = V → ⊥.
+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 elim (destruct_tpair_tpair … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+| #J #W #U #IHW #_ #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
(* Basic_1: was: thead_x_y_y *)
-lemma discr_tpair_xy_y: ∀I,V,T. ②{I} V. T = T → ⊥.
+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 elim (destruct_tpair_tpair … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+| #J #W #U #_ #IHU #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2.
- (②{I} V1. T1 = ②{I} V2. T2 → ⊥) →
+ (②{I}V1.T1 = ②{I}V2.T2 → ⊥) →
(V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)).
#I #V1 #T1 #V2 #T2 #H
elim (eq_term_dec V1 V2) /3 width=1 by or_introl/ #HV12 destruct
qed-.
lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2.
- (②{I} V1. T1 = ②{I} V2. T2 → ⊥) →
+ (②{I} V1. T1 = ②{I}V2.T2 → ⊥) →
(T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)).
#I #V1 #T1 #V2 #T2 #H
elim (eq_term_dec T1 T2) /3 width=1 by or_introl/ #HT12 destruct