(* Basic properties *********************************************************)
-lemma abst_dec (X): ∨∨ ∃∃p,W,T. X = ⓛ{p}W.T
- | (∀p,W,T. X = ⓛ{p}W.T → ⊥).
+lemma abst_dec (X): ∨∨ ∃∃p,W,T. X = ⓛ[p]W.T
+ | (∀p,W,T. X = ⓛ[p]W.T → ⊥).
* [ #I | * [ #p * | #I ] #V #T ]
[3: /3 width=4 by ex1_3_intro, or_introl/ ]
@or_intror #q #W #U #H destruct
(* Basic inversion lemmas ***************************************************)
-fact destruct_tatom_tatom_aux: ∀I1,I2. ⓪{I1} = ⓪{I2} → I1 = I2.
+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 →
+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_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_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