(* Basic properties *********************************************************)
-axiom lenv_eq_dec: ∀L1,L2:lenv. Decidable (L1 = L2).
+lemma eq_lenv_dec: ∀L1,L2:lenv. Decidable (L1 = L2).
+#L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ] * [2,4: #L2 #I2 #V2 ]
+[1,4: @or_intror #H destruct
+| elim (eq_bind2_dec I1 I2) #HI
+ [ elim (eq_term_dec V1 V2) #HV
+ [ elim (IHL1 L2) -IHL1 /2 width=1 by or_introl/ #HL ]
+ ]
+ @or_intror #H destruct /2 width=1 by/
+| /2 width=1 by or_introl/
+]
+qed-.
(* Basic inversion lemmas ***************************************************)
-lemma destruct_lpair_lpair: ∀I1,I2,L1,L2,V1,V2. L1.ⓑ{I1}V1 = L2.ⓑ{I2}V2 →
- ∧∧L1 = L2 & I1 = I2 & V1 = V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #H destruct /2 width=1/
+fact destruct_lpair_lpair_aux: ∀I1,I2,L1,L2,V1,V2. L1.ⓑ{I1}V1 = L2.ⓑ{I2}V2 →
+ ∧∧L1 = L2 & I1 = I2 & V1 = V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
qed-.
lemma discr_lpair_x_xy: ∀I,V,L. L = L.ⓑ{I}V → ⊥.
#I #V #L elim L -L
[ #H destruct
| #L #J #W #IHL #H
- elim (destruct_lpair_lpair … H) -H #H1 #H2 #H3 destruct /2 width=1/ (**) (* destruct lemma needed *)
+ elim (destruct_lpair_lpair_aux … H) -H #H1 #H2 #H3 destruct /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
-(* Basic_1: removed theorems 2: chead_ctail c_tail_ind *)
+lemma discr_lpair_xy_x: ∀I,V,L. L.ⓑ{I}V = L→ ⊥.
+/2 width=4 by discr_lpair_x_xy/ qed-.