include "basics/jmeq.ma".
include "basics/types.ma".
include "basics/list.ma".

definition inject : ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ λA,P,a,p. dp … a p.
definition eject : ∀A.∀P: A → Prop.(Σx:A.P x) → A ≝ λA,P,c.match c with [ dp w p ⇒ w].

coercion inject nocomposites: ∀A.∀P:A → Prop.∀a.∀p:P a.Σx:A.P x ≝ inject on a:? to Σx:?.?.
coercion eject nocomposites: ∀A.∀P:A → Prop.∀c:Σx:A.P x.A ≝ eject on _c:Σx:?.? to ?.

(*axiom VOID: Type[0].
axiom assert_false: VOID.
definition bigbang: ∀A:Type[0].False → VOID → A.
 #A #abs cases abs
qed.

coercion bigbang nocomposites: ∀A:Type[0].False → ∀v:VOID.A ≝ bigbang on _v:VOID to ?.*)

lemma sig2: ∀A.∀P:A → Prop. ∀p:Σx:A.P x. P (eject … p).
 #A #P #p cases p #w #q @q
qed.

(* END RUSSELL **)