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

notation > "hvbox(a break ≃ b)"
  non associative with precedence 45
for @{ 'jmeq ? $a ? $b }.

notation < "hvbox(term 46 a break maction (≃) (≃\sub(t,u)) term 46 b)"
  non associative with precedence 45
for @{ 'jmeq $t $a $u $b }.

interpretation "john major's equality" 'jmeq t x u y = (jmeq t x u y).

lemma eq_to_jmeq:
  ∀A: Type[0].
  ∀x, y: A.
    x = y → x ≃ y.
  //
qed.

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.

lemma jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. x≃y → x=y.
 #A #x #y #JMEQ @(jmeq_elim ? x … JMEQ) %
qed.

coercion jmeq_to_eq: ∀A:Type[0]. ∀x,y:A. ∀p:x≃y.x=y ≝ jmeq_to_eq on _p:?≃? to ?=?.

(* END RUSSELL **)