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(** * Classical logic and proof-irrelevance *)

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(** ** CC |- prop ext + A inhabited -> (A = A->A) -> A has fixpoint *)

(** We successively show that:

   [prop_extensionality]
     implies equality of [A] and [A->A] for inhabited [A], which
     implies the existence of a (trivial) retract from [A->A] to [A]
        (just take the identity), which
     implies the existence of a fixpoint operator in [A]
        (e.g. take the Y combinator of lambda-calculus)

*)

Local Notation inhabited A := A (only parsing).

Lemma prop_ext_A_eq_A_imp_A : prop_extensionality -> forall A:Prop, inhabited A -> (A -> A) = A.