(** Remark: [prop_extensionality] can be replaced in lemma [ext_prop_fixpoint] by the weakest property [provable_prop_extensionality]. *)
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(** ** CC |- prop_ext /\ dep elim on bool -> proof-irrelevance  *)
(** [proof_irrelevance] asserts equality of all proofs of a given formula *)
Definition proof_irrelevance := forall (A:Prop) (a1 a2:A), a1 = a2.
(** Assume that we have booleans with the property that there is at most 2 booleans (which is equivalent to dependent case analysis). Consider the fixpoint of the negation function: it is either true or false by dependent case analysis, but also the opposite by fixpoint. Hence proof-irrelevance.  We then map equality of boolean proofs to proof irrelevance in all propositions. *)
Section Proof_irrelevance_gen.
Variable bool : Prop.
Variable true : bool.
Variable false : bool.
Hypothesis bool_elim : forall C:Prop, C -> C -> bool -> C.
Hypothesis
bool_elim_redl : forall (C:Prop) (c1 c2:C), c1 = bool_elim C c1 c2 true.
Hypothesis
bool_elim_redr : forall (C:Prop) (c1 c2:C), c2 = bool_elim C c1 c2 false.
Let bool_dep_induction :=
forall P:bool -> Prop, P true -> P false -> forall b:bool, P b.

Lemma ext_prop_dep_proof_irrel_gen :
prop_extensionality -> bool_dep_induction -> proof_irrelevance.