ProofIrrelevanceFacts.v

Built with Alectryon, running Coq+SerAPI v8.10.0+0.7.0. Coq sources are in this panel; goals and messages will appear in the other. Bubbles () indicate interactive fragments: hover for details, tap to reveal contents. Use Ctrl+↑ Ctrl+↓ to navigate, Ctrl+🖱️ to focus.

(************************************************************************)
(*         *   The Coq Proof Assistant / The Coq Development Team       *)
(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2018       *)
(* <O___,, *       (see CREDITS file for the list of authors)           *)
(*   \VV/  **************************************************************)
(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
(*         *     (see LICENSE file for the text of the license)         *)
(************************************************************************)

This defines the functor that build consequences of proof-irrelevance

Require Export EqdepFacts.

Module Type ProofIrrelevance.

  Axiom proof_irrelevance : forall (P:Prop) (p1 p2:P), p1 = p2.

End ProofIrrelevance.

Module ProofIrrelevanceTheory (M:ProofIrrelevance).

Proof-irrelevance implies uniqueness of reflexivity proofs

  Module Eq_rect_eq.
    Lemma eq_rect_eq :
      forall (U:Type) (p:U) (Q:U -> Type) (x:Q p) (h:p = p),
        x = eq_rect p Q x p h.forall (U : Type) (p : U) (Q : U -> Type) (x : Q p)
  (h : p = p), x = eq_rect p Q x p h
    Proof.forall (U : Type) (p : U) (Q : U -> Type) (x : Q p)
  (h : p = p), x = eq_rect p Q x p h
      intros; rewrite M.proof_irrelevance with (p1:=h) (p2:=eq_refl p).U:Type
p:U
Q:U -> Type
x:Q p
h:p = p
x = eq_rect p Q x p eq_refl
      reflexivity.
    Qed.
  End Eq_rect_eq.

Export the theory of injective dependent elimination

  Module EqdepTheory := EqdepTheory(Eq_rect_eq).
  Export EqdepTheory.

  Scheme eq_indd := Induction for eq Sort Prop.

We derive the irrelevance of the membership property for subsets

  Lemma subset_eq_compat :
    forall (U:Type) (P:U->Prop) (x y:U) (p:P x) (q:P y),
      x = y -> exist P x p = exist P y q.forall (U : Type) (P : U -> Prop) (x y : U) (p : P x)
  (q : P y), x = y -> exist P x p = exist P y q
  Proof.forall (U : Type) (P : U -> Prop) (x y : U) (p : P x)
  (q : P y), x = y -> exist P x p = exist P y q
    intros.U:Type
P:U -> Prop
x, y:U
p:P x
q:P y
H:x = y
exist P x p = exist P y q
    rewrite M.proof_irrelevance with (p1:=q) (p2:=eq_rect x P p y H).U:Type
P:U -> Prop
x, y:U
p:P x
q:P y
H:x = y
exist P x p = exist P y (eq_rect x P p y H)
    elim H using eq_indd.U:Type
P:U -> Prop
x, y:U
p:P x
q:P y
H:x = y
exist P x p = exist P x (eq_rect x P p x eq_refl)
    reflexivity.
  Qed.

  Lemma subsetT_eq_compat :
    forall (U:Type) (P:U->Prop) (x y:U) (p:P x) (q:P y),
      x = y -> existT P x p = existT P y q.forall (U : Type) (P : U -> Prop) (x y : U) (p : P x)
  (q : P y), x = y -> existT P x p = existT P y q
  Proof.forall (U : Type) (P : U -> Prop) (x y : U) (p : P x)
  (q : P y), x = y -> existT P x p = existT P y q
    intros.U:Type
P:U -> Prop
x, y:U
p:P x
q:P y
H:x = y
existT P x p = existT P y q
    rewrite M.proof_irrelevance with (p1:=q) (p2:=eq_rect x P p y H).U:Type
P:U -> Prop
x, y:U
p:P x
q:P y
H:x = y
existT P x p = existT P y (eq_rect x P p y H)
    elim H using eq_indd.U:Type
P:U -> Prop
x, y:U
p:P x
q:P y
H:x = y
existT P x p = existT P x (eq_rect x P p x eq_refl)
    reflexivity.
  Qed.

End ProofIrrelevanceTheory.