From c690a68ba0b78b913cbcb60f7508d431071d4c79 Mon Sep 17 00:00:00 2001
From: Vincent Semeria
Date: Fri, 4 Oct 2019 18:29:58 +0200
Subject: Replace classical reals quotient axioms by functional extensionality.
 Define homotopy propositions and homotopy sets. Rename local variable R in
 test Nsatz, to avoid a name collision with the type of real numbers.

---
 theories/Logic/HLevels.v | 146 +++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 146 insertions(+)
 create mode 100644 theories/Logic/HLevels.v

(limited to 'theories/Logic')

diff --git a/theories/Logic/HLevels.v b/theories/Logic/HLevels.v
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+++ b/theories/Logic/HLevels.v
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+(************************************************************************)
+(*         *   The Coq Proof Assistant / The Coq Development Team       *)
+(*  v      *   INRIA, CNRS and contributors - Copyright 1999-2019       *)
+(* <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)         *)
+(************************************************************************)
+
+(** The first three levels of homotopy type theory: homotopy propositions,
+    homotopy sets and homotopy one types. For more information,
+    https://github.com/HoTT/HoTT
+    and
+    https://homotopytypetheory.org/book
+
+    Univalence is not assumed here, and equality is Coq's usual inductive
+    type eq in sort Prop. This is a little different from HoTT, where
+    sort Prop does not exist and equality is directly in sort Type. *)
+
+
+(* It is almost impossible to prove that a type is a homotopy proposition
+   without funext, so we assume it here. *)
+Require Import Coq.Logic.FunctionalExtensionality.
+
+(* A homotopy proposition is a type that has at most one element.
+   Its unique inhabitant, when it exists, is to be interpreted as the
+   proof of the homotopy proposition.
+   Homotopy propositions are therefore an alternative to the sort Prop,
+   to select which types represent mathematical propositions. *)
+Definition IsHProp (P : Type) : Prop
+  := forall p q : P, p = q.
+
+(* A homotopy set is a type such as each equality type x = y is a
+   homotopy proposition. For example, any type which equality is
+   decidable in sort Prop is a homotopy set, as shown in file
+   Coq.Logic.Eqdep_dec.v. *)
+Definition IsHSet (X : Type) : Prop
+  := forall (x y : X) (p q : x = y), p = q.
+
+Definition IsHOneType (X : Type) : Prop
+  := forall (x y : X) (p q : x = y) (r s : p = q), r = s.
+
+Lemma forall_hprop : forall (A : Type) (P : A -> Prop),
+    (forall x:A, IsHProp (P x))
+    -> IsHProp (forall x:A, P x).
+Proof.
+  intros A P H p q. apply functional_extensionality_dep.
+  intro x. apply H.
+Qed.
+
+(* Homotopy propositions are stable by conjunction, but not by disjunction,
+   which can have a proof by the left and another proof by the right. *)
+Lemma and_hprop : forall P Q : Prop,
+    IsHProp P -> IsHProp Q -> IsHProp (P /\ Q).
+Proof.
+  intros. intros p q. destruct p,q.
+  replace p0 with p. replace q0 with q. reflexivity.
+  apply H0. apply H.
+Qed.
+
+Lemma impl_hprop : forall P Q : Prop,
+    IsHProp Q -> IsHProp (P -> Q).
+Proof.
+  intros P Q H p q. apply functional_extensionality.
+  intros. apply H.
+Qed.
+
+Lemma false_hprop : IsHProp False.
+Proof.
+  intros p q. contradiction.
+Qed.
+
+Lemma true_hprop : IsHProp True.
+Proof.
+  intros p q. destruct p,q. reflexivity.
+Qed.
+
+(* All negations are homotopy propositions. *)
+Lemma not_hprop : forall P : Type, IsHProp (P -> False).
+Proof.
+  intros P p q. apply functional_extensionality.
+  intros. contradiction.
+Qed.
+
+(* Homotopy propositions are included in homotopy sets.
+   They are the first 2 levels of a cumulative hierarchy of types
+   indexed by the natural numbers. In homotopy type theory,
+   homotopy propositions are call (-1)-types and homotopy
+   sets 0-types. *)
+Lemma hset_hprop : forall X : Type,
+    IsHProp X -> IsHSet X.
+Proof.
+  intros X H.
+  assert (forall (x y z:X) (p : y = z), eq_trans (H x y) p = H x z).
+  { intros. unfold eq_trans, eq_ind. destruct p. reflexivity. }
+  assert (forall (x y z:X) (p : y = z),
+             p = eq_trans (eq_sym (H x y)) (H x z)).
+  { intros. rewrite <- (H0 x y z p). unfold eq_trans, eq_sym, eq_ind.
+    destruct p, (H x y). reflexivity. }
+  intros x y p q.
+  rewrite (H1 x x y p), (H1 x x y q). reflexivity.
+Qed.
+
+Lemma eq_trans_cancel : forall {X : Type} {x y z : X} (p : x = y) (q r : y = z),
+  (eq_trans p q = eq_trans p r) -> q = r.
+Proof.
+  intros. destruct p. simpl in H. destruct r.
+  simpl in H. rewrite eq_trans_refl_l in H. exact H.
+Qed.
+
+Lemma hset_hOneType : forall X : Type,
+    IsHSet X -> IsHOneType X.
+Proof.
+  intros X f x y p q.
+  pose (fun a => f x y p a) as g.
+  assert (forall a (r : q = a), eq_trans (g q) r = g a).
+  { intros. destruct a. subst q. reflexivity. }
+  intros r s. pose proof (H p (eq_sym r)).
+  pose proof (H p (eq_sym s)).
+  rewrite <- H1 in H0. apply eq_trans_cancel in H0.
+  rewrite <- eq_sym_involutive. rewrite <- (eq_sym_involutive r).
+  rewrite H0. reflexivity.
+Qed.
+
+(* "IsHProp X" sounds like a proposition, because it asserts
+   a property of the type X. And indeed: *)
+Lemma hprop_hprop : forall X : Type,
+    IsHProp (IsHProp X).
+Proof.
+  intros X p q.
+  apply forall_hprop. intro x.
+  apply forall_hprop. intro y. intros f g.
+  apply (hset_hprop X p).
+Qed.
+
+Lemma hprop_hset : forall X : Type,
+    IsHProp (IsHSet X).
+Proof.
+  intros X f g.
+  apply functional_extensionality_dep. intro x.
+  apply functional_extensionality_dep. intro y.
+  apply functional_extensionality_dep. intro a.
+  apply functional_extensionality_dep. intro b.
+  apply (hset_hOneType). exact f.
+Qed.
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