The Coq Proof Assistant

# Library Coq.Logic.Hurkens

Exploiting Hurkens's paradox [Hurkens95] for system U- so as to derive various contradictory contexts.
The file is divided into various sub-modules which all follow the same structure: a section introduces the contradictory hypotheses and a theorem named paradox concludes the module with a proof of False.

- The Generic module contains the actual Hurkens's paradox for a postulated shallow encoding of system U- in Coq. This is an adaptation by Arnaud Spiwack of a previous, more restricted implementation by Herman Geuvers. It is used to derive every other special cases of the paradox in this file.
- The NoRetractToImpredicativeUniverse module contains a simple and effective formulation by Herman Geuvers [Geuvers01] of a result by Thierry Coquand [Coquand90]. It states that no impredicative sort can contain a type of which it is a retract. This result implies that Coq with classical logic stated in impredicative Set is inconsistent and that classical logic stated in Prop implies proof-irrelevance (see ClassicalFacts.v)
- The NoRetractFromSmallPropositionToProp module is a specialisation of the NoRetractToImpredicativeUniverse module to the case where the impredicative sort is Prop.
- The NoRetractToModalProposition module is a strengthening of the NoRetractFromSmallPropositionToProp module. It shows that given a monadic modality (aka closure operator) M, the type of modal propositions (i.e. such that M A -> A) cannot be a retract of a modal proposition. It is an example of use of the paradox where the universes of system U- are not mapped to universes of Coq.
- The NoRetractToNegativeProp module is the specialisation of the NoRetractFromSmallPropositionToProp module where the modality is double-negation. This result implies that the principle of weak excluded middle (forall A, ~~A\/~A) implies a weak variant of proof irrelevance.
- The NoRetractFromTypeToProp module proves that Prop cannot be a retract of a larger type.
- The TypeNeqSmallType module proves that Type is different from any smaller type.
- The PropNeqType module proves that Prop is different from any larger Type. It is an instance of the previous result.

References:

- [Coquand90] T. Coquand, "Metamathematical Investigations of a Calculus of Constructions", Proceedings of Logic in Computer Science (LICS'90), 1990.
- [Hurkens95] A. J. Hurkens, "A simplification of Girard's paradox", Proceedings of the 2nd international conference Typed Lambda-Calculi and Applications (TLCA'95), 1995.
- [Geuvers01] H. Geuvers, "Inconsistency of Classical Logic in Type Theory", 2001, revised 2007 (see external link http://www.cs.ru.nl/~herman/PUBS/newnote.ps.gz).

# A modular proof of Hurkens's paradox.

It relies on an axiomatisation of a shallow embedding of system U- (i.e. types of U- are interpreted by types of Coq). The universes are encoded in a style, due to Martin-Löf, where they are given by a set of names and a family El:Name->Type which interprets each name into a type. This allows the encoding of universe to be decoupled from Coq's universes. Dependent products and abstractions are similarly postulated rather than encoded as Coq's dependent products and abstractions.
Module Generic.
Section Paradox.

## Axiomatisation of impredicative universes in a Martin-Löf style

System U- has two impredicative universes. In the proof of the paradox they are slightly asymmetric (in particular the reduction rules of the small universe are not needed). Therefore, the axioms are duplicated allowing for a weaker requirement than the actual system U-.

### Large universe

Variable U1 : Type.
Variable El1 : U1 -> Type.

#### Closure by small product

Variable Forall1 : forall u:U1, (El1 u -> U1) -> U1.
Notation "'∀₁' x : A , B" := (Forall1 A (fun x => B)).
Notation "A '⟶₁' B" := (Forall1 A (fun _ => B)).
Variable lam1 : forall u B, (forall x:El1 u, El1 (B x)) -> El1 (∀₁ x:u, B x).
Notation "'λ₁' x , u" := (lam1 _ _ (fun x => u)).
Variable app1 : forall u B (f:El1 (Forall1 u B)) (x:El1 u), El1 (B x).
Notation "f '·₁' x" := (app1 _ _ f x).
Variable beta1 : forall u B (f:forall x:El1 u, El1 (B x)) x,
(λ₁ y, f y) ·₁ x = f x.

#### Closure by large products

U1 only needs to quantify over itself.
Variable ForallU1 : (U1->U1) -> U1.
Notation "'∀₂' A , F" := (ForallU1 (fun A => F)).
Variable lamU1 : forall F, (forall A:U1, El1 (F A)) -> El1 (∀₂ A, F A).
Notation "'λ₂' x , u" := (lamU1 _ (fun x => u)).
Variable appU1 : forall F (f:El1(∀₂ A,F A)) (A:U1), El1 (F A).
Notation "f '·₁' [ A ]" := (appU1 _ f A).
Variable betaU1 : forall F (f:forall A:U1, El1 (F A)) A,
(λ₂ x, f x) ·₁ [ A ] = f A.

### Small universe

The small universe is an element of the large one.
Variable u0 : U1.
Notation U0 := (El1 u0).
Variable El0 : U0 -> Type.

#### Closure by small product

U0 does not need reduction rules
Variable Forall0 : forall u:U0, (El0 u -> U0) -> U0.
Notation "'∀₀' x : A , B" := (Forall0 A (fun x => B)).
Notation "A '⟶₀' B" := (Forall0 A (fun _ => B)).
Variable lam0 : forall u B, (forall x:El0 u, El0 (B x)) -> El0 (∀₀ x:u, B x).
Notation "'λ₀' x , u" := (lam0 _ _ (fun x => u)).
Variable app0 : forall u B (f:El0 (Forall0 u B)) (x:El0 u), El0 (B x).
Notation "f '·₀' x" := (app0 _ _ f x).

#### Closure by large products

Variable ForallU0 : forall u:U1, (El1 u->U0) -> U0.
Notation "'∀₀¹' A : U , F" := (ForallU0 U (fun A => F)).
Variable lamU0 : forall U F, (forall A:El1 U, El0 (F A)) -> El0 (∀₀¹ A:U, F A).
Notation "'λ₀¹' x , u" := (lamU0 _ _ (fun x => u)).
Variable appU0 : forall U F (f:El0(∀₀¹ A:U,F A)) (A:El1 U), El0 (F A).
Notation "f '·₀' [ A ]" := (appU0 _ _ f A).

## Automating the rewrite rules of our encoding.

Local Ltac simplify :=
(repeat rewrite ?beta1, ?betaU1);
lazy beta.
Local Ltac simplify_in h :=
(repeat rewrite ?beta1, ?betaU1 in h);
lazy beta in h.

## Hurkens's paradox.

An inhabitant of U0 standing for False.
Variable F:U0.

### Preliminary definitions

Definition V : U1 := ∀₂ A, ((A ⟶₁ u0) ⟶₁ A ⟶₁ u0) ⟶₁ A ⟶₁ u0.
Definition U : U1 := V ⟶₁ u0.
Definition sb (z:El1 V) : El1 V := λ₂ A, λ₁ r, λ₁ a, r ·₁ (z·₁[A]·₁r) ·₁ a.
Definition le (i:El1 (U⟶₁u0)) (x:El1 U) : U0 :=
x ·₁ (λ₂ A, λ₁ r, λ₁ a, i ·₁ (λ₁ v, (sb v) ·₁ [A] ·₁ r ·₁ a)).
Definition le' : El1 ((U⟶₁u0) ⟶₁ U ⟶₁ u0) := λ₁ i, λ₁ x, le i x.
Definition induct (i:El1 (U⟶₁u0)) : U0 :=
∀₀¹ x:U, le i x ⟶₀ i ·₁ x.
Definition WF : El1 U := λ₁ z, (induct (z·₁[U] ·₁ le')).
Definition I (x:El1 U) : U0 :=
(∀₀¹ i:U⟶₁u0, le i x ⟶₀ i ·₁ (λ₁ v, (sb v) ·₁ [U] ·₁ le' ·₁ x)) ⟶₀ F
.

### Proof

Lemma Omega : El0 (∀₀¹ i:U⟶₁u0, induct i ⟶₀ i ·₁ WF).
Proof.
refine (λ₀¹ i, λ₀ y, _).
refine (y·₀[_]·₀_).
unfold le,WF,induct. simplify.
refine (λ₀¹ x, λ₀ h0, _). simplify.
refine (y·₀[_]·₀_).
unfold le. simplify.
unfold sb at 1. simplify.
unfold le' at 1. simplify.
exact h0.
Qed.
Lemma lemma1 : El0 (induct (λ₁ u, I u)).
Proof.
unfold induct.
refine (λ₀¹ x, λ₀ p, _). simplify.
refine (λ₀ q,_).
assert (El0 (I (λ₁ v, (sb v)·₁[U]·₁le'·₁x))) as h.
{ generalize (q·₀[λ₁ u, I u]·₀p). simplify.
intros q'.
exact q'. }
refine (h·₀_).
refine (λ₀¹ i,_).
refine (λ₀ h', _).
generalize (q·₀[λ₁ y, i ·₁ (λ₁ v, (sb v)·₁[U] ·₁ le' ·₁ y)]). simplify.
intros q'.
refine (q'·₀_). clear q'.
unfold le at 1 in h'. simplify_in h'.
unfold sb at 1 in h'. simplify_in h'.
unfold le' at 1 in h'. simplify_in h'.
exact h'.
Qed.
Lemma lemma2 : El0 ((∀₀¹i:U⟶₁u0, induct i ⟶₀ i·₁WF) ⟶₀ F).
Proof.
refine (λ₀ x, _).
assert (El0 (I WF)) as h.
{ generalize (x·₀[λ₁ u, I u]·₀lemma1). simplify.
intros q.
exact q. }
refine (h·₀_). clear h.
refine (λ₀¹ i, λ₀ h0, _).
generalize (x·₀[λ₁ y, i·₁(λ₁ v, (sb v)·₁[U]·₁le'·₁y)]). simplify.
intros q.
refine (q·₀_). clear q.
unfold le in h0. simplify_in h0.
unfold WF in h0. simplify_in h0.
exact h0.
Qed.
Theorem paradox : El0 F.
Proof.
exact (lemma2·₀Omega).
Qed.
End Paradox.
The paradox tactic can be called as a shortcut to use the paradox.
Ltac paradox h :=
unshelve (refine ((fun h => _) (paradox _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ))).
End Generic.

# Impredicative universes are not retracts.

There can be no retract to an impredicative Coq universe from a smaller type. In this version of the proof, the impredicativity of the universe is postulated with a pair of functions from the universe to its type and back which commute with dependent product in an appropriate way.
Module NoRetractToImpredicativeUniverse.
Section Paradox.
Let U2 := Type.
Let U1:U2 := Type.
Variable U0:U1.

### U1 is impredicative

Variable u22u1 : U2 -> U1.
Hypothesis u22u1_unit : forall (c:U2), c -> u22u1 c.
u22u1_counit and u22u1_coherent only apply to dependent product so that the equations happen in the smaller U1 rather than U2. Indeed, it is not generally the case that one can project from a large universe to an impredicative universe and then get back the original type again. It would be too strong a hypothesis to require (in particular, it is not true of Prop). The formulation is reminiscent of the monadic characteristic of the projection from a large type to Prop.
Hypothesis u22u1_counit : forall (F:U1->U1), u22u1 (forall A,F A) -> (forall A,F A).
Hypothesis u22u1_coherent : forall (F:U1 -> U1) (f:forall x:U1, F x) (x:U1),
u22u1_counit _ (u22u1_unit _ f) x = f x.

### U0 is a retract of U1

Variable u02u1 : U0 -> U1.
Variable u12u0 : U1 -> U0.
Hypothesis u12u0_unit : forall (b:U1), b -> u02u1 (u12u0 b).
Hypothesis u12u0_counit : forall (b:U1), u02u1 (u12u0 b) -> b.

## Paradox

Theorem paradox : forall F:U1, F.
Proof.
intros F.
Generic.paradox h.
Large universe
+ exact U1.
+ exact (fun X => X).
+ cbn. exact (fun u F => forall x:u, F x).
+ cbn. exact (fun _ _ x => x).
+ cbn. exact (fun _ _ x => x).
+ cbn. exact (fun F => u22u1 (forall x, F x)).
+ cbn. exact (fun _ x => u22u1_unit _ x).
+ cbn. exact (fun _ x => u22u1_counit _ x).
Small universe
+ exact U0.
The interpretation of the small universe is the image of U0 in U1.
+ cbn. exact (fun X => u02u1 X).
+ cbn. exact (fun u F => u12u0 (forall x:(u02u1 u), u02u1 (F x))).
+ cbn. exact (fun u F => u12u0 (forall x:u, u02u1 (F x))).
+ cbn. exact (u12u0 F).
+ cbn in h.
exact (u12u0_counit _ h).
+ cbn. easy.
+ cbn. intros **. now rewrite u22u1_coherent.
+ cbn. intros * x. exact (u12u0_unit _ x).
+ cbn. intros * x. exact (u12u0_counit _ x).
+ cbn. intros * x. exact (u12u0_unit _ x).
+ cbn. intros * x. exact (u12u0_counit _ x).
Qed.
End Paradox.
End NoRetractToImpredicativeUniverse.

# Modal fragments of Prop are not retracts

In presence of a a monadic modality on Prop, we can define a subset of Prop of modal propositions which is also a complete Heyting algebra. These cannot be a retract of a modal proposition. This is a case where the universe in system U- are not encoded as Coq universes.
Module NoRetractToModalProposition.

## Monadic modality

Section Paradox.
Variable M : Prop -> Prop.
Hypothesis incr : forall A B:Prop, (A->B) -> M A -> M B.
Lemma strength: forall A (P:A->Prop), M(forall x:A,P x) -> forall x:A,M(P x).
Proof.
intros A P h x.
eapply incr in h; eauto.
Qed.

## The universe of modal propositions

Definition MProp := { P:Prop | M P -> P }.
Definition El : MProp -> Prop := @proj1_sig _ _.
Lemma modal : forall P:MProp, M(El P) -> El P.
Proof.
intros [P m]. cbn.
exact m.
Qed.
Definition Forall {A:Type} (P:A->MProp) : MProp.
Proof.
unshelve (refine (exist _ _ _)).
+ exact (forall x:A, El (P x)).
+ intros h x.
eapply strength in h.
eauto using modal.
Defined.

## Retract of the modal fragment of Prop in a small type

The retract is axiomatized using logical equivalence as the equality on propositions.
Variable bool : MProp.
Variable p2b : MProp -> El bool.
Variable b2p : El bool -> MProp.
Hypothesis p2p1 : forall A:MProp, El (b2p (p2b A)) -> El A.
Hypothesis p2p2 : forall A:MProp, El A -> El (b2p (p2b A)).

## Paradox

Theorem paradox : forall B:MProp, El B.
Proof.
intros B.
Generic.paradox h.
Large universe
+ exact MProp.
+ exact El.
+ exact (fun _ => Forall).
+ cbn. exact (fun _ _ f => f).
+ cbn. exact (fun _ _ f => f).
+ exact Forall.
+ cbn. exact (fun _ f => f).
+ cbn. exact (fun _ f => f).
Small universe
+ exact bool.
+ exact (fun b => El (b2p b)).
+ cbn. exact (fun _ F => p2b (Forall (fun x => b2p (F x)))).
+ exact (fun _ F => p2b (Forall (fun x => b2p (F x)))).
+ apply p2b.
exact B.
+ cbn in h. auto.
+ cbn. easy.
+ cbn. easy.
+ cbn. auto.
+ cbn. intros * f.
apply p2p1 in f. cbn in f.
exact f.
+ cbn. auto.
+ cbn. intros * f.
apply p2p1 in f. cbn in f.
exact f.
Qed.
End Paradox.
End NoRetractToModalProposition.

# The negative fragment of Prop is not a retract

The existence in the pure Calculus of Constructions of a retract from the negative fragment of Prop into a negative proposition is inconsistent. This is an instance of the previous result.
Module NoRetractToNegativeProp.

## The universe of negative propositions.

Definition NProp := { P:Prop | ~~P -> P }.
Definition El : NProp -> Prop := @proj1_sig _ _.
Section Paradox.

## Retract of the negative fragment of Prop in a small type

The retract is axiomatized using logical equivalence as the equality on propositions.
Variable bool : NProp.
Variable p2b : NProp -> El bool.
Variable b2p : El bool -> NProp.
Hypothesis p2p1 : forall A:NProp, El (b2p (p2b A)) -> El A.
Hypothesis p2p2 : forall A:NProp, El A -> El (b2p (p2b A)).

## Paradox

Theorem paradox : forall B:NProp, El B.
Proof.
intros B.
unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _))).
+ exact (fun P => ~~P).
+ exact bool.
+ exact p2b.
+ exact b2p.
+ exact B.
+ exact h.
+ cbn. auto.
+ cbn. auto.
+ cbn. auto.
Qed.
End Paradox.
End NoRetractToNegativeProp.

# Prop is not a retract

The existence in the pure Calculus of Constructions of a retract from Prop into a small type of Prop is inconsistent. This is a special case of the previous result.
Module NoRetractFromSmallPropositionToProp.

## The universe of propositions.

Definition NProp := { P:Prop | P -> P}.
Definition El : NProp -> Prop := @proj1_sig _ _.
Section MParadox.

## Retract of Prop in a small type, using the identity modality.

Variable bool : NProp.
Variable p2b : NProp -> El bool.
Variable b2p : El bool -> NProp.
Hypothesis p2p1 : forall A:NProp, El (b2p (p2b A)) -> El A.
Hypothesis p2p2 : forall A:NProp, El A -> El (b2p (p2b A)).

## Paradox

Theorem mparadox : forall B:NProp, El B.
Proof.
intros B.
unshelve (refine ((fun h => _) (NoRetractToModalProposition.paradox _ _ _ _ _ _ _ _))).
+ exact (fun P => P).
+ exact bool.
+ exact p2b.
+ exact b2p.
+ exact B.
+ exact h.
+ cbn. auto.
+ cbn. auto.
+ cbn. auto.
Qed.
End MParadox.
Section Paradox.

## Retract of Prop in a small type

The retract is axiomatized using logical equivalence as the equality on propositions.
Variable bool : Prop.
Variable p2b : Prop -> bool.
Variable b2p : bool -> Prop.
Hypothesis p2p1 : forall A:Prop, b2p (p2b A) -> A.
Hypothesis p2p2 : forall A:Prop, A -> b2p (p2b A).

## Paradox

Theorem paradox : forall B:Prop, B.
Proof.
intros B.
unshelve (refine (mparadox (exist _ bool (fun x => x)) _ _ _ _
(exist _ B (fun x => x)))).
+ intros p. red. red. exact (p2b (El p)).
+ cbn. intros b. red. exists (b2p b). exact (fun x => x).
+ cbn. intros [A H]. cbn. apply p2p1.
+ cbn. intros [A H]. cbn. apply p2p2.
Qed.
End Paradox.
End NoRetractFromSmallPropositionToProp.

# Large universes are not retracts of Prop.

The existence in the Calculus of Constructions with universes of a retract from some Type universe into Prop is inconsistent.
Module NoRetractFromTypeToProp.
Definition Type2 := Type.
Definition Type1 := Type : Type2.
Section Paradox.

## Assumption of a retract from Type into Prop

Variable down : Type1 -> Prop.
Variable up : Prop -> Type1.
Hypothesis up_down : forall (A:Type1), up (down A) = A :> Type1.

## Paradox

Theorem paradox : forall P:Prop, P.
Proof.
intros P.
Generic.paradox h.
Large universe.
+ exact Type1.
+ exact (fun X => X).
+ cbn. exact (fun u F => forall x, F x).
+ cbn. exact (fun _ _ x => x).
+ cbn. exact (fun _ _ x => x).
+ exact (fun F => forall A:Prop, F(up A)).
+ cbn. exact (fun F f A => f (up A)).
+ cbn.
intros F f A.
specialize (f (down A)).
rewrite up_down in f.
exact f.
+ exact Prop.
+ cbn. exact (fun X => X).
+ cbn. exact (fun A P => forall x:A, P x).
+ cbn. exact (fun A P => forall x:A, P x).
+ cbn. exact P.
+ exact h.
+ cbn. easy.
+ cbn.
intros F f A.
destruct (up_down A). cbn.
reflexivity.
+ cbn. exact (fun _ _ x => x).
+ cbn. exact (fun _ _ x => x).
+ cbn. exact (fun _ _ x => x).
+ cbn. exact (fun _ _ x => x).
Qed.
End Paradox.
End NoRetractFromTypeToProp.

# A<>Type

No Coq universe can be equal to one of its elements.
Module TypeNeqSmallType.
Unset Universe Polymorphism.
Section Paradox.

## Universe U is equal to one of its elements.

Let U := Type.
Variable A:U.
Hypothesis h : U=A.

## Universe U is a retract of A

The following context is actually sufficient for the paradox to hold. The hypothesis h:U=A is only used to define down, up and up_down.
Let down (X:U) : A := @eq_rect _ _ (fun X => X) X _ h.
Let up (X:A) : U := @eq_rect_r _ _ (fun X => X) X _ h.
Lemma up_down : forall (X:U), up (down X) = X.
Proof.
unfold up,down.
rewrite <- h.
reflexivity.
Qed.
Theorem paradox : False.
Proof.
Generic.paradox p.
Large universe
+ exact U.
+ exact (fun X=>X).
+ cbn. exact (fun X F => forall x:X, F x).
+ cbn. exact (fun _ _ x => x).
+ cbn. exact (fun _ _ x => x).
+ exact (fun F => forall x:A, F (up x)).
+ cbn. exact (fun _ f => fun x:A => f (up x)).
+ cbn. intros * f X.
specialize (f (down X)).
rewrite up_down in f.
exact f.
Small universe
+ exact A.
The interpretation of A as a universe is U.
+ cbn. exact up.
+ cbn. exact (fun _ F => down (forall x, up (F x))).
+ cbn. exact (fun _ F => down (forall x, up (F x))).
+ cbn. exact (down False).
+ rewrite up_down in p.
exact p.
+ cbn. easy.
+ cbn. intros ? f X.
destruct (up_down X). cbn.
reflexivity.
+ cbn. intros ? ? f.
rewrite up_down.
exact f.
+ cbn. intros ? ? f.
rewrite up_down in f.
exact f.
+ cbn. intros ? ? f.
rewrite up_down.
exact f.
+ cbn. intros ? ? f.
rewrite up_down in f.
exact f.
Qed.
End Paradox.
End TypeNeqSmallType.

# Prop<>Type.

Special case of TypeNeqSmallType.
Module PropNeqType.
Theorem paradox : Prop <> Type.
Proof.
intros h.
unshelve (refine (TypeNeqSmallType.paradox _ _)).
+ exact Prop.
+ easy.
Qed.
End PropNeqType.