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(* Check that Burali-Forti paradox does not go through *)
(* Source: contrib/Rocq/PARADOX/{Logics,BuraliForti},v *)
(* Some properties about relations on objects in Type *)
Inductive ACC (A : Type) (R : A -> A -> Prop) : A -> Prop :=
ACC_intro :
forall x : A, (forall y : A, R y x -> ACC A R y) -> ACC A R x.
Lemma ACC_nonreflexive :
forall (A : Type) (R : A -> A -> Prop) (x : A),
ACC A R x -> R x x -> False.
simple induction 1; intros.
exact (H1 x0 H2 H2).
Qed.
Definition WF (A : Type) (R : A -> A -> Prop) := forall x : A, ACC A R x.
Section Inverse_Image.
Variables (A B : Type) (R : B -> B -> Prop) (f : A -> B).
Definition Rof (x y : A) : Prop := R (f x) (f y).
Remark ACC_lemma :
forall y : B, ACC B R y -> forall x : A, y = f x -> ACC A Rof x.
simple induction 1; intros.
constructor; intros.
apply (H1 (f y0)); trivial.
elim H2 using eq_ind_r; trivial.
Qed.
Lemma ACC_inverse_image : forall x : A, ACC B R (f x) -> ACC A Rof x.
intros; apply (ACC_lemma (f x)); trivial.
Qed.
Lemma WF_inverse_image : WF B R -> WF A Rof.
red in |- *; intros; apply ACC_inverse_image; auto.
Qed.
End Inverse_Image.
(* Remark: the paradox is written in Type, but also works in Prop or Set. *)
Section Burali_Forti_Paradox.
Definition morphism (A : Type) (R : A -> A -> Prop)
(B : Type) (S : B -> B -> Prop) (f : A -> B) :=
forall x y : A, R x y -> S (f x) (f y).
(* The hypothesis of the paradox:
assumes there exists an universal system of notations, i.e:
- A type A0
- An injection i0 from relations on any type into A0
- The proof that i0 is injective modulo morphism
*)
Variable A0 : Type. (* Type_i *)
Variable i0 : forall X : Type, (X -> X -> Prop) -> A0. (* X: Type_j *)
Hypothesis
inj :
forall (X1 : Type) (R1 : X1 -> X1 -> Prop) (X2 : Type)
(R2 : X2 -> X2 -> Prop),
i0 X1 R1 = i0 X2 R2 -> exists f : X1 -> X2, morphism X1 R1 X2 R2 f.
(* Embedding of x in y: x and y are images of 2 well founded relations
R1 and R2, the ordinal of R2 being strictly greater than that of R1.
*)
Record emb (x y : A0) : Prop :=
{X1 : Type;
R1 : X1 -> X1 -> Prop;
eqx : x = i0 X1 R1;
X2 : Type;
R2 : X2 -> X2 -> Prop;
eqy : y = i0 X2 R2;
W2 : WF X2 R2;
f : X1 -> X2;
fmorph : morphism X1 R1 X2 R2 f;
maj : X2;
majf : forall z : X1, R2 (f z) maj}.
Lemma emb_trans : forall x y z : A0, emb x y -> emb y z -> emb x z.
intros.
case H; intros.
case H0; intros.
generalize eqx0; clear eqx0.
elim eqy using eq_ind_r; intro.
case (inj _ _ _ _ eqx0); intros.
exists X1 R1 X3 R3 (fun x : X1 => f0 (x0 (f x))) maj0; trivial.
red in |- *; auto.
Defined.
Lemma ACC_emb :
forall (X : Type) (R : X -> X -> Prop) (x : X),
ACC X R x ->
forall (Y : Type) (S : Y -> Y -> Prop) (f : Y -> X),
morphism Y S X R f -> (forall y : Y, R (f y) x) -> ACC A0 emb (i0 Y S).
simple induction 1; intros.
constructor; intros.
case H4; intros.
elim eqx using eq_ind_r.
case (inj X2 R2 Y S).
apply sym_eq; assumption.
intros.
apply H1 with (y := f (x1 maj)) (f := fun x : X1 => f (x1 (f0 x)));
try red in |- *; auto.
Defined.
(* The embedding relation is well founded *)
Lemma WF_emb : WF A0 emb.
constructor; intros.
case H; intros.
elim eqx using eq_ind_r.
apply ACC_emb with (X := X2) (R := R2) (x := maj) (f := f); trivial.
Defined.
(* The following definition enforces Type_j >= Type_i *)
Definition Omega : A0 := i0 A0 emb.
Section Subsets.
Variable a : A0.
(* We define the type of elements of A0 smaller than a w.r.t embedding.
The Record is in Type, but it is possible to avoid such structure. *)
Record sub : Type := {witness : A0; emb_wit : emb witness a}.
(* F is its image through i0 *)
Definition F : A0 := i0 sub (Rof _ _ emb witness).
(* F is embedded in Omega:
- the witness projection is a morphism
- a is an upper bound because emb_wit proves that witness is
smaller than a.
*)
Lemma F_emb_Omega : emb F Omega.
exists sub (Rof _ _ emb witness) A0 emb witness a; trivial.
exact WF_emb.
red in |- *; trivial.
exact emb_wit.
Defined.
End Subsets.
Definition fsub (a b : A0) (H : emb a b) (x : sub a) :
sub b := Build_sub _ (witness _ x) (emb_trans _ _ _ (emb_wit _ x) H).
(* F is a morphism: a < b => F(a) < F(b)
- the morphism from F(a) to F(b) is fsub above
- the upper bound is a, which is in F(b) since a < b
*)
Lemma F_morphism : morphism A0 emb A0 emb F.
red in |- *; intros.
exists
(sub x)
(Rof _ _ emb (witness x))
(sub y)
(Rof _ _ emb (witness y))
(fsub x y H)
(Build_sub _ x H); trivial.
apply WF_inverse_image.
exact WF_emb.
unfold morphism, Rof, fsub in |- *; simpl in |- *; intros.
trivial.
unfold Rof, fsub in |- *; simpl in |- *; intros.
apply emb_wit.
Defined.
(* Omega is embedded in itself:
- F is a morphism
- Omega is an upper bound of the image of F
*)
Lemma Omega_refl : emb Omega Omega.
exists A0 emb A0 emb F Omega; trivial.
exact WF_emb.
exact F_morphism.
exact F_emb_Omega.
Defined.
(* The paradox is that Omega cannot be embedded in itself, since
the embedding relation is well founded.
*)
Theorem Burali_Forti : False.
apply ACC_nonreflexive with A0 emb Omega.
apply WF_emb.
exact Omega_refl.
Defined.
End Burali_Forti_Paradox.
(* The following type seems to satisfy the hypothesis of the paradox.
But it does not!
*)
Record A0 : Type := (* Type_i' *)
i0 {X0 : Type; R0 : X0 -> X0 -> Prop}. (* X0: Type_j' *)
(* Note: this proof uses a large elimination of A0. *)
Lemma inj :
forall (X1 : Type) (R1 : X1 -> X1 -> Prop) (X2 : Type)
(R2 : X2 -> X2 -> Prop),
i0 X1 R1 = i0 X2 R2 -> exists f : X1 -> X2, morphism X1 R1 X2 R2 f.
intros.
change
match i0 X1 R1, i0 X2 R2 with
| i0 x1 r1, i0 x2 r2 => exists f : _, morphism x1 r1 x2 r2 f
end in |- *.
case H; simpl in |- *.
exists (fun x : X1 => x).
red in |- *; trivial.
Defined.
(* The following command raises 'Error: Universe Inconsistency'.
To allow large elimination of A0, i0 must not be a large constructor.
Hence, the constraint Type_j' < Type_i' is added, which is incompatible
with the constraint j >= i in the paradox.
*)
Definition Paradox : False := Burali_Forti A0 i0 inj.
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