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|
(************************************************************************)
(* * The Rocq Prover / The Rocq 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) *)
(************************************************************************)
(************************************************************************)
(** Proof that LPO and the excluded middle for negations imply
the existence of least upper bounds for all non-empty and bounded
subsets of the real numbers.
WARNING: this file is experimental and likely to change in future releases.
*)
From Stdlib Require Import Znat QArith_base Qabs.
From Stdlib Require Import ConstructiveReals.
From Stdlib Require Import ConstructiveAbs.
From Stdlib Require Import ConstructiveLimits.
From Stdlib Require Import ConstructiveEpsilon.
Local Open Scope ConstructiveReals.
Definition sig_forall_dec_T : Type
:= forall (P : nat -> Prop), (forall n, {P n} + {~P n})
-> {n | ~P n} + {forall n, P n}.
Definition sig_not_dec_T : Type := forall P : Prop, { ~~P } + { ~P }.
Definition is_upper_bound {R : ConstructiveReals}
(E:CRcarrier R -> Prop) (m:CRcarrier R)
:= forall x:CRcarrier R, E x -> x <= m.
Definition is_lub {R : ConstructiveReals}
(E:CRcarrier R -> Prop) (m:CRcarrier R) :=
is_upper_bound E m /\ (forall b:CRcarrier R, is_upper_bound E b -> m <= b).
Lemma CRlt_lpo_dec : forall {R : ConstructiveReals} (x y : CRcarrier R),
(forall (P : nat -> Prop), (forall n, {P n} + {~P n})
-> {n | ~P n} + {forall n, P n})
-> sum (x < y) (y <= x).
Proof.
intros R x y lpo.
assert (forall (z:CRcarrier R) (n : nat), z < z + CR_of_Q R (1 # Pos.of_nat (S n))).
{ intros. apply (CRle_lt_trans _ (z+0)).
- rewrite CRplus_0_r. apply CRle_refl.
- apply CRplus_lt_compat_l.
apply CR_of_Q_pos. reflexivity. }
pose (fun n:nat => let (q,_) := CR_Q_dense
R x (x + CR_of_Q R (1 # Pos.of_nat (S n))) (H x n)
in q)
as xn.
pose (fun n:nat => let (q,_) := CR_Q_dense
R y (y + CR_of_Q R (1 # Pos.of_nat (S n))) (H y n)
in q)
as yn.
destruct (lpo (fun n => Qle (yn n) (xn n + (1 # Pos.of_nat (S n))))).
- intro n. destruct (Q_dec (yn n) (xn n + (1 # Pos.of_nat (S n)))).
+ destruct s.
* left. apply Qlt_le_weak, q.
* right. apply (Qlt_not_le _ _ q).
+ left.
rewrite q. apply Qle_refl.
- left. destruct s as [n nmaj]. apply Qnot_le_lt in nmaj.
apply (CRlt_le_trans _ (CR_of_Q R (xn n))).
+ unfold xn.
destruct (CR_Q_dense R x (x + CR_of_Q R (1 # Pos.of_nat (S n))) (H x n)).
exact (fst p).
+ apply (CRle_trans _ (CR_of_Q R (yn n - (1 # Pos.of_nat (S n))))).
* apply CR_of_Q_le. rewrite <- (Qplus_le_l _ _ (1# Pos.of_nat (S n))).
ring_simplify. apply Qlt_le_weak, nmaj.
* unfold yn.
destruct (CR_Q_dense R y (y + CR_of_Q R (1 # Pos.of_nat (S n))) (H y n)).
unfold Qminus. rewrite CR_of_Q_plus, CR_of_Q_opp.
apply (CRplus_le_reg_r (CR_of_Q R (1 # Pos.of_nat (S n)))).
rewrite CRplus_assoc, CRplus_opp_l, CRplus_0_r.
apply CRlt_asym, (snd p).
- right. apply (CR_cv_le (fun n => CR_of_Q R (yn n))
(fun n => CR_of_Q R (xn n) + CR_of_Q R (1 # Pos.of_nat (S n)))).
+ intro n. rewrite <- CR_of_Q_plus. apply CR_of_Q_le. exact (q n).
+ intro p. exists (Pos.to_nat p). intros.
unfold yn.
destruct (CR_Q_dense R y (y + CR_of_Q R (1 # Pos.of_nat (S i))) (H y i)).
rewrite CRabs_right.
* apply (CRplus_le_reg_r y).
unfold CRminus. rewrite CRplus_assoc, CRplus_opp_l, CRplus_0_r.
rewrite CRplus_comm.
apply (CRle_trans _ (y + CR_of_Q R (1 # Pos.of_nat (S i)))).
-- apply CRlt_asym, (snd p0).
-- apply CRplus_le_compat_l.
apply CR_of_Q_le. unfold Qle, Qnum, Qden.
rewrite Z.mul_1_l, Z.mul_1_l. apply Pos2Z.pos_le_pos.
apply Pos2Nat.inj_le. rewrite Nat2Pos.id.
++ apply le_S, H0.
++ discriminate.
* rewrite <- (CRplus_opp_r y).
apply CRplus_le_compat_r, CRlt_asym, p0.
+ apply (CR_cv_proper _ (x+0)). 2: rewrite CRplus_0_r; reflexivity.
apply CR_cv_plus.
* intro p. exists (Pos.to_nat p). intros.
unfold xn.
destruct (CR_Q_dense R x (x + CR_of_Q R (1 # Pos.of_nat (S i))) (H x i)).
rewrite CRabs_right.
-- apply (CRplus_le_reg_r x).
unfold CRminus. rewrite CRplus_assoc, CRplus_opp_l, CRplus_0_r.
rewrite CRplus_comm.
apply (CRle_trans _ (x + CR_of_Q R (1 # Pos.of_nat (S i)))).
++ apply CRlt_asym, (snd p0).
++ apply CRplus_le_compat_l.
apply CR_of_Q_le. unfold Qle, Qnum, Qden.
rewrite Z.mul_1_l, Z.mul_1_l. apply Pos2Z.pos_le_pos.
apply Pos2Nat.inj_le. rewrite Nat2Pos.id.
** apply le_S, H0.
** discriminate.
-- rewrite <- (CRplus_opp_r x).
apply CRplus_le_compat_r, CRlt_asym, p0.
* intro p. exists (Pos.to_nat p). intros.
unfold CRminus. rewrite CRopp_0, CRplus_0_r, CRabs_right.
-- apply CR_of_Q_le. unfold Qle, Qnum, Qden.
rewrite Z.mul_1_l, Z.mul_1_l. apply Pos2Z.pos_le_pos.
apply Pos2Nat.inj_le. rewrite Nat2Pos.id.
++ apply le_S, H0.
++ discriminate.
-- apply CR_of_Q_le. discriminate.
Qed.
Lemma is_upper_bound_dec :
forall {R : ConstructiveReals} (E:CRcarrier R -> Prop) (x:CRcarrier R),
sig_forall_dec_T
-> sig_not_dec_T
-> { is_upper_bound E x } + { ~is_upper_bound E x }.
Proof.
intros R E x lpo sig_not_dec.
destruct (sig_not_dec (~exists y:CRcarrier R, E y /\ CRltProp R x y)).
- left. intros y H.
destruct (CRlt_lpo_dec x y lpo). 2: exact c.
exfalso. apply n. intro abs. apply abs. clear abs.
exists y. split.
+ exact H.
+ apply CRltForget. exact c.
- right. intro abs. apply n. intros [y [H H0]].
specialize (abs y H). apply CRltEpsilon in H0. contradiction.
Qed.
Lemma is_upper_bound_epsilon :
forall {R : ConstructiveReals} (E:CRcarrier R -> Prop),
sig_forall_dec_T
-> sig_not_dec_T
-> (exists x:CRcarrier R, is_upper_bound E x)
-> { n:nat | is_upper_bound E (CR_of_Q R (Z.of_nat n # 1)) }.
Proof.
intros R E lpo sig_not_dec Ebound.
apply constructive_indefinite_ground_description_nat.
- intro n. apply is_upper_bound_dec.
+ exact lpo.
+ exact sig_not_dec.
- destruct Ebound as [x H]. destruct (CRup_nat x) as [n nmaj]. exists n.
intros y ey. specialize (H y ey).
apply (CRle_trans _ x _ H). apply CRlt_asym, nmaj.
Qed.
Lemma is_upper_bound_not_epsilon :
forall {R : ConstructiveReals} (E:CRcarrier R -> Prop),
sig_forall_dec_T
-> sig_not_dec_T
-> (exists x : CRcarrier R, E x)
-> { m:nat | ~is_upper_bound E (-CR_of_Q R (Z.of_nat m # 1)) }.
Proof.
intros R E lpo sig_not_dec H.
apply constructive_indefinite_ground_description_nat.
- intro n.
destruct (is_upper_bound_dec E (-CR_of_Q R (Z.of_nat n # 1)) lpo sig_not_dec).
+ right. intro abs. contradiction.
+ left. exact n0.
- destruct H as [x H]. destruct (CRup_nat (-x)) as [n H0].
exists n. intro abs. specialize (abs x H).
apply abs. rewrite <- (CRopp_involutive x).
apply CRopp_gt_lt_contravar. exact H0.
Qed.
(* Decidable Dedekind cuts are Cauchy reals. *)
Record DedekindDecCut : Type :=
{
DDupcut : Q -> Prop;
DDproper : forall q r : Q, (q == r -> DDupcut q -> DDupcut r)%Q;
DDlow : Q;
DDhigh : Q;
DDdec : forall q:Q, { DDupcut q } + { ~DDupcut q };
DDinterval : forall q r : Q, Qle q r -> DDupcut q -> DDupcut r;
DDhighProp : DDupcut DDhigh;
DDlowProp : ~DDupcut DDlow;
}.
Lemma DDlow_below_up : forall (upcut : DedekindDecCut) (a b : Q),
DDupcut upcut a -> ~DDupcut upcut b -> Qlt b a.
Proof.
intros. destruct (Qlt_le_dec b a).
- exact q.
- exfalso. apply H0. apply (DDinterval upcut a).
+ exact q.
+ exact H.
Qed.
Fixpoint DDcut_limit_fix (upcut : DedekindDecCut) (r : Q) (n : nat) :
Qlt 0 r
-> (DDupcut upcut (DDlow upcut + (Z.of_nat n#1) * r))
-> { q : Q | DDupcut upcut q /\ ~DDupcut upcut (q - r) }.
Proof.
destruct n.
- intros. exfalso. simpl in H0.
apply (DDproper upcut _ (DDlow upcut)) in H0. 2: ring.
exact (DDlowProp upcut H0).
- intros. destruct (DDdec upcut (DDlow upcut + (Z.of_nat n # 1) * r)).
+ exact (DDcut_limit_fix upcut r n H d).
+ exists (DDlow upcut + (Z.of_nat (S n) # 1) * r)%Q. split.
* exact H0.
* intro abs.
apply (DDproper upcut _ (DDlow upcut + (Z.of_nat n # 1) * r)) in abs.
-- contradiction.
-- rewrite Nat2Z.inj_succ. unfold Z.succ. rewrite <- Qinv_plus_distr.
ring.
Qed.
Lemma DDcut_limit : forall (upcut : DedekindDecCut) (r : Q),
Qlt 0 r
-> { q : Q | DDupcut upcut q /\ ~DDupcut upcut (q - r) }.
Proof.
intros.
destruct (Qarchimedean ((DDhigh upcut - DDlow upcut)/r)) as [n nmaj].
apply (DDcut_limit_fix upcut r (Pos.to_nat n) H).
apply (Qmult_lt_r _ _ r) in nmaj. 2: exact H.
unfold Qdiv in nmaj.
rewrite <- Qmult_assoc, (Qmult_comm (/r)), Qmult_inv_r, Qmult_1_r in nmaj.
- apply (DDinterval upcut (DDhigh upcut)). 2: exact (DDhighProp upcut).
apply Qlt_le_weak. apply (Qplus_lt_r _ _ (-DDlow upcut)).
rewrite Qplus_assoc, <- (Qplus_comm (DDlow upcut)), Qplus_opp_r,
Qplus_0_l, Qplus_comm.
rewrite positive_nat_Z. exact nmaj.
- intros abs. rewrite abs in H. exact (Qlt_irrefl 0 H).
Qed.
Lemma glb_dec_Q : forall {R : ConstructiveReals} (upcut : DedekindDecCut),
{ x : CRcarrier R
| forall r:Q, (x < CR_of_Q R r -> DDupcut upcut r)
/\ (CR_of_Q R r < x -> ~DDupcut upcut r) }.
Proof.
intros.
assert (forall a b : Q, Qle a b -> Qle (-b) (-a)).
{ intros. apply (Qplus_le_l _ _ (a+b)). ring_simplify. exact H. }
assert (CR_cauchy R (fun n:nat => CR_of_Q R (proj1_sig (DDcut_limit
upcut (1#Pos.of_nat n) (eq_refl _))))).
{ intros p. exists (Pos.to_nat p). intros i j pi pj.
destruct (DDcut_limit upcut (1 # Pos.of_nat i) eq_refl),
(DDcut_limit upcut (1 # Pos.of_nat j) eq_refl); unfold proj1_sig.
apply (CRabs_le). split.
- intros. unfold CRminus.
rewrite <- CR_of_Q_opp, <- CR_of_Q_opp, <- CR_of_Q_plus.
apply CR_of_Q_le.
apply (Qplus_le_l _ _ x0). ring_simplify.
setoid_replace (-1 * (1 # p) + x0)%Q with (x0 - (1 # p))%Q.
2: ring. apply (Qle_trans _ (x0- (1#Pos.of_nat j))).
+ apply Qplus_le_r. apply H.
apply Z2Nat.inj_le.
* discriminate.
* discriminate.
* simpl.
rewrite Nat2Pos.id.
-- exact pj.
-- intro abs.
subst j. inversion pj. pose proof (Pos2Nat.is_pos p).
rewrite H1 in H0. inversion H0.
+ apply Qlt_le_weak, (DDlow_below_up upcut).
* apply a.
* apply a0.
- unfold CRminus. rewrite <- CR_of_Q_opp, <- CR_of_Q_plus.
apply CR_of_Q_le.
apply (Qplus_le_l _ _ (x0-(1#p))). ring_simplify.
setoid_replace (x -1 * (1 # p))%Q with (x - (1 # p))%Q.
2: ring. apply (Qle_trans _ (x- (1#Pos.of_nat i))).
+ apply Qplus_le_r. apply H.
apply Z2Nat.inj_le.
* discriminate.
* discriminate.
* simpl.
rewrite Nat2Pos.id.
-- exact pi.
-- intro abs.
subst i. inversion pi. pose proof (Pos2Nat.is_pos p).
rewrite H1 in H0. inversion H0.
+ apply Qlt_le_weak, (DDlow_below_up upcut).
* apply a0.
* apply a. }
apply CR_complete in H0. destruct H0 as [l lcv].
exists l. split.
- intros. (* find an upper point between the limit and r *)
destruct (CR_cv_open_above _ (CR_of_Q R r) l lcv H0) as [p pmaj].
specialize (pmaj p (Nat.le_refl p)).
unfold proj1_sig in pmaj.
destruct (DDcut_limit upcut (1 # Pos.of_nat p) eq_refl) as [q qmaj].
apply (DDinterval upcut q). 2: apply qmaj.
destruct (Q_dec q r).
+ destruct s.
* apply Qlt_le_weak, q0.
* exfalso. apply (CR_of_Q_lt R) in q0. exact (CRlt_asym _ _ pmaj q0).
+ rewrite q0. apply Qle_refl.
- intros H0 abs.
assert ((CR_of_Q R r+l) * CR_of_Q R (1#2) < l).
{ apply (CRmult_lt_reg_r (CR_of_Q R 2)).
- apply CR_of_Q_pos. reflexivity.
- rewrite CRmult_assoc, <- CR_of_Q_mult, (CR_of_Q_plus R 1 1).
setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity.
rewrite CRmult_plus_distr_l, CRmult_1_r, CRmult_1_r.
apply CRplus_lt_compat_r. exact H0. }
destruct (CR_cv_open_below _ _ l lcv H1) as [p pmaj].
assert (0 < (l-CR_of_Q R r) * CR_of_Q R (1#2)).
{ apply CRmult_lt_0_compat.
- rewrite <- (CRplus_opp_r (CR_of_Q R r)).
apply CRplus_lt_compat_r. exact H0.
- apply CR_of_Q_pos. reflexivity. }
destruct (CRup_nat (CRinv R _ (inr H2))) as [i imaj].
destruct i.
+ exfalso. simpl in imaj.
exact (CRlt_asym _ _ imaj (CRinv_0_lt_compat R _ (inr H2) H2)).
+ specialize (pmaj (max (S i) (S p)) (Nat.le_trans p (S p) _ (le_S p p (Nat.le_refl p)) (Nat.le_max_r (S i) (S p)))).
unfold proj1_sig in pmaj.
destruct (DDcut_limit upcut (1 # Pos.of_nat (max (S i) (S p))) eq_refl)
as [q qmaj].
destruct qmaj. apply H4. clear H4.
apply (DDinterval upcut r). 2: exact abs.
apply (Qplus_le_l _ _ (1 # Pos.of_nat (Init.Nat.max (S i) (S p)))).
ring_simplify. apply (Qle_trans _ (r + (1 # Pos.of_nat (S i)))).
* rewrite Qplus_le_r. unfold Qle,Qnum,Qden.
rewrite Z.mul_1_l, Z.mul_1_l. apply Pos2Z.pos_le_pos.
apply Pos2Nat.inj_le. rewrite Nat2Pos.id, Nat2Pos.id.
-- apply Nat.le_max_l.
-- discriminate.
-- discriminate.
* apply (CRmult_lt_compat_l ((l - CR_of_Q R r) * CR_of_Q R (1 # 2))) in imaj.
2: exact H2.
rewrite CRinv_r in imaj.
destruct (Q_dec (r+(1#Pos.of_nat (S i))) q);[|rewrite q0; apply Qle_refl].
destruct s.
{ apply Qlt_le_weak, q0. }
exfalso. apply (CR_of_Q_lt R) in q0.
apply (CRlt_asym _ _ pmaj). apply (CRlt_le_trans _ _ _ q0).
apply (CRplus_le_reg_l (-CR_of_Q R r)).
rewrite CR_of_Q_plus, <- CRplus_assoc, CRplus_opp_l, CRplus_0_l.
apply (CRmult_lt_compat_r (CR_of_Q R (1 # Pos.of_nat (S i)))) in imaj.
-- rewrite CRmult_1_l in imaj.
apply (CRle_trans _ (
(l - CR_of_Q R r) * CR_of_Q R (1 # 2) * CR_of_Q R (Z.of_nat (S i) # 1) *
CR_of_Q R (1 # Pos.of_nat (S i)))).
++ apply CRlt_asym, imaj.
++ rewrite CRmult_assoc, <- CR_of_Q_mult.
setoid_replace ((Z.of_nat (S i) # 1) * (1 # Pos.of_nat (S i)))%Q with 1%Q.
** rewrite CRmult_1_r.
unfold CRminus. rewrite CRmult_plus_distr_r, (CRplus_comm (-CR_of_Q R r)).
rewrite (CRplus_comm (CR_of_Q R r)), CRmult_plus_distr_r.
rewrite CRplus_assoc. apply CRplus_le_compat_l.
rewrite <- CR_of_Q_mult, <- CR_of_Q_opp, <- CR_of_Q_mult, <- CR_of_Q_plus.
apply CR_of_Q_le. ring_simplify. apply Qle_refl.
** unfold Qeq, Qmult, Qnum, Qden. rewrite Z.mul_1_r, Z.mul_1_r.
rewrite Z.mul_1_l, Pos.mul_1_l. unfold Z.of_nat.
apply f_equal. apply Pos.of_nat_succ.
-- apply CR_of_Q_pos. reflexivity.
Qed.
Lemma is_upper_bound_glb :
forall {R : ConstructiveReals} (E:CRcarrier R -> Prop),
sig_not_dec_T
-> sig_forall_dec_T
-> (exists x : CRcarrier R, E x)
-> (exists x : CRcarrier R, is_upper_bound E x)
-> { x : CRcarrier R
| forall r:Q, (x < CR_of_Q R r -> is_upper_bound E (CR_of_Q R r))
/\ (CR_of_Q R r < x -> ~is_upper_bound E (CR_of_Q R r)) }.
Proof.
intros R E sig_not_dec lpo Einhab Ebound.
destruct (is_upper_bound_epsilon E lpo sig_not_dec Ebound) as [a luba].
destruct (is_upper_bound_not_epsilon E lpo sig_not_dec Einhab) as [b glbb].
pose (fun q => is_upper_bound E (CR_of_Q R q)) as upcut.
assert (forall q:Q, { upcut q } + { ~upcut q } ).
{ intro q. apply is_upper_bound_dec.
- exact lpo.
- exact sig_not_dec. }
assert (forall q r : Q, (q <= r)%Q -> upcut q -> upcut r).
{ intros. intros x Ex. specialize (H1 x Ex). intro abs.
apply H1. apply (CRle_lt_trans _ (CR_of_Q R r)). 2: exact abs.
apply CR_of_Q_le. exact H0. }
assert (upcut (Z.of_nat a # 1)%Q).
{ intros x Ex. exact (luba x Ex). }
assert (~upcut (- Z.of_nat b # 1)%Q).
{ intros abs. apply glbb. intros x Ex.
specialize (abs x Ex). rewrite <- CR_of_Q_opp.
exact abs. }
assert (forall q r : Q, (q == r)%Q -> upcut q -> upcut r).
{ intros. intros x Ex. specialize (H4 x Ex). rewrite <- H3. exact H4. }
destruct (@glb_dec_Q R (Build_DedekindDecCut
upcut H3 (-Z.of_nat b # 1)%Q (Z.of_nat a # 1)
H H0 H1 H2)).
simpl in a0. exists x. intro r. split.
- intros. apply a0. exact H4.
- intros H6 abs. specialize (a0 r) as [_ a0]. apply a0.
+ exact H6.
+ exact abs.
Qed.
Lemma is_upper_bound_closed :
forall {R : ConstructiveReals}
(E:CRcarrier R -> Prop) (sig_forall_dec : sig_forall_dec_T)
(sig_not_dec : sig_not_dec_T)
(Einhab : exists x : CRcarrier R, E x)
(Ebound : exists x : CRcarrier R, is_upper_bound E x),
is_lub
E (proj1_sig (is_upper_bound_glb
E sig_not_dec sig_forall_dec Einhab Ebound)).
Proof.
intros. split.
- intros x Ex.
destruct (is_upper_bound_glb E sig_not_dec sig_forall_dec Einhab Ebound); simpl.
intro abs. destruct (CR_Q_dense R x0 x abs) as [q [qmaj H]].
specialize (a q) as [a _]. specialize (a qmaj x Ex).
contradiction.
- intros.
destruct (is_upper_bound_glb E sig_not_dec sig_forall_dec Einhab Ebound); simpl.
intro abs. destruct (CR_Q_dense R b x abs) as [q [qmaj H0]].
specialize (a q) as [_ a]. apply a.
+ exact H0.
+ intros y Ey. specialize (H y Ey). intro abs2.
apply H. exact (CRlt_trans _ (CR_of_Q R q) _ qmaj abs2).
Qed.
Lemma sig_lub :
forall {R : ConstructiveReals} (E:CRcarrier R -> Prop),
sig_forall_dec_T
-> sig_not_dec_T
-> (exists x : CRcarrier R, E x)
-> (exists x : CRcarrier R, is_upper_bound E x)
-> { u : CRcarrier R | is_lub E u }.
Proof.
intros R E sig_forall_dec sig_not_dec Einhab Ebound.
pose proof (is_upper_bound_closed E sig_forall_dec sig_not_dec Einhab Ebound).
destruct (is_upper_bound_glb
E sig_not_dec sig_forall_dec Einhab Ebound); simpl in H.
exists x. exact H.
Qed.
Definition CRis_upper_bound {R : ConstructiveReals} (E:CRcarrier R -> Prop) (m:CRcarrier R)
:= forall x:CRcarrier R, E x -> CRlt R m x -> False.
Lemma CR_sig_lub :
forall {R : ConstructiveReals} (E:CRcarrier R -> Prop),
(forall x y : CRcarrier R, CReq R x y -> (E x <-> E y))
-> sig_forall_dec_T
-> sig_not_dec_T
-> (exists x : CRcarrier R, E x)
-> (exists x : CRcarrier R, CRis_upper_bound E x)
-> { u : CRcarrier R | CRis_upper_bound E u /\
forall y:CRcarrier R, CRis_upper_bound E y -> CRlt R y u -> False }.
Proof.
intros. exact (sig_lub E X X0 H0 H1).
Qed.
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