Library UniMath.Foundations.Sets
Generalities on hSet. Vladimir Voevodsky. Feb. - Sep. 2011
Contents
- The type of sets i.e. of types of h-level 2 in UU
- hProp as a set
- Booleans as a set
- Types X which satisfy "weak" axiom of choice for all families P : X → UU
- The type of monic subtypes of a type (subsets of the set of connected
components)
- General definitions
- Direct product of two subtypes
- A subtype with paths between any two elements is an hProp
- Relations on types (or equivalently relations on the sets of connected
components)
- Relations and boolean relations
- Standard properties of relations
- Elementary implications between properties of relations
- Standard properties of relations and logical equivalences
- Preorderings, partial orderings, and associated types
- Equivalence relations and associated types
- Direct product of two relations
- Negation of a relation and its properties
- Boolean representation of decidable equality
- Boolean representation of decidable relations
- Restriction of a relation to a subtype
- Equivalence classes with respect to a given relation
- Direct product of equivalence classes
- Surjections to sets are epimorphisms
- Epimorphisms are surjections
- Universal property enjoyed by surjections
- Set quotients of types
- Set quotients defined in terms of equivalence classes
- Universal property of setquot R for functions to sets satisfying compatibility condition iscomprelfun
- Functoriality of setquot for functions mapping one relation to another
- Universal property of setquot for predicates of one and several variables
- The case when the function between quotients defined by setquotfun is a surjection, inclusion or a weak equivalence
- setquot with respect to the product of two relations
- Universal property of setquot for functions of two variables
- Functoriality of setquot for functions of two variables mapping one relation to another
- Set quotients with respect to decidable equivalence relations have decidable equality
- Relations on quotient sets
- Subtypes of quotients and quotients of subtypes
- The set of connected components of a type
- Set quotients. Construction 2 (Unfinished)
- Consequences of univalence
Preamble
The type of sets i.e. of types of h-level 2 in UU
Definition hSet : UU := total2 (λ X : UU, isaset X).
Definition make_hSet (X : UU) (i : isaset X) := tpair isaset X i : hSet.
Definition pr1hSet : hSet → UU := @pr1 UU (λ X : UU, isaset X).
Coercion pr1hSet: hSet >-> UU.
Definition eqset {X : hSet} (x x' : X) : hProp
:= make_hProp (x = x') (pr2 X x x').
Declare Scope set.
Notation "a = b" := (eqset a b) (at level 70, no associativity) : set.
Definition neqset {X : hSet} (x x' : X) : hProp
:= make_hProp (x != x') (isapropneg _). Notation "a != b" := (neqset a b) (at level 70, no associativity) : set.
Delimit Scope set with set.
Definition setproperty (X : hSet) := pr2 X.
Definition setdirprod (X Y : hSet) : hSet.
Proof.
intros. ∃ (X × Y).
apply (isofhleveldirprod 2); apply setproperty.
Defined.
Definition setcoprod (X Y : hSet) : hSet.
Proof.
intros. ∃ (X ⨿ Y). apply isasetcoprod; apply setproperty.
Defined.
Lemma isaset_total2_hSet (X : hSet) (Y : X → hSet) : isaset (∑ x, Y x).
Proof.
intros. apply isaset_total2.
- apply setproperty.
- intro x. apply setproperty.
Defined.
Definition total2_hSet {X : hSet} (Y : X → hSet) : hSet
:= make_hSet (∑ x, Y x) (isaset_total2_hSet X Y).
Definition hfiber_hSet {X Y : hSet} (f : X → Y) (y : Y) : hSet
:= make_hSet (hfiber f y) (isaset_hfiber f y (pr2 X) (pr2 Y)).
Delimit Scope set with set.
Notation "'∑' x .. y , P" := (total2_hSet (λ x,.. (total2_hSet (λ y, P))..))
(at level 200, x binder, y binder, right associativity) : set.
Lemma isaset_forall_hSet (X : UU) (Y : X → hSet) : isaset (∏ x, Y x).
Proof.
intros. apply impred_isaset. intro x. apply setproperty.
Defined.
Definition forall_hSet {X : UU} (Y : X → hSet) : hSet
:= make_hSet (∏ x, Y x) (isaset_forall_hSet X Y).
Notation "'∏' x .. y , P" := (forall_hSet (λ x,.. (forall_hSet (λ y, P))..))
(at level 200, x binder, y binder, right associativity) : set.
Definition unitset : hSet := make_hSet unit isasetunit.
Definition dirprod_hSet (X Y : hSet) : hSet.
Proof.
∃ (X × Y).
abstract (exact (isasetdirprod _ _ (setproperty X) (setproperty Y))).
Defined.
Notation "A × B" := (dirprod_hSet A B) (at level 75, right associativity) : set.
hProp as a set
Definition hPropset : hSet := tpair _ hProp isasethProp.
Definition hProp_to_hSet (P : hProp) : hSet
:= make_hSet P (isasetaprop (propproperty P)).
Definition boolset : hSet := make_hSet bool isasetbool.
Definition isInjectiveFunction {X Y : hSet} (f : X → Y) : hProp.
Proof.
intros. ∃ (∏ (x x': X), f x = f x' → x = x').
abstract (
intros; apply impred; intro x; apply impred; intro y;
apply impred; intro e; apply setproperty)
using isaprop_isInjectiveFunction.
Defined.
Types X which satisfy "weak" axiom of choice for all families P : X → UU
Definition ischoicebase_uu1 (X : UU)
:= ∏ P : X → UU, (∏ x : X, ishinh (P x)) → ishinh (∏ x : X, P x).
Uses RR1
Lemma isapropischoicebase (X : UU) : isaprop (ischoicebase_uu1 X).
Proof.
apply impred.
intro P. apply impred.
intro fs. apply (pr2 (ishinh _)).
Defined.
Definition ischoicebase (X : UU) : hProp := make_hProp _ (isapropischoicebase X).
Lemma ischoicebaseweqf {X Y : UU} (w : X ≃ Y) (is : ischoicebase X) :
ischoicebase Y.
Proof.
intros. unfold ischoicebase.
intros Q fs.
apply (hinhfun (invweq (weqonsecbase Q w))).
apply (is (funcomp w Q) (λ x : X, fs (w x))).
Defined.
Lemma ischoicebaseweqb {X Y : UU} (w : X ≃ Y) (is : ischoicebase Y) :
ischoicebase X.
Proof.
intros. apply (ischoicebaseweqf (invweq w) is).
Defined.
Lemma ischoicebaseunit : ischoicebase unit.
Proof.
unfold ischoicebase. intros P fs.
apply (hinhfun (tosecoverunit P)).
apply (fs tt).
Defined.
Lemma ischoicebasecontr {X : UU} (is : iscontr X) : ischoicebase X.
Proof.
intros.
apply (ischoicebaseweqb (weqcontrtounit is) ischoicebaseunit).
Defined.
Lemma ischoicebaseempty : ischoicebase empty.
Proof.
unfold ischoicebase. intros P fs.
apply (hinhpr (λ x : empty, fromempty x)).
Defined.
Lemma ischoicebaseempty2 {X : UU} (is : ¬ X) : ischoicebase X.
Proof.
intros.
apply (ischoicebaseweqb (weqtoempty is) ischoicebaseempty).
Defined.
Lemma ischoicebasecoprod {X Y : UU}
(isx : ischoicebase X) (isy : ischoicebase Y) : ischoicebase (coprod X Y).
Proof.
intros. unfold ischoicebase.
intros P fs. apply (hinhfun (invweq (weqsecovercoprodtoprod P))).
apply hinhand.
apply (isx _ (λ x : X, fs (ii1 x))).
apply (isy _ (λ y : Y, fs (ii2 y))).
Defined.
Proof.
apply impred.
intro P. apply impred.
intro fs. apply (pr2 (ishinh _)).
Defined.
Definition ischoicebase (X : UU) : hProp := make_hProp _ (isapropischoicebase X).
Lemma ischoicebaseweqf {X Y : UU} (w : X ≃ Y) (is : ischoicebase X) :
ischoicebase Y.
Proof.
intros. unfold ischoicebase.
intros Q fs.
apply (hinhfun (invweq (weqonsecbase Q w))).
apply (is (funcomp w Q) (λ x : X, fs (w x))).
Defined.
Lemma ischoicebaseweqb {X Y : UU} (w : X ≃ Y) (is : ischoicebase Y) :
ischoicebase X.
Proof.
intros. apply (ischoicebaseweqf (invweq w) is).
Defined.
Lemma ischoicebaseunit : ischoicebase unit.
Proof.
unfold ischoicebase. intros P fs.
apply (hinhfun (tosecoverunit P)).
apply (fs tt).
Defined.
Lemma ischoicebasecontr {X : UU} (is : iscontr X) : ischoicebase X.
Proof.
intros.
apply (ischoicebaseweqb (weqcontrtounit is) ischoicebaseunit).
Defined.
Lemma ischoicebaseempty : ischoicebase empty.
Proof.
unfold ischoicebase. intros P fs.
apply (hinhpr (λ x : empty, fromempty x)).
Defined.
Lemma ischoicebaseempty2 {X : UU} (is : ¬ X) : ischoicebase X.
Proof.
intros.
apply (ischoicebaseweqb (weqtoempty is) ischoicebaseempty).
Defined.
Lemma ischoicebasecoprod {X Y : UU}
(isx : ischoicebase X) (isy : ischoicebase Y) : ischoicebase (coprod X Y).
Proof.
intros. unfold ischoicebase.
intros P fs. apply (hinhfun (invweq (weqsecovercoprodtoprod P))).
apply hinhand.
apply (isx _ (λ x : X, fs (ii1 x))).
apply (isy _ (λ y : Y, fs (ii2 y))).
Defined.
The type of monic subtypes of a type (subsets of the set of connected components)
General definitions
Definition hsubtype (X : UU) : UU := X → hProp.
Identity Coercion id_hsubtype : hsubtype >-> Funclass.
Definition carrier {X : UU} (A : hsubtype X) := total2 A.
Coercion carrier : hsubtype >-> Sortclass.
Definition make_carrier {X : UU} (A : hsubtype X) :
∏ t : X, A t → ∑ x : X, A x := tpair A.
Definition pr1carrier {X : UU} (A : hsubtype X) := @pr1 _ _ : carrier A → X.
Lemma isaset_carrier_subset (X : hSet) (Y : hsubtype X) : isaset (∑ x, Y x).
Proof.
intros. apply isaset_total2.
- apply setproperty.
- intro x. apply isasetaprop, propproperty.
Defined.
Definition carrier_subset {X : hSet} (Y : hsubtype X) : hSet
:= make_hSet (∑ x, Y x) (isaset_carrier_subset X Y).
Declare Scope subset.
Notation "'∑' x .. y , P"
:= (carrier_subset (λ x,.. (carrier_subset (λ y, P))..))
(at level 200, x binder, y binder, right associativity) : subset.
Delimit Scope subset with subset.
Lemma isinclpr1carrier {X : UU} (A : hsubtype X) : isincl (@pr1carrier X A).
Proof.
intros. apply (isinclpr1 A (λ x : _, pr2 (A x))).
Defined.
Lemma isasethsubtype (X : UU) : isaset (hsubtype X).
Proof.
change (isofhlevel 2 (hsubtype X)).
apply impred; intro x.
exact isasethProp.
Defined.
Definition totalsubtype (X : UU) : hsubtype X := λ x, htrue.
Definition weqtotalsubtype (X : UU) : totalsubtype X ≃ X.
Proof.
apply weqpr1. intro. apply iscontrunit.
Defined.
Definition weq_subtypes {X Y : UU} (w : X ≃ Y)
(S : hsubtype X) (T : hsubtype Y) :
(∏ x, S x ↔ T (w x)) → carrier S ≃ carrier T.
Proof.
intros eq. apply (weqbandf w). intro x. apply weqiff.
- apply eq.
- apply propproperty.
- apply propproperty.
Defined.
Definition subtypesdirprod {X Y : UU} (A : hsubtype X) (B : hsubtype Y) :
hsubtype (X × Y) := λ xy : _, hconj (A (pr1 xy)) (B (pr2 xy)).
Definition fromdsubtypesdirprodcarrier {X Y : UU}
(A : hsubtype X) (B : hsubtype Y)
(xyis : subtypesdirprod A B) : dirprod A B.
Proof.
intros.
set (xy := pr1 xyis). set (is := pr2 xyis).
set (x := pr1 xy). set (y := pr2 xy).
simpl in is. simpl in y.
apply (make_dirprod (tpair A x (pr1 is)) (tpair B y (pr2 is))).
Defined.
Definition tosubtypesdirprodcarrier {X Y : UU}
(A : hsubtype X) (B : hsubtype Y)
(xisyis : dirprod A B) : subtypesdirprod A B.
Proof.
intros.
set (xis := pr1 xisyis). set (yis := pr2 xisyis).
set (x := pr1 xis). set (isx := pr2 xis).
set (y := pr1 yis). set (isy := pr2 yis).
simpl in isx. simpl in isy.
apply (tpair (subtypesdirprod A B) (make_dirprod x y) (make_dirprod isx isy)).
Defined.
Lemma weqsubtypesdirprod {X Y : UU} (A : hsubtype X) (B : hsubtype Y) :
subtypesdirprod A B ≃ A × B.
Proof.
intros.
set (f := fromdsubtypesdirprodcarrier A B).
set (g := tosubtypesdirprodcarrier A B).
split with f.
assert (egf : ∏ a : _, paths (g (f a)) a).
{
intro a.
induction a as [ xy is ].
induction xy as [ x y ].
induction is as [ isx isy ].
apply idpath.
}
assert (efg : ∏ a : _, paths (f (g a)) a).
{
intro a.
induction a as [ xis yis ].
induction xis as [ x isx ].
induction yis as [ y isy ].
apply idpath.
}
apply (isweq_iso _ _ egf efg).
Defined.
Lemma ishinhsubtypedirprod {X Y : UU} (A : hsubtype X) (B : hsubtype Y)
(isa : ishinh A) (isb : ishinh B) : ishinh (subtypesdirprod A B).
Proof.
intros.
apply (hinhfun (invweq (weqsubtypesdirprod A B))).
apply hinhand. apply isa. apply isb.
Defined.
A subtype with paths between any two elements is an hProp.
Lemma isapropsubtype {X : UU} (A : hsubtype X)
(is : ∏ (x1 x2 : X), A x1 → A x2 → x1 = x2) : isaprop (carrier A).
Proof.
intros. apply invproofirrelevance.
intros x x'.
assert (X0 : isincl (@pr1 _ A)).
{
apply isinclpr1.
intro x0.
apply (pr2 (A x0)).
}
apply (invmaponpathsincl (@pr1 _ A) X0).
induction x as [ x0 is0 ].
induction x' as [ x0' is0' ].
simpl.
apply (is x0 x0' is0 is0').
Defined.
Definition squash_pairs_to_set {Y : UU} (F : Y → UU) :
(isaset Y) → (∏ y y', F y → F y' → y = y') → (∃ y, F y) → Y.
Proof.
intros is e.
set (P := ∑ y, ∥ F y ∥).
assert (iP : isaprop P).
{
apply isapropsubtype. intros y y' f f'.
apply (squash_to_prop f). apply is. clear f; intro f.
apply (squash_to_prop f'). apply is. clear f'; intro f'.
apply e.
- assumption.
- assumption.
}
intros w.
assert (p : P).
{
apply (squash_to_prop w). exact iP. clear w; intro w.
exact (pr1 w,,hinhpr (pr2 w)).
}
clear w.
exact (pr1 p).
Defined.
Definition squash_to_set {X Y : UU} (is : isaset Y) (f : X → Y) :
(∏ x x', f x = f x') → ∥ X ∥ → Y.
Proof.
intros e w.
set (P := ∑ y, ∃ x, f x = y).
assert (j : isaprop P).
{
apply isapropsubtype; intros y y' j j'.
apply (squash_to_prop j). apply is. clear j; intros [j k].
apply (squash_to_prop j'). apply is. clear j'; intros [j' k'].
intermediate_path (f j). exact (!k).
intermediate_path (f j'). apply e. exact k'.
}
assert (p : P).
{
apply (squash_to_prop w). exact j. intro x0.
∃ (f x0). apply hinhpr. ∃ x0. apply idpath.
}
exact (pr1 p).
Defined.
Relations on types (or equivalently relations on the sets of connected components)
Relations and boolean relations
Definition hrel (X : UU) : UU := X → X → hProp.
Identity Coercion idhrel : hrel >-> Funclass.
Definition brel (X : UU) : UU := X → X → bool.
Identity Coercion idbrel : brel >-> Funclass.
Definition istrans {X : UU} (R : hrel X) : UU
:= ∏ (x1 x2 x3 : X), R x1 x2 → R x2 x3 → R x1 x3.
Definition isrefl {X : UU} (R : hrel X) : UU
:= ∏ x : X, R x x.
Definition issymm {X : UU} (R : hrel X) : UU
:= ∏ (x1 x2 : X), R x1 x2 → R x2 x1.
Definition ispreorder {X : UU} (R : hrel X) : UU := istrans R × isrefl R.
Definition iseqrel {X : UU} (R : hrel X) := ispreorder R × issymm R.
Definition iseqrelconstr {X : UU} {R : hrel X}
(trans0 : istrans R) (refl0 : isrefl R) (symm0 : issymm R) :
iseqrel R := make_dirprod (make_dirprod trans0 refl0) symm0.
Definition isirrefl {X : UU} (R : hrel X) : UU := ∏ x : X, ¬ R x x.
Definition isasymm {X : UU} (R : hrel X) : UU
:= ∏ (x1 x2 : X), R x1 x2 → R x2 x1 → empty.
Definition iscoasymm {X : UU} (R : hrel X) : UU := ∏ x1 x2, ¬ R x1 x2 → R x2 x1.
Definition istotal {X : UU} (R : hrel X) : UU := ∏ x1 x2, R x1 x2 ∨ R x2 x1.
Definition isdectotal {X : UU} (R : hrel X) : UU := ∏ x1 x2, R x1 x2 ⨿ R x2 x1.
Definition iscotrans {X : UU} (R : hrel X) : UU
:= ∏ x1 x2 x3, R x1 x3 → R x1 x2 ∨ R x2 x3.
Definition isdeccotrans {X : UU} (R : hrel X) : UU
:= ∏ x1 x2 x3, R x1 x3 → R x1 x2 ⨿ R x2 x3.
Definition isdecrel {X : UU} (R : hrel X) : UU := ∏ x1 x2, R x1 x2 ⨿ ¬ R x1 x2.
Definition isnegrel {X : UU} (R : hrel X) : UU
:= ∏ x1 x2, ¬ ¬ R x1 x2 → R x1 x2.
Note that the property of being (co-)antisymmetric is different from other
properties of relations which we consider due to the presence of paths in
its formulation. As a consequence it behaves differently relative to the
quotients of types - the quotient relation can be (co-)antisymmetric while
the original relation was not.
Definition isantisymm {X : UU} (R : hrel X) : UU
:= ∏ (x1 x2 : X), R x1 x2 → R x2 x1 → x1 = x2.
Definition isPartialOrder {X : UU} (R : hrel X) : UU
:= ispreorder R × isantisymm R.
Ltac unwrap_isPartialOrder i :=
induction i as [transrefl antisymm]; induction transrefl as [trans refl].
Definition isantisymmneg {X : UU} (R : hrel X) : UU
:= ∏ (x1 x2 : X), ¬ R x1 x2 → ¬ R x2 x1 → x1 = x2.
Definition iscoantisymm {X : UU} (R : hrel X) : UU
:= ∏ x1 x2, ¬ R x1 x2 → R x2 x1 ⨿ (x1 = x2).
Note that the following condition on a relation is different from all the
other which we have considered since it is not a property but a structure,
i.e. it is in general unclear whether isaprop (neqchoice R) is provable.
proofs that the properties are propositions
Lemma isaprop_istrans {X : hSet} (R : hrel X) : isaprop (istrans R).
Proof.
intros. repeat (apply impred;intro). apply propproperty.
Defined.
Lemma isaprop_isrefl {X : hSet} (R : hrel X) : isaprop (isrefl R).
Proof.
intros. apply impred; intro. apply propproperty.
Defined.
Lemma isaprop_istotal {X : hSet} (R : hrel X) : isaprop (istotal R).
Proof.
intros. unfold istotal.
apply impred; intro x.
apply impred; intro y.
apply propproperty.
Defined.
Lemma isaprop_isantisymm {X : hSet} (R : hrel X) : isaprop (isantisymm R).
Proof.
intros. unfold isantisymm. apply impred; intro x. apply impred; intro y.
apply impred; intro r. apply impred; intro s. apply setproperty.
Defined.
Lemma isaprop_ispreorder {X : hSet} (R : hrel X) : isaprop (ispreorder R).
Proof.
intros.
unfold ispreorder.
apply isapropdirprod.
{ apply isaprop_istrans. }
{ apply isaprop_isrefl. }
Defined.
Lemma isaprop_isPartialOrder {X : hSet} (R : hrel X) :
isaprop (isPartialOrder R).
Proof.
intros.
unfold isPartialOrder.
apply isapropdirprod.
{ apply isaprop_ispreorder. }
{ apply isaprop_isantisymm. }
Defined.
the relations on a set form a set
Definition isaset_hrel (X : hSet) : isaset (hrel X).
intros. unfold hrel.
apply impred_isaset; intro x.
apply impred_isaset; intro y.
exact isasethProp.
Defined.
Lemma istransandirrefltoasymm {X : UU} {R : hrel X}
(is1 : istrans R) (is2 : isirrefl R) : isasymm R.
Proof.
intros. intros a b rab rba. apply (is2 _ (is1 _ _ _ rab rba)).
Defined.
Lemma istotaltoiscoasymm {X : UU} {R : hrel X} (is : istotal R) : iscoasymm R.
Proof.
intros. intros x1 x2. apply (hdisjtoimpl (is _ _)).
Defined.
Lemma isdecreltoisnegrel {X : UU} {R : hrel X} (is : isdecrel R) : isnegrel R.
Proof.
intros. intros x1 x2.
induction (is x1 x2) as [ r | nr ].
- intro. apply r.
- intro nnr. induction (nnr nr).
Defined.
Lemma isantisymmnegtoiscoantisymm {X : UU} {R : hrel X}
(isdr : isdecrel R) (isr : isantisymmneg R) : iscoantisymm R.
Proof.
intros. intros x1 x2 nrx12.
induction (isdr x2 x1) as [ r | nr ].
apply (ii1 r). apply ii2. apply (isr _ _ nrx12 nr).
Defined.
Lemma rtoneq {X : UU} {R : hrel X} (is : isirrefl R) {a b : X} (r : R a b) :
a != b.
Proof.
intros. intro e. rewrite e in r. apply (is b r).
Defined.
Definition hrellogeq {X : UU} (L R : hrel X) : UU
:= ∏ x1 x2, (L x1 x2 ↔ R x1 x2).
Definition istranslogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : istrans L) : istrans R.
Proof.
intros. intros x1 x2 x3 r12 r23.
apply ((pr1 (lg _ _)) (isl _ _ _ ((pr2 (lg _ _)) r12) ((pr2 (lg _ _)) r23))).
Defined.
Definition isrefllogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : isrefl L) : isrefl R.
Proof.
intros. intro x. apply (pr1 (lg _ _) (isl x)).
Defined.
Definition issymmlogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : issymm L) : issymm R.
Proof.
intros. intros x1 x2 r12.
apply (pr1 (lg _ _) (isl _ _ (pr2 (lg _ _) r12))).
Defined.
Definition ispologeqf {X : UU} {L R : hrel X} (lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2)
(isl : ispreorder L) : ispreorder R.
Proof.
intros.
apply (make_dirprod (istranslogeqf lg (pr1 isl)) (isrefllogeqf lg (pr2 isl))).
Defined.
Definition iseqrellogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : iseqrel L) : iseqrel R.
Proof.
intros.
apply (make_dirprod (ispologeqf lg (pr1 isl)) (issymmlogeqf lg (pr2 isl))).
Defined.
Definition isirrefllogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : isirrefl L) : isirrefl R.
Proof.
intros. intros x r. apply (isl _ (pr2 (lg x x) r)).
Defined.
Definition isasymmlogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : isasymm L) : isasymm R.
Proof.
intros. intros x1 x2 r12 r21.
apply (isl _ _ (pr2 (lg _ _) r12) (pr2 (lg _ _) r21)).
Defined.
Definition iscoasymmlogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : iscoasymm L) : iscoasymm R.
Proof.
intros. intros x1 x2 r12.
apply ((pr1 (lg _ _)) (isl _ _ (negf (pr1 (lg _ _)) r12))).
Defined.
Definition istotallogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : istotal L) : istotal R.
Proof.
intros. intros x1 x2. set (int := isl x1 x2).
generalize int. clear int. simpl. apply hinhfun.
apply (coprodf (pr1 (lg x1 x2)) (pr1 (lg x2 x1))).
Defined.
Definition iscotranslogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : iscotrans L) : iscotrans R.
Proof.
intros. intros x1 x2 x3 r13.
set (int := isl x1 x2 x3 (pr2 (lg _ _) r13)). generalize int.
clear int. simpl. apply hinhfun.
apply (coprodf (pr1 (lg x1 x2)) (pr1 (lg x2 x3))).
Defined.
Definition isdecrellogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : isdecrel L) : isdecrel R.
Proof.
intros. intros x1 x2.
induction (isl x1 x2) as [ l | nl ].
- apply (ii1 (pr1 (lg _ _) l)).
- apply (ii2 (negf (pr2 (lg _ _)) nl)).
Defined.
Definition isnegrellogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : isnegrel L) : isnegrel R.
Proof.
intros. intros x1 x2 nnr.
apply ((pr1 (lg _ _)) (isl _ _ (negf (negf (pr2 (lg _ _))) nnr))).
Defined.
Definition isantisymmlogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : isantisymm L) :
isantisymm R.
Proof.
intros. intros x1 x2 r12 r21.
apply (isl _ _ (pr2 (lg _ _) r12) (pr2 (lg _ _) r21)).
Defined.
Definition isantisymmneglogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : isantisymmneg L) :
isantisymmneg R.
Proof.
intros. intros x1 x2 nr12 nr21.
apply (isl _ _ (negf (pr1 (lg _ _)) nr12) (negf (pr1 (lg _ _)) nr21)).
Defined.
Definition iscoantisymmlogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : iscoantisymm L) :
iscoantisymm R.
Proof.
intros. intros x1 x2 r12.
set (int := isl _ _ (negf (pr1 (lg _ _)) r12)). generalize int. clear int.
simpl. apply (coprodf (pr1 (lg _ _)) (idfun _)).
Defined.
Definition neqchoicelogeqf {X : UU} {L R : hrel X}
(lg : ∏ x1 x2, L x1 x2 ↔ R x1 x2) (isl : neqchoice L) : neqchoice R.
Proof.
intros. intros x1 x2 ne.
apply (coprodf (pr1 (lg x1 x2)) (pr1 (lg x2 x1)) (isl _ _ ne)).
Defined.
Definition po (X : UU) : UU := ∑ R : hrel X, ispreorder R.
Definition make_po {X : UU} (R : hrel X) (is : ispreorder R) : po X
:= tpair ispreorder R is.
Definition carrierofpo (X : UU) : po X → (X → X → hProp) := @pr1 _ ispreorder.
Coercion carrierofpo : po >-> Funclass.
Definition PreorderedSet : UU := ∑ X : hSet, po X.
Definition PreorderedSetPair (X : hSet) (R :po X) : PreorderedSet
:= tpair _ X R.
Definition carrierofPreorderedSet : PreorderedSet → hSet := pr1.
Coercion carrierofPreorderedSet : PreorderedSet >-> hSet.
Definition PreorderedSetRelation (X : PreorderedSet) : hrel X := pr1 (pr2 X).
Definition PartialOrder (X : hSet) : UU := ∑ R : hrel X, isPartialOrder R.
Definition make_PartialOrder {X : hSet} (R : hrel X) (is : isPartialOrder R) :
PartialOrder X
:= tpair isPartialOrder R is.
Definition carrierofPartialOrder {X : hSet} : PartialOrder X → hrel X := pr1.
Coercion carrierofPartialOrder : PartialOrder >-> hrel.
Definition Poset : UU := ∑ X, PartialOrder X.
Definition make_Poset (X : hSet) (R : PartialOrder X) : Poset
:= tpair PartialOrder X R.
Definition carrierofposet : Poset → hSet := pr1.
Coercion carrierofposet : Poset >-> hSet.
Definition posetRelation (X : Poset) : hrel X := pr1 (pr2 X).
Lemma isrefl_posetRelation (X : Poset) : isrefl (posetRelation X).
Proof.
intros x. exact (pr2 (pr1 (pr2 (pr2 X))) x).
Defined.
Lemma istrans_posetRelation (X : Poset) : istrans (posetRelation X).
Proof.
intros x y z l m. exact (pr1 (pr1 (pr2 (pr2 X))) x y z l m).
Defined.
Lemma isantisymm_posetRelation (X : Poset) : isantisymm (posetRelation X).
Proof.
intros x y l m. exact (pr2 (pr2 (pr2 X)) x y l m).
Defined.
Declare Scope poset.
Delimit Scope poset with poset.
Notation "m ≤ n" := (posetRelation _ m n) (no associativity, at level 70) :
poset.
Definition isaposetmorphism {X Y : Poset} (f : X → Y)
:= (∏ x x' : X, x ≤ x' → f x ≤ f x')%poset.
Definition posetmorphism (X Y : Poset) : UU
:= total2 (fun f : X → Y ⇒ isaposetmorphism f).
Definition make_posetmorphism (X Y : Poset) :
∏ t : X → Y, isaposetmorphism t → ∑ f : X → Y, isaposetmorphism f
:= tpair (fun f : X → Y ⇒ isaposetmorphism f).
Definition carrierofposetmorphism (X Y : Poset) : posetmorphism X Y → (X → Y)
:= @pr1 _ _.
Coercion carrierofposetmorphism : posetmorphism >-> Funclass.
Definition isdec_ordering (X : Poset) : UU
:= ∏ (x y : X), decidable (x ≤ y)%poset.
Lemma isaprop_isaposetmorphism {X Y : Poset} (f : X → Y) :
isaprop (isaposetmorphism f).
Proof.
intros. apply impredtwice; intros. apply impred_prop.
Defined.
the preorders on a set form a set
Definition isaset_po (X : hSet) : isaset (po X).
intros.
unfold po.
apply (isofhleveltotal2 2).
{ apply isaset_hrel. }
intros x. apply hlevelntosn. apply isaprop_ispreorder.
Defined.
the partial orders on a set form a set
Definition isaset_PartialOrder X : isaset (PartialOrder X).
intros.
unfold PartialOrder.
apply (isofhleveltotal2 2).
{ apply isaset_hrel. }
intros x. apply hlevelntosn. apply isaprop_isPartialOrder.
Defined.
poset equivalences
Definition isPosetEquivalence {X Y : Poset} (f : X ≃ Y) :=
isaposetmorphism f × isaposetmorphism (invmap f).
Lemma isaprop_isPosetEquivalence {X Y : Poset} (f : X ≃ Y) :
isaprop (isPosetEquivalence f).
Proof.
intros. unfold isPosetEquivalence.
apply isapropdirprod; apply isaprop_isaposetmorphism.
Defined.
Definition isPosetEquivalence_idweq (X : Poset) : isPosetEquivalence (idweq X).
Proof.
intros. split.
- intros x y le. exact le.
- intros x y le. exact le.
Defined.
Definition PosetEquivalence (X Y : Poset) : UU
:= ∑ f : X ≃ Y, isPosetEquivalence f.
Local Open Scope poset.
Notation "X ≅ Y" := (PosetEquivalence X Y) (at level 60, no associativity) :
poset.
Definition posetUnderlyingEquivalence {X Y : Poset} : X ≅ Y → X ≃ Y := pr1.
Coercion posetUnderlyingEquivalence : PosetEquivalence >-> weq.
Definition identityPosetEquivalence (X : Poset) : PosetEquivalence X X.
Proof.
intros. ∃ (idweq X). apply isPosetEquivalence_idweq.
Defined.
Lemma isincl_pr1_PosetEquivalence (X Y : Poset) : isincl (pr1 : X ≅ Y → X ≃ Y).
Proof.
intros. apply isinclpr1. apply isaprop_isPosetEquivalence.
Defined.
Lemma isinj_pr1_PosetEquivalence (X Y : Poset) :
isInjective (pr1 : X ≅ Y → X ≃ Y).
Proof.
intros f g. apply isweqonpathsincl. apply isincl_pr1_PosetEquivalence.
Defined.
poset concepts
Notation "m < n" := (m ≤ n × m != n)%poset (only parsing) : poset.
Definition isMinimal {X : Poset} (x : X) : UU := ∏ y, x ≤ y.
Definition isMaximal {X : Poset} (x : X) : UU := ∏ y, y ≤ x.
Definition consecutive {X : Poset} (x y : X) : UU
:= x < y × ∏ z, ¬ (x < z × z < y).
Lemma isaprop_isMinimal {X : Poset} (x : X) : isaprop (isMaximal x).
Proof.
intros. unfold isMaximal. apply impred_prop.
Defined.
Lemma isaprop_isMaximal {X : Poset} (x : X) : isaprop (isMaximal x).
Proof.
intros. unfold isMaximal. apply impred_prop.
Defined.
Lemma isaprop_consecutive {X : Poset} (x y : X) : isaprop (consecutive x y).
Proof.
intros. unfold consecutive. apply isapropdirprod.
- apply isapropdirprod. { apply pr2. } simpl. apply isapropneg.
- apply impred; intro z. apply isapropneg.
Defined.
Definition eqrel (X : UU) : UU := total2 (λ R : hrel X, iseqrel R).
Definition make_eqrel {X : UU} (R : hrel X) (is : iseqrel R) : eqrel X
:= tpair (λ R : hrel X, iseqrel R) R is.
Definition eqrelconstr {X : UU} (R : hrel X)
(is1 : istrans R) (is2 : isrefl R) (is3 : issymm R) : eqrel X
:= make_eqrel R (make_dirprod (make_dirprod is1 is2) is3).
Definition pr1eqrel (X : UU) : eqrel X → (X → (X → hProp)) := @pr1 _ _.
Coercion pr1eqrel : eqrel >-> Funclass.
Definition eqreltrans {X : UU} (R : eqrel X) : istrans R := pr1 (pr1 (pr2 R)).
Definition eqrelrefl {X : UU} (R : eqrel X) : isrefl R := pr2 (pr1 (pr2 R)).
Definition eqrelsymm {X : UU} (R : eqrel X) : issymm R := pr2 (pr2 R).
Definition hreldirprod {X Y : UU} (RX : hrel X) (RY : hrel Y) :
hrel (X × Y)
:= λ xy xy' : dirprod X Y, hconj (RX (pr1 xy) (pr1 xy'))
(RY (pr2 xy) (pr2 xy')).
Definition istransdirprod {X Y : UU} (RX : hrel X) (RY : hrel Y)
(isx : istrans RX) (isy : istrans RY) :
istrans (hreldirprod RX RY)
:= λ xy1 xy2 xy3 : _,
λ is12 : _ ,
λ is23 : _,
make_dirprod (isx _ _ _ (pr1 is12) (pr1 is23))
(isy _ _ _ (pr2 is12) (pr2 is23)).
Definition isrefldirprod {X Y : UU} (RX : hrel X) (RY : hrel Y)
(isx : isrefl RX) (isy : isrefl RY) : isrefl (hreldirprod RX RY)
:= λ xy : _, make_dirprod (isx _) (isy _).
Definition issymmdirprod {X Y : UU} (RX : hrel X) (RY : hrel Y)
(isx : issymm RX) (isy : issymm RY) : issymm (hreldirprod RX RY)
:= λ xy1 xy2 : _, λ is12 : _, make_dirprod (isx _ _ (pr1 is12))
(isy _ _ (pr2 is12)).
Definition eqreldirprod {X Y : UU} (RX : eqrel X) (RY : eqrel Y) :
eqrel (X × Y)
:= eqrelconstr (hreldirprod RX RY)
(istransdirprod _ _ (eqreltrans RX) (eqreltrans RY))
(isrefldirprod _ _ (eqrelrefl RX) (eqrelrefl RY))
(issymmdirprod _ _ (eqrelsymm RX) (eqrelsymm RY)).
Definition negrel {X : UU} (R : hrel X) : hrel X
:= λ x x', make_hProp (¬ R x x') (isapropneg _).
Lemma istransnegrel {X : UU} (R : hrel X) (isr : iscotrans R) :
istrans (negrel R).
Proof.
intros. intros x1 x2 x3 r12 r23.
apply (negf (isr x1 x2 x3)).
apply (toneghdisj (make_dirprod r12 r23)).
Defined.
Lemma isasymmnegrel {X : UU} (R : hrel X) (isr : iscoasymm R) :
isasymm (negrel R).
Proof.
intros. intros x1 x2 r12 r21. apply (r21 (isr _ _ r12)).
Defined.
Lemma iscoasymmgenrel {X : UU} (R : hrel X) (isr : isasymm R) :
iscoasymm (negrel R).
Proof.
intros. intros x1 x2 nr12. apply (negf (isr _ _) nr12).
Defined.
Lemma isdecnegrel {X : UU} (R : hrel X) (isr : isdecrel R) :
isdecrel (negrel R).
Proof.
intros. intros x1 x2.
induction (isr x1 x2) as [ r | nr ].
- apply ii2. apply (todneg _ r).
- apply (ii1 nr).
Defined.
Lemma isnegnegrel {X : UU} (R : hrel X) : isnegrel (negrel R).
Proof.
intros. intros x1 x2.
apply (negf (todneg (R x1 x2))).
Defined.
Lemma isantisymmnegrel {X : UU} (R : hrel X) (isr : isantisymmneg R) :
isantisymm (negrel R).
Proof.
intros. apply isr.
Defined.
Definition eqh {X : UU} (is : isdeceq X) : hrel X
:= λ x x', make_hProp (booleq is x x' = true)
(isasetbool (booleq is x x') true).
Definition neqh {X : UU} (is : isdeceq X) : hrel X
:= λ x x', make_hProp (booleq is x x' = false)
(isasetbool (booleq is x x') false).
Lemma isrefleqh {X : UU} (is : isdeceq X) : isrefl (eqh is).
Proof.
intros. unfold eqh. unfold booleq.
intro x. induction (is x x) as [ e | ne ].
- simpl. apply idpath.
- induction (ne (idpath x)).
Defined.
Definition weqeqh {X : UU} (is : isdeceq X) (x x' : X) :
(x = x') ≃ (eqh is x x').
Proof.
intros. apply weqimplimpl.
- intro e. induction e. apply isrefleqh.
- intro e. unfold eqh in e. unfold booleq in e.
induction (is x x') as [ e' | ne' ].
+ apply e'.
+ induction (nopathsfalsetotrue e).
- unfold isaprop. unfold isofhlevel. apply (isasetifdeceq X is x x').
- unfold eqh. simpl. unfold isaprop. unfold isofhlevel.
apply (isasetbool _ true).
Defined.
Definition weqneqh {X : UU} (is : isdeceq X) (x x' : X) :
(x != x') ≃ (neqh is x x').
Proof.
intros. unfold neqh. unfold booleq. apply weqimplimpl.
- induction (is x x') as [ e | ne ].
+ intro ne. induction (ne e).
+ intro ne'. simpl. apply idpath.
- induction (is x x') as [ e | ne ].
+ intro tf. induction (nopathstruetofalse tf).
+ intro. exact ne.
- apply (isapropneg).
- simpl. unfold isaprop. unfold isofhlevel. apply (isasetbool _ false).
Defined.
Definition decrel (X : UU) : UU := total2 (λ R : hrel X, isdecrel R).
Definition pr1decrel (X : UU) : decrel X → hrel X := @pr1 _ _.
Definition make_decrel {X : UU} {R : hrel X} (is : isdecrel R) : decrel X
:= tpair _ R is.
Coercion pr1decrel : decrel >-> hrel.
Definition decreltobrel {X : UU} (R : decrel X) : brel X.
Proof.
intros. intros x x'. induction ((pr2 R) x x').
- apply true.
- apply false.
Defined.
Definition breltodecrel {X : UU} (B : brel X) : decrel X
:= @make_decrel _ (λ x x', make_hProp ((B x x') = true) (isasetbool _ _))
(λ x x', (isdeceqbool _ _)).
Definition pathstor {X : UU} (R : decrel X) (x x' : X)
(e : decreltobrel R x x' = true) : R x x'.
Proof.
unfold decreltobrel in e.
induction (pr2 R x x') as [ e' | ne ].
- apply e'.
- induction (nopathsfalsetotrue e).
Defined.
Definition rtopaths {X : UU} (R : decrel X) (x x' : X) (r : R x x') :
decreltobrel R x x' = true.
Proof.
unfold decreltobrel. intros. induction ((pr2 R) x x') as [ r' | nr ].
- apply idpath.
- induction (nr r).
Defined.
Definition pathstonegr {X : UU} (R : decrel X) (x x' : X)
(e : decreltobrel R x x' = false) : neg (R x x').
Proof.
unfold decreltobrel in e. induction (pr2 R x x') as [ e' | ne ].
- induction (nopathstruetofalse e).
- apply ne.
Defined.
Definition negrtopaths {X : UU} (R : decrel X) (x x' : X) (nr : neg (R x x')) :
decreltobrel R x x' = false.
Proof.
unfold decreltobrel. intros.
induction (pr2 R x x') as [ r | nr' ].
- induction (nr r).
- apply idpath.
Defined.
The following construction of "ct" ("canonical term") is inspired by the
ideas of George Gonthier. The expression ct (R, x, y) where R is in
hrel X for some X and has a canonical structure of a decidable relation
and x, y are closed terms of type X such that R x y is inhabited is the
term of type R x y which relizes the canonical term in isdecrel R x y.
Definition pathstor_comp {X : UU} (R : decrel X) (x x' : X)
(e : (decreltobrel R x x') = true) : R x x'.
Proof. unfold decreltobrel. intros. induction (pr2 R x x') as e' | ne .
apply e'. induction (nopathsfalsetotrue e).
Defined.
Notation " 'ct' (R, x, y) " := ((pathstor_comp _ x y (idpath true)) : R x y)
(at level 70).
Definition ctlong {X : UU} (R : hrel X) (is : isdecrel R) (x x' : X)
(e : decreltobrel (make_decrel is) x x' = true) : R x x'.
Proof.
unfold decreltobrel in e. simpl in e. induction (is x x') as [ e' | ne ].
- apply e'.
- induction (nopathsfalsetotrue e).
Defined.
Notation " 'ct' ( R , is , x , y ) " := (ctlong R is x y (idpath true))
(at level 70).
Definition deceq_to_decrel {X:UU} : isdeceq X → decrel X.
Proof.
intros i. use make_decrel.
- intros x y. ∃ (x=y). apply isasetifdeceq. assumption.
- exact i.
Defined.
Definition confirm_equal {X : UU} (i : isdeceq X) (x x' : X)
(e : decreltobrel (deceq_to_decrel i) x x' = true) : x = x'.
Proof.
intros.
exact (pathstor (deceq_to_decrel i) _ _ e).
Defined.
Definition confirm_not_equal {X : UU} (i : isdeceq X) (x x' : X)
(e : decreltobrel (deceq_to_decrel i) x x' = false) : x != x'.
Proof.
intros.
exact (pathstonegr (deceq_to_decrel i) _ _ e).
Defined.
Ltac confirm_yes d x y := exact_op (pathstor d x y (idpath true)).
Ltac confirm_no d x y := exact_op (pathstonegr d x y (idpath false)).
Ltac confirm_equal i := match goal with |- ?x = ?y ⇒ confirm_yes (deceq_to_decrel i) x y end.
Ltac confirm_not_equal i := match goal with |- ?x != ?y ⇒ confirm_no (deceq_to_decrel i) x y end.
Ltac confirm_equal_absurd i := match goal with |- ?x = ?y → ∅ ⇒ confirm_no (deceq_to_decrel i) x y end.
Definition resrel {X : UU} (L : hrel X) (P : hsubtype X) : hrel P
:= λ p1 p2, L (pr1 p1) (pr1 p2).
Definition istransresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : istrans L) : istrans (resrel L P).
Proof.
intros. intros x1 x2 x3 r12 r23.
apply (isl _ (pr1 x2) _ r12 r23).
Defined.
Definition isreflresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : isrefl L) : isrefl (resrel L P).
Proof.
intros. intro x. apply isl.
Defined.
Definition issymmresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : issymm L) : issymm (resrel L P).
Proof.
intros. intros x1 x2 r12. apply isl. apply r12.
Defined.
Definition isporesrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : ispreorder L) : ispreorder (resrel L P).
Proof.
intros.
apply (make_dirprod (istransresrel L P (pr1 isl))
(isreflresrel L P (pr2 isl))).
Defined.
Definition iseqrelresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : iseqrel L) : iseqrel (resrel L P).
Proof.
intros.
apply (make_dirprod (isporesrel L P (pr1 isl)) (issymmresrel L P (pr2 isl))).
Defined.
Definition isirreflresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : isirrefl L) : isirrefl (resrel L P).
Proof.
intros. intros x r. apply (isl _ r).
Defined.
Definition isasymmresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : isasymm L) : isasymm (resrel L P).
Proof.
intros. intros x1 x2 r12 r21. apply (isl _ _ r12 r21).
Defined.
Definition iscoasymmresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : iscoasymm L) : iscoasymm (resrel L P).
Proof.
intros. intros x1 x2 r12. apply (isl _ _ r12).
Defined.
Definition istotalresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : istotal L) : istotal (resrel L P).
Proof.
intros. intros x1 x2. apply isl.
Defined.
Definition iscotransresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : iscotrans L) : iscotrans (resrel L P).
Proof.
intros. intros x1 x2 x3 r13. apply (isl _ _ _ r13).
Defined.
Definition isdecrelresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : isdecrel L) : isdecrel (resrel L P).
Proof.
intros. intros x1 x2. apply isl.
Defined.
Definition isnegrelresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : isnegrel L) : isnegrel (resrel L P).
Proof.
intros. intros x1 x2 nnr. apply (isl _ _ nnr).
Defined.
Definition isantisymmresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : isantisymm L) : isantisymm (resrel L P).
Proof.
intros. intros x1 x2 r12 r21.
apply (invmaponpathsincl _ (isinclpr1carrier _) _ _ (isl _ _ r12 r21)).
Defined.
Definition isantisymmnegresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : isantisymmneg L) : isantisymmneg (resrel L P).
Proof.
intros. intros x1 x2 nr12 nr21.
apply (invmaponpathsincl _ (isinclpr1carrier _) _ _ (isl _ _ nr12 nr21)).
Defined.
Definition iscoantisymmresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : iscoantisymm L) : iscoantisymm (resrel L P).
Proof.
intros. intros x1 x2 r12. induction (isl _ _ r12) as [ l | e ].
- apply (ii1 l).
- apply ii2. apply (invmaponpathsincl _ (isinclpr1carrier _) _ _ e).
Defined.
Definition neqchoiceresrel {X : UU} (L : hrel X) (P : hsubtype X)
(isl : neqchoice L) : neqchoice (resrel L P).
Proof.
intros. intros x1 x2 ne.
set (int := negf (invmaponpathsincl _ (isinclpr1carrier P) _ _) ne).
apply (isl _ _ int).
Defined.
Definition iseqclass {X : UU} (R : hrel X) (A : hsubtype X) : UU
:= dirprod (ishinh (carrier A))
(dirprod (∏ x1 x2 : X, R x1 x2 → A x1 → A x2)
(∏ x1 x2 : X, A x1 → A x2 → R x1 x2)).
Definition iseqclassconstr {X : UU} (R : hrel X) {A : hsubtype X}
(ax0 : ishinh (carrier A))
(ax1 : ∏ x1 x2 : X, R x1 x2 → A x1 → A x2)
(ax2 : ∏ x1 x2 : X, A x1 → A x2 → R x1 x2) : iseqclass R A
:= make_dirprod ax0 (make_dirprod ax1 ax2).
Definition eqax0 {X : UU} {R : hrel X} {A : hsubtype X} :
iseqclass R A → ishinh (carrier A) := λ is : iseqclass R A, pr1 is.
Definition eqax1 {X : UU} {R : hrel X} {A : hsubtype X} :
iseqclass R A → ∏ x1 x2 : X, R x1 x2 → A x1 → A x2
:= λ is : iseqclass R A, pr1 (pr2 is).
Definition eqax2 {X : UU} {R : hrel X} {A : hsubtype X} :
iseqclass R A → ∏ x1 x2 : X, A x1 → A x2 → R x1 x2
:= λ is : iseqclass R A, pr2 (pr2 is).
Lemma isapropiseqclass {X : UU} (R : hrel X) (A : hsubtype X) :
isaprop (iseqclass R A).
Proof.
apply isofhleveldirprod.
- exact (isapropishinh (carrier A)).
- apply isofhleveldirprod.
+ repeat (apply impred; intro).
exact (pr2 (A t0)).
+ repeat (apply impred; intro).
exact (pr2 (R t t0)).
Defined.
Lemma iseqclassdirprod {X Y : UU} {R : hrel X} {Q : hrel Y}
{A : hsubtype X} {B : hsubtype Y}
(isa : iseqclass R A) (isb : iseqclass Q B) :
iseqclass (hreldirprod R Q) (subtypesdirprod A B).
Proof.
intros.
set (XY := dirprod X Y).
set (AB := subtypesdirprod A B).
set (RQ := hreldirprod R Q).
set (ax0 := ishinhsubtypedirprod A B (eqax0 isa) (eqax0 isb)).
assert (ax1 : ∏ xy1 xy2 : XY, RQ xy1 xy2 → AB xy1 → AB xy2).
{
intros xy1 xy2 rq ab1.
apply (make_dirprod (eqax1 isa _ _ (pr1 rq) (pr1 ab1))
(eqax1 isb _ _ (pr2 rq) (pr2 ab1))).
}
assert (ax2 : ∏ xy1 xy2 : XY, AB xy1 → AB xy2 → RQ xy1 xy2).
{
intros xy1 xy2 ab1 ab2.
apply (make_dirprod (eqax2 isa _ _ (pr1 ab1) (pr1 ab2))
(eqax2 isb _ _ (pr2 ab1) (pr2 ab2))).
}
apply (iseqclassconstr _ ax0 ax1 ax2).
Defined.
Theorem surjectionisepitosets {X Y Z : UU} (f : X → Y) (g1 g2 : Y → Z)
(is1 : issurjective f) (is2 : isaset Z)
(isf : ∏ x : X, g1 (f x) = g2 (f x)) : ∏ y : Y, g1 y = g2 y.
Proof.
intros.
set (P1:= make_hProp (paths (g1 y) (g2 y)) (is2 (g1 y) (g2 y))).
unfold issurjective in is1.
assert (s1: (hfiber f y)-> paths (g1 y) (g2 y)).
{
intro X1. induction X1 as [t x ]. induction x. apply (isf t).
}
assert (s2: ishinh (paths (g1 y) (g2 y)))
by apply (hinhfun s1 (is1 y)).
set (is3 := is2 (g1 y) (g2 y)).
simpl in is3.
apply (@hinhuniv (paths (g1 y) (g2 y)) (make_hProp _ is3)).
- intro X1. assumption.
- assumption.
Defined.
Epimorphisms are surjections to sets
Lemma isaset_set_fun_space A (B : hSet) : isaset (A → B).
Proof.
intros.
change isaset with (isofhlevel 2).
apply impred.
apply (λ _, (pr2 B)).
Qed.
TODO find a proof without univalence for propositions (if possible)
Lemma epiissurjectiontosets {A B : UU} (p : A → B) (isB:isaset B)
(epip : ∏ (C:hSet) (g1 g2:B→C), (∏ x : A, g1 (p x) = g2 (p x)) →
(∏ y : B, g1 y = g2 y)) : issurjective p.
Proof.
intros.
assert(pred_set : isaset (B → hProp)).
{ apply (isaset_set_fun_space _ (make_hSet _ isasethProp)). }
specialize (epip (make_hSet _ pred_set)
(λ b x, ∥ ∑ y : hfiber p b, x = p (pr1 y) ∥ )
(λ b x, make_hProp (x = b) (isB x b))
).
lapply epip.
- intro h.
intro y.
specialize (h y).
apply toforallpaths in h.
specialize (h y).
cbn in h.
match type of h with _ = ?type_witn ⇒ set (typ:= type_witn) in h end.
assert (witness:typ ).
{ apply idpath. }
revert witness.
rewrite <- h.
apply hinhfun.
intro h'.
exact (pr1 h').
- intro b.
apply funextfun.
intro x; cbn.
apply weqtopathshProp.
apply logeqweq.
+ apply hinhuniv.
intros [y eqx].
rewrite eqx.
apply (hfiberpr2 _ _ y).
+ intro eqx.
apply hinhpr.
use tpair.
× ∃ b. apply idpath.
× exact eqx.
Qed.
(epip : ∏ (C:hSet) (g1 g2:B→C), (∏ x : A, g1 (p x) = g2 (p x)) →
(∏ y : B, g1 y = g2 y)) : issurjective p.
Proof.
intros.
assert(pred_set : isaset (B → hProp)).
{ apply (isaset_set_fun_space _ (make_hSet _ isasethProp)). }
specialize (epip (make_hSet _ pred_set)
(λ b x, ∥ ∑ y : hfiber p b, x = p (pr1 y) ∥ )
(λ b x, make_hProp (x = b) (isB x b))
).
lapply epip.
- intro h.
intro y.
specialize (h y).
apply toforallpaths in h.
specialize (h y).
cbn in h.
match type of h with _ = ?type_witn ⇒ set (typ:= type_witn) in h end.
assert (witness:typ ).
{ apply idpath. }
revert witness.
rewrite <- h.
apply hinhfun.
intro h'.
exact (pr1 h').
- intro b.
apply funextfun.
intro x; cbn.
apply weqtopathshProp.
apply logeqweq.
+ apply hinhuniv.
intros [y eqx].
rewrite eqx.
apply (hfiberpr2 _ _ y).
+ intro eqx.
apply hinhpr.
use tpair.
× ∃ b. apply idpath.
× exact eqx.
Qed.
Universal property enjoyed by surjections
f
A ---> C
|
| p
|
v
B
Section LiftSurjection.
Context {A B C :UU}.
Hypothesis hsc:isaset C.
Variables (p : A → B ) (f: A → C ).
Hypothesis comp_f_epi: ∏ x y, p x = p y → f x = f y.
Hypothesis surjectivep : issurjective p.
Lemma surjective_iscontr_im : ∏ b : B, iscontr
(image (λ (x:hfiber p b), f (pr1 x))).
Proof.
intro b.
apply (squash_to_prop (surjectivep b)).
{ apply isapropiscontr. }
intro H.
apply iscontraprop1.
- apply isapropsubtype.
intros x1 x2.
apply (@hinhuniv2 _ _ (make_hProp _ (hsc _ _))).
simpl; intros y1 y2; simpl.
induction y1 as [ [z1 h1] h1' ].
induction y2 as [ [z2 h2] h2' ].
rewrite <- h1' ,<-h2'.
apply comp_f_epi;simpl.
rewrite h1,h2.
apply idpath.
- apply prtoimage. apply H.
Defined.
Definition univ_surj : B → C :=
λ b, (pr1 (pr1 (surjective_iscontr_im b))).
Lemma univ_surj_ax : ∏ x, univ_surj (p x) = f x.
Proof.
intro x.
apply pathsinv0.
apply path_to_ctr.
apply (squash_to_prop (surjectivep (p x))).
{ apply isapropishinh. }
intro r. apply hinhpr.
∃ r.
apply comp_f_epi.
apply (pr2 r).
Qed.
Lemma univ_surj_unique : ∏ (g : B → C) (H : ∏ a : A, g (p a) = f a)
(b : B), g b = univ_surj b.
Proof.
intros g H b.
apply (surjectionisepitosets p); [assumption|assumption|].
intro x.
rewrite H,univ_surj_ax. apply idpath.
Qed.
End LiftSurjection.
Context {A B C :UU}.
Hypothesis hsc:isaset C.
Variables (p : A → B ) (f: A → C ).
Hypothesis comp_f_epi: ∏ x y, p x = p y → f x = f y.
Hypothesis surjectivep : issurjective p.
Lemma surjective_iscontr_im : ∏ b : B, iscontr
(image (λ (x:hfiber p b), f (pr1 x))).
Proof.
intro b.
apply (squash_to_prop (surjectivep b)).
{ apply isapropiscontr. }
intro H.
apply iscontraprop1.
- apply isapropsubtype.
intros x1 x2.
apply (@hinhuniv2 _ _ (make_hProp _ (hsc _ _))).
simpl; intros y1 y2; simpl.
induction y1 as [ [z1 h1] h1' ].
induction y2 as [ [z2 h2] h2' ].
rewrite <- h1' ,<-h2'.
apply comp_f_epi;simpl.
rewrite h1,h2.
apply idpath.
- apply prtoimage. apply H.
Defined.
Definition univ_surj : B → C :=
λ b, (pr1 (pr1 (surjective_iscontr_im b))).
Lemma univ_surj_ax : ∏ x, univ_surj (p x) = f x.
Proof.
intro x.
apply pathsinv0.
apply path_to_ctr.
apply (squash_to_prop (surjectivep (p x))).
{ apply isapropishinh. }
intro r. apply hinhpr.
∃ r.
apply comp_f_epi.
apply (pr2 r).
Qed.
Lemma univ_surj_unique : ∏ (g : B → C) (H : ∏ a : A, g (p a) = f a)
(b : B), g b = univ_surj b.
Proof.
intros g H b.
apply (surjectionisepitosets p); [assumption|assumption|].
intro x.
rewrite H,univ_surj_ax. apply idpath.
Qed.
End LiftSurjection.
Set quotients of types.
Setquotient defined in terms of equivalence classes
Definition setquot {X : UU} (R : hrel X) : UU
:= total2 (λ A : _, iseqclass R A).
Definition make_setquot {X : UU} (R : hrel X) (A : hsubtype X)
(is : iseqclass R A) : setquot R := tpair _ A is.
Definition pr1setquot {X : UU} (R : hrel X) : setquot R → (hsubtype X)
:= @pr1 _ (λ A : _, iseqclass R A).
Coercion pr1setquot : setquot >-> hsubtype.
Lemma isinclpr1setquot {X : UU} (R : hrel X) : isincl (pr1setquot R).
Proof.
apply isinclpr1. intro x0. apply isapropiseqclass.
Defined.
Theorem isasetsetquot {X : UU} (R : hrel X) : isaset (setquot R).
Proof.
apply (isasetsubset (@pr1 _ _) (isasethsubtype X)).
apply isinclpr1; intro x.
apply isapropiseqclass.
Defined.
Definition setquotinset {X : UU} (R : hrel X) : hSet :=
make_hSet _ (isasetsetquot R).
Theorem setquotpr {X : UU} (R : eqrel X) : X → setquot R.
Proof.
intros X0.
set (rax := eqrelrefl R).
set (sax := eqrelsymm R).
set (tax := eqreltrans R).
apply (tpair _ (λ x : X, R X0 x)).
split.
- exact (hinhpr (tpair _ X0 (rax X0))).
- split; intros x1 x2 X1 X2.
+ exact (tax X0 x1 x2 X2 X1).
+ exact (tax x1 X0 x2 (sax X0 x1 X1) X2).
Defined.
Lemma setquotl0 {X : UU} (R : eqrel X) (c : setquot R) (x : c) :
setquotpr R (pr1 x) = c.
Proof.
apply (invmaponpathsincl _ (isinclpr1setquot R)).
apply funextsec; intro x0.
apply hPropUnivalence; intro r.
- exact (eqax1 (pr2 c) (pr1 x) x0 r (pr2 x)).
- exact (eqax2 (pr2 c) (pr1 x) x0 (pr2 x) r).
Defined.
Theorem issurjsetquotpr {X : UU} (R : eqrel X) : issurjective (setquotpr R).
Proof.
intros. unfold issurjective.
intro c. apply (@hinhuniv (carrier (pr1 c))).
intro x. apply hinhpr.
split with (pr1 x).
- apply setquotl0.
- apply (eqax0 (pr2 c)).
Defined.
Lemma iscompsetquotpr {X : UU} (R : eqrel X) (x x' : X) (a : R x x') :
setquotpr R x = setquotpr R x'.
Proof.
intros. apply (invmaponpathsincl _ (isinclpr1setquot R)).
simpl. apply funextsec.
intro x0. apply hPropUnivalence.
intro r0. apply (eqreltrans R _ _ _ (eqrelsymm R _ _ a) r0).
intro x0'. apply (eqreltrans R _ _ _ a x0').
Defined.
Universal property of seqtquot R for functions to sets satisfying compatibility condition iscomprelfun
Definition iscomprelfun {X Y : UU} (R : hrel X) (f : X → Y) : UU
:= ∏ x x' : X, R x x' → f x = f x'.
Lemma iscomprelfunlogeqf {X Y : UU} {R L : hrel X} (lg : hrellogeq L R)
(f : X → Y) (is : iscomprelfun L f) : iscomprelfun R f.
Proof.
intros. intros x x' r. apply (is _ _ (pr2 (lg _ _) r)).
Defined.
Lemma isapropimeqclass {X : UU} (R : hrel X) (Y : hSet) (f : X → Y)
(is : iscomprelfun R f) (c : setquot R) :
isaprop (image (λ x : c, f (pr1 x))).
Proof.
intros. apply isapropsubtype.
intros y1 y2. simpl.
apply (@hinhuniv2 _ _ (make_hProp (y1 = y2) (pr2 Y y1 y2))).
intros x1 x2. simpl.
induction c as [ A iseq ].
induction x1 as [ x1 is1 ]. induction x2 as [ x2 is2 ].
induction x1 as [ x1 is1' ]. induction x2 as [ x2 is2' ].
simpl in is1. simpl in is2. simpl in is1'. simpl in is2'.
assert (r : R x1 x2) by apply (eqax2 iseq _ _ is1' is2').
apply (pathscomp0 (pathsinv0 is1) (pathscomp0 (is _ _ r) is2)).
Defined.
Global Opaque isapropimeqclass.
Theorem setquotuniv {X : UU} (R : hrel X) (Y : hSet) (f : X → Y)
(is : iscomprelfun R f) (c : setquot R) : Y.
Proof.
intros.
apply (pr1image (λ x : c, f (pr1 x))).
apply (@hinhuniv (pr1 c) (make_hProp _ (isapropimeqclass R Y f is c))
(prtoimage (λ x : c, f (pr1 x)))).
apply (eqax0 (pr2 c)).
Defined.
Note : the axioms rax, sax and trans are not used in the proof of
setquotuniv. If we consider a relation which is not an equivalence relation
then setquot will still be the set of subsets which are equivalence classes.
Now however such subsets need not to cover all of the type. In fact their set
can be empty. Nevertheless setquotuniv will apply.
Theorem setquotunivcomm {X : UU} (R : eqrel X) (Y : hSet) (f : X → Y)
(is : iscomprelfun R f) :
∏ x : X, setquotuniv R Y f is (setquotpr R x) = f x.
Proof.
intros. unfold setquotuniv. unfold setquotpr.
simpl. apply idpath.
Defined.
Theorem weqpathsinsetquot {X : UU} (R : eqrel X) (x x' : X) :
R x x' ≃ setquotpr R x = setquotpr R x'.
Proof.
intros. split with (iscompsetquotpr R x x').
apply isweqimplimpl.
- intro e.
set (e' := maponpaths (pr1setquot R) e).
unfold pr1setquot in e'. unfold setquotpr in e'. simpl in e'.
set (e'' := maponpaths (λ f : _, f x') e'). simpl in e''.
apply (eqweqmaphProp (pathsinv0 e'') (eqrelrefl R x')).
- apply (pr2 (R x x')).
- set (int := isasetsetquot R (setquotpr R x) (setquotpr R x')). assumption.
Defined.
Functoriality of setquot for functions mapping one relation to another
Definition iscomprelrelfun {X Y : UU} (RX : hrel X) (RY : hrel Y) (f : X → Y)
: UU := ∏ x x' : X, RX x x' → RY (f x) (f x').
Lemma iscomprelfunlogeqf1 {X Y : UU} {LX RX : hrel X} (RY : hrel Y)
(lg : hrellogeq LX RX) (f : X → Y) (is : iscomprelrelfun LX RY f) :
iscomprelrelfun RX RY f.
Proof.
intros. intros x x' r. apply (is _ _ (pr2 (lg _ _) r)).
Defined.
Lemma iscomprelfunlogeqf2 {X Y : UU} (RX : hrel X) {LY RY : hrel Y}
(lg : hrellogeq LY RY) (f : X → Y) (is : iscomprelrelfun RX LY f) :
iscomprelrelfun RX RY f.
Proof.
intros. intros x x' r. apply ((pr1 (lg _ _)) (is _ _ r)).
Defined.
Definition setquotfun {X Y : UU} (RX : hrel X) (RY : eqrel Y) (f : X → Y)
(is : iscomprelrelfun RX RY f) (cx : setquot RX) : setquot RY.
Proof.
intros.
set (ff := funcomp f (setquotpr RY)).
assert (isff : iscomprelfun RX ff).
{
intros x x'. intro r.
apply (weqpathsinsetquot RY (f x) (f x')).
apply is. apply r.
}
apply (setquotuniv RX (setquotinset RY) ff isff cx).
Defined.
Definition setquotfuncomm {X Y : UU} (RX : eqrel X) (RY : eqrel Y)
(f : X → Y) (is : iscomprelrelfun RX RY f) :
∏ x : X, setquotfun RX RY f is (setquotpr RX x) = setquotpr RY (f x).
Proof.
intros. simpl. apply idpath.
Defined.
Universal property of setquot for predicates of one and several variables
Theorem setquotunivprop {X : UU} (R : eqrel X) (P : setquot (pr1 R) → hProp)
(ps : ∏ x : X, pr1 (P (setquotpr R x))) : ∏ c : setquot (pr1 R), pr1 (P c).
Proof.
intros c.
apply (@hinhuniv (carrier (pr1 c)) (P c)).
- intro x.
set (e := setquotl0 R c x).
apply (eqweqmaphProp (maponpaths P e)).
exact (ps (pr1 x)).
- exact (eqax0 (pr2 c)).
Defined.
Theorem setquotuniv2prop {X : UU} (R : eqrel X)
(P : setquot R → setquot R → hProp)
(is : ∏ x x' : X, P (setquotpr R x) (setquotpr R x')) :
∏ c c' : setquot R, P c c'.
Proof.
intros.
assert (int1 : ∏ c0' : _, P c c0').
{
apply (setquotunivprop R (λ c0', P c c0')).
intro x. apply (setquotunivprop R (λ c0 : _, P c0 (setquotpr R x))).
intro x0. apply (is x0 x).
}
apply (int1 c').
Defined.
Theorem setquotuniv3prop {X : UU} (R : eqrel X)
(P : setquot R → setquot R → setquot R → hProp)
(is : ∏ x x' x'' : X, P (setquotpr R x) (setquotpr R x')
(setquotpr R x'')) :
∏ c c' c'' : setquot R, P c c' c''.
Proof.
intros.
assert (int1 : ∏ c0' c0'' : _, P c c0' c0'').
{
apply (setquotuniv2prop R (λ c0' c0'', P c c0' c0'')).
intros x x'.
apply (setquotunivprop R (λ c0 : _, P c0 (setquotpr R x)
(setquotpr R x'))).
intro x0. apply (is x0 x x').
}
apply (int1 c' c'').
Defined.
Theorem setquotuniv4prop {X : UU} (R : eqrel X)
(P : setquot R → setquot R → setquot R → setquot R → hProp)
(is : ∏ x x' x'' x''' : X, P (setquotpr R x) (setquotpr R x')
(setquotpr R x'') (setquotpr R x''')) :
∏ c c' c'' c''' : setquot R, P c c' c'' c'''.
Proof.
intros.
assert (int1 : ∏ c0 c0' c0'' : _, P c c0 c0' c0'').
{
apply (setquotuniv3prop R (λ c0 c0' c0'', P c c0 c0' c0'')).
intros x x' x''.
apply (setquotunivprop R (λ c0 : _, P c0 (setquotpr R x) (setquotpr R x')
(setquotpr R x''))).
intro x0. apply (is x0 x x' x'').
}
apply (int1 c' c'' c''').
Defined.
Important note : theorems proved above can not be used (al least at the
moment) to construct terms whose complete normalization (evaluation) is
important. For example they should not be used * directly * to construct
isdeceq property of setquot since isdeceq is in turn used to construct
boolean equality booleq and evaluation of booleq x y is important for
computational purposes. Terms produced using these universality theorems will
not fully normalize even in simple cases due to the following steps in the
proof of setquotunivprop. As a part of the proof term of this theorem there
appears the composition of an application of hPropUnivalence, transfer of
the resulting term of the identity type by maponpaths along P followed by
the reconstruction of a equivalence (two directional implication) between the
corresponding propositions through eqweqmaphProp. The resulting
implications are "opaque" and the proofs of disjunctions P ∨ Q produced
with the use of such implications can not be evaluated to one of the summands
of the disjunction. An example is given by the following theorem
isdeceqsetquot_non_constr which, as simple experiments show, can not be used
to compute the value of isdeceqsetquot. Below we give another proof of
isdeceq (setquot R) using the same assumptions which is "constructive"
i.e. usable for the evaluation purposes.
The case when setquotfun is a surjection, inclusion or a weak equivalence
Lemma issurjsetquotfun {X Y : UU} (RX : eqrel X) (RY : eqrel Y) (f : X → Y)
(is : issurjective f) (is1 : iscomprelrelfun RX RY f) :
issurjective (setquotfun RX RY f is1).
Proof.
intros. apply (issurjtwooutof3b (setquotpr RX)).
apply (issurjcomp f (setquotpr RY) is (issurjsetquotpr RY)).
Defined.
Lemma isinclsetquotfun {X Y : UU} (RX : eqrel X) (RY : eqrel Y) (f : X → Y)
(is1 : iscomprelrelfun RX RY f)
(is2 : ∏ x x' : X, RY (f x) (f x') → RX x x') :
isincl (setquotfun RX RY f is1).
Proof.
intros. apply isinclbetweensets.
- apply isasetsetquot.
- apply isasetsetquot.
- assert (is : ∏ (x x' : setquot RX),
isaprop (paths (setquotfun RX RY f is1 x)
(setquotfun RX RY f is1 x') → x = x')).
{
intros.
apply impred. intro.
apply isasetsetquot.
}
apply (setquotuniv2prop RX (λ x x', make_hProp _ (is x x'))).
simpl. intros x x'. intro e.
set (e' := invweq (weqpathsinsetquot RY (f x) (f x')) e).
apply (weqpathsinsetquot RX _ _ (is2 x x' e')).
Defined.
Definition setquotincl {X Y : UU} (RX : eqrel X) (RY : eqrel Y) (f : X → Y)
(is1 : iscomprelrelfun RX RY f)
(is2 : ∏ x x' : X, RY (f x) (f x') → RX x x') :
incl (setquot RX) (setquot RY)
:= make_incl (setquotfun RX RY f is1) (isinclsetquotfun RX RY f is1 is2).
Definition weqsetquotweq {X Y : UU} (RX : eqrel X) (RY : eqrel Y) (f : X ≃ Y)
(is1 : iscomprelrelfun RX RY f)
(is2 : ∏ x x' : X, RY (f x) (f x') → RX x x') :
(setquot RX) ≃ (setquot RY).
Proof.
intros.
set (ff := setquotfun RX RY f is1). split with ff.
assert (is2' : ∏ y y' : Y, RY y y' → RX (invmap f y) (invmap f y')).
intros y y'.
rewrite (pathsinv0 (homotweqinvweq f y)).
rewrite (pathsinv0 (homotweqinvweq f y')).
rewrite (homotinvweqweq f (invmap f y)).
rewrite (homotinvweqweq f (invmap f y')).
apply (is2 _ _). set (gg := setquotfun RY RX (invmap f) is2').
assert (egf : ∏ a, paths (gg (ff a)) a).
{
apply (setquotunivprop
RX (λ a0, make_hProp _ (isasetsetquot RX (gg (ff a0)) a0))).
simpl. intro x. unfold ff. unfold gg.
apply (maponpaths (setquotpr RX) (homotinvweqweq f x)).
}
assert (efg : ∏ a, paths (ff (gg a)) a).
{
apply (setquotunivprop
RY (λ a0, make_hProp _ (isasetsetquot RY (ff (gg a0)) a0))).
simpl. intro x. unfold ff. unfold gg.
apply (maponpaths (setquotpr RY) (homotweqinvweq f x)).
}
apply (isweq_iso _ _ egf efg).
Defined.
Definition weqsetquotsurj {X Y : UU} (RX : eqrel X) (RY : eqrel Y) (f : X → Y)
(is : issurjective f) (is1 : iscomprelrelfun RX RY f)
(is2 : ∏ x x' : X, RY (f x) (f x') → RX x x') :
(setquot RX) ≃ (setquot RY).
Proof.
intros.
set (ff := setquotfun RX RY f is1).
split with ff.
apply (@isweqinclandsurj (setquotinset RX) (setquotinset RY) ff).
apply (isinclsetquotfun RX RY f is1 is2).
apply (issurjsetquotfun RX RY f is is1).
Defined.
setquot with respect to the product of two relations
Definition setquottodirprod {X Y : UU} (RX : eqrel X) (RY : eqrel Y)
(cc : setquot (eqreldirprod RX RY)) :
(setquot RX) × (setquot RY).
Proof.
intros.
set (RXY := eqreldirprod RX RY).
apply (make_dirprod
(setquotuniv RXY (setquotinset RX)
(funcomp (@pr1 _ (λ x : _, Y)) (setquotpr RX))
(λ xy xy' : dirprod X Y,
λ rr : RXY xy xy',
iscompsetquotpr RX _ _ (pr1 rr)) cc)
(setquotuniv RXY (setquotinset RY) (funcomp (@pr2 _ (λ x : _, Y))
(setquotpr RY))
(λ xy xy' : dirprod X Y,
λ rr : RXY xy xy',
iscompsetquotpr RY _ _ (pr2 rr)) cc)).
Defined.
Definition dirprodtosetquot {X Y : UU} (RX : hrel X) (RY : hrel Y)
(cd : (setquot RX) × (setquot RY)) :
setquot (hreldirprod RX RY)
:= make_setquot _ _ (iseqclassdirprod (pr2 (pr1 cd)) (pr2 (pr2 cd))).
Theorem weqsetquottodirprod {X Y : UU} (RX : eqrel X) (RY : eqrel Y) :
weq (setquot (eqreldirprod RX RY)) ((setquot RX) × (setquot RY)).
Proof.
intros.
set (f := setquottodirprod RX RY).
set (g := dirprodtosetquot RX RY).
split with f.
assert (egf : ∏ a : _, paths (g (f a)) a).
{
apply (setquotunivprop _ (λ a : _, (make_hProp _ (isasetsetquot _ (g (f a))
a)))).
intro xy. induction xy as [ x y ]. simpl.
apply (invmaponpathsincl _ (isinclpr1setquot _)).
simpl. apply funextsec. intro xy'.
induction xy' as [ x' y' ]. apply idpath.
}
assert (efg : ∏ a : _, paths (f (g a)) a).
{
intro a. induction a as [ ax ay ]. apply pathsdirprod.
generalize ax. clear ax.
apply (setquotunivprop RX (λ ax : _, (make_hProp _ (isasetsetquot _ _ _)))).
intro x. simpl. generalize ay. clear ay.
apply (setquotunivprop RY (λ ay : _, (make_hProp _ (isasetsetquot _ _ _)))).
intro y. simpl.
apply (invmaponpathsincl _ (isinclpr1setquot _)). apply funextsec.
intro x0. simpl. apply idpath. generalize ax. clear ax.
apply (setquotunivprop RX (λ ax : _, (make_hProp _ (isasetsetquot _ _ _)))).
intro x. simpl. generalize ay. clear ay.
apply (setquotunivprop RY (λ ay : _, (make_hProp _ (isasetsetquot _ _ _)))).
intro y. simpl.
apply (invmaponpathsincl _ (isinclpr1setquot _)). apply funextsec.
intro x0. simpl. apply idpath.
}
apply (isweq_iso _ _ egf efg).
Defined.
Universal property of setquot for functions of two variables
Definition iscomprelfun2 {X Y : UU} (R : hrel X) (f : X → X → Y) : UU
:= ∏ x x' x0 x0' : X, R x x' → R x0 x0' → f x x0 = f x' x0'.
Lemma iscomprelfun2if {X Y : UU} (R : hrel X) (f : X → X → Y)
(is1 : ∏ x x' x0 : X, R x x' → f x x0 = f x' x0)
(is2 : ∏ x x0 x0' : X, R x0 x0' → f x x0 = f x x0') : iscomprelfun2 R f.
Proof.
intros. intros x x' x0 x0'. intros r r'.
set (e := is1 x x' x0 r).
set (e' := is2 x' x0 x0' r').
apply (pathscomp0 e e').
Defined.
Lemma iscomprelfun2logeqf {X Y : UU} {L R : hrel X} (lg : hrellogeq L R)
(f : X → X → Y) (is : iscomprelfun2 L f) : iscomprelfun2 R f.
Proof.
intros. intros x x' x0 x0' r r0.
apply (is _ _ _ _ ((pr2 (lg _ _)) r) ((pr2 (lg _ _)) r0)).
Defined.
Local Lemma setquotuniv2_iscomprelfun {X : UU} (R : hrel X) (Y : hSet) (f : X → X → Y)
(is : iscomprelfun2 R f) (c c0 : setquot R) :
iscomprelfun (hreldirprod R R) (λ xy : dirprod X X, f (pr1 xy) (pr2 xy)).
Proof.
intros xy x'y'. simpl. intro dp. induction dp as [ r r'].
apply (is _ _ _ _ r r').
Defined.
Global Opaque setquotuniv2_iscomprelfun.
Definition setquotuniv2 {X : UU} (R : hrel X) (Y : hSet) (f : X → X → Y)
(is : iscomprelfun2 R f) (c c0 : setquot R) : Y.
Proof.
intros.
set (ff := λ xy : dirprod X X, f (pr1 xy) (pr2 xy)).
set (RR := hreldirprod R R).
apply (setquotuniv RR Y ff (setquotuniv2_iscomprelfun R Y f is c c0)
(dirprodtosetquot R R (make_dirprod c c0))).
Defined.
Theorem setquotuniv2comm {X : UU} (R : eqrel X) (Y : hSet) (f : X → X → Y)
(is : iscomprelfun2 R f) :
∏ x x' : X, setquotuniv2 R Y f is (setquotpr R x) (setquotpr R x') = f x x'.
Proof.
intros. apply idpath.
Defined.
Functoriality of setquot for functions of two variables mapping one relation to another
Definition iscomprelrelfun2 {X Y : UU} (RX : hrel X) (RY : hrel Y)
(f : X → X → Y) : UU
:= ∏ x x' x0 x0' : X, RX x x' → RX x0 x0' → RY (f x x0) (f x' x0').
Lemma iscomprelrelfun2if {X Y : UU} (RX : hrel X) (RY : eqrel Y)
(f : X → X → Y)
(is1 : ∏ x x' x0 : X, RX x x' → RY (f x x0) (f x' x0))
(is2 : ∏ x x0 x0' : X, RX x0 x0' → RY (f x x0) (f x x0')) :
iscomprelrelfun2 RX RY f.
Proof.
intros. intros x x' x0 x0'. intros r r'.
set (e := is1 x x' x0 r). set (e' := is2 x' x0 x0' r').
apply (eqreltrans RY _ _ _ e e').
Defined.
Lemma iscomprelrelfun2logeqf1 {X Y : UU} {LX RX : hrel X} (RY : hrel Y)
(lg : hrellogeq LX RX) (f : X → X → Y) (is : iscomprelrelfun2 LX RY f) :
iscomprelrelfun2 RX RY f.
Proof.
intros. intros x x' x0 x0' r r0.
apply (is _ _ _ _ ((pr2 (lg _ _)) r) ((pr2 (lg _ _)) r0)).
Defined.
Lemma iscomprelrelfun2logeqf2 {X Y : UU} (RX : hrel X) {LY RY : hrel Y}
(lg : hrellogeq LY RY) (f : X → X → Y) (is : iscomprelrelfun2 RX LY f) :
iscomprelrelfun2 RX RY f.
Proof.
intros. intros x x' x0 x0' r r0.
apply ((pr1 (lg _ _)) (is _ _ _ _ r r0)).
Defined.
Local Lemma setquotfun2_iscomprelfun2 {X Y : UU} (RX : hrel X) (RY : eqrel Y)
(f : X → X → Y) (is : iscomprelrelfun2 RX RY f)
(cx cx0 : setquot RX) : iscomprelfun2 RX (λ x x0 : X, setquotpr RY (f x x0)).
Proof.
intros x x' x0 x0'. intros r r0.
apply (weqpathsinsetquot RY (f x x0) (f x' x0')).
apply is. apply r. apply r0.
Defined.
Global Opaque setquotfun2_iscomprelfun2.
Definition setquotfun2 {X Y : UU} (RX : hrel X) (RY : eqrel Y)
(f : X → X → Y) (is : iscomprelrelfun2 RX RY f)
(cx cx0 : setquot RX) : setquot RY.
Proof.
intros.
set (ff := λ x x0 : X, setquotpr RY (f x x0)).
exact (setquotuniv2 RX (setquotinset RY) ff (setquotfun2_iscomprelfun2 RX RY f is cx cx0) cx cx0).
Defined.
Theorem setquotfun2comm {X Y : UU} (RX : eqrel X) (RY : eqrel Y)
(f : X → X → Y) (is : iscomprelrelfun2 RX RY f) :
∏ (x x' : X), setquotfun2 RX RY f is (setquotpr RX x) (setquotpr RX x')
= setquotpr RY (f x x').
Proof.
intros. apply idpath.
Defined.
Theorem isdeceqsetquot_non_constr {X : UU} (R : eqrel X)
(is : ∏ x x' : X, isdecprop (R x x')) : isdeceq (setquot R).
Proof.
intros. apply isdeceqif. intros x x'.
apply (setquotuniv2prop
R (λ x0 x0', make_hProp _ (isapropisdecprop (x0 = x0')))).
intros x0 x0'. simpl.
apply (isdecpropweqf (weqpathsinsetquot R x0 x0') (is x0 x0')).
Defined.
Definition setquotbooleqint {X : UU} (R : eqrel X)
(is : ∏ x x' : X, isdecprop (R x x')) (x x' : X) : bool.
Proof.
intros. induction (pr1 (is x x')). apply true. apply false.
Defined.
Lemma setquotbooleqintcomp {X : UU} (R : eqrel X)
(is : ∏ x x' : X, isdecprop (R x x')) :
iscomprelfun2 R (setquotbooleqint R is).
Proof.
intros. unfold iscomprelfun2.
intros x x' x0 x0' r r0. unfold setquotbooleqint.
induction (pr1 (is x x0)) as [ r1 | nr1 ].
- induction (pr1 (is x' x0')) as [ r1' | nr1' ].
+ apply idpath.
+ induction (nr1' (eqreltrans
R _ _ _ (eqreltrans
R _ _ _ (eqrelsymm R _ _ r) r1) r0)).
- induction (pr1 (is x' x0')) as [ r1' | nr1' ].
+ induction (nr1 (eqreltrans
R _ _ _ r (eqreltrans
R _ _ _ r1' (eqrelsymm R _ _ r0)))).
+ apply idpath.
Defined.
Definition setquotbooleq {X : UU} (R : eqrel X)
(is : ∏ x x' : X, isdecprop (R x x')) :
setquot R → setquot R → bool
:= setquotuniv2 R (make_hSet _ (isasetbool)) (setquotbooleqint R is)
(setquotbooleqintcomp R is).
Lemma setquotbooleqtopaths {X : UU} (R : eqrel X)
(is : ∏ x x' : X, isdecprop (R x x')) (x x' : setquot R) :
setquotbooleq R is x x' = true → x = x'.
Proof.
revert x x'.
assert (isp : ∏ (x x' : setquot R),
isaprop ((setquotbooleq R is x x') = true → x = x')).
{
intros x x'. apply impred. intro. apply (isasetsetquot R x x').
}
apply (setquotuniv2prop R (λ x x', make_hProp _ (isp x x'))). simpl.
intros x x'.
change ((setquotbooleqint R is x x') = true
→ paths (setquotpr R x) (setquotpr R x')).
unfold setquotbooleqint. induction (pr1 (is x x')) as [ i1 | i2 ].
- intro. apply (weqpathsinsetquot R _ _ i1).
- intro H. induction (nopathsfalsetotrue H).
Defined.
Lemma setquotpathstobooleq {X : UU} (R : eqrel X)
(is : ∏ x x' : X, isdecprop (R x x')) (x x' : setquot R) :
x = x' → setquotbooleq R is x x' = true.
Proof.
intros e. induction e. generalize x.
apply (setquotunivprop
R (λ x, make_hProp _ (isasetbool (setquotbooleq R is x x) true))).
simpl. intro x0.
change ((setquotbooleqint R is x0 x0) = true).
unfold setquotbooleqint. induction (pr1 (is x0 x0)) as [ i1 | i2 ].
- apply idpath.
- induction (i2 (eqrelrefl R x0)).
Defined.
Definition isdeceqsetquot {X : UU} (R : eqrel X)
(is : ∏ x x' : X, isdecprop (R x x')) : isdeceq (setquot R).
Proof.
intros. intros x x'.
induction (boolchoice (setquotbooleq R is x x')) as [ i | ni ].
- apply (ii1 (setquotbooleqtopaths R is x x' i)).
- apply ii2. intro e.
induction (falsetonegtrue _ ni (setquotpathstobooleq R is x x' e)).
Defined.
Relations on quotient sets
Definition iscomprelrel {X : UU} (R : hrel X) (L : hrel X) : UU
:= iscomprelfun2 R L.
Lemma iscomprelrelif {X : UU} {R : hrel X} (L : hrel X) (isr : issymm R)
(i1 : ∏ x x' y, R x x' → L x y → L x' y)
(i2 : ∏ x y y', R y y' → L x y → L x y') : iscomprelrel R L.
Proof.
intros. intros x x' y y' rx ry.
set (rx' := isr _ _ rx). set (ry' := isr _ _ ry).
apply hPropUnivalence.
- intro lxy. set (int1 := i1 _ _ _ rx lxy). apply (i2 _ _ _ ry int1).
- intro lxy'. set (int1 := i1 _ _ _ rx' lxy'). apply (i2 _ _ _ ry' int1).
Defined.
Lemma iscomprelrellogeqf1 {X : UU} {R R' : hrel X} (L : hrel X)
(lg : hrellogeq R R') (is : iscomprelrel R L) : iscomprelrel R' L.
Proof.
intros. apply (iscomprelfun2logeqf lg L is).
Defined.
Lemma iscomprelrellogeqf2 {X : UU} (R : hrel X) {L L' : hrel X}
(lg : hrellogeq L L') (is : iscomprelrel R L) : iscomprelrel R L'.
Proof.
intros. intros x x' x0 x0' r r0.
assert (e : paths (L x x0) (L' x x0)).
{
apply hPropUnivalence.
- apply (pr1 (lg _ _)).
- apply (pr2 (lg _ _)).
}
assert (e' : paths (L x' x0') (L' x' x0')).
{
apply hPropUnivalence.
- apply (pr1 (lg _ _)).
- apply (pr2 (lg _ _)).
}
induction e. induction e'.
apply (is _ _ _ _ r r0).
Defined.
Definition quotrel {X : UU} {R L : hrel X} (is : iscomprelrel R L) :
hrel (setquot R) := setquotuniv2 R hPropset L is.
Lemma istransquotrel {X : UU} {R : eqrel X} {L : hrel X}
(is : iscomprelrel R L) (isl : istrans L) : istrans (quotrel is).
Proof.
intros. unfold istrans.
assert (int : ∏ (x1 x2 x3 : setquot R),
isaprop (quotrel is x1 x2 → quotrel is x2 x3
→ quotrel is x1 x3)).
{
intros x1 x2 x3.
apply impred. intro.
apply impred. intro.
apply (pr2 (quotrel is x1 x3)).
}
apply (setquotuniv3prop R (λ x1 x2 x3, make_hProp _ (int x1 x2 x3))).
intros x x' x''. intros r r'.
apply (isl x x' x'' r r').
Defined.
Lemma issymmquotrel {X : UU} {R : eqrel X} {L : hrel X} (is : iscomprelrel R L)
(isl : issymm L) : issymm (quotrel is).
Proof.
intros. unfold issymm.
assert (int : ∏ (x1 x2 : setquot R),
isaprop (quotrel is x1 x2 → quotrel is x2 x1)).
{
intros x1 x2.
apply impred. intro.
apply (pr2 (quotrel is x2 x1)).
}
apply (setquotuniv2prop R (λ x1 x2, make_hProp _ (int x1 x2))).
intros x x'. intros r.
apply (isl x x' r).
Defined.
Lemma isreflquotrel {X : UU} {R : eqrel X} {L : hrel X} (is : iscomprelrel R L)
(isl : isrefl L) : isrefl (quotrel is).
Proof.
intros. unfold isrefl. apply (setquotunivprop R).
intro x. apply (isl x).
Defined.
Lemma ispoquotrel {X : UU} {R : eqrel X} {L : hrel X} (is : iscomprelrel R L)
(isl : ispreorder L) : ispreorder (quotrel is).
Proof.
intros. split with (istransquotrel is (pr1 isl)).
apply (isreflquotrel is (pr2 isl)).
Defined.
Lemma iseqrelquotrel {X : UU} {R : eqrel X} {L : hrel X} (is : iscomprelrel R L)
(isl : iseqrel L) : iseqrel (quotrel is).
Proof.
intros. split with (ispoquotrel is (pr1 isl)).
apply (issymmquotrel is (pr2 isl)).
Defined.
Lemma isirreflquotrel {X : UU} {R : eqrel X} {L : hrel X}
(is : iscomprelrel R L) (isl : isirrefl L) : isirrefl (quotrel is).
Proof.
intros. unfold isirrefl.
apply (setquotunivprop R (λ x, make_hProp _ (isapropneg (quotrel is x x)))).
intro x. apply (isl x).
Defined.
Lemma isasymmquotrel {X : UU} {R : eqrel X} {L : hrel X} (is : iscomprelrel R L)
(isl : isasymm L) : isasymm (quotrel is).
Proof.
intros. unfold isasymm.
assert (int : ∏ (x1 x2 : setquot R),
isaprop (quotrel is x1 x2 → quotrel is x2 x1 → empty)).
{
intros x1 x2.
apply impred. intro.
apply impred. intro.
apply isapropempty.
}
apply (setquotuniv2prop R (λ x1 x2, make_hProp _ (int x1 x2))).
intros x x'. intros r r'.
apply (isl x x' r r').
Defined.
Lemma iscoasymmquotrel {X : UU} {R : eqrel X} {L : hrel X}
(is : iscomprelrel R L) (isl : iscoasymm L) : iscoasymm (quotrel is).
Proof.
intros. unfold iscoasymm.
assert (int : ∏ (x1 x2 : setquot R),
isaprop (neg (quotrel is x1 x2) → quotrel is x2 x1)).
{
intros x1 x2.
apply impred. intro.
apply (pr2 _).
}
apply (setquotuniv2prop R (λ x1 x2, make_hProp _ (int x1 x2))).
intros x x'. intros r.
apply (isl x x' r).
Defined.
Lemma istotalquotrel {X : UU} {R : eqrel X} {L : hrel X}
(is : iscomprelrel R L) (isl : istotal L) : istotal (quotrel is).
Proof.
intros. unfold istotal.
apply (setquotuniv2prop R (λ x1 x2, hdisj _ _)).
intros x x'. intros r r'.
apply (isl x x' r r').
Defined.
Lemma iscotransquotrel {X : UU} {R : eqrel X} {L : hrel X}
(is : iscomprelrel R L) (isl : iscotrans L) : iscotrans (quotrel is).
Proof.
intros. unfold iscotrans.
assert (int : ∏ (x1 x2 x3 : setquot R),
isaprop (quotrel is x1 x3 → hdisj (quotrel is x1 x2)
(quotrel is x2 x3))).
{
intros.
apply impred. intro.
apply (pr2 _).
}
apply (setquotuniv3prop R (λ x1 x2 x3, make_hProp _ (int x1 x2 x3))).
intros x x' x''. intros r.
apply (isl x x' x'' r).
Defined.
Lemma isantisymmquotrel {X : UU} {R : eqrel X} {L : hrel X}
(is : iscomprelrel R L) (isl : isantisymm L) : isantisymm (quotrel is).
Proof.
intros. unfold isantisymm.
assert (int : ∏ (x1 x2 : setquot R),
isaprop (quotrel is x1 x2 → quotrel is x2 x1 → x1 = x2)).
{
intros x1 x2.
apply impred. intro.
apply impred. intro.
apply (isasetsetquot R x1 x2).
}
apply (setquotuniv2prop R (λ x1 x2, make_hProp _ (int x1 x2))).
intros x x'. intros r r'.
apply (maponpaths (setquotpr R) (isl x x' r r')).
Defined.
Lemma isantisymmnegquotrel {X : UU} {R : eqrel X} {L : hrel X}
(is : iscomprelrel R L) (isl : isantisymmneg L) :
isantisymmneg (quotrel is).
Proof.
intros. unfold isantisymmneg.
assert (int : ∏ (x1 x2 : setquot R),
isaprop (neg (quotrel is x1 x2) → neg (quotrel is x2 x1)
→ x1 = x2)).
{
intros x1 x2.
apply impred. intro.
apply impred. intro.
apply (isasetsetquot R x1 x2).
}
apply (setquotuniv2prop R (λ x1 x2, make_hProp _ (int x1 x2))).
intros x x'. intros r r'.
apply (maponpaths (setquotpr R) (isl x x' r r')).
Defined.
We do not have a lemma for neqchoicequotrel since neqchoice is not a
property and since even when it is a property such as under the additional
condition isasymm on the relation it still carrier computational content
(similarly to isdec) which would be lost under our current approach of taking
quotients. How to best define neqchoicequotrel remains at the moment an open
question.
Lemma quotrelimpl {X : UU} {R : eqrel X} {L L' : hrel X} (is : iscomprelrel R L)
(is' : iscomprelrel R L') (impl : ∏ x x', L x x' → L' x x')
(x x' : setquot R) (ql : quotrel is x x') : quotrel is' x x'.
Proof.
intros. generalize x x' ql.
assert (int : ∏ (x0 x0' : setquot R),
isaprop (quotrel is x0 x0' → quotrel is' x0 x0')).
{
intros x0 x0'.
apply impred. intro.
apply (pr2 _).
}
apply (setquotuniv2prop _ (λ x0 x0', make_hProp _ (int x0 x0'))).
intros x0 x0'. change (L x0 x0' → L' x0 x0').
apply (impl x0 x0').
Defined.
Lemma quotrellogeq {X : UU} {R : eqrel X} {L L' : hrel X}
(is : iscomprelrel R L) (is' : iscomprelrel R L')
(lg : ∏ x x', L x x' ↔ L' x x') (x x' : setquot R) :
(quotrel is x x') ↔ (quotrel is' x x').
Proof.
intros. split.
- apply (quotrelimpl is is' (λ x0 x0', pr1 (lg x0 x0')) x x').
- apply (quotrelimpl is' is (λ x0 x0', pr2 (lg x0 x0')) x x').
Defined.
Constructive proof of decidability of the quotient relation
Definition quotdecrelint {X : UU} {R : hrel X} (L : decrel X)
(is : iscomprelrel R (pr1 L)) : brel (setquot R).
Proof.
intros.
set (f := decreltobrel L). unfold brel.
apply (setquotuniv2 R boolset f).
intros x x' x0 x0' r r0. unfold f. unfold decreltobrel in ×.
induction (pr2 L x x0') as [ l | nl ].
- induction (pr2 L x' x0') as [ l' | nl' ].
+ induction (pr2 L x x0) as [ l'' | nl'' ].
× apply idpath.
× set (e := is x x' x0 x0' r r0).
induction e. induction (nl'' l').
+ induction (pr2 L x x0) as [ l'' | nl'' ].
× set (e := is x x' x0 x0' r r0). induction e. induction (nl' l'').
× apply idpath.
- induction (pr2 L x x0) as [ l' | nl' ].
+ induction (pr2 L x' x0') as [ l'' | nl'' ].
× apply idpath.
× set (e := is x x' x0 x0' r r0). induction e. induction (nl'' l').
+ induction (pr2 L x' x0') as [ l'' | nl'' ].
× set (e := is x x' x0 x0' r r0). induction e. induction (nl' l'').
× apply idpath.
Defined.
Definition quotdecrelintlogeq {X : UU} {R : eqrel X} (L : decrel X)
(is : iscomprelrel R (pr1 L)) (x x' : setquot R) :
breltodecrel (quotdecrelint L is) x x' ↔ quotrel is x x'.
Proof.
revert x x'.
assert (int : ∏ (x x' : setquot R),
isaprop ((quotdecrelint L is x x') = true
↔ (quotrel is x x'))).
{
intros x x'. apply isapropdirprod.
- apply impred. intro. apply (pr2 (quotrel _ _ _)).
- apply impred. intro. apply isasetbool.
}
apply (setquotuniv2prop R (λ x x', make_hProp _ (int x x'))).
intros x x'. simpl. split.
- apply (pathstor L x x').
- apply (rtopaths L x x').
Defined.
Lemma isdecquotrel {X : UU} {R : eqrel X} {L : hrel X}
(is : iscomprelrel R L) (isl : isdecrel L) : isdecrel (quotrel is).
Proof.
intros.
apply (isdecrellogeqf
(quotdecrelintlogeq (make_decrel isl) is)
(pr2 (breltodecrel (quotdecrelint (make_decrel isl) is)))).
Defined.
Definition decquotrel {X : UU} {R : eqrel X} (L : decrel X)
(is : iscomprelrel R L) : decrel (setquot R)
:= make_decrel (isdecquotrel is (pr2 L)).
Definition reseqrel {X : UU} (R : eqrel X) (P : hsubtype X) : eqrel P :=
make_eqrel _ (iseqrelresrel R P (pr2 R)).
Lemma iseqclassresrel {X : UU} (R : hrel X) (P Q : hsubtype X)
(is : iseqclass R Q) (is' : ∏ x, Q x → P x) :
iseqclass (resrel R P) (λ x : P, Q (pr1 x)).
Proof.
intros. split.
- set (l1 := pr1 is). generalize l1. clear l1. simpl. apply hinhfun.
intro q. split with (make_carrier P (pr1 q) (is' (pr1 q) (pr2 q))).
apply (pr2 q).
- split.
+ intros p1 p2 r12 q1. apply ((pr1 (pr2 is)) _ _ r12 q1).
+ intros p1 p2 q1 q2. apply ((pr2 (pr2 is)) _ _ q1 q2).
Defined.
Definition fromsubquot {X : UU} (R : eqrel X) (P : hsubtype (setquot R))
(p : P) : setquot (resrel R (funcomp (setquotpr R) P)).
Proof.
intros.
split with (fun rp : carrier (funcomp (setquotpr R) P) ⇒ (pr1 p) (pr1 rp)).
apply (iseqclassresrel R (funcomp (setquotpr R) P) _ (pr2 (pr1 p))).
intros x px. set (e := setquotl0 R _ (make_carrier _ x px)).
simpl in e. unfold funcomp. rewrite e. apply (pr2 p).
Defined.
Definition tosubquot {X : UU} (R : eqrel X) (P : hsubtype (setquot R)) :
setquot (resrel R (funcomp (setquotpr R) P)) → P.
Proof.
assert (int : isaset P).
{
apply (isasetsubset (@pr1 _ P)).
- apply (setproperty (setquotinset R)).
- refine (isinclpr1carrier _).
}
apply (setquotuniv _ (make_hSet _ int) (λ xp, make_carrier
P (setquotpr R (pr1 xp))
(pr2 xp))).
intros xp1 xp2 rp12.
apply (invmaponpathsincl _ (isinclpr1carrier P) _ _).
simpl. apply (iscompsetquotpr). apply rp12.
Defined.
Definition weqsubquot {X : UU} (R : eqrel X) (P : hsubtype (setquot R)) :
weq P (setquot (resrel R (funcomp (setquotpr R) P))).
Proof.
intros.
set (f := fromsubquot R P). set (g := tosubquot R P). split with f.
assert (int0 : isaset P).
{
apply (isasetsubset (@pr1 _ P)).
- apply (setproperty (setquotinset R)).
- refine (isinclpr1carrier _).
}
assert (egf : ∏ (a : P), paths (g (f a)) a).
{
intros a. induction a as [ p isp ].
generalize isp. generalize p. clear isp. clear p.
assert (int : ∏ (p : setquot R),
isaprop (∏ isp : P p, paths (g (f (tpair _ p isp)))
(tpair _ p isp))).
{
intro p.
apply impred. intro.
apply (int0 _ _).
}
apply (setquotunivprop _ (λ a, make_hProp _ (int a))).
simpl. intros x isp. apply (invmaponpathsincl _ (isinclpr1carrier P) _ _).
apply idpath.
}
assert (efg : ∏ (a : setquot (resrel R (P ∘ setquotpr R))),
paths (f (g a)) a).
{
assert (int : ∏ a, isaprop (paths (f (g a)) a)).
{
intro a.
apply (setproperty (setquotinset (resrel R (funcomp (setquotpr R) P)))).
}
set (Q := reseqrel R (funcomp (setquotpr R) P)).
apply (setquotunivprop
Q (fun a : setquot (resrel R (funcomp (setquotpr R) P)) ⇒
make_hProp _ (int a))).
intro a. simpl. unfold f. unfold g. unfold fromsubquot. unfold tosubquot.
apply (invmaponpathsincl _ (isinclpr1 _ (λ a, isapropiseqclass _ a))).
apply idpath.
}
apply (isweq_iso _ _ egf efg).
Defined.
Comment: unfortunetely weqsubquot is not as useful as it should be at
moment due to the failure of the following code to work:
Lemma test (X : UU) (R : eqrel X) (P : hsubtype (setquot R)) (x : X)
(is : P (setquotpr R x)) :
paths (setquotpr (reseqrel R (funcomp (setquotpr R) P)) (tpair _ x is))
(fromsubquot R P (tpair _ (setquotpr R x) is)).
Proof. intros. apply idpath. Defined.
As one of the consequences we are forced to use a "hack" in the definition of
multiplicative inverses for rationals in hqmultinv.
The issue which arises here is the same one which arises in several other
places in the work with quotients. It can be traced back first to the failure
of invmaponpathsincl to map idpath to idpath and then to the fact that
for (X : UU) (is : isaprop X) the term
t := proofirrelevance is : ∏ x1 x2 : X, x1 = x2 does not satisfy
(definitionally) the condition t x x == idpath x.
It can and probably should be fixed by the addition of a new componenet to
CIC in the form of a term constructor:
tfc (X : UU) (E : X -> UU) (is : ∏ x, iscontr (E x)) (x0 : X) (e0 : E x0) :
∏ x : X, E x.
and a computation rule
tfc_comp (tfc X E is x0 e0 x0) == e0
(with both tfc and tfc_comp definable in an arbitrary context)
Such an extension will be compatible with the univalent models and should not,
as far as I can see, provide any problems for normalization or for the
decidability of typing. Using tfc one can give a construction of
proofirrelevance as follows (recall that
isaprop := ∏ x1 x2, iscontr (x1 = x2)) :
Lemma proofirrelevance {X : UU} (is : isaprop X) : ∏ x1 x2, x1 = x2.
Proof.
intros X is x1.
apply (tfc X (λ x2, x1 = x2) is x1 (idpath x1)).
Defined.
Defined in this way proofirrelevance will have the required property and
will enable to define invmaponpathsincl in such a way that the existing
proofs of setquotl0 and fromsubquot and weqsubquot will provide the
desired behavior of fromsubquot on terms of the form
(tpair _ (setquotpr R x) is).
The set of connected components of type.
Definition pathshrel (X : UU) : X → X → hProp := λ x x' : X, ishinh (x = x').
Definition istranspathshrel (X : UU) : istrans (pathshrel X)
:= λ x x' x'' : _,
λ a : _,
λ b : _,
hinhfun2 (fun e1 : x = x' ⇒ fun e2 : x' = x'' ⇒ e1 @ e2) a b.
Definition isreflpathshrel (X : UU) : isrefl (pathshrel X)
:= λ x : _, hinhpr (idpath x).
Definition issymmpathshrel (X : UU) : issymm (pathshrel X)
:= λ x x': _, λ a : _, hinhfun (fun e : x = x' ⇒ ! e) a.
Definition pathseqrel (X : UU) : eqrel X
:= eqrelconstr (pathshrel X) (istranspathshrel X) (isreflpathshrel X)
(issymmpathshrel X).
Definition pi0 (X : UU) : UU := setquot (pathshrel X).
Definition pi0pr (X : UU) : X → setquot (pathseqrel X)
:= setquotpr (pathseqrel X).
Set quotients. Construction 2.
Functions compatible with a relation
Definition compfun {X : UU} (R : hrel X) (S : UU) : UU
:= total2 (fun F : X → S ⇒ iscomprelfun R F).
Definition make_compfun {X : UU} (R : hrel X) {S : UU} (f : X → S)
(is : iscomprelfun R f) : compfun R S := tpair _ f is.
Definition pr1compfun (X : UU) (R : hrel X) (S : UU) :
@compfun X R S → (X → S) := @pr1 _ _.
Coercion pr1compfun : compfun >-> Funclass.
Definition compevmapset {X : UU} (R : hrel X) :
X → ∏ S : hSet, (compfun R S) → S
:= λ x : X, λ S : _, λ f : compfun R S, pr1 f x.
Definition compfuncomp {X : UU} (R : hrel X) {S S' : UU} (f : compfun R S)
(g : S → S') : compfun R S'.
Proof.
intros. split with (funcomp f g). intros x x' r.
apply (maponpaths g (pr2 f x x' r)).
Defined.
Tests
Definition F (X Y : UU) (R : hrel X) := (compfun R Y) -> Y.
Definition Fi (X Y : UU) (R : hrel X) : X -> F X Y R := λ x, λ f, f x.
Lemma iscompFi {X Y : UU} (R : hrel X) : iscomprelfun R (Fi X Y R).
Proof.
intros. intros x x' r. unfold Fi. apply funextfun.
intro f. apply (pr2 f x x' r).
Defined.
Definition Fv {X Y : UU} (R : hrel X) (f : compfun R Y) (phi : F X Y R) : Y
:= phi f.
Definition qeq {X Y : UU} (R : hrel X)
:= total2 (λ phi : F X Y R,
∏ psi : F X Y R -> Y,
paths (psi phi)
(Fv R (compfuncomp
R (make_compfun R _ (iscompFi R)) psi)
phi)).
Lemma isinclpr1qeq {X : UU} (R : hrel X) (Y : hSet) :
isincl (@pr1 _ (λ phi : F X Y R,
∏ psi : F X Y R -> Y,
paths
(psi phi)
(Fv R (compfuncomp
R (make_compfun R _ (iscompFi R))
psi) phi))).
Proof.
intros. apply isinclpr1.
intro phi.
apply impred. intro psi.
apply (pr2 Y).
Defined.
Definition toqeq {X Y : UU} (R : hrel X) (x : X) : @qeq X Y R.
Proof.
intros. split with (Fi X Y R x). intro psi. apply idpath.
Defined.
Lemma iscomptoqeq {X : UU} (Y : hSet) (R : hrel X) :
iscomprelfun R (@toqeq X Y R).
Proof.
intros. intros x x' r. unfold toqeq.
apply (invmaponpathsincl _ (isinclpr1qeq R Y)).
apply (@iscompFi X Y R x x' r).
Defined.
Definition qequniv {X : UU} (Y : hSet) (R : hrel X) (f : compfun R Y)
(phi : @qeq X Y R) : Y.
Proof.
intros. apply (Fv R f (pr1 phi)).
Defined.
Lemma qequnivandpr {X : UU} (Y : hSet) (R : hrel X) (f : compfun R Y)
(x : X) : paths (qequniv Y R f (toqeq R x)) (f x).
Proof.
intros. apply idpath.
Defined.
Lemma etaqeq {X : UU} (Y : hSet) (R : hrel X) (psi : qeq R -> Y)
(phi : qeq R) : paths (psi phi)
(qequniv Y R (compfuncomp
R (make_compfun
R _ (iscomptoqeq Y R))
psi) phi).
Proof.
intros. apply (pr2 phi psi).
Definition Fd1 {X Y : UU} : F X Y R -> (F (F X Y) Y) := Fi (F X Y) Y.
Definition Fd2 {X Y : UU} (R : hrel X) (phi : F X Y R) (psi : F X Y R -> Y) :
Y := (Fv R (funcomp (Fi X Y R) psi) phi).
Definition Ffunct {X1 X2 : UU} (f : X1 -> X2) (Y : UU) : F X1 Y -> F X2 Y
:= λ phi, λ g, phi (funcomp f g).
Lemma testd1 {X Y : UU} (psi : F X Y -> Y) (phi : F X Y) :
paths (psi phi) (Fd1 phi psi).
Proof.
intros. apply idpath.
Defined.
Lemma testd2 {X Y : UU} (psi : F X Y -> Y) (phi : F X Y) :
paths (Fv (funcomp (Fi X Y) psi) phi) (Fd2 phi psi).
Proof.
intros. apply idpath.
Defined.
Definition F (X Y : UU) := (X -> Y) -> Y.
Definition Ffunct {X1 X2 : UU} (f : X1 -> X2) (Y : UU) : F X1 Y -> F X2 Y
:= λ phi, λ g, phi (funcomp f g).
Definition Fi (X Y : UU) : X -> F X Y := λ x, λ f, f x.
Definition Fd1 {X Y : UU} : F X Y -> (F (F X Y) Y) := Fi (F X Y) Y.
Definition Fd2 {X Y : UU} : F X Y -> (F (F X Y) Y) := Ffunct (Fi X Y) Y.
Definition Fv {X Y : UU} (f : X -> Y) (phi : F X Y) : Y := phi f.
Lemma testd1 {X Y : UU} (psi : F X Y -> Y) (phi : F X Y) :
paths (psi phi) (Fd1 phi psi).
Proof.
intros. apply idpath.
Defined.
Lemma testd2 {X Y : UU} (psi : F X Y -> Y) (phi : F X Y) :
paths (Fv (funcomp (Fi X Y) psi) phi) (Fd2 phi psi).
Proof.
intros. apply idpath.
Defined.
Lemma Xineq (X Y : UU) (x : X) : paths (Fd1 (Fi X Y x)) (Fd2 (Fi X Y x)).
Proof.
intros. apply idpath.
Defined.
Lemma test (X Y : UU) (phi : F X Y) (f : X -> Y) :
paths (Fd1 phi (Fi (X -> Y) Y f)) (Fd2 phi (Fi (X -> Y) Y f)).
Proof.
intros. unfold Fd1. unfold Fd2. unfold Fi. unfold Ffunct. unfold funcomp.
simpl. apply (maponpaths phi). apply etacorrection.
Defined.
Inductive try0 (T : UU) (t : T) :
∏ (t1 t2 : T) (e1 : t = t1) (e2 : t = t2), UU
:= idconstr : ∏ (t' : T) (e' : t = t'), try0 T t t' t' e' e'.
Definition try0map1 (T : UU) (t : T) (t1 t2 : T) (e1 : t = t1)
(e2 : t = t2) (X : try0 T t t1 t2 e1 e2) : t1 = t2.
Proof.
intros. induction X. apply idpath.
Defined.
Definition try0map2 (T : UU) (t : T) (t1 t2 : T) (e1 : t = t1)
(e2 : t = t2) : try0 T t t1 t2 e1 e2.
Proof.
Lemma test (X : UU) (t : X) :
paths (pr2 (iscontrcoconustot X t) (pr1 (iscontrcoconustot X t)))
(idpath _).
Proof.
intros. apply idpath.
Lemma test {X : UU} (is : iscontr X) :
paths (pr2 (iscontrcor is) (pr1 (iscontrcor is))) (idpath _).
Proof.
intros. apply idpath.
Lemma test {X : UU} (R : eqrel X) (Y : hSet) (f : setquot R -> Y) :
paths f (setquotuniv R Y (funcomp (setquotpr R) f)
(λ x x' : X, λ r : R x x',
maponpaths f (iscompsetquotpr R x x' r))).
Proof.
intros. apply funextfun. intro c. simpl. induction c as A iseq . simpl.
The quotient set of a type by a relation.
Definition setquot2 {X : UU} (R : hrel X) : UU := image (compevmapset R).
Theorem isasetsetquot2 {X : UU} (R : hrel X) : isaset (setquot2 R).
Proof.
intros.
assert (is1: isofhlevel 2 (∏ S : hSet, (compfun R S) → S)).
{
apply impred. intro.
apply impred. intro X0.
apply (pr2 t).
}
apply (isasetsubset _ is1 (isinclpr1image _)).
Defined.
Definition setquot2inset {X : UU} (R : hrel X) : hSet
:= make_hSet _ (isasetsetquot2 R).
We will be asuming below that setquot2 is in UU. In the future it should
be proved using issurjsetquot2pr below and a resizing axiom. The
appropriate resizing axiom for this should say that if X -> Y is a surjection,
Y is an hset and X : UU then Y : UU.
Definition setquot2pr {X : UU} (R : hrel X) : X → setquot2 R
:= λ x : X, make_image (compevmapset R) _
(hinhpr (make_hfiber (compevmapset R) x (idpath _))).
Lemma issurjsetquot2pr {X : UU} (R : hrel X) : issurjective (setquot2pr R).
Proof.
intros. apply issurjprtoimage.
Defined.
Lemma iscompsetquot2pr {X : UU} (R : hrel X) : iscomprelfun R (setquot2pr R).
Proof.
intros. intros x x' r.
assert (e1: paths (compevmapset R x) (compevmapset R x')).
{
apply funextsec. intro S.
apply funextsec. intro f.
unfold compfun in f.
apply (pr2 f x x' r).
}
apply (invmaponpathsincl _ (isinclpr1image (compevmapset R))
(setquot2pr R x) (setquot2pr R x') e1).
Defined.
Universal property of seqtquot2 R for functions to sets satisfying compatibility condition iscomprelfun
Definition setquot2univ {X : UU} (R : hrel X) (Y : hSet) (F : X → Y)
(is : iscomprelfun R F) (c : setquot2 R) : Y
:= pr1 c Y (make_compfun _ F is).
Theorem setquot2univcomm {X : UU} (R : hrel X) (Y : hSet) (F : X → Y)
(iscomp : iscomprelfun R F) (x : X) :
setquot2univ _ _ F iscomp (setquot2pr R x) = F x.
Proof.
intros. apply idpath.
Defined.
Weak equivalence from R x x' to paths (setquot2pr R x) (setquot2pr R x')
Lemma weqpathssetquot2l1 {X : UU} (R : eqrel X) (x : X) :
iscomprelfun R (λ x', R x x').
Proof.
intros. intros x' x''. intro r.
apply hPropUnivalence.
intro r'.
apply (eqreltrans R _ _ _ r' r).
intro r''.
apply (eqreltrans R _ _ _ r'' (eqrelsymm R _ _ r)).
Defined.
Theorem weqpathsinsetquot2 {X : UU} (R : eqrel X) (x x' : X) :
(R x x') ≃ (setquot2pr R x = setquot2pr R x').
Proof.
intros. apply weqimplimpl. apply iscompsetquot2pr.
set (int := setquot2univ R hPropset (λ x'', R x x'')
(weqpathssetquot2l1 R x)).
intro e. change (pr1 (int (setquot2pr R x'))). induction e.
change (R x x). apply (eqrelrefl R). apply (pr2 (R x x')).
apply (isasetsetquot2).
Defined.
Definition setquottosetquot2 (X : UU) (R : hrel X) (is : iseqrel R) :
setquot R → setquot2 R.
Proof.
intros X0.
apply (setquotuniv R (make_hSet _ (isasetsetquot2 R)) (setquot2pr R)
(iscompsetquot2pr R) X0).
Defined.
Require Import UniMath.Foundations.UnivalenceAxiom.
Definition hSet_univalence_map (X Y : hSet) : (X = Y) → (pr1 X ≃ pr1 Y).
Proof.
intros e. exact (eqweqmap (maponpaths pr1hSet e)).
Defined.
Theorem hSet_univalence (X Y : hSet) : (X = Y) ≃ (X ≃ Y).
Proof.
Set Printing Coercions.
intros.
set (f := hSet_univalence_map X Y).
∃ f.
set (g := @eqweqmap (pr1 X) (pr1 Y)).
set (h := λ e : X = Y, maponpaths pr1hSet e).
assert (comp : f = g ∘ h).
{
apply funextfun; intro e. induction e. apply idpath.
}
induction (!comp). apply twooutof3c.
- apply isweqonpathsincl. apply isinclpr1. exact isapropisaset.
- apply univalenceAxiom.
Unset Printing Coercions.
Defined.
Theorem hSet_rect (X Y : hSet) (P : X ≃ Y → UU) :
(∏ e : X=Y, P (hSet_univalence _ _ e)) → ∏ f, P f.
Proof.
intros ih ?.
Set Printing Coercions.
set (p := ih (invmap (hSet_univalence _ _) f)).
set (h := homotweqinvweq (hSet_univalence _ _) f).
exact (transportf P h p).
Unset Printing Coercions.
Defined.
Ltac hSet_induction f e := generalize f; apply UU_rect; intro e; clear f.