# Library UniMath.MoreFoundations.Propositions

Require Export UniMath.MoreFoundations.Notations.
Require Export UniMath.MoreFoundations.PartA.
Require Export UniMath.MoreFoundations.Tactics.
Require Export UniMath.MoreFoundations.DecidablePropositions.

Local Open Scope logic.

Lemma ishinh_irrel {X:UU} (x:X) (x':X) : hinhpr x = x'.
Proof.
apply proofirrelevance. apply propproperty.
Defined.

Corollary squash_to_hProp {X:UU} {Q:hProp} : X (X Q) Q.
Proof. intros h f. exact (hinhuniv f h). Defined.

Lemma hdisj_impl_1 {P Q : hProp} : PQ (QP) P.
Proof.
intros o f. apply (squash_to_hProp o).
intros [p|q].
- exact p.
- exact (f q).
Defined.

Lemma hdisj_impl_2 {P Q : hProp} : PQ (PQ) Q.
Proof.
intros o f. apply (squash_to_hProp o).
intros [p|q].
- exact (f p).
- exact q.
Defined.

Definition weqlogeq (P Q : hProp) : (P = Q) (P Q).
Proof.
intros.
apply weqimplimpl.
- intro e. induction e. apply isrefl_logeq.
- intro c. apply hPropUnivalence.
+ exact (pr1 c).
+ exact (pr2 c).
- apply isasethProp.
- apply propproperty.
Defined.

Lemma decidable_proof_by_contradiction {P:hProp} : decidable P ¬ ¬ P P.
Proof.
intros dec nnp. induction dec as [p|np].
- exact p.
- apply fromempty. exact (nnp np).
Defined.

Lemma proof_by_contradiction {P:hProp} : LEM ¬ ¬ P P.
Proof.
intro lem.
Defined.

Lemma dneg_elim_to_LEM : ( P:hProp, ¬ ¬ P P) LEM.
Proof.
intros dne. intros P. simple refine (dne (_,,_) _).
simpl. intros n.
assert (q : ¬ (P ¬ P)).
{ now apply weqnegtonegishinh. }
assert (r := fromnegcoprod_prop q).
exact (pr2 r (pr1 r)).
Defined.

Lemma negforall_to_existsneg {X:UU} (P:XhProp) : LEM (¬ x, P x) ( x, ¬ (P x)).
Proof.
intros lem nf. apply (proof_by_contradiction lem); intro c. use nf; clear nf. intro x.
assert (q := neghexisttoforallneg _ c x); clear c; simpl in q.
Defined.

Lemma negimpl_to_conj (P Q:hProp) : LEM ( ¬ (P Q) P ¬ Q ).
Proof.
intros lem ni. assert (r := negforall_to_existsneg _ lem ni); clear lem ni.
apply (squash_to_hProp r); clear r; intros [p nq]. exact (p,,nq).
Defined.

Definition hrel_set (X : hSet) : hSet := make_hSet (hrel X) (isaset_hrel X).

Lemma isaprop_assume_it_is {X : UU} : (X isaprop X) isaprop X.
Proof.
intros f. apply invproofirrelevance; intros x y.
apply proofirrelevance. now apply f.
Defined.

Lemma isaproptotal2 {X : UU} (P : X UU) :
isPredicate P
( x y : X, P x P y x = y)
isaprop ( x : X, P x).
Proof.
intros HP Heq.
apply invproofirrelevance.
intros [x p] [y q].
induction (Heq x y p q).
induction (iscontrpr1 (HP x p q)).
apply idpath.
Defined.

Lemma squash_rec {X : UU} (P : X hProp) : ( x, P (hinhpr x)) ( x, P x).
Proof.
intros xp x'. simple refine (hinhuniv _ x'). intro x.
assert (q : hinhpr x = x').
{ apply propproperty. }
induction q. exact (xp x).
Defined.

Lemma logeq_if_both_true (P Q : hProp) : P Q ( P Q ).
Proof.
intros p q.
split.
- intros _. exact q.
- intros _. exact p.
Defined.

Lemma logeq_if_both_false (P Q : hProp) : ¬P ¬Q ( P Q ).
Proof.
intros np nq.
split.
- intros p. apply fromempty. exact (np p).
- intros q. apply fromempty. exact (nq q).
Defined.

Definition proofirrelevance_hProp (X : hProp) : isProofIrrelevant X
:= proofirrelevance X (propproperty X).

Ltac induction_hProp x y := induction (proofirrelevance_hProp _ x y).

Definition iscontr_hProp (X:UU) : hProp := make_hProp (iscontr X) (isapropiscontr X).

Notation "'∃!' x .. y , P"
:= (iscontr_hProp ( x, .. ( y, P) ..))
(at level 200, x binder, y binder, right associativity) : logic.

Various algebraic properties of hProp
Section hProp_logic.

We first state the algebraic properties as bi-implications

Lemma islogeqassochconj {P Q R : hProp} : (P Q) R P (Q R).
Proof.
split.
- intros PQR.
exact (pr1 (pr1 PQR),,(pr2 (pr1 PQR),,pr2 PQR)).
- intros PQR.
exact ((pr1 PQR,,pr1 (pr2 PQR)),,pr2 (pr2 PQR)).
Defined.

Lemma islogeqcommhconj {P Q : hProp} : P Q Q P.
Proof.
split.
- intros PQ.
exact (pr2 PQ,,pr1 PQ).
- intros QP.
exact (pr2 QP,,pr1 QP).
Defined.

Lemma islogeqassochdisj {P Q R : hProp} : (P Q) R P (Q R).
Proof.
split.
- apply hinhuniv; intros hPQR.
induction hPQR as [hPQ|hR].
+ use (hinhuniv _ hPQ); clear hPQ; intros hPQ.
induction hPQ as [hP|hQ].
× exact (hinhpr (ii1 hP)).
× exact (hinhpr (ii2 (hinhpr (ii1 hQ)))).
+ exact (hinhpr (ii2 (hinhpr (ii2 hR)))).
- apply hinhuniv; intros hPQR.
induction hPQR as [hP|hQR].
+ exact (hinhpr (ii1 (hinhpr (ii1 hP)))).
+ use (hinhuniv _ hQR); clear hQR; intros hQR.
induction hQR as [hQ|hR].
× exact (hinhpr (ii1 (hinhpr (ii2 hQ)))).
× exact (hinhpr (ii2 hR)).
Defined.

Lemma islogeqhconj_absorb_hdisj {P Q : hProp} : P (P Q) P.
Proof.
split.
- intros hPPQ; apply (pr1 hPPQ).
- intros hP.
split; [ apply hP | apply (hinhpr (ii1 hP)) ].
Defined.

Lemma islogeqhdisj_absorb_hconj {P Q : hProp} : P (P Q) P.
Proof.
split.
- apply hinhuniv; intros hPPQ.
induction hPPQ as [hP|hPQ].
+ exact hP.
+ exact (pr1 hPQ).
- intros hP; apply (hinhpr (ii1 hP)).
Defined.

Lemma islogeqhfalse_hdisj {P : hProp} : P P.
Proof.
split.
- apply hinhuniv; intros hPPQ.
induction hPPQ as [hF|hP].
+ induction hF.
+ exact hP.
- intros hP; apply (hinhpr (ii2 hP)).
Defined.

Lemma islogeqhhtrue_hconj {P : hProp} : htrue P P.
Proof.
split.
- intros hP; apply (pr2 hP).
- intros hP.
split; [ apply tt | apply hP ].
Defined.

We now turn these into equalities using univalence for propositions
Lemma isassoc_hconj (P Q R : hProp) : ((P Q) R) = (P (Q R)).
Proof.
now apply hPropUnivalence; apply islogeqassochconj.
Qed.

Lemma iscomm_hconj (P Q : hProp) : (P Q) = (Q P).
Proof.
now apply hPropUnivalence; apply islogeqcommhconj.
Qed.

Lemma isassoc_hdisj (P Q R : hProp) : ((P Q) R) = (P (Q R)).
Proof.
now apply hPropUnivalence; apply islogeqassochdisj.
Qed.

Lemma iscomm_hdisj (P Q : hProp) : (P Q) = (Q P).
Proof.
now apply hPropUnivalence; apply islogeqcommhdisj.
Qed.

Lemma hconj_absorb_hdisj (P Q : hProp) : (P (P Q)) = P.
Proof.
now apply hPropUnivalence; apply islogeqhconj_absorb_hdisj.
Qed.

Lemma hdisj_absorb_hconj (P Q : hProp) : (P (P Q)) = P.
Proof.
now apply hPropUnivalence; apply islogeqhdisj_absorb_hconj.
Qed.

Lemma hfalse_hdisj (P : hProp) : ( P) = P.
Proof.
now apply hPropUnivalence; apply islogeqhfalse_hdisj.
Qed.

Lemma htrue_hconj (P : hProp) : (htrue P) = P.
Proof.
now apply hPropUnivalence; apply islogeqhhtrue_hconj.
Qed.

End hProp_logic.

## Factoring maps through squash

Lemma squash_uniqueness {X} (x:X) (h: X ) : squash_element x = h.
Proof. intros. apply propproperty. Qed.

Goal X Q (i:isaprop Q) (f:X Q) (x:X),
factor_through_squash i f (squash_element x) = f x.
Proof. reflexivity. Defined.

Lemma factor_dep_through_squash {X} {Q: X UU} :
( h, isaprop (Q h))
( x, Q(squash_element x))
( h, Q h).
Proof.
intros i f ?. apply (h (make_hProp (Q h) (i h))).
intro x. simpl. induction (squash_uniqueness x h). exact (f x).
Defined.

Lemma factor_through_squash_hProp {X} : hQ:hProp, (X hQ) X hQ.
Proof. intros [Q i] f h. refine (h _ _). assumption. Defined.

Lemma funspace_isaset {X Y} : isaset Y isaset (X Y).
Proof. intros is. apply (impredfun 2). assumption. Defined.

Lemma squash_map_uniqueness {X S} (ip : isaset S) (g g' : X S) :
g squash_element ¬ g' squash_element g ¬ g'.
Proof.
intros h.
set ( Q := λ y, g y = g' y ).
unfold homot.
apply (@factor_dep_through_squash X). intros y. apply ip.
intro x. apply h.
Qed.

Lemma squash_map_epi {X S} (ip : isaset S) (g g' : X S) :
g squash_element = g' squash_element g = g'.
Proof.
intros e.
apply funextsec.
apply squash_map_uniqueness. exact ip.
intro x. induction e. apply idpath.
Qed.

Definition squash_section_to_hProp {X:UU} {P:X hProp} :
( x, P (hinhpr x)) ( x', P x').
Proof.
intros f.
assert (i := isaprop_total2 _ P).
assert (h := λ x, tpair (λ x', P x') (hinhpr x) (f x)).
intros x'.
assert (q := squash_to_prop x' i h).
assert (e := iscontrpr1 (propproperty (X) (pr1 q) x')).
assert (w := make_hfiber pr1 q e).
assert (g := invweq (ezweqpr1 P x')).
exact (g w).
Defined.

Goal {X:UU} {P:X hProp} (h : x, P (hinhpr x)) (x:X),
squash_section_to_hProp h (hinhpr x) = h x.
Proof.
Fail reflexivity. Abort.

Goal (X:Type)
(P := λ x':X, x, x' = hinhpr x)
(h := λ x, (hinhpr (x,,idpath _) : P (hinhpr x)))
(x:X),
squash_section_to_hProp h (hinhpr x) = h x.
Proof.
Fail reflexivity. Abort.

Lemma uniqueExists {A : UU} {P : A UU} {a b : A}
(Hexists : ∃! a, P a) (Ha : P a) (Hb : P b) : a = b.
Proof.
assert (H : tpair _ _ Ha = tpair _ _ Hb).
{ now apply proofirrelevance, isapropifcontr. }
exact (base_paths _ _ H).
Defined.

## Connected types

Definition isConnected X : hProp := X (x y:X), x = y .

Lemma predicateOnConnectedType {X:Type} (i : isConnected X) {P:XhProp} (x0:X) (p:P x0) :
x, P x.
Proof.
intros x. apply (squash_to_hProp (pr2 i x x0)); intros e. now induction e.
Defined.

Definition isBaseConnected (X:PointedType) : hProp := (y:X), basepoint X = y .

Lemma isConnected_isBaseConnected (X:PointedType) : isConnected X isBaseConnected X.
Proof.
split.
- intros [_ ic] x. use ic.
- intros ibc. split.
+ exact (hinhpr (basepoint X)).
+ intros x y.
apply (squash_to_hProp (ibc x)); intros p; apply (squash_to_hProp (ibc y)); intros q.
exact (hinhpr (!p @ q)).
Defined.

Definition BasePointComponent (X:PointedType) : PointedType :=
pointedType ( (y:X), basepoint X = y ) (basepoint X,, hinhpr (idpath (basepoint X))).

Definition basePointComponent_inclusion {X:PointedType} (x : BasePointComponent X) : X
:= pr1 x.

Lemma BasePointComponent_isBaseConnected (X:PointedType) : isBaseConnected (BasePointComponent X).
Proof.
intros [x' c'].
change (basepoint (BasePointComponent X))
with (tpair (λ (y:X), basepoint X = y ) (basepoint X) (hinhpr (idpath (basepoint X)))).
use (hinhfun _ c'); intro q. induction q.
apply maponpaths. apply propproperty.
Defined.

Lemma BasePointComponent_isincl {X:PointedType} : isincl (@basePointComponent_inclusion X).
Proof.
use isinclpr1. intros x. apply propproperty.
Defined.

Lemma BasePointComponent_isweq {X:PointedType} (bc : isBaseConnected X) :
isweq (@basePointComponent_inclusion X).
Proof.
use isweqpr1.
intros x.
apply iscontraprop1.
- apply propproperty.
- exact (bc x).
Defined.

Definition BasePointComponent_weq {X:PointedType} (bc : isBaseConnected X) :
BasePointComponent X X
:= make_weq (@basePointComponent_inclusion X) (BasePointComponent_isweq bc).

Lemma baseConnectedness X : isBaseConnected X isConnected X.
Proof.
intros p. split.
- exact (hinhpr (basepoint X)).
- intros x y. assert (a := p x); assert (b := p y); clear p.
apply (squash_to_prop a); [apply propproperty|]; clear a; intros a.
apply (squash_to_prop b); [apply propproperty|]; clear b; intros b.
apply hinhpr. exact (!a@b).
Defined.

Lemma predicateOnBaseConnectedType (X:PointedType) (b:isBaseConnected X)
(P:XhProp) (p:P (basepoint X)) :
x, P x.
Proof.
intros x. apply (squash_to_hProp (b x)); intros e. now induction e.
Defined.

Goal (X:PointedType) (b:isBaseConnected X) (P:XhProp) (p:P (basepoint X)),
predicateOnBaseConnectedType X b P p (basepoint X) = p.
Proof.
Fail reflexivity.
intros.
Abort.

Lemma predicateOnBasePointComponent
(X:PointedType) (X' := BasePointComponent X) (pt' := basepoint X')
(P:X'hProp) (p:P pt') :
x, P x.
Proof.
intros x.
apply (squash_to_hProp (BasePointComponent_isBaseConnected _ x)); intros e.
now induction e.
Defined.

Goal (X:PointedType)
(X' := BasePointComponent X) (pt' := basepoint X')
(P:X'hProp) (p:P pt'),
predicateOnBasePointComponent X P p pt' = p.
Proof.
Fail reflexivity.
intros.
unfold pt', basepoint, X', BasePointComponent, pointedType, pr2; cbn beta.
Abort.