Library HoTT.DProp


Decidable propositions


Require Import HoTT.Basics HoTT.Types.
Require Import TruncType HProp.
Require Import HoTT.Truncations.

Local Open Scope path_scope.

Definitions

A decidable proposition is, morally speaking, an hprop that is decidable. However, we only require that it be an hprop under the additional assumption of Funext; this enables decidable propositions to usually be used without Funext hypotheses.

Record DProp :=
  { dprop_type : Type ;
    ishprop_dprop : Funext IsHProp dprop_type ;
    dec_dprop : Decidable dprop_type
  }.

A fancier definition, which would have the property that negation is judgmentally involutive, would be
Record DProp :=
  { dprop_holds : Type ;
    ishprop_holds : Funext -> IsHProp dprop_holds ;
    dprop_denies : Type ;
    ishprop_denies : Funext -> IsHProp dprop_denies ;
    holds_or_denies : dprop_holds + dprop_denies ;
    denies_or_holds : dprop_denies + dprop_holds ;
    not_holds_and_denies : dprop_holds -> dprop_denies -> Empty
  }.
At some point we may want to go that route, but it would be more work. In particualar, Instances of Decidable wouldn't be automatically computed for us, and the characterization of the homotopy type of DProp itself would be a lot harder.

Coercion dprop_type : DProp >-> Sortclass.
Global Existing Instance ishprop_dprop.
Global Existing Instance dec_dprop.

Sometimes, however, we have decidable props that are hprops without funext, and we want to remember that.

Record DHProp :=
  { dhprop_hprop : hProp ;
    dec_dhprop : Decidable dhprop_hprop
  }.

Coercion dhprop_hprop : DHProp >-> hProp.
Global Existing Instance dec_dhprop.

Definition dhprop_to_dprop : DHProp DProp
  := fun PBuild_DProp P (fun __) _.

Coercion dhprop_to_dprop : DHProp >-> DProp.

In particular, True and False are always hprops.

Definition True : DHProp
  := Build_DHProp Unit_hp (inl tt).

Definition False : DHProp
  := Build_DHProp False_hp (inr idmap).

Decidable props can be coerced to Bool.

Definition dprop_to_bool (P : DProp) : Bool
  := if dec P then true else false.

Coercion dprop_to_bool : DProp >-> Bool.

And back again, but we don't declare that as a coercion.

Definition bool_to_dhprop (b : Bool) : DHProp
  := if b then True else False.

The type of decidable props

Definition issig_dprop
  : { X : Type & { _ : Funext IsHProp X & Decidable X } } <~> DProp.
Proof.
  issig.
Defined.

Definition equiv_path_dprop `{Funext} (P Q : DProp)
: (P = Q :> Type) <~> (P = Q :> DProp).
Proof.
  destruct P as [P hP dP]. destruct Q as [Q hQ dQ].
  refine (((equiv_ap' issig_dprop^-1 _ _)^-1)
            oE _); cbn.
  refine ((equiv_ap' (equiv_sigma_assoc' _ _)^-1
                     ((P;hP);dP) ((Q;hQ);dQ))
            oE _).
  refine (equiv_path_sigma_hprop _ _ oE _); cbn.
  { intros [X hX]; exact _. }
  refine (equiv_path_sigma_hprop (P;hP) (Q;hQ)).
Defined.

Definition path_dprop `{Funext} {P Q : DProp}
: (P = Q :> Type) (P = Q :> DProp)
  := equiv_path_dprop P Q.

Definition issig_dhprop
  : { X : hProp & Decidable X } <~> DHProp.
Proof.
  issig.
Defined.

Definition equiv_path_dhprop' `{Funext} (P Q : DHProp)
: (P = Q :> hProp) <~> (P = Q :> DHProp).
Proof.
  destruct P as [P dP]. destruct Q as [Q dQ].
  refine (((equiv_ap' issig_dhprop^-1 _ _)^-1)
            oE _); cbn.
  refine ((equiv_path_sigma_hprop (P; dP) (Q; dQ))).
Defined.

Definition equiv_path_dhprop `{Univalence} (P Q : DHProp)
: (P = Q :> Type) <~> (P = Q :> DHProp).
Proof.
  assert (eq_type_hprop : (P = Q :> Type) <~> (P = Q :> hProp)) by apply equiv_path_trunctype'.
  assert (eq_hprop_dhprop : (P = Q :> hProp) <~> (P = Q :> DHProp)) by apply equiv_path_dhprop'.
  refine (eq_hprop_dhprop oE eq_type_hprop).
Defined.

Definition path_dhprop `{Univalence} {P Q : DHProp}
: (P = Q :> Type) (P = Q :> DHProp)
  := equiv_path_dhprop P Q.

Global Instance ishset_dprop `{Univalence} : IsHSet DProp.
Proof.
  intros P Q.
  refine (trunc_equiv' _ (n := -1) (equiv_path_dprop P Q)).
Defined.

Global Instance isequiv_dprop_to_bool `{Univalence}
: IsEquiv dprop_to_bool.
Proof.
  refine (isequiv_adjointify dprop_to_bool bool_to_dhprop _ _).
  - intros []; reflexivity.
  - intros P; unfold dprop_to_bool.
    destruct (dec P); symmetry; apply path_dprop, path_universe_uncurried.
    + apply if_hprop_then_equiv_Unit; [ exact _ | assumption ].
    + apply if_not_hprop_then_equiv_Empty; [ exact _ | assumption ].
Defined.

Definition equiv_dprop_to_bool `{Univalence}
: DProp <~> Bool
  := Build_Equiv _ _ dprop_to_bool _.

Operations

We define the logical operations on decidable hprops to be the operations on ordinary hprops, with decidability carrying over. For the operations which preserve hprops without funext, we define separate versions that act on DHProp.

Definition dand (b1 b2 : DProp) : DProp
  := Build_DProp (b1 × b2) _ _.

Definition dhand (b1 b2 : DHProp) : DHProp
  := Build_DHProp (BuildhProp (b1 × b2)) _.

Definition dor (b1 b2 : DProp) : DProp
  := Build_DProp (hor b1 b2) _ _.

Definition dhor (b1 b2 : DHProp) : DHProp
  := Build_DHProp (BuildhProp (hor b1 b2)) _.

Definition dneg (b : DProp) : DProp
  := Build_DProp (¬b) _ _.

Definition dimpl (b1 b2 : DProp) : DProp
  := Build_DProp (b1 b2) _ _.

Declare Scope dprop_scope.
Delimit Scope dprop_scope with dprop.
Bind Scope dprop_scope with DProp.
Declare Scope dhprop_scope.
Delimit Scope dhprop_scope with dhprop.
Bind Scope dhprop_scope with DHProp.

Infix "&&" := dand : dprop_scope.
Infix "&&" := dhand : dhprop_scope.
Infix "||" := dor : dprop_scope.
Infix "||" := dhor : dhprop_scope.
Infix "→" := dimpl : dprop_scope.
Notation "!! P" := (dneg P) : dprop_scope.

Local Open Scope dprop_scope.

Computation

In order to be able to "compute" with DProps like booleans, we define a couple of typeclasses.

Class IsTrue (P : DProp) :=
  dprop_istrue : P.

Class IsFalse (P : DProp) :=
  dprop_isfalse : ¬ P.

Note that we are not using Typeclasses Strict Resolution for IsTrue and IsFalse; this enables us to write simply dprop_istrue as a proof of some true dprop, and Coq will infer from context what the dprop is that we're proving.

Global Instance true_istrue : IsTrue True := tt.

Global Instance false_isfalse : IsFalse False := idmap.

Global Instance dand_true_true {P Q} `{IsTrue P} `{IsTrue Q}
: IsTrue (P && Q).
Proof.
  exact (dprop_istrue, dprop_istrue).
Defined.

Global Instance dand_false_l {P Q} `{IsFalse P}
: IsFalse (P && Q).
Proof.
  intros [p q].
  exact (dprop_isfalse p).
Defined.

Global Instance dand_false_r {P Q} `{IsFalse Q}
: IsFalse (P && Q).
Proof.
  intros [p q].
  exact (dprop_isfalse q).
Defined.

Global Instance dhand_true_true {P Q : DHProp} `{IsTrue P} `{IsTrue Q}
: IsTrue (P && Q)%dhprop.
Proof.
We have to give P as an explicit argument here. This is apparently because with two IsTrue instances in the context, when we write simply dprop_istrue, Coq guesses one of them during typeclass resolution, and isn't willing to backtrack once it realizes that that choice fails to be what's needed to solve the goal. Coq currently seems to consistently guess Q rather than P, so that we don't have to give the argument Q to the second call to dprop_istrue; but rather than depend on such behavior, we give both arguments explicitly. (The problem doesn't arise with dand_true_true because in that case, unification, which fires before typeclass search, is able to guess that the argument must be P.)
  exact (@dprop_istrue P _, @dprop_istrue Q _).
Defined.

Global Instance dhand_false_l {P Q : DHProp} `{IsFalse P}
: IsFalse (P && Q)%dhprop.
Proof.
  intros [p q].
  exact (dprop_isfalse p).
Defined.

Global Instance dhand_false_r {P Q : DHProp} `{IsFalse Q}
: IsFalse (P && Q)%dhprop.
Proof.
  intros [p q].
  exact (dprop_isfalse q).
Defined.

Global Instance dor_true_l {P Q} `{IsTrue P}
: IsTrue (P || Q).
Proof.
  exact (tr (inl Q dprop_istrue)).
Defined.

Global Instance dor_true_r {P Q} `{IsTrue Q}
: IsTrue (P || Q).
Proof.
  exact (tr (inr P dprop_istrue)).
Defined.

Global Instance dor_false_false {P Q} `{IsFalse P} `{IsFalse Q}
: IsFalse (P || Q).
Proof.
  intros pq. strip_truncations. destruct pq as [p|q].
  - exact (dprop_isfalse p).
  - exact (dprop_isfalse q).
Defined.

Global Instance dhor_true_l {P Q : DHProp} `{IsTrue P}
: IsTrue (P || Q)%dhprop.
Proof.
  exact (tr (inl Q dprop_istrue)).
Defined.

Global Instance dhor_true_r {P Q : DHProp} `{IsTrue Q}
: IsTrue (P || Q)%dhprop.
Proof.
  exact (tr (inr P dprop_istrue)).
Defined.

Global Instance dhor_false_false {P Q : DHProp} `{IsFalse P} `{IsFalse Q}
: IsFalse (P || Q)%dhprop.
Proof.
  intros pq. strip_truncations. destruct pq as [p|q].
See comment in the proof of dhand_true_true.
  - exact (@dprop_isfalse P _ p).
  - exact (@dprop_isfalse Q _ q).
Defined.

Global Instance dneg_true {P} `{IsTrue P}
: IsFalse (!! P).
Proof.
  intros np; exact (np dprop_istrue).
Defined.

Global Instance dneg_false {P} `{IsFalse P}
: IsTrue (!! P).
Proof.
  exact dprop_isfalse.
Defined.

Global Instance dimpl_true_r {P Q} `{IsTrue Q}
: IsTrue (P Q).
Proof.
  intros p. exact dprop_istrue.
Defined.

Global Instance dimpl_false_l {P Q} `{IsFalse P}
: IsTrue (P Q).
Proof.
  intros p. elim (dprop_isfalse p).
Defined.

Global Instance dimpl_true_false {P Q} `{IsTrue P} `{IsFalse Q}
: IsFalse (P Q).
Proof.
  intros f. exact (dprop_isfalse (f dprop_istrue)).
Defined.

Lemma path_dec (A :Type) `{IsHProp A} `{Decidable A} `{Univalence} : A = is_inl (dec A).
Proof.
  refine (path_universe_uncurried _).
  apply equiv_iff_hprop_uncurried.
  split.
  - intros b. destruct (dec A); simpl; auto.
  - destruct (dec A); simpl; auto.
    intros [].
Defined.