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Solve exercise 3.15 and 3.17 #1678

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Solve exercise 3.15 and 3.17 #1678

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da9b309
Solve exercise 3.15 and 3.17
HyunggyuJang Nov 4, 2022
0910287
Reflect reviews
HyunggyuJang Nov 30, 2022
fea4f10
Reflect review
HyunggyuJang Nov 30, 2022
2b657ab
Make propositional resizing assumption explicit
HyunggyuJang Dec 2, 2022
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41 changes: 40 additions & 1 deletion contrib/HoTTBookExercises.v
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Original file line number Diff line number Diff line change
Expand Up @@ -939,7 +939,36 @@ End Book_3_14.
(* ================================================== ex:impred-brck *)
(** Exercise 3.15 *)

Section Book_3_15.
Definition Book_3_15_trunc@{u +} `{PropResizing} `{Funext} (A : Type@{u})
: Type@{u}.
Proof.
exact (resize_hprop@{_ u} (forall P : HProp@{u}, (A -> P) -> P)).
Defined.

Definition Book_3_15_ishprop `{PropResizing} `{Funext} (A : Type)
: IsHProp (Book_3_15_trunc A)
:= _.

Definition Book_3_15_rec `{PropResizing} `{Funext} {A B} `{IsHProp B}
: (A -> B) -> (Book_3_15_trunc A) -> B.
Proof.
intros f trA.
set (B' := Build_HProp B).
apply equiv_resize_hprop in trA.
specialize (trA B').
apply trA.
assumption.
Defined.

Lemma Book_3_15_eq `{PropResizing} `{Funext} {A B} `{IsHProp B} (f : A -> B)
: forall a, f a = Book_3_15_rec f (equiv_resize_hprop _ (fun _ f' => f' a)).
Proof.
intro a.
(* Now judgemental computation rule does not hold in Coq, as in Coq, propositional resizing isn't definitional; judgemental equality doesn't hold. *)
apply path_ishprop.
Qed.
End Book_3_15.

(* ================================================== ex:lem-impl-dn-commutes *)
(** Exercise 3.16 *)
Expand All @@ -949,7 +978,17 @@ End Book_3_14.
(* ================================================== ex:prop-trunc-ind *)
(** Exercise 3.17 *)


Section Book_3_17.
Theorem prop_trunc_ind (A: Type) (B : merely A -> Type) `{forall x, IsHProp (B x)}
: (forall a, B (tr a)) -> forall x, B x.
Proof.
intros base x.
refine (Trunc_rec _ x).
intro a.
assert (H': tr a = x) by (apply path_ishprop).
destruct H'. exact (base a).
Defined.
End Book_3_17.

(* ================================================== ex:lem-ldn *)
(** Exercise 3.18 *)
Expand Down