Require Import UniMath.Foundations.All.

Inductive empty : Type := .

Definition slide16_exercise (t : empty): bool :=
  match t with end.

Definition slide18_exercise (t : nat): bool :=
  match t with
  | O => true
  | S _ => false
  end.

(* Slide 22, Exercise 1*)

Definition fst {A B : Type} (p : A × B): A.
Proof.
  induction p.
  apply pr1.
Defined.
(* Show Proof. *)

Definition snd {A B : Type} (p : A × B): B.
Proof.
  induction p.
  apply pr2.
Defined.

Definition swap {A B : Type} (p : A × B): B × A :=
  match p with (a ,, b) => (b ,, a) end.
(* Definition swap {A B : Type} (p : A × B): B × A.
Proof.
  induction p.
  constructor.
  - apply pr2. 
  - apply pr1.
Defined. *)

(* Slide 38, Exercise: transport *)

Definition transport {A : Type} {B : A -> Type}
  {x y : A} (p : paths x y) (bx : B x): B y.
  induction p. 
  apply bx.
Defined.
(* Show Proof. *)

(* Slide 39, Exercise: swap involutive *)

Definition slide39_exercise {A B : Type} (t : A × B) : paths (swap (swap t)) t.
  reflexivity.
Show Proof.

(* Slide 44, Exercise: neg *)

(* Definition slide44_exercise {A : Type} : not  *)