unimath2024/exercises_tactics.v
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(** Exercise sheet for lecture 4: Tactics in Coq.
prepared using material by Ralph Matthes
*)
(** * Exercise 1
Formalize the following types in UniMath and construct elements for them interactively -
if they exist. Give a counter-example otherwise, i.e., give specific parameters and a
proof of the negation of the statement.
[[
1. A × (B + C) → A × B + A × C, given types A, B, C
2. (A → A)(A → A), given type A (for extra credit, write down five elements of this type)
3. Id_nat (0, succ 0)
4. ∑ (A : Universe) (A → empty) → empty
5. ∏ (n : nat), ∑ (m : nat), Id_nat n (2 * m) + Id_nat n (2 * m + 1),
assuming you have got arithmetic
6. ((x : A) B × P x) → B ×(x : A) P x, given types A, B, and P : A → Universe
7. B → (B → A) → A, given types A and B
8. B → ∏ (A : Universe) (B → A) → A, given type B
9. ((A : Universe) (B → A) → A) → B, given type B
10. x = y → y = x, for elements x and y of some type A
11. x = y → y = z → x = z, for elements x, y, and z of some type A
]]
More precise instructions and hints:
1. Use [⨿] in place of the + and pay attention to operator precedence.
2. Write a function that provides different elements for any natural number argument,
not just five elements; for extra credits: state correctly that they are different -
for a good choice of [A]; for more extra credits: prove that they are different.
3. An auxiliary function may be helpful (a well-known trick).
4. The symbol for Sigma-types is [∑], not [Σ].
5. Same as 1; and there is need for module [UniMath.Foundations.NaturalNumbers], e.g.,
for Lemma [natpluscomm].
6.-9. no further particulars
*)
Require Import UniMath.Foundations.All.
Require Import UniMath.MoreFoundations.All.
Require Import UniMath.Foundations.Preamble.
Require Import UniMath.Foundations.PartA.
Require Import UniMath.Foundations.NaturalNumbers.
(* 1. A × (B + C) → A × B + A × C, given types A, B, C *)
Goal (A B C : UU), A × (B ⨿ C) -> (A × B) ⨿ (A × C).
Proof.
intros.
induction X.
induction pr2.
- apply ii1. apply tpair. apply pr1. apply a.
- apply ii2. apply tpair. apply pr1. apply b.
Defined.
(* 2. (A → A)(A → A), given type A (for extra credit, write down five elements of this type) *)
Goal (A : UU), (A -> A) -> (A -> A).
Proof.
intros.
apply X0.
Defined.
(* 3. Id_nat (0, succ 0) *)
Goal (O = S O) -> empty.
Proof.
intros.
exact (transportf (fun c => match c with O => unit | S _ => empty end) X tt).
Defined.
Locate "".
Print total2.
(* 4. ∑ (A : Universe) (A → empty) → empty *)
Goal (A : UU), (A -> empty) -> empty.
Proof.
use tpair.
- exact unit.
- simpl. intros. apply X. exact tt.
Defined.
(* 5. ∏ (n : nat), ∑ (m : nat), Id_nat n (2 * m) + Id_nat n (2 * m + 1), *)
(* assuming you have got arithmetic *)
Goal (n : nat), (m : nat), (paths n (2 * m)) ⨿ (paths n (1 + (2 * m))).
Proof.
intros.
induction n.
- use tpair.
+ exact 0.
+ simpl. apply inl. reflexivity.
- induction IHn. induction pr2.
* use tpair. exact pr1. simpl. apply inr. rewrite a. reflexivity.
* use tpair. exact (S pr1). simpl. apply inl. rewrite b. simpl.
assert (S (pr1 + pr1) = S pr1 + pr1).
+ reflexivity.
+ rewrite X. rewrite natpluscomm. reflexivity.
Defined.
(* 6. ((x : A) B × P x) → B ×(x : A) P x, given types A, B, and P : A → Universe *)
Goal (A B : UU) (P : A -> UU), ( (x : A), B × P x) -> B × (x : A), P x.
Proof.
intros.
induction X. induction pr2.
use tpair.
* exact pr0.
* simpl. use tpair.
+ exact pr1.
+ simpl. exact pr2.
Defined.
(* 7. B → (B → A) → A, given types A and B *)
Goal (A B : UU), B -> (B -> A) -> A.
Proof.
intros.
apply X0. exact X.
Defined.
(* 8. B → ∏ (A : Universe) (B → A) → A, given type B *)
Goal (B : UU), B -> (A : UU), (B -> A) -> A.
Proof.
intros.
apply X0. exact X.
Defined.
(* 9. ((A : Universe) (B → A) → A) → B, given type B *)
Goal (B : UU), ( (A : UU), (B -> A) -> A) -> B.
Proof.
intros.
exact (X B (fun x => x)).
Defined.
(* 10. x = y → y = x, for elements x and y of some type A *)
Goal (A : UU) (x y : A), x = y -> y = x.
Proof.
intros.
induction X.
apply idpath.
Defined.
(* 11. x = y → y = z → x = z, for elements x, y, and z of some type A *)
Goal (A : UU) (x y z : A), x = y -> y = z -> x = z.
Proof.
intros.
induction X. induction X0.
apply idpath.
Defined.
(** * Exercise 2
Define two computable strict comparison operators for natural numbers based on the fact
that [m < n] iff [n - m <> 0] iff [(m+1) - n = 0]. Prove that the two operators are
equal (using function extensionality, i.e., [funextfunStatement] in the UniMath library).
It may be helpful to use the definitions of the exercises for lecture 2.
The following lemmas on substraction [sub] in the natural numbers may be
useful:
a) [sub n (S m) = pred (sub n m)]
b) [sub 0 n = 0]
c) [pred (sub 1 n) = 0]
d) [sub (S n) (S m) = sub n m]
*)
(** from exercises to Lecture 2: *)
Definition ifbool (A : UU) (x y : A) : bool -> A :=
bool_rect (λ _ : bool, A) x y.
Definition negbool : bool -> bool := ifbool bool false true.
Definition nat_rec (A : UU) (a : A) (f : nat -> A -> A) : nat -> A :=
nat_rect (λ _ : nat, A) a f.
Definition pred : nat -> nat := nat_rec nat 0 (fun x _ => x).
Definition is_zero : nat -> bool := nat_rec bool true (λ _ _, false).
Definition iter (A : UU) (a : A) (f : A A) : nat A :=
nat_rec A a (λ _ y, f y).
Notation "f ̂ n" := (λ x, iter _ x f n) (at level 10).
Definition sub (m n : nat) : nat := pred ̂ n m.
Definition lt1 (m n : nat) : bool := negb (is_zero (sub m n)).
Definition lt2 (m n : nat) : bool := is_zero (sub (S m) n).
Definition lt1_eq_lt2 : lt1 = lt2.
Proof.
rewrite (funextfun lt1 lt2).
- apply idpath.
- intro x. rewrite (funextfun (lt1 x) (lt2 x)).
+ apply idpath.
+ intro y. unfold lt1. unfold lt2.
induction x.
induction y.
{ auto. }
{
assert ( (y : nat), sub 0 y = 0) as Sub0.
{ intros. induction y0. { auto. } { simpl. unfold pred. rewrite IHy0. auto. } }
rewrite Sub0. cbn.
}