Library UniMath.Foundations.Preamble

Introduction. Vladimir Voevodsky . Feb. 2010 - Sep. 2011

This is the first in the group of files which contain the (current state of) the mathematical library for the proof assistant Coq based on the Univalent Foundations. It contains some new notations for constructions defined in Coq.Init library as well as the definition of dependent sum.

Initial setup unrelated to Univalent Foundations

Require Export UniMath.Foundations.Init.

Universe structure

Definition UU := Type.

Identity Coercion fromUUtoType : UU >-> Sortclass.

The empty type

Inductive empty : UU := .

Notation "∅" := empty.

The one-element type

Inductive unit : UU :=
tt : unit.

The two-element type

Inductive bool : UU :=
| true : bool
| false : bool.

Definition negb (b:bool) := if b then false else true.

The coproduct of two types

Inductive coprod (A B:UU) : UU :=
| ii1 : A → coprod A B
| ii2 : B → coprod A B.

Arguments coprod_rect {_ _} _ _ _ _.
Arguments ii1 {_ _} _.
Arguments ii2 {_ _} _.

Notation inl := ii1. Notation inr := ii2.
Notation "X ⨿ Y" := (coprod X Y).

The natural numbers

Inductive nat : UU :=
  | O : nat
  | S : nat → nat.

Definition succ := S.

Declare Scope nat_scope.
Delimit Scope nat_scope with nat.
Bind Scope nat_scope with nat.
Arguments S _%nat.
Open Scope nat_scope.

Fixpoint add n m :=
  match n with
  | O ⇒ m
  | S p ⇒ S (p + m)
  end
where "n + m" := (add n m) : nat_scope.

Fixpoint sub n m :=
  match n, m with
  | S k, S l ⇒ k - l
  | _, _ ⇒ n
  end
where "n - m" := (sub n m) : nat_scope.

Definition mul : nat → nat → nat.
Proof.
  intros n m.
  induction n as [|p pm].
  - exact O.
  - exact (pm + m).
Defined.

Notation "n * m" := (mul n m) : nat_scope.

Fixpoint max n m :=
  match n, m with
    | O, _ ⇒ m
    | S n', O ⇒ n
    | S n', S m' ⇒ S (max n' m')
  end.

Fixpoint min n m :=
  match n, m with
    | O, _ ⇒ O
    | S n', O ⇒ O
    | S n', S m' ⇒ S (min n' m')
  end.

Notation "0" := (O) : nat_scope.
Notation "1" := (S 0) : nat_scope.
Notation "2" := (S 1) : nat_scope.
Notation "3" := (S 2) : nat_scope.
Notation "4" := (S 3) : nat_scope.
Notation "5" := (S 4) : nat_scope.
Notation "6" := (S 5) : nat_scope.
Notation "7" := (S 6) : nat_scope.
Notation "8" := (S 7) : nat_scope.
Notation "9" := (S 8) : nat_scope.
Notation "10" := (S 9) : nat_scope.
Notation "11" := (S 10) : nat_scope.
Notation "12" := (S 11) : nat_scope.
Notation "13" := (S 12) : nat_scope.
Notation "14" := (S 13) : nat_scope.
Notation "15" := (S 14) : nat_scope.
Notation "16" := (S 15) : nat_scope.
Notation "17" := (S 16) : nat_scope.
Notation "18" := (S 17) : nat_scope.
Notation "19" := (S 18) : nat_scope.
Notation "20" := (S 19) : nat_scope.
Notation "21" := (S 20) : nat_scope.
Notation "22" := (S 21) : nat_scope.
Notation "23" := (S 22) : nat_scope.
Notation "24" := (S 23) : nat_scope.
Notation "100" := (10 × 10) : nat_scope.
Notation "1000" := (10 × 100) : nat_scope.

Identity Types

Inductive paths {A:UU} (a:A) : A → UU := paths_refl : paths a a.
Hint Resolve paths_refl : core .
Notation "a = b" := (paths a b) : type_scope.
Notation idpath := paths_refl .

Dependent sums.

One can not use a new record each time one needs it because the general theorems about this construction would not apply to new instances of "Record" due to the "generativity" of inductive definitions in Coq.

We use "Record", which is equivalent to "Structure", instead of "Inductive" here, so we can take advantage of the "primitive projections" feature of Coq, which introduces η-reduction for pairs, by adding the option "Set Primitive Projections". It also speeds up compilation by 56 percent.

The terms produced by the "induction" tactic, when we define "total2" as a record, contain the "match" construction instead appealing to the eliminator. However, assuming the eliminator will be justified mathematically, the way to justify the the "match" construction is to point out that it can be regarded as an abbreviation for the eliminator that omits explicit mention of the first two parameters (X:Type) and (Y:X->Type).

I.e., whenever you see

match w as t0 return TYPE with | tpair _ _ x y ⇒ BODY end

in a proof term, just mentally replace it by

@total2_rect _ _ (λ t0, TYPE) (λ x y, BODY) w

Set Primitive Projections.
Set Nonrecursive Elimination Schemes.

Record total2 { T: UU } ( P: T → UU ) := tpair { pr1 : T; pr2 : P pr1 }.

Arguments tpair {_} _ _ _.
Arguments pr1 {_ _} _.
Arguments pr2 {_ _} _.

Notation "'∑' x .. y , P" := (total2 (λ x, .. (total2 (λ y, P)) ..))
(at level 200, x binder, y binder, right associativity) : type_scope.

Notation "x ,, y" := (tpair _ x y).