Library HoTT.Tests

Require Import HoTT.

Fail Check Type0 : Type0.
Check Susp nat : Type0.
Fail Check Susp Type0 : Type0.

Fail Check (fun (P : interval Type) (a : P (b : P
                (p p' : seg # a = b)
            ⇒ idpath : interval_ind P a b p = interval_rect P a b p').

Local Open Scope nat_scope.
Fail Check Lift nat : Type0.
Check 1 : Lift nat.

Check lift'@{i j}.
Check lower'@{i j}.
Check isequiv_lift'@{i j}.

Check that ispointed_susp doesn't require just a Set
Check (fun A : Type_ : IsPointed (Susp A)).
Check (@ispointed_susp Type).
Check (@ispointed_susp Set).

Check that nested sigma-type notation didn't get clobbered by surreal cuts
Check ({ l : Unit & { n : Unit & Unit }}).

Test 1 from issue 754
Module Issue754_1.
  Inductive nat : Type1 :=
  | O : nat
  | S : nat nat.
  Fixpoint code_nat (m n : nat) {struct m} : DProp.DHProp :=
    match m with
      | Omatch n with
               | ODProp.True
               | S _DProp.False
      | S m'match n with
                  | ODProp.False
                  | S n'code_nat m' n'

  Local Set Warnings Append "-notation-overridden".
  Notation "x =n y" := (code_nat x y) : nat_scope.
  Local Set Warnings Append "notation-overridden".
  Bind Scope nat_scope with nat.
  Axiom equiv_path_nat :
     n m : nat,
      Trunc.trunctype_type (DProp.dhprop_hprop (n =n m)) <~> n = m.

  Definition nat_discr `{Funext} {n: nat}: O S n.
    intro H'.
    equiv_induction (@equiv_path_nat O (S n)).
End Issue754_1.

Module Issue_1358.

  Axiom A@{i} : Type@{i}.

  Axiom foo@{i} : A@{i} <~> A@{i}.

  Definition bar@{i} : A@{i} <~> A@{i}.

  Definition bar'@{i} : A@{i} <~> A@{i}.
    exact equiv_idmap.

End Issue_1358.

Module Issue_973.

  Inductive vec (A : Type) : nat Type :=
  | nil : vec A 0
  | cons : n : nat, A vec A n vec A (S n).

  Definition hd (A : Type) (n : nat) (v : vec A (S n)) : A :=
  match v in (vec _ (S n)) return A with
  | cons _ h _h

End Issue_973.

Module PR_1382.

  Goal O = S O Empty.
   discriminate 1.

  Goal H : O = S O, H = H.
   discriminate H.

  Goal O = S O Unit.
  intros H. discriminate H. Qed.
  Goal O = S O Unit.
  intros H. Ltac g x := discriminate x. g H. Qed.

  Goal ( x y : nat, x = y x = S y) Unit.
  try discriminate (H O) || exact tt.

  Goal ( x y : nat, x = y x = S y) Unit.
  intros H. ediscriminate (H O). instantiate (1:=O). Abort.

  Inductive A := B (T := Unit) (x y : Bool) (z := x).
  Goal x y, B x true = B y false Empty.

End PR_1382.