(* QuickChick Prelude *)
Set Warnings "-extraction-opaque-accessed,-extraction".
Set Warnings "-notation-overridden,-parsing".
Set Bullet Behavior "Strict Subproofs".
(* End prelude *)
Require Export Tactics.
Example and_exercise :
forall n m : nat, n + m = 0 -> n = 0 /\ m = 0.
Admitted. (* OUT-OF-SCOPE *)
Instance testSuchThat_Conj {A B : Type} (P : A -> Prop) (Q : B -> Prop)
(prop : A -> B -> Prop)
`{Checkable (forall a, P a -> forall b, Q b -> prop a b)} :
Checkable (forall a b, P a /\ Q b -> prop a b) :=
{| checker f :=
checker (fun a P b Q => f a b _ ) |}.
Admitted. (* QuickChick and_example2''. *)
Admitted. (* Higher Order *)
Admitted. (* Higher Order *)
Admitted. (* Higher Order *)
Instance testSuchThat_Disj {A B : Type} (P : A -> Prop) (Q : B -> Prop)
(prop : A -> B -> Prop)
`{Checkable (forall a, P a -> forall b, prop a b)}
`{Checkable (forall b, Q b -> forall a, prop a b)} :
Checkable (forall a b, P a \/ Q b -> prop a b) :=
{| checker f :=
disjoin (Datatypes.cons (checker (fun a P b => f a b _ ))
(Datatypes.cons (checker (fun b Q a => f a b _))
Datatypes.nil)) |}.
Admitted. (* Higher Order *)
Admitted. (* QuickChick zero_or_succ. *)
Admitted. (* OUT-OF-SCOPE *)
Admitted. (* Higher Order *)
Fact not_implies_our_not : forall (P:Prop),
~ P -> (forall (Q:Prop), P -> Q).
Admitted. (* Higher Order *)
Admitted. (* Higher Order *)
Admitted. (* Higher Order *)
Admitted. (* Higher Order *)
Admitted. (* QuickChick not_true_is_false. *)
Admitted. (* QuickChick not_true_is_false'. *)
Admitted. (* Higher Order *)
Admitted. (* Leo: TODO *)
Admitted. (* Higher Order *)
Admitted. (* Higher Order *)

Theorem or_distributes_over_and : forall P Q R : Prop,
P \/ (Q /\ R) <-> (P \/ Q) /\ (P \/ R).