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Require Import Corelib.Setoids.Setoid.
Require Import Ltac2.Ltac2.
Axiom f: nat -> nat.
Definition g := f.
Axiom Foo1: nat -> Prop.
Axiom Foo2: nat -> Prop.
Axiom Impl: forall n: nat, Foo1 (f n) -> Foo2 (f n).
Create HintDb foo discriminated.
#[export] Hint Constants Opaque : foo.
#[export] Hint Resolve Impl : foo.
Goal forall x, Foo1 (f x) -> Foo2 (g x).
Proof.
auto with foo.
#[export] Hint Transparent g : foo.
auto with foo.
Qed.
Goal forall (x: nat), exists y, f x = g y.
Proof.
intros.
eexists.
unify f g.
lazy_match! goal with
| [ |- ?a ?b = ?rhs ] => unify ($a $b) $rhs
end.
Abort.
Goal forall (x: nat), exists y, f x = g y.
Proof.
intros.
eexists.
Unification.unify TransparentState.full 'f 'g.
lazy_match! goal with
| [ |- ?a ?b = ?rhs ] => Unification.unify_with_full_ts '($a $b) rhs
end.
Abort.
Goal True.
Proof.
Fail Unification.unify TransparentState.empty '(1 + 1) '2.
Unification.unify TransparentState.full '(1 + 1) '2.
Unification.unify (TransparentState.current ()) '(1 + 1) '2.
Opaque Nat.add.
Fail Unification.unify (TransparentState.current ()) '(1 + 1) '2.
Succeed Unification.unify TransparentState.full '(1 + 1) '2.
exact I.
Qed.
(* Test that by clause of assert doesn't eat all semicolons:
https://github.com/coq/coq/issues/17491 *)
Goal forall (a: nat), a = a.
Proof.
intros.
assert (a = a) by Std.reflexivity ();
assumption.
Qed.
(* Test that notations in by clause still work: *)
Goal forall (a: nat), a = a.
Proof.
intros.
assert (a = a) by exact eq_refl;
assumption.
Qed.
Goal forall x, (forall (y : unit), y = x) -> forall (x: unit), x = x.
Proof.
intros x H y.
rewrite -> H at 1 2.
reflexivity.
Qed.
Goal forall x, (forall (y : unit), x = y) -> forall (x: unit), x = x.
Proof.
intros x H y.
rewrite <- H at 1 2.
reflexivity.
Qed.
Goal forall x, (forall y : unit, y = x) -> forall y : unit, y = y.
Proof.
intros x H y.
setoid_rewrite H at 1 2.
setoid_rewrite <- (H y) at 1 2.
setoid_rewrite H.
reflexivity.
Qed.
Axiom x : unit.
Axiom H : forall y : unit, y = x.
Goal forall y : unit, y = y.
Proof.
assert (H2 : tt = x).
{ setoid_rewrite H with (y := tt).
reflexivity. }
setoid_rewrite H with (y := tt) at 1 in H2.
intros y.
setoid_rewrite H with (y := y) at 1.
setoid_rewrite H with (y := y) at 1.
reflexivity.
Qed.
(* view intro patterns *)
Axiom lem : forall x:nat, bool -> x = x.
Axiom lem' : 3 = 3 -> False.
Goal bool -> 2 = 2.
Proof.
Fail intros a%lem.
eintros a%lem.
revert a.
intros b%lem'.
destruct b.
Qed.
(* assert interns and typechecks the type as a type
(bug #15094) *)
Inductive my_type :=
| my_el : my_type.
Definition my_coercion (_ : my_type) := True.
Coercion my_coercion : my_type >-> Sortclass.
Goal False.
Proof.
(* expected type is sortclass -> coercion inserted *)
assert my_el by exact I.
(* interned as with type scope locally open *)
assert (H' : nat * (0 * 0 = 0)) by (split; first [exact 0 | reflexivity]).
Abort.
Goal nat.
Proof.
Std.apply true false [fun () => Control.plus (fun () => 'I) (fun _ => '0), Std.NoBindings] None.
Qed.
(* eassumption *)
Goal forall (x : nat) y z, x = y -> y = z -> x = z.
Proof.
intros **.
etransitivity; eassumption.
Qed.
(* cycle *)
Goal nat * bool.
Proof.
split.
all: cycle 1.
- exact true.
- exact 0.
Qed.
(* focus *)
Goal bool * nat * nat.
Proof.
repeat split.
all: Control.focus 2 3 (fun () => exact 0).
exact true.
Qed.
(* exfalso *)
Goal False -> nat * nat.
Proof.
intros oops.
split.
all: exfalso; assumption.
Qed.
|
