path: root/test-suite/success/apply.v

(* Test apply in *)

Goal (forall x y, x = S y -> y=y) -> 2 = 4 -> 3=3.
intros H H0.
apply H in H0.
assumption.
Qed.

Require Import ZArith.
Goal (forall x y z, ~ z <= 0 -> x * z < y * z -> x <= y)%Z.
intros; apply Znot_le_gt, Zgt_lt in H.
apply Zmult_lt_reg_r, Zlt_le_weak in H0; auto.
Qed.

(* Test application under tuples *)

Goal (forall x, x=0 <-> 0=x) -> 1=0 -> 0=1.
intros H H'.
apply H in H'.
exact H'.
Qed.

(* Test as clause *)

Goal (forall x, x=0 <-> (0=x /\ True)) -> 1=0 -> True.
intros H H'.
apply H in H' as (_,H').
exact H'.
Qed.

(* Test application modulo conversion *)

Goal (forall x, id x = 0 -> 0 = x) -> 1 = id 0 -> 0 = 1.
intros H H'.
apply H in H'.
exact H'.
Qed.

(* Check apply/eapply distinction in presence of open terms *)

Parameter h : forall x y z : nat, x = z -> x = y.
Implicit Arguments h [[x] [y]].
Goal 1 = 0 -> True.
intro H.
apply h in H || exact I.
Qed.

Goal False -> 1 = 0.
intro H.
apply h || contradiction.
Qed.

(* Check if it unfolds when there are not enough premises *)

Goal forall n, n = S n -> False.
intros.
apply n_Sn in H.
assumption.
Qed.

(* Check naming in with bindings; printing used to be inconsistent before *)
(* revision 9450 *)

Notation S':=S (only parsing).
Goal (forall S, S = S' S) -> (forall S, S = S' S).
intros.
apply H with (S0 := S).
Qed.

(* Check inference of implicit arguments in bindings *)

Goal exists y : nat -> Type, y 0 = y 0.
exists (fun x => True).
trivial.
Qed.

(* Check universe handling in typed unificationn *)

Definition E := Type.
Goal exists y : E, y = y.
exists Prop.
trivial.
Qed.

Variable Eq : Prop = (Prop -> Prop) :> E.
Goal Prop.
rewrite Eq.
Abort.

(* Check insertion of coercions in bindings *)

Coercion eq_true : bool >-> Sortclass.
Goal exists A:Prop, A = A.
exists true.
trivial.
Qed.

(* Check use of unification of bindings types in specialize *)

Variable P : nat -> Prop.
Variable L : forall (l : nat), P l -> P l.
Goal P 0 -> True.
intros.
specialize L with (1:=H).
Abort.
Reset P.

(* Two examples that show that hnf_constr is used when unifying types
   of bindings (a simplification of a script from Field_Theory) *)

Require Import List.
Open Scope list_scope.
Fixpoint P (l : list nat) : Prop :=
  match l with
  | nil => True
  | e1 :: nil => e1 = e1
  | e1 :: l1 => e1 = e1 /\ P l1
  end.
Variable L : forall n l, P (n::l) -> P l.

Goal forall (x:nat) l, P (x::l) -> P l.
intros.
apply L with (1:=H).
Qed.

Goal forall (x:nat) l, match l with nil => x=x | _::_ => x=x /\ P l end -> P l.
intros.
apply L with (1:=H).
Qed.

(* The following call to auto fails if the type of the clause
   associated to the H is not beta-reduced [but apply H works]
   (a simplification of a script from FSetAVL) *)

Definition apply (f:nat->Prop) := forall x, f x.
Goal apply (fun n => n=0) -> 1=0.
intro H.
auto.
Qed.

(* The following fails if the coercion Zpos is not introduced around p
   before trying a subterm that matches the left-hand-side of the equality
   (a simplication of an example taken from Nijmegen/QArith) *)

Require Import ZArith.
Coercion Zpos : positive >-> Z.
Variable f : Z -> Z -> Z.
Variable g : forall q1 q2 p : Z, f (f q1 p) (f q2 p) = Z0.
Goal forall p q1 q2, f (f q1 (Zpos p)) (f q2 (Zpos p)) = Z0.
intros; rewrite g with (p:=p).
reflexivity.
Qed.

(* A funny example where the behavior differs depending on which of a
   multiple solution to a unification problem is chosen (an instance
   of this case can be found in the proof of Buchberger.BuchRed.nf_divp) *)

Definition succ x := S x.
Goal forall (I : nat -> Set) (P : nat -> Prop) (Q : forall n:nat, I n -> Prop),
  (forall x y, P x -> Q x y) ->
  (forall x, P (S x)) -> forall y: I (S 0), Q (succ 0) y.
intros.
apply H with (y:=y).
(* [x] had two possible instances: [S 0], coming from unifying the
   type of [y] with [I ?n] and [succ 0] coming from the unification with
   the goal; only the first one allows to make the next apply (which
   does not work modulo delta) working *)
apply H0.
Qed.

(* A similar example with a arbitrary long conversion between the two
   possible instances *)

Fixpoint compute_succ x :=
  match x with O => S 0 | S n => S (compute_succ n) end.

Goal forall (I : nat -> Set) (P : nat -> Prop) (Q : forall n:nat, I n -> Prop),
  (forall x y, P x -> Q x y) ->
  (forall x, P (S x)) -> forall y: I (S 100), Q (compute_succ 100) y.
intros.
apply H with (y:=y).
apply H0.
Qed.

(* Another example with multiple convertible solutions to the same
   metavariable (extracted from Algebra.Hom_module.Hom_module, 10th
   subgoal which precisely fails) *)

Definition ID (A:Type) := A.
Goal forall f:Type -> Type,
  forall (P : forall A:Type, A -> Prop),
  (forall (B:Type) x, P (f B) x -> P (f B) x) ->
  (forall (A:Type) x, P (f (f A)) x) ->
  forall (A:Type) (x:f (f A)), P (f (ID (f A))) x.
intros.
apply H.
(* The parameter [B] had two possible instances: [ID (f A)] by direct
   unification and [f A] by unification of the type of [x]; only the
   first choice makes the next command fail, as it was
   (unfortunately?) in Hom_module *)
try apply H.
unfold ID; apply H0.
Qed.

(* Test coercion below product and on non meta-free terms in with bindings *)
(* Cf wishes #1408 from E. Makarov *)

Parameter bool_Prop :> bool -> Prop.
Parameter r : bool -> bool -> bool.
Axiom ax : forall (A : Set) (R : A -> A -> Prop) (x y : A), R x y.

Theorem t : r true false.
apply ax with (R := r).
Qed.

(* Check verification of type at unification (submitted by Stéphane Lengrand):
   without verification, the first "apply" works what leads to the incorrect
   instantiation of x by Prop *)

Theorem u : ~(forall x:Prop, ~x).
unfold not.
intro.
eapply H.
apply (forall B:Prop,B->B) || (instantiate (1:=True); exact I).
Defined.

(* Fine-tuning coercion insertion in presence of unfolding (bug #1883) *)

Parameter name : Set.
Definition atom := name.

Inductive exp : Set :=
  | var : atom -> exp.

Coercion var : atom >-> exp.

Axiom silly_axiom : forall v : exp, v = v -> False.

Lemma silly_lemma : forall x : atom, False.
intros x.
apply silly_axiom with (v := x).  (* fails *)
reflexivity.
Qed.

(* Check that unification does not commit too early to a representative
   of an eta-equivalence class that would be incompatible with other
   unification constraints *)

Lemma eta : forall f : (forall P, P 1),
  (forall P, f P = f P) ->
   forall Q, f (fun x => Q x) = f (fun x => Q x).
intros.
apply H.
Qed.

(* Test propagation of evars from subgoal to brother subgoals *)

  (* This works because unfold calls clos_norm_flags which calls nf_evar *)

Lemma eapply_evar_unfold : let x:=O in O=x -> 0=O.
intros x H; eapply trans_equal;
[apply H | unfold x;match goal with |- ?x = ?x => reflexivity end].
Qed.

(* Test non-regression of (temporary) bug 1981 *)

Goal exists n : nat, True.
eapply ex_intro.
exact O.
trivial.
Qed.

(* Test non-regression of (temporary) bug 1980 *)

Goal True.
try eapply ex_intro.
trivial.
Qed.

(* Check pattern-unification on evars in apply unification *)

Lemma evar : exists f : nat -> nat, forall x, f x = 0 -> x = 0.
Proof.
eexists; intros x H.
apply H.
Qed.

(* Check that "as" clause applies to main premise only and leave the
   side conditions away *)

Lemma side_condition : 
  forall (A:Type) (B:Prop) x, (True -> B -> x=0) -> B -> x=x.
Proof.
intros.
apply H in H0 as ->.
reflexivity.
exact I.
Qed.

(* Check chaining of "apply in" on the last subgoal (assuming that
   side conditions come first) *)

Lemma chaining :
  forall B C D : Prop, (True -> B -> C) -> (C -> D) -> B -> D.
Proof.
intros.
apply H, H0 in H1; auto.
Qed.