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Require Import Program.Tactics.
Module Backtracking.
Class A := { foo : nat }.
Instance A_1 : A | 2 := { foo := 42 }.
Instance A_0 : A | 1 := { foo := 0 }.
Lemma aeq (a : A) : foo = foo.
reflexivity.
Qed.
Arguments foo A : clear implicits.
Example find42 : exists n, n = 42.
Proof.
eexists.
eapply eq_trans.
evar (a : A). subst a.
refine (@aeq ?a).
Unshelve. all:cycle 1.
typeclasses eauto.
Fail reflexivity.
Undo 2.
(* Without multiple successes it fails *)
Set Typeclasses Debug Verbosity 2.
Fail all:((once (typeclasses eauto with typeclass_instances))
+ apply eq_refl).
(* Does backtrack if other goals fail *)
all:[> typeclasses eauto + reflexivity .. ].
Undo 1.
all:(typeclasses eauto + reflexivity). (* Note "+" is a focussing combinator *)
Show Proof.
Qed.
Print find42.
Hint Extern 0 (_ = _) => reflexivity : equality.
Goal exists n, n = 42.
eexists.
eapply eq_trans.
evar (a : A). subst a.
refine (@aeq ?a).
Unshelve. all:cycle 1.
typeclasses eauto.
Fail reflexivity.
Undo 2.
(* Does backtrack between individual goals *)
Set Typeclasses Debug.
all:(typeclasses eauto with typeclass_instances equality).
Qed.
Unset Typeclasses Debug.
Module Leivant.
Axiom A : Type.
Existing Class A.
Axioms a b c d e: A.
Existing Instances a b c d e.
Ltac get_value H := eval cbv delta [H] in H.
Goal True.
Fail refine (let H := _ : A in _); let v := get_value H in idtac v; fail.
Admitted.
Goal exists x:A, x=a.
unshelve evar (t : A). all:cycle 1.
refine (@ex_intro _ _ t _).
all:cycle 1.
all:(typeclasses eauto + reflexivity).
Qed.
End Leivant.
End Backtracking.
Hint Resolve 100 eq_sym eq_trans : core.
Hint Cut [(_)* eq_sym eq_sym] : core.
Hint Cut [_* eq_trans eq_trans] : core.
Hint Cut [_* eq_trans eq_sym eq_trans] : core.
Goal forall x y z : nat, x = y -> z = y -> x = z.
Proof.
intros.
typeclasses eauto with core.
Qed.
Module Hierarchies.
Class A := mkA { data : nat }.
Class B := mkB { aofb :> A }.
Existing Instance mkB.
Definition makeB (a : A) : B := _.
Definition makeA (a : B) : A := _.
Fail Timeout 1 Definition makeA' : A := _.
Hint Cut [_* mkB aofb] : typeclass_instances.
Fail Definition makeA' : A := _.
Fail Definition makeB' : B := _.
End Hierarchies.
(** Hint modes *)
Class Equality (A : Type) := { eqp : A -> A -> Prop }.
Check (eqp 0%nat 0).
Instance nat_equality : Equality nat := { eqp := eq }.
Instance default_equality A : Equality A | 1000 :=
{ eqp := eq }.
Check (eqp 0%nat 0).
(* Defaulting *)
Check (fun x y => eqp x y).
(* No more defaulting, reduce "trigger-happiness" *)
Definition ambiguous x y := eqp x y.
Hint Mode Equality ! : typeclass_instances.
Fail Definition ambiguous' x y := eqp x y.
Definition nonambiguous (x y : nat) := eqp x y.
(** Typical looping instances with defaulting: *)
Definition flip {A B C} (f : A -> B -> C) := fun x y => f y x.
Class SomeProp {A : Type} (f : A -> A -> A) :=
{ prf : forall x y, f x y = f x y }.
Instance propflip (A : Type) (f : A -> A -> A) :
SomeProp f -> SomeProp (flip f).
Proof.
intros []. constructor. reflexivity.
Qed.
Fail Timeout 1 Check prf.
Hint Mode SomeProp + + : typeclass_instances.
Check prf.
Check (fun H : SomeProp plus => _ : SomeProp (flip plus)).
(** Iterative deepening / breadth-first search *)
Module IterativeDeepening.
Class A.
Class B.
Class C.
Instance: B -> A | 0.
Instance: C -> A | 0.
Instance: C -> B -> A | 0.
Instance: A -> A | 0.
Goal C -> A.
intros.
Fail Timeout 1 typeclasses eauto.
Set Typeclasses Iterative Deepening.
Fail typeclasses eauto 1.
typeclasses eauto 2.
Undo.
Unset Typeclasses Iterative Deepening.
Fail Timeout 1 typeclasses eauto.
Set Typeclasses Iterative Deepening.
Typeclasses eauto := debug 3.
typeclasses eauto.
Qed.
End IterativeDeepening.
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