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Require Import Setoid.
Parameter A : Set.
Axiom eq_dec : forall a b : A, {a = b} + {a <> b}.
Inductive set : Set :=
| Empty : set
| Add : A -> set -> set.
Fixpoint In (a : A) (s : set) {struct s} : Prop :=
match s with
| Empty => False
| Add b s' => a = b \/ In a s'
end.
Definition same (s t : set) : Prop := forall a : A, In a s <-> In a t.
Lemma setoid_set : Setoid_Theory set same.
unfold same in |- *; split ; red.
red in |- *; auto.
red in |- *.
intros.
elim (H a); auto.
intros.
elim (H a); elim (H0 a).
split; auto.
Qed.
Add Setoid set same setoid_set as setsetoid.
Add Morphism In : In_ext.
unfold same in |- *; intros a s t H; elim (H a); auto.
Qed.
Lemma add_aux :
forall s t : set,
same s t -> forall a b : A, In a (Add b s) -> In a (Add b t).
unfold same in |- *; simple induction 2; intros.
rewrite H1.
simpl in |- *; left; reflexivity.
elim (H a).
intros.
simpl in |- *; right.
apply (H2 H1).
Qed.
Add Morphism Add : Add_ext.
split; apply add_aux.
assumption.
rewrite H.
reflexivity.
Qed.
Fixpoint remove (a : A) (s : set) {struct s} : set :=
match s with
| Empty => Empty
| Add b t =>
match eq_dec a b with
| left _ => remove a t
| right _ => Add b (remove a t)
end
end.
Lemma in_rem_not : forall (a : A) (s : set), ~ In a (remove a (Add a Empty)).
intros.
setoid_replace (remove a (Add a Empty)) with Empty.
auto.
unfold same in |- *.
split.
simpl in |- *.
case (eq_dec a a).
intros e ff; elim ff.
intros; absurd (a = a); trivial.
simpl in |- *.
intro H; elim H.
Qed.
Parameter P : set -> Prop.
Parameter P_ext : forall s t : set, same s t -> P s -> P t.
Add Morphism P : P_extt.
intros; split; apply P_ext; (assumption || apply (Seq_sym _ _ setoid_set); assumption).
Qed.
Lemma test_rewrite :
forall (a : A) (s t : set), same s t -> P (Add a s) -> P (Add a t).
intros.
rewrite <- H.
rewrite H.
setoid_rewrite <- H.
setoid_rewrite H.
setoid_rewrite <- H.
trivial.
Qed.
(* Unifying the domain up to delta-conversion (example from emakarov) *)
Definition id: Set -> Set := fun A => A.
Definition rel : forall A : Set, relation (id A) := @eq.
Definition f: forall A : Set, A -> A := fun A x => x.
Add Relation (id A) (rel A) as eq_rel.
Add Morphism (@f A) : f_morph.
Proof.
unfold rel, f. trivial.
Qed.
(* Submitted by Nicolas Tabareau *)
(* Needs unification.ml to support environments with de Bruijn *)
Goal forall
(f : Prop -> Prop)
(Q : (nat -> Prop) -> Prop)
(H : forall (h : nat -> Prop), Q (fun x : nat => f (h x)) <-> True)
(h:nat -> Prop),
Q (fun x : nat => f (Q (fun b : nat => f (h x)))) <-> True.
intros f0 Q H.
setoid_rewrite H.
tauto.
Qed.
(** Check proper refreshing of the lemma application for multiple
different instances in a single setoid rewrite. *)
Section mult.
Context (fold : forall {A} {B}, (A -> B) -> A -> B).
Context (add : forall A, A -> A).
Context (fold_lemma : forall {A B f} {eqA : relation B} x, eqA (fold A B f (add A x)) (fold _ _ f x)).
Context (ab : forall B, A -> B).
Context (anat : forall A, nat -> A).
Goal forall x, (fold _ _ (fun x => ab A x) (add A x) = anat _ (fold _ _ (ab nat) (add _ x))).
Proof. intros.
setoid_rewrite fold_lemma.
change (fold A A (fun x0 : A => ab A x0) x = anat A (fold A nat (ab nat) x)).
Abort.
End mult.
(** Current semantics for rewriting with typeclass constraints in the lemma
does not fix the instance at the first unification, use [at], or simply rewrite for
this semantics. *)
Require Import Arith.
Class Foo (A : Type) := {foo_neg : A -> A ; foo_prf : forall x : A, x = foo_neg x}.
Instance: Foo nat. admit. Defined.
Instance: Foo bool. admit. Defined.
Goal forall (x : nat) (y : bool), beq_nat (foo_neg x) 0 = foo_neg y.
Proof. intros. setoid_rewrite <- foo_prf. change (beq_nat x 0 = y). Abort.
Goal forall (x : nat) (y : bool), beq_nat (foo_neg x) 0 = foo_neg y.
Proof. intros. setoid_rewrite <- @foo_prf at 1. change (beq_nat x 0 = foo_neg y). Abort.
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