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Require Import
Coq.Classes.Morphisms Coq.Classes.RelationClasses Coq.Relations.Relation_Definitions Coq.Lists.List
MathClasses.interfaces.abstract_algebra MathClasses.interfaces.universal_algebra MathClasses.theory.ua_homomorphisms MathClasses.interfaces.canonical_names MathClasses.theory.ua_subalgebraT MathClasses.misc.util.
Require MathClasses.theory.ua_products.
Require MathClasses.theory.categories.
(* Remove this *)
Local Hint Transparent Equiv : typeclass_instances.
Section contents.
Context σ `{@Algebra σ v ve vo}.
Notation op_type := (op_type (sorts σ)).
Notation vv := (ua_products.carrier σ bool (λ _: bool, v)).
Instance hint:
@Algebra σ vv (ua_products.product_e σ bool (fun _ : bool => v) (fun _ : bool => ve)) _
:= @ua_products.product_algebra σ bool (λ _, v) _ _ _.
(* Given an equivalence on the algebra's carrier that respects its setoid equality... *)
Instance hint' (a: sorts σ): Equiv (ua_products.carrier σ bool (fun _: bool => v) a).
Proof. apply products.dep_prod_equiv. intro. apply _. Defined.
Context (e: ∀ s, relation (v s)).
Section for_nice_e.
Context
(e_e: ∀ s, Equivalence (e s))
(e_proper: ∀ s, Proper ((=) ==> (=) ==> iff) (e s)).
(* We can show that that equivalence lifted to arbitrary operation types still respects the setoid equality: *)
Let Q s x := e s (x true) (x false).
Let lifted_e := @op_type_equiv (sorts σ) v e.
Let lifted_normal := @op_type_equiv (sorts σ) v ve.
Instance lifted_e_proper o: Proper ((=) ==> (=) ==> iff) (lifted_e o).
Proof with intuition.
induction o; simpl. intuition.
repeat intro.
unfold respectful.
split; intros.
assert (x x1 = y x1). apply H0...
assert (x0 y1 = y0 y1). apply H1...
apply (IHo (x x1) (y x1) H4 (x0 y1) (y0 y1) H5)...
assert (x x1 = y x1). apply H0...
assert (x0 y1 = y0 y1). apply H1...
apply (IHo (x x1) (y x1) H4 (x0 y1) (y0 y1) H5)...
Qed. (* todo: clean up *)
(* With that out of the way, we show that there are two equivalent ways of stating compatibility with the
algebra's operations: *)
(* 1: the direct way with Algebra; *)
Let eAlgebra := @Algebra σ v e _.
(* 2: the indirect way of saying that the relation as a set of pairs is a subalgebra in the product algebra: *)
Let eSub := @ua_subalgebraT.ClosedSubset σ vv _ _ Q.
Lemma eAlgebra_eSub: eAlgebra → eSub.
Proof with intuition.
intros.
constructor.
unfold Q.
repeat intro.
constructor; intro.
rewrite <- (H1 true), <- (H1 false)...
rewrite (H1 true), (H1 false)...
intro o.
simpl.
unfold algebra_op, ua_products.product_ops, algebra_op.
set (f := λ _: bool, vo o).
assert (∀ b, Proper (=) (f b)).
intro.
unfold f.
apply algebra_propers.
assert (lifted_e _ (f true) (f false)). unfold f.
unfold lifted_e.
destruct H0.
apply algebra_propers.
assert (∀ b, Proper (lifted_e (σ o)) (f b))...
clearbody f.
induction (σ o)...
simpl in *...
apply IHo0...
apply H1...
Qed. (* todo: clean up *)
Lemma eSub_eAlgebra: eSub → eAlgebra.
Proof with intuition.
intros [proper closed].
constructor. unfold abstract_algebra.Setoid. apply _.
intro o.
generalize (closed o). clear closed. (* todo: must be a cleaner way *)
unfold algebra_op.
simpl.
unfold ua_products.product_ops.
intro.
change (lifted_e _ (algebra_op o) (algebra_op o)).
set (f := λ _: bool, algebra_op o) in *.
assert (∀ b, lifted_normal _ (f b) (f b)). intros.
subst f. simpl.
apply algebra_propers...
change (lifted_e (σ o) (f true) (f false)).
clearbody f.
induction (σ o)...
simpl in *.
repeat intro.
unfold respectful in H0.
apply (IHo0 (λ b, f b (if b then x else y)))...
Qed. (* todo: clean up *)
Lemma back_and_forth: iffT eSub eAlgebra.
Proof. split; intro; [apply eSub_eAlgebra | apply eAlgebra_eSub]; assumption. Qed.
End for_nice_e.
(* This justifies the following definition of a congruence: *)
Class Congruence: Prop :=
{ congruence_proper:: ∀ s: sorts σ, Proper ((=) ==> (=) ==> iff) (e s)
; congruence_quotient:: Algebra σ v (e:=e) }.
End contents.
(* A variety for an equational theory none of whose laws have
premises is closed under quotients generated by congruences: *)
Lemma quotient_variety
(et: EquationalTheory) `{InVariety et v}
(e': ∀ s, relation (v s)) `{!Congruence et e'}
(no_premises: ∀ s, et_laws et s → entailment_premises _ s ≡ nil):
InVariety et v (e:=e').
(* Todo: Can this no-premises condition be weakened? Does it occur in this form in the literature? *)
Proof.
constructor. apply _.
intros l law vars.
pose proof (variety_laws l law vars) as E.
pose proof (no_premises l law).
destruct l as [prems [conc ?]]. simpl in *. subst. simpl in *.
unfold equiv. rewrite E.
pose proof (_: Equivalence (e' conc)).
reflexivity.
Qed.
Section in_domain.
Context {A B} {R: Equiv B} (f: A → B).
Definition in_domain: Equiv A := λ x y, f x = f y. (* todo: use pointwise thingy *)
Global Instance in_domain_equivalence: Equivalence R → Equivalence in_domain.
Proof with intuition.
constructor; repeat intro; unfold equiv, in_domain in *...
Qed.
End in_domain.
Section first_iso.
(* "If A and B are algebras, and f is a homomorphism from A to B, then
the equivalence relation Φ on A defined by a~b if and only if f(a)=f(b) is
a congruence on A, and the algebra A/Φ is isomorphic to the image
of f, which is a subalgebra of B." *)
Context `{Algebra σ A} `{Algebra σ B} `{!HomoMorphism σ A B f}.
Notation Φ := (λ s, in_domain (f s)).
Lemma square o0 x x' y y':
Preservation σ A B f x x' →
Preservation σ A B f y y' →
op_type_equiv (sorts σ) B o0 x' y' →
@op_type_equiv (sorts σ) A (λ s: sorts σ, @in_domain (A s) (B s) (e0 s) (f s)) o0 x y.
Proof.
induction o0.
simpl.
intros.
unfold in_domain.
rewrite H3, H4.
assumption.
simpl in *.
repeat intro.
unfold in_domain in H6.
unfold respectful in H5.
simpl in *.
pose proof (H3 x0).
pose proof (H3 y0). clear H3.
pose proof (H4 x0).
pose proof (H4 y0). clear H4.
apply (IHo0 (x x0) (x' (f _ x0)) (y y0) (y' (f _ y0)) H7 H9).
apply H5.
assumption.
Qed.
Instance co: Congruence σ Φ.
Proof with intuition.
constructor.
repeat intro.
unfold in_domain.
rewrite H3, H4...
constructor; intro.
unfold abstract_algebra.Setoid. apply _.
unfold algebra_op.
generalize (preserves σ A B f o).
generalize (@algebra_propers σ B _ _ _ o).
unfold algebra_op.
generalize (H o), (H1 o).
induction (σ o); simpl in *; repeat intro.
apply _.
apply (square _ _ _ _ _ (H4 x) (H4 y))...
Qed.
Definition image s (b: B s): Type := sigT (λ a, f s a = b).
Lemma image_proper: ∀ s (x0 x' : B s), x0 = x' → iffT (image s x0) (image s x').
Proof. intros ??? E. split; intros [v ?]; exists v; rewrite E in *; assumption. Qed.
Instance: ClosedSubset image.
Proof with intuition.
constructor; repeat intro.
split; intros [q p]; exists q; rewrite p...
generalize (preserves σ A B f o).
generalize (@algebra_propers σ B _ _ _ o).
unfold algebra_op.
generalize (H1 o), (H o).
induction (σ o); simpl; intros.
exists o1...
destruct X.
apply (@op_closed_proper σ B _ _ _ image image_proper _ (o1 z) (o1 (f _ x))).
apply H3...
apply IHo0 with (o2 x)...
apply _.
Qed.
Definition quot_obj: algebras.Object σ := algebras.object σ A (algebra_equiv:=Φ). (* A/Φ *)
Definition subobject: algebras.Object σ := algebras.object σ (ua_subalgebraT.carrier image).
Program Definition back: subobject ⟶ quot_obj := λ _ X, projT1 (projT2 X).
Next Obligation. Proof with try apply _; intuition.
repeat constructor...
intros [x [i E']] [y [j E'']] E.
change (x = y) in E.
change (f a i = f a j).
rewrite E', E''...
unfold ua_subalgebraT.impl.
generalize (subset_closed image o).
unfold algebra_op.
generalize (H o) (H1 o) (preserves σ A B f o)
(_: Proper (=) (H o)) (_: Proper (=) (H1 o)).
induction (σ o); simpl; intros ? ? ? ? ?.
intros [? E]. change (f _ x = f _ o0). rewrite E...
intros ? [x [? E]]. apply IHo0... simpl in *. rewrite <- E...
Defined.
Program Definition forth: quot_obj ⟶ subobject :=
λ a X, existT _ (f a X) (existT _ X (reflexivity _)).
Next Obligation. Proof with try apply _; intuition.
repeat constructor...
unfold ua_subalgebraT.impl.
generalize (subset_closed image o).
unfold algebra_op.
generalize (H o) (H1 o) (preserves σ A B f o)
(_: Proper (=) (H o)) (_: Proper (=) (H1 o)).
induction (σ o); simpl...
apply IHo0...
Qed.
Theorem first_iso: categories.iso_arrows back forth.
Proof.
split. intro. reflexivity.
intros ? [? [? ?]]. assumption.
Qed.
End first_iso.
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