Merge pull request #1443 from andreaslyn/infinitary-universal-algebra

Term algebra and congruence

Author: andreaslyn

_CoqProject

@@ -354,6 +354,8 @@ theories/Spectra/Coinductive.v
 theories/Algebra/Universal/Algebra.v
 theories/Algebra/Universal/Homomorphism.v
 theories/Algebra/Universal/Operation.v
+theories/Algebra/Universal/Congruence.v
+theories/Algebra/Universal/TermAlgebra.v
 
 #
 #   Algebra

theories/Algebra/Universal/Algebra.v

@@ -95,6 +95,8 @@ Global Existing Instance hset_algebra.
 
 Global Coercion carriers : Algebra >-> Funclass.
 
+Bind Scope Algebra_scope with Algebra.
+
 Definition SigAlgebra (σ : Signature) : Type
   := {c : Carriers σ
         | { _ : forall (u : Symbol σ), Operation c (σ u)
@@ -158,8 +160,7 @@ Defined.
 Arguments path_path_algebra {_} {σ} {A B}%Algebra_scope (p q r)%path_scope.
 
 Module notations_algebra.
-  (** Conflicts with inverse notation so cannot be reserved *)
-  Global Notation "u ^^ A" := (operations A u)
-                              (at level 6, no associativity)
-                              : Algebra_scope.
+
+  Global Notation "u .# A" := (operations A u) : Algebra_scope.
+
 End notations_algebra.

theories/Algebra/Universal/Congruence.v

@@ -0,0 +1,91 @@
+(** This file implements algebra congruence relation. It serves as a
+    universal algebra generalization of normal subgroup, ring ideal, etc.
+
+    Congruence is used to construct quotients, in similarity with how
+    normal subgroup and ring ideal is used to construct quotients. *)
+
+Require Export HoTT.Algebra.Universal.Algebra.
+
+Require Import
+  HoTT.Basics
+  HoTT.Types
+  HoTT.HProp
+  HoTT.Classes.interfaces.canonical_names
+  HoTT.Algebra.Universal.Homomorphism.
+
+Unset Elimination Schemes.
+
+Import notations_algebra.
+
+Section congruence.
+  Context {σ : Signature} (A : Algebra σ) (Φ : forall s, Relation (A s)).
+
+(** A finitary operation [f : A s1 * A s2 * ... * A sn -> A t]
+    satisfies [OpCompatible f] iff
+
+    <<
+      Φ s1 x1 y1 * Φ s2 x2 y2 * ... * Φ sn xn yn
+    >>
+
+    implies
+
+    <<
+      Φ t (f (x1, x2, ..., xn)) (f (y1, y2, ..., yn)).
+    >>
+
+    The below definition generalizes this to infinitary operations.
+*)
+
+  Definition OpCompatible {w : SymbolType σ} (f : Operation A w) : Type
+    := forall (a b : DomOperation A w),
+       (forall i : Arity w, Φ (sorts_dom w i) (a i) (b i)) ->
+       Φ (sort_cod w) (f a) (f b).
+
+  Class OpsCompatible : Type
+    := ops_compatible : forall (u : Symbol σ), OpCompatible u.#A.
+
+  Global Instance trunc_ops_compatible `{Funext} {n : trunc_index}
+    `{!forall s x y, IsTrunc n (Φ s x y)}
+    : IsTrunc n OpsCompatible.
+  Proof.
+    apply trunc_forall.
+  Defined.
+
+  (** A family of relations [Φ] is a congruence iff it is a family of
+      mere equivalence relations and [OpsCompatible A Φ] holds. *)
+
+  Class IsCongruence : Type := Build_IsCongruence
+   { is_mere_relation_cong : forall (s : Sort σ), is_mere_relation (A s) (Φ s)
+   ; equiv_rel_cong : forall (s : Sort σ), EquivRel (Φ s)
+   ; ops_compatible_cong : OpsCompatible }.
+
+  Global Arguments Build_IsCongruence {is_mere_relation_cong}
+                                      {equiv_rel_cong}
+                                      {ops_compatible_cong}.
+
+  Global Existing Instance is_mere_relation_cong.
+
+  Global Existing Instance equiv_rel_cong.
+
+  Global Existing Instance ops_compatible_cong.
+
+  Global Instance hprop_is_congruence `{Funext} : IsHProp IsCongruence.
+  Proof.
+    apply (equiv_hprop_allpath _)^-1.
+    intros [C1 C2 C3] [D1 D2 D3].
+    by destruct (path_ishprop C1 D1),
+                (path_ishprop C2 D2),
+                (path_ishprop C3 D3).
+  Defined.
+
+End congruence.
+
+(** A homomorphism [f : forall s, A s -> B s] is compatible
+    with a congruence [Φ] iff [Φ s x y] implies [f s x = f s y]. *)
+
+Definition HomCompatible {σ : Signature} {A B : Algebra σ}
+  (Φ : forall s, Relation (A s)) `{!IsCongruence A Φ}
+  (f : forall s, A s -> B s) `{!IsHomomorphism f}
+  : Type
+  := forall s (x y : A s), Φ s x y -> f s x = f s y.
+

theories/Algebra/Universal/Homomorphism.v

@@ -28,7 +28,7 @@ Section is_homomorphism.
   Qed.
 
   Class IsHomomorphism : Type
-    := oppreserving_hom : forall (u : Symbol σ), OpPreserving u^^A u^^B.
+    := oppreserving_hom : forall (u : Symbol σ), OpPreserving u.#A u.#B.
 
   Global Instance hprop_is_homomorphism `{Funext}
     : IsHProp IsHomomorphism.