From b21e69503571272a5f35cf84c731994f3e921a3a Mon Sep 17 00:00:00 2001
From: Greg Brown <greg.brown01@ed.ac.uk>
Date: Tue, 29 Nov 2022 14:36:13 +0000
Subject: Move indexed setoids and functions out of Soat.

---
 soat.ipkg                             |  5 ++--
 src/Data/Morphism/Indexed.idr         | 13 ++++++++++
 src/Data/Setoid/Indexed.idr           | 36 +++++++++++++++++++++++++++
 src/Soat/Data/Product.idr             |  4 +--
 src/Soat/FirstOrder/Algebra.idr       |  7 +++---
 src/Soat/FirstOrder/Term.idr          |  5 ++--
 src/Soat/Relation.idr                 | 46 -----------------------------------
 src/Soat/SecondOrder/Algebra.idr      |  3 ++-
 src/Soat/SecondOrder/Algebra/Lift.idr |  3 ++-
 9 files changed, 63 insertions(+), 59 deletions(-)
 create mode 100644 src/Data/Morphism/Indexed.idr
 create mode 100644 src/Data/Setoid/Indexed.idr
 delete mode 100644 src/Soat/Relation.idr

diff --git a/soat.ipkg b/soat.ipkg
index c42996f..7d584ac 100644
--- a/soat.ipkg
+++ b/soat.ipkg
@@ -2,13 +2,14 @@ package soat
 authors = "Greg Brown"
 sourcedir = "src"
 
-modules = Data.Setoid
+modules = Data.Morphism.Indexed
+        , Data.Setoid
+        , Data.Setoid.Indexed
         , Soat.Data.Product
         , Soat.Data.Sublist
         , Soat.FirstOrder.Algebra
         , Soat.FirstOrder.Signature
         , Soat.FirstOrder.Term
-        , Soat.Relation
         , Soat.SecondOrder.Algebra
         , Soat.SecondOrder.Algebra.Lift
         , Soat.SecondOrder.Signature
diff --git a/src/Data/Morphism/Indexed.idr b/src/Data/Morphism/Indexed.idr
new file mode 100644
index 0000000..c271c90
--- /dev/null
+++ b/src/Data/Morphism/Indexed.idr
@@ -0,0 +1,13 @@
+module Data.Morphism.Indexed
+
+import Data.Setoid.Indexed
+
+public export
+IFunc : {a : Type} -> (x, y : a -> Type) -> Type
+IFunc {a} x y = (i : a) -> x i -> y i
+
+public export
+record IFunction {a : Type} (x, y : ISetoid a) where
+  constructor MkIFunction
+  func : IFunc x.U y.U
+  cong : (i : a) -> {u, v : x.U i} -> x.relation i u v -> y.relation i (func i u) (func i v)
diff --git a/src/Data/Setoid/Indexed.idr b/src/Data/Setoid/Indexed.idr
new file mode 100644
index 0000000..df5606d
--- /dev/null
+++ b/src/Data/Setoid/Indexed.idr
@@ -0,0 +1,36 @@
+module Data.Setoid.Indexed
+
+import public Data.Setoid
+
+%default total
+
+public export
+IRel : {a : Type} -> (a -> Type) -> Type
+IRel {a = a} x = (i : a) -> x i -> x i -> Type
+
+public export
+IReflexive : {a : Type} -> (x : a -> Type) -> IRel x -> Type
+IReflexive x rel = (i : a) -> Reflexive (x i) (rel i)
+
+public export
+ISymmetric : {a : Type} -> (x : a -> Type) -> IRel x -> Type
+ISymmetric x rel = (i : a) -> Symmetric (x i) (rel i)
+
+public export
+ITransitive : {a : Type} -> (x : a -> Type) -> IRel x -> Type
+ITransitive x rel = (i : a) -> Transitive (x i) (rel i)
+
+public export
+IEquivalence : {a : Type} -> (x : a -> Type) -> IRel x -> Type
+IEquivalence x rel = (i : a) -> Setoid.Equivalence (x i) (rel i)
+
+public export
+record ISetoid (a : Type) where
+  constructor MkISetoid
+  0 U         : a -> Type
+  0 relation  : IRel U
+  equivalence : IEquivalence U relation
+
+public export
+isetoid : (a -> Type) -> ISetoid a
+isetoid u = MkISetoid u (\_ => Equal) (\_ => equiv)
diff --git a/src/Soat/Data/Product.idr b/src/Soat/Data/Product.idr
index 1582d02..f392a2d 100644
--- a/src/Soat/Data/Product.idr
+++ b/src/Soat/Data/Product.idr
@@ -1,8 +1,8 @@
 module Soat.Data.Product
 
-import Control.Relation
 import Data.List.Elem
-import Soat.Relation
+import Data.Morphism.Indexed
+import Data.Setoid.Indexed
 
 %default total
 
diff --git a/src/Soat/FirstOrder/Algebra.idr b/src/Soat/FirstOrder/Algebra.idr
index 4b40cee..6f25f59 100644
--- a/src/Soat/FirstOrder/Algebra.idr
+++ b/src/Soat/FirstOrder/Algebra.idr
@@ -1,10 +1,9 @@
 module Soat.FirstOrder.Algebra
 
-import Control.Relation
-
-import Soat.Data.Product
+import Data.Morphism.Indexed
+import Data.Setoid.Indexed
+import public Soat.Data.Product
 import Soat.FirstOrder.Signature
-import Soat.Relation
 
 %default total
 %hide Control.Relation.Equivalence
diff --git a/src/Soat/FirstOrder/Term.idr b/src/Soat/FirstOrder/Term.idr
index 2712c88..07ab248 100644
--- a/src/Soat/FirstOrder/Term.idr
+++ b/src/Soat/FirstOrder/Term.idr
@@ -1,14 +1,13 @@
 module Soat.FirstOrder.Term
 
-import Control.Relation
+import Data.Morphism.Indexed
+import Data.Setoid.Indexed
 
 import Soat.Data.Product
 import Soat.FirstOrder.Algebra
 import Soat.FirstOrder.Signature
-import Soat.Relation
 
 %default total
-%hide Control.Relation.Equivalence
 
 public export
 data Term : (0 sig : Signature) -> (0 _ : sig.T -> Type) -> sig.T -> Type where
diff --git a/src/Soat/Relation.idr b/src/Soat/Relation.idr
deleted file mode 100644
index 9826d60..0000000
--- a/src/Soat/Relation.idr
+++ /dev/null
@@ -1,46 +0,0 @@
-module Soat.Relation
-
-import public Data.Setoid
-
-%default total
-
-public export
-IRel : {a : Type} -> (a -> Type) -> Type
-IRel {a = a} x = (i : a) -> x i -> x i -> Type
-
-public export
-IReflexive : {a : Type} -> (x : a -> Type) -> IRel x -> Type
-IReflexive x rel = (i : a) -> Reflexive (x i) (rel i)
-
-public export
-ISymmetric : {a : Type} -> (x : a -> Type) -> IRel x -> Type
-ISymmetric x rel = (i : a) -> Symmetric (x i) (rel i)
-
-public export
-ITransitive : {a : Type} -> (x : a -> Type) -> IRel x -> Type
-ITransitive x rel = (i : a) -> Transitive (x i) (rel i)
-
-public export
-IEquivalence : {a : Type} -> (x : a -> Type) -> IRel x -> Type
-IEquivalence x rel = (i : a) -> Setoid.Equivalence (x i) (rel i)
-
-public export
-record ISetoid (a : Type) where
-  constructor MkISetoid
-  0 U         : a -> Type
-  0 relation  : IRel U
-  equivalence : IEquivalence U relation
-
-public export
-isetoid : (a -> Type) -> ISetoid a
-isetoid u = MkISetoid u (\_ => Equal) (\_ => equiv)
-
-public export
-IFunc : {a : Type} -> (x, y : a -> Type) -> Type
-IFunc {a} x y = (i : a) -> x i -> y i
-
-public export
-record IFunction {a : Type} (x, y : ISetoid a) where
-  constructor MkIFunction
-  func : IFunc x.U y.U
-  cong : (i : a) -> {u, v : x.U i} -> x.relation i u v -> y.relation i (func i u) (func i v)
diff --git a/src/Soat/SecondOrder/Algebra.idr b/src/Soat/SecondOrder/Algebra.idr
index 50b65e0..c506c85 100644
--- a/src/Soat/SecondOrder/Algebra.idr
+++ b/src/Soat/SecondOrder/Algebra.idr
@@ -1,10 +1,11 @@
 module Soat.SecondOrder.Algebra
 
+import Data.Morphism.Indexed
+import Data.Setoid.Indexed
 import Data.List.Elem
 
 import Soat.Data.Product
 import Soat.Data.Sublist
-import Soat.Relation
 import Soat.SecondOrder.Signature
 
 public export
diff --git a/src/Soat/SecondOrder/Algebra/Lift.idr b/src/Soat/SecondOrder/Algebra/Lift.idr
index 00d453a..b08eb0e 100644
--- a/src/Soat/SecondOrder/Algebra/Lift.idr
+++ b/src/Soat/SecondOrder/Algebra/Lift.idr
@@ -1,12 +1,13 @@
 module Soat.SecondOrder.Algebra.Lift
 
 import Data.List.Elem
+import Data.Morphism.Indexed
+import Data.Setoid.Indexed
 
 import Soat.Data.Product
 import Soat.Data.Sublist
 import Soat.FirstOrder.Algebra
 import Soat.FirstOrder.Term
-import Soat.Relation
 import Soat.SecondOrder.Algebra
 import Soat.SecondOrder.Signature.Lift
 
-- 
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