diff --git a/Makefile b/Makefile
index db28a1e8ae..e2ed2f507d 100644
--- a/Makefile
+++ b/Makefile
@@ -62,7 +62,7 @@ src/everything.lagda.md: agdaFiles
| cut -c 5- \
| cut -f1 -d'.' \
| sed 's/\//\./g' \
- | awk 'BEGIN { FS = "."; OFS = "."; lastdir = "" } { if ($$1 != lastdir) { print ""; lastdir = $$1 } print "open import " $$0 }' \
+ | awk 'BEGIN { FS = "."; OFS = "."; lastdir = "" } { if ($$1 != lastdir) { print ""; lastdir = $$1 } print "open import " $$0 " public"}' \
>> $@ ;\
echo "\`\`\`" >> $@ ;
diff --git a/src/finite-algebra.lagda.md b/src/finite-algebra.lagda.md
index 27ed892454..359239c09d 100644
--- a/src/finite-algebra.lagda.md
+++ b/src/finite-algebra.lagda.md
@@ -8,13 +8,8 @@ module finite-algebra where
open import finite-algebra.commutative-finite-rings public
open import finite-algebra.dependent-products-commutative-finite-rings public
open import finite-algebra.dependent-products-finite-rings public
-open import finite-algebra.finite-abelian-groups public
-open import finite-algebra.finite-commutative-monoids public
open import finite-algebra.finite-fields public
-open import finite-algebra.finite-groups public
-open import finite-algebra.finite-monoids public
open import finite-algebra.finite-rings public
-open import finite-algebra.finite-semigroups public
open import finite-algebra.homomorphisms-commutative-finite-rings public
open import finite-algebra.homomorphisms-finite-rings public
open import finite-algebra.products-commutative-finite-rings public
diff --git a/src/finite-algebra/commutative-finite-rings.lagda.md b/src/finite-algebra/commutative-finite-rings.lagda.md
index dbb33bd468..52394614ad 100644
--- a/src/finite-algebra/commutative-finite-rings.lagda.md
+++ b/src/finite-algebra/commutative-finite-rings.lagda.md
@@ -591,7 +591,7 @@ module _
preserves-concat-add-list-Ring-π½ finite-ring-Commutative-Ring-π½
```
-### Equip a finite type with a structure of commutative finite ring
+### Equipping a finite type with the structure of a commutative finite ring
```agda
module _
@@ -603,14 +603,14 @@ module _
UU l1
structure-commutative-ring-π½ =
Ξ£ ( structure-ring-π½ X)
- ( Ξ» r β is-commutative-Ring-π½ (compute-structure-ring-π½ X r))
+ ( Ξ» r β is-commutative-Ring-π½ (finite-ring-structure-ring-π½ X r))
- compute-structure-commutative-ring-π½ :
+ finite-commutative-ring-structure-commutative-ring-π½ :
structure-commutative-ring-π½ β
Commutative-Ring-π½ l1
- pr1 (compute-structure-commutative-ring-π½ (r , c)) =
- compute-structure-ring-π½ X r
- pr2 (compute-structure-commutative-ring-π½ (r , c)) = c
+ pr1 (finite-commutative-ring-structure-commutative-ring-π½ (r , c)) =
+ finite-ring-structure-ring-π½ X r
+ pr2 (finite-commutative-ring-structure-commutative-ring-π½ (r , c)) = c
is-finite-structure-commutative-ring-π½ :
is-finite structure-commutative-ring-π½
diff --git a/src/finite-algebra/dependent-products-finite-rings.lagda.md b/src/finite-algebra/dependent-products-finite-rings.lagda.md
index 18b15b2f33..bcf62b5a1c 100644
--- a/src/finite-algebra/dependent-products-finite-rings.lagda.md
+++ b/src/finite-algebra/dependent-products-finite-rings.lagda.md
@@ -156,5 +156,5 @@ module _
ring-Ξ -Ring-π½ = Ξ -Ring (type-π½ I) (ring-Ring-π½ β A)
Ξ -Ring-π½ : Ring-π½ (l1 β l2)
- Ξ -Ring-π½ = compute-ring-π½ ring-Ξ -Ring-π½ is-finite-type-Ξ -Ring-π½
+ Ξ -Ring-π½ = finite-ring-is-finite-Ring ring-Ξ -Ring-π½ is-finite-type-Ξ -Ring-π½
```
diff --git a/src/finite-algebra/finite-groups.lagda.md b/src/finite-algebra/finite-groups.lagda.md
deleted file mode 100644
index 6e5c9cda1d..0000000000
--- a/src/finite-algebra/finite-groups.lagda.md
+++ /dev/null
@@ -1,217 +0,0 @@
-# Abstract finite groups
-
-```agda
-module finite-algebra.finite-groups where
-```
-
-Imports
-
-```agda
-open import finite-algebra.finite-monoids
-open import finite-algebra.finite-semigroups
-
-open import foundation.identity-types
-open import foundation.propositions
-open import foundation.sets
-open import foundation.universe-levels
-
-open import group-theory.commuting-elements-groups
-open import group-theory.groups
-open import group-theory.monoids
-open import group-theory.semigroups
-
-open import structured-types.pointed-types
-
-open import univalent-combinatorics.cartesian-product-types
-open import univalent-combinatorics.dependent-function-types
-open import univalent-combinatorics.dependent-pair-types
-open import univalent-combinatorics.equality-finite-types
-open import univalent-combinatorics.finite-types
-```
-
-Imports
-
-```agda
-open import finite-algebra.finite-semigroups
-
-open import foundation.identity-types
-open import foundation.propositions
-open import foundation.sets
-open import foundation.unital-binary-operations
-open import foundation.universe-levels
-
-open import group-theory.monoids
-open import group-theory.semigroups
-
-open import univalent-combinatorics.cartesian-product-types
-open import univalent-combinatorics.dependent-function-types
-open import univalent-combinatorics.dependent-pair-types
-open import univalent-combinatorics.equality-finite-types
-open import univalent-combinatorics.finite-types
-```
-
-Imports
-
-```agda
-open import foundation.decidable-propositions
-open import foundation.identity-types
-open import foundation.sets
-open import foundation.universe-levels
-
-open import group-theory.semigroups
-
-open import univalent-combinatorics.dependent-function-types
-open import univalent-combinatorics.dependent-pair-types
-open import univalent-combinatorics.equality-finite-types
-open import univalent-combinatorics.finite-types
-```
-
-Imports
```agda
-open import finite-algebra.finite-groups
+open import finite-group-theory.finite-groups
open import foundation.equivalences
open import foundation.identity-types
@@ -57,10 +57,11 @@ is-prop-is-abelian-Group-π½ G =
Ab-π½ : (l : Level) β UU (lsuc l)
Ab-π½ l = Ξ£ (Group-π½ l) is-abelian-Group-π½
-compute-abelian-group-π½ :
+finite-abelian-group-is-finite-Ab :
{l : Level} β (A : Ab l) β is-finite (type-Ab A) β Ab-π½ l
-pr1 (compute-abelian-group-π½ A f) = compute-group-π½ (group-Ab A) f
-pr2 (compute-abelian-group-π½ A f) = pr2 A
+pr1 (finite-abelian-group-is-finite-Ab A f) =
+ finite-group-is-finite-Group (group-Ab A) f
+pr2 (finite-abelian-group-is-finite-Ab A f) = pr2 A
module _
{l : Level} (A : Ab-π½ l)
@@ -204,7 +205,7 @@ module _
## Properties
-### There is a finite number of ways to equip a finite type with a structure of abelian group
+### There is a finite number of ways to equip a finite type with the structure of an abelian group
```agda
module _
@@ -215,13 +216,13 @@ module _
structure-abelian-group-π½ : UU l
structure-abelian-group-π½ =
Ξ£ ( structure-group-π½ X)
- ( Ξ» g β is-abelian-Group-π½ (compute-structure-group-π½ X g))
+ ( Ξ» g β is-abelian-Group-π½ (finite-group-structure-group-π½ X g))
- compute-structure-abelian-group-π½ :
+ finite-abelian-group-structure-abelian-group-π½ :
structure-abelian-group-π½ β Ab-π½ l
- pr1 (compute-structure-abelian-group-π½ (m , c)) =
- compute-structure-group-π½ X m
- pr2 (compute-structure-abelian-group-π½ (m , c)) = c
+ pr1 (finite-abelian-group-structure-abelian-group-π½ (m , c)) =
+ finite-group-structure-group-π½ X m
+ pr2 (finite-abelian-group-structure-abelian-group-π½ (m , c)) = c
is-finite-structure-abelian-group-π½ :
is-finite structure-abelian-group-π½
diff --git a/src/finite-algebra/finite-commutative-monoids.lagda.md b/src/finite-group-theory/finite-commutative-monoids.lagda.md
similarity index 93%
rename from src/finite-algebra/finite-commutative-monoids.lagda.md
rename to src/finite-group-theory/finite-commutative-monoids.lagda.md
index c6d35c3d73..02ffea267b 100644
--- a/src/finite-algebra/finite-commutative-monoids.lagda.md
+++ b/src/finite-group-theory/finite-commutative-monoids.lagda.md
@@ -1,13 +1,13 @@
# Finite Commutative monoids
```agda
-module finite-algebra.finite-commutative-monoids where
+module finite-group-theory.finite-commutative-monoids where
```
Imports
```agda
-open import finite-algebra.finite-monoids
+open import finite-group-theory.finite-monoids
open import foundation.identity-types
open import foundation.sets
@@ -179,7 +179,7 @@ module _
## Properties
-### There is a finite number of ways to equip a finite type with a structure of finite commutative monoids
+### There is a finite number of ways to equip a finite type with the structure of a finite commutative monoid
```agda
module _
@@ -190,13 +190,13 @@ module _
structure-commutative-monoid-π½ : UU l
structure-commutative-monoid-π½ =
Ξ£ ( structure-monoid-π½ X)
- ( Ξ» m β is-commutative-Monoid-π½ (compute-structure-monoid-π½ X m))
+ ( Ξ» m β is-commutative-Monoid-π½ (finite-monoid-structure-monoid-π½ X m))
- compute-structure-commutative-monoid-π½ :
+ finite-commutative-monoid-structure-commutative-monoid-π½ :
structure-commutative-monoid-π½ β Commutative-Monoid-π½ l
- pr1 (compute-structure-commutative-monoid-π½ (m , c)) =
- compute-structure-monoid-π½ X m
- pr2 (compute-structure-commutative-monoid-π½ (m , c)) = c
+ pr1 (finite-commutative-monoid-structure-commutative-monoid-π½ (m , c)) =
+ finite-monoid-structure-monoid-π½ X m
+ pr2 (finite-commutative-monoid-structure-commutative-monoid-π½ (m , c)) = c
is-finite-structure-commutative-monoid-π½ :
is-finite structure-commutative-monoid-π½
diff --git a/src/finite-group-theory/finite-groups.lagda.md b/src/finite-group-theory/finite-groups.lagda.md
index ac29136492..fbb7a1421a 100644
--- a/src/finite-group-theory/finite-groups.lagda.md
+++ b/src/finite-group-theory/finite-groups.lagda.md
@@ -9,6 +9,7 @@ module finite-group-theory.finite-groups where
```agda
open import elementary-number-theory.natural-numbers
+open import finite-group-theory.finite-monoids
open import finite-group-theory.finite-semigroups
open import foundation.binary-embeddings
@@ -30,6 +31,7 @@ open import foundation.sets
open import foundation.type-arithmetic-dependent-pair-types
open import foundation.universe-levels
+open import group-theory.commuting-elements-groups
open import group-theory.groups
open import group-theory.monoids
open import group-theory.semigroups
@@ -43,6 +45,7 @@ open import univalent-combinatorics.decidable-dependent-function-types
open import univalent-combinatorics.decidable-dependent-pair-types
open import univalent-combinatorics.decidable-propositions
open import univalent-combinatorics.dependent-function-types
+open import univalent-combinatorics.dependent-pair-types
open import univalent-combinatorics.equality-finite-types
open import univalent-combinatorics.finite-types
open import univalent-combinatorics.function-types
@@ -54,17 +57,37 @@ open import univalent-combinatorics.standard-finite-types
## Idea
-A finite group is a group of which the underlying type is finite.
+An {{#concept "(abstract) finite group" Agda=Group-π½}} is a finite group in the
+usual algebraic sense, i.e., it consists of a
+[finite type](univalent-combinatorics.finite-types.md)
+[equipped](foundation.structure.md) with a unit element `e`, a binary operation
+`x, y β¦ xy`, and an inverse operation `x β¦ xβ»ΒΉ` satisfying the
+[group](group-theory.groups.md) laws
+
+```text
+ (xy)z = x(yz) (associativity)
+ ex = x (left unit law)
+ xe = x (right unit law)
+ xβ»ΒΉx = e (left inverse law)
+ xxβ»ΒΉ = e (right inverse law)
+```
## Definitions
+### The condition that a finite semigroup is a finite group
+
+```agda
+is-group-π½ :
+ {l : Level} (G : Semigroup-π½ l) β UU l
+is-group-π½ G = is-group (semigroup-Semigroup-π½ G)
+```
+
### The type of finite groups
```agda
Group-π½ :
(l : Level) β UU (lsuc l)
-Group-π½ l =
- Ξ£ (Semigroup-π½ l) (Ξ» G β is-group (semigroup-Semigroup-π½ G))
+Group-π½ l = Ξ£ (Semigroup-π½ l) (is-group-π½)
module _
{l : Level} (G : Group-π½ l)
@@ -282,6 +305,23 @@ module _
inv-inv-Group-π½ :
(x : type-Group-π½) β inv-Group-π½ (inv-Group-π½ x) οΌ x
inv-inv-Group-π½ = inv-inv-Group group-Group-π½
+
+finite-group-is-finite-Group :
+ {l : Level} β (G : Group l) β is-finite (type-Group G) β Group-π½ l
+pr1 (finite-group-is-finite-Group G f) =
+ finite-semigroup-is-finite-Semigroup (semigroup-Group G) f
+pr2 (finite-group-is-finite-Group G f) = is-group-Group G
+
+module _
+ {l : Level} (G : Group-π½ l)
+ where
+
+ commute-Group-π½ : type-Group-π½ G β type-Group-π½ G β UU l
+ commute-Group-π½ = commute-Group (group-Group-π½ G)
+
+ finite-monoid-Group-π½ : Monoid-π½ l
+ pr1 finite-monoid-Group-π½ = finite-semigroup-Group-π½ G
+ pr2 finite-monoid-Group-π½ = is-unital-Group-π½ G
```
### Groups of fixed finite order
@@ -374,3 +414,43 @@ mere-equiv-number-of-groups-of-order n =
mere-equiv-number-of-connected-components
( is-Ο-finite-Group-of-Order {lzero} zero-β n)
```
+
+### There is a finite number of ways to equip a finite type with the structure of a group
+
+```agda
+module _
+ {l : Level}
+ (X : π½ l)
+ where
+
+ structure-group-π½ : UU l
+ structure-group-π½ =
+ Ξ£ (structure-semigroup-π½ X) (Ξ» s β is-group-π½ (X , s))
+
+ finite-group-structure-group-π½ :
+ structure-group-π½ β Group-π½ l
+ pr1 (finite-group-structure-group-π½ (s , g)) = (X , s)
+ pr2 (finite-group-structure-group-π½ (s , g)) = g
+
+ is-finite-structure-group-π½ :
+ is-finite (structure-group-π½)
+ is-finite-structure-group-π½ =
+ is-finite-Ξ£
+ ( is-finite-structure-semigroup-π½ X)
+ ( Ξ» s β
+ is-finite-Ξ£
+ ( is-finite-is-unital-Semigroup-π½ (X , s))
+ ( Ξ» u β
+ is-finite-Ξ£
+ ( is-finite-Ξ
+ ( is-finite-type-π½ X)
+ ( Ξ» _ β is-finite-type-π½ X))
+ ( Ξ» i β
+ is-finite-product
+ ( is-finite-Ξ
+ ( is-finite-type-π½ X)
+ ( Ξ» x β is-finite-eq-π½ X))
+ ( is-finite-Ξ
+ ( is-finite-type-π½ X)
+ ( Ξ» x β is-finite-eq-π½ X)))))
+```
diff --git a/src/finite-group-theory/finite-monoids.lagda.md b/src/finite-group-theory/finite-monoids.lagda.md
index e9d8d415a3..a9fa4daee5 100644
--- a/src/finite-group-theory/finite-monoids.lagda.md
+++ b/src/finite-group-theory/finite-monoids.lagda.md
@@ -15,18 +15,26 @@ open import foundation.decidable-propositions
open import foundation.decidable-types
open import foundation.dependent-pair-types
open import foundation.equivalences
+open import foundation.identity-types
open import foundation.mere-equivalences
open import foundation.propositional-truncations
+open import foundation.propositions
open import foundation.set-truncations
+open import foundation.sets
open import foundation.type-arithmetic-dependent-pair-types
+open import foundation.unital-binary-operations
open import foundation.universe-levels
open import group-theory.monoids
open import group-theory.semigroups
+open import univalent-combinatorics.cartesian-product-types
open import univalent-combinatorics.counting
open import univalent-combinatorics.decidable-dependent-function-types
open import univalent-combinatorics.decidable-dependent-pair-types
+open import univalent-combinatorics.dependent-function-types
+open import univalent-combinatorics.dependent-pair-types
+open import univalent-combinatorics.equality-finite-types
open import univalent-combinatorics.finite-types
open import univalent-combinatorics.pi-finite-types
open import univalent-combinatorics.standard-finite-types
@@ -40,7 +48,78 @@ A finite monoid is a monoid of which the underlying type is finite.
## Definition
-### Monoids of order n
+### The type of finite monoids
+
+```agda
+is-unital-Semigroup-π½ :
+ {l : Level} β Semigroup-π½ l β UU l
+is-unital-Semigroup-π½ G = is-unital (mul-Semigroup-π½ G)
+
+Monoid-π½ :
+ (l : Level) β UU (lsuc l)
+Monoid-π½ l = Ξ£ (Semigroup-π½ l) is-unital-Semigroup-π½
+
+module _
+ {l : Level} (M : Monoid-π½ l)
+ where
+
+ finite-semigroup-Monoid-π½ : Semigroup-π½ l
+ finite-semigroup-Monoid-π½ = pr1 M
+
+ semigroup-Monoid-π½ : Semigroup l
+ semigroup-Monoid-π½ = semigroup-Semigroup-π½ finite-semigroup-Monoid-π½
+
+ finite-type-Monoid-π½ : π½ l
+ finite-type-Monoid-π½ = finite-type-Semigroup-π½ finite-semigroup-Monoid-π½
+
+ type-Monoid-π½ : UU l
+ type-Monoid-π½ = type-Semigroup-π½ finite-semigroup-Monoid-π½
+
+ is-finite-type-Monoid-π½ : is-finite type-Monoid-π½
+ is-finite-type-Monoid-π½ = is-finite-type-Semigroup-π½ finite-semigroup-Monoid-π½
+
+ set-Monoid-π½ : Set l
+ set-Monoid-π½ = set-Semigroup semigroup-Monoid-π½
+
+ is-set-type-Monoid-π½ : is-set type-Monoid-π½
+ is-set-type-Monoid-π½ = is-set-type-Semigroup semigroup-Monoid-π½
+
+ mul-Monoid-π½ : type-Monoid-π½ β type-Monoid-π½ β type-Monoid-π½
+ mul-Monoid-π½ = mul-Semigroup semigroup-Monoid-π½
+
+ mul-Monoid-π½' : type-Monoid-π½ β type-Monoid-π½ β type-Monoid-π½
+ mul-Monoid-π½' y x = mul-Monoid-π½ x y
+
+ ap-mul-Monoid-π½ :
+ {x x' y y' : type-Monoid-π½} β
+ x οΌ x' β y οΌ y' β mul-Monoid-π½ x y οΌ mul-Monoid-π½ x' y'
+ ap-mul-Monoid-π½ = ap-mul-Semigroup semigroup-Monoid-π½
+
+ associative-mul-Monoid-π½ :
+ (x y z : type-Monoid-π½) β
+ mul-Monoid-π½ (mul-Monoid-π½ x y) z οΌ mul-Monoid-π½ x (mul-Monoid-π½ y z)
+ associative-mul-Monoid-π½ = associative-mul-Semigroup semigroup-Monoid-π½
+
+ has-unit-Monoid-π½ : is-unital mul-Monoid-π½
+ has-unit-Monoid-π½ = pr2 M
+
+ monoid-Monoid-π½ : Monoid l
+ pr1 monoid-Monoid-π½ = semigroup-Monoid-π½
+ pr2 monoid-Monoid-π½ = has-unit-Monoid-π½
+
+ unit-Monoid-π½ : type-Monoid-π½
+ unit-Monoid-π½ = unit-Monoid monoid-Monoid-π½
+
+ left-unit-law-mul-Monoid-π½ :
+ (x : type-Monoid-π½) β mul-Monoid-π½ unit-Monoid-π½ x οΌ x
+ left-unit-law-mul-Monoid-π½ = left-unit-law-mul-Monoid monoid-Monoid-π½
+
+ right-unit-law-mul-Monoid-π½ :
+ (x : type-Monoid-π½) β mul-Monoid-π½ x unit-Monoid-π½ οΌ x
+ right-unit-law-mul-Monoid-π½ = right-unit-law-mul-Monoid monoid-Monoid-π½
+```
+
+### Monoids of order `n`
```agda
Monoid-of-Order : (l : Level) (n : β) β UU (lsuc l)
@@ -49,7 +128,7 @@ Monoid-of-Order l n = Ξ£ (Monoid l) (Ξ» M β mere-equiv (Fin n) (type-Monoid M)
## Properties
-### For any semigroup of order n, the type of multiplicative units is finite
+### For any semigroup of order `n`, the type of multiplicative units is finite
```agda
is-finite-is-unital-Semigroup :
@@ -101,7 +180,7 @@ is-Ο-finite-Monoid-of-Order {l} k n =
e = equiv-right-swap-Ξ£
```
-### The function that returns for any n the number of monoids of order n up to isomorphism
+### The function that returns for any `n` the number of monoids of order `n` up to isomorphism
```agda
number-of-monoids-of-order : β β β
@@ -118,3 +197,56 @@ mere-equiv-number-of-monoids-of-order n =
mere-equiv-number-of-connected-components
( is-Ο-finite-Monoid-of-Order {lzero} zero-β n)
```
+
+### For any finite semigroup `G`, being unital is a property
+
+```agda
+abstract
+ is-prop-is-unital-Semigroup-π½ :
+ {l : Level} (G : Semigroup-π½ l) β is-prop (is-unital-Semigroup-π½ G)
+ is-prop-is-unital-Semigroup-π½ G =
+ is-prop-is-unital-Semigroup (semigroup-Semigroup-π½ G)
+
+is-unital-Semigroup-π½-Prop : {l : Level} (G : Semigroup-π½ l) β Prop l
+pr1 (is-unital-Semigroup-π½-Prop G) = is-unital-Semigroup-π½ G
+pr2 (is-unital-Semigroup-π½-Prop G) = is-prop-is-unital-Semigroup-π½ G
+```
+
+### For any finite semigroup `G`, being unital is finite
+
+```agda
+is-finite-is-unital-Semigroup-π½ :
+ {l : Level} (G : Semigroup-π½ l) β is-finite (is-unital-Semigroup-π½ G)
+is-finite-is-unital-Semigroup-π½ G =
+ is-finite-Ξ£
+ ( is-finite-type-Semigroup-π½ G)
+ ( Ξ» e β
+ is-finite-product
+ ( is-finite-Ξ
+ ( is-finite-type-Semigroup-π½ G)
+ ( Ξ» x β is-finite-eq-π½ (finite-type-Semigroup-π½ G)))
+ ( is-finite-Ξ
+ ( is-finite-type-Semigroup-π½ G)
+ ( Ξ» x β is-finite-eq-π½ (finite-type-Semigroup-π½ G))))
+```
+
+### There is a finite number of ways to equip a finite type with the structure of a monoid
+
+```agda
+structure-monoid-π½ :
+ {l1 : Level} β π½ l1 β UU l1
+structure-monoid-π½ X =
+ Ξ£ (structure-semigroup-π½ X) (Ξ» p β is-unital-Semigroup-π½ (X , p))
+
+finite-monoid-structure-monoid-π½ :
+ {l : Level} β (X : π½ l) β structure-monoid-π½ X β Monoid-π½ l
+pr1 (finite-monoid-structure-monoid-π½ X (a , u)) = X , a
+pr2 (finite-monoid-structure-monoid-π½ X (a , u)) = u
+
+is-finite-structure-monoid-π½ :
+ {l : Level} β (X : π½ l) β is-finite (structure-monoid-π½ X)
+is-finite-structure-monoid-π½ X =
+ is-finite-Ξ£
+ ( is-finite-structure-semigroup-π½ X)
+ ( Ξ» m β is-finite-is-unital-Semigroup-π½ (X , m))
+```
diff --git a/src/finite-group-theory/finite-semigroups.lagda.md b/src/finite-group-theory/finite-semigroups.lagda.md
index 95c63559fb..14bbbfe4f2 100644
--- a/src/finite-group-theory/finite-semigroups.lagda.md
+++ b/src/finite-group-theory/finite-semigroups.lagda.md
@@ -9,6 +9,7 @@ module finite-group-theory.finite-semigroups where
```agda
open import elementary-number-theory.natural-numbers
+open import foundation.decidable-propositions
open import foundation.equivalences
open import foundation.function-types
open import foundation.functoriality-dependent-pair-types
@@ -39,12 +40,18 @@ Finite semigroups are semigroups of which the underlying type is finite.
## Definitions
+### The predicate of having an associative multiplication operation on finite types
+
+```agda
+has-associative-mul-π½ : {l : Level} (X : π½ l) β UU l
+has-associative-mul-π½ X = has-associative-mul (type-π½ X)
+```
+
### Finite semigroups
```agda
Semigroup-π½ : (l : Level) β UU (lsuc l)
-Semigroup-π½ l =
- Ξ£ (π½ l) (Ξ» X β has-associative-mul (type-π½ X))
+Semigroup-π½ l = Ξ£ (π½ l) (has-associative-mul-π½)
module _
{l : Level} (G : Semigroup-π½ l)
@@ -89,9 +96,24 @@ module _
( mul-Semigroup-π½ x (mul-Semigroup-π½ y z))
associative-mul-Semigroup-π½ =
associative-mul-Semigroup semigroup-Semigroup-π½
+
+finite-semigroup-is-finite-Semigroup :
+ {l : Level} β (G : Semigroup l) β is-finite (type-Semigroup G) β Semigroup-π½ l
+pr1 (pr1 (finite-semigroup-is-finite-Semigroup G f)) = type-Semigroup G
+pr2 (pr1 (finite-semigroup-is-finite-Semigroup G f)) = f
+pr2 (finite-semigroup-is-finite-Semigroup G f) = has-associative-mul-Semigroup G
+
+module _
+ {l : Level} (G : Semigroup-π½ l)
+ where
+
+ ap-mul-Semigroup-π½ :
+ {x x' y y' : type-Semigroup-π½ G} β
+ x οΌ x' β y οΌ y' β mul-Semigroup-π½ G x y οΌ mul-Semigroup-π½ G x' y'
+ ap-mul-Semigroup-π½ = ap-mul-Semigroup (semigroup-Semigroup-π½ G)
```
-### Semigroups of order n
+### Semigroups of order `n`
```agda
Semigroup-of-Order' : (l : Level) (n : β) β UU (lsuc l)
@@ -124,7 +146,7 @@ is-finite-has-associative-mul H =
is-finite-eq (has-decidable-equality-is-finite H)))))
```
-### The type of semigroups of order n is Ο-finite
+### The type of semigroups of order `n` is Ο-finite
```agda
is-Ο-finite-Semigroup-of-Order' :
@@ -176,3 +198,39 @@ mere-equiv-number-of-semi-groups-of-order n =
mere-equiv-number-of-connected-components
( is-Ο-finite-Semigroup-of-Order {lzero} zero-β n)
```
+
+### There is a finite number of ways to equip a finite type with the structure of a semigroup
+
+```agda
+structure-semigroup-π½ :
+ {l1 : Level} β π½ l1 β UU l1
+structure-semigroup-π½ = has-associative-mul-π½
+
+is-finite-structure-semigroup-π½ :
+ {l : Level} β (X : π½ l) β is-finite (structure-semigroup-π½ X)
+is-finite-structure-semigroup-π½ X =
+ is-finite-Ξ£
+ ( is-finite-Ξ
+ ( is-finite-type-π½ X)
+ ( Ξ» _ β is-finite-Ξ (is-finite-type-π½ X) (Ξ» _ β is-finite-type-π½ X)))
+ ( Ξ» m β
+ is-finite-Ξ
+ ( is-finite-type-π½ X)
+ ( Ξ» x β
+ is-finite-Ξ
+ ( is-finite-type-π½ X)
+ ( Ξ» y β
+ is-finite-Ξ
+ ( is-finite-type-π½ X)
+ ( Ξ» z β
+ is-finite-is-decidable-Prop
+ ( (m (m x y) z οΌ m x (m y z)) ,
+ is-set-is-finite
+ ( is-finite-type-π½ X)
+ ( m (m x y) z)
+ ( m x (m y z)))
+ ( has-decidable-equality-is-finite
+ ( is-finite-type-π½ X)
+ ( m (m x y) z)
+ ( m x (m y z)))))))
+```
diff --git a/src/foundation/path-algebra.lagda.md b/src/foundation/path-algebra.lagda.md
index 96a439ae35..bb110a0312 100644
--- a/src/foundation/path-algebra.lagda.md
+++ b/src/foundation/path-algebra.lagda.md
@@ -37,12 +37,12 @@ type is often called a _type of 2-paths_.
Since 2-paths are just identifications, they have the usual operations and
coherences on paths/identifications. In the context of 2-paths, this famliar
-concatination operation is called vertical concatination (see
+concatenation operation is called vertical concatenation (see
`vertical-concat-IdΒ²` below). However, 2-paths have novel operations and
coherences derived from the operations and coherences of the boundary 1-paths
-(these are `p` and `q` in the example above). Since concatination of 1-paths is
+(these are `p` and `q` in the example above). Since concatenation of 1-paths is
a functor, it has an induced action on paths. We call this operation horizontal
-concatination (see `horizontal-concat-IdΒ²` below). It comes with the standard
+concatenation (see `horizontal-concat-IdΒ²` below). It comes with the standard
coherences of an action on paths function, as well as coherences induced by
coherences on the boundary 1-paths.
@@ -187,7 +187,7 @@ right-unit-law-horizontal-concat-IdΒ² :
right-unit-law-horizontal-concat-IdΒ² = right-unit-ap-binary (_β_)
```
-Horizontal concatination satisfies an additional "2-dimensional" unit law (on
+Horizontal concatenation satisfies an additional "2-dimensional" unit law (on
both the left and right) induced by the unit laws on the boundary 1-paths.
```agda
@@ -318,7 +318,7 @@ unit-law-Ξ΄-interchange-IdΒ² p refl = refl
3-paths are identifications of 2-paths. In symbols, a type of 3-paths is a type
of the form `Ξ± οΌ Ξ²` where `Ξ± Ξ² : p οΌ q` and `p q : x οΌ y`.
-### Concatination in a type of 3-paths
+### Concatenation in a type of 3-paths
Like with 2-paths, 3-paths have the standard operations on equalties, plus the
operations induced by the operations on 1-paths. But 3-paths also have
@@ -327,7 +327,7 @@ in triple identity types. We name the three concatenations of triple identity
types x-, y-, and z-concatenation, after the standard names for the three axis
in 3-dimensional space.
-The x-concatenation operation corresponds the standard concatination of
+The x-concatenation operation corresponds the standard concatenation of
equalities.
```agda
@@ -338,7 +338,7 @@ x-concat-IdΒ³ Ο Ο = vertical-concat-IdΒ² Ο Ο
```
The y-concatenation operation corresponds the operation induced by the
-concatination on 1-paths.
+concatenation on 1-paths.
```agda
y-concat-IdΒ³ :
@@ -347,8 +347,8 @@ y-concat-IdΒ³ :
y-concat-IdΒ³ = horizontal-concat-IdΒ²
```
-The z-concatenation operation corresponds the concatination induced by the
-horizontal concatination on 2-paths.
+The z-concatenation operation corresponds the concatenation induced by the
+horizontal concatenation on 2-paths.
```agda
z-concat-IdΒ³ :
@@ -436,7 +436,7 @@ after the standard names for the quaternions `i`, `j`, and `k`.
### Concatenation of four paths
-#### The standard concatination
+#### The standard concatenation
```agda
concat-Idβ΄ :
@@ -445,7 +445,7 @@ concat-Idβ΄ :
concat-Idβ΄ Ο Ο = x-concat-IdΒ³ Ο Ο
```
-#### Concatination induced by concatination of boundary 1-paths
+#### Concatenation induced by concatenation of boundary 1-paths
```agda
i-concat-Idβ΄ :
@@ -455,7 +455,7 @@ i-concat-Idβ΄ :
i-concat-Idβ΄ Ο Ο = y-concat-IdΒ³ Ο Ο
```
-#### Concatination induced by concatination of boundary 2-paths
+#### Concatenation induced by concatenation of boundary 2-paths
```agda
j-concat-Idβ΄ :
@@ -465,7 +465,7 @@ j-concat-Idβ΄ :
j-concat-Idβ΄ Ο Ο = z-concat-IdΒ³ Ο Ο
```
-#### Concatination induced by concatination of boundary 3-paths
+#### Concatenation induced by concatenation of boundary 3-paths
```agda
k-concat-Idβ΄ :
@@ -482,7 +482,7 @@ module _
{l : Level} {A : UU l} {x000 x001 x010 x100 x011 x101 x110 x111 : A}
where
- cube :
+ coherence-cube-identifications :
(p000Μ : x000 οΌ x001) (p00Μ0 : x000 οΌ x010) (p0Μ00 : x000 οΌ x100)
(p00Μ1 : x001 οΌ x011) (p0Μ01 : x001 οΌ x101) (p010Μ : x010 οΌ x011)
(p0Μ10 : x010 οΌ x110) (p100Μ : x100 οΌ x101) (p10Μ0 : x100 οΌ x110)
@@ -493,7 +493,7 @@ module _
(p0Μ0Μ1 : coherence-square-identifications p0Μ01 p00Μ1 p10Μ1 p0Μ11)
(p0Μ10Μ : coherence-square-identifications p0Μ10 p010Μ p110Μ p0Μ11)
(p10Μ0Μ : coherence-square-identifications p10Μ0 p100Μ p110Μ p10Μ1) β UU l
- cube
+ coherence-cube-identifications
p000Μ p00Μ0 p0Μ00 p00Μ1 p0Μ01 p010Μ p0Μ10 p100Μ p10Μ0 p0Μ11 p10Μ1 p110Μ
p00Μ0Μ p0Μ00Μ p0Μ0Μ0 p0Μ0Μ1 p0Μ10Μ p10Μ0Μ =
Id
diff --git a/src/group-theory/abelian-groups.lagda.md b/src/group-theory/abelian-groups.lagda.md
index 22fb7110a0..518213fe02 100644
--- a/src/group-theory/abelian-groups.lagda.md
+++ b/src/group-theory/abelian-groups.lagda.md
@@ -240,12 +240,13 @@ module _
structure-abelian-group :
{l1 : Level} β UU l1 β UU l1
structure-abelian-group X =
- Ξ£ (structure-group X) (Ξ» p β is-abelian-Group (compute-structure-group X p))
+ Ξ£ (structure-group X) (Ξ» p β is-abelian-Group (group-structure-group X p))
-compute-structure-abelian-group :
+abelian-group-structure-abelian-group :
{l1 : Level} β (X : UU l1) β structure-abelian-group X β Ab l1
-pr1 (compute-structure-abelian-group X (p , q)) = compute-structure-group X p
-pr2 (compute-structure-abelian-group X (p , q)) = q
+pr1 (abelian-group-structure-abelian-group X (p , q)) =
+ group-structure-group X p
+pr2 (abelian-group-structure-abelian-group X (p , q)) = q
```
### Conjugation in an abelian group
diff --git a/src/group-theory/groups.lagda.md b/src/group-theory/groups.lagda.md
index 9376350274..513db7f496 100644
--- a/src/group-theory/groups.lagda.md
+++ b/src/group-theory/groups.lagda.md
@@ -216,12 +216,12 @@ module _
structure-group :
{l1 : Level} β UU l1 β UU l1
structure-group X =
- Ξ£ (structure-semigroup X) (Ξ» p β is-group (compute-structure-semigroup X p))
+ Ξ£ (structure-semigroup X) (Ξ» p β is-group (semigroup-structure-semigroup X p))
-compute-structure-group :
+group-structure-group :
{l1 : Level} β (X : UU l1) β structure-group X β Group l1
-pr1 (compute-structure-group X (p , q)) = compute-structure-semigroup X p
-pr2 (compute-structure-group X (p , q)) = q
+pr1 (group-structure-group X (p , q)) = semigroup-structure-semigroup X p
+pr2 (group-structure-group X (p , q)) = q
```
## Properties
diff --git a/src/group-theory/semigroups.lagda.md b/src/group-theory/semigroups.lagda.md
index d90ca26f2f..be720c9094 100644
--- a/src/group-theory/semigroups.lagda.md
+++ b/src/group-theory/semigroups.lagda.md
@@ -108,8 +108,8 @@ structure-semigroup :
structure-semigroup X =
Ξ£ (is-set X) (Ξ» p β has-associative-mul-Set (X , p))
-compute-structure-semigroup :
+semigroup-structure-semigroup :
{l1 : Level} β (X : UU l1) β structure-semigroup X β Semigroup l1
-pr1 (compute-structure-semigroup X (s , g)) = X , s
-pr2 (compute-structure-semigroup X (s , g)) = g
+pr1 (semigroup-structure-semigroup X (s , g)) = X , s
+pr2 (semigroup-structure-semigroup X (s , g)) = g
```
diff --git a/src/orthogonal-factorization-systems/higher-modalities.lagda.md b/src/orthogonal-factorization-systems/higher-modalities.lagda.md
index 1ace39ec54..b44ca06b76 100644
--- a/src/orthogonal-factorization-systems/higher-modalities.lagda.md
+++ b/src/orthogonal-factorization-systems/higher-modalities.lagda.md
@@ -47,8 +47,8 @@ defining map (propositionally).
Lastly, higher modalities must also be **identity closed** in the sense that for
every type `X` the identity types `(x' οΌ y')` are modal for all terms
`x' y' : β X`. In other words, `β X` is
-[`β`-separated](orthogonal-factorization-systems.separated-types.md). Because of
-this, higher modalities in their most general form only make sense for
+[`β`-separated](foundation.separated-types.md). Because of this, higher
+modalities in their most general form only make sense for
[locally small modal operators](orthogonal-factorization-systems.locally-small-modal-operators.md).
## Definition
diff --git a/src/orthogonal-factorization-systems/separated-types.lagda.md b/src/orthogonal-factorization-systems/separated-types.lagda.md
index d773fc900e..965dcad305 100644
--- a/src/orthogonal-factorization-systems/separated-types.lagda.md
+++ b/src/orthogonal-factorization-systems/separated-types.lagda.md
@@ -28,11 +28,11 @@ module _
{l1 l2 l3 : Level} {X : UU l1} {Y : UU l2} (f : X β Y)
where
- is-separated-family : (X β UU l3) β UU (l1 β l2 β l3)
- is-separated-family A = (x : X) (y z : A x) β is-local f (y οΌ z)
+ is-map-separated-family : (X β UU l3) β UU (l1 β l2 β l3)
+ is-map-separated-family A = (x : X) (y z : A x) β is-local f (y οΌ z)
- is-separated : UU l3 β UU (l1 β l2 β l3)
- is-separated A = is-separated-family (Ξ» _ β A)
+ is-map-separated : UU l3 β UU (l1 β l2 β l3)
+ is-map-separated A = is-map-separated-family (Ξ» _ β A)
```
## References
diff --git a/src/reflection/abstractions.lagda.md b/src/reflection/abstractions.lagda.md
index 9dc0d8b673..6395c53b4c 100644
--- a/src/reflection/abstractions.lagda.md
+++ b/src/reflection/abstractions.lagda.md
@@ -16,14 +16,14 @@ open import primitives.strings
## Idea
-The `Abs` type represents a lambda abstraction.
+The `Abstraction-Agda` type represents a lambda abstraction.
## Definition
```agda
-data Abs {l} (A : UU l) : UU l where
- abs : String β A β Abs A
+data Abstraction-Agda {l : Level} (A : UU l) : UU l where
+ cons-Abstraction-Agda : String β A β Abstraction-Agda A
-{-# BUILTIN ABS Abs #-}
-{-# BUILTIN ABSABS abs #-}
+{-# BUILTIN ABS Abstraction-Agda #-}
+{-# BUILTIN ABSABS cons-Abstraction-Agda #-}
```
diff --git a/src/reflection/arguments.lagda.md b/src/reflection/arguments.lagda.md
index c225b10bf4..176d7be2c4 100644
--- a/src/reflection/arguments.lagda.md
+++ b/src/reflection/arguments.lagda.md
@@ -23,8 +23,8 @@ argument has three properties:
3. Quantity: whether they are run-time relevant or not (see,
[docs](https://agda.readthedocs.io/en/latest/language/runtime-irrelevance.html))
-The properties of `Relevance` and `Quantity` are combined in one, called
-`Modality`.
+The properties of `Relevance-Argument-Agda` and `Quantity-Argument-Agda` are
+combined in one, called `Modality-Argument-Agda`.
For concrete examples, see
[`reflection.definitions`](reflection.definitions.md).
@@ -32,49 +32,55 @@ For concrete examples, see
## Definitions
```agda
-data Visibility : UU lzero where
- visible hidden instanceβ² : Visibility
+data Visibility-Argument-Agda : UU lzero where
+ visible-Visibility-Argument-Agda : Visibility-Argument-Agda
+ hidden-Visibility-Argument-Agda : Visibility-Argument-Agda
+ instance-Visibility-Argument-Agda : Visibility-Argument-Agda
-data Relevance : UU lzero where
- relevant irrelevant : Relevance
+data Relevance-Argument-Agda : UU lzero where
+ relevant-Relevance-Argument-Agda : Relevance-Argument-Agda
+ irrelevant-Relevance-Argument-Agda : Relevance-Argument-Agda
-data Quantity : UU lzero where
- quantity-0 quantity-Ο : Quantity
+data Quantity-Argument-Agda : UU lzero where
+ zero-Quantity-Argument-Agda : Quantity-Argument-Agda
+ omega-Quantity-Argument-Agda : Quantity-Argument-Agda
-data Modality : UU lzero where
- modality : (r : Relevance) (q : Quantity) β Modality
+data Modality-Argument-Agda : UU lzero where
+ cons-Modality-Argument-Agda :
+ Relevance-Argument-Agda β Quantity-Argument-Agda β Modality-Argument-Agda
-data ArgInfo : UU lzero where
- arg-info : (v : Visibility) (m : Modality) β ArgInfo
+data Info-Argument-Agda : UU lzero where
+ cons-Info-Argument-Agda :
+ Visibility-Argument-Agda β Modality-Argument-Agda β Info-Argument-Agda
-data Arg {l} (A : UU l) : UU l where
- arg : (i : ArgInfo) (x : A) β Arg A
+data Argument-Agda {l : Level} (A : UU l) : UU l where
+ cons-Argument-Agda : Info-Argument-Agda β A β Argument-Agda A
```
Bindings
```agda
-{-# BUILTIN HIDING Visibility #-}
-{-# BUILTIN VISIBLE visible #-}
-{-# BUILTIN HIDDEN hidden #-}
-{-# BUILTIN INSTANCE instanceβ² #-}
+{-# BUILTIN HIDING Visibility-Argument-Agda #-}
+{-# BUILTIN VISIBLE visible-Visibility-Argument-Agda #-}
+{-# BUILTIN HIDDEN hidden-Visibility-Argument-Agda #-}
+{-# BUILTIN INSTANCE instance-Visibility-Argument-Agda #-}
-{-# BUILTIN RELEVANCE Relevance #-}
-{-# BUILTIN RELEVANT relevant #-}
-{-# BUILTIN IRRELEVANT irrelevant #-}
+{-# BUILTIN RELEVANCE Relevance-Argument-Agda #-}
+{-# BUILTIN RELEVANT relevant-Relevance-Argument-Agda #-}
+{-# BUILTIN IRRELEVANT irrelevant-Relevance-Argument-Agda #-}
-{-# BUILTIN QUANTITY Quantity #-}
-{-# BUILTIN QUANTITY-0 quantity-0 #-}
-{-# BUILTIN QUANTITY-Ο quantity-Ο #-}
+{-# BUILTIN QUANTITY Quantity-Argument-Agda #-}
+{-# BUILTIN QUANTITY-0 zero-Quantity-Argument-Agda #-}
+{-# BUILTIN QUANTITY-Ο omega-Quantity-Argument-Agda #-}
-{-# BUILTIN MODALITY Modality #-}
-{-# BUILTIN MODALITY-CONSTRUCTOR modality #-}
+{-# BUILTIN MODALITY Modality-Argument-Agda #-}
+{-# BUILTIN MODALITY-CONSTRUCTOR cons-Modality-Argument-Agda #-}
-{-# BUILTIN ARGINFO ArgInfo #-}
-{-# BUILTIN ARGARGINFO arg-info #-}
+{-# BUILTIN ARGINFO Info-Argument-Agda #-}
+{-# BUILTIN ARGARGINFO cons-Info-Argument-Agda #-}
-{-# BUILTIN ARG Arg #-}
-{-# BUILTIN ARGARG arg #-}
+{-# BUILTIN ARG Argument-Agda #-}
+{-# BUILTIN ARGARG cons-Argument-Agda #-}
```
Bindings
+## Bindings
```agda
-{-# BUILTIN AGDATERM Term #-}
-{-# BUILTIN AGDASORT Sort #-}
-{-# BUILTIN AGDAPATTERN Pattern #-}
-{-# BUILTIN AGDACLAUSE Clause #-}
-
-{-# BUILTIN AGDATERMVAR var #-}
-{-# BUILTIN AGDATERMCON con #-}
-{-# BUILTIN AGDATERMDEF def #-}
-{-# BUILTIN AGDATERMMETA meta #-}
-{-# BUILTIN AGDATERMLAM lam #-}
-{-# BUILTIN AGDATERMEXTLAM pat-lam #-}
-{-# BUILTIN AGDATERMPI pi #-}
-{-# BUILTIN AGDATERMSORT agda-sort #-}
-{-# BUILTIN AGDATERMLIT lit #-}
-{-# BUILTIN AGDATERMUNSUPPORTED unknown #-}
-
-{-# BUILTIN AGDASORTSET set #-}
-{-# BUILTIN AGDASORTLIT lit #-}
-{-# BUILTIN AGDASORTPROP prop #-}
-{-# BUILTIN AGDASORTPROPLIT propLit #-}
-{-# BUILTIN AGDASORTINF inf #-}
-{-# BUILTIN AGDASORTUNSUPPORTED unknown #-}
-
-{-# BUILTIN AGDAPATCON con #-}
-{-# BUILTIN AGDAPATDOT dot #-}
-{-# BUILTIN AGDAPATVAR var #-}
-{-# BUILTIN AGDAPATLIT lit #-}
-{-# BUILTIN AGDAPATPROJ proj #-}
-{-# BUILTIN AGDAPATABSURD absurd #-}
-
-{-# BUILTIN AGDACLAUSECLAUSE clause #-}
-{-# BUILTIN AGDACLAUSEABSURD absurd-clause #-}
+{-# BUILTIN AGDATERM Term-Agda #-}
+{-# BUILTIN AGDASORT Sort-Agda #-}
+{-# BUILTIN AGDAPATTERN Pattern-Agda #-}
+{-# BUILTIN AGDACLAUSE Clause-Agda #-}
+
+{-# BUILTIN AGDATERMVAR variable-Term-Agda #-}
+{-# BUILTIN AGDATERMCON constructor-Term-Agda #-}
+{-# BUILTIN AGDATERMDEF definition-Term-Agda #-}
+{-# BUILTIN AGDATERMMETA metavariable-Term-Agda #-}
+{-# BUILTIN AGDATERMLAM lambda-Term-Agda #-}
+{-# BUILTIN AGDATERMEXTLAM pattern-lambda-Term-Agda #-}
+{-# BUILTIN AGDATERMPI dependent-product-Term-Agda #-}
+{-# BUILTIN AGDATERMSORT sort-Term-Agda #-}
+{-# BUILTIN AGDATERMLIT literal-Term-Agda #-}
+{-# BUILTIN AGDATERMUNSUPPORTED unknown-Term-Agda #-}
+
+{-# BUILTIN AGDASORTSET universe-Sort-Agda #-}
+{-# BUILTIN AGDASORTLIT fixed-universe-Sort-Agda #-}
+{-# BUILTIN AGDASORTPROP prop-Sort-Agda #-}
+{-# BUILTIN AGDASORTPROPLIT fixed-prop-Sort-Agda #-}
+{-# BUILTIN AGDASORTINF fixed-large-universe-Sort-Agda #-}
+{-# BUILTIN AGDASORTUNSUPPORTED unknown-Sort-Agda #-}
+
+{-# BUILTIN AGDAPATCON constructor-Term-Agda #-}
+{-# BUILTIN AGDAPATDOT dot-Pattern-Agda #-}
+{-# BUILTIN AGDAPATVAR variable-Term-Agda #-}
+{-# BUILTIN AGDAPATLIT literal-Term-Agda #-}
+{-# BUILTIN AGDAPATPROJ projection-Pattern-Agda #-}
+{-# BUILTIN AGDAPATABSURD absurd-Pattern-Agda #-}
+
+{-# BUILTIN AGDACLAUSECLAUSE clause-Clause-Agda #-}
+{-# BUILTIN AGDACLAUSEABSURD absurd-Clause-Agda #-}
```
-Bindings
+## Bindings
```agda
-{-# BUILTIN AGDAERRORPART ErrorPart #-}
-{-# BUILTIN AGDAERRORPARTSTRING strErr #-}
-{-# BUILTIN AGDAERRORPARTTERM termErr #-}
-{-# BUILTIN AGDAERRORPARTPATT pattErr #-}
-{-# BUILTIN AGDAERRORPARTNAME nameErr #-}
-
-{-# BUILTIN AGDATCM TC #-}
-{-# BUILTIN AGDATCMRETURN returnTC #-}
-{-# BUILTIN AGDATCMBIND bindTC #-}
+{-# BUILTIN AGDAERRORPART Error-Part #-}
+{-# BUILTIN AGDAERRORPARTSTRING string-Error-Part #-}
+{-# BUILTIN AGDAERRORPARTTERM term-Error-Part #-}
+{-# BUILTIN AGDAERRORPARTPATT pattern-Error-Part #-}
+{-# BUILTIN AGDAERRORPARTNAME name-Error-Part #-}
+
+{-# BUILTIN AGDATCM type-Type-Checker #-}
+{-# BUILTIN AGDATCMRETURN return-Type-Checker #-}
+{-# BUILTIN AGDATCMBIND bind-Type-Checker #-}
{-# BUILTIN AGDATCMUNIFY unify #-}
-{-# BUILTIN AGDATCMTYPEERROR typeError #-}
-{-# BUILTIN AGDATCMINFERTYPE inferType #-}
-{-# BUILTIN AGDATCMCHECKTYPE checkType #-}
-{-# BUILTIN AGDATCMNORMALISE normalise #-}
+{-# BUILTIN AGDATCMTYPEERROR type-error #-}
+{-# BUILTIN AGDATCMINFERTYPE infer-type #-}
+{-# BUILTIN AGDATCMCHECKTYPE check-type #-}
+{-# BUILTIN AGDATCMNORMALISE normalize #-}
{-# BUILTIN AGDATCMREDUCE reduce #-}
-{-# BUILTIN AGDATCMCATCHERROR catchTC #-}
-{-# BUILTIN AGDATCMQUOTETERM quoteTC #-}
-{-# BUILTIN AGDATCMUNQUOTETERM unquoteTC #-}
-{-# BUILTIN AGDATCMQUOTEOMEGATERM quoteΟTC #-}
-{-# BUILTIN AGDATCMGETCONTEXT getContext #-}
-{-# BUILTIN AGDATCMEXTENDCONTEXT extendContext #-}
-{-# BUILTIN AGDATCMINCONTEXT inContext #-}
-{-# BUILTIN AGDATCMFRESHNAME freshName #-}
-{-# BUILTIN AGDATCMDECLAREDEF declareDef #-}
-{-# BUILTIN AGDATCMDECLAREPOSTULATE declarePostulate #-}
-{-# BUILTIN AGDATCMDEFINEFUN defineFun #-}
-{-# BUILTIN AGDATCMGETTYPE getType #-}
-{-# BUILTIN AGDATCMGETDEFINITION getDefinition #-}
-{-# BUILTIN AGDATCMBLOCK blockTC #-}
-{-# BUILTIN AGDATCMCOMMIT commitTC #-}
-{-# BUILTIN AGDATCMISMACRO isMacro #-}
-{-# BUILTIN AGDATCMWITHNORMALISATION withNormalisation #-}
-{-# BUILTIN AGDATCMFORMATERRORPARTS formatErrorParts #-}
-{-# BUILTIN AGDATCMDEBUGPRINT debugPrint #-}
-{-# BUILTIN AGDATCMWITHRECONSTRUCTED withReconstructed #-}
-{-# BUILTIN AGDATCMWITHEXPANDLAST withExpandLast #-}
-{-# BUILTIN AGDATCMWITHREDUCEDEFS withReduceDefs #-}
-{-# BUILTIN AGDATCMASKNORMALISATION askNormalisation #-}
-{-# BUILTIN AGDATCMASKRECONSTRUCTED askReconstructed #-}
-{-# BUILTIN AGDATCMASKEXPANDLAST askExpandLast #-}
-{-# BUILTIN AGDATCMASKREDUCEDEFS askReduceDefs #-}
-{-# BUILTIN AGDATCMNOCONSTRAINTS noConstraints #-}
-{-# BUILTIN AGDATCMRUNSPECULATIVE runSpeculative #-}
-{-# BUILTIN AGDATCMGETINSTANCES getInstances #-}
-{-# BUILTIN AGDATCMDECLAREDATA declareData #-}
-{-# BUILTIN AGDATCMDEFINEDATA defineData #-}
+{-# BUILTIN AGDATCMCATCHERROR catch-Type-Checker #-}
+{-# BUILTIN AGDATCMQUOTETERM quote-Type-Checker #-}
+{-# BUILTIN AGDATCMUNQUOTETERM unquote-Type-Checker #-}
+{-# BUILTIN AGDATCMQUOTEOMEGATERM quoteΟ-Type-Checker #-}
+{-# BUILTIN AGDATCMGETCONTEXT get-context #-}
+{-# BUILTIN AGDATCMEXTENDCONTEXT extend-context #-}
+{-# BUILTIN AGDATCMINCONTEXT in-context #-}
+{-# BUILTIN AGDATCMFRESHNAME fresh-name #-}
+{-# BUILTIN AGDATCMDECLAREDEF declare-definition #-}
+{-# BUILTIN AGDATCMDECLAREPOSTULATE declare-postulate #-}
+{-# BUILTIN AGDATCMDEFINEFUN define-function #-}
+{-# BUILTIN AGDATCMGETTYPE get-type #-}
+{-# BUILTIN AGDATCMGETDEFINITION get-definition #-}
+{-# BUILTIN AGDATCMBLOCK block-Type-Checker #-}
+{-# BUILTIN AGDATCMCOMMIT commit-Type-Checker #-}
+{-# BUILTIN AGDATCMISMACRO is-macro #-}
+{-# BUILTIN AGDATCMWITHNORMALISATION with-normalization #-}
+{-# BUILTIN AGDATCMFORMATERRORPARTS format-error #-}
+{-# BUILTIN AGDATCMDEBUGPRINT debug-print #-}
+{-# BUILTIN AGDATCMWITHRECONSTRUCTED with-reconstructed #-}
+{-# BUILTIN AGDATCMWITHEXPANDLAST with-expand-last #-}
+{-# BUILTIN AGDATCMWITHREDUCEDEFS with-reduce-definitions #-}
+{-# BUILTIN AGDATCMASKNORMALISATION ask-normalization #-}
+{-# BUILTIN AGDATCMASKRECONSTRUCTED ask-reconstructed #-}
+{-# BUILTIN AGDATCMASKEXPANDLAST ask-expand-last #-}
+{-# BUILTIN AGDATCMASKREDUCEDEFS ask-reduce-definitions #-}
+{-# BUILTIN AGDATCMNOCONSTRAINTS no-constraints #-}
+{-# BUILTIN AGDATCMRUNSPECULATIVE run-speculative #-}
+{-# BUILTIN AGDATCMGETINSTANCES get-instances #-}
+{-# BUILTIN AGDATCMDECLAREDATA declare-data #-}
+{-# BUILTIN AGDATCMDEFINEDATA define-data #-}
```
-