### view free-monoid.agda @ 790:1e7319868d77

Sets is CCC
author Shinji KONO Fri, 19 Apr 2019 23:42:19 +0900 3d41a8edbf63
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-- Free Monoid and it's Universal Mapping
---- using Sets and forgetful functor

-- Shinji KONO <kono@ie.u-ryukyu.ac.jp>

open import Category -- https://github.com/konn/category-agda
open import Level
module free-monoid { c c₁ c₂ ℓ : Level }   where

open import Category.Sets
open import Category.Cat
open import HomReasoning
open import cat-utility
open import Relation.Binary.Core
open import universal-mapping
open import  Relation.Binary.PropositionalEquality

-- from https://github.com/danr/Agda-projects/blob/master/Category-Theory/FreeMonoid.agda

open import Algebra.FunctionProperties using (Op₁; Op₂)
open import Algebra.Structures
open import Data.Product

record ≡-Monoid : Set (suc c) where
infixr 9 _∙_
field
Carrier  : Set c
_∙_      : Op₂ Carrier
ε        : Carrier
isMonoid : IsMonoid _≡_ _∙_ ε

open ≡-Monoid

≡-cong = Relation.Binary.PropositionalEquality.cong

-- module ≡-reasoning (m : ≡-Monoid ) where

infixr 40 _::_
data  List  (A : Set c ) : Set c where
[] : List A
_::_ : A → List A → List A

infixl 30 _++_
_++_ :   {A : Set c } → List A → List A → List A
[]        ++ ys = ys
(x :: xs) ++ ys = x :: (xs ++ ys)

list-id-l : { A : Set c } → { x : List A } →  [] ++ x ≡ x
list-id-l {_} {_} = refl
list-id-r : { A : Set c } → { x : List A } →  x ++ [] ≡ x
list-id-r {_} {[]} =   refl
list-id-r {A} {x :: xs} =  ≡-cong ( λ y → x :: y ) ( list-id-r {A} {xs} )
list-assoc : {A : Set c} → { xs ys zs : List A } →
( xs ++ ( ys ++ zs ) ) ≡ ( ( xs ++ ys ) ++ zs )
list-assoc  {_} {[]} {_} {_} = refl
list-assoc  {A} {x :: xs}  {ys} {zs} = ≡-cong  ( λ y  → x :: y )
( list-assoc {A} {xs} {ys} {zs} )
list-o-resp-≈ :  {A : Set c} →  {f g : List A } → {h i : List A } →
f ≡  g → h ≡  i → (h ++ f) ≡  (i ++ g)
list-o-resp-≈  {A} refl refl = refl
list-isEquivalence : {A : Set c} → IsEquivalence {_} {_} {List A }  _≡_
list-isEquivalence {A} =      -- this is the same function as A's equivalence but has different types
record { refl  = refl
; sym   = sym
; trans = trans
}

_<_∙_> :  (m : ≡-Monoid)  →   Carrier m → Carrier m → Carrier m
m < x ∙ y > = _∙_ m x y

infixr 9 _<_∙_>

record Monomorph ( a b : ≡-Monoid )  : Set c where
field
morph : Carrier a → Carrier b
identity     :  morph (ε a)  ≡  ε b
homo :  ∀{x y} → morph ( a < x ∙ y > ) ≡ b < morph  x  ∙ morph  y >

open Monomorph

_+_ : { a b c : ≡-Monoid } → Monomorph b c → Monomorph a b → Monomorph a c
_+_ {a} {b} {c} f g =  record {
morph = λ x →  morph  f ( morph g x ) ;
identity  = identity1 ;
homo  = homo1
} where
identity1 : morph f ( morph g (ε a) )  ≡  ε c
identity1 = let open ≡-Reasoning in begin
morph f (morph g (ε a))
≡⟨  ≡-cong (morph f ) ( identity g )  ⟩
morph f (ε b)
≡⟨  identity f  ⟩
ε c
∎
homo1 :  ∀{x y} → morph f ( morph g ( a < x ∙  y > )) ≡ ( c <   (morph f (morph  g x )) ∙(morph f (morph  g y) ) > )
homo1 {x} {y} = let open ≡-Reasoning in begin
morph f (morph g (a < x ∙ y >))
≡⟨  ≡-cong (morph f ) ( homo g) ⟩
morph f (b < morph g x ∙ morph g y >)
≡⟨  homo f ⟩
c < morph f (morph g x) ∙ morph f (morph g y) >
∎

M-id : { a : ≡-Monoid } → Monomorph a a
M-id = record { morph = λ x → x  ; identity = refl ; homo = refl }

_==_ : { a b : ≡-Monoid } → Monomorph a b → Monomorph a b → Set c
_==_  f g =  morph f ≡ morph g

-- Functional Extensionality Axiom (We cannot prove this in Agda / Coq, just assumming )
-- postulate extensionality : { a b : Obj A } {f g : Hom A a b } → (∀ {x} → (f x ≡ g x))  → ( f ≡ g )
postulate extensionality : Relation.Binary.PropositionalEquality.Extensionality c c

isMonoids : IsCategory ≡-Monoid Monomorph _==_  _+_  (M-id)
isMonoids  = record  { isEquivalence =  isEquivalence1
; identityL =  refl
; identityR =  refl
; associative = refl
; o-resp-≈ =    λ {a} {b} {c} {f} {g} {h} {i} → o-resp-≈ {a} {b} {c} {f} {g} {h} {i}
}
where
isEquivalence1 : { a b : ≡-Monoid } → IsEquivalence {_} {_} {Monomorph a b}  _==_
isEquivalence1  =      -- this is the same function as A's equivalence but has different types
record { refl  = refl
; sym   = sym
; trans = trans
}
o-resp-≈ :  {a b c :  ≡-Monoid } {f g : Monomorph a b } → {h i : Monomorph b c }  →
f ==  g → h ==  i → (h + f) ==  (i + g)
o-resp-≈ {a} {b} {c} {f} {g} {h} {i} f==g h==i =  let open ≡-Reasoning in begin
morph ( h + f )
≡⟨ ≡-cong ( λ  g → ( ( λ (x : Carrier a ) →  g x ) )) (extensionality {Carrier a} lemma11) ⟩
morph ( i + g )
∎
where
lemma11 : (x : Carrier a) → morph (h + f) x ≡ morph (i + g) x
lemma11 x =   let open ≡-Reasoning in begin
morph ( h + f ) x
≡⟨⟩
morph h ( ( morph f ) x )
≡⟨  ≡-cong ( \y -> morph h ( y x ) )  f==g ⟩
morph h ( morph g x )
≡⟨  ≡-cong ( \y -> y ( morph g  x ) )  h==i ⟩
morph i ( morph g x )
≡⟨⟩
morph ( i + g ) x
∎

Monoids : Category _ _ _
Monoids  =
record { Obj =  ≡-Monoid
; Hom = Monomorph
; _o_ = _+_
; _≈_ = _==_
; Id  =  M-id
; isCategory = isMonoids
}

A = Sets {c}
B = Monoids

open Functor

U : Functor B A
U = record {
FObj = λ m → Carrier m ;
FMap = λ f → morph f ;
isFunctor = record {
≈-cong   = λ x → x
; identity = refl
; distr    = refl
}
}

-- FObj
list  : (a : Set c) → ≡-Monoid
list a = record {
Carrier    =  List a
;  _∙_ = _++_
; ε        = []
; isMonoid = record {
identity = ( ( λ x → list-id-l {a}  ) , ( λ x → list-id-r {a} ) ) ;
isSemigroup = record {
assoc =  λ x → λ y → λ z → sym ( list-assoc {a} {x} {y} {z} )
; isEquivalence = list-isEquivalence
; ∙-cong = λ x → λ y →  list-o-resp-≈ y x
}
}
}

Generator : Obj A → Obj B
Generator s =  list s

-- solution

--   [a,b,c] → f(a) ∙ f(b) ∙ f(c)
Φ :  {a : Obj A } {b : Obj B} ( f : Hom A a (FObj U b) ) →  List a → Carrier b
Φ {a} {b} f [] = ε b
Φ {a} {b} f ( x :: xs ) = b < ( f x ) ∙ (Φ {a} {b} f xs ) >

solution : (a : Obj A ) (b : Obj B) ( f : Hom A a (FObj U b) ) →  Hom B (Generator a ) b
solution a b f = record {
morph = λ (l : List a) → Φ f l   ;
identity = refl;
homo = λ {x y} → homo1 x y
} where
_*_ : Carrier b → Carrier b → Carrier b
_*_  f g = b <  f ∙ g >
homo1 : (x y : Carrier (Generator a)) → Φ f ( (Generator a) < x ∙  y > ) ≡ (Φ f x) * (Φ {a} {b} f y )
homo1 [] y = sym (proj₁ ( IsMonoid.identity ( isMonoid b) ) (Φ f y))
homo1 (x :: xs) y = let open ≡-Reasoning in
sym ( begin
(Φ {a} {b} f (x :: xs)) * (Φ f y)
≡⟨⟩
((f x) * (Φ f xs)) * (Φ f y)
≡⟨ ( IsMonoid.assoc ( isMonoid b )) _ _ _ ⟩
(f x) * ( (Φ f xs)  * (Φ f y) )
≡⟨ sym ( (IsMonoid.∙-cong (isMonoid b)) refl (homo1 xs y )) ⟩
(f x) * ( Φ f ( xs ++ y ) )
≡⟨⟩
Φ {a} {b} f ( x :: ( xs ++ y ) )
≡⟨⟩
Φ {a} {b} f ( (x ::  xs) ++ y )
≡⟨⟩
Φ {a} {b} f ((Generator a) < ( x :: xs) ∙ y > )
∎ )

eta :  (a : Obj A) → Hom A a ( FObj U (Generator a) )
eta  a = λ ( x : a ) →  x :: []

FreeMonoidUniveralMapping : UniversalMapping A B U
FreeMonoidUniveralMapping =
record {
F = Generator ;
η = eta ;
_* = λ {a b} → λ f → solution a b f ;
isUniversalMapping = record {
universalMapping = λ {a b f} → universalMapping {a} {b} {f} ;
uniquness = λ {a b f g} → uniquness {a} {b} {f} {g}
}
} where
universalMapping :  {a : Obj A } {b : Obj B} { f : Hom A a (FObj U b) } →  FMap U ( solution a b f  ) o eta a   ≡ f
universalMapping {a} {b} {f} = let open ≡-Reasoning in
begin
FMap U ( solution a b f ) o eta a
≡⟨⟩
( λ (x : a ) → Φ {a} {b} f (eta a x) )
≡⟨⟩
( λ (x : a ) → Φ {a} {b} f ( x :: [] ) )
≡⟨⟩
( λ (x : a ) →  b < ( f x ) ∙ (Φ {a} {b} f [] ) > )
≡⟨⟩
( λ (x : a ) →  b <    ( f x ) ∙ ( ε b ) > )
≡⟨ ≡-cong ( λ  g → ( ( λ (x : a ) →  g x ) )) (extensionality {a} lemma-ext1) ⟩
( λ (x : a ) →  f x  )
≡⟨⟩
f
∎  where
lemma-ext1 : ∀( x : a ) →  b <    ( f x ) ∙ ( ε b ) >  ≡ f x
lemma-ext1 x = ( proj₂ ( IsMonoid.identity ( isMonoid b) ) ) (f x)
uniquness :  {a : Obj A } {b : Obj B} { f : Hom A a (FObj U b) } →
{ g :  Hom B (Generator a) b } → (FMap U g) o (eta a )  ≡ f  → B [ solution a b f ≈ g ]
uniquness {a} {b} {f} {g} eq =
extensionality  lemma-ext2 where
lemma-ext2 : ( ∀( x : List a ) → (morph ( solution a b f)) x  ≡ (morph g) x  )
-- (morph ( solution a b f)) []  ≡ (morph g) []  )
lemma-ext2 [] = let open ≡-Reasoning in
begin
(morph ( solution a b f)) []
≡⟨ identity ( solution a b f) ⟩
ε b
≡⟨ sym ( identity g ) ⟩
(morph g) []
∎
lemma-ext2 (x :: xs)  = let open ≡-Reasoning in
begin
(morph ( solution a b f)) ( x :: xs )
≡⟨ homo ( solution a b f) {x :: []} {xs} ⟩
b <   ((morph ( solution a b f)) ( x :: []) ) ∙ ((morph ( solution a b f)) xs ) >
≡⟨  ≡-cong ( λ  k → (b <   ((morph ( solution a b f)) ( x :: []) ) ∙ k > )) (lemma-ext2 xs )   ⟩
b <   ((morph ( solution a b f)) ( x :: []) )  ∙((morph g) ( xs )) >
≡⟨  ≡-cong ( λ k → ( b <   ( k x ) ∙ ((morph g) ( xs )) >  )) (
begin
( λ x → (morph ( solution a b f)) ( x :: [] ) )
≡⟨ extensionality {a} lemma-ext3 ⟩
( λ x → (morph g) ( x :: [] ) )
∎
) ⟩
b <   ((morph g) ( x :: [] )) ∙((morph g) ( xs )) >
≡⟨ sym ( homo g ) ⟩
(morph g) ( x :: xs )
∎   where
lemma-ext3 :  ∀( x : a ) → (morph ( solution a b f)) (x :: [])  ≡ (morph g) ( x :: []  )
lemma-ext3 x = let open ≡-Reasoning in
begin
(morph ( solution a b f)) (x :: [])
≡⟨  ( proj₂  ( IsMonoid.identity ( isMonoid b) )(f x) ) ⟩
f x
≡⟨  sym ( ≡-cong (λ f → f x )  eq  ) ⟩
(morph g) ( x :: []  )
∎

open NTrans
--   fm-ε b = Φ

fm-ε :  NTrans B B ( ( FunctorF A B FreeMonoidUniveralMapping) ○ U) identityFunctor
fm-ε =  nat-ε A B FreeMonoidUniveralMapping
--   TMap = λ a  →  solution (FObj U a) a (λ x → x ) ;
--   isNTrans = record {
--        commute =  commute1
--     }
--   } where
--     commute1 :  {a b : Obj B} {f : Hom B a b} → let open ≈-Reasoning B  renaming (_o_ to _*_ ) in
--                ( FMap (identityFunctor {_} {_} {_} {B}) f * solution (FObj U a) a (λ x → x) ) ≈
--                (  solution (FObj U b) b (λ x → x) * FMap (FunctorF A B FreeMonoidUniveralMapping ○ U) f )
--     commute1 {a} {b} {f} = let open ≡-Reasoning in begin
--              morph ((B ≈-Reasoning.o FMap identityFunctor f) (solution (FObj U a) a (λ x → x)))
--         ≡⟨ {!!}  ⟩
--             morph ((B ≈-Reasoning.o solution (FObj U b) b (λ x → x)) (FMap (FunctorF A B FreeMonoidUniveralMapping ○ U) f))
--         ∎

fm-η :  NTrans A A identityFunctor ( U ○ ( FunctorF A B FreeMonoidUniveralMapping) )
fm-η =  record {
TMap = λ a →  λ (x : a) → x :: [] ;
isNTrans =  record {
commute = commute1
}
} where
commute1 :  {a b : Obj A} {f : Hom A a b} → let open ≈-Reasoning A  renaming (_o_ to _*_ ) in
(( FMap (U ○ FunctorF A B FreeMonoidUniveralMapping) f ) * (λ x → x :: []) ) ≈ ( (λ x → x :: []) * (λ z → FMap (identityFunctor {_} {_} {_} {A}) f z ) )
commute1 {a} {b} {f} =  refl   --    λ (x : a ) → f x :: []

-- A = Sets {c}
-- B = Monoids
-- U   underline funcotr
-- F   generator = x :: []
-- Eta          x :: []
-- Epsiron      morph = Φ