Mercurial > hg > Members > kono > Proof > category

annotate comparison-functor.agda @ 98:b0ba34a27783

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
generated version of comparison functor
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 29 Jul 2013 16:49:11 +0900
parents
children bd542a1caf08
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
98
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 -- -- -- -- -- -- -- --
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 -- Comparison Functor of Kelisli Category
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 -- defines U_K and F_K as a resolution of Monad
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4 -- checks Adjointness
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 --
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp>
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7 -- -- -- -- -- -- -- --
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 open import Category -- https://github.com/konn/category-agda
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 open import Level
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 --open import Category.HomReasoning
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 open import HomReasoning
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13 open import cat-utility
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14 open import Category.Cat
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15 open import Relation.Binary.Core
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17 module comparison-functor
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18 { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19 { T : Functor A A }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
20 { η : NTrans A A identityFunctor T }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21 { μ : NTrans A A (T ○ T) T }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 { M' : Monad A T η μ }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
23 { K' : Kleisli A T η μ M' }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
24 {c₁' c₂' ℓ' : Level} ( B : Category c₁' c₂' ℓ' )
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
25 { U_K : Functor B A } { F_K : Functor A B }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26 { η_K : NTrans A A identityFunctor ( U_K ○ F_K ) }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27 { ε_K : NTrans B B ( F_K ○ U_K ) identityFunctor }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28 { μ_K' : NTrans A A (( U_K ○ F_K ) ○ ( U_K ○ F_K )) ( U_K ○ F_K ) }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29 ( AdjK : Adjunction A B U_K F_K η_K ε_K )
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 ( RK : MResolution A B T U_K F_K {η_K} {ε_K} {μ_K'} AdjK )
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
31 where
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
32
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
33 open import adj-monad
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
34
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
35 μ_K : NTrans A A (( U_K ○ F_K ) ○ ( U_K ○ F_K )) ( U_K ○ F_K )
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
36 μ_K = UεF A B U_K F_K ε_K
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
37
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
38 M : Monad A (U_K ○ F_K ) η_K μ_K
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
39 -- M = Adj2Monad A B {U_K} {F_K} {η_K} {ε_K} AdjK
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
40 M = Adj2Monad A B {U_K} {F_K} {η_K} {ε_K} AdjK
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
41
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
42 K : Kleisli A (U_K ○ F_K ) η_K μ_K M
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
43 K = record {}
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
44
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
45 open import nat {c₁} {c₂} {ℓ} {A} { U_K ○ F_K } { η_K } { μ_K } { M } { K }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
46
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
47 open Functor
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
48 open NTrans
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
49 open Kleisli
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
50 open KleisliHom
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
51 open Adjunction
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
52 open MResolution
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
53
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
54 kfmap : {a b : Obj A} (f : KHom a b) -> Hom B (FObj F_K a) (FObj F_K b)
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
55 kfmap {_} {b} f = B [ TMap ε_K (FObj F_K b) o FMap F_K (KMap f) ]
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
56
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
57 K_T : Functor KleisliCategory B
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
58 K_T = record {
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
59 FObj = FObj F_K
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
60 ; FMap = kfmap
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
61 ; isFunctor = record
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
62 { ≈-cong = ≈-cong
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
63 ; identity = identity
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
64 ; distr = distr1
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
65 }
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
66 } where
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
67 identity : {a : Obj A} → B [ kfmap (K-id {a}) ≈ id1 B (FObj F_K a) ]
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
68 identity {a} = let open ≈-Reasoning (B) in
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
69 begin
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
70 kfmap (K-id {a})
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
71 ≈⟨⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
72 TMap ε_K (FObj F_K a) o FMap F_K (KMap (K-id {a}))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
73 ≈⟨⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
74 TMap ε_K (FObj F_K a) o FMap F_K (TMap η_K a)
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
75 ≈⟨ IsAdjunction.adjoint2 (isAdjunction AdjK) ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
76 id1 B (FObj F_K a)
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
77 ∎
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
78 ≈-cong : {a b : Obj A} -> {f g : KHom a b} → A [ KMap f ≈ KMap g ] → B [ kfmap f ≈ kfmap g ]
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
79 ≈-cong {a} {b} {f} {g} f≈g = let open ≈-Reasoning (B) in
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
80 begin
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
81 kfmap f
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
82 ≈⟨⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
83 TMap ε_K (FObj F_K b) o FMap F_K (KMap f)
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
84 ≈⟨ cdr ( fcong F_K f≈g) ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
85 TMap ε_K (FObj F_K b) o FMap F_K (KMap g)
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
86 ≈⟨⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
87 kfmap g
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
88 ∎
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
89 distr1 : {a b c : Obj A} {f : KHom a b} {g : KHom b c} → B [ kfmap (g * f) ≈ (B [ kfmap g o kfmap f ] )]
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
90 distr1 {a} {b} {c} {f} {g} = let open ≈-Reasoning (B) in
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
91 begin
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
92 kfmap (g * f)
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
93 ≈⟨⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
94 TMap ε_K (FObj F_K c) o FMap F_K (KMap (g * f))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
95 ≈⟨⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
96 TMap ε_K (FObj F_K c) o FMap F_K (A [ TMap μ_K c o A [ FMap ( U_K ○ F_K ) (KMap g) o KMap f ] ] )
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
97 ≈⟨ cdr ( distr F_K ) ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
98 TMap ε_K (FObj F_K c) o ( FMap F_K (TMap μ_K c) o ( FMap F_K (A [ FMap ( U_K ○ F_K ) (KMap g) o KMap f ])))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
99 ≈⟨ cdr (cdr ( distr F_K )) ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
100 TMap ε_K (FObj F_K c) o ( FMap F_K (TMap μ_K c) o (( FMap F_K (FMap ( U_K ○ F_K ) (KMap g))) o (FMap F_K (KMap f))))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
101 ≈⟨ cdr assoc ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
102 TMap ε_K (FObj F_K c) o ((( FMap F_K (TMap μ_K c) o ( FMap F_K (FMap (U_K ○ F_K) (KMap g))))) o (FMap F_K (KMap f)))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
103 ≈⟨⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
104 TMap ε_K (FObj F_K c) o (( FMap F_K ( FMap U_K ( TMap ε_K ( FObj F_K c ) )) o
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
105 ( FMap F_K (FMap (U_K ○ F_K) (KMap g)))) o (FMap F_K (KMap f)))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
106 ≈⟨ sym (cdr assoc) ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
107 TMap ε_K (FObj F_K c) o (( FMap F_K ( FMap U_K ( TMap ε_K ( FObj F_K c ) ))) o
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
108 (( FMap F_K (FMap (U_K ○ F_K) (KMap g))) o (FMap F_K (KMap f))))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
109 ≈⟨ assoc ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
110 (TMap ε_K (FObj F_K c) o ( FMap F_K ( FMap U_K ( TMap ε_K ( FObj F_K c ) )))) o
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
111 (( FMap F_K (FMap (U_K ○ F_K) (KMap g))) o (FMap F_K (KMap f)))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
112 ≈⟨ car (sym (nat ε_K)) ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
113 (TMap ε_K (FObj F_K c) o ( TMap ε_K (FObj (F_K ○ U_K) (FObj F_K c)))) o
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
114 (( FMap F_K (FMap (U_K ○ F_K) (KMap g))) o (FMap F_K (KMap f)))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
115 ≈⟨ sym assoc ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
116 TMap ε_K (FObj F_K c) o (( TMap ε_K (FObj (F_K ○ U_K) (FObj F_K c))) o
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
117 ((( FMap F_K (FMap (U_K ○ F_K) (KMap g)))) o (FMap F_K (KMap f))))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
118 ≈⟨ cdr assoc ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
119 TMap ε_K (FObj F_K c) o ((( TMap ε_K (FObj (F_K ○ U_K) (FObj F_K c))) o
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
120 (( FMap F_K (FMap (U_K ○ F_K) (KMap g))))) o (FMap F_K (KMap f)))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
121 ≈⟨ cdr ( car (
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
122 begin
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
123 TMap ε_K (FObj (F_K ○ U_K) (FObj F_K c)) o ((FMap F_K (FMap (U_K ○ F_K) (KMap g))))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
124 ≈⟨⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
125 TMap ε_K (FObj (F_K ○ U_K) (FObj F_K c)) o (FMap (F_K ○ U_K) (FMap F_K (KMap g)))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
126 ≈⟨ sym (nat ε_K) ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
127 ( FMap F_K (KMap g)) o (TMap ε_K (FObj F_K b))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
128 ∎
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
129 )) ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
130 TMap ε_K (FObj F_K c) o ((( FMap F_K (KMap g)) o (TMap ε_K (FObj F_K b))) o FMap F_K (KMap f))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
131 ≈⟨ cdr (sym assoc) ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
132 TMap ε_K (FObj F_K c) o (( FMap F_K (KMap g)) o (TMap ε_K (FObj F_K b) o FMap F_K (KMap f)))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
133 ≈⟨ assoc ⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
134 (TMap ε_K (FObj F_K c) o FMap F_K (KMap g)) o (TMap ε_K (FObj F_K b) o FMap F_K (KMap f))
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
135 ≈⟨⟩
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
136 kfmap g o kfmap f
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
137 ∎
b0ba34a27783 generated version of comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
138