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library
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 09 Jan 2021 10:18:08 +0900
parents Gutil.agda@c68956f6c3ad
children c209aebeab2a
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254:a5b3061f15ee 255:6d1619d9f880
1 open import Level hiding ( suc ; zero )
2 open import Algebra
3 module Gutil {n m : Level} (G : Group n m ) where
4
5 open import Data.Unit
6 open import Function.Inverse as Inverse using (_↔_; Inverse; _InverseOf_)
7 open import Function
8 open import Data.Nat hiding (_⊔_) -- using (ℕ; suc; zero)
9 open import Relation.Nullary
10 open import Data.Empty
11 open import Data.Product
12 open import Relation.Binary.PropositionalEquality hiding ( [_] )
13
14
15 open Group G
16
17 import Relation.Binary.Reasoning.Setoid as EqReasoning
18
19 gsym = Algebra.Group.sym G
20 grefl = Algebra.Group.refl G
21 gtrans = Algebra.Group.trans G
22
23 lemma3 : ε ≈ ε ⁻¹
24 lemma3 = begin
25 ε ≈⟨ gsym (proj₁ inverse _) ⟩
26 ε ⁻¹ ∙ ε ≈⟨ proj₂ identity _ ⟩
27 ε ⁻¹
28 ∎ where open EqReasoning (Algebra.Group.setoid G)
29
30 lemma6 : {f : Carrier } → ( f ⁻¹ ) ⁻¹ ≈ f
31 lemma6 {f} = begin
32 ( f ⁻¹ ) ⁻¹ ≈⟨ gsym ( proj₁ identity _) ⟩
33 ε ∙ ( f ⁻¹ ) ⁻¹ ≈⟨ ∙-cong (gsym ( proj₂ inverse _ )) grefl ⟩
34 (f ∙ f ⁻¹ ) ∙ ( f ⁻¹ ) ⁻¹ ≈⟨ assoc _ _ _ ⟩
35 f ∙ ( f ⁻¹ ∙ ( f ⁻¹ ) ⁻¹ ) ≈⟨ ∙-cong grefl (proj₂ inverse _) ⟩
36 f ∙ ε ≈⟨ proj₂ identity _ ⟩
37 f
38 ∎ where open EqReasoning (Algebra.Group.setoid G)
39
40 ≡→≈ : {f g : Carrier } → f ≡ g → f ≈ g
41 ≡→≈ refl = grefl
42
43 ---
44 -- to avoid assoc storm, flatten multiplication according to the template
45 --
46
47 data MP : Carrier → Set (Level.suc n) where
48 am : (x : Carrier ) → MP x
49 _o_ : {x y : Carrier } → MP x → MP y → MP ( x ∙ y )
50
51 mpf : {x : Carrier } → (m : MP x ) → Carrier → Carrier
52 mpf (am x) y = x ∙ y
53 mpf (m o m₁) y = mpf m ( mpf m₁ y )
54
55 mp-flatten : {x : Carrier } → (m : MP x ) → Carrier
56 mp-flatten m = mpf m ε
57
58 mpl1 : Carrier → {x : Carrier } → MP x → Carrier
59 mpl1 x (am y) = x ∙ y
60 mpl1 x (y o y1) = mpl1 ( mpl1 x y ) y1
61
62 mpl : {x z : Carrier } → MP x → MP z → Carrier
63 mpl (am x) m = mpl1 x m
64 mpl (m o m1) m2 = mpl m (m1 o m2)
65
66 mp-flattenl : {x : Carrier } → (m : MP x ) → Carrier
67 mp-flattenl m = mpl m (am ε)
68
69 test1 : ( f g : Carrier ) → Carrier
70 test1 f g = mp-flattenl ((am (g ⁻¹) o am (f ⁻¹) ) o ( (am f o am g) o am ((f ∙ g) ⁻¹ )))
71
72 test2 : ( f g : Carrier ) → test1 f g ≡ g ⁻¹ ∙ f ⁻¹ ∙ f ∙ g ∙ (f ∙ g) ⁻¹ ∙ ε
73 test2 f g = _≡_.refl
74
75 test3 : ( f g : Carrier ) → Carrier
76 test3 f g = mp-flatten ((am (g ⁻¹) o am (f ⁻¹) ) o ( (am f o am g) o am ((f ∙ g) ⁻¹ )))
77
78 test4 : ( f g : Carrier ) → test3 f g ≡ g ⁻¹ ∙ (f ⁻¹ ∙ (f ∙ (g ∙ ((f ∙ g) ⁻¹ ∙ ε))))
79 test4 f g = _≡_.refl
80
81
82 ∙-flatten : {x : Carrier } → (m : MP x ) → x ≈ mp-flatten m
83 ∙-flatten {x} (am x) = gsym (proj₂ identity _)
84 ∙-flatten {_} (am x o q) = ∙-cong grefl ( ∙-flatten q )
85 ∙-flatten (_o_ {_} {z} (_o_ {x} {y} p q) r) = lemma9 _ _ _ ( ∙-flatten {x ∙ y } (p o q )) ( ∙-flatten {z} r ) where
86 mp-cong : {p q r : Carrier} → (P : MP p) → q ≈ r → mpf P q ≈ mpf P r
87 mp-cong (am x) q=r = ∙-cong grefl q=r
88 mp-cong (P o P₁) q=r = mp-cong P ( mp-cong P₁ q=r )
89 mp-assoc : {p q r : Carrier} → (P : MP p) → mpf P q ∙ r ≈ mpf P (q ∙ r )
90 mp-assoc (am x) = assoc _ _ _
91 mp-assoc {p} {q} {r} (P o P₁) = begin
92 mpf P (mpf P₁ q) ∙ r ≈⟨ mp-assoc P ⟩
93 mpf P (mpf P₁ q ∙ r) ≈⟨ mp-cong P (mp-assoc P₁) ⟩ mpf P ((mpf P₁) (q ∙ r))
94 ∎ where open EqReasoning (Algebra.Group.setoid G)
95 lemma9 : (x y z : Carrier) → x ∙ y ≈ mpf p (mpf q ε) → z ≈ mpf r ε → x ∙ y ∙ z ≈ mp-flatten ((p o q) o r)
96 lemma9 x y z t s = begin
97 x ∙ y ∙ z ≈⟨ ∙-cong t grefl ⟩
98 mpf p (mpf q ε) ∙ z ≈⟨ mp-assoc p ⟩
99 mpf p (mpf q ε ∙ z) ≈⟨ mp-cong p (mp-assoc q ) ⟩
100 mpf p (mpf q (ε ∙ z)) ≈⟨ mp-cong p (mp-cong q (proj₁ identity _ )) ⟩
101 mpf p (mpf q z) ≈⟨ mp-cong p (mp-cong q s) ⟩
102 mpf p (mpf q (mpf r ε))
103 ∎ where open EqReasoning (Algebra.Group.setoid G)
104
105 grepl : { x y0 y1 z : Carrier } → x ∙ y0 ≈ y1 → x ∙ ( y0 ∙ z ) ≈ y1 ∙ z
106 grepl eq = gtrans (gsym (assoc _ _ _ )) (∙-cong eq grefl )
107
108 grm : { x y0 y1 z : Carrier } → x ∙ y0 ≈ ε → x ∙ ( y0 ∙ z ) ≈ z
109 grm eq = gtrans ( gtrans (gsym (assoc _ _ _ )) (∙-cong eq grefl )) ( proj₁ identity _ )
110
111 -- ∙-flattenl : {x : Carrier } → (m : MP x ) → x ≈ mp-flattenl m
112 -- ∙-flattenl (am x) = gsym (proj₂ identity _)
113 -- ∙-flattenl (q o am x) with ∙-flattenl q -- x₁ ∙ x ≈ mpl q (am x o am ε) , t : x₁ ≈ mpl q (am ε)
114 -- ... | t = {!!}
115 -- ∙-flattenl (q o (x o y )) with ∙-flattenl q
116 -- ... | t = gtrans (gsym (assoc _ _ _ )) {!!}
117
118 lemma5 : (f g : Carrier ) → g ⁻¹ ∙ f ⁻¹ ≈ (f ∙ g) ⁻¹
119 lemma5 f g = begin
120 g ⁻¹ ∙ f ⁻¹ ≈⟨ gsym (proj₂ identity _) ⟩
121 g ⁻¹ ∙ f ⁻¹ ∙ ε ≈⟨ gsym (∙-cong grefl (proj₂ inverse _ )) ⟩
122 g ⁻¹ ∙ f ⁻¹ ∙ ( (f ∙ g) ∙ (f ∙ g) ⁻¹ ) ≈⟨ ∙-flatten ((am (g ⁻¹) o am (f ⁻¹) ) o ( (am f o am g) o am ((f ∙ g) ⁻¹ ))) ⟩
123 g ⁻¹ ∙ (f ⁻¹ ∙ (f ∙ (g ∙ ((f ∙ g) ⁻¹ ∙ ε)))) ≈⟨ ∙-cong grefl (gsym (assoc _ _ _ )) ⟩
124 g ⁻¹ ∙ ((f ⁻¹ ∙ f) ∙ (g ∙ ((f ∙ g) ⁻¹ ∙ ε))) ≈⟨ ∙-cong grefl (gtrans (∙-cong (proj₁ inverse _ ) grefl) (proj₁ identity _)) ⟩
125 g ⁻¹ ∙ (g ∙ ((f ∙ g) ⁻¹ ∙ ε)) ≈⟨ gsym (assoc _ _ _) ⟩
126 (g ⁻¹ ∙ g ) ∙ ((f ∙ g) ⁻¹ ∙ ε) ≈⟨ gtrans (∙-cong (proj₁ inverse _ ) grefl) (proj₁ identity _) ⟩
127 (f ∙ g) ⁻¹ ∙ ε ≈⟨ proj₂ identity _ ⟩
128 (f ∙ g) ⁻¹
129 ∎ where open EqReasoning (Algebra.Group.setoid G)
130