annotate Solvable.agda @ 10:04f40fc4eb69

...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 17 Aug 2020 14:02:04 +0900
parents 6bbd861e9ae8
children 9dae7ef74342
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
0
dc677bac3c54 Permutation group
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 open import Level hiding ( suc ; zero )
dc677bac3c54 Permutation group
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 open import Algebra
4
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
3 module Solvable {n m : Level} (G : Group n m ) where
0
dc677bac3c54 Permutation group
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4
4
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
5 open import Data.Unit
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
6 open import Function.Inverse as Inverse using (_↔_; Inverse; _InverseOf_)
6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
7 open import Function
7
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 6
diff changeset
8 open import Data.Nat hiding (_⊔_) -- using (ℕ; suc; zero)
4
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
9 open import Relation.Nullary
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
10 open import Data.Empty
5
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
11 open import Data.Product
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
12 open import Relation.Binary.PropositionalEquality hiding ( [_] )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
13
0
dc677bac3c54 Permutation group
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14
4
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
15 open Group G
0
dc677bac3c54 Permutation group
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16
4
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
17 [_,_] : Carrier → Carrier → Carrier
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
18 [ g , h ] = g ⁻¹ ∙ h ⁻¹ ∙ g ∙ h
0
dc677bac3c54 Permutation group
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19
5
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
20 data Commutator (P : Carrier → Set (Level.suc n ⊔ m)) : (f : Carrier) → Set (Level.suc n ⊔ m) where
6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
21 uni : Commutator P ε
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
22 comm : {g h : Carrier} → P g → P h → Commutator P [ g , h ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
23 gen : {f g : Carrier} → Commutator P f → Commutator P g → Commutator P ( f ∙ g )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
24 ccong : {f g : Carrier} → f ≈ g → Commutator P f → Commutator P g
5
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
25
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
26 deriving : ( i : ℕ ) → Carrier → Set (Level.suc n ⊔ m)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
27 deriving 0 x = Lift (Level.suc n ⊔ m) ⊤
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
28 deriving (suc i) x = Commutator (deriving i) x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
29
4
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
30 record Solvable : Set (Level.suc n ⊔ m) where
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
31 field
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
32 dervied-length : ℕ
121213cfc85a add Solvable
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
33 end : (x : Carrier ) → deriving dervied-length x → x ≈ ε
6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
34
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
35 -- deriving stage is closed on multiplication and inversion
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
36
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
37 import Relation.Binary.Reasoning.Setoid as EqReasoning
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
38
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
39 gsym = Algebra.Group.sym G
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
40 grefl = Algebra.Group.refl G
8
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
41
6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
42 lemma3 : ε ≈ ε ⁻¹
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
43 lemma3 = begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
44 ε ≈⟨ gsym (proj₁ inverse _) ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
45 ε ⁻¹ ∙ ε ≈⟨ proj₂ identity _ ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
46 ε ⁻¹
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
47 ∎ where open EqReasoning (Algebra.Group.setoid G)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
48
8
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
49 lemma6 : {f : Carrier } → ( f ⁻¹ ) ⁻¹ ≈ f
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
50 lemma6 {f} = begin
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
51 ( f ⁻¹ ) ⁻¹ ≈⟨ gsym ( proj₁ identity _) ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
52 ε ∙ ( f ⁻¹ ) ⁻¹ ≈⟨ ∙-cong (gsym ( proj₂ inverse _ )) grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
53 (f ∙ f ⁻¹ ) ∙ ( f ⁻¹ ) ⁻¹ ≈⟨ assoc _ _ _ ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
54 f ∙ ( f ⁻¹ ∙ ( f ⁻¹ ) ⁻¹ ) ≈⟨ ∙-cong grefl (proj₂ inverse _) ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
55 f ∙ ε ≈⟨ proj₂ identity _ ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
56 f
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
57 ∎ where open EqReasoning (Algebra.Group.setoid G)
6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
58
9
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
59 data MP : Carrier → Set (Level.suc n) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
60 am : (x : Carrier ) → MP x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
61 _o_ : {x y : Carrier } → MP x → MP y → MP ( x ∙ y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
62
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
63 mpf : {x : Carrier } → (m : MP x ) → Carrier → Carrier
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
64 mpf {x} (am x) y = x ∙ y
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
65 mpf (m o m₁) y = mpf m ( mpf m₁ y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
66
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
67 mp-flatten : {x : Carrier } → (m : MP x ) → Carrier
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
68 mp-flatten {x} m = mpf {x} m ε
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
69
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
70 ∙-flatten : {x : Carrier } → (m : MP x ) → x ≈ mp-flatten m
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
71 ∙-flatten {x} (am x) = gsym (proj₂ identity _)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
72 ∙-flatten {_} (am x o q) = ∙-cong grefl ( ∙-flatten q )
10
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
73 ∙-flatten (_o_ {_} {z} (_o_ {x} {y} p q) r) with ∙-flatten (p o q ) -- t : x ∙ y ≈ mpf p (mpf q ε)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
74 ... | t = begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
75 x ∙ y ∙ z ≈⟨ ∙-cong t grefl ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
76 mpf p (mpf q ε) ∙ z ≈⟨ {!!} ⟩
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
77 mpf p (mpf q (mpf r ε))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 9
diff changeset
78 ∎ where open EqReasoning (Algebra.Group.setoid G)
9
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
79
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
80
8
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
81 lemma5 : (f g : Carrier ) → g ⁻¹ ∙ f ⁻¹ ≈ (f ∙ g) ⁻¹
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
82 lemma5 f g = begin
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
83 g ⁻¹ ∙ f ⁻¹ ≈⟨ gsym (proj₂ identity _) ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
84 g ⁻¹ ∙ f ⁻¹ ∙ ε ≈⟨ gsym (∙-cong grefl (proj₂ inverse _ )) ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
85 g ⁻¹ ∙ f ⁻¹ ∙ ( (f ∙ g) ∙ (f ∙ g) ⁻¹ ) ≈⟨ assoc _ _ _ ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
86 g ⁻¹ ∙ (f ⁻¹ ∙ (f ∙ g ∙ (f ∙ g) ⁻¹)) ≈⟨ ∙-cong grefl (gsym (assoc _ _ _)) ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
87 g ⁻¹ ∙ (f ⁻¹ ∙ (f ∙ g) ∙ (f ∙ g) ⁻¹) ≈⟨ gsym ( assoc _ _ _) ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
88 g ⁻¹ ∙ (f ⁻¹ ∙ (f ∙ g)) ∙ (f ∙ g) ⁻¹ ≈⟨ ∙-cong (gsym (assoc _ _ _ )) grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
89 (g ⁻¹ ∙ f ⁻¹) ∙ (f ∙ g) ∙ (f ∙ g) ⁻¹ ≈⟨ ∙-cong (assoc _ _ _ ) grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
90 (g ⁻¹ ∙ (f ⁻¹ ∙ (f ∙ g))) ∙ (f ∙ g) ⁻¹ ≈⟨ ∙-cong (∙-cong grefl (gsym (assoc _ _ _ )) ) grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
91 (g ⁻¹ ∙ ((f ⁻¹ ∙ f) ∙ g)) ∙ (f ∙ g) ⁻¹ ≈⟨ ∙-cong (gsym (assoc _ _ _ )) grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
92 (g ⁻¹ ∙ (f ⁻¹ ∙ f) ∙ g) ∙ (f ∙ g) ⁻¹ ≈⟨ ∙-cong (assoc _ _ _) grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
93 g ⁻¹ ∙ ((f ⁻¹ ∙ f) ∙ g) ∙ (f ∙ g) ⁻¹ ≈⟨ ∙-cong (∙-cong grefl (∙-cong (proj₁ inverse _ ) grefl )) grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
94 g ⁻¹ ∙ (ε ∙ g) ∙ (f ∙ g) ⁻¹ ≈⟨ ∙-cong (∙-cong grefl ( proj₁ identity _) ) grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
95 g ⁻¹ ∙ g ∙ (f ∙ g) ⁻¹ ≈⟨ ∙-cong (proj₁ inverse _) grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
96 ε ∙ (f ∙ g) ⁻¹ ≈⟨ proj₁ identity _ ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
97 (f ∙ g) ⁻¹
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
98 ∎ where open EqReasoning (Algebra.Group.setoid G)
6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
99
8
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
100 lemma4 : (g h : Carrier ) → [ h , g ] ≈ [ g , h ] ⁻¹
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
101 lemma4 g h = begin
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
102 [ h , g ] ≈⟨ grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
103 (h ⁻¹ ∙ g ⁻¹ ∙ h ) ∙ g ≈⟨ assoc _ _ _ ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
104 h ⁻¹ ∙ g ⁻¹ ∙ (h ∙ g) ≈⟨ ∙-cong grefl (gsym (∙-cong lemma6 lemma6)) ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
105 h ⁻¹ ∙ g ⁻¹ ∙ ((h ⁻¹) ⁻¹ ∙ (g ⁻¹) ⁻¹) ≈⟨ ∙-cong grefl (lemma5 _ _ ) ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
106 h ⁻¹ ∙ g ⁻¹ ∙ (g ⁻¹ ∙ h ⁻¹) ⁻¹ ≈⟨ assoc _ _ _ ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
107 h ⁻¹ ∙ (g ⁻¹ ∙ (g ⁻¹ ∙ h ⁻¹) ⁻¹) ≈⟨ ∙-cong grefl (lemma5 (g ⁻¹ ∙ h ⁻¹ ) g ) ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
108 h ⁻¹ ∙ (g ⁻¹ ∙ h ⁻¹ ∙ g) ⁻¹ ≈⟨ lemma5 (g ⁻¹ ∙ h ⁻¹ ∙ g) h ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
109 (g ⁻¹ ∙ h ⁻¹ ∙ g ∙ h) ⁻¹ ≈⟨ grefl ⟩
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
110 [ g , h ] ⁻¹
4e275f918e63 deriving-inv done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 7
diff changeset
111 ∎ where open EqReasoning (Algebra.Group.setoid G)
6
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
112
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
113 deriving-mul : { i : ℕ } → { x y : Carrier } → deriving i x → deriving i y → deriving i ( x ∙ y )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
114 deriving-mul {zero} {x} {y} _ _ = lift tt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
115 deriving-mul {suc i} {x} {y} ix iy = gen ix iy
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
116
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
117 deriving-inv : { i : ℕ } → { x : Carrier } → deriving i x → deriving i ( x ⁻¹ )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
118 deriving-inv {zero} {x} (lift tt) = lift tt
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
119 deriving-inv {suc i} {ε} uni = ccong lemma3 uni
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
120 deriving-inv {suc i} {_} (comm x x₁ ) = ccong (lemma4 _ _) (comm x₁ x) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
121 deriving-inv {suc i} {_} (gen x x₁ ) = ccong (lemma5 _ _ ) ( gen (deriving-inv x₁) (deriving-inv x)) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
122 deriving-inv {suc i} {x} (ccong eq ix ) = ccong (⁻¹-cong eq) ( deriving-inv ix )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 5
diff changeset
123