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comparison constructible-set.agda @ 14:e11e95d5ddee

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separete constructible set
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 14 May 2019 03:38:26 +0900
parents zf.agda@2df90eb0896c
children 497152f625ee
comparison
equal deleted inserted replaced
11:2df90eb0896c 14:e11e95d5ddee
1 module constructible-set where
2
3 open import Level
4 open import zf
5
6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat )
7
8 open import Relation.Binary.PropositionalEquality
9
10 data Ordinal {n : Level } : (lv : Nat) → Set n where
11 Φ : {lv : Nat} → Ordinal {n} lv
12 T-suc : {lv : Nat} → Ordinal {n} lv → Ordinal lv
13 ℵ_ : (lv : Nat) → Ordinal (Suc lv)
14
15 data _o<_ {n : Level } : {lx ly : Nat} → Ordinal {n} lx → Ordinal {n} ly → Set n where
16 l< : {lx ly : Nat } → {x : Ordinal {n} lx } → {y : Ordinal {n} ly } → lx < ly → x o< y
17 Φ< : {lx : Nat} → {x : Ordinal {n} lx} → Φ {n} {lx} o< T-suc {n} {lx} x
18 s< : {lx : Nat} → {x y : Ordinal {n} lx} → x o< y → T-suc {n} {lx} x o< T-suc {n} {lx} y
19 ℵΦ< : {lx : Nat} → {x : Ordinal {n} (Suc lx) } → Φ {n} {Suc lx} o< (ℵ lx)
20 ℵ< : {lx : Nat} → {x : Ordinal {n} (Suc lx) } → T-suc {n} {Suc lx} x o< (ℵ lx)
21
22 open import Data.Nat.Properties
23 open import Data.Empty
24 open import Relation.Nullary
25
26 open import Relation.Binary
27 open import Relation.Binary.Core
28
29
30 nat< : { x y : Nat } → x ≡ y → x < y → ⊥
31 nat< {Zero} {Zero} refl ()
32 nat< {Suc x} {.(Suc x)} refl (s≤s t) = nat< {x} {x} refl t
33
34 x≤x : { x : Nat } → x ≤ x
35 x≤x {Zero} = z≤n
36 x≤x {Suc x} = s≤s ( x≤x )
37
38 x<>y : { x y : Nat } → x > y → x < y → ⊥
39 x<>y {.(Suc _)} {.(Suc _)} (s≤s lt) (s≤s lt1) = x<>y lt lt1
40
41 triO> : {n : Level } → {lx ly : Nat} {x : Ordinal {n} lx } { y : Ordinal {n} ly } → ly < lx → x o< y → ⊥
42 triO> {n} {lx} {ly} {x} {y} y<x xo<y with <-cmp lx ly
43 triO> {n} {lx} {ly} {x} {y} y<x xo<y | tri< a ¬b ¬c = ¬c y<x
44 triO> {n} {lx} {ly} {x} {y} y<x xo<y | tri≈ ¬a b ¬c = ¬c y<x
45 triO> {n} {lx} {ly} {x} {y} y<x (l< x₁) | tri> ¬a ¬b c = ¬a x₁
46 triO> {n} {lx} {ly} {Φ} {T-suc _} y<x Φ< | tri> ¬a ¬b c = ¬b refl
47 triO> {n} {lx} {ly} {T-suc px} {T-suc py} y<x (s< w) | tri> ¬a ¬b c = triO> y<x w
48 triO> {n} {lx} {ly} {Φ {u}} {ℵ w} y<x ℵΦ< | tri> ¬a ¬b c = ¬b refl
49 triO> {n} {lx} {ly} {(T-suc _)} {ℵ u} y<x ℵ< | tri> ¬a ¬b c = ¬b refl
50
51 trio! : {n : Level } → {lv : Nat} → {x : Ordinal {n} lv } → x o< x → ⊥
52 trio! {n} {lx} {x} (l< y) = nat< refl y
53 trio! {n} {lx} {T-suc y} (s< t) = trio! t
54
55 trio<> : {n : Level } → {lx : Nat} {x : Ordinal {n} lx } { y : Ordinal {n} lx } → y o< x → x o< y → ⊥
56 trio<> {n} {lx} {x} {y} (l< lt) _ = nat< refl lt
57 trio<> {n} {lx} {x} {y} _ (l< lt) = nat< refl lt
58 trio<> {n} {lx} {.(T-suc _)} {.(T-suc _)} (s< s) (s< t) =
59 trio<> s t
60
61 trio<≡ : {n : Level } → {lx : Nat} {x : Ordinal {n} lx } { y : Ordinal {n} lx } → x ≡ y → x o< y → ⊥
62 trio<≡ refl = trio!
63
64 trio>≡ : {n : Level } → {lx : Nat} {x : Ordinal {n} lx } { y : Ordinal {n} lx } → x ≡ y → y o< x → ⊥
65 trio>≡ refl = trio!
66
67 triO : {n : Level } → {lx ly : Nat} → Ordinal {n} lx → Ordinal {n} ly → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
68 triO {n} {lx} {ly} x y = <-cmp lx ly
69
70 triOonSameLevel : {n : Level } → {lx : Nat} → Trichotomous _≡_ ( _o<_ {n} {lx} {lx} )
71 triOonSameLevel {n} {lv} Φ Φ = tri≈ trio! refl trio!
72 triOonSameLevel {n} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ trio! refl trio!
73 triOonSameLevel {n} {lv} Φ (T-suc y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
74 triOonSameLevel {n} {.(Suc lv)} Φ (ℵ lv) = tri< (ℵΦ< {n} {lv} {Φ} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {n} {lv} {Φ} )) )
75 triOonSameLevel {n} {Suc lv} (ℵ lv) Φ = tri> ( λ lt → trio<> lt (ℵΦ< {n} {lv} {Φ} ) ) (λ ()) (ℵΦ< {n} {lv} {Φ} )
76 triOonSameLevel {n} {Suc lv} (ℵ lv) (T-suc y) = tri> ( λ lt → trio<> lt (ℵ< {n} {lv} {y} ) ) (λ ()) (ℵ< {n} {lv} {y} )
77 triOonSameLevel {n} {lv} (T-suc x) Φ = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
78 triOonSameLevel {n} {.(Suc lv)} (T-suc x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
79 triOonSameLevel {n} {lv} (T-suc x) (T-suc y) with triOonSameLevel x y
80 triOonSameLevel {n} {lv} (T-suc x) (T-suc y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) ) ( λ lt → trio<> lt (s< a) )
81 triOonSameLevel {n} {lv} (T-suc x) (T-suc x) | tri≈ ¬a refl ¬c = tri≈ trio! refl trio!
82 triOonSameLevel {n} {lv} (T-suc x) (T-suc y) | tri> ¬a ¬b c = tri> ( λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c)
83
84
85 max : (x y : Nat) → Nat
86 max Zero Zero = Zero
87 max Zero (Suc x) = (Suc x)
88 max (Suc x) Zero = (Suc x)
89 max (Suc x) (Suc y) = Suc ( max x y )
90
91 lvconv : { lx ly : Nat } → lx > ly → lx ≡ (max lx ly)
92 lvconv {Zero} {Zero} ()
93 lvconv {Zero} {Suc ly} ()
94 lvconv {Suc lx} {Zero} (s≤s lt) = refl
95 lvconv {Suc lx} {Suc ly} (s≤s lt) = cong ( λ x → Suc x ) ( lvconv lt )
96
97 olconv : {n : Level } → { lx ly : Nat } → Ordinal {n} ly → lx < ly → Ordinal {n} (max lx ly)
98 olconv {n} {lx} {ly} (Φ {x}) lt = Φ
99 olconv {n} {lx} {ly} (T-suc x) lt = T-suc (olconv x lt)
100 olconv {n} {lx} {Suc lv} (ℵ lv) lt = {!!}
101
102 maxα : {n : Level } → { lx ly : Nat } → Ordinal {n} lx → Ordinal {n} ly → Ordinal {n} (max lx ly)
103 maxα x y with triO x y
104 maxα Φ y | tri< a ¬b ¬c = Φ
105 maxα (T-suc x) y | tri< a ¬b ¬c = olconv x a
106 maxα (ℵ lv) y | tri< a ¬b ¬c = {!!}
107 maxα x y | tri> ¬a ¬b c = {!!}
108 maxα Φ y | tri≈ ¬a b ¬c = Φ
109 maxα (T-suc x) y | tri≈ ¬a b ¬c = T-suc {!!}
110 maxα (ℵ lv) y | tri≈ ¬a b ¬c = {!!}
111
112 -- X' = { x ∈ X | ψ x } ∪ X , Mα = ( ∪ (β < α) Mβ ) '
113
114 data Constructible {n : Level } {lv : Nat} ( α : Ordinal {n} lv ) : Set (suc n) where
115 fsub : ( ψ : Ordinal {n} lv → Set n ) → Constructible α
116 xself : Ordinal {n} lv → Constructible α
117
118 record ConstructibleSet {n : Level } : Set (suc n) where
119 field
120 level : Nat
121 α : Ordinal {n} level
122 constructible : Constructible α
123
124 open ConstructibleSet
125
126 data _c∋_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } →
127 Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where
128 c> : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' }
129 (ta : Constructible {n} {lv} α ) ( tx : Constructible {n} {lv'} α' ) → α' o< α → ta c∋ tx
130 xself-fsub : {lv : Nat} {α : Ordinal {n} lv }
131 (ta : Ordinal {n} lv ) ( ψ : Ordinal {n} lv → Set n ) → _c∋_ {n} {_} {_} {α} {α} (xself ta ) ( fsub ψ)
132 fsub-fsub : {lv lv' : Nat} {α : Ordinal {n} lv }
133 ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n ) →
134 ( ∀ ( x : Ordinal {n} lv ) → ψ x → ψ₁ x ) → _c∋_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁)
135
136 _∋_ : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n
137 a ∋ x = constructible a c∋ constructible x
138
139 data _c≈_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } →
140 Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where
141 crefl : {lv : Nat} {α : Ordinal {n} lv } → _c≈_ {n} {_} {_} {α} {α} (xself α ) (xself α )
142 feq : {lv : Nat} {α : Ordinal {n} lv }
143 → ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n )
144 → (∀ ( x : Ordinal {n} lv ) → ψ x ⇔ ψ₁ x ) → _c≈_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁)
145
146 _≈_ : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n
147 a ≈ x = constructible a c≈ constructible x
148
149 ConstructibleSet→ZF : {n : Level } → ZF {suc n} {n}
150 ConstructibleSet→ZF {n} = record {
151 ZFSet = ConstructibleSet
152 ; _∋_ = _∋_
153 ; _≈_ = _≈_
154 ; ∅ = record { level = Zero ; α = Φ ; constructible = xself Φ }
155 ; _×_ = {!!}
156 ; Union = {!!}
157 ; Power = {!!}
158 ; Select = {!!}
159 ; Replace = {!!}
160 ; infinite = {!!}
161 ; isZF = {!!}
162 }