Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 36:4d64509067d0
transitive
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Thu, 23 May 2019 02:32:02 +0900 |
parents | c9ad0d97ce41 |
children | f10ceee99d00 |
rev | line source |
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16 | 1 open import Level |
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2 module ordinal-definable where |
3 | 3 |
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4 open import zf |
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5 open import ordinal |
3 | 6 |
23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
3 | 8 |
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9 open import Relation.Binary.PropositionalEquality |
3 | 10 |
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11 open import Data.Nat.Properties |
6 | 12 open import Data.Empty |
13 open import Relation.Nullary | |
14 | |
15 open import Relation.Binary | |
16 open import Relation.Binary.Core | |
17 | |
27 | 18 -- Ordinal Definable Set |
11 | 19 |
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20 record OD {n : Level} : Set (suc n) where |
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21 field |
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22 def : (x : Ordinal {n} ) → Set n |
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23 |
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24 open OD |
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25 open import Data.Unit |
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26 |
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27 postulate |
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28 od→ord : {n : Level} → OD {n} → Ordinal {n} |
36 | 29 ord→od : {n : Level} → Ordinal {n} → OD {n} |
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30 |
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31 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n |
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32 _∋_ {n} a x = def a ( od→ord x ) |
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33 |
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34 _c<_ : { n : Level } → ( a x : OD {n} ) → Set n |
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35 x c< a = a ∋ x |
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36 |
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37 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) |
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38 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) |
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39 |
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40 postulate |
36 | 41 c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord y |
42 o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od y | |
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43 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x |
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44 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x |
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45 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} |
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46 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → ψ x c< sup-od ψ |
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47 |
28 | 48 HOD = OD |
49 | |
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50 od∅ : {n : Level} → HOD {n} |
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51 od∅ {n} = record { def = λ _ → Lift n ⊥ } |
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52 |
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53 ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) |
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54 ∅1 {n} x (lift ()) |
28 | 55 |
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56 ∅3 : {n : Level} → ( x : Ordinal {n}) → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
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57 ∅3 {n} x = TransFinite {n} c1 c2 c3 x where |
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58 c0 : Nat → Ordinal {n} → Set n |
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59 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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60 c1 : ∀ (lx : Nat ) → c0 lx (record { lv = Suc lx ; ord = ℵ lx } ) |
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61 c1 lx not with not ( record { lv = lx ; ord = Φ lx } ) |
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62 ... | t with t (case1 ≤-refl ) |
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63 c1 lx not | t | () |
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64 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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65 c2 Zero not = refl |
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66 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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67 ... | t with t (case1 ≤-refl ) |
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68 c2 (Suc lx) not | t | () |
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69 c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ }) |
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70 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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71 ... | t with t (case2 Φ< ) |
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72 c3 lx (Φ .lx) d not | t | () |
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73 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
34 | 74 ... | t with t (case2 (s< s<refl ) ) |
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75 c3 lx (OSuc .lx x₁) d not | t | () |
34 | 76 c3 (Suc lx) (ℵ lx) d not with not ( record { lv = Suc lx ; ord = OSuc (Suc lx) (Φ (Suc lx)) } ) |
77 ... | t with t (case2 (s< (ℵΦ< {_} {_} {Φ (Suc lx)}))) | |
78 c3 .(Suc lx) (ℵ lx) d not | t | () | |
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79 |
36 | 80 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x |
81 def-subst df refl refl = df | |
82 | |
83 transitive : {n : Level } { x y z : OD {n} } → y ∋ x → z ∋ y → z ∋ x | |
84 transitive {n} {x} {y} {z} x∋y z∋y with ordtrans ( c<→o< {n} {x} {y} x∋y ) ( c<→o< {n} {y} {z} z∋y ) | |
85 ... | t = lemma0 (lemma t) where | |
86 lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) | |
87 lemma xo<z = o<→c< xo<z | |
88 lemma0 : def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) → def z (od→ord x) | |
89 lemma0 dz = def-subst {n} { ord→od ( od→ord z )} { od→ord ( ord→od ( od→ord x))} dz (oiso) (diso) | |
90 | |
91 open Ordinal | |
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92 |
28 | 93 HOD→ZF : {n : Level} → ZF {suc n} {suc n} |
94 HOD→ZF {n} = record { | |
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95 ZFSet = OD {n} |
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96 ; _∋_ = λ a x → Lift (suc n) ( a ∋ x ) |
28 | 97 ; _≈_ = _≡_ |
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98 ; ∅ = od∅ |
28 | 99 ; _,_ = _,_ |
100 ; Union = Union | |
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101 ; Power = Power |
28 | 102 ; Select = Select |
103 ; Replace = Replace | |
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104 ; infinite = record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } } |
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105 ; isZF = isZF |
28 | 106 } where |
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107 Replace : OD {n} → (OD {n} → OD {n} ) → OD {n} |
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108 Replace X ψ = sup-od ψ |
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109 Select : OD {n} → (OD {n} → Set (suc n) ) → OD {n} |
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110 Select X ψ = record { def = λ x → select ( ord→od x ) } where |
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111 select : OD {n} → Set n |
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112 select x with ψ x |
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113 ... | t = Lift n ⊤ |
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114 _,_ : OD {n} → OD {n} → OD {n} |
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115 x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) } |
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116 Union : OD {n} → OD {n} |
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117 Union x = record { def = λ y → {z : Ordinal {n}} → def x z → def (ord→od z) y } |
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118 Power : OD {n} → OD {n} |
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119 Power x = record { def = λ y → (z : Ordinal {n} ) → ( def x y ∧ def (ord→od z) y ) } |
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120 ZFSet = OD {n} |
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121 _∈_ : ( A B : ZFSet ) → Set n |
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122 A ∈ B = B ∋ A |
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123 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set n |
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124 _⊆_ A B {x} = A ∋ x → B ∋ x |
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125 _∩_ : ( A B : ZFSet ) → ZFSet |
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126 A ∩ B = Select (A , B) ( λ x → (Lift (suc n) ( A ∋ x )) ∧ (Lift (suc n) ( B ∋ x ) )) |
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127 _∪_ : ( A B : ZFSet ) → ZFSet |
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128 A ∪ B = Select (A , B) ( λ x → (Lift (suc n) ( A ∋ x )) ∨ (Lift (suc n) ( B ∋ x ) )) |
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129 infixr 200 _∈_ |
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130 infixr 230 _∩_ _∪_ |
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131 infixr 220 _⊆_ |
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132 isZF : IsZF (OD {n}) (λ a x → Lift (suc n) ( a ∋ x )) _≡_ od∅ _,_ Union Power Select Replace (record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } }) |
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133 isZF = record { |
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134 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } |
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135 ; pair = pair |
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136 ; union→ = {!!} |
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137 ; union← = {!!} |
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138 ; empty = empty |
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139 ; power→ = {!!} |
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140 ; power← = {!!} |
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141 ; extentionality = {!!} |
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142 ; minimul = minimul |
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143 ; regularity = {!!} |
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144 ; infinity∅ = {!!} |
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145 ; infinity = {!!} |
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146 ; selection = {!!} |
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147 ; replacement = {!!} |
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148 } where |
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149 open _∧_ |
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150 pair : (A B : OD {n} ) → Lift (suc n) ((A , B) ∋ A) ∧ Lift (suc n) ((A , B) ∋ B) |
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151 proj1 (pair A B ) = lift ( case1 refl ) |
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152 proj2 (pair A B ) = lift ( case2 refl ) |
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153 empty : (x : OD {n} ) → ¬ Lift (suc n) (od∅ ∋ x) |
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154 empty x (lift (lift ())) |
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155 union→ : (X x y : OD {n} ) → Lift (suc n) (X ∋ x) → Lift (suc n) (x ∋ y) → Lift (suc n) (Union X ∋ y) |
36 | 156 union→ X x y (lift X∋x) (lift x∋y) = lift {!!} where |
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157 lemma : {z : Ordinal {n} } → def X z → z ≡ od→ord y |
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158 lemma {z} X∋z = {!!} |
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159 minimul : OD {n} → ( OD {n} ∧ OD {n} ) |
36 | 160 minimul x = record { proj1 = record { def = {!!} } ; proj2 = record { def = {!!} } } |
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161 regularity : (x : OD) → ¬ x ≡ od∅ → |
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162 Lift (suc n) (x ∋ proj1 (minimul x)) ∧ |
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163 (Select (proj1 (minimul x ) , x) (λ x₁ → Lift (suc n) (proj1 ( minimul x ) ∋ x₁) ∧ Lift (suc n) (x ∋ x₁)) ≡ od∅) |
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164 proj1 ( regularity x non ) = lift lemma where |
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165 lemma : def x (od→ord (proj1 (minimul x))) |
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166 lemma = {!!} |
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167 proj2 ( regularity x non ) = {!!} |