Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 257:53b7acd63481
move product to OD
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Fri, 30 Aug 2019 15:37:04 +0900 |
| parents | 2ea2a19f9cd6 |
| children | 63df67b6281c |
| rev | line source |
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| 16 | 1 open import Level |
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2 open import Ordinals |
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3 module OD {n : Level } (O : Ordinals {n} ) where |
| 3 | 4 |
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5 open import zf |
| 23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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7 open import Relation.Binary.PropositionalEquality |
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8 open import Data.Nat.Properties |
| 6 | 9 open import Data.Empty |
| 10 open import Relation.Nullary | |
| 11 open import Relation.Binary | |
| 12 open import Relation.Binary.Core | |
| 13 | |
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14 open import logic |
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15 open import nat |
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16 |
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17 open inOrdinal O |
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18 |
| 27 | 19 -- Ordinal Definable Set |
| 11 | 20 |
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21 record OD : Set (suc n ) where |
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22 field |
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23 def : (x : Ordinal ) → Set n |
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24 |
| 141 | 25 open OD |
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26 |
| 120 | 27 open _∧_ |
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28 open _∨_ |
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29 open Bool |
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30 |
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31 record _==_ ( a b : OD ) : Set n where |
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32 field |
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33 eq→ : ∀ { x : Ordinal } → def a x → def b x |
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34 eq← : ∀ { x : Ordinal } → def b x → def a x |
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35 |
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36 id : {A : Set n} → A → A |
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37 id x = x |
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38 |
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39 eq-refl : { x : OD } → x == x |
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40 eq-refl {x} = record { eq→ = id ; eq← = id } |
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41 |
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42 open _==_ |
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43 |
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44 eq-sym : { x y : OD } → x == y → y == x |
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45 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } |
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46 |
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47 eq-trans : { x y z : OD } → x == y → y == z → x == z |
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48 eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y t) ; eq← = λ t → eq← x=y ( eq← y=z t) } |
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49 |
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50 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y |
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51 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m |
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52 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m |
| 120 | 53 |
| 179 | 54 -- Ordinal in OD ( and ZFSet ) Transitive Set |
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55 Ord : ( a : Ordinal ) → OD |
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56 Ord a = record { def = λ y → y o< a } |
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57 |
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58 od∅ : OD |
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59 od∅ = Ord o∅ |
| 40 | 60 |
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61 postulate |
| 141 | 62 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) |
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63 od→ord : OD → Ordinal |
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64 ord→od : Ordinal → OD |
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65 c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y |
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66 oiso : {x : OD } → ord→od ( od→ord x ) ≡ x |
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67 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x |
| 150 | 68 -- we should prove this in agda, but simply put here |
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69 ==→o≡ : { x y : OD } → (x == y) → x ≡ y |
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70 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set |
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71 -- o<→c< : {n : Level} {x y : Ordinal } → x o< y → def (ord→od y) x |
| 159 | 72 -- ord→od x ≡ Ord x results the same |
| 100 | 73 -- supermum as Replacement Axiom |
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74 sup-o : ( Ordinal → Ordinal ) → Ordinal |
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75 sup-o< : { ψ : Ordinal → Ordinal } → ∀ {x : Ordinal } → ψ x o< sup-o ψ |
| 111 | 76 -- contra-position of mimimulity of supermum required in Power Set Axiom |
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77 -- sup-x : {n : Level } → ( Ordinal → Ordinal ) → Ordinal |
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78 -- sup-lb : {n : Level } → { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) |
| 183 | 79 -- mimimul and x∋minimul is an Axiom of choice |
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80 minimul : (x : OD ) → ¬ (x == od∅ )→ OD |
| 117 | 81 -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) |
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82 x∋minimul : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) |
| 187 | 83 -- minimulity (may proved by ε-induction ) |
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84 minimul-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) ) |
| 123 | 85 |
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86 _∋_ : ( a x : OD ) → Set n |
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87 _∋_ a x = def a ( od→ord x ) |
| 95 | 88 |
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89 _c<_ : ( x a : OD ) → Set n |
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90 x c< a = a ∋ x |
| 103 | 91 |
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92 _c≤_ : OD → OD → Set (suc n) |
| 95 | 93 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) |
| 94 | |
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95 cseq : {n : Level} → OD → OD |
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96 cseq x = record { def = λ y → def x (osuc y) } where |
| 113 | 97 |
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98 def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x |
| 95 | 99 def-subst df refl refl = df |
| 100 | |
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101 sup-od : ( OD → OD ) → OD |
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102 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) |
| 95 | 103 |
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104 sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x )) |
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105 sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )} |
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106 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where |
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107 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x))) |
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108 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) ) |
| 28 | 109 |
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110 otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y |
| 187 | 111 otrans x<a y<x = ordtrans y<x x<a |
| 123 | 112 |
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113 def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X |
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114 def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso |
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115 |
| 51 | 116 -- avoiding lv != Zero error |
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117 orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 118 orefl refl = refl |
| 119 | |
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120 ==-iso : { x y : OD } → ord→od (od→ord x) == ord→od (od→ord y) → x == y |
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121 ==-iso {x} {y} eq = record { |
| 51 | 122 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; |
| 123 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
| 124 where | |
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125 lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z |
| 51 | 126 lemma {x} {z} d = def-subst d oiso refl |
| 127 | |
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128 =-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y) |
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129 =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso) |
| 57 | 130 |
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131 ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y |
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132 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
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133 lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy) |
| 51 | 134 lemma ox ox refl = eq-refl |
| 135 | |
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136 o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y |
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137 o≡→== {x} {.x} refl = eq-refl |
| 51 | 138 |
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139 c≤-refl : {n : Level} → ( x : OD ) → x c≤ x |
| 51 | 140 c≤-refl x = case1 refl |
| 141 | |
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142 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
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143 o∅≡od∅ = ==→o≡ lemma where |
| 150 | 144 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x |
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145 lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (def-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso |
| 150 | 146 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x |
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147 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
| 150 | 148 lemma : ord→od o∅ == od∅ |
| 149 lemma = record { eq→ = lemma0 ; eq← = lemma1 } | |
| 150 | |
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151 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
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152 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
| 80 | 153 |
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154 ∅0 : record { def = λ x → Lift n ⊥ } == od∅ |
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155 eq→ ∅0 {w} (lift ()) |
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156 eq← ∅0 {w} lt = lift (¬x<0 lt) |
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157 |
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158 ∅< : { x y : OD } → def x (od→ord y ) → ¬ ( x == od∅ ) |
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159 ∅< {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d |
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160 ∅< {x} {y} d eq | lift () |
| 57 | 161 |
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162 ∅6 : { x : OD } → ¬ ( x ∋ x ) -- no Russel paradox |
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163 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
| 51 | 164 |
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165 def-iso : {A B : OD } {x y : Ordinal } → x ≡ y → (def A y → def B y) → def A x → def B x |
| 76 | 166 def-iso refl t = t |
| 167 | |
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168 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
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169 is-o∅ x with trio< x o∅ |
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170 is-o∅ x | tri< a ¬b ¬c = no ¬b |
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171 is-o∅ x | tri≈ ¬a b ¬c = yes b |
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172 is-o∅ x | tri> ¬a ¬b c = no ¬b |
| 57 | 173 |
| 254 | 174 _,_ : OD → OD → OD |
| 175 x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y)) | |
| 176 <_,_> : (x y : OD) → OD | |
| 177 < x , y > = (x , x ) , (x , y ) | |
| 178 | |
| 179 exg-pair : { x y : OD } → (x , y ) == ( y , x ) | |
| 180 exg-pair {x} {y} = record { eq→ = left ; eq← = right } where | |
| 181 left : {z : Ordinal} → def (x , y) z → def (y , x) z | |
| 182 left (case1 t) = case2 t | |
| 183 left (case2 t) = case1 t | |
| 184 right : {z : Ordinal} → def (y , x) z → def (x , y) z | |
| 185 right (case1 t) = case2 t | |
| 186 right (case2 t) = case1 t | |
| 187 | |
| 188 ==-trans : { x y z : OD } → x == y → y == z → x == z | |
| 189 ==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) } | |
| 190 | |
| 191 ==-sym : { x y : OD } → x == y → y == x | |
| 192 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } | |
| 193 | |
| 194 ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y | |
| 195 ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq ) | |
| 196 | |
| 197 od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y | |
| 198 od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq ) | |
| 199 | |
| 200 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' > | |
| 201 eq-prod refl refl = refl | |
| 202 | |
| 203 prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' ) | |
| 204 prod-eq {x} {x'} {y} {y'} eq = record { proj1 = lemmax ; proj2 = lemmay } where | |
| 205 lemma0 : {x y z : OD } → ( x , x ) == ( z , y ) → x ≡ y | |
| 206 lemma0 {x} {y} eq with trio< (od→ord x) (od→ord y) | |
| 207 lemma0 {x} {y} eq | tri< a ¬b ¬c with eq← eq {od→ord y} (case2 refl) | |
| 208 lemma0 {x} {y} eq | tri< a ¬b ¬c | case1 s = ⊥-elim ( o<¬≡ (sym s) a ) | |
| 209 lemma0 {x} {y} eq | tri< a ¬b ¬c | case2 s = ⊥-elim ( o<¬≡ (sym s) a ) | |
| 210 lemma0 {x} {y} eq | tri≈ ¬a b ¬c = ord≡→≡ b | |
| 211 lemma0 {x} {y} eq | tri> ¬a ¬b c with eq← eq {od→ord y} (case2 refl) | |
| 212 lemma0 {x} {y} eq | tri> ¬a ¬b c | case1 s = ⊥-elim ( o<¬≡ s c ) | |
| 213 lemma0 {x} {y} eq | tri> ¬a ¬b c | case2 s = ⊥-elim ( o<¬≡ s c ) | |
| 214 lemma2 : {x y z : OD } → ( x , x ) == ( z , y ) → z ≡ y | |
| 215 lemma2 {x} {y} {z} eq = trans (sym (lemma0 lemma3 )) ( lemma0 eq ) where | |
| 216 lemma3 : ( x , x ) == ( y , z ) | |
| 217 lemma3 = ==-trans eq exg-pair | |
| 218 lemma1 : {x y : OD } → ( x , x ) == ( y , y ) → x ≡ y | |
| 219 lemma1 {x} {y} eq with eq← eq {od→ord y} (case2 refl) | |
| 220 lemma1 {x} {y} eq | case1 s = ord≡→≡ (sym s) | |
| 221 lemma1 {x} {y} eq | case2 s = ord≡→≡ (sym s) | |
| 222 lemma4 : {x y z : OD } → ( x , y ) == ( x , z ) → y ≡ z | |
| 223 lemma4 {x} {y} {z} eq with eq← eq {od→ord z} (case2 refl) | |
| 224 lemma4 {x} {y} {z} eq | case1 s with ord≡→≡ s -- x ≡ z | |
| 225 ... | refl with lemma2 (==-sym eq ) | |
| 226 ... | refl = refl | |
| 227 lemma4 {x} {y} {z} eq | case2 s = ord≡→≡ (sym s) -- y ≡ z | |
| 228 lemmax : x ≡ x' | |
| 229 lemmax with eq→ eq {od→ord (x , x)} (case1 refl) | |
| 230 lemmax | case1 s = lemma1 (ord→== s ) -- (x,x)≡(x',x') | |
| 231 lemmax | case2 s with lemma2 (ord→== s ) -- (x,x)≡(x',y') with x'≡y' | |
| 232 ... | refl = lemma1 (ord→== s ) | |
| 233 lemmay : y ≡ y' | |
| 234 lemmay with lemmax | |
| 235 ... | refl with lemma4 eq -- with (x,y)≡(x,y') | |
| 236 ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 )) | |
| 237 | |
| 257 | 238 data ord-pair : (p : Ordinal) → Set n where |
| 239 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) | |
| 240 | |
| 241 ZFProduct : OD | |
| 242 ZFProduct = record { def = λ x → ord-pair x } | |
| 243 | |
| 244 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) | |
| 245 -- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y' | |
| 246 -- eq-pair refl refl = HE.refl | |
| 247 | |
| 248 pi1 : { p : Ordinal } → ord-pair p → Ordinal | |
| 249 pi1 ( pair x y) = x | |
| 250 | |
| 251 π1 : { p : OD } → ZFProduct ∋ p → OD | |
| 252 π1 lt = ord→od (pi1 lt ) | |
| 253 | |
| 254 pi2 : { p : Ordinal } → ord-pair p → Ordinal | |
| 255 pi2 ( pair x y ) = y | |
| 256 | |
| 257 π2 : { p : OD } → ZFProduct ∋ p → OD | |
| 258 π2 lt = ord→od (pi2 lt ) | |
| 259 | |
| 260 op-cons : { ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy > | |
| 261 op-cons {ox} {oy} = pair ox oy | |
| 262 | |
| 263 p-cons : ( x y : OD ) → ZFProduct ∋ < x , y > | |
| 264 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl ( | |
| 265 let open ≡-Reasoning in begin | |
| 266 od→ord < ord→od (od→ord x) , ord→od (od→ord y) > | |
| 267 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ | |
| 268 od→ord < x , y > | |
| 269 ∎ ) | |
| 270 | |
| 271 op-iso : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op | |
| 272 op-iso (pair ox oy) = refl | |
| 273 | |
| 274 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < π1 p , π2 p > ≡ x | |
| 275 p-iso {x} p = ord≡→≡ (op-iso p) | |
| 276 | |
| 277 p-pi1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≡ x | |
| 278 p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) )) | |
| 279 | |
| 280 p-pi2 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π2 p ≡ y | |
| 281 p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p))) | |
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282 |
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283 -- |
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284 -- Axiom of choice in intutionistic logic implies the exclude middle |
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285 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog |
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286 -- |
| 257 | 287 |
| 288 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p | |
| 289 ppp {p} {a} d = d | |
| 290 | |
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291 p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice |
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292 p∨¬p p with is-o∅ ( od→ord ( record { def = λ x → p } )) |
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293 p∨¬p p | yes eq = case2 (¬p eq) where |
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294 ps = record { def = λ x → p } |
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295 lemma : ps == od∅ → p → ⊥ |
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296 lemma eq p0 = ¬x<0 {od→ord ps} (eq→ eq p0 ) |
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297 ¬p : (od→ord ps ≡ o∅) → p → ⊥ |
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298 ¬p eq = lemma ( subst₂ (λ j k → j == k ) oiso o∅≡od∅ ( o≡→== eq )) |
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299 p∨¬p p | no ¬p = case1 (ppp {p} {minimul ps (λ eq → ¬p (eqo∅ eq))} lemma) where |
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300 ps = record { def = λ x → p } |
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301 eqo∅ : ps == od∅ → od→ord ps ≡ o∅ |
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302 eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) |
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303 lemma : ps ∋ minimul ps (λ eq → ¬p (eqo∅ eq)) |
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304 lemma = x∋minimul ps (λ eq → ¬p (eqo∅ eq)) |
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305 |
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306 decp : ( p : Set n ) → Dec p -- assuming axiom of choice |
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307 decp p with p∨¬p p |
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308 decp p | case1 x = yes x |
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309 decp p | case2 x = no x |
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310 |
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311 double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic |
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312 double-neg-eilm {A} notnot with decp A -- assuming axiom of choice |
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313 ... | yes p = p |
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314 ... | no ¬p = ⊥-elim ( notnot ¬p ) |
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315 |
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316 OrdP : ( x : Ordinal ) ( y : OD ) → Dec ( Ord x ∋ y ) |
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317 OrdP x y with trio< x (od→ord y) |
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318 OrdP x y | tri< a ¬b ¬c = no ¬c |
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319 OrdP x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) |
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320 OrdP x y | tri> ¬a ¬b c = yes c |
| 119 | 321 |
| 79 | 322 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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323 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) |
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324 |
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325 in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD |
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326 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } |
| 141 | 327 |
| 96 | 328 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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329 |
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330 ZFSubset : (A x : OD ) → OD |
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331 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set |
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332 |
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333 Def : (A : OD ) → OD |
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334 Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
| 96 | 335 |
| 190 | 336 |
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337 _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n |
| 190 | 338 _⊆_ A B {x} = A ∋ x → B ∋ x |
| 339 | |
| 340 infixr 220 _⊆_ | |
| 341 | |
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342 subset-lemma : {A x y : OD } → ( x ∋ y → ZFSubset A x ∋ y ) ⇔ ( _⊆_ x A {y} ) |
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343 subset-lemma {A} {x} {y} = record { |
| 190 | 344 proj1 = λ z lt → proj1 (z lt) |
| 345 ; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt } | |
| 346 } | |
| 347 | |
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348 OD→ZF : ZF |
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349 OD→ZF = record { |
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350 ZFSet = OD |
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351 ; _∋_ = _∋_ |
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352 ; _≈_ = _==_ |
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353 ; ∅ = od∅ |
| 28 | 354 ; _,_ = _,_ |
| 355 ; Union = Union | |
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356 ; Power = Power |
| 28 | 357 ; Select = Select |
| 358 ; Replace = Replace | |
| 161 | 359 ; infinite = infinite |
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360 ; isZF = isZF |
| 28 | 361 } where |
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362 ZFSet = OD |
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363 Select : (X : OD ) → ((x : OD ) → Set n ) → OD |
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364 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } |
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365 Replace : OD → (OD → OD ) → OD |
| 141 | 366 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x } |
| 144 | 367 _∩_ : ( A B : ZFSet ) → ZFSet |
| 145 | 368 A ∩ B = record { def = λ x → def A x ∧ def B x } |
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369 Union : OD → OD |
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370 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } |
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371 _∈_ : ( A B : ZFSet ) → Set n |
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372 A ∈ B = B ∋ A |
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373 Power : OD → OD |
| 129 | 374 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) |
| 103 | 375 ｛_｝ : ZFSet → ZFSet |
| 376 ｛ x ｝ = ( x , x ) | |
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377 |
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378 data infinite-d : ( x : Ordinal ) → Set n where |
| 161 | 379 iφ : infinite-d o∅ |
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380 isuc : {x : Ordinal } → infinite-d x → |
| 161 | 381 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) |
| 382 | |
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383 infinite : OD |
| 161 | 384 infinite = record { def = λ x → infinite-d x } |
| 385 | |
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386 infixr 200 _∈_ |
| 96 | 387 -- infixr 230 _∩_ _∪_ |
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388 isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite |
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389 isZF = record { |
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390 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
| 247 | 391 ; pair→ = pair→ |
| 392 ; pair← = pair← | |
| 72 | 393 ; union→ = union→ |
| 394 ; union← = union← | |
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395 ; empty = empty |
| 129 | 396 ; power→ = power→ |
| 76 | 397 ; power← = power← |
| 186 | 398 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
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399 -- ; ε-induction = {!!} |
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400 ; infinity∅ = infinity∅ |
| 160 | 401 ; infinity = infinity |
| 116 | 402 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 135 | 403 ; replacement← = replacement← |
| 404 ; replacement→ = replacement→ | |
| 183 | 405 ; choice-func = choice-func |
| 406 ; choice = choice | |
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407 } where |
| 129 | 408 |
| 247 | 409 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y ) |
| 410 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x )) | |
| 411 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y )) | |
| 412 | |
| 413 pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t | |
| 414 pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x)) | |
| 415 pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y)) | |
| 416 | |
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417 empty : (x : OD ) → ¬ (od∅ ∋ x) |
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418 empty x = ¬x<0 |
| 129 | 419 |
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420 o<→c< : {x y : Ordinal } {z : OD }→ x o< y → _⊆_ (Ord x) (Ord y) {z} |
| 155 | 421 o<→c< lt lt1 = ordtrans lt1 lt |
| 422 | |
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423 ⊆→o< : {x y : Ordinal } → (∀ (z : OD) → _⊆_ (Ord x) (Ord y) {z} ) → x o< osuc y |
| 155 | 424 ⊆→o< {x} {y} lt with trio< x y |
| 425 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 426 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
| 427 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl ) | |
| 428 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) | |
| 151 | 429 |
| 144 | 430 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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431 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
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432 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) |
| 159 | 433 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) |
| 166 | 434 union← X z UX∋z = TransFiniteExists _ lemma UX∋z where |
| 165 | 435 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
| 436 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
| 144 | 437 |
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438 ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y |
| 144 | 439 ψiso {ψ} t refl = t |
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440 selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) |
| 144 | 441 selection {ψ} {X} {y} = record { |
| 442 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 443 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 444 } | |
| 445 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x | |
| 446 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where | |
| 447 lemma : def (in-codomain X ψ) (od→ord (ψ x)) | |
| 150 | 448 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
| 144 | 449 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) |
| 150 | 450 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
| 451 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) | |
| 452 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) | |
| 144 | 453 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
| 150 | 454 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) |
| 144 | 455 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) |
| 150 | 456 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) |
| 457 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) | |
| 144 | 458 |
| 459 --- | |
| 460 --- Power Set | |
| 461 --- | |
| 462 --- First consider ordinals in OD | |
| 100 | 463 --- |
| 464 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
| 465 --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A | |
| 466 -- | |
| 467 -- | |
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468 ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) |
| 142 | 469 ∩-≡ {a} {b} inc = record { |
| 470 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
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471 proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; |
| 142 | 472 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
| 100 | 473 -- |
| 474 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t | |
| 475 -- Power A is a sup of ZFSubset A t, so Power A ∋ t | |
| 476 -- | |
| 141 | 477 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t |
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478 ord-power← a t t→A = def-subst {_} {_} {Def (Ord a)} {od→ord t} |
| 127 | 479 lemma refl (lemma1 lemma-eq )where |
| 129 | 480 lemma-eq : ZFSubset (Ord a) t == t |
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481 eq→ lemma-eq {z} w = proj2 w |
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482 eq← lemma-eq {z} w = record { proj2 = w ; |
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483 proj1 = def-subst {_} {_} {(Ord a)} {z} |
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484 ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
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485 lemma1 : {a : Ordinal } { t : OD } |
| 129 | 486 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t |
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487 lemma1 {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq )) |
| 129 | 488 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x))) |
| 98 | 489 lemma = sup-o< |
| 129 | 490 |
| 144 | 491 -- |
| 492 -- Every set in OD is a subset of Ordinals | |
| 493 -- | |
| 142 | 494 -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y ) |
| 166 | 495 |
| 496 -- we have oly double negation form because of the replacement axiom | |
| 497 -- | |
| 498 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
| 499 power→ A t P∋t {x} t∋x = TransFiniteExists _ lemma5 lemma4 where | |
| 142 | 500 a = od→ord A |
| 501 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y))) | |
| 502 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t | |
| 166 | 503 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) |
| 504 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) | |
| 142 | 505 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) |
| 506 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) | |
| 166 | 507 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) |
| 508 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not | |
| 509 | |
| 142 | 510 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
| 511 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where | |
| 512 a = od→ord A | |
| 513 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x | |
| 514 lemma0 {x} t∋x = c<→o< (t→A t∋x) | |
| 515 lemma3 : Def (Ord a) ∋ t | |
| 516 lemma3 = ord-power← a t lemma0 | |
| 152 | 517 lt1 : od→ord (A ∩ ord→od (od→ord t)) o< sup-o (λ x → od→ord (A ∩ ord→od x)) |
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518 lt1 = sup-o< {λ x → od→ord (A ∩ ord→od x)} {od→ord t} |
| 152 | 519 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t |
| 520 lemma4 = let open ≡-Reasoning in begin | |
| 521 A ∩ ord→od (od→ord t) | |
| 522 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
| 523 A ∩ t | |
| 524 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ | |
| 525 t | |
| 526 ∎ | |
| 142 | 527 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ ord→od x)) |
| 152 | 528 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ ord→od x))) |
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529 lemma4 (sup-o< {λ x → od→ord (A ∩ ord→od x)} {od→ord t}) |
| 142 | 530 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
| 151 | 531 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
| 532 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
| 533 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) | |
| 142 | 534 |
| 190 | 535 -- assuming axiom of choice |
| 141 | 536 regularity : (x : OD) (not : ¬ (x == od∅)) → |
| 115 | 537 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) |
| 117 | 538 proj1 (regularity x not ) = x∋minimul x not |
| 539 proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where | |
| 540 lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁ | |
| 541 lemma1 {x₁} s = ⊥-elim ( minimul-1 x not (ord→od x₁) lemma3 ) where | |
| 542 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁)) | |
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543 lemma3 = record { proj1 = def-subst {_} {_} {minimul x not} {_} (proj1 s) refl (sym diso) |
| 142 | 544 ; proj2 = proj2 (proj2 s) } |
| 117 | 545 lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ |
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546 lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) )) |
| 129 | 547 |
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548 extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
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549 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d |
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550 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d |
| 186 | 551 |
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552 extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) |
| 186 | 553 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d |
| 554 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
| 129 | 555 |
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556 infinity∅ : infinite ∋ od∅ |
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557 infinity∅ = def-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where |
| 161 | 558 lemma : o∅ ≡ od→ord od∅ |
| 559 lemma = let open ≡-Reasoning in begin | |
| 560 o∅ | |
| 561 ≡⟨ sym diso ⟩ | |
| 562 od→ord ( ord→od o∅ ) | |
| 563 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
| 564 od→ord od∅ | |
| 565 ∎ | |
| 566 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) | |
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567 infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where |
| 161 | 568 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) |
| 569 ≡ od→ord (Union (x , (x , x))) | |
| 570 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
| 571 | |
| 179 | 572 -- Axiom of choice ( is equivalent to the existence of minimul in our case ) |
| 162 | 573 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] |
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574 choice-func : (X : OD ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD |
| 162 | 575 choice-func X {x} not X∋x = minimul x not |
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576 choice : (X : OD ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A |
| 162 | 577 choice X {A} X∋A not = x∋minimul A not |
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578 |
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579 --- |
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580 --- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice |
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581 --- |
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582 record choiced ( X : OD) : Set (suc n) where |
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583 field |
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584 a-choice : OD |
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585 is-in : X ∋ a-choice |
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586 |
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587 choice-func' : (X : OD ) → (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X |
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588 choice-func' X p∨¬p not = have_to_find where |
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589 ψ : ( ox : Ordinal ) → Set (suc n) |
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590 ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ def X x )) ∨ choiced X |
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591 lemma-ord : ( ox : Ordinal ) → ψ ox |
| 235 | 592 lemma-ord ox = TransFinite {ψ} induction ox where |
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593 ∋-p : (A x : OD ) → Dec ( A ∋ x ) |
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594 ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x )) |
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595 ∋-p A x | case1 (lift t) = yes t |
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596 ∋-p A x | case2 t = no (λ x → t (lift x )) |
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597 ∀-imply-or : {A : Ordinal → Set n } {B : Set (suc n) } |
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598 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B |
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599 ∀-imply-or {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x)) |
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600 ∀-imply-or {A} {B} ∀AB | case1 (lift t) = case1 t |
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601 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x (lift not ))) where |
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602 lemma : ¬ ((x : Ordinal ) → A x) → B |
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603 lemma not with p∨¬p B |
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604 lemma not | case1 b = b |
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605 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) |
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606 induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x |
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607 induction x prev with ∋-p X ( ord→od x) |
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608 ... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } ) |
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609 ... | no ¬p = lemma where |
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610 lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X |
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611 lemma1 y with ∋-p X (ord→od y) |
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612 lemma1 y | yes y<X = case2 ( record { a-choice = ord→od y ; is-in = y<X } ) |
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613 lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → def X k ) (sym diso) y<X ) ) |
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614 lemma : ((y : Ordinals.ord O) → (O Ordinals.o< y) x → def X y → ⊥) ∨ choiced X |
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615 lemma = ∀-imply-or lemma1 |
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616 have_to_find : choiced X |
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617 have_to_find with lemma-ord (od→ord X ) |
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618 have_to_find | t = dont-or t ¬¬X∋x where |
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619 ¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → def X x → ⊥) |
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620 ¬¬X∋x nn = not record { |
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621 eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt) |
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622 ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt ) |
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623 } |
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624 |
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625 |
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626 Union = ZF.Union OD→ZF |
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627 Power = ZF.Power OD→ZF |
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628 Select = ZF.Select OD→ZF |
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629 Replace = ZF.Replace OD→ZF |
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630 isZF = ZF.isZF OD→ZF |