Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate ordinal-definable.agda @ 104:d92411bed18c
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sun, 16 Jun 2019 02:06:09 +0900 |
| parents | c8b79d303867 |
| children | ec6235ce0215 |
| rev | line source |
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| 16 | 1 open import Level |
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posturate OD is isomorphic to Ordinal
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2 module ordinal-definable where |
| 3 | 3 |
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e11e95d5ddee
separete constructible set
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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⊔_ ) |
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8 open import Relation.Binary.PropositionalEquality |
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separete constructible set
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9 open import Data.Nat.Properties |
| 6 | 10 open import Data.Empty |
| 11 open import Relation.Nullary | |
| 12 open import Relation.Binary | |
| 13 open import Relation.Binary.Core | |
| 14 | |
| 27 | 15 -- Ordinal Definable Set |
| 11 | 16 |
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17 record OD {n : Level} : Set (suc n) where |
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18 field |
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19 def : (x : Ordinal {n} ) → Set n |
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20 |
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21 open OD |
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22 open import Data.Unit |
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23 |
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fcac01485f32
od→lv : {n : Level} → OD {n} → Nat
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24 open Ordinal |
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od→lv : {n : Level} → OD {n} → Nat
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25 |
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equalitu and internal parametorisity
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26 record _==_ {n : Level} ( a b : OD {n} ) : Set n where |
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27 field |
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28 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x |
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29 eq← : ∀ { x : Ordinal {n} } → def b x → def a x |
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30 |
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31 id : {n : Level} {A : Set n} → A → A |
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32 id x = x |
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33 |
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34 eq-refl : {n : Level} { x : OD {n} } → x == x |
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35 eq-refl {n} {x} = record { eq→ = id ; eq← = id } |
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36 |
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37 open _==_ |
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38 |
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39 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x |
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40 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } |
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41 |
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42 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z |
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43 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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44 |
| 40 | 45 od∅ : {n : Level} → OD {n} |
| 46 od∅ {n} = record { def = λ _ → Lift n ⊥ } | |
| 47 | |
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posturate OD is isomorphic to Ordinal
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48 postulate |
| 100 | 49 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) |
| 95 | 50 od→ord : {n : Level} → OD {n} → Ordinal {n} |
| 51 ord→od : {n : Level} → Ordinal {n} → OD {n} | |
| 52 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x | |
| 53 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x | |
| 100 | 54 -- supermum as Replacement Axiom |
| 95 | 55 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 98 | 56 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ |
| 103 | 57 -- a contra-position of minimality of supermum |
| 98 | 58 sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 59 sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 104 | 60 -- sup-min : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → {z : Ordinal {n}} → ψ z o< z → sup-o ψ o< osuc z |
| 61 minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} | |
| 62 x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) | |
| 63 minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) ) | |
| 64 | |
| 95 | 65 |
| 66 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n | |
| 67 _∋_ {n} a x = def a ( od→ord x ) | |
| 68 | |
| 104 | 69 Ord : { n : Level } → ( a : Ordinal {suc n} ) → OD {suc n} |
| 70 Ord {n} a = record { def = λ y → y o< a } | |
| 71 | |
| 72 _c<_ : { n : Level } → ( x a : Ordinal {n} ) → Set n | |
| 73 x c< a = Ord a ∋ Ord x | |
| 74 | |
| 75 c<→o< : { n : Level } → { x a : OD {n} } → record { def = λ y → y o< od→ord a } ∋ x → od→ord x o< od→ord a | |
| 76 c<→o< lt = lt | |
| 77 | |
| 78 o<→c< : { n : Level } → { x a : OD {n} } → od→ord x o< od→ord a → record { def = λ y → y o< od→ord a } ∋ x | |
| 79 o<→c< lt = lt | |
| 80 | |
| 81 ==→o≡' : {n : Level} → { x y : Ordinal {suc n} } → Ord x == Ord y → x ≡ y | |
| 82 ==→o≡' {n} {x} {y} eq with trio< {n} x y | |
| 83 ==→o≡' {n} {x} {y} eq | tri< a ¬b ¬c with eq← eq {x} a | |
| 84 ... | t = ⊥-elim ( o<¬≡ x x refl t ) | |
| 85 ==→o≡' {n} {x} {y} eq | tri≈ ¬a refl ¬c = refl | |
| 86 ==→o≡' {n} {x} {y} eq | tri> ¬a ¬b c with eq→ eq {y} c | |
| 87 ... | t = ⊥-elim ( o<¬≡ y y refl t ) | |
| 103 | 88 |
| 104 | 89 ∅∨ : { n : Level } → { x y : Ordinal {suc n} } → ( Ord {n} x == Ord y ) ∨ ( ¬ ( Ord x == Ord y ) ) |
| 90 ∅∨ {n} {x} {y} with trio< x y | |
| 91 ∅∨ {n} {x} {y} | tri< a ¬b ¬c = case2 ( λ eq → ¬b ( ==→o≡' eq ) ) | |
| 92 ∅∨ {n} {x} {y} | tri≈ ¬a refl ¬c = case1 ( record { eq→ = id ; eq← = id } ) | |
| 93 ∅∨ {n} {x} {y} | tri> ¬a ¬b c = case2 ( λ eq → ¬b ( ==→o≡' eq ) ) | |
| 94 | |
| 95 ¬x∋x' : { n : Level } → { x : Ordinal {n} } → ¬ ( record { def = λ y → y o< x } ∋ record { def = λ y → y o< x } ) | |
| 96 ¬x∋x' {n} {record { lv = Zero ; ord = ord }} (case1 ()) | |
| 97 ¬x∋x' {n} {record { lv = Suc lx ; ord = Φ .(Suc lx) }} (case1 (s≤s x)) = ¬x∋x' {n} {record { lv = lx ; ord = Φ lx }} (case1 {!!}) | |
| 98 ¬x∋x' {n} {record { lv = Suc lx ; ord = OSuc (Suc lx) ox }} (case1 (s≤s x)) = ¬x∋x' {n} {record { lv = Suc lx ; ord = ox}} (case1 {!!}) | |
| 99 ¬x∋x' {n} {record { lv = lv ; ord = Φ (lv) }} (case2 ()) | |
| 100 ¬x∋x' {n} {record { lv = lv ; ord = OSuc (lv) ox }} (case2 x) = | |
| 101 ¬x∋x' {n} {record { lv = lv ; ord = ox }} (case2 {!!}) | |
| 102 | |
| 103 ¬x∋x : { n : Level } → { x : OD {n} } → ¬ x ∋ x | |
| 104 ¬x∋x = {!!} | |
| 105 | |
| 106 oc-lemma : { n : Level } → { x : OD {n} } → { oa : Ordinal {n} } → def (record { def = λ y → y o< oa }) oa → ⊥ | |
| 107 oc-lemma {n} {x} {oa} lt = o<¬≡ oa oa refl lt | |
| 108 | |
| 109 oc-lemma1 : { n : Level } → { x : OD {n} } → { oa : Ordinal {n} } → od→ord (record { def = λ y → y o< oa }) o< oa → ⊥ | |
| 110 oc-lemma1 {n} {x} {oa} lt = ¬x∋x' {n} lt -- lt : def (record { def = λ y → y o< oa }) (record { def = λ y → y o< oa }) | |
| 111 | |
| 112 oc-lemma2 : { n : Level } → { x a : OD {n} } → { oa : Ordinal {n} } → oa o< od→ord (record { def = λ y → y o< oa }) → ⊥ | |
| 113 oc-lemma2 {n} {x} {oa} lt = {!!} | |
| 95 | 114 |
| 115 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) | |
| 116 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) | |
| 117 | |
| 118 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 | |
| 119 def-subst df refl refl = df | |
| 120 | |
| 104 | 121 o<-def : {n : Level } {x y : Ordinal {n} } → x o< y → def (record { def = λ x → x o< y }) x |
| 122 o<-def x<y = x<y | |
| 123 | |
| 124 def-o< : {n : Level } {x y : Ordinal {n} } → def (record { def = λ x → x o< y }) x → x o< y | |
| 125 def-o< x<y = x<y | |
| 103 | 126 |
| 95 | 127 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} |
| 128 sup-od ψ = ord→od ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) | |
| 129 | |
| 130 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) | |
| 131 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {sup-od ψ} {od→ord ( ψ x )} | |
| 103 | 132 {!!} refl (cong ( λ k → od→ord (ψ k) ) oiso) |
| 46 | 133 |
| 104 | 134 od∅' : {n : Level} → OD {n} |
| 135 od∅' = record { def = λ x → x o< o∅ } | |
| 136 | |
| 137 ∅0 : {n : Level} → od∅ {suc n} == record { def = λ x → x o< o∅ } | |
| 138 eq→ ∅0 {w} (lift ()) | |
| 139 eq← ∅0 {w} (case1 ()) | |
| 140 eq← ∅0 {w} (case2 ()) | |
| 141 | |
| 142 ∅1 : {n : Level} → ( x : Ordinal {n} ) → ¬ ( x c< o∅ {n} ) | |
| 143 ∅1 {n} x lt = {!!} | |
| 28 | 144 |
| 37 | 145 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
| 81 | 146 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
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problem on Ordinal ( OSuc ℵ )
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147 c0 : Nat → Ordinal {n} → Set n |
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148 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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149 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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150 c2 Zero not = refl |
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151 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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152 ... | t with t (case1 ≤-refl ) |
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153 c2 (Suc lx) not | t | () |
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problem on Ordinal ( OSuc ℵ )
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154 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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155 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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156 ... | t with t (case2 Φ< ) |
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problem on Ordinal ( OSuc ℵ )
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157 c3 lx (Φ .lx) d not | t | () |
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problem on Ordinal ( OSuc ℵ )
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158 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
| 34 | 159 ... | t with t (case2 (s< s<refl ) ) |
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160 c3 lx (OSuc .lx x₁) d not | t | () |
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161 |
| 57 | 162 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
| 163 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
| 164 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
| 165 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
| 37 | 166 |
| 46 | 167 ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y } |
| 168 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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od→lv : {n : Level} → OD {n} → Nat
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169 |
| 51 | 170 -- avoiding lv != Zero error |
| 171 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y | |
| 172 orefl refl = refl | |
| 173 | |
| 174 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y | |
| 175 ==-iso {n} {x} {y} eq = record { | |
| 176 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
| 177 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
| 178 where | |
| 179 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z | |
| 180 lemma {x} {z} d = def-subst d oiso refl | |
| 181 | |
| 57 | 182 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
| 183 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) | |
| 184 | |
| 51 | 185 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
| 186 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where | |
| 187 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
| 188 lemma ox ox refl = eq-refl | |
| 189 | |
| 190 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
| 191 o≡→== {n} {x} {.x} refl = eq-refl | |
| 192 | |
| 193 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
| 194 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
| 195 | |
| 196 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x | |
| 197 c≤-refl x = case1 refl | |
| 198 | |
| 54 | 199 o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ |
| 52 | 200 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with |
| 103 | 201 yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} {!!} oiso refl ) |
| 202 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl {!!} | |
| 52 | 203 ... | () |
| 204 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with | |
| 103 | 205 yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} {!!} oiso refl ) |
| 206 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl {!!} | |
| 52 | 207 ... | () |
| 208 | |
| 79 | 209 ==→o≡ : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y |
| 210 ==→o≡ {n} {x} {y} eq with trio< {n} x y | |
| 211 ==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso ))) | |
| 212 ==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b | |
| 213 ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) | |
| 214 | |
| 90 | 215 ≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } ) |
| 216 ≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where | |
| 217 lemma : ord→od x == record { def = λ z → z o< x } | |
| 104 | 218 eq→ lemma {w} lt = {!!} |
| 219 -- ?subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where | |
| 220 --t : (od→ord ( ord→od w)) o< (od→ord (ord→od x)) | |
| 221 --t = o<-subst lt ? ? | |
| 222 eq← lemma {w} lt = def-subst {suc n} {_} {_} {ord→od x} {w} {!!} refl refl | |
| 90 | 223 |
| 224 od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } | |
| 225 od≡-def {n} {x} = subst (λ k → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} )) | |
| 226 | |
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227 ==→o≡1 : {n : Level} → { x y : OD {suc n} } → x == y → od→ord x ≡ od→ord y |
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228 ==→o≡1 eq = ==→o≡ (subst₂ (λ k j → k == j ) (sym oiso) (sym oiso) eq ) |
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229 |
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230 ==-def-l : {n : Level } {x y : Ordinal {suc n} } { z : OD {suc n} }→ (ord→od x == ord→od y) → def z x → def z y |
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231 ==-def-l {n} {x} {y} {z} eq z>x = subst ( λ k → def z k ) (==→o≡ eq) z>x |
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232 |
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233 ==-def-r : {n : Level } {x y : OD {suc n} } { z : Ordinal {suc n} }→ (x == y) → def x z → def y z |
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234 ==-def-r {n} {x} {y} {z} eq z>x = subst (λ k → def k z ) (subst₂ (λ j k → j ≡ k ) oiso oiso (cong (λ k → ord→od k) (==→o≡1 eq))) z>x |
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235 |
| 91 | 236 ∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a |
| 237 ∋→o< {n} {a} {x} lt = t where | |
| 238 t : (od→ord x) o< (od→ord a) | |
| 103 | 239 t = {!!} |
| 91 | 240 |
| 241 o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x | |
| 95 | 242 o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t where |
| 243 t : def (ord→od (od→ord a)) (od→ord x) | |
| 103 | 244 t = {!!} |
| 91 | 245 |
| 104 | 246 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅' {suc n} |
| 247 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅' {suc n} )) | |
| 80 | 248 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where |
| 104 | 249 lemma : o∅ {suc n } o< (od→ord (od∅' {suc n} )) → ⊥ |
| 250 lemma lt with def-subst {suc n} {_} {_} {_} {_} ( o<→c< ( o<-subst lt (sym diso) refl ) ) refl diso | |
| 103 | 251 lemma lt | t = {!!} |
| 80 | 252 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso |
| 253 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 254 | |
| 51 | 255 o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) |
| 52 | 256 o<→¬== {n} {x} {y} lt eq = o<→o> eq lt |
| 51 | 257 |
| 104 | 258 o<→¬c> : {n : Level} → { x y : Ordinal {n} } → x o< y → ¬ (y c< x ) |
| 103 | 259 o<→¬c> {n} {x} {y} olt clt = o<> olt {!!} where |
| 51 | 260 |
| 104 | 261 o≡→¬c< : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ¬ x c< y |
| 262 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ x y {!!} {!!} | |
| 51 | 263 |
| 104 | 264 tri-c< : {n : Level} → Trichotomous _≡_ (_c<_ {suc n}) |
| 265 tri-c< {n} x y with trio< {n} x y | |
| 266 tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst {!!} oiso refl) {!!} ( o<→¬c> a ) | |
| 267 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) {!!} (o≡→¬c< (sym b)) | |
| 268 tri-c< {n} x y | tri> ¬a ¬b c = tri> ( o<→¬c> c) (λ eq → {!!} ) (def-subst {!!} oiso refl) | |
| 51 | 269 |
| 104 | 270 c<> : {n : Level } { x y : Ordinal {suc n}} → x c< y → y c< x → ⊥ |
| 54 | 271 c<> {n} {x} {y} x<y y<x with tri-c< x y |
| 272 c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x | |
| 104 | 273 c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> {!!} {!!} |
| 54 | 274 c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y |
| 275 | |
| 60 | 276 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
| 277 ∅< {n} {x} {y} d eq with eq→ eq d | |
| 278 ∅< {n} {x} {y} d eq | lift () | |
| 57 | 279 |
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280 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
| 104 | 281 ∅6 {n} {x} x∋x = c<> {n} {{!!}} {{!!}} {!!} {!!} |
| 51 | 282 |
| 76 | 283 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x |
| 284 def-iso refl t = t | |
| 285 | |
| 57 | 286 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
| 287 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
| 288 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
| 289 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
| 290 | |
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291 open _∧_ |
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292 |
| 66 | 293 ¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} |
| 294 ¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where | |
| 295 lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} | |
| 296 lemma ox ne with is-o∅ ox | |
| 297 lemma ox ne | yes refl with ne ( ord→== lemma1 ) where | |
| 298 lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ | |
| 104 | 299 lemma1 = cong ( λ k → od→ord k ) {!!} |
| 66 | 300 lemma o∅ ne | yes refl | () |
| 104 | 301 lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) {!!} {!!} |
| 69 | 302 |
| 79 | 303 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 94 | 304 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
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305 |
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306 csuc : {n : Level} → OD {suc n} → OD {suc n} |
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307 csuc x = ord→od ( osuc ( od→ord x )) |
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308 |
| 96 | 309 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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310 |
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311 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} |
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312 ZFSubset A x = record { def = λ y → def A y ∧ def x y } |
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313 |
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314 Def : {n : Level} → (A : OD {suc n}) → OD {suc n} |
| 104 | 315 Def {n} A = record { def = λ y → y o< ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )))) } |
| 96 | 316 |
| 317 -- Constructible Set on α | |
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318 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n} |
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319 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ |
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320 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) |
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321 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) |
| 104 | 322 record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) } |
| 89 | 323 |
| 54 | 324 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
| 40 | 325 OD→ZF {n} = record { |
| 54 | 326 ZFSet = OD {suc n} |
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327 ; _∋_ = _∋_ |
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328 ; _≈_ = _==_ |
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329 ; ∅ = od∅ |
| 28 | 330 ; _,_ = _,_ |
| 331 ; Union = Union | |
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332 ; Power = Power |
| 28 | 333 ; Select = Select |
| 334 ; Replace = Replace | |
| 81 | 335 ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } ) |
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336 ; isZF = isZF |
| 28 | 337 } where |
| 54 | 338 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} |
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339 Replace X ψ = sup-od ψ |
| 54 | 340 Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} |
| 341 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } | |
| 342 _,_ : OD {suc n} → OD {suc n} → OD {suc n} | |
| 84 | 343 x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) } |
| 54 | 344 Union : OD {suc n} → OD {suc n} |
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345 Union U = record { def = λ y → osuc y o< (od→ord U) } |
| 77 | 346 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) |
| 54 | 347 Power : OD {suc n} → OD {suc n} |
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348 Power A = Def A |
| 54 | 349 ZFSet = OD {suc n} |
| 350 _∈_ : ( A B : ZFSet ) → Set (suc n) | |
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351 A ∈ B = B ∋ A |
| 54 | 352 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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353 _⊆_ A B {x} = A ∋ x → B ∋ x |
| 103 | 354 _∩_ : ( A B : ZFSet ) → ZFSet |
| 355 A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) | |
| 96 | 356 -- _∪_ : ( A B : ZFSet ) → ZFSet |
| 357 -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) | |
| 103 | 358 ｛_｝ : ZFSet → ZFSet |
| 359 ｛ x ｝ = ( x , x ) | |
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360 infixr 200 _∈_ |
| 96 | 361 -- infixr 230 _∩_ _∪_ |
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362 infixr 220 _⊆_ |
| 81 | 363 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} )) |
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364 isZF = record { |
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365 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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366 ; pair = pair |
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367 ; union-u = λ _ z _ → csuc z |
| 72 | 368 ; union→ = union→ |
| 369 ; union← = union← | |
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370 ; empty = empty |
| 76 | 371 ; power→ = power→ |
| 372 ; power← = power← | |
| 373 ; extensionality = extensionality | |
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374 ; minimul = minimul |
| 51 | 375 ; regularity = regularity |
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376 ; infinity∅ = infinity∅ |
| 93 | 377 ; infinity = λ _ → infinity |
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378 ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} |
| 93 | 379 ; replacement = replacement |
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380 } where |
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381 open _∧_ |
| 54 | 382 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
| 87 | 383 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
| 384 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
| 54 | 385 empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
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386 empty x () |
| 100 | 387 --- |
| 388 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
| 389 --- 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 | |
| 390 -- | |
| 103 | 391 -- if Power A ∋ t, from a minimulity of sup, there is osuc ZFSubset A ∋ t |
| 100 | 392 -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x |
| 393 -- In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity | |
| 394 -- | |
| 76 | 395 power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x |
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396 power→ A t P∋t {x} t∋x = proj1 lemma-s where |
| 98 | 397 minsup : OD |
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398 minsup = ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) |
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399 lemma-t : csuc minsup ∋ t |
| 103 | 400 lemma-t = {!!} |
| 98 | 401 lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) ∋ x |
| 103 | 402 lemma-s = {!!} |
| 100 | 403 -- |
| 404 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t | |
| 405 -- Power A is a sup of ZFSubset A t, so Power A ∋ t | |
| 406 -- | |
| 77 | 407 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
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408 power← A t t→A = def-subst {suc n} {_} {_} {Power A} {od→ord t} |
| 103 | 409 {!!} refl lemma1 where |
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410 lemma-eq : ZFSubset A t == t |
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411 eq→ lemma-eq {z} w = proj2 w |
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412 eq← lemma-eq {z} w = record { proj2 = w ; |
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413 proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
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414 lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t |
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415 lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) (==→o≡1 (lemma-eq)) |
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416 lemma : od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x))) |
| 98 | 417 lemma = sup-o< |
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418 union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z |
| 72 | 419 union-lemma-u {X} {z} U>z = lemma <-osuc where |
| 420 lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz | |
| 103 | 421 lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} {!!} refl refl |
| 73 | 422 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
| 72 | 423 union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) |
| 103 | 424 union→ X y u xx | tri< a ¬b ¬c with osuc-< a {!!} |
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425 union→ X y u xx | tri< a ¬b ¬c | () |
| 103 | 426 union→ X y u xx | tri≈ ¬a b ¬c = lemma b {!!} where |
| 73 | 427 lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX |
| 428 lemma refl lt = lt | |
| 103 | 429 union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c {!!} |
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430 union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z ) |
| 103 | 431 union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} {!!} oiso (sym diso) ; proj2 = union-lemma-u X∋z } |
| 54 | 432 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y |
| 433 ψiso {ψ} t refl = t | |
| 434 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
| 435 selection {ψ} {X} {y} = record { | |
| 436 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 437 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 438 } | |
| 93 | 439 replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x |
| 440 replacement {ψ} X x = sup-c< ψ {x} | |
| 60 | 441 ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) |
| 442 ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq | |
| 57 | 443 regularity : (x : OD) (not : ¬ (x == od∅)) → |
| 444 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) | |
| 104 | 445 proj1 (regularity x not ) = x∋minimul x not |
| 66 | 446 proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } where |
| 447 reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y | |
| 104 | 448 reg {y} t with minimul-1 x not (ord→od y) (proj2 t ) |
| 449 ... | t1 = lift t1 | |
| 76 | 450 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
| 451 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
| 452 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 89 | 453 xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } |
| 454 xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x)) | |
| 455 xxx-union : {x : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))} | |
| 456 xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where | |
| 91 | 457 lemma1 : {x : OD {suc n}} → od→ord x o< od→ord (x , x) |
| 103 | 458 lemma1 {x} = {!!} |
| 91 | 459 lemma2 : {x : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x) |
| 460 lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) (sym ≡-def) | |
| 89 | 461 lemma : {x : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x)) |
| 91 | 462 lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 ) |
| 90 | 463 uxxx-union : {x : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } |
| 464 uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where | |
| 465 lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x)) | |
| 91 | 466 lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def ) |
| 467 uxxx-2 : {x : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) } | |
| 468 eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt | |
| 469 eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt | |
| 470 uxxx-ord : {x : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x) | |
| 471 uxxx-ord {x} = trans (cong (λ k → od→ord k ) uxxx-union) (==→o≡ (subst₂ (λ j k → j == k ) (sym oiso) (sym od≡-def ) uxxx-2 )) | |
| 472 omega = record { lv = Suc Zero ; ord = Φ 1 } | |
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473 infinite : OD {suc n} |
| 91 | 474 infinite = ord→od ( omega ) |
| 475 infinity∅ : ord→od ( omega ) ∋ od∅ {suc n} | |
| 95 | 476 infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅} |
| 104 | 477 {!!} refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k → od→ord k) {!!} )) |
| 91 | 478 infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega |
| 479 infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where | |
| 480 t : od→ord x o< od→ord (ord→od (omega)) | |
| 481 t = ∋→o< {n} {infinite} {x} lt | |
| 482 infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x )) | |
| 483 infinite∋uxxx x lt = o<∋→ t where | |
| 484 t : od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega)) | |
| 485 t = subst (λ k → od→ord (Union (x , (x , x))) o< k ) (sym diso ) ( subst ( λ k → k o< omega ) ( sym (uxxx-ord {x} ) ) lt ) | |
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486 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
| 91 | 487 infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt )) where |
| 488 lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega | |
| 489 lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
| 490 lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) | |
| 491 lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) | |
| 492 lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) | |
| 493 lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 | |
| 494 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl | |
| 103 | 495 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set |
| 496 -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ] | |
| 497 record Choice (z : OD {suc n}) : Set (suc (suc n)) where | |
| 498 field | |
| 499 u : {x : OD {suc n}} ( x∈z : x ∈ z ) → OD {suc n} | |
| 500 t : {x : OD {suc n}} ( x∈z : x ∈ z ) → (x : OD {suc n} ) → OD {suc n} | |
| 501 choice : { x : OD {suc n} } → ( x∈z : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝ | |
| 502 -- choice : {x : OD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) → | |
| 503 -- axiom-of-choice : { X : OD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : OD } → (A∈X : A ∈ X ) → choice ¬x∅ A∈X ∈ A | |
| 504 -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!} | |
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| 103 | 506 |