Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate HOD.agda @ 179:aa89d1b8ce96
fix comments
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sat, 20 Jul 2019 08:21:54 +0900 |
| parents | ecb329ba38ac |
| children | 11490a3170d4 |
| rev | line source |
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| 16 | 1 open import Level |
| 112 | 2 module HOD where |
| 3 | 3 |
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4 open import zf |
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5 open import ordinal |
| 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 | |
| 27 | 14 -- Ordinal Definable Set |
| 11 | 15 |
| 141 | 16 record OD {n : Level} : Set (suc n) where |
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17 field |
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18 def : (x : Ordinal {n} ) → Set n |
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19 |
| 141 | 20 open OD |
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21 |
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22 open Ordinal |
| 120 | 23 open _∧_ |
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24 |
| 141 | 25 record _==_ {n : Level} ( a b : OD {n} ) : Set n where |
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26 field |
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27 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x |
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28 eq← : ∀ { x : Ordinal {n} } → def b x → def a x |
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29 |
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30 id : {n : Level} {A : Set n} → A → A |
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31 id x = x |
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32 |
| 141 | 33 eq-refl : {n : Level} { x : OD {n} } → x == x |
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34 eq-refl {n} {x} = record { eq→ = id ; eq← = id } |
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35 |
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36 open _==_ |
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37 |
| 141 | 38 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x |
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39 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } |
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40 |
| 141 | 41 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z |
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42 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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43 |
| 141 | 44 ⇔→== : {n : Level} { x y : OD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔ def y z) → x == y |
| 120 | 45 eq→ ( ⇔→== {n} {x} {y} eq ) {z} m = proj1 eq m |
| 46 eq← ( ⇔→== {n} {x} {y} eq ) {z} m = proj2 eq m | |
| 47 | |
| 179 | 48 -- Ordinal in OD ( and ZFSet ) Transitive Set |
| 141 | 49 Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} |
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50 Ord {n} a = record { def = λ y → y o< a } |
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51 |
| 141 | 52 od∅ : {n : Level} → OD {n} |
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53 od∅ {n} = Ord o∅ |
| 40 | 54 |
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55 postulate |
| 141 | 56 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) |
| 57 od→ord : {n : Level} → OD {n} → Ordinal {n} | |
| 58 ord→od : {n : Level} → Ordinal {n} → OD {n} | |
| 166 | 59 c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y |
| 60 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x | |
| 113 | 61 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x |
| 150 | 62 -- we should prove this in agda, but simply put here |
| 141 | 63 ==→o≡ : {n : Level} → { x y : OD {suc n} } → (x == y) → x ≡ y |
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64 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set |
| 159 | 65 -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x |
| 66 -- ord→od x ≡ Ord x results the same | |
| 100 | 67 -- supermum as Replacement Axiom |
| 95 | 68 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 98 | 69 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ |
| 111 | 70 -- contra-position of mimimulity of supermum required in Power Set Axiom |
| 165 | 71 -- sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 72 -- sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
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73 -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) |
| 167 | 74 -- mimimul and x∋minimul is a weaker form of Axiom of choice |
| 141 | 75 minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
| 117 | 76 -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) |
| 141 | 77 x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) |
| 78 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) ) | |
| 123 | 79 |
| 141 | 80 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n |
| 95 | 81 _∋_ {n} a x = def a ( od→ord x ) |
| 82 | |
| 141 | 83 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n |
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84 x c< a = a ∋ x |
| 103 | 85 |
| 141 | 86 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) |
| 95 | 87 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) |
| 88 | |
| 141 | 89 cseq : {n : Level} → OD {n} → OD {n} |
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90 cseq x = record { def = λ y → def x (osuc y) } where |
| 113 | 91 |
| 141 | 92 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 |
| 95 | 93 def-subst df refl refl = df |
| 94 | |
| 141 | 95 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} |
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96 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) |
| 95 | 97 |
| 141 | 98 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) |
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99 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )} |
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100 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where |
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101 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x))) |
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102 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) ) |
| 28 | 103 |
| 142 | 104 otrans : {n : Level} {a x : Ordinal {n} } → def (Ord a) x → { y : Ordinal {n} } → y o< x → def (Ord a) y |
| 105 otrans {n} {a} {x} x<a {y} y<x = ordtrans y<x x<a | |
| 123 | 106 |
| 37 | 107 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} |
| 81 | 108 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
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109 c0 : Nat → Ordinal {n} → Set n |
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110 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} |
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111 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) |
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112 c2 Zero not = refl |
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113 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) |
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114 ... | t with t (case1 ≤-refl ) |
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115 c2 (Suc lx) not | t | () |
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116 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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117 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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118 ... | t with t (case2 Φ< ) |
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119 c3 lx (Φ .lx) d not | t | () |
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120 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
| 34 | 121 ... | t with t (case2 (s< s<refl ) ) |
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122 c3 lx (OSuc .lx x₁) d not | t | () |
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123 |
| 57 | 124 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
| 125 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
| 126 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
| 127 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
| 37 | 128 |
| 46 | 129 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 } |
| 130 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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131 |
| 51 | 132 -- avoiding lv != Zero error |
| 141 | 133 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 134 orefl refl = refl |
| 135 | |
| 141 | 136 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y |
| 51 | 137 ==-iso {n} {x} {y} eq = record { |
| 138 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
| 139 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
| 140 where | |
| 141 | 141 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z |
| 51 | 142 lemma {x} {z} d = def-subst d oiso refl |
| 143 | |
| 141 | 144 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
| 57 | 145 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) |
| 146 | |
| 141 | 147 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
| 51 | 148 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
| 149 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
| 150 lemma ox ox refl = eq-refl | |
| 151 | |
| 152 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
| 153 o≡→== {n} {x} {.x} refl = eq-refl | |
| 154 | |
| 155 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) | |
| 156 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x | |
| 157 | |
| 141 | 158 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x |
| 51 | 159 c≤-refl x = case1 refl |
| 160 | |
| 141 | 161 ∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a |
| 91 | 162 ∋→o< {n} {a} {x} lt = t where |
| 163 t : (od→ord x) o< (od→ord a) | |
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164 t = c<→o< {suc n} {x} {a} lt |
| 91 | 165 |
| 150 | 166 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} |
| 167 o∅≡od∅ {n} = ==→o≡ lemma where | |
| 168 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x | |
| 169 lemma0 {x} lt = o<-subst (c<→o< {suc n} {ord→od x} {ord→od o∅} (def-subst {suc n} {ord→od o∅} {x} lt refl (sym diso)) ) diso diso | |
| 170 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x | |
| 171 lemma1 (case1 ()) | |
| 172 lemma1 (case2 ()) | |
| 173 lemma : ord→od o∅ == od∅ | |
| 174 lemma = record { eq→ = lemma0 ; eq← = lemma1 } | |
| 175 | |
| 176 ord-od∅ : {n : Level} → od→ord (od∅ {suc n}) ≡ o∅ {suc n} | |
| 177 ord-od∅ {n} = sym ( subst (λ k → k ≡ od→ord (od∅ {suc n}) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) | |
| 80 | 178 |
| 141 | 179 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) |
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180 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where |
| 51 | 181 |
| 141 | 182 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y |
| 111 | 183 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (orefl oeq ) (c<→o< lt) |
| 54 | 184 |
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185 ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ } == od∅ {n} |
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186 eq→ ∅0 {w} (lift ()) |
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187 eq← ∅0 {w} (case1 ()) |
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188 eq← ∅0 {w} (case2 ()) |
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189 |
| 141 | 190 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
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191 ∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d |
| 60 | 192 ∅< {n} {x} {y} d eq | lift () |
| 57 | 193 |
| 141 | 194 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
| 120 | 195 ∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {x} {x} x∋x ) |
| 51 | 196 |
| 141 | 197 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 |
| 76 | 198 def-iso refl t = t |
| 199 | |
| 57 | 200 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
| 201 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
| 202 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
| 203 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
| 204 | |
| 167 | 205 OrdP : {n : Level} → ( x : Ordinal {suc n} ) ( y : OD {suc n} ) → Dec ( Ord x ∋ y ) |
| 206 OrdP {n} x y with trio< x (od→ord y) | |
| 207 OrdP {n} x y | tri< a ¬b ¬c = no ¬c | |
| 208 OrdP {n} x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) | |
| 209 OrdP {n} x y | tri> ¬a ¬b c = yes c | |
| 119 | 210 |
| 79 | 211 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 94 | 212 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
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213 |
| 141 | 214 in-codomain : {n : Level} → (X : OD {suc n} ) → ( ψ : OD {suc n} → OD {suc n} ) → OD {suc n} |
| 148 | 215 in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } |
| 141 | 216 |
| 96 | 217 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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218 |
| 141 | 219 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} |
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220 ZFSubset A x = record { def = λ y → def A y ∧ def x y } where |
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221 |
| 141 | 222 Def : {n : Level} → (A : OD {suc n}) → OD {suc n} |
| 154 | 223 Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) -- Ord x does not help ord-power→ |
| 96 | 224 |
| 225 -- Constructible Set on α | |
| 122 | 226 -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x } |
| 227 -- L (Φ 0) = Φ | |
| 228 -- L (OSuc lv n) = { Def ( L n ) } | |
| 229 -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) ) | |
| 141 | 230 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n} |
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231 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ |
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232 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) |
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233 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) |
| 121 | 234 cseq ( Ord (od→ord (L {n} (record { lv = lx ; ord = Φ lx })))) |
| 89 | 235 |
| 167 | 236 -- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α |
| 141 | 237 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x |
| 122 | 238 |
| 170 | 239 |
| 141 | 240 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
| 241 OD→ZF {n} = record { | |
| 242 ZFSet = OD {suc n} | |
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243 ; _∋_ = _∋_ |
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244 ; _≈_ = _==_ |
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245 ; ∅ = od∅ |
| 28 | 246 ; _,_ = _,_ |
| 247 ; Union = Union | |
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248 ; Power = Power |
| 28 | 249 ; Select = Select |
| 250 ; Replace = Replace | |
| 161 | 251 ; infinite = infinite |
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252 ; isZF = isZF |
| 28 | 253 } where |
| 144 | 254 ZFSet = OD {suc n} |
| 141 | 255 Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → Set (suc n) ) → OD {suc n} |
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256 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } |
| 141 | 257 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} |
| 258 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 } | |
| 259 _,_ : OD {suc n} → OD {suc n} → OD {suc n} | |
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260 x , y = Ord (omax (od→ord x) (od→ord y)) |
| 144 | 261 _∩_ : ( A B : ZFSet ) → ZFSet |
| 145 | 262 A ∩ B = record { def = λ x → def A x ∧ def B x } |
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263 Union : OD {suc n} → OD {suc n} |
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264 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } |
| 54 | 265 _∈_ : ( A B : ZFSet ) → Set (suc n) |
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266 A ∈ B = B ∋ A |
| 54 | 267 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) |
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268 _⊆_ A B {x} = A ∋ x → B ∋ x |
| 141 | 269 Power : OD {suc n} → OD {suc n} |
| 129 | 270 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) |
| 103 | 271 ｛_｝ : ZFSet → ZFSet |
| 272 ｛ x ｝ = ( x , x ) | |
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273 |
| 161 | 274 data infinite-d : ( x : Ordinal {suc n} ) → Set (suc n) where |
| 275 iφ : infinite-d o∅ | |
| 276 isuc : {x : Ordinal {suc n} } → infinite-d x → | |
| 277 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) | |
| 278 | |
| 279 infinite : OD {suc n} | |
| 280 infinite = record { def = λ x → infinite-d x } | |
| 281 | |
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282 infixr 200 _∈_ |
| 96 | 283 -- infixr 230 _∩_ _∪_ |
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284 infixr 220 _⊆_ |
| 161 | 285 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite |
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286 isZF = record { |
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287 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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288 ; pair = pair |
| 72 | 289 ; union→ = union→ |
| 290 ; union← = union← | |
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291 ; empty = empty |
| 129 | 292 ; power→ = power→ |
| 76 | 293 ; power← = power← |
| 294 ; extensionality = extensionality | |
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295 ; minimul = minimul |
| 51 | 296 ; regularity = regularity |
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297 ; infinity∅ = infinity∅ |
| 160 | 298 ; infinity = infinity |
| 116 | 299 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 135 | 300 ; replacement← = replacement← |
| 301 ; replacement→ = replacement→ | |
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302 } where |
| 129 | 303 |
| 141 | 304 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
| 87 | 305 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
| 306 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
| 129 | 307 |
| 167 | 308 empty : {n : Level } (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
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309 empty x (case1 ()) |
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310 empty x (case2 ()) |
| 129 | 311 |
| 151 | 312 ord-⊆ : ( t x : OD {suc n} ) → _⊆_ t (Ord (od→ord t )) {x} |
| 313 ord-⊆ t x lt = c<→o< lt | |
| 154 | 314 o<→c< : {x y : Ordinal {suc n}} {z : OD {suc n}}→ x o< y → _⊆_ (Ord x) (Ord y) {z} |
| 155 | 315 o<→c< lt lt1 = ordtrans lt1 lt |
| 316 | |
| 317 ⊆→o< : {x y : Ordinal {suc n}} → (∀ (z : OD) → _⊆_ (Ord x) (Ord y) {z} ) → x o< osuc y | |
| 318 ⊆→o< {x} {y} lt with trio< x y | |
| 319 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 320 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
| 321 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl ) | |
| 322 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) | |
| 151 | 323 |
| 144 | 324 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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325 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
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326 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) |
| 159 | 327 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) |
| 166 | 328 union← X z UX∋z = TransFiniteExists _ lemma UX∋z where |
| 165 | 329 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
| 330 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
| 144 | 331 |
| 332 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y | |
| 333 ψiso {ψ} t refl = t | |
| 334 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
| 335 selection {ψ} {X} {y} = record { | |
| 336 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 337 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 338 } | |
| 339 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x | |
| 340 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where | |
| 341 lemma : def (in-codomain X ψ) (od→ord (ψ x)) | |
| 150 | 342 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
| 144 | 343 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) |
| 150 | 344 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
| 345 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) | |
| 346 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) | |
| 144 | 347 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
| 150 | 348 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) |
| 144 | 349 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) |
| 150 | 350 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) |
| 351 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) | |
| 144 | 352 |
| 353 --- | |
| 354 --- Power Set | |
| 355 --- | |
| 356 --- First consider ordinals in OD | |
| 100 | 357 --- |
| 358 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
| 359 --- 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 | |
| 360 -- | |
| 361 -- | |
| 142 | 362 ∩-≡ : { a b : OD {suc n} } → ({x : OD {suc n} } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) |
| 363 ∩-≡ {a} {b} inc = record { | |
| 364 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
| 365 proj1 = def-subst {suc n} {_} {_} {b} {x} (inc (def-subst {suc n} {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; | |
| 366 eq← = λ {x} x<a∩b → proj2 x<a∩b } | |
| 100 | 367 -- |
| 368 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t | |
| 369 -- Power A is a sup of ZFSubset A t, so Power A ∋ t | |
| 370 -- | |
| 141 | 371 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t |
| 129 | 372 ord-power← a t t→A = def-subst {suc n} {_} {_} {Def (Ord a)} {od→ord t} |
| 127 | 373 lemma refl (lemma1 lemma-eq )where |
| 129 | 374 lemma-eq : ZFSubset (Ord a) t == t |
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375 eq→ lemma-eq {z} w = proj2 w |
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376 eq← lemma-eq {z} w = record { proj2 = w ; |
| 129 | 377 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z} |
| 126 | 378 ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
| 141 | 379 lemma1 : {n : Level } {a : Ordinal {suc n}} { t : OD {suc n}} |
| 129 | 380 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t |
| 150 | 381 lemma1 {n} {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 | 382 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 | 383 lemma = sup-o< |
| 129 | 384 |
| 179 | 385 -- double-neg-eilm : {n : Level } {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic |
| 144 | 386 -- |
| 387 -- Every set in OD is a subset of Ordinals | |
| 388 -- | |
| 142 | 389 -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y ) |
| 166 | 390 |
| 391 -- we have oly double negation form because of the replacement axiom | |
| 392 -- | |
| 393 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
| 394 power→ A t P∋t {x} t∋x = TransFiniteExists _ lemma5 lemma4 where | |
| 142 | 395 a = od→ord A |
| 396 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y))) | |
| 397 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t | |
| 166 | 398 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) |
| 399 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) | |
| 142 | 400 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) |
| 401 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) | |
| 166 | 402 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) |
| 403 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not | |
| 404 | |
| 142 | 405 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
| 406 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where | |
| 407 a = od→ord A | |
| 408 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x | |
| 409 lemma0 {x} t∋x = c<→o< (t→A t∋x) | |
| 410 lemma3 : Def (Ord a) ∋ t | |
| 411 lemma3 = ord-power← a t lemma0 | |
| 152 | 412 lt1 : od→ord (A ∩ ord→od (od→ord t)) o< sup-o (λ x → od→ord (A ∩ ord→od x)) |
| 413 lt1 = sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t} | |
| 414 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t | |
| 415 lemma4 = let open ≡-Reasoning in begin | |
| 416 A ∩ ord→od (od→ord t) | |
| 417 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
| 418 A ∩ t | |
| 419 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ | |
| 420 t | |
| 421 ∎ | |
| 142 | 422 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ ord→od x)) |
| 152 | 423 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ ord→od x))) |
| 424 lemma4 (sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t}) | |
| 142 | 425 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
| 151 | 426 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
| 427 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
| 428 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) | |
| 142 | 429 |
| 141 | 430 regularity : (x : OD) (not : ¬ (x == od∅)) → |
| 115 | 431 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) |
| 117 | 432 proj1 (regularity x not ) = x∋minimul x not |
| 433 proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where | |
| 434 lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁ | |
| 435 lemma1 {x₁} s = ⊥-elim ( minimul-1 x not (ord→od x₁) lemma3 ) where | |
| 436 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁)) | |
| 142 | 437 lemma3 = record { proj1 = def-subst {suc n} {_} {_} {minimul x not} {_} (proj1 s) refl (sym diso) |
| 438 ; proj2 = proj2 (proj2 s) } | |
| 117 | 439 lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ |
| 440 lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) )) | |
| 129 | 441 |
| 141 | 442 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
| 76 | 443 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d |
| 444 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 129 | 445 |
| 161 | 446 infinity∅ : infinite ∋ od∅ {suc n} |
| 447 infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} iφ refl lemma where | |
| 448 lemma : o∅ ≡ od→ord od∅ | |
| 449 lemma = let open ≡-Reasoning in begin | |
| 450 o∅ | |
| 451 ≡⟨ sym diso ⟩ | |
| 452 od→ord ( ord→od o∅ ) | |
| 453 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
| 454 od→ord od∅ | |
| 455 ∎ | |
| 456 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) | |
| 457 infinity x lt = def-subst {suc n} {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
| 458 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) | |
| 459 ≡ od→ord (Union (x , (x , x))) | |
| 460 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
| 461 | |
| 179 | 462 -- Axiom of choice ( is equivalent to the existence of minimul in our case ) |
| 162 | 463 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] |
| 464 choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD | |
| 465 choice-func X {x} not X∋x = minimul x not | |
| 466 choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A | |
| 467 choice X {A} X∋A not = x∋minimul A not | |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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468 |
| 176 | 469 -- another form of regularity |
| 470 -- | |
| 471 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} | |
| 472 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) | |
| 473 → (x : OD {suc n} ) → ψ x | |
| 474 ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where | |
| 475 ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly } | |
| 476 → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) ) | |
| 477 ε-induction-ord Zero (Φ 0) (case1 ()) | |
| 478 ε-induction-ord Zero (Φ 0) (case2 ()) | |
| 479 ε-induction-ord lx (OSuc lx ox) {ly} {oy} y<x = | |
| 480 ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where | |
| 481 lemma : (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox } | |
| 482 lemma y lt with osuc-≡< y<x | |
| 483 lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso | |
| 484 lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 | |
| 485 ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) = | |
| 486 ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt ) where | |
| 179 | 487 -- |
| 488 -- if lv of z if less than x Ok | |
| 489 -- else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma | |
| 490 -- | |
| 491 -- lx Suc lx (1) lz(a) <lx by case1 | |
| 492 -- ly(1) ly(2) (2) lz(b) <lx by case1 | |
| 493 -- lz(a) lz(b) lz(c) lz(c) <lx by case2 ( ly==lz==lx) | |
| 494 -- | |
| 176 | 495 lemma0 : { lx ly : Nat } → ly < Suc lx → lx < ly → ⊥ |
| 496 lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2 | |
| 497 lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly | |
| 498 lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin | |
| 499 lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) | |
| 500 ≡⟨ cong ( λ k → lv k ) diso ⟩ | |
| 501 lv (record { lv = ly ; ord = oy }) | |
| 502 ≡⟨⟩ | |
| 503 ly | |
| 504 ∎ | |
| 505 lemma2 : { lx : Nat } → lx < Suc lx | |
| 506 lemma2 {Zero} = s≤s z≤n | |
| 507 lemma2 {Suc lx} = s≤s (lemma2 {lx}) | |
| 508 lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z | |
| 509 lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt | |
| 510 lemma z lt | case1 lz<ly with <-cmp lx ly | |
| 511 lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen | |
| 512 lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c = -- (1) | |
| 513 subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) )) | |
| 514 lemma z lt | case1 lz<ly | tri> ¬a ¬b c = -- z(a) | |
| 515 subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c)))) | |
| 516 lemma z lt | case2 lz=ly with <-cmp lx ly | |
| 517 lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen | |
| 518 lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- z(b) | |
| 179 | 519 ... | eq = subst (λ k → ψ k ) oiso |
| 520 (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c ))) | |
| 521 lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly -- z(c) | |
| 522 ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where | |
| 176 | 523 lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z |
| 524 lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt ) | |
| 525 lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly } → | |
| 179 | 526 lx ≡ ly → ly ≡ lv (od→ord z) → ψ z |
| 176 | 527 lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl) |
| 528 |