Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 216:5b9b6ef971dd
Cardinal start
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sun, 04 Aug 2019 18:09:00 +0900 |
| parents | f15eaa7c5932 |
| children | d5668179ee69 |
| rev | line source |
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| 16 | 1 open import Level |
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2 module OD 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 | |
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14 open import logic |
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15 open import nat |
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16 |
| 27 | 17 -- Ordinal Definable Set |
| 11 | 18 |
| 141 | 19 record OD {n : Level} : Set (suc n) where |
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20 field |
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21 def : (x : Ordinal {n} ) → Set n |
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22 |
| 141 | 23 open OD |
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24 |
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25 open Ordinal |
| 120 | 26 open _∧_ |
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27 open _∨_ |
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28 open Bool |
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29 |
| 141 | 30 record _==_ {n : Level} ( a b : OD {n} ) : Set n where |
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31 field |
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32 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x |
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33 eq← : ∀ { x : Ordinal {n} } → def b x → def a x |
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34 |
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35 id : {n : Level} {A : Set n} → A → A |
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36 id x = x |
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37 |
| 141 | 38 eq-refl : {n : Level} { x : OD {n} } → x == x |
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39 eq-refl {n} {x} = record { eq→ = id ; eq← = id } |
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40 |
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41 open _==_ |
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42 |
| 141 | 43 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x |
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44 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } |
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45 |
| 141 | 46 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z |
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47 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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48 |
| 141 | 49 ⇔→== : {n : Level} { x y : OD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔ def y z) → x == y |
| 120 | 50 eq→ ( ⇔→== {n} {x} {y} eq ) {z} m = proj1 eq m |
| 51 eq← ( ⇔→== {n} {x} {y} eq ) {z} m = proj2 eq m | |
| 52 | |
| 179 | 53 -- Ordinal in OD ( and ZFSet ) Transitive Set |
| 141 | 54 Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} |
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55 Ord {n} a = record { def = λ y → y o< a } |
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56 |
| 141 | 57 od∅ : {n : Level} → OD {n} |
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58 od∅ {n} = Ord o∅ |
| 40 | 59 |
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60 postulate |
| 141 | 61 -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) |
| 62 od→ord : {n : Level} → OD {n} → Ordinal {n} | |
| 63 ord→od : {n : Level} → Ordinal {n} → OD {n} | |
| 166 | 64 c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y |
| 65 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x | |
| 113 | 66 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x |
| 150 | 67 -- we should prove this in agda, but simply put here |
| 141 | 68 ==→o≡ : {n : Level} → { x y : OD {suc n} } → (x == y) → x ≡ y |
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69 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set |
| 159 | 70 -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x |
| 71 -- ord→od x ≡ Ord x results the same | |
| 100 | 72 -- supermum as Replacement Axiom |
| 95 | 73 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 98 | 74 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ |
| 111 | 75 -- contra-position of mimimulity of supermum required in Power Set Axiom |
| 165 | 76 -- sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} |
| 77 -- sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 183 | 78 -- mimimul and x∋minimul is an Axiom of choice |
| 141 | 79 minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} |
| 117 | 80 -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) |
| 141 | 81 x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) |
| 187 | 82 -- minimulity (may proved by ε-induction ) |
| 141 | 83 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 | 84 |
| 141 | 85 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n |
| 95 | 86 _∋_ {n} a x = def a ( od→ord x ) |
| 87 | |
| 141 | 88 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n |
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89 x c< a = a ∋ x |
| 103 | 90 |
| 141 | 91 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) |
| 95 | 92 a c≤ b = (a ≡ b) ∨ ( b ∋ a ) |
| 93 | |
| 141 | 94 cseq : {n : Level} → OD {n} → OD {n} |
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95 cseq x = record { def = λ y → def x (osuc y) } where |
| 113 | 96 |
| 141 | 97 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 | 98 def-subst df refl refl = df |
| 99 | |
| 141 | 100 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} |
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101 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) |
| 95 | 102 |
| 141 | 103 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) |
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104 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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105 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where |
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106 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x))) |
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107 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) ) |
| 28 | 108 |
| 187 | 109 otrans : {n : Level} {a x y : Ordinal {n} } → def (Ord a) x → def (Ord x) y → def (Ord a) y |
| 110 otrans x<a y<x = ordtrans y<x x<a | |
| 123 | 111 |
| 205 | 112 def→o< : {n : Level } {X : OD {suc n}} → {x : Ordinal {suc n}} → def X x → x o< od→ord X |
| 113 def→o< {n} {X} {x} lt = o<-subst {suc n} {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {suc n} {X} {x} lt (sym oiso) (sym diso) )) diso diso | |
| 114 | |
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115 ∅3 : {n : Level} → { x : Ordinal {suc n}} → ( ∀(y : Ordinal {suc n}) → ¬ (y o< x ) ) → x ≡ o∅ {suc n} |
| 81 | 116 ∅3 {n} {x} = TransFinite {n} c2 c3 x where |
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117 c0 : Nat → Ordinal {suc n} → Set (suc n) |
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118 c0 lx x = (∀(y : Ordinal {suc n}) → ¬ (y o< x)) → x ≡ o∅ {suc n} |
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119 c2 : (lx : Nat) → ((x₁ : Ordinal) → x₁ o< ordinal lx (Φ lx) → c0 (lv x₁) (record { lv = lv x₁ ; ord = ord x₁ }))→ c0 lx (record { lv = lx ; ord = Φ lx } ) |
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120 c2 Zero _ not = refl |
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121 c2 (Suc lx) _ not with not ( record { lv = lx ; ord = Φ lx } ) |
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122 ... | t with t (case1 ≤-refl ) |
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123 c2 (Suc lx) _ not | t | () |
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124 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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125 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) |
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126 ... | t with t (case2 Φ< ) |
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127 c3 lx (Φ .lx) d not | t | () |
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128 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) |
| 34 | 129 ... | t with t (case2 (s< s<refl ) ) |
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130 c3 lx (OSuc .lx x₁) d not | t | () |
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131 |
| 57 | 132 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x |
| 133 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) | |
| 134 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< | |
| 135 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) | |
| 37 | 136 |
| 46 | 137 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 } |
| 138 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso | |
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139 |
| 51 | 140 -- avoiding lv != Zero error |
| 141 | 141 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y |
| 51 | 142 orefl refl = refl |
| 143 | |
| 141 | 144 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y |
| 51 | 145 ==-iso {n} {x} {y} eq = record { |
| 146 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; | |
| 147 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } | |
| 148 where | |
| 141 | 149 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z |
| 51 | 150 lemma {x} {z} d = def-subst d oiso refl |
| 151 | |
| 141 | 152 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) |
| 57 | 153 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) |
| 154 | |
| 141 | 155 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y |
| 51 | 156 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
| 157 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) | |
| 158 lemma ox ox refl = eq-refl | |
| 159 | |
| 160 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y | |
| 161 o≡→== {n} {x} {.x} refl = eq-refl | |
| 162 | |
| 141 | 163 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x |
| 51 | 164 c≤-refl x = case1 refl |
| 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 |
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179 ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ } == od∅ {n} |
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180 eq→ ∅0 {w} (lift ()) |
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181 eq← ∅0 {w} (case1 ()) |
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182 eq← ∅0 {w} (case2 ()) |
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183 |
| 141 | 184 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) |
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185 ∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d |
| 60 | 186 ∅< {n} {x} {y} d eq | lift () |
| 57 | 187 |
| 141 | 188 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox |
| 120 | 189 ∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {x} {x} x∋x ) |
| 51 | 190 |
| 141 | 191 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 | 192 def-iso refl t = t |
| 193 | |
| 57 | 194 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) |
| 195 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl | |
| 196 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) | |
| 197 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) | |
| 198 | |
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199 ppp : { n : Level } → { p : Set (suc n) } { a : OD {suc n} } → record { def = λ x → p } ∋ a → p |
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200 ppp {n} {p} {a} d = d |
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201 |
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202 -- |
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203 -- Axiom of choice in intutionistic logic implies the exclude middle |
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204 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog |
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205 -- |
| 190 | 206 p∨¬p : { n : Level } → ( p : Set (suc n) ) → p ∨ ( ¬ p ) -- assuming axiom of choice |
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207 p∨¬p {n} p with is-o∅ ( od→ord ( record { def = λ x → p } )) |
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208 p∨¬p {n} p | yes eq = case2 (¬p eq) where |
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209 ps = record { def = λ x → p } |
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210 lemma : ps == od∅ → p → ⊥ |
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211 lemma eq p0 = ¬x<0 {n} {od→ord ps} (eq→ eq p0 ) |
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212 ¬p : (od→ord ps ≡ o∅) → p → ⊥ |
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213 ¬p eq = lemma ( subst₂ (λ j k → j == k ) oiso o∅≡od∅ ( o≡→== eq )) |
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214 p∨¬p {n} p | no ¬p = case1 (ppp {n} {p} {minimul ps (λ eq → ¬p (eqo∅ eq))} lemma) where |
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215 ps = record { def = λ x → p } |
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216 eqo∅ : ps == od∅ {suc n} → od→ord ps ≡ o∅ {suc n} |
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217 eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) |
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218 lemma : ps ∋ minimul ps (λ eq → ¬p (eqo∅ eq)) |
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219 lemma = x∋minimul ps (λ eq → ¬p (eqo∅ eq)) |
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220 |
| 190 | 221 ∋-p : { n : Level } → ( p : Set (suc n) ) → Dec p -- assuming axiom of choice |
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222 ∋-p {n} p with p∨¬p p |
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223 ∋-p {n} p | case1 x = yes x |
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224 ∋-p {n} p | case2 x = no x |
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225 |
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226 double-neg-eilm : {n : Level } {A : Set (suc n)} → ¬ ¬ A → A -- we don't have this in intutionistic logic |
| 190 | 227 double-neg-eilm {n} {A} notnot with ∋-p A -- assuming axiom of choice |
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228 ... | yes p = p |
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229 ... | no ¬p = ⊥-elim ( notnot ¬p ) |
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230 |
| 167 | 231 OrdP : {n : Level} → ( x : Ordinal {suc n} ) ( y : OD {suc n} ) → Dec ( Ord x ∋ y ) |
| 232 OrdP {n} x y with trio< x (od→ord y) | |
| 233 OrdP {n} x y | tri< a ¬b ¬c = no ¬c | |
| 234 OrdP {n} x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) | |
| 235 OrdP {n} x y | tri> ¬a ¬b c = yes c | |
| 119 | 236 |
| 79 | 237 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 94 | 238 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) |
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239 |
| 141 | 240 in-codomain : {n : Level} → (X : OD {suc n} ) → ( ψ : OD {suc n} → OD {suc n} ) → OD {suc n} |
| 148 | 241 in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } |
| 141 | 242 |
| 96 | 243 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) |
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244 |
| 141 | 245 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} |
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246 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set |
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247 |
| 141 | 248 Def : {n : Level} → (A : OD {suc n}) → OD {suc n} |
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249 Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
| 96 | 250 |
| 190 | 251 |
| 252 _⊆_ : {n : Level} ( A B : OD {suc n} ) → ∀{ x : OD {suc n} } → Set (suc n) | |
| 253 _⊆_ A B {x} = A ∋ x → B ∋ x | |
| 254 | |
| 255 infixr 220 _⊆_ | |
| 256 | |
| 257 subset-lemma : {n : Level} → {A x y : OD {suc n} } → ( x ∋ y → ZFSubset A x ∋ y ) ⇔ ( _⊆_ x A {y} ) | |
| 258 subset-lemma {n} {A} {x} {y} = record { | |
| 259 proj1 = λ z lt → proj1 (z lt) | |
| 260 ; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt } | |
| 261 } | |
| 262 | |
| 96 | 263 -- Constructible Set on α |
| 122 | 264 -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x } |
| 265 -- L (Φ 0) = Φ | |
| 266 -- L (OSuc lv n) = { Def ( L n ) } | |
| 267 -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) ) | |
| 141 | 268 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n} |
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269 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ |
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270 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) |
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271 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) |
| 121 | 272 cseq ( Ord (od→ord (L {n} (record { lv = lx ; ord = Φ lx })))) |
| 89 | 273 |
| 167 | 274 -- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α |
| 141 | 275 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x |
| 122 | 276 |
| 170 | 277 |
| 141 | 278 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} |
| 279 OD→ZF {n} = record { | |
| 280 ZFSet = OD {suc n} | |
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281 ; _∋_ = _∋_ |
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282 ; _≈_ = _==_ |
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283 ; ∅ = od∅ |
| 28 | 284 ; _,_ = _,_ |
| 285 ; Union = Union | |
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286 ; Power = Power |
| 28 | 287 ; Select = Select |
| 288 ; Replace = Replace | |
| 161 | 289 ; infinite = infinite |
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290 ; isZF = isZF |
| 28 | 291 } where |
| 144 | 292 ZFSet = OD {suc n} |
| 141 | 293 Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → Set (suc n) ) → OD {suc n} |
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294 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } |
| 141 | 295 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} |
| 296 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 } | |
| 297 _,_ : OD {suc n} → OD {suc n} → OD {suc n} | |
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298 x , y = Ord (omax (od→ord x) (od→ord y)) |
| 144 | 299 _∩_ : ( A B : ZFSet ) → ZFSet |
| 145 | 300 A ∩ B = record { def = λ x → def A x ∧ def B x } |
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301 Union : OD {suc n} → OD {suc n} |
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302 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } |
| 54 | 303 _∈_ : ( A B : ZFSet ) → Set (suc n) |
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304 A ∈ B = B ∋ A |
| 141 | 305 Power : OD {suc n} → OD {suc n} |
| 129 | 306 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x ) |
| 103 | 307 ｛_｝ : ZFSet → ZFSet |
| 308 ｛ x ｝ = ( x , x ) | |
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309 |
| 161 | 310 data infinite-d : ( x : Ordinal {suc n} ) → Set (suc n) where |
| 311 iφ : infinite-d o∅ | |
| 312 isuc : {x : Ordinal {suc n} } → infinite-d x → | |
| 313 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) | |
| 314 | |
| 315 infinite : OD {suc n} | |
| 316 infinite = record { def = λ x → infinite-d x } | |
| 317 | |
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318 infixr 200 _∈_ |
| 96 | 319 -- infixr 230 _∩_ _∪_ |
| 161 | 320 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite |
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321 isZF = record { |
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322 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } |
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323 ; pair = pair |
| 72 | 324 ; union→ = union→ |
| 325 ; union← = union← | |
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326 ; empty = empty |
| 129 | 327 ; power→ = power→ |
| 76 | 328 ; power← = power← |
| 186 | 329 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
| 183 | 330 ; ε-induction = ε-induction |
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331 ; infinity∅ = infinity∅ |
| 160 | 332 ; infinity = infinity |
| 116 | 333 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
| 135 | 334 ; replacement← = replacement← |
| 335 ; replacement→ = replacement→ | |
| 183 | 336 ; choice-func = choice-func |
| 337 ; choice = choice | |
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338 } where |
| 129 | 339 |
| 141 | 340 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) |
| 87 | 341 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) |
| 342 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) | |
| 129 | 343 |
| 167 | 344 empty : {n : Level } (x : OD {suc n} ) → ¬ (od∅ ∋ x) |
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345 empty x (case1 ()) |
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346 empty x (case2 ()) |
| 129 | 347 |
| 154 | 348 o<→c< : {x y : Ordinal {suc n}} {z : OD {suc n}}→ x o< y → _⊆_ (Ord x) (Ord y) {z} |
| 155 | 349 o<→c< lt lt1 = ordtrans lt1 lt |
| 350 | |
| 351 ⊆→o< : {x y : Ordinal {suc n}} → (∀ (z : OD) → _⊆_ (Ord x) (Ord y) {z} ) → x o< osuc y | |
| 352 ⊆→o< {x} {y} lt with trio< x y | |
| 353 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
| 354 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
| 355 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl ) | |
| 356 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) | |
| 151 | 357 |
| 144 | 358 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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359 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
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360 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) |
| 159 | 361 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) |
| 166 | 362 union← X z UX∋z = TransFiniteExists _ lemma UX∋z where |
| 165 | 363 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
| 364 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
| 144 | 365 |
| 366 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y | |
| 367 ψiso {ψ} t refl = t | |
| 368 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
| 369 selection {ψ} {X} {y} = record { | |
| 370 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
| 371 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
| 372 } | |
| 373 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x | |
| 374 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where | |
| 375 lemma : def (in-codomain X ψ) (od→ord (ψ x)) | |
| 150 | 376 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
| 144 | 377 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) |
| 150 | 378 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
| 379 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) | |
| 380 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) | |
| 144 | 381 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
| 150 | 382 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) |
| 144 | 383 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) |
| 150 | 384 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) |
| 385 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) | |
| 144 | 386 |
| 387 --- | |
| 388 --- Power Set | |
| 389 --- | |
| 390 --- First consider ordinals in OD | |
| 100 | 391 --- |
| 392 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A | |
| 393 --- 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 | |
| 394 -- | |
| 395 -- | |
| 142 | 396 ∩-≡ : { a b : OD {suc n} } → ({x : OD {suc n} } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) |
| 397 ∩-≡ {a} {b} inc = record { | |
| 398 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
| 399 proj1 = def-subst {suc n} {_} {_} {b} {x} (inc (def-subst {suc n} {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; | |
| 400 eq← = λ {x} x<a∩b → proj2 x<a∩b } | |
| 100 | 401 -- |
| 402 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t | |
| 403 -- Power A is a sup of ZFSubset A t, so Power A ∋ t | |
| 404 -- | |
| 141 | 405 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t |
| 129 | 406 ord-power← a t t→A = def-subst {suc n} {_} {_} {Def (Ord a)} {od→ord t} |
| 127 | 407 lemma refl (lemma1 lemma-eq )where |
| 129 | 408 lemma-eq : ZFSubset (Ord a) t == t |
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410 eq← lemma-eq {z} w = record { proj2 = w ; |
| 129 | 411 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z} |
| 126 | 412 ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } |
| 141 | 413 lemma1 : {n : Level } {a : Ordinal {suc n}} { t : OD {suc n}} |
| 129 | 414 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t |
| 150 | 415 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 | 416 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 | 417 lemma = sup-o< |
| 129 | 418 |
| 144 | 419 -- |
| 420 -- Every set in OD is a subset of Ordinals | |
| 421 -- | |
| 142 | 422 -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y ) |
| 166 | 423 |
| 424 -- we have oly double negation form because of the replacement axiom | |
| 425 -- | |
| 426 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
| 427 power→ A t P∋t {x} t∋x = TransFiniteExists _ lemma5 lemma4 where | |
| 142 | 428 a = od→ord A |
| 429 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y))) | |
| 430 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t | |
| 166 | 431 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) |
| 432 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) | |
| 142 | 433 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) |
| 434 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) | |
| 166 | 435 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) |
| 436 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not | |
| 437 | |
| 142 | 438 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
| 439 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where | |
| 440 a = od→ord A | |
| 441 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x | |
| 442 lemma0 {x} t∋x = c<→o< (t→A t∋x) | |
| 443 lemma3 : Def (Ord a) ∋ t | |
| 444 lemma3 = ord-power← a t lemma0 | |
| 152 | 445 lt1 : od→ord (A ∩ ord→od (od→ord t)) o< sup-o (λ x → od→ord (A ∩ ord→od x)) |
| 446 lt1 = sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t} | |
| 447 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t | |
| 448 lemma4 = let open ≡-Reasoning in begin | |
| 449 A ∩ ord→od (od→ord t) | |
| 450 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
| 451 A ∩ t | |
| 452 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ | |
| 453 t | |
| 454 ∎ | |
| 142 | 455 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ ord→od x)) |
| 152 | 456 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ ord→od x))) |
| 457 lemma4 (sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t}) | |
| 142 | 458 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
| 151 | 459 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
| 460 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
| 461 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) | |
| 142 | 462 |
| 190 | 463 -- assuming axiom of choice |
| 141 | 464 regularity : (x : OD) (not : ¬ (x == od∅)) → |
| 115 | 465 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) |
| 117 | 466 proj1 (regularity x not ) = x∋minimul x not |
| 467 proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where | |
| 468 lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁ | |
| 469 lemma1 {x₁} s = ⊥-elim ( minimul-1 x not (ord→od x₁) lemma3 ) where | |
| 470 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁)) | |
| 142 | 471 lemma3 = record { proj1 = def-subst {suc n} {_} {_} {minimul x not} {_} (proj1 s) refl (sym diso) |
| 472 ; proj2 = proj2 (proj2 s) } | |
| 117 | 473 lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ |
| 474 lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) )) | |
| 129 | 475 |
| 186 | 476 extensionality0 : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B |
| 477 eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
| 478 eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
| 479 | |
| 480 extensionality : {A B w : OD {suc n} } → ((z : OD {suc n}) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) | |
| 481 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d | |
| 482 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
| 129 | 483 |
| 161 | 484 infinity∅ : infinite ∋ od∅ {suc n} |
| 485 infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} iφ refl lemma where | |
| 486 lemma : o∅ ≡ od→ord od∅ | |
| 487 lemma = let open ≡-Reasoning in begin | |
| 488 o∅ | |
| 489 ≡⟨ sym diso ⟩ | |
| 490 od→ord ( ord→od o∅ ) | |
| 491 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
| 492 od→ord od∅ | |
| 493 ∎ | |
| 494 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) | |
| 495 infinity x lt = def-subst {suc n} {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
| 496 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) | |
| 497 ≡ od→ord (Union (x , (x , x))) | |
| 498 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
| 499 | |
| 179 | 500 -- Axiom of choice ( is equivalent to the existence of minimul in our case ) |
| 162 | 501 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] |
| 502 choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD | |
| 503 choice-func X {x} not X∋x = minimul x not | |
| 504 choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A | |
| 505 choice X {A} X∋A not = x∋minimul A not | |
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507 -- |
| 176 | 508 -- another form of regularity |
| 509 -- | |
| 510 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} | |
| 511 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) | |
| 512 → (x : OD {suc n} ) → ψ x | |
| 513 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 | |
| 514 ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly } | |
| 515 → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) ) | |
| 516 ε-induction-ord lx (OSuc lx ox) {ly} {oy} y<x = | |
| 517 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 | |
| 197 | 518 lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → od→ord z o< record { lv = lx ; ord = ox } |
| 519 lemma z lt with osuc-≡< y<x | |
| 520 lemma z lt | case1 refl = o<-subst (c<→o< lt) refl diso | |
| 521 lemma z lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 | |
| 176 | 522 ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) = |
| 523 ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt ) where | |
| 179 | 524 -- |
| 525 -- if lv of z if less than x Ok | |
| 526 -- else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma | |
| 527 -- | |
| 528 -- lx Suc lx (1) lz(a) <lx by case1 | |
| 529 -- ly(1) ly(2) (2) lz(b) <lx by case1 | |
| 530 -- lz(a) lz(b) lz(c) lz(c) <lx by case2 ( ly==lz==lx) | |
| 531 -- | |
| 176 | 532 lemma0 : { lx ly : Nat } → ly < Suc lx → lx < ly → ⊥ |
| 533 lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2 | |
| 534 lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly | |
| 535 lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin | |
| 536 lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) | |
| 537 ≡⟨ cong ( λ k → lv k ) diso ⟩ | |
| 538 lv (record { lv = ly ; ord = oy }) | |
| 539 ≡⟨⟩ | |
| 540 ly | |
| 541 ∎ | |
| 542 lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z | |
| 543 lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt | |
| 544 lemma z lt | case1 lz<ly with <-cmp lx ly | |
| 545 lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen | |
| 180 | 546 lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c = -- ly(1) |
| 176 | 547 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 ) )) |
| 180 | 548 lemma z lt | case1 lz<ly | tri> ¬a ¬b c = -- lz(a) |
| 176 | 549 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)))) |
| 550 lemma z lt | case2 lz=ly with <-cmp lx ly | |
| 551 lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen | |
| 180 | 552 lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- lz(b) |
| 179 | 553 ... | eq = subst (λ k → ψ k ) oiso |
| 554 (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c ))) | |
| 180 | 555 lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly -- lz(c) |
| 179 | 556 ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where |
| 176 | 557 lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z |
| 558 lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt ) | |
| 559 lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly } → | |
| 179 | 560 lx ≡ ly → ly ≡ lv (od→ord z) → ψ z |
| 176 | 561 lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl) |
| 562 | |
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563 --- |
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564 --- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice |
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565 --- |
| 196 | 566 record choiced {n : Level} ( X : OD {suc n}) : Set (suc (suc n)) where |
| 567 field | |
| 568 a-choice : OD {suc n} | |
| 569 is-in : X ∋ a-choice | |
| 209 | 570 choice-func' : (X : OD {suc n} ) → (p∨¬p : { n : Level } → ( p : Set (suc n) ) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X |
| 571 choice-func' X p∨¬p not = have_to_find | |
| 205 | 572 where |
| 207 | 573 ψ : ( ox : Ordinal {suc n}) → Set (suc (suc n)) |
| 209 | 574 ψ ox = (( x : Ordinal {suc n}) → x o< ox → ( ¬ def X x )) ∨ choiced X |
| 207 | 575 lemma-ord : ( ox : Ordinal {suc n} ) → ψ ox |
| 576 lemma-ord ox = TransFinite {n} {suc (suc n)} {ψ} caseΦ caseOSuc ox where | |
| 209 | 577 ∋-p' : (A x : OD {suc n} ) → Dec ( A ∋ x ) |
| 578 ∋-p' A x with p∨¬p ( A ∋ x ) | |
| 579 ∋-p' A x | case1 t = yes t | |
| 580 ∋-p' A x | case2 t = no t | |
| 581 ∀-imply-or : {n : Level} {A : Ordinal {suc n} → Set (suc n) } {B : Set (suc (suc n)) } | |
| 582 → ((x : Ordinal {suc n}) → A x ∨ B) → ((x : Ordinal {suc n}) → A x) ∨ B | |
| 583 ∀-imply-or {n} {A} {B} ∀AB with p∨¬p ((x : Ordinal {suc n}) → A x) | |
| 584 ∀-imply-or {n} {A} {B} ∀AB | case1 t = case1 t | |
| 585 ∀-imply-or {n} {A} {B} ∀AB | case2 x = case2 (lemma x) where | |
| 586 lemma : ¬ ((x : Ordinal {suc n}) → A x) → B | |
| 587 lemma not with p∨¬p B | |
| 588 lemma not | case1 b = b | |
| 589 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
| 207 | 590 caseΦ : (lx : Nat) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x) → ψ (ordinal lx (Φ lx) ) |
| 209 | 591 caseΦ lx prev with ∋-p' X ( ord→od (ordinal lx (Φ lx) )) |
| 592 caseΦ lx prev | yes p = case2 ( record { a-choice = ord→od (ordinal lx (Φ lx)) ; is-in = p } ) | |
| 593 caseΦ lx prev | no ¬p = lemma where | |
| 594 lemma1 : (x : Ordinal) → (((Suc (lv x) ≤ lx) ∨ (ord x d< Φ lx) → def X x → ⊥) ∨ choiced X) | |
| 595 lemma1 x with trio< x (ordinal lx (Φ lx)) | |
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596 lemma1 x | tri< a ¬b ¬c with prev (osuc x) (lemma2 a) where |
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597 lemma2 : x o< (ordinal lx (Φ lx)) → osuc x o< ordinal lx (Φ lx) |
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598 lemma2 (case1 lt) = case1 lt |
| 209 | 599 lemma1 x | tri< a ¬b ¬c | case2 found = case2 found |
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600 lemma1 x | tri< a ¬b ¬c | case1 not_found = case1 ( λ lt df → not_found x <-osuc df ) |
| 209 | 601 lemma1 x | tri≈ ¬a refl ¬c = case1 ( λ lt → ⊥-elim (o<¬≡ refl lt )) |
| 602 lemma1 x | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim (o<> lt c )) | |
| 603 lemma : ((x : Ordinal) → (Suc (lv x) ≤ lx) ∨ (ord x d< Φ lx) → def X x → ⊥) ∨ choiced X | |
| 604 lemma = ∀-imply-or lemma1 | |
| 207 | 605 caseOSuc : (lx : Nat) (x : OrdinalD lx) → ψ ( ordinal lx x ) → ψ ( ordinal lx (OSuc lx x) ) |
| 209 | 606 caseOSuc lx x prev with ∋-p' X (ord→od record { lv = lx ; ord = x } ) |
| 607 caseOSuc lx x prev | yes p = case2 (record { a-choice = ord→od record { lv = lx ; ord = x } ; is-in = p }) | |
| 608 caseOSuc lx x (case1 not_found) | no ¬p = case1 lemma where | |
| 609 lemma : (y : Ordinal) → (Suc (lv y) ≤ lx) ∨ (ord y d< OSuc lx x) → def X y → ⊥ | |
| 610 lemma y lt with trio< y (ordinal lx x ) | |
| 611 lemma y lt | tri< a ¬b ¬c = not_found y a | |
| 612 lemma y lt | tri≈ ¬a refl ¬c = subst (λ k → def X k → ⊥ ) diso ¬p | |
| 613 lemma y lt | tri> ¬a ¬b c with osuc-≡< lt | |
| 614 lemma y lt | tri> ¬a ¬b c | case1 refl = ⊥-elim ( ¬b refl ) | |
| 615 lemma y lt | tri> ¬a ¬b c | case2 lt1 = ⊥-elim (o<> c lt1 ) | |
| 616 caseOSuc lx x (case2 found) | no ¬p = case2 found | |
| 206 | 617 have_to_find : choiced X |
| 618 have_to_find with lemma-ord (od→ord X ) | |
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619 have_to_find | t = dont-or t ¬¬X∋x where |
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620 ¬¬X∋x : ¬ ((x : Ordinal) → (Suc (lv x) ≤ lv (od→ord X)) ∨ (ord x d< ord (od→ord X)) → def X x → ⊥) |
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621 ¬¬X∋x nn = not record { |
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622 eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt) |
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623 ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt ) |
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624 } |
| 205 | 625 |
| 216 | 626 ------------ |
| 627 -- | |
| 628 -- Onto map | |
| 629 -- def X x -> xmap | |
| 630 -- X ---------------------------> Y | |
| 631 -- ymap <- def Y y | |
| 632 -- | |
| 633 record Onto {n : Level } (X Y : OD {suc n}) : Set (suc (suc n)) where | |
| 634 field | |
| 635 xmap : (x : Ordinal {suc n}) → Ordinal {suc n} | |
| 636 ymap : (y : Ordinal {suc n}) → Ordinal {suc n} | |
| 637 xmap-on-Y : (x : Ordinal {suc n} ) → def X x → def Y (xmap x) | |
| 638 ymap-on-X : (y : Ordinal {suc n} ) → def Y y → def X (ymap y) | |
| 639 onto-iso : (y : Ordinal {suc n} ) → def Y y → xmap ( ymap y ) ≡ y | |
| 640 | |
| 641 record Cardinal {n : Level } (X : OD {suc n}) : Set (suc (suc n)) where | |
| 642 field | |
| 643 cardinal : Ordinal {suc n} | |
| 644 conto : Onto (Ord cardinal) X | |
| 645 cmax : ( y : Ordinal {suc n} ) → cardinal o< y → ¬ Onto (Ord y) X | |
| 646 | |
| 647 cardinal : {n : Level } (X : OD {suc n}) → Cardinal X | |
| 648 cardinal {n} X = record { | |
| 649 cardinal = sup-o ( λ x → cardinal-p x ) | |
| 650 ; conto = {!!} | |
| 651 ; cmax = {!!} | |
| 652 } where | |
| 653 cardinal-p : (x : Ordinal {suc n}) → Ordinal {suc n} | |
| 654 cardinal-p x with p∨¬p ( Onto (Ord x) X ) | |
| 655 cardinal-p x | case1 True = x | |
| 656 cardinal-p x | case2 False = o∅ |