Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 358:811152bf2f47
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 14 Jul 2020 12:39:21 +0900 |
parents | 08d94fec239c |
children | 2a8a51375e49 |
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16 | 1 open import Level |
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2 open import Ordinals |
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3 module OD {n : Level } (O : Ordinals {n} ) where |
3 | 4 |
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5 open import zf |
23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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7 open import Relation.Binary.PropositionalEquality |
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8 open import Data.Nat.Properties |
6 | 9 open import Data.Empty |
10 open import Relation.Nullary | |
11 open import Relation.Binary | |
12 open import Relation.Binary.Core | |
13 | |
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14 open import logic |
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15 open import nat |
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16 |
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17 open inOrdinal O |
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18 |
27 | 19 -- Ordinal Definable Set |
11 | 20 |
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21 record OD : Set (suc n ) where |
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22 field |
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23 def : (x : Ordinal ) → Set n |
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24 |
141 | 25 open OD |
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26 |
120 | 27 open _∧_ |
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28 open _∨_ |
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29 open Bool |
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30 |
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31 record _==_ ( a b : OD ) : Set n where |
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32 field |
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33 eq→ : ∀ { x : Ordinal } → def a x → def b x |
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34 eq← : ∀ { x : Ordinal } → def b x → def a x |
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35 |
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36 id : {A : Set n} → A → A |
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37 id x = x |
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38 |
271 | 39 ==-refl : { x : OD } → x == x |
40 ==-refl {x} = record { eq→ = id ; eq← = id } | |
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41 |
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42 open _==_ |
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43 |
271 | 44 ==-trans : { x y z : OD } → x == y → y == z → x == z |
45 ==-trans x=y y=z = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← = λ {m} t → eq← x=y (eq← y=z t) } | |
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46 |
271 | 47 ==-sym : { x y : OD } → x == y → y == x |
48 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } | |
49 | |
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50 |
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51 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y |
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52 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m |
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53 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m |
120 | 54 |
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55 -- next assumptions are our axiom |
322 | 56 -- |
57 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one | |
58 -- correspondence to the OD then the OD looks like a ZF Set. | |
59 -- | |
60 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e. | |
61 -- bbounded ODs are ZF Set. Unbounded ODs are classes. | |
62 -- | |
290 | 63 -- In classical Set Theory, HOD is used, as a subset of OD, |
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64 -- HOD = { x | TC x ⊆ OD } |
290 | 65 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x. |
322 | 66 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD. |
290 | 67 -- |
309 | 68 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks. |
322 | 69 -- There two contraints on the HOD order, one is ∋, the other one is ⊂. |
70 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary | |
71 -- bound on each HOD. | |
290 | 72 -- |
322 | 73 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic, |
290 | 74 -- we need explict assumption on sup. |
309 | 75 -- |
76 -- ==→o≡ is necessary to prove axiom of extensionality. | |
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77 |
303 | 78 data One : Set n where |
79 OneObj : One | |
80 | |
81 -- Ordinals in OD , the maximum | |
82 Ords : OD | |
83 Ords = record { def = λ x → One } | |
84 | |
85 record HOD : Set (suc n) where | |
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86 field |
303 | 87 od : OD |
304 | 88 odmax : Ordinal |
308 | 89 <odmax : {y : Ordinal} → def od y → y o< odmax |
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90 |
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91 open HOD |
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92 |
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93 record ODAxiom : Set (suc n) where |
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94 field |
304 | 95 -- HOD is isomorphic to Ordinal (by means of Goedel number) |
303 | 96 od→ord : HOD → Ordinal |
97 ord→od : Ordinal → HOD | |
98 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
335 | 99 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) |
303 | 100 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x |
322 | 101 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x |
335 | 102 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y |
103 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal | |
306 | 104 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ |
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105 |
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106 postulate odAxiom : ODAxiom |
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107 open ODAxiom odAxiom |
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108 |
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109 -- possible order restriction |
339 | 110 hod-ord< : {x : HOD } → Set n |
111 hod-ord< {x} = od→ord x o< next (odmax x) | |
112 | |
335 | 113 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD |
114 ¬OD-order : ( od→ord : OD → Ordinal ) → ( ord→od : Ordinal → OD ) → ( { x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y) → ⊥ | |
115 ¬OD-order od→ord ord→od c<→o< = osuc-< <-osuc (c<→o< {Ords} OneObj ) | |
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116 |
335 | 117 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup |
118 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥ | |
119 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where | |
120 next-ord : Ordinal → Ordinal | |
121 next-ord x = osuc x | |
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122 |
179 | 123 -- Ordinal in OD ( and ZFSet ) Transitive Set |
303 | 124 Ord : ( a : Ordinal ) → HOD |
304 | 125 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where |
126 lemma : {x : Ordinal} → x o< a → x o< a | |
127 lemma {x} lt = lt | |
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128 |
303 | 129 od∅ : HOD |
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130 od∅ = Ord o∅ |
40 | 131 |
303 | 132 odef : HOD → Ordinal → Set n |
133 odef A x = def ( od A ) x | |
123 | 134 |
335 | 135 -- If we have reverse of c<→o<, everything becomes Ordinal |
136 o<→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x) | |
303 | 137 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where |
138 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y | |
139 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → odef x k ) (sym diso) lt)) | |
140 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y | |
141 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
95 | 142 |
303 | 143 _∋_ : ( a x : HOD ) → Set n |
144 _∋_ a x = odef a ( od→ord x ) | |
145 | |
146 _c<_ : ( x a : HOD ) → Set n | |
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147 x c< a = a ∋ x |
103 | 148 |
303 | 149 cseq : {n : Level} → HOD → HOD |
308 | 150 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where |
151 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) | |
152 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) | |
95 | 153 |
303 | 154 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x |
155 odef-subst df refl refl = df | |
95 | 156 |
303 | 157 otrans : {n : Level} {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y |
187 | 158 otrans x<a y<x = ordtrans y<x x<a |
123 | 159 |
303 | 160 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X |
161 odef→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( odef-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso | |
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162 |
51 | 163 -- avoiding lv != Zero error |
303 | 164 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
51 | 165 orefl refl = refl |
166 | |
303 | 167 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y |
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168 ==-iso {x} {y} eq = record { |
303 | 169 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ; |
170 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } | |
51 | 171 where |
303 | 172 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z |
173 lemma {x} {z} d = odef-subst d oiso refl | |
51 | 174 |
303 | 175 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y) |
176 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso) | |
57 | 177 |
303 | 178 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y |
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179 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
303 | 180 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy) |
271 | 181 lemma ox ox refl = ==-refl |
51 | 182 |
303 | 183 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y) |
271 | 184 o≡→== {x} {.x} refl = ==-refl |
51 | 185 |
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186 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
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187 o∅≡od∅ = ==→o≡ lemma where |
303 | 188 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x |
189 lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (odef-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso | |
190 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x | |
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191 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
303 | 192 lemma : od (ord→od o∅) == od od∅ |
150 | 193 lemma = record { eq→ = lemma0 ; eq← = lemma1 } |
194 | |
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195 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
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196 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
80 | 197 |
303 | 198 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅ |
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199 eq→ ∅0 {w} (lift ()) |
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200 eq← ∅0 {w} lt = lift (¬x<0 lt) |
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201 |
303 | 202 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ ) |
271 | 203 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d |
223
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204 ∅< {x} {y} d eq | lift () |
57 | 205 |
303 | 206 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox |
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207 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
51 | 208 |
303 | 209 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x |
210 odef-iso refl t = t | |
76 | 211 |
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212 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
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213 is-o∅ x with trio< x o∅ |
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214 is-o∅ x | tri< a ¬b ¬c = no ¬b |
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215 is-o∅ x | tri≈ ¬a b ¬c = yes b |
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216 is-o∅ x | tri> ¬a ¬b c = no ¬b |
57 | 217 |
335 | 218 -- the pair |
338 | 219 _,_ : HOD → HOD → HOD |
308 | 220 x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; odmax = omax (od→ord x) (od→ord y) ; <odmax = lemma } where |
221 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y) | |
222 lemma {t} (case1 refl) = omax-x _ _ | |
223 lemma {t} (case2 refl) = omax-y _ _ | |
224 | |
343 | 225 pair-xx<xy : {x y : HOD} → od→ord (x , x) o< osuc (od→ord (x , y) ) |
226 pair-xx<xy {x} {y} = ⊆→o≤ lemma where | |
227 lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z | |
228 lemma {z} (case1 refl) = case1 refl | |
229 lemma {z} (case2 refl) = case1 refl | |
230 | |
339 | 231 -- another form of infinite |
343 | 232 -- pair-ord< : {x : Ordinal } → Set n |
233 pair-ord< : {x : HOD } → ( {y : HOD } → od→ord y o< next (odmax y) ) → od→ord ( x , x ) o< next (od→ord x) | |
234 pair-ord< {x} ho< = subst (λ k → od→ord (x , x) o< k ) lemmab0 lemmab1 where | |
235 lemmab0 : next (odmax (x , x)) ≡ next (od→ord x) | |
236 lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡) | |
237 lemmab1 : od→ord (x , x) o< next ( odmax (x , x)) | |
238 lemmab1 = ho< | |
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239 |
344
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240 pair<y : {x y : HOD } → y ∋ x → od→ord (x , x) o< osuc (od→ord y) |
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241 pair<y {x} {y} y∋x = ⊆→o≤ lemma where |
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242 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z |
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243 lemma (case1 refl) = y∋x |
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244 lemma (case2 refl) = y∋x |
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245 |
79 | 246 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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247 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) |
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248 |
318 | 249 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD |
250 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } | |
141 | 251 |
303 | 252 ZFSubset : (A x : HOD ) → HOD |
308 | 253 ZFSubset A x = record { od = record { def = λ y → odef A y ∧ odef x y } ; odmax = omin (odmax A) (odmax x) ; <odmax = lemma } where -- roughly x = A → Set |
254 lemma : {y : Ordinal} → def (od A) y ∧ def (od x) y → y o< omin (odmax A) (odmax x) | |
255 lemma {y} and = min1 (<odmax A (proj1 and)) (<odmax x (proj2 and)) | |
256 | |
303 | 257 record _⊆_ ( A B : HOD ) : Set (suc n) where |
271 | 258 field |
303 | 259 incl : { x : HOD } → A ∋ x → B ∋ x |
271 | 260 |
261 open _⊆_ | |
190 | 262 infixr 220 _⊆_ |
263 | |
335 | 264 od⊆→o≤ : {x y : HOD } → x ⊆ y → od→ord x o< osuc (od→ord y) |
265 od⊆→o≤ {x} {y} lt = ⊆→o≤ {x} {y} (λ {z} x>z → subst (λ k → def (od y) k ) diso (incl lt (subst (λ k → def (od x) k ) (sym diso) x>z ))) | |
266 | |
267 -- if we have od→ord (x , x) ≡ osuc (od→ord x), ⊆→o≤ → c<→o< | |
338 | 268 ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) ) |
269 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) ) | |
270 → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
271 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y) | |
272 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a | |
273 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (peq {x}) (pair<y (subst (λ k → k ∋ x) (sym ( ==→o≡ {x} {y} (ord→== b))) y∋x ))) | |
274 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c = | |
275 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where | |
276 lemma : {z : Ordinal} → (z ≡ od→ord x) ∨ (z ≡ od→ord x) → od→ord x ≡ z | |
277 lemma (case1 refl) = refl | |
278 lemma (case2 refl) = refl | |
279 y⊆x,x : {z : Ordinals.ord O} → def (od (x , x)) z → def (od y) z | |
280 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x | |
281 lemma1 : osuc (od→ord y) o< od→ord (x , x) | |
282 lemma1 = subst (λ k → osuc (od→ord y) o< k ) (sym (peq {x})) (osucc c ) | |
335 | 283 |
303 | 284 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) |
271 | 285 subset-lemma {A} {x} = record { |
286 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
287 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
190 | 288 } |
289 | |
312 | 290 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x |
291 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where | |
292 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y)) | |
293 lemma y x∋y = incl x⊆A (subst (λ k → def (od x) k ) (sym diso) x∋y ) | |
294 | |
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295 open import Data.Unit |
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296 |
324 | 297 ε-induction : { ψ : HOD → Set n} |
303 | 298 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) |
299 → (x : HOD ) → ψ x | |
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300 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
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301 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
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302 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) |
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303 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
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304 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
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305 |
335 | 306 -- level trick (what'a shame) |
330 | 307 ε-induction1 : { ψ : HOD → Set (suc n)} |
308 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) | |
309 → (x : HOD ) → ψ x | |
310 ε-induction1 {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where | |
311 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) | |
312 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) | |
313 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) | |
314 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy | |
315 | |
303 | 316 HOD→ZF : ZF |
317 HOD→ZF = record { | |
318 ZFSet = HOD | |
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319 ; _∋_ = _∋_ |
303 | 320 ; _≈_ = _=h=_ |
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321 ; ∅ = od∅ |
28 | 322 ; _,_ = _,_ |
323 ; Union = Union | |
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324 ; Power = Power |
28 | 325 ; Select = Select |
326 ; Replace = Replace | |
161 | 327 ; infinite = infinite |
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328 ; isZF = isZF |
28 | 329 } where |
303 | 330 ZFSet = HOD -- is less than Ords because of maxod |
331 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD | |
308 | 332 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( ord→od x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) } |
310 | 333 Replace : HOD → (HOD → HOD) → HOD |
318 | 334 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) ∧ def (in-codomain X ψ) x } |
335 ; odmax = rmax ; <odmax = rmax<} where | |
336 rmax : Ordinal | |
337 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y))) | |
338 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax | |
339 rmax< lt = proj1 lt | |
144 | 340 _∩_ : ( A B : ZFSet ) → ZFSet |
318 | 341 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } |
342 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))} | |
303 | 343 Union : HOD → HOD |
318 | 344 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } |
345 ; odmax = osuc (od→ord U) ; <odmax = umax< } where | |
346 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U) | |
319 | 347 umax< {y} not = lemma (FExists _ lemma1 not ) where |
348 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x | |
349 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) diso diso (c<→o< (subst (λ k → def (od (ord→od x)) k) (sym diso) x<y)) | |
350 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U | |
351 lemma2 {x} x<U = subst (λ k → k o< od→ord U ) diso (c<→o< (subst (λ k → def (od U) k) (sym diso) x<U)) | |
352 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y) | |
353 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) | |
354 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U) | |
355 lemma not with trio< y (od→ord U) | |
356 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc | |
357 lemma not | tri≈ ¬a refl ¬c = <-osuc | |
358 lemma not | tri> ¬a ¬b c = ⊥-elim (not c) | |
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359 _∈_ : ( A B : ZFSet ) → Set n |
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360 A ∈ B = B ∋ A |
312 | 361 |
362 OPwr : (A : HOD ) → HOD | |
363 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( ZFSubset A (ord→od x)) ) ) | |
364 | |
303 | 365 Power : HOD → HOD |
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366 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) |
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367 -- {_} : ZFSet → ZFSet |
335 | 368 -- { x } = ( x , x ) -- better to use (x , x) directly |
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370 data infinite-d : ( x : Ordinal ) → Set n where |
161 | 371 iφ : infinite-d o∅ |
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372 isuc : {x : Ordinal } → infinite-d x → |
161 | 373 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) |
374 | |
328 | 375 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. |
338 | 376 -- We simply assumes infinite-d y has a maximum. |
328 | 377 -- |
338 | 378 -- This means that many of OD may not be HODs because of the od→ord mapping divergence. |
346 | 379 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x). |
328 | 380 -- |
381 postulate | |
382 ωmax : Ordinal | |
383 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax | |
384 | |
303 | 385 infinite : HOD |
328 | 386 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } |
303 | 387 |
339 | 388 infinite' : ({x : HOD} → od→ord x o< next (odmax x)) → HOD |
389 infinite' ho< = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where | |
390 u : (y : Ordinal ) → HOD | |
391 u y = Union (ord→od y , (ord→od y , ord→od y)) | |
392 lemma : {y : Ordinal} → infinite-d y → y o< next o∅ | |
348
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393 lemma {o∅} iφ = x<nx |
339 | 394 lemma (isuc {y} x) = lemma2 where |
358 | 395 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z |
339 | 396 lemma0 : y o< next o∅ |
397 lemma0 = lemma x | |
340 | 398 lemma8 : od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y)) |
399 lemma8 = ho< | |
343 | 400 --- (x,y) < next (omax x y) < next (osuc y) = next y |
341 | 401 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y) |
343 | 402 lemmaa {x} {y} x<y = subst (λ k → od→ord (x , y) o< k ) (sym nexto≡) (subst (λ k → od→ord (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< ) |
340 | 403 lemma81 : od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y)) |
404 lemma81 = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8) | |
405 lemma9 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y)) | |
342 | 406 lemma9 = lemmaa (c<→o< (case1 refl)) |
340 | 407 lemma71 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y)) |
408 lemma71 = next< lemma81 lemma9 | |
339 | 409 lemma1 : od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y)))) |
410 lemma1 = ho< | |
411 lemma2 : od→ord (u y) o< next o∅ | |
340 | 412 lemma2 = next< lemma0 (next< (subst (λ k → od→ord (ord→od y , (ord→od y , ord→od y)) o< next k) diso lemma71 ) (nexto=n lemma1)) |
338 | 413 |
414 | |
303 | 415 _=h=_ : (x y : HOD) → Set n |
416 x =h= y = od x == od y | |
161 | 417 |
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418 infixr 200 _∈_ |
96 | 419 -- infixr 230 _∩_ _∪_ |
303 | 420 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite |
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421 isZF = record { |
271 | 422 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } |
247 | 423 ; pair→ = pair→ |
424 ; pair← = pair← | |
72 | 425 ; union→ = union→ |
426 ; union← = union← | |
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427 ; empty = empty |
129 | 428 ; power→ = power→ |
76 | 429 ; power← = power← |
186 | 430 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
274 | 431 ; ε-induction = ε-induction |
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432 ; infinity∅ = infinity∅ |
160 | 433 ; infinity = infinity |
116 | 434 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
135 | 435 ; replacement← = replacement← |
317 | 436 ; replacement→ = λ {ψ} → replacement→ {ψ} |
274 | 437 -- ; choice-func = choice-func |
438 -- ; choice = choice | |
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439 } where |
129 | 440 |
303 | 441 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) |
442 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) | |
443 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) | |
247 | 444 |
303 | 445 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t |
446 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) | |
447 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) | |
247 | 448 |
303 | 449 empty : (x : HOD ) → ¬ (od∅ ∋ x) |
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450 empty x = ¬x<0 |
129 | 451 |
271 | 452 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) |
453 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
155 | 454 |
271 | 455 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y |
155 | 456 ⊆→o< {x} {y} lt with trio< x y |
457 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
458 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
271 | 459 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) |
155 | 460 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) |
151 | 461 |
303 | 462 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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463 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
303 | 464 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) |
465 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) | |
258 | 466 union← X z UX∋z = FExists _ lemma UX∋z where |
303 | 467 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
468 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
144 | 469 |
303 | 470 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y |
144 | 471 ψiso {ψ} t refl = t |
303 | 472 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) |
144 | 473 selection {ψ} {X} {y} = record { |
474 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
475 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
476 } | |
311 | 477 sup-c< : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → od→ord (ψ x) o< (sup-o X (λ y X∋y → od→ord (ψ (ord→od y)))) |
478 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt ) | |
303 | 479 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x |
311 | 480 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where |
318 | 481 lemma : def (in-codomain X ψ) (od→ord (ψ x)) |
150 | 482 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
303 | 483 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) |
150 | 484 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
303 | 485 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) |
486 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) | |
144 | 487 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
303 | 488 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) |
489 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) | |
490 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) | |
491 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) | |
144 | 492 |
493 --- | |
494 --- Power Set | |
495 --- | |
303 | 496 --- First consider ordinals in HOD |
100 | 497 --- |
303 | 498 --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of A |
100 | 499 -- |
500 -- | |
303 | 501 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) |
142 | 502 ∩-≡ {a} {b} inc = record { |
503 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
303 | 504 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; |
142 | 505 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
100 | 506 -- |
258 | 507 -- Transitive Set case |
303 | 508 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t =h= t |
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509 -- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t |
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510 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) |
100 | 511 -- |
303 | 512 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t |
513 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} | |
127 | 514 lemma refl (lemma1 lemma-eq )where |
303 | 515 lemma-eq : ZFSubset (Ord a) t =h= t |
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516 eq→ lemma-eq {z} w = proj2 w |
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517 eq← lemma-eq {z} w = record { proj2 = w ; |
303 | 518 proj1 = odef-subst {_} {_} {(Ord a)} {z} |
519 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } | |
520 lemma1 : {a : Ordinal } { t : HOD } | |
521 → (eq : ZFSubset (Ord a) t =h= t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t | |
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522 lemma1 {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq )) |
312 | 523 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a))) |
524 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) diso (t→A (subst (λ k → def (od t) k) (sym diso) x<t))) | |
525 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (Ord (osuc (od→ord (Ord a)))) (λ x lt → od→ord (ZFSubset (Ord a) (ord→od x))) | |
311 | 526 lemma = sup-o< _ lemma2 |
129 | 527 |
144 | 528 -- |
303 | 529 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first |
258 | 530 -- then replace of all elements of the Power set by A ∩ y |
144 | 531 -- |
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532 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) |
166 | 533 |
534 -- we have oly double negation form because of the replacement axiom | |
535 -- | |
303 | 536 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) |
258 | 537 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where |
142 | 538 a = od→ord A |
303 | 539 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) |
317 | 540 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t |
303 | 541 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) |
166 | 542 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) |
303 | 543 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) |
544 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) | |
545 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) | |
166 | 546 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not |
547 | |
303 | 548 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
311 | 549 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where |
142 | 550 a = od→ord A |
303 | 551 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x |
142 | 552 lemma0 {x} t∋x = c<→o< (t→A t∋x) |
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553 lemma3 : OPwr (Ord a) ∋ t |
142 | 554 lemma3 = ord-power← a t lemma0 |
152 | 555 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t |
556 lemma4 = let open ≡-Reasoning in begin | |
557 A ∩ ord→od (od→ord t) | |
558 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
559 A ∩ t | |
317 | 560 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩ |
152 | 561 t |
562 ∎ | |
317 | 563 sup1 : Ordinal |
314 | 564 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord (ZFSubset (Ord (od→ord A)) (ord→od x))) |
313 | 565 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A))) |
566 lemma9 = <-osuc | |
315 | 567 lemmab : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) )))) o< sup1 |
568 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9 | |
569 lemmad : Ord (osuc (od→ord A)) ∋ t | |
317 | 570 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) diso (t→A (subst (λ k → def (od t) k ) (sym diso) lt))) |
315 | 571 lemmac : ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A) ))) =h= Ord (od→ord A) |
317 | 572 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where |
573 lemmaf : {x : Ordinal} → def (od (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A)))))) x → def (od (Ord (od→ord A))) x | |
574 lemmaf {x} lt = proj1 lt | |
575 lemmag : {x : Ordinal} → def (od (Ord (od→ord A))) x → def (od (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A)))))) x | |
576 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt } | |
315 | 577 lemmae : od→ord (ZFSubset (Ord (od→ord A)) (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A)) |
578 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac) | |
311 | 579 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t) |
315 | 580 lemma7 with osuc-≡< lemmad |
581 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab ) | |
317 | 582 lemma7 | case1 eq with osuc-≡< (⊆→o≤ {ord→od (od→ord t)} {ord→od (od→ord (Ord (od→ord t)))} (λ {x} lt → lemmah lt )) where |
583 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x | |
584 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t)) | |
585 diso | |
586 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt ))) | |
587 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where | |
588 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t | |
589 lemmai = let open ≡-Reasoning in begin | |
590 od→ord (Ord (od→ord A)) | |
591 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩ | |
592 od→ord (Ord (od→ord t)) | |
593 ≡⟨ sym diso ⟩ | |
594 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
595 ≡⟨ sym eq1 ⟩ | |
596 od→ord (ord→od (od→ord t)) | |
597 ≡⟨ diso ⟩ | |
598 od→ord t | |
599 ∎ | |
600 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where | |
601 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A)) | |
602 lemmak = let open ≡-Reasoning in begin | |
603 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
604 ≡⟨ diso ⟩ | |
605 od→ord (Ord (od→ord t)) | |
606 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩ | |
607 od→ord (Ord (od→ord A)) | |
608 ∎ | |
609 lemmaj : od→ord t o< od→ord (Ord (od→ord A)) | |
610 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt | |
310 | 611 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))) |
612 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))) | |
311 | 613 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 ) |
318 | 614 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
151 | 615 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
616 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
317 | 617 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A ))) |
142 | 618 |
311 | 619 |
271 | 620 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) |
621 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
303 | 622 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y |
271 | 623 lemma lt y<x with osuc-≡< lt |
624 lemma lt y<x | case1 refl = c<→o< y<x | |
625 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
262 | 626 |
276 | 627 continuum-hyphotheis : (a : Ordinal) → Set (suc n) |
628 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
129 | 629 |
303 | 630 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B |
631 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
632 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
186 | 633 |
303 | 634 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) |
186 | 635 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d |
636 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
129 | 637 |
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638 infinity∅ : infinite ∋ od∅ |
303 | 639 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where |
161 | 640 lemma : o∅ ≡ od→ord od∅ |
641 lemma = let open ≡-Reasoning in begin | |
642 o∅ | |
643 ≡⟨ sym diso ⟩ | |
644 od→ord ( ord→od o∅ ) | |
645 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
646 od→ord od∅ | |
647 ∎ | |
303 | 648 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
649 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
161 | 650 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) |
651 ≡ od→ord (Union (x , (x , x))) | |
652 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
653 | |
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654 |
303 | 655 Union = ZF.Union HOD→ZF |
656 Power = ZF.Power HOD→ZF | |
657 Select = ZF.Select HOD→ZF | |
658 Replace = ZF.Replace HOD→ZF | |
659 isZF = ZF.isZF HOD→ZF |