Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 363:aad9249d1e8f
hω2
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 18 Jul 2020 10:36:32 +0900 |
parents | 4cbcf71b09c4 |
children | 67580311cc8e |
rev | line source |
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16 | 1 open import Level |
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2 open import Ordinals |
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3 module OD {n : Level } (O : Ordinals {n} ) where |
3 | 4 |
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5 open import zf |
23 | 6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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7 open import Relation.Binary.PropositionalEquality |
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8 open import Data.Nat.Properties |
6 | 9 open import Data.Empty |
10 open import Relation.Nullary | |
11 open import Relation.Binary | |
12 open import Relation.Binary.Core | |
13 | |
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14 open import logic |
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15 open import nat |
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16 |
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17 open inOrdinal O |
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18 |
27 | 19 -- Ordinal Definable Set |
11 | 20 |
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21 record OD : Set (suc n ) where |
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22 field |
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23 def : (x : Ordinal ) → Set n |
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24 |
141 | 25 open OD |
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26 |
120 | 27 open _∧_ |
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28 open _∨_ |
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29 open Bool |
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30 |
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31 record _==_ ( a b : OD ) : Set n where |
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32 field |
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33 eq→ : ∀ { x : Ordinal } → def a x → def b x |
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34 eq← : ∀ { x : Ordinal } → def b x → def a x |
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35 |
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36 id : {A : Set n} → A → A |
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37 id x = x |
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38 |
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 |
363 | 109 -- odmax minimality |
110 -- | |
111 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD. | |
112 -- We can calculate the minimum using sup but it is tedius. | |
113 -- Only Select has non minimum odmax. | |
114 -- We have the same problem on 'def' itself, but we leave it. | |
115 | |
116 odmaxmin : Set (suc n) | |
117 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z | |
118 | |
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119 -- possible order restriction |
339 | 120 hod-ord< : {x : HOD } → Set n |
121 hod-ord< {x} = od→ord x o< next (odmax x) | |
122 | |
335 | 123 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD |
124 ¬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) → ⊥ | |
125 ¬OD-order od→ord ord→od c<→o< = osuc-< <-osuc (c<→o< {Ords} OneObj ) | |
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126 |
335 | 127 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup |
128 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥ | |
129 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where | |
130 next-ord : Ordinal → Ordinal | |
131 next-ord x = osuc x | |
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132 |
179 | 133 -- Ordinal in OD ( and ZFSet ) Transitive Set |
303 | 134 Ord : ( a : Ordinal ) → HOD |
304 | 135 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where |
136 lemma : {x : Ordinal} → x o< a → x o< a | |
137 lemma {x} lt = lt | |
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138 |
303 | 139 od∅ : HOD |
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140 od∅ = Ord o∅ |
40 | 141 |
303 | 142 odef : HOD → Ordinal → Set n |
143 odef A x = def ( od A ) x | |
123 | 144 |
335 | 145 -- If we have reverse of c<→o<, everything becomes Ordinal |
146 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 | 147 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where |
148 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y | |
149 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)) | |
150 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y | |
151 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
95 | 152 |
303 | 153 _∋_ : ( a x : HOD ) → Set n |
154 _∋_ a x = odef a ( od→ord x ) | |
155 | |
156 _c<_ : ( x a : HOD ) → Set n | |
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157 x c< a = a ∋ x |
103 | 158 |
361 | 159 cseq : HOD → HOD |
308 | 160 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where |
161 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) | |
162 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) | |
95 | 163 |
303 | 164 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x |
165 odef-subst df refl refl = df | |
95 | 166 |
361 | 167 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y |
187 | 168 otrans x<a y<x = ordtrans y<x x<a |
123 | 169 |
303 | 170 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X |
171 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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172 |
51 | 173 -- avoiding lv != Zero error |
303 | 174 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
51 | 175 orefl refl = refl |
176 | |
303 | 177 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y |
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178 ==-iso {x} {y} eq = record { |
303 | 179 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ; |
180 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } | |
51 | 181 where |
303 | 182 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z |
183 lemma {x} {z} d = odef-subst d oiso refl | |
51 | 184 |
303 | 185 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y) |
186 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso) | |
57 | 187 |
303 | 188 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y |
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189 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
303 | 190 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy) |
271 | 191 lemma ox ox refl = ==-refl |
51 | 192 |
303 | 193 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y) |
271 | 194 o≡→== {x} {.x} refl = ==-refl |
51 | 195 |
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196 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
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197 o∅≡od∅ = ==→o≡ lemma where |
303 | 198 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x |
199 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 | |
200 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x | |
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201 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
303 | 202 lemma : od (ord→od o∅) == od od∅ |
150 | 203 lemma = record { eq→ = lemma0 ; eq← = lemma1 } |
204 | |
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205 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
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206 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
80 | 207 |
303 | 208 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅ |
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209 eq→ ∅0 {w} (lift ()) |
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210 eq← ∅0 {w} lt = lift (¬x<0 lt) |
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211 |
303 | 212 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ ) |
271 | 213 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d |
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214 ∅< {x} {y} d eq | lift () |
57 | 215 |
303 | 216 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox |
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217 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
51 | 218 |
303 | 219 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x |
220 odef-iso refl t = t | |
76 | 221 |
223
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222 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
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223 is-o∅ x with trio< x o∅ |
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224 is-o∅ x | tri< a ¬b ¬c = no ¬b |
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225 is-o∅ x | tri≈ ¬a b ¬c = yes b |
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226 is-o∅ x | tri> ¬a ¬b c = no ¬b |
57 | 227 |
335 | 228 -- the pair |
338 | 229 _,_ : HOD → HOD → HOD |
308 | 230 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 |
231 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y) | |
232 lemma {t} (case1 refl) = omax-x _ _ | |
233 lemma {t} (case2 refl) = omax-y _ _ | |
234 | |
343 | 235 pair-xx<xy : {x y : HOD} → od→ord (x , x) o< osuc (od→ord (x , y) ) |
236 pair-xx<xy {x} {y} = ⊆→o≤ lemma where | |
237 lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z | |
238 lemma {z} (case1 refl) = case1 refl | |
239 lemma {z} (case2 refl) = case1 refl | |
240 | |
339 | 241 -- another form of infinite |
343 | 242 -- pair-ord< : {x : Ordinal } → Set n |
243 pair-ord< : {x : HOD } → ( {y : HOD } → od→ord y o< next (odmax y) ) → od→ord ( x , x ) o< next (od→ord x) | |
244 pair-ord< {x} ho< = subst (λ k → od→ord (x , x) o< k ) lemmab0 lemmab1 where | |
245 lemmab0 : next (odmax (x , x)) ≡ next (od→ord x) | |
246 lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡) | |
247 lemmab1 : od→ord (x , x) o< next ( odmax (x , x)) | |
248 lemmab1 = ho< | |
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249 |
344
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250 pair<y : {x y : HOD } → y ∋ x → od→ord (x , x) o< osuc (od→ord y) |
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251 pair<y {x} {y} y∋x = ⊆→o≤ lemma where |
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252 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z |
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253 lemma (case1 refl) = y∋x |
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254 lemma (case2 refl) = y∋x |
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255 |
361 | 256 -- another possible restriction. We reqest no minimality on odmax, so it may arbitrary larger. |
257 odmax<od→ord : { x y : HOD } → x ∋ y → Set n | |
258 odmax<od→ord {x} {y} x∋y = odmax x o< od→ord x | |
259 | |
79 | 260 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
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261 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) |
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262 |
318 | 263 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD |
264 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } | |
141 | 265 |
360 | 266 _∩_ : ( A B : HOD ) → HOD |
267 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } | |
268 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))} | |
308 | 269 |
303 | 270 record _⊆_ ( A B : HOD ) : Set (suc n) where |
271 | 271 field |
303 | 272 incl : { x : HOD } → A ∋ x → B ∋ x |
271 | 273 |
274 open _⊆_ | |
190 | 275 infixr 220 _⊆_ |
276 | |
335 | 277 od⊆→o≤ : {x y : HOD } → x ⊆ y → od→ord x o< osuc (od→ord y) |
278 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 ))) | |
279 | |
280 -- if we have od→ord (x , x) ≡ osuc (od→ord x), ⊆→o≤ → c<→o< | |
338 | 281 ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) ) |
282 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) ) | |
283 → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
284 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y) | |
285 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a | |
286 ⊆→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 ))) | |
287 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c = | |
288 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where | |
289 lemma : {z : Ordinal} → (z ≡ od→ord x) ∨ (z ≡ od→ord x) → od→ord x ≡ z | |
290 lemma (case1 refl) = refl | |
291 lemma (case2 refl) = refl | |
292 y⊆x,x : {z : Ordinals.ord O} → def (od (x , x)) z → def (od y) z | |
293 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x | |
294 lemma1 : osuc (od→ord y) o< od→ord (x , x) | |
295 lemma1 = subst (λ k → osuc (od→ord y) o< k ) (sym (peq {x})) (osucc c ) | |
335 | 296 |
360 | 297 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → (A ∩ x ) ∋ y ) ⇔ ( x ⊆ A ) |
271 | 298 subset-lemma {A} {x} = record { |
299 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
300 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
190 | 301 } |
302 | |
312 | 303 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x |
304 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where | |
305 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y)) | |
306 lemma y x∋y = incl x⊆A (subst (λ k → def (od x) k ) (sym diso) x∋y ) | |
307 | |
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308 open import Data.Unit |
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309 |
324 | 310 ε-induction : { ψ : HOD → Set n} |
303 | 311 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) |
312 → (x : HOD ) → ψ x | |
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313 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
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314 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
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315 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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316 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
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317 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
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318 |
335 | 319 -- level trick (what'a shame) |
330 | 320 ε-induction1 : { ψ : HOD → Set (suc n)} |
321 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) | |
322 → (x : HOD ) → ψ x | |
323 ε-induction1 {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where | |
324 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) | |
325 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) | |
326 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) | |
327 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy | |
328 | |
303 | 329 HOD→ZF : ZF |
330 HOD→ZF = record { | |
331 ZFSet = HOD | |
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332 ; _∋_ = _∋_ |
363 | 333 ; _≈_ = hod→zf._=h=_ |
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334 ; ∅ = od∅ |
28 | 335 ; _,_ = _,_ |
363 | 336 ; Union = hod→zf.Union |
337 ; Power = hod→zf.Power | |
338 ; Select = hod→zf.Select | |
339 ; Replace = hod→zf.Replace | |
340 ; infinite = hod→zf.infinite | |
341 ; isZF = hod→zf.isZF | |
28 | 342 } where |
363 | 343 module hod→zf where |
303 | 344 ZFSet = HOD -- is less than Ords because of maxod |
345 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD | |
308 | 346 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 | 347 Replace : HOD → (HOD → HOD) → HOD |
318 | 348 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 } |
349 ; odmax = rmax ; <odmax = rmax<} where | |
350 rmax : Ordinal | |
351 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y))) | |
352 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax | |
353 rmax< lt = proj1 lt | |
303 | 354 Union : HOD → HOD |
318 | 355 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } |
356 ; odmax = osuc (od→ord U) ; <odmax = umax< } where | |
357 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U) | |
319 | 358 umax< {y} not = lemma (FExists _ lemma1 not ) where |
359 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x | |
360 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)) | |
361 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U | |
362 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)) | |
363 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y) | |
364 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) | |
365 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U) | |
366 lemma not with trio< y (od→ord U) | |
367 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc | |
368 lemma not | tri≈ ¬a refl ¬c = <-osuc | |
369 lemma not | tri> ¬a ¬b c = ⊥-elim (not c) | |
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370 _∈_ : ( A B : ZFSet ) → Set n |
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371 A ∈ B = B ∋ A |
312 | 372 |
373 OPwr : (A : HOD ) → HOD | |
360 | 374 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( A ∩ (ord→od x)) ) ) |
312 | 375 |
303 | 376 Power : HOD → HOD |
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377 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) |
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378 -- {_} : ZFSet → ZFSet |
335 | 379 -- { x } = ( x , x ) -- better to use (x , x) directly |
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380 |
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381 data infinite-d : ( x : Ordinal ) → Set n where |
161 | 382 iφ : infinite-d o∅ |
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383 isuc : {x : Ordinal } → infinite-d x → |
161 | 384 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) |
385 | |
328 | 386 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. |
338 | 387 -- We simply assumes infinite-d y has a maximum. |
328 | 388 -- |
338 | 389 -- This means that many of OD may not be HODs because of the od→ord mapping divergence. |
346 | 390 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x). |
328 | 391 -- |
392 postulate | |
393 ωmax : Ordinal | |
394 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax | |
395 | |
303 | 396 infinite : HOD |
328 | 397 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } |
303 | 398 |
339 | 399 infinite' : ({x : HOD} → od→ord x o< next (odmax x)) → HOD |
400 infinite' ho< = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where | |
401 u : (y : Ordinal ) → HOD | |
402 u y = Union (ord→od y , (ord→od y , ord→od y)) | |
403 lemma : {y : Ordinal} → infinite-d y → y o< next o∅ | |
348
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404 lemma {o∅} iφ = x<nx |
339 | 405 lemma (isuc {y} x) = lemma2 where |
358 | 406 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z |
339 | 407 lemma0 : y o< next o∅ |
408 lemma0 = lemma x | |
340 | 409 lemma8 : od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y)) |
410 lemma8 = ho< | |
343 | 411 --- (x,y) < next (omax x y) < next (osuc y) = next y |
341 | 412 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y) |
343 | 413 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 | 414 lemma81 : od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y)) |
415 lemma81 = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8) | |
416 lemma9 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y)) | |
342 | 417 lemma9 = lemmaa (c<→o< (case1 refl)) |
340 | 418 lemma71 : od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y)) |
419 lemma71 = next< lemma81 lemma9 | |
339 | 420 lemma1 : od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y)))) |
421 lemma1 = ho< | |
422 lemma2 : od→ord (u y) o< next o∅ | |
340 | 423 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)) |
363 | 424 |
425 nat→ω : Nat → HOD | |
426 nat→ω Zero = od∅ | |
427 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y)) | |
428 | |
429 ω→nat : (n : HOD) → infinite ∋ n → Nat | |
430 ω→nat n = lemma where | |
431 lemma : {y : Ordinal} → infinite-d y → Nat | |
432 lemma iφ = Zero | |
433 lemma (isuc lt) = Suc (lemma lt) | |
338 | 434 |
303 | 435 _=h=_ : (x y : HOD) → Set n |
436 x =h= y = od x == od y | |
161 | 437 |
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438 infixr 200 _∈_ |
96 | 439 -- infixr 230 _∩_ _∪_ |
303 | 440 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite |
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441 isZF = record { |
271 | 442 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } |
247 | 443 ; pair→ = pair→ |
444 ; pair← = pair← | |
72 | 445 ; union→ = union→ |
446 ; union← = union← | |
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447 ; empty = empty |
129 | 448 ; power→ = power→ |
76 | 449 ; power← = power← |
186 | 450 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} |
274 | 451 ; ε-induction = ε-induction |
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452 ; infinity∅ = infinity∅ |
160 | 453 ; infinity = infinity |
116 | 454 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
135 | 455 ; replacement← = replacement← |
317 | 456 ; replacement→ = λ {ψ} → replacement→ {ψ} |
274 | 457 -- ; choice-func = choice-func |
458 -- ; choice = choice | |
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459 } where |
129 | 460 |
303 | 461 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) |
462 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) | |
463 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) | |
247 | 464 |
303 | 465 pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t |
466 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) | |
467 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) | |
247 | 468 |
303 | 469 empty : (x : HOD ) → ¬ (od∅ ∋ x) |
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470 empty x = ¬x<0 |
129 | 471 |
271 | 472 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) |
473 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
155 | 474 |
271 | 475 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y |
155 | 476 ⊆→o< {x} {y} lt with trio< x y |
477 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
478 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
271 | 479 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) |
155 | 480 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) |
151 | 481 |
303 | 482 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
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483 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx |
303 | 484 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) |
485 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) | |
258 | 486 union← X z UX∋z = FExists _ lemma UX∋z where |
303 | 487 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) |
488 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
144 | 489 |
303 | 490 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y |
144 | 491 ψiso {ψ} t refl = t |
303 | 492 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) |
144 | 493 selection {ψ} {X} {y} = record { |
494 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
495 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
496 } | |
311 | 497 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)))) |
498 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt ) | |
303 | 499 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x |
311 | 500 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where |
318 | 501 lemma : def (in-codomain X ψ) (od→ord (ψ x)) |
150 | 502 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) |
303 | 503 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) |
150 | 504 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where |
303 | 505 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) |
506 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) | |
144 | 507 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
303 | 508 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) |
509 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) | |
510 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) | |
511 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) | |
144 | 512 |
513 --- | |
514 --- Power Set | |
515 --- | |
303 | 516 --- First consider ordinals in HOD |
100 | 517 --- |
360 | 518 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A |
100 | 519 -- |
520 -- | |
303 | 521 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) |
142 | 522 ∩-≡ {a} {b} inc = record { |
523 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
303 | 524 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; |
142 | 525 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
100 | 526 -- |
258 | 527 -- Transitive Set case |
360 | 528 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t |
529 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t | |
530 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( A ∩ (ord→od x )) ) ) | |
100 | 531 -- |
303 | 532 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t |
533 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} | |
127 | 534 lemma refl (lemma1 lemma-eq )where |
360 | 535 lemma-eq : ((Ord a) ∩ t) =h= t |
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536 eq→ lemma-eq {z} w = proj2 w |
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537 eq← lemma-eq {z} w = record { proj2 = w ; |
303 | 538 proj1 = odef-subst {_} {_} {(Ord a)} {z} |
539 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } | |
540 lemma1 : {a : Ordinal } { t : HOD } | |
360 | 541 → (eq : ((Ord a) ∩ t) =h= t) → od→ord ((Ord a) ∩ (ord→od (od→ord t))) ≡ od→ord t |
542 lemma1 {a} {t} eq = subst (λ k → od→ord ((Ord a) ∩ k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq )) | |
312 | 543 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a))) |
544 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))) | |
360 | 545 lemma : od→ord ((Ord a) ∩ (ord→od (od→ord t)) ) o< sup-o (Ord (osuc (od→ord (Ord a)))) (λ x lt → od→ord ((Ord a) ∩ (ord→od x))) |
311 | 546 lemma = sup-o< _ lemma2 |
129 | 547 |
144 | 548 -- |
303 | 549 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first |
258 | 550 -- then replace of all elements of the Power set by A ∩ y |
144 | 551 -- |
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552 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) |
166 | 553 |
554 -- we have oly double negation form because of the replacement axiom | |
555 -- | |
303 | 556 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) |
258 | 557 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where |
142 | 558 a = od→ord A |
303 | 559 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) |
317 | 560 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t |
303 | 561 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) |
166 | 562 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) |
303 | 563 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) |
564 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) | |
565 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) | |
166 | 566 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not |
567 | |
303 | 568 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t |
311 | 569 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where |
142 | 570 a = od→ord A |
303 | 571 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x |
142 | 572 lemma0 {x} t∋x = c<→o< (t→A t∋x) |
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573 lemma3 : OPwr (Ord a) ∋ t |
142 | 574 lemma3 = ord-power← a t lemma0 |
152 | 575 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t |
576 lemma4 = let open ≡-Reasoning in begin | |
577 A ∩ ord→od (od→ord t) | |
578 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
579 A ∩ t | |
317 | 580 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩ |
152 | 581 t |
582 ∎ | |
317 | 583 sup1 : Ordinal |
360 | 584 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord ((Ord (od→ord A)) ∩ (ord→od x))) |
313 | 585 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A))) |
586 lemma9 = <-osuc | |
360 | 587 lemmab : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) o< sup1 |
315 | 588 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9 |
589 lemmad : Ord (osuc (od→ord A)) ∋ t | |
317 | 590 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))) |
360 | 591 lemmac : ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) =h= Ord (od→ord A) |
317 | 592 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where |
360 | 593 lemmaf : {x : Ordinal} → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x → def (od (Ord (od→ord A))) x |
317 | 594 lemmaf {x} lt = proj1 lt |
360 | 595 lemmag : {x : Ordinal} → def (od (Ord (od→ord A))) x → def (od ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A)))))) x |
317 | 596 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt } |
360 | 597 lemmae : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A)) |
315 | 598 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac) |
311 | 599 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t) |
315 | 600 lemma7 with osuc-≡< lemmad |
601 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab ) | |
317 | 602 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 |
603 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x | |
604 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t)) | |
605 diso | |
606 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt ))) | |
607 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where | |
608 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t | |
609 lemmai = let open ≡-Reasoning in begin | |
610 od→ord (Ord (od→ord A)) | |
611 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩ | |
612 od→ord (Ord (od→ord t)) | |
613 ≡⟨ sym diso ⟩ | |
614 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
615 ≡⟨ sym eq1 ⟩ | |
616 od→ord (ord→od (od→ord t)) | |
617 ≡⟨ diso ⟩ | |
618 od→ord t | |
619 ∎ | |
620 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where | |
621 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A)) | |
622 lemmak = let open ≡-Reasoning in begin | |
623 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
624 ≡⟨ diso ⟩ | |
625 od→ord (Ord (od→ord t)) | |
626 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩ | |
627 od→ord (Ord (od→ord A)) | |
628 ∎ | |
629 lemmaj : od→ord t o< od→ord (Ord (od→ord A)) | |
630 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt | |
310 | 631 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))) |
632 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 | 633 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 ) |
318 | 634 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
151 | 635 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where |
636 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) | |
317 | 637 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A ))) |
142 | 638 |
311 | 639 |
271 | 640 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) |
641 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
303 | 642 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y |
271 | 643 lemma lt y<x with osuc-≡< lt |
644 lemma lt y<x | case1 refl = c<→o< y<x | |
645 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
262 | 646 |
276 | 647 continuum-hyphotheis : (a : Ordinal) → Set (suc n) |
648 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
129 | 649 |
303 | 650 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B |
651 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
652 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
186 | 653 |
303 | 654 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) |
186 | 655 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d |
656 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
129 | 657 |
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658 infinity∅ : infinite ∋ od∅ |
303 | 659 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where |
161 | 660 lemma : o∅ ≡ od→ord od∅ |
661 lemma = let open ≡-Reasoning in begin | |
662 o∅ | |
663 ≡⟨ sym diso ⟩ | |
664 od→ord ( ord→od o∅ ) | |
665 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
666 od→ord od∅ | |
667 ∎ | |
303 | 668 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) |
669 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
161 | 670 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) |
671 ≡ od→ord (Union (x , (x , x))) | |
672 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
673 | |
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674 |
303 | 675 Union = ZF.Union HOD→ZF |
676 Power = ZF.Power HOD→ZF | |
677 Select = ZF.Select HOD→ZF | |
678 Replace = ZF.Replace HOD→ZF | |
363 | 679 infinite = ZF.infinite HOD→ZF |
303 | 680 isZF = ZF.isZF HOD→ZF |
363 | 681 |