Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 388:19687f3304c9
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 25 Jul 2020 12:54:28 +0900 |
parents | 8b0715e28b33 |
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364 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
16 | 2 open import Level |
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3 open import Ordinals |
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4 module OD {n : Level } (O : Ordinals {n} ) where |
3 | 5 |
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6 open import zf |
23 | 7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
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8 open import Relation.Binary.PropositionalEquality |
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9 open import Data.Nat.Properties |
6 | 10 open import Data.Empty |
11 open import Relation.Nullary | |
12 open import Relation.Binary | |
13 open import Relation.Binary.Core | |
14 | |
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15 open import logic |
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16 open import nat |
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17 |
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18 open inOrdinal O |
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19 |
27 | 20 -- Ordinal Definable Set |
11 | 21 |
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22 record OD : Set (suc n ) where |
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23 field |
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24 def : (x : Ordinal ) → Set n |
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25 |
141 | 26 open OD |
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27 |
120 | 28 open _∧_ |
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29 open _∨_ |
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30 open Bool |
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31 |
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32 record _==_ ( a b : OD ) : Set n where |
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33 field |
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34 eq→ : ∀ { x : Ordinal } → def a x → def b x |
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35 eq← : ∀ { x : Ordinal } → def b x → def a x |
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36 |
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37 id : {A : Set n} → A → A |
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38 id x = x |
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39 |
271 | 40 ==-refl : { x : OD } → x == x |
41 ==-refl {x} = record { eq→ = id ; eq← = id } | |
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42 |
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43 open _==_ |
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44 |
271 | 45 ==-trans : { x y z : OD } → x == y → y == z → x == z |
46 ==-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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47 |
271 | 48 ==-sym : { x y : OD } → x == y → y == x |
49 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } | |
50 | |
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51 |
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52 ⇔→== : { x y : OD } → ( {z : Ordinal } → def x z ⇔ def y z) → x == y |
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53 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m |
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54 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m |
120 | 55 |
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56 -- next assumptions are our axiom |
322 | 57 -- |
58 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one | |
59 -- correspondence to the OD then the OD looks like a ZF Set. | |
60 -- | |
61 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e. | |
62 -- bbounded ODs are ZF Set. Unbounded ODs are classes. | |
63 -- | |
290 | 64 -- In classical Set Theory, HOD is used, as a subset of OD, |
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65 -- HOD = { x | TC x ⊆ OD } |
290 | 66 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x. |
322 | 67 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD. |
290 | 68 -- |
309 | 69 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks. |
322 | 70 -- There two contraints on the HOD order, one is ∋, the other one is ⊂. |
71 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary | |
72 -- bound on each HOD. | |
290 | 73 -- |
322 | 74 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic, |
290 | 75 -- we need explict assumption on sup. |
309 | 76 -- |
77 -- ==→o≡ is necessary to prove axiom of extensionality. | |
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78 |
303 | 79 -- Ordinals in OD , the maximum |
80 Ords : OD | |
81 Ords = record { def = λ x → One } | |
82 | |
83 record HOD : Set (suc n) where | |
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84 field |
303 | 85 od : OD |
304 | 86 odmax : Ordinal |
308 | 87 <odmax : {y : Ordinal} → def od y → y o< odmax |
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88 |
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89 open HOD |
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90 |
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91 record ODAxiom : Set (suc n) where |
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92 field |
304 | 93 -- HOD is isomorphic to Ordinal (by means of Goedel number) |
303 | 94 od→ord : HOD → Ordinal |
95 ord→od : Ordinal → HOD | |
96 c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
335 | 97 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) |
303 | 98 oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x |
322 | 99 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x |
335 | 100 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y |
101 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal | |
306 | 102 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 |
367 | 103 -- possible order restriction |
104 ho< : {x : HOD} → od→ord x o< next (odmax x) | |
105 | |
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106 |
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107 postulate odAxiom : ODAxiom |
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108 open ODAxiom odAxiom |
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109 |
363 | 110 -- odmax minimality |
111 -- | |
112 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD. | |
113 -- We can calculate the minimum using sup but it is tedius. | |
114 -- Only Select has non minimum odmax. | |
115 -- We have the same problem on 'def' itself, but we leave it. | |
116 | |
117 odmaxmin : Set (suc n) | |
118 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z | |
119 | |
335 | 120 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD |
121 ¬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) → ⊥ | |
122 ¬OD-order od→ord ord→od c<→o< = osuc-< <-osuc (c<→o< {Ords} OneObj ) | |
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123 |
335 | 124 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup |
125 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥ | |
126 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where | |
127 next-ord : Ordinal → Ordinal | |
128 next-ord x = osuc x | |
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129 |
179 | 130 -- Ordinal in OD ( and ZFSet ) Transitive Set |
303 | 131 Ord : ( a : Ordinal ) → HOD |
304 | 132 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where |
133 lemma : {x : Ordinal} → x o< a → x o< a | |
134 lemma {x} lt = lt | |
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135 |
303 | 136 od∅ : HOD |
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137 od∅ = Ord o∅ |
40 | 138 |
303 | 139 odef : HOD → Ordinal → Set n |
140 odef A x = def ( od A ) x | |
123 | 141 |
335 | 142 -- If we have reverse of c<→o<, everything becomes Ordinal |
143 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 | 144 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where |
145 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y | |
146 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)) | |
147 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y | |
148 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
95 | 149 |
303 | 150 _∋_ : ( a x : HOD ) → Set n |
151 _∋_ a x = odef a ( od→ord x ) | |
152 | |
153 _c<_ : ( x a : HOD ) → Set n | |
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154 x c< a = a ∋ x |
103 | 155 |
361 | 156 cseq : HOD → HOD |
308 | 157 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where |
158 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) | |
159 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) | |
95 | 160 |
303 | 161 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x |
162 odef-subst df refl refl = df | |
95 | 163 |
361 | 164 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y |
187 | 165 otrans x<a y<x = ordtrans y<x x<a |
123 | 166 |
303 | 167 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X |
168 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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169 |
388 | 170 odefo→o< : {X y : Ordinal } → odef (ord→od X) y → y o< X |
171 odefo→o< {X} {y} lt = subst₂ (λ j k → j o< k ) diso diso ( c<→o< (subst (λ k → odef (ord→od X) k ) (sym diso ) lt )) | |
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 ()) |
223
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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 |
223
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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 | |
376 | 260 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
261 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) | |
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262 |
376 | 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 | |
387 | 277 trans-⊆ : { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C |
278 trans-⊆ A⊆B B⊆C = record { incl = λ x → incl B⊆C (incl A⊆B x) } | |
279 | |
280 refl-⊆ : {A : HOD} → A ⊆ A | |
281 refl-⊆ {A} = record { incl = λ x → x } | |
282 | |
335 | 283 od⊆→o≤ : {x y : HOD } → x ⊆ y → od→ord x o< osuc (od→ord y) |
284 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 ))) | |
285 | |
286 -- if we have od→ord (x , x) ≡ osuc (od→ord x), ⊆→o≤ → c<→o< | |
338 | 287 ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) ) |
288 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) ) | |
289 → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
290 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y) | |
291 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a | |
292 ⊆→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 ))) | |
293 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c = | |
294 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where | |
295 lemma : {z : Ordinal} → (z ≡ od→ord x) ∨ (z ≡ od→ord x) → od→ord x ≡ z | |
296 lemma (case1 refl) = refl | |
297 lemma (case2 refl) = refl | |
298 y⊆x,x : {z : Ordinals.ord O} → def (od (x , x)) z → def (od y) z | |
299 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x | |
300 lemma1 : osuc (od→ord y) o< od→ord (x , x) | |
301 lemma1 = subst (λ k → osuc (od→ord y) o< k ) (sym (peq {x})) (osucc c ) | |
335 | 302 |
360 | 303 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → (A ∩ x ) ∋ y ) ⇔ ( x ⊆ A ) |
271 | 304 subset-lemma {A} {x} = record { |
305 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
306 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
190 | 307 } |
308 | |
312 | 309 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x |
310 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where | |
311 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y)) | |
312 lemma y x∋y = incl x⊆A (subst (λ k → def (od x) k ) (sym diso) x∋y ) | |
313 | |
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314 open import Data.Unit |
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315 |
324 | 316 ε-induction : { ψ : HOD → Set n} |
303 | 317 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) |
318 → (x : HOD ) → ψ x | |
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319 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
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320 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
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321 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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322 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
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323 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
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324 |
388 | 325 -- level trick (what'a shame) for LEM / minimal |
326 ε-induction1 : { ψ : HOD → Set (suc n)} | |
327 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) | |
328 → (x : HOD ) → ψ x | |
329 ε-induction1 {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where | |
330 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) | |
331 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) | |
332 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) | |
333 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy | |
330 | 334 |
376 | 335 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD |
336 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) } | |
377 | 337 |
376 | 338 Replace : HOD → (HOD → HOD) → HOD |
339 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 } | |
365 | 340 ; odmax = rmax ; <odmax = rmax<} where |
341 rmax : Ordinal | |
376 | 342 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y))) |
365 | 343 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax |
344 rmax< lt = proj1 lt | |
377 | 345 |
346 d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (ord→od x) | |
347 d→∋ a lt = subst (λ k → odef a k ) (sym diso) lt | |
348 | |
378 | 349 -- |
350 -- If we have LEM, Replace' is equivalent to Replace | |
351 -- | |
377 | 352 in-codomain' : (X : HOD ) → ((x : HOD) → X ∋ x → HOD) → OD |
353 in-codomain' X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ((lt : odef X y) → x ≡ od→ord (ψ (ord→od y ) (d→∋ X lt) )))) } | |
354 Replace' : (X : HOD) → ((x : HOD) → X ∋ x → HOD) → HOD | |
355 Replace' X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → od→ord (ψ (ord→od y) (d→∋ X X∋y) ))) ∧ def (in-codomain' X ψ) x } | |
356 ; odmax = rmax ; <odmax = rmax< } where | |
357 rmax : Ordinal | |
358 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y) (d→∋ X X∋y))) | |
359 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain' X ψ) y → y o< rmax | |
360 rmax< lt = proj1 lt | |
361 | |
365 | 362 Union : HOD → HOD |
363 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } | |
364 ; odmax = osuc (od→ord U) ; <odmax = umax< } where | |
365 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U) | |
366 umax< {y} not = lemma (FExists _ lemma1 not ) where | |
367 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x | |
368 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)) | |
369 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U | |
370 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)) | |
371 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y) | |
372 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) | |
373 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U) | |
374 lemma not with trio< y (od→ord U) | |
375 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc | |
376 lemma not | tri≈ ¬a refl ¬c = <-osuc | |
377 lemma not | tri> ¬a ¬b c = ⊥-elim (not c) | |
378 _∈_ : ( A B : HOD ) → Set n | |
379 A ∈ B = B ∋ A | |
380 | |
381 OPwr : (A : HOD ) → HOD | |
382 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( A ∩ (ord→od x)) ) ) | |
383 | |
384 Power : HOD → HOD | |
376 | 385 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) |
365 | 386 -- {_} : ZFSet → ZFSet |
387 -- { x } = ( x , x ) -- better to use (x , x) directly | |
388 | |
389 data infinite-d : ( x : Ordinal ) → Set n where | |
390 iφ : infinite-d o∅ | |
391 isuc : {x : Ordinal } → infinite-d x → | |
392 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) | |
393 | |
394 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. | |
395 -- We simply assumes infinite-d y has a maximum. | |
396 -- | |
397 -- This means that many of OD may not be HODs because of the od→ord mapping divergence. | |
398 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x). | |
399 -- | |
367 | 400 -- postulate |
401 -- ωmax : Ordinal | |
402 -- <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax | |
403 -- | |
404 -- infinite : HOD | |
405 -- infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } | |
365 | 406 |
407 infinite : HOD | |
367 | 408 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where |
365 | 409 u : (y : Ordinal ) → HOD |
410 u y = Union (ord→od y , (ord→od y , ord→od y)) | |
411 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z | |
412 lemma8 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y)) | |
413 lemma8 = ho< | |
414 --- (x,y) < next (omax x y) < next (osuc y) = next y | |
415 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y) | |
416 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< ) | |
417 lemma81 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y)) | |
418 lemma81 {y} = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8) | |
419 lemma9 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y)) | |
420 lemma9 = lemmaa (c<→o< (case1 refl)) | |
421 lemma71 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y)) | |
422 lemma71 = next< lemma81 lemma9 | |
423 lemma1 : {y : Ordinal} → od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y)))) | |
424 lemma1 = ho< | |
425 --- main recursion | |
426 lemma : {y : Ordinal} → infinite-d y → y o< next o∅ | |
427 lemma {o∅} iφ = x<nx | |
428 lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → od→ord (ord→od y , (ord→od y , ord→od y)) o< next k) diso lemma71 ) (nexto=n lemma1)) | |
429 | |
430 ω<next-o∅ : ({x : HOD} → od→ord x o< next (odmax x)) → {y : Ordinal} → infinite-d y → y o< next o∅ | |
367 | 431 ω<next-o∅ ho< {y} lt = <odmax infinite lt |
365 | 432 |
433 nat→ω : Nat → HOD | |
434 nat→ω Zero = od∅ | |
435 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y)) | |
436 | |
437 ω→nat : (n : HOD) → infinite ∋ n → Nat | |
438 ω→nat n = lemma where | |
439 lemma : {y : Ordinal} → infinite-d y → Nat | |
440 lemma iφ = Zero | |
441 lemma (isuc lt) = Suc (lemma lt) | |
442 | |
443 ω∋nat→ω : {n : Nat} → def (od infinite) (od→ord (nat→ω n)) | |
367 | 444 ω∋nat→ω {Zero} = subst (λ k → def (od infinite) k) (sym ord-od∅) iφ |
445 ω∋nat→ω {Suc n} = subst (λ k → def (od infinite) k) lemma (isuc ( ω∋nat→ω {n})) where | |
446 lemma : od→ord (Union (ord→od (od→ord (nat→ω n)) , (ord→od (od→ord (nat→ω n)) , ord→od (od→ord (nat→ω n))))) ≡ od→ord (nat→ω (Suc n)) | |
447 lemma = subst (λ k → od→ord (Union (k , ( k , k ))) ≡ od→ord (nat→ω (Suc n))) (sym oiso) refl | |
365 | 448 |
449 _=h=_ : (x y : HOD) → Set n | |
450 x =h= y = od x == od y | |
451 | |
452 infixr 200 _∈_ | |
453 -- infixr 230 _∩_ _∪_ | |
454 | |
455 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) | |
456 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) | |
457 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) | |
458 | |
459 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t | |
460 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) | |
461 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) | |
462 | |
463 empty : (x : HOD ) → ¬ (od∅ ∋ x) | |
464 empty x = ¬x<0 | |
465 | |
466 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) | |
467 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
468 | |
469 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y | |
470 ⊆→o< {x} {y} lt with trio< x y | |
471 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
472 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
473 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) | |
474 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) | |
475 | |
476 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z | |
477 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx | |
478 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) | |
479 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) | |
480 union← X z UX∋z = FExists _ lemma UX∋z where | |
481 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) | |
482 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } | |
483 | |
376 | 484 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y |
485 ψiso {ψ} t refl = t | |
486 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
487 selection {ψ} {X} {y} = record { | |
488 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
489 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
490 } | |
377 | 491 |
376 | 492 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)))) |
493 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt ) | |
494 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x | |
495 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where | |
496 lemma : def (in-codomain X ψ) (od→ord (ψ x)) | |
497 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) | |
498 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) | |
499 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where | |
500 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) | |
501 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) | |
365 | 502 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
376 | 503 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) |
365 | 504 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) |
376 | 505 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) |
506 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) | |
365 | 507 |
508 --- | |
509 --- Power Set | |
510 --- | |
511 --- First consider ordinals in HOD | |
512 --- | |
513 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A | |
514 -- | |
515 -- | |
516 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) | |
517 ∩-≡ {a} {b} inc = record { | |
518 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
519 proj1 = odef-subst {_} {_} {b} {x} (inc (odef-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; | |
520 eq← = λ {x} x<a∩b → proj2 x<a∩b } | |
521 -- | |
522 -- Transitive Set case | |
523 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t | |
524 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t | |
525 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( A ∩ (ord→od x )) ) ) | |
526 -- | |
527 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t | |
528 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} | |
529 lemma refl (lemma1 lemma-eq )where | |
530 lemma-eq : ((Ord a) ∩ t) =h= t | |
531 eq→ lemma-eq {z} w = proj2 w | |
532 eq← lemma-eq {z} w = record { proj2 = w ; | |
533 proj1 = odef-subst {_} {_} {(Ord a)} {z} | |
534 ( t→A (odef-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } | |
535 lemma1 : {a : Ordinal } { t : HOD } | |
536 → (eq : ((Ord a) ∩ t) =h= t) → od→ord ((Ord a) ∩ (ord→od (od→ord t))) ≡ od→ord t | |
537 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 )) | |
538 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a))) | |
539 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))) | |
540 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))) | |
541 lemma = sup-o< _ lemma2 | |
542 | |
543 -- | |
544 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first | |
545 -- then replace of all elements of the Power set by A ∩ y | |
546 -- | |
547 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) | |
548 | |
549 -- we have oly double negation form because of the replacement axiom | |
550 -- | |
551 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) | |
552 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where | |
553 a = od→ord A | |
554 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) | |
376 | 555 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t |
365 | 556 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) |
557 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) | |
558 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) | |
559 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) | |
560 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) | |
561 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not | |
562 | |
563 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t | |
564 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where | |
565 a = od→ord A | |
566 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x | |
567 lemma0 {x} t∋x = c<→o< (t→A t∋x) | |
568 lemma3 : OPwr (Ord a) ∋ t | |
569 lemma3 = ord-power← a t lemma0 | |
570 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t | |
571 lemma4 = let open ≡-Reasoning in begin | |
572 A ∩ ord→od (od→ord t) | |
573 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
574 A ∩ t | |
575 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩ | |
576 t | |
577 ∎ | |
578 sup1 : Ordinal | |
579 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord ((Ord (od→ord A)) ∩ (ord→od x))) | |
580 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A))) | |
581 lemma9 = <-osuc | |
582 lemmab : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) o< sup1 | |
583 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9 | |
584 lemmad : Ord (osuc (od→ord A)) ∋ t | |
585 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))) | |
586 lemmac : ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) =h= Ord (od→ord A) | |
587 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where | |
588 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 | |
589 lemmaf {x} lt = proj1 lt | |
590 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 | |
591 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt } | |
592 lemmae : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A)) | |
593 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac) | |
594 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t) | |
595 lemma7 with osuc-≡< lemmad | |
596 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab ) | |
597 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 | |
598 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x | |
599 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t)) | |
600 diso | |
601 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt ))) | |
602 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where | |
603 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t | |
604 lemmai = let open ≡-Reasoning in begin | |
605 od→ord (Ord (od→ord A)) | |
606 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩ | |
607 od→ord (Ord (od→ord t)) | |
608 ≡⟨ sym diso ⟩ | |
609 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
610 ≡⟨ sym eq1 ⟩ | |
611 od→ord (ord→od (od→ord t)) | |
612 ≡⟨ diso ⟩ | |
613 od→ord t | |
614 ∎ | |
615 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where | |
616 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A)) | |
617 lemmak = let open ≡-Reasoning in begin | |
618 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
619 ≡⟨ diso ⟩ | |
620 od→ord (Ord (od→ord t)) | |
621 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩ | |
622 od→ord (Ord (od→ord A)) | |
623 ∎ | |
624 lemmaj : od→ord t o< od→ord (Ord (od→ord A)) | |
625 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt | |
626 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))) | |
627 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))) | |
628 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 ) | |
376 | 629 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
630 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where | |
365 | 631 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) |
632 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A ))) | |
633 | |
634 | |
635 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) | |
636 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
637 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y | |
638 lemma lt y<x with osuc-≡< lt | |
639 lemma lt y<x | case1 refl = c<→o< y<x | |
640 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
641 | |
642 continuum-hyphotheis : (a : Ordinal) → Set (suc n) | |
643 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
644 | |
645 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B | |
646 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
647 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
648 | |
649 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) | |
650 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d | |
651 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
652 | |
653 infinity∅ : infinite ∋ od∅ | |
654 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where | |
655 lemma : o∅ ≡ od→ord od∅ | |
656 lemma = let open ≡-Reasoning in begin | |
657 o∅ | |
658 ≡⟨ sym diso ⟩ | |
659 od→ord ( ord→od o∅ ) | |
660 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
661 od→ord od∅ | |
662 ∎ | |
663 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) | |
664 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
665 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) | |
666 ≡ od→ord (Union (x , (x , x))) | |
667 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
668 | |
669 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite | |
670 isZF = record { | |
671 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } | |
672 ; pair→ = pair→ | |
673 ; pair← = pair← | |
674 ; union→ = union→ | |
675 ; union← = union← | |
676 ; empty = empty | |
677 ; power→ = power→ | |
678 ; power← = power← | |
679 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} | |
680 ; ε-induction = ε-induction | |
681 ; infinity∅ = infinity∅ | |
682 ; infinity = infinity | |
376 | 683 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
684 ; replacement← = replacement← | |
685 ; replacement→ = λ {ψ} → replacement→ {ψ} | |
365 | 686 -- ; choice-func = choice-func |
687 -- ; choice = choice | |
688 } | |
689 | |
303 | 690 HOD→ZF : ZF |
691 HOD→ZF = record { | |
692 ZFSet = HOD | |
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
693 ; _∋_ = _∋_ |
365 | 694 ; _≈_ = _=h=_ |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
695 ; ∅ = od∅ |
28 | 696 ; _,_ = _,_ |
365 | 697 ; Union = Union |
698 ; Power = Power | |
699 ; Select = Select | |
700 ; Replace = Replace | |
701 ; infinite = infinite | |
702 ; isZF = isZF | |
703 } | |
704 | |
144 | 705 |