Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate OD.agda @ 404:f7b844af9a50
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 28 Jul 2020 13:34:25 +0900 |
parents | ce2ce3f62023 |
children | 85b328d3b96b |
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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 |
303 | 142 _∋_ : ( a x : HOD ) → Set n |
143 _∋_ a x = odef a ( od→ord x ) | |
144 | |
145 _c<_ : ( x a : HOD ) → Set n | |
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146 x c< a = a ∋ x |
103 | 147 |
396 | 148 d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (ord→od x) |
149 d→∋ a lt = subst (λ k → odef a k ) (sym diso) lt | |
150 | |
361 | 151 cseq : HOD → HOD |
308 | 152 cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where |
153 lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x) | |
154 lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc ) | |
95 | 155 |
303 | 156 odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x |
157 odef-subst df refl refl = df | |
95 | 158 |
361 | 159 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y |
187 | 160 otrans x<a y<x = ordtrans y<x x<a |
123 | 161 |
303 | 162 odef→o< : {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X |
163 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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164 |
388 | 165 odefo→o< : {X y : Ordinal } → odef (ord→od X) y → y o< X |
166 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 )) | |
167 | |
396 | 168 -- If we have reverse of c<→o<, everything becomes Ordinal |
169 o<→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x) | |
170 o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where | |
171 lemma1 : {y : Ordinal} → odef x y → odef (Ord (od→ord x)) y | |
172 lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (d→∋ x lt)) | |
173 lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y | |
174 lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) | |
175 | |
51 | 176 -- avoiding lv != Zero error |
303 | 177 orefl : { x : HOD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y |
51 | 178 orefl refl = refl |
179 | |
303 | 180 ==-iso : { x y : HOD } → od (ord→od (od→ord x)) == od (ord→od (od→ord y)) → od x == od y |
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181 ==-iso {x} {y} eq = record { |
303 | 182 eq→ = λ d → lemma ( eq→ eq (odef-subst d (sym oiso) refl )) ; |
183 eq← = λ d → lemma ( eq← eq (odef-subst d (sym oiso) refl )) } | |
51 | 184 where |
303 | 185 lemma : {x : HOD } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z |
186 lemma {x} {z} d = odef-subst d oiso refl | |
51 | 187 |
303 | 188 =-iso : {x y : HOD } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y) |
189 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym oiso) | |
57 | 190 |
303 | 191 ord→== : { x y : HOD } → od→ord x ≡ od→ord y → od x == od y |
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192 ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where |
303 | 193 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (ord→od ox) == od (ord→od oy) |
271 | 194 lemma ox ox refl = ==-refl |
51 | 195 |
303 | 196 o≡→== : { x y : Ordinal } → x ≡ y → od (ord→od x) == od (ord→od y) |
271 | 197 o≡→== {x} {.x} refl = ==-refl |
51 | 198 |
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199 o∅≡od∅ : ord→od (o∅ ) ≡ od∅ |
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200 o∅≡od∅ = ==→o≡ lemma where |
303 | 201 lemma0 : {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x |
202 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 | |
203 lemma1 : {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x | |
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204 lemma1 {x} lt = ⊥-elim (¬x<0 lt) |
303 | 205 lemma : od (ord→od o∅) == od od∅ |
150 | 206 lemma = record { eq→ = lemma0 ; eq← = lemma1 } |
207 | |
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208 ord-od∅ : od→ord (od∅ ) ≡ o∅ |
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209 ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) |
80 | 210 |
303 | 211 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅ |
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212 eq→ ∅0 {w} (lift ()) |
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213 eq← ∅0 {w} lt = lift (¬x<0 lt) |
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214 |
303 | 215 ∅< : { x y : HOD } → odef x (od→ord y ) → ¬ ( od x == od od∅ ) |
271 | 216 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d |
223
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217 ∅< {x} {y} d eq | lift () |
57 | 218 |
303 | 219 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox |
223
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220 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x ) |
51 | 221 |
303 | 222 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x |
223 odef-iso refl t = t | |
76 | 224 |
223
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225 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ ) |
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226 is-o∅ x with trio< x o∅ |
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227 is-o∅ x | tri< a ¬b ¬c = no ¬b |
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228 is-o∅ x | tri≈ ¬a b ¬c = yes b |
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229 is-o∅ x | tri> ¬a ¬b c = no ¬b |
57 | 230 |
335 | 231 -- the pair |
338 | 232 _,_ : HOD → HOD → HOD |
308 | 233 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 |
234 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y) | |
235 lemma {t} (case1 refl) = omax-x _ _ | |
236 lemma {t} (case2 refl) = omax-y _ _ | |
237 | |
343 | 238 pair-xx<xy : {x y : HOD} → od→ord (x , x) o< osuc (od→ord (x , y) ) |
239 pair-xx<xy {x} {y} = ⊆→o≤ lemma where | |
240 lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z | |
241 lemma {z} (case1 refl) = case1 refl | |
242 lemma {z} (case2 refl) = case1 refl | |
243 | |
393 | 244 pair-<xy : {x y : HOD} → {n : Ordinal} → od→ord x o< next n → od→ord y o< next n → od→ord (x , y) o< next n |
245 pair-<xy {x} {y} {o} x<nn y<nn with trio< (od→ord x) (od→ord y) | inspect (omax (od→ord x)) (od→ord y) | |
246 ... | tri< a ¬b ¬c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (sym eq1) (osuc<nx y<nn)) ho< | |
247 ... | tri> ¬a ¬b c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (sym eq1) (osuc<nx x<nn)) ho< | |
248 ... | tri≈ ¬a b ¬c | record { eq = eq1 } = next< (subst (λ k → k o< next o ) (omax≡ _ _ b) (subst (λ k → osuc k o< next o) b (osuc<nx x<nn))) ho< | |
249 | |
339 | 250 -- another form of infinite |
343 | 251 -- pair-ord< : {x : Ordinal } → Set n |
252 pair-ord< : {x : HOD } → ( {y : HOD } → od→ord y o< next (odmax y) ) → od→ord ( x , x ) o< next (od→ord x) | |
253 pair-ord< {x} ho< = subst (λ k → od→ord (x , x) o< k ) lemmab0 lemmab1 where | |
254 lemmab0 : next (odmax (x , x)) ≡ next (od→ord x) | |
255 lemmab0 = trans (cong (λ k → next k) (omxx _)) (sym nexto≡) | |
256 lemmab1 : od→ord (x , x) o< next ( odmax (x , x)) | |
257 lemmab1 = ho< | |
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258 |
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259 pair<y : {x y : HOD } → y ∋ x → od→ord (x , x) o< osuc (od→ord y) |
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260 pair<y {x} {y} y∋x = ⊆→o≤ lemma where |
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261 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z |
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262 lemma (case1 refl) = y∋x |
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263 lemma (case2 refl) = y∋x |
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264 |
361 | 265 -- another possible restriction. We reqest no minimality on odmax, so it may arbitrary larger. |
266 odmax<od→ord : { x y : HOD } → x ∋ y → Set n | |
267 odmax<od→ord {x} {y} x∋y = odmax x o< od→ord x | |
268 | |
376 | 269 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD |
270 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } | |
141 | 271 |
360 | 272 _∩_ : ( A B : HOD ) → HOD |
273 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } | |
274 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))} | |
308 | 275 |
303 | 276 record _⊆_ ( A B : HOD ) : Set (suc n) where |
271 | 277 field |
303 | 278 incl : { x : HOD } → A ∋ x → B ∋ x |
271 | 279 |
280 open _⊆_ | |
190 | 281 infixr 220 _⊆_ |
282 | |
387 | 283 trans-⊆ : { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C |
284 trans-⊆ A⊆B B⊆C = record { incl = λ x → incl B⊆C (incl A⊆B x) } | |
285 | |
286 refl-⊆ : {A : HOD} → A ⊆ A | |
287 refl-⊆ {A} = record { incl = λ x → x } | |
288 | |
335 | 289 od⊆→o≤ : {x y : HOD } → x ⊆ y → od→ord x o< osuc (od→ord y) |
396 | 290 od⊆→o≤ {x} {y} lt = ⊆→o≤ {x} {y} (λ {z} x>z → subst (λ k → def (od y) k ) diso (incl lt (d→∋ x x>z))) |
335 | 291 |
292 -- if we have od→ord (x , x) ≡ osuc (od→ord x), ⊆→o≤ → c<→o< | |
338 | 293 ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) ) |
294 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) ) | |
295 → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y | |
296 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y) | |
297 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a | |
298 ⊆→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 ))) | |
299 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c = | |
300 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where | |
301 lemma : {z : Ordinal} → (z ≡ od→ord x) ∨ (z ≡ od→ord x) → od→ord x ≡ z | |
302 lemma (case1 refl) = refl | |
303 lemma (case2 refl) = refl | |
304 y⊆x,x : {z : Ordinals.ord O} → def (od (x , x)) z → def (od y) z | |
305 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x | |
306 lemma1 : osuc (od→ord y) o< od→ord (x , x) | |
307 lemma1 = subst (λ k → osuc (od→ord y) o< k ) (sym (peq {x})) (osucc c ) | |
335 | 308 |
360 | 309 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → (A ∩ x ) ∋ y ) ⇔ ( x ⊆ A ) |
271 | 310 subset-lemma {A} {x} = record { |
311 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } | |
312 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } | |
190 | 313 } |
314 | |
312 | 315 power< : {A x : HOD } → x ⊆ A → Ord (osuc (od→ord A)) ∋ x |
316 power< {A} {x} x⊆A = ⊆→o≤ (λ {y} x∋y → subst (λ k → def (od A) k) diso (lemma y x∋y ) ) where | |
317 lemma : (y : Ordinal) → def (od x) y → def (od A) (od→ord (ord→od y)) | |
396 | 318 lemma y x∋y = incl x⊆A (d→∋ x x∋y) |
312 | 319 |
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320 open import Data.Unit |
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321 |
324 | 322 ε-induction : { ψ : HOD → Set n} |
303 | 323 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) |
324 → (x : HOD ) → ψ x | |
261
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325 ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where |
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326 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) |
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327 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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328 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) |
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329 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy |
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330 |
388 | 331 -- level trick (what'a shame) for LEM / minimal |
332 ε-induction1 : { ψ : HOD → Set (suc n)} | |
333 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) | |
334 → (x : HOD ) → ψ x | |
335 ε-induction1 {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where | |
336 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) | |
337 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) | |
338 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) | |
339 ε-induction-ord ox {oy} lt = TransFinite1 {λ oy → ψ (ord→od oy)} induction oy | |
330 | 340 |
376 | 341 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD |
342 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 | 343 |
376 | 344 Replace : HOD → (HOD → HOD) → HOD |
345 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 | 346 ; odmax = rmax ; <odmax = rmax<} where |
347 rmax : Ordinal | |
376 | 348 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y))) |
365 | 349 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax |
350 rmax< lt = proj1 lt | |
377 | 351 |
378 | 352 -- |
353 -- If we have LEM, Replace' is equivalent to Replace | |
354 -- | |
377 | 355 in-codomain' : (X : HOD ) → ((x : HOD) → X ∋ x → HOD) → OD |
356 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) )))) } | |
357 Replace' : (X : HOD) → ((x : HOD) → X ∋ x → HOD) → HOD | |
358 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 } | |
359 ; odmax = rmax ; <odmax = rmax< } where | |
360 rmax : Ordinal | |
361 rmax = sup-o X (λ y X∋y → od→ord (ψ (ord→od y) (d→∋ X X∋y))) | |
362 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain' X ψ) y → y o< rmax | |
363 rmax< lt = proj1 lt | |
364 | |
365 | 365 Union : HOD → HOD |
366 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } | |
367 ; odmax = osuc (od→ord U) ; <odmax = umax< } where | |
368 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (ord→od u)) y) → y o< osuc (od→ord U) | |
369 umax< {y} not = lemma (FExists _ lemma1 not ) where | |
370 lemma0 : {x : Ordinal} → def (od (ord→od x)) y → y o< x | |
396 | 371 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) diso diso (c<→o< (d→∋ (ord→od x) x<y )) |
365 | 372 lemma2 : {x : Ordinal} → def (od U) x → x o< od→ord U |
396 | 373 lemma2 {x} x<U = subst (λ k → k o< od→ord U ) diso (c<→o< (d→∋ U x<U)) |
365 | 374 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (ord→od x)) y → ¬ (od→ord U o< y) |
375 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) ) | |
376 lemma : ¬ ((od→ord U) o< y ) → y o< osuc (od→ord U) | |
377 lemma not with trio< y (od→ord U) | |
378 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc | |
379 lemma not | tri≈ ¬a refl ¬c = <-osuc | |
380 lemma not | tri> ¬a ¬b c = ⊥-elim (not c) | |
381 _∈_ : ( A B : HOD ) → Set n | |
382 A ∈ B = B ∋ A | |
383 | |
384 OPwr : (A : HOD ) → HOD | |
385 OPwr A = Ord ( sup-o (Ord (osuc (od→ord A))) ( λ x A∋x → od→ord ( A ∩ (ord→od x)) ) ) | |
386 | |
387 Power : HOD → HOD | |
376 | 388 Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) |
365 | 389 -- {_} : ZFSet → ZFSet |
390 -- { x } = ( x , x ) -- better to use (x , x) directly | |
391 | |
402 | 392 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z |
393 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx | |
394 ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) | |
395 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) | |
396 union← X z UX∋z = FExists _ lemma UX∋z where | |
397 lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) | |
398 lemma {y} xx not = not (ord→od y) record { proj1 = d→∋ X (proj1 xx) ; proj2 = proj2 xx } | |
399 | |
365 | 400 data infinite-d : ( x : Ordinal ) → Set n where |
401 iφ : infinite-d o∅ | |
402 isuc : {x : Ordinal } → infinite-d x → | |
403 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) | |
404 | |
405 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. | |
406 -- We simply assumes infinite-d y has a maximum. | |
407 -- | |
408 -- This means that many of OD may not be HODs because of the od→ord mapping divergence. | |
409 -- We should have some axioms to prevent this such as od→ord x o< next (odmax x). | |
410 -- | |
367 | 411 -- postulate |
412 -- ωmax : Ordinal | |
413 -- <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax | |
414 -- | |
415 -- infinite : HOD | |
416 -- infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } | |
365 | 417 |
400 | 418 odsuc : (y : HOD) → HOD |
419 odsuc y = Union (y , (y , y)) | |
420 | |
365 | 421 infinite : HOD |
367 | 422 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where |
365 | 423 u : (y : Ordinal ) → HOD |
424 u y = Union (ord→od y , (ord→od y , ord→od y)) | |
425 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z | |
426 lemma8 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (odmax (ord→od y , ord→od y)) | |
427 lemma8 = ho< | |
428 --- (x,y) < next (omax x y) < next (osuc y) = next y | |
429 lemmaa : {x y : HOD} → od→ord x o< od→ord y → od→ord (x , y) o< next (od→ord y) | |
430 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< ) | |
431 lemma81 : {y : Ordinal} → od→ord (ord→od y , ord→od y) o< next (od→ord (ord→od y)) | |
432 lemma81 {y} = nexto=n (subst (λ k → od→ord (ord→od y , ord→od y) o< k ) (cong (λ k → next k) (omxx _)) lemma8) | |
433 lemma9 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y , ord→od y)) | |
434 lemma9 = lemmaa (c<→o< (case1 refl)) | |
435 lemma71 : {y : Ordinal} → od→ord (ord→od y , (ord→od y , ord→od y)) o< next (od→ord (ord→od y)) | |
436 lemma71 = next< lemma81 lemma9 | |
437 lemma1 : {y : Ordinal} → od→ord (u y) o< next (osuc (od→ord (ord→od y , (ord→od y , ord→od y)))) | |
438 lemma1 = ho< | |
439 --- main recursion | |
440 lemma : {y : Ordinal} → infinite-d y → y o< next o∅ | |
441 lemma {o∅} iφ = x<nx | |
442 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)) | |
443 | |
393 | 444 ω<next-o∅ : {y : Ordinal} → infinite-d y → y o< next o∅ |
445 ω<next-o∅ {y} lt = <odmax infinite lt | |
365 | 446 |
447 nat→ω : Nat → HOD | |
448 nat→ω Zero = od∅ | |
449 nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y)) | |
450 | |
400 | 451 ω→nato : {y : Ordinal} → infinite-d y → Nat |
452 ω→nato iφ = Zero | |
453 ω→nato (isuc lt) = Suc (ω→nato lt) | |
454 | |
365 | 455 ω→nat : (n : HOD) → infinite ∋ n → Nat |
400 | 456 ω→nat n = ω→nato |
365 | 457 |
458 ω∋nat→ω : {n : Nat} → def (od infinite) (od→ord (nat→ω n)) | |
367 | 459 ω∋nat→ω {Zero} = subst (λ k → def (od infinite) k) (sym ord-od∅) iφ |
460 ω∋nat→ω {Suc n} = subst (λ k → def (od infinite) k) lemma (isuc ( ω∋nat→ω {n})) where | |
461 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)) | |
462 lemma = subst (λ k → od→ord (Union (k , ( k , k ))) ≡ od→ord (nat→ω (Suc n))) (sym oiso) refl | |
365 | 463 |
464 _=h=_ : (x y : HOD) → Set n | |
465 x =h= y = od x == od y | |
466 | |
467 infixr 200 _∈_ | |
468 -- infixr 230 _∩_ _∪_ | |
469 | |
470 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) | |
471 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) | |
472 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) | |
473 | |
474 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t | |
475 pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) | |
476 pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) | |
477 | |
402 | 478 pair1 : { x y : HOD } → (x , y ) ∋ x |
479 pair1 = case1 refl | |
480 | |
481 pair2 : { x y : HOD } → (x , y ) ∋ y | |
482 pair2 = case2 refl | |
483 | |
365 | 484 empty : (x : HOD ) → ¬ (od∅ ∋ x) |
485 empty x = ¬x<0 | |
486 | |
487 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) | |
488 o<→c< lt = record { incl = λ z → ordtrans z lt } | |
489 | |
490 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y | |
491 ⊆→o< {x} {y} lt with trio< x y | |
492 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc | |
493 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc | |
494 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) | |
495 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) | |
496 | |
404 | 497 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
402 | 498 -- postulate f-extensionality : { n m : Level} → HE.Extensionality n m |
499 | |
500 nat→ω-iso : {i : HOD} → (lt : infinite ∋ i ) → nat→ω ( ω→nat i lt ) ≡ i | |
501 nat→ω-iso {i} = ε-induction1 {λ i → (lt : infinite ∋ i ) → nat→ω ( ω→nat i lt ) ≡ i } ind i where | |
502 ind : {x : HOD} → ({y : HOD} → x ∋ y → (lt : infinite ∋ y) → nat→ω (ω→nat y lt) ≡ y) → | |
503 (lt : infinite ∋ x) → nat→ω (ω→nat x lt) ≡ x | |
504 ind {x} prev lt = ind1 lt oiso where | |
505 ind1 : {ox : Ordinal } → (ltd : infinite-d ox ) → ord→od ox ≡ x → nat→ω (ω→nato ltd) ≡ x | |
506 ind1 {o∅} iφ refl = sym o∅≡od∅ | |
507 ind1 (isuc {x₁} ltd) ox=x = begin | |
508 nat→ω (ω→nato (isuc ltd) ) | |
509 ≡⟨⟩ | |
510 Union (nat→ω (ω→nato ltd) , (nat→ω (ω→nato ltd) , nat→ω (ω→nato ltd))) | |
511 ≡⟨ cong (λ k → Union (k , (k , k ))) lemma ⟩ | |
512 Union (ord→od x₁ , (ord→od x₁ , ord→od x₁)) | |
513 ≡⟨ trans ( sym oiso) ox=x ⟩ | |
514 x | |
515 ∎ where | |
516 open ≡-Reasoning | |
517 lemma0 : x ∋ ord→od x₁ | |
518 lemma0 = subst (λ k → odef k (od→ord (ord→od x₁))) (trans (sym oiso) ox=x) (λ not → not | |
519 (od→ord (ord→od x₁ , ord→od x₁)) record {proj1 = pair2 ; proj2 = subst (λ k → odef k (od→ord (ord→od x₁))) (sym oiso) pair1 } ) | |
520 lemma1 : infinite ∋ ord→od x₁ | |
521 lemma1 = subst (λ k → odef infinite k) (sym diso) ltd | |
404 | 522 lemma5 : {x y : Ordinal} → od→ord (Union (ord→od y , (ord→od y , ord→od y))) ≡ od→ord (Union (ord→od x , (ord→od x , ord→od x))) → x ≡ y |
523 lemma5 {x} {y} eq = {!!} | |
524 lemma3 : {x y : Ordinal} → (ltd : infinite-d x ) (ltd1 : infinite-d y ) → y ≡ x → ltd ≅ ltd1 | |
525 lemma3 iφ iφ refl = HE.refl | |
526 lemma3 iφ (isuc ltd1) eq = {!!} | |
527 lemma3 (isuc ltd) iφ eq = {!!} | |
528 lemma3 (isuc {x} ltd) (isuc {y} ltd1) eq with lemma3 ltd ltd1 (lemma5 (sym eq)) | |
529 ... | t = HE.cong₂ (λ j k → isuc {j} k ) (HE.≡-to-≅ (lemma5 eq)) t | |
530 lemma2 : {x y : Ordinal} → (ltd : infinite-d x ) (ltd1 : infinite-d y ) → y ≡ x → ω→nato ltd ≡ ω→nato ltd1 | |
531 lemma2 {x} {y} ltd ltd1 eq = lemma6 eq (lemma3 {x} {y} ltd ltd1 eq) where | |
532 lemma6 : {x y : Ordinal} → {ltd : infinite-d x } {ltd1 : infinite-d y } → y ≡ x → ltd ≅ ltd1 → ω→nato ltd ≡ ω→nato ltd1 | |
533 lemma6 refl HE.refl = refl | |
402 | 534 lemma : nat→ω (ω→nato ltd) ≡ ord→od x₁ |
404 | 535 lemma = trans (cong (λ k → nat→ω k) (lemma2 {x₁} {_} ltd (subst (λ k → infinite-d k ) (sym diso) ltd) diso ) ) ( prev {ord→od x₁} lemma0 lemma1 ) |
536 | |
537 ω→nat-iso : {i : Nat} → ω→nat ( nat→ω i ) (ω∋nat→ω {i}) ≡ i | |
538 ω→nat-iso {Zero} = {!!} | |
539 ω→nat-iso {Suc i} = {!!} | |
365 | 540 |
376 | 541 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y |
542 ψiso {ψ} t refl = t | |
543 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) | |
544 selection {ψ} {X} {y} = record { | |
545 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } | |
546 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } | |
547 } | |
377 | 548 |
395 | 549 selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y |
550 selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt) | |
551 | |
376 | 552 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)))) |
553 sup-c< ψ {X} {x} lt = subst (λ k → od→ord (ψ k) o< _ ) oiso (sup-o< X lt ) | |
554 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x | |
555 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {X} {x} lt ; proj2 = lemma } where | |
556 lemma : def (in-codomain X ψ) (od→ord (ψ x)) | |
557 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) | |
558 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y)) | |
559 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where | |
560 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) | |
561 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y))) | |
365 | 562 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where |
376 | 563 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) =h= ψ (ord→od y)) |
365 | 564 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) |
376 | 565 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) |
566 lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) | |
365 | 567 |
568 --- | |
569 --- Power Set | |
570 --- | |
571 --- First consider ordinals in HOD | |
572 --- | |
573 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A | |
574 -- | |
575 -- | |
576 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) | |
577 ∩-≡ {a} {b} inc = record { | |
578 eq→ = λ {x} x<a → record { proj2 = x<a ; | |
396 | 579 proj1 = odef-subst {_} {_} {b} {x} (inc (d→∋ a x<a)) refl diso } ; |
365 | 580 eq← = λ {x} x<a∩b → proj2 x<a∩b } |
581 -- | |
582 -- Transitive Set case | |
583 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t | |
584 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t | |
585 -- OPwr A = Ord ( sup-o ( λ x → od→ord ( A ∩ (ord→od x )) ) ) | |
586 -- | |
587 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t | |
588 ord-power← a t t→A = odef-subst {_} {_} {OPwr (Ord a)} {od→ord t} | |
589 lemma refl (lemma1 lemma-eq )where | |
590 lemma-eq : ((Ord a) ∩ t) =h= t | |
591 eq→ lemma-eq {z} w = proj2 w | |
592 eq← lemma-eq {z} w = record { proj2 = w ; | |
593 proj1 = odef-subst {_} {_} {(Ord a)} {z} | |
396 | 594 ( t→A (d→∋ t w)) refl diso } |
365 | 595 lemma1 : {a : Ordinal } { t : HOD } |
596 → (eq : ((Ord a) ∩ t) =h= t) → od→ord ((Ord a) ∩ (ord→od (od→ord t))) ≡ od→ord t | |
597 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 )) | |
598 lemma2 : (od→ord t) o< (osuc (od→ord (Ord a))) | |
396 | 599 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) diso (t→A (d→∋ t x<t))) |
365 | 600 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))) |
601 lemma = sup-o< _ lemma2 | |
602 | |
603 -- | |
604 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first | |
605 -- then replace of all elements of the Power set by A ∩ y | |
606 -- | |
607 -- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y ) | |
608 | |
609 -- we have oly double negation form because of the replacement axiom | |
610 -- | |
611 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x) | |
612 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where | |
613 a = od→ord A | |
614 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y))) | |
376 | 615 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (od→ord A))) t P∋t |
365 | 616 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x) |
617 lemma3 y eq not = not (proj1 (eq→ eq t∋x)) | |
618 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y))) | |
619 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 )) | |
620 lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) → ¬ ¬ (odef A (od→ord x)) | |
621 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not | |
622 | |
623 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t | |
624 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where | |
625 a = od→ord A | |
626 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x | |
627 lemma0 {x} t∋x = c<→o< (t→A t∋x) | |
628 lemma3 : OPwr (Ord a) ∋ t | |
629 lemma3 = ord-power← a t lemma0 | |
630 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t | |
631 lemma4 = let open ≡-Reasoning in begin | |
632 A ∩ ord→od (od→ord t) | |
633 ≡⟨ cong (λ k → A ∩ k) oiso ⟩ | |
634 A ∩ t | |
635 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩ | |
636 t | |
637 ∎ | |
638 sup1 : Ordinal | |
639 sup1 = sup-o (Ord (osuc (od→ord (Ord (od→ord A))))) (λ x A∋x → od→ord ((Ord (od→ord A)) ∩ (ord→od x))) | |
640 lemma9 : def (od (Ord (Ordinals.osuc O (od→ord (Ord (od→ord A)))))) (od→ord (Ord (od→ord A))) | |
641 lemma9 = <-osuc | |
642 lemmab : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) o< sup1 | |
643 lemmab = sup-o< (Ord (osuc (od→ord (Ord (od→ord A))))) lemma9 | |
644 lemmad : Ord (osuc (od→ord A)) ∋ t | |
396 | 645 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) diso (t→A (d→∋ t lt))) |
365 | 646 lemmac : ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A) )))) =h= Ord (od→ord A) |
647 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where | |
648 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 | |
649 lemmaf {x} lt = proj1 lt | |
650 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 | |
651 lemmag {x} lt = record { proj1 = lt ; proj2 = subst (λ k → def (od k) x) (sym oiso) lt } | |
652 lemmae : od→ord ((Ord (od→ord A)) ∩ (ord→od (od→ord (Ord (od→ord A))))) ≡ od→ord (Ord (od→ord A)) | |
653 lemmae = cong (λ k → od→ord k ) ( ==→o≡ lemmac) | |
654 lemma7 : def (od (OPwr (Ord (od→ord A)))) (od→ord t) | |
655 lemma7 with osuc-≡< lemmad | |
656 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab ) | |
657 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 | |
658 lemmah : {x : Ordinal } → def (od (ord→od (od→ord t))) x → def (od (ord→od (od→ord (Ord (od→ord t))))) x | |
659 lemmah {x} lt = subst (λ k → def (od k) x ) (sym oiso) (subst (λ k → k o< (od→ord t)) | |
660 diso | |
661 (c<→o< (subst₂ (λ j k → def (od j) k) oiso (sym diso) lt ))) | |
662 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where | |
663 lemmai : od→ord (Ord (od→ord A)) ≡ od→ord t | |
664 lemmai = let open ≡-Reasoning in begin | |
665 od→ord (Ord (od→ord A)) | |
666 ≡⟨ sym (cong (λ k → od→ord (Ord k)) eq) ⟩ | |
667 od→ord (Ord (od→ord t)) | |
668 ≡⟨ sym diso ⟩ | |
669 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
670 ≡⟨ sym eq1 ⟩ | |
671 od→ord (ord→od (od→ord t)) | |
672 ≡⟨ diso ⟩ | |
673 od→ord t | |
674 ∎ | |
675 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where | |
676 lemmak : od→ord (ord→od (od→ord (Ord (od→ord t)))) ≡ od→ord (Ord (od→ord A)) | |
677 lemmak = let open ≡-Reasoning in begin | |
678 od→ord (ord→od (od→ord (Ord (od→ord t)))) | |
679 ≡⟨ diso ⟩ | |
680 od→ord (Ord (od→ord t)) | |
681 ≡⟨ cong (λ k → od→ord (Ord k)) eq ⟩ | |
682 od→ord (Ord (od→ord A)) | |
683 ∎ | |
684 lemmaj : od→ord t o< od→ord (Ord (od→ord A)) | |
685 lemmaj = subst₂ (λ j k → j o< k ) diso lemmak lt | |
686 lemma1 : od→ord t o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x))) | |
687 lemma1 = subst (λ k → od→ord k o< sup-o (OPwr (Ord (od→ord A))) (λ x lt → od→ord (A ∩ (ord→od x)))) | |
688 lemma4 (sup-o< (OPwr (Ord (od→ord A))) lemma7 ) | |
376 | 689 lemma2 : def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t) |
690 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where | |
365 | 691 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) |
692 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym oiso) ( ∩-≡ {t} {A} t→A ))) | |
693 | |
694 | |
695 ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) | |
696 ord⊆power a = record { incl = λ {x} lt → power← (Ord a) x (lemma lt) } where | |
697 lemma : {x y : HOD} → od→ord x o< osuc a → x ∋ y → Ord a ∋ y | |
698 lemma lt y<x with osuc-≡< lt | |
699 lemma lt y<x | case1 refl = c<→o< y<x | |
700 lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a | |
701 | |
702 continuum-hyphotheis : (a : Ordinal) → Set (suc n) | |
703 continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) | |
704 | |
705 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B | |
706 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d | |
707 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d | |
708 | |
709 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) | |
710 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d | |
711 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d | |
712 | |
713 infinity∅ : infinite ∋ od∅ | |
714 infinity∅ = odef-subst {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where | |
715 lemma : o∅ ≡ od→ord od∅ | |
716 lemma = let open ≡-Reasoning in begin | |
717 o∅ | |
718 ≡⟨ sym diso ⟩ | |
719 od→ord ( ord→od o∅ ) | |
720 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ | |
721 od→ord od∅ | |
722 ∎ | |
723 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) | |
724 infinity x lt = odef-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where | |
725 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) | |
726 ≡ od→ord (Union (x , (x , x))) | |
727 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso | |
728 | |
729 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite | |
730 isZF = record { | |
731 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } | |
732 ; pair→ = pair→ | |
733 ; pair← = pair← | |
734 ; union→ = union→ | |
735 ; union← = union← | |
736 ; empty = empty | |
737 ; power→ = power→ | |
738 ; power← = power← | |
739 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w} | |
740 ; ε-induction = ε-induction | |
741 ; infinity∅ = infinity∅ | |
742 ; infinity = infinity | |
376 | 743 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y} |
744 ; replacement← = replacement← | |
745 ; replacement→ = λ {ψ} → replacement→ {ψ} | |
365 | 746 -- ; choice-func = choice-func |
747 -- ; choice = choice | |
748 } | |
749 | |
303 | 750 HOD→ZF : ZF |
751 HOD→ZF = record { | |
752 ZFSet = HOD | |
43
0d9b9db14361
equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
42
diff
changeset
|
753 ; _∋_ = _∋_ |
365 | 754 ; _≈_ = _=h=_ |
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
755 ; ∅ = od∅ |
28 | 756 ; _,_ = _,_ |
365 | 757 ; Union = Union |
758 ; Power = Power | |
759 ; Select = Select | |
760 ; Replace = Replace | |
761 ; infinite = infinite | |
762 ; isZF = isZF | |
763 } | |
764 | |
144 | 765 |