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reorganiztion for apkg
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 21 Dec 2020 10:23:37 +0900
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children 3681cac8d6a8 81691a6b352b
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430:28c7be8f252c 431:a5f8084b8368
1 {-# OPTIONS --allow-unsolved-metas #-}
2 open import Level
3 open import Ordinals
4 module OD {n : Level } (O : Ordinals {n} ) where
5
6 open import zf
7 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
8 open import Relation.Binary.PropositionalEquality hiding ( [_] )
9 open import Data.Nat.Properties
10 open import Data.Empty
11 open import Relation.Nullary
12 open import Relation.Binary hiding (_⇔_)
13 open import Relation.Binary.Core hiding (_⇔_)
14
15 open import logic
16 import OrdUtil
17 open import nat
18
19 open Ordinals.Ordinals O
20 open Ordinals.IsOrdinals isOrdinal
21 open Ordinals.IsNext isNext
22 open OrdUtil O
23
24 -- Ordinal Definable Set
25
26 record OD : Set (suc n ) where
27 field
28 def : (x : Ordinal ) → Set n
29
30 open OD
31
32 open _∧_
33 open _∨_
34 open Bool
35
36 record _==_ ( a b : OD ) : Set n where
37 field
38 eq→ : ∀ { x : Ordinal } → def a x → def b x
39 eq← : ∀ { x : Ordinal } → def b x → def a x
40
41 ==-refl : { x : OD } → x == x
42 ==-refl {x} = record { eq→ = λ x → x ; eq← = λ x → x }
43
44 open _==_
45
46 ==-trans : { x y z : OD } → x == y → y == z → x == z
47 ==-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) }
48
49 ==-sym : { x y : OD } → x == y → y == x
50 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
51
52
53 ⇔→== : { x y : OD } → ( {z : Ordinal } → (def x z ⇔ def y z)) → x == y
54 eq→ ( ⇔→== {x} {y} eq ) {z} m = proj1 eq m
55 eq← ( ⇔→== {x} {y} eq ) {z} m = proj2 eq m
56
57 -- next assumptions are our axiom
58 --
59 -- OD is an equation on Ordinals, so it contains Ordinals. If these Ordinals have one-to-one
60 -- correspondence to the OD then the OD looks like a ZF Set.
61 --
62 -- If all ZF Set have supreme upper bound, the solutions of OD have to be bounded, i.e.
63 -- bbounded ODs are ZF Set. Unbounded ODs are classes.
64 --
65 -- In classical Set Theory, HOD is used, as a subset of OD,
66 -- HOD = { x | TC x ⊆ OD }
67 -- where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
68 -- This is not possible because we don't have V yet. So we assumes HODs are bounded OD.
69 --
70 -- We also assumes HODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
71 -- There two contraints on the HOD order, one is ∋, the other one is ⊂.
72 -- ODs have an ovbious maximum, but Ordinals are not, but HOD has no maximum because there is an aribtrary
73 -- bound on each HOD.
74 --
75 -- In classical Set Theory, sup is defined by Uion, since we are working on constructive logic,
76 -- we need explict assumption on sup.
77 --
78 -- ==→o≡ is necessary to prove axiom of extensionality.
79
80 -- Ordinals in OD , the maximum
81 Ords : OD
82 Ords = record { def = λ x → One }
83
84 record HOD : Set (suc n) where
85 field
86 od : OD
87 odmax : Ordinal
88 <odmax : {y : Ordinal} → def od y → y o< odmax
89
90 open HOD
91
92 record ODAxiom : Set (suc n) where
93 field
94 -- HOD is isomorphic to Ordinal (by means of Goedel number)
95 & : HOD → Ordinal
96 * : Ordinal → HOD
97 c<→o< : {x y : HOD } → def (od y) ( & x ) → & x o< & y
98 ⊆→o≤ : {y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z)
99 *iso : {x : HOD } → * ( & x ) ≡ x
100 &iso : {x : Ordinal } → & ( * x ) ≡ x
101 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
102 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal
103 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
104 -- possible order restriction
105 ho< : {x : HOD} → & x o< next (odmax x)
106
107
108 postulate odAxiom : ODAxiom
109 open ODAxiom odAxiom
110
111 -- odmax minimality
112 --
113 -- since we have ==→o≡ , so odmax have to be unique. We should have odmaxmin in HOD.
114 -- We can calculate the minimum using sup but it is tedius.
115 -- Only Select has non minimum odmax.
116 -- We have the same problem on 'def' itself, but we leave it.
117
118 odmaxmin : Set (suc n)
119 odmaxmin = (y : HOD) (z : Ordinal) → ((x : Ordinal)→ def (od y) x → x o< z) → odmax y o< osuc z
120
121 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD
122 ¬OD-order : ( & : OD → Ordinal ) → ( * : Ordinal → OD ) → ( { x y : OD } → def y ( & x ) → & x o< & y) → ⊥
123 ¬OD-order & * c<→o< = osuc-< <-osuc (c<→o< {Ords} OneObj )
124
125 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup
126 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥
127 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where
128 next-ord : Ordinal → Ordinal
129 next-ord x = osuc x
130
131 -- Ordinal in OD ( and ZFSet ) Transitive Set
132 Ord : ( a : Ordinal ) → HOD
133 Ord a = record { od = record { def = λ y → y o< a } ; odmax = a ; <odmax = lemma } where
134 lemma : {x : Ordinal} → x o< a → x o< a
135 lemma {x} lt = lt
136
137 od∅ : HOD
138 od∅ = Ord o∅
139
140 odef : HOD → Ordinal → Set n
141 odef A x = def ( od A ) x
142
143 _∋_ : ( a x : HOD ) → Set n
144 _∋_ a x = odef a ( & x )
145
146 -- _c<_ : ( x a : HOD ) → Set n
147 -- x c< a = a ∋ x
148
149 d→∋ : ( a : HOD ) { x : Ordinal} → odef a x → a ∋ (* x)
150 d→∋ a lt = subst (λ k → odef a k ) (sym &iso) lt
151
152 -- odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x
153 -- odef-subst df refl refl = df
154
155 otrans : {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y
156 otrans x<a y<x = ordtrans y<x x<a
157
158 -- If we have reverse of c<→o<, everything becomes Ordinal
159 ∈→c<→HOD=Ord : ( o<→c< : {x y : Ordinal } → x o< y → odef (* y) x ) → {x : HOD } → x ≡ Ord (& x)
160 ∈→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
161 lemma1 : {y : Ordinal} → odef x y → odef (Ord (& x)) y
162 lemma1 {y} lt = subst ( λ k → k o< & x ) &iso (c<→o< {* y} {x} (d→∋ x lt))
163 lemma2 : {y : Ordinal} → odef (Ord (& x)) y → odef x y
164 lemma2 {y} lt = subst (λ k → odef k y ) *iso (o<→c< {y} {& x} lt )
165
166 -- avoiding lv != Zero error
167 orefl : { x : HOD } → { y : Ordinal } → & x ≡ y → & x ≡ y
168 orefl refl = refl
169
170 ==-iso : { x y : HOD } → od (* (& x)) == od (* (& y)) → od x == od y
171 ==-iso {x} {y} eq = record {
172 eq→ = λ {z} d → lemma ( eq→ eq (subst (λ k → odef k z ) (sym *iso) d )) ;
173 eq← = λ {z} d → lemma ( eq← eq (subst (λ k → odef k z ) (sym *iso) d )) }
174 where
175 lemma : {x : HOD } {z : Ordinal } → odef (* (& x)) z → odef x z
176 lemma {x} {z} d = subst (λ k → odef k z) (*iso) d
177
178 =-iso : {x y : HOD } → (od x == od y) ≡ (od (* (& x)) == od y)
179 =-iso {_} {y} = cong ( λ k → od k == od y ) (sym *iso)
180
181 ord→== : { x y : HOD } → & x ≡ & y → od x == od y
182 ord→== {x} {y} eq = ==-iso (lemma (& x) (& y) (orefl eq)) where
183 lemma : ( ox oy : Ordinal ) → ox ≡ oy → od (* ox) == od (* oy)
184 lemma ox ox refl = ==-refl
185
186 o≡→== : { x y : Ordinal } → x ≡ y → od (* x) == od (* y)
187 o≡→== {x} {.x} refl = ==-refl
188
189 o∅≡od∅ : * (o∅ ) ≡ od∅
190 o∅≡od∅ = ==→o≡ lemma where
191 lemma0 : {x : Ordinal} → odef (* o∅) x → odef od∅ x
192 lemma0 {x} lt with c<→o< {* x} {* o∅} (subst (λ k → odef (* o∅) k ) (sym &iso) lt)
193 ... | t = subst₂ (λ j k → j o< k ) &iso &iso t
194 lemma1 : {x : Ordinal} → odef od∅ x → odef (* o∅) x
195 lemma1 {x} lt = ⊥-elim (¬x<0 lt)
196 lemma : od (* o∅) == od od∅
197 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
198
199 ord-od∅ : & (od∅ ) ≡ o∅
200 ord-od∅ = sym ( subst (λ k → k ≡ & (od∅ ) ) &iso (cong ( λ k → & k ) o∅≡od∅ ) )
201
202 ≡o∅→=od∅ : {x : HOD} → & x ≡ o∅ → od x == od od∅
203 ≡o∅→=od∅ {x} eq = record { eq→ = λ {y} lt → ⊥-elim ( ¬x<0 {y} (subst₂ (λ j k → j o< k ) &iso eq ( c<→o< {* y} {x} (d→∋ x lt))))
204 ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )}
205
206 =od∅→≡o∅ : {x : HOD} → od x == od od∅ → & x ≡ o∅
207 =od∅→≡o∅ {x} eq = trans (cong (λ k → & k ) (==→o≡ {x} {od∅} eq)) ord-od∅
208
209 ∅0 : record { def = λ x → Lift n ⊥ } == od od∅
210 eq→ ∅0 {w} (lift ())
211 eq← ∅0 {w} lt = lift (¬x<0 lt)
212
213 ∅< : { x y : HOD } → odef x (& y ) → ¬ ( od x == od od∅ )
214 ∅< {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
215 ∅< {x} {y} d eq | lift ()
216
217 ∅6 : { x : HOD } → ¬ ( x ∋ x ) -- no Russel paradox
218 ∅6 {x} x∋x = o<¬≡ refl ( c<→o< {x} {x} x∋x )
219
220 odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y → (odef A y → odef B y) → odef A x → odef B x
221 odef-iso refl t = t
222
223 is-o∅ : ( x : Ordinal ) → Dec ( x ≡ o∅ )
224 is-o∅ x with trio< x o∅
225 is-o∅ x | tri< a ¬b ¬c = no ¬b
226 is-o∅ x | tri≈ ¬a b ¬c = yes b
227 is-o∅ x | tri> ¬a ¬b c = no ¬b
228
229 -- the pair
230 _,_ : HOD → HOD → HOD
231 x , y = record { od = record { def = λ t → (t ≡ & x ) ∨ ( t ≡ & y ) } ; odmax = omax (& x) (& y) ; <odmax = lemma } where
232 lemma : {t : Ordinal} → (t ≡ & x) ∨ (t ≡ & y) → t o< omax (& x) (& y)
233 lemma {t} (case1 refl) = omax-x _ _
234 lemma {t} (case2 refl) = omax-y _ _
235
236 pair<y : {x y : HOD } → y ∋ x → & (x , x) o< osuc (& y)
237 pair<y {x} {y} y∋x = ⊆→o≤ lemma where
238 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z
239 lemma (case1 refl) = y∋x
240 lemma (case2 refl) = y∋x
241
242 -- another possible restriction. We reqest no minimality on odmax, so it may arbitrary larger.
243 odmax<& : { x y : HOD } → x ∋ y → Set n
244 odmax<& {x} {y} x∋y = odmax x o< & x
245
246 in-codomain : (X : HOD ) → ( ψ : HOD → HOD ) → OD
247 in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ & (ψ (* y ))))) }
248
249 _∩_ : ( A B : HOD ) → HOD
250 A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x }
251 ; odmax = omin (odmax A) (odmax B) ; <odmax = λ y → min1 (<odmax A (proj1 y)) (<odmax B (proj2 y))}
252
253 record _⊆_ ( A B : HOD ) : Set (suc n) where
254 field
255 incl : { x : HOD } → A ∋ x → B ∋ x
256
257 open _⊆_
258 infixr 220 _⊆_
259
260 -- if we have & (x , x) ≡ osuc (& x), ⊆→o≤ → c<→o<
261 ⊆→o≤→c<→o< : ({x : HOD} → & (x , x) ≡ osuc (& x) )
262 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → & y o< osuc (& z) )
263 → {x y : HOD } → def (od y) ( & x ) → & x o< & y
264 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (& x) (& y)
265 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a
266 ⊆→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 )))
267 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c =
268 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where
269 lemma : {z : Ordinal} → (z ≡ & x) ∨ (z ≡ & x) → & x ≡ z
270 lemma (case1 refl) = refl
271 lemma (case2 refl) = refl
272 y⊆x,x : {z : Ordinal} → def (od (x , x)) z → def (od y) z
273 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x
274 lemma1 : osuc (& y) o< & (x , x)
275 lemma1 = subst (λ k → osuc (& y) o< k ) (sym (peq {x})) (osucc c )
276
277 ε-induction : { ψ : HOD → Set (suc n)}
278 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
279 → (x : HOD ) → ψ x
280 ε-induction {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
281 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
282 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso )))
283 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
284 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy
285
286 Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD
287 Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( * x )) } ; odmax = odmax X ; <odmax = λ y → <odmax X (proj1 y) }
288
289 Replace : HOD → (HOD → HOD) → HOD
290 Replace X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → & (ψ (* y)))) ∧ def (in-codomain X ψ) x }
291 ; odmax = rmax ; <odmax = rmax<} where
292 rmax : Ordinal
293 rmax = sup-o X (λ y X∋y → & (ψ (* y)))
294 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain X ψ) y → y o< rmax
295 rmax< lt = proj1 lt
296
297 --
298 -- If we have LEM, Replace' is equivalent to Replace
299 --
300 in-codomain' : (X : HOD ) → ((x : HOD) → X ∋ x → HOD) → OD
301 in-codomain' X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ((lt : odef X y) → x ≡ & (ψ (* y ) (d→∋ X lt) )))) }
302 Replace' : (X : HOD) → ((x : HOD) → X ∋ x → HOD) → HOD
303 Replace' X ψ = record { od = record { def = λ x → (x o< sup-o X (λ y X∋y → & (ψ (* y) (d→∋ X X∋y) ))) ∧ def (in-codomain' X ψ) x }
304 ; odmax = rmax ; <odmax = rmax< } where
305 rmax : Ordinal
306 rmax = sup-o X (λ y X∋y → & (ψ (* y) (d→∋ X X∋y)))
307 rmax< : {y : Ordinal} → (y o< rmax) ∧ def (in-codomain' X ψ) y → y o< rmax
308 rmax< lt = proj1 lt
309
310 Union : HOD → HOD
311 Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (* u) x))) }
312 ; odmax = osuc (& U) ; <odmax = umax< } where
313 umax< : {y : Ordinal} → ¬ ((u : Ordinal) → ¬ def (od U) u ∧ def (od (* u)) y) → y o< osuc (& U)
314 umax< {y} not = lemma (FExists _ lemma1 not ) where
315 lemma0 : {x : Ordinal} → def (od (* x)) y → y o< x
316 lemma0 {x} x<y = subst₂ (λ j k → j o< k ) &iso &iso (c<→o< (d→∋ (* x) x<y ))
317 lemma2 : {x : Ordinal} → def (od U) x → x o< & U
318 lemma2 {x} x<U = subst (λ k → k o< & U ) &iso (c<→o< (d→∋ U x<U))
319 lemma1 : {x : Ordinal} → def (od U) x ∧ def (od (* x)) y → ¬ (& U o< y)
320 lemma1 {x} lt u<y = o<> u<y (ordtrans (lemma0 (proj2 lt)) (lemma2 (proj1 lt)) )
321 lemma : ¬ ((& U) o< y ) → y o< osuc (& U)
322 lemma not with trio< y (& U)
323 lemma not | tri< a ¬b ¬c = ordtrans a <-osuc
324 lemma not | tri≈ ¬a refl ¬c = <-osuc
325 lemma not | tri> ¬a ¬b c = ⊥-elim (not c)
326 _∈_ : ( A B : HOD ) → Set n
327 A ∈ B = B ∋ A
328
329 OPwr : (A : HOD ) → HOD
330 OPwr A = Ord ( sup-o (Ord (osuc (& A))) ( λ x A∋x → & ( A ∩ (* x)) ) )
331
332 Power : HOD → HOD
333 Power A = Replace (OPwr (Ord (& A))) ( λ x → A ∩ x )
334 -- ｛_｝ : ZFSet → ZFSet
335 -- ｛ x ｝ = ( x , x ) -- better to use (x , x) directly
336
337 union→ : (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
338 union→ X z u xx not = ⊥-elim ( not (& u) ( ⟪ proj1 xx
339 , subst ( λ k → odef k (& z)) (sym *iso) (proj2 xx) ⟫ ))
340 union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
341 union← X z UX∋z = FExists _ lemma UX∋z where
342 lemma : {y : Ordinal} → odef X y ∧ odef (* y) (& z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
343 lemma {y} xx not = not (* y) ⟪ d→∋ X (proj1 xx) , proj2 xx ⟫
344
345 data infinite-d : ( x : Ordinal ) → Set n where
346 iφ : infinite-d o∅
347 isuc : {x : Ordinal } → infinite-d x →
348 infinite-d (& ( Union (* x , (* x , * x ) ) ))
349
350 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
351 -- We simply assumes infinite-d y has a maximum.
352 --
353 -- This means that many of OD may not be HODs because of the & mapping divergence.
354 -- We should have some axioms to prevent this such as & x o< next (odmax x).
355 --
356 -- postulate
357 -- ωmax : Ordinal
358 -- <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
359 --
360 -- infinite : HOD
361 -- infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax }
362
363 infinite : HOD
364 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where
365 u : (y : Ordinal ) → HOD
366 u y = Union (* y , (* y , * y))
367 -- next< : {x y z : Ordinal} → x o< next z → y o< next x → y o< next z
368 lemma8 : {y : Ordinal} → & (* y , * y) o< next (odmax (* y , * y))
369 lemma8 = ho<
370 --- (x,y) < next (omax x y) < next (osuc y) = next y
371 lemmaa : {x y : HOD} → & x o< & y → & (x , y) o< next (& y)
372 lemmaa {x} {y} x<y = subst (λ k → & (x , y) o< k ) (sym nexto≡) (subst (λ k → & (x , y) o< next k ) (sym (omax< _ _ x<y)) ho< )
373 lemma81 : {y : Ordinal} → & (* y , * y) o< next (& (* y))
374 lemma81 {y} = nexto=n (subst (λ k → & (* y , * y) o< k ) (cong (λ k → next k) (omxx _)) lemma8)
375 lemma9 : {y : Ordinal} → & (* y , (* y , * y)) o< next (& (* y , * y))
376 lemma9 = lemmaa (c<→o< (case1 refl))
377 lemma71 : {y : Ordinal} → & (* y , (* y , * y)) o< next (& (* y))
378 lemma71 = next< lemma81 lemma9
379 lemma1 : {y : Ordinal} → & (u y) o< next (osuc (& (* y , (* y , * y))))
380 lemma1 = ho<
381 --- main recursion
382 lemma : {y : Ordinal} → infinite-d y → y o< next o∅
383 lemma {o∅} iφ = x<nx
384 lemma (isuc {y} x) = next< (lemma x) (next< (subst (λ k → & (* y , (* y , * y)) o< next k) &iso lemma71 ) (nexto=n lemma1))
385
386 empty : (x : HOD ) → ¬ (od∅ ∋ x)
387 empty x = ¬x<0
388
389 _=h=_ : (x y : HOD) → Set n
390 x =h= y = od x == od y
391
392 pair→ : ( x y t : HOD ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y )
393 pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡x ))
394 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) *iso *iso (o≡→== t≡y ))
395
396 pair← : ( x y t : HOD ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t
397 pair← x y t (case1 t=h=x) = case1 (cong (λ k → & k ) (==→o≡ t=h=x))
398 pair← x y t (case2 t=h=y) = case2 (cong (λ k → & k ) (==→o≡ t=h=y))
399
400 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
401 o<→c< lt = record { incl = λ z → ordtrans z lt }
402
403 ⊆→o< : {x y : Ordinal } → (Ord x) ⊆ (Ord y) → x o< osuc y
404 ⊆→o< {x} {y} lt with trio< x y
405 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
406 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
407 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym &iso) refl )
408 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt &iso refl ))
409
410 ψiso : {ψ : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y
411 ψiso {ψ} t refl = t
412 selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
413 selection {ψ} {X} {y} = ⟪
414 ( λ cond → ⟪ proj1 cond , ψiso {ψ} (proj2 cond) (sym *iso) ⟫ )
415 , ( λ select → ⟪ proj1 select , ψiso {ψ} (proj2 select) *iso ⟫ )
416 ⟫
417
418 selection-in-domain : {ψ : HOD → Set n} {X y : HOD} → Select X ψ ∋ y → X ∋ y
419 selection-in-domain {ψ} {X} {y} lt = proj1 ((proj2 (selection {ψ} {X} )) lt)
420
421 sup-c< : (ψ : HOD → HOD) → {X x : HOD} → X ∋ x → & (ψ x) o< (sup-o X (λ y X∋y → & (ψ (* y))))
422 sup-c< ψ {X} {x} lt = subst (λ k → & (ψ k) o< _ ) *iso (sup-o< X lt )
423 replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x
424 replacement← {ψ} X x lt = ⟪ sup-c< ψ {X} {x} lt , lemma ⟫ where
425 lemma : def (in-codomain X ψ) (& (ψ x))
426 lemma not = ⊥-elim ( not ( & x ) ⟪ lt , cong (λ k → & (ψ k)) (sym *iso)⟫ )
427 replacement→ : {ψ : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= ψ y))
428 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
429 lemma2 : ¬ ((y : Ordinal) → ¬ odef X y ∧ ((& x) ≡ & (ψ (* y))))
430 → ¬ ((y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y)))
431 lemma2 not not2 = not ( λ y d → not2 y ⟪ proj1 d , lemma3 (proj2 d)⟫) where
432 lemma3 : {y : Ordinal } → (& x ≡ & (ψ (* y))) → (* (& x) =h= ψ (* y))
433 lemma3 {y} eq = subst (λ k → * (& x) =h= k ) *iso (o≡→== eq )
434 lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (* (& x) =h= ψ (* y)) )
435 lemma not y not2 = not (* y) (subst (λ k → k =h= ψ (* y)) *iso ( proj2 not2 ))
436
437 ---
438 --- Power Set
439 ---
440 --- First consider ordinals in HOD
441 ---
442 --- A ∩ x = record { def = λ y → odef A y ∧ odef x y } subset of A
443 --
444 --
445 ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a )
446 ∩-≡ {a} {b} inc = record {
447 eq→ = λ {x} x<a → ⟪ (subst (λ k → odef b k ) &iso (inc (d→∋ a x<a))) , x<a ⟫ ;
448 eq← = λ {x} x<a∩b → proj2 x<a∩b }
449 --
450 -- Transitive Set case
451 -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is (Ord a) ∩ t =h= t
452 -- OPwr (Ord a) is a sup of (Ord a) ∩ t, so OPwr (Ord a) ∋ t
453 -- OPwr A = Ord ( sup-o ( λ x → & ( A ∩ (* x )) ) )
454 --
455 ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
456 ord-power← a t t→A = subst (λ k → odef (OPwr (Ord a)) k ) (lemma1 lemma-eq) lemma where
457 lemma-eq : ((Ord a) ∩ t) =h= t
458 eq→ lemma-eq {z} w = proj2 w
459 eq← lemma-eq {z} w = ⟪ subst (λ k → odef (Ord a) k ) &iso ( t→A (d→∋ t w)) , w ⟫
460 lemma1 : {a : Ordinal } { t : HOD }
461 → (eq : ((Ord a) ∩ t) =h= t) → & ((Ord a) ∩ (* (& t))) ≡ & t
462 lemma1 {a} {t} eq = subst (λ k → & ((Ord a) ∩ k) ≡ & t ) (sym *iso) (cong (λ k → & k ) (==→o≡ eq ))
463 lemma2 : (& t) o< (osuc (& (Ord a)))
464 lemma2 = ⊆→o≤ {t} {Ord a} (λ {x} x<t → subst (λ k → def (od (Ord a)) k) &iso (t→A (d→∋ t x<t)))
465 lemma : & ((Ord a) ∩ (* (& t)) ) o< sup-o (Ord (osuc (& (Ord a)))) (λ x lt → & ((Ord a) ∩ (* x)))
466 lemma = sup-o< _ lemma2
467
468 --
469 -- Every set in HOD is a subset of Ordinals, so make OPwr (Ord (& A)) first
470 -- then replace of all elements of the Power set by A ∩ y
471 --
472 -- Power A = Replace (OPwr (Ord (& A))) ( λ y → A ∩ y )
473
474 -- we have oly double negation form because of the replacement axiom
475 --
476 power→ : ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → ¬ ¬ (A ∋ x)
477 power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
478 a = & A
479 lemma2 : ¬ ( (y : HOD) → ¬ (t =h= (A ∩ y)))
480 lemma2 = replacement→ {λ x → A ∩ x} (OPwr (Ord (& A))) t P∋t
481 lemma3 : (y : HOD) → t =h= ( A ∩ y ) → ¬ ¬ (A ∋ x)
482 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
483 lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ * y)))
484 lemma4 not = lemma2 ( λ y not1 → not (& y) (subst (λ k → t =h= ( A ∩ k )) (sym *iso) not1 ))
485 lemma5 : {y : Ordinal} → t =h= (A ∩ * y) → ¬ ¬ (odef A (& x))
486 lemma5 {y} eq not = (lemma3 (* y) eq) not
487
488 power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
489 power← A t t→A = ⟪ lemma1 , lemma2 ⟫ where
490 a = & A
491 lemma0 : {x : HOD} → t ∋ x → Ord a ∋ x
492 lemma0 {x} t∋x = c<→o< (t→A t∋x)
493 lemma3 : OPwr (Ord a) ∋ t
494 lemma3 = ord-power← a t lemma0
495 lemma4 : (A ∩ * (& t)) ≡ t
496 lemma4 = let open ≡-Reasoning in begin
497 A ∩ * (& t)
498 ≡⟨ cong (λ k → A ∩ k) *iso ⟩
499 A ∩ t
500 ≡⟨ sym (==→o≡ ( ∩-≡ {t} {A} t→A )) ⟩
501 t
502 ∎
503 sup1 : Ordinal
504 sup1 = sup-o (Ord (osuc (& (Ord (& A))))) (λ x A∋x → & ((Ord (& A)) ∩ (* x)))
505 lemma9 : def (od (Ord (Ordinals.osuc O (& (Ord (& A)))))) (& (Ord (& A)))
506 lemma9 = <-osuc
507 lemmab : & ((Ord (& A)) ∩ (* (& (Ord (& A) )))) o< sup1
508 lemmab = sup-o< (Ord (osuc (& (Ord (& A))))) lemma9
509 lemmad : Ord (osuc (& A)) ∋ t
510 lemmad = ⊆→o≤ (λ {x} lt → subst (λ k → def (od A) k ) &iso (t→A (d→∋ t lt)))
511 lemmac : ((Ord (& A)) ∩ (* (& (Ord (& A) )))) =h= Ord (& A)
512 lemmac = record { eq→ = lemmaf ; eq← = lemmag } where
513 lemmaf : {x : Ordinal} → def (od ((Ord (& A)) ∩ (* (& (Ord (& A)))))) x → def (od (Ord (& A))) x
514 lemmaf {x} lt = proj1 lt
515 lemmag : {x : Ordinal} → def (od (Ord (& A))) x → def (od ((Ord (& A)) ∩ (* (& (Ord (& A)))))) x
516 lemmag {x} lt = ⟪ lt , subst (λ k → def (od k) x) (sym *iso) lt ⟫
517 lemmae : & ((Ord (& A)) ∩ (* (& (Ord (& A))))) ≡ & (Ord (& A))
518 lemmae = cong (λ k → & k ) ( ==→o≡ lemmac)
519 lemma7 : def (od (OPwr (Ord (& A)))) (& t)
520 lemma7 with osuc-≡< lemmad
521 lemma7 | case2 lt = ordtrans (c<→o< lt) (subst (λ k → k o< sup1) lemmae lemmab )
522 lemma7 | case1 eq with osuc-≡< (⊆→o≤ {* (& t)} {* (& (Ord (& t)))} (λ {x} lt → lemmah lt )) where
523 lemmah : {x : Ordinal } → def (od (* (& t))) x → def (od (* (& (Ord (& t))))) x
524 lemmah {x} lt = subst (λ k → def (od k) x ) (sym *iso) (subst (λ k → k o< (& t))
525 &iso
526 (c<→o< (subst₂ (λ j k → def (od j) k) *iso (sym &iso) lt )))
527 lemma7 | case1 eq | case1 eq1 = subst (λ k → k o< sup1) (trans lemmae lemmai) lemmab where
528 lemmai : & (Ord (& A)) ≡ & t
529 lemmai = let open ≡-Reasoning in begin
530 & (Ord (& A))
531 ≡⟨ sym (cong (λ k → & (Ord k)) eq) ⟩
532 & (Ord (& t))
533 ≡⟨ sym &iso ⟩
534 & (* (& (Ord (& t))))
535 ≡⟨ sym eq1 ⟩
536 & (* (& t))
537 ≡⟨ &iso ⟩
538 & t
539 ∎
540 lemma7 | case1 eq | case2 lt = ordtrans lemmaj (subst (λ k → k o< sup1) lemmae lemmab ) where
541 lemmak : & (* (& (Ord (& t)))) ≡ & (Ord (& A))
542 lemmak = let open ≡-Reasoning in begin
543 & (* (& (Ord (& t))))
544 ≡⟨ &iso ⟩
545 & (Ord (& t))
546 ≡⟨ cong (λ k → & (Ord k)) eq ⟩
547 & (Ord (& A))
548 ∎
549 lemmaj : & t o< & (Ord (& A))
550 lemmaj = subst₂ (λ j k → j o< k ) &iso lemmak lt
551 lemma1 : & t o< sup-o (OPwr (Ord (& A))) (λ x lt → & (A ∩ (* x)))
552 lemma1 = subst (λ k → & k o< sup-o (OPwr (Ord (& A))) (λ x lt → & (A ∩ (* x))))
553 lemma4 (sup-o< (OPwr (Ord (& A))) lemma7 )
554 lemma2 : def (in-codomain (OPwr (Ord (& A))) (_∩_ A)) (& t)
555 lemma2 not = ⊥-elim ( not (& t) ⟪ lemma3 , lemma6 ⟫ ) where
556 lemma6 : & t ≡ & (A ∩ * (& t))
557 lemma6 = cong ( λ k → & k ) (==→o≡ (subst (λ k → t =h= (A ∩ k)) (sym *iso) ( ∩-≡ {t} {A} t→A )))
558
559
560 extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
561 eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso {A} {B} (sym &iso) (proj1 (eq (* x))) d
562 eq← (extensionality0 {A} {B} eq ) {x} d = odef-iso {B} {A} (sym &iso) (proj2 (eq (* x))) d
563
564 extensionality : {A B w : HOD } → ((z : HOD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
565 proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
566 proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d
567
568 infinity∅ : infinite ∋ od∅
569 infinity∅ = subst (λ k → odef infinite k ) lemma iφ where
570 lemma : o∅ ≡ & od∅
571 lemma = let open ≡-Reasoning in begin
572 o∅
573 ≡⟨ sym &iso ⟩
574 & ( * o∅ )
575 ≡⟨ cong ( λ k → & k ) o∅≡od∅ ⟩
576 & od∅
577 ∎
578 infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
579 infinity x lt = subst (λ k → odef infinite k ) lemma (isuc {& x} lt) where
580 lemma : & (Union (* (& x) , (* (& x) , * (& x))))
581 ≡ & (Union (x , (x , x)))
582 lemma = cong (λ k → & (Union ( k , ( k , k ) ))) *iso
583
584 isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite
585 isZF = record {
586 isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
587 ; pair→ = pair→
588 ; pair← = pair←
589 ; union→ = union→
590 ; union← = union←
591 ; empty = empty
592 ; power→ = power→
593 ; power← = power←
594 ; extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
595 ; ε-induction = ε-induction
596 ; infinity∅ = infinity∅
597 ; infinity = infinity
598 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
599 ; replacement← = replacement←
600 ; replacement→ = λ {ψ} → replacement→ {ψ}
601 }
602
603 HOD→ZF : ZF
604 HOD→ZF = record {
605 ZFSet = HOD
606 ; _∋_ = _∋_
607 ; _≈_ = _=h=_
608 ; ∅ = od∅
609 ; _,_ = _,_
610 ; Union = Union
611 ; Power = Power
612 ; Select = Select
613 ; Replace = Replace
614 ; infinite = infinite
615 ; isZF = isZF
616 }
617
618