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comparison OD.agda @ 182:9f3c0e0b2bc9

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remove ordinal-definable
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 21 Jul 2019 12:11:50 +0900
parents HOD.agda@11490a3170d4
children de3d87b7494f
comparison
equal deleted inserted replaced
181:7012158bf2d9 182:9f3c0e0b2bc9
1 open import Level
2 module OD where
3
4 open import zf
5 open import ordinal
6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
7 open import Relation.Binary.PropositionalEquality
8 open import Data.Nat.Properties
9 open import Data.Empty
10 open import Relation.Nullary
11 open import Relation.Binary
12 open import Relation.Binary.Core
13
14 -- Ordinal Definable Set
15
16 record OD {n : Level} : Set (suc n) where
17 field
18 def : (x : Ordinal {n} ) → Set n
19
20 open OD
21
22 open Ordinal
23 open _∧_
24
25 record _==_ {n : Level} ( a b : OD {n} ) : Set n where
26 field
27 eq→ : ∀ { x : Ordinal {n} } → def a x → def b x
28 eq← : ∀ { x : Ordinal {n} } → def b x → def a x
29
30 id : {n : Level} {A : Set n} → A → A
31 id x = x
32
33 eq-refl : {n : Level} { x : OD {n} } → x == x
34 eq-refl {n} {x} = record { eq→ = id ; eq← = id }
35
36 open _==_
37
38 eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x
39 eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq }
40
41 eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z
42 eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y t) ; eq← = λ t → eq← x=y ( eq← y=z t) }
43
44 ⇔→== : {n : Level} { x y : OD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔ def y z) → x == y
45 eq→ ( ⇔→== {n} {x} {y} eq ) {z} m = proj1 eq m
46 eq← ( ⇔→== {n} {x} {y} eq ) {z} m = proj2 eq m
47
48 -- Ordinal in OD ( and ZFSet ) Transitive Set
49 Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n}
50 Ord {n} a = record { def = λ y → y o< a }
51
52 od∅ : {n : Level} → OD {n}
53 od∅ {n} = Ord o∅
54
55 postulate
56 -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
57 od→ord : {n : Level} → OD {n} → Ordinal {n}
58 ord→od : {n : Level} → Ordinal {n} → OD {n}
59 c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y
60 oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x
61 diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
62 -- we should prove this in agda, but simply put here
63 ==→o≡ : {n : Level} → { x y : OD {suc n} } → (x == y) → x ≡ y
64 -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set
65 -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) x
66 -- ord→od x ≡ Ord x results the same
67 -- supermum as Replacement Axiom
68 sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
69 sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ
70 -- contra-position of mimimulity of supermum required in Power Set Axiom
71 -- sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n}
72 -- sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
73 -- sup-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ )
74 -- mimimul and x∋minimul is a weaker form of Axiom of choice
75 minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n}
76 -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x )
77 x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) )
78 minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) )
79
80 _∋_ : { n : Level } → ( a x : OD {n} ) → Set n
81 _∋_ {n} a x = def a ( od→ord x )
82
83 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n
84 x c< a = a ∋ x
85
86 _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n)
87 a c≤ b = (a ≡ b) ∨ ( b ∋ a )
88
89 cseq : {n : Level} → OD {n} → OD {n}
90 cseq x = record { def = λ y → def x (osuc y) } where
91
92 def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x
93 def-subst df refl refl = df
94
95 sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n}
96 sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )
97
98 sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x ))
99 sup-c< {n} ψ {x} = def-subst {n} {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )}
100 lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
101 lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x)))
102 lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) )
103
104 otrans : {n : Level} {a x : Ordinal {n} } → def (Ord a) x → { y : Ordinal {n} } → y o< x → def (Ord a) y
105 otrans {n} {a} {x} x<a {y} y<x = ordtrans y<x x<a
106
107 ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
108 ∅3 {n} {x} = TransFinite {n} c2 c3 x where
109 c0 : Nat → Ordinal {n} → Set n
110 c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n}
111 c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } )
112 c2 Zero not = refl
113 c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } )
114 ... | t with t (case1 ≤-refl )
115 c2 (Suc lx) not | t | ()
116 c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ })
117 c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } )
118 ... | t with t (case2 Φ< )
119 c3 lx (Φ .lx) d not | t | ()
120 c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } )
121 ... | t with t (case2 (s< s<refl ) )
122 c3 lx (OSuc .lx x₁) d not | t | ()
123
124 ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x
125 ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl)
126 ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
127 ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n)
128
129 ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y }
130 ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso
131
132 -- avoiding lv != Zero error
133 orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
134 orefl refl = refl
135
136 ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y
137 ==-iso {n} {x} {y} eq = record {
138 eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ;
139 eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) }
140 where
141 lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z
142 lemma {x} {z} d = def-subst d oiso refl
143
144 =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y)
145 =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
146
147 ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y
148 ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
149 lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy)
150 lemma ox ox refl = eq-refl
151
152 o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y
153 o≡→== {n} {x} {.x} refl = eq-refl
154
155 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x )
156 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
157
158 c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x
159 c≤-refl x = case1 refl
160
161 ∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a
162 ∋→o< {n} {a} {x} lt = t where
163 t : (od→ord x) o< (od→ord a)
164 t = c<→o< {suc n} {x} {a} lt
165
166 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
167 o∅≡od∅ {n} = ==→o≡ lemma where
168 lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x
169 lemma0 {x} lt = o<-subst (c<→o< {suc n} {ord→od x} {ord→od o∅} (def-subst {suc n} {ord→od o∅} {x} lt refl (sym diso)) ) diso diso
170 lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x
171 lemma1 (case1 ())
172 lemma1 (case2 ())
173 lemma : ord→od o∅ == od∅
174 lemma = record { eq→ = lemma0 ; eq← = lemma1 }
175
176 ord-od∅ : {n : Level} → od→ord (od∅ {suc n}) ≡ o∅ {suc n}
177 ord-od∅ {n} = sym ( subst (λ k → k ≡ od→ord (od∅ {suc n}) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) )
178
179 o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x )
180 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where
181
182 o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y
183 o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (orefl oeq ) (c<→o< lt)
184
185 ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ } == od∅ {n}
186 eq→ ∅0 {w} (lift ())
187 eq← ∅0 {w} (case1 ())
188 eq← ∅0 {w} (case2 ())
189
190 ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} )
191 ∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d
192 ∅< {n} {x} {y} d eq | lift ()
193
194 ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox
195 ∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {x} {x} x∋x )
196
197 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x
198 def-iso refl t = t
199
200 is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} )
201 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
202 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
203 is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
204
205 OrdP : {n : Level} → ( x : Ordinal {suc n} ) ( y : OD {suc n} ) → Dec ( Ord x ∋ y )
206 OrdP {n} x y with trio< x (od→ord y)
207 OrdP {n} x y | tri< a ¬b ¬c = no ¬c
208 OrdP {n} x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl )
209 OrdP {n} x y | tri> ¬a ¬b c = yes c
210
211 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
212 -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))
213
214 in-codomain : {n : Level} → (X : OD {suc n} ) → ( ψ : OD {suc n} → OD {suc n} ) → OD {suc n}
215 in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
216
217 -- Power Set of X ( or constructible by λ y → def X (od→ord y )
218
219 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n}
220 ZFSubset A x = record { def = λ y → def A y ∧ def x y } where
221
222 Def : {n : Level} → (A : OD {suc n}) → OD {suc n}
223 Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) -- Ord x does not help ord-power→
224
225 -- Constructible Set on α
226 -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x }
227 -- L (Φ 0) = Φ
228 -- L (OSuc lv n) = { Def ( L n ) }
229 -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) )
230 L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n}
231 L {n} record { lv = Zero ; ord = (Φ .0) } = od∅
232 L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) )
233 L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α )
234 cseq ( Ord (od→ord (L {n} (record { lv = lx ; ord = Φ lx }))))
235
236 -- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α
237 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x
238
239
240 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
241 OD→ZF {n} = record {
242 ZFSet = OD {suc n}
243 ; _∋_ = _∋_
244 ; _≈_ = _==_
245 ; ∅ = od∅
246 ; _,_ = _,_
247 ; Union = Union
248 ; Power = Power
249 ; Select = Select
250 ; Replace = Replace
251 ; infinite = infinite
252 ; isZF = isZF
253 } where
254 ZFSet = OD {suc n}
255 Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → Set (suc n) ) → OD {suc n}
256 Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) }
257 Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n}
258 Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x }
259 _,_ : OD {suc n} → OD {suc n} → OD {suc n}
260 x , y = Ord (omax (od→ord x) (od→ord y))
261 _∩_ : ( A B : ZFSet ) → ZFSet
262 A ∩ B = record { def = λ x → def A x ∧ def B x }
263 Union : OD {suc n} → OD {suc n}
264 Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) }
265 _∈_ : ( A B : ZFSet ) → Set (suc n)
266 A ∈ B = B ∋ A
267 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n)
268 _⊆_ A B {x} = A ∋ x → B ∋ x
269 Power : OD {suc n} → OD {suc n}
270 Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
271 ｛_｝ : ZFSet → ZFSet
272 ｛ x ｝ = ( x , x )
273
274 data infinite-d : ( x : Ordinal {suc n} ) → Set (suc n) where
275 iφ : infinite-d o∅
276 isuc : {x : Ordinal {suc n} } → infinite-d x →
277 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
278
279 infinite : OD {suc n}
280 infinite = record { def = λ x → infinite-d x }
281
282 infixr 200 _∈_
283 -- infixr 230 _∩_ _∪_
284 infixr 220 _⊆_
285 isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite
286 isZF = record {
287 isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
288 ; pair = pair
289 ; union→ = union→
290 ; union← = union←
291 ; empty = empty
292 ; power→ = power→
293 ; power← = power←
294 ; extensionality = extensionality
295 ; minimul = minimul
296 ; regularity = regularity
297 ; infinity∅ = infinity∅
298 ; infinity = infinity
299 ; selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
300 ; replacement← = replacement←
301 ; replacement→ = replacement→
302 } where
303
304 pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B)
305 proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B)
306 proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B)
307
308 empty : {n : Level } (x : OD {suc n} ) → ¬ (od∅ ∋ x)
309 empty x (case1 ())
310 empty x (case2 ())
311
312 ord-⊆ : ( t x : OD {suc n} ) → _⊆_ t (Ord (od→ord t )) {x}
313 ord-⊆ t x lt = c<→o< lt
314 o<→c< : {x y : Ordinal {suc n}} {z : OD {suc n}}→ x o< y → _⊆_ (Ord x) (Ord y) {z}
315 o<→c< lt lt1 = ordtrans lt1 lt
316
317 ⊆→o< : {x y : Ordinal {suc n}} → (∀ (z : OD) → _⊆_ (Ord x) (Ord y) {z} ) → x o< osuc y
318 ⊆→o< {x} {y} lt with trio< x y
319 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
320 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
321 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl )
322 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
323
324 union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
325 union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
326 ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
327 union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
328 union← X z UX∋z = TransFiniteExists _ lemma UX∋z where
329 lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
330 lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }
331
332 ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y
333 ψiso {ψ} t refl = t
334 selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
335 selection {ψ} {X} {y} = record {
336 proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) }
337 ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso }
338 }
339 replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x
340 replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where
341 lemma : def (in-codomain X ψ) (od→ord (ψ x))
342 lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
343 replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y))
344 replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
345 lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
346 → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)))
347 lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
348 lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y))
349 lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq )
350 lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) )
351 lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 ))
352
353 ---
354 --- Power Set
355 ---
356 --- First consider ordinals in OD
357 ---
358 --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A
359 --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A
360 --
361 --
362 ∩-≡ : { a b : OD {suc n} } → ({x : OD {suc n} } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
363 ∩-≡ {a} {b} inc = record {
364 eq→ = λ {x} x<a → record { proj2 = x<a ;
365 proj1 = def-subst {suc n} {_} {_} {b} {x} (inc (def-subst {suc n} {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ;
366 eq← = λ {x} x<a∩b → proj2 x<a∩b }
367 --
368 -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t
369 -- Power A is a sup of ZFSubset A t, so Power A ∋ t
370 --
371 ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
372 ord-power← a t t→A = def-subst {suc n} {_} {_} {Def (Ord a)} {od→ord t}
373 lemma refl (lemma1 lemma-eq )where
374 lemma-eq : ZFSubset (Ord a) t == t
375 eq→ lemma-eq {z} w = proj2 w
376 eq← lemma-eq {z} w = record { proj2 = w ;
377 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z}
378 ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso }
379 lemma1 : {n : Level } {a : Ordinal {suc n}} { t : OD {suc n}}
380 → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
381 lemma1 {n} {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
382 lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x)))
383 lemma = sup-o<
384
385 -- double-neg-eilm : {n : Level } {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic
386 --
387 -- Every set in OD is a subset of Ordinals
388 --
389 -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y )
390
391 -- we have oly double negation form because of the replacement axiom
392 --
393 power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
394 power→ A t P∋t {x} t∋x = TransFiniteExists _ lemma5 lemma4 where
395 a = od→ord A
396 lemma2 : ¬ ( (y : OD) → ¬ (t == (A ∩ y)))
397 lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t
398 lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
399 lemma3 y eq not = not (proj1 (eq→ eq t∋x))
400 lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y)))
401 lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 ))
402 lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x))
403 lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
404
405 power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
406 power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
407 a = od→ord A
408 lemma0 : {x : OD} → t ∋ x → Ord a ∋ x
409 lemma0 {x} t∋x = c<→o< (t→A t∋x)
410 lemma3 : Def (Ord a) ∋ t
411 lemma3 = ord-power← a t lemma0
412 lt1 : od→ord (A ∩ ord→od (od→ord t)) o< sup-o (λ x → od→ord (A ∩ ord→od x))
413 lt1 = sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t}
414 lemma4 : (A ∩ ord→od (od→ord t)) ≡ t
415 lemma4 = let open ≡-Reasoning in begin
416 A ∩ ord→od (od→ord t)
417 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
418 A ∩ t
419 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩
420 t
421 ∎
422 lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ ord→od x))
423 lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ ord→od x)))
424 lemma4 (sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t})
425 lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t)
426 lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
427 lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
428 lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A )))
429
430 regularity : (x : OD) (not : ¬ (x == od∅)) →
431 (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
432 proj1 (regularity x not ) = x∋minimul x not
433 proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where
434 lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁
435 lemma1 {x₁} s = ⊥-elim ( minimul-1 x not (ord→od x₁) lemma3 ) where
436 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁))
437 lemma3 = record { proj1 = def-subst {suc n} {_} {_} {minimul x not} {_} (proj1 s) refl (sym diso)
438 ; proj2 = proj2 (proj2 s) }
439 lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁
440 lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) ))
441
442 extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
443 eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
444 eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
445
446 infinity∅ : infinite ∋ od∅ {suc n}
447 infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} iφ refl lemma where
448 lemma : o∅ ≡ od→ord od∅
449 lemma = let open ≡-Reasoning in begin
450 o∅
451 ≡⟨ sym diso ⟩
452 od→ord ( ord→od o∅ )
453 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
454 od→ord od∅
455 ∎
456 infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
457 infinity x lt = def-subst {suc n} {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
458 lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
459 ≡ od→ord (Union (x , (x , x)))
460 lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso
461
462 -- Axiom of choice ( is equivalent to the existence of minimul in our case )
463 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
464 choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD
465 choice-func X {x} not X∋x = minimul x not
466 choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A
467 choice X {A} X∋A not = x∋minimul A not
468
469 -- another form of regularity
470 --
471 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m}
472 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x )
473 → (x : OD {suc n} ) → ψ x
474 ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where
475 ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly }
476 → (ly < lx) ∨ (oy d< ox ) → ψ (ord→od (record { lv = ly ; ord = oy } ) )
477 ε-induction-ord Zero (Φ 0) (case1 ())
478 ε-induction-ord Zero (Φ 0) (case2 ())
479 ε-induction-ord lx (OSuc lx ox) {ly} {oy} y<x =
480 ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where
481 lemma : (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox }
482 lemma y lt with osuc-≡< y<x
483 lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso
484 lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1
485 ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) =
486 ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt ) where
487 --
488 -- if lv of z if less than x Ok
489 -- else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma
490 --
491 -- lx Suc lx (1) lz(a) <lx by case1
492 -- ly(1) ly(2) (2) lz(b) <lx by case1
493 -- lz(a) lz(b) lz(c) lz(c) <lx by case2 ( ly==lz==lx)
494 --
495 lemma0 : { lx ly : Nat } → ly < Suc lx → lx < ly → ⊥
496 lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2
497 lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly
498 lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin
499 lv (od→ord (ord→od (record { lv = ly ; ord = oy })))
500 ≡⟨ cong ( λ k → lv k ) diso ⟩
501 lv (record { lv = ly ; ord = oy })
502 ≡⟨⟩
503 ly
504 ∎
505 lemma : (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z
506 lemma z lt with c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt
507 lemma z lt | case1 lz<ly with <-cmp lx ly
508 lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen
509 lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c = -- ly(1)
510 subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) ))
511 lemma z lt | case1 lz<ly | tri> ¬a ¬b c = -- lz(a)
512 subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c))))
513 lemma z lt | case2 lz=ly with <-cmp lx ly
514 lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen
515 lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- lz(b)
516 ... | eq = subst (λ k → ψ k ) oiso
517 (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c )))
518 lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly -- lz(c)
519 ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡ k) lemma1 eq)) where
520 lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z
521 lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt )
522 lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly } →
523 lx ≡ ly → ly ≡ lv (od→ord z) → ψ z
524 lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl)
525