Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison OD.agda @ 338:bca043423554

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 12 Jul 2020 12:32:42 +0900
parents daafa2213dd2
children feb0fcc430a9
comparison
equal deleted inserted replaced
337:de2c472bcbcd 338:bca043423554
101 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x 101 diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x
102 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y 102 ==→o≡ : {x y : HOD } → (od x == od y) → x ≡ y
103 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal 103 sup-o : (A : HOD) → ( ( x : Ordinal ) → def (od A) x → Ordinal ) → Ordinal
104 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ 104 sup-o< : (A : HOD) → { ψ : ( x : Ordinal ) → def (od A) x → Ordinal } → ∀ {x : Ordinal } → (lt : def (od A) x ) → ψ x lt o< sup-o A ψ
105 105
106 -- another form of infinite
107 -- pair-ord< : {x : Ordinal } → od→ord ( ord→od x , ord→od x ) o< next (od→ord x)
108
106 postulate odAxiom : ODAxiom 109 postulate odAxiom : ODAxiom
107 open ODAxiom odAxiom 110 open ODAxiom odAxiom
108 111
109 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD 112 -- OD ⇔ Ordinal leads a contradiction, so we need bounded HOD
110 ¬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) → ⊥ 113 ¬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) → ⊥
210 is-o∅ x | tri< a ¬b ¬c = no ¬b 213 is-o∅ x | tri< a ¬b ¬c = no ¬b
211 is-o∅ x | tri≈ ¬a b ¬c = yes b 214 is-o∅ x | tri≈ ¬a b ¬c = yes b
212 is-o∅ x | tri> ¬a ¬b c = no ¬b 215 is-o∅ x | tri> ¬a ¬b c = no ¬b
213 216
214 -- the pair 217 -- the pair
215 _,_ : HOD → HOD → HOD 218 _,_ : HOD → HOD → HOD
216 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 219 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
217 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y) 220 lemma : {t : Ordinal} → (t ≡ od→ord x) ∨ (t ≡ od→ord y) → t o< omax (od→ord x) (od→ord y)
218 lemma {t} (case1 refl) = omax-x _ _ 221 lemma {t} (case1 refl) = omax-x _ _
219 lemma {t} (case2 refl) = omax-y _ _ 222 lemma {t} (case2 refl) = omax-y _ _
220 223
245 pair<y {x} {y} y∋x = ⊆→o≤ lemma where 248 pair<y {x} {y} y∋x = ⊆→o≤ lemma where
246 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z 249 lemma : {z : Ordinal} → def (od (x , x)) z → def (od y) z
247 lemma (case1 refl) = y∋x 250 lemma (case1 refl) = y∋x
248 lemma (case2 refl) = y∋x 251 lemma (case2 refl) = y∋x
249 252
250 -- ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) ) 253 ⊆→o≤→c<→o< : ({x : HOD} → od→ord (x , x) ≡ osuc (od→ord x) )
251 -- → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) ) 254 → ({y z : HOD } → ({x : Ordinal} → def (od y) x → def (od z) x ) → od→ord y o< osuc (od→ord z) )
252 -- → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y 255 → {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y
253 -- ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y) 256 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x with trio< (od→ord x) (od→ord y)
254 -- ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a 257 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri< a ¬b ¬c = a
255 -- ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (peq {x}) (pair<y (subst (λ k → def (od k) (od→ord x)) {!!} y∋x))) 258 ⊆→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 )))
256 -- ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ (peq {x}) (pair<y (subst (λ k → def (od k) (od→ord x)) {!!} y∋x))) 259 ⊆→o≤→c<→o< peq ⊆→o≤ {x} {y} y∋x | tri> ¬a ¬b c =
260 ⊥-elim ( o<> (⊆→o≤ {x , x} {y} y⊆x,x ) lemma1 ) where
261 lemma : {z : Ordinal} → (z ≡ od→ord x) ∨ (z ≡ od→ord x) → od→ord x ≡ z
262 lemma (case1 refl) = refl
263 lemma (case2 refl) = refl
264 y⊆x,x : {z : Ordinals.ord O} → def (od (x , x)) z → def (od y) z
265 y⊆x,x {z} lt = subst (λ k → def (od y) k ) (lemma lt) y∋x
266 lemma1 : osuc (od→ord y) o< od→ord (x , x)
267 lemma1 = subst (λ k → osuc (od→ord y) o< k ) (sym (peq {x})) (osucc c )
257 268
258 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) 269 subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A )
259 subset-lemma {A} {x} = record { 270 subset-lemma {A} {x} = record {
260 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } 271 proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) }
261 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } 272 ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt }
345 iφ : infinite-d o∅ 356 iφ : infinite-d o∅
346 isuc : {x : Ordinal } → infinite-d x → 357 isuc : {x : Ordinal } → infinite-d x →
347 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) 358 infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
348 359
349 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair. 360 -- ω can be diverged in our case, since we have no restriction on the corresponding ordinal of a pair.
350 -- We simply assumes nfinite-d y has a maximum. 361 -- We simply assumes infinite-d y has a maximum.
351 -- 362 --
352 -- This means that many of OD cannot be HODs because of the od→ord mapping divergence. 363 -- This means that many of OD may not be HODs because of the od→ord mapping divergence.
353 -- We should have some axioms to prevent this, but it may complicate thins. 364 -- We should have some axioms to prevent this.
354 -- 365 --
355 postulate 366 postulate
356 ωmax : Ordinal 367 ωmax : Ordinal
357 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax 368 <ωmax : {y : Ordinal} → infinite-d y → y o< ωmax
358 369
359 infinite : HOD 370 infinite : HOD
360 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax } 371 infinite = record { od = record { def = λ x → infinite-d x } ; odmax = ωmax ; <odmax = <ωmax }
372
373 -- infinite' : HOD
374 -- infinite' = record { od = record { def = λ x → infinite-d x } ; odmax = next o∅ ; <odmax = lemma } where
375 -- u : (y : Ordinal ) → HOD
376 -- u y = Union (ord→od y , (ord→od y , ord→od y))
377 -- lemma : {y : Ordinal} → infinite-d y → y o< next o∅
378 -- lemma {o∅} iφ = {!!}
379 -- lemma (isuc {y} x) = {!!} where
380 -- lemma1 : od→ord (Union (ord→od y , (ord→od y , ord→od y))) o< od→ord (Union (u y , (u y , u y )))
381 -- lemma1 = {!!}
382
361 383
362 _=h=_ : (x y : HOD) → Set n 384 _=h=_ : (x y : HOD) → Set n
363 x =h= y = od x == od y 385 x =h= y = od x == od y
364 386
365 infixr 200 _∈_ 387 infixr 200 _∈_