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

comparison OD.agda @ 191:9eb6a8691f02

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choice function cannot jump between ordinal level
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 28 Jul 2019 14:07:08 +0900
parents 6e778b0a7202
children 38ecc75d93ce
comparison
equal deleted inserted replaced
190:6e778b0a7202 191:9eb6a8691f02
236 in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } 236 in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) }
237 237
238 -- Power Set of X ( or constructible by λ y → def X (od→ord y ) 238 -- Power Set of X ( or constructible by λ y → def X (od→ord y )
239 239
240 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} 240 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n}
241 ZFSubset A x = record { def = λ y → def A y ∧ def x y } where 241 ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set
242 242
243 Def : {n : Level} → (A : OD {suc n}) → OD {suc n} 243 Def : {n : Level} → (A : OD {suc n}) → OD {suc n}
244 Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) -- Ord x does not help ord-power→ 244 Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
245 245
246 246
247 _⊆_ : {n : Level} ( A B : OD {suc n} ) → ∀{ x : OD {suc n} } → Set (suc n) 247 _⊆_ : {n : Level} ( A B : OD {suc n} ) → ∀{ x : OD {suc n} } → Set (suc n)
248 _⊆_ A B {x} = A ∋ x → B ∋ x 248 _⊆_ A B {x} = A ∋ x → B ∋ x
249 249
253 subset-lemma {n} {A} {x} {y} = record { 253 subset-lemma {n} {A} {x} {y} = record {
254 proj1 = λ z lt → proj1 (z lt) 254 proj1 = λ z lt → proj1 (z lt)
255 ; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt } 255 ; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt }
256 } 256 }
257 257
258 Def=A→Set : {n : Level} → (A : Ordinal {suc n}) → Def (Ord A) == record { def = λ x → x o< A → Set n }
259 Def=A→Set {n} A = record {
260 eq→ = λ {y} DAx y<A → {!!}
261 ; eq← = {!!} -- λ {y} f → {!!}
262 } where
263 258
264 -- Constructible Set on α 259 -- Constructible Set on α
265 -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x } 260 -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x }
266 -- L (Φ 0) = Φ 261 -- L (Φ 0) = Φ
267 -- L (OSuc lv n) = { Def ( L n ) } 262 -- L (OSuc lv n) = { Def ( L n ) }
503 choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD 498 choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD
504 choice-func X {x} not X∋x = minimul x not 499 choice-func X {x} not X∋x = minimul x not
505 choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A 500 choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A
506 choice X {A} X∋A not = x∋minimul A not 501 choice X {A} X∋A not = x∋minimul A not
507 502
508 -- choice-func' : (X : OD {suc n} ) → (∋-p : (A x : OD {suc n} ) → Dec ( A ∋ x ) ) → ¬ ( X == od∅ ) → OD {suc n} 503 choice-func' : (X : OD {suc n} ) → (∋-p : (A x : OD {suc n} ) → Dec ( A ∋ x ) ) → ¬ ( X == od∅ ) → OD {suc n}
504 choice-func' X ∋-p not = lemma (lv (od→ord X )) (ord (od→ord X)) where
505 lemma : (lx : Nat ) ( ox : OrdinalD lx ) → OD {suc n}
506 lemma lx ox with ∋-p X (ord→od record { lv = lx ; ord = ox })
507 lemma lx ox | yes p = (ord→od record { lv = lx ; ord = ox })
508 lemma Zero (Φ 0) | no ¬p = od∅
509 lemma Zero (OSuc 0 ox) | no ¬p = lemma Zero ox
510 lemma (Suc lx) (Φ (Suc lx)) | no ¬p = {!!}
511 lemma (Suc lx) (OSuc (Suc lx) ox) | no ¬p = lemma ( Suc lx ) ox
512
509 -- 513 --
510 -- another form of regularity 514 -- another form of regularity
511 -- 515 --
512 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} 516 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m}
513 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) 517 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x )