Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 54:33fb8228ace9
fix selection axiom
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 27 May 2019 21:58:17 +0900 |
| parents | d13a10a1723e |
| children | 9c0a5e28a572 |
| files | ordinal-definable.agda zf.agda |
| diffstat | 2 files changed, 58 insertions(+), 33 deletions(-) [+] |
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--- a/ordinal-definable.agda Mon May 27 16:14:35 2019 +0900 +++ b/ordinal-definable.agda Mon May 27 21:58:17 2019 +0900 @@ -186,11 +186,11 @@ o<> ox oy (case2 x<y) (case2 y<x) with d<→lv x<y ... | refl = trio<> x<y y<x -o<¬≡ : {n : Level } ( ox oy : Ordinal {n}) → ox ≡ oy → ox o< oy → ⊥ +o<¬≡ : {n : Level } ( ox oy : Ordinal {n}) → ox ≡ oy → ox o< oy → ⊥ o<¬≡ ox ox refl (case1 lt) = =→¬< lt o<¬≡ ox ox refl (case2 (s< lt)) = trio<≡ refl lt -o<→o> : {n : Level} → { x y : OD {n} } → (x == y ) → (od→ord x ) o< ( od→ord y) → ⊥ +o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case1 lt )) oiso diso ) ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) @@ -215,6 +215,12 @@ tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b)) tri-c< {n} x y | tri> ¬a ¬b c = tri> ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso diso) +c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ +c<> {n} {x} {y} x<y y<x with tri-c< x y +c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x +c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) +c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y + ∅2 : {n : Level} → { x : OD {n} } → o∅ {n} o< od→ord x → ¬ ( x == od∅ {n} ) ∅2 {n} {x} lt record { eq→ = eq→ ; eq← = eq← } with ominimal (od→ord x ) lt ... | min with eq→ ( def-subst (o<→c< (Minimumo.min<x min)) oiso refl ) @@ -241,9 +247,9 @@ lemma : ¬ od→ord x ≡ o∅ lemma eq = not ( ∅7 eq ) -OD→ZF : {n : Level} → ZF {suc n} {n} +OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} OD→ZF {n} = record { - ZFSet = OD {n} + ZFSet = OD {suc n} ; _∋_ = _∋_ ; _≈_ = _==_ ; ∅ = od∅ @@ -255,20 +261,20 @@ ; infinite = record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } } ; isZF = isZF } where - Replace : OD {n} → (OD {n} → OD {n} ) → OD {n} + Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} Replace X ψ = sup-od ψ - Select : OD {n} → (OD {n} → Set n ) → OD {n} - Select X ψ = record { def = λ x → def X x → ψ ( ord→od x ) } - _,_ : OD {n} → OD {n} → OD {n} + Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} + Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } + _,_ : OD {suc n} → OD {suc n} → OD {suc n} x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) } - Union : OD {n} → OD {n} - Union x = record { def = λ y → {z : Ordinal {n}} → def x z → def (ord→od z) y } - Power : OD {n} → OD {n} - Power x = record { def = λ y → (z : Ordinal {n} ) → ( def x y ∧ def (ord→od z) y ) } - ZFSet = OD {n} - _∈_ : ( A B : ZFSet ) → Set n + Union : OD {suc n} → OD {suc n} + Union x = record { def = λ y → {z : Ordinal {suc n}} → def x z → def (ord→od z) y } + Power : OD {suc n} → OD {suc n} + Power x = record { def = λ y → (z : Ordinal {suc n} ) → ( def x y ∧ def (ord→od z) y ) } + ZFSet = OD {suc n} + _∈_ : ( A B : ZFSet ) → Set (suc n) A ∈ B = B ∋ A - _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set n + _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) _⊆_ A B {x} = A ∋ x → B ∋ x _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) @@ -277,7 +283,7 @@ infixr 200 _∈_ infixr 230 _∩_ _∪_ infixr 220 _⊆_ - isZF : IsZF (OD {n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } }) + isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } }) isZF = record { isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } ; pair = pair @@ -296,35 +302,54 @@ } where open _∧_ open Minimumo - pair : (A B : OD {n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) + pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) proj1 (pair A B ) = case1 refl proj2 (pair A B ) = case2 refl - empty : (x : OD {n} ) → ¬ (od∅ ∋ x) + empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) empty x () - union→ : (X x y : OD {n} ) → (X ∋ x) → (x ∋ y) → (Union X ∋ y) + union→ : (X x y : OD {suc n} ) → (X ∋ x) → (x ∋ y) → (Union X ∋ y) union→ X x y X∋x x∋y = {!!} where - lemma : {z : Ordinal {n} } → def X z → z ≡ od→ord y + lemma : {z : Ordinal {suc n} } → def X z → z ≡ od→ord y lemma {z} X∋z = {!!} - selection : {ψ : OD → Set n} (X y : OD) → Select X ψ ∋ y → ψ y - selection = {!!} - minord : (x : OD {n} ) → ¬ (x == od∅ )→ Minimumo (od→ord x) + ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y + ψiso {ψ} t refl = t + selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) + selection {ψ} {X} {y} = record { + proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } + ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } + } + minord : (x : OD {suc n} ) → ¬ (x == od∅ )→ Minimumo (od→ord x) minord x not = ominimal (od→ord x) (∅9 x not) - minimul : (x : OD {n} ) → ¬ (x == od∅ )→ OD {n} + minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} minimul x not = ord→od ( mino (minord x not)) - minimul<x : (x : OD {n} ) → (not : ¬ x == od∅ ) → x ∋ minimul x not + minimul<x : (x : OD {suc n} ) → (not : ¬ x == od∅ ) → x ∋ minimul x not minimul<x x not = lemma0 (min<x (minord x not)) where lemma0 : mino (minord x not) o< (od→ord x) → def x (od→ord (ord→od (mino (minord x not)))) - lemma0 m<x = def-subst {n} {ord→od (od→ord x)} {od→ord (ord→od (mino (minord x not)))} (o<→c< m<x) oiso refl + lemma0 m<x = def-subst {suc n} {ord→od (od→ord x)} {od→ord (ord→od (mino (minord x not)))} (o<→c< m<x) oiso refl regularity : (x : OD) (not : ¬ (x == od∅)) → (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) proj1 ( regularity x non ) = minimul<x x non proj2 ( regularity x non ) = reg1 where - reg2 : {y : Ordinal} → ( def (minimul x non) y ) - → ( def (minimul x non) y → (minimul x non ∋ ord→od y) ∧ (x ∋ ord→od y) ) → ⊥ - reg2 = {!!} - reg0 : {y : Ordinal {n}} → Minimumo (od→ord x) → def (Select (minimul x non) (λ z → (minimul x non ∋ z) ∧ (x ∋ z))) y → def od∅ y - reg0 {y} m t with {!!} y (mino (minord x non)) - ... | lt = {!!} + reg2 : {y : Ordinal} → ( def (minimul x non) y ∧ (minimul x non ∋ ord→od y) ∧ (x ∋ ord→od y) ) → ⊥ + reg2 {y} t with proj1 t | proj1 (proj2 t) | proj2 (proj2 t) + ... | p1 | p2 | p3 = ? + reg0 : {y : Ordinal {suc n}} → Minimumo (od→ord x) → def (Select (minimul x non) (λ z → (minimul x non ∋ z) ∧ (x ∋ z))) y → def od∅ y + reg0 {y} m t with trio< y (mino (minord x non)) + reg0 {y} m t | tri< a ¬b ¬c with reg2 {y} ( record { + proj1 = (def-subst {suc n} {minimul x non} (o<→c< a) refl diso ) + ; proj2 = record { proj1 = {!!} + ; proj2 = {!!} + }} ) + ... | () + reg0 {y} m t | tri≈ ¬a refl ¬c = lemma y ( mino (minord x non) ) {!!} where + lemma : ( ox oy : Ordinal {suc n} ) → ord→od ox c< ord→od oy → Lift (suc n) ⊥ + lemma ox oy lt = lift ( o≡→¬c< {suc n} refl lt ) + -- o≡→¬c< {suc n} refl ( c<→o< {suc n} {ord→od y} {ord→od (od→ord (minimul x non))} (proj1 t) ) + reg0 {y} m t | tri> ¬a ¬b c = lemma y ( mino (minord x non) ) {!!} {!!} where + lemma : ( ox oy : Ordinal {suc n} ) → ox o< oy → ord→od oy c< ord→od ox → Lift (suc n) ⊥ + lemma ox oy (case1 x) yc<x = {!!} + lemma ox oy (case2 lt) yc<x with d<→lv lt + ... | refl = lift ( o<→¬c> {suc n} (case2 {!!}) yc<x ) reg1 : Select (minimul x non) (λ x₁ → (minimul x non ∋ x₁) ∧ (x ∋ x₁)) == od∅ reg1 = record { eq→ = reg0 (minord x non) ; eq← = λ () }
--- a/zf.agda Mon May 27 16:14:35 2019 +0900 +++ b/zf.agda Mon May 27 21:58:17 2019 +0900 @@ -76,7 +76,7 @@ -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) infinity∅ : ∅ ∈ infinite infinity : ∀( X x : ZFSet ) → x ∈ infinite → ( x ∪ Select X ( λ y → x ≈ y )) ∈ infinite - selection : { ψ : ZFSet → Set m } → ∀ ( X y : ZFSet ) → ( y ∈ Select X ψ ) → ψ y + selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) replacement : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( ψ x ∈ Replace X ψ )