Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 247:d09437fcfc7c
fix pair
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 26 Aug 2019 12:27:20 +0900 |
| parents | 3506f53c7d83 |
| children | 9fd85b954477 |
| files | OD.agda cardinal.agda zf.agda |
| diffstat | 3 files changed, 55 insertions(+), 9 deletions(-) [+] |
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--- a/OD.agda Mon Aug 26 02:50:16 2019 +0900 +++ b/OD.agda Mon Aug 26 12:27:20 2019 +0900 @@ -269,7 +269,7 @@ Replace : OD → (OD → OD ) → OD 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 } _,_ : OD → OD → OD - x , y = Ord (omax (od→ord x) (od→ord y)) + x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y)) _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = record { def = λ x → def A x ∧ def B x } Union : OD → OD @@ -294,7 +294,8 @@ isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite isZF = record { isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } - ; pair = pair + ; pair→ = pair→ + ; pair← = pair← ; union→ = union→ ; union← = union← ; empty = empty @@ -311,9 +312,17 @@ ; choice = choice } where - pair : (A B : OD ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) - proj1 (pair A B ) = omax-x (od→ord A) (od→ord B) - proj2 (pair A B ) = omax-y (od→ord A) (od→ord B) + pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y ) + pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x )) + pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y )) + + pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t + pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x)) + pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y)) + + -- pair0 : (A B : OD ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) + -- proj1 (pair A B ) = omax-x (od→ord A) (od→ord B) + -- proj2 (pair A B ) = omax-y (od→ord A) (od→ord B) empty : (x : OD ) → ¬ (od∅ ∋ x) empty x = ¬x<0
--- a/cardinal.agda Mon Aug 26 02:50:16 2019 +0900 +++ b/cardinal.agda Mon Aug 26 12:27:20 2019 +0900 @@ -61,13 +61,49 @@ eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' > eq-prod refl refl = refl +-- prod-eq : { x x' y y' : OD } → < x , y > ≡ < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' ) +-- prod-eq {x} {x'} {y} {y'} eq = {!!} where +-- lemma : < x , y > ≡ < x , y' > → y ≡ y' +-- lemma eq1 with trio< (od→ord x) (od→ord y) +-- lemma eq1 | tri< a ¬b ¬c = {!!} +-- lemma eq1 | tri≈ ¬a b ¬c = {!!} +-- lemma eq1 | tri> ¬a ¬b c = {!!} + postulate def-eq : { P Q p q : OD } → P ≡ Q → p ≡ q → (pt : P ∋ p ) → (qt : Q ∋ q ) → pt ≅ qt +∈-to-ord : {p : Ordinal } → ( ZFProduct ∋ ord→od p ) → ord-pair p +∈-to-ord {p} lt = def-subst {ZFProduct} {(od→ord (ord→od p))} {_} {_} lt refl diso + +ord-to-∈ : {p : Ordinal } → ord-pair p → ZFProduct ∋ ord→od p +ord-to-∈ {p} lt = def-subst {_} {_} {ZFProduct} {(od→ord (ord→od p))} lt refl (sym diso) + +lemma333 : { A a : OD } → { x : A ∋ a } → def-subst {A} {od→ord a} (def-subst {A} {od→ord a} x refl refl ) refl refl ≡ x +lemma333 = refl + +lemma334 : { A B : OD } → {a b : Ordinal} → { x : A ∋ ord→od a } → { y : B ∋ ord→od b } → (f1 : A ≡ B) → (f2 : a ≡ b) + → def-subst {B} {od→ord (ord→od b)} (def-subst {A} { od→ord (ord→od a)} x f1 (cong (λ k → od→ord (ord→od k)) f2 )) refl refl ≅ x +lemma334 {A} {A} {a} {a} {x} {y} refl refl with def-eq {A} {A} {ord→od a} {ord→od a} refl refl x y +... | HE.refl = HE.refl + +lemma335 : { A B C : OD } → {a b c : Ordinal} → { x : A ∋ ord→od a } → { y : C ∋ ord→od c } → (f1 : A ≡ B) → (f2 : a ≡ b) → (g1 : B ≡ C) → (g2 : b ≡ c) + → def-subst {B} {od→ord (ord→od b)} (def-subst {A} { od→ord (ord→od a)} x f1 (cong (λ k → od→ord (ord→od k)) f2 )) g1 (cong (λ k → od→ord (ord→od k)) g2 ) + ≅ def-subst {A} { od→ord (ord→od a)} {C } { od→ord (ord→od c)} x (trans f1 g1) + (trans (cong (λ k → od→ord (ord→od k)) f2 ) (cong (λ k → od→ord (ord→od k)) g2 )) +lemma335 {A} {A} {A} {a} {a} {a} {x} {y} refl refl refl refl with def-eq {A} {A} {ord→od a} {ord→od a} refl refl x y +... | HE.refl = HE.refl + +∈-to-ord-oiso : { p : Ordinal } → { x : ord-pair p } → ∈-to-ord (ord-to-∈ x) ≡ x +∈-to-ord-oiso {p} {x} = {!!} where + lemma : def-subst {_} {_} {ZFProduct} {{!!}} (def-subst {_} {_} {ZFProduct} {{!!}} x refl (sym diso)) refl diso ≡ x + lemma = {!!} + lemma34 : { p q : Ordinal } → (x : ord-pair p ) → (y : ord-pair q ) → p ≡ q → x ≅ y -lemma34 {p} {q} x y eq = {!!} where +lemma34 {p} {q} x y refl = subst₂ (λ j k → j ≅ k) ∈-to-ord-oiso ∈-to-ord-oiso (HE.cong (λ k → ∈-to-ord k) lemma1 ) where lemma : (pt : ZFProduct ∋ ord→od p ) → (qt : ZFProduct ∋ ord→od q ) → p ≡ q → pt ≅ qt - lemma pt qt eq = def-eq {ZFProduct} {ZFProduct} refl {!!} {!!} {!!} + lemma pt qt eq = def-eq {ZFProduct} {ZFProduct} refl (cong (λ k → ord→od k) eq) pt qt + lemma1 : (ord-to-∈ x) ≅ (ord-to-∈ y ) + lemma1 = lemma (ord-to-∈ x) (ord-to-∈ y ) refl π1-cong : { p q : OD } → p ≡ q → (pt : ZFProduct ∋ p ) → (qt : ZFProduct ∋ q ) → π1 pt ≅ π1 qt π1-cong {p} {q} refl s t = HE.cong (λ k → pi1 k ) (def-eq {ZFProduct} {ZFProduct} refl refl s t )
--- a/zf.agda Mon Aug 26 02:50:16 2019 +0900 +++ b/zf.agda Mon Aug 26 12:27:20 2019 +0900 @@ -22,8 +22,9 @@ : Set (suc (n ⊔ m)) where field isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ - -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) - pair : ( A B : ZFSet ) → ( (A , B) ∋ A ) ∧ ( (A , B) ∋ B ) + -- ∀ x ∀ y ∃ z ∀ t ( z ∋ t → t ≈ x ∨ t ≈ y) + pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y ) + pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u)) union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z )))