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

comparison cardinal.agda @ 247:d09437fcfc7c

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fix pair
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 26 Aug 2019 12:27:20 +0900
parents 3506f53c7d83
children 9fd85b954477
comparison
equal deleted inserted replaced
246:3506f53c7d83 247:d09437fcfc7c
59 eq-pair refl refl = HE.refl 59 eq-pair refl refl = HE.refl
60 60
61 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' > 61 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
62 eq-prod refl refl = refl 62 eq-prod refl refl = refl
63 63
64 -- prod-eq : { x x' y y' : OD } → < x , y > ≡ < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
65 -- prod-eq {x} {x'} {y} {y'} eq = {!!} where
66 -- lemma : < x , y > ≡ < x , y' > → y ≡ y'
67 -- lemma eq1 with trio< (od→ord x) (od→ord y)
68 -- lemma eq1 | tri< a ¬b ¬c = {!!}
69 -- lemma eq1 | tri≈ ¬a b ¬c = {!!}
70 -- lemma eq1 | tri> ¬a ¬b c = {!!}
71
64 postulate 72 postulate
65 def-eq : { P Q p q : OD } → P ≡ Q → p ≡ q → (pt : P ∋ p ) → (qt : Q ∋ q ) → pt ≅ qt 73 def-eq : { P Q p q : OD } → P ≡ Q → p ≡ q → (pt : P ∋ p ) → (qt : Q ∋ q ) → pt ≅ qt
66 74
75 ∈-to-ord : {p : Ordinal } → ( ZFProduct ∋ ord→od p ) → ord-pair p
76 ∈-to-ord {p} lt = def-subst {ZFProduct} {(od→ord (ord→od p))} {_} {_} lt refl diso
77
78 ord-to-∈ : {p : Ordinal } → ord-pair p → ZFProduct ∋ ord→od p
79 ord-to-∈ {p} lt = def-subst {_} {_} {ZFProduct} {(od→ord (ord→od p))} lt refl (sym diso)
80
81 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
82 lemma333 = refl
83
84 lemma334 : { A B : OD } → {a b : Ordinal} → { x : A ∋ ord→od a } → { y : B ∋ ord→od b } → (f1 : A ≡ B) → (f2 : a ≡ b)
85 → 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
86 lemma334 {A} {A} {a} {a} {x} {y} refl refl with def-eq {A} {A} {ord→od a} {ord→od a} refl refl x y
87 ... | HE.refl = HE.refl
88
89 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)
90 → 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 )
91 ≅ def-subst {A} { od→ord (ord→od a)} {C } { od→ord (ord→od c)} x (trans f1 g1)
92 (trans (cong (λ k → od→ord (ord→od k)) f2 ) (cong (λ k → od→ord (ord→od k)) g2 ))
93 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
94 ... | HE.refl = HE.refl
95
96 ∈-to-ord-oiso : { p : Ordinal } → { x : ord-pair p } → ∈-to-ord (ord-to-∈ x) ≡ x
97 ∈-to-ord-oiso {p} {x} = {!!} where
98 lemma : def-subst {_} {_} {ZFProduct} {{!!}} (def-subst {_} {_} {ZFProduct} {{!!}} x refl (sym diso)) refl diso ≡ x
99 lemma = {!!}
100
67 lemma34 : { p q : Ordinal } → (x : ord-pair p ) → (y : ord-pair q ) → p ≡ q → x ≅ y 101 lemma34 : { p q : Ordinal } → (x : ord-pair p ) → (y : ord-pair q ) → p ≡ q → x ≅ y
68 lemma34 {p} {q} x y eq = {!!} where 102 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
69 lemma : (pt : ZFProduct ∋ ord→od p ) → (qt : ZFProduct ∋ ord→od q ) → p ≡ q → pt ≅ qt 103 lemma : (pt : ZFProduct ∋ ord→od p ) → (qt : ZFProduct ∋ ord→od q ) → p ≡ q → pt ≅ qt
70 lemma pt qt eq = def-eq {ZFProduct} {ZFProduct} refl {!!} {!!} {!!} 104 lemma pt qt eq = def-eq {ZFProduct} {ZFProduct} refl (cong (λ k → ord→od k) eq) pt qt
105 lemma1 : (ord-to-∈ x) ≅ (ord-to-∈ y )
106 lemma1 = lemma (ord-to-∈ x) (ord-to-∈ y ) refl
71 107
72 π1-cong : { p q : OD } → p ≡ q → (pt : ZFProduct ∋ p ) → (qt : ZFProduct ∋ q ) → π1 pt ≅ π1 qt 108 π1-cong : { p q : OD } → p ≡ q → (pt : ZFProduct ∋ p ) → (qt : ZFProduct ∋ q ) → π1 pt ≅ π1 qt
73 π1-cong {p} {q} refl s t = HE.cong (λ k → pi1 k ) (def-eq {ZFProduct} {ZFProduct} refl refl s t ) 109 π1-cong {p} {q} refl s t = HE.cong (λ k → pi1 k ) (def-eq {ZFProduct} {ZFProduct} refl refl s t )
74 110
75 π1--iso : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≅ od→ord x 111 π1--iso : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≅ od→ord x