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

comparison cardinal.agda @ 251:9e0125b06e76

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Thu, 29 Aug 2019 01:04:52 +0900
parents 08428a661677
children 8a58e2cd1f55
comparison
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250:08428a661677 251:9e0125b06e76
47 ==-sym : { x y : OD } → x == y → y == x 47 ==-sym : { x y : OD } → x == y → y == x
48 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t } 48 ==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← = λ {m} t → eq→ x=y t }
49 49
50 ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y 50 ord≡→≡ : { x y : OD } → od→ord x ≡ od→ord y → x ≡ y
51 ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq ) 51 ord≡→≡ eq = subst₂ (λ j k → j ≡ k ) oiso oiso ( cong ( λ k → ord→od k ) eq )
52
53 od≡→≡ : { x y : Ordinal } → ord→od x ≡ ord→od y → x ≡ y
54 od≡→≡ eq = subst₂ (λ j k → j ≡ k ) diso diso ( cong ( λ k → od→ord k ) eq )
52 55
53 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' > 56 eq-prod : { x x' y y' : OD } → x ≡ x' → y ≡ y' → < x , y > ≡ < x' , y' >
54 eq-prod refl refl = refl 57 eq-prod refl refl = refl
55 58
56 prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' ) 59 prod-eq : { x x' y y' : OD } → < x , y > == < x' , y' > → (x ≡ x' ) ∧ ( y ≡ y' )
129 lemma66 = refl 132 lemma66 = refl
130 133
131 lemma77 : {ox oy : Ordinal } → ZFProduct ∋ < ord→od (pi1 ( pair ox oy )) , ord→od (pi2 ( pair ox oy )) > ≡ ZFProduct ∋ < ord→od ox , ord→od oy > 134 lemma77 : {ox oy : Ordinal } → ZFProduct ∋ < ord→od (pi1 ( pair ox oy )) , ord→od (pi2 ( pair ox oy )) > ≡ ZFProduct ∋ < ord→od ox , ord→od oy >
132 lemma77 = refl 135 lemma77 = refl
133 136
137
138 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ x
139 p-iso {x} p = {!!} where
140
141 pair-iso : {op ox oy : Ordinal} (x : ord-pair (od→ord < ord→od ox , ord→od oy >) ) → pi1 x ≡ ox → pi2 x ≡ oy → x ≡ pair ox oy
142 pair-iso (pair ox oy) = {!!}
143
134 p-iso3 : { ox oy : Ordinal } → (p : ZFProduct ∋ < ord→od ox , ord→od oy > ) → p ≡ pair ox oy 144 p-iso3 : { ox oy : Ordinal } → (p : ZFProduct ∋ < ord→od ox , ord→od oy > ) → p ≡ pair ox oy
135 p-iso3 p = {!!} where 145 p-iso3 {ox} {oy} p with p-iso p
136 lemma0 : {ox oy : Ordinal } → ord-pair (od→ord < ord→od ox , ord→od oy >) ≡ ZFProduct ∋ < ord→od ox , ord→od oy > 146 ... | eq with prod-eq ( ord→== (cong (λ k → od→ord k) eq ) )
137 lemma0 = refl 147 ... | record { proj1 = eq1 ; proj2 = eq2 } = lemma eq1 eq2 where
138 lemma1 : {op ox oy : Ordinal } → op ≡ od→ord < ord→od ox , ord→od oy > → ord-pair op ≡ ZFProduct ∋ < ord→od ox , ord→od oy > 148 lemma : ord→od (pi1 p) ≡ ord→od ox → ord→od (pi2 p) ≡ ord→od oy → p ≡ pair ox oy
139 lemma1 refl = refl 149 lemma eq1 eq2 with od≡→≡ eq1 | od≡→≡ eq2
140 lemma : {op ox oy : Ordinal } → (p : ord-pair op ) → od→ord < ord→od ox , ord→od oy > ≡ op → p ≅ pair ox oy 150 ... | eq1' | eq2' = pair-iso {od→ord < ord→od ox , ord→od oy >} {ox} {oy} p eq1' eq2'
141 lemma {op} {ox} {oy} (pair ox' oy') eq = {!!}
142
143 151
144 p-iso2 : { ox oy : Ordinal } → p-cons (ord→od ox) (ord→od oy) ≡ pair ox oy 152 p-iso2 : { ox oy : Ordinal } → p-cons (ord→od ox) (ord→od oy) ≡ pair ox oy
145 p-iso2 = subst₂ ( λ j k → j ≡ k ) {!!} {!!} refl 153 p-iso2 {ox} {oy} = p-iso3 (p-cons (ord→od ox) (ord→od oy))
146 154
147 p-iso1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ < x , y > 155 p-iso1 : { ox oy : Ordinal } → (p : ZFProduct ∋ < ord→od ox , ord→od oy > ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ < ord→od ox , ord→od oy >
148 p-iso1 {x} {y} p with p-cons (ord→od (π1 p)) (ord→od (π2 p)) 156 p-iso1 {x} {y} p with p-cons (ord→od (π1 p)) (ord→od (π2 p))
149 ... | t = {!!} 157 ... | t with p-iso3 p | p-iso3 t
158 ... | refl | refl = refl
150 159
151
152 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ x
153 p-iso {x} p = {!!}
154
155 ∋-p : (A x : OD ) → Dec ( A ∋ x ) 160 ∋-p : (A x : OD ) → Dec ( A ∋ x )
156 ∋-p A x with p∨¬p ( A ∋ x ) 161 ∋-p A x with p∨¬p ( A ∋ x )
157 ∋-p A x | case1 t = yes t 162 ∋-p A x | case1 t = yes t
158 ∋-p A x | case2 t = no t 163 ∋-p A x | case2 t = no t
159 164