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

comparison cardinal.agda @ 250:08428a661677

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 28 Aug 2019 23:52:54 +0900
parents 2ecda48298e3
children 9e0125b06e76
comparison
equal deleted inserted replaced
249:2ecda48298e3 250:08428a661677
86 lemmay : y ≡ y' 86 lemmay : y ≡ y'
87 lemmay with lemmax 87 lemmay with lemmax
88 ... | refl with lemma4 eq -- with (x,y)≡(x,y') 88 ... | refl with lemma4 eq -- with (x,y)≡(x,y')
89 ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 )) 89 ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 ))
90 90
91
91 data ord-pair : (p : Ordinal) → Set n where 92 data ord-pair : (p : Ordinal) → Set n where
92 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) 93 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) )
93 94
94 ZFProduct : OD 95 ZFProduct : OD
95 ZFProduct = record { def = λ x → ord-pair x } 96 ZFProduct = record { def = λ x → ord-pair x }
116 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ 117 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩
117 od→ord < x , y > 118 od→ord < x , y >
118 ∎ ) 119 ∎ )
119 120
120 121
121 p-iso-1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≡ od→ord x 122 lemma44 : {ox oy : Ordinal } → ord-pair (od→ord < ord→od ox , ord→od oy >)
122 p-iso-1 {x} {y} p = lemma1 {od→ord < x , y >} {od→ord x} {od→ord y} p (cong₂ (λ j k → ord-pair (od→ord < j , k >)) (sym oiso) (sym oiso) ) where 123 lemma44 {ox} {oy} = pair ox oy
123 lemma1 : {op ox oy : Ordinal } → ( p : ord-pair op ) → ord-pair op ≡ ord-pair (od→ord ( < ord→od ox , ord→od oy > )) → pi1 p ≡ ox 124
124 lemma1 (pair ox oy) eq = {!!} 125 lemma55 : {ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy >
125 lemma2 : {op ox oy : Ordinal } → ord-pair op ≡ ord-pair (od→ord ( < ord→od ox , ord→od oy > )) 126 lemma55 {ox} {oy} = pair ox oy
126 lemma2 = {!!} 127
127 lemma0 : {op ox oy : Ordinal } → ( p : ord-pair (od→ord ( < ord→od ox , ord→od oy > ))) → pi1 p ≡ ox 128 lemma66 : {ox oy : Ordinal } → pair ( pi1 ( pair ox oy )) ( pi2 ( pair ox oy )) ≡ pair ox oy
128 lemma0 = {!!} 129 lemma66 = refl
129 lemma3 : {ox oy : Ordinal } ( p : ord-pair (od→ord ( < ord→od ox , ord→od oy > )) ) → pi1 p ≡ ox 130
130 lemma3 {ox} {oy} p = {!!} 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 >
131 lemma4 : {ox oy : Ordinal } → ord-pair (od→ord < ord→od ox , ord→od oy > ) ≡ ZFProduct ∋ < ord→od ox , ord→od oy > 132 lemma77 = refl
132 lemma4 = refl 133
133 lemma : {p : OD } → ord-pair (od→ord p) ≡ ZFProduct ∋ p 134 p-iso3 : { ox oy : Ordinal } → (p : ZFProduct ∋ < ord→od ox , ord→od oy > ) → p ≡ pair ox oy
134 lemma = refl 135 p-iso3 p = {!!} where
136 lemma0 : {ox oy : Ordinal } → ord-pair (od→ord < ord→od ox , ord→od oy >) ≡ ZFProduct ∋ < ord→od ox , ord→od oy >
137 lemma0 = refl
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 >
139 lemma1 refl = refl
140 lemma : {op ox oy : Ordinal } → (p : ord-pair op ) → od→ord < ord→od ox , ord→od oy > ≡ op → p ≅ pair ox oy
141 lemma {op} {ox} {oy} (pair ox' oy') eq = {!!}
142
143
144 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
146
147 p-iso1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ < x , y >
148 p-iso1 {x} {y} p with p-cons (ord→od (π1 p)) (ord→od (π2 p))
149 ... | t = {!!}
150
135 151
136 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ x 152 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ x
137 p-iso {x} p = {!!} 153 p-iso {x} p = {!!}
138 154
139 ∋-p : (A x : OD ) → Dec ( A ∋ x ) 155 ∋-p : (A x : OD ) → Dec ( A ∋ x )