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

comparison cardinal.agda @ 242:c10048d69614

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 25 Aug 2019 18:44:41 +0900
parents ccc84f289c98
children f97a2e4df451
comparison
equal deleted inserted replaced
241:ccc84f289c98 242:c10048d69614
31 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) 31 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) )
32 32
33 ZFProduct : OD 33 ZFProduct : OD
34 ZFProduct = record { def = λ x → ord-pair x } 34 ZFProduct = record { def = λ x → ord-pair x }
35 35
36 pi1 : { p : Ordinal } → ord-pair p → Ordinal
37 pi1 ( pair x y ) = x
38
36 π1 : { p : OD } → ZFProduct ∋ p → Ordinal 39 π1 : { p : OD } → ZFProduct ∋ p → Ordinal
37 π1 lt = pi1 lt where 40 π1 lt = pi1 lt
38 pi1 : { p : Ordinal } → ord-pair p → Ordinal 41
39 pi1 ( pair x y ) = x 42 pi2 : { p : Ordinal } → ord-pair p → Ordinal
43 pi2 ( pair x y ) = y
40 44
41 π2 : { p : OD } → ZFProduct ∋ p → Ordinal 45 π2 : { p : OD } → ZFProduct ∋ p → Ordinal
42 π2 lt = pi2 lt where 46 π2 lt = pi2 lt
43 pi2 : { p : Ordinal } → ord-pair p → Ordinal 47
44 pi2 ( pair x y ) = y 48 p-cons : ( x y : OD ) → ZFProduct ∋ < x , y >
45 49 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl (
46 p-cons : { x y : OD } → ZFProduct ∋ < x , y >
47 p-cons {x} {y} = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl (
48 let open ≡-Reasoning in begin 50 let open ≡-Reasoning in begin
49 od→ord < ord→od (od→ord x) , ord→od (od→ord y) > 51 od→ord < ord→od (od→ord x) , ord→od (od→ord y) >
50 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ 52 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩
51 od→ord < x , y > 53 od→ord < x , y >
52 ∎ ) 54 ∎ )
55
56 π1-iso : { x y : OD } → π1 ( p-cons x y ) ≡ od→ord x
57 π1-iso {x} {y} = {!!} where
58 lemma : {ox oy : Ordinal} → pi1 ( pair ox oy ) ≡ ox
59 lemma = refl
60 lemma2 : {ox oy : Ordinal} →
61 def-subst {ZFProduct} {_} (pair (od→ord (ord→od ox)) (od→ord (ord→od oy))) refl (trans (cong₂ (λ j k → od→ord < j , k >) oiso oiso) refl) ≡ pair ox oy
62 lemma2 {ox} {oy} = let open ≡-Reasoning in begin
63 def-subst {ZFProduct} {_} (pair (od→ord (ord→od ox)) (od→ord (ord→od oy))) refl (trans (cong₂ (λ j k → od→ord < j , k >) oiso oiso) refl)
64 ≡⟨ ? ⟩
65 pair ox oy
66 ∎
67 lemma1 : {ox oy : Ordinal} → π1 ( p-cons (ord→od ox) (ord→od oy) ) ≡ ox
68 lemma1 {ox} {oy} = let open ≡-Reasoning in begin
69 π1 ( p-cons (ord→od ox) (ord→od oy) )
70 ≡⟨⟩
71 pi1 (pair (pi1 (def-subst {ZFProduct} {_} (pair (od→ord (ord→od ox)) (od→ord (ord→od oy))) refl (trans (cong₂ (λ j k → od→ord < j , k >) oiso oiso) refl))) oy )
72 ≡⟨ cong (λ k → pi1 k) lemma2 ⟩
73 pi1 (pair ox oy )
74 ≡⟨ lemma {ox} {oy} ⟩
75 ox
76 ∎
77
78 p-iso : { x : OD } → {p : ZFProduct ∋ x } → < ord→od (π1 p) , ord→od (π2 p) > ≡ x
79 p-iso {x} {p} = pi p pc where
80 pc : ZFProduct ∋ < ord→od (π1 p) , ord→od (π2 p) >
81 pc = p-cons (ord→od (π1 p)) (ord→od (π2 p))
82 pi : { prod prod1 : Ordinal } → ord-pair prod → ord-pair prod1 → {!!}
83 pi (pair p1 p2) (pair q1 q2) = {!!}
84
53 85
54 86
55 ∋-p : (A x : OD ) → Dec ( A ∋ x ) 87 ∋-p : (A x : OD ) → Dec ( A ∋ x )
56 ∋-p A x with p∨¬p ( A ∋ x ) 88 ∋-p A x with p∨¬p ( A ∋ x )
57 ∋-p A x | case1 t = yes t 89 ∋-p A x | case1 t = yes t
60 _⊗_ : (A B : OD) → OD 92 _⊗_ : (A B : OD) → OD
61 A ⊗ B = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } where 93 A ⊗ B = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } where
62 checkAB : { p : Ordinal } → def ZFProduct p → Set n 94 checkAB : { p : Ordinal } → def ZFProduct p → Set n
63 checkAB (pair x y) = def A x ∧ def B y 95 checkAB (pair x y) = def A x ∧ def B y
64 96
97 func→od0 : (f : Ordinal → Ordinal ) → OD
98 func→od0 f = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkfunc p ) } where
99 checkfunc : { p : Ordinal } → def ZFProduct p → Set n
100 checkfunc (pair x y) = f x ≡ y
101
65 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) 102 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B )
66 103
67 Func : ( A B : OD ) → OD 104 Func : ( A B : OD ) → OD
68 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } 105 Func A B = record { def = λ x → def (Power (A ⊗ B)) x }
69 106
71 108
72 109
73 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD 110 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD
74 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) 111 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > )
75 112
76 record Func←cd { dom cod : OD } {f : Ordinal } (f<F : def (Func dom cod ) f) : Set n where 113 record Func←cd { dom cod : OD } {f : Ordinal } : Set n where
77 field 114 field
78 func-1 : Ordinal → Ordinal 115 func-1 : Ordinal → Ordinal
79 func→od∈Func-1 : (Func dom (Ord (sup-o (λ x → func-1 x)) )) ∋ func→od func-1 dom 116 func→od∈Func-1 : Func dom cod ∋ func→od func-1 dom
80 117
81 od→func : { dom cod : OD } → {f : Ordinal } → (f<F : def (Func dom cod ) f ) → Func←cd {dom} {cod} {f} f<F 118 od→func : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → Func←cd {dom} {cod} {f}
82 od→func {dom} {cod} {f} lt = record { func-1 = λ x → sup-o ( λ y → lemma x y ) ; func→od∈Func-1 = record { proj1 = {!!} ; proj2 = {!!} } } where 119 od→func {dom} {cod} {f} lt = record { func-1 = λ x → sup-o ( λ y → lemma x y ) ; func→od∈Func-1 = record { proj1 = {!!} ; proj2 = {!!} } } where
83 lemma : Ordinal → Ordinal → Ordinal 120 lemma : Ordinal → Ordinal → Ordinal
84 lemma x y with IsZF.power→ isZF (dom ⊗ cod) (ord→od f) (subst (λ k → def (Power (dom ⊗ cod)) k ) (sym diso) lt ) | ∋-p (ord→od f) (ord→od y) 121 lemma x y with IsZF.power→ isZF (dom ⊗ cod) (ord→od f) (subst (λ k → def (Power (dom ⊗ cod)) k ) (sym diso) lt ) | ∋-p (ord→od f) (ord→od y)
85 lemma x y | p | no n = o∅ 122 lemma x y | p | no n = o∅
86 lemma x y | p | yes f∋y = lemma2 (proj1 (double-neg-eilm ( p {ord→od y} f∋y ))) where -- p : {y : OD} → f ∋ y → ¬ ¬ (dom ⊗ cod ∋ y) 123 lemma x y | p | yes f∋y = lemma2 (proj1 (double-neg-eilm ( p {ord→od y} f∋y ))) where -- p : {y : OD} → f ∋ y → ¬ ¬ (dom ⊗ cod ∋ y)
87 lemma2 : {p : Ordinal} → ord-pair p → Ordinal 124 lemma2 : {p : Ordinal} → ord-pair p → Ordinal
88 lemma2 (pair x1 y1) with decp ( x1 ≡ x) 125 lemma2 (pair x1 y1) with decp ( x1 ≡ x)
89 lemma2 (pair x1 y1) | yes p = y1 126 lemma2 (pair x1 y1) | yes p = y1
90 lemma2 (pair x1 y1) | no ¬p = o∅ 127 lemma2 (pair x1 y1) | no ¬p = o∅
128 fod : OD
129 fod = Replace dom ( λ x → < x , ord→od (sup-o ( λ y → lemma (od→ord x) y )) > )
91 130
92 131
93 open Func←cd 132 open Func←cd
94
95 func→od∈Func : (f : Ordinal → Ordinal ) ( dom : OD ) → (Func dom (Ord (sup-o (λ x → f x)) )) ∋ func→od f dom
96 func→od∈Func f dom = record { proj1 = {!!} ; proj2 = {!!} } where
97 f<F : def (Func dom (Ord (sup-o (λ x → f x)))) (od→ord (func→od f dom))
98 f<F = {!!}
99 odfunc : Func←cd {dom} f<F
100 odfunc = ( od→func {dom} {Ord (sup-o (λ x → f x))} {od→ord (func→od f dom)} f<F )
101 lemma : Func dom (Ord (sup-o ( func-1 odfunc ) )) ∋ func→od (func-1 odfunc ) dom
102 lemma = func→od∈Func-1 odfunc
103 133
104 -- contra position of sup-o< 134 -- contra position of sup-o<
105 -- 135 --
106 136
107 -- postulate 137 -- postulate