Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate 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 |
| rev | line source |
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| 16 | 1 open import Level |
| 224 | 2 open import Ordinals |
| 3 module cardinal {n : Level } (O : Ordinals {n}) where | |
| 3 | 4 |
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5 open import zf |
| 219 | 6 open import logic |
| 224 | 7 import OD |
| 23 | 8 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 224 | 9 open import Relation.Binary.PropositionalEquality |
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10 open import Data.Nat.Properties |
| 6 | 11 open import Data.Empty |
| 12 open import Relation.Nullary | |
| 13 open import Relation.Binary | |
| 14 open import Relation.Binary.Core | |
| 15 | |
| 224 | 16 open inOrdinal O |
| 17 open OD O | |
| 219 | 18 open OD.OD |
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posturate OD is isomorphic to Ordinal
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19 |
| 120 | 20 open _∧_ |
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21 open _∨_ |
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22 open Bool |
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od→lv : {n : Level} → OD {n} → Nat
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23 |
| 230 | 24 -- we have to work on Ordinal to keep OD Level n |
| 25 -- since we use p∨¬p which works only on Level n | |
| 225 | 26 |
| 233 | 27 <_,_> : (x y : OD) → OD |
| 28 < x , y > = (x , x ) , (x , y ) | |
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29 |
| 238 | 30 data ord-pair : (p : Ordinal) → Set n where |
| 31 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) | |
| 32 | |
| 33 ZFProduct : OD | |
| 34 ZFProduct = record { def = λ x → ord-pair x } | |
| 35 | |
| 242 | 36 pi1 : { p : Ordinal } → ord-pair p → Ordinal |
| 37 pi1 ( pair x y ) = x | |
| 38 | |
| 239 | 39 π1 : { p : OD } → ZFProduct ∋ p → Ordinal |
| 242 | 40 π1 lt = pi1 lt |
| 41 | |
| 42 pi2 : { p : Ordinal } → ord-pair p → Ordinal | |
| 43 pi2 ( pair x y ) = y | |
| 237 | 44 |
| 239 | 45 π2 : { p : OD } → ZFProduct ∋ p → Ordinal |
| 242 | 46 π2 lt = pi2 lt |
| 237 | 47 |
| 242 | 48 p-cons : ( x y : OD ) → ZFProduct ∋ < x , y > |
| 49 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl ( | |
| 238 | 50 let open ≡-Reasoning in begin |
| 51 od→ord < ord→od (od→ord x) , ord→od (od→ord y) > | |
| 52 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ | |
| 53 od→ord < x , y > | |
| 54 ∎ ) | |
| 242 | 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 | |
| 238 | 85 |
| 86 | |
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87 ∋-p : (A x : OD ) → Dec ( A ∋ x ) |
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88 ∋-p A x with p∨¬p ( A ∋ x ) |
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89 ∋-p A x | case1 t = yes t |
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90 ∋-p A x | case2 t = no t |
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91 |
| 233 | 92 _⊗_ : (A B : OD) → OD |
| 239 | 93 A ⊗ B = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } where |
| 94 checkAB : { p : Ordinal } → def ZFProduct p → Set n | |
| 95 checkAB (pair x y) = def A x ∧ def B y | |
| 233 | 96 |
| 242 | 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 | |
| 233 | 102 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) |
| 225 | 103 |
| 233 | 104 Func : ( A B : OD ) → OD |
| 105 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } | |
| 106 | |
| 107 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
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108 |
| 236 | 109 |
| 233 | 110 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD |
| 111 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
| 112 | |
| 242 | 113 record Func←cd { dom cod : OD } {f : Ordinal } : Set n where |
| 236 | 114 field |
| 115 func-1 : Ordinal → Ordinal | |
| 242 | 116 func→od∈Func-1 : Func dom cod ∋ func→od func-1 dom |
| 236 | 117 |
| 242 | 118 od→func : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → Func←cd {dom} {cod} {f} |
| 240 | 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 |
| 236 | 120 lemma : Ordinal → Ordinal → Ordinal |
| 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) | |
| 122 lemma x y | p | no n = o∅ | |
| 240 | 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) |
| 124 lemma2 : {p : Ordinal} → ord-pair p → Ordinal | |
| 125 lemma2 (pair x1 y1) with decp ( x1 ≡ x) | |
| 126 lemma2 (pair x1 y1) | yes p = y1 | |
| 127 lemma2 (pair x1 y1) | no ¬p = o∅ | |
| 242 | 128 fod : OD |
| 129 fod = Replace dom ( λ x → < x , ord→od (sup-o ( λ y → lemma (od→ord x) y )) > ) | |
| 240 | 130 |
| 131 | |
| 132 open Func←cd | |
| 236 | 133 |
| 227 | 134 -- contra position of sup-o< |
| 135 -- | |
| 136 | |
| 235 | 137 -- postulate |
| 138 -- -- contra-position of mimimulity of supermum required in Cardinal | |
| 139 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
| 140 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 227 | 141 |
| 219 | 142 ------------ |
| 143 -- | |
| 144 -- Onto map | |
| 145 -- def X x -> xmap | |
| 146 -- X ---------------------------> Y | |
| 147 -- ymap <- def Y y | |
| 148 -- | |
| 224 | 149 record Onto (X Y : OD ) : Set n where |
| 219 | 150 field |
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151 xmap : Ordinal |
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152 ymap : Ordinal |
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153 xfunc : def (Func X Y) xmap |
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154 yfunc : def (Func Y X) ymap |
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155 onto-iso : {y : Ordinal } → (lty : def Y y ) → |
| 240 | 156 func-1 ( od→func {X} {Y} {xmap} xfunc ) ( func-1 (od→func yfunc) y ) ≡ y |
| 230 | 157 |
| 158 open Onto | |
| 159 | |
| 160 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z | |
| 161 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { | |
| 162 xmap = xmap1 | |
| 163 ; ymap = zmap | |
| 164 ; xfunc = xfunc1 | |
| 165 ; yfunc = zfunc | |
| 166 ; onto-iso = onto-iso1 | |
| 167 } where | |
| 168 xmap1 : Ordinal | |
| 169 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
| 170 zmap : Ordinal | |
| 171 zmap = {!!} | |
| 172 xfunc1 : def (Func X Z) xmap1 | |
| 173 xfunc1 = {!!} | |
| 174 zfunc : def (Func Z X) zmap | |
| 175 zfunc = {!!} | |
| 240 | 176 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func-1 (od→func xfunc1 ) (func-1 (od→func zfunc ) z ) ≡ z |
| 230 | 177 onto-iso1 = {!!} |
| 178 | |
| 51 | 179 |
| 224 | 180 record Cardinal (X : OD ) : Set n where |
| 219 | 181 field |
| 224 | 182 cardinal : Ordinal |
| 230 | 183 conto : Onto X (Ord cardinal) |
| 184 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
| 151 | 185 |
| 224 | 186 cardinal : (X : OD ) → Cardinal X |
| 187 cardinal X = record { | |
| 219 | 188 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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189 ; conto = onto |
| 219 | 190 ; cmax = cmax |
| 191 } where | |
| 230 | 192 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
| 193 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
| 194 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 219 | 195 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
| 229 | 196 S = sup-o (λ x → proj1 (cardinal-p x)) |
| 230 | 197 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
| 198 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
| 229 | 199 lemma1 x prev with trio< x (osuc S) |
| 200 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
| 230 | 201 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
| 202 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
| 203 lemma2 : Onto X (Ord x) | |
| 204 lemma2 with prev {!!} {!!} | |
| 205 ... | lift t = t {!!} | |
| 229 | 206 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
| 207 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
| 230 | 208 onto : Onto X (Ord S) |
| 209 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
| 210 ... | lift t = t <-osuc | |
| 211 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
| 229 | 212 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
| 224 | 213 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
| 219 | 214 lemma : proj1 (cardinal-p y) ≡ y |
| 230 | 215 lemma with p∨¬p ( Onto X (Ord y) ) |
| 219 | 216 lemma | case1 x = refl |
| 217 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 218 |
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219 |
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220 ----- |
| 219 | 221 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 222 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 223 |
| 224 | |
| 225 |