Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 241:ccc84f289c98
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 22 Aug 2019 12:41:41 +0900 |
| parents | c76c812de395 |
| children | c10048d69614 |
| 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 | |
| 239 | 36 π1 : { p : OD } → ZFProduct ∋ p → Ordinal |
| 37 π1 lt = pi1 lt where | |
| 238 | 38 pi1 : { p : Ordinal } → ord-pair p → Ordinal |
| 39 pi1 ( pair x y ) = x | |
| 237 | 40 |
| 239 | 41 π2 : { p : OD } → ZFProduct ∋ p → Ordinal |
| 42 π2 lt = pi2 lt where | |
| 238 | 43 pi2 : { p : Ordinal } → ord-pair p → Ordinal |
| 44 pi2 ( pair x y ) = y | |
| 237 | 45 |
| 238 | 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 | |
| 49 od→ord < ord→od (od→ord x) , ord→od (od→ord y) > | |
| 50 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ | |
| 51 od→ord < x , y > | |
| 52 ∎ ) | |
| 53 | |
| 54 | |
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55 ∋-p : (A x : OD ) → Dec ( A ∋ x ) |
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56 ∋-p A x with p∨¬p ( A ∋ x ) |
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57 ∋-p A x | case1 t = yes t |
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58 ∋-p A x | case2 t = no t |
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59 |
| 233 | 60 _⊗_ : (A B : OD) → OD |
| 239 | 61 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 | |
| 63 checkAB (pair x y) = def A x ∧ def B y | |
| 233 | 64 |
| 65 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) | |
| 225 | 66 |
| 233 | 67 Func : ( A B : OD ) → OD |
| 68 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } | |
| 69 | |
| 70 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
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71 |
| 236 | 72 |
| 233 | 73 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD |
| 74 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
| 75 | |
| 236 | 76 record Func←cd { dom cod : OD } {f : Ordinal } (f<F : def (Func dom cod ) f) : Set n where |
| 77 field | |
| 78 func-1 : Ordinal → Ordinal | |
| 79 func→od∈Func-1 : (Func dom (Ord (sup-o (λ x → func-1 x)) )) ∋ func→od func-1 dom | |
| 80 | |
| 240 | 81 od→func : { dom cod : OD } → {f : Ordinal } → (f<F : def (Func dom cod ) f ) → Func←cd {dom} {cod} {f} f<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 | |
| 236 | 83 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) | |
| 85 lemma x y | p | no n = o∅ | |
| 240 | 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) |
| 87 lemma2 : {p : Ordinal} → ord-pair p → Ordinal | |
| 88 lemma2 (pair x1 y1) with decp ( x1 ≡ x) | |
| 89 lemma2 (pair x1 y1) | yes p = y1 | |
| 90 lemma2 (pair x1 y1) | no ¬p = o∅ | |
| 91 | |
| 92 | |
| 93 open Func←cd | |
| 236 | 94 |
| 95 func→od∈Func : (f : Ordinal → Ordinal ) ( dom : OD ) → (Func dom (Ord (sup-o (λ x → f x)) )) ∋ func→od f dom | |
| 240 | 96 func→od∈Func f dom = record { proj1 = {!!} ; proj2 = {!!} } where |
| 241 | 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 | |
| 225 | 103 |
| 227 | 104 -- contra position of sup-o< |
| 105 -- | |
| 106 | |
| 235 | 107 -- postulate |
| 108 -- -- contra-position of mimimulity of supermum required in Cardinal | |
| 109 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
| 110 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 227 | 111 |
| 219 | 112 ------------ |
| 113 -- | |
| 114 -- Onto map | |
| 115 -- def X x -> xmap | |
| 116 -- X ---------------------------> Y | |
| 117 -- ymap <- def Y y | |
| 118 -- | |
| 224 | 119 record Onto (X Y : OD ) : Set n where |
| 219 | 120 field |
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121 xmap : Ordinal |
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122 ymap : Ordinal |
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123 xfunc : def (Func X Y) xmap |
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124 yfunc : def (Func Y X) ymap |
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125 onto-iso : {y : Ordinal } → (lty : def Y y ) → |
| 240 | 126 func-1 ( od→func {X} {Y} {xmap} xfunc ) ( func-1 (od→func yfunc) y ) ≡ y |
| 230 | 127 |
| 128 open Onto | |
| 129 | |
| 130 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z | |
| 131 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { | |
| 132 xmap = xmap1 | |
| 133 ; ymap = zmap | |
| 134 ; xfunc = xfunc1 | |
| 135 ; yfunc = zfunc | |
| 136 ; onto-iso = onto-iso1 | |
| 137 } where | |
| 138 xmap1 : Ordinal | |
| 139 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
| 140 zmap : Ordinal | |
| 141 zmap = {!!} | |
| 142 xfunc1 : def (Func X Z) xmap1 | |
| 143 xfunc1 = {!!} | |
| 144 zfunc : def (Func Z X) zmap | |
| 145 zfunc = {!!} | |
| 240 | 146 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func-1 (od→func xfunc1 ) (func-1 (od→func zfunc ) z ) ≡ z |
| 230 | 147 onto-iso1 = {!!} |
| 148 | |
| 51 | 149 |
| 224 | 150 record Cardinal (X : OD ) : Set n where |
| 219 | 151 field |
| 224 | 152 cardinal : Ordinal |
| 230 | 153 conto : Onto X (Ord cardinal) |
| 154 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
| 151 | 155 |
| 224 | 156 cardinal : (X : OD ) → Cardinal X |
| 157 cardinal X = record { | |
| 219 | 158 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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159 ; conto = onto |
| 219 | 160 ; cmax = cmax |
| 161 } where | |
| 230 | 162 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
| 163 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
| 164 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 219 | 165 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
| 229 | 166 S = sup-o (λ x → proj1 (cardinal-p x)) |
| 230 | 167 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
| 168 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
| 229 | 169 lemma1 x prev with trio< x (osuc S) |
| 170 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
| 230 | 171 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
| 172 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
| 173 lemma2 : Onto X (Ord x) | |
| 174 lemma2 with prev {!!} {!!} | |
| 175 ... | lift t = t {!!} | |
| 229 | 176 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
| 177 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
| 230 | 178 onto : Onto X (Ord S) |
| 179 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
| 180 ... | lift t = t <-osuc | |
| 181 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
| 229 | 182 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
| 224 | 183 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
| 219 | 184 lemma : proj1 (cardinal-p y) ≡ y |
| 230 | 185 lemma with p∨¬p ( Onto X (Ord y) ) |
| 219 | 186 lemma | case1 x = refl |
| 187 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 188 |
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189 |
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190 ----- |
| 219 | 191 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 192 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 193 |
| 194 | |
| 195 |