Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 403:ce2ce3f62023
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Tue, 28 Jul 2020 10:51:08 +0900 |
| parents | 853ead1b56b8 |
| children | b737a2e0b46e |
| 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 |
| 276 | 8 import ODC |
| 274 | 9 import OPair |
| 23 | 10 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 224 | 11 open import Relation.Binary.PropositionalEquality |
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12 open import Data.Nat.Properties |
| 6 | 13 open import Data.Empty |
| 14 open import Relation.Nullary | |
| 15 open import Relation.Binary | |
| 16 open import Relation.Binary.Core | |
| 17 | |
| 224 | 18 open inOrdinal O |
| 19 open OD O | |
| 219 | 20 open OD.OD |
| 274 | 21 open OPair O |
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22 open ODAxiom odAxiom |
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23 |
| 120 | 24 open _∧_ |
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25 open _∨_ |
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26 open Bool |
| 254 | 27 open _==_ |
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od→lv : {n : Level} → OD {n} → Nat
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28 |
| 230 | 29 -- we have to work on Ordinal to keep OD Level n |
| 30 -- since we use p∨¬p which works only on Level n | |
| 250 | 31 |
| 331 | 32 ∋-p : (A x : HOD ) → Dec ( A ∋ x ) |
| 276 | 33 ∋-p A x with ODC.p∨¬p O ( A ∋ x ) |
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34 ∋-p A x | case1 t = yes t |
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35 ∋-p A x | case2 t = no t |
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36 |
| 376 | 37 --_⊗_ : (A B : HOD) → HOD |
| 38 --A ⊗ B = record { od = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } } where | |
| 39 -- checkAB : { p : Ordinal } → def ZFProduct p → Set n | |
| 40 -- checkAB (pair x y) = odef A x ∧ odef B y | |
| 233 | 41 |
| 331 | 42 func→od0 : (f : Ordinal → Ordinal ) → HOD |
| 43 func→od0 f = record { od = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkfunc p ) }} where | |
| 242 | 44 checkfunc : { p : Ordinal } → def ZFProduct p → Set n |
| 45 checkfunc (pair x y) = f x ≡ y | |
| 46 | |
| 233 | 47 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) |
| 225 | 48 |
| 331 | 49 Func : ( A B : HOD ) → HOD |
| 50 Func A B = record { od = record { def = λ x → odef (Power (A ⊗ B)) x } } | |
| 233 | 51 |
| 52 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
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53 |
| 331 | 54 func→od : (f : Ordinal → Ordinal ) → ( dom : HOD ) → HOD |
| 233 | 55 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) |
| 56 | |
| 331 | 57 record Func←cd { dom cod : HOD } {f : Ordinal } : Set n where |
| 236 | 58 field |
| 59 func-1 : Ordinal → Ordinal | |
| 242 | 60 func→od∈Func-1 : Func dom cod ∋ func→od func-1 dom |
| 236 | 61 |
| 331 | 62 od→func : { dom cod : HOD } → {f : Ordinal } → odef (Func dom cod ) f → Func←cd {dom} {cod} {f} |
| 63 od→func {dom} {cod} {f} lt = record { func-1 = λ x → sup-o {!!} ( λ y lt → lemma x {!!} ) ; func→od∈Func-1 = record { proj1 = {!!} ; proj2 = {!!} } } where | |
| 236 | 64 lemma : Ordinal → Ordinal → Ordinal |
| 331 | 65 lemma x y with IsZF.power→ isZF (dom ⊗ cod) (ord→od f) (subst (λ k → odef (Power (dom ⊗ cod)) k ) (sym diso) lt ) | ∋-p (ord→od f) (ord→od y) |
| 236 | 66 lemma x y | p | no n = o∅ |
| 378 | 67 lemma x y | p | yes f∋y = lemma2 {!!} where -- (ODC.double-neg-eilm O ( p {ord→od y} f∋y ))) where -- p : {y : OD} → f ∋ y → ¬ ¬ (dom ⊗ cod ∋ y) |
| 240 | 68 lemma2 : {p : Ordinal} → ord-pair p → Ordinal |
| 276 | 69 lemma2 (pair x1 y1) with ODC.decp O ( x1 ≡ x) |
| 240 | 70 lemma2 (pair x1 y1) | yes p = y1 |
| 71 lemma2 (pair x1 y1) | no ¬p = o∅ | |
| 331 | 72 fod : HOD |
| 73 fod = Replace dom ( λ x → < x , ord→od (sup-o {!!} ( λ y lt → lemma (od→ord x) {!!} )) > ) | |
| 240 | 74 |
| 75 | |
| 76 open Func←cd | |
| 236 | 77 |
| 227 | 78 -- contra position of sup-o< |
| 79 -- | |
| 80 | |
| 235 | 81 -- postulate |
| 82 -- -- contra-position of mimimulity of supermum required in Cardinal | |
| 83 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
| 84 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 227 | 85 |
| 219 | 86 ------------ |
| 87 -- | |
| 88 -- Onto map | |
| 89 -- def X x -> xmap | |
| 90 -- X ---------------------------> Y | |
| 91 -- ymap <- def Y y | |
| 92 -- | |
| 331 | 93 record Onto (X Y : HOD ) : Set n where |
| 219 | 94 field |
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95 xmap : Ordinal |
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96 ymap : Ordinal |
| 331 | 97 xfunc : odef (Func X Y) xmap |
| 98 yfunc : odef (Func Y X) ymap | |
| 99 onto-iso : {y : Ordinal } → (lty : odef Y y ) → | |
| 240 | 100 func-1 ( od→func {X} {Y} {xmap} xfunc ) ( func-1 (od→func yfunc) y ) ≡ y |
| 230 | 101 |
| 102 open Onto | |
| 103 | |
| 331 | 104 onto-restrict : {X Y Z : HOD} → Onto X Y → Z ⊆ Y → Onto X Z |
| 230 | 105 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { |
| 106 xmap = xmap1 | |
| 107 ; ymap = zmap | |
| 108 ; xfunc = xfunc1 | |
| 109 ; yfunc = zfunc | |
| 110 ; onto-iso = onto-iso1 | |
| 111 } where | |
| 112 xmap1 : Ordinal | |
| 113 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
| 114 zmap : Ordinal | |
| 115 zmap = {!!} | |
| 331 | 116 xfunc1 : odef (Func X Z) xmap1 |
| 230 | 117 xfunc1 = {!!} |
| 331 | 118 zfunc : odef (Func Z X) zmap |
| 230 | 119 zfunc = {!!} |
| 331 | 120 onto-iso1 : {z : Ordinal } → (ltz : odef Z z ) → func-1 (od→func xfunc1 ) (func-1 (od→func zfunc ) z ) ≡ z |
| 230 | 121 onto-iso1 = {!!} |
| 122 | |
| 51 | 123 |
| 331 | 124 record Cardinal (X : HOD ) : Set n where |
| 219 | 125 field |
| 224 | 126 cardinal : Ordinal |
| 230 | 127 conto : Onto X (Ord cardinal) |
| 128 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
| 151 | 129 |
| 331 | 130 cardinal : (X : HOD ) → Cardinal X |
| 224 | 131 cardinal X = record { |
| 331 | 132 cardinal = sup-o {!!} ( λ x lt → proj1 ( cardinal-p {!!}) ) |
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133 ; conto = onto |
| 219 | 134 ; cmax = cmax |
| 135 } where | |
| 230 | 136 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
| 276 | 137 cardinal-p x with ODC.p∨¬p O ( Onto X (Ord x) ) |
| 230 | 138 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } |
| 219 | 139 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
| 331 | 140 S = sup-o {!!} (λ x lt → proj1 (cardinal-p {!!})) |
| 141 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → (y o< (osuc S) → Onto X (Ord y))) → | |
| 142 (x o< (osuc S) → Onto X (Ord x) ) | |
| 229 | 143 lemma1 x prev with trio< x (osuc S) |
| 144 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
| 331 | 145 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = ( λ lt → {!!} ) |
| 146 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = ( λ lt → lemma2 ) where | |
| 230 | 147 lemma2 : Onto X (Ord x) |
| 148 lemma2 with prev {!!} {!!} | |
| 331 | 149 ... | t = {!!} |
| 150 lemma1 x prev | tri≈ ¬a b ¬c = ( λ lt → ⊥-elim ( o<¬≡ b lt )) | |
| 151 lemma1 x prev | tri> ¬a ¬b c = ( λ lt → ⊥-elim ( o<> c lt )) | |
| 230 | 152 onto : Onto X (Ord S) |
| 331 | 153 onto with TransFinite {λ x → ( x o< osuc S → Onto X (Ord x) ) } lemma1 S |
| 154 ... | t = t <-osuc | |
| 230 | 155 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) |
| 331 | 156 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} {!!} lemma refl ) where |
| 157 -- (sup-o< ? {λ x lt → proj1 ( cardinal-p {!!})}{{!!}} ) lemma refl ) where | |
| 219 | 158 lemma : proj1 (cardinal-p y) ≡ y |
| 276 | 159 lemma with ODC.p∨¬p O ( Onto X (Ord y) ) |
| 219 | 160 lemma | case1 x = refl |
| 161 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 162 |
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163 |
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164 ----- |
| 219 | 165 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 166 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 167 |
| 168 | |
| 169 |