Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 274:29a85a427ed2
ε-induction
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sat, 25 Apr 2020 15:09:07 +0900 |
| parents | 985a1af11bce |
| children | 6f10c47e4e7a |
| 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 |
| 274 | 8 import OPair |
| 23 | 9 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 224 | 10 open import Relation.Binary.PropositionalEquality |
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11 open import Data.Nat.Properties |
| 6 | 12 open import Data.Empty |
| 13 open import Relation.Nullary | |
| 14 open import Relation.Binary | |
| 15 open import Relation.Binary.Core | |
| 16 | |
| 224 | 17 open inOrdinal O |
| 18 open OD O | |
| 219 | 19 open OD.OD |
| 274 | 20 open OPair O |
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21 |
| 120 | 22 open _∧_ |
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23 open _∨_ |
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24 open Bool |
| 254 | 25 open _==_ |
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od→lv : {n : Level} → OD {n} → Nat
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26 |
| 230 | 27 -- we have to work on Ordinal to keep OD Level n |
| 28 -- since we use p∨¬p which works only on Level n | |
| 250 | 29 |
| 30 | |
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31 ∋-p : (A x : OD ) → Dec ( A ∋ x ) |
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32 ∋-p A x with p∨¬p ( A ∋ x ) |
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33 ∋-p A x | case1 t = yes t |
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34 ∋-p A x | case2 t = no t |
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35 |
| 233 | 36 _⊗_ : (A B : OD) → OD |
| 239 | 37 A ⊗ B = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } where |
| 38 checkAB : { p : Ordinal } → def ZFProduct p → Set n | |
| 39 checkAB (pair x y) = def A x ∧ def B y | |
| 233 | 40 |
| 242 | 41 func→od0 : (f : Ordinal → Ordinal ) → OD |
| 42 func→od0 f = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkfunc p ) } where | |
| 43 checkfunc : { p : Ordinal } → def ZFProduct p → Set n | |
| 44 checkfunc (pair x y) = f x ≡ y | |
| 45 | |
| 233 | 46 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) |
| 225 | 47 |
| 233 | 48 Func : ( A B : OD ) → OD |
| 49 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } | |
| 50 | |
| 51 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
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52 |
| 233 | 53 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD |
| 54 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
| 55 | |
| 242 | 56 record Func←cd { dom cod : OD } {f : Ordinal } : Set n where |
| 236 | 57 field |
| 58 func-1 : Ordinal → Ordinal | |
| 242 | 59 func→od∈Func-1 : Func dom cod ∋ func→od func-1 dom |
| 236 | 60 |
| 242 | 61 od→func : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → Func←cd {dom} {cod} {f} |
| 240 | 62 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 | 63 lemma : Ordinal → Ordinal → Ordinal |
| 64 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) | |
| 65 lemma x y | p | no n = o∅ | |
| 240 | 66 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) |
| 67 lemma2 : {p : Ordinal} → ord-pair p → Ordinal | |
| 68 lemma2 (pair x1 y1) with decp ( x1 ≡ x) | |
| 69 lemma2 (pair x1 y1) | yes p = y1 | |
| 70 lemma2 (pair x1 y1) | no ¬p = o∅ | |
| 242 | 71 fod : OD |
| 72 fod = Replace dom ( λ x → < x , ord→od (sup-o ( λ y → lemma (od→ord x) y )) > ) | |
| 240 | 73 |
| 74 | |
| 75 open Func←cd | |
| 236 | 76 |
| 227 | 77 -- contra position of sup-o< |
| 78 -- | |
| 79 | |
| 235 | 80 -- postulate |
| 81 -- -- contra-position of mimimulity of supermum required in Cardinal | |
| 82 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
| 83 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 227 | 84 |
| 219 | 85 ------------ |
| 86 -- | |
| 87 -- Onto map | |
| 88 -- def X x -> xmap | |
| 89 -- X ---------------------------> Y | |
| 90 -- ymap <- def Y y | |
| 91 -- | |
| 224 | 92 record Onto (X Y : OD ) : Set n where |
| 219 | 93 field |
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94 xmap : Ordinal |
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95 ymap : Ordinal |
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96 xfunc : def (Func X Y) xmap |
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97 yfunc : def (Func Y X) ymap |
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98 onto-iso : {y : Ordinal } → (lty : def Y y ) → |
| 240 | 99 func-1 ( od→func {X} {Y} {xmap} xfunc ) ( func-1 (od→func yfunc) y ) ≡ y |
| 230 | 100 |
| 101 open Onto | |
| 102 | |
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103 onto-restrict : {X Y Z : OD} → Onto X Y → Z ⊆ Y → Onto X Z |
| 230 | 104 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { |
| 105 xmap = xmap1 | |
| 106 ; ymap = zmap | |
| 107 ; xfunc = xfunc1 | |
| 108 ; yfunc = zfunc | |
| 109 ; onto-iso = onto-iso1 | |
| 110 } where | |
| 111 xmap1 : Ordinal | |
| 112 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
| 113 zmap : Ordinal | |
| 114 zmap = {!!} | |
| 115 xfunc1 : def (Func X Z) xmap1 | |
| 116 xfunc1 = {!!} | |
| 117 zfunc : def (Func Z X) zmap | |
| 118 zfunc = {!!} | |
| 240 | 119 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func-1 (od→func xfunc1 ) (func-1 (od→func zfunc ) z ) ≡ z |
| 230 | 120 onto-iso1 = {!!} |
| 121 | |
| 51 | 122 |
| 224 | 123 record Cardinal (X : OD ) : Set n where |
| 219 | 124 field |
| 224 | 125 cardinal : Ordinal |
| 230 | 126 conto : Onto X (Ord cardinal) |
| 127 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
| 151 | 128 |
| 224 | 129 cardinal : (X : OD ) → Cardinal X |
| 130 cardinal X = record { | |
| 219 | 131 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
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132 ; conto = onto |
| 219 | 133 ; cmax = cmax |
| 134 } where | |
| 230 | 135 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
| 136 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
| 137 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 219 | 138 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
| 229 | 139 S = sup-o (λ x → proj1 (cardinal-p x)) |
| 230 | 140 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
| 141 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
| 229 | 142 lemma1 x prev with trio< x (osuc S) |
| 143 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
| 230 | 144 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
| 145 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
| 146 lemma2 : Onto X (Ord x) | |
| 147 lemma2 with prev {!!} {!!} | |
| 148 ... | lift t = t {!!} | |
| 229 | 149 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
| 150 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
| 230 | 151 onto : Onto X (Ord S) |
| 152 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
| 153 ... | lift t = t <-osuc | |
| 154 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
| 229 | 155 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
| 224 | 156 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
| 219 | 157 lemma : proj1 (cardinal-p y) ≡ y |
| 230 | 158 lemma with p∨¬p ( Onto X (Ord y) ) |
| 219 | 159 lemma | case1 x = refl |
| 160 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 161 |
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162 |
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163 ----- |
| 219 | 164 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 165 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 166 |
| 167 | |
| 168 |