Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate cardinal.agda @ 254:2ea2a19f9cd6
ordered pair clean up
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 29 Aug 2019 16:16:51 +0900 |
| parents | 0446b6c5e7bc |
| children | 53b7acd63481 |
| rev | line source |
|---|---|
| 16 | 1 open import Level |
| 224 | 2 open import Ordinals |
| 3 module cardinal {n : Level } (O : Ordinals {n}) where | |
| 3 | 4 |
|
14
e11e95d5ddee
separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
11
diff
changeset
|
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 |
|
14
e11e95d5ddee
separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
11
diff
changeset
|
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 |
|
29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
28
diff
changeset
|
19 |
| 120 | 20 open _∧_ |
|
213
22d435172d1a
separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
210
diff
changeset
|
21 open _∨_ |
|
22d435172d1a
separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
210
diff
changeset
|
22 open Bool |
| 254 | 23 open _==_ |
|
44
fcac01485f32
od→lv : {n : Level} → OD {n} → Nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
43
diff
changeset
|
24 |
| 230 | 25 -- we have to work on Ordinal to keep OD Level n |
| 26 -- since we use p∨¬p which works only on Level n | |
| 254 | 27 -- < x , y > = (x , x) , (x , y) |
| 250 | 28 |
| 249 | 29 data ord-pair : (p : Ordinal) → Set n where |
| 30 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) ) | |
| 247 | 31 |
| 249 | 32 ZFProduct : OD |
| 33 ZFProduct = record { def = λ x → ord-pair x } | |
| 247 | 34 |
| 254 | 35 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 36 -- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y' | |
| 37 -- eq-pair refl refl = HE.refl | |
| 247 | 38 |
| 249 | 39 pi1 : { p : Ordinal } → ord-pair p → Ordinal |
| 40 pi1 ( pair x y) = x | |
| 41 | |
| 42 π1 : { p : OD } → ZFProduct ∋ p → Ordinal | |
| 43 π1 lt = pi1 lt | |
| 247 | 44 |
| 249 | 45 pi2 : { p : Ordinal } → ord-pair p → Ordinal |
| 46 pi2 ( pair x y ) = y | |
| 47 | |
| 48 π2 : { p : OD } → ZFProduct ∋ p → Ordinal | |
| 49 π2 lt = pi2 lt | |
| 247 | 50 |
| 249 | 51 p-cons : ( x y : OD ) → ZFProduct ∋ < x , y > |
| 52 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl ( | |
| 53 let open ≡-Reasoning in begin | |
| 54 od→ord < ord→od (od→ord x) , ord→od (od→ord y) > | |
| 55 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩ | |
| 56 od→ord < x , y > | |
|
252
8a58e2cd1f55
give up product uniquness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
251
diff
changeset
|
57 ∎ ) |
| 250 | 58 |
| 251 | 59 |
|
252
8a58e2cd1f55
give up product uniquness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
251
diff
changeset
|
60 p-iso1 : { ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy > |
|
8a58e2cd1f55
give up product uniquness
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
251
diff
changeset
|
61 p-iso1 {ox} {oy} = pair ox oy |
| 251 | 62 |
|
253
0446b6c5e7bc
proudct uniquness done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
252
diff
changeset
|
63 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < ord→od (π1 p) , ord→od (π2 p) > ≡ x |
| 254 | 64 p-iso {x} p = ord≡→≡ (lemma p) where |
|
253
0446b6c5e7bc
proudct uniquness done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
252
diff
changeset
|
65 lemma : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op |
|
0446b6c5e7bc
proudct uniquness done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
252
diff
changeset
|
66 lemma (pair ox oy) = refl |
| 250 | 67 |
| 68 | |
|
234
e06b76e5b682
ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
233
diff
changeset
|
69 ∋-p : (A x : OD ) → Dec ( A ∋ x ) |
|
e06b76e5b682
ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
233
diff
changeset
|
70 ∋-p A x with p∨¬p ( A ∋ x ) |
|
e06b76e5b682
ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
233
diff
changeset
|
71 ∋-p A x | case1 t = yes t |
|
e06b76e5b682
ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
233
diff
changeset
|
72 ∋-p A x | case2 t = no t |
|
e06b76e5b682
ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
233
diff
changeset
|
73 |
| 233 | 74 _⊗_ : (A B : OD) → OD |
| 239 | 75 A ⊗ B = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkAB p ) } where |
| 76 checkAB : { p : Ordinal } → def ZFProduct p → Set n | |
| 77 checkAB (pair x y) = def A x ∧ def B y | |
| 233 | 78 |
| 242 | 79 func→od0 : (f : Ordinal → Ordinal ) → OD |
| 80 func→od0 f = record { def = λ x → def ZFProduct x ∧ ( { x : Ordinal } → (p : def ZFProduct x ) → checkfunc p ) } where | |
| 81 checkfunc : { p : Ordinal } → def ZFProduct p → Set n | |
| 82 checkfunc (pair x y) = f x ≡ y | |
| 83 | |
| 233 | 84 -- Power (Power ( A ∪ B )) ∋ ( A ⊗ B ) |
| 225 | 85 |
| 233 | 86 Func : ( A B : OD ) → OD |
| 87 Func A B = record { def = λ x → def (Power (A ⊗ B)) x } | |
| 88 | |
| 89 -- power→ : ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x) | |
|
226
176ff97547b4
set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
225
diff
changeset
|
90 |
| 236 | 91 |
| 233 | 92 func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD |
| 93 func→od f dom = Replace dom ( λ x → < x , ord→od (f (od→ord x)) > ) | |
| 94 | |
| 242 | 95 record Func←cd { dom cod : OD } {f : Ordinal } : Set n where |
| 236 | 96 field |
| 97 func-1 : Ordinal → Ordinal | |
| 242 | 98 func→od∈Func-1 : Func dom cod ∋ func→od func-1 dom |
| 236 | 99 |
| 242 | 100 od→func : { dom cod : OD } → {f : Ordinal } → def (Func dom cod ) f → Func←cd {dom} {cod} {f} |
| 240 | 101 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 | 102 lemma : Ordinal → Ordinal → Ordinal |
| 103 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) | |
| 104 lemma x y | p | no n = o∅ | |
| 240 | 105 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) |
| 106 lemma2 : {p : Ordinal} → ord-pair p → Ordinal | |
| 107 lemma2 (pair x1 y1) with decp ( x1 ≡ x) | |
| 108 lemma2 (pair x1 y1) | yes p = y1 | |
| 109 lemma2 (pair x1 y1) | no ¬p = o∅ | |
| 242 | 110 fod : OD |
| 111 fod = Replace dom ( λ x → < x , ord→od (sup-o ( λ y → lemma (od→ord x) y )) > ) | |
| 240 | 112 |
| 113 | |
| 114 open Func←cd | |
| 236 | 115 |
| 227 | 116 -- contra position of sup-o< |
| 117 -- | |
| 118 | |
| 235 | 119 -- postulate |
| 120 -- -- contra-position of mimimulity of supermum required in Cardinal | |
| 121 -- sup-x : ( Ordinal → Ordinal ) → Ordinal | |
| 122 -- sup-lb : { ψ : Ordinal → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) | |
| 227 | 123 |
| 219 | 124 ------------ |
| 125 -- | |
| 126 -- Onto map | |
| 127 -- def X x -> xmap | |
| 128 -- X ---------------------------> Y | |
| 129 -- ymap <- def Y y | |
| 130 -- | |
| 224 | 131 record Onto (X Y : OD ) : Set n where |
| 219 | 132 field |
|
226
176ff97547b4
set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
225
diff
changeset
|
133 xmap : Ordinal |
|
176ff97547b4
set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
225
diff
changeset
|
134 ymap : Ordinal |
|
176ff97547b4
set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
225
diff
changeset
|
135 xfunc : def (Func X Y) xmap |
|
176ff97547b4
set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
225
diff
changeset
|
136 yfunc : def (Func Y X) ymap |
|
234
e06b76e5b682
ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
233
diff
changeset
|
137 onto-iso : {y : Ordinal } → (lty : def Y y ) → |
| 240 | 138 func-1 ( od→func {X} {Y} {xmap} xfunc ) ( func-1 (od→func yfunc) y ) ≡ y |
| 230 | 139 |
| 140 open Onto | |
| 141 | |
| 142 onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z | |
| 143 onto-restrict {X} {Y} {Z} onto Z⊆Y = record { | |
| 144 xmap = xmap1 | |
| 145 ; ymap = zmap | |
| 146 ; xfunc = xfunc1 | |
| 147 ; yfunc = zfunc | |
| 148 ; onto-iso = onto-iso1 | |
| 149 } where | |
| 150 xmap1 : Ordinal | |
| 151 xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) | |
| 152 zmap : Ordinal | |
| 153 zmap = {!!} | |
| 154 xfunc1 : def (Func X Z) xmap1 | |
| 155 xfunc1 = {!!} | |
| 156 zfunc : def (Func Z X) zmap | |
| 157 zfunc = {!!} | |
| 240 | 158 onto-iso1 : {z : Ordinal } → (ltz : def Z z ) → func-1 (od→func xfunc1 ) (func-1 (od→func zfunc ) z ) ≡ z |
| 230 | 159 onto-iso1 = {!!} |
| 160 | |
| 51 | 161 |
| 224 | 162 record Cardinal (X : OD ) : Set n where |
| 219 | 163 field |
| 224 | 164 cardinal : Ordinal |
| 230 | 165 conto : Onto X (Ord cardinal) |
| 166 cmax : ( y : Ordinal ) → cardinal o< y → ¬ Onto X (Ord y) | |
| 151 | 167 |
| 224 | 168 cardinal : (X : OD ) → Cardinal X |
| 169 cardinal X = record { | |
| 219 | 170 cardinal = sup-o ( λ x → proj1 ( cardinal-p x) ) |
|
226
176ff97547b4
set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
225
diff
changeset
|
171 ; conto = onto |
| 219 | 172 ; cmax = cmax |
| 173 } where | |
| 230 | 174 cardinal-p : (x : Ordinal ) → ( Ordinal ∧ Dec (Onto X (Ord x) ) ) |
| 175 cardinal-p x with p∨¬p ( Onto X (Ord x) ) | |
| 176 cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True } | |
| 219 | 177 cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False } |
| 229 | 178 S = sup-o (λ x → proj1 (cardinal-p x)) |
| 230 | 179 lemma1 : (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) → |
| 180 Lift (suc n) (x o< (osuc S) → Onto X (Ord x) ) | |
| 229 | 181 lemma1 x prev with trio< x (osuc S) |
| 182 lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a | |
| 230 | 183 lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} ) |
| 184 lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where | |
| 185 lemma2 : Onto X (Ord x) | |
| 186 lemma2 with prev {!!} {!!} | |
| 187 ... | lift t = t {!!} | |
| 229 | 188 lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt )) |
| 189 lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt )) | |
| 230 | 190 onto : Onto X (Ord S) |
| 191 onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S | |
| 192 ... | lift t = t <-osuc | |
| 193 cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) | |
| 229 | 194 cmax y lt ontoy = o<> lt (o<-subst {_} {_} {y} {S} |
| 224 | 195 (sup-o< {λ x → proj1 ( cardinal-p x)}{y} ) lemma refl ) where |
| 219 | 196 lemma : proj1 (cardinal-p y) ≡ y |
| 230 | 197 lemma with p∨¬p ( Onto X (Ord y) ) |
| 219 | 198 lemma | case1 x = refl |
| 199 lemma | case2 not = ⊥-elim ( not ontoy ) | |
| 217 | 200 |
|
226
176ff97547b4
set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
225
diff
changeset
|
201 |
|
176ff97547b4
set theortic function definition using sup
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
225
diff
changeset
|
202 ----- |
| 219 | 203 -- All cardinal is ℵ0, since we are working on Countable Ordinal, |
| 204 -- Power ω is larger than ℵ0, so it has no cardinal. | |
| 218 | 205 |
| 206 | |
| 207 |