Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 515:5faeae7cfe22
ε-induction does not work on Zorn
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 16 Apr 2022 12:10:09 +0900 |
| parents | 97c8abf28706 |
| children | 286016848403 |
| rev | line source |
|---|---|
| 478 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 508 | 2 open import Level hiding ( suc ; zero ) |
| 431 | 3 open import Ordinals |
| 497 | 4 import OD |
| 5 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) where | |
| 431 | 6 |
| 7 open import zf | |
| 477 | 8 open import logic |
| 9 -- open import partfunc {n} O | |
| 10 | |
| 11 open import Relation.Nullary | |
| 12 open import Relation.Binary | |
| 13 open import Data.Empty | |
| 431 | 14 open import Relation.Binary |
| 15 open import Relation.Binary.Core | |
| 477 | 16 open import Relation.Binary.PropositionalEquality |
| 17 import BAlgbra | |
| 431 | 18 |
| 19 | |
| 20 open inOrdinal O | |
| 21 open OD O | |
| 22 open OD.OD | |
| 23 open ODAxiom odAxiom | |
| 477 | 24 import OrdUtil |
| 25 import ODUtil | |
| 431 | 26 open Ordinals.Ordinals O |
| 27 open Ordinals.IsOrdinals isOrdinal | |
| 28 open Ordinals.IsNext isNext | |
| 29 open OrdUtil O | |
| 477 | 30 open ODUtil O |
| 31 | |
| 32 | |
| 33 import ODC | |
| 34 | |
| 35 | |
| 36 open _∧_ | |
| 37 open _∨_ | |
| 38 open Bool | |
| 431 | 39 |
| 40 | |
| 41 open HOD | |
| 42 | |
| 508 | 43 record Element (A : HOD) : Set (Level.suc n) where |
| 469 | 44 field |
| 45 elm : HOD | |
| 46 is-elm : A ∋ elm | |
| 47 | |
| 48 open Element | |
| 49 | |
| 509 | 50 _<A_ : {A : HOD} → (x y : Element A ) → Set n |
| 51 x <A y = elm x < elm y | |
| 52 _≡A_ : {A : HOD} → (x y : Element A ) → Set (Level.suc n) | |
| 53 x ≡A y = elm x ≡ elm y | |
| 54 | |
| 508 | 55 IsPartialOrderSet : ( A : HOD ) → Set (Level.suc n) |
| 509 | 56 IsPartialOrderSet A = IsStrictPartialOrder (_≡A_ {A}) _<A_ |
| 490 | 57 |
| 492 | 58 open _==_ |
| 59 open _⊆_ | |
| 60 | |
| 495 | 61 isA : { A B : HOD } → B ⊆ A → (x : Element B) → Element A |
| 62 isA B⊆A x = record { elm = elm x ; is-elm = incl B⊆A (is-elm x) } | |
| 494 | 63 |
| 497 | 64 ⊆-IsPartialOrderSet : { A B : HOD } → B ⊆ A → IsPartialOrderSet A → IsPartialOrderSet B |
| 65 ⊆-IsPartialOrderSet {A} {B} B⊆A PA = record { | |
| 66 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; trans = λ {x} {y} {z} → trans1 {x} {y} {z} | |
| 498 | 67 ; irrefl = λ {x} {y} → irrefl1 {x} {y} ; <-resp-≈ = resp0 |
| 493 | 68 } where |
| 495 | 69 _<B_ : (x y : Element B ) → Set n |
| 70 x <B y = elm x < elm y | |
| 71 trans1 : {x y z : Element B} → x <B y → y <B z → x <B z | |
| 72 trans1 {x} {y} {z} x<y y<z = IsStrictPartialOrder.trans PA {isA B⊆A x} {isA B⊆A y} {isA B⊆A z} x<y y<z | |
| 73 irrefl1 : {x y : Element B} → elm x ≡ elm y → ¬ ( x <B y ) | |
| 74 irrefl1 {x} {y} x=y x<y = IsStrictPartialOrder.irrefl PA {isA B⊆A x} {isA B⊆A y} x=y x<y | |
| 75 open import Data.Product | |
| 76 resp0 : ({x y z : Element B} → elm y ≡ elm z → x <B y → x <B z) × ({x y z : Element B} → elm y ≡ elm z → y <B x → z <B x) | |
| 77 resp0 = Data.Product._,_ (λ {x} {y} {z} → proj₁ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z }) | |
| 78 (λ {x} {y} {z} → proj₂ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z }) | |
| 492 | 79 |
| 497 | 80 -- open import Relation.Binary.Properties.Poset as Poset |
| 496 | 81 |
| 508 | 82 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) |
| 509 | 83 IsTotalOrderSet A = IsStrictTotalOrder (_≡A_ {A}) _<A_ |
| 490 | 84 |
| 469 | 85 me : { A a : HOD } → A ∋ a → Element A |
| 86 me {A} {a} lt = record { elm = a ; is-elm = lt } | |
| 87 | |
| 504 | 88 A∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (A ∋ x) ≡ (A ∋ y ) |
| 89 A∋x-irr A {x} {y} refl = refl | |
| 90 | |
| 506 | 91 me-elm-refl : (A : HOD) → (x : Element A) → elm (me {A} (d→∋ A (is-elm x))) ≡ elm x |
| 92 me-elm-refl A record { elm = ex ; is-elm = ax } = *iso | |
| 504 | 93 |
| 94 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) | |
| 95 | |
| 96 postulate | |
| 97 ∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (ax : A ∋ x) (ay : A ∋ y ) → ax ≅ ay | |
| 98 odef-irr : (A : OD) {x y : Ordinal} → x ≡ y → (ax : def A x) (ay : def A y ) → ax ≅ ay | |
| 99 | |
| 100 is-elm-irr : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y | |
| 101 is-elm-irr A {x} {y} eq = ∋x-irr A eq (is-elm x) (is-elm y ) | |
| 102 | |
| 103 El-irr2 : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y → x ≡ y | |
| 104 El-irr2 A {x} {y} refl HE.refl = refl | |
| 105 | |
| 106 El-irr : {A : HOD} → {x y : Element A } → elm x ≡ elm y → x ≡ y | |
| 107 El-irr {A} {x} {y} eq = El-irr2 A eq ( is-elm-irr A eq ) | |
| 108 | |
| 508 | 109 record ZChain ( A : HOD ) (y : Ordinal) : Set (Level.suc n) where |
| 464 | 110 field |
| 497 | 111 max : HOD |
| 112 A∋max : A ∋ max | |
| 113 y<max : y o< & max | |
| 114 chain : HOD | |
| 115 chain⊆A : chain ⊆ A | |
| 116 total : IsTotalOrderSet chain | |
| 117 chain-max : (x : HOD) → chain ∋ x → (x ≡ max ) ∨ ( x < max ) | |
| 498 | 118 |
| 119 data IChain (A : HOD) : Ordinal → Set n where | |
| 120 ifirst : {ox : Ordinal} → odef A ox → IChain A ox | |
| 121 inext : {ox oy : Ordinal} → odef A oy → * ox < * oy → IChain A ox → IChain A oy | |
| 122 | |
| 506 | 123 -- * ox < .. < * oy |
| 124 ic-connect : {A : HOD} {oy : Ordinal} → (ox : Ordinal) → (iy : IChain A oy) → Set n | |
| 125 ic-connect {A} ox (ifirst {oy} ay) = Lift n ⊥ | |
| 126 ic-connect {A} ox (inext {oy} {oz} ay y<z iz) = (ox ≡ oy) ∨ ic-connect ox iz | |
| 127 | |
| 128 ic→odef : {A : HOD} {ox : Ordinal} → IChain A ox → odef A ox | |
| 129 ic→odef {A} {ox} (ifirst ax) = ax | |
| 130 ic→odef {A} {ox} (inext ax x<y ic) = ax | |
| 131 | |
| 132 ic→< : {A : HOD} → (IsPartialOrderSet A) → (x : Ordinal) → odef A x → {y : Ordinal} → (iy : IChain A y) → ic-connect {A} {y} x iy → * x < * y | |
| 133 ic→< {A} PO x ax {y} (ifirst ay) () | |
| 134 ic→< {A} PO x ax {y} (inext ay x<y iy) (case1 refl) = x<y | |
| 135 ic→< {A} PO x ax {y} (inext {oy} ay x<y iy) (case2 ic) = IsStrictPartialOrder.trans PO | |
| 136 {me (subst (λ k → odef A k) (sym &iso) ax )} {me (subst (λ k → odef A k) (sym &iso) (ic→odef {A} {oy} iy) ) } {me (subst (λ k → odef A k) (sym &iso) ay) } | |
| 137 (ic→< {A} PO x ax iy ic ) x<y | |
| 138 | |
| 139 record IChained (A : HOD) (x y : Ordinal) : Set n where | |
| 140 field | |
| 141 iy : IChain A y | |
| 142 ic : ic-connect x iy | |
| 498 | 143 |
| 144 IChainSet : {A : HOD} → Element A → HOD | |
| 506 | 145 IChainSet {A} ax = record { od = record { def = λ y → odef A y ∧ IChained A (& (elm ax)) y } |
| 498 | 146 ; odmax = & A ; <odmax = λ {y} lt → subst (λ k → k o< & A) &iso (c<→o< (subst (λ k → odef A k) (sym &iso) (proj1 lt))) } |
| 147 | |
| 512 | 148 IChainSet⊆A : {A : HOD} → (x : Element A ) → IChainSet x ⊆ A |
| 149 IChainSet⊆A {A} x = record { incl = λ {oy} y → proj1 y } | |
| 150 | |
| 498 | 151 -- there is a y, & y > & x |
| 152 | |
| 501 | 153 record OSup> (A : HOD) {x : Ordinal} (ax : A ∋ * x) : Set n where |
| 498 | 154 field |
| 501 | 155 y : Ordinal |
| 156 icy : odef (IChainSet {A} (me ax)) y | |
| 157 y>x : x o< y | |
| 498 | 158 |
| 505 | 159 record IChainSup> (A : HOD) {x : Ordinal} (ax : A ∋ * x) : Set n where |
| 160 field | |
| 161 y : Ordinal | |
| 162 A∋y : odef A y | |
| 163 y>x : * x < * y | |
| 164 | |
| 498 | 165 -- finite IChain |
| 166 | |
| 511 | 167 ic→A∋y : (A : HOD) {x y : Ordinal} (ax : A ∋ * x) → odef (IChainSet {A} (me ax)) y → A ∋ * y |
| 168 ic→A∋y A {x} {y} ax ⟪ ay , _ ⟫ = subst (λ k → odef A k) (sym &iso) ay | |
| 169 | |
| 501 | 170 record InFiniteIChain (A : HOD) {x : Ordinal} (ax : A ∋ * x) : Set n where |
| 498 | 171 field |
| 501 | 172 chain<x : (y : Ordinal ) → odef (IChainSet {A} (me ax)) y → y o< x |
| 173 c-infinite : (y : Ordinal ) → (cy : odef (IChainSet {A} (me ax)) y ) | |
| 511 | 174 → IChainSup> A (ic→A∋y A ax cy) |
| 509 | 175 |
| 510 | 176 open import Data.Nat hiding (_<_) |
| 177 import Data.Nat.Properties as NP | |
| 178 open import nat | |
| 179 | |
| 514 | 180 data Chain (A : HOD) (s : Ordinal) (next : Ordinal → Ordinal ) : ( x : Ordinal ) → Set n where |
| 513 | 181 cfirst : odef A s → Chain A s next s |
| 514 | 182 csuc : (x : Ordinal ) → (ax : odef A x ) → Chain A s next x → odef A (next x) → Chain A s next (next x ) |
| 512 | 183 |
| 514 | 184 ct∈A : (A : HOD ) (s : Ordinal) → (next : Ordinal → Ordinal ) → {x : Ordinal} → Chain A s next x → odef A x |
| 513 | 185 ct∈A A s next {x} (cfirst x₁) = x₁ |
| 514 | 186 ct∈A A s next {.(next x )} (csuc x ax t anx) = anx |
| 509 | 187 |
| 514 | 188 ChainClosure : (A : HOD) (s : Ordinal) → (next : Ordinal → Ordinal ) → HOD |
| 513 | 189 ChainClosure A s next = record { od = record { def = λ x → Chain A s next x } ; odmax = & A ; <odmax = cc01 } where |
| 190 cc01 : {y : Ordinal} → Chain A s next y → y o< & A | |
| 191 cc01 {y} cy = subst (λ k → k o< & A ) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso) ( ct∈A A s next cy ) ) ) | |
| 509 | 192 |
| 514 | 193 cton0 : (A : HOD ) (s : Ordinal) → (next : Ordinal → Ordinal ) {y : Ordinal } → Chain A s next y → ℕ |
| 513 | 194 cton0 A s next (cfirst _) = zero |
| 195 cton0 A s next (csuc x ax z _) = suc (cton0 A s next z) | |
| 514 | 196 cton : (A : HOD ) (s : Ordinal) → (next : Ordinal → Ordinal ) → Element (ChainClosure A s next) → ℕ |
| 513 | 197 cton A s next y = cton0 A s next (is-elm y) |
| 510 | 198 |
| 514 | 199 cinext : (A : HOD) {x : Ordinal } → (ax : A ∋ * x ) → (ifc : InFiniteIChain A ax ) → Ordinal → Ordinal |
| 200 cinext A ax ifc y with ODC.∋-p O (IChainSet (me ax)) (* y) | |
| 201 ... | yes ics-y = IChainSup>.y ( InFiniteIChain.c-infinite ifc y (subst (λ k → odef (IChainSet (me ax)) k) &iso ics-y )) | |
| 202 ... | no _ = o∅ | |
| 203 | |
| 509 | 204 InFCSet : (A : HOD) → {x : Ordinal} (ax : A ∋ * x) |
| 205 → (ifc : InFiniteIChain A ax ) → HOD | |
| 514 | 206 InFCSet A {x} ax ifc = ChainClosure (IChainSet {A} (me ax)) x {!!} -- (λ y → IChainSup>.y ( InFiniteIChain.c-infinite ifc y ay ) ) |
| 509 | 207 |
| 512 | 208 InFCSet⊆A : (A : HOD) → {x : Ordinal} (ax : A ∋ * x) → (ifc : InFiniteIChain A ax ) → InFCSet A ax ifc ⊆ A |
| 209 InFCSet⊆A A {x} ax ifc = record { incl = λ {y} lt → incl (IChainSet⊆A (me ax)) ( | |
| 514 | 210 ct∈A (IChainSet {A} (me ax)) x {!!} lt ) } |
| 211 -- ct∈A (IChainSet {A} (me ax)) x (λ y ay → IChainSup>.y ( InFiniteIChain.c-infinite ifc y ay ) ) lt ) } | |
| 512 | 212 |
| 509 | 213 ChainClosure-is-total : (A : HOD) → {x : Ordinal} (ax : A ∋ * x) |
| 214 → IsPartialOrderSet A | |
| 215 → (ifc : InFiniteIChain A ax ) | |
| 216 → IsTotalOrderSet ( InFCSet A ax ifc ) | |
| 217 ChainClosure-is-total A {x} ax PO ifc = record { isEquivalence = IsStrictPartialOrder.isEquivalence IPO | |
| 218 ; trans = λ {x} {y} {z} → IsStrictPartialOrder.trans IPO {x} {y} {z} ; compare = cmp } where | |
| 219 IPO : IsPartialOrderSet (InFCSet A ax ifc ) | |
| 512 | 220 IPO = ⊆-IsPartialOrderSet record { incl = λ {y} lt → incl (InFCSet⊆A A {x} ax ifc) lt} PO |
| 511 | 221 B = IChainSet {A} (me ax) |
| 514 | 222 cnext = cinext A ax ifc |
| 513 | 223 ct02 : {oy : Ordinal} → (y : Chain B x cnext oy ) → A ∋ * oy |
| 224 ct02 y = incl (IChainSet⊆A {A} (me ax)) (subst (λ k → odef (IChainSet (me ax)) k) (sym &iso) (ct∈A B x cnext y) ) | |
| 225 ct-inject : {ox oy : Ordinal} → (x1 : Chain B x cnext ox ) → (y : Chain B x cnext oy ) | |
| 226 → (cton0 B x cnext x1) ≡ (cton0 B x cnext y) → ox ≡ oy | |
| 227 ct-inject {ox} {ox} (cfirst x) (cfirst x₁) refl = refl | |
| 514 | 228 ct-inject {.(cnext x₀ )} {.(cnext x₃ )} (csuc x₀ ax x₁ x₂) (csuc x₃ ax₁ y x₄) eq = cong cnext ct05 where |
| 513 | 229 ct06 : {x y : ℕ} → suc x ≡ suc y → x ≡ y |
| 230 ct06 refl = refl | |
| 231 ct05 : x₀ ≡ x₃ | |
| 232 ct05 = ct-inject x₁ y (ct06 eq) | |
| 233 ct-monotonic : {ox oy : Ordinal} → (x1 : Chain B x cnext ox ) → (y : Chain B x cnext oy ) | |
| 234 → (cton0 B x cnext x1) Data.Nat.< (cton0 B x cnext y) → * ox < * oy | |
| 235 ct-monotonic {ox} {oy} x1 (csuc oy1 ay y _) (s≤s lt) with NP.<-cmp ( cton0 B x cnext x1 ) ( cton0 B x cnext y ) | |
| 514 | 236 ... | tri< a ¬b ¬c = ct07 where |
| 237 ct07 : * ox < * (cnext oy1) | |
| 238 ct07 with ODC.∋-p O (IChainSet {A} (me ax)) (* oy1) | |
| 239 ... | no not = ⊥-elim ( not (subst (λ k → odef (IChainSet {A} (me ax)) k ) (sym &iso) ay ) ) | |
| 240 ... | yes ay1 = IsStrictPartialOrder.trans PO {me (ct02 x1) } {me (ct02 y)} {me ct031 } (ct-monotonic x1 y a ) ct011 where | |
| 241 ct031 : A ∋ * (IChainSup>.y (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 ) )) | |
| 242 ct031 = subst (λ k → odef A k ) (sym &iso) ( | |
| 243 IChainSup>.A∋y (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 )) ) | |
| 244 ct011 : * oy1 < * ( IChainSup>.y (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 )) ) | |
| 245 ct011 = IChainSup>.y>x (InFiniteIChain.c-infinite ifc oy1 (subst (λ k → odef (IChainSet {A} (me ax)) k) &iso ay1 )) | |
| 511 | 246 ... | tri≈ ¬a b ¬c = {!!} |
| 247 ... | tri> ¬a ¬b c = {!!} | |
| 509 | 248 cmp : Trichotomous _ _ |
| 513 | 249 cmp x1 y with NP.<-cmp (cton B x cnext x1) (cton B x cnext y) |
| 250 ... | tri< a ¬b ¬c = tri< ct04 {!!} {!!} where | |
| 251 ct04 : elm x1 < elm y | |
| 252 ct04 = subst₂ (λ j k → j < k ) *iso *iso (ct-monotonic (is-elm x1) (is-elm y) a) | |
| 510 | 253 ... | tri≈ ¬a b ¬c = {!!} |
| 254 ... | tri> ¬a ¬b c = {!!} | |
| 509 | 255 |
| 501 | 256 |
| 502 | 257 record IsFC (A : HOD) {x : Ordinal} (ax : A ∋ * x) (y : Ordinal) : Set n where |
| 501 | 258 field |
| 259 icy : odef (IChainSet {A} (me ax)) y | |
| 505 | 260 c-finite : ¬ IChainSup> A ax |
| 497 | 261 |
| 508 | 262 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 503 | 263 field |
| 264 maximal : HOD | |
| 265 A∋maximal : A ∋ maximal | |
| 266 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 267 | |
| 268 -- | |
| 269 -- possible three cases in a limit ordinal step | |
| 270 -- | |
| 507 | 271 -- case 1) < goes > x (will contradic in the transfinite induction ) |
| 503 | 272 -- case 2) no > x in some chain ( maximal ) |
| 507 | 273 -- case 3) countably infinite chain below x (will be prohibited by sup condtion ) |
| 503 | 274 -- |
| 275 Zorn-lemma-3case : { A : HOD } | |
| 498 | 276 → o∅ o< & A |
| 277 → IsPartialOrderSet A | |
| 501 | 278 → (x : Element A) → OSup> A (d→∋ A (is-elm x)) ∨ Maximal A ∨ InFiniteIChain A (d→∋ A (is-elm x)) |
| 503 | 279 Zorn-lemma-3case {A} 0<A PO x = zc2 where |
| 499 | 280 Gtx : HOD |
| 281 Gtx = record { od = record { def = λ y → odef ( IChainSet x ) y ∧ ( & (elm x) o< y ) } ; odmax = & A | |
| 501 | 282 ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (d→∋ A (proj1 (proj1 lt)))) } |
| 283 HG : HOD | |
| 502 | 284 HG = record { od = record { def = λ y → odef A y ∧ IsFC A (d→∋ A (is-elm x) ) y } ; odmax = & A |
| 501 | 285 ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (d→∋ A (proj1 lt) )) } |
| 286 zc2 : OSup> A (d→∋ A (is-elm x)) ∨ Maximal A ∨ InFiniteIChain A (d→∋ A (is-elm x)) | |
| 499 | 287 zc2 with is-o∅ (& Gtx) |
| 504 | 288 ... | no not = case1 record { y = & y ; icy = zc4 ; y>x = proj2 zc3 } where |
| 289 y : HOD | |
| 290 y = ODC.minimal O Gtx (λ eq → not (=od∅→≡o∅ eq)) | |
| 291 zc3 : odef ( IChainSet x ) (& y) ∧ ( & (elm x) o< (& y )) | |
| 292 zc3 = ODC.x∋minimal O Gtx (λ eq → not (=od∅→≡o∅ eq)) | |
| 293 zc4 : odef (IChainSet (me (subst (OD.def (od A)) (sym &iso) (is-elm x)))) (& y) | |
| 506 | 294 zc4 = ⟪ proj1 (proj1 zc3) , subst (λ k → IChained A (& k) (& y) ) (sym *iso) (proj2 (proj1 zc3)) ⟫ |
| 501 | 295 ... | yes nogt with is-o∅ (& HG) |
| 505 | 296 ... | no finite-chain = case2 (case1 record { maximal = y ; A∋maximal = proj1 zc3 ; ¬maximal<x = zc4 } ) where |
| 504 | 297 y : HOD |
| 505 | 298 y = ODC.minimal O HG (λ eq → finite-chain (=od∅→≡o∅ eq)) |
| 504 | 299 zc3 : odef A (& y) ∧ IsFC A (d→∋ A (is-elm x) ) (& y) |
| 505 | 300 zc3 = ODC.x∋minimal O HG (λ eq → finite-chain (=od∅→≡o∅ eq)) |
| 504 | 301 zc5 : odef (IChainSet {A} (me (d→∋ A (is-elm x) ))) (& y) |
| 302 zc5 = IsFC.icy (proj2 zc3) | |
| 303 zc4 : {z : HOD} → A ∋ z → ¬ (y < z) | |
| 506 | 304 zc4 {z} az y<z = IsFC.c-finite (proj2 zc3) record { y = & z ; A∋y = az ; y>x = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) zc6 } where |
| 305 zc8 : ic-connect (& (* (& (elm x)))) (IChained.iy (proj2 zc5)) | |
| 306 zc8 = IChained.ic (proj2 zc5) | |
| 307 zc7 : elm x < y | |
| 308 zc7 = subst₂ (λ j k → j < k ) *iso *iso ( ic→< {A} PO (& (elm x)) (is-elm x) (IChained.iy (proj2 zc5)) | |
| 309 (subst (λ k → ic-connect (& k) (IChained.iy (proj2 zc5)) ) (me-elm-refl A x) (IChained.ic (proj2 zc5)) ) ) | |
| 505 | 310 zc6 : elm x < z |
| 506 | 311 zc6 = IsStrictPartialOrder.trans PO {x} {me (proj1 zc3)} {me az} zc7 y<z |
| 507 | 312 ... | yes inifite = case2 (case2 record { chain<x = {!!} ; c-infinite = {!!} } ) |
| 499 | 313 |
| 498 | 314 |
| 508 | 315 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 503 | 316 field |
| 317 sup : HOD | |
| 318 A∋maximal : A ∋ sup | |
| 319 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 320 | |
| 497 | 321 Zorn-lemma : { A : HOD } |
| 464 | 322 → o∅ o< & A |
| 497 | 323 → IsPartialOrderSet A |
| 324 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition | |
| 325 → Maximal A | |
| 507 | 326 Zorn-lemma {A} 0<A PO supP = zorn04 where |
| 493 | 327 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 496 | 328 z01 {a} {b} A∋a A∋b (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO {me A∋b} {me A∋a} (sym a=b) b<a |
| 329 z01 {a} {b} A∋a A∋b (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO {me A∋b} {me A∋b} refl (IsStrictPartialOrder.trans PO {me A∋b} {me A∋a} {me A∋b} b<a a<b) | |
| 507 | 330 z02 : {x : Ordinal } → (ax : A ∋ * x ) → InFiniteIChain A ax → ⊥ |
| 331 z02 {x} ax ic = zc5 ic where | |
| 332 FC : HOD | |
| 333 FC = IChainSet {A} (me ax) | |
| 334 zc6 : InFiniteIChain A ax → ¬ SUP A FC | |
| 335 zc6 inf = {!!} | |
| 336 FC-is-total : IsTotalOrderSet FC | |
| 337 FC-is-total = {!!} | |
| 338 FC⊆A : FC ⊆ A | |
| 339 FC⊆A = record { incl = λ {x} lt → proj1 lt } | |
| 340 zc5 : InFiniteIChain A ax → ⊥ | |
| 341 zc5 x = zc6 x ( supP FC FC⊆A FC-is-total ) | |
| 478 | 342 -- ZChain is not compatible with the SUP condition |
| 497 | 343 ind : (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y ∨ Maximal A ) |
| 344 → ZChain A x ∨ Maximal A | |
| 345 ind x prev with Oprev-p x | |
| 477 | 346 ... | yes op with ODC.∋-p O A (* x) |
| 498 | 347 ... | no ¬Ax = zc1 where |
| 476 | 348 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain |
| 471 | 349 px = Oprev.oprev op |
| 498 | 350 zc1 : ZChain A x ∨ Maximal A |
| 497 | 351 zc1 with prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) |
| 498 | 352 ... | case2 x = case2 x -- we have the Maximal |
| 353 ... | case1 z with trio< x (& (ZChain.max z)) | |
| 354 ... | tri< a ¬b ¬c = case1 record { max = ZChain.max z ; y<max = a } | |
| 355 ... | tri≈ ¬a b ¬c = {!!} -- x = max so ¬ A ∋ max | |
| 356 ... | tri> ¬a ¬b c = {!!} -- can't happen | |
| 503 | 357 ... | yes ax = zc4 where -- we have previous ordinal and A ∋ x |
| 472 | 358 px = Oprev.oprev op |
| 503 | 359 zc1 : OSup> A (subst (OD.def (od A)) (sym &iso) ax) → ZChain A x ∨ Maximal A |
| 360 zc1 os with prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) | |
| 498 | 361 ... | case2 x = case2 x |
| 507 | 362 ... | case1 x = {!!} |
| 503 | 363 zc4 : ZChain A x ∨ Maximal A |
| 364 zc4 with Zorn-lemma-3case 0<A PO (me ax) | |
| 365 ... | case1 y>x = zc1 y>x | |
| 366 ... | case2 (case1 x) = case2 x | |
| 367 ... | case2 (case2 x) = ⊥-elim (zc5 x) where | |
| 368 FC : HOD | |
| 369 FC = IChainSet {A} (me ax) | |
| 511 | 370 B : InFiniteIChain A ax → HOD |
| 371 B ifc = InFCSet A ax ifc | |
| 372 zc6 : (ifc : InFiniteIChain A ax ) → ¬ SUP A (B ifc) | |
| 503 | 373 zc6 = {!!} |
| 511 | 374 FC-is-total : (ifc : InFiniteIChain A ax) → IsTotalOrderSet (B ifc) |
| 375 FC-is-total ifc = ChainClosure-is-total A ax PO ifc | |
| 376 B⊆A : (ifc : InFiniteIChain A ax) → B ifc ⊆ A | |
| 377 B⊆A = {!!} | |
| 378 ifc : InFiniteIChain A (subst (OD.def (od A)) (sym &iso) ax) → InFiniteIChain A ax | |
| 512 | 379 ifc record { chain<x = chain<x ; c-infinite = c-infinite } = record { chain<x = {!!} ; c-infinite = {!!} } where |
| 380 ifc01 : {!!} -- me (subst (OD.def (od A)) (sym &iso) ax) | |
| 381 ifc01 = {!!} | |
| 382 -- (y : Ordinal) → odef (IChainSet (me (subst (OD.def (od A)) (sym &iso) ax))) y → y o< & (* x₁) | |
| 383 -- (y : Ordinal) → odef (IChainSet (me ax)) y → y o< x₁ | |
| 503 | 384 zc5 : InFiniteIChain A (subst (OD.def (od A)) (sym &iso) ax) → ⊥ |
| 511 | 385 zc5 x = zc6 (ifc x) ( supP (B (ifc x)) (B⊆A (ifc x)) (FC-is-total (ifc x) )) |
| 497 | 386 ind x prev | no ¬ox with trio< (& A) x --- limit ordinal case |
| 483 | 387 ... | tri< a ¬b ¬c = {!!} where |
| 497 | 388 zc1 : ZChain A (& A) |
| 389 zc1 with prev (& A) a | |
| 390 ... | t = {!!} | |
| 483 | 391 ... | tri≈ ¬a b ¬c = {!!} where |
| 478 | 392 ... | tri> ¬a ¬b c with ODC.∋-p O A (* x) |
| 393 ... | no ¬Ax = {!!} where | |
| 507 | 394 ... | yes ax = {!!} |
| 497 | 395 zorn03 : (x : Ordinal) → ZChain A x ∨ Maximal A |
| 507 | 396 zorn03 x = TransFinite ind x |
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397 zorn07 : (x : HOD) → (ax : A ∋ x ) → OSup> A (d→∋ A ax) ∨ Maximal A ∨ InFiniteIChain A (d→∋ A ax) |
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398 zorn07 x = ε-induction (λ {x} prev ax → Zorn-lemma-3case 0<A PO (me {A} {x} ax ) ) x |
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399 zorn05 : Maximal A |
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400 zorn05 with zorn07 A {!!} |
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401 ... | case1 chain = {!!} |
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402 ... | case2 (case1 m) = m |
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403 ... | case2 (case2 chain) = {!!} |
| 497 | 404 zorn04 : Maximal A |
| 405 zorn04 with zorn03 (& A) | |
| 507 | 406 ... | case1 chain = ⊥-elim ( o<> (c<→o< {ZChain.max chain} {A} (ZChain.A∋max chain)) (ZChain.y<max chain) ) |
| 497 | 407 ... | case2 m = m |
| 464 | 408 |
| 497 | 409 -- _⊆'_ : ( A B : HOD ) → Set n |
| 410 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 411 |
| 497 | 412 -- MaximumSubset : {L P : HOD} |
| 413 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 414 -- → IsPartialOrderSet P _⊆'_ | |
| 415 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 416 -- → Maximal P (_⊆'_) | |
| 417 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |