Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 547:379bd9b4610c
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 27 Apr 2022 23:21:21 +0900 |
| parents | 3234a5f6bfcf |
| children | 5ad7a31df4f4 |
| 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 | |
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528
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
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changeset
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43 _≤_ : (x y : HOD) → Set (Level.suc n) |
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8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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44 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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45 |
| 508 | 46 record Element (A : HOD) : Set (Level.suc n) where |
| 469 | 47 field |
| 48 elm : HOD | |
| 49 is-elm : A ∋ elm | |
| 50 | |
| 51 open Element | |
| 52 | |
| 509 | 53 _<A_ : {A : HOD} → (x y : Element A ) → Set n |
| 54 x <A y = elm x < elm y | |
| 55 _≡A_ : {A : HOD} → (x y : Element A ) → Set (Level.suc n) | |
| 56 x ≡A y = elm x ≡ elm y | |
| 57 | |
| 508 | 58 IsPartialOrderSet : ( A : HOD ) → Set (Level.suc n) |
| 509 | 59 IsPartialOrderSet A = IsStrictPartialOrder (_≡A_ {A}) _<A_ |
| 490 | 60 |
| 492 | 61 open _==_ |
| 62 open _⊆_ | |
| 63 | |
| 495 | 64 isA : { A B : HOD } → B ⊆ A → (x : Element B) → Element A |
| 65 isA B⊆A x = record { elm = elm x ; is-elm = incl B⊆A (is-elm x) } | |
| 494 | 66 |
| 497 | 67 ⊆-IsPartialOrderSet : { A B : HOD } → B ⊆ A → IsPartialOrderSet A → IsPartialOrderSet B |
| 68 ⊆-IsPartialOrderSet {A} {B} B⊆A PA = record { | |
| 69 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; trans = λ {x} {y} {z} → trans1 {x} {y} {z} | |
| 498 | 70 ; irrefl = λ {x} {y} → irrefl1 {x} {y} ; <-resp-≈ = resp0 |
| 493 | 71 } where |
| 495 | 72 _<B_ : (x y : Element B ) → Set n |
| 73 x <B y = elm x < elm y | |
| 74 trans1 : {x y z : Element B} → x <B y → y <B z → x <B z | |
| 75 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 | |
| 76 irrefl1 : {x y : Element B} → elm x ≡ elm y → ¬ ( x <B y ) | |
| 77 irrefl1 {x} {y} x=y x<y = IsStrictPartialOrder.irrefl PA {isA B⊆A x} {isA B⊆A y} x=y x<y | |
| 78 open import Data.Product | |
| 79 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) | |
| 80 resp0 = Data.Product._,_ (λ {x} {y} {z} → proj₁ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z }) | |
| 81 (λ {x} {y} {z} → proj₂ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z }) | |
| 492 | 82 |
| 497 | 83 -- open import Relation.Binary.Properties.Poset as Poset |
| 496 | 84 |
| 508 | 85 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) |
| 509 | 86 IsTotalOrderSet A = IsStrictTotalOrder (_≡A_ {A}) _<A_ |
| 490 | 87 |
| 469 | 88 me : { A a : HOD } → A ∋ a → Element A |
| 89 me {A} {a} lt = record { elm = a ; is-elm = lt } | |
| 90 | |
| 504 | 91 A∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (A ∋ x) ≡ (A ∋ y ) |
| 92 A∋x-irr A {x} {y} refl = refl | |
| 93 | |
| 506 | 94 me-elm-refl : (A : HOD) → (x : Element A) → elm (me {A} (d→∋ A (is-elm x))) ≡ elm x |
| 95 me-elm-refl A record { elm = ex ; is-elm = ax } = *iso | |
| 504 | 96 |
| 537 | 97 -- <-induction : (A : HOD) { ψ : (x : HOD) → A ∋ x → Set (Level.suc n)} |
| 98 -- → IsPartialOrderSet A | |
| 99 -- → ( {x : HOD } → A ∋ x → ({ y : HOD } → A ∋ y → y < x → ψ y ) → ψ x ) | |
| 100 -- → {x0 x : HOD } → A ∋ x0 → A ∋ x → x0 < x → ψ x | |
| 101 -- <-induction A {ψ} PO ind ax0 ax x0<a = subst (λ k → ψ k ) *iso (<-induction-ord (osuc (& x)) <-osuc ) where | |
| 102 -- -- y < * ox → & y o< ox | |
| 103 -- induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox) | |
| 104 -- induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) {!!})) | |
| 105 -- <-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy) | |
| 106 -- <-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy | |
| 107 | |
| 108 | |
| 504 | 109 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) |
| 110 | |
| 526 | 111 -- Don't use Element other than Order, you'll be in a trouble |
| 517 | 112 -- postulate -- may be proved by transfinite induction and functional extentionality |
| 113 -- ∋x-irr : (A : HOD) {x y : HOD} → x ≡ y → (ax : A ∋ x) (ay : A ∋ y ) → ax ≅ ay | |
| 114 -- odef-irr : (A : OD) {x y : Ordinal} → x ≡ y → (ax : def A x) (ay : def A y ) → ax ≅ ay | |
| 504 | 115 |
| 517 | 116 -- is-elm-irr : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y |
| 117 -- is-elm-irr A {x} {y} eq = ∋x-irr A eq (is-elm x) (is-elm y ) | |
| 504 | 118 |
| 119 El-irr2 : (A : HOD) → {x y : Element A } → elm x ≡ elm y → is-elm x ≅ is-elm y → x ≡ y | |
| 120 El-irr2 A {x} {y} refl HE.refl = refl | |
| 121 | |
| 517 | 122 -- El-irr : {A : HOD} → {x y : Element A } → elm x ≡ elm y → x ≡ y |
| 123 -- El-irr {A} {x} {y} eq = El-irr2 A eq ( is-elm-irr A eq ) | |
| 504 | 124 |
| 527 | 125 record _Set≈_ (A B : Ordinal ) : Set n where |
| 526 | 126 field |
| 127 fun← : {x : Ordinal } → odef (* A) x → Ordinal | |
| 128 fun→ : {x : Ordinal } → odef (* B) x → Ordinal | |
| 129 funB : {x : Ordinal } → ( lt : odef (* A) x ) → odef (* B) ( fun← lt ) | |
| 130 funA : {x : Ordinal } → ( lt : odef (* B) x ) → odef (* A) ( fun→ lt ) | |
| 131 fiso← : {x : Ordinal } → ( lt : odef (* B) x ) → fun← ( funA lt ) ≡ x | |
| 132 fiso→ : {x : Ordinal } → ( lt : odef (* A) x ) → fun→ ( funB lt ) ≡ x | |
| 133 | |
| 527 | 134 open _Set≈_ |
| 135 record _OS≈_ {A B : HOD} (TA : IsTotalOrderSet A ) (TB : IsTotalOrderSet B ) : Set (Level.suc n) where | |
| 526 | 136 field |
| 527 | 137 iso : (& A ) Set≈ (& B) |
| 526 | 138 fmap : {x y : Ordinal} → (ax : odef A x) → (ay : odef A y) → * x < * y |
| 139 → * (fun← iso (subst (λ k → odef k x) (sym *iso) ax)) < * (fun← iso (subst (λ k → odef k y) (sym *iso) ay)) | |
| 140 | |
| 141 Cut< : ( A x : HOD ) → HOD | |
| 142 Cut< A x = record { od = record { def = λ y → ( odef A y ) ∧ ( x < * y ) } ; odmax = & A | |
| 143 ; <odmax = λ lt → subst (λ k → k o< & A) &iso (c<→o< (subst (λ k → odef A k ) (sym &iso) (proj1 lt))) } | |
| 144 | |
| 527 | 145 Cut<T : {A : HOD} → (TA : IsTotalOrderSet A ) ( x : HOD )→ IsTotalOrderSet ( Cut< A x ) |
| 146 Cut<T {A} TA x = record { isEquivalence = record { refl = refl ; trans = trans ; sym = sym } | |
| 526 | 147 ; trans = λ {x} {y} {z} → IsStrictTotalOrder.trans TA {me (proj1 (is-elm x))} {me (proj1 (is-elm y))} {me (proj1 (is-elm z))} ; |
| 527 | 148 compare = λ x y → IsStrictTotalOrder.compare TA (me (proj1 (is-elm x))) (me (proj1 (is-elm y))) } |
| 526 | 149 |
| 527 | 150 record _OS<_ {A B : HOD} (TA : IsTotalOrderSet A ) (TB : IsTotalOrderSet B ) : Set (Level.suc n) where |
| 151 field | |
| 152 x : HOD | |
| 153 iso : TA OS≈ (Cut<T TA x) | |
| 154 | |
| 529 | 155 -- OS<-cmp : {x : HOD} → Trichotomous {_} {IsTotalOrderSet x} _OS≈_ _OS<_ |
| 156 -- OS<-cmp A B = {!!} | |
| 498 | 157 |
| 497 | 158 |
| 508 | 159 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 503 | 160 field |
| 161 maximal : HOD | |
| 162 A∋maximal : A ∋ maximal | |
| 163 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 164 | |
| 530 | 165 -- |
| 166 -- inductive maxmum tree from x | |
| 167 -- tree structure | |
| 168 -- | |
| 169 | |
| 170 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 171 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 172 | |
| 533 | 173 record Indirect< (A : HOD) {x y : Ordinal } (xa : odef A x) (ya : odef A y) (z : Ordinal) : Set n where |
| 529 | 174 field |
| 533 | 175 az : odef A z |
| 176 x<z : * x < * z | |
| 177 z<y : * z < * y | |
| 178 | |
| 541 | 179 record Prev< (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 180 field |
| 534 | 181 y : Ordinal |
| 541 | 182 ay : odef B y |
| 534 | 183 x=fy : x ≡ f y |
| 529 | 184 |
| 508 | 185 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 503 | 186 field |
| 187 sup : HOD | |
| 188 A∋maximal : A ∋ sup | |
| 189 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 190 | |
| 533 | 191 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
| 192 SupCond A B _ _ = SUP A B | |
| 193 | |
| 546 | 194 record ZChain ( A : HOD ) {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) |
| 547 | 195 (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where |
| 533 | 196 field |
| 197 chain : HOD | |
| 198 chain⊆A : chain ⊆ A | |
| 538 | 199 chain∋x : odef chain x |
| 533 | 200 f-total : IsTotalOrderSet chain |
| 546 | 201 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 541 | 202 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) |
| 547 | 203 is-max : {a b : Ordinal } → (ca : odef chain a ) → a o< z → (ba : odef A b) |
| 541 | 204 → Prev< A chain ba f |
| 205 ∨ (sup (& chain) (subst (λ k → k ⊆ A) (sym *iso) chain⊆A) (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b ) | |
| 534 | 206 → * a < * b → odef chain b |
| 533 | 207 |
| 497 | 208 Zorn-lemma : { A : HOD } |
| 464 | 209 → o∅ o< & A |
| 497 | 210 → IsPartialOrderSet A |
| 211 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition | |
| 212 → Maximal A | |
| 530 | 213 Zorn-lemma {A} 0<A PO supP = zorn00 where |
| 535 | 214 supO : (C : Ordinal ) → (* C) ⊆ A → IsTotalOrderSet (* C) → Ordinal |
| 215 supO C C⊆A TC = & ( SUP.sup ( supP (* C) C⊆A TC )) | |
| 493 | 216 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 496 | 217 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 |
| 524 | 218 z01 {a} {b} A∋a A∋b (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO {me A∋b} {me A∋b} refl |
| 219 (IsStrictPartialOrder.trans PO {me A∋b} {me A∋a} {me A∋b} b<a a<b) | |
| 537 | 220 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 221 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 222 s : HOD |
| 223 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 224 sa : A ∋ * ( & s ) | |
| 225 sa = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) | |
| 547 | 226 s<A : & s o< & A |
| 227 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso sa ) | |
| 530 | 228 HasMaximal : HOD |
| 537 | 229 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 230 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 231 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 232 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 233 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 234 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 235 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 236 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 237 x-is-maximal : ¬ Maximal A → {x : Ordinal} → (ax : odef A x) → & (Gtx (subst (λ k → odef A k ) (sym &iso) ax)) ≡ o∅ → (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m)) | |
| 238 x-is-maximal nmx {x} ax nogt m am = ⟪ subst (λ k → odef A k) &iso (subst (λ k → odef A k ) (sym &iso) ax) , ¬x<m ⟫ where | |
| 239 ¬x<m : ¬ (* x < * m) | |
| 240 ¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) | |
| 543 | 241 |
| 242 -- Uncountable acending chain by axiom of choice | |
| 530 | 243 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 244 cf nmx x with ODC.∋-p O A (* x) |
| 245 ... | no _ = o∅ | |
| 246 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 247 ... | yes nogt = -- no larger element, so it is maximal |
| 248 ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) | |
| 532 | 249 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 250 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 251 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 252 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 253 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 254 ... | yes nogt = ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ ) | |
| 255 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 530 | 256 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 257 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 258 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 259 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 260 |
| 547 | 261 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A sa f mf supO (& A) ) → SUP A (ZChain.chain zc) |
| 533 | 262 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
| 547 | 263 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) ) |
| 533 | 264 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
| 265 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
| 547 | 266 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) → SUP A (* (& (ZChain.chain zc))) |
| 538 | 267 sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) |
| 268 (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) ) | |
| 543 | 269 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 270 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 271 | |
| 272 --- | |
| 273 --- sup is fix point in maximum chain | |
| 274 --- | |
| 547 | 275 z03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) |
| 546 | 276 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) |
| 538 | 277 z03 f mf zc = z14 where |
| 278 chain = ZChain.chain zc | |
| 279 sp1 = sp0 f mf zc | |
| 547 | 280 z10 : {a b : Ordinal } → (ca : odef chain a ) → a o< & A → (ab : odef A b ) |
| 541 | 281 → Prev< A chain ab f |
| 282 ∨ (supO (& chain) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc)) ≡ b ) | |
| 538 | 283 → * a < * b → odef chain b |
| 284 z10 = ZChain.is-max zc | |
| 543 | 285 z11 : & (SUP.sup sp1) o< & A |
| 286 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 287 z12 : odef chain (& (SUP.sup sp1)) |
| 288 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 289 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
| 547 | 290 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) s<A (SUP.A∋maximal sp1) (case2 refl ) z13 where |
| 538 | 291 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 292 z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc )) | |
| 293 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) | |
| 294 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 295 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) | |
| 546 | 296 z14 with IsStrictTotalOrder.compare (ZChain.f-total zc ) (me (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 ))) (me z12 ) |
| 538 | 297 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 298 z16 : ⊥ | |
| 299 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 300 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 301 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 302 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 303 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 304 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 546 | 305 z15 = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (ZChain.f-next zc z12 )) |
| 538 | 306 z17 : ⊥ |
| 307 z17 with z15 | |
| 308 ... | case1 eq = ¬b eq | |
| 309 ... | case2 lt = ¬a lt | |
| 547 | 310 -- ZChain requires the Maximal |
| 311 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥ | |
| 537 | 312 z04 nmx zc = z01 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) |
| 538 | 313 (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
| 314 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) | |
| 315 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where | |
| 546 | 316 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc |
| 538 | 317 c = & (SUP.sup sp1) |
| 547 | 318 premax : {x y : Ordinal} → y o< x → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 : ZChain A sa f mf supO y ) |
| 543 | 319 → {a b : Ordinal} (ca : odef (ZChain.chain zc0) a) → (ab : odef A b) → a o< y |
| 320 → Prev< A (ZChain.chain zc0) ab f ∨ (supO (& (ZChain.chain zc0)) | |
| 321 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 322 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b) | |
| 541 | 323 → * a < * b → odef (ZChain.chain zc0) b |
| 547 | 324 premax {x} {y} y<x f mf zc0 {a} {b} ca ab a<y P a<b = ZChain.is-max zc0 ca a<y ab P a<b -- ca ab y P a<b |
| 540 | 325 -- Union of ZFChain |
| 326 UZFChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (B : Ordinal) | |
| 547 | 327 → ( (z : Ordinal) → z o< B → {y : Ordinal} → (ya : odef A y) → ZChain A ya f mf supO z ) → HOD |
| 546 | 328 UZFChain f mf B prev = record { od = record { def = λ y → odef A y ∧ (y o< B) ∧ ( (y<b : y o< B) → (ya : odef A y) → odef (ZChain.chain (prev y y<b ya )) y) } |
| 540 | 329 ; odmax = & A ; <odmax = z07 } |
| 547 | 330 -- create all ZChains under o< x |
| 546 | 331 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → |
| 547 | 332 ((z : Ordinal) → z o< x → {y : Ordinal} → (ya : odef A y) → ZChain A ya f mf supO z ) → { y : Ordinal } → (ya : odef A y) → ZChain A ya f mf supO x |
| 333 ind f mf x prev {y} ay with Oprev-p x | |
| 546 | 334 ... | yes op with ODC.∋-p O A (* (Oprev.oprev op)) |
| 335 ... | yes apx = zc4 where -- we have previous ordinal and A ∋ op | |
| 530 | 336 px = Oprev.oprev op |
| 546 | 337 apx0 = subst (λ k → odef A k ) &iso apx |
| 547 | 338 zc0 : ZChain A ay f mf supO (Oprev.oprev op) |
| 339 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay | |
| 340 zc1 : {y : Ordinal } → (ay : odef A y ) → ZChain A ay f mf supO (Oprev.oprev op) | |
| 341 zc1 {y} ay = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay | |
| 342 ay0 : odef A (& (* y)) | |
| 343 ay0 = (subst (λ k → odef A k ) (sym &iso) ay ) | |
| 543 | 344 Afx : { x : Ordinal } → A ∋ * x → A ∋ * (f x) |
| 345 Afx {x} ax = (subst (λ k → odef A k ) (sym &iso) (proj2 (mf x (subst (λ k → odef A k ) &iso ax)))) | |
| 533 | 346 -- x is in the previous chain, use the same |
| 347 -- x has some y which y < x ∧ f y ≡ x | |
| 348 -- x has no y which y < x | |
| 547 | 349 zc4 : ZChain A ay f mf supO x |
| 350 zc4 with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) ay f ) | |
| 351 ... | case1 pr = zc7 where -- we have previous < | |
| 541 | 352 chain = ZChain.chain zc0 |
| 547 | 353 zc7 : ZChain A ay f mf supO x |
| 540 | 354 zc7 with ODC.∋-p O (ZChain.chain zc0) (* ( f x ) ) |
| 547 | 355 ... | yes pr = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 |
| 546 | 356 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = {!!} -- ZChain.chain∋x zc0 |
| 357 ; is-max = {!!} } where -- no extention | |
| 544 | 358 z22 : {a : Ordinal} → x o< osuc a → ¬ odef (ZChain.chain zc0) a |
| 546 | 359 z22 {a} x<oa = {!!} |
| 544 | 360 zc20 : {P : Ordinal → Set n} → ({a : Ordinal} → odef (ZChain.chain zc0) a → a o< px → P a) |
| 361 → {a : Ordinal} → (za : odef (ZChain.chain zc0) a ) → (a<x : a o< x) → P a | |
| 362 zc20 {P} prev {a} za a<x with trio< a px | |
| 363 ... | tri< a₁ ¬b ¬c = prev za a₁ | |
| 546 | 364 ... | tri≈ ¬a b ¬c = {!!} |
| 365 ... | tri> ¬a ¬b c = {!!} | |
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366 z21 : {a : Ordinal} → odef (ZChain.chain zc0) a → a o< x → odef (ZChain.chain zc0) (f a) |
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367 z21 {a} za a<x with trio< a x |
| 546 | 368 ... | tri< a₁ ¬b ¬c = ZChain.f-next zc0 za |
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369 ... | tri≈ ¬a b ¬c = {!!} |
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370 ... | tri> ¬a ¬b c = ⊥-elim ( o<> c a<x ) |
| 540 | 371 ... | no not = record { chain = zc5 ; chain⊆A = ⊆-zc5 |
| 546 | 372 ; f-total = zc6 ; f-next = {!!} ; f-immediate = {!!} ; chain∋x = case1 {!!} ; ¬chain∋x>z = {!!} ; is-max = {!!} } where |
| 542 | 373 -- extend with f x -- cahin ∋ y ∧ chain ∋ f y ∧ x ≡ f ( f y ) |
| 540 | 374 zc5 : HOD |
| 547 | 375 zc5 = record { od = record { def = λ z → odef (ZChain.chain zc0) z ∨ (z ≡ f y) } ; odmax = & A ; <odmax = {!!} } |
| 540 | 376 ⊆-zc5 : zc5 ⊆ A |
| 543 | 377 ⊆-zc5 = record { incl = λ {y} lt → zc15 lt } where |
| 547 | 378 zc15 : {z : Ordinal } → ( odef (ZChain.chain zc0) z ∨ (z ≡ f y) ) → odef A z |
| 543 | 379 zc15 {z} (case1 zz) = subst (λ k → odef A k ) &iso ( incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k ) (sym &iso) zz ) ) |
| 547 | 380 zc15 (case2 refl) = proj2 ( mf y (subst (λ k → odef A k ) &iso {!!} ) ) |
| 540 | 381 IPO = ⊆-IsPartialOrderSet ⊆-zc5 PO |
| 542 | 382 zc8 : { A B x : HOD } → (ax : A ∋ x ) → (P : Prev< A B ax f ) → * (f (& (* (Prev<.y P)))) ≡ x |
| 383 zc8 {A} {B} {x} ax P = subst₂ (λ j k → * ( f j ) ≡ k ) (sym &iso) *iso (sym (cong (*) ( Prev<.x=fy P))) | |
| 547 | 384 fx=zc : odef (ZChain.chain zc0) y → Tri (* (f y) < * y ) (* (f y) ≡ * y) (* y < * (f y) ) |
| 385 fx=zc c with mf y (subst (λ k → odef A k) &iso ay0 ) | |
| 386 ... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ay0 (Afx ay0) (case1 (sym zc13))) zc13 (z01 (Afx ay0) ay0 (case1 zc13)) where | |
| 387 zc13 : * (f y) ≡ * y | |
| 388 zc13 = subst (λ k → k ≡ * y ) (subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym &iso))) (sym ( x=fx )) | |
| 389 ... | ⟪ case2 x<fx , afx ⟫ = tri> (z01 ay0 (Afx ay0) (case2 zc14)) (λ lt → z01 (Afx ay0) ay0 (case1 lt) zc14) zc14 where | |
| 390 zc14 : * y < * (f y) | |
| 391 zc14 = subst (λ k → * y < k ) (subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx | |
| 540 | 392 cmp : Trichotomous _ _ |
| 541 | 393 cmp record { elm = a ; is-elm = aa } record { elm = b ; is-elm = ab } with aa | ab |
| 394 ... | case1 x | case1 x₁ = IsStrictTotalOrder.compare (ZChain.f-total zc0) (me x) (me x₁) | |
| 542 | 395 ... | case2 fx | case2 fx₁ = tri≈ {!!} (subst₂ (λ j k → j ≡ k ) *iso *iso (trans (cong (*) fx) (sym (cong (*) fx₁ ) ))) {!!} |
| 396 ... | case1 c | case2 fx = {!!} -- subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) {!!} {!!} ( fx>zc (subst (λ k → odef chain k) {!!} c )) | |
| 397 ... | case2 fx | case1 c with ODC.p∨¬p O ( Prev< A chain (incl (ZChain.chain⊆A zc0) c) f ) | |
| 398 ... | case2 n = {!!} | |
| 547 | 399 ... | case1 fb with IsStrictTotalOrder.compare (ZChain.f-total zc0) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay pr))) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay fb))) |
| 542 | 400 ... | tri< a₁ ¬b ¬c = {!!} |
| 543 | 401 ... | tri≈ ¬a b₁ ¬c = subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) zc11 zc10 ( fx=zc zc12 ) where |
| 547 | 402 zc10 : * y ≡ b |
| 403 zc10 = subst₂ (λ j k → j ≡ k ) (zc8 ay {!!} ) (zc8 (incl ( ZChain.chain⊆A zc0 ) c) fb) (cong (λ k → * ( f ( & k ))) b₁) | |
| 404 zc11 : * (f y) ≡ a | |
| 405 zc11 = subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym fx)) | |
| 406 zc12 : odef chain y | |
| 543 | 407 zc12 = subst (λ k → odef chain k ) (subst (λ k → & b ≡ k ) &iso (sym (cong (&) zc10))) c |
| 542 | 408 ... | tri> ¬a ¬b c₁ = {!!} |
| 540 | 409 zc6 : IsTotalOrderSet zc5 |
| 410 zc6 = record { isEquivalence = IsStrictPartialOrder.isEquivalence IPO ; trans = λ {x} {y} {z} → IsStrictPartialOrder.trans IPO {x} {y} {z} | |
| 411 ; compare = cmp } | |
| 412 ... | case2 not with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) {!!} {!!} ) )) | |
| 547 | 413 ... | case1 pr = {!!} -- x is sup |
| 542 | 414 ... | case2 not = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = {!!} |
| 546 | 415 ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} } -- no extention |
| 416 ... | no noapx = {!!} -- we have previous ordinal but ¬ A ∋ op | |
| 417 ind f mf x prev ya | no ¬ox with trio< (& A) x --- limit ordinal case | |
| 418 ... | tri< a ¬b ¬c = {!!} where | |
| 533 | 419 zc0 = prev (& A) a |
| 420 ... | tri≈ ¬a b ¬c = {!!} | |
| 546 | 421 ... | tri> ¬a ¬b c = {!!} where |
| 543 | 422 uzc : HOD |
| 423 uzc = UZFChain f mf x prev | |
| 530 | 424 zorn00 : Maximal A |
| 531 | 425 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM |
| 530 | 426 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where |
| 427 -- yes we have the maximal | |
| 428 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 429 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 430 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 531 | 431 zorn01 = proj1 zorn03 |
| 530 | 432 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) |
| 531 | 433 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) |
| 546 | 434 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where |
| 530 | 435 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 533 | 436 nmx : ¬ Maximal A |
| 437 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 531 | 438 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) |
| 439 zc5 = ⟪ Maximal.A∋maximal mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫ | |
| 547 | 440 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO (& A) |
| 441 zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain A ya f mf supO z } (ind f mf) (& A) | |
| 442 zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) | |
| 546 | 443 zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa ) |
| 464 | 444 |
| 516 | 445 -- usage (see filter.agda ) |
| 446 -- | |
| 497 | 447 -- _⊆'_ : ( A B : HOD ) → Set n |
| 448 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 449 |
| 497 | 450 -- MaximumSubset : {L P : HOD} |
| 451 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 452 -- → IsPartialOrderSet P _⊆'_ | |
| 453 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 454 -- → Maximal P (_⊆'_) | |
| 455 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |