Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 538:854908eb70f2
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 24 Apr 2022 14:10:06 +0900 |
| parents | e12add1519ec |
| children | adbac273d2a6 |
| 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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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
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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 | |
| 529 | 165 |
| 530 | 166 -- |
| 167 -- inductive maxmum tree from x | |
| 168 -- tree structure | |
| 169 -- | |
| 170 | |
| 171 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 172 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 173 | |
| 533 | 174 record Indirect< (A : HOD) {x y : Ordinal } (xa : odef A x) (ya : odef A y) (z : Ordinal) : Set n where |
| 529 | 175 field |
| 533 | 176 az : odef A z |
| 177 x<z : * x < * z | |
| 178 z<y : * z < * y | |
| 179 | |
| 534 | 180 record Prev< (A : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 181 field |
| 534 | 182 y : Ordinal |
| 183 ay : odef A y | |
| 184 x=fy : x ≡ f y | |
| 529 | 185 |
| 508 | 186 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 503 | 187 field |
| 188 sup : HOD | |
| 189 A∋maximal : A ∋ sup | |
| 190 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 191 | |
| 533 | 192 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
| 193 SupCond A B _ _ = SUP A B | |
| 194 | |
| 195 record ZChain ( A : HOD ) {x : Ordinal} (ax : A ∋ * x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) | |
| 535 | 196 (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) (z : Ordinal) : Set (Level.suc n) where |
| 533 | 197 field |
| 198 chain : HOD | |
| 199 chain⊆A : chain ⊆ A | |
| 538 | 200 chain∋x : odef chain x |
| 533 | 201 f-total : IsTotalOrderSet chain |
| 202 f-next : {a : Ordinal } → odef chain a → odef chain (f a) | |
| 534 | 203 is-max : {a b : Ordinal } → (ca : odef chain a ) → odef A b → a o< z |
| 535 | 204 → ( Prev< A (incl chain⊆A (subst (λ k → odef chain k ) (sym &iso) ca)) 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 )) ) | |
| 226 HasMaximal : HOD | |
| 537 | 227 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 228 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 229 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 230 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 231 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 232 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 233 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 234 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 235 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)) | |
| 236 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 | |
| 237 ¬x<m : ¬ (* x < * m) | |
| 238 ¬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) | |
| 530 | 239 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 240 cf nmx x with ODC.∋-p O A (* x) |
| 241 ... | no _ = o∅ | |
| 242 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 243 ... | yes nogt = -- no larger element, so it is maximal |
| 244 ⊥-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 | 245 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 246 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 247 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 248 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 249 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 250 ... | 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 ⟫ ) | |
| 251 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 530 | 252 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 253 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 254 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 255 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 535 | 256 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A sa f mf supO (& A)) → SUP A (ZChain.chain zc) |
| 533 | 257 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
| 535 | 258 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) |
| 533 | 259 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
| 260 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
| 538 | 261 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A sa f mf supO (& A)) → SUP A (* (& (ZChain.chain zc))) |
| 262 sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) | |
| 263 (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) ) | |
| 264 z03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A sa f mf supO (& A)) | |
| 265 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) | |
| 266 z03 f mf zc = z14 where | |
| 267 chain = ZChain.chain zc | |
| 268 sp1 = sp0 f mf zc | |
| 269 z10 : {a b : Ordinal } → (ca : odef chain a ) → odef A b → a o< (& A) | |
| 270 → ( Prev< A (incl (ZChain.chain⊆A zc) (subst (λ k → odef chain k ) (sym &iso) ca)) f | |
| 271 ∨ (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 )) | |
| 272 → * a < * b → odef chain b | |
| 273 z10 = ZChain.is-max zc | |
| 274 z12 : odef chain (& (SUP.sup sp1)) | |
| 275 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 276 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
| 277 ... | no ne = z10 {& s} {& (SUP.sup sp1)} (ZChain.chain∋x zc) (SUP.A∋maximal sp1) (c<→o< (subst (λ k → odef A (& k) ) *iso sa) ) (case2 refl ) z13 where | |
| 278 z13 : * (& s) < * (& (SUP.sup sp1)) | |
| 279 z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc )) | |
| 280 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) | |
| 281 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 282 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) | |
| 283 z14 with IsStrictTotalOrder.compare (ZChain.f-total zc ) (me (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12))) (me z12 ) | |
| 284 ... | tri< a ¬b ¬c = ⊥-elim z16 where | |
| 285 z16 : ⊥ | |
| 286 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 287 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 288 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 289 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 290 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 291 c1 : SUP.sup sp1 < * (f ( & ( SUP.sup sp1 ))) | |
| 292 c1 = c | |
| 293 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 294 z15 = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (ZChain.f-next zc z12) ) | |
| 295 z17 : ⊥ | |
| 296 z17 with z15 | |
| 297 ... | case1 eq = ¬b eq | |
| 298 ... | case2 lt = ¬a lt | |
| 535 | 299 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥ |
| 537 | 300 z04 nmx zc = z01 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) |
| 538 | 301 (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
| 302 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) | |
| 303 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where | |
| 304 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc | |
| 305 c = & (SUP.sup sp1) | |
| 478 | 306 -- ZChain is not compatible with the SUP condition |
| 535 | 307 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A sa f mf supO y ) |
| 308 → ZChain A sa f mf supO x | |
| 533 | 309 ind f mf x prev with Oprev-p x |
| 530 | 310 ... | yes op with ODC.∋-p O A (* x) |
| 311 ... | no ¬Ax = zc1 where | |
| 312 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain | |
| 313 px = Oprev.oprev op | |
| 535 | 314 zc0 : ZChain A sa f mf supO (Oprev.oprev op) |
| 533 | 315 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) |
| 535 | 316 zc1 : ZChain A sa f mf supO x |
| 533 | 317 zc1 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 ; is-max = {!!} } |
| 530 | 318 ... | yes ax = zc4 where -- we have previous ordinal and A ∋ x |
| 319 px = Oprev.oprev op | |
| 535 | 320 zc0 : ZChain A sa f mf supO (Oprev.oprev op) |
| 533 | 321 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) |
| 322 -- x is in the previous chain, use the same | |
| 323 -- x has some y which y < x ∧ f y ≡ x | |
| 324 -- x has no y which y < x | |
| 535 | 325 zc4 : ZChain A sa f mf supO x |
| 533 | 326 zc4 = record { chain = {!!} ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} ; is-max = {!!} } |
| 327 ind f mf x prev | no ¬ox with trio< (& A) x --- limit ordinal case | |
| 328 ... | tri< a ¬b ¬c = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 329 ; is-max = {!!} } where | |
| 330 zc0 = prev (& A) a | |
| 331 ... | tri≈ ¬a b ¬c = {!!} | |
| 332 ... | tri> ¬a ¬b c = {!!} | |
| 530 | 333 zorn00 : Maximal A |
| 531 | 334 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM |
| 530 | 335 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where |
| 336 -- yes we have the maximal | |
| 337 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 338 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 339 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 531 | 340 zorn01 = proj1 zorn03 |
| 530 | 341 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) |
| 531 | 342 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) |
| 533 | 343 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (zorn03 (cf nmx) (cf-is-≤-monotonic nmx))) where |
| 530 | 344 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 533 | 345 nmx : ¬ Maximal A |
| 346 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 531 | 347 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) |
| 348 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) ) ⟫ | |
| 535 | 349 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → ZChain A sa f mf supO (& A) |
| 533 | 350 zorn03 f mf = TransFinite (ind f mf) (& A) |
| 464 | 351 |
| 516 | 352 -- usage (see filter.agda ) |
| 353 -- | |
| 497 | 354 -- _⊆'_ : ( A B : HOD ) → Set n |
| 355 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 356 |
| 497 | 357 -- MaximumSubset : {L P : HOD} |
| 358 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 359 -- → IsPartialOrderSet P _⊆'_ | |
| 360 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 361 -- → Maximal P (_⊆'_) | |
| 362 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |