Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 553:567a1a9f3e0a
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 28 Apr 2022 19:00:40 +0900 |
| parents | fb210e812eba |
| children | 0687736285ce |
| rev | line source |
|---|---|
| 478 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 508 | 2 open import Level hiding ( suc ; zero ) |
| 431 | 3 open import Ordinals |
| 552 | 4 open import Relation.Binary |
| 5 open import Relation.Binary.Core | |
| 6 open import Relation.Binary.PropositionalEquality | |
| 497 | 7 import OD |
| 552 | 8 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where |
| 431 | 9 |
| 10 open import zf | |
| 477 | 11 open import logic |
| 12 -- open import partfunc {n} O | |
| 13 | |
| 14 open import Relation.Nullary | |
| 15 open import Data.Empty | |
| 16 import BAlgbra | |
| 431 | 17 |
| 18 | |
| 19 open inOrdinal O | |
| 20 open OD O | |
| 21 open OD.OD | |
| 22 open ODAxiom odAxiom | |
| 477 | 23 import OrdUtil |
| 24 import ODUtil | |
| 431 | 25 open Ordinals.Ordinals O |
| 26 open Ordinals.IsOrdinals isOrdinal | |
| 27 open Ordinals.IsNext isNext | |
| 28 open OrdUtil O | |
| 477 | 29 open ODUtil O |
| 30 | |
| 31 | |
| 32 import ODC | |
| 33 | |
| 34 | |
| 35 open _∧_ | |
| 36 open _∨_ | |
| 37 open Bool | |
| 431 | 38 |
| 39 | |
| 40 open HOD | |
| 41 | |
|
528
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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42 _≤_ : (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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43 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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44 |
| 490 | 45 |
| 492 | 46 open _==_ |
| 47 open _⊆_ | |
| 48 | |
| 497 | 49 -- open import Relation.Binary.Properties.Poset as Poset |
| 496 | 50 |
| 552 | 51 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) |
| 52 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 498 | 53 |
| 497 | 54 |
| 508 | 55 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 503 | 56 field |
| 57 maximal : HOD | |
| 58 A∋maximal : A ∋ maximal | |
| 59 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 60 | |
| 530 | 61 -- |
| 62 -- inductive maxmum tree from x | |
| 63 -- tree structure | |
| 64 -- | |
| 65 | |
| 66 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 67 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 68 | |
| 551 | 69 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 70 init : {x : Ordinal} → odef A s → FClosure A f s s | |
| 71 fsuc : {x : Ordinal} ( p : FClosure A f s x ) → FClosure A f s (f x) | |
| 533 | 72 |
| 541 | 73 record Prev< (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 74 field |
| 534 | 75 y : Ordinal |
| 541 | 76 ay : odef B y |
| 534 | 77 x=fy : x ≡ f y |
| 529 | 78 |
| 508 | 79 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 503 | 80 field |
| 81 sup : HOD | |
| 82 A∋maximal : A ∋ sup | |
| 83 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 84 | |
| 533 | 85 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
| 86 SupCond A B _ _ = SUP A B | |
| 87 | |
| 546 | 88 record ZChain ( A : HOD ) {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) |
| 547 | 89 (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where |
| 533 | 90 field |
| 91 chain : HOD | |
| 92 chain⊆A : chain ⊆ A | |
| 538 | 93 chain∋x : odef chain x |
| 533 | 94 f-total : IsTotalOrderSet chain |
| 546 | 95 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 541 | 96 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) |
| 548 | 97 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< z → (ba : odef A b) |
| 541 | 98 → Prev< A chain ba f |
| 99 ∨ (sup (& chain) (subst (λ k → k ⊆ A) (sym *iso) chain⊆A) (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b ) | |
| 534 | 100 → * a < * b → odef chain b |
| 533 | 101 |
| 497 | 102 Zorn-lemma : { A : HOD } |
| 464 | 103 → o∅ o< & A |
| 497 | 104 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 105 → Maximal A | |
| 552 | 106 Zorn-lemma {A} 0<A supP = zorn00 where |
| 535 | 107 supO : (C : Ordinal ) → (* C) ⊆ A → IsTotalOrderSet (* C) → Ordinal |
| 108 supO C C⊆A TC = & ( SUP.sup ( supP (* C) C⊆A TC )) | |
| 493 | 109 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 552 | 110 z01 {a} {b} A∋a A∋b (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a |
| 111 z01 {a} {b} A∋a A∋b (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 112 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 537 | 113 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 114 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 115 s : HOD |
| 116 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 117 sa : A ∋ * ( & s ) | |
| 118 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 | 119 s<A : & s o< & A |
| 120 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso sa ) | |
| 530 | 121 HasMaximal : HOD |
| 537 | 122 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 123 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 124 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 125 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 126 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 127 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 128 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 129 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 130 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)) | |
| 131 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 | |
| 132 ¬x<m : ¬ (* x < * m) | |
| 133 ¬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 | 134 |
| 135 -- Uncountable acending chain by axiom of choice | |
| 530 | 136 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 137 cf nmx x with ODC.∋-p O A (* x) |
| 138 ... | no _ = o∅ | |
| 139 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 140 ... | yes nogt = -- no larger element, so it is maximal |
| 141 ⊥-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 | 142 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 143 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 144 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 145 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 146 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 147 ... | 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 ⟫ ) | |
| 148 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 530 | 149 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 150 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 151 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 152 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 153 |
| 547 | 154 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A sa f mf supO (& A) ) → SUP A (ZChain.chain zc) |
| 533 | 155 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
| 547 | 156 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) ) |
| 533 | 157 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
| 158 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
| 547 | 159 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 | 160 sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) |
| 161 (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) ) | |
| 543 | 162 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 163 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 164 | |
| 165 --- | |
| 166 --- sup is fix point in maximum chain | |
| 167 --- | |
| 547 | 168 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 | 169 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) |
| 538 | 170 z03 f mf zc = z14 where |
| 171 chain = ZChain.chain zc | |
| 172 sp1 = sp0 f mf zc | |
| 548 | 173 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
| 541 | 174 → Prev< A chain ab f |
| 175 ∨ (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 | 176 → * a < * b → odef chain b |
| 177 z10 = ZChain.is-max zc | |
| 543 | 178 z11 : & (SUP.sup sp1) o< & A |
| 179 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 180 z12 : odef chain (& (SUP.sup sp1)) |
| 181 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 182 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
| 548 | 183 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) z11 (SUP.A∋maximal sp1) (case2 refl ) z13 where |
| 538 | 184 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 185 z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc )) | |
| 186 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) | |
| 187 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 188 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) | |
| 552 | 189 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 538 | 190 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 191 z16 : ⊥ | |
| 192 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 193 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 194 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 195 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 196 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 197 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 546 | 198 z15 = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (ZChain.f-next zc z12 )) |
| 538 | 199 z17 : ⊥ |
| 200 z17 with z15 | |
| 201 ... | case1 eq = ¬b eq | |
| 202 ... | case2 lt = ¬a lt | |
| 547 | 203 -- ZChain requires the Maximal |
| 204 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥ | |
| 537 | 205 z04 nmx zc = z01 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) |
| 538 | 206 (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
| 207 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) | |
| 208 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where | |
| 546 | 209 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc |
| 538 | 210 c = & (SUP.sup sp1) |
| 548 | 211 |
| 550 | 212 -- 3cases : {x y : Ordinal} → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) |
| 213 -- → (ax : odef A x )→ (ay : odef A y ) | |
| 214 -- → (zc0 : ZChain A ay f mf supO x ) | |
| 215 -- → Prev< A (ZChain.chain zc0) ax f | |
| 216 -- ∨ (supO (& (ZChain.chain zc0)) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ x) | |
| 217 -- ∨ ( ( z : Ordinal) → odef (ZChain.chain zc0) z → ¬ ( * z < * x )) | |
| 218 -- 3cases {x} {y} f mf ax ay zc0 = {!!} | |
| 547 | 219 -- create all ZChains under o< x |
| 546 | 220 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → |
| 547 | 221 ((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 |
| 222 ind f mf x prev {y} ay with Oprev-p x | |
| 548 | 223 ... | yes op = zc4 where |
| 530 | 224 px = Oprev.oprev op |
| 547 | 225 zc0 : ZChain A ay f mf supO (Oprev.oprev op) |
| 226 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay | |
| 227 zc4 : ZChain A ay f mf supO x | |
| 551 | 228 zc4 with ODC.∋-p O A (* px) |
| 229 ... | no noapx = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 230 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 } where -- no extention | |
| 231 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
| 232 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
| 233 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 234 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) ≡ b) → | |
| 235 * a < * b → odef (ZChain.chain zc0) b | |
| 236 zc11 {a} {b} za b<x ba P a<b with osuc-≡< (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x) | |
| 237 ... | case1 eq = ⊥-elim ( noapx (subst (λ k → odef A k) (trans eq (sym &iso) ) ba )) | |
| 238 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
| 239 ... | yes apx with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) apx f ) -- we have to check adding x preserve is-max ZChain A ay f mf supO px | |
| 549 | 240 ... | case1 pr = zc9 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 551 | 241 chain = ZChain.chain zc0 |
| 242 zc17 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
| 243 Prev< A (ZChain.chain zc0) ba f ∨ (supO (& (ZChain.chain zc0)) | |
| 244 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 245 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b) → | |
| 246 * a < * b → odef (ZChain.chain zc0) b | |
| 247 zc17 {a} {b} za b<x ba P a<b with osuc-≡< (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x) | |
| 248 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
| 249 ... | case1 b=px = subst (λ k → odef chain k ) (trans (sym (Prev<.x=fy pr )) (trans &iso (sym b=px))) ( ZChain.f-next zc0 (Prev<.ay pr)) | |
| 549 | 250 zc9 : ZChain A ay f mf supO x |
| 251 zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 551 | 252 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 } -- no extention |
| 253 ... | case2 ¬fy<x with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) )) | |
| 552 | 254 ... | case1 x=sup = record { chain = schain ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} |
| 255 ; f-immediate = {!!} ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup | |
| 551 | 256 sp = SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) |
| 257 chain = ZChain.chain zc0 | |
| 552 | 258 schain : HOD |
| 259 schain = record { od = record { def = λ x → odef chain x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = {!!} } | |
| 260 ... | case2 ¬x=sup = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 261 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = {!!} } where -- no extention | |
| 262 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
| 263 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
| 264 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 265 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) | |
| 266 ≡ b) → | |
| 267 * a < * b → odef (ZChain.chain zc0) b | |
| 268 z18 {a} {b} za b<x ba (case1 p) a<b = {!!} | |
| 269 z18 {a} {b} za b<x ba (case2 p) a<b = {!!} | |
| 553 | 270 ... | no ¬ox = {!!} where --- limit ordinal case |
| 271 -- Union of ZFChain | |
| 272 record UZFChain (y : Ordinal) : Set n where | |
| 273 field | |
| 274 u : Ordinal | |
| 275 u<x : u o< x | |
| 276 zuy : odef (ZChain.chain (prev u u<x ay )) y | |
| 551 | 277 uzc : HOD |
| 553 | 278 uzc = record { od = record { def = λ y → UZFChain y } ; odmax = & A ; <odmax = {!!} } |
| 279 u-total : IsTotalOrderSet uzc | |
| 280 u-total {x} {y} ux uy = {!!} | |
| 281 | |
| 551 | 282 zorn00 : Maximal A |
| 283 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 284 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 285 -- yes we have the maximal | |
| 286 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 287 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 288 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 289 zorn01 = proj1 zorn03 | |
| 290 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 291 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 292 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where | |
| 293 -- if we have no maximal, make ZChain, which contradict SUP condition | |
| 294 nmx : ¬ Maximal A | |
| 295 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 296 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 297 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) ) ⟫ | |
| 298 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO (& A) | |
| 299 zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain A ya f mf supO z } (ind f mf) (& A) | |
| 300 zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) | |
| 301 zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa ) | |
| 302 | |
| 303 zorn99 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) → (x y : Ordinal) (ay : odef A y) → (zc0 : ZChain A {!!} f mf supO x) → Prev< A (ZChain.chain zc0) {!!} f → {!!} | |
| 304 zorn99 f mf x y ay zc0 pr = {!!} where | |
| 548 | 305 ay0 : odef A (& (* y)) |
| 306 ay0 = (subst (λ k → odef A k ) (sym &iso) ay ) | |
| 307 Afx : { x : Ordinal } → A ∋ * x → A ∋ * (f x) | |
| 308 Afx {x} ax = (subst (λ k → odef A k ) (sym &iso) (proj2 (mf x (subst (λ k → odef A k ) &iso ax)))) | |
| 541 | 309 chain = ZChain.chain zc0 |
| 547 | 310 zc7 : ZChain A ay f mf supO x |
| 549 | 311 zc7 with trio< (Prev<.y pr) x |
| 312 ... | tri< a ¬b ¬c = {!!} -- already x ∈ chain because of is-max | |
| 313 ... | tri≈ ¬a b ¬c = {!!} -- x ≡ z ∈ chain | |
| 314 ... | tri> ¬a ¬b x<z = record { chain = zc5 ; chain⊆A = ⊆-zc5 --- | |
| 315 ; f-total = zc6 ; f-next = {!!} ; f-immediate = {!!} ; chain∋x = case1 {!!} ; is-max = {!!} } where | |
| 316 -- extend with x ≡ f z where cahin ∋ z | |
| 540 | 317 zc5 : HOD |
| 549 | 318 zc5 = record { od = record { def = λ z → odef (ZChain.chain zc0) z ∨ (z ≡ f x) } ; odmax = & A ; <odmax = {!!} } |
| 540 | 319 ⊆-zc5 : zc5 ⊆ A |
| 543 | 320 ⊆-zc5 = record { incl = λ {y} lt → zc15 lt } where |
| 549 | 321 zc15 : {z : Ordinal } → ( odef (ZChain.chain zc0) z ∨ (z ≡ f x) ) → odef A z |
| 543 | 322 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 ) ) |
| 549 | 323 zc15 (case2 refl) = proj2 ( mf x (subst (λ k → odef A k ) &iso {!!} ) ) |
| 542 | 324 zc8 : { A B x : HOD } → (ax : A ∋ x ) → (P : Prev< A B ax f ) → * (f (& (* (Prev<.y P)))) ≡ x |
| 325 zc8 {A} {B} {x} ax P = subst₂ (λ j k → * ( f j ) ≡ k ) (sym &iso) *iso (sym (cong (*) ( Prev<.x=fy P))) | |
| 547 | 326 fx=zc : odef (ZChain.chain zc0) y → Tri (* (f y) < * y ) (* (f y) ≡ * y) (* y < * (f y) ) |
| 327 fx=zc c with mf y (subst (λ k → odef A k) &iso ay0 ) | |
| 328 ... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ay0 (Afx ay0) (case1 (sym zc13))) zc13 (z01 (Afx ay0) ay0 (case1 zc13)) where | |
| 329 zc13 : * (f y) ≡ * y | |
| 330 zc13 = subst (λ k → k ≡ * y ) (subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym &iso))) (sym ( x=fx )) | |
| 331 ... | ⟪ case2 x<fx , afx ⟫ = tri> (z01 ay0 (Afx ay0) (case2 zc14)) (λ lt → z01 (Afx ay0) ay0 (case1 lt) zc14) zc14 where | |
| 332 zc14 : * y < * (f y) | |
| 333 zc14 = subst (λ k → * y < k ) (subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx | |
| 552 | 334 zc6 : IsTotalOrderSet zc5 |
| 335 zc6 {a} {b} ( case1 x ) ( case1 x₁ ) = ZChain.f-total zc0 x x₁ | |
| 336 zc6 {a} {b} ( case2 fx ) ( case2 fx₁ ) = tri≈ {!!} (subst₂ (λ j k → j ≡ k ) *iso *iso (trans (cong (*) fx) (sym (cong (*) fx₁ ) ))) {!!} | |
| 337 zc6 {a} {b} ( case1 c ) ( case2 fx ) = {!!} -- subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) {!!} {!!} ( fx>zc (subst (λ k → odef chain k) {!!} c )) | |
| 338 zc6 {a} {b} ( case2 fx ) ( case1 c ) with ODC.p∨¬p O ( Prev< A chain (incl (ZChain.chain⊆A zc0) c) f ) | |
| 542 | 339 ... | case2 n = {!!} |
| 552 | 340 ... | case1 fb with ZChain.f-total zc0 (subst (λ k → odef chain k) (sym &iso) (Prev<.ay pr)) (subst (λ k → odef chain k) (sym &iso) (Prev<.ay fb)) |
| 542 | 341 ... | tri< a₁ ¬b ¬c = {!!} |
| 543 | 342 ... | tri≈ ¬a b₁ ¬c = subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) zc11 zc10 ( fx=zc zc12 ) where |
| 547 | 343 zc10 : * y ≡ b |
| 344 zc10 = subst₂ (λ j k → j ≡ k ) (zc8 ay {!!} ) (zc8 (incl ( ZChain.chain⊆A zc0 ) c) fb) (cong (λ k → * ( f ( & k ))) b₁) | |
| 345 zc11 : * (f y) ≡ a | |
| 549 | 346 zc11 = subst (λ k → * (f y) ≡ k ) *iso (cong (*) (sym {!!} )) |
| 547 | 347 zc12 : odef chain y |
| 543 | 348 zc12 = subst (λ k → odef chain k ) (subst (λ k → & b ≡ k ) &iso (sym (cong (&) zc10))) c |
| 542 | 349 ... | tri> ¬a ¬b c₁ = {!!} |
| 516 | 350 -- usage (see filter.agda ) |
| 351 -- | |
| 497 | 352 -- _⊆'_ : ( A B : HOD ) → Set n |
| 353 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 354 |
| 497 | 355 -- MaximumSubset : {L P : HOD} |
| 356 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 357 -- → IsPartialOrderSet P _⊆'_ | |
| 358 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 359 -- → Maximal P (_⊆'_) | |
| 360 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |