Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 543:f0b45446c7ea
...
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Wed, 27 Apr 2022 04:04:30 +0900 |
parents | 3826221c61a6 |
children | 27bb51b7f012 |
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 | |
528
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
43 _≤_ : (x y : HOD) → Set (Level.suc n) |
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
44 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
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 | |
194 record ZChain ( A : HOD ) {x : Ordinal} (ax : A ∋ * x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) | |
535 | 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 |
543 | 200 ¬chain∋x>z : { a : Ordinal } → z o< osuc a → ¬ odef chain a |
533 | 201 f-total : IsTotalOrderSet chain |
542 | 202 f-next : {a : Ordinal } → odef chain a → a o< z → odef chain (f a) |
541 | 203 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) |
204 is-max : {a b : Ordinal } → (ca : odef chain a ) → (ba : odef A b) → a o< z | |
205 → Prev< A chain ba f | |
206 ∨ (sup (& chain) (subst (λ k → k ⊆ A) (sym *iso) chain⊆A) (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b ) | |
534 | 207 → * a < * b → odef chain b |
533 | 208 |
497 | 209 Zorn-lemma : { A : HOD } |
464 | 210 → o∅ o< & A |
497 | 211 → IsPartialOrderSet A |
212 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition | |
213 → Maximal A | |
530 | 214 Zorn-lemma {A} 0<A PO supP = zorn00 where |
535 | 215 supO : (C : Ordinal ) → (* C) ⊆ A → IsTotalOrderSet (* C) → Ordinal |
216 supO C C⊆A TC = & ( SUP.sup ( supP (* C) C⊆A TC )) | |
493 | 217 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
496 | 218 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 | 219 z01 {a} {b} A∋a A∋b (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO {me A∋b} {me A∋b} refl |
220 (IsStrictPartialOrder.trans PO {me A∋b} {me A∋a} {me A∋b} b<a a<b) | |
537 | 221 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
222 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
530 | 223 s : HOD |
224 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
225 sa : A ∋ * ( & s ) | |
226 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 )) ) | |
227 HasMaximal : HOD | |
537 | 228 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
229 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
230 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
532 | 231 Gtx : { x : HOD} → A ∋ x → HOD |
537 | 232 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
233 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
234 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
235 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
236 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)) | |
237 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 | |
238 ¬x<m : ¬ (* x < * m) | |
239 ¬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 | 240 |
241 -- Uncountable acending chain by axiom of choice | |
530 | 242 cf : ¬ Maximal A → Ordinal → Ordinal |
532 | 243 cf nmx x with ODC.∋-p O A (* x) |
244 ... | no _ = o∅ | |
245 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
538 | 246 ... | yes nogt = -- no larger element, so it is maximal |
247 ⊥-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 | 248 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
537 | 249 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
250 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
251 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
252 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
253 ... | 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 ⟫ ) | |
254 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
530 | 255 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
537 | 256 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
530 | 257 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
532 | 258 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
543 | 259 |
535 | 260 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A sa f mf supO (& A)) → SUP A (ZChain.chain zc) |
533 | 261 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
535 | 262 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) |
533 | 263 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
264 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
538 | 265 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A sa f mf supO (& A)) → SUP A (* (& (ZChain.chain zc))) |
266 sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) | |
267 (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) ) | |
543 | 268 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
269 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
270 | |
271 --- | |
272 --- sup is fix point in maximum chain | |
273 --- | |
538 | 274 z03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A sa f mf supO (& A)) |
275 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) | |
276 z03 f mf zc = z14 where | |
277 chain = ZChain.chain zc | |
278 sp1 = sp0 f mf zc | |
541 | 279 z10 : {a b : Ordinal } → (ca : odef chain a ) → (ab : odef A b ) → a o< (& A) |
280 → Prev< A chain ab f | |
281 ∨ (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 | 282 → * a < * b → odef chain b |
283 z10 = ZChain.is-max zc | |
543 | 284 z11 : & (SUP.sup sp1) o< & A |
285 z11 = c<→o< ( SUP.A∋maximal sp1) | |
538 | 286 z12 : odef chain (& (SUP.sup sp1)) |
287 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
288 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
289 ... | 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 | |
290 z13 : * (& s) < * (& (SUP.sup sp1)) | |
291 z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc )) | |
292 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) | |
293 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
294 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) | |
543 | 295 z14 with IsStrictTotalOrder.compare (ZChain.f-total zc ) (me (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 z11 ))) (me z12 ) |
538 | 296 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
297 z16 : ⊥ | |
298 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
299 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
300 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
301 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
302 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
303 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
543 | 304 z15 = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (ZChain.f-next zc z12 z11 ) ) |
538 | 305 z17 : ⊥ |
306 z17 with z15 | |
307 ... | case1 eq = ¬b eq | |
308 ... | case2 lt = ¬a lt | |
535 | 309 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥ |
537 | 310 z04 nmx zc = z01 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) |
538 | 311 (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
312 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) | |
313 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where | |
314 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc | |
315 c = & (SUP.sup sp1) | |
541 | 316 premax : {x y : Ordinal} → y o< x → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 : ZChain A sa f mf supO y ) |
543 | 317 → {a b : Ordinal} (ca : odef (ZChain.chain zc0) a) → (ab : odef A b) → a o< y |
318 → Prev< A (ZChain.chain zc0) ab f ∨ (supO (& (ZChain.chain zc0)) | |
319 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
320 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b) | |
541 | 321 → * a < * b → odef (ZChain.chain zc0) b |
543 | 322 premax {x} {y} y<x f mf zc0 {a} {b} ca ab a<y P a<b = ZChain.is-max zc0 ca ab a<y P a<b |
540 | 323 -- Union of ZFChain |
324 UZFChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (B : Ordinal) | |
325 → ( (y : Ordinal) → y o< B → ZChain A sa f mf supO y ) → HOD | |
326 UZFChain f mf B prev = record { od = record { def = λ y → odef A y ∧ (y o< B) ∧ ( (y<b : y o< B) → odef (ZChain.chain (prev y y<b)) y) } | |
327 ; odmax = & A ; <odmax = z07 } | |
478 | 328 -- ZChain is not compatible with the SUP condition |
535 | 329 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A sa f mf supO y ) |
330 → ZChain A sa f mf supO x | |
533 | 331 ind f mf x prev with Oprev-p x |
530 | 332 ... | yes op with ODC.∋-p O A (* x) |
333 ... | no ¬Ax = zc1 where | |
334 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain | |
335 px = Oprev.oprev op | |
535 | 336 zc0 : ZChain A sa f mf supO (Oprev.oprev op) |
533 | 337 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) |
535 | 338 zc1 : ZChain A sa f mf supO x |
543 | 339 zc1 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 |
340 ; f-next = zc20 (ZChain.f-next zc0) ; f-immediate = ZChain.f-immediate zc0 | |
341 ; ¬chain∋x>z = λ {a} x<oa → ZChain.¬chain∋x>z zc0 (ordtrans (subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc ) x<oa ) | |
342 ; chain∋x = ZChain.chain∋x zc0 ; is-max = λ za ba a<x → zc20 (λ za a<x → ZChain.is-max zc0 za ba a<x ) za a<x } where | |
343 zc20 : {P : Ordinal → Set n} → ({a : Ordinal} → odef (ZChain.chain zc0) a → a o< px → P a) | |
344 → {a : Ordinal} → (za : odef (ZChain.chain zc0) a ) → (a<x : a o< x) → P a | |
345 zc20 {P} prev {a} za a<x with trio< a px | |
346 ... | tri< a₁ ¬b ¬c = prev za a₁ | |
347 ... | tri≈ ¬a b ¬c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (subst (λ k → k o< osuc a) b <-osuc ) za ) | |
348 ... | tri> ¬a ¬b c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (ordtrans c <-osuc ) za ) | |
530 | 349 ... | yes ax = zc4 where -- we have previous ordinal and A ∋ x |
350 px = Oprev.oprev op | |
535 | 351 zc0 : ZChain A sa f mf supO (Oprev.oprev op) |
533 | 352 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) |
543 | 353 Afx : { x : Ordinal } → A ∋ * x → A ∋ * (f x) |
354 Afx {x} ax = (subst (λ k → odef A k ) (sym &iso) (proj2 (mf x (subst (λ k → odef A k ) &iso ax)))) | |
533 | 355 -- x is in the previous chain, use the same |
356 -- x has some y which y < x ∧ f y ≡ x | |
357 -- x has no y which y < x | |
535 | 358 zc4 : ZChain A sa f mf supO x |
541 | 359 zc4 with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) ax f ) |
542 | 360 ... | case1 y = zc7 where -- we have previous < |
541 | 361 chain = ZChain.chain zc0 |
540 | 362 zc7 : ZChain A sa f mf supO x |
363 zc7 with ODC.∋-p O (ZChain.chain zc0) (* ( f x ) ) | |
542 | 364 ... | yes y = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = {!!} |
541 | 365 ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} } -- no extention |
540 | 366 ... | no not = record { chain = zc5 ; chain⊆A = ⊆-zc5 |
541 | 367 ; f-total = zc6 ; f-next = {!!} ; f-immediate = {!!} ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where |
542 | 368 -- extend with f x -- cahin ∋ y ∧ chain ∋ f y ∧ x ≡ f ( f y ) |
540 | 369 zc5 : HOD |
541 | 370 zc5 = record { od = record { def = λ z → odef (ZChain.chain zc0) z ∨ (z ≡ f x) } ; odmax = & A ; <odmax = {!!} } |
540 | 371 ⊆-zc5 : zc5 ⊆ A |
543 | 372 ⊆-zc5 = record { incl = λ {y} lt → zc15 lt } where |
373 zc15 : {z : Ordinal } → ( odef (ZChain.chain zc0) z ∨ (z ≡ f x) ) → odef A z | |
374 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 ) ) | |
375 zc15 (case2 refl) = proj2 ( mf x (subst (λ k → odef A k ) &iso ax ) ) | |
540 | 376 IPO = ⊆-IsPartialOrderSet ⊆-zc5 PO |
542 | 377 zc8 : { A B x : HOD } → (ax : A ∋ x ) → (P : Prev< A B ax f ) → * (f (& (* (Prev<.y P)))) ≡ x |
378 zc8 {A} {B} {x} ax P = subst₂ (λ j k → * ( f j ) ≡ k ) (sym &iso) *iso (sym (cong (*) ( Prev<.x=fy P))) | |
379 fx=zc : odef (ZChain.chain zc0) x → Tri (* (f x) < * x ) (* (f x) ≡ * x) (* x < * (f x) ) | |
380 fx=zc c with mf x (subst (λ k → odef A k) &iso ax ) | |
543 | 381 ... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ax (Afx ax) (case1 (sym zc13))) zc13 (z01 (Afx ax) ax (case1 zc13)) where |
542 | 382 zc13 : * (f x) ≡ * x |
383 zc13 = subst (λ k → k ≡ * x ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso))) (sym ( x=fx )) | |
543 | 384 ... | ⟪ case2 x<fx , afx ⟫ = tri> (z01 ax (Afx ax) (case2 zc14)) (λ lt → z01 (Afx ax) ax (case1 lt) zc14) zc14 where |
542 | 385 zc14 : * x < * (f x) |
543 | 386 zc14 = subst (λ k → * x < k ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx |
540 | 387 cmp : Trichotomous _ _ |
541 | 388 cmp record { elm = a ; is-elm = aa } record { elm = b ; is-elm = ab } with aa | ab |
389 ... | case1 x | case1 x₁ = IsStrictTotalOrder.compare (ZChain.f-total zc0) (me x) (me x₁) | |
542 | 390 ... | case2 fx | case2 fx₁ = tri≈ {!!} (subst₂ (λ j k → j ≡ k ) *iso *iso (trans (cong (*) fx) (sym (cong (*) fx₁ ) ))) {!!} |
391 ... | case1 c | case2 fx = {!!} -- subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) {!!} {!!} ( fx>zc (subst (λ k → odef chain k) {!!} c )) | |
392 ... | case2 fx | case1 c with ODC.p∨¬p O ( Prev< A chain (incl (ZChain.chain⊆A zc0) c) f ) | |
393 ... | case2 n = {!!} | |
394 ... | case1 fb with IsStrictTotalOrder.compare (ZChain.f-total zc0) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay y))) (me (subst (λ k → odef chain k) (sym &iso) (Prev<.ay fb))) | |
395 ... | tri< a₁ ¬b ¬c = {!!} | |
543 | 396 ... | tri≈ ¬a b₁ ¬c = subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) zc11 zc10 ( fx=zc zc12 ) where |
542 | 397 zc10 : * x ≡ b |
398 zc10 = subst₂ (λ j k → j ≡ k ) (zc8 ax y ) (zc8 (incl ( ZChain.chain⊆A zc0 ) c) fb) (cong (λ k → * ( f ( & k ))) b₁) | |
543 | 399 zc11 : * (f x) ≡ a |
400 zc11 = subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym fx)) | |
401 zc12 : odef chain x | |
402 zc12 = subst (λ k → odef chain k ) (subst (λ k → & b ≡ k ) &iso (sym (cong (&) zc10))) c | |
542 | 403 ... | tri> ¬a ¬b c₁ = {!!} |
540 | 404 zc6 : IsTotalOrderSet zc5 |
405 zc6 = record { isEquivalence = IsStrictPartialOrder.isEquivalence IPO ; trans = λ {x} {y} {z} → IsStrictPartialOrder.trans IPO {x} {y} {z} | |
406 ; compare = cmp } | |
407 ... | case2 not with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) {!!} {!!} ) )) | |
542 | 408 ... | case1 y = {!!} -- x is sup |
409 ... | case2 not = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = {!!} | |
541 | 410 ; f-immediate = {!!} ; chain∋x = ZChain.chain∋x zc0 ; is-max = {!!} } -- no extention |
533 | 411 ind f mf x prev | no ¬ox with trio< (& A) x --- limit ordinal case |
543 | 412 ... | tri< a ¬b ¬c = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 |
413 ; f-next = {!!} | |
414 ; f-immediate = {!!} ; chain∋x = ZChain.chain∋x zc0 ; is-max = {!!} } where | |
533 | 415 zc0 = prev (& A) a |
416 ... | tri≈ ¬a b ¬c = {!!} | |
543 | 417 ... | tri> ¬a ¬b c = record { chain = uzc ; chain⊆A = record { incl = λ {x} lt → proj1 lt } ; f-total = {!!} ; f-next = {!!} |
418 ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} } where | |
419 uzc : HOD | |
420 uzc = UZFChain f mf x prev | |
530 | 421 zorn00 : Maximal A |
531 | 422 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM |
530 | 423 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where |
424 -- yes we have the maximal | |
425 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
426 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
427 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
531 | 428 zorn01 = proj1 zorn03 |
530 | 429 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) |
531 | 430 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) |
533 | 431 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (zorn03 (cf nmx) (cf-is-≤-monotonic nmx))) where |
530 | 432 -- if we have no maximal, make ZChain, which contradict SUP condition |
533 | 433 nmx : ¬ Maximal A |
434 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
531 | 435 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) |
436 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 | 437 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → ZChain A sa f mf supO (& A) |
533 | 438 zorn03 f mf = TransFinite (ind f mf) (& A) |
464 | 439 |
516 | 440 -- usage (see filter.agda ) |
441 -- | |
497 | 442 -- _⊆'_ : ( A B : HOD ) → Set n |
443 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
482 | 444 |
497 | 445 -- MaximumSubset : {L P : HOD} |
446 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
447 -- → IsPartialOrderSet P _⊆'_ | |
448 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
449 -- → Maximal P (_⊆'_) | |
450 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |