Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 563:d94f90607a7c
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 30 Apr 2022 17:07:35 +0900 |
parents | 42ad203ff913 |
children | b8eb708dec3c |
rev | line source |
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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 |
560 | 10 -- |
11 -- Zorn-lemma : { A : HOD } | |
12 -- → o∅ o< & A | |
13 -- → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition | |
14 -- → Maximal A | |
15 -- | |
16 | |
431 | 17 open import zf |
477 | 18 open import logic |
19 -- open import partfunc {n} O | |
20 | |
21 open import Relation.Nullary | |
22 open import Data.Empty | |
23 import BAlgbra | |
431 | 24 |
555 | 25 open import Data.Nat hiding ( _<_ ; _≤_ ) |
26 open import Data.Nat.Properties | |
27 open import nat | |
28 | |
431 | 29 |
30 open inOrdinal O | |
31 open OD O | |
32 open OD.OD | |
33 open ODAxiom odAxiom | |
477 | 34 import OrdUtil |
35 import ODUtil | |
431 | 36 open Ordinals.Ordinals O |
37 open Ordinals.IsOrdinals isOrdinal | |
38 open Ordinals.IsNext isNext | |
39 open OrdUtil O | |
477 | 40 open ODUtil O |
41 | |
42 | |
43 import ODC | |
44 | |
45 | |
46 open _∧_ | |
47 open _∨_ | |
48 open Bool | |
431 | 49 |
50 | |
51 open HOD | |
52 | |
560 | 53 -- |
54 -- Partial Order on HOD ( possibly limited in A ) | |
55 -- | |
56 | |
528
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TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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57 _≤_ : (x y : HOD) → Set (Level.suc n) |
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TransitiveClosure with x <= f x is possible
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parents:
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58 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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59 |
554 | 60 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
61 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
62 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
63 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
64 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
65 | |
556 | 66 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
67 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
68 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
69 (IsStrictPartialOrder.trans PO b<a a<b) | |
490 | 70 |
561 | 71 ptrans = IsStrictPartialOrder.trans PO |
72 | |
492 | 73 open _==_ |
74 open _⊆_ | |
75 | |
530 | 76 -- |
560 | 77 -- Closure of ≤-monotonic function f has total order |
530 | 78 -- |
79 | |
80 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
81 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
82 | |
560 | 83 -- immieate-f : (A : HOD) → ( f : Ordinal → Ordinal ) → Set n |
84 -- immieate-f A f = { x y : Ordinal } → odef A x → odef A y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
556 | 85 |
551 | 86 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
554 | 87 init : odef A s → FClosure A f s s |
555 | 88 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
554 | 89 |
556 | 90 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y |
91 A∋fc {A} s f mf (init as) = as | |
92 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) | |
555 | 93 |
556 | 94 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
95 s≤fc {A} s {.s} f mf (init x) = case1 refl | |
96 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) | |
97 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
98 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
99 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
100 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
555 | 101 |
557 | 102 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
103 fcn s mf (init as) = zero | |
558 | 104 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) |
105 ... | case1 eq = fcn s mf p | |
106 ... | case2 y<fy = suc (fcn s mf p ) | |
557 | 107 |
558 | 108 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
109 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
559 | 110 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where |
111 fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y | |
112 fc00 zero zero refl (init _) (init x₁) i=x i=y = refl | |
113 fc00 zero zero refl (init sa) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
114 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init sa) cy i=x i=y ) | |
115 fc00 zero zero refl (fsuc x cx) (init sa) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
116 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init sa) i=x i=y ) | |
117 fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
118 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) | |
119 fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
120 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) | |
121 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
122 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
123 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
560 | 124 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x |
559 | 125 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where |
126 fc04 : * x1 ≡ * y | |
127 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
128 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
129 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
130 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
131 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
132 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
133 fc05 : * x ≡ * y1 | |
134 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
135 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) | |
557 | 136 |
137 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
138 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
558 | 139 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where |
140 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y | |
141 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
142 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
143 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
144 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
145 ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy | |
146 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
147 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
148 fc03 eq = cong pred eq | |
149 fc02 : * x < * y1 | |
150 fc02 = fc01 i cx cy (fc03 i=y ) a | |
557 | 151 |
559 | 152 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
554 | 153 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
559 | 154 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
155 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
156 fc11 : * x < * y | |
157 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
158 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
159 fc10 : * x ≡ * y | |
160 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
161 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
162 fc12 : * y < * x | |
163 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
164 | |
562 | 165 fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
166 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
563 | 167 fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where |
168 fc20 : fcn s mf cy Data.Nat.< suc (fcn s mf cx) → (fcn s mf cy ≡ fcn s mf cx) ∨ ( fcn s mf cy Data.Nat.< fcn s mf cx ) | |
169 fc20 y<sx with <-cmp ( fcn s mf cy ) (fcn s mf cx ) | |
170 ... | tri< a ¬b ¬c = case2 a | |
171 ... | tri≈ ¬a b ¬c = case1 b | |
172 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> y<sx (s≤s c)) | |
173 fc17 : {x y : Ordinal } → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → suc (fcn s mf cx) ≡ fcn s mf cy → * (f x ) ≡ * y | |
174 fc17 {x} {y} cx cy sx=y = fc18 (fcn s mf cy) cx cy refl sx=y where | |
175 fc18 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → suc (fcn s mf cx) ≡ i → * (f x) ≡ * y | |
176 fc18 (suc i) {y} cx (fsuc y1 cy) i=y sx=i with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
177 ... | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy ( fc18 (suc i) {y1} cx cy i=y sx=i) -- dereference | |
178 ... | case2 y<fy = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k) ) ) fc19) where | |
179 fc19 : * x ≡ * y1 | |
180 fc19 = fcn-inject s mf cx cy (cong pred ( trans sx=i i=y )) | |
181 fc21 : ⊥ | |
182 fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy ) | |
183 ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy | |
184 cxx : FClosure A f s (f x) | |
185 cxx = fsuc x cx | |
186 fc16 : (x : Ordinal ) → (cx : FClosure A f s x) → (fcn s mf cx ≡ fcn s mf (fsuc x cx)) ∨ ( suc (fcn s mf cx ) ≡ fcn s mf (fsuc x cx)) | |
187 fc16 x (init sa) with proj1 (mf s sa ) | |
188 ... | case1 _ = case1 refl | |
189 ... | case2 _ = case2 refl | |
190 fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) | |
191 ... | case1 _ = case1 refl | |
192 ... | case2 _ = case2 refl | |
193 fc22 : (suc ( fcn s mf cx )) Data.Nat.< (fcn s mf cy ) → * (f x) < * y | |
194 fc22 a with fc16 x cx | |
195 ... | case1 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq (<-trans a<sa a)) | |
196 ... | case2 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq a ) | |
197 ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx | |
198 ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx | |
199 ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y | |
200 ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) | |
201 | |
562 | 202 |
560 | 203 -- open import Relation.Binary.Properties.Poset as Poset |
204 | |
205 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
206 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
207 | |
208 | |
209 record Maximal ( A : HOD ) : Set (Level.suc n) where | |
210 field | |
211 maximal : HOD | |
212 A∋maximal : A ∋ maximal | |
213 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
214 | |
215 -- | |
216 -- inductive maxmum tree from x | |
217 -- tree structure | |
218 -- | |
554 | 219 |
541 | 220 record Prev< (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
533 | 221 field |
534 | 222 y : Ordinal |
541 | 223 ay : odef B y |
534 | 224 x=fy : x ≡ f y |
529 | 225 |
508 | 226 record SUP ( A B : HOD ) : Set (Level.suc n) where |
503 | 227 field |
228 sup : HOD | |
229 A∋maximal : A ∋ sup | |
230 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
231 | |
533 | 232 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
233 SupCond A B _ _ = SUP A B | |
234 | |
546 | 235 record ZChain ( A : HOD ) {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) |
547 | 236 (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where |
533 | 237 field |
238 chain : HOD | |
239 chain⊆A : chain ⊆ A | |
538 | 240 chain∋x : odef chain x |
561 | 241 initial : {y : Ordinal } → odef chain y → * x ≤ * y |
533 | 242 f-total : IsTotalOrderSet chain |
546 | 243 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
541 | 244 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) |
548 | 245 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< z → (ba : odef A b) |
541 | 246 → Prev< A chain ba f |
247 ∨ (sup (& chain) (subst (λ k → k ⊆ A) (sym *iso) chain⊆A) (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b ) | |
534 | 248 → * a < * b → odef chain b |
533 | 249 |
497 | 250 Zorn-lemma : { A : HOD } |
464 | 251 → o∅ o< & A |
497 | 252 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
253 → Maximal A | |
552 | 254 Zorn-lemma {A} 0<A supP = zorn00 where |
535 | 255 supO : (C : Ordinal ) → (* C) ⊆ A → IsTotalOrderSet (* C) → Ordinal |
256 supO C C⊆A TC = & ( SUP.sup ( supP (* C) C⊆A TC )) | |
493 | 257 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
556 | 258 z01 {a} {b} A∋a A∋b = <-irr |
537 | 259 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
260 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
530 | 261 s : HOD |
262 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
263 sa : A ∋ * ( & s ) | |
264 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 | 265 s<A : & s o< & A |
266 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso sa ) | |
530 | 267 HasMaximal : HOD |
537 | 268 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
269 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
270 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
532 | 271 Gtx : { x : HOD} → A ∋ x → HOD |
537 | 272 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
273 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
274 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
275 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
276 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)) | |
277 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 | |
278 ¬x<m : ¬ (* x < * m) | |
279 ¬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 | 280 |
560 | 281 -- Uncountable ascending chain by axiom of choice |
530 | 282 cf : ¬ Maximal A → Ordinal → Ordinal |
532 | 283 cf nmx x with ODC.∋-p O A (* x) |
284 ... | no _ = o∅ | |
285 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
538 | 286 ... | yes nogt = -- no larger element, so it is maximal |
287 ⊥-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 | 288 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
537 | 289 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
290 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
291 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
292 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
293 ... | 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 ⟫ ) | |
294 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
530 | 295 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
537 | 296 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
530 | 297 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
532 | 298 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
543 | 299 |
547 | 300 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A sa f mf supO (& A) ) → SUP A (ZChain.chain zc) |
533 | 301 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
547 | 302 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) ) |
533 | 303 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
304 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
547 | 305 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 | 306 sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) |
307 (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) ) | |
543 | 308 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
309 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
310 | |
311 --- | |
560 | 312 --- the maximum chain has fix point of any ≤-monotonic function |
543 | 313 --- |
547 | 314 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 | 315 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) |
538 | 316 z03 f mf zc = z14 where |
317 chain = ZChain.chain zc | |
318 sp1 = sp0 f mf zc | |
548 | 319 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
541 | 320 → Prev< A chain ab f |
321 ∨ (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 | 322 → * a < * b → odef chain b |
323 z10 = ZChain.is-max zc | |
543 | 324 z11 : & (SUP.sup sp1) o< & A |
325 z11 = c<→o< ( SUP.A∋maximal sp1) | |
538 | 326 z12 : odef chain (& (SUP.sup sp1)) |
327 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
328 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
548 | 329 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) z11 (SUP.A∋maximal sp1) (case2 refl ) z13 where |
538 | 330 z13 : * (& s) < * (& (SUP.sup sp1)) |
331 z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc )) | |
332 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) | |
333 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
334 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) | |
552 | 335 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
538 | 336 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
337 z16 : ⊥ | |
338 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
339 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
340 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
341 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
342 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
343 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
546 | 344 z15 = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (ZChain.f-next zc z12 )) |
538 | 345 z17 : ⊥ |
346 z17 with z15 | |
347 ... | case1 eq = ¬b eq | |
348 ... | case2 lt = ¬a lt | |
560 | 349 |
350 -- ZChain contradicts ¬ Maximal | |
351 -- | |
352 -- ZChain forces fix point on any ≤-monotonic function (z03) | |
353 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
354 -- | |
547 | 355 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥ |
537 | 356 z04 nmx zc = z01 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) |
538 | 357 (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
358 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) | |
359 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where | |
546 | 360 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc |
538 | 361 c = & (SUP.sup sp1) |
548 | 362 |
560 | 363 -- |
547 | 364 -- create all ZChains under o< x |
560 | 365 -- |
546 | 366 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → |
547 | 367 ((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 |
368 ind f mf x prev {y} ay with Oprev-p x | |
548 | 369 ... | yes op = zc4 where |
560 | 370 -- |
371 -- we have previous ordinal to use induction | |
372 -- | |
530 | 373 px = Oprev.oprev op |
547 | 374 zc0 : ZChain A ay f mf supO (Oprev.oprev op) |
375 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay | |
376 zc4 : ZChain A ay f mf supO x | |
551 | 377 zc4 with ODC.∋-p O A (* px) |
560 | 378 ... | no noapx = -- ¬ A ∋ px, just skip |
379 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 | |
554 | 380 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 |
551 | 381 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 } where -- no extention |
382 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
383 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
384 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
385 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) ≡ b) → | |
386 * a < * b → odef (ZChain.chain zc0) b | |
387 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) | |
388 ... | case1 eq = ⊥-elim ( noapx (subst (λ k → odef A k) (trans eq (sym &iso) ) ba )) | |
389 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
390 ... | 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 | 391 ... | case1 pr = zc9 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
551 | 392 chain = ZChain.chain zc0 |
393 zc17 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
394 Prev< A (ZChain.chain zc0) ba f ∨ (supO (& (ZChain.chain zc0)) | |
395 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
396 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b) → | |
397 * a < * b → odef (ZChain.chain zc0) b | |
398 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) | |
399 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
400 ... | 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 | 401 zc9 : ZChain A ay f mf supO x |
402 zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
554 | 403 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 } -- no extention |
551 | 404 ... | case2 ¬fy<x with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) )) |
560 | 405 ... | case1 x=sup = -- previous ordinal is a sup of a smaller ZChain |
561 | 406 record { chain = schain ; chain⊆A = record { incl = A∋schain } ; f-total = scmp ; f-next = scnext |
562 | 407 ; initial = scinit ; f-immediate = simm ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup |
561 | 408 sup0 = supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) |
409 sp = SUP.sup sup0 | |
551 | 410 chain = ZChain.chain zc0 |
561 | 411 sc<A : {y : Ordinal} → odef chain y ∨ FClosure A f (& sp) y → y o< & A |
412 sc<A {y} (case1 zx) = subst (λ k → k o< (& A)) &iso ( c<→o< ( incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k) (sym &iso) zx ))) | |
413 sc<A {y} (case2 fx) = subst (λ k → k o< (& A)) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso) (A∋fc (& sp) f mf fx )) ) | |
552 | 414 schain : HOD |
561 | 415 schain = record { od = record { def = λ x → odef chain x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = λ {y} sy → sc<A {y} sy } |
416 A∋schain : {x : HOD } → schain ∋ x → A ∋ x | |
417 A∋schain (case1 zx ) = subst (λ k → odef A k ) &iso (incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k) (sym &iso) zx )) | |
418 A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx | |
419 cmp : {a b : HOD} (za : odef chain (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) | |
420 cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb | |
421 ... | case1 sp=a | case1 sp=b = tri≈ (λ lt → <-irr (case1 (sym eq)) lt ) eq (λ lt → <-irr (case1 eq) lt ) where | |
422 eq : a ≡ b | |
423 eq = trans sp=a (subst₂ (λ j k → j ≡ k ) *iso *iso sp=b ) | |
424 ... | case1 sp=a | case2 sp<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b) lt ) where | |
425 a<b : a < b | |
426 a<b = subst (λ k → k < b ) (sym sp=a) (subst₂ (λ j k → j < k ) *iso *iso sp<b ) | |
427 ... | case2 a<sp | case1 sp=b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b) lt ) where | |
428 a<b : a < b | |
429 a<b = subst (λ k → a < k ) (trans sp=b *iso ) (subst (λ k → a < k ) (sym *iso) a<sp ) | |
430 ... | case2 a<sp | case2 sp<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b) lt ) where | |
431 a<b : a < b | |
432 a<b = ptrans (subst (λ k → a < k ) (sym *iso) a<sp ) ( subst₂ (λ j k → j < k ) refl *iso sp<b ) | |
433 scmp : {a b : HOD} → odef schain (& a) → odef schain (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
434 scmp (case1 za) (case1 zb) = ZChain.f-total zc0 za zb | |
435 scmp {a} {b} (case1 za) (case2 fb) = cmp za fb | |
436 scmp (case2 fa) (case1 zb) with cmp zb fa | |
437 ... | tri< a ¬b ¬c = tri> ¬c (λ eq → ¬b (sym eq)) a | |
438 ... | tri≈ ¬a b ¬c = tri≈ ¬c (sym b) ¬a | |
439 ... | tri> ¬a ¬b c = tri< c (λ eq → ¬b (sym eq)) ¬a | |
440 scmp (case2 fa) (case2 fb) = subst₂ (λ a b → Tri (a < b) (a ≡ b) (b < a ) ) *iso *iso (fcn-cmp (& sp) f mf fa fb) | |
441 scnext : {a : Ordinal} → odef schain a → odef schain (f a) | |
442 scnext {x} (case1 zx) = case1 (ZChain.f-next zc0 zx) | |
443 scnext {x} (case2 sx) = case2 ( fsuc x sx ) | |
444 scinit : {x : Ordinal} → odef schain x → * y ≤ * x | |
445 scinit {x} (case1 zx) = ZChain.initial zc0 zx | |
446 scinit {x} (case2 sx) with (s≤fc (& sp) f mf sx ) | SUP.x<sup sup0 (subst (λ k → odef chain k ) (sym &iso) ( ZChain.chain∋x zc0 ) ) | |
562 | 447 ... | case1 sp=x | case1 y=sp = case1 (trans y=sp (subst (λ k → k ≡ * x ) *iso sp=x ) ) |
448 ... | case1 sp=x | case2 y<sp = case2 (subst (λ k → * y < k ) (trans (sym *iso) sp=x) y<sp ) | |
449 ... | case2 sp<x | case1 y=sp = case2 (subst (λ k → k < * x ) (trans *iso (sym y=sp )) sp<x ) | |
450 ... | case2 sp<x | case2 y<sp = case2 (ptrans y<sp (subst (λ k → k < * x ) *iso sp<x) ) | |
451 A∋za : {a : Ordinal } → odef chain a → odef A a | |
452 A∋za za = (subst (λ k → odef A k ) &iso (incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k ) (sym &iso) za) ) ) | |
453 za<sup : {a : Ordinal } → odef chain a → ( * a ≡ sp ) ∨ ( * a < sp ) | |
454 za<sup za = SUP.x<sup sup0 (subst (λ k → odef chain k ) (sym &iso) za ) | |
455 simm : {a b : Ordinal} → odef schain a → odef schain b → ¬ (* a < * b) ∧ (* b < * (f a)) | |
456 simm {a} {b} (case1 za) (case1 zb) = ZChain.f-immediate zc0 za zb | |
457 simm {a} {b} (case1 za) (case2 sb) p with proj1 (mf a (A∋za za) ) | |
458 ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) | |
459 ... | case2 a<fa with za<sup ( ZChain.f-next zc0 za ) | s≤fc (& sp) f mf sb | |
460 ... | case1 fa=sp | case1 sp=b = <-irr (case1 (trans fa=sp (trans (sym *iso) sp=b )) ) ( proj2 p ) | |
461 ... | case2 fa<sp | case1 sp=b = <-irr (case2 fa<sp) (subst (λ k → k < * (f a) ) (trans (sym sp=b) *iso) (proj2 p ) ) | |
462 ... | case1 fa=sp | case2 sp<b = <-irr (case2 (proj2 p )) (subst (λ k → k < * b) (sym fa=sp) (subst (λ k → k < * b ) *iso sp<b ) ) | |
463 ... | case2 fa<sp | case2 sp<b = <-irr (case2 (proj2 p )) (ptrans fa<sp (subst (λ k → k < * b ) *iso sp<b ) ) | |
464 simm {a} {b} (case2 sa) (case1 zb) p with proj1 (mf a ( subst (λ k → odef A k) &iso ( A∋schain (case2 (subst (λ k → FClosure A f (& sp) k ) (sym &iso) sa) )) ) ) | |
465 ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) | |
466 ... | case2 b<fb with s≤fc (& sp) f mf sa | za<sup zb | |
467 ... | case1 sp=a | case1 b=sp = <-irr (case1 (trans b=sp (trans (sym *iso) sp=a )) ) (proj1 p ) | |
468 ... | case1 sp=a | case2 b<sp = <-irr (case2 (subst (λ k → * b < k ) (trans (sym *iso) sp=a) b<sp ) ) (proj1 p ) | |
469 ... | case2 sp<a | case1 b=sp = <-irr (case2 (subst ( λ k → k < * a ) (trans *iso (sym b=sp)) sp<a )) (proj1 p ) | |
470 ... | case2 sp<a | case2 b<sp = <-irr (case2 (ptrans b<sp (subst (λ k → k < * a) *iso sp<a ))) (proj1 p ) | |
471 simm {a} {b} (case2 sa) (case2 sb) p = {!!} | |
560 | 472 ... | case2 ¬x=sup = -- x is not f y' nor sup of former ZChain from y |
473 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
554 | 474 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = {!!} } where -- no extention |
552 | 475 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → |
476 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
477 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
478 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) | |
479 ≡ b) → | |
480 * a < * b → odef (ZChain.chain zc0) b | |
481 z18 {a} {b} za b<x ba (case1 p) a<b = {!!} | |
482 z18 {a} {b} za b<x ba (case2 p) a<b = {!!} | |
553 | 483 ... | no ¬ox = {!!} where --- limit ordinal case |
554 | 484 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x |
553 | 485 field |
486 u : Ordinal | |
487 u<x : u o< x | |
554 | 488 zuy : odef (ZChain.chain (prev u u<x {y} ay )) z |
489 Uz : HOD | |
490 Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A ; <odmax = {!!} } | |
491 u-total : IsTotalOrderSet Uz | |
553 | 492 u-total {x} {y} ux uy = {!!} |
560 | 493 --- ux ⊆ uy ∨ uy ⊆ ux |
553 | 494 |
551 | 495 zorn00 : Maximal A |
496 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
497 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
498 -- yes we have the maximal | |
499 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
500 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
501 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
502 zorn01 = proj1 zorn03 | |
503 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
504 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
505 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where | |
506 -- if we have no maximal, make ZChain, which contradict SUP condition | |
507 nmx : ¬ Maximal A | |
508 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
509 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
510 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) ) ⟫ | |
511 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO (& A) | |
512 zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain A ya f mf supO z } (ind f mf) (& A) | |
513 zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) | |
514 zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa ) | |
515 | |
516 | 516 -- usage (see filter.agda ) |
517 -- | |
497 | 518 -- _⊆'_ : ( A B : HOD ) → Set n |
519 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
482 | 520 |
497 | 521 -- MaximumSubset : {L P : HOD} |
522 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
523 -- → IsPartialOrderSet P _⊆'_ | |
524 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
525 -- → Maximal P (_⊆'_) | |
526 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |