Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 839:710574600659
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 28 Aug 2022 14:46:18 +0900 |
parents | 1a5bb940fceb |
children | 52bff0b4cb37 |
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 open _∧_ | |
46 open _∨_ | |
47 open Bool | |
431 | 48 |
49 open HOD | |
50 | |
560 | 51 -- |
52 -- Partial Order on HOD ( possibly limited in A ) | |
53 -- | |
54 | |
571 | 55 _<<_ : (x y : Ordinal ) → Set n -- Set n order |
570 | 56 x << y = * x < * y |
57 | |
765 | 58 _<=_ : (x y : Ordinal ) → Set n -- Set n order |
59 x <= y = (x ≡ y ) ∨ ( * x < * y ) | |
60 | |
570 | 61 POO : IsStrictPartialOrder _≡_ _<<_ |
62 POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
63 ; trans = IsStrictPartialOrder.trans PO | |
64 ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y | |
65 ; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } } | |
66 | |
528
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67 _≤_ : (x y : HOD) → Set (Level.suc n) |
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68 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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69 |
554 | 70 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
71 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
72 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
73 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
74 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
75 | |
779 | 76 <-ftrans : {x y z : Ordinal } → x <= y → y <= z → x <= z |
77 <-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
78 <-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
79 <-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
80 <-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
81 | |
770 | 82 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y |
83 <=to≤ (case1 eq) = case1 (cong (*) eq) | |
84 <=to≤ (case2 lt) = case2 lt | |
85 | |
779 | 86 ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y |
87 ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) | |
88 ≤to<= (case2 lt) = case2 lt | |
89 | |
556 | 90 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
91 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
92 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
93 (IsStrictPartialOrder.trans PO b<a a<b) | |
490 | 94 |
561 | 95 ptrans = IsStrictPartialOrder.trans PO |
96 | |
492 | 97 open _==_ |
98 open _⊆_ | |
99 | |
530 | 100 -- |
560 | 101 -- Closure of ≤-monotonic function f has total order |
530 | 102 -- |
103 | |
104 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
105 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
106 | |
551 | 107 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
783 | 108 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
555 | 109 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
554 | 110 |
556 | 111 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 |
783 | 112 A∋fc {A} s f mf (init as refl ) = as |
556 | 113 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
555 | 114 |
714 | 115 A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s |
783 | 116 A∋fcs {A} s f mf (init as refl) = as |
714 | 117 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
118 | |
556 | 119 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
783 | 120 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
556 | 121 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
122 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
123 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
124 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
125 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
555 | 126 |
800 | 127 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
128 fcn s mf (init as refl) = zero | |
129 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
130 ... | case1 eq = fcn s mf p | |
131 ... | case2 y<fy = suc (fcn s mf p ) | |
132 | |
133 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
134 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
135 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 | |
136 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
137 fc06 {x} {y} refl {j} not = fc08 not where | |
138 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
139 fc08 () | |
140 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
141 fc07 {x} (init as refl) eq = refl | |
142 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
143 ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) | |
144 -- ... | case2 x<fx = ? | |
145 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 | |
146 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
147 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
148 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
149 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
150 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y ) | |
151 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
152 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) i=x i=y ) | |
153 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 ) ) | |
154 ... | 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 ) | |
155 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 ) ) | |
156 ... | 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 ) | |
157 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
158 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
159 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
160 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
161 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x | |
162 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
163 fc04 : * x1 ≡ * y | |
164 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
165 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
166 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
167 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
168 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
169 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
170 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
171 fc05 : * x ≡ * y1 | |
172 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
173 ... | 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))) | |
174 | |
175 | |
176 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
177 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
178 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
179 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
180 fc06 {x} {y} refl {j} not = fc08 not where | |
181 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
182 fc08 () | |
183 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 | |
184 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
185 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
186 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
187 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
188 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
189 ... | 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 | |
190 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
191 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
192 fc03 eq = cong pred eq | |
193 fc02 : * x < * y1 | |
194 fc02 = fc01 i cx cy (fc03 i=y ) a | |
195 | |
557 | 196 |
559 | 197 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
554 | 198 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
800 | 199 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
200 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
201 fc11 : * x < * y | |
202 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
203 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
204 fc10 : * x ≡ * y | |
205 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
206 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
207 fc12 : * y < * x | |
208 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
600 | 209 |
563 | 210 |
729 | 211 |
560 | 212 -- open import Relation.Binary.Properties.Poset as Poset |
213 | |
214 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
215 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
216 | |
567 | 217 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
568 | 218 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
567 | 219 |
568 | 220 _⊆'_ : ( A B : HOD ) → Set n |
221 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
560 | 222 |
223 -- | |
224 -- inductive maxmum tree from x | |
225 -- tree structure | |
226 -- | |
554 | 227 |
836 | 228 record HasPrev (A B : HOD) (x : Ordinal ) ( f : Ordinal → Ordinal ) : Set n where |
533 | 229 field |
836 | 230 ax : odef A x |
534 | 231 y : Ordinal |
541 | 232 ay : odef B y |
534 | 233 x=fy : x ≡ f y |
529 | 234 |
570 | 235 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
654 | 236 field |
779 | 237 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
568 | 238 |
656 | 239 record SUP ( A B : HOD ) : Set (Level.suc n) where |
240 field | |
241 sup : HOD | |
804 | 242 as : A ∋ sup |
656 | 243 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive |
244 | |
690 | 245 -- |
246 -- sup and its fclosure is in a chain HOD | |
247 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
248 -- whole chain is a union of separated Chain | |
803 | 249 -- minimum index is sup of y not ϕ |
690 | 250 -- |
251 | |
787 | 252 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where |
690 | 253 field |
765 | 254 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
828 | 255 order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
256 supu=u : supf u ≡ u | |
694 | 257 |
748 | 258 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
259 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where | |
260 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
791 | 261 ch-is-sup : (u : Ordinal) {z : Ordinal } (u≤x : u o≤ x) ( is-sup : ChainP A f mf ay supf u ) |
748 | 262 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z |
694 | 263 |
264 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A | |
265 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
266 | |
803 | 267 -- Union of supf z which o< x |
268 -- | |
694 | 269 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
270 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
271 UnionCF A f mf ay supf x | |
272 = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x z } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } | |
662 | 273 |
703 | 274 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
783 | 275 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
655 | 276 field |
694 | 277 supf : Ordinal → Ordinal |
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278 chain : HOD |
703 | 279 chain = UnionCF A f mf ay supf z |
568 | 280 field |
281 chain⊆A : chain ⊆' A | |
783 | 282 chain∋init : odef chain y |
283 initial : {z : Ordinal } → odef chain z → * y ≤ * z | |
568 | 284 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
654 | 285 f-total : IsTotalOrderSet chain |
756 | 286 |
832 | 287 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
817 | 288 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z → IsSup A (UnionCF A f mf ay supf b) ab → supf b ≡ b |
832 | 289 supf-is-sup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ & (SUP.sup (sup x≤z) ) |
825 | 290 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y |
828 | 291 csupf : {b : Ordinal } → b o≤ z → odef (UnionCF A f mf ay supf (supf b)) (supf b) |
825 | 292 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
293 supf-inject {x} {y} sx<sy with trio< x y | |
294 ... | tri< a ¬b ¬c = a | |
295 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
296 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
297 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
298 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
798 | 299 |
803 | 300 -- ordering is proved here for totality and sup |
301 | |
813 | 302 fcy<sup : {u w : Ordinal } → u o< z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf |
832 | 303 fcy<sup {u} {w} u<z fc with SUP.x<sup (sup (o<→≤ u<z)) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) |
798 | 304 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ |
832 | 305 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans (cong (&) eq) (sym (supf-is-sup (o<→≤ u<z) ) ) )) |
306 ... | case2 lt = case2 (subst (λ k → * w < k ) (subst (λ k → k ≡ _ ) *iso (cong (*) (sym (supf-is-sup (o<→≤ u<z) ))) ) lt ) | |
825 | 307 |
832 | 308 csupf-fc : {b s z1 : Ordinal} → b o≤ z → supf s o< supf b → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1 |
309 csupf-fc {b} {s} {z1} b≤z ss<sb (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc05 where | |
828 | 310 s<b : s o< b |
311 s<b = supf-inject ss<sb | |
830 | 312 s≤<z : s o≤ z |
831 | 313 s≤<z = ordtrans s<b b≤z |
825 | 314 zc04 : odef (UnionCF A f mf ay supf (supf s)) (supf s) |
830 | 315 zc04 = csupf s≤<z |
825 | 316 zc05 : odef (UnionCF A f mf ay supf b) (supf s) |
317 zc05 with zc04 | |
318 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ | |
828 | 319 ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ as , ch-is-sup u (zc09 u≤x ) is-sup fc ⟫ where |
825 | 320 zc06 : supf u ≡ u |
828 | 321 zc06 = ChainP.supu=u is-sup |
322 zc09 : u o≤ supf s → u o≤ b | |
826 | 323 zc09 u<s with osuc-≡< (subst (λ k → k o≤ supf s) (sym zc06) u<s) |
324 ... | case1 su=ss = zc08 where | |
825 | 325 zc07 : supf u o≤ supf b |
326 zc07 = subst (λ k → k o≤ supf b) (sym su=ss) (supf-mono (o<→≤ s<b) ) | |
828 | 327 zc08 : u o≤ b |
825 | 328 zc08 with osuc-≡< zc07 |
829 | 329 ... | case1 su=sb = ⊥-elim ( o<¬≡ (trans (sym su=ss) su=sb ) ss<sb ) |
828 | 330 ... | case2 lt = o<→≤ (supf-inject lt ) |
331 ... | case2 lt = o<→≤ (ordtrans (supf-inject lt) s<b) | |
831 | 332 csupf-fc {b} {s} {z1} b<z ss≤sb (fsuc x fc) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc04 where |
785 | 333 zc04 : odef (UnionCF A f mf ay supf b) (f x) |
831 | 334 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb fc ) |
785 | 335 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ |
791 | 336 ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ |
832 | 337 order : {b s z1 : Ordinal} → b o< z → supf s o< supf b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b) |
338 order {b} {s} {z1} b<z ss<sb fc = zc04 where | |
339 zc00 : ( * z1 ≡ SUP.sup (sup (o<→≤ b<z) )) ∨ ( * z1 < SUP.sup ( sup (o<→≤ b<z) ) ) | |
340 zc00 = SUP.x<sup (sup (o<→≤ b<z) ) (csupf-fc (o<→≤ b<z) ss<sb fc ) | |
797 | 341 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) |
342 zc04 with zc00 | |
832 | 343 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso (sym (supf-is-sup (o<→≤ b<z)) ) (cong (&) eq) ) |
344 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) refl (subst₂ (λ j k → j ≡ k ) *iso refl (cong (*) (sym (supf-is-sup (o<→≤ b<z) ) ))) lt ) | |
756 | 345 |
728 | 346 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
783 | 347 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
728 | 348 field |
349 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay (ZChain.supf zc) z) a ) → b o< z → (ab : odef A b) | |
836 | 350 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) z) b f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) z) ab |
728 | 351 → * a < * b → odef ((UnionCF A f mf ay (ZChain.supf zc) z)) b |
352 | |
837 | 353 initial-segment : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
354 {a b y : Ordinal} (ay : odef A y) (za : ZChain A f mf ay a ) (zb : ZChain A f mf ay b ) | |
355 → {z : Ordinal } → a o≤ b → z o≤ a | |
356 → ZChain.supf za z ≡ ZChain.supf zb z | |
357 initial-segment A f mf {a} {b} {y} ay za zb {z} a≤b z≤a = TransFinite0 { λ x → x o≤ a → ZChain.supf za x ≡ ZChain.supf zb x } ind z z≤a where | |
358 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → z o≤ a → ZChain.supf za z ≡ ZChain.supf zb z ) → | |
359 x o≤ a → ZChain.supf za x ≡ ZChain.supf zb x | |
360 ind x prev x≤a = ? where | |
361 supfa = ZChain.supf za | |
362 supfb = ZChain.supf zb | |
363 zc10 : {w : Ordinal } → w o< z → UnionCF A f mf ay supfa w ≡ UnionCF A f mf ay supfb w | |
364 zc10 = ? | |
365 -- w o< z → supfa w ≡ supfb w | |
366 supa : SUP A (UnionCF A f mf ay supfa x) | |
367 supa = ZChain.sup za x≤a | |
368 supb : SUP A (UnionCF A f mf ay supfb x) | |
369 supb = ZChain.sup zb (OrdTrans x≤a a≤b) | |
370 zc13 : UnionCF A f mf ay supfa x ≡ UnionCF A f mf ay supfb x | |
371 zc13 = ? -- | |
372 -- if x is sup of UCF px (or Union o< x ) , then supfa x ≡ x supfb x | |
373 -- if x is not sup of UCF px (or Union o< x ) or HasPrev, UCF x ≡ UCF px (or Union o< x) | |
374 zc15 : {B : HOD} → (a b : SUP A B) → SUP.sup a ≡ SUP.sup b | |
375 zc15 = ? | |
376 zc14 : supfa x ≡ supfb x | |
377 zc14 = begin | |
378 supfa x ≡⟨ ? ⟩ | |
379 & (SUP.sup supa) ≡⟨ ? ⟩ | |
380 & (SUP.sup supb) ≡⟨ ? ⟩ | |
381 supfb x ∎ where open ≡-Reasoning | |
382 | |
568 | 383 record Maximal ( A : HOD ) : Set (Level.suc n) where |
384 field | |
385 maximal : HOD | |
804 | 386 as : A ∋ maximal |
568 | 387 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative |
567 | 388 |
748 | 389 -- data UChain is total |
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390 |
694 | 391 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
748 | 392 {s s1 a b : Ordinal } ( ca : UChain A f mf ay supf s a ) ( cb : UChain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) |
694 | 393 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where |
748 | 394 ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) |
395 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb | |
791 | 396 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca |
766 | 397 ... | case1 eq with s≤fc (supf ub) f mf fcb |
398 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
399 ct00 : * a ≡ * b | |
400 ct00 = trans (cong (*) eq) eq1 | |
765 | 401 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where |
766 | 402 ct01 : * a < * b |
403 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
791 | 404 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where |
748 | 405 ct00 : * a < * (supf ub) |
765 | 406 ct00 = lt |
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407 ct01 : * a < * b |
748 | 408 ct01 with s≤fc (supf ub) f mf fcb |
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409 ... | case1 eq = subst (λ k → * a < k ) eq ct00 |
34650e39e553
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410 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
791 | 411 ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb |
766 | 412 ... | case1 eq with s≤fc (supf ua) f mf fca |
413 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
414 ct00 : * a ≡ * b | |
415 ct00 = sym (trans (cong (*) eq) eq1 ) | |
765 | 416 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where |
766 | 417 ct01 : * b < * a |
418 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
791 | 419 ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where |
749 | 420 ct00 : * b < * (supf ua) |
765 | 421 ct00 = lt |
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422 ct01 : * b < * a |
749 | 423 ct01 with s≤fc (supf ua) f mf fca |
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424 ... | case1 eq = subst (λ k → * b < k ) eq ct00 |
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425 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
800 | 426 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub |
829 | 427 ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁) fca |
766 | 428 ... | case1 eq with s≤fc (supf ub) f mf fcb |
429 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
430 ct00 : * a ≡ * b | |
431 ct00 = trans (cong (*) eq) eq1 | |
432 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
433 ct02 : * a < * b | |
434 ct02 = subst (λ k → * k < * b ) (sym eq) lt | |
788 | 435 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where |
748 | 436 ct03 : * a < * (supf ub) |
765 | 437 ct03 = lt |
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34650e39e553
Chain is not strictly positive
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parents:
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438 ct02 : * a < * b |
748 | 439 ct02 with s≤fc (supf ub) f mf fcb |
689
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440 ... | case1 eq = subst (λ k → * a < k ) eq ct03 |
34650e39e553
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441 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt |
788 | 442 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri≈ ¬a eq ¬c |
800 | 443 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) |
829 | 444 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) fcb |
766 | 445 ... | case1 eq with s≤fc (supf ua) f mf fca |
446 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
447 ct00 : * a ≡ * b | |
448 ct00 = sym (trans (cong (*) eq) eq1) | |
449 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
450 ct02 : * b < * a | |
451 ct02 = subst (λ k → * k < * a ) (sym eq) lt | |
788 | 452 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where |
749 | 453 ct05 : * b < * (supf ua) |
765 | 454 ct05 = lt |
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Chain is not strictly positive
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parents:
688
diff
changeset
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455 ct04 : * b < * a |
749 | 456 ct04 with s≤fc (supf ua) f mf fca |
689
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Chain is not strictly positive
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parents:
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diff
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457 ... | case1 eq = subst (λ k → * b < k ) eq ct05 |
34650e39e553
Chain is not strictly positive
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parents:
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458 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt |
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no transfinite on data Chain trichotomos
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
683
diff
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459 |
743 | 460 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
461 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
783 | 462 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ |
743 | 463 |
497 | 464 Zorn-lemma : { A : HOD } |
464 | 465 → o∅ o< & A |
568 | 466 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
497 | 467 → Maximal A |
552 | 468 Zorn-lemma {A} 0<A supP = zorn00 where |
571 | 469 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
470 <-irr0 {a} {b} A∋a A∋b = <-irr | |
788 | 471 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
472 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
760 | 473 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A |
474 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
530 | 475 s : HOD |
476 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
568 | 477 as : A ∋ * ( & s ) |
478 as = 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 )) ) | |
608
6655f03984f9
mutual tranfinite in zorn
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parents:
607
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479 as0 : odef A (& s ) |
6655f03984f9
mutual tranfinite in zorn
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480 as0 = subst (λ k → odef A k ) &iso as |
547 | 481 s<A : & s o< & A |
568 | 482 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
530 | 483 HasMaximal : HOD |
537 | 484 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
485 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
486 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
532 | 487 Gtx : { x : HOD} → A ∋ x → HOD |
537 | 488 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
489 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
804 | 490 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
537 | 491 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
492 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)) | |
493 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 | |
494 ¬x<m : ¬ (* x < * m) | |
495 ¬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 | 496 |
560 | 497 -- Uncountable ascending chain by axiom of choice |
530 | 498 cf : ¬ Maximal A → Ordinal → Ordinal |
532 | 499 cf nmx x with ODC.∋-p O A (* x) |
500 ... | no _ = o∅ | |
501 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
538 | 502 ... | yes nogt = -- no larger element, so it is maximal |
503 ⊥-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 | 504 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
537 | 505 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
506 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
507 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
508 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
509 ... | 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 ⟫ ) | |
510 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
606 | 511 |
512 --- | |
513 --- infintie ascention sequence of f | |
514 --- | |
530 | 515 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
537 | 516 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
530 | 517 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
532 | 518 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
543 | 519 |
793 | 520 chain-mono : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
521 {a b c : Ordinal} → a o≤ b | |
522 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c | |
523 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = | |
524 ⟪ ua , ch-init fc ⟫ | |
525 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = | |
526 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (osucc a≤b )) is-sup fc ⟫ | |
527 | |
703 | 528 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
653 | 529 (total : IsTotalOrderSet (ZChain.chain zc) ) → SUP A (ZChain.chain zc) |
703 | 530 sp0 f mf zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total |
543 | 531 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
532 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
533 | |
803 | 534 -- |
535 -- Second TransFinite Pass for maximality | |
536 -- | |
537 | |
793 | 538 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
728 | 539 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
793 | 540 SZ1 f mf {y} ay zc x = TransFinite { λ x → ZChain1 A f mf ay zc x } zc1 x where |
541 chain-mono1 : {a b c : Ordinal} → a o≤ b | |
788 | 542 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
793 | 543 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) a≤b |
735 | 544 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
545 b o< x → (ab : odef A b) → | |
836 | 546 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) b f → |
735 | 547 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
749 | 548 is-max-hp x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
549 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ | |
791 | 550 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , |
749 | 551 subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) |
791 | 552 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ |
728 | 553 zc1 : (x : Ordinal) → ((y₁ : Ordinal) → y₁ o< x → ZChain1 A f mf ay zc y₁) → ZChain1 A f mf ay zc x |
732
ddeb107b6f71
bchain can be reached from upwords by f. so it is worng.
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parents:
729
diff
changeset
|
554 zc1 x prev with Oprev-p x |
756 | 555 ... | yes op = record { is-max = is-max } where |
732
ddeb107b6f71
bchain can be reached from upwords by f. so it is worng.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
729
diff
changeset
|
556 px = Oprev.oprev op |
789 | 557 zc-b<x : {b : Ordinal } → b o< x → b o< osuc px |
558 zc-b<x {b} lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
728 | 559 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
560 b o< x → (ab : odef A b) → | |
836 | 561 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) b f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → |
728 | 562 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
735 | 563 is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b |
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
564 is-max {a} {b} ua b<x ab (case2 is-sup) a<b |
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
789
diff
changeset
|
565 = ⟪ ab , ch-is-sup b (o<→≤ b<x) m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ where |
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
566 b<A : b o< & A |
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
567 b<A = z09 ab |
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
568 m05 : b ≡ ZChain.supf zc b |
814 | 569 m05 = sym ( ZChain.sup=u zc ab (o<→≤ (z09 ab) ) |
817 | 570 record { x<sup = λ {z} uz → IsSup.x<sup is-sup (chain-mono1 (o<→≤ b<x) uz ) } ) |
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
571 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
813 | 572 m08 {z} fcz = ZChain.fcy<sup zc b<A fcz |
828 | 573 m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
825 | 574 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
832 | 575 m09 {s} {z} s<b fcz = ZChain.order zc b<A s<b fcz |
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
576 m06 : ChainP A f mf ay (ZChain.supf zc) b |
828 | 577 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = ZChain.sup=u zc ab (o<→≤ b<A ) |
578 record { x<sup = λ {z} uz → IsSup.x<sup is-sup (chain-mono1 (o<→≤ b<x) uz ) } } | |
756 | 579 ... | no lim = record { is-max = is-max } where |
734 | 580 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
581 b o< x → (ab : odef A b) → | |
836 | 582 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) b f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → |
734 | 583 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
735 | 584 is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b |
743 | 585 is-max {a} {b} ua b<x ab (case2 is-sup) a<b with IsSup.x<sup is-sup (init-uchain A f mf ay ) |
789 | 586 ... | case1 b=y = ⟪ subst (λ k → odef A k ) b=y ay , ch-init (subst (λ k → FClosure A f y k ) b=y (init ay refl )) ⟫ |
793 | 587 ... | case2 y<b = chain-mono1 (osucc b<x) |
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
588 ⟪ ab , ch-is-sup b (ordtrans o≤-refl <-osuc ) m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ where |
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
589 m09 : b o< & A |
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
789
diff
changeset
|
590 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) |
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
591 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
813 | 592 m07 {z} fc = ZChain.fcy<sup zc m09 fc |
828 | 593 m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
825 | 594 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
832 | 595 m08 {s} {z1} s<b fc = ZChain.order zc m09 s<b fc |
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
596 m05 : b ≡ ZChain.supf zc b |
814 | 597 m05 = sym (ZChain.sup=u zc ab (o<→≤ m09) |
817 | 598 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono1 (o<→≤ b<x) lt )} ) -- ZChain on x |
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
599 m06 : ChainP A f mf ay (ZChain.supf zc) b |
828 | 600 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = ZChain.sup=u zc ab (o<→≤ m09) |
601 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono1 (o<→≤ b<x) lt )} } | |
727 | 602 |
543 | 603 --- |
560 | 604 --- the maximum chain has fix point of any ≤-monotonic function |
543 | 605 --- |
703 | 606 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
633
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
607 → (total : IsTotalOrderSet (ZChain.chain zc) ) |
703 | 608 → f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total)) |
609 fixpoint f mf zc total = z14 where | |
538 | 610 chain = ZChain.chain zc |
703 | 611 sp1 = sp0 f mf zc total |
712 | 612 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
836 | 613 → HasPrev A chain b f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
538 | 614 → * a < * b → odef chain b |
793 | 615 z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) ) |
543 | 616 z11 : & (SUP.sup sp1) o< & A |
804 | 617 z11 = c<→o< ( SUP.as sp1) |
538 | 618 z12 : odef chain (& (SUP.sup sp1)) |
619 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
653 | 620 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
804 | 621 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z11 (SUP.as sp1) |
570 | 622 (case2 z19 ) z13 where |
538 | 623 z13 : * (& s) < * (& (SUP.sup sp1)) |
653 | 624 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc ) |
538 | 625 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
626 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
804 | 627 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.as sp1) |
571 | 628 z19 = record { x<sup = z20 } where |
629 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
630 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
570 | 631 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
632 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
633 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
703 | 634 z14 : f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total )) |
633
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ZChain1 is not strictly positive
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parents:
631
diff
changeset
|
635 z14 with total (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
631 | 636 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
637 z16 : ⊥ | |
804 | 638 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.as sp1 )) |
631 | 639 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) |
640 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
641 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
642 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
643 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
644 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) | |
645 z17 : ⊥ | |
646 z17 with z15 | |
647 ... | case1 eq = ¬b eq | |
648 ... | case2 lt = ¬a lt | |
560 | 649 |
650 -- ZChain contradicts ¬ Maximal | |
651 -- | |
571 | 652 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
560 | 653 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
654 -- | |
697 | 655 z04 : (nmx : ¬ Maximal A ) |
703 | 656 → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) |
664 | 657 → IsTotalOrderSet (ZChain.chain zc) → ⊥ |
804 | 658 z04 nmx zc total = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.as sp1 )))) |
659 (subst (λ k → odef A (& k)) (sym *iso) (SUP.as sp1) ) | |
703 | 660 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc total ))) -- x ≡ f x ̄ |
804 | 661 (proj1 (cf-is-<-monotonic nmx c (SUP.as sp1 ))) where -- x < f x |
633
6cd4a483122c
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
662 sp1 : SUP A (ZChain.chain zc) |
703 | 663 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc total |
538 | 664 c = & (SUP.sup sp1) |
548 | 665 |
757 | 666 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
667 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = | |
668 λ {z} cz → subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (A∋fc y f mf cz ))) } | |
669 | |
670 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
671 → IsTotalOrderSet (uchain f mf ay) | |
672 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
673 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
674 uz01 = fcn-cmp y f mf ca cb | |
675 | |
676 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
677 → SUP A (uchain f mf ay) | |
678 ysup f mf {y} ay = supP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) | |
679 | |
793 | 680 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
804 | 681 SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x<sup = λ lt → SUP.x<sup sup (B⊆C lt) } |
711 | 682 |
833 | 683 record xSUP (B : HOD) (x : Ordinal) : Set n where |
684 field | |
685 ax : odef A x | |
686 is-sup : IsSup A B ax | |
687 | |
560 | 688 -- |
547 | 689 -- create all ZChains under o< x |
560 | 690 -- |
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
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|
691 |
674 | 692 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
703 | 693 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
707 | 694 ind f mf {y} ay x prev with Oprev-p x |
697 | 695 ... | yes op = zc4 where |
682 | 696 -- |
697 -- we have previous ordinal to use induction | |
698 -- | |
699 px = Oprev.oprev op | |
703 | 700 zc : ZChain A f mf ay (Oprev.oprev op) |
682 | 701 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
702 px<x : px o< x | |
703 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
709 | 704 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
705 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
697 | 706 |
703 | 707 pchain : HOD |
830 | 708 pchain = UnionCF A f mf ay (ZChain.supf zc) px |
703 | 709 ptotal : IsTotalOrderSet pchain |
710 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
711 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
748 | 712 uz01 = chain-total A f mf ay (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb)) |
704 | 713 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y |
714 pchain⊆A {y} ny = proj1 ny | |
715 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
749 | 716 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
797 | 717 pnext {a} ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u≤x is-sup (fsuc _ fc ) ⟫ |
704 | 718 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
748 | 719 pinit {a} ⟪ aa , ua ⟫ with ua |
720 ... | ch-init fc = s≤fc y f mf fc | |
791 | 721 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
765 | 722 zc7 : y <= (ZChain.supf zc) u |
783 | 723 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
704 | 724 pcy : odef pchain y |
783 | 725 pcy = ⟪ ay , ch-init (init ay refl) ⟫ |
703 | 726 |
754 | 727 supf0 = ZChain.supf zc |
728 | |
835 | 729 sup1 : SUP A (UnionCF A f mf ay supf0 px) |
730 sup1 = supP pchain pchain⊆A ptotal | |
731 sp1 = & (SUP.sup sup1 ) | |
732 supf1 : Ordinal → Ordinal | |
733 supf1 z with trio< z px | |
734 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
735 ... | tri≈ ¬a b ¬c = sp1 | |
736 ... | tri> ¬a ¬b c = sp1 | |
737 | |
738 pchain1 : HOD | |
739 pchain1 = UnionCF A f mf ay supf1 x | |
740 | |
611 | 741 -- if previous chain satisfies maximality, we caan reuse it |
742 -- | |
791 | 743 -- supf0 px is sup of UnionCF px , supf0 x is sup of UnionCF x |
805 | 744 |
836 | 745 no-extension : (¬ xSUP (UnionCF A f mf ay supf0 px) x ) ∨ HasPrev A pchain x f → ZChain A f mf ay x |
746 no-extension ¬sp=x = record { supf = supf1 ; sup = ? ; supf-mono = {!!} | |
833 | 747 ; initial = ? ; chain∋init = ? ; sup=u = ? ; supf-is-sup = ? ; csupf = {!!} |
748 ; chain⊆A = λ lt → proj1 lt ; f-next = ? ; f-total = ? } where | |
838 | 749 pchain0=1 : pchain ≡ pchain1 |
750 pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
751 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z | |
752 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
753 zc10 {z} ⟪ az , ch-is-sup u u≤px is-sup fc ⟫ = zc12 fc where | |
754 zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z | |
755 zc12 (fsuc x fc) with zc12 fc | |
756 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
757 ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ | |
839 | 758 zc12 (init asu su=z ) with trio< u px |
759 ... | tri< a ¬b ¬c = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u (ordtrans u≤px (osucc (pxo<x op)) ) | |
760 record { fcy<sup = ? ; order = ? ; supu=u = ? } (init ? ? ) ⟫ | |
761 ... | tri≈ ¬a b ¬c = ? | |
762 ... | tri> ¬a ¬b c = ? | |
838 | 763 zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z |
764 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
765 zc11 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc13 fc where | |
766 zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z | |
767 zc13 (fsuc x fc) with zc13 fc | |
768 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
769 ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ | |
770 zc13 (init asu su=z ) with trio< u x | |
771 ... | tri< a ¬b ¬c = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u (subst (λ k → u o< k) (sym (Oprev.oprev=x op)) a) ? (init ? ? ) ⟫ | |
772 ... | tri≈ ¬a b ¬c = ? | |
773 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> u≤x c ) | |
833 | 774 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf0 z) |
832 | 775 sup {z} z≤x with trio< z px |
833 | 776 ... | tri< a ¬b ¬c = SUP⊆ ? ( ZChain.sup zc (o<→≤ a) ) |
777 ... | tri≈ ¬a b ¬c = record { sup = SUP.sup ? ; as = SUP.as ? ; x<sup = ? } where | |
778 zc61 : {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup ?) ∨ (w < SUP.sup ? ) | |
830 | 779 zc61 {w} lt = ? -- SUP.x<sup sup1 (UnionCFR⊆ (o<→≤ z<x) z<x lt ) |
832 | 780 ... | tri> ¬a ¬b px<z = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) ? ⟫ ) |
803 | 781 sup=u : {b : Ordinal} (ab : odef A b) → |
833 | 782 b o≤ x → IsSup A (UnionCF A f mf ay supf0 b) ab → supf0 b ≡ b |
814 | 783 sup=u {b} ab b≤x is-sup with trio< b px |
833 | 784 ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) record { x<sup = λ lt → IsSup.x<sup is-sup ? } |
830 | 785 ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u zc ab (o≤-refl0 b) record { x<sup = λ lt → IsSup.x<sup is-sup (UnionCF⊆ o≤-refl (o≤-refl0 b) lt) } |
833 | 786 ... | tri> ¬a ¬b px<b = ? where -- ⊥-elim (¬sp=x zcsup ) where |
815 | 787 zc30 : x ≡ b |
788 zc30 with osuc-≡< b≤x | |
789 ... | case1 eq = sym (eq) | |
790 ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
833 | 791 zcsup : ? |
792 zcsup = ? -- with zc30 | |
793 -- ... | refl = case1 record { ax = ab ; is-sup = record { x<sup = λ lt → IsSup.x<sup is-sup ? } } | |
794 csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay supf0 b) (supf0 b) | |
795 csupf {b} b<x with trio< b px | inspect supf0 b | |
796 ... | tri< a ¬b ¬c | _ = ? -- UnionCF⊆ o≤-refl a {!!} | |
830 | 797 ... | tri≈ ¬a refl ¬c | _ = {!!} -- UnionCF⊆ o≤-refl o≤-refl ( ZChain.csupf zc o≤-refl ) |
798 ... | tri> ¬a ¬b px<b | record { eq = eq1 } = {!!} -- UnionCF⊆ (o<→≤ px<b) o≤-refl ( ZChain.csupf zc o≤-refl ) | |
833 | 799 sis : {z : Ordinal} (z≤x : z o≤ x) → supf0 z ≡ & (SUP.sup (sup z≤x)) |
823 | 800 sis {z} z<x with trio< z px |
832 | 801 ... | tri< a ¬b ¬c = ZChain.supf-is-sup zc (o<→≤ a ) |
830 | 802 ... | tri≈ ¬a b ¬c = {!!} |
832 | 803 ... | tri> ¬a ¬b px<z = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) ? ⟫ ) |
833 | 804 |
703 | 805 zc4 : ZChain A f mf ay x |
793 | 806 zc4 with ODC.∋-p O A (* x) |
796 | 807 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
836 | 808 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc ) x f ) |
703 | 809 -- we have to check adding x preserve is-max ZChain A y f mf x |
796 | 810 ... | case1 pr = no-extension {!!} -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
793 | 811 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc ) ax ) |
682 | 812 ... | case1 is-sup = -- x is a sup of zc |
830 | 813 record { supf = psupf1 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} ; csupf = {!!} ; sup=u = {!!} ; supf-mono = {!!} |
800 | 814 ; initial = {!!} ; chain∋init = {!!} ; sup = {!!} ; supf-is-sup = {!!} } where |
793 | 815 supx : SUP A (UnionCF A f mf ay supf0 x) |
804 | 816 supx = record { sup = * x ; as = subst (λ k → odef A k ) {!!} ax ; x<sup = {!!} } |
793 | 817 spx = & (SUP.sup supx ) |
818 x=spx : x ≡ spx | |
807 | 819 x=spx = sym &iso |
750 | 820 psupf1 : Ordinal → Ordinal |
821 psupf1 z with trio< z x | |
822 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
823 ... | tri≈ ¬a b ¬c = x | |
824 ... | tri> ¬a ¬b c = x | |
822 | 825 csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay psupf1 b) (psupf1 b) |
804 | 826 csupf {b} b≤x with trio< b px | inspect psupf1 b |
811
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827 ... | tri< a ¬b ¬c | record { eq = eq1 } = ⟪ {!!} , {!!} ⟫ |
808 | 828 ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ {!!} , {!!} ⟫ |
829 ... | tri> ¬a ¬b c | record { eq = eq1 } = {!!} where -- b ≡ x, supf x ≡ sp | |
804 | 830 zc30 : x ≡ b |
831 zc30 with trio< x b | |
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832 ... | tri< a ¬b ¬c = {!!} |
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833 ... | tri≈ ¬a b ¬c = {!!} |
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834 ... | tri> ¬a ¬b c = {!!} |
793 | 835 |
796 | 836 ... | case2 ¬x=sup = no-extension {!!} -- px is not f y' nor sup of former ZChain from y -- no extention |
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837 |
728 | 838 ... | no lim = zc5 where |
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839 |
703 | 840 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
841 pzc z z<x = prev z z<x | |
726
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842 |
835 | 843 ysp = & (SUP.sup (ysup f mf ay)) |
755 | 844 |
837 | 845 initial-segment0 : {a b z : Ordinal } → (a<x : a o< x) ( b<x : b o< x) → a o< b → z o≤ a |
836 | 846 → ZChain.supf (pzc (osuc a) (ob<x lim a<x )) z ≡ ZChain.supf (pzc (osuc b) (ob<x lim b<x )) z |
837 | 847 initial-segment0 = ? |
834 | 848 |
835 | 849 supf0 : Ordinal → Ordinal |
850 supf0 z with trio< z x | |
851 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
836 | 852 ... | tri≈ ¬a b ¬c = ysp |
853 ... | tri> ¬a ¬b c = ysp | |
835 | 854 |
838 | 855 pchain : HOD |
856 pchain = UnionCF A f mf ay supf0 x | |
835 | 857 |
838 | 858 ptotal0 : IsTotalOrderSet pchain |
835 | 859 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
860 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
861 uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb)) | |
862 | |
838 | 863 usup : SUP A pchain |
864 usup = supP pchain (λ lt → proj1 lt) ptotal0 | |
835 | 865 spu = & (SUP.sup usup) |
834 | 866 |
794 | 867 supf1 : Ordinal → Ordinal |
835 | 868 supf1 z with trio< z x |
869 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
836 | 870 ... | tri≈ ¬a b ¬c = spu |
871 ... | tri> ¬a ¬b c = spu | |
755 | 872 |
838 | 873 pchain1 : HOD |
874 pchain1 = UnionCF A f mf ay supf1 x | |
704 | 875 |
838 | 876 pchain⊆A : {y : Ordinal} → odef pchain1 y → odef A y |
704 | 877 pchain⊆A {y} ny = proj1 ny |
838 | 878 pnext : {a : Ordinal} → odef pchain1 a → odef pchain1 (f a) |
750 | 879 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-init (fsuc _ fc) ⟫ |
794 | 880 pnext {a} ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ |
838 | 881 pinit : {y₁ : Ordinal} → odef pchain1 y₁ → * y ≤ * y₁ |
748 | 882 pinit {a} ⟪ aa , ua ⟫ with ua |
883 ... | ch-init fc = s≤fc y f mf fc | |
791 | 884 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
794 | 885 zc7 : y <= supf1 _ |
783 | 886 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
838 | 887 pcy : odef pchain1 y |
783 | 888 pcy = ⟪ ay , ch-init (init ay refl) ⟫ |
838 | 889 ptotal : IsTotalOrderSet pchain1 |
755 | 890 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
891 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
794 | 892 uz01 = chain-total A f mf ay supf1 ( (proj2 ca)) ( (proj2 cb)) |
754 | 893 |
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894 is-max-hp : (supf : Ordinal → Ordinal) (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
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895 b o< x → (ab : odef A b) → |
836 | 896 HasPrev A (UnionCF A f mf ay supf x) b f → |
758
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897 * a < * b → odef (UnionCF A f mf ay supf x) b |
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898 is-max-hp supf x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
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899 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
791 | 900 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , |
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901 subst (λ k → UChain A f mf ay supf x k ) |
794 | 902 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ |
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903 |
836 | 904 no-extension : ¬ ( xSUP (UnionCF A f mf ay supf0 x) x ) ∨ HasPrev A pchain x f → ZChain A f mf ay x |
802 | 905 no-extension ¬sp=x = record { initial = pinit ; chain∋init = pcy ; supf = supf1 ; sup=u = sup=u |
830 | 906 ; sup = sup ; supf-is-sup = sis ; supf-mono = {!!} |
907 ; csupf = {!!} ; chain⊆A = pchain⊆A ; f-next = pnext ; f-total = ptotal } where | |
795
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908 supfu : {u : Ordinal } → ( a : u o< x ) → (z : Ordinal) → Ordinal |
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909 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z |
838 | 910 pchain0=1 : pchain ≡ pchain1 |
911 pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
912 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z | |
913 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
914 zc10 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc12 fc where | |
915 zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z | |
916 zc12 (fsuc x fc) with zc12 fc | |
917 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
918 ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ | |
919 zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u u≤x ? (init ? ? ) ⟫ | |
920 zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z | |
921 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
922 zc11 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc13 fc where | |
923 zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z | |
924 zc13 (fsuc x fc) with zc13 fc | |
925 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
926 ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ | |
927 zc13 (init asu su=z ) with trio< u x | |
928 ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u u≤x ? (init ? ? ) ⟫ | |
929 ... | tri≈ ¬a b ¬c = ? | |
930 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> u≤x c ) | |
832 | 931 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
797 | 932 sup {z} z≤x with trio< z x |
838 | 933 ... | tri< a ¬b ¬c = SUP⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} ) |
815 | 934 ... | tri≈ ¬a b ¬c = {!!} |
935 ... | tri> ¬a ¬b c = {!!} | |
832 | 936 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup {!!})) |
813 | 937 sis {z} z<x with trio< z x |
800 | 938 ... | tri< a ¬b ¬c = {!!} where |
815 | 939 zc8 = ZChain.supf-is-sup (pzc z a) {!!} |
940 ... | tri≈ ¬a b ¬c = {!!} | |
941 ... | tri> ¬a ¬b c = {!!} | |
817 | 942 sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsSup A (UnionCF A f mf ay supf1 b) ab → supf1 b ≡ b |
797 | 943 sup=u {b} ab b<x is-sup with trio< b x |
833 | 944 ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x<sup = {!!} } |
815 | 945 ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?) ?) ab {!!} record { x<sup = {!!} } |
800 | 946 ... | tri> ¬a ¬b c = {!!} |
822 | 947 csupf : {z : Ordinal} → z o≤ x → odef (UnionCF A f mf ay supf1 z) (supf1 z) |
804 | 948 csupf {z} z≤x with trio< z x |
833 | 949 ... | tri< a ¬b ¬c = ? where |
797 | 950 zc9 : odef (UnionCF A f mf ay supf1 z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z) |
800 | 951 zc9 = {!!} |
797 | 952 zc8 : odef (UnionCF A f mf ay (supfu a) z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z) |
830 | 953 zc8 = {!!} -- ZChain.csupf (pzc (osuc z) (ob<x lim a)) ? -- (o<→≤ <-osuc ) |
808 | 954 ... | tri≈ ¬a b ¬c = {!!} |
830 | 955 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z {!!} )) |
797 | 956 |
703 | 957 zc5 : ZChain A f mf ay x |
697 | 958 zc5 with ODC.∋-p O A (* x) |
796 | 959 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
836 | 960 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain x f ) |
703 | 961 -- we have to check adding x preserve is-max ZChain A y f mf x |
796 | 962 ... | case1 pr = no-extension {!!} |
704 | 963 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A pchain ax ) |
794 | 964 ... | case1 is-sup = record { initial = {!!} ; chain∋init = {!!} ; supf = supf1 ; sup=u = {!!} |
830 | 965 ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!} |
808 | 966 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} ; csupf = {!!} } -- where -- x is a sup of (zc ?) |
796 | 967 ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention |
553 | 968 |
703 | 969 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay (& A) |
970 SZ f mf {y} ay = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) (& A) | |
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
971 |
551 | 972 zorn00 : Maximal A |
973 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
804 | 974 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
551 | 975 -- yes we have the maximal |
976 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
606 | 977 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
551 | 978 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
979 zorn01 = proj1 zorn03 | |
980 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
981 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
703 | 982 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04 total ) where |
551 | 983 -- if we have no maximal, make ZChain, which contradict SUP condition |
984 nmx : ¬ Maximal A | |
985 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
986 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
804 | 987 zc5 = ⟪ Maximal.as 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) ) ⟫ |
703 | 988 zorn04 : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) |
653 | 989 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
634 | 990 total : IsTotalOrderSet (ZChain.chain zorn04) |
654 | 991 total {a} {b} = zorn06 where |
992 zorn06 : odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a) | |
993 zorn06 = ZChain.f-total (SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as) ) | |
551 | 994 |
516 | 995 -- usage (see filter.agda ) |
996 -- | |
497 | 997 -- _⊆'_ : ( A B : HOD ) → Set n |
998 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
482 | 999 |
497 | 1000 -- MaximumSubset : {L P : HOD} |
1001 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
1002 -- → IsPartialOrderSet P _⊆'_ | |
1003 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
1004 -- → Maximal P (_⊆'_) | |
1005 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |