Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 721:562ddd33fe21
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 16 Jul 2022 04:05:18 +0900 |
parents | 6c9fed204440 |
children | 0dd8cc755ec9 |
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 | |
58 POO : IsStrictPartialOrder _≡_ _<<_ | |
59 POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
60 ; trans = IsStrictPartialOrder.trans PO | |
61 ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y | |
62 ; <-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 } } | |
63 | |
528
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TransitiveClosure with x <= f x is possible
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64 _≤_ : (x y : HOD) → Set (Level.suc n) |
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TransitiveClosure with x <= f x is possible
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65 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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TransitiveClosure with x <= f x is possible
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66 |
554 | 67 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
68 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
69 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
70 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
71 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
72 | |
556 | 73 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
74 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
75 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
76 (IsStrictPartialOrder.trans PO b<a a<b) | |
490 | 77 |
561 | 78 ptrans = IsStrictPartialOrder.trans PO |
79 | |
492 | 80 open _==_ |
81 open _⊆_ | |
82 | |
530 | 83 -- |
560 | 84 -- Closure of ≤-monotonic function f has total order |
530 | 85 -- |
86 | |
87 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
88 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
89 | |
551 | 90 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
600 | 91 init : odef A s → FClosure A f s s |
555 | 92 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
554 | 93 |
556 | 94 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 |
600 | 95 A∋fc {A} s f mf (init as) = as |
556 | 96 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
555 | 97 |
714 | 98 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 |
99 A∋fcs {A} s f mf (init as) = as | |
100 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy | |
101 | |
556 | 102 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
600 | 103 s≤fc {A} s {.s} f mf (init x) = case1 refl |
556 | 104 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
105 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
106 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
107 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
108 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
555 | 109 |
557 | 110 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
600 | 111 fcn s mf (init as) = zero |
558 | 112 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) |
113 ... | case1 eq = fcn s mf p | |
114 ... | case2 y<fy = suc (fcn s mf p ) | |
557 | 115 |
558 | 116 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
117 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
559 | 118 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 |
119 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 | |
600 | 120 fc00 zero zero refl (init _) (init x₁) i=x i=y = refl |
121 fc00 zero zero refl (init as) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
122 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as) cy i=x i=y ) | |
123 fc00 zero zero refl (fsuc x cx) (init as) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
124 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as) i=x i=y ) | |
559 | 125 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 ) ) |
126 ... | 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 ) | |
127 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 ) ) | |
128 ... | 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 ) | |
129 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
130 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
131 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
560 | 132 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x |
559 | 133 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where |
134 fc04 : * x1 ≡ * y | |
135 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
136 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
137 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
138 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
139 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
140 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
141 fc05 : * x ≡ * y1 | |
142 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
143 ... | 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 | 144 |
600 | 145 |
557 | 146 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
147 → (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 | 148 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where |
149 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 | |
150 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
151 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
152 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
153 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
154 ... | 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 | |
155 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
156 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
157 fc03 eq = cong pred eq | |
158 fc02 : * x < * y1 | |
159 fc02 = fc01 i cx cy (fc03 i=y ) a | |
557 | 160 |
559 | 161 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
554 | 162 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
559 | 163 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
164 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
165 fc11 : * x < * y | |
166 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
167 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
168 fc10 : * x ≡ * y | |
169 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
170 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
171 fc12 : * y < * x | |
172 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
173 | |
600 | 174 |
562 | 175 fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
176 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
563 | 177 fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where |
178 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 ) | |
179 fc20 y<sx with <-cmp ( fcn s mf cy ) (fcn s mf cx ) | |
180 ... | tri< a ¬b ¬c = case2 a | |
181 ... | tri≈ ¬a b ¬c = case1 b | |
182 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> y<sx (s≤s c)) | |
183 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 | |
184 fc17 {x} {y} cx cy sx=y = fc18 (fcn s mf cy) cx cy refl sx=y where | |
185 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 | |
186 fc18 (suc i) {y} cx (fsuc y1 cy) i=y sx=i with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
187 ... | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy ( fc18 (suc i) {y1} cx cy i=y sx=i) -- dereference | |
188 ... | case2 y<fy = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k) ) ) fc19) where | |
189 fc19 : * x ≡ * y1 | |
190 fc19 = fcn-inject s mf cx cy (cong pred ( trans sx=i i=y )) | |
191 fc21 : ⊥ | |
192 fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy ) | |
193 ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy | |
194 cxx : FClosure A f s (f x) | |
195 cxx = fsuc x cx | |
196 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)) | |
600 | 197 fc16 x (init as) with proj1 (mf s as ) |
563 | 198 ... | case1 _ = case1 refl |
199 ... | case2 _ = case2 refl | |
200 fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) | |
201 ... | case1 _ = case1 refl | |
202 ... | case2 _ = case2 refl | |
203 fc22 : (suc ( fcn s mf cx )) Data.Nat.< (fcn s mf cy ) → * (f x) < * y | |
204 fc22 a with fc16 x cx | |
205 ... | case1 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq (<-trans a<sa a)) | |
206 ... | case2 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq a ) | |
207 ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx | |
208 ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx | |
209 ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y | |
210 ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) | |
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 |
567 | 228 record HasPrev (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
533 | 229 field |
534 | 230 y : Ordinal |
541 | 231 ay : odef B y |
534 | 232 x=fy : x ≡ f y |
529 | 233 |
570 | 234 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
654 | 235 field |
571 | 236 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
568 | 237 |
656 | 238 record SUP ( A B : HOD ) : Set (Level.suc n) where |
239 field | |
240 sup : HOD | |
241 A∋maximal : A ∋ sup | |
242 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
243 | |
690 | 244 -- |
245 -- sup and its fclosure is in a chain HOD | |
246 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
247 -- whole chain is a union of separated Chain | |
248 -- minimum index is y not ϕ | |
249 -- | |
250 | |
714 | 251 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u z : Ordinal) : Set n where |
690 | 252 field |
695 | 253 y-init : supf o∅ ≡ y |
714 | 254 au : odef A u |
721 | 255 ¬u=fx : {x : Ordinal} → ¬ ( u ≡ f x ) |
695 | 256 asup : (x : Ordinal) → odef A (supf x) |
714 | 257 fcy<sup : {z : Ordinal } → FClosure A f y z → z << supf u |
258 csupz : FClosure A f (supf u) z | |
259 order : {sup1 z1 : Ordinal} → (lt : sup1 o< u ) → FClosure A f (supf sup1 ) z1 → z1 << supf u | |
694 | 260 |
261 data Chain (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) : Ordinal → Ordinal → Set n where | |
711 | 262 ch-init : (z : Ordinal) → FClosure A f y z → Chain A f mf ay supf o∅ z |
694 | 263 ch-is-sup : {sup z : Ordinal } |
264 → ( is-sup : ChainP A f mf ay supf sup z) | |
265 → ( fc : FClosure A f (supf sup) z ) → Chain A f mf ay supf sup z | |
266 | |
267 -- Union of supf z which o< x | |
268 -- | |
690 | 269 |
694 | 270 record UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
271 (supf : Ordinal → Ordinal) (x : Ordinal) (z : Ordinal) : Set n where | |
272 field | |
273 u : Ordinal | |
711 | 274 u<x : (u o< x ) ∨ ( u ≡ o∅) |
707 | 275 uchain : Chain A f mf ay supf u z |
694 | 276 |
277 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A | |
278 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
279 | |
280 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) | |
281 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
282 UnionCF A f mf ay supf x | |
283 = 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 | 284 |
703 | 285 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
286 {init : Ordinal} (ay : odef A init) ( z : Ordinal ) : Set (Level.suc n) where | |
655 | 287 field |
694 | 288 supf : Ordinal → Ordinal |
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289 chain : HOD |
703 | 290 chain = UnionCF A f mf ay supf z |
568 | 291 field |
292 chain⊆A : chain ⊆' A | |
653 | 293 chain∋init : odef chain init |
294 initial : {y : Ordinal } → odef chain y → * init ≤ * y | |
568 | 295 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
654 | 296 f-total : IsTotalOrderSet chain |
712 | 297 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< z → (ab : odef A b) |
574 | 298 → HasPrev A chain ab f ∨ IsSup A chain ab |
568 | 299 → * a < * b → odef chain b |
653 | 300 |
568 | 301 record Maximal ( A : HOD ) : Set (Level.suc n) where |
302 field | |
303 maximal : HOD | |
304 A∋maximal : A ∋ maximal | |
305 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
567 | 306 |
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307 -- data Chain (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) : Ordinal → Ordinal → Set n where |
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308 -- |
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309 -- data Chain is total |
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310 |
694 | 311 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
312 {s s1 a b : Ordinal } ( ca : Chain A f mf ay supf s a ) ( cb : Chain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) | |
313 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where | |
314 ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → Chain A f mf ay supf xa a → Chain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) | |
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315 ct-ind xa xb {a} {b} (ch-init a fca) (ch-init b fcb) = fcn-cmp y f mf fca fcb |
690 | 316 ct-ind xa xb {a} {b} (ch-init a fca) (ch-is-sup supb fcb) = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where |
695 | 317 ct00 : * a < * (supf xb) |
318 ct00 = ChainP.fcy<sup supb fca | |
689
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319 ct01 : * a < * b |
695 | 320 ct01 with s≤fc (supf xb) f mf fcb |
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321 ... | case1 eq = subst (λ k → * a < k ) eq ct00 |
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322 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
690 | 323 ct-ind xa xb {a} {b} (ch-is-sup supa fca) (ch-init b fcb)= tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where |
695 | 324 ct00 : * b < * (supf xa) |
325 ct00 = ChainP.fcy<sup supa fcb | |
689
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326 ct01 : * b < * a |
695 | 327 ct01 with s≤fc (supf xa) f mf fca |
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328 ... | case1 eq = subst (λ k → * b < k ) eq ct00 |
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329 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
690 | 330 ct-ind xa xb {a} {b} (ch-is-sup supa fca) (ch-is-sup supb fcb) with trio< xa xb |
685 | 331 ... | tri< a₁ ¬b ¬c = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where |
695 | 332 ct03 : * a < * (supf xb) |
333 ct03 = ChainP.order supb a₁ (ChainP.csupz supa) | |
689
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334 ct02 : * a < * b |
695 | 335 ct02 with s≤fc (supf xb) f mf fcb |
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336 ... | case1 eq = subst (λ k → * a < k ) eq ct03 |
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337 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt |
695 | 338 ... | tri≈ ¬a refl ¬c = fcn-cmp (supf xa) f mf fca fcb |
685 | 339 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where |
695 | 340 ct05 : * b < * (supf xa) |
341 ct05 = ChainP.order supa c (ChainP.csupz supb) | |
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342 ct04 : * b < * a |
695 | 343 ct04 with s≤fc (supf xa) f mf fca |
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344 ... | case1 eq = subst (λ k → * b < k ) eq ct05 |
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345 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt |
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346 |
698 | 347 ChainP-next : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
348 → {x z : Ordinal } → ChainP A f mf ay supf x z → ChainP A f mf ay supf x (f z ) | |
714 | 349 ChainP-next A f mf {y} ay supf {x} {z} cp = record { y-init = ChainP.y-init cp ; asup = ChainP.asup cp ; au = ChainP.au cp |
698 | 350 ; fcy<sup = ChainP.fcy<sup cp ; csupz = fsuc _ (ChainP.csupz cp) ; order = ChainP.order cp } |
351 | |
497 | 352 Zorn-lemma : { A : HOD } |
464 | 353 → o∅ o< & A |
568 | 354 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
497 | 355 → Maximal A |
552 | 356 Zorn-lemma {A} 0<A supP = zorn00 where |
571 | 357 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
358 <-irr0 {a} {b} A∋a A∋b = <-irr | |
537 | 359 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
360 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
530 | 361 s : HOD |
362 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
568 | 363 as : A ∋ * ( & s ) |
364 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
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365 as0 : odef A (& s ) |
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366 as0 = subst (λ k → odef A k ) &iso as |
547 | 367 s<A : & s o< & A |
568 | 368 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
530 | 369 HasMaximal : HOD |
537 | 370 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
371 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
372 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
532 | 373 Gtx : { x : HOD} → A ∋ x → HOD |
537 | 374 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
375 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
376 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
377 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
378 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)) | |
379 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 | |
380 ¬x<m : ¬ (* x < * m) | |
381 ¬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 | 382 |
560 | 383 -- Uncountable ascending chain by axiom of choice |
530 | 384 cf : ¬ Maximal A → Ordinal → Ordinal |
532 | 385 cf nmx x with ODC.∋-p O A (* x) |
386 ... | no _ = o∅ | |
387 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
538 | 388 ... | yes nogt = -- no larger element, so it is maximal |
389 ⊥-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 | 390 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
537 | 391 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
392 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
393 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
394 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
395 ... | 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 ⟫ ) | |
396 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
606 | 397 |
398 --- | |
399 --- infintie ascention sequence of f | |
400 --- | |
530 | 401 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
537 | 402 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
530 | 403 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
532 | 404 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
543 | 405 |
703 | 406 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
653 | 407 (total : IsTotalOrderSet (ZChain.chain zc) ) → SUP A (ZChain.chain zc) |
703 | 408 sp0 f mf zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total |
543 | 409 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
410 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
411 | |
412 --- | |
560 | 413 --- the maximum chain has fix point of any ≤-monotonic function |
543 | 414 --- |
703 | 415 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
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parents:
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416 → (total : IsTotalOrderSet (ZChain.chain zc) ) |
703 | 417 → f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total)) |
418 fixpoint f mf zc total = z14 where | |
538 | 419 chain = ZChain.chain zc |
703 | 420 sp1 = sp0 f mf zc total |
712 | 421 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
570 | 422 → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
538 | 423 → * a < * b → odef chain b |
424 z10 = ZChain.is-max zc | |
543 | 425 z11 : & (SUP.sup sp1) o< & A |
426 z11 = c<→o< ( SUP.A∋maximal sp1) | |
538 | 427 z12 : odef chain (& (SUP.sup sp1)) |
428 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
653 | 429 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
712 | 430 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z11 (SUP.A∋maximal sp1) |
570 | 431 (case2 z19 ) z13 where |
538 | 432 z13 : * (& s) < * (& (SUP.sup sp1)) |
653 | 433 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc ) |
538 | 434 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
435 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
570 | 436 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) |
571 | 437 z19 = record { x<sup = z20 } where |
438 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
439 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
570 | 440 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
441 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
442 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
703 | 443 z14 : f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total )) |
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parents:
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444 z14 with total (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
631 | 445 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
446 z16 : ⊥ | |
447 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
448 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
449 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
450 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
451 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
452 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
453 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) | |
454 z17 : ⊥ | |
455 z17 with z15 | |
456 ... | case1 eq = ¬b eq | |
457 ... | case2 lt = ¬a lt | |
560 | 458 |
459 -- ZChain contradicts ¬ Maximal | |
460 -- | |
571 | 461 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
560 | 462 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
463 -- | |
697 | 464 z04 : (nmx : ¬ Maximal A ) |
703 | 465 → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) |
664 | 466 → IsTotalOrderSet (ZChain.chain zc) → ⊥ |
703 | 467 z04 nmx zc total = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal sp1 )))) |
571 | 468 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) |
703 | 469 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc total ))) -- x ≡ f x ̄ |
633
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470 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1 ))) where -- x < f x |
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471 sp1 : SUP A (ZChain.chain zc) |
703 | 472 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc total |
538 | 473 c = & (SUP.sup sp1) |
548 | 474 |
711 | 475 inititalChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → ZChain A f mf ay o∅ |
476 inititalChain f mf {y} ay = record { supf = isupf ; chain⊆A = λ lt → proj1 lt ; chain∋init = cy | |
477 ; initial = isy ; f-next = inext ; f-total = itotal ; is-max = imax } where | |
478 isupf : Ordinal → Ordinal | |
479 isupf z = y | |
480 cy : odef (UnionCF A f mf ay isupf o∅) y | |
481 cy = ⟪ ay , record { u = o∅ ; u<x = case2 refl ; uchain = ch-init _ (init ay) } ⟫ | |
482 isy : {z : Ordinal } → odef (UnionCF A f mf ay isupf o∅) z → * y ≤ * z | |
483 isy {z} ⟪ az , uz ⟫ with UChain.uchain uz | |
484 ... | ch-init z fc = s≤fc y f mf fc | |
485 ... | ch-is-sup is-sup fc = ⊥-elim ( <-irr (case1 refl) ( ChainP.fcy<sup is-sup (init ay) ) ) | |
486 inext : {a : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a → odef (UnionCF A f mf ay isupf o∅) (f a) | |
487 inext {a} ua with UChain.uchain (proj2 ua) | |
488 ... | ch-init a fc = ⟪ proj2 (mf _ (proj1 ua)) , record { u = o∅ ; u<x = case2 refl ; uchain = ch-init _ (fsuc _ fc ) } ⟫ | |
489 ... | ch-is-sup is-sup fc = ⊥-elim ( <-irr (case1 refl) ( ChainP.fcy<sup is-sup (init ay) ) ) | |
490 itotal : IsTotalOrderSet (UnionCF A f mf ay isupf o∅) | |
491 itotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
492 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
493 uz01 = chain-total A f mf ay isupf (UChain.uchain (proj2 ca)) (UChain.uchain (proj2 cb)) | |
494 imax : {a b : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a → | |
712 | 495 b o< o∅ → (ab : odef A b) → |
711 | 496 HasPrev A (UnionCF A f mf ay isupf o∅) ab f ∨ IsSup A (UnionCF A f mf ay isupf o∅) ab → |
497 * a < * b → odef (UnionCF A f mf ay isupf o∅) b | |
714 | 498 imax {a} {b} ua b<x ab (case1 hasp) a<b = subst (λ k → odef (UnionCF A f mf ay isupf o∅) k ) (sym (HasPrev.x=fy hasp)) ( inext (HasPrev.ay hasp) ) |
499 imax {a} {b} ua b<x ab (case2 sup) a<b = ⊥-elim ( ¬x<0 b<x ) | |
711 | 500 |
716 | 501 -- exor-sup : (B : HOD) |
502 -- → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → | |
503 -- → {x : Ordinal } (xa : odef A x) → HasPrev A B xa → IsSup A B xa → ⊥ | |
504 -- exor-sup B f mf {y} ay {x} xa hasp is-sup with trio< | |
505 | |
560 | 506 -- |
547 | 507 -- create all ZChains under o< x |
560 | 508 -- |
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
509 |
674 | 510 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
703 | 511 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
707 | 512 ind f mf {y} ay x prev with Oprev-p x |
697 | 513 ... | yes op = zc4 where |
682 | 514 -- |
515 -- we have previous ordinal to use induction | |
516 -- | |
517 px = Oprev.oprev op | |
703 | 518 zc : ZChain A f mf ay (Oprev.oprev op) |
682 | 519 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
520 px<x : px o< x | |
521 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
709 | 522 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
523 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
697 | 524 |
703 | 525 pchain : HOD |
526 pchain = UnionCF A f mf ay (ZChain.supf zc) x | |
527 ptotal : IsTotalOrderSet pchain | |
528 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
529 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
707 | 530 uz01 = chain-total A f mf ay (ZChain.supf zc) (UChain.uchain (proj2 ca)) (UChain.uchain (proj2 cb)) |
704 | 531 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y |
532 pchain⊆A {y} ny = proj1 ny | |
533 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
707 | 534 pnext {a} ⟪ aa , ua ⟫ = ⟪ afa , record { u = UChain.u ua ; u<x = UChain.u<x ua ; uchain = fua } ⟫ where |
704 | 535 afa : odef A ( f a ) |
536 afa = proj2 ( mf a aa ) | |
707 | 537 fua : Chain A f mf ay (ZChain.supf zc) (UChain.u ua) (f a) |
538 fua with UChain.uchain ua | |
704 | 539 ... | ch-init a fc = ch-init (f a) ( fsuc _ fc ) |
707 | 540 ... | ch-is-sup is-sup fc = ch-is-sup (ChainP-next A f mf ay _ is-sup ) (fsuc _ fc ) |
704 | 541 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
707 | 542 pinit {a} ⟪ aa , ua ⟫ with UChain.uchain ua |
704 | 543 ... | ch-init a fc = s≤fc y f mf fc |
707 | 544 ... | ch-is-sup is-sup fc = ≤-ftrans (case2 zc7) (s≤fc _ f mf fc) where |
545 zc7 : y << (ZChain.supf zc) (UChain.u ua) | |
546 zc7 = ChainP.fcy<sup is-sup (init ay) | |
704 | 547 pcy : odef pchain y |
711 | 548 pcy = ⟪ ay , record { u = o∅ ; u<x = case2 refl ; uchain = ch-init _ (init ay) } ⟫ |
703 | 549 |
611 | 550 -- if previous chain satisfies maximality, we caan reuse it |
551 -- | |
721 | 552 no-extension : ( {a b : Ordinal} → odef pchain a → b o< x → (ab : odef A b) → |
707 | 553 HasPrev A pchain ab f ∨ IsSup A pchain ab → |
554 * a < * b → odef pchain b ) → ZChain A f mf ay x | |
721 | 555 no-extension is-max = record { initial = pinit ; chain∋init = pcy |
707 | 556 ; chain⊆A = pchain⊆A ; f-next = pnext ; f-total = ptotal ; is-max = is-max } |
610 | 557 |
710 | 558 zcp : {a b : Ordinal} → odef pchain a |
559 → pchain ≡ UnionCF A f mf ay (ZChain.supf zc) (Oprev.oprev op ) | |
712 | 560 → b o< px → (ab : odef A b) |
710 | 561 → HasPrev A pchain ab f ∨ IsSup A pchain ab |
562 → * a < * b → odef pchain b | |
563 zcp {a} {b} za cheq b<x ab P a<b = subst (λ k → odef k b ) (sym cheq) ( | |
712 | 564 ZChain.is-max zc (subst (λ k → odef k a) cheq za) b<x ab |
710 | 565 (subst (λ k → HasPrev A k ab f ∨ IsSup A k ab ) cheq P) a<b ) |
709 | 566 |
710 | 567 chain-mono : UnionCF A f mf ay (ZChain.supf zc) (Oprev.oprev op ) ⊆' pchain |
568 chain-mono {a} za = ⟪ proj1 za , record { u = UChain.u (proj2 za) ; u<x = zc11 ; uchain = UChain.uchain (proj2 za) } ⟫ where | |
711 | 569 zc11 : (UChain.u (proj2 za) o< x) ∨ (UChain.u (proj2 za) ≡ o∅) |
710 | 570 zc11 with UChain.u<x (proj2 za) |
571 ... | case1 z<x = case1 (ordtrans z<x px<x ) | |
711 | 572 ... | case2 z=0 = case2 z=0 |
573 | |
574 chain-≡ : pchain ⊆' UnionCF A f mf ay (ZChain.supf zc) (Oprev.oprev op ) | |
575 → pchain ≡ UnionCF A f mf ay (ZChain.supf zc) (Oprev.oprev op ) | |
576 chain-≡ lt = ==→o≡ record { eq→ = lt ; eq← = chain-mono } | |
709 | 577 |
721 | 578 chain-x : ( {z : Ordinal} → (az : odef pchain z) → ¬ ( UChain.u (proj2 az) ≡ px )) → pchain ⊆' UnionCF A f mf ay (ZChain.supf zc) px |
579 chain-x ne {z} ⟪ az , record { u = u ; u<x = case2 u=y ; uchain = uc } ⟫ = | |
714 | 580 ⟪ az , record { u = u ; u<x = case2 u=y ; uchain = uc } ⟫ |
721 | 581 chain-x ne {z} ⟪ az , record { u = u ; u<x = case1 u<x ; uchain = ch-init z fc } ⟫ with trio< o∅ px |
714 | 582 ... | tri< a ¬b ¬c = ⟪ az , record { u = u ; u<x = case1 a ; uchain = ch-init z fc } ⟫ |
583 ... | tri≈ ¬a b ¬c = ⟪ az , record { u = u ; u<x = case2 refl ; uchain = ch-init z fc } ⟫ | |
584 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) | |
721 | 585 chain-x ne {z} uz@record { proj1 = az ; proj2 = record { u = u ; u<x = case1 u<x ; uchain = ch-is-sup is-sup fc } } with trio< u px |
714 | 586 ... | tri< a ¬b ¬c = ⟪ az , record { u = u ; u<x = case1 a ; uchain = ch-is-sup is-sup fc } ⟫ |
587 ... | tri≈ ¬a b ¬c = ⊥-elim ( ne uz b ) | |
588 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x ⟫ ) | |
589 | |
703 | 590 zc4 : ZChain A f mf ay x |
713 | 591 zc4 with ODC.∋-p O A (* px) |
721 | 592 ... | no noapx = no-extension zc1 where -- ¬ A ∋ p, just skip |
712 | 593 zc1 : {a b : Ordinal} → odef pchain a → b o< x → (ab : odef A b) → |
707 | 594 HasPrev A pchain ab f ∨ IsSup A pchain ab → |
595 * a < * b → odef pchain b | |
713 | 596 zc1 {a} {b} za b<x ab P a<b with trio< b px |
721 | 597 ... | tri< lt ¬b ¬c = zcp za (chain-≡ (chain-x zc14)) lt ab P a<b where |
598 zc14 : {z : Ordinal} (az : odef pchain z) → ¬ UChain.u (proj2 az) ≡ px | |
599 zc14 {z} ⟪ az , record { u = u ; u<x = case1 a ; uchain = ch-is-sup is-sup fc } ⟫ upx | |
600 = ⊥-elim ( noapx ( subst (λ k → odef A k ) (trans upx (sym &iso) ) (ChainP.au is-sup ) )) | |
714 | 601 ... | tri≈ ¬a b=px ¬c = ⊥-elim ( noapx (subst (λ k → odef A k ) (trans b=px (sym &iso)) ab ) ) |
713 | 602 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) |
603 ... | yes apx with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc ) apx f ) | |
703 | 604 -- we have to check adding x preserve is-max ZChain A y f mf x |
721 | 605 ... | case1 pr = no-extension zc7 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
712 | 606 zc7 : {a b : Ordinal} → odef pchain a → b o< x → (ab : odef A b) → |
707 | 607 HasPrev A pchain ab f ∨ IsSup A pchain ab → |
608 * a < * b → odef pchain b | |
714 | 609 zc7 {a} {b} za b<x ab P a<b with trio< b px |
721 | 610 ... | tri< lt ¬b ¬c = zcp za (chain-≡ (chain-x zc14)) lt ab P a<b where |
611 zc14 : {z : Ordinal} (az : odef pchain z) → ¬ UChain.u (proj2 az) ≡ px | |
612 zc14 {z} ⟪ az , record { u = u ; u<x = case1 a ; uchain = ch-is-sup is-sup fc } ⟫ upx | |
613 = ChainP.¬u=fx is-sup (subst (λ k → k ≡ _ ) (trans &iso (sym upx) ) (HasPrev.x=fy pr) ) | |
714 | 614 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) |
715 | 615 ... | tri≈ ¬a b=px ¬c = zc15 where |
616 zc14 : f (HasPrev.y pr) ≡ b | |
617 zc14 = begin f (HasPrev.y pr) ≡⟨ sym (HasPrev.x=fy pr) ⟩ | |
618 & (* px) ≡⟨ &iso ⟩ | |
619 px ≡⟨ sym b=px ⟩ | |
620 b ∎ where open ≡-Reasoning | |
621 zc15 : odef pchain b | |
622 zc15 with ZChain.f-next zc (HasPrev.ay pr) | |
623 ... | ⟪ az , record { u = u ; u<x = case2 refl ; uchain = ch-init z fc } ⟫ | |
624 = ⟪ subst (λ k → odef A k ) zc14 az , record { u = u ; u<x = case2 refl | |
625 ; uchain = ch-init _ (subst (λ k → FClosure A f y k ) zc14 fc) } ⟫ | |
626 ... | ⟪ az , record { u = u ; u<x = case1 u<x ; uchain = ch-init z fc } ⟫ | |
627 = ⟪ subst (λ k → odef A k ) zc14 az , record { u = u ; u<x = case1 (b<x→0<x b<x ) | |
628 ; uchain = ch-init _ (subst (λ k → FClosure A f y k ) zc14 fc) } ⟫ | |
713 | 629 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc ) apx ) |
682 | 630 ... | case1 is-sup = -- x is a sup of zc |
698 | 631 record { chain⊆A = pchain⊆A ; f-next = pnext ; f-total = ptotal |
632 ; initial = pinit ; chain∋init = pcy ; is-max = p-ismax } where | |
633 p-ismax : {a b : Ordinal} → odef pchain a → | |
712 | 634 b o< x → (ab : odef A b) → |
698 | 635 ( HasPrev A pchain ab f ∨ IsSup A pchain ab ) → |
636 * a < * b → odef pchain b | |
714 | 637 p-ismax {a} {b} za b<x ab P a<b with trio< b px |
638 ... | tri< lt ¬b ¬c = zcp za (chain-≡ ? ) lt ab P a<b | |
639 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
640 ... | tri≈ ¬a b=px ¬c with P | |
641 ... | case1 hasp = ? | |
642 ... | case2 sup = ? | |
721 | 643 ... | case2 ¬x=sup = no-extension z18 where -- px is not f y' nor sup of former ZChain from y -- no extention |
712 | 644 z18 : {a b : Ordinal} → odef pchain a → b o< x → (ab : odef A b) → |
707 | 645 HasPrev A pchain ab f ∨ IsSup A pchain ab → |
646 * a < * b → odef pchain b | |
721 | 647 z18 {a} {b} za b<x ab P a<b with P |
648 ... | case1 pr = subst (λ k → odef pchain k ) (sym (HasPrev.x=fy pr)) ( pnext (HasPrev.ay pr) ) | |
649 ... | case2 b=sup with trio< b px | |
714 | 650 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) |
721 | 651 ... | tri≈ ¬a b=px ¬c = ⊥-elim ( ¬x=sup record { |
652 x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=px (sym &iso)) | |
710 | 653 (IsSup.x<sup b=sup (chain-mono zy) ) } ) |
721 | 654 ... | tri< b<px ¬b ¬c = chain-mono (ZChain.is-max zc pa b<px ab (case2 record { x<sup = sup1 }) a<b) where |
655 z19 : {z : Ordinal} (az : odef pchain z) → ¬ UChain.u (proj2 az) ≡ px | |
656 z19 {z} za@record {proj1 = az ; proj2 = record { u = u ; u<x = case1 a ; uchain = ch-is-sup is-sup fc } } | |
657 with trio< (UChain.u (proj2 za)) px | |
658 ... | tri> ¬a ¬b₁ c = ¬b₁ | |
659 ... | tri< a₁ ¬b₁ ¬c₁ = ¬b₁ | |
660 ... | tri≈ ¬a u=px ¬c₁ with IsSup.x<sup b=sup ⟪ az , record { u = u ; u<x = case1 a ; uchain = ch-is-sup is-sup fc } ⟫ | |
661 ... | case1 z=b = ? | |
662 ... | case2 z<b = ? | |
663 pa : odef (UnionCF A f mf ay (ZChain.supf zc) (Oprev.oprev op)) a | |
664 pa = chain-x z19 za | |
665 sup1 : {z : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) (Oprev.oprev op)) z → (z ≡ b) ∨ (z << b) | |
666 sup1 {z} uz = IsSup.x<sup b=sup ( chain-mono uz ) | |
682 | 667 ... | no op = zc5 where |
703 | 668 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
669 pzc z z<x = prev z z<x | |
670 psupf0 : (z : Ordinal) → Ordinal | |
671 psupf0 z with trio< z x | |
672 ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z | |
704 | 673 ... | tri≈ ¬a b ¬c = y |
674 ... | tri> ¬a ¬b c = y | |
675 pchain : HOD | |
676 pchain = UnionCF A f mf ay psupf0 x | |
677 ptotal : IsTotalOrderSet pchain | |
678 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
703 | 679 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
707 | 680 uz01 = chain-total A f mf ay psupf0 (UChain.uchain (proj2 ca)) (UChain.uchain (proj2 cb)) |
704 | 681 |
682 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y | |
683 pchain⊆A {y} ny = proj1 ny | |
684 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
707 | 685 pnext {a} ⟪ aa , ua ⟫ = ⟪ afa , record { u = UChain.u ua ; u<x = UChain.u<x ua ; uchain = fua } ⟫ where |
704 | 686 afa : odef A ( f a ) |
687 afa = proj2 ( mf a aa ) | |
688 fua : Chain A f mf ay psupf0 (UChain.u ua) (f a) | |
707 | 689 fua with UChain.uchain ua |
704 | 690 ... | ch-init a fc = ch-init (f a) ( fsuc _ fc ) |
707 | 691 ... | ch-is-sup is-sup fc = ch-is-sup (ChainP-next A f mf ay _ is-sup ) (fsuc _ fc ) |
704 | 692 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
707 | 693 pinit {a} ⟪ aa , ua ⟫ with UChain.uchain ua |
704 | 694 ... | ch-init a fc = s≤fc y f mf fc |
707 | 695 ... | ch-is-sup is-sup fc = ≤-ftrans (case2 zc7) (s≤fc _ f mf fc) where |
704 | 696 zc7 : y << psupf0 (UChain.u ua) |
707 | 697 zc7 = ChainP.fcy<sup is-sup (init ay) |
704 | 698 pcy : odef pchain y |
711 | 699 pcy = ⟪ ay , record { u = o∅ ; u<x = case2 refl ; uchain = ch-init _ (init ay) } ⟫ |
704 | 700 |
721 | 701 no-extension : ( {a b : Ordinal} → odef pchain a → b o< x → (ab : odef A b) → |
709 | 702 HasPrev A pchain ab f ∨ IsSup A pchain ab → |
703 * a < * b → odef pchain b ) → ZChain A f mf ay x | |
721 | 704 no-extension is-max = record { initial = pinit ; chain∋init = pcy |
709 | 705 ; chain⊆A = pchain⊆A ; f-next = pnext ; f-total = ptotal ; is-max = is-max } |
706 | |
704 | 707 usup : SUP A pchain |
708 usup = supP pchain (λ lt → proj1 lt) ptotal | |
703 | 709 spu = & (SUP.sup usup) |
710 psupf : Ordinal → Ordinal | |
711 psupf z with trio< z x | |
712 ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z | |
713 ... | tri≈ ¬a b ¬c = spu | |
714 ... | tri> ¬a ¬b c = spu | |
704 | 715 |
711 | 716 uzc : {a : Ordinal } → (za : odef pchain a) → ZChain A f mf ay (UChain.u (proj2 za)) |
717 uzc {a} za with UChain.u<x (proj2 za) | |
718 ... | case1 u<x = pzc _ u<x | |
713 | 719 ... | case2 u=0 = subst (λ k → ZChain A f mf ay k ) (sym u=0) (inititalChain f mf {y} ay ) |
711 | 720 |
721 zcp : {a b : Ordinal} → (za : odef pchain a ) | |
713 | 722 → pchain ≡ UnionCF A f mf ay psupf x |
710 | 723 → b o< x → (ab : odef A b) |
724 → HasPrev A pchain ab f ∨ IsSup A pchain ab | |
725 → * a < * b → odef pchain b | |
713 | 726 zcp {a} {b} za cheq b<x ab P a<b = subst (λ k → odef k b) (sym cheq) zc12 where |
727 zc13 : odef (UnionCF A f mf ay (ZChain.supf (uzc za)) (UChain.u (proj2 za))) a | |
728 zc13 = ⟪ proj1 za , record { u = UChain.u (proj2 za) ; u<x = ? ; uchain = ? } ⟫ | |
729 zc14 : b o< UChain.u (proj2 za) | |
730 zc14 = ? | |
731 zc12 : odef (UnionCF A f mf ay psupf x) b | |
732 zc12 = ⟪ ab , record { u = UChain.u (proj2 za) ; u<x = ? ; uchain = ? } ⟫ | |
733 -- ZChain.is-max (uzc za) ? ? ab (subst (λ k → HasPrev A k ab f ∨ IsSup A k ab ) cheq P) a<b | |
710 | 734 |
735 chain-mono : pchain ⊆' UnionCF A f mf ay psupf x | |
721 | 736 chain-mono {a} ⟪ az , record { u = u ; u<x = u<x ; uchain = ch-init a x } ⟫ = |
737 ⟪ az , record { u = u ; u<x = u<x ; uchain = ch-init a x } ⟫ | |
738 chain-mono {.(psupf0 u)} ⟪ az , record { u = u ; u<x = u<x ; uchain = ch-is-sup is-sup (init x) } ⟫ = | |
739 ⟪ az , record { u = u ; u<x = u<x ; uchain = ch-is-sup ? ? } ⟫ | |
740 chain-mono {.(f x)} ⟪ az , record { u = u ; u<x = u<x ; uchain = ch-is-sup is-sup (fsuc x fc) } ⟫ = | |
741 ⟪ az , record { u = u ; u<x = u<x ; uchain = ch-is-sup ? (fsuc x ?) } ⟫ | |
742 | |
743 chain-x : {z : Ordinal } → ¬ ( z << spu ) → odef (UnionCF A f mf ay psupf x) z → odef pchain z | |
744 chain-x = ? | |
710 | 745 |
711 | 746 chain-≡ : UnionCF A f mf ay psupf x ⊆' pchain |
747 → UnionCF A f mf ay psupf x ≡ pchain | |
748 chain-≡ lt = ==→o≡ record { eq→ = lt ; eq← = chain-mono } | |
749 | |
703 | 750 zc5 : ZChain A f mf ay x |
697 | 751 zc5 with ODC.∋-p O A (* x) |
721 | 752 ... | no noax = no-extension ? where -- ¬ A ∋ p, just skip |
704 | 753 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain ax f ) |
703 | 754 -- we have to check adding x preserve is-max ZChain A y f mf x |
721 | 755 ... | case1 pr = no-extension ? |
756 -- subst (λ k → odef pchain k ) (sym (HasPrev.x=fy pr)) ( pnext (HasPrev.ay pr) ) | |
704 | 757 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A pchain ax ) |
689
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
758 ... | case1 is-sup = {!!} -- x is a sup of (zc ?) |
721 | 759 ... | case2 ¬x=sup = no-extension z18 where -- x is not f y' nor sup of former ZChain from y -- no extention |
712 | 760 z18 : {a b : Ordinal} → odef pchain a → b o< x → (ab : odef A b) → |
711 | 761 HasPrev A pchain ab f ∨ IsSup A pchain ab → |
762 * a < * b → odef pchain b | |
721 | 763 z18 {a} {b} za b<x ab P a<b with P |
711 | 764 ... | case1 pr = subst (λ k → odef pchain k ) (sym (HasPrev.x=fy pr)) ( pnext (HasPrev.ay pr) ) |
721 | 765 ... | case2 b=sup = ? where |
766 z30 = ZChain.is-max (uzc za) ? ? ab (case2 ?) a<b | |
767 | |
768 --⊥-elim ( ¬x=sup record { | |
769 -- x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) ? | |
770 -- (IsSup.x<sup b=sup ? ) } ) | |
682 | 771 |
553 | 772 |
703 | 773 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay (& A) |
774 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
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775 |
551 | 776 zorn00 : Maximal A |
777 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
778 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
779 -- yes we have the maximal | |
780 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
606 | 781 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
551 | 782 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
783 zorn01 = proj1 zorn03 | |
784 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
785 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
703 | 786 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04 total ) where |
551 | 787 -- if we have no maximal, make ZChain, which contradict SUP condition |
788 nmx : ¬ Maximal A | |
789 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
790 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
791 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) ) ⟫ | |
703 | 792 zorn04 : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) |
653 | 793 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
634 | 794 total : IsTotalOrderSet (ZChain.chain zorn04) |
654 | 795 total {a} {b} = zorn06 where |
796 zorn06 : odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a) | |
797 zorn06 = ZChain.f-total (SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as) ) | |
551 | 798 |
516 | 799 -- usage (see filter.agda ) |
800 -- | |
497 | 801 -- _⊆'_ : ( A B : HOD ) → Set n |
802 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
482 | 803 |
497 | 804 -- MaximumSubset : {L P : HOD} |
805 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
806 -- → IsPartialOrderSet P _⊆'_ | |
807 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
808 -- → Maximal P (_⊆'_) | |
809 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |