Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 799:c8a166abcae0
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 07 Aug 2022 18:39:18 +0900 |
| parents | 9cf74877efab |
| children | 06eedb0d26a0 |
| rev | line source |
|---|---|
| 478 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 508 | 2 open import Level hiding ( suc ; zero ) |
| 431 | 3 open import Ordinals |
| 552 | 4 open import Relation.Binary |
| 5 open import Relation.Binary.Core | |
| 6 open import Relation.Binary.PropositionalEquality | |
| 497 | 7 import OD |
| 552 | 8 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where |
| 431 | 9 |
| 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 | |
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TransitiveClosure with x <= f x is possible
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67 _≤_ : (x y : HOD) → Set (Level.suc n) |
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TransitiveClosure with x <= f x is possible
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parents:
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68 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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TransitiveClosure with x <= f x is possible
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parents:
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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 |
| 557 | 127 |
| 559 | 128 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 129 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 796 | 130 fcn-cmp-1 : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 131 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → * x < * y → (* (f x) ≡ * y) ∨ ( * (f x) < * y ) | |
| 132 fcn-cmp-1 {A} s f mf (init x refl) (init x₁ refl) x<y = {!!} | |
| 133 fcn-cmp-1 {A} s f mf (init x refl) (fsuc x₁ cy) x<y = {!!} | |
| 134 fcn-cmp-1 {A} s f mf (fsuc x cx) (init ay refl) x<y = {!!} | |
| 135 fcn-cmp-1 {A} s f mf (fsuc x cx) (fsuc y cy) x<y with proj1 (mf x (A∋fc s f mf cx)) | proj1 (mf y (A∋fc s f mf cy)) | |
| 136 ... | case1 eqx | case1 eqy = {!!} | |
| 137 ... | case1 eqx | case2 lt = {!!} | |
| 138 ... | case2 lt | case1 eqy = {!!} | |
| 139 ... | case2 ltx | case2 lty = {!!} | |
| 600 | 140 |
| 796 | 141 fcn-cmp {A} s {.s} {y} f mf (init ax refl) (init ay refl) = {!!} |
| 142 fcn-cmp {A} s {.s} {.(f x)} f mf (init ax refl) (fsuc x cy) = {!!} | |
| 143 fcn-cmp {A} s {.(f x)} {y} f mf (fsuc x cx) (init ay refl) = {!!} | |
| 144 fcn-cmp {A} s {.(f x)} {.(f y)} f mf (fsuc x cx) (fsuc y cy) with proj1 (mf x (A∋fc s f mf cx)) | proj1 (mf y (A∋fc s f mf cy)) | |
| 145 ... | case1 eqx | case1 eqy = {!!} | |
| 146 ... | case1 eqx | case2 lt = {!!} | |
| 147 ... | case2 lt | case1 eqy = {!!} | |
| 148 ... | case2 ltx | case2 lty = {!!} | |
| 563 | 149 |
| 729 | 150 |
| 560 | 151 -- open import Relation.Binary.Properties.Poset as Poset |
| 152 | |
| 153 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 154 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 155 | |
| 567 | 156 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 157 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 158 |
| 568 | 159 _⊆'_ : ( A B : HOD ) → Set n |
| 160 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 161 |
| 162 -- | |
| 163 -- inductive maxmum tree from x | |
| 164 -- tree structure | |
| 165 -- | |
| 554 | 166 |
| 567 | 167 record HasPrev (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 168 field |
| 534 | 169 y : Ordinal |
| 541 | 170 ay : odef B y |
| 534 | 171 x=fy : x ≡ f y |
| 529 | 172 |
| 570 | 173 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 654 | 174 field |
| 779 | 175 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 568 | 176 |
| 656 | 177 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 178 field | |
| 179 sup : HOD | |
| 180 A∋maximal : A ∋ sup | |
| 181 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 182 | |
| 690 | 183 -- |
| 184 -- sup and its fclosure is in a chain HOD | |
| 185 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 186 -- whole chain is a union of separated Chain | |
| 187 -- minimum index is y not ϕ | |
| 188 -- | |
| 189 | |
| 787 | 190 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 | 191 field |
| 765 | 192 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
| 793 | 193 order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
| 799 | 194 supu=u : supf u ≡ u |
| 694 | 195 |
| 196 -- Union of supf z which o< x | |
| 197 -- | |
| 690 | 198 |
| 748 | 199 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 200 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where | |
| 201 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 791 | 202 ch-is-sup : (u : Ordinal) {z : Ordinal } (u≤x : u o≤ x) ( is-sup : ChainP A f mf ay supf u ) |
| 748 | 203 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z |
| 694 | 204 |
| 205 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A | |
| 206 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 207 | |
| 208 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) | |
| 209 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
| 210 UnionCF A f mf ay supf x | |
| 211 = 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 | 212 |
| 703 | 213 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 214 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 215 field |
| 694 | 216 supf : Ordinal → Ordinal |
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mutual tranfinite in zorn
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217 chain : HOD |
| 703 | 218 chain = UnionCF A f mf ay supf z |
| 568 | 219 field |
| 220 chain⊆A : chain ⊆' A | |
| 783 | 221 chain∋init : odef chain y |
| 222 initial : {z : Ordinal } → odef chain z → * y ≤ * z | |
| 568 | 223 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 654 | 224 f-total : IsTotalOrderSet chain |
| 756 | 225 |
| 791 | 226 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
| 761 | 227 sup=u : {b : Ordinal} → (ab : odef A b) → b o< z → IsSup A (UnionCF A f mf ay supf (osuc b)) ab → supf b ≡ b |
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csupf depends on order cyclicly
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228 supf-is-sup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ & (SUP.sup (sup x≤z) ) |
| 797 | 229 csupf : {b : Ordinal } → b o≤ z → odef (UnionCF A f mf ay supf b) (supf b) |
| 230 supf≤x :{x : Ordinal } → z o≤ x → supf z ≡ supf x | |
| 798 | 231 |
| 799 | 232 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 |
| 233 fcy<sup {u} {w} u≤z fc with SUP.x<sup (sup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) | |
| 798 | 234 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ |
| 799 | 235 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans (cong (&) eq) (sym (supf-is-sup u≤z ) ) )) |
| 236 ... | case2 lt = case2 (subst (λ k → * w < k ) (subst (λ k → k ≡ _ ) *iso (cong (*) (sym (supf-is-sup u≤z ))) ) lt ) | |
| 798 | 237 |
| 799 | 238 supf-mono : {x y1 : Ordinal } → x o< y1 → supf x o≤ supf y1 |
| 239 supf-mono {x} {y1} x<y1 = sf<sy1 where | |
| 240 -- supf x << supf y1 → supf x o< supf y1 | |
| 241 -- x o< y1 → supf x <= supf y1 | |
| 242 -- z o≤ x → supf x ≡ supf y1 ≡ supf z | |
| 243 -- x o< z → z o< y1 → supf x ≡ supf y1 ≡ supf z | |
| 244 supy : {x : Ordinal } → x o≤ z → FClosure A f y (supf x) → supf x ≡ y | |
| 245 supy {x} x≤z fcx = ? where | |
| 246 zc06 : ( * y ≡ SUP.sup (sup x≤z )) ∨ ( * y < SUP.sup ( sup x≤z ) ) | |
| 247 zc06 = SUP.x<sup (sup x≤z) ⟪ subst (λ k → odef A k ) (sym &iso) ay , ch-init (init ay (sym &iso) ) ⟫ | |
| 248 sf<sy1 : supf x o≤ supf y1 | |
| 249 sf<sy1 with trio< x z | |
| 250 ... | tri> ¬a ¬b c = o≤-refl0 (( trans (sym (supf≤x (o<→≤ c))) (supf≤x (ordtrans (ordtrans c x<y1 ) <-osuc ) ) )) | |
| 251 ... | tri≈ ¬a b ¬c = o≤-refl0 (trans (sym (supf≤x (o≤-refl0 (sym b)))) (supf≤x (subst (λ k → k o< osuc y1) b (o<→≤ x<y1)))) | |
| 252 ... | tri< x<z ¬b ¬c with trio< (supf x) (supf y1) | |
| 798 | 253 ... | tri< a ¬b ¬c = o<→≤ a |
| 254 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 799 | 255 ... | tri> ¬a ¬b sy1<sx with trio< z y1 |
| 798 | 256 ... | tri< a ¬b ¬c = ? |
| 797 | 257 ... | tri≈ ¬a b ¬c = ? |
| 799 | 258 ... | tri> ¬a ¬b y1<z = ? |
| 259 zc04 : x o< z → y1 o≤ z → supf x o≤ supf y1 | |
| 260 zc04 x<z y1≤z with csupf (o<→≤ x<z) | csupf y1≤z | |
| 261 ... | ⟪ ax , ch-init fcx ⟫ | ⟪ ay1 , ch-init fcy1 ⟫ with fcy<sup (o<→≤ x<z) fcy1 | |
| 798 | 262 ... | case1 eq = o≤-refl0 (sym eq) |
| 799 | 263 ... | case2 lt with fcy<sup y1≤z fcx |
| 798 | 264 ... | case1 eq = o≤-refl0 eq |
| 265 ... | case2 lt1 = ⊥-elim ( <-irr (case2 lt) lt1 ) | |
| 799 | 266 zc04 x<z y1≤z | ⟪ ax , ch-is-sup ux ux≤z is-sup-x fcx ⟫ | ⟪ ay1 , ch-init fcy1 ⟫ with fcy<sup (o<→≤ x<z) fcy1 |
| 798 | 267 ... | case1 eq = o≤-refl0 (sym eq) |
| 799 | 268 ... | case2 lt with trio< (supf x) (supf y1) |
| 269 ... | tri< a ¬b ¬c = o<→≤ a | |
| 270 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 271 ... | tri> ¬a ¬b y1<z = ? where | |
| 272 zc05 : ( supf y1 ≡ supf ux ) ∨ (supf y1 << supf ux ) | |
| 273 zc05 = ChainP.fcy<sup is-sup-x fcy1 | |
| 274 zc04 x<z y1≤z | ⟪ ax , ch-is-sup ux ux≤z is-sup-x fcx ⟫ | ⟪ ay1 , ch-init fcy1 ⟫ | case2 lt1 = ? -- sy1 << sx | |
| 275 zc04 x<z y1≤z | ⟪ ax , ch-init fcx ⟫ | ⟪ ay1 , ch-is-sup uy1 uy1≤z is-sup-y1 fcy1 ⟫ = ? | |
| 276 zc04 x<z y1≤z | ⟪ ax , ch-is-sup ux ux≤z is-sup-x fcx ⟫ | ⟪ ay1 , ch-is-sup uy1 uy1≤z is-sup-y1 fcy1 ⟫ = ? | |
| 797 | 277 -- ... | tri< a ¬b ¬c = csupf (o<→≤ a) |
| 278 -- ... | tri≈ ¬a b ¬c = csupf (o≤-refl0 b) | |
| 799 | 279 -- ... | tri> ¬a ¬b c = subst (λ k → odef (UnionCF A f mf ay1 supf x) k ) ? (csupf ? ) |
| 280 -- csy1 : odef (UnionCF A f mf ay1 supf y1) (supf y1) | |
| 281 -- csy1 = csupf ? | |
| 797 | 282 |
| 784 | 283 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) |
| 785 | 284 order {b} {s} {z1} b<z sf<sb fc = zc04 where |
| 797 | 285 zc01 : {z1 : Ordinal } → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1 |
| 286 zc01 (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc03 where | |
| 788 | 287 s<b : s o< b |
| 288 s<b with trio< s b | |
| 785 | 289 ... | tri< a ¬b ¬c = a |
| 788 | 290 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (cong supf b) sf<sb ) |
| 785 | 291 ... | tri> ¬a ¬b c with osuc-≡< ( supf-mono c ) |
| 788 | 292 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sf<sb ) |
| 293 ... | case2 lt = ⊥-elim ( o<> lt sf<sb ) | |
| 294 s<z : s o< z | |
| 295 s<z = ordtrans s<b b<z | |
| 785 | 296 zc03 : odef (UnionCF A f mf ay supf b) (supf s) |
| 797 | 297 zc03 with csupf (o<→≤ s<z ) |
| 785 | 298 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ |
| 791 | 299 ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ as , ch-is-sup u (ordtrans u≤x (osucc s<b)) is-sup fc ⟫ |
| 797 | 300 zc01 (fsuc x fc) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc04 where |
| 785 | 301 zc04 : odef (UnionCF A f mf ay supf b) (f x) |
| 302 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (zc01 fc ) | |
| 303 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ | |
| 791 | 304 ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ |
| 797 | 305 zc00 : ( * z1 ≡ SUP.sup (sup (o<→≤ b<z) )) ∨ ( * z1 < SUP.sup ( sup (o<→≤ b<z) ) ) |
| 306 zc00 = SUP.x<sup (sup (o<→≤ b<z)) (zc01 fc ) | |
| 307 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) | |
| 308 zc04 with zc00 | |
| 309 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso (sym (supf-is-sup (o<→≤ b<z) ) ) (cong (&) eq) ) | |
| 310 ... | 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 | 311 |
| 653 | 312 |
| 728 | 313 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 314 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
| 728 | 315 field |
| 316 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay (ZChain.supf zc) z) a ) → b o< z → (ab : odef A b) | |
| 317 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) z) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) z) ab | |
| 318 → * a < * b → odef ((UnionCF A f mf ay (ZChain.supf zc) z)) b | |
| 319 | |
| 568 | 320 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 321 field | |
| 322 maximal : HOD | |
| 323 A∋maximal : A ∋ maximal | |
| 324 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 567 | 325 |
| 748 | 326 -- data UChain is total |
|
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no transfinite on data Chain trichotomos
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parents:
683
diff
changeset
|
327 |
| 694 | 328 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 748 | 329 {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 | 330 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where |
| 748 | 331 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) |
| 332 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb | |
| 791 | 333 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca |
| 766 | 334 ... | case1 eq with s≤fc (supf ub) f mf fcb |
| 335 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 336 ct00 : * a ≡ * b | |
| 337 ct00 = trans (cong (*) eq) eq1 | |
| 765 | 338 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where |
| 766 | 339 ct01 : * a < * b |
| 340 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
| 791 | 341 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 | 342 ct00 : * a < * (supf ub) |
| 765 | 343 ct00 = lt |
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34650e39e553
Chain is not strictly positive
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parents:
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diff
changeset
|
344 ct01 : * a < * b |
| 748 | 345 ct01 with s≤fc (supf ub) f mf fcb |
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34650e39e553
Chain is not strictly positive
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parents:
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diff
changeset
|
346 ... | case1 eq = subst (λ k → * a < k ) eq ct00 |
|
34650e39e553
Chain is not strictly positive
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parents:
688
diff
changeset
|
347 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 791 | 348 ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb |
| 766 | 349 ... | case1 eq with s≤fc (supf ua) f mf fca |
| 350 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 351 ct00 : * a ≡ * b | |
| 352 ct00 = sym (trans (cong (*) eq) eq1 ) | |
| 765 | 353 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where |
| 766 | 354 ct01 : * b < * a |
| 355 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
| 791 | 356 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 | 357 ct00 : * b < * (supf ua) |
| 765 | 358 ct00 = lt |
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Chain is not strictly positive
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parents:
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diff
changeset
|
359 ct01 : * b < * a |
| 749 | 360 ct01 with s≤fc (supf ua) f mf fca |
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689
34650e39e553
Chain is not strictly positive
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parents:
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diff
changeset
|
361 ... | case1 eq = subst (λ k → * b < k ) eq ct00 |
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34650e39e553
Chain is not strictly positive
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parents:
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diff
changeset
|
362 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 788 | 363 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< (supf ua) (supf ub) |
| 775 | 364 ... | tri< a₁ ¬b ¬c with ChainP.order supb a₁ fca |
| 766 | 365 ... | case1 eq with s≤fc (supf ub) f mf fcb |
| 366 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 367 ct00 : * a ≡ * b | |
| 368 ct00 = trans (cong (*) eq) eq1 | |
| 369 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 370 ct02 : * a < * b | |
| 371 ct02 = subst (λ k → * k < * b ) (sym eq) lt | |
| 788 | 372 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 | 373 ct03 : * a < * (supf ub) |
| 765 | 374 ct03 = lt |
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689
34650e39e553
Chain is not strictly positive
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parents:
688
diff
changeset
|
375 ct02 : * a < * b |
| 748 | 376 ct02 with s≤fc (supf ub) f mf fcb |
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689
34650e39e553
Chain is not strictly positive
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parents:
688
diff
changeset
|
377 ... | case1 eq = subst (λ k → * a < k ) eq ct03 |
|
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
378 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt |
| 788 | 379 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 |
| 769 | 380 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (sym eq) fcb ) |
| 788 | 381 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 c fcb |
| 766 | 382 ... | case1 eq with s≤fc (supf ua) f mf fca |
| 383 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 384 ct00 : * a ≡ * b | |
| 385 ct00 = sym (trans (cong (*) eq) eq1) | |
| 386 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
| 387 ct02 : * b < * a | |
| 388 ct02 = subst (λ k → * k < * a ) (sym eq) lt | |
| 788 | 389 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 | 390 ct05 : * b < * (supf ua) |
| 765 | 391 ct05 = lt |
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689
34650e39e553
Chain is not strictly positive
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parents:
688
diff
changeset
|
392 ct04 : * b < * a |
| 749 | 393 ct04 with s≤fc (supf ua) f mf fca |
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689
34650e39e553
Chain is not strictly positive
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parents:
688
diff
changeset
|
394 ... | case1 eq = subst (λ k → * b < k ) eq ct05 |
|
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
395 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt |
|
684
822fce8af579
no transfinite on data Chain trichotomos
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
683
diff
changeset
|
396 |
| 743 | 397 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 398 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 783 | 399 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ |
| 743 | 400 |
| 497 | 401 Zorn-lemma : { A : HOD } |
| 464 | 402 → o∅ o< & A |
| 568 | 403 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 404 → Maximal A |
| 552 | 405 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 406 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 407 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 408 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 409 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 760 | 410 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A |
| 411 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 530 | 412 s : HOD |
| 413 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 414 as : A ∋ * ( & s ) |
| 415 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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
416 as0 : odef A (& s ) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
changeset
|
417 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 418 s<A : & s o< & A |
| 568 | 419 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 420 HasMaximal : HOD |
| 537 | 421 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 422 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 423 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 424 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 425 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 426 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 427 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 428 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 429 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)) | |
| 430 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 | |
| 431 ¬x<m : ¬ (* x < * m) | |
| 432 ¬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 | 433 |
| 560 | 434 -- Uncountable ascending chain by axiom of choice |
| 530 | 435 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 436 cf nmx x with ODC.∋-p O A (* x) |
| 437 ... | no _ = o∅ | |
| 438 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 439 ... | yes nogt = -- no larger element, so it is maximal |
| 440 ⊥-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 | 441 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 442 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 443 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 444 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 445 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 446 ... | 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 ⟫ ) | |
| 447 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 448 |
| 449 --- | |
| 450 --- infintie ascention sequence of f | |
| 451 --- | |
| 530 | 452 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 453 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 454 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 455 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 456 |
| 793 | 457 chain-mono : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 458 {a b c : Ordinal} → a o≤ b | |
| 459 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c | |
| 460 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = | |
| 461 ⟪ ua , ch-init fc ⟫ | |
| 462 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = | |
| 463 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (osucc a≤b )) is-sup fc ⟫ | |
| 464 | |
| 703 | 465 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 653 | 466 (total : IsTotalOrderSet (ZChain.chain zc) ) → SUP A (ZChain.chain zc) |
| 703 | 467 sp0 f mf zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total |
| 543 | 468 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 469 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 470 | |
| 793 | 471 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 728 | 472 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
| 793 | 473 SZ1 f mf {y} ay zc x = TransFinite { λ x → ZChain1 A f mf ay zc x } zc1 x where |
| 474 chain-mono1 : {a b c : Ordinal} → a o≤ b | |
| 788 | 475 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
| 793 | 476 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) a≤b |
| 735 | 477 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 478 b o< x → (ab : odef A b) → | |
| 479 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f → | |
| 480 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b | |
| 749 | 481 is-max-hp x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
| 482 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ | |
| 791 | 483 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , |
| 749 | 484 subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) |
| 791 | 485 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ |
| 728 | 486 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.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
729
diff
changeset
|
487 zc1 x prev with Oprev-p x |
| 756 | 488 ... | 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
|
489 px = Oprev.oprev op |
| 789 | 490 zc-b<x : {b : Ordinal } → b o< x → b o< osuc px |
| 491 zc-b<x {b} lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 728 | 492 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 493 b o< x → (ab : odef A b) → | |
| 494 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → | |
| 495 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b | |
| 735 | 496 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
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parents:
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497 is-max {a} {b} ua b<x ab (case2 is-sup) a<b |
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parents:
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|
498 = ⟪ ab , ch-is-sup b (o<→≤ b<x) m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ where |
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201b66da4e69
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parents:
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diff
changeset
|
499 b<A : b o< & A |
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remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
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|
500 b<A = z09 ab |
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parents:
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|
501 m05 : b ≡ ZChain.supf zc b |
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201b66da4e69
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parents:
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|
502 m05 = sym ( ZChain.sup=u zc ab (z09 ab) |
| 793 | 503 record { x<sup = λ {z} uz → IsSup.x<sup is-sup (chain-mono1 (osucc b<x) uz ) } ) |
|
790
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parents:
789
diff
changeset
|
504 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
| 799 | 505 m08 {z} fcz = ZChain.fcy<sup zc (o<→≤ b<A) fcz |
|
790
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remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
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changeset
|
506 m09 : {sup1 z1 : Ordinal} → (ZChain.supf zc sup1) o< (ZChain.supf zc b) |
| 769 | 507 → FClosure A f (ZChain.supf zc sup1) z1 → z1 <= ZChain.supf zc b |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
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changeset
|
508 m09 {sup1} {z} s<b fcz = ZChain.order zc b<A s<b fcz |
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parents:
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|
509 m06 : ChainP A f mf ay (ZChain.supf zc) b |
| 799 | 510 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = ZChain.sup=u zc ab b<A {!!} } |
| 756 | 511 ... | no lim = record { is-max = is-max } where |
| 734 | 512 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 513 b o< x → (ab : odef A b) → | |
| 514 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → | |
| 515 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b | |
| 735 | 516 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 | 517 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 | 518 ... | 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 | 519 ... | case2 y<b = chain-mono1 (osucc b<x) |
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790
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parents:
789
diff
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|
520 ⟪ ab , ch-is-sup b (ordtrans o≤-refl <-osuc ) m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ where |
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201b66da4e69
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parents:
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|
521 m09 : b o< & A |
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parents:
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522 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) |
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parents:
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523 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
| 799 | 524 m07 {z} fc = ZChain.fcy<sup zc (o<→≤ m09) fc |
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parents:
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diff
changeset
|
525 m08 : {sup1 z1 : Ordinal} → (ZChain.supf zc sup1) o< (ZChain.supf zc b) |
| 769 | 526 → FClosure A f (ZChain.supf zc sup1) z1 → z1 <= ZChain.supf zc b |
|
790
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parents:
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|
527 m08 {sup1} {z1} s<b fc = ZChain.order zc m09 s<b fc |
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201b66da4e69
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parents:
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changeset
|
528 m05 : b ≡ ZChain.supf zc b |
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201b66da4e69
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parents:
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changeset
|
529 m05 = sym (ZChain.sup=u zc ab m09 |
| 793 | 530 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono1 (osucc b<x) lt )} ) -- ZChain on x |
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790
201b66da4e69
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parents:
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changeset
|
531 m06 : ChainP A f mf ay (ZChain.supf zc) b |
| 799 | 532 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = ZChain.sup=u zc ab m09 {!!} } |
| 727 | 533 |
| 543 | 534 --- |
| 560 | 535 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 536 --- |
| 703 | 537 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
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633
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parents:
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538 → (total : IsTotalOrderSet (ZChain.chain zc) ) |
| 703 | 539 → f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total)) |
| 540 fixpoint f mf zc total = z14 where | |
| 538 | 541 chain = ZChain.chain zc |
| 703 | 542 sp1 = sp0 f mf zc total |
| 712 | 543 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
| 570 | 544 → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 545 → * a < * b → odef chain b |
| 793 | 546 z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) ) |
| 543 | 547 z11 : & (SUP.sup sp1) o< & A |
| 548 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 549 z12 : odef chain (& (SUP.sup sp1)) |
| 550 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 653 | 551 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 712 | 552 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z11 (SUP.A∋maximal sp1) |
| 570 | 553 (case2 z19 ) z13 where |
| 538 | 554 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 653 | 555 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc ) |
| 538 | 556 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 557 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 570 | 558 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) |
| 571 | 559 z19 = record { x<sup = z20 } where |
| 560 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
| 561 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
| 570 | 562 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
| 563 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
| 564 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
| 703 | 565 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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566 z14 with total (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 631 | 567 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 568 z16 : ⊥ | |
| 569 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 570 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 571 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 572 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 573 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 574 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 575 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) | |
| 576 z17 : ⊥ | |
| 577 z17 with z15 | |
| 578 ... | case1 eq = ¬b eq | |
| 579 ... | case2 lt = ¬a lt | |
| 560 | 580 |
| 581 -- ZChain contradicts ¬ Maximal | |
| 582 -- | |
| 571 | 583 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
| 560 | 584 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
| 585 -- | |
| 697 | 586 z04 : (nmx : ¬ Maximal A ) |
| 703 | 587 → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) |
| 664 | 588 → IsTotalOrderSet (ZChain.chain zc) → ⊥ |
| 703 | 589 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 | 590 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) |
| 703 | 591 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc total ))) -- x ≡ f x ̄ |
|
633
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parents:
631
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|
592 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1 ))) where -- x < f x |
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593 sp1 : SUP A (ZChain.chain zc) |
| 703 | 594 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc total |
| 538 | 595 c = & (SUP.sup sp1) |
| 548 | 596 |
| 757 | 597 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 598 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = | |
| 599 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 ))) } | |
| 600 | |
| 601 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 602 → IsTotalOrderSet (uchain f mf ay) | |
| 603 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 604 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 605 uz01 = fcn-cmp y f mf ca cb | |
| 606 | |
| 607 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 608 → SUP A (uchain f mf ay) | |
| 609 ysup f mf {y} ay = supP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) | |
| 610 | |
| 793 | 611 initChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → ZChain A f mf ay o∅ |
| 796 | 612 initChain f mf {y} ay = record { supf = isupf ; chain⊆A = λ lt → proj1 lt ; chain∋init = cy ; sup = {!!} ; supf-is-sup = {!!} |
| 613 ; initial = isy ; f-next = inext ; f-total = itotal ; sup=u = λ _ b<0 → ⊥-elim (¬x<0 b<0) ; supf-mono = mono ; csupf = {!!} } where | |
| 764 | 614 spi = & (SUP.sup (ysup f mf ay)) |
| 711 | 615 isupf : Ordinal → Ordinal |
| 768 | 616 isupf z = spi |
| 763 | 617 sp = ysup f mf ay |
| 767 | 618 asi = SUP.A∋maximal sp |
| 711 | 619 cy : odef (UnionCF A f mf ay isupf o∅) y |
| 783 | 620 cy = ⟪ ay , ch-init (init ay refl) ⟫ |
| 759 | 621 y<sup : * y ≤ SUP.sup (ysup f mf ay) |
| 783 | 622 y<sup = SUP.x<sup (ysup f mf ay) (subst (λ k → FClosure A f y k ) (sym &iso) (init ay refl)) |
| 786 | 623 sup : {x : Ordinal} → x o< o∅ → SUP A (UnionCF A f mf ay isupf x) |
| 624 sup {x} lt = ⊥-elim ( ¬x<0 lt ) | |
| 711 | 625 isy : {z : Ordinal } → odef (UnionCF A f mf ay isupf o∅) z → * y ≤ * z |
| 748 | 626 isy {z} ⟪ az , uz ⟫ with uz |
| 627 ... | ch-init fc = s≤fc y f mf fc | |
| 791 | 628 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (subst (λ k → * y ≤ k) (sym *iso) y<sup) (s≤fc (& (SUP.sup (ysup f mf ay))) f mf fc ) |
| 711 | 629 inext : {a : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a → odef (UnionCF A f mf ay isupf o∅) (f a) |
| 748 | 630 inext {a} ua with (proj2 ua) |
| 631 ... | ch-init fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-init (fsuc _ fc ) ⟫ | |
| 791 | 632 ... | ch-is-sup u u≤x is-sup fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ |
| 711 | 633 itotal : IsTotalOrderSet (UnionCF A f mf ay isupf o∅) |
| 634 itotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 635 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 763 | 636 uz01 = chain-total A f mf ay isupf (proj2 ca) (proj2 cb) |
| 786 | 637 mono : {x : Ordinal} {z : Ordinal} → x o< z → isupf x o≤ isupf z |
| 638 mono {x} {z} x<z = o≤-refl | |
| 639 csupf : {z : Ordinal} → z o≤ o∅ → odef (UnionCF A f mf ay isupf z ) (isupf z) | |
| 789 | 640 csupf {z} z≤0 = ⟪ asi , ch-is-sup o∅ o∅≤z uz02 (init asi refl) ⟫ where |
| 768 | 641 uz03 : {z : Ordinal } → FClosure A f y z → (z ≡ isupf spi) ∨ (z << isupf spi) |
| 767 | 642 uz03 {z} fc with SUP.x<sup sp (subst (λ k → FClosure A f y k ) (sym &iso) fc ) |
| 643 ... | case1 eq = case1 ( begin | |
| 644 z ≡⟨ sym &iso ⟩ | |
| 645 & (* z) ≡⟨ cong (&) eq ⟩ | |
| 646 spi ∎ ) where open ≡-Reasoning | |
| 647 ... | case2 lt = case2 (subst (λ k → * z < k ) (sym *iso) lt ) | |
| 769 | 648 uz04 : {sup1 z1 : Ordinal} → isupf sup1 o< isupf spi → FClosure A f (isupf sup1) z1 → (z1 ≡ isupf spi) ∨ (z1 << isupf spi) |
| 649 uz04 {s} {z} s<spi fcz = ⊥-elim ( o<¬≡ refl s<spi ) | |
| 789 | 650 uz02 : ChainP A f mf ay isupf o∅ |
| 799 | 651 uz02 = record { fcy<sup = uz03 ; order = λ {s} {z} → uz04 {s} {z} ; supu=u = ? } |
| 767 | 652 |
| 793 | 653 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 654 SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; A∋maximal = SUP.A∋maximal sup ; x<sup = λ lt → SUP.x<sup sup (B⊆C lt) } | |
| 711 | 655 |
| 560 | 656 -- |
| 547 | 657 -- create all ZChains under o< x |
| 560 | 658 -- |
|
608
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parents:
607
diff
changeset
|
659 |
| 674 | 660 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 661 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 707 | 662 ind f mf {y} ay x prev with Oprev-p x |
| 697 | 663 ... | yes op = zc4 where |
| 682 | 664 -- |
| 665 -- we have previous ordinal to use induction | |
| 666 -- | |
| 667 px = Oprev.oprev op | |
| 703 | 668 zc : ZChain A f mf ay (Oprev.oprev op) |
| 682 | 669 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 670 px<x : px o< x | |
| 671 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 709 | 672 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 673 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 697 | 674 |
| 703 | 675 pchain : HOD |
| 676 pchain = UnionCF A f mf ay (ZChain.supf zc) 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 | |
| 679 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 748 | 680 uz01 = chain-total A f mf ay (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb)) |
| 704 | 681 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y |
| 682 pchain⊆A {y} ny = proj1 ny | |
| 683 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
| 749 | 684 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
| 797 | 685 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 | 686 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
| 748 | 687 pinit {a} ⟪ aa , ua ⟫ with ua |
| 688 ... | ch-init fc = s≤fc y f mf fc | |
| 791 | 689 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 765 | 690 zc7 : y <= (ZChain.supf zc) u |
| 783 | 691 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
| 704 | 692 pcy : odef pchain y |
| 783 | 693 pcy = ⟪ ay , ch-init (init ay refl) ⟫ |
| 703 | 694 |
| 754 | 695 supf0 = ZChain.supf zc |
| 696 | |
| 793 | 697 sup1 : SUP A (UnionCF A f mf ay supf0 x) |
| 698 sup1 = supP pchain pchain⊆A ptotal | |
| 699 sp1 = & (SUP.sup sup1 ) | |
| 700 supf1 : Ordinal → Ordinal | |
| 701 supf1 z with trio< z px | |
| 702 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
| 703 ... | tri≈ ¬a b ¬c = ZChain.supf zc z | |
| 704 ... | tri> ¬a ¬b c = sp1 | |
| 705 | |
| 611 | 706 -- if previous chain satisfies maximality, we caan reuse it |
| 707 -- | |
| 791 | 708 -- supf0 px is sup of UnionCF px , supf0 x is sup of UnionCF x |
| 793 | 709 no-extension : ¬ sp1 ≡ x → ZChain A f mf ay x |
| 796 | 710 no-extension ¬sp=x = record { supf = supf1 ; supf-mono = {!!} ; sup = sup |
| 711 ; initial = {!!} ; chain∋init = {!!} ; sup=u = {!!} ; supf-is-sup = {!!} ; csupf = {!!} | |
| 712 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} } where | |
| 791 | 713 UnionCF⊆ : UnionCF A f mf ay supf1 x ⊆' UnionCF A f mf ay supf0 x |
| 714 UnionCF⊆ ⟪ as , ch-init fc ⟫ = UnionCF⊆ ⟪ as , ch-init fc ⟫ | |
| 793 | 715 UnionCF⊆ ⟪ as , ch-is-sup u {z} u≤x record { fcy<sup = f1 ; order = o1 ; supu=u = su=u1 } fc ⟫ with trio< u px |
| 796 | 716 ... | tri< a ¬b ¬c = ⟪ as , ch-is-sup u {z} u≤x record { fcy<sup = f1 ; order = order0 ; supu=u = {!!} } fc ⟫ where |
| 791 | 717 order0 : {s z1 : Ordinal} → supf0 s o< supf0 u → FClosure A f (supf0 s) z1 |
| 718 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u) | |
| 792 | 719 order0 {s} {z1} ss<su fc with trio< s px | inspect supf1 s |
| 720 ... | tri< a ¬b ¬c | record {eq = eq1} = o1 {s} {z1} (subst (λ k → k o< supf0 u) (sym eq1) ss<su ) | |
| 721 (subst (λ k → FClosure A f k z1 ) (sym eq1) fc ) | |
| 722 ... | tri≈ ¬a b ¬c | record {eq = eq1} = o1 {s} {z1} (subst (λ k → k o< supf0 u) (sym eq1) ss<su ) | |
| 723 (subst (λ k → FClosure A f k z1 ) (sym eq1) fc ) | |
| 796 | 724 ... | tri> ¬a ¬b c | record {eq = eq1} = {!!} |
| 725 ... | tri≈ ¬a b ¬c = ⟪ as , ch-is-sup u {z} u≤x record { fcy<sup = f1 ; order = order0 ; supu=u = {!!}} fc ⟫ where | |
| 792 | 726 order0 : {s z1 : Ordinal} → supf0 s o< supf0 u → FClosure A f (supf0 s) z1 |
| 727 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u) | |
| 728 order0 {s} {z1} ss<su fc with trio< s px | inspect supf1 s | |
| 729 ... | tri< a ¬b ¬c | record {eq = eq1} = o1 {s} {z1} (subst (λ k → k o< supf0 u) (sym eq1) ss<su ) | |
| 730 (subst (λ k → FClosure A f k z1 ) (sym eq1) fc ) | |
| 731 ... | tri≈ ¬a b ¬c | record {eq = eq1} = o1 {s} {z1} (subst (λ k → k o< supf0 u) (sym eq1) ss<su ) | |
| 732 (subst (λ k → FClosure A f k z1 ) (sym eq1) fc ) | |
| 796 | 733 ... | tri> ¬a ¬b px<s | record {eq = eq1} = ⊥-elim ( ¬sp=x (subst (λ k → sp1 ≡ k ) u=x {!!} )) where |
| 793 | 734 s≤u : s o≤ u |
| 796 | 735 s≤u = {!!} |
| 793 | 736 u=x : u ≡ x |
| 737 u=x with trio< u x | |
| 796 | 738 ... | tri< a ¬b ¬c = {!!} |
| 793 | 739 ... | tri≈ ¬a b ¬c = b |
| 796 | 740 ... | tri> ¬a ¬b c = {!!} |
| 799 | 741 ... | tri> ¬a ¬b c = ⊥-elim ( ¬sp=x (subst (λ k → sp1 ≡ k ) u=x su=u1 )) where |
| 793 | 742 u=x : u ≡ x |
| 743 u=x with trio< u x | |
| 796 | 744 ... | tri< a ¬b ¬c = {!!} |
| 793 | 745 ... | tri≈ ¬a b ¬c = b |
| 796 | 746 ... | tri> ¬a ¬b c = {!!} |
| 791 | 747 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 748 sup {z} z≤x with trio< z px | |
| 796 | 749 ... | tri< a ¬b ¬c = SUP⊆ {!!} (ZChain.sup zc (o<→≤ a)) |
| 750 ... | tri≈ ¬a b ¬c = SUP⊆ {!!} (ZChain.sup zc (o≤-refl0 b)) | |
| 751 ... | tri> ¬a ¬b c = SUP⊆ {!!} sup1 | |
| 709 | 752 |
| 703 | 753 zc4 : ZChain A f mf ay x |
| 793 | 754 zc4 with ODC.∋-p O A (* x) |
| 796 | 755 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
| 793 | 756 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc ) ax f ) |
| 703 | 757 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 796 | 758 ... | case1 pr = no-extension {!!} -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 793 | 759 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc ) ax ) |
| 682 | 760 ... | case1 is-sup = -- x is a sup of zc |
| 786 | 761 record { supf = psupf1 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} ; csupf = {!!} ; sup=u = {!!} |
| 796 | 762 ; supf-mono = {!!} ; initial = {!!} ; chain∋init = {!!} ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!} } where |
| 793 | 763 supx : SUP A (UnionCF A f mf ay supf0 x) |
| 796 | 764 supx = record { sup = * x ; A∋maximal = subst (λ k → odef A k ) {!!} ax ; x<sup = {!!} } |
| 793 | 765 spx = & (SUP.sup supx ) |
| 766 x=spx : x ≡ spx | |
| 796 | 767 x=spx = {!!} |
| 750 | 768 psupf1 : Ordinal → Ordinal |
| 769 psupf1 z with trio< z x | |
| 770 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
| 771 ... | tri≈ ¬a b ¬c = x | |
| 772 ... | tri> ¬a ¬b c = x | |
| 793 | 773 |
| 796 | 774 ... | case2 ¬x=sup = no-extension {!!} -- px is not f y' nor sup of former ZChain from y -- no extention |
|
758
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diff
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|
775 |
| 728 | 776 ... | no lim = zc5 where |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
777 |
| 703 | 778 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
| 779 pzc z z<x = prev z z<x | |
| 794 | 780 ysp = & (SUP.sup (ysup f mf ay)) |
|
726
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psupf-mono and is-max conflict
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parents:
725
diff
changeset
|
781 |
| 703 | 782 psupf0 : (z : Ordinal) → Ordinal |
| 783 psupf0 z with trio< z x | |
| 755 | 784 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z |
| 794 | 785 ... | tri≈ ¬a b ¬c = ysp |
| 786 ... | tri> ¬a ¬b c = ysp | |
| 755 | 787 |
| 788 pchain0 : HOD | |
| 789 pchain0 = UnionCF A f mf ay psupf0 x | |
| 790 | |
| 791 ptotal0 : IsTotalOrderSet pchain0 | |
| 792 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 793 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 794 uz01 = chain-total A f mf ay psupf0 ( (proj2 ca)) ( (proj2 cb)) | |
| 795 | |
| 796 usup : SUP A pchain0 | |
| 797 usup = supP pchain0 (λ lt → proj1 lt) ptotal0 | |
| 798 spu = & (SUP.sup usup) | |
| 799 | |
| 794 | 800 supf1 : Ordinal → Ordinal |
| 801 supf1 z with trio< z x | |
| 755 | 802 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z |
| 794 | 803 ... | tri≈ ¬a b ¬c = spu |
| 804 ... | tri> ¬a ¬b c = spu | |
| 755 | 805 |
| 704 | 806 pchain : HOD |
| 794 | 807 pchain = UnionCF A f mf ay supf1 x |
| 704 | 808 |
| 809 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y | |
| 810 pchain⊆A {y} ny = proj1 ny | |
| 811 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
| 750 | 812 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-init (fsuc _ fc) ⟫ |
| 794 | 813 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 | 814 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
| 748 | 815 pinit {a} ⟪ aa , ua ⟫ with ua |
| 816 ... | ch-init fc = s≤fc y f mf fc | |
| 791 | 817 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 794 | 818 zc7 : y <= supf1 _ |
| 783 | 819 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
| 704 | 820 pcy : odef pchain y |
| 783 | 821 pcy = ⟪ ay , ch-init (init ay refl) ⟫ |
| 755 | 822 ptotal : IsTotalOrderSet pchain |
| 823 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 824 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 794 | 825 uz01 = chain-total A f mf ay supf1 ( (proj2 ca)) ( (proj2 cb)) |
| 754 | 826 |
|
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is-max on first transfinite induction is not good
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parents:
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diff
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|
827 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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parents:
757
diff
changeset
|
828 b o< x → (ab : odef A b) → |
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829 HasPrev A (UnionCF A f mf ay supf x) ab f → |
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|
830 * a < * b → odef (UnionCF A f mf ay supf x) b |
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parents:
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831 is-max-hp supf x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
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parents:
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|
832 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 791 | 833 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , |
|
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parents:
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diff
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|
834 subst (λ k → UChain A f mf ay supf x k ) |
| 794 | 835 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ |
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|
836 |
| 794 | 837 no-extension : ¬ spu ≡ x → ZChain A f mf ay x |
| 838 no-extension ¬sp=x = record { initial = pinit ; chain∋init = pcy ; supf = supf1 ; sup=u = {!!} | |
| 797 | 839 ; sup = sup ; supf-is-sup = sis |
| 840 ; csupf = csupf ; chain⊆A = pchain⊆A ; f-next = pnext ; f-total = ptotal ; supf-mono = {!!} } where | |
|
795
408e7e8a3797
csupf depends on order cyclicly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
794
diff
changeset
|
841 supfu : {u : Ordinal } → ( a : u o< x ) → (z : Ordinal) → Ordinal |
|
408e7e8a3797
csupf depends on order cyclicly
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parents:
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842 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z |
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408e7e8a3797
csupf depends on order cyclicly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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|
843 UnionCF⊆ : {u : Ordinal} → (a : u o< x ) → UnionCF A f mf ay supf1 x ⊆' UnionCF A f mf ay (supfu a) x |
| 796 | 844 UnionCF⊆ = {!!} |
| 797 | 845 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 846 sup {z} z≤x with trio< z x | |
| 847 ... | tri< a ¬b ¬c = SUP⊆ {!!} (ZChain.sup (pzc z a) o≤-refl ) | |
| 848 ... | tri≈ ¬a b ¬c = SUP⊆ {!!} usup | |
| 849 ... | tri> ¬a ¬b c = SUP⊆ {!!} usup | |
| 850 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup x≤z)) | |
| 851 sis {z} z≤x with trio< z x | |
| 852 ... | tri< a ¬b ¬c = ? where | |
| 853 zc8 = ZChain.supf-is-sup (pzc z a) o≤-refl | |
| 854 ... | tri≈ ¬a b ¬c = refl | |
| 855 ... | tri> ¬a ¬b c with osuc-≡< z≤x | |
| 856 ... | case1 eq = ⊥-elim ( ¬b eq ) | |
| 857 ... | case2 lt = ⊥-elim ( ¬a lt ) | |
| 858 sup=u : {b : Ordinal} (ab : odef A b) → b o< x → IsSup A (UnionCF A f mf ay supf1 (osuc b)) ab → supf1 b ≡ b | |
| 859 sup=u {b} ab b<x is-sup with trio< b x | |
| 860 ... | tri< a ¬b ¬c = ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab <-osuc record { x<sup = ? } | |
| 861 ... | tri≈ ¬a b ¬c = ? | |
| 862 ... | tri> ¬a ¬b c = ? | |
| 863 csupf : {z : Ordinal} → z o≤ x → odef (UnionCF A f mf ay supf1 z) (supf1 z) | |
| 864 csupf {z} z<x with trio< z x | |
| 865 ... | tri< a ¬b ¬c = zc9 where | |
| 866 zc9 : odef (UnionCF A f mf ay supf1 z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z) | |
| 867 zc9 = ? | |
| 868 zc8 : odef (UnionCF A f mf ay (supfu a) z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z) | |
| 869 zc8 = ZChain.csupf (pzc (osuc z) (ob<x lim a)) (o<→≤ <-osuc ) | |
| 870 ... | tri≈ ¬a b ¬c = ? -- ⊥-elim (¬a z<x) | |
| 871 ... | tri> ¬a ¬b c = ? -- ⊥-elim (¬a z<x) | |
| 872 supf-mono : {a b : Ordinal} → a o< b → supf1 a o≤ supf1 b | |
| 873 supf-mono {a0} {b0} a<b = zc10 where | |
| 874 -- x o≤ a → supf1 a ≡ supf1 b ≡ spu | |
| 875 -- x o≤ b → supf1 b ≡ spu | |
| 876 -- a o< x → b o≤ x → supf1 (supf1 a) ≡ supf1 a | |
| 877 -- supf1 (supf1 b) ≡ supf1 b | |
| 878 usa : odef (UnionCF A f mf ay (supfu ?) (osuc a0)) (supf1 a0) | |
| 879 usa = ? | |
| 880 usb : odef (UnionCF A f mf ay (supfu ?) (osuc b0)) (supf1 b0) | |
| 881 usb = ? | |
| 882 zc10 : supf1 a0 o≤ supf1 b0 | |
| 883 zc10 with trio< a0 x | trio< b0 x | |
| 884 ... | tri< a ¬b ¬c | tri< a' ¬b' ¬c' = ? where | |
| 885 zc11 = ZChain.supf-mono (pzc (osuc a0) (ob<x lim a)) a<b | |
| 886 zc12 = ZChain.supf-mono (pzc (osuc b0) (ob<x lim a')) a<b | |
| 887 ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c' = ? | |
| 888 ... | tri< a ¬b ¬c | tri> ¬a ¬b' c = ? | |
| 889 ... | tri≈ ¬a b ¬c | tri< a ¬b ¬c' = ? | |
| 890 ... | tri≈ ¬a b ¬c | tri≈ ¬a' b' ¬c' = ? | |
| 891 ... | tri≈ ¬a b ¬c | tri> ¬a' ¬b c = ? | |
| 892 ... | tri> ¬a ¬b c | _ = ? | |
| 893 | |
| 703 | 894 zc5 : ZChain A f mf ay x |
| 697 | 895 zc5 with ODC.∋-p O A (* x) |
| 796 | 896 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
| 704 | 897 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain ax f ) |
| 703 | 898 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 796 | 899 ... | case1 pr = no-extension {!!} |
| 704 | 900 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A pchain ax ) |
| 794 | 901 ... | case1 is-sup = record { initial = {!!} ; chain∋init = {!!} ; supf = supf1 ; sup=u = {!!} |
| 796 | 902 ; sup = {!!} ; supf-is-sup = {!!} |
| 903 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} ; csupf = {!!} ; supf-mono = {!!} } where -- x is a sup of (zc ?) | |
| 904 ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention | |
| 553 | 905 |
| 703 | 906 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay (& A) |
| 907 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
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parents:
607
diff
changeset
|
908 |
| 551 | 909 zorn00 : Maximal A |
| 910 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 911 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 912 -- yes we have the maximal | |
| 913 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 914 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 915 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 916 zorn01 = proj1 zorn03 | |
| 917 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 918 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 703 | 919 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04 total ) where |
| 551 | 920 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 921 nmx : ¬ Maximal A | |
| 922 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 923 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 924 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 | 925 zorn04 : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) |
| 653 | 926 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
| 634 | 927 total : IsTotalOrderSet (ZChain.chain zorn04) |
| 654 | 928 total {a} {b} = zorn06 where |
| 929 zorn06 : odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a) | |
| 930 zorn06 = ZChain.f-total (SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as) ) | |
| 551 | 931 |
| 516 | 932 -- usage (see filter.agda ) |
| 933 -- | |
| 497 | 934 -- _⊆'_ : ( A B : HOD ) → Set n |
| 935 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 936 |
| 497 | 937 -- MaximumSubset : {L P : HOD} |
| 938 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 939 -- → IsPartialOrderSet P _⊆'_ | |
| 940 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 941 -- → Maximal P (_⊆'_) | |
| 942 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |