Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 803:7c6612b753b9
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 09 Aug 2022 12:52:57 +0900 |
| parents | 358c33d3a2bd |
| children | 2d84411a636e |
| 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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changeset
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67 _≤_ : (x y : HOD) → Set (Level.suc n) |
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TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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68 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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8facdd7cc65a
TransitiveClosure with x <= f x is possible
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parents:
527
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69 |
| 554 | 70 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 71 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 72 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 73 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 74 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 75 | |
| 779 | 76 <-ftrans : {x y z : Ordinal } → x <= y → y <= z → x <= z |
| 77 <-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 78 <-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 79 <-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 80 <-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 81 | |
| 770 | 82 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y |
| 83 <=to≤ (case1 eq) = case1 (cong (*) eq) | |
| 84 <=to≤ (case2 lt) = case2 lt | |
| 85 | |
| 779 | 86 ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y |
| 87 ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) | |
| 88 ≤to<= (case2 lt) = case2 lt | |
| 89 | |
| 556 | 90 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 91 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 92 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 93 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 94 |
| 561 | 95 ptrans = IsStrictPartialOrder.trans PO |
| 96 | |
| 492 | 97 open _==_ |
| 98 open _⊆_ | |
| 99 | |
| 530 | 100 -- |
| 560 | 101 -- Closure of ≤-monotonic function f has total order |
| 530 | 102 -- |
| 103 | |
| 104 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 105 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 106 | |
| 551 | 107 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 783 | 108 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
| 555 | 109 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 110 |
| 556 | 111 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y |
| 783 | 112 A∋fc {A} s f mf (init as refl ) = as |
| 556 | 113 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 114 |
| 714 | 115 A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s |
| 783 | 116 A∋fcs {A} s f mf (init as refl) = as |
| 714 | 117 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
| 118 | |
| 556 | 119 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 783 | 120 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
| 556 | 121 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 122 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 123 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 124 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 125 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 126 |
| 800 | 127 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 128 fcn s mf (init as refl) = zero | |
| 129 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
| 130 ... | case1 eq = fcn s mf p | |
| 131 ... | case2 y<fy = suc (fcn s mf p ) | |
| 132 | |
| 133 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 134 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 135 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where | |
| 136 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 137 fc06 {x} {y} refl {j} not = fc08 not where | |
| 138 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 139 fc08 () | |
| 140 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
| 141 fc07 {x} (init as refl) eq = refl | |
| 142 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 143 ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) | |
| 144 -- ... | case2 x<fx = ? | |
| 145 fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y | |
| 146 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
| 147 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
| 148 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
| 149 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 150 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y ) | |
| 151 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 152 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) i=x i=y ) | |
| 153 fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 154 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) | |
| 155 fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 156 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) | |
| 157 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 158 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 159 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 160 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 161 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x | |
| 162 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
| 163 fc04 : * x1 ≡ * y | |
| 164 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 165 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 166 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 167 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 168 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 169 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 170 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 171 fc05 : * x ≡ * y1 | |
| 172 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 173 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) | |
| 174 | |
| 175 | |
| 176 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 177 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 178 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
| 179 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 180 fc06 {x} {y} refl {j} not = fc08 not where | |
| 181 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 182 fc08 () | |
| 183 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y | |
| 184 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
| 185 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 186 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 187 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 188 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 189 ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy | |
| 190 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 191 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 192 fc03 eq = cong pred eq | |
| 193 fc02 : * x < * y1 | |
| 194 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 195 | |
| 557 | 196 |
| 559 | 197 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 198 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 800 | 199 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 200 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 201 fc11 : * x < * y | |
| 202 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 203 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 204 fc10 : * x ≡ * y | |
| 205 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 206 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 207 fc12 : * y < * x | |
| 208 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 600 | 209 |
| 563 | 210 |
| 729 | 211 |
| 560 | 212 -- open import Relation.Binary.Properties.Poset as Poset |
| 213 | |
| 214 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 215 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 216 | |
| 567 | 217 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 218 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 219 |
| 568 | 220 _⊆'_ : ( A B : HOD ) → Set n |
| 221 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 222 |
| 223 -- | |
| 224 -- inductive maxmum tree from x | |
| 225 -- tree structure | |
| 226 -- | |
| 554 | 227 |
| 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 |
| 779 | 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 | |
| 803 | 248 -- minimum index is sup of y not ϕ |
| 690 | 249 -- |
| 250 | |
| 787 | 251 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 | 252 field |
| 765 | 253 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
| 800 | 254 order : {s z1 : Ordinal} → (lt : s o< u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
| 799 | 255 supu=u : supf u ≡ u |
| 694 | 256 |
| 748 | 257 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 258 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where | |
| 259 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 791 | 260 ch-is-sup : (u : Ordinal) {z : Ordinal } (u≤x : u o≤ x) ( is-sup : ChainP A f mf ay supf u ) |
| 748 | 261 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z |
| 694 | 262 |
| 263 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A | |
| 264 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 265 | |
| 803 | 266 -- Union of supf z which o< x |
| 267 -- | |
| 694 | 268 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 269 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
| 270 UnionCF A f mf ay supf x | |
| 271 = 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 | 272 |
| 703 | 273 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 274 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 275 field |
| 694 | 276 supf : Ordinal → Ordinal |
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6655f03984f9
mutual tranfinite in zorn
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277 chain : HOD |
| 703 | 278 chain = UnionCF A f mf ay supf z |
| 568 | 279 field |
| 280 chain⊆A : chain ⊆' A | |
| 783 | 281 chain∋init : odef chain y |
| 282 initial : {z : Ordinal } → odef chain z → * y ≤ * z | |
| 568 | 283 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 654 | 284 f-total : IsTotalOrderSet chain |
| 756 | 285 |
| 791 | 286 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
| 803 | 287 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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288 supf-is-sup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ & (SUP.sup (sup x≤z) ) |
| 797 | 289 csupf : {b : Ordinal } → b o≤ z → odef (UnionCF A f mf ay supf b) (supf b) |
| 798 | 290 |
| 803 | 291 -- ordering is proved here for totality and sup |
| 292 | |
| 799 | 293 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 |
| 294 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 | 295 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ |
| 799 | 296 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans (cong (&) eq) (sym (supf-is-sup u≤z ) ) )) |
| 297 ... | case2 lt = case2 (subst (λ k → * w < k ) (subst (λ k → k ≡ _ ) *iso (cong (*) (sym (supf-is-sup u≤z ))) ) lt ) | |
| 797 | 298 |
| 800 | 299 order : {b s z1 : Ordinal} → b o< z → s o< b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b) |
| 300 order {b} {s} {z1} b<z s<b fc = zc04 where | |
| 797 | 301 zc01 : {z1 : Ordinal } → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1 |
| 302 zc01 (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc03 where | |
| 788 | 303 s<z : s o< z |
| 304 s<z = ordtrans s<b b<z | |
| 785 | 305 zc03 : odef (UnionCF A f mf ay supf b) (supf s) |
| 797 | 306 zc03 with csupf (o<→≤ s<z ) |
| 785 | 307 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ |
| 791 | 308 ... | ⟪ 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 | 309 zc01 (fsuc x fc) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc04 where |
| 785 | 310 zc04 : odef (UnionCF A f mf ay supf b) (f x) |
| 311 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (zc01 fc ) | |
| 312 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ | |
| 791 | 313 ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ |
| 797 | 314 zc00 : ( * z1 ≡ SUP.sup (sup (o<→≤ b<z) )) ∨ ( * z1 < SUP.sup ( sup (o<→≤ b<z) ) ) |
| 315 zc00 = SUP.x<sup (sup (o<→≤ b<z)) (zc01 fc ) | |
| 316 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) | |
| 317 zc04 with zc00 | |
| 318 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso (sym (supf-is-sup (o<→≤ b<z) ) ) (cong (&) eq) ) | |
| 319 ... | 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 | 320 |
| 728 | 321 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 322 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
| 728 | 323 field |
| 324 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay (ZChain.supf zc) z) a ) → b o< z → (ab : odef A b) | |
| 325 → 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 | |
| 326 → * a < * b → odef ((UnionCF A f mf ay (ZChain.supf zc) z)) b | |
| 327 | |
| 568 | 328 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 329 field | |
| 330 maximal : HOD | |
| 331 A∋maximal : A ∋ maximal | |
| 332 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 567 | 333 |
| 748 | 334 -- data UChain is total |
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no transfinite on data Chain trichotomos
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parents:
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diff
changeset
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335 |
| 694 | 336 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 748 | 337 {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 | 338 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where |
| 748 | 339 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) |
| 340 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb | |
| 791 | 341 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca |
| 766 | 342 ... | case1 eq with s≤fc (supf ub) f mf fcb |
| 343 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 344 ct00 : * a ≡ * b | |
| 345 ct00 = trans (cong (*) eq) eq1 | |
| 765 | 346 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where |
| 766 | 347 ct01 : * a < * b |
| 348 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
| 791 | 349 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 | 350 ct00 : * a < * (supf ub) |
| 765 | 351 ct00 = lt |
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Chain is not strictly positive
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changeset
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352 ct01 : * a < * b |
| 748 | 353 ct01 with s≤fc (supf ub) f mf fcb |
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354 ... | case1 eq = subst (λ k → * a < k ) eq ct00 |
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parents:
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355 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 791 | 356 ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb |
| 766 | 357 ... | case1 eq with s≤fc (supf ua) f mf fca |
| 358 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 359 ct00 : * a ≡ * b | |
| 360 ct00 = sym (trans (cong (*) eq) eq1 ) | |
| 765 | 361 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where |
| 766 | 362 ct01 : * b < * a |
| 363 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
| 791 | 364 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 | 365 ct00 : * b < * (supf ua) |
| 765 | 366 ct00 = lt |
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Chain is not strictly positive
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367 ct01 : * b < * a |
| 749 | 368 ct01 with s≤fc (supf ua) f mf fca |
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369 ... | case1 eq = subst (λ k → * b < k ) eq ct00 |
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Chain is not strictly positive
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370 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 800 | 371 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub |
| 775 | 372 ... | tri< a₁ ¬b ¬c with ChainP.order supb a₁ fca |
| 766 | 373 ... | case1 eq with s≤fc (supf ub) f mf fcb |
| 374 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 375 ct00 : * a ≡ * b | |
| 376 ct00 = trans (cong (*) eq) eq1 | |
| 377 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 378 ct02 : * a < * b | |
| 379 ct02 = subst (λ k → * k < * b ) (sym eq) lt | |
| 788 | 380 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 | 381 ct03 : * a < * (supf ub) |
| 765 | 382 ct03 = lt |
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Chain is not strictly positive
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parents:
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diff
changeset
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383 ct02 : * a < * b |
| 748 | 384 ct02 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
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385 ... | case1 eq = subst (λ k → * a < k ) eq ct03 |
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34650e39e553
Chain is not strictly positive
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parents:
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diff
changeset
|
386 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt |
| 788 | 387 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri≈ ¬a eq ¬c |
| 800 | 388 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) |
| 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 with ChainP.order supa c fcb |
| 766 | 390 ... | case1 eq with s≤fc (supf ua) f mf fca |
| 391 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 392 ct00 : * a ≡ * b | |
| 393 ct00 = sym (trans (cong (*) eq) eq1) | |
| 394 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
| 395 ct02 : * b < * a | |
| 396 ct02 = subst (λ k → * k < * a ) (sym eq) lt | |
| 788 | 397 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 | 398 ct05 : * b < * (supf ua) |
| 765 | 399 ct05 = lt |
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Chain is not strictly positive
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parents:
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diff
changeset
|
400 ct04 : * b < * a |
| 749 | 401 ct04 with s≤fc (supf ua) f mf fca |
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Chain is not strictly positive
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parents:
688
diff
changeset
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402 ... | case1 eq = subst (λ k → * b < k ) eq ct05 |
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34650e39e553
Chain is not strictly positive
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parents:
688
diff
changeset
|
403 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt |
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no transfinite on data Chain trichotomos
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
683
diff
changeset
|
404 |
| 743 | 405 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 406 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 783 | 407 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ |
| 743 | 408 |
| 497 | 409 Zorn-lemma : { A : HOD } |
| 464 | 410 → o∅ o< & A |
| 568 | 411 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 412 → Maximal A |
| 552 | 413 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 414 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 415 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 416 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 417 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 760 | 418 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A |
| 419 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 530 | 420 s : HOD |
| 421 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 422 as : A ∋ * ( & s ) |
| 423 as = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) | |
|
608
6655f03984f9
mutual tranfinite in zorn
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parents:
607
diff
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424 as0 : odef A (& s ) |
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6655f03984f9
mutual tranfinite in zorn
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parents:
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diff
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425 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 426 s<A : & s o< & A |
| 568 | 427 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 428 HasMaximal : HOD |
| 537 | 429 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 430 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 431 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 432 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 433 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 434 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 435 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 436 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 437 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)) | |
| 438 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 | |
| 439 ¬x<m : ¬ (* x < * m) | |
| 440 ¬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 | 441 |
| 560 | 442 -- Uncountable ascending chain by axiom of choice |
| 530 | 443 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 444 cf nmx x with ODC.∋-p O A (* x) |
| 445 ... | no _ = o∅ | |
| 446 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 447 ... | yes nogt = -- no larger element, so it is maximal |
| 448 ⊥-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 | 449 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 450 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 451 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 452 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 453 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 454 ... | 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 ⟫ ) | |
| 455 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 456 |
| 457 --- | |
| 458 --- infintie ascention sequence of f | |
| 459 --- | |
| 530 | 460 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 461 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 462 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 463 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 464 |
| 793 | 465 chain-mono : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 466 {a b c : Ordinal} → a o≤ b | |
| 467 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c | |
| 468 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = | |
| 469 ⟪ ua , ch-init fc ⟫ | |
| 470 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = | |
| 471 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (osucc a≤b )) is-sup fc ⟫ | |
| 472 | |
| 703 | 473 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 653 | 474 (total : IsTotalOrderSet (ZChain.chain zc) ) → SUP A (ZChain.chain zc) |
| 703 | 475 sp0 f mf zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total |
| 543 | 476 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 477 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 478 | |
| 803 | 479 -- |
| 480 -- Second TransFinite Pass for maximality | |
| 481 -- | |
| 482 | |
| 793 | 483 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 728 | 484 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
| 793 | 485 SZ1 f mf {y} ay zc x = TransFinite { λ x → ZChain1 A f mf ay zc x } zc1 x where |
| 486 chain-mono1 : {a b c : Ordinal} → a o≤ b | |
| 788 | 487 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
| 793 | 488 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) a≤b |
| 735 | 489 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 490 b o< x → (ab : odef A b) → | |
| 491 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f → | |
| 492 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b | |
| 749 | 493 is-max-hp x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
| 494 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ | |
| 791 | 495 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , |
| 749 | 496 subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) |
| 791 | 497 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ |
| 728 | 498 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
|
499 zc1 x prev with Oprev-p x |
| 756 | 500 ... | 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
|
501 px = Oprev.oprev op |
| 789 | 502 zc-b<x : {b : Ordinal } → b o< x → b o< osuc px |
| 503 zc-b<x {b} lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 728 | 504 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 505 b o< x → (ab : odef A b) → | |
| 506 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 → | |
| 507 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b | |
| 735 | 508 is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
509 is-max {a} {b} ua b<x ab (case2 is-sup) a<b |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
510 = ⟪ ab , ch-is-sup b (o<→≤ b<x) m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ where |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
511 b<A : b o< & A |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
512 b<A = z09 ab |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
513 m05 : b ≡ ZChain.supf zc b |
| 803 | 514 m05 = sym ( ZChain.sup=u zc ab (o<→≤ (z09 ab)) |
| 793 | 515 record { x<sup = λ {z} uz → IsSup.x<sup is-sup (chain-mono1 (osucc b<x) uz ) } ) |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
516 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
| 799 | 517 m08 {z} fcz = ZChain.fcy<sup zc (o<→≤ b<A) fcz |
| 800 | 518 m09 : {sup1 z1 : Ordinal} → sup1 o< b |
| 769 | 519 → 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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
520 m09 {sup1} {z} s<b fcz = ZChain.order zc b<A s<b fcz |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
521 m06 : ChainP A f mf ay (ZChain.supf zc) b |
| 803 | 522 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = ZChain.sup=u zc ab (o<→≤ b<A ) |
| 801 | 523 record { x<sup = λ {z} uz → IsSup.x<sup is-sup (chain-mono1 (osucc b<x) uz ) } } |
| 756 | 524 ... | no lim = record { is-max = is-max } where |
| 734 | 525 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 526 b o< x → (ab : odef A b) → | |
| 527 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 → | |
| 528 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b | |
| 735 | 529 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 | 530 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 | 531 ... | 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 | 532 ... | case2 y<b = chain-mono1 (osucc b<x) |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
533 ⟪ ab , ch-is-sup b (ordtrans o≤-refl <-osuc ) m06 (subst (λ k → FClosure A f k b) m05 (init ab refl)) ⟫ where |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
534 m09 : b o< & A |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
789
diff
changeset
|
535 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
536 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
| 799 | 537 m07 {z} fc = ZChain.fcy<sup zc (o<→≤ m09) fc |
| 800 | 538 m08 : {sup1 z1 : Ordinal} → sup1 o< b |
| 769 | 539 → 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
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
540 m08 {sup1} {z1} s<b fc = ZChain.order zc m09 s<b fc |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
541 m05 : b ≡ ZChain.supf zc b |
| 803 | 542 m05 = sym (ZChain.sup=u zc ab (o<→≤ m09) |
| 793 | 543 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono1 (osucc b<x) lt )} ) -- ZChain on x |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
544 m06 : ChainP A f mf ay (ZChain.supf zc) b |
| 803 | 545 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = ZChain.sup=u zc ab (o<→≤ m09) |
| 801 | 546 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono1 (osucc b<x) lt )} } |
| 727 | 547 |
| 543 | 548 --- |
| 560 | 549 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 550 --- |
| 703 | 551 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
|
633
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
552 → (total : IsTotalOrderSet (ZChain.chain zc) ) |
| 703 | 553 → f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total)) |
| 554 fixpoint f mf zc total = z14 where | |
| 538 | 555 chain = ZChain.chain zc |
| 703 | 556 sp1 = sp0 f mf zc total |
| 712 | 557 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
| 570 | 558 → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 559 → * a < * b → odef chain b |
| 793 | 560 z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) ) |
| 543 | 561 z11 : & (SUP.sup sp1) o< & A |
| 562 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 563 z12 : odef chain (& (SUP.sup sp1)) |
| 564 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 653 | 565 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 712 | 566 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z11 (SUP.A∋maximal sp1) |
| 570 | 567 (case2 z19 ) z13 where |
| 538 | 568 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 653 | 569 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc ) |
| 538 | 570 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 571 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 570 | 572 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) |
| 571 | 573 z19 = record { x<sup = z20 } where |
| 574 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
| 575 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
| 570 | 576 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
| 577 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
| 578 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
| 703 | 579 z14 : f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total )) |
|
633
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
580 z14 with total (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 631 | 581 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 582 z16 : ⊥ | |
| 583 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 584 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 585 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 586 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 587 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 588 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 589 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) | |
| 590 z17 : ⊥ | |
| 591 z17 with z15 | |
| 592 ... | case1 eq = ¬b eq | |
| 593 ... | case2 lt = ¬a lt | |
| 560 | 594 |
| 595 -- ZChain contradicts ¬ Maximal | |
| 596 -- | |
| 571 | 597 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
| 560 | 598 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
| 599 -- | |
| 697 | 600 z04 : (nmx : ¬ Maximal A ) |
| 703 | 601 → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) |
| 664 | 602 → IsTotalOrderSet (ZChain.chain zc) → ⊥ |
| 703 | 603 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 | 604 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) |
| 703 | 605 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc total ))) -- x ≡ f x ̄ |
|
633
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
606 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1 ))) where -- x < f x |
|
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
607 sp1 : SUP A (ZChain.chain zc) |
| 703 | 608 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc total |
| 538 | 609 c = & (SUP.sup sp1) |
| 548 | 610 |
| 757 | 611 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 612 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = | |
| 613 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 ))) } | |
| 614 | |
| 615 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 616 → IsTotalOrderSet (uchain f mf ay) | |
| 617 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 618 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 619 uz01 = fcn-cmp y f mf ca cb | |
| 620 | |
| 621 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 622 → SUP A (uchain f mf ay) | |
| 623 ysup f mf {y} ay = supP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) | |
| 624 | |
| 793 | 625 initChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → ZChain A f mf ay o∅ |
| 796 | 626 initChain f mf {y} ay = record { supf = isupf ; chain⊆A = λ lt → proj1 lt ; chain∋init = cy ; sup = {!!} ; supf-is-sup = {!!} |
| 803 | 627 ; initial = isy ; f-next = inext ; f-total = itotal ; sup=u = λ _ b<0 → ? ; csupf = {!!} } where |
| 764 | 628 spi = & (SUP.sup (ysup f mf ay)) |
| 711 | 629 isupf : Ordinal → Ordinal |
| 768 | 630 isupf z = spi |
| 763 | 631 sp = ysup f mf ay |
| 767 | 632 asi = SUP.A∋maximal sp |
| 711 | 633 cy : odef (UnionCF A f mf ay isupf o∅) y |
| 783 | 634 cy = ⟪ ay , ch-init (init ay refl) ⟫ |
| 759 | 635 y<sup : * y ≤ SUP.sup (ysup f mf ay) |
| 783 | 636 y<sup = SUP.x<sup (ysup f mf ay) (subst (λ k → FClosure A f y k ) (sym &iso) (init ay refl)) |
| 786 | 637 sup : {x : Ordinal} → x o< o∅ → SUP A (UnionCF A f mf ay isupf x) |
| 638 sup {x} lt = ⊥-elim ( ¬x<0 lt ) | |
| 711 | 639 isy : {z : Ordinal } → odef (UnionCF A f mf ay isupf o∅) z → * y ≤ * z |
| 748 | 640 isy {z} ⟪ az , uz ⟫ with uz |
| 641 ... | ch-init fc = s≤fc y f mf fc | |
| 791 | 642 ... | 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 | 643 inext : {a : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a → odef (UnionCF A f mf ay isupf o∅) (f a) |
| 748 | 644 inext {a} ua with (proj2 ua) |
| 645 ... | ch-init fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-init (fsuc _ fc ) ⟫ | |
| 791 | 646 ... | ch-is-sup u u≤x is-sup fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ |
| 711 | 647 itotal : IsTotalOrderSet (UnionCF A f mf ay isupf o∅) |
| 648 itotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 649 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 763 | 650 uz01 = chain-total A f mf ay isupf (proj2 ca) (proj2 cb) |
| 786 | 651 csupf : {z : Ordinal} → z o≤ o∅ → odef (UnionCF A f mf ay isupf z ) (isupf z) |
| 789 | 652 csupf {z} z≤0 = ⟪ asi , ch-is-sup o∅ o∅≤z uz02 (init asi refl) ⟫ where |
| 768 | 653 uz03 : {z : Ordinal } → FClosure A f y z → (z ≡ isupf spi) ∨ (z << isupf spi) |
| 767 | 654 uz03 {z} fc with SUP.x<sup sp (subst (λ k → FClosure A f y k ) (sym &iso) fc ) |
| 655 ... | case1 eq = case1 ( begin | |
| 656 z ≡⟨ sym &iso ⟩ | |
| 657 & (* z) ≡⟨ cong (&) eq ⟩ | |
| 658 spi ∎ ) where open ≡-Reasoning | |
| 659 ... | case2 lt = case2 (subst (λ k → * z < k ) (sym *iso) lt ) | |
| 769 | 660 uz04 : {sup1 z1 : Ordinal} → isupf sup1 o< isupf spi → FClosure A f (isupf sup1) z1 → (z1 ≡ isupf spi) ∨ (z1 << isupf spi) |
| 661 uz04 {s} {z} s<spi fcz = ⊥-elim ( o<¬≡ refl s<spi ) | |
| 789 | 662 uz02 : ChainP A f mf ay isupf o∅ |
| 800 | 663 uz02 = record { fcy<sup = uz03 ; order = λ {s} {z} → {!!} ; supu=u = {!!} } |
| 767 | 664 |
| 793 | 665 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 666 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 | 667 |
| 560 | 668 -- |
| 547 | 669 -- create all ZChains under o< x |
| 560 | 670 -- |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
671 |
| 674 | 672 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 673 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 707 | 674 ind f mf {y} ay x prev with Oprev-p x |
| 697 | 675 ... | yes op = zc4 where |
| 682 | 676 -- |
| 677 -- we have previous ordinal to use induction | |
| 678 -- | |
| 679 px = Oprev.oprev op | |
| 703 | 680 zc : ZChain A f mf ay (Oprev.oprev op) |
| 682 | 681 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 682 px<x : px o< x | |
| 683 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 709 | 684 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 685 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 697 | 686 |
| 703 | 687 pchain : HOD |
| 688 pchain = UnionCF A f mf ay (ZChain.supf zc) x | |
| 689 ptotal : IsTotalOrderSet pchain | |
| 690 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 691 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 748 | 692 uz01 = chain-total A f mf ay (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb)) |
| 704 | 693 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y |
| 694 pchain⊆A {y} ny = proj1 ny | |
| 695 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
| 749 | 696 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
| 797 | 697 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 | 698 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
| 748 | 699 pinit {a} ⟪ aa , ua ⟫ with ua |
| 700 ... | ch-init fc = s≤fc y f mf fc | |
| 791 | 701 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 765 | 702 zc7 : y <= (ZChain.supf zc) u |
| 783 | 703 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
| 704 | 704 pcy : odef pchain y |
| 783 | 705 pcy = ⟪ ay , ch-init (init ay refl) ⟫ |
| 703 | 706 |
| 754 | 707 supf0 = ZChain.supf zc |
| 708 | |
| 793 | 709 sup1 : SUP A (UnionCF A f mf ay supf0 x) |
| 710 sup1 = supP pchain pchain⊆A ptotal | |
| 711 sp1 = & (SUP.sup sup1 ) | |
| 712 supf1 : Ordinal → Ordinal | |
| 713 supf1 z with trio< z px | |
| 714 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
| 715 ... | tri≈ ¬a b ¬c = ZChain.supf zc z | |
| 716 ... | tri> ¬a ¬b c = sp1 | |
| 717 | |
| 803 | 718 pchain1 : HOD |
| 719 pchain1 = UnionCF A f mf ay supf1 x | |
| 720 pcy1 : odef pchain1 y | |
| 721 pcy1 = ⟪ ay , ch-init (init ay refl) ⟫ | |
| 722 pinit1 : {y₁ : Ordinal} → odef pchain1 y₁ → * y ≤ * y₁ | |
| 723 pinit1 {a} ⟪ aa , ua ⟫ with ua | |
| 724 ... | ch-init fc = s≤fc y f mf fc | |
| 725 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where | |
| 726 zc7 : y <= supf1 u | |
| 727 zc7 = ChainP.fcy<sup is-sup (init ay refl) | |
| 728 pnext1 : {a : Ordinal} → odef pchain1 a → odef pchain1 (f a) | |
| 729 pnext1 {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ | |
| 730 pnext1 {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 ) ⟫ | |
| 731 | |
| 611 | 732 -- if previous chain satisfies maximality, we caan reuse it |
| 733 -- | |
| 791 | 734 -- supf0 px is sup of UnionCF px , supf0 x is sup of UnionCF x |
| 793 | 735 no-extension : ¬ sp1 ≡ x → ZChain A f mf ay x |
| 800 | 736 no-extension ¬sp=x = record { supf = supf1 ; sup = sup |
| 803 | 737 ; initial = pinit1 ; chain∋init = pcy1 ; sup=u = sup=u ; supf-is-sup = {!!} ; csupf = {!!} |
| 738 ; chain⊆A = λ lt → proj1 lt ; f-next = pnext1 ; f-total = {!!} } where | |
| 802 | 739 UnionCF⊆ : {z : Ordinal } → z o≤ x → UnionCF A f mf ay supf1 z ⊆' UnionCF A f mf ay supf0 x |
| 740 UnionCF⊆ {z} z≤x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ | |
| 741 UnionCF⊆ {z} z≤x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (init au1 refl) ⟫ = ⟪ au , ch-is-sup u1 ? ? (init ? ?) ⟫ | |
| 742 UnionCF⊆ {z} z≤x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (fsuc xp fcu1) ⟫ with | |
| 743 UnionCF⊆ {z} z≤x ⟪ A∋fc _ f mf fcu1 , ch-is-sup u1 u1≤x u1-is-sup fcu1 ⟫ | |
| 744 ... | ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf _ aa ) , ch-init (fsuc _ fc) ⟫ | |
| 745 ... | ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 ( mf _ aa ) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ | |
| 791 | 746 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 747 sup {z} z≤x with trio< z px | |
| 802 | 748 ... | tri< a ¬b ¬c = SUP⊆ (UnionCF⊆ ? ) (ZChain.sup zc ? ) |
| 749 ... | tri≈ ¬a b ¬c = SUP⊆ (UnionCF⊆ ? ) (ZChain.sup zc ? ) | |
| 750 ... | tri> ¬a ¬b c = SUP⊆ (λ lt → chain-mono f mf ay _ ? (UnionCF⊆ ? lt )) sup1 | |
| 803 | 751 sup=u : {b : Ordinal} (ab : odef A b) → |
| 752 b o≤ x → IsSup A (UnionCF A f mf ay supf1 (osuc b)) ab → supf1 b ≡ b | |
| 753 sup=u {b} ab b<x is-sup with trio< b px | |
| 754 ... | tri< a ¬b ¬c = ? where | |
| 755 zc11 = ZChain.sup=u zc ab ? ? | |
| 756 ... | tri≈ ¬a b ¬c = ? | |
| 757 ... | tri> ¬a ¬b c = ? | |
| 709 | 758 |
| 703 | 759 zc4 : ZChain A f mf ay x |
| 793 | 760 zc4 with ODC.∋-p O A (* x) |
| 796 | 761 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
| 793 | 762 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc ) ax f ) |
| 703 | 763 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 796 | 764 ... | case1 pr = no-extension {!!} -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 793 | 765 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc ) ax ) |
| 682 | 766 ... | case1 is-sup = -- x is a sup of zc |
| 786 | 767 record { supf = psupf1 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} ; csupf = {!!} ; sup=u = {!!} |
| 800 | 768 ; initial = {!!} ; chain∋init = {!!} ; sup = {!!} ; supf-is-sup = {!!} } where |
| 793 | 769 supx : SUP A (UnionCF A f mf ay supf0 x) |
| 796 | 770 supx = record { sup = * x ; A∋maximal = subst (λ k → odef A k ) {!!} ax ; x<sup = {!!} } |
| 793 | 771 spx = & (SUP.sup supx ) |
| 772 x=spx : x ≡ spx | |
| 796 | 773 x=spx = {!!} |
| 750 | 774 psupf1 : Ordinal → Ordinal |
| 775 psupf1 z with trio< z x | |
| 776 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
| 777 ... | tri≈ ¬a b ¬c = x | |
| 778 ... | tri> ¬a ¬b c = x | |
| 793 | 779 |
| 796 | 780 ... | case2 ¬x=sup = no-extension {!!} -- px is not f y' nor sup of former ZChain from y -- no extention |
|
758
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parents:
757
diff
changeset
|
781 |
| 728 | 782 ... | no lim = zc5 where |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
783 |
| 703 | 784 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
| 785 pzc z z<x = prev z z<x | |
| 794 | 786 ysp = & (SUP.sup (ysup f mf ay)) |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
787 |
| 703 | 788 psupf0 : (z : Ordinal) → Ordinal |
| 789 psupf0 z with trio< z x | |
| 755 | 790 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z |
| 794 | 791 ... | tri≈ ¬a b ¬c = ysp |
| 792 ... | tri> ¬a ¬b c = ysp | |
| 755 | 793 |
| 794 pchain0 : HOD | |
| 795 pchain0 = UnionCF A f mf ay psupf0 x | |
| 796 | |
| 797 ptotal0 : IsTotalOrderSet pchain0 | |
| 798 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 799 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 800 uz01 = chain-total A f mf ay psupf0 ( (proj2 ca)) ( (proj2 cb)) | |
| 801 | |
| 802 usup : SUP A pchain0 | |
| 803 usup = supP pchain0 (λ lt → proj1 lt) ptotal0 | |
| 804 spu = & (SUP.sup usup) | |
| 805 | |
| 794 | 806 supf1 : Ordinal → Ordinal |
| 807 supf1 z with trio< z x | |
| 755 | 808 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z |
| 794 | 809 ... | tri≈ ¬a b ¬c = spu |
| 810 ... | tri> ¬a ¬b c = spu | |
| 755 | 811 |
| 704 | 812 pchain : HOD |
| 794 | 813 pchain = UnionCF A f mf ay supf1 x |
| 704 | 814 |
| 815 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y | |
| 816 pchain⊆A {y} ny = proj1 ny | |
| 817 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
| 750 | 818 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-init (fsuc _ fc) ⟫ |
| 794 | 819 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 | 820 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
| 748 | 821 pinit {a} ⟪ aa , ua ⟫ with ua |
| 822 ... | ch-init fc = s≤fc y f mf fc | |
| 791 | 823 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 794 | 824 zc7 : y <= supf1 _ |
| 783 | 825 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
| 704 | 826 pcy : odef pchain y |
| 783 | 827 pcy = ⟪ ay , ch-init (init ay refl) ⟫ |
| 755 | 828 ptotal : IsTotalOrderSet pchain |
| 829 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 830 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 794 | 831 uz01 = chain-total A f mf ay supf1 ( (proj2 ca)) ( (proj2 cb)) |
| 754 | 832 |
|
758
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is-max on first transfinite induction is not good
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parents:
757
diff
changeset
|
833 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
|
834 b o< x → (ab : odef A b) → |
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parents:
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|
835 HasPrev A (UnionCF A f mf ay supf x) ab f → |
|
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parents:
757
diff
changeset
|
836 * a < * b → odef (UnionCF A f mf ay supf x) b |
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is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
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|
837 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:
757
diff
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|
838 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 791 | 839 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , |
|
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parents:
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840 subst (λ k → UChain A f mf ay supf x k ) |
| 794 | 841 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ |
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parents:
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|
842 |
| 794 | 843 no-extension : ¬ spu ≡ x → ZChain A f mf ay x |
| 802 | 844 no-extension ¬sp=x = record { initial = pinit ; chain∋init = pcy ; supf = supf1 ; sup=u = sup=u |
| 797 | 845 ; sup = sup ; supf-is-sup = sis |
| 800 | 846 ; csupf = csupf ; chain⊆A = pchain⊆A ; f-next = pnext ; f-total = ptotal } where |
|
795
408e7e8a3797
csupf depends on order cyclicly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
794
diff
changeset
|
847 supfu : {u : Ordinal } → ( a : u o< x ) → (z : Ordinal) → Ordinal |
|
408e7e8a3797
csupf depends on order cyclicly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
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|
848 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z |
| 802 | 849 UnionCF⊆ : {u : Ordinal} → (a : u o< x ) → UnionCF A f mf ay supf1 u ⊆' UnionCF A f mf ay (supfu a) x |
| 850 UnionCF⊆ {u} u<x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ | |
| 851 UnionCF⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (init au1 refl) ⟫ = ⟪ au , ch-is-sup u1 ? ? (init ? ?) ⟫ | |
| 852 UnionCF⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (fsuc xp fcu1) ⟫ with | |
| 853 UnionCF⊆ {u} u<x ⟪ A∋fc _ f mf fcu1 , ch-is-sup u1 u1≤x u1-is-sup fcu1 ⟫ | |
| 854 ... | ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf _ aa ) , ch-init (fsuc _ fc) ⟫ | |
| 855 ... | ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 ( mf _ aa ) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ | |
| 856 UnionCF0⊆ : {z : Ordinal} → (a : z o≤ x ) → UnionCF A f mf ay supf1 z ⊆' UnionCF A f mf ay psupf0 x | |
| 857 UnionCF0⊆ {u} u<x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ | |
| 858 UnionCF0⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (init au1 refl) ⟫ = ⟪ au , ch-is-sup u1 ? ? (init ? ?) ⟫ | |
| 859 UnionCF0⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (fsuc xp fcu1) ⟫ with | |
| 860 UnionCF0⊆ {u} u<x ⟪ A∋fc _ f mf fcu1 , ch-is-sup u1 u1≤x u1-is-sup fcu1 ⟫ | |
| 861 ... | ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf _ aa ) , ch-init (fsuc _ fc) ⟫ | |
| 862 ... | ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 ( mf _ aa ) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ | |
| 797 | 863 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 864 sup {z} z≤x with trio< z x | |
| 802 | 865 ... | tri< a ¬b ¬c = SUP⊆ (UnionCF⊆ a) (ZChain.sup (pzc (osuc z) ?) ? ) |
| 866 ... | tri≈ ¬a b ¬c = SUP⊆ (UnionCF0⊆ ?) usup | |
| 867 ... | tri> ¬a ¬b c = SUP⊆ (UnionCF0⊆ ?) usup | |
| 797 | 868 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup x≤z)) |
| 869 sis {z} z≤x with trio< z x | |
| 800 | 870 ... | tri< a ¬b ¬c = {!!} where |
| 797 | 871 zc8 = ZChain.supf-is-sup (pzc z a) o≤-refl |
| 872 ... | tri≈ ¬a b ¬c = refl | |
| 873 ... | tri> ¬a ¬b c with osuc-≡< z≤x | |
| 874 ... | case1 eq = ⊥-elim ( ¬b eq ) | |
| 875 ... | case2 lt = ⊥-elim ( ¬a lt ) | |
| 803 | 876 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 |
| 797 | 877 sup=u {b} ab b<x is-sup with trio< b x |
| 803 | 878 ... | tri< a ¬b ¬c = ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab ? record { x<sup = {!!} } |
| 800 | 879 ... | tri≈ ¬a b ¬c = {!!} |
| 880 ... | tri> ¬a ¬b c = {!!} | |
| 797 | 881 csupf : {z : Ordinal} → z o≤ x → odef (UnionCF A f mf ay supf1 z) (supf1 z) |
| 882 csupf {z} z<x with trio< z x | |
| 883 ... | tri< a ¬b ¬c = zc9 where | |
| 884 zc9 : odef (UnionCF A f mf ay supf1 z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z) | |
| 800 | 885 zc9 = {!!} |
| 797 | 886 zc8 : odef (UnionCF A f mf ay (supfu a) z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z) |
| 887 zc8 = ZChain.csupf (pzc (osuc z) (ob<x lim a)) (o<→≤ <-osuc ) | |
| 800 | 888 ... | tri≈ ¬a b ¬c = {!!} -- ⊥-elim (¬a z<x) |
| 889 ... | tri> ¬a ¬b c = {!!} -- ⊥-elim (¬a z<x) | |
| 797 | 890 |
| 703 | 891 zc5 : ZChain A f mf ay x |
| 697 | 892 zc5 with ODC.∋-p O A (* x) |
| 796 | 893 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
| 704 | 894 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain ax f ) |
| 703 | 895 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 796 | 896 ... | case1 pr = no-extension {!!} |
| 704 | 897 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A pchain ax ) |
| 794 | 898 ... | case1 is-sup = record { initial = {!!} ; chain∋init = {!!} ; supf = supf1 ; sup=u = {!!} |
| 796 | 899 ; sup = {!!} ; supf-is-sup = {!!} |
| 800 | 900 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} ; csupf = {!!} } where -- x is a sup of (zc ?) |
| 796 | 901 ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention |
| 553 | 902 |
| 703 | 903 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay (& A) |
| 904 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
|
905 |
| 551 | 906 zorn00 : Maximal A |
| 907 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 908 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 909 -- yes we have the maximal | |
| 910 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 911 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 912 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 913 zorn01 = proj1 zorn03 | |
| 914 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 915 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 703 | 916 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04 total ) where |
| 551 | 917 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 918 nmx : ¬ Maximal A | |
| 919 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 920 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 921 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 | 922 zorn04 : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) |
| 653 | 923 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
| 634 | 924 total : IsTotalOrderSet (ZChain.chain zorn04) |
| 654 | 925 total {a} {b} = zorn06 where |
| 926 zorn06 : odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a) | |
| 927 zorn06 = ZChain.f-total (SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as) ) | |
| 551 | 928 |
| 516 | 929 -- usage (see filter.agda ) |
| 930 -- | |
| 497 | 931 -- _⊆'_ : ( A B : HOD ) → Set n |
| 932 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 933 |
| 497 | 934 -- MaximumSubset : {L P : HOD} |
| 935 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 936 -- → IsPartialOrderSet P _⊆'_ | |
| 937 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 938 -- → Maximal P (_⊆'_) | |
| 939 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |