Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 843:ef0433f41e55
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 30 Aug 2022 14:30:36 +0900 |
| parents | 962a9f3dbd3c |
| children | 0855fce6ee92 |
| 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 | |
|
528
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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67 _≤_ : (x y : HOD) → Set (Level.suc n) |
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8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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68 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
69 |
| 554 | 70 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 71 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 72 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 73 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 74 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 75 | |
| 779 | 76 <-ftrans : {x y z : Ordinal } → x <= y → y <= z → x <= z |
| 77 <-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 78 <-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 79 <-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 80 <-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 81 | |
| 770 | 82 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y |
| 83 <=to≤ (case1 eq) = case1 (cong (*) eq) | |
| 84 <=to≤ (case2 lt) = case2 lt | |
| 85 | |
| 779 | 86 ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y |
| 87 ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) | |
| 88 ≤to<= (case2 lt) = case2 lt | |
| 89 | |
| 556 | 90 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 91 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 92 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 93 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 94 |
| 561 | 95 ptrans = IsStrictPartialOrder.trans PO |
| 96 | |
| 492 | 97 open _==_ |
| 98 open _⊆_ | |
| 99 | |
| 530 | 100 -- |
| 560 | 101 -- Closure of ≤-monotonic function f has total order |
| 530 | 102 -- |
| 103 | |
| 104 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 105 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 106 | |
| 551 | 107 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 783 | 108 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
| 555 | 109 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 110 |
| 556 | 111 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y |
| 783 | 112 A∋fc {A} s f mf (init as refl ) = as |
| 556 | 113 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 114 |
| 714 | 115 A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s |
| 783 | 116 A∋fcs {A} s f mf (init as refl) = as |
| 714 | 117 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
| 118 | |
| 556 | 119 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 783 | 120 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
| 556 | 121 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 122 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 123 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 124 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 125 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 126 |
| 800 | 127 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 128 fcn s mf (init as refl) = zero | |
| 129 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
| 130 ... | case1 eq = fcn s mf p | |
| 131 ... | case2 y<fy = suc (fcn s mf p ) | |
| 132 | |
| 133 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 134 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 135 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where | |
| 136 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 137 fc06 {x} {y} refl {j} not = fc08 not where | |
| 138 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 139 fc08 () | |
| 140 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
| 141 fc07 {x} (init as refl) eq = refl | |
| 142 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 143 ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) | |
| 144 -- ... | case2 x<fx = ? | |
| 145 fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y | |
| 146 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
| 147 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
| 148 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
| 149 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 150 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y ) | |
| 151 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 152 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) i=x i=y ) | |
| 153 fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 154 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) | |
| 155 fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 156 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) | |
| 157 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 158 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 159 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 160 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 161 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x | |
| 162 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
| 163 fc04 : * x1 ≡ * y | |
| 164 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 165 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 166 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 167 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 168 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 169 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 170 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 171 fc05 : * x ≡ * y1 | |
| 172 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 173 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) | |
| 174 | |
| 175 | |
| 176 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 177 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 178 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
| 179 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 180 fc06 {x} {y} refl {j} not = fc08 not where | |
| 181 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 182 fc08 () | |
| 183 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y | |
| 184 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
| 185 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 186 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 187 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 188 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 189 ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy | |
| 190 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 191 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 192 fc03 eq = cong pred eq | |
| 193 fc02 : * x < * y1 | |
| 194 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 195 | |
| 557 | 196 |
| 559 | 197 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 198 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 800 | 199 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 200 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 201 fc11 : * x < * y | |
| 202 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 203 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 204 fc10 : * x ≡ * y | |
| 205 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 206 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 207 fc12 : * y < * x | |
| 208 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 600 | 209 |
| 563 | 210 |
| 729 | 211 |
| 560 | 212 -- open import Relation.Binary.Properties.Poset as Poset |
| 213 | |
| 214 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 215 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 216 | |
| 567 | 217 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 218 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 219 |
| 568 | 220 _⊆'_ : ( A B : HOD ) → Set n |
| 221 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 222 |
| 223 -- | |
| 224 -- inductive maxmum tree from x | |
| 225 -- tree structure | |
| 226 -- | |
| 554 | 227 |
| 836 | 228 record HasPrev (A B : HOD) (x : Ordinal ) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 229 field |
| 836 | 230 ax : odef A x |
| 534 | 231 y : Ordinal |
| 541 | 232 ay : odef B y |
| 534 | 233 x=fy : x ≡ f y |
| 529 | 234 |
| 570 | 235 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 654 | 236 field |
| 779 | 237 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 568 | 238 |
| 656 | 239 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 240 field | |
| 241 sup : HOD | |
| 804 | 242 as : A ∋ sup |
| 656 | 243 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive |
| 244 | |
| 690 | 245 -- |
| 246 -- sup and its fclosure is in a chain HOD | |
| 247 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 248 -- whole chain is a union of separated Chain | |
| 803 | 249 -- minimum index is sup of y not ϕ |
| 690 | 250 -- |
| 251 | |
| 787 | 252 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where |
| 690 | 253 field |
| 765 | 254 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
| 828 | 255 order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
| 256 supu=u : supf u ≡ u | |
| 694 | 257 |
| 748 | 258 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 259 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where | |
| 260 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 791 | 261 ch-is-sup : (u : Ordinal) {z : Ordinal } (u≤x : u o≤ x) ( is-sup : ChainP A f mf ay supf u ) |
| 748 | 262 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z |
| 694 | 263 |
| 264 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A | |
| 265 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 266 | |
| 803 | 267 -- Union of supf z which o< x |
| 268 -- | |
| 694 | 269 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 270 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
| 271 UnionCF A f mf ay supf x | |
| 272 = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x z } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } | |
| 662 | 273 |
| 842 | 274 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) |
| 275 → supf x o< supf y → x o< y | |
| 276 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y | |
| 277 ... | tri< a ¬b ¬c = a | |
| 278 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 279 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 280 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 281 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 282 | |
| 703 | 283 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 284 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 285 field |
| 694 | 286 supf : Ordinal → Ordinal |
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287 chain : HOD |
| 703 | 288 chain = UnionCF A f mf ay supf z |
| 568 | 289 field |
| 290 chain⊆A : chain ⊆' A | |
| 783 | 291 chain∋init : odef chain y |
| 292 initial : {z : Ordinal } → odef chain z → * y ≤ * z | |
| 568 | 293 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 654 | 294 f-total : IsTotalOrderSet chain |
| 756 | 295 |
| 832 | 296 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
| 817 | 297 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z → IsSup A (UnionCF A f mf ay supf b) ab → supf b ≡ b |
| 832 | 298 supf-is-sup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ & (SUP.sup (sup x≤z) ) |
| 825 | 299 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y |
| 828 | 300 csupf : {b : Ordinal } → b o≤ z → odef (UnionCF A f mf ay supf (supf b)) (supf b) |
| 825 | 301 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
| 302 supf-inject {x} {y} sx<sy with trio< x y | |
| 303 ... | tri< a ¬b ¬c = a | |
| 304 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 305 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 306 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 307 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 798 | 308 |
| 803 | 309 -- ordering is proved here for totality and sup |
| 310 | |
| 813 | 311 fcy<sup : {u w : Ordinal } → u o< z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf |
| 832 | 312 fcy<sup {u} {w} u<z fc with SUP.x<sup (sup (o<→≤ u<z)) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) |
| 798 | 313 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ |
| 832 | 314 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans (cong (&) eq) (sym (supf-is-sup (o<→≤ u<z) ) ) )) |
| 315 ... | case2 lt = case2 (subst (λ k → * w < k ) (subst (λ k → k ≡ _ ) *iso (cong (*) (sym (supf-is-sup (o<→≤ u<z) ))) ) lt ) | |
| 825 | 316 |
| 832 | 317 csupf-fc : {b s z1 : Ordinal} → b o≤ z → supf s o< supf b → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1 |
| 318 csupf-fc {b} {s} {z1} b≤z ss<sb (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc05 where | |
| 828 | 319 s<b : s o< b |
| 320 s<b = supf-inject ss<sb | |
| 830 | 321 s≤<z : s o≤ z |
| 831 | 322 s≤<z = ordtrans s<b b≤z |
| 825 | 323 zc04 : odef (UnionCF A f mf ay supf (supf s)) (supf s) |
| 830 | 324 zc04 = csupf s≤<z |
| 825 | 325 zc05 : odef (UnionCF A f mf ay supf b) (supf s) |
| 326 zc05 with zc04 | |
| 327 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ | |
| 828 | 328 ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ as , ch-is-sup u (zc09 u≤x ) is-sup fc ⟫ where |
| 825 | 329 zc06 : supf u ≡ u |
| 828 | 330 zc06 = ChainP.supu=u is-sup |
| 331 zc09 : u o≤ supf s → u o≤ b | |
| 826 | 332 zc09 u<s with osuc-≡< (subst (λ k → k o≤ supf s) (sym zc06) u<s) |
| 333 ... | case1 su=ss = zc08 where | |
| 825 | 334 zc07 : supf u o≤ supf b |
| 335 zc07 = subst (λ k → k o≤ supf b) (sym su=ss) (supf-mono (o<→≤ s<b) ) | |
| 828 | 336 zc08 : u o≤ b |
| 825 | 337 zc08 with osuc-≡< zc07 |
| 829 | 338 ... | case1 su=sb = ⊥-elim ( o<¬≡ (trans (sym su=ss) su=sb ) ss<sb ) |
| 828 | 339 ... | case2 lt = o<→≤ (supf-inject lt ) |
| 340 ... | case2 lt = o<→≤ (ordtrans (supf-inject lt) s<b) | |
| 831 | 341 csupf-fc {b} {s} {z1} b<z ss≤sb (fsuc x fc) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc04 where |
| 785 | 342 zc04 : odef (UnionCF A f mf ay supf b) (f x) |
| 831 | 343 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb fc ) |
| 785 | 344 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ |
| 791 | 345 ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ |
| 832 | 346 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) |
| 347 order {b} {s} {z1} b<z ss<sb fc = zc04 where | |
| 348 zc00 : ( * z1 ≡ SUP.sup (sup (o<→≤ b<z) )) ∨ ( * z1 < SUP.sup ( sup (o<→≤ b<z) ) ) | |
| 349 zc00 = SUP.x<sup (sup (o<→≤ b<z) ) (csupf-fc (o<→≤ b<z) ss<sb fc ) | |
| 797 | 350 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) |
| 351 zc04 with zc00 | |
| 832 | 352 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso (sym (supf-is-sup (o<→≤ b<z)) ) (cong (&) eq) ) |
| 353 ... | 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 | 354 |
| 728 | 355 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 356 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
| 728 | 357 field |
| 358 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay (ZChain.supf zc) z) a ) → b o< z → (ab : odef A b) | |
| 836 | 359 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) z) b f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) z) ab |
| 728 | 360 → * a < * b → odef ((UnionCF A f mf ay (ZChain.supf zc) z)) b |
| 361 | |
| 837 | 362 initial-segment : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 363 {a b y : Ordinal} (ay : odef A y) (za : ZChain A f mf ay a ) (zb : ZChain A f mf ay b ) | |
| 364 → {z : Ordinal } → a o≤ b → z o≤ a | |
| 365 → ZChain.supf za z ≡ ZChain.supf zb z | |
| 366 initial-segment A f mf {a} {b} {y} ay za zb {z} a≤b z≤a = TransFinite0 { λ x → x o≤ a → ZChain.supf za x ≡ ZChain.supf zb x } ind z z≤a where | |
| 367 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → z o≤ a → ZChain.supf za z ≡ ZChain.supf zb z ) → | |
| 368 x o≤ a → ZChain.supf za x ≡ ZChain.supf zb x | |
| 369 ind x prev x≤a = ? where | |
| 370 supfa = ZChain.supf za | |
| 371 supfb = ZChain.supf zb | |
| 372 zc10 : {w : Ordinal } → w o< z → UnionCF A f mf ay supfa w ≡ UnionCF A f mf ay supfb w | |
| 373 zc10 = ? | |
| 374 -- w o< z → supfa w ≡ supfb w | |
| 375 supa : SUP A (UnionCF A f mf ay supfa x) | |
| 376 supa = ZChain.sup za x≤a | |
| 377 supb : SUP A (UnionCF A f mf ay supfb x) | |
| 378 supb = ZChain.sup zb (OrdTrans x≤a a≤b) | |
| 379 zc13 : UnionCF A f mf ay supfa x ≡ UnionCF A f mf ay supfb x | |
| 380 zc13 = ? -- | |
| 381 -- if x is sup of UCF px (or Union o< x ) , then supfa x ≡ x supfb x | |
| 382 -- if x is not sup of UCF px (or Union o< x ) or HasPrev, UCF x ≡ UCF px (or Union o< x) | |
| 383 zc15 : {B : HOD} → (a b : SUP A B) → SUP.sup a ≡ SUP.sup b | |
| 384 zc15 = ? | |
| 385 zc14 : supfa x ≡ supfb x | |
| 386 zc14 = begin | |
| 387 supfa x ≡⟨ ? ⟩ | |
| 388 & (SUP.sup supa) ≡⟨ ? ⟩ | |
| 389 & (SUP.sup supb) ≡⟨ ? ⟩ | |
| 390 supfb x ∎ where open ≡-Reasoning | |
| 391 | |
| 568 | 392 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 393 field | |
| 394 maximal : HOD | |
| 804 | 395 as : A ∋ maximal |
| 568 | 396 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative |
| 567 | 397 |
| 748 | 398 -- data UChain is total |
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parents:
683
diff
changeset
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399 |
| 694 | 400 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 748 | 401 {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 | 402 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where |
| 748 | 403 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) |
| 404 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb | |
| 791 | 405 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca |
| 766 | 406 ... | case1 eq with s≤fc (supf ub) f mf fcb |
| 407 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 408 ct00 : * a ≡ * b | |
| 409 ct00 = trans (cong (*) eq) eq1 | |
| 765 | 410 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where |
| 766 | 411 ct01 : * a < * b |
| 412 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
| 791 | 413 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 | 414 ct00 : * a < * (supf ub) |
| 765 | 415 ct00 = 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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416 ct01 : * a < * b |
| 748 | 417 ct01 with s≤fc (supf ub) f mf fcb |
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parents:
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418 ... | case1 eq = subst (λ k → * a < k ) eq ct00 |
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Chain is not strictly positive
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parents:
688
diff
changeset
|
419 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 791 | 420 ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb |
| 766 | 421 ... | case1 eq with s≤fc (supf ua) f mf fca |
| 422 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 423 ct00 : * a ≡ * b | |
| 424 ct00 = sym (trans (cong (*) eq) eq1 ) | |
| 765 | 425 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where |
| 766 | 426 ct01 : * b < * a |
| 427 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
| 791 | 428 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 | 429 ct00 : * b < * (supf ua) |
| 765 | 430 ct00 = lt |
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Chain is not strictly positive
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parents:
688
diff
changeset
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431 ct01 : * b < * a |
| 749 | 432 ct01 with s≤fc (supf ua) f mf fca |
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parents:
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diff
changeset
|
433 ... | 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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434 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 800 | 435 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub |
| 829 | 436 ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁) fca |
| 766 | 437 ... | case1 eq with s≤fc (supf ub) f mf fcb |
| 438 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 439 ct00 : * a ≡ * b | |
| 440 ct00 = trans (cong (*) eq) eq1 | |
| 441 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 442 ct02 : * a < * b | |
| 443 ct02 = subst (λ k → * k < * b ) (sym eq) lt | |
| 788 | 444 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 | 445 ct03 : * a < * (supf ub) |
| 765 | 446 ct03 = lt |
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689
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
447 ct02 : * a < * b |
| 748 | 448 ct02 with s≤fc (supf ub) f mf fcb |
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689
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
449 ... | 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
|
450 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt |
| 788 | 451 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 | 452 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) |
| 829 | 453 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) fcb |
| 766 | 454 ... | case1 eq with s≤fc (supf ua) f mf fca |
| 455 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 456 ct00 : * a ≡ * b | |
| 457 ct00 = sym (trans (cong (*) eq) eq1) | |
| 458 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
| 459 ct02 : * b < * a | |
| 460 ct02 = subst (λ k → * k < * a ) (sym eq) lt | |
| 788 | 461 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 | 462 ct05 : * b < * (supf ua) |
| 765 | 463 ct05 = lt |
|
689
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
464 ct04 : * b < * a |
| 749 | 465 ct04 with s≤fc (supf ua) f mf fca |
|
689
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
466 ... | 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
|
467 ... | 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
|
468 |
| 743 | 469 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 470 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 783 | 471 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ |
| 743 | 472 |
| 497 | 473 Zorn-lemma : { A : HOD } |
| 464 | 474 → o∅ o< & A |
| 568 | 475 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 476 → Maximal A |
| 552 | 477 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 478 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 479 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 480 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 481 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 760 | 482 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A |
| 483 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 530 | 484 s : HOD |
| 485 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 486 as : A ∋ * ( & s ) |
| 487 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
|
488 as0 : odef A (& s ) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
489 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 490 s<A : & s o< & A |
| 568 | 491 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 492 HasMaximal : HOD |
| 537 | 493 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 494 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 495 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 496 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 497 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 498 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 804 | 499 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
| 537 | 500 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
| 501 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)) | |
| 502 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 | |
| 503 ¬x<m : ¬ (* x < * m) | |
| 504 ¬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 | 505 |
| 560 | 506 -- Uncountable ascending chain by axiom of choice |
| 530 | 507 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 508 cf nmx x with ODC.∋-p O A (* x) |
| 509 ... | no _ = o∅ | |
| 510 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 511 ... | yes nogt = -- no larger element, so it is maximal |
| 512 ⊥-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 | 513 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 514 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 515 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 516 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 517 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 518 ... | 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 ⟫ ) | |
| 519 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 520 |
| 521 --- | |
| 522 --- infintie ascention sequence of f | |
| 523 --- | |
| 530 | 524 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 525 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 526 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 527 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 528 |
| 793 | 529 chain-mono : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 530 {a b c : Ordinal} → a o≤ b | |
| 531 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c | |
| 532 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = | |
| 533 ⟪ ua , ch-init fc ⟫ | |
| 534 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = | |
| 535 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (osucc a≤b )) is-sup fc ⟫ | |
| 536 | |
| 703 | 537 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 653 | 538 (total : IsTotalOrderSet (ZChain.chain zc) ) → SUP A (ZChain.chain zc) |
| 703 | 539 sp0 f mf zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total |
| 543 | 540 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 541 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 542 | |
| 803 | 543 -- |
| 544 -- Second TransFinite Pass for maximality | |
| 545 -- | |
| 546 | |
| 793 | 547 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 728 | 548 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
| 793 | 549 SZ1 f mf {y} ay zc x = TransFinite { λ x → ZChain1 A f mf ay zc x } zc1 x where |
| 550 chain-mono1 : {a b c : Ordinal} → a o≤ b | |
| 788 | 551 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
| 793 | 552 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) a≤b |
| 735 | 553 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 554 b o< x → (ab : odef A b) → | |
| 836 | 555 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) b f → |
| 735 | 556 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
| 749 | 557 is-max-hp x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
| 558 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ | |
| 791 | 559 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , |
| 749 | 560 subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) |
| 791 | 561 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ |
| 728 | 562 zc1 : (x : Ordinal) → ((y₁ : Ordinal) → y₁ o< x → ZChain1 A f mf ay zc y₁) → ZChain1 A f mf ay zc x |
|
732
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bchain can be reached from upwords by f. so it is worng.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
729
diff
changeset
|
563 zc1 x prev with Oprev-p x |
| 756 | 564 ... | yes op = record { is-max = is-max } where |
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parents:
729
diff
changeset
|
565 px = Oprev.oprev op |
| 789 | 566 zc-b<x : {b : Ordinal } → b o< x → b o< osuc px |
| 567 zc-b<x {b} lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 728 | 568 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 569 b o< x → (ab : odef A b) → | |
| 836 | 570 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) b f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → |
| 728 | 571 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
| 735 | 572 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:
789
diff
changeset
|
573 is-max {a} {b} ua b<x ab (case2 is-sup) a<b |
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201b66da4e69
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parents:
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|
574 = ⟪ 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
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|
575 b<A : b o< & A |
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201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
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diff
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|
576 b<A = z09 ab |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
789
diff
changeset
|
577 m05 : b ≡ ZChain.supf zc b |
| 814 | 578 m05 = sym ( ZChain.sup=u zc ab (o<→≤ (z09 ab) ) |
| 817 | 579 record { x<sup = λ {z} uz → IsSup.x<sup is-sup (chain-mono1 (o<→≤ b<x) uz ) } ) |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
789
diff
changeset
|
580 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
| 813 | 581 m08 {z} fcz = ZChain.fcy<sup zc b<A fcz |
| 828 | 582 m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 583 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
| 832 | 584 m09 {s} {z} s<b fcz = ZChain.order zc b<A s<b fcz |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
585 m06 : ChainP A f mf ay (ZChain.supf zc) b |
| 828 | 586 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = ZChain.sup=u zc ab (o<→≤ b<A ) |
| 587 record { x<sup = λ {z} uz → IsSup.x<sup is-sup (chain-mono1 (o<→≤ b<x) uz ) } } | |
| 756 | 588 ... | no lim = record { is-max = is-max } where |
| 734 | 589 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 590 b o< x → (ab : odef A b) → | |
| 836 | 591 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) b f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → |
| 734 | 592 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
| 735 | 593 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 | 594 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 | 595 ... | 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 | 596 ... | case2 y<b = chain-mono1 (osucc b<x) |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
789
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changeset
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597 ⟪ 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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598 m09 : b o< & A |
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201b66da4e69
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parents:
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599 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) |
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201b66da4e69
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parents:
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|
600 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
| 813 | 601 m07 {z} fc = ZChain.fcy<sup zc m09 fc |
| 828 | 602 m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 603 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
| 832 | 604 m08 {s} {z1} s<b fc = ZChain.order zc m09 s<b fc |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
605 m05 : b ≡ ZChain.supf zc b |
| 814 | 606 m05 = sym (ZChain.sup=u zc ab (o<→≤ m09) |
| 817 | 607 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono1 (o<→≤ b<x) lt )} ) -- ZChain on x |
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790
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remove unnesesary part in SZ1 the second TransFinite induction for is-max
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parents:
789
diff
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608 m06 : ChainP A f mf ay (ZChain.supf zc) b |
| 828 | 609 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = ZChain.sup=u zc ab (o<→≤ m09) |
| 610 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono1 (o<→≤ b<x) lt )} } | |
| 727 | 611 |
| 543 | 612 --- |
| 560 | 613 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 614 --- |
| 703 | 615 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
|
633
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parents:
631
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|
616 → (total : IsTotalOrderSet (ZChain.chain zc) ) |
| 703 | 617 → f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total)) |
| 618 fixpoint f mf zc total = z14 where | |
| 538 | 619 chain = ZChain.chain zc |
| 703 | 620 sp1 = sp0 f mf zc total |
| 712 | 621 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
| 836 | 622 → HasPrev A chain b f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 623 → * a < * b → odef chain b |
| 793 | 624 z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) ) |
| 543 | 625 z11 : & (SUP.sup sp1) o< & A |
| 804 | 626 z11 = c<→o< ( SUP.as sp1) |
| 538 | 627 z12 : odef chain (& (SUP.sup sp1)) |
| 628 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 653 | 629 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 804 | 630 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z11 (SUP.as sp1) |
| 570 | 631 (case2 z19 ) z13 where |
| 538 | 632 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 653 | 633 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc ) |
| 538 | 634 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 635 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 804 | 636 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.as sp1) |
| 571 | 637 z19 = record { x<sup = z20 } where |
| 638 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
| 639 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
| 570 | 640 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
| 641 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
| 642 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
| 703 | 643 z14 : f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total )) |
|
633
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parents:
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644 z14 with total (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 631 | 645 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 646 z16 : ⊥ | |
| 804 | 647 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.as sp1 )) |
| 631 | 648 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) |
| 649 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 650 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 651 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 652 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 653 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) | |
| 654 z17 : ⊥ | |
| 655 z17 with z15 | |
| 656 ... | case1 eq = ¬b eq | |
| 657 ... | case2 lt = ¬a lt | |
| 560 | 658 |
| 659 -- ZChain contradicts ¬ Maximal | |
| 660 -- | |
| 571 | 661 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
| 560 | 662 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
| 663 -- | |
| 697 | 664 z04 : (nmx : ¬ Maximal A ) |
| 703 | 665 → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) |
| 664 | 666 → IsTotalOrderSet (ZChain.chain zc) → ⊥ |
| 804 | 667 z04 nmx zc total = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.as sp1 )))) |
| 668 (subst (λ k → odef A (& k)) (sym *iso) (SUP.as sp1) ) | |
| 703 | 669 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc total ))) -- x ≡ f x ̄ |
| 804 | 670 (proj1 (cf-is-<-monotonic nmx c (SUP.as sp1 ))) where -- x < f x |
|
633
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parents:
631
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671 sp1 : SUP A (ZChain.chain zc) |
| 703 | 672 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc total |
| 538 | 673 c = & (SUP.sup sp1) |
| 548 | 674 |
| 757 | 675 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 676 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = | |
| 677 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 ))) } | |
| 678 | |
| 679 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 680 → IsTotalOrderSet (uchain f mf ay) | |
| 681 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 682 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 683 uz01 = fcn-cmp y f mf ca cb | |
| 684 | |
| 685 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 686 → SUP A (uchain f mf ay) | |
| 687 ysup f mf {y} ay = supP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) | |
| 688 | |
| 793 | 689 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 804 | 690 SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x<sup = λ lt → SUP.x<sup sup (B⊆C lt) } |
| 711 | 691 |
| 833 | 692 record xSUP (B : HOD) (x : Ordinal) : Set n where |
| 693 field | |
| 694 ax : odef A x | |
| 695 is-sup : IsSup A B ax | |
| 696 | |
| 560 | 697 -- |
| 547 | 698 -- create all ZChains under o< x |
| 560 | 699 -- |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
700 |
| 674 | 701 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 702 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 707 | 703 ind f mf {y} ay x prev with Oprev-p x |
| 697 | 704 ... | yes op = zc4 where |
| 682 | 705 -- |
| 706 -- we have previous ordinal to use induction | |
| 707 -- | |
| 708 px = Oprev.oprev op | |
| 703 | 709 zc : ZChain A f mf ay (Oprev.oprev op) |
| 682 | 710 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 711 px<x : px o< x | |
| 712 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 709 | 713 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 714 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 697 | 715 |
| 703 | 716 pchain : HOD |
| 830 | 717 pchain = UnionCF A f mf ay (ZChain.supf zc) px |
| 703 | 718 ptotal : IsTotalOrderSet pchain |
| 719 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 720 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 748 | 721 uz01 = chain-total A f mf ay (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb)) |
| 704 | 722 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y |
| 723 pchain⊆A {y} ny = proj1 ny | |
| 724 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
| 749 | 725 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
| 797 | 726 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 | 727 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
| 748 | 728 pinit {a} ⟪ aa , ua ⟫ with ua |
| 729 ... | ch-init fc = s≤fc y f mf fc | |
| 791 | 730 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 765 | 731 zc7 : y <= (ZChain.supf zc) u |
| 783 | 732 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
| 704 | 733 pcy : odef pchain y |
| 783 | 734 pcy = ⟪ ay , ch-init (init ay refl) ⟫ |
| 703 | 735 |
| 754 | 736 supf0 = ZChain.supf zc |
| 737 | |
| 835 | 738 sup1 : SUP A (UnionCF A f mf ay supf0 px) |
| 739 sup1 = supP pchain pchain⊆A ptotal | |
| 740 sp1 = & (SUP.sup sup1 ) | |
| 741 supf1 : Ordinal → Ordinal | |
| 742 supf1 z with trio< z px | |
| 743 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
| 840 | 744 ... | tri≈ ¬a b ¬c = ZChain.supf zc z |
| 835 | 745 ... | tri> ¬a ¬b c = sp1 |
| 746 | |
| 747 pchain1 : HOD | |
| 748 pchain1 = UnionCF A f mf ay supf1 x | |
| 749 | |
| 840 | 750 ptotal1 : IsTotalOrderSet pchain1 |
| 751 ptotal1 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 752 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 753 uz01 = chain-total A f mf ay supf1 ( (proj2 ca)) ( (proj2 cb)) | |
| 754 pchain⊆A1 : {y : Ordinal} → odef pchain1 y → odef A y | |
| 755 pchain⊆A1 {y} ny = proj1 ny | |
| 756 pnext1 : {a : Ordinal} → odef pchain1 a → odef pchain1 (f a) | |
| 757 pnext1 {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ | |
| 758 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 ) ⟫ | |
| 759 pinit1 : {y₁ : Ordinal} → odef pchain1 y₁ → * y ≤ * y₁ | |
| 760 pinit1 {a} ⟪ aa , ua ⟫ with ua | |
| 761 ... | ch-init fc = s≤fc y f mf fc | |
| 762 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where | |
| 763 zc7 : y <= supf1 u | |
| 764 zc7 = ChainP.fcy<sup is-sup (init ay refl) | |
| 765 pcy1 : odef pchain1 y | |
| 766 pcy1 = ⟪ ay , ch-init (init ay refl) ⟫ | |
| 767 | |
| 843 | 768 supf1≤sp1 : {a : Ordinal } → supf1 a o≤ sp1 |
| 769 supf1≤sp1 = ? | |
| 770 | |
| 771 supf-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b | |
| 772 supf-mono = ? | |
| 773 | |
| 840 | 774 -- zc100 : xSUP (UnionCF A f mf ay supf0 px) x → x ≡ sp1 |
| 775 -- zc100 = ? | |
| 776 | |
| 611 | 777 -- if previous chain satisfies maximality, we caan reuse it |
| 778 -- | |
| 791 | 779 -- supf0 px is sup of UnionCF px , supf0 x is sup of UnionCF x |
| 805 | 780 |
| 836 | 781 no-extension : (¬ xSUP (UnionCF A f mf ay supf0 px) x ) ∨ HasPrev A pchain x f → ZChain A f mf ay x |
| 842 | 782 no-extension ¬sp=x = record { supf = supf1 ; sup = sup ; supf-mono = supf-mono |
| 840 | 783 ; initial = pinit1 ; chain∋init = pcy1 ; sup=u = sup=u ; supf-is-sup = sis ; csupf = csupf |
| 784 ; chain⊆A = λ lt → proj1 lt ; f-next = pnext1 ; f-total = ptotal1 } where | |
| 838 | 785 pchain0=1 : pchain ≡ pchain1 |
| 786 pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
| 787 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z | |
| 788 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 841 | 789 zc10 {z} ⟪ az , ch-is-sup u1 u1≤x u1-is-sup fc ⟫ = zc12 fc where |
| 790 zc12 : {z : Ordinal} → FClosure A f (supf0 u1) z → odef pchain1 z | |
| 838 | 791 zc12 (fsuc x fc) with zc12 fc |
| 792 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 841 | 793 ... | ⟪ ua1 , ch-is-sup u u≤x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x u1-is-sup (fsuc _ fc₁) ⟫ |
| 794 zc12 (init asp refl ) with trio< u1 px | inspect supf1 u1 | |
| 842 | 795 ... | tri< a ¬b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc , ch-is-sup u1 (OrdTrans u1≤x (o<→≤ px<x) ) |
| 841 | 796 record { fcy<sup = fcy<sup ; order = order ; supu=u = trans eq1 (ChainP.supu=u u1-is-sup) } (init (subst (λ k → odef A k ) (sym eq1) asp) eq1 ) ⟫ where |
| 797 fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf1 u1) ∨ (z << supf1 u1 ) | |
| 798 fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) (sym eq1) ( ChainP.fcy<sup u1-is-sup fc ) | |
| 799 order : {s : Ordinal} {z2 : Ordinal} → supf1 s o< supf1 u1 → FClosure A f (supf1 s) z2 → | |
| 800 (z2 ≡ supf1 u1) ∨ (z2 << supf1 u1) | |
| 842 | 801 order {s} {z2} s<u1 fc with trio< s px | inspect supf1 s |
| 802 ... | tri< a ¬b ¬c | _ = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) ( ChainP.order u1-is-sup (subst₂ (λ j k → j o< k) refl eq1 s<u1) fc ) | |
| 803 ... | tri≈ ¬a b ¬c | _ = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) ( ChainP.order u1-is-sup (subst₂ (λ j k → j o< k) refl eq1 s<u1) fc ) | |
| 804 ... | tri> ¬a ¬b px<s | record { eq = eq2 } = ⊥-elim ( o<¬≡ refl (ordtrans px<s (ordtrans zc14 a) )) where -- px o< s < u1 < px | |
| 805 zc14 : s o< u1 | |
| 806 zc14 = supf-inject0 supf-mono (subst₂ (λ j k → j o< k ) (sym eq2) refl s<u1 ) | |
| 807 --- s ≡ sp1, px<s = px o< sp1 | |
| 808 ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc , ch-is-sup u1 (OrdTrans u1≤x (o<→≤ px<x) ) | |
| 841 | 809 record { fcy<sup = fcy<sup ; order = order ; supu=u = trans eq1 (ChainP.supu=u u1-is-sup) } (init (subst (λ k → odef A k ) (sym eq1) asp) eq1 ) ⟫ where |
| 810 fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf1 u1) ∨ (z << supf1 u1 ) | |
| 811 fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) (sym eq1) ( ChainP.fcy<sup u1-is-sup fc ) | |
| 812 order : {s : Ordinal} {z2 : Ordinal} → supf1 s o< supf1 u1 → FClosure A f (supf1 s) z2 → | |
| 813 (z2 ≡ supf1 u1) ∨ (z2 << supf1 u1) | |
| 842 | 814 order {s} {z2} s<u1 fc with trio< s px | inspect supf1 s |
| 815 ... | tri< a ¬b ¬c | _ = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) ( ChainP.order u1-is-sup (subst₂ (λ j k → j o< k) refl eq1 s<u1) fc ) | |
| 816 ... | tri≈ ¬a b ¬c | _ = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) ( ChainP.order u1-is-sup (subst₂ (λ j k → j o< k) refl eq1 s<u1) fc ) | |
| 817 ... | tri> ¬a ¬b px<s | record { eq = eq2 } = ⊥-elim ( o<¬≡ refl (ordtrans px<s (subst (λ k → s o< k) b zc14 ) )) where -- px o< s < u1 = px | |
| 818 zc14 : s o< u1 | |
| 819 zc14 = supf-inject0 supf-mono (subst₂ (λ j k → j o< k ) (sym eq2) refl s<u1 ) | |
| 841 | 820 ... | tri> ¬a ¬b px<u1 | record { eq = eq1 } with osuc-≡< u1≤x |
| 821 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) px<u1 ) | |
| 822 ... | case2 lt = ⊥-elim ( o<> lt px<u1 ) | |
| 823 | |
| 838 | 824 zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z |
| 825 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 841 | 826 zc11 {z} ⟪ az , ch-is-sup u1 u1≤x u1-is-sup fc ⟫ = zc13 fc where |
| 827 zc13 : {z : Ordinal} → FClosure A f (supf1 u1) z → odef pchain z | |
| 838 | 828 zc13 (fsuc x fc) with zc13 fc |
| 829 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 830 ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ | |
| 841 | 831 zc13 (init asp refl ) with trio< u1 px | inspect supf1 u1 |
| 843 | 832 ... | tri< a ¬b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc , ch-is-sup u1 (o<→≤ a) |
| 833 record { fcy<sup = fcy<sup ; order = order ; supu=u = trans (sym eq1) (ChainP.supu=u u1-is-sup) } (init (A∋fcs _ f mf fc) refl) ⟫ where | |
| 841 | 834 fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf0 u1) ∨ (z << supf0 u1 ) |
| 843 | 835 fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) eq1 ( ChainP.fcy<sup u1-is-sup fc ) |
| 836 order : {s : Ordinal} {z2 : Ordinal} → supf0 s o< supf0 u1 → FClosure A f (supf0 s) z2 → | |
| 837 (z2 ≡ supf0 u1) ∨ (z2 << supf0 u1) | |
| 838 order {s} {z2} s<u1 fc with trio< s px | inspect supf1 s | |
| 839 ... | tri< a ¬b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) eq1 ( ChainP.order u1-is-sup (subst₂ (λ j k → j o< k) (sym eq2) (sym eq1) s<u1) (subst (λ k → FClosure A f k z2) (sym eq2) fc )) | |
| 840 ... | tri≈ ¬a b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) eq1 ( ChainP.order u1-is-sup (subst₂ (λ j k → j o< k) (sym eq2) (sym eq1) s<u1) (subst (λ k → FClosure A f k z2) (sym eq2) fc )) | |
| 841 ... | tri> ¬a ¬b px<s | record { eq = eq2 } = ⊥-elim ( o<¬≡ refl (ordtrans px<s (ordtrans zc14 a ))) where -- px o< s < u1 < px | |
| 842 zc14 : s o< u1 | |
| 843 zc14 = ZChain.supf-inject zc s<u1 | |
| 844 ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc , ch-is-sup u1 (o≤-refl0 b) | |
| 845 record { fcy<sup = fcy<sup ; order = order ; supu=u = trans (sym eq1) (ChainP.supu=u u1-is-sup) } (init (A∋fcs _ f mf fc) refl ) ⟫ where | |
| 846 fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf0 u1) ∨ (z << supf0 u1 ) | |
| 847 fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) eq1 ( ChainP.fcy<sup u1-is-sup fc ) | |
| 841 | 848 order : {s : Ordinal} {z2 : Ordinal} → supf0 s o< supf0 u1 → FClosure A f (supf0 s) z2 → |
| 849 (z2 ≡ supf0 u1) ∨ (z2 << supf0 u1) | |
| 850 order {s} {z2} s<u1 fc with trio< s px | inspect supf1 s | |
| 843 | 851 ... | tri< a ¬b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) eq1 ( ChainP.order u1-is-sup (subst₂ (λ j k → j o< k) (sym eq2) (sym eq1) s<u1) (subst (λ k → FClosure A f k z2) (sym eq2) fc )) |
| 852 ... | tri≈ ¬a b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) eq1 ( ChainP.order u1-is-sup (subst₂ (λ j k → j o< k) (sym eq2) (sym eq1) s<u1) (subst (λ k → FClosure A f k z2) (sym eq2) fc )) | |
| 853 ... | tri> ¬a ¬b px<s | _ = ⊥-elim ( o<¬≡ refl (ordtrans px<s (subst (λ k → s o< k ) b zc14))) where -- px o< s < u1 = px | |
| 854 zc14 : s o< u1 | |
| 855 zc14 = ZChain.supf-inject zc s<u1 | |
| 856 ... | tri> ¬a ¬b px<u1 | record { eq = eq2 } = ⊥-elim ? where | |
| 857 zc31 : x ≡ u1 | |
| 858 zc31 with trio< x u1 | |
| 859 ... | tri≈ ¬a b ¬c = b | |
| 860 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ px<u1 , subst (λ k → u1 o< k) (sym (Oprev.oprev=x op)) c ⟫ ) | |
| 861 zc31 | tri< a ¬b ¬c with osuc-≡< (subst (λ k → u1 o≤ k ) refl u1≤x ) -- px<u1 u1≤x, | |
| 862 ... | case1 u1=x = ⊥-elim ( ¬b (sym u1=x) ) | |
| 863 ... | case2 u1<x = ⊥-elim ( o<> u1<x a ) | |
| 864 zc33 : supf1 u1 ≡ u1 -- u1 ≡ supf1 u1 ≡ supf1 x ≡ sp1 | |
| 865 zc33 = ChainP.supu=u u1-is-sup | |
| 866 zc32 : sp1 ≡ x | |
| 867 zc32 = begin | |
| 868 sp1 ≡⟨ sym eq2 ⟩ | |
| 869 supf1 u1 ≡⟨ zc33 ⟩ | |
| 870 u1 ≡⟨ sym zc31 ⟩ | |
| 871 x ∎ where open ≡-Reasoning | |
| 872 zc34 : {z : Ordinal} → odef (UnionCF A f mf ay supf0 px) z → (z ≡ x) ∨ (z << x) | |
| 873 zc34 {z} lt with SUP.x<sup sup1 (subst (λ k → odef (UnionCF A f mf ay supf0 px) k ) (sym &iso) lt ) | |
| 874 ... | case1 eq = case1 ( begin | |
| 875 z ≡⟨ sym &iso ⟩ | |
| 876 & (* z) ≡⟨ cong (&) eq ⟩ | |
| 877 sp1 ≡⟨ zc32 ⟩ | |
| 878 x ∎ ) where open ≡-Reasoning | |
| 879 ... | case2 lt = case2 ( subst (λ k → * z < k ) (trans (sym *iso) (cong (*) zc32 )) lt ) | |
| 880 zcsup : xSUP (UnionCF A f mf ay supf0 px) x | |
| 881 zcsup = record { ax = subst (λ k → odef A k) zc32 asp ; is-sup = record { x<sup = zc34 } } | |
| 882 | |
| 840 | 883 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 884 sup {z} z≤x with trio< z px | inspect supf1 z | |
| 885 ... | tri< a ¬b ¬c | record { eq = eq1} = ? -- ZChain.sup zc (o<→≤ a) | |
| 886 ... | tri≈ ¬a b ¬c | record { eq = eq1} = ? -- ZChain.sup zc (o≤-refl0 b) | |
| 887 ... | tri> ¬a ¬b px<z | record { eq = eq1} = record { sup = SUP.sup sup1 ; as = SUP.as sup1 ; x<sup = zc31 } where | |
| 888 zc30 : z ≡ x | |
| 889 zc30 with osuc-≡< z≤x | |
| 890 ... | case1 eq = eq | |
| 891 ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 892 zc31 : {w : HOD} → UnionCF A f mf ay supf1 z ∋ w → (w ≡ SUP.sup sup1) ∨ (w < SUP.sup sup1) | |
| 893 zc31 = ? | |
| 803 | 894 sup=u : {b : Ordinal} (ab : odef A b) → |
| 840 | 895 b o≤ x → IsSup A (UnionCF A f mf ay supf1 b) ab → supf1 b ≡ b |
| 814 | 896 sup=u {b} ab b≤x is-sup with trio< b px |
| 833 | 897 ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) record { x<sup = λ lt → IsSup.x<sup is-sup ? } |
| 840 | 898 ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (o≤-refl0 b) record { x<sup = λ lt → IsSup.x<sup is-sup ? } |
| 833 | 899 ... | tri> ¬a ¬b px<b = ? where -- ⊥-elim (¬sp=x zcsup ) where |
| 815 | 900 zc30 : x ≡ b |
| 901 zc30 with osuc-≡< b≤x | |
| 902 ... | case1 eq = sym (eq) | |
| 903 ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
| 833 | 904 zcsup : ? |
| 905 zcsup = ? -- with zc30 | |
| 906 -- ... | refl = case1 record { ax = ab ; is-sup = record { x<sup = λ lt → IsSup.x<sup is-sup ? } } | |
| 840 | 907 csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay supf1 (supf1 b)) (supf1 b) |
| 908 csupf {b} b≤x with trio< b px | inspect supf0 b | |
| 909 ... | tri< a ¬b ¬c | _ = ? -- ZChain.csupf zc (o<→≤ a ) | |
| 910 ... | tri≈ ¬a refl ¬c | _ = ? -- ZChain.csupf zc o≤-refl | |
| 911 ... | tri> ¬a ¬b px<b | record { eq = eq1 } = ? where | |
| 912 zc30 : x ≡ b | |
| 913 zc30 with osuc-≡< b≤x | |
| 914 ... | case1 eq = sym (eq) | |
| 915 ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
| 916 sis : {z : Ordinal} (z≤x : z o≤ x) → supf1 z ≡ & (SUP.sup (sup z≤x)) | |
| 917 sis {z} z≤x = zc40 where | |
| 841 | 918 zc40 : supf1 z ≡ & (SUP.sup (sup z≤x)) -- direct with statment causes error |
| 840 | 919 zc40 with trio< z px | inspect supf1 z | inspect sup z≤x |
| 920 ... | tri< a ¬b ¬c | record { eq = eq1 } | record { eq = eq2 } = ? | |
| 921 ... | tri≈ ¬a b ¬c | record { eq = eq1 } | record { eq = eq2 } = ? | |
| 922 ... | tri> ¬a ¬b c | record { eq = eq1 } | record { eq = eq2 } = ? | |
| 833 | 923 |
| 703 | 924 zc4 : ZChain A f mf ay x |
| 793 | 925 zc4 with ODC.∋-p O A (* x) |
| 796 | 926 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
| 836 | 927 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc ) x f ) |
| 703 | 928 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 796 | 929 ... | case1 pr = no-extension {!!} -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 793 | 930 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc ) ax ) |
| 682 | 931 ... | case1 is-sup = -- x is a sup of zc |
| 830 | 932 record { supf = psupf1 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} ; csupf = {!!} ; sup=u = {!!} ; supf-mono = {!!} |
| 800 | 933 ; initial = {!!} ; chain∋init = {!!} ; sup = {!!} ; supf-is-sup = {!!} } where |
| 793 | 934 supx : SUP A (UnionCF A f mf ay supf0 x) |
| 804 | 935 supx = record { sup = * x ; as = subst (λ k → odef A k ) {!!} ax ; x<sup = {!!} } |
| 793 | 936 spx = & (SUP.sup supx ) |
| 937 x=spx : x ≡ spx | |
| 807 | 938 x=spx = sym &iso |
| 750 | 939 psupf1 : Ordinal → Ordinal |
| 940 psupf1 z with trio< z x | |
| 941 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
| 942 ... | tri≈ ¬a b ¬c = x | |
| 943 ... | tri> ¬a ¬b c = x | |
| 822 | 944 csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay psupf1 b) (psupf1 b) |
| 804 | 945 csupf {b} b≤x with trio< b px | inspect psupf1 b |
|
811
e09ba00c9b85
nvim-agda bug in zorn.agda
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
810
diff
changeset
|
946 ... | tri< a ¬b ¬c | record { eq = eq1 } = ⟪ {!!} , {!!} ⟫ |
| 808 | 947 ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ {!!} , {!!} ⟫ |
| 948 ... | tri> ¬a ¬b c | record { eq = eq1 } = {!!} where -- b ≡ x, supf x ≡ sp | |
| 804 | 949 zc30 : x ≡ b |
| 950 zc30 with trio< x b | |
|
811
e09ba00c9b85
nvim-agda bug in zorn.agda
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
810
diff
changeset
|
951 ... | tri< a ¬b ¬c = {!!} |
|
e09ba00c9b85
nvim-agda bug in zorn.agda
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
810
diff
changeset
|
952 ... | tri≈ ¬a b ¬c = {!!} |
|
e09ba00c9b85
nvim-agda bug in zorn.agda
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
810
diff
changeset
|
953 ... | tri> ¬a ¬b c = {!!} |
| 793 | 954 |
| 796 | 955 ... | case2 ¬x=sup = no-extension {!!} -- px is not f y' nor sup of former ZChain from y -- no extention |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
956 |
| 728 | 957 ... | no lim = zc5 where |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
958 |
| 703 | 959 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
| 960 pzc z z<x = prev z z<x | |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
961 |
| 835 | 962 ysp = & (SUP.sup (ysup f mf ay)) |
| 755 | 963 |
| 837 | 964 initial-segment0 : {a b z : Ordinal } → (a<x : a o< x) ( b<x : b o< x) → a o< b → z o≤ a |
| 836 | 965 → ZChain.supf (pzc (osuc a) (ob<x lim a<x )) z ≡ ZChain.supf (pzc (osuc b) (ob<x lim b<x )) z |
| 837 | 966 initial-segment0 = ? |
| 834 | 967 |
| 835 | 968 supf0 : Ordinal → Ordinal |
| 969 supf0 z with trio< z x | |
| 970 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 971 ... | tri≈ ¬a b ¬c = ysp |
| 972 ... | tri> ¬a ¬b c = ysp | |
| 835 | 973 |
| 840 | 974 |
| 975 -- Union of UnionCF z, z o< x undef initial-segment condition | |
| 976 -- this is not a ZChain because supf0 is not monotonic | |
| 838 | 977 pchain : HOD |
| 978 pchain = UnionCF A f mf ay supf0 x | |
| 835 | 979 |
| 838 | 980 ptotal0 : IsTotalOrderSet pchain |
| 835 | 981 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 982 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 983 uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb)) | |
| 984 | |
| 838 | 985 usup : SUP A pchain |
| 986 usup = supP pchain (λ lt → proj1 lt) ptotal0 | |
| 835 | 987 spu = & (SUP.sup usup) |
| 834 | 988 |
| 794 | 989 supf1 : Ordinal → Ordinal |
| 835 | 990 supf1 z with trio< z x |
| 991 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 992 ... | tri≈ ¬a b ¬c = spu |
| 993 ... | tri> ¬a ¬b c = spu | |
| 755 | 994 |
| 838 | 995 pchain1 : HOD |
| 996 pchain1 = UnionCF A f mf ay supf1 x | |
| 704 | 997 |
| 838 | 998 pchain⊆A : {y : Ordinal} → odef pchain1 y → odef A y |
| 704 | 999 pchain⊆A {y} ny = proj1 ny |
| 838 | 1000 pnext : {a : Ordinal} → odef pchain1 a → odef pchain1 (f a) |
| 750 | 1001 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-init (fsuc _ fc) ⟫ |
| 794 | 1002 pnext {a} ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ |
| 838 | 1003 pinit : {y₁ : Ordinal} → odef pchain1 y₁ → * y ≤ * y₁ |
| 748 | 1004 pinit {a} ⟪ aa , ua ⟫ with ua |
| 1005 ... | ch-init fc = s≤fc y f mf fc | |
| 791 | 1006 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 794 | 1007 zc7 : y <= supf1 _ |
| 783 | 1008 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
| 838 | 1009 pcy : odef pchain1 y |
| 783 | 1010 pcy = ⟪ ay , ch-init (init ay refl) ⟫ |
| 838 | 1011 ptotal : IsTotalOrderSet pchain1 |
| 755 | 1012 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 1013 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 794 | 1014 uz01 = chain-total A f mf ay supf1 ( (proj2 ca)) ( (proj2 cb)) |
| 754 | 1015 |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1016 is-max-hp : (supf : Ordinal → Ordinal) (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
|
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1017 b o< x → (ab : odef A b) → |
| 836 | 1018 HasPrev A (UnionCF A f mf ay supf x) b f → |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1019 * a < * b → odef (UnionCF A f mf ay supf x) b |
|
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1020 is-max-hp supf x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
|
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1021 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 791 | 1022 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1023 subst (λ k → UChain A f mf ay supf x k ) |
| 794 | 1024 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1025 |
| 836 | 1026 no-extension : ¬ ( xSUP (UnionCF A f mf ay supf0 x) x ) ∨ HasPrev A pchain x f → ZChain A f mf ay x |
| 802 | 1027 no-extension ¬sp=x = record { initial = pinit ; chain∋init = pcy ; supf = supf1 ; sup=u = sup=u |
| 830 | 1028 ; sup = sup ; supf-is-sup = sis ; supf-mono = {!!} |
| 843 | 1029 ; 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
|
1030 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:
794
diff
changeset
|
1031 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z |
| 838 | 1032 pchain0=1 : pchain ≡ pchain1 |
| 1033 pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
| 1034 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z | |
| 1035 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 1036 zc10 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc12 fc where | |
| 1037 zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z | |
| 1038 zc12 (fsuc x fc) with zc12 fc | |
| 1039 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 1040 ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ | |
| 1041 zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u u≤x ? (init ? ? ) ⟫ | |
| 1042 zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z | |
| 1043 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 1044 zc11 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc13 fc where | |
| 1045 zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z | |
| 1046 zc13 (fsuc x fc) with zc13 fc | |
| 1047 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 1048 ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ | |
| 1049 zc13 (init asu su=z ) with trio< u x | |
| 1050 ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u u≤x ? (init ? ? ) ⟫ | |
| 1051 ... | tri≈ ¬a b ¬c = ? | |
| 1052 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> u≤x c ) | |
| 832 | 1053 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 797 | 1054 sup {z} z≤x with trio< z x |
| 838 | 1055 ... | tri< a ¬b ¬c = SUP⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} ) |
| 815 | 1056 ... | tri≈ ¬a b ¬c = {!!} |
| 843 | 1057 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 832 | 1058 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup {!!})) |
| 843 | 1059 sis {z} z≤x with trio< z x |
| 800 | 1060 ... | tri< a ¬b ¬c = {!!} where |
| 815 | 1061 zc8 = ZChain.supf-is-sup (pzc z a) {!!} |
| 1062 ... | tri≈ ¬a b ¬c = {!!} | |
| 843 | 1063 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 817 | 1064 sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsSup A (UnionCF A f mf ay supf1 b) ab → supf1 b ≡ b |
| 843 | 1065 sup=u {z} ab z≤x is-sup with trio< z x |
| 833 | 1066 ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x<sup = {!!} } |
| 815 | 1067 ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?) ?) ab {!!} record { x<sup = {!!} } |
| 843 | 1068 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 1069 csupf : {z : Ordinal} → z o≤ x → odef (UnionCF A f mf ay supf1 (supf1 z)) (supf1 z) | |
| 804 | 1070 csupf {z} z≤x with trio< z x |
| 833 | 1071 ... | tri< a ¬b ¬c = ? where |
| 797 | 1072 zc9 : odef (UnionCF A f mf ay supf1 z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z) |
| 800 | 1073 zc9 = {!!} |
| 797 | 1074 zc8 : odef (UnionCF A f mf ay (supfu a) z) (ZChain.supf (pzc (osuc z) (ob<x lim a)) z) |
| 830 | 1075 zc8 = {!!} -- ZChain.csupf (pzc (osuc z) (ob<x lim a)) ? -- (o<→≤ <-osuc ) |
| 808 | 1076 ... | tri≈ ¬a b ¬c = {!!} |
| 843 | 1077 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 797 | 1078 |
| 703 | 1079 zc5 : ZChain A f mf ay x |
| 697 | 1080 zc5 with ODC.∋-p O A (* x) |
| 796 | 1081 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
| 836 | 1082 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain x f ) |
| 703 | 1083 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 796 | 1084 ... | case1 pr = no-extension {!!} |
| 704 | 1085 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A pchain ax ) |
| 794 | 1086 ... | case1 is-sup = record { initial = {!!} ; chain∋init = {!!} ; supf = supf1 ; sup=u = {!!} |
| 830 | 1087 ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!} |
| 808 | 1088 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} ; csupf = {!!} } -- where -- x is a sup of (zc ?) |
| 796 | 1089 ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention |
| 553 | 1090 |
| 703 | 1091 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay (& A) |
| 1092 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
|
1093 |
| 551 | 1094 zorn00 : Maximal A |
| 1095 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 804 | 1096 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
| 551 | 1097 -- yes we have the maximal |
| 1098 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 1099 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 1100 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 1101 zorn01 = proj1 zorn03 | |
| 1102 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 1103 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 703 | 1104 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04 total ) where |
| 551 | 1105 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 1106 nmx : ¬ Maximal A | |
| 1107 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 1108 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 804 | 1109 zc5 = ⟪ Maximal.as mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫ |
| 703 | 1110 zorn04 : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) |
| 653 | 1111 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
| 634 | 1112 total : IsTotalOrderSet (ZChain.chain zorn04) |
| 654 | 1113 total {a} {b} = zorn06 where |
| 1114 zorn06 : odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a) | |
| 1115 zorn06 = ZChain.f-total (SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as) ) | |
| 551 | 1116 |
| 516 | 1117 -- usage (see filter.agda ) |
| 1118 -- | |
| 497 | 1119 -- _⊆'_ : ( A B : HOD ) → Set n |
| 1120 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 1121 |
| 497 | 1122 -- MaximumSubset : {L P : HOD} |
| 1123 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 1124 -- → IsPartialOrderSet P _⊆'_ | |
| 1125 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 1126 -- → Maximal P (_⊆'_) | |
| 1127 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |