Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 870:f9fc8da87b5a
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 15 Sep 2022 03:32:06 +0900 |
| parents | 1ca338c3f09c |
| children | 2eaa61279c36 |
| 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
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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
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69 |
| 554 | 70 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 71 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 72 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 73 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 74 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 75 | |
| 779 | 76 <-ftrans : {x y z : Ordinal } → x <= y → y <= z → x <= z |
| 77 <-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 78 <-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 79 <-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 80 <-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 81 | |
| 770 | 82 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y |
| 83 <=to≤ (case1 eq) = case1 (cong (*) eq) | |
| 84 <=to≤ (case2 lt) = case2 lt | |
| 85 | |
| 779 | 86 ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y |
| 87 ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) | |
| 88 ≤to<= (case2 lt) = case2 lt | |
| 89 | |
| 556 | 90 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 91 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 92 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 93 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 94 |
| 561 | 95 ptrans = IsStrictPartialOrder.trans PO |
| 96 | |
| 492 | 97 open _==_ |
| 98 open _⊆_ | |
| 99 | |
| 530 | 100 -- |
| 560 | 101 -- Closure of ≤-monotonic function f has total order |
| 530 | 102 -- |
| 103 | |
| 104 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 105 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 106 | |
| 551 | 107 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 783 | 108 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
| 555 | 109 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 110 |
| 556 | 111 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y |
| 783 | 112 A∋fc {A} s f mf (init as refl ) = as |
| 556 | 113 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 114 |
| 714 | 115 A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s |
| 783 | 116 A∋fcs {A} s f mf (init as refl) = as |
| 714 | 117 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
| 118 | |
| 556 | 119 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 783 | 120 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
| 556 | 121 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 122 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 123 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 124 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 125 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 126 |
| 800 | 127 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 128 fcn s mf (init as refl) = zero | |
| 129 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
| 130 ... | case1 eq = fcn s mf p | |
| 131 ... | case2 y<fy = suc (fcn s mf p ) | |
| 132 | |
| 133 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 134 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 135 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where | |
| 136 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 137 fc06 {x} {y} refl {j} not = fc08 not where | |
| 138 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 139 fc08 () | |
| 140 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
| 141 fc07 {x} (init as refl) eq = refl | |
| 142 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 143 ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) | |
| 144 -- ... | case2 x<fx = ? | |
| 145 fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y | |
| 146 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
| 147 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
| 148 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
| 149 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 150 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y ) | |
| 151 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 152 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) i=x i=y ) | |
| 153 fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 154 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) | |
| 155 fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 156 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) | |
| 157 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 158 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 159 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 160 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 161 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x | |
| 162 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
| 163 fc04 : * x1 ≡ * y | |
| 164 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 165 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 166 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 167 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 168 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 169 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 170 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 171 fc05 : * x ≡ * y1 | |
| 172 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 173 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) | |
| 174 | |
| 175 | |
| 176 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 177 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 178 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
| 179 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 180 fc06 {x} {y} refl {j} not = fc08 not where | |
| 181 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 182 fc08 () | |
| 183 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y | |
| 184 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
| 185 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 186 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 187 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 188 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 189 ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy | |
| 190 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 191 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 192 fc03 eq = cong pred eq | |
| 193 fc02 : * x < * y1 | |
| 194 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 195 | |
| 557 | 196 |
| 559 | 197 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 198 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 800 | 199 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 200 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 201 fc11 : * x < * y | |
| 202 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 203 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 204 fc10 : * x ≡ * y | |
| 205 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 206 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 207 fc12 : * y < * x | |
| 208 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 600 | 209 |
| 563 | 210 |
| 729 | 211 |
| 560 | 212 -- open import Relation.Binary.Properties.Poset as Poset |
| 213 | |
| 214 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 215 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 216 | |
| 567 | 217 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 218 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 219 |
| 568 | 220 _⊆'_ : ( A B : HOD ) → Set n |
| 221 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 222 |
| 223 -- | |
| 224 -- inductive maxmum tree from x | |
| 225 -- tree structure | |
| 226 -- | |
| 554 | 227 |
| 836 | 228 record HasPrev (A B : HOD) (x : Ordinal ) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 229 field |
| 836 | 230 ax : odef A x |
| 534 | 231 y : Ordinal |
| 541 | 232 ay : odef B y |
| 534 | 233 x=fy : x ≡ f y |
| 529 | 234 |
| 570 | 235 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 654 | 236 field |
| 779 | 237 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 568 | 238 |
| 656 | 239 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 240 field | |
| 241 sup : HOD | |
| 804 | 242 as : A ∋ sup |
| 656 | 243 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive |
| 244 | |
| 690 | 245 -- |
| 246 -- sup and its fclosure is in a chain HOD | |
| 247 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 248 -- whole chain is a union of separated Chain | |
| 803 | 249 -- minimum index is sup of y not ϕ |
| 690 | 250 -- |
| 251 | |
| 787 | 252 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where |
| 690 | 253 field |
| 765 | 254 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
| 828 | 255 order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
| 256 supu=u : supf u ≡ u | |
| 694 | 257 |
| 748 | 258 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 259 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where | |
| 260 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 863 | 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 |
| 861 | 264 -- data UChain is total |
| 265 | |
| 266 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) | |
| 267 {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 ) | |
| 268 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where | |
| 269 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) | |
| 270 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb | |
| 271 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca | |
| 272 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 273 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 274 ct00 : * a ≡ * b | |
| 275 ct00 = trans (cong (*) eq) eq1 | |
| 276 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where | |
| 277 ct01 : * a < * b | |
| 278 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
| 279 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 | |
| 280 ct00 : * a < * (supf ub) | |
| 281 ct00 = lt | |
| 282 ct01 : * a < * b | |
| 283 ct01 with s≤fc (supf ub) f mf fcb | |
| 284 ... | case1 eq = subst (λ k → * a < k ) eq ct00 | |
| 285 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 286 ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb | |
| 287 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 288 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 289 ct00 : * a ≡ * b | |
| 290 ct00 = sym (trans (cong (*) eq) eq1 ) | |
| 291 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where | |
| 292 ct01 : * b < * a | |
| 293 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
| 294 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 | |
| 295 ct00 : * b < * (supf ua) | |
| 296 ct00 = lt | |
| 297 ct01 : * b < * a | |
| 298 ct01 with s≤fc (supf ua) f mf fca | |
| 299 ... | case1 eq = subst (λ k → * b < k ) eq ct00 | |
| 300 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 301 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 | |
| 302 ... | 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 | |
| 303 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 304 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 305 ct00 : * a ≡ * b | |
| 306 ct00 = trans (cong (*) eq) eq1 | |
| 307 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 308 ct02 : * a < * b | |
| 309 ct02 = subst (λ k → * k < * b ) (sym eq) lt | |
| 310 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 | |
| 311 ct03 : * a < * (supf ub) | |
| 312 ct03 = lt | |
| 313 ct02 : * a < * b | |
| 314 ct02 with s≤fc (supf ub) f mf fcb | |
| 315 ... | case1 eq = subst (λ k → * a < k ) eq ct03 | |
| 316 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt | |
| 317 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 | |
| 318 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) | |
| 319 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 | |
| 320 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 321 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 322 ct00 : * a ≡ * b | |
| 323 ct00 = sym (trans (cong (*) eq) eq1) | |
| 324 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
| 325 ct02 : * b < * a | |
| 326 ct02 = subst (λ k → * k < * a ) (sym eq) lt | |
| 327 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 | |
| 328 ct05 : * b < * (supf ua) | |
| 329 ct05 = lt | |
| 330 ct04 : * b < * a | |
| 331 ct04 with s≤fc (supf ua) f mf fca | |
| 332 ... | case1 eq = subst (λ k → * b < k ) eq ct05 | |
| 333 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt | |
| 334 | |
| 694 | 335 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 336 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 337 | |
| 803 | 338 -- Union of supf z which o< x |
| 339 -- | |
| 694 | 340 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 341 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
| 342 UnionCF A f mf ay supf x | |
| 343 = 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 | 344 |
| 842 | 345 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) |
| 346 → supf x o< supf y → x o< y | |
| 347 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y | |
| 348 ... | tri< a ¬b ¬c = a | |
| 349 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 350 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 351 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 352 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 353 | |
| 866 | 354 zsupf0 : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) |
| 355 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition | |
| 356 → { z : Ordinal } → (odef A z) → SUP A (UnionCF A f mf ay (λ _ → z) z) | |
| 357 zsupf0 A f mf ay supP {z} az = supP chain (λ lt → proj1 lt ) f-total where | |
| 358 chain = UnionCF A f mf ay (λ _ → z) z | |
| 359 f-total : IsTotalOrderSet chain | |
| 360 f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 361 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 362 uz01 = chain-total A f mf ay (λ _ → z) ( (proj2 ca)) ( (proj2 cb)) | |
| 363 | |
| 364 record ZSupf ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) | |
| 865 | 365 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 366 field | |
| 866 | 367 supf : Ordinal → Ordinal |
| 868 | 368 pchain : HOD |
| 369 pchain = UnionCF A f mf ay supf z | |
| 370 field | |
| 866 | 371 sup : (x : Ordinal ) → SUP A (UnionCF A f mf ay (λ _ → z) x) |
| 372 supf-is-sup : {x : Ordinal } → supf x ≡ & (SUP.sup (sup x) ) | |
| 373 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y | |
| 374 | |
| 867 | 375 -- |
| 376 -- f (f ( ... (sup y))) f (f ( ... (sup z1))) | |
| 377 -- / | / | | |
| 378 -- / | / | | |
| 379 -- sup y < sup z1 < sup z2 | |
| 380 -- o< o< | |
| 381 | |
| 866 | 382 zsupf : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {init : Ordinal} (ay : odef A init) |
| 383 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition | |
| 384 → (x : Ordinal) → ZSupf A f mf ay x | |
| 385 zsupf A f mf {y} ay supP x = TransFinite { λ x → ZSupf A f mf ay x } zc1 x where | |
| 386 | |
| 387 zc1 : (x : Ordinal) → ((y₁ : Ordinal) → y₁ o< x → ZSupf A f mf ay y₁) → ZSupf A f mf ay x | |
| 388 zc1 x prev with Oprev-p x | |
| 867 | 389 ... | yes op = record { supf = ? ; sup = ? ; supf-is-sup = ? ; supf-mono = ? } where |
| 866 | 390 px = Oprev.oprev op |
| 868 | 391 pchain0 : HOD |
| 392 pchain0 = UnionCF A f mf ay (ZSupf.supf ( prev px (pxo<x op) )) px | |
| 866 | 393 zc-b<x : {b : Ordinal } → b o< x → b o< osuc px |
| 394 zc-b<x {b} lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 867 | 395 |
| 866 | 396 supf : Ordinal → Ordinal |
| 397 supf z with trio< z px | |
| 867 | 398 ... | tri< a ¬b ¬c = ZSupf.supf (prev z (ordtrans a (pxo<x op))) z |
| 399 ... | tri≈ ¬a b ¬c = ZSupf.supf (prev z (subst (λ k → k o< x ) (sym b) (pxo<x op))) z | |
| 400 ... | tri> ¬a ¬b px<x = ZSupf.supf (prev px (pxo<x op) ) px | |
| 401 | |
| 868 | 402 sp1 = & (SUP.sup (zsupf0 A f mf ay supP ? )) |
| 867 | 403 |
| 404 zc2 : ZSupf A f mf ay x | |
| 405 zc2 with ODC.∋-p O A (* x) | |
| 406 ... | no noax = ? | |
| 868 | 407 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain0 ? f ) |
| 867 | 408 ... | case1 pr = ? |
| 868 | 409 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A pchain0 ax ) |
| 867 | 410 ... | case2 ¬x=sup = ? |
| 866 | 411 ... | case1 is-sup = ? |
| 412 | |
| 413 ... | no lim = ? where | |
| 414 | |
| 415 -- Range of supf is total order set, so it has SUP | |
| 416 supfmax : Ordinal | |
| 417 supfmax = ? | |
| 418 | |
| 419 supfx : Ordinal | |
| 420 supfx with ODC.∋-p O A (* x) | |
| 421 ... | no noax = supfmax | |
| 867 | 422 ... | yes ax with ODC.p∨¬p O ( HasPrev A ? x f ) |
| 866 | 423 ... | case1 pr = supfmax |
| 867 | 424 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A ? ax ) |
| 866 | 425 ... | case2 ¬x=sup = supfmax |
| 426 ... | case1 is-sup = ? | |
| 427 | |
| 428 supf : Ordinal → Ordinal | |
| 429 supf z with trio< z x | |
| 867 | 430 ... | tri< a ¬b ¬c = ZSupf.supf (prev z a) z |
| 866 | 431 ... | tri≈ ¬a b ¬c = supfx |
| 432 ... | tri> ¬a ¬b px<x = supfx | |
| 433 | |
| 865 | 434 |
| 703 | 435 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 436 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 437 field |
| 694 | 438 supf : Ordinal → Ordinal |
| 868 | 439 asupf : {x : Ordinal } → odef A (supf x) |
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440 chain : HOD |
| 703 | 441 chain = UnionCF A f mf ay supf z |
| 861 | 442 chain⊆A : chain ⊆' A |
| 443 chain⊆A = λ lt → proj1 lt | |
| 568 | 444 field |
| 832 | 445 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
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446 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z |
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447 → IsSup A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) b f) → supf b ≡ b |
| 832 | 448 supf-is-sup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ & (SUP.sup (sup x≤z) ) |
| 825 | 449 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y |
| 870 | 450 csupf : {b : Ordinal } → supf b o≤ z → odef (UnionCF A f mf ay supf z) (supf b) |
| 869 | 451 |
| 857 | 452 |
| 861 | 453 chain∋init : odef chain y |
| 454 chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ | |
| 455 f-next : {a : Ordinal} → odef chain a → odef chain (f a) | |
| 456 f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ | |
| 457 f-next {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 ) ⟫ | |
| 458 initial : {z : Ordinal } → odef chain z → * y ≤ * z | |
| 459 initial {a} ⟪ aa , ua ⟫ with ua | |
| 460 ... | ch-init fc = s≤fc y f mf fc | |
| 461 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where | |
| 462 zc7 : y <= supf u | |
| 463 zc7 = ChainP.fcy<sup is-sup (init ay refl) | |
| 464 f-total : IsTotalOrderSet chain | |
| 465 f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 466 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 467 uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) | |
| 468 | |
| 825 | 469 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
| 470 supf-inject {x} {y} sx<sy with trio< x y | |
| 471 ... | tri< a ¬b ¬c = a | |
| 472 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 473 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 474 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 475 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 798 | 476 |
| 803 | 477 -- ordering is proved here for totality and sup |
| 478 | |
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479 supf∈A : {b : Ordinal} → b o≤ z → odef A (supf b) |
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480 supf∈A {b} b≤z = subst (λ k → odef A k ) (sym (supf-is-sup b≤z)) ( SUP.as ( sup b≤z ) ) |
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481 |
| 813 | 482 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 | 483 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 | 484 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ |
| 832 | 485 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans (cong (&) eq) (sym (supf-is-sup (o<→≤ u<z) ) ) )) |
| 486 ... | case2 lt = case2 (subst (λ k → * w < k ) (subst (λ k → k ≡ _ ) *iso (cong (*) (sym (supf-is-sup (o<→≤ u<z) ))) ) lt ) | |
| 825 | 487 |
| 756 | 488 |
| 728 | 489 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 490 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
| 869 | 491 supf = ZChain.supf zc |
| 728 | 492 field |
| 869 | 493 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → b o< z → (ab : odef A b) |
| 494 → HasPrev A (UnionCF A f mf ay supf z) b f ∨ IsSup A (UnionCF A f mf ay supf z) ab | |
| 495 → * a < * b → odef ((UnionCF A f mf ay supf z)) b | |
| 728 | 496 |
| 837 | 497 initial-segment : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 498 {a b y : Ordinal} (ay : odef A y) (za : ZChain A f mf ay a ) (zb : ZChain A f mf ay b ) | |
| 499 → {z : Ordinal } → a o≤ b → z o≤ a | |
| 500 → ZChain.supf za z ≡ ZChain.supf zb z | |
| 501 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 | |
| 502 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → z o≤ a → ZChain.supf za z ≡ ZChain.supf zb z ) → | |
| 503 x o≤ a → ZChain.supf za x ≡ ZChain.supf zb x | |
| 504 ind x prev x≤a = ? where | |
| 505 supfa = ZChain.supf za | |
| 506 supfb = ZChain.supf zb | |
| 507 zc10 : {w : Ordinal } → w o< z → UnionCF A f mf ay supfa w ≡ UnionCF A f mf ay supfb w | |
| 508 zc10 = ? | |
| 509 -- w o< z → supfa w ≡ supfb w | |
| 510 supa : SUP A (UnionCF A f mf ay supfa x) | |
| 511 supa = ZChain.sup za x≤a | |
| 512 supb : SUP A (UnionCF A f mf ay supfb x) | |
| 513 supb = ZChain.sup zb (OrdTrans x≤a a≤b) | |
| 514 zc13 : UnionCF A f mf ay supfa x ≡ UnionCF A f mf ay supfb x | |
| 515 zc13 = ? -- | |
| 516 -- if x is sup of UCF px (or Union o< x ) , then supfa x ≡ x supfb x | |
| 517 -- if x is not sup of UCF px (or Union o< x ) or HasPrev, UCF x ≡ UCF px (or Union o< x) | |
| 518 zc15 : {B : HOD} → (a b : SUP A B) → SUP.sup a ≡ SUP.sup b | |
| 519 zc15 = ? | |
| 520 zc14 : supfa x ≡ supfb x | |
| 521 zc14 = begin | |
| 522 supfa x ≡⟨ ? ⟩ | |
| 523 & (SUP.sup supa) ≡⟨ ? ⟩ | |
| 524 & (SUP.sup supb) ≡⟨ ? ⟩ | |
| 525 supfb x ∎ where open ≡-Reasoning | |
| 526 | |
| 568 | 527 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 528 field | |
| 529 maximal : HOD | |
| 804 | 530 as : A ∋ maximal |
| 568 | 531 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative |
| 567 | 532 |
| 743 | 533 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 534 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 783 | 535 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ |
| 743 | 536 |
| 497 | 537 Zorn-lemma : { A : HOD } |
| 464 | 538 → o∅ o< & A |
| 568 | 539 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 540 → Maximal A |
| 552 | 541 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 542 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 543 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 544 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 545 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 760 | 546 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A |
| 547 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 530 | 548 s : HOD |
| 549 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 550 as : A ∋ * ( & s ) |
| 551 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
|
552 as0 : odef A (& s ) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
553 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 554 s<A : & s o< & A |
| 568 | 555 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 556 HasMaximal : HOD |
| 537 | 557 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 558 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 559 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 560 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 561 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 562 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 804 | 563 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
| 537 | 564 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
| 565 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)) | |
| 566 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 | |
| 567 ¬x<m : ¬ (* x < * m) | |
| 568 ¬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 | 569 |
| 560 | 570 -- Uncountable ascending chain by axiom of choice |
| 530 | 571 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 572 cf nmx x with ODC.∋-p O A (* x) |
| 573 ... | no _ = o∅ | |
| 574 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 575 ... | yes nogt = -- no larger element, so it is maximal |
| 576 ⊥-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 | 577 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 578 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 579 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 580 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 581 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 582 ... | 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 ⟫ ) | |
| 583 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 584 |
| 585 --- | |
| 586 --- infintie ascention sequence of f | |
| 587 --- | |
| 530 | 588 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 589 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 590 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 591 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 592 |
| 793 | 593 chain-mono : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 594 {a b c : Ordinal} → a o≤ b | |
| 595 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c | |
| 596 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = | |
| 597 ⟪ ua , ch-init fc ⟫ | |
| 598 chain-mono f mf ay supf {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = | |
| 863 | 599 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x a≤b ) is-sup fc ⟫ |
| 793 | 600 |
| 703 | 601 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 653 | 602 (total : IsTotalOrderSet (ZChain.chain zc) ) → SUP A (ZChain.chain zc) |
| 703 | 603 sp0 f mf zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total |
| 543 | 604 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 605 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 606 | |
| 803 | 607 -- |
| 608 -- Second TransFinite Pass for maximality | |
| 609 -- | |
| 610 | |
| 793 | 611 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 728 | 612 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
| 793 | 613 SZ1 f mf {y} ay zc x = TransFinite { λ x → ZChain1 A f mf ay zc x } zc1 x where |
| 614 chain-mono1 : {a b c : Ordinal} → a o≤ b | |
| 788 | 615 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
| 793 | 616 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) a≤b |
| 735 | 617 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 618 b o< x → (ab : odef A b) → | |
| 836 | 619 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) b f → |
| 735 | 620 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
| 749 | 621 is-max-hp x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
| 622 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ | |
| 791 | 623 ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , |
| 749 | 624 subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) |
| 791 | 625 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc)) ⟫ |
| 868 | 626 |
| 869 | 627 supf = ZChain.supf zc |
| 628 | |
| 629 csupf-fc : {b s z1 : Ordinal} → b o≤ & A → supf s o< supf b → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1 | |
| 630 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 | |
| 631 s<b : s o< b | |
| 632 s<b = ZChain.supf-inject zc ss<sb | |
| 633 s≤<z : s o≤ & A | |
| 634 s≤<z = ordtrans s<b b≤z | |
| 870 | 635 zc04 : odef (UnionCF A f mf ay supf (& A)) (supf s) |
| 636 zc04 = ZChain.csupf zc (o<→≤ (z09 (ZChain.asupf zc))) | |
| 869 | 637 zc05 : odef (UnionCF A f mf ay supf b) (supf s) |
| 638 zc05 with zc04 | |
| 639 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ | |
| 870 | 640 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u ? is-sup fc ⟫ where |
| 641 zc07 : FClosure A f (supf u) (supf s) -- supf u ≤ supf s → supf u o≤ supf s | |
| 642 zc07 = fc | |
| 869 | 643 zc06 : supf u ≡ u |
| 644 zc06 = ChainP.supu=u is-sup | |
| 870 | 645 zc09 : u o≤ supf s → u o< b |
| 646 zc09 u≤s with osuc-≡< u≤s | |
| 647 ... | case1 u=ss = ZChain.supf-inject zc (subst (λ k → k o< supf b) (sym (trans zc06 u=ss)) ss<sb ) | |
| 648 ... | case2 u<ss = ordtrans (ZChain.supf-inject zc (subst (λ k → k o< supf s) (sym zc06) u<ss)) s<b | |
| 869 | 649 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 |
| 650 zc04 : odef (UnionCF A f mf ay supf b) (f x) | |
| 651 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb fc ) | |
| 652 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ | |
| 653 ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫ | |
| 654 order : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b) | |
| 655 order {b} {s} {z1} b<z ss<sb fc = zc04 where | |
| 656 zc00 : ( * z1 ≡ SUP.sup (ZChain.sup zc (o<→≤ b<z) )) ∨ ( * z1 < SUP.sup ( ZChain.sup zc (o<→≤ b<z) ) ) | |
| 657 zc00 = SUP.x<sup (ZChain.sup zc (o<→≤ b<z) ) (csupf-fc (o<→≤ b<z) ss<sb fc ) | |
| 870 | 658 -- supf (supf b) ≡ supf b |
| 869 | 659 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) |
| 660 zc04 with zc00 | |
| 661 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso (sym (ZChain.supf-is-sup zc (o<→≤ b<z)) ) (cong (&) eq) ) | |
| 662 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) refl (subst₂ (λ j k → j ≡ k ) *iso refl (cong (*) (sym (ZChain.supf-is-sup zc (o<→≤ b<z) ) ))) lt ) | |
| 868 | 663 |
| 728 | 664 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
|
665 zc1 x prev with Oprev-p x |
| 756 | 666 ... | 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
|
667 px = Oprev.oprev op |
| 789 | 668 zc-b<x : {b : Ordinal } → b o< x → b o< osuc px |
| 669 zc-b<x {b} lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 728 | 670 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 671 b o< x → (ab : odef A b) → | |
| 869 | 672 HasPrev A (UnionCF A f mf ay supf x) b f ∨ IsSup A (UnionCF A f mf ay supf x) ab → |
| 673 * a < * b → odef (UnionCF A f mf ay supf x) b | |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
674 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
675 is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua b<x ab has-prev a<b |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
676 is-max {a} {b} ua b<x ab P a<b | case2 is-sup |
| 863 | 677 = ⟪ ab , ch-is-sup b b<x m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where |
|
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
|
678 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
|
679 b<A = z09 ab |
| 869 | 680 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) b f |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
681 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
682 ; x=fy = HasPrev.x=fy nhp } ) |
| 859 | 683 m05 : ZChain.supf zc b ≡ b |
| 684 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) | |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
685 ⟪ record { x<sup = λ {z} uz → IsSup.x<sup (proj2 is-sup) (chain-mono1 (o<→≤ b<x) uz ) } , m04 ⟫ |
|
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
|
686 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
| 813 | 687 m08 {z} fcz = ZChain.fcy<sup zc b<A fcz |
| 828 | 688 m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 689 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
| 869 | 690 m09 {s} {z} s<b fcz = order b<A s<b fcz |
| 691 m06 : ChainP A f mf ay supf b | |
| 859 | 692 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = m05 } |
| 756 | 693 ... | no lim = record { is-max = is-max } where |
| 869 | 694 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
| 734 | 695 b o< x → (ab : odef A b) → |
| 869 | 696 HasPrev A (UnionCF A f mf ay supf x) b f ∨ IsSup A (UnionCF A f mf ay supf x) ab → |
| 697 * a < * b → odef (UnionCF A f mf ay supf x) b | |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
698 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
699 is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua b<x ab has-prev a<b |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
700 is-max {a} {b} ua b<x ab P a<b | case2 is-sup with IsSup.x<sup (proj2 is-sup) (init-uchain A f mf ay ) |
| 789 | 701 ... | 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 | 702 ... | case2 y<b = chain-mono1 (osucc b<x) |
| 863 | 703 ⟪ ab , ch-is-sup b <-osuc m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where |
|
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
|
704 m09 : 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
|
705 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
|
706 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
| 813 | 707 m07 {z} fc = ZChain.fcy<sup zc m09 fc |
| 828 | 708 m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 709 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
| 869 | 710 m08 {s} {z1} s<b fc = order m09 s<b fc |
| 711 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) b f | |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
712 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
713 ; x=fy = HasPrev.x=fy nhp } ) |
| 859 | 714 m05 : ZChain.supf zc b ≡ b |
| 715 m05 = ZChain.sup=u zc ab (o<→≤ m09) | |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
716 ⟪ record { x<sup = λ lt → IsSup.x<sup (proj2 is-sup) (chain-mono1 (o<→≤ b<x) lt )} , m04 ⟫ -- ZChain on x |
| 869 | 717 m06 : ChainP A f mf ay supf b |
| 859 | 718 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = m05 } |
| 727 | 719 |
| 543 | 720 --- |
| 560 | 721 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 722 --- |
| 703 | 723 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
|
724 → (total : IsTotalOrderSet (ZChain.chain zc) ) |
| 703 | 725 → f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total)) |
| 726 fixpoint f mf zc total = z14 where | |
| 538 | 727 chain = ZChain.chain zc |
| 703 | 728 sp1 = sp0 f mf zc total |
| 712 | 729 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
| 836 | 730 → HasPrev A chain b f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 731 → * a < * b → odef chain b |
| 793 | 732 z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) ) |
| 543 | 733 z11 : & (SUP.sup sp1) o< & A |
| 804 | 734 z11 = c<→o< ( SUP.as sp1) |
| 538 | 735 z12 : odef chain (& (SUP.sup sp1)) |
| 736 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 653 | 737 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 804 | 738 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z11 (SUP.as sp1) |
| 570 | 739 (case2 z19 ) z13 where |
| 538 | 740 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 653 | 741 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc ) |
| 538 | 742 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 743 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 804 | 744 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.as sp1) |
| 571 | 745 z19 = record { x<sup = z20 } where |
| 746 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
| 747 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
| 570 | 748 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
| 749 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
| 750 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
| 703 | 751 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
|
752 z14 with total (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 631 | 753 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 754 z16 : ⊥ | |
| 804 | 755 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.as sp1 )) |
| 631 | 756 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) |
| 757 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 758 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 759 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 760 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 761 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) | |
| 762 z17 : ⊥ | |
| 763 z17 with z15 | |
| 764 ... | case1 eq = ¬b eq | |
| 765 ... | case2 lt = ¬a lt | |
| 560 | 766 |
| 767 -- ZChain contradicts ¬ Maximal | |
| 768 -- | |
| 571 | 769 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
| 560 | 770 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
| 771 -- | |
| 697 | 772 z04 : (nmx : ¬ Maximal A ) |
| 703 | 773 → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) |
| 664 | 774 → IsTotalOrderSet (ZChain.chain zc) → ⊥ |
| 804 | 775 z04 nmx zc total = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.as sp1 )))) |
| 776 (subst (λ k → odef A (& k)) (sym *iso) (SUP.as sp1) ) | |
| 703 | 777 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc total ))) -- x ≡ f x ̄ |
| 804 | 778 (proj1 (cf-is-<-monotonic nmx c (SUP.as sp1 ))) where -- x < f x |
|
633
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
779 sp1 : SUP A (ZChain.chain zc) |
| 703 | 780 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc total |
| 538 | 781 c = & (SUP.sup sp1) |
| 548 | 782 |
| 757 | 783 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 784 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = | |
| 785 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 ))) } | |
| 786 | |
| 787 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 788 → IsTotalOrderSet (uchain f mf ay) | |
| 789 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 790 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 791 uz01 = fcn-cmp y f mf ca cb | |
| 792 | |
| 793 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 794 → SUP A (uchain f mf ay) | |
| 795 ysup f mf {y} ay = supP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) | |
| 796 | |
| 793 | 797 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 804 | 798 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 | 799 |
| 833 | 800 record xSUP (B : HOD) (x : Ordinal) : Set n where |
| 801 field | |
| 802 ax : odef A x | |
| 803 is-sup : IsSup A B ax | |
| 804 | |
| 560 | 805 -- |
| 547 | 806 -- create all ZChains under o< x |
| 560 | 807 -- |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
808 |
| 674 | 809 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 810 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 707 | 811 ind f mf {y} ay x prev with Oprev-p x |
| 697 | 812 ... | yes op = zc4 where |
| 682 | 813 -- |
| 814 -- we have previous ordinal to use induction | |
| 815 -- | |
| 816 px = Oprev.oprev op | |
| 703 | 817 zc : ZChain A f mf ay (Oprev.oprev op) |
| 682 | 818 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 819 px<x : px o< x | |
| 820 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 709 | 821 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 822 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 697 | 823 |
| 754 | 824 supf0 = ZChain.supf zc |
| 869 | 825 pchain : HOD |
| 826 pchain = UnionCF A f mf ay supf0 px | |
| 835 | 827 pchain1 : HOD |
| 857 | 828 pchain1 = UnionCF A f mf ay supf0 x |
| 835 | 829 |
| 857 | 830 supfx : {z : Ordinal } → x o≤ z → supf0 px ≡ supf0 z |
| 831 supfx {z} x≤z with trio< z px | |
| 832 ... | tri< a ¬b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → z o< k ) (Oprev.oprev=x op) (ordtrans a <-osuc ))) | |
| 833 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) (pxo<x op))) | |
| 865 | 834 ... | tri> ¬a ¬b c = ? -- ZChain.supf-max zc (o<→≤ c) |
| 844 | 835 |
| 857 | 836 supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b |
| 837 supf-mono = ZChain.supf-mono zc | |
| 844 | 838 |
| 857 | 839 supf-max : {z : Ordinal} → x o≤ z → supf0 x ≡ supf0 z |
| 840 supf-max {z} z≤x = trans ( sym zc02) zc01 where | |
| 841 zc02 : supf0 px ≡ supf0 x | |
| 865 | 842 zc02 = ? -- ZChain.supf-max zc (o<→≤ (pxo<x op)) |
| 857 | 843 zc01 : supf0 px ≡ supf0 z |
| 865 | 844 zc01 = ? -- ZChain.supf-max zc (OrdTrans (o<→≤ (pxo<x op)) z≤x) |
| 868 | 845 |
| 861 | 846 zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) |
| 847 zc04 {b} b≤x with trio< b px | |
| 848 ... | tri< a ¬b ¬c = case1 (o<→≤ a) | |
| 849 ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) | |
| 850 ... | tri> ¬a ¬b px<b with osuc-≡< b≤x | |
| 851 ... | case1 eq = case2 eq | |
| 852 ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
| 840 | 853 |
| 611 | 854 -- if previous chain satisfies maximality, we caan reuse it |
| 855 -- | |
| 863 | 856 -- UninCF supf0 px previous chain u o< px, supf0 px is not included |
| 857 -- UninCF supf0 x supf0 px is included | |
| 858 -- supf0 px ≡ px | |
| 859 -- supf0 px ≡ supf0 x | |
| 805 | 860 |
| 858 | 861 no-extension : ( (¬ xSUP (UnionCF A f mf ay supf0 px) x ) ∨ HasPrev A pchain x f) → ZChain A f mf ay x |
| 869 | 862 no-extension P = record { supf = supf0 ; asupf = ZChain.asupf zc ; sup = λ lt → STMP.tsup (sup lt ) ; supf-mono = supf-mono |
| 870 | 863 ; sup=u = sup=u ; supf-is-sup = λ lt → STMP.tsup=sup (sup lt) } where |
| 863 | 864 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z |
| 865 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 866 zc10 {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 (ordtrans u1<x (pxo<x op)) u1-is-sup fc ⟫ | |
| 867 zc11 : (¬ xSUP (UnionCF A f mf ay supf0 px) x ) ∨ (HasPrev A pchain x f ) | |
| 864 | 868 → {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z ∨ ( (supf0 px ≡ px) ∧ FClosure A f px z ) |
| 863 | 869 zc11 P {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
| 870 zc11 P {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ with trio< u1 px | |
| 871 ... | tri< u1<px ¬b ¬c = case1 ⟪ az , ch-is-sup u1 u1<px u1-is-sup fc ⟫ | |
| 864 | 872 ... | tri≈ ¬a eq ¬c = case2 ⟪ subst (λ k → supf0 k ≡ k) eq s1u=u , subst (λ k → FClosure A f k z) zc12 fc ⟫ where |
| 863 | 873 s1u=u : supf0 u1 ≡ u1 |
| 874 s1u=u = ChainP.supu=u u1-is-sup | |
| 864 | 875 zc12 : supf0 u1 ≡ px |
| 876 zc12 = trans (ChainP.supu=u u1-is-sup) eq | |
| 863 | 877 zc11 (case1 ¬sp=x) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = ⊥-elim (¬sp=x zcsup) where |
| 878 eq : u1 ≡ x | |
| 879 eq with trio< u1 x | |
| 880 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) | |
| 881 ... | tri≈ ¬a b ¬c = b | |
| 882 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x c ) | |
| 883 s1u=x : supf0 u1 ≡ x | |
| 884 s1u=x = trans (ChainP.supu=u u1-is-sup) eq | |
| 885 zc13 : osuc px o< osuc u1 | |
| 886 zc13 = o≤-refl0 ( trans (Oprev.oprev=x op) (sym eq ) ) | |
| 887 x<sup : {w : Ordinal} → odef (UnionCF A f mf ay supf0 px) w → (w ≡ x) ∨ (w << x) | |
| 888 x<sup {w} ⟪ az , ch-init {w} fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x | |
| 851 | 889 ( ChainP.fcy<sup u1-is-sup {w} fc ) |
| 863 | 890 x<sup {w} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u<x) zc13 )) |
| 891 ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 (o<→≤ u<x) ) where | |
| 851 | 892 zc14 : u ≡ osuc px |
| 893 zc14 = begin | |
| 894 u ≡⟨ sym ( ChainP.supu=u is-sup) ⟩ | |
| 857 | 895 supf0 u ≡⟨ eq1 ⟩ |
| 896 supf0 u1 ≡⟨ s1u=x ⟩ | |
| 851 | 897 x ≡⟨ sym (Oprev.oprev=x op) ⟩ |
| 898 osuc px ∎ where open ≡-Reasoning | |
| 863 | 899 ... | case2 lt = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ( ChainP.order u1-is-sup lt fc ) |
| 900 zc12 : supf0 x ≡ u1 | |
| 901 zc12 = subst (λ k → supf0 k ≡ u1) eq (ChainP.supu=u u1-is-sup) | |
| 902 zcsup : xSUP (UnionCF A f mf ay supf0 px) x | |
| 868 | 903 zcsup = record { ax = subst (λ k → odef A k) (trans zc12 eq) (ZChain.asupf zc) |
| 851 | 904 ; is-sup = record { x<sup = x<sup } } |
| 864 | 905 zc11 (case2 hp) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = case1 (zc20 fc ) where |
| 863 | 906 eq : u1 ≡ x |
| 864 | 907 eq with trio< u1 x |
| 908 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) | |
| 909 ... | tri≈ ¬a b ¬c = b | |
| 910 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x c ) | |
| 858 | 911 zc20 : {z : Ordinal} → FClosure A f (supf0 u1) z → OD.def (od pchain) z |
| 912 zc20 {z} (init asu su=z ) = zc13 where | |
| 913 zc14 : x ≡ z | |
| 914 zc14 = begin | |
| 915 x ≡⟨ sym eq ⟩ | |
| 916 u1 ≡⟨ sym (ChainP.supu=u u1-is-sup ) ⟩ | |
| 917 supf0 u1 ≡⟨ su=z ⟩ | |
| 918 z ∎ where open ≡-Reasoning | |
| 919 zc13 : odef pchain z | |
| 920 zc13 = subst (λ k → odef pchain k) (trans (sym (HasPrev.x=fy hp)) zc14) ( ZChain.f-next zc (HasPrev.ay hp) ) | |
| 921 zc20 {.(f w)} (fsuc w fc) = ZChain.f-next zc (zc20 fc) | |
| 857 | 922 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
| 923 field | |
| 924 tsup : SUP A (UnionCF A f mf ay supf0 z) | |
| 925 tsup=sup : supf0 z ≡ & (SUP.sup tsup ) | |
| 926 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x | |
| 927 sup {z} z≤x with trio< z px | |
| 928 ... | tri< a ¬b ¬c = record { tsup = ZChain.sup zc (o<→≤ a) ; tsup=sup = ZChain.supf-is-sup zc (o<→≤ a) } | |
| 929 ... | tri≈ ¬a b ¬c = record { tsup = ZChain.sup zc (o≤-refl0 b) ; tsup=sup = ZChain.supf-is-sup zc (o≤-refl0 b) } | |
| 865 | 930 ... | tri> ¬a ¬b px<z = zc35 where |
| 840 | 931 zc30 : z ≡ x |
| 932 zc30 with osuc-≡< z≤x | |
| 933 ... | case1 eq = eq | |
| 934 ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 865 | 935 zc32 = ZChain.sup zc o≤-refl |
| 936 zc34 : ¬ (supf0 px ≡ px) → {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup zc32) ∨ (w < SUP.sup zc32) | |
| 937 zc34 ne {w} lt with zc11 P (subst (λ k → odef (UnionCF A f mf ay supf0 k) (& w)) zc30 lt) | |
| 864 | 938 ... | case1 lt = SUP.x<sup zc32 lt |
| 865 | 939 ... | case2 ⟪ spx=px , fc ⟫ = ⊥-elim ( ne spx=px ) |
| 857 | 940 zc33 : supf0 z ≡ & (SUP.sup zc32) |
| 868 | 941 zc33 = ? -- trans (sym (supfx (o≤-refl0 (sym zc30)))) ( ZChain.supf-is-sup zc o≤-refl ) |
| 865 | 942 zc36 : ¬ (supf0 px ≡ px) → STMP z≤x |
| 943 zc36 ne = record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x<sup = zc34 ne } ; tsup=sup = zc33 } | |
| 944 zc35 : STMP z≤x | |
| 945 zc35 with trio< (supf0 px) px | |
| 946 ... | tri< a ¬b ¬c = zc36 ¬b | |
| 947 ... | tri> ¬a ¬b c = zc36 ¬b | |
| 948 ... | tri≈ ¬a b ¬c = record { tsup = zc37 ; tsup=sup = ? } where | |
| 949 zc37 : SUP A (UnionCF A f mf ay supf0 z) | |
| 950 zc37 = record { sup = ? ; as = ? ; x<sup = ? } | |
| 803 | 951 sup=u : {b : Ordinal} (ab : odef A b) → |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
952 b o≤ x → IsSup A (UnionCF A f mf ay supf0 b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf0 b) b f ) → supf0 b ≡ b |
| 814 | 953 sup=u {b} ab b≤x is-sup with trio< b px |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
954 ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) ⟪ record { x<sup = λ lt → IsSup.x<sup (proj1 is-sup) lt } , proj2 is-sup ⟫ |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
955 ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (o≤-refl0 b) ⟪ record { x<sup = λ lt → IsSup.x<sup (proj1 is-sup) lt } , proj2 is-sup ⟫ |
| 858 | 956 ... | tri> ¬a ¬b px<b = zc31 P where |
| 815 | 957 zc30 : x ≡ b |
| 958 zc30 with osuc-≡< b≤x | |
| 959 ... | case1 eq = sym (eq) | |
| 960 ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
| 859 | 961 zcsup : xSUP (UnionCF A f mf ay supf0 px) x |
| 962 zcsup with zc30 | |
| 963 ... | refl = record { ax = ab ; is-sup = record { x<sup = λ {w} lt → | |
| 863 | 964 IsSup.x<sup (proj1 is-sup) (zc10 lt)} } |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
965 zc31 : ( (¬ xSUP (UnionCF A f mf ay supf0 px) x ) ∨ HasPrev A (UnionCF A f mf ay supf0 px) x f) → supf0 b ≡ b |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
966 zc31 (case1 ¬sp=x) with zc30 |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
967 ... | refl = ⊥-elim (¬sp=x zcsup ) |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
968 zc31 (case2 hasPrev ) with zc30 |
| 863 | 969 ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev |
| 970 ; ay = zc10 (HasPrev.ay hasPrev) ; x=fy = HasPrev.x=fy hasPrev } ) | |
| 833 | 971 |
| 703 | 972 zc4 : ZChain A f mf ay x |
| 793 | 973 zc4 with ODC.∋-p O A (* x) |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
974 ... | no noax = no-extension (case1 ( λ s → noax (subst (λ k → odef A k ) (sym &iso) (xSUP.ax s) ))) -- ¬ A ∋ p, just skip |
| 836 | 975 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc ) x f ) |
| 703 | 976 -- we have to check adding x preserve is-max ZChain A y f mf x |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
977 ... | case1 pr = no-extension (case2 pr) -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 793 | 978 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc ) ax ) |
| 682 | 979 ... | case1 is-sup = -- x is a sup of zc |
| 868 | 980 record { supf = supf1 |
| 981 ; sup=u = {!!} ; supf-mono = {!!} | |
| 861 | 982 ; sup = {!!} ; supf-is-sup = {!!} } where |
| 857 | 983 supf1 : Ordinal → Ordinal |
| 984 supf1 z with trio< z px | |
| 862 | 985 ... | tri< a ¬b ¬c = ZChain.supf zc z |
| 986 ... | tri≈ ¬a b ¬c = ZChain.supf zc z | |
| 987 ... | tri> ¬a ¬b c = x --- SUP A (UnionCF A f mf ay supf0 px) ≡ SUP A (UnionCF A f mf ay supf1 px) | |
| 861 | 988 |
| 989 pchainx : HOD | |
| 990 pchainx = UnionCF A f mf ay supf1 x | |
| 991 | |
| 862 | 992 supf0px=x : (ax : odef A x) → IsSup A (ZChain.chain zc ) ax → x ≡ & (SUP.sup (ZChain.sup zc o≤-refl ) ) |
| 993 supf0px=x is-sup = ? where | |
| 861 | 994 zc50 : supf0 px ≡ & (SUP.sup (ZChain.sup zc o≤-refl ) ) |
| 862 | 995 zc50 = ZChain.supf-is-sup zc ? |
| 861 | 996 |
| 997 supf-monox : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b | |
| 998 supf-monox {i} {j} i≤j with trio< i px | trio< j px | |
| 999 ... | tri< a ¬b ¬c | tri< ja ¬jb ¬jc = ? | |
| 1000 ... | tri< a ¬b ¬c | tri≈ ¬ja jb ¬jc = ? | |
| 1001 ... | tri< a ¬b ¬c | tri> ¬ja ¬jb jc = ? | |
| 1002 ... | tri≈ ¬a b ¬c | tri< ja ¬jb ¬jc = ? | |
| 1003 ... | tri≈ ¬a b ¬c | tri≈ ¬ja jb ¬jc = ? | |
| 1004 ... | tri≈ ¬a b ¬c | tri> ¬ja ¬jb jc = ? | |
| 1005 ... | tri> ¬a ¬b c | tri< ja ¬jb ¬jc = ? | |
| 1006 ... | tri> ¬a ¬b c | tri≈ ¬ja jb ¬jc = ? | |
| 1007 ... | tri> ¬a ¬b c | tri> ¬ja ¬jb jc = ? | |
| 1008 | |
| 1009 pchain⊆x : UnionCF A f mf ay supf0 px ⊆' UnionCF A f mf ay supf1 x | |
| 1010 pchain⊆x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ | |
| 1011 pchain⊆x ⟪ au , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ au , ch-is-sup u ? ? ? ⟫ | |
| 1012 | |
| 1013 supfx1 : {z : Ordinal } → x o≤ z → supf1 z ≡ x | |
| 1014 supfx1 {z} x≤z with trio< z px | |
| 1015 ... | tri< a ¬b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → z o< k ) (Oprev.oprev=x op) (ordtrans a <-osuc ))) | |
| 1016 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) (pxo<x op))) | |
| 1017 ... | tri> ¬a ¬b c = refl | |
| 1018 | |
| 862 | 1019 pchain0=x : UnionCF A f mf ay supf0 px ≡ UnionCF A f mf ay supf1 px |
| 1020 pchain0=x = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
| 1021 zc10 : {z : Ordinal} → OD.def (od pchain) z → odef (UnionCF A f mf ay supf1 px) z | |
| 1022 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 1023 zc10 {z} ⟪ az , ch-is-sup u1 u1≤x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ | |
| 1024 zc11 : {z : Ordinal} → odef (UnionCF A f mf ay supf1 px) z → OD.def (od pchain) z | |
| 1025 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 1026 zc11 {z} ⟪ az , ch-is-sup u1 u1≤x u1-is-sup fc ⟫ = ? | |
| 1027 | |
| 793 | 1028 |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1029 ... | case2 ¬x=sup = no-extension (case1 nsup) where -- px is not f y' nor sup of former ZChain from y -- no extention |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1030 nsup : ¬ xSUP (UnionCF A f mf ay supf0 px) x |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1031 nsup s = ¬x=sup z12 where |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1032 z12 : IsSup A (UnionCF A f mf ay supf0 px) ax |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1033 z12 = record { x<sup = λ {z} lt → subst (λ k → (z ≡ k) ∨ (z << k )) (sym &iso) ( IsSup.x<sup ( xSUP.is-sup s ) lt ) } |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1034 |
| 728 | 1035 ... | no lim = zc5 where |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1036 |
| 703 | 1037 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
| 1038 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
|
1039 |
| 835 | 1040 ysp = & (SUP.sup (ysup f mf ay)) |
| 755 | 1041 |
| 837 | 1042 initial-segment0 : {a b z : Ordinal } → (a<x : a o< x) ( b<x : b o< x) → a o< b → z o≤ a |
| 836 | 1043 → ZChain.supf (pzc (osuc a) (ob<x lim a<x )) z ≡ ZChain.supf (pzc (osuc b) (ob<x lim b<x )) z |
| 837 | 1044 initial-segment0 = ? |
| 834 | 1045 |
| 835 | 1046 supf0 : Ordinal → Ordinal |
| 1047 supf0 z with trio< z x | |
| 1048 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 1049 ... | tri≈ ¬a b ¬c = ysp |
| 1050 ... | tri> ¬a ¬b c = ysp | |
| 835 | 1051 |
| 840 | 1052 |
| 1053 -- Union of UnionCF z, z o< x undef initial-segment condition | |
| 1054 -- this is not a ZChain because supf0 is not monotonic | |
| 838 | 1055 pchain : HOD |
| 1056 pchain = UnionCF A f mf ay supf0 x | |
| 835 | 1057 |
| 838 | 1058 ptotal0 : IsTotalOrderSet pchain |
| 835 | 1059 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 1060 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 1061 uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb)) | |
| 844 | 1062 |
| 838 | 1063 usup : SUP A pchain |
| 1064 usup = supP pchain (λ lt → proj1 lt) ptotal0 | |
| 835 | 1065 spu = & (SUP.sup usup) |
| 834 | 1066 |
| 794 | 1067 supf1 : Ordinal → Ordinal |
| 835 | 1068 supf1 z with trio< z x |
| 1069 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 1070 ... | tri≈ ¬a b ¬c = spu |
| 1071 ... | tri> ¬a ¬b c = spu | |
| 755 | 1072 |
| 838 | 1073 pchain1 : HOD |
| 1074 pchain1 = UnionCF A f mf ay supf1 x | |
| 704 | 1075 |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1076 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
|
1077 b o< x → (ab : odef A b) → |
| 836 | 1078 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
|
1079 * 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
|
1080 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
|
1081 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 791 | 1082 ... | ⟪ 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
|
1083 subst (λ k → UChain A f mf ay supf x k ) |
| 794 | 1084 (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
|
1085 |
| 844 | 1086 zc70 : HasPrev A pchain x f → ¬ xSUP pchain x |
| 1087 zc70 pr xsup = ? | |
| 1088 | |
| 1089 no-extension : ¬ ( xSUP (UnionCF A f mf ay supf0 x) x ) → ZChain A f mf ay x | |
| 870 | 1090 no-extension ¬sp=x = record { supf = supf1 ; sup=u = sup=u |
| 868 | 1091 ; sup = sup ; supf-is-sup = sis ; supf-mono = {!!} ; asupf = ? } where |
|
795
408e7e8a3797
csupf depends on order cyclicly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
794
diff
changeset
|
1092 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
|
1093 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z |
| 838 | 1094 pchain0=1 : pchain ≡ pchain1 |
| 1095 pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
| 1096 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z | |
| 1097 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 1098 zc10 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc12 fc where | |
| 1099 zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z | |
| 1100 zc12 (fsuc x fc) with zc12 fc | |
| 1101 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 1102 ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ | |
| 1103 zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u u≤x ? (init ? ? ) ⟫ | |
| 1104 zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z | |
| 1105 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 863 | 1106 zc11 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc13 fc where |
| 838 | 1107 zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z |
| 1108 zc13 (fsuc x fc) with zc13 fc | |
| 1109 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 863 | 1110 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ |
| 838 | 1111 zc13 (init asu su=z ) with trio< u x |
| 863 | 1112 ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u u<x ? (init ? ? ) ⟫ |
| 838 | 1113 ... | tri≈ ¬a b ¬c = ? |
| 863 | 1114 ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u<x c ) |
| 832 | 1115 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 797 | 1116 sup {z} z≤x with trio< z x |
| 838 | 1117 ... | tri< a ¬b ¬c = SUP⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} ) |
| 815 | 1118 ... | tri≈ ¬a b ¬c = {!!} |
| 843 | 1119 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 832 | 1120 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup {!!})) |
| 843 | 1121 sis {z} z≤x with trio< z x |
| 800 | 1122 ... | tri< a ¬b ¬c = {!!} where |
| 815 | 1123 zc8 = ZChain.supf-is-sup (pzc z a) {!!} |
| 1124 ... | tri≈ ¬a b ¬c = {!!} | |
| 843 | 1125 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1126 sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsSup A (UnionCF A f mf ay supf1 b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf1 b) b f ) → supf1 b ≡ b |
| 843 | 1127 sup=u {z} ab z≤x is-sup with trio< z x |
| 833 | 1128 ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x<sup = {!!} } |
| 815 | 1129 ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?) ?) ab {!!} record { x<sup = {!!} } |
| 843 | 1130 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 797 | 1131 |
| 703 | 1132 zc5 : ZChain A f mf ay x |
| 697 | 1133 zc5 with ODC.∋-p O A (* x) |
| 796 | 1134 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
| 836 | 1135 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain x f ) |
| 703 | 1136 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 796 | 1137 ... | case1 pr = no-extension {!!} |
| 704 | 1138 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A pchain ax ) |
| 861 | 1139 ... | case1 is-sup = record { supf = supf1 ; sup=u = {!!} |
| 868 | 1140 ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!}; asupf = {!!} } -- where -- x is a sup of (zc ?) |
| 796 | 1141 ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention |
| 553 | 1142 |
| 703 | 1143 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay (& A) |
| 1144 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
|
1145 |
| 551 | 1146 zorn00 : Maximal A |
| 1147 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 804 | 1148 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
| 551 | 1149 -- yes we have the maximal |
| 1150 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 1151 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 1152 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 1153 zorn01 = proj1 zorn03 | |
| 1154 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 1155 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 703 | 1156 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04 total ) where |
| 551 | 1157 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 1158 nmx : ¬ Maximal A | |
| 1159 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 1160 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 804 | 1161 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 | 1162 zorn04 : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) |
| 653 | 1163 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
| 634 | 1164 total : IsTotalOrderSet (ZChain.chain zorn04) |
| 654 | 1165 total {a} {b} = zorn06 where |
| 1166 zorn06 : odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a) | |
| 1167 zorn06 = ZChain.f-total (SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as) ) | |
| 551 | 1168 |
| 516 | 1169 -- usage (see filter.agda ) |
| 1170 -- | |
| 497 | 1171 -- _⊆'_ : ( A B : HOD ) → Set n |
| 1172 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 1173 |
| 497 | 1174 -- MaximumSubset : {L P : HOD} |
| 1175 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 1176 -- → IsPartialOrderSet P _⊆'_ | |
| 1177 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 1178 -- → Maximal P (_⊆'_) | |
| 1179 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |