Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn1.agda @ 933:409ac0af7b3b
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 24 Oct 2022 09:15:49 +0900 |
| parents | b1899e33e2c7 |
| children |
| 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 |
| 930 | 8 module zorn1 {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 | |
| 872 | 55 _<<_ : (x y : Ordinal ) → Set n |
| 570 | 56 x << y = * x < * y |
| 57 | |
| 872 | 58 _<=_ : (x y : Ordinal ) → Set n -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain |
| 765 | 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
|
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 | |
| 879 | 100 -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A |
| 101 -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x | |
| 102 -- <-TransFinite = ? | |
| 103 | |
| 530 | 104 -- |
| 560 | 105 -- Closure of ≤-monotonic function f has total order |
| 530 | 106 -- |
| 107 | |
| 108 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 109 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 110 | |
| 551 | 111 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 783 | 112 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
| 555 | 113 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 114 |
| 556 | 115 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 | 116 A∋fc {A} s f mf (init as refl ) = as |
| 556 | 117 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 118 |
| 714 | 119 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 | 120 A∋fcs {A} s f mf (init as refl) = as |
| 714 | 121 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
| 122 | |
| 556 | 123 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 | 124 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
| 556 | 125 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 126 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 127 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 128 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 129 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 130 |
| 800 | 131 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 132 fcn s mf (init as refl) = zero | |
| 133 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
| 134 ... | case1 eq = fcn s mf p | |
| 135 ... | case2 y<fy = suc (fcn s mf p ) | |
| 136 | |
| 137 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 138 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 139 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 | |
| 140 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 141 fc06 {x} {y} refl {j} not = fc08 not where | |
| 142 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 143 fc08 () | |
| 144 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
| 145 fc07 {x} (init as refl) eq = refl | |
| 146 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 147 ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) | |
| 148 -- ... | case2 x<fx = ? | |
| 149 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 | |
| 150 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
| 151 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
| 152 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
| 153 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 154 ... | 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 ) | |
| 155 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 156 ... | 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 ) | |
| 157 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 ) ) | |
| 158 ... | 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 ) | |
| 159 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 ) ) | |
| 160 ... | 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 ) | |
| 161 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 162 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 163 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 164 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 165 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x | |
| 166 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
| 167 fc04 : * x1 ≡ * y | |
| 168 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 169 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 170 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 171 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 172 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 173 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 174 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 175 fc05 : * x ≡ * y1 | |
| 176 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 177 ... | 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))) | |
| 178 | |
| 179 | |
| 180 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 181 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 182 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
| 183 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 184 fc06 {x} {y} refl {j} not = fc08 not where | |
| 185 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 186 fc08 () | |
| 187 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 | |
| 188 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
| 189 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 190 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 191 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 192 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 193 ... | 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 | |
| 194 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 195 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 196 fc03 eq = cong pred eq | |
| 197 fc02 : * x < * y1 | |
| 198 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 199 | |
| 557 | 200 |
| 559 | 201 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 202 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 800 | 203 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 204 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 205 fc11 : * x < * y | |
| 206 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 207 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 208 fc10 : * x ≡ * y | |
| 209 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 210 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 211 fc12 : * y < * x | |
| 212 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 600 | 213 |
| 563 | 214 |
| 729 | 215 |
| 560 | 216 -- open import Relation.Binary.Properties.Poset as Poset |
| 217 | |
| 218 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 219 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 220 | |
| 567 | 221 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 222 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 223 |
| 568 | 224 _⊆'_ : ( A B : HOD ) → Set n |
| 225 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 226 |
| 227 -- | |
| 228 -- inductive maxmum tree from x | |
| 229 -- tree structure | |
| 230 -- | |
| 554 | 231 |
| 836 | 232 record HasPrev (A B : HOD) (x : Ordinal ) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 233 field |
| 836 | 234 ax : odef A x |
| 534 | 235 y : Ordinal |
| 541 | 236 ay : odef B y |
| 534 | 237 x=fy : x ≡ f y |
| 529 | 238 |
| 570 | 239 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 654 | 240 field |
| 779 | 241 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 568 | 242 |
| 656 | 243 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 244 field | |
| 245 sup : HOD | |
| 804 | 246 as : A ∋ sup |
| 656 | 247 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive |
| 248 | |
| 690 | 249 -- |
| 250 -- sup and its fclosure is in a chain HOD | |
| 251 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 252 -- whole chain is a union of separated Chain | |
| 803 | 253 -- minimum index is sup of y not ϕ |
| 690 | 254 -- |
| 255 | |
| 787 | 256 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 | 257 field |
| 765 | 258 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
| 828 | 259 order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
| 260 supu=u : supf u ≡ u | |
| 694 | 261 |
| 748 | 262 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 263 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where | |
| 264 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 919 | 265 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : supf u o< supf x) ( is-sup : ChainP A f mf ay supf u ) |
| 748 | 266 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z |
| 694 | 267 |
| 878 | 268 -- |
| 269 -- f (f ( ... (sup y))) f (f ( ... (sup z1))) | |
| 270 -- / | / | | |
| 271 -- / | / | | |
| 272 -- sup y < sup z1 < sup z2 | |
| 273 -- o< o< | |
| 861 | 274 -- data UChain is total |
| 275 | |
| 276 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) | |
| 277 {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 ) | |
| 278 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where | |
| 279 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) | |
| 280 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb | |
| 281 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca | |
| 282 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 283 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 284 ct00 : * a ≡ * b | |
| 285 ct00 = trans (cong (*) eq) eq1 | |
| 286 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where | |
| 287 ct01 : * a < * b | |
| 288 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
| 289 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 | |
| 290 ct00 : * a < * (supf ub) | |
| 291 ct00 = lt | |
| 292 ct01 : * a < * b | |
| 293 ct01 with s≤fc (supf ub) f mf fcb | |
| 294 ... | case1 eq = subst (λ k → * a < k ) eq ct00 | |
| 295 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 296 ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb | |
| 297 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 298 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 299 ct00 : * a ≡ * b | |
| 300 ct00 = sym (trans (cong (*) eq) eq1 ) | |
| 301 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where | |
| 302 ct01 : * b < * a | |
| 303 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
| 304 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 | |
| 305 ct00 : * b < * (supf ua) | |
| 306 ct00 = lt | |
| 307 ct01 : * b < * a | |
| 308 ct01 with s≤fc (supf ua) f mf fca | |
| 309 ... | case1 eq = subst (λ k → * b < k ) eq ct00 | |
| 310 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 311 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 | |
| 312 ... | 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 | |
| 313 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 314 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 315 ct00 : * a ≡ * b | |
| 316 ct00 = trans (cong (*) eq) eq1 | |
| 317 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 318 ct02 : * a < * b | |
| 319 ct02 = subst (λ k → * k < * b ) (sym eq) lt | |
| 320 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 | |
| 321 ct03 : * a < * (supf ub) | |
| 322 ct03 = lt | |
| 323 ct02 : * a < * b | |
| 324 ct02 with s≤fc (supf ub) f mf fcb | |
| 325 ... | case1 eq = subst (λ k → * a < k ) eq ct03 | |
| 326 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 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 eq ¬c | |
| 328 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) | |
| 329 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 | |
| 330 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 331 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 332 ct00 : * a ≡ * b | |
| 333 ct00 = sym (trans (cong (*) eq) eq1) | |
| 334 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
| 335 ct02 : * b < * a | |
| 336 ct02 = subst (λ k → * k < * a ) (sym eq) lt | |
| 337 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 | |
| 338 ct05 : * b < * (supf ua) | |
| 339 ct05 = lt | |
| 340 ct04 : * b < * a | |
| 341 ct04 with s≤fc (supf ua) f mf fca | |
| 342 ... | case1 eq = subst (λ k → * b < k ) eq ct05 | |
| 343 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt | |
| 344 | |
| 694 | 345 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 346 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 347 | |
| 803 | 348 -- Union of supf z which o< x |
| 349 -- | |
| 694 | 350 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 351 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
| 352 UnionCF A f mf ay supf x | |
| 894 | 353 = 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 | 354 |
| 842 | 355 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) |
| 356 → supf x o< supf y → x o< y | |
| 357 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y | |
| 358 ... | tri< a ¬b ¬c = a | |
| 359 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 360 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 361 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 362 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 363 | |
| 879 | 364 record MinSUP ( A B : HOD ) : Set n where |
| 365 field | |
| 366 sup : Ordinal | |
| 367 asm : odef A sup | |
| 368 x<sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) | |
| 369 minsup : { sup1 : Ordinal } → odef A sup1 | |
| 370 → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 | |
| 371 | |
| 372 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A | |
| 373 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 374 | |
| 880 | 375 M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } |
| 376 → (supf : Ordinal → Ordinal ) | |
| 377 → MinSUP A (UnionCF A f mf ay supf x) | |
| 378 → SUP A (UnionCF A f mf ay supf x) | |
| 379 M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) | |
| 380 ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x<sup = ms00 } where | |
| 381 msup = MinSUP.sup ms | |
| 382 ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup) | |
| 383 ms00 {z} uz with MinSUP.x<sup ms uz | |
| 384 ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq)) | |
| 385 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso refl lt ) | |
| 386 | |
| 867 | 387 |
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388 chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 919 | 389 (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b |
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390 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c |
| 919 | 391 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = |
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392 ⟪ ua , ch-init fc ⟫ |
| 919 | 393 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = |
| 394 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (supf-mono a≤b ) ) is-sup fc ⟫ | |
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395 |
| 703 | 396 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 397 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 398 field |
| 694 | 399 supf : Ordinal → Ordinal |
| 880 | 400 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z |
| 401 → IsSup A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) b f) → supf b ≡ b | |
| 402 | |
| 868 | 403 asupf : {x : Ordinal } → odef A (supf x) |
| 880 | 404 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y |
| 405 supf-< : {x y : Ordinal } → supf x o< supf y → supf x << supf y | |
| 891 | 406 supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z |
| 880 | 407 |
| 408 minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) | |
| 891 | 409 supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z ) |
| 880 | 410 csupf : {b : Ordinal } → supf b o< z → odef (UnionCF A f mf ay supf z) (supf b) -- supf z is not an element of this chain |
| 411 | |
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412 chain : HOD |
| 703 | 413 chain = UnionCF A f mf ay supf z |
| 861 | 414 chain⊆A : chain ⊆' A |
| 415 chain⊆A = λ lt → proj1 lt | |
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416 |
| 879 | 417 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
| 880 | 418 sup {x} x≤z = M→S supf (minsup x≤z) |
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419 |
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420 s=ms : {x : Ordinal } → (x≤z : x o≤ z ) → & (SUP.sup (sup x≤z)) ≡ MinSUP.sup (minsup x≤z) |
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421 s=ms {x} x≤z = &iso |
| 878 | 422 |
| 861 | 423 chain∋init : odef chain y |
| 424 chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ | |
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425 f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a) |
| 861 | 426 f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
| 427 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 ) ⟫ | |
| 428 initial : {z : Ordinal } → odef chain z → * y ≤ * z | |
| 429 initial {a} ⟪ aa , ua ⟫ with ua | |
| 430 ... | ch-init fc = s≤fc y f mf fc | |
| 431 ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where | |
| 432 zc7 : y <= supf u | |
| 433 zc7 = ChainP.fcy<sup is-sup (init ay refl) | |
| 434 f-total : IsTotalOrderSet chain | |
| 435 f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 436 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 437 uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) | |
| 438 | |
| 871 | 439 supf-<= : {x y : Ordinal } → supf x <= supf y → supf x o≤ supf y |
| 440 supf-<= {x} {y} (case1 sx=sy) = o≤-refl0 sx=sy | |
| 441 supf-<= {x} {y} (case2 sx<sy) with trio< (supf x) (supf y) | |
| 442 ... | tri< a ¬b ¬c = o<→≤ a | |
| 443 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 444 ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) ) | |
| 445 | |
| 825 | 446 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
| 447 supf-inject {x} {y} sx<sy with trio< x y | |
| 448 ... | tri< a ¬b ¬c = a | |
| 449 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 450 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 451 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 452 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 798 | 453 |
| 872 | 454 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 |
| 891 | 455 fcy<sup {u} {w} u≤z fc with MinSUP.x<sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) |
| 798 | 456 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ |
| 892 | 457 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) |
| 458 ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) | |
| 825 | 459 |
| 871 | 460 -- ordering is not proved here but in ZChain1 |
| 756 | 461 |
| 728 | 462 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 463 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
| 869 | 464 supf = ZChain.supf zc |
| 728 | 465 field |
| 919 | 466 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → supf b o< supf z → (ab : odef A b) |
| 869 | 467 → HasPrev A (UnionCF A f mf ay supf z) b f ∨ IsSup A (UnionCF A f mf ay supf z) ab |
| 468 → * a < * b → odef ((UnionCF A f mf ay supf z)) b | |
| 728 | 469 |
| 568 | 470 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 471 field | |
| 472 maximal : HOD | |
| 804 | 473 as : A ∋ maximal |
| 568 | 474 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative |
| 567 | 475 |
| 743 | 476 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 477 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 783 | 478 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ |
| 743 | 479 |
| 497 | 480 Zorn-lemma : { A : HOD } |
| 464 | 481 → o∅ o< & A |
| 568 | 482 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 483 → Maximal A |
| 552 | 484 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 485 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 486 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 487 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 488 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 489 s : HOD |
| 490 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 491 as : A ∋ * ( & s ) |
| 492 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 )) ) | |
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493 as0 : odef A (& s ) |
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494 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 495 s<A : & s o< & A |
| 568 | 496 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 497 HasMaximal : HOD |
| 537 | 498 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 499 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 500 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 501 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 502 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 503 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 804 | 504 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
| 537 | 505 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
| 506 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)) | |
| 507 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 | |
| 508 ¬x<m : ¬ (* x < * m) | |
| 509 ¬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 | 510 |
| 879 | 511 minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B |
| 512 minsupP B B⊆A total = m02 where | |
| 513 xsup : (sup : Ordinal ) → Set n | |
| 514 xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) | |
| 515 ∀-imply-or : {A : Ordinal → Set n } {B : Set n } | |
| 516 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B | |
| 517 ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM | |
| 518 ∀-imply-or {A} {B} ∀AB | case1 t = case1 t | |
| 519 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where | |
| 520 lemma : ¬ ((x : Ordinal ) → A x) → B | |
| 521 lemma not with ODC.p∨¬p O B | |
| 522 lemma not | case1 b = b | |
| 523 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
| 524 m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B | |
| 525 m00 x = TransFinite0 ind x where | |
| 526 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) | |
| 527 → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B | |
| 528 ind x prev = ∀-imply-or m01 where | |
| 529 m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B | |
| 530 m01 z with trio< z x | |
| 531 ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 532 ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 533 ... | tri< a ¬b ¬c with prev z a | |
| 534 ... | case2 mins = case2 mins | |
| 535 ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) | |
| 536 ... | case1 mins = case2 record { sup = z ; asm = proj1 mins ; x<sup = proj2 mins ; minsup = m04 } where | |
| 537 m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 | |
| 538 m04 {s} as lt with trio< z s | |
| 539 ... | tri< a ¬b ¬c = o<→≤ a | |
| 540 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 541 ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) | |
| 542 ... | case2 notz = case1 (λ _ → notz ) | |
| 543 m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) | |
| 544 m03 not = ⊥-elim ( not s1 (z09 (SUP.as S)) ⟪ SUP.as S , m05 ⟫ ) where | |
| 545 S : SUP A B | |
| 546 S = supP B B⊆A total | |
| 547 s1 = & (SUP.sup S) | |
| 548 m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) | |
| 549 m05 {w} bw with SUP.x<sup S {* w} (subst (λ k → odef B k) (sym &iso) bw ) | |
| 550 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) | |
| 551 ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) | |
| 552 m02 : MinSUP A B | |
| 553 m02 = dont-or (m00 (& A)) m03 | |
| 554 | |
| 560 | 555 -- Uncountable ascending chain by axiom of choice |
| 530 | 556 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 557 cf nmx x with ODC.∋-p O A (* x) |
| 558 ... | no _ = o∅ | |
| 559 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 560 ... | yes nogt = -- no larger element, so it is maximal |
| 561 ⊥-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 | 562 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 563 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 564 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 565 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 566 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 567 ... | 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 ⟫ ) | |
| 568 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 569 |
| 570 --- | |
| 571 --- infintie ascention sequence of f | |
| 572 --- | |
| 530 | 573 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 574 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 575 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 576 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 577 |
| 803 | 578 -- |
| 579 -- Second TransFinite Pass for maximality | |
| 580 -- | |
| 581 | |
| 793 | 582 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 728 | 583 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
| 930 | 584 SZ1 f mf {y} ay zc x = ? |
| 727 | 585 |
| 757 | 586 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 587 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = | |
| 588 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 ))) } | |
| 589 | |
| 590 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 591 → IsTotalOrderSet (uchain f mf ay) | |
| 592 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 593 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 594 uz01 = fcn-cmp y f mf ca cb | |
| 595 | |
| 596 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 928 | 597 → MinSUP A (uchain f mf ay) |
| 598 ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) | |
| 757 | 599 |
| 793 | 600 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 804 | 601 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 | 602 |
| 833 | 603 record xSUP (B : HOD) (x : Ordinal) : Set n where |
| 604 field | |
| 605 ax : odef A x | |
| 606 is-sup : IsSup A B ax | |
| 607 | |
| 560 | 608 -- |
| 547 | 609 -- create all ZChains under o< x |
| 560 | 610 -- |
|
608
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611 |
| 674 | 612 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 613 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 930 | 614 ind f mf {y} ay x prev = ? |
| 553 | 615 |
| 921 | 616 --- |
| 617 --- the maximum chain has fix point of any ≤-monotonic function | |
| 618 --- | |
| 619 | |
| 620 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf ay x | |
| 621 SZ f mf {y} ay x = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) x | |
| 622 | |
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623 data ZChainP ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) |
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624 ( supf : Ordinal → Ordinal ) (z : Ordinal) : Set n where |
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625 zchain : (uz : Ordinal ) → odef (UnionCF A f mf ay supf uz) z → ZChainP f mf ay supf z |
| 925 | 626 |
| 627 auzc : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) | |
| 628 (supf : Ordinal → Ordinal ) → {x : Ordinal } → ZChainP f mf ay supf x → odef A x | |
| 629 auzc f mf {y} ay supf {x} (zchain uz ucf) = proj1 ucf | |
| 630 | |
| 926 | 631 zp-uz : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) |
| 632 (supf : Ordinal → Ordinal ) → {x : Ordinal } → ZChainP f mf ay supf x → Ordinal | |
| 633 zp-uz f mf ay supf (zchain uz _) = uz | |
| 634 | |
| 635 uzc⊆zc : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) | |
| 636 (supf : Ordinal → Ordinal ) → {x : Ordinal } → (zp : ZChainP f mf ay supf x ) → UChain A f mf ay supf (zp-uz f mf ay supf zp) x | |
| 637 uzc⊆zc f mf {y} ay supf {x} (zchain uz ⟪ ua , ch-init fc ⟫) = ch-init fc | |
| 638 uzc⊆zc f mf {y} ay supf {x} (zchain uz ⟪ ua , ch-is-sup u u<x is-sup fc ⟫) with ChainP.supu=u is-sup | |
| 639 ... | eq = ch-is-sup u u<x is-sup fc | |
| 640 | |
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641 UnionZF : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) |
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642 (supf : Ordinal → Ordinal ) → HOD |
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643 UnionZF f mf {y} ay supf = record { od = record { def = λ x → ZChainP f mf ay supf x } |
| 925 | 644 ; odmax = & A ; <odmax = λ lt → ∈∧P→o< ⟪ auzc f mf ay supf lt , lift true ⟫ } |
| 921 | 645 |
| 925 | 646 uzctotal : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) |
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647 → ( supf : Ordinal → Ordinal ) |
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648 → IsTotalOrderSet (UnionZF f mf ay supf ) |
| 926 | 649 uzctotal f mf ay supf {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (uz01 ca cb) where |
| 650 uz01 : {ua ub : Ordinal } → ZChainP f mf ay supf ua → ZChainP f mf ay supf ub | |
| 651 → Tri (* ua < * ub) (* ua ≡ * ub) (* ub < * ua ) | |
| 652 uz01 {ua} {ub} (zchain uza uca) (zchain uzb ucb) = chain-total A f mf ay supf (proj2 uca) (proj2 ucb) | |
| 921 | 653 |
| 931 | 654 msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
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655 → (zc : ZChain A f mf ay x ) |
| 931 | 656 → MinSUP A (UnionCF A f mf ay (ZChain.supf zc) x) |
| 657 msp0 f mf {x} ay zc = minsupP (UnionCF A f mf ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) ztotal where | |
| 922 | 658 ztotal : IsTotalOrderSet (ZChain.chain zc) |
| 659 ztotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 660 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 661 uz01 = chain-total A f mf ay (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb)) | |
| 921 | 662 |
| 931 | 663 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
| 664 → (zc : ZChain A f mf ay x ) | |
| 665 → SUP A (UnionCF A f mf ay (ZChain.supf zc) x) | |
| 666 sp0 f mf ay zc = M→S (ZChain.supf zc) (msp0 f mf ay zc ) | |
| 667 | |
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668 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 927 | 669 → ZChain.supf zc (& (SUP.sup (sp0 f mf as0 zc))) o< ZChain.supf zc (& A) |
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670 → f (& (SUP.sup (sp0 f mf as0 zc ))) ≡ & (SUP.sup (sp0 f mf as0 zc )) |
| 930 | 671 fixpoint f mf zc ss<sa = ? |
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672 |
| 921 | 673 |
| 674 -- ZChain contradicts ¬ Maximal | |
| 675 -- | |
| 676 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) | |
| 677 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
| 678 -- | |
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679 |
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680 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥ |
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681 z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.as sp1 )))) |
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682 (subst (λ k → odef A (& k)) (sym *iso) (SUP.as sp1) ) |
| 927 | 683 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc ss<sa ))) -- x ≡ f x ̄ |
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684 (proj1 (cf-is-<-monotonic nmx c (SUP.as sp1 ))) where -- x < f x |
| 927 | 685 supf = ZChain.supf zc |
| 931 | 686 msp1 : MinSUP A (ZChain.chain zc) |
| 687 msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc | |
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688 sp1 : SUP A (ZChain.chain zc) |
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689 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc |
| 931 | 690 c : Ordinal |
| 691 c = & ( SUP.sup sp1 ) | |
| 692 mc = MinSUP.sup msp1 | |
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693 c=mc : c ≡ mc |
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694 c=mc = &iso |
| 931 | 695 z20 : mc << cf nmx mc |
| 696 z20 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1) ) | |
| 697 asc : odef A (supf mc) | |
| 928 | 698 asc = ZChain.asupf zc |
| 699 spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) | |
| 700 spd = ysup (cf nmx) (cf-is-≤-monotonic nmx) asc | |
| 701 d = MinSUP.sup spd | |
| 702 d<A : d o< & A | |
| 703 d<A = ∈∧P→o< ⟪ MinSUP.asm spd , lift true ⟫ | |
| 929 | 704 msup : MinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) |
| 705 msup = ZChain.minsup zc (o<→≤ d<A) | |
| 928 | 706 sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) ) |
| 707 sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A) | |
| 930 | 708 -- z26 : {x : Ordinal } → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) x |
| 709 -- → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) c) x ∨ odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x | |
| 710 -- z26 = ? | |
| 929 | 711 is-sup : IsSup A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) (MinSUP.asm spd) |
| 712 is-sup = record { x<sup = z22 } where | |
| 713 z23 : {z : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z → (z ≡ MinSUP.sup spd) ∨ (z << MinSUP.sup spd) | |
| 714 z23 lt = MinSUP.x<sup spd lt | |
| 715 z22 : {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y → | |
| 716 (y ≡ MinSUP.sup spd) ∨ (y << MinSUP.sup spd) | |
| 930 | 717 z22 {a} ⟪ aa , ch-init fc ⟫ = ? |
| 718 z22 {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ? | |
| 719 -- u<x : ZChain.supf zc u o< ZChain.supf zc d | |
| 720 -- supf u o< spuf c → order | |
| 929 | 721 not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) d (cf nmx) |
| 722 not-hasprev hp = ? where | |
| 723 y : Ordinal | |
| 724 y = HasPrev.y hp | |
| 725 z24 : y << d | |
| 726 z24 = subst (λ k → y << k) (sym (HasPrev.x=fy hp)) ( proj1 (cf-is-<-monotonic nmx y (proj1 (HasPrev.ay hp) ) )) | |
| 930 | 727 -- z26 : {x : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x → (x ≡ d ) ∨ (x << d ) |
| 728 -- z26 lt with MinSUP.x<sup spd (subst (λ k → odef _ k ) ? lt) | |
| 729 -- ... | case1 eq = ? | |
| 730 -- ... | case2 lt = ? | |
| 731 -- z25 : {x : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x → (x ≡ y ) ∨ (x << y ) | |
| 732 -- z25 {x} (init au eq ) = ? -- sup c = x, cf y ≡ d, sup c =< d | |
| 733 -- z25 (fsuc x lt) = ? -- cf (sup c) | |
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734 |
| 928 | 735 sd=d : supf d ≡ d |
| 929 | 736 sd=d = ZChain.sup=u zc (MinSUP.asm spd) (o<→≤ d<A) ⟪ is-sup , not-hasprev ⟫ |
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737 |
| 933 | 738 sc<<d : {mc : Ordinal } → {asc : odef A (supf mc)} → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) |
| 739 → supf mc << MinSUP.sup spd | |
| 740 sc<<d {mc} {asc} spd = z25 where | |
| 741 d1 : Ordinal | |
| 742 d1 = MinSUP.sup spd | |
| 743 z24 : (supf mc ≡ d1) ∨ ( supf mc << d1 ) | |
| 744 z24 = MinSUP.x<sup spd (init asc refl) | |
| 745 z25 : supf mc << d1 | |
| 746 z25 with z24 | |
| 747 ... | case2 lt = lt | |
| 748 ... | case1 eq = ? | |
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749 |
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750 sc<sd : {mc d : Ordinal } → supf mc << supf d → supf mc o< supf d |
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751 sc<sd {mc} {d} sc<<sd with osuc-≡< ( ZChain.supf-<= zc (case2 sc<<sd ) ) |
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752 ... | case1 eq = ⊥-elim ( <-irr (case1 (cong (*) (sym eq) )) sc<<sd ) |
| 931 | 753 ... | case2 lt = lt |
| 754 | |
| 755 sms<sa : supf mc o< supf (& A) | |
| 756 sms<sa with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫) )) | |
| 757 ... | case2 lt = lt | |
| 933 | 758 ... | case1 eq = ⊥-elim ( o<¬≡ eq ( ordtrans<-≤ (sc<sd (subst (λ k → supf mc << k ) (sym sd=d) (sc<<d {mc} {asc} spd)) ) |
| 759 ( ZChain.supf-mono zc (o<→≤ d<A )))) | |
| 928 | 760 |
| 927 | 761 ss<sa : supf c o< supf (& A) |
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762 ss<sa = subst (λ k → supf k o< supf (& A)) (sym c=mc) sms<sa |
| 931 | 763 |
| 551 | 764 zorn00 : Maximal A |
| 765 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 804 | 766 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
| 551 | 767 -- yes we have the maximal |
| 768 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 769 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 770 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 771 zorn01 = proj1 zorn03 | |
| 772 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 773 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 927 | 774 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where |
| 551 | 775 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 776 nmx : ¬ Maximal A | |
| 777 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 778 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 804 | 779 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) ) ⟫ |
| 551 | 780 |
| 516 | 781 -- usage (see filter.agda ) |
| 782 -- | |
| 497 | 783 -- _⊆'_ : ( A B : HOD ) → Set n |
| 784 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 785 |
| 497 | 786 -- MaximumSubset : {L P : HOD} |
| 787 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 788 -- → IsPartialOrderSet P _⊆'_ | |
| 789 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 790 -- → Maximal P (_⊆'_) | |
| 791 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |