Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 625:886e1f82cca0
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 20 Jun 2022 08:43:23 +0900 |
| parents | d0938f220648 |
| children | 35d8aca1a2b7 |
| rev | line source |
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| 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 | |
| 46 open _∧_ | |
| 47 open _∨_ | |
| 48 open Bool | |
| 431 | 49 |
| 50 | |
| 51 open HOD | |
| 52 | |
| 560 | 53 -- |
| 54 -- Partial Order on HOD ( possibly limited in A ) | |
| 55 -- | |
| 56 | |
| 571 | 57 _<<_ : (x y : Ordinal ) → Set n -- Set n order |
| 570 | 58 x << y = * x < * y |
| 59 | |
| 60 POO : IsStrictPartialOrder _≡_ _<<_ | |
| 61 POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 62 ; trans = IsStrictPartialOrder.trans PO | |
| 63 ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y | |
| 64 ; <-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 } } | |
| 65 | |
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66 _≤_ : (x y : HOD) → Set (Level.suc n) |
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67 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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68 |
| 554 | 69 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 70 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 71 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 72 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 73 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 74 | |
| 556 | 75 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 76 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 77 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 78 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 79 |
| 561 | 80 ptrans = IsStrictPartialOrder.trans PO |
| 81 | |
| 492 | 82 open _==_ |
| 83 open _⊆_ | |
| 84 | |
| 530 | 85 -- |
| 560 | 86 -- Closure of ≤-monotonic function f has total order |
| 530 | 87 -- |
| 88 | |
| 89 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 90 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 91 | |
| 551 | 92 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 600 | 93 init : odef A s → FClosure A f s s |
| 555 | 94 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 95 |
| 556 | 96 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 |
| 600 | 97 A∋fc {A} s f mf (init as) = as |
| 556 | 98 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 99 |
| 556 | 100 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 600 | 101 s≤fc {A} s {.s} f mf (init x) = case1 refl |
| 556 | 102 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 103 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 104 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 105 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 106 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 107 |
| 557 | 108 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 600 | 109 fcn s mf (init as) = zero |
| 558 | 110 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) |
| 111 ... | case1 eq = fcn s mf p | |
| 112 ... | case2 y<fy = suc (fcn s mf p ) | |
| 557 | 113 |
| 558 | 114 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 115 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 559 | 116 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 |
| 117 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 | |
| 600 | 118 fc00 zero zero refl (init _) (init x₁) i=x i=y = refl |
| 119 fc00 zero zero refl (init as) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 120 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as) cy i=x i=y ) | |
| 121 fc00 zero zero refl (fsuc x cx) (init as) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 122 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as) i=x i=y ) | |
| 559 | 123 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 ) ) |
| 124 ... | 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 ) | |
| 125 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 ) ) | |
| 126 ... | 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 ) | |
| 127 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 128 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 129 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 560 | 130 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x |
| 559 | 131 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where |
| 132 fc04 : * x1 ≡ * y | |
| 133 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 134 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 135 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 136 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 137 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 138 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 139 fc05 : * x ≡ * y1 | |
| 140 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 141 ... | 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))) | |
| 557 | 142 |
| 600 | 143 |
| 557 | 144 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 145 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 558 | 146 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where |
| 147 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 | |
| 148 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 149 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 150 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 151 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 152 ... | 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 | |
| 153 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 154 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 155 fc03 eq = cong pred eq | |
| 156 fc02 : * x < * y1 | |
| 157 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 557 | 158 |
| 559 | 159 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 160 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 559 | 161 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 162 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 163 fc11 : * x < * y | |
| 164 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 165 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 166 fc10 : * x ≡ * y | |
| 167 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 168 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 169 fc12 : * y < * x | |
| 170 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 171 | |
| 600 | 172 |
| 562 | 173 fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 174 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
| 563 | 175 fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where |
| 176 fc20 : fcn s mf cy Data.Nat.< suc (fcn s mf cx) → (fcn s mf cy ≡ fcn s mf cx) ∨ ( fcn s mf cy Data.Nat.< fcn s mf cx ) | |
| 177 fc20 y<sx with <-cmp ( fcn s mf cy ) (fcn s mf cx ) | |
| 178 ... | tri< a ¬b ¬c = case2 a | |
| 179 ... | tri≈ ¬a b ¬c = case1 b | |
| 180 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> y<sx (s≤s c)) | |
| 181 fc17 : {x y : Ordinal } → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → suc (fcn s mf cx) ≡ fcn s mf cy → * (f x ) ≡ * y | |
| 182 fc17 {x} {y} cx cy sx=y = fc18 (fcn s mf cy) cx cy refl sx=y where | |
| 183 fc18 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → suc (fcn s mf cx) ≡ i → * (f x) ≡ * y | |
| 184 fc18 (suc i) {y} cx (fsuc y1 cy) i=y sx=i with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 185 ... | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy ( fc18 (suc i) {y1} cx cy i=y sx=i) -- dereference | |
| 186 ... | case2 y<fy = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k) ) ) fc19) where | |
| 187 fc19 : * x ≡ * y1 | |
| 188 fc19 = fcn-inject s mf cx cy (cong pred ( trans sx=i i=y )) | |
| 189 fc21 : ⊥ | |
| 190 fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy ) | |
| 191 ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy | |
| 192 cxx : FClosure A f s (f x) | |
| 193 cxx = fsuc x cx | |
| 194 fc16 : (x : Ordinal ) → (cx : FClosure A f s x) → (fcn s mf cx ≡ fcn s mf (fsuc x cx)) ∨ ( suc (fcn s mf cx ) ≡ fcn s mf (fsuc x cx)) | |
| 600 | 195 fc16 x (init as) with proj1 (mf s as ) |
| 563 | 196 ... | case1 _ = case1 refl |
| 197 ... | case2 _ = case2 refl | |
| 198 fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) | |
| 199 ... | case1 _ = case1 refl | |
| 200 ... | case2 _ = case2 refl | |
| 201 fc22 : (suc ( fcn s mf cx )) Data.Nat.< (fcn s mf cy ) → * (f x) < * y | |
| 202 fc22 a with fc16 x cx | |
| 203 ... | case1 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq (<-trans a<sa a)) | |
| 204 ... | case2 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq a ) | |
| 205 ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx | |
| 206 ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx | |
| 207 ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y | |
| 208 ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) | |
| 209 | |
| 560 | 210 -- open import Relation.Binary.Properties.Poset as Poset |
| 211 | |
| 212 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 213 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 214 | |
| 567 | 215 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 216 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 217 |
| 568 | 218 _⊆'_ : ( A B : HOD ) → Set n |
| 219 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 220 |
| 221 -- | |
| 222 -- inductive maxmum tree from x | |
| 223 -- tree structure | |
| 224 -- | |
| 554 | 225 |
| 567 | 226 record HasPrev (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 227 field |
| 534 | 228 y : Ordinal |
| 541 | 229 ay : odef B y |
| 534 | 230 x=fy : x ≡ f y |
| 529 | 231 |
| 570 | 232 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 567 | 233 field |
| 571 | 234 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 568 | 235 |
| 624 | 236 record ZChain ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) (supf : Ordinal → HOD) ( z : Ordinal ) : Set (Level.suc n) where |
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237 chain : HOD |
| 624 | 238 chain = supf z |
| 568 | 239 field |
| 240 chain⊆A : chain ⊆' A | |
| 241 chain∋x : odef chain x | |
| 242 initial : {y : Ordinal } → odef chain y → * x ≤ * y | |
| 243 f-next : {a : Ordinal } → odef chain a → odef chain (f a) | |
| 244 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
| 245 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc z → (ab : odef A b) | |
| 574 | 246 → HasPrev A chain ab f ∨ IsSup A chain ab |
| 568 | 247 → * a < * b → odef chain b |
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248 |
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249 record ZChain1 ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) ( z : Ordinal ) : Set (Level.suc n) where |
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250 field |
| 624 | 251 supf : Ordinal → HOD |
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252 zc : ZChain A x f supf z |
| 624 | 253 chain-mono : {x y : Ordinal} → x o≤ y → y o≤ z → supf x ⊆' supf y |
| 254 f-total : {x y : Ordinal} → x o≤ z → IsTotalOrderSet (supf x) | |
| 595 | 255 |
| 568 | 256 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 257 field | |
| 258 maximal : HOD | |
| 259 A∋maximal : A ∋ maximal | |
| 260 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 567 | 261 |
| 508 | 262 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 503 | 263 field |
| 264 sup : HOD | |
| 265 A∋maximal : A ∋ sup | |
| 266 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 267 | |
| 533 | 268 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
| 269 SupCond A B _ _ = SUP A B | |
| 270 | |
| 497 | 271 Zorn-lemma : { A : HOD } |
| 464 | 272 → o∅ o< & A |
| 568 | 273 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 274 → Maximal A |
| 552 | 275 Zorn-lemma {A} 0<A supP = zorn00 where |
| 568 | 276 supO : (C : HOD ) → C ⊆' A → IsTotalOrderSet C → Ordinal |
| 566 | 277 supO C C⊆A TC = & ( SUP.sup ( supP C C⊆A TC )) |
| 571 | 278 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 279 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 537 | 280 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 281 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 282 s : HOD |
| 283 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 284 as : A ∋ * ( & s ) |
| 285 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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286 as0 : odef A (& s ) |
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287 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 288 s<A : & s o< & A |
| 568 | 289 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 290 HasMaximal : HOD |
| 537 | 291 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 292 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 293 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 294 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 295 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 296 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 297 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 298 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 299 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)) | |
| 300 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 | |
| 301 ¬x<m : ¬ (* x < * m) | |
| 302 ¬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 | 303 |
| 560 | 304 -- Uncountable ascending chain by axiom of choice |
| 530 | 305 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 306 cf nmx x with ODC.∋-p O A (* x) |
| 307 ... | no _ = o∅ | |
| 308 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 309 ... | yes nogt = -- no larger element, so it is maximal |
| 310 ⊥-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 | 311 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 312 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 313 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 314 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 315 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 316 ... | 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 ⟫ ) | |
| 317 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 318 |
| 319 --- | |
| 320 --- infintie ascention sequence of f | |
| 321 --- | |
| 530 | 322 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 323 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 324 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 325 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 326 |
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327 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc1 : ZChain1 A (& s) f (& A) ) → SUP A (ZChain.chain (ZChain1.zc zc1)) |
| 625 | 328 zsup f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain1.f-total zc1 {& A} {& A} o≤-refl ) where |
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329 zc = ZChain1.zc zc1 |
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330 A∋zsup : (nmx : ¬ Maximal A ) (zc1 : ZChain1 A (& s) (cf nmx) (& A) ) |
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331 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc1 ))) |
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332 A∋zsup nmx zc1 = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc1 ) ) |
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333 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc1 : ZChain1 A (& s) f (& A) ) → SUP A (ZChain.chain (ZChain1.zc zc1)) |
| 625 | 334 sp0 f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain1.f-total zc1 {& A} {& A} o≤-refl ) where |
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335 zc = ZChain1.zc zc1 |
| 543 | 336 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 337 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 338 | |
| 339 --- | |
| 560 | 340 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 341 --- |
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342 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc1 : ZChain1 A (& s) f (& A) ) |
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343 → f (& (SUP.sup (sp0 f mf zc1 ))) ≡ & (SUP.sup (sp0 f mf zc1 )) |
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344 fixpoint f mf zc1 = z14 where |
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345 zc = ZChain1.zc zc1 |
| 538 | 346 chain = ZChain.chain zc |
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347 sp1 = sp0 f mf zc1 |
| 565 | 348 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc (& A) → (ab : odef A b ) |
| 570 | 349 → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 350 → * a < * b → odef chain b |
| 351 z10 = ZChain.is-max zc | |
| 543 | 352 z11 : & (SUP.sup sp1) o< & A |
| 353 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 354 z12 : odef chain (& (SUP.sup sp1)) |
| 355 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 356 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
| 569 | 357 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) (ordtrans z11 <-osuc ) (SUP.A∋maximal sp1) |
| 570 | 358 (case2 z19 ) z13 where |
| 538 | 359 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 566 | 360 z13 with SUP.x<sup sp1 ( ZChain.chain∋x zc ) |
| 538 | 361 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 362 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 570 | 363 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) |
| 571 | 364 z19 = record { x<sup = z20 } where |
| 365 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
| 366 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
| 570 | 367 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
| 368 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
| 369 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
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370 z14 : f (& (SUP.sup (sp0 f mf zc1))) ≡ & (SUP.sup (sp0 f mf zc1)) |
| 625 | 371 z14 with ZChain1.f-total zc1 {& A} {& A} o≤-refl (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 538 | 372 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 373 z16 : ⊥ | |
| 374 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 375 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 376 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 377 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 378 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 379 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 566 | 380 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) |
| 538 | 381 z17 : ⊥ |
| 382 z17 with z15 | |
| 383 ... | case1 eq = ¬b eq | |
| 384 ... | case2 lt = ¬a lt | |
| 560 | 385 |
| 386 -- ZChain contradicts ¬ Maximal | |
| 387 -- | |
| 571 | 388 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
| 560 | 389 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
| 390 -- | |
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391 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain1 A (& s) (cf nmx) (& A)) → ⊥ |
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392 z04 nmx zc1 = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
| 571 | 393 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) |
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394 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc1 ))) -- x ≡ f x ̄ |
| 571 | 395 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where -- x < f x |
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396 zc = ZChain1.zc zc1 |
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397 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc1 |
| 538 | 398 c = & (SUP.sup sp1) |
| 548 | 399 |
| 560 | 400 -- |
| 547 | 401 -- create all ZChains under o< x |
| 560 | 402 -- |
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403 |
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404 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → ((y : Ordinal) → y o< x → { y₁ : Ordinal} (ay : odef A y₁) |
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405 → ZChain1 A y₁ f y) → {y : Ordinal} (ay : odef A y) → ZChain1 A y f x |
| 547 | 406 ind f mf x prev {y} ay with Oprev-p x |
| 548 | 407 ... | yes op = zc4 where |
| 560 | 408 -- |
| 409 -- we have previous ordinal to use induction | |
| 410 -- | |
| 604 | 411 open ZChain |
| 412 | |
| 530 | 413 px = Oprev.oprev op |
| 624 | 414 supf : Ordinal → HOD |
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415 supf = ZChain1.supf (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay) |
| 610 | 416 zc1 : ZChain1 A y f (Oprev.oprev op) |
| 417 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay | |
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418 zc0 : ZChain A y f (ZChain1.supf (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay)) (Oprev.oprev op) |
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419 zc0 = ZChain1.zc (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay) |
| 569 | 420 zc0-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 421 zc0-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 604 | 422 px<x : px o< x |
| 423 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 569 | 424 |
| 611 | 425 -- if previous chain satisfies maximality, we caan reuse it |
| 426 -- | |
| 610 | 427 no-extenion : ( {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → |
| 428 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → | |
| 429 * a < * b → odef (ZChain.chain zc0) b ) → ZChain1 A y f x | |
| 430 no-extenion is-max = record { supf = supf0 ; zc = record { chain⊆A = subst (λ k → k ⊆' A ) seq (ZChain.chain⊆A zc0) | |
| 431 ; initial = subst (λ k → {y₁ : Ordinal} → odef k y₁ → * y ≤ * y₁ ) seq (ZChain.initial zc0) | |
| 432 ; f-next = subst (λ k → {a : Ordinal} → odef k a → odef k (f a) ) seq (ZChain.f-next zc0) | |
| 433 ; f-immediate = subst (λ k → {x₁ : Ordinal} {y₁ : Ordinal} → odef k x₁ → odef k y₁ → | |
| 434 ¬ (* x₁ < * y₁) ∧ (* y₁ < * (f x₁)) ) seq (ZChain.f-immediate zc0) ; chain∋x = subst (λ k → odef k y ) seq (ZChain.chain∋x zc0) | |
| 435 ; is-max = subst (λ k → {a b : Ordinal} → odef k a → b o< osuc x → (ab : odef A b) → | |
| 436 HasPrev A k ab f ∨ IsSup A k ab → * a < * b → odef k b ) seq is-max } | |
| 625 | 437 ; chain-mono = mono ; f-total = {!!} } where |
| 624 | 438 supf0 : Ordinal → HOD |
| 610 | 439 supf0 z with trio< z x |
| 440 ... | tri< a ¬b ¬c = supf z | |
| 624 | 441 ... | tri≈ ¬a b ¬c = ZChain.chain zc0 |
| 442 ... | tri> ¬a ¬b c = ZChain.chain zc0 | |
| 443 seq : ZChain.chain zc0 ≡ supf0 x | |
| 610 | 444 seq with trio< x x |
| 445 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) | |
| 624 | 446 ... | tri≈ ¬a b ¬c = refl |
| 447 ... | tri> ¬a ¬b c = refl | |
| 448 seq<x : {b : Ordinal } → b o< x → supf b ≡ supf0 b | |
| 611 | 449 seq<x {b} b<x with trio< b x |
| 450 ... | tri< a ¬b ¬c = refl | |
| 451 ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) | |
| 452 ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) | |
| 453 mono : {a b : Ordinal} → a o≤ b → b o< osuc x → | |
| 624 | 454 supf0 a ⊆' supf0 b |
| 611 | 455 mono {a} {b} a≤b b<ox with osuc-≡< a≤b |
| 456 ... | case1 refl = λ x → x | |
| 457 ... | case2 a<b with osuc-≡< b<ox | |
| 458 ... | case1 b=x = subst₂ (λ j k → j ⊆' k ) (seq<x a<x) nc00 ( ZChain1.chain-mono zc1 a≤px <-osuc ) where | |
| 459 a<x : a o< x | |
| 460 a<x with osuc-≡< b<ox | |
| 461 ... | case1 b=x = subst (λ k → a o< k ) b=x a<b | |
| 462 ... | case2 b<x = ordtrans a<b b<x | |
| 463 a≤px : a o≤ px | |
| 464 a≤px = subst (λ k → a o< k ) (sym (Oprev.oprev=x op)) a<x | |
| 624 | 465 nc00 : supf px ≡ supf0 b |
| 611 | 466 nc00 with trio< b x |
| 467 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b b=x ) | |
| 624 | 468 ... | tri≈ ¬a b ¬c = refl |
| 611 | 469 ... | tri> ¬a ¬b c = ⊥-elim ( ¬b b=x ) |
| 470 ... | case2 b<x = subst₂ (λ j k → j ⊆' k ) (seq<x a<x ) (seq<x b<x ) | |
| 471 ( ZChain1.chain-mono zc1 a≤b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x ) ) | |
| 472 where | |
| 473 a<x : a o< x | |
| 474 a<x with osuc-≡< b<ox | |
| 475 ... | case1 b=x = subst (λ k → a o< k ) b=x a<b | |
| 476 ... | case2 b<x = ordtrans a<b b<x | |
| 610 | 477 |
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478 zc4 : ZChain1 A y f x |
| 565 | 479 zc4 with ODC.∋-p O A (* x) |
| 611 | 480 ... | no noax = no-extenion zc11 where -- ¬ A ∋ p, just skip |
| 568 | 481 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → |
| 570 | 482 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → |
| 551 | 483 * a < * b → odef (ZChain.chain zc0) b |
| 568 | 484 zc11 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox |
| 485 ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) ab ) ) | |
| 569 | 486 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b |
| 582 | 487 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc0) ax f ) -- we have to check adding x preserve is-max ZChain A y f mf supO x |
| 611 | 488 ... | case1 pr = no-extenion zc17 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 604 | 489 chain0 = ZChain.chain zc0 |
| 568 | 490 zc17 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → |
| 570 | 491 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → |
| 551 | 492 * a < * b → odef (ZChain.chain zc0) b |
| 568 | 493 zc17 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox |
| 569 | 494 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b |
| 604 | 495 ... | case1 b=x = subst (λ k → odef chain0 k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr)) |
| 570 | 496 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax ) |
| 571 | 497 ... | case1 is-sup = -- x is a sup of zc0 |
| 625 | 498 record { zc = record { chain⊆A = {!!} ; f-next = {!!} |
| 499 ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} } ; supf = supf0 ; chain-mono = mono ; f-total = {!!} } where | |
| 570 | 500 sup0 : SUP A (ZChain.chain zc0) |
| 571 | 501 sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where |
| 502 x21 : {y : HOD} → ZChain.chain zc0 ∋ y → (y ≡ * x) ∨ (y < * x) | |
| 503 x21 {y} zy with IsSup.x<sup is-sup zy | |
| 504 ... | case1 y=x = case1 ( subst₂ (λ j k → j ≡ * k ) *iso &iso ( cong (*) y=x) ) | |
| 505 ... | case2 y<x = case2 (subst₂ (λ j k → j < * k ) *iso &iso y<x ) | |
| 570 | 506 sp : HOD |
| 561 | 507 sp = SUP.sup sup0 |
| 570 | 508 x=sup : x ≡ & sp |
| 509 x=sup = sym &iso | |
| 604 | 510 chain0 = ZChain.chain zc0 |
| 511 sc<A : {y : Ordinal} → odef chain0 y ∨ FClosure A f (& sp) y → y o< & A | |
| 512 sc<A {y} (case1 zx) = subst (λ k → k o< (& A)) &iso ( c<→o< (ZChain.chain⊆A zc0 (subst (λ k → odef chain0 k) (sym &iso) zx ))) | |
| 561 | 513 sc<A {y} (case2 fx) = subst (λ k → k o< (& A)) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso) (A∋fc (& sp) f mf fx )) ) |
| 552 | 514 schain : HOD |
| 604 | 515 schain = record { od = record { def = λ x → odef chain0 x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = λ {y} sy → sc<A {y} sy } |
| 561 | 516 A∋schain : {x : HOD } → schain ∋ x → A ∋ x |
| 569 | 517 A∋schain (case1 zx ) = ZChain.chain⊆A zc0 zx |
| 561 | 518 A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx |
| 569 | 519 s⊆A : schain ⊆' A |
| 520 s⊆A {x} (case1 zx) = ZChain.chain⊆A zc0 zx | |
| 521 s⊆A {x} (case2 fx) = A∋fc (& sp) f mf fx | |
| 604 | 522 cmp : {a b : HOD} (za : odef chain0 (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) |
| 561 | 523 cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb |
| 524 ... | case1 sp=a | case1 sp=b = tri≈ (λ lt → <-irr (case1 (sym eq)) lt ) eq (λ lt → <-irr (case1 eq) lt ) where | |
| 525 eq : a ≡ b | |
| 526 eq = trans sp=a (subst₂ (λ j k → j ≡ k ) *iso *iso sp=b ) | |
| 527 ... | case1 sp=a | case2 sp<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b) lt ) where | |
| 528 a<b : a < b | |
| 529 a<b = subst (λ k → k < b ) (sym sp=a) (subst₂ (λ j k → j < k ) *iso *iso sp<b ) | |
| 530 ... | case2 a<sp | case1 sp=b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b) lt ) where | |
| 531 a<b : a < b | |
| 532 a<b = subst (λ k → a < k ) (trans sp=b *iso ) (subst (λ k → a < k ) (sym *iso) a<sp ) | |
| 533 ... | case2 a<sp | case2 sp<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b) lt ) where | |
| 534 a<b : a < b | |
| 535 a<b = ptrans (subst (λ k → a < k ) (sym *iso) a<sp ) ( subst₂ (λ j k → j < k ) refl *iso sp<b ) | |
| 536 scmp : {a b : HOD} → odef schain (& a) → odef schain (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 625 | 537 scmp {a} {b} (case1 za) (case1 zb) = ZChain1.f-total zc1 {px} {px} o≤-refl za zb |
| 561 | 538 scmp {a} {b} (case1 za) (case2 fb) = cmp za fb |
| 539 scmp (case2 fa) (case1 zb) with cmp zb fa | |
| 540 ... | tri< a ¬b ¬c = tri> ¬c (λ eq → ¬b (sym eq)) a | |
| 541 ... | tri≈ ¬a b ¬c = tri≈ ¬c (sym b) ¬a | |
| 542 ... | tri> ¬a ¬b c = tri< c (λ eq → ¬b (sym eq)) ¬a | |
| 543 scmp (case2 fa) (case2 fb) = subst₂ (λ a b → Tri (a < b) (a ≡ b) (b < a ) ) *iso *iso (fcn-cmp (& sp) f mf fa fb) | |
| 544 scnext : {a : Ordinal} → odef schain a → odef schain (f a) | |
| 545 scnext {x} (case1 zx) = case1 (ZChain.f-next zc0 zx) | |
| 546 scnext {x} (case2 sx) = case2 ( fsuc x sx ) | |
| 547 scinit : {x : Ordinal} → odef schain x → * y ≤ * x | |
| 548 scinit {x} (case1 zx) = ZChain.initial zc0 zx | |
| 604 | 549 scinit {x} (case2 sx) with (s≤fc (& sp) f mf sx ) | SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) ( ZChain.chain∋x zc0 ) ) |
| 562 | 550 ... | case1 sp=x | case1 y=sp = case1 (trans y=sp (subst (λ k → k ≡ * x ) *iso sp=x ) ) |
| 551 ... | case1 sp=x | case2 y<sp = case2 (subst (λ k → * y < k ) (trans (sym *iso) sp=x) y<sp ) | |
| 552 ... | case2 sp<x | case1 y=sp = case2 (subst (λ k → k < * x ) (trans *iso (sym y=sp )) sp<x ) | |
| 553 ... | case2 sp<x | case2 y<sp = case2 (ptrans y<sp (subst (λ k → k < * x ) *iso sp<x) ) | |
| 604 | 554 A∋za : {a : Ordinal } → odef chain0 a → odef A a |
| 569 | 555 A∋za za = ZChain.chain⊆A zc0 za |
| 604 | 556 za<sup : {a : Ordinal } → odef chain0 a → ( * a ≡ sp ) ∨ ( * a < sp ) |
| 557 za<sup za = SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) za ) | |
| 562 | 558 simm : {a b : Ordinal} → odef schain a → odef schain b → ¬ (* a < * b) ∧ (* b < * (f a)) |
| 559 simm {a} {b} (case1 za) (case1 zb) = ZChain.f-immediate zc0 za zb | |
| 560 simm {a} {b} (case1 za) (case2 sb) p with proj1 (mf a (A∋za za) ) | |
| 561 ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) | |
| 562 ... | case2 a<fa with za<sup ( ZChain.f-next zc0 za ) | s≤fc (& sp) f mf sb | |
| 563 ... | case1 fa=sp | case1 sp=b = <-irr (case1 (trans fa=sp (trans (sym *iso) sp=b )) ) ( proj2 p ) | |
| 564 ... | case2 fa<sp | case1 sp=b = <-irr (case2 fa<sp) (subst (λ k → k < * (f a) ) (trans (sym sp=b) *iso) (proj2 p ) ) | |
| 565 ... | case1 fa=sp | case2 sp<b = <-irr (case2 (proj2 p )) (subst (λ k → k < * b) (sym fa=sp) (subst (λ k → k < * b ) *iso sp<b ) ) | |
| 566 ... | case2 fa<sp | case2 sp<b = <-irr (case2 (proj2 p )) (ptrans fa<sp (subst (λ k → k < * b ) *iso sp<b ) ) | |
| 567 simm {a} {b} (case2 sa) (case1 zb) p with proj1 (mf a ( subst (λ k → odef A k) &iso ( A∋schain (case2 (subst (λ k → FClosure A f (& sp) k ) (sym &iso) sa) )) ) ) | |
| 568 ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) | |
| 569 ... | case2 b<fb with s≤fc (& sp) f mf sa | za<sup zb | |
| 570 ... | case1 sp=a | case1 b=sp = <-irr (case1 (trans b=sp (trans (sym *iso) sp=a )) ) (proj1 p ) | |
| 571 ... | case1 sp=a | case2 b<sp = <-irr (case2 (subst (λ k → * b < k ) (trans (sym *iso) sp=a) b<sp ) ) (proj1 p ) | |
| 572 ... | case2 sp<a | case1 b=sp = <-irr (case2 (subst ( λ k → k < * a ) (trans *iso (sym b=sp)) sp<a )) (proj1 p ) | |
| 573 ... | case2 sp<a | case2 b<sp = <-irr (case2 (ptrans b<sp (subst (λ k → k < * a) *iso sp<a ))) (proj1 p ) | |
| 564 | 574 simm {a} {b} (case2 sa) (case2 sb) p = fcn-imm {A} (& sp) {a} {b} f mf sa sb p |
| 571 | 575 s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) |
| 576 → HasPrev A schain ab f ∨ IsSup A schain ab | |
| 569 | 577 → * a < * b → odef schain b |
| 571 | 578 s-ismax {a} {b} sa b<ox ab p a<b with osuc-≡< b<ox -- b is x? |
| 600 | 579 ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) |
| 571 | 580 s-ismax {a} {b} (case1 za) b<ox ab (case1 p) a<b | case2 b<x = z21 p where -- has previous |
| 568 | 581 z21 : HasPrev A schain ab f → odef schain b |
| 567 | 582 z21 record { y = y ; ay = (case1 zy) ; x=fy = x=fy } = |
| 569 | 583 case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) |
| 567 | 584 z21 record { y = y ; ay = (case2 sy) ; x=fy = x=fy } = subst (λ k → odef schain k) (sym x=fy) (case2 (fsuc y sy) ) |
| 571 | 585 s-ismax {a} {b} (case1 za) b<ox ab (case2 p) a<b | case2 b<x = case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case2 z22) a<b ) where -- previous sup |
| 586 z22 : IsSup A (ZChain.chain zc0) ab | |
| 587 z22 = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } | |
| 588 s-ismax {a} {b} (case2 sa) b<ox ab (case1 p) a<b | case2 b<x with HasPrev.ay p | |
| 604 | 589 ... | case1 zy = case1 (subst (λ k → odef chain0 k ) (sym (HasPrev.x=fy p)) (ZChain.f-next zc0 zy )) -- in previous closure of f |
| 571 | 590 ... | case2 sy = case2 (subst (λ k → FClosure A f (& (* x)) k ) (sym (HasPrev.x=fy p)) (fsuc (HasPrev.y p) sy )) -- in current closure of f |
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591 s-ismax {a} {b} (case2 sa) b<ox ab (case2 p) a<b | case2 b<x = case1 z23 where -- sup o< x is already in zc0 |
| 571 | 592 z24 : IsSup A schain ab → IsSup A (ZChain.chain zc0) ab |
| 593 z24 p = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } | |
| 604 | 594 z23 : odef chain0 b |
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595 z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋x zc0 ) |
| 604 | 596 ... | case1 y=b = subst (λ k → odef chain0 k ) y=b ( ZChain.chain∋x zc0 ) |
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597 ... | case2 y<b = ZChain.is-max zc0 (ZChain.chain∋x zc0 ) (zc0-b<x b b<x) ab (case2 (z24 p)) y<b |
| 624 | 598 supf0 : Ordinal → HOD |
| 611 | 599 supf0 z with trio< z x |
| 600 ... | tri< a ¬b ¬c = supf z | |
| 624 | 601 ... | tri≈ ¬a b ¬c = schain |
| 602 ... | tri> ¬a ¬b c = schain | |
| 603 seq : schain ≡ supf0 x | |
| 611 | 604 seq with trio< x x |
| 605 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) | |
| 624 | 606 ... | tri≈ ¬a b ¬c = refl |
| 607 ... | tri> ¬a ¬b c = refl | |
| 608 seq<x : {b : Ordinal } → b o< x → supf b ≡ supf0 b | |
| 611 | 609 seq<x {b} b<x with trio< b x |
| 610 ... | tri< a ¬b ¬c = refl | |
| 611 ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) | |
| 612 ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) | |
| 624 | 613 mono : {a b : Ordinal} → a o≤ b → b o< osuc x → supf0 a ⊆' supf0 b |
| 611 | 614 mono {a} {b} a≤b b<ox = {!!} |
| 615 | |
| 616 ... | case2 ¬x=sup = no-extenion z18 where -- x is not f y' nor sup of former ZChain from y -- no extention | |
| 568 | 617 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → |
| 570 | 618 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → |
| 552 | 619 * a < * b → odef (ZChain.chain zc0) b |
| 568 | 620 z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x |
| 569 | 621 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab p a<b |
| 565 | 622 ... | case1 b=x with p |
| 567 | 623 ... | case1 pr = ⊥-elim ( ¬fy<x record {y = HasPrev.y pr ; ay = HasPrev.ay pr ; x=fy = trans (trans &iso (sym b=x) ) (HasPrev.x=fy pr ) } ) |
| 571 | 624 ... | case2 b=sup = ⊥-elim ( ¬x=sup record { |
| 625 x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy) } ) | |
| 625 | 626 ... | no ¬ox = record { supf = supf0 ; chain-mono = {!!} ; f-total = u-total |
| 627 ; zc = record { chain⊆A = {!!} ; f-next = {!!} | |
| 611 | 628 ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} } } where --- limit ordinal case |
| 554 | 629 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x |
| 553 | 630 field |
| 631 u : Ordinal | |
| 632 u<x : u o< x | |
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633 chain∋z : odef (ZChain.chain (ZChain1.zc (prev u u<x {y} ay ))) z |
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634 Uz⊆A : {z : Ordinal} → UZFChain z → odef A z |
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635 Uz⊆A {z} u = ZChain.chain⊆A (ZChain1.zc ( prev (UZFChain.u u) (UZFChain.u<x u) {y} ay )) (UZFChain.chain∋z u) |
| 609 | 636 uzc1 : {z : Ordinal} → (u : UZFChain z) → ZChain1 A y f (UZFChain.u u) |
| 637 uzc1 {z} u = prev (UZFChain.u u) (UZFChain.u<x u) {y} ay | |
| 638 uzc : {z : Ordinal} → (u : UZFChain z) → ZChain A y f (ZChain1.supf (uzc1 u)) (UZFChain.u u) | |
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639 uzc {z} u = ZChain1.zc (prev (UZFChain.u u) (UZFChain.u<x u) {y} ay) |
| 554 | 640 Uz : HOD |
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641 Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A |
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642 ; <odmax = λ lt → subst (λ k → k o< & A ) &iso (c<→o< (subst (λ k → odef A k ) (sym &iso) (Uz⊆A lt))) } |
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643 u-next : {z : Ordinal} → odef Uz z → odef Uz (f z) |
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644 u-next {z} u = record { u = UZFChain.u u ; u<x = UZFChain.u<x u ; chain∋z = ZChain.f-next ( uzc u ) (UZFChain.chain∋z u) } |
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645 u-initial : {z : Ordinal} → odef Uz z → * y ≤ * z |
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646 u-initial {z} u = ZChain.initial ( uzc u ) (UZFChain.chain∋z u) |
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647 u-chain∋x : odef Uz y |
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648 u-chain∋x = record { u = y ; u<x = {!!} ; chain∋z = ZChain.chain∋x (ZChain1.zc (prev y {!!} ay )) } |
| 624 | 649 supf0 : Ordinal → HOD |
| 611 | 650 supf0 z with trio< z x |
| 651 ... | tri< a ¬b ¬c = ZChain1.supf (prev z a {y} ay) z | |
| 624 | 652 ... | tri≈ ¬a b ¬c = Uz |
| 653 ... | tri> ¬a ¬b c = Uz | |
| 654 seq : Uz ≡ supf0 x | |
| 611 | 655 seq with trio< x x |
| 656 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) | |
| 624 | 657 ... | tri≈ ¬a b ¬c = refl |
| 658 ... | tri> ¬a ¬b c = refl | |
| 659 seq<x : {b : Ordinal } → (b<x : b o< x ) → ZChain1.supf (prev b b<x {y} ay) b ≡ supf0 b | |
| 611 | 660 seq<x {b} b<x with trio< b x |
| 624 | 661 ... | tri< a ¬b ¬c = cong (λ k → ZChain1.supf (prev b k {y} ay) b) o<-irr -- b<x ≡ a |
| 611 | 662 ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) |
| 663 ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) | |
| 625 | 664 u-mono : {z : Ordinal} {y : Ordinal} → z o≤ y → y o≤ x → supf0 z ⊆' supf0 y |
| 665 u-mono {z} {y} z≤y y≤x with trio< z x | trio< y x | |
| 666 ... | tri< a ¬b ¬c | t = {!!} | |
| 667 ... | tri≈ ¬a b ¬c | t = {!!} | |
| 668 ... | tri> ¬a ¬b c | t = ⊥-elim ( osuc-< y≤x {!!} ) | |
| 669 u-total : {z : Ordinal} → z o≤ x → IsTotalOrderSet (supf0 z) | |
| 670 u-total {z} z<x ux uy with trio< z x | |
| 671 ... | t = {!!} | |
| 672 -- with trio< (UZFChain.u ux) (UZFChain.u uy) | |
| 673 -- ... | tri< a ¬b ¬c = ZChain1.f-total (uzc1 uy) {!!} (u-mono (UZFChain.u ux) (UZFChain.u uy) | |
| 674 -- (UZFChain.u<x uy) (ordtrans a <-osuc ) (uzc1 ux) (uzc1 uy) (UZFChain.chain∋z ux)) (UZFChain.chain∋z uy) | |
| 675 -- ... | tri≈ ¬a b ¬c = ZChain1.f-total (uzc1 ux) {!!} (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) | |
| 676 -- (UZFChain.u<x ux) (subst (λ k → k o< osuc (UZFChain.u ux)) b <-osuc) (uzc1 uy) (uzc1 ux) (UZFChain.chain∋z uy)) | |
| 677 -- ... | tri> ¬a ¬b c = ZChain1.f-total (uzc1 ux) {!!} (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) | |
| 678 -- (UZFChain.u<x ux) (ordtrans c <-osuc) (uzc1 uy) (uzc1 ux) (UZFChain.chain∋z uy)) | |
| 553 | 679 |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
680 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ya : odef A y) → ZChain1 A y f (& A) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
681 SZ f mf = TransFinite {λ z → {y : Ordinal } → (ay : odef A y ) → ZChain1 A y f z } (ind f mf) (& A) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
682 |
| 551 | 683 zorn00 : Maximal A |
| 684 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 685 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 686 -- yes we have the maximal | |
| 687 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 688 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 689 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 690 zorn01 = proj1 zorn03 | |
| 691 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 692 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 693 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where | |
| 694 -- if we have no maximal, make ZChain, which contradict SUP condition | |
| 695 nmx : ¬ Maximal A | |
| 696 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 697 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 698 zc5 = ⟪ Maximal.A∋maximal mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫ | |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
699 zorn04 : ZChain1 A (& s) (cf nmx) (& A) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
700 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
| 551 | 701 |
| 516 | 702 -- usage (see filter.agda ) |
| 703 -- | |
| 497 | 704 -- _⊆'_ : ( A B : HOD ) → Set n |
| 705 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 706 |
| 497 | 707 -- MaximumSubset : {L P : HOD} |
| 708 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 709 -- → IsPartialOrderSet P _⊆'_ | |
| 710 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 711 -- → Maximal P (_⊆'_) | |
| 712 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |