Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 704:01a88eeb9d00
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 12 Jul 2022 23:05:31 +0900 |
| parents | 0278f0d151f2 |
| children | 799963f352d6 |
| rev | line source |
|---|---|
| 478 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 508 | 2 open import Level hiding ( suc ; zero ) |
| 431 | 3 open import Ordinals |
| 552 | 4 open import Relation.Binary |
| 5 open import Relation.Binary.Core | |
| 6 open import Relation.Binary.PropositionalEquality | |
| 497 | 7 import OD |
| 552 | 8 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where |
| 431 | 9 |
| 560 | 10 -- |
| 11 -- Zorn-lemma : { A : HOD } | |
| 12 -- → o∅ o< & A | |
| 13 -- → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition | |
| 14 -- → Maximal A | |
| 15 -- | |
| 16 | |
| 431 | 17 open import zf |
| 477 | 18 open import logic |
| 19 -- open import partfunc {n} O | |
| 20 | |
| 21 open import Relation.Nullary | |
| 22 open import Data.Empty | |
| 23 import BAlgbra | |
| 431 | 24 |
| 555 | 25 open import Data.Nat hiding ( _<_ ; _≤_ ) |
| 26 open import Data.Nat.Properties | |
| 27 open import nat | |
| 28 | |
| 431 | 29 |
| 30 open inOrdinal O | |
| 31 open OD O | |
| 32 open OD.OD | |
| 33 open ODAxiom odAxiom | |
| 477 | 34 import OrdUtil |
| 35 import ODUtil | |
| 431 | 36 open Ordinals.Ordinals O |
| 37 open Ordinals.IsOrdinals isOrdinal | |
| 38 open Ordinals.IsNext isNext | |
| 39 open OrdUtil O | |
| 477 | 40 open ODUtil O |
| 41 | |
| 42 | |
| 43 import ODC | |
| 44 | |
| 45 | |
| 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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TransitiveClosure with x <= f x is possible
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66 _≤_ : (x y : HOD) → Set (Level.suc n) |
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TransitiveClosure with x <= f x is possible
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67 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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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 |
| 654 | 233 field |
| 571 | 234 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 568 | 235 |
| 656 | 236 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 237 field | |
| 238 sup : HOD | |
| 239 A∋maximal : A ∋ sup | |
| 240 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 241 | |
| 690 | 242 -- |
| 243 -- sup and its fclosure is in a chain HOD | |
| 244 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 245 -- whole chain is a union of separated Chain | |
| 246 -- minimum index is y not ϕ | |
| 247 -- | |
| 248 | |
| 694 | 249 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (sup z : Ordinal) : Set n where |
| 690 | 250 field |
| 695 | 251 y-init : supf o∅ ≡ y |
| 252 asup : (x : Ordinal) → odef A (supf x) | |
| 694 | 253 fcy<sup : {z : Ordinal } → FClosure A f y z → z << supf sup |
| 695 | 254 csupz : FClosure A f (supf sup) z |
| 255 order : {sup1 z1 : Ordinal} → (lt : sup1 o< sup ) → FClosure A f (supf sup1 ) z1 → z1 << supf sup | |
| 694 | 256 |
| 257 data Chain (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) : Ordinal → Ordinal → Set n where | |
| 258 ch-init : (z : Ordinal) → FClosure A f y z → Chain A f mf ay supf y z | |
| 259 ch-is-sup : {sup z : Ordinal } | |
| 260 → ( is-sup : ChainP A f mf ay supf sup z) | |
| 261 → ( fc : FClosure A f (supf sup) z ) → Chain A f mf ay supf sup z | |
| 262 | |
| 263 -- Union of supf z which o< x | |
| 264 -- | |
| 690 | 265 |
| 694 | 266 record UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 267 (supf : Ordinal → Ordinal) (x : Ordinal) (z : Ordinal) : Set n where | |
| 268 field | |
| 269 u : Ordinal | |
| 270 u<x : u o< x | |
| 271 chain∋z : Chain A f mf ay supf u z | |
| 272 | |
| 273 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A | |
| 274 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 275 | |
| 276 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) | |
| 277 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
| 278 UnionCF A f mf ay supf x | |
| 279 = 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 | 280 |
| 703 | 281 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 282 {init : Ordinal} (ay : odef A init) ( z : Ordinal ) : Set (Level.suc n) where | |
| 655 | 283 field |
| 694 | 284 supf : Ordinal → Ordinal |
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mutual tranfinite in zorn
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285 chain : HOD |
| 703 | 286 chain = UnionCF A f mf ay supf z |
| 568 | 287 field |
| 288 chain⊆A : chain ⊆' A | |
| 653 | 289 chain∋init : odef chain init |
| 290 initial : {y : Ordinal } → odef chain y → * init ≤ * y | |
| 568 | 291 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 654 | 292 f-total : IsTotalOrderSet chain |
| 568 | 293 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc z → (ab : odef A b) |
| 574 | 294 → HasPrev A chain ab f ∨ IsSup A chain ab |
| 568 | 295 → * a < * b → odef chain b |
| 653 | 296 |
| 568 | 297 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 298 field | |
| 299 maximal : HOD | |
| 300 A∋maximal : A ∋ maximal | |
| 301 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 567 | 302 |
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303 -- data Chain (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) : Ordinal → Ordinal → Set n where |
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parents:
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304 -- |
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305 -- data Chain is total |
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306 |
| 694 | 307 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 308 {s s1 a b : Ordinal } ( ca : Chain A f mf ay supf s a ) ( cb : Chain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) | |
| 309 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where | |
| 310 ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → Chain A f mf ay supf xa a → Chain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) | |
|
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311 ct-ind xa xb {a} {b} (ch-init a fca) (ch-init b fcb) = fcn-cmp y f mf fca fcb |
| 690 | 312 ct-ind xa xb {a} {b} (ch-init a fca) (ch-is-sup supb fcb) = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where |
| 695 | 313 ct00 : * a < * (supf xb) |
| 314 ct00 = ChainP.fcy<sup supb fca | |
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315 ct01 : * a < * b |
| 695 | 316 ct01 with s≤fc (supf xb) f mf fcb |
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317 ... | case1 eq = subst (λ k → * a < k ) eq ct00 |
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318 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 690 | 319 ct-ind xa xb {a} {b} (ch-is-sup supa fca) (ch-init b fcb)= tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where |
| 695 | 320 ct00 : * b < * (supf xa) |
| 321 ct00 = ChainP.fcy<sup supa fcb | |
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322 ct01 : * b < * a |
| 695 | 323 ct01 with s≤fc (supf xa) f mf fca |
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324 ... | case1 eq = subst (λ k → * b < k ) eq ct00 |
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325 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 690 | 326 ct-ind xa xb {a} {b} (ch-is-sup supa fca) (ch-is-sup supb fcb) with trio< xa xb |
| 685 | 327 ... | tri< a₁ ¬b ¬c = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where |
| 695 | 328 ct03 : * a < * (supf xb) |
| 329 ct03 = ChainP.order supb a₁ (ChainP.csupz supa) | |
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330 ct02 : * a < * b |
| 695 | 331 ct02 with s≤fc (supf xb) f mf fcb |
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parents:
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332 ... | case1 eq = subst (λ k → * a < k ) eq ct03 |
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333 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt |
| 695 | 334 ... | tri≈ ¬a refl ¬c = fcn-cmp (supf xa) f mf fca fcb |
| 685 | 335 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where |
| 695 | 336 ct05 : * b < * (supf xa) |
| 337 ct05 = ChainP.order supa c (ChainP.csupz supb) | |
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338 ct04 : * b < * a |
| 695 | 339 ct04 with s≤fc (supf xa) f mf fca |
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parents:
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340 ... | case1 eq = subst (λ k → * b < k ) eq ct05 |
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Chain is not strictly positive
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parents:
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341 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt |
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parents:
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342 |
| 698 | 343 ChainP-next : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 344 → {x z : Ordinal } → ChainP A f mf ay supf x z → ChainP A f mf ay supf x (f z ) | |
| 345 ChainP-next A f mf {y} ay supf {x} {z} cp = record { y-init = ChainP.y-init cp ; asup = ChainP.asup cp | |
| 346 ; fcy<sup = ChainP.fcy<sup cp ; csupz = fsuc _ (ChainP.csupz cp) ; order = ChainP.order cp } | |
| 347 | |
| 497 | 348 Zorn-lemma : { A : HOD } |
| 464 | 349 → o∅ o< & A |
| 568 | 350 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 351 → Maximal A |
| 552 | 352 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 353 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 354 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 537 | 355 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 356 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 357 s : HOD |
| 358 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 359 as : A ∋ * ( & s ) |
| 360 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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361 as0 : odef A (& s ) |
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362 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 363 s<A : & s o< & A |
| 568 | 364 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 365 HasMaximal : HOD |
| 537 | 366 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 367 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 368 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 369 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 370 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 371 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 372 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 373 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 374 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)) | |
| 375 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 | |
| 376 ¬x<m : ¬ (* x < * m) | |
| 377 ¬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 | 378 |
| 560 | 379 -- Uncountable ascending chain by axiom of choice |
| 530 | 380 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 381 cf nmx x with ODC.∋-p O A (* x) |
| 382 ... | no _ = o∅ | |
| 383 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 384 ... | yes nogt = -- no larger element, so it is maximal |
| 385 ⊥-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 | 386 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 387 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 388 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 389 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 390 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 391 ... | 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 ⟫ ) | |
| 392 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 393 |
| 394 --- | |
| 395 --- infintie ascention sequence of f | |
| 396 --- | |
| 530 | 397 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 398 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 399 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 400 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 401 |
| 703 | 402 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 653 | 403 (total : IsTotalOrderSet (ZChain.chain zc) ) → SUP A (ZChain.chain zc) |
| 703 | 404 sp0 f mf zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total |
| 543 | 405 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 406 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 407 | |
| 408 --- | |
| 560 | 409 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 410 --- |
| 703 | 411 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
|
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ZChain1 is not strictly positive
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parents:
631
diff
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412 → (total : IsTotalOrderSet (ZChain.chain zc) ) |
| 703 | 413 → f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total)) |
| 414 fixpoint f mf zc total = z14 where | |
| 538 | 415 chain = ZChain.chain zc |
| 703 | 416 sp1 = sp0 f mf zc total |
| 565 | 417 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc (& A) → (ab : odef A b ) |
| 570 | 418 → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 419 → * a < * b → odef chain b |
| 420 z10 = ZChain.is-max zc | |
| 543 | 421 z11 : & (SUP.sup sp1) o< & A |
| 422 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 423 z12 : odef chain (& (SUP.sup sp1)) |
| 424 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 653 | 425 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 426 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) (ordtrans z11 <-osuc ) (SUP.A∋maximal sp1) | |
| 570 | 427 (case2 z19 ) z13 where |
| 538 | 428 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 653 | 429 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc ) |
| 538 | 430 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 431 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 570 | 432 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) |
| 571 | 433 z19 = record { x<sup = z20 } where |
| 434 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
| 435 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
| 570 | 436 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
| 437 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
| 438 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
| 703 | 439 z14 : f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total )) |
|
633
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ZChain1 is not strictly positive
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parents:
631
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440 z14 with total (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 631 | 441 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 442 z16 : ⊥ | |
| 443 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 444 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 445 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 446 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 447 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 448 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 449 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) | |
| 450 z17 : ⊥ | |
| 451 z17 with z15 | |
| 452 ... | case1 eq = ¬b eq | |
| 453 ... | case2 lt = ¬a lt | |
| 560 | 454 |
| 455 -- ZChain contradicts ¬ Maximal | |
| 456 -- | |
| 571 | 457 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
| 560 | 458 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
| 459 -- | |
| 697 | 460 z04 : (nmx : ¬ Maximal A ) |
| 703 | 461 → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) |
| 664 | 462 → IsTotalOrderSet (ZChain.chain zc) → ⊥ |
| 703 | 463 z04 nmx zc total = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal sp1 )))) |
| 571 | 464 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) |
| 703 | 465 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc total ))) -- x ≡ f x ̄ |
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633
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parents:
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466 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1 ))) where -- x < f x |
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467 sp1 : SUP A (ZChain.chain zc) |
| 703 | 468 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc total |
| 538 | 469 c = & (SUP.sup sp1) |
| 548 | 470 |
| 560 | 471 -- |
| 547 | 472 -- create all ZChains under o< x |
| 560 | 473 -- |
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parents:
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474 |
| 674 | 475 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 476 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 477 ind f mf {y} ay x prev with trio< y x | |
| 478 ... | tri> ¬a ¬b c = ? | |
| 479 ... | tri≈ ¬a refl ¬c = ? | |
| 704 | 480 ... | tri< y<x ¬b ¬c with Oprev-p x |
| 697 | 481 ... | yes op = zc4 where |
| 682 | 482 -- |
| 483 -- we have previous ordinal to use induction | |
| 484 -- | |
| 485 px = Oprev.oprev op | |
| 703 | 486 zc : ZChain A f mf ay (Oprev.oprev op) |
| 682 | 487 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 488 px<x : px o< x | |
| 489 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 697 | 490 zc0-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 491 zc0-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 492 | |
| 703 | 493 pchain : HOD |
| 494 pchain = UnionCF A f mf ay (ZChain.supf zc) x | |
| 495 ptotal : IsTotalOrderSet pchain | |
| 496 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 497 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 498 uz01 = chain-total A f mf ay (ZChain.supf zc) (UChain.chain∋z (proj2 ca)) (UChain.chain∋z (proj2 cb)) | |
| 704 | 499 psupf : Ordinal → Ordinal |
| 500 psupf z with trio< z x | |
| 501 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
| 502 ... | tri≈ ¬a b ¬c = x | |
| 503 ... | tri> ¬a ¬b c = x | |
| 504 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y | |
| 505 pchain⊆A {y} ny = proj1 ny | |
| 506 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
| 507 pnext {a} ⟪ aa , ua ⟫ = ⟪ afa , record { u = UChain.u ua ; u<x = UChain.u<x ua ; chain∋z = ? } ⟫ where | |
| 508 afa : odef A ( f a ) | |
| 509 afa = proj2 ( mf a aa ) | |
| 510 fua : Chain A f mf ay psupf (UChain.u ua) (f a) | |
| 511 fua with UChain.chain∋z ua | |
| 512 ... | ch-init a fc = ch-init (f a) ( fsuc _ fc ) | |
| 513 ... | ch-is-sup is-sup fc = ch-is-sup (ChainP-next A f mf ay _ ? ) (fsuc _ ? ) | |
| 514 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ | |
| 515 pinit {a} ⟪ aa , ua ⟫ with UChain.chain∋z ua | |
| 516 ... | ch-init a fc = s≤fc y f mf fc | |
| 517 ... | ch-is-sup is-sup fc = ≤-ftrans (case2 zc7) ? where -- (s≤fc _ f mf fc) where | |
| 518 zc7 : y << psupf (UChain.u ua) | |
| 519 zc7 = ? -- ChainP.fcy<sup is-sup (init ay) | |
| 520 pcy : odef pchain y | |
| 521 pcy = ⟪ ay , record { u = y ; u<x = y<x ; chain∋z = ch-init _ (init ay) } ⟫ | |
| 703 | 522 |
| 611 | 523 -- if previous chain satisfies maximality, we caan reuse it |
| 524 -- | |
| 696 | 525 no-extenion : ( {a b z : Ordinal} → (z<x : z o< x) → odef (ZChain.chain zc) a → b o< osuc x → (ab : odef A b) → |
| 526 HasPrev A (ZChain.chain zc ) ab f ∨ IsSup A (ZChain.chain zc ) ab → | |
| 703 | 527 * a < * b → odef (ZChain.chain zc ) b ) → ZChain A f mf ay x |
| 528 no-extenion is-max = record { initial = ? ; chain∋init = ? } | |
| 610 | 529 |
| 703 | 530 zc4 : ZChain A f mf ay x |
| 697 | 531 zc4 with ODC.∋-p O A (* x) |
| 626 | 532 ... | no noax = no-extenion zc1 where -- ¬ A ∋ p, just skip |
| 682 | 533 zc1 : {a b z : Ordinal} → (z<x : z o< x) → odef (ZChain.chain zc ) a → b o< osuc x → (ab : odef A b) → |
| 534 HasPrev A (ZChain.chain zc ) ab f ∨ IsSup A (ZChain.chain zc ) ab → | |
| 535 * a < * b → odef (ZChain.chain zc ) b | |
| 675 | 536 zc1 {a} {b} z<x za b<ox ab P a<b with osuc-≡< b<ox |
| 568 | 537 ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) ab ) ) |
| 697 | 538 ... | case2 lt = ZChain.is-max zc za (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) lt ) ab P a<b |
| 682 | 539 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc ) ax f ) |
| 703 | 540 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 626 | 541 ... | case1 pr = no-extenion zc7 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 682 | 542 chain0 = ZChain.chain zc |
| 543 zc7 : {a b z : Ordinal} → (z<x : z o< x) → odef (ZChain.chain zc ) a → b o< osuc x → (ab : odef A b) → | |
| 544 HasPrev A (ZChain.chain zc ) ab f ∨ IsSup A (ZChain.chain zc ) ab → | |
| 545 * a < * b → odef (ZChain.chain zc ) b | |
| 675 | 546 zc7 {a} {b} z<x za b<ox ab P a<b with osuc-≡< b<ox |
| 696 | 547 ... | case2 lt = ZChain.is-max zc za (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) lt ) ab P a<b |
| 548 ... | case1 b=x = subst (λ k → odef chain0 k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc (HasPrev.ay pr)) | |
| 682 | 549 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc ) ax ) |
| 550 ... | case1 is-sup = -- x is a sup of zc | |
| 698 | 551 record { chain⊆A = pchain⊆A ; f-next = pnext ; f-total = ptotal |
| 552 ; initial = pinit ; chain∋init = pcy ; is-max = p-ismax } where | |
| 553 p-ismax : {a b : Ordinal} → odef pchain a → | |
| 554 b o< osuc x → (ab : odef A b) → | |
| 555 ( HasPrev A pchain ab f ∨ IsSup A pchain ab ) → | |
| 556 * a < * b → odef pchain b | |
| 557 p-ismax {a} {b} ua b<ox ab (case1 hasp) a<b = ? | |
| 558 p-ismax {a} {b} ua b<ox ab (case2 sup) a<b = ? | |
| 611 | 559 |
| 560 ... | case2 ¬x=sup = no-extenion z18 where -- x is not f y' nor sup of former ZChain from y -- no extention | |
| 682 | 561 z18 : {a b z : Ordinal} → (z<x : z o< x) → odef (ZChain.chain zc ) a → b o< osuc x → (ab : odef A b) → |
| 562 HasPrev A (ZChain.chain zc ) ab f ∨ IsSup A (ZChain.chain zc ) ab → | |
| 563 * a < * b → odef (ZChain.chain zc ) b | |
| 675 | 564 z18 {a} {b} z<x za b<x ab p a<b with osuc-≡< b<x |
| 697 | 565 ... | case2 lt = ZChain.is-max zc za (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) lt ) ab p a<b |
| 565 | 566 ... | case1 b=x with p |
| 697 | 567 ... | 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 | 568 ... | case2 b=sup = ⊥-elim ( ¬x=sup record { |
| 697 | 569 x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy ) } ) |
| 682 | 570 ... | no op = zc5 where |
| 703 | 571 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
| 572 pzc z z<x = prev z z<x | |
| 573 psupf0 : (z : Ordinal) → Ordinal | |
| 574 psupf0 z with trio< z x | |
| 575 ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z | |
| 704 | 576 ... | tri≈ ¬a b ¬c = y |
| 577 ... | tri> ¬a ¬b c = y | |
| 578 pchain : HOD | |
| 579 pchain = UnionCF A f mf ay psupf0 x | |
| 580 ptotal : IsTotalOrderSet pchain | |
| 581 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 703 | 582 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 583 uz01 = chain-total A f mf ay psupf0 (UChain.chain∋z (proj2 ca)) (UChain.chain∋z (proj2 cb)) | |
| 704 | 584 |
| 585 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y | |
| 586 pchain⊆A {y} ny = proj1 ny | |
| 587 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
| 588 pnext {a} ⟪ aa , ua ⟫ = ⟪ afa , record { u = UChain.u ua ; u<x = UChain.u<x ua ; chain∋z = ? } ⟫ where | |
| 589 afa : odef A ( f a ) | |
| 590 afa = proj2 ( mf a aa ) | |
| 591 fua : Chain A f mf ay psupf0 (UChain.u ua) (f a) | |
| 592 fua with UChain.chain∋z ua | |
| 593 ... | ch-init a fc = ch-init (f a) ( fsuc _ fc ) | |
| 594 ... | ch-is-sup is-sup fc = ch-is-sup (ChainP-next A f mf ay _ ? ) (fsuc _ ? ) | |
| 595 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ | |
| 596 pinit {a} ⟪ aa , ua ⟫ with UChain.chain∋z ua | |
| 597 ... | ch-init a fc = s≤fc y f mf fc | |
| 598 ... | ch-is-sup is-sup fc = ≤-ftrans (case2 zc7) ? where -- (s≤fc _ f mf fc) where | |
| 599 zc7 : y << psupf0 (UChain.u ua) | |
| 600 zc7 = ? -- ChainP.fcy<sup is-sup (init ay) | |
| 601 pcy : odef pchain y | |
| 602 pcy = ⟪ ay , record { u = y ; u<x = y<x ; chain∋z = ch-init _ (init ay) } ⟫ | |
| 603 | |
| 604 usup : SUP A pchain | |
| 605 usup = supP pchain (λ lt → proj1 lt) ptotal | |
| 703 | 606 spu = & (SUP.sup usup) |
| 607 psupf : Ordinal → Ordinal | |
| 608 psupf z with trio< z x | |
| 609 ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z | |
| 610 ... | tri≈ ¬a b ¬c = spu | |
| 611 ... | tri> ¬a ¬b c = spu | |
| 704 | 612 |
| 703 | 613 zc5 : ZChain A f mf ay x |
| 697 | 614 zc5 with ODC.∋-p O A (* x) |
|
689
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
615 ... | no noax = {!!} where -- ¬ A ∋ p, just skip |
| 704 | 616 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain ax f ) |
| 703 | 617 -- we have to check adding x preserve is-max ZChain A y f mf x |
|
689
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
618 ... | case1 pr = {!!} where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 704 | 619 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A pchain ax ) |
|
689
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
620 ... | case1 is-sup = {!!} -- x is a sup of (zc ?) |
|
34650e39e553
Chain is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
688
diff
changeset
|
621 ... | case2 ¬x=sup = {!!} -- no-extenion z18 where -- x is not f y' nor sup of former ZChain from y -- no extention |
| 682 | 622 |
| 553 | 623 |
| 703 | 624 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay (& A) |
| 625 SZ f mf {y} ay = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) (& A) | |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
626 |
| 551 | 627 zorn00 : Maximal A |
| 628 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 629 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 630 -- yes we have the maximal | |
| 631 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 632 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 633 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 634 zorn01 = proj1 zorn03 | |
| 635 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 636 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 703 | 637 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04 total ) where |
| 551 | 638 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 639 nmx : ¬ Maximal A | |
| 640 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 641 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 642 zc5 = ⟪ Maximal.A∋maximal mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫ | |
| 703 | 643 zorn04 : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) |
| 653 | 644 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
| 634 | 645 total : IsTotalOrderSet (ZChain.chain zorn04) |
| 654 | 646 total {a} {b} = zorn06 where |
| 647 zorn06 : odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a) | |
| 648 zorn06 = ZChain.f-total (SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as) ) | |
| 551 | 649 |
| 516 | 650 -- usage (see filter.agda ) |
| 651 -- | |
| 497 | 652 -- _⊆'_ : ( A B : HOD ) → Set n |
| 653 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 654 |
| 497 | 655 -- MaximumSubset : {L P : HOD} |
| 656 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 657 -- → IsPartialOrderSet P _⊆'_ | |
| 658 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 659 -- → Maximal P (_⊆'_) | |
| 660 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |