Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 567:4d8a54e2861e
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 01 May 2022 05:35:36 +0900 |
| parents | a64dad8d2e93 |
| children | 666377324edd |
| 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 | |
|
528
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
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changeset
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57 _≤_ : (x y : HOD) → Set (Level.suc n) |
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8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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58 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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59 |
| 554 | 60 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 61 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 62 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 63 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 64 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 65 | |
| 556 | 66 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 67 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 68 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 69 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 70 |
| 561 | 71 ptrans = IsStrictPartialOrder.trans PO |
| 72 | |
| 492 | 73 open _==_ |
| 74 open _⊆_ | |
| 75 | |
| 530 | 76 -- |
| 560 | 77 -- Closure of ≤-monotonic function f has total order |
| 530 | 78 -- |
| 79 | |
| 80 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 81 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 82 | |
| 560 | 83 -- immieate-f : (A : HOD) → ( f : Ordinal → Ordinal ) → Set n |
| 84 -- immieate-f A f = { x y : Ordinal } → odef A x → odef A y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
| 556 | 85 |
| 551 | 86 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 554 | 87 init : odef A s → FClosure A f s s |
| 555 | 88 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 89 |
| 556 | 90 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 |
| 91 A∋fc {A} s f mf (init as) = as | |
| 92 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) | |
| 555 | 93 |
| 556 | 94 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 95 s≤fc {A} s {.s} f mf (init x) = case1 refl | |
| 96 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) | |
| 97 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 98 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 99 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 100 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 101 |
| 557 | 102 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 103 fcn s mf (init as) = zero | |
| 558 | 104 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) |
| 105 ... | case1 eq = fcn s mf p | |
| 106 ... | case2 y<fy = suc (fcn s mf p ) | |
| 557 | 107 |
| 558 | 108 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 109 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 559 | 110 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 |
| 111 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 | |
| 112 fc00 zero zero refl (init _) (init x₁) i=x i=y = refl | |
| 113 fc00 zero zero refl (init sa) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 114 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init sa) cy i=x i=y ) | |
| 115 fc00 zero zero refl (fsuc x cx) (init sa) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 116 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init sa) i=x i=y ) | |
| 117 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 ) ) | |
| 118 ... | 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 ) | |
| 119 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 ) ) | |
| 120 ... | 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 ) | |
| 121 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 122 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 123 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 560 | 124 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x |
| 559 | 125 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where |
| 126 fc04 : * x1 ≡ * y | |
| 127 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 128 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 129 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 130 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 131 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 132 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 133 fc05 : * x ≡ * y1 | |
| 134 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 135 ... | 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 | 136 |
| 137 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 138 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 558 | 139 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where |
| 140 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 | |
| 141 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 142 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 143 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 144 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 145 ... | 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 | |
| 146 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 147 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 148 fc03 eq = cong pred eq | |
| 149 fc02 : * x < * y1 | |
| 150 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 557 | 151 |
| 559 | 152 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 153 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 559 | 154 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 155 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 156 fc11 : * x < * y | |
| 157 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 158 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 159 fc10 : * x ≡ * y | |
| 160 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 161 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 162 fc12 : * y < * x | |
| 163 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 164 | |
| 562 | 165 fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 166 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
| 563 | 167 fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where |
| 168 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 ) | |
| 169 fc20 y<sx with <-cmp ( fcn s mf cy ) (fcn s mf cx ) | |
| 170 ... | tri< a ¬b ¬c = case2 a | |
| 171 ... | tri≈ ¬a b ¬c = case1 b | |
| 172 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> y<sx (s≤s c)) | |
| 173 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 | |
| 174 fc17 {x} {y} cx cy sx=y = fc18 (fcn s mf cy) cx cy refl sx=y where | |
| 175 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 | |
| 176 fc18 (suc i) {y} cx (fsuc y1 cy) i=y sx=i with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 177 ... | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy ( fc18 (suc i) {y1} cx cy i=y sx=i) -- dereference | |
| 178 ... | case2 y<fy = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k) ) ) fc19) where | |
| 179 fc19 : * x ≡ * y1 | |
| 180 fc19 = fcn-inject s mf cx cy (cong pred ( trans sx=i i=y )) | |
| 181 fc21 : ⊥ | |
| 182 fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy ) | |
| 183 ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy | |
| 184 cxx : FClosure A f s (f x) | |
| 185 cxx = fsuc x cx | |
| 186 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)) | |
| 187 fc16 x (init sa) with proj1 (mf s sa ) | |
| 188 ... | case1 _ = case1 refl | |
| 189 ... | case2 _ = case2 refl | |
| 190 fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) | |
| 191 ... | case1 _ = case1 refl | |
| 192 ... | case2 _ = case2 refl | |
| 193 fc22 : (suc ( fcn s mf cx )) Data.Nat.< (fcn s mf cy ) → * (f x) < * y | |
| 194 fc22 a with fc16 x cx | |
| 195 ... | case1 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq (<-trans a<sa a)) | |
| 196 ... | case2 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq a ) | |
| 197 ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx | |
| 198 ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx | |
| 199 ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y | |
| 200 ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) | |
| 201 | |
| 562 | 202 |
| 560 | 203 -- open import Relation.Binary.Properties.Poset as Poset |
| 204 | |
| 205 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 206 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 207 | |
| 567 | 208 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 209 ⊆-IsTotalOrderSet = {!!} | |
| 210 | |
| 560 | 211 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 212 field | |
| 213 maximal : HOD | |
| 214 A∋maximal : A ∋ maximal | |
| 215 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 216 | |
| 217 -- | |
| 218 -- inductive maxmum tree from x | |
| 219 -- tree structure | |
| 220 -- | |
| 554 | 221 |
| 567 | 222 record HasPrev (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 223 field |
| 534 | 224 y : Ordinal |
| 541 | 225 ay : odef B y |
| 534 | 226 x=fy : x ≡ f y |
| 529 | 227 |
| 567 | 228 _⊆'_ : ( A B : HOD ) → Set n |
| 229 _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 230 | |
| 231 record IsSup (A : HOD) (T : IsTotalOrderSet A) {x : Ordinal } | |
| 232 (xa : odef A x) (sup : (C : HOD ) → ( C ⊆ A) → IsTotalOrderSet C → Ordinal) ( f : Ordinal → Ordinal ) : Set n where | |
| 233 field | |
| 234 chain : Ordinal | |
| 235 chain⊆A : (* chain) ⊆' A | |
| 236 -- ¬prev : ¬ HasPrev A (* chain) xa f | |
| 237 x=sup : x ≡ sup (* chain) record { incl = λ {x} → chain⊆A (& x) } ( ⊆-IsTotalOrderSet {A} {* chain} record { incl = λ {x} → chain⊆A (& x) } T ) | |
| 238 | |
| 508 | 239 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 503 | 240 field |
| 241 sup : HOD | |
| 242 A∋maximal : A ∋ sup | |
| 243 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 244 | |
| 533 | 245 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
| 246 SupCond A B _ _ = SUP A B | |
| 247 | |
| 546 | 248 record ZChain ( A : HOD ) {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) |
| 566 | 249 (sup : (C : HOD ) → ( C ⊆ A) → IsTotalOrderSet C → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where |
| 533 | 250 field |
| 251 chain : HOD | |
| 252 chain⊆A : chain ⊆ A | |
| 538 | 253 chain∋x : odef chain x |
| 561 | 254 initial : {y : Ordinal } → odef chain y → * x ≤ * y |
| 533 | 255 f-total : IsTotalOrderSet chain |
| 546 | 256 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 541 | 257 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) |
| 565 | 258 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc z → (ba : odef A b) |
| 567 | 259 → HasPrev A chain ba f ∨ ((sup chain chain⊆A f-total) ≡ b ) |
| 534 | 260 → * a < * b → odef chain b |
| 533 | 261 |
| 497 | 262 Zorn-lemma : { A : HOD } |
| 464 | 263 → o∅ o< & A |
| 497 | 264 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 265 → Maximal A | |
| 552 | 266 Zorn-lemma {A} 0<A supP = zorn00 where |
| 566 | 267 supO : (C : HOD ) → C ⊆ A → IsTotalOrderSet C → Ordinal |
| 268 supO C C⊆A TC = & ( SUP.sup ( supP C C⊆A TC )) | |
| 493 | 269 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 556 | 270 z01 {a} {b} A∋a A∋b = <-irr |
| 537 | 271 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 272 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 273 s : HOD |
| 274 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 275 sa : A ∋ * ( & s ) | |
| 276 sa = 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 )) ) | |
| 547 | 277 s<A : & s o< & A |
| 278 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso sa ) | |
| 530 | 279 HasMaximal : HOD |
| 537 | 280 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 281 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 282 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 283 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 284 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 285 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 286 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 287 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 288 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)) | |
| 289 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 | |
| 290 ¬x<m : ¬ (* x < * m) | |
| 291 ¬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 | 292 |
| 560 | 293 -- Uncountable ascending chain by axiom of choice |
| 530 | 294 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 295 cf nmx x with ODC.∋-p O A (* x) |
| 296 ... | no _ = o∅ | |
| 297 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 298 ... | yes nogt = -- no larger element, so it is maximal |
| 299 ⊥-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 | 300 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 301 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 302 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 303 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 304 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 305 ... | 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 ⟫ ) | |
| 306 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 530 | 307 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 308 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 309 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 310 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 311 |
| 547 | 312 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A sa f mf supO (& A) ) → SUP A (ZChain.chain zc) |
| 533 | 313 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
| 547 | 314 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) ) |
| 533 | 315 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
| 316 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
| 566 | 317 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) → SUP A (ZChain.chain zc) |
| 318 sp0 f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain.f-total zc) | |
| 543 | 319 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 320 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 321 | |
| 322 --- | |
| 560 | 323 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 324 --- |
| 547 | 325 z03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) |
| 546 | 326 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) |
| 538 | 327 z03 f mf zc = z14 where |
| 328 chain = ZChain.chain zc | |
| 329 sp1 = sp0 f mf zc | |
| 565 | 330 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc (& A) → (ab : odef A b ) |
| 567 | 331 → HasPrev A chain ab f ∨ (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 332 → * a < * b → odef chain b |
| 333 z10 = ZChain.is-max zc | |
| 543 | 334 z11 : & (SUP.sup sp1) o< & A |
| 335 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 336 z12 : odef chain (& (SUP.sup sp1)) |
| 337 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 338 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
| 565 | 339 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) (ordtrans z11 <-osuc ) (SUP.A∋maximal sp1) (case2 refl ) z13 where |
| 538 | 340 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 566 | 341 z13 with SUP.x<sup sp1 ( ZChain.chain∋x zc ) |
| 538 | 342 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 343 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 344 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) | |
| 552 | 345 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 538 | 346 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 347 z16 : ⊥ | |
| 348 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 349 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 350 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 351 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 352 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 353 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 566 | 354 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) |
| 538 | 355 z17 : ⊥ |
| 356 z17 with z15 | |
| 357 ... | case1 eq = ¬b eq | |
| 358 ... | case2 lt = ¬a lt | |
| 560 | 359 |
| 360 -- ZChain contradicts ¬ Maximal | |
| 361 -- | |
| 362 -- ZChain forces fix point on any ≤-monotonic function (z03) | |
| 363 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
| 364 -- | |
| 547 | 365 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥ |
| 537 | 366 z04 nmx zc = z01 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) |
| 538 | 367 (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
| 368 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) | |
| 369 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where | |
| 546 | 370 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc |
| 538 | 371 c = & (SUP.sup sp1) |
| 548 | 372 |
| 560 | 373 -- |
| 547 | 374 -- create all ZChains under o< x |
| 560 | 375 -- |
| 546 | 376 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → |
| 547 | 377 ((z : Ordinal) → z o< x → {y : Ordinal} → (ya : odef A y) → ZChain A ya f mf supO z ) → { y : Ordinal } → (ya : odef A y) → ZChain A ya f mf supO x |
| 378 ind f mf x prev {y} ay with Oprev-p x | |
| 548 | 379 ... | yes op = zc4 where |
| 560 | 380 -- |
| 381 -- we have previous ordinal to use induction | |
| 382 -- | |
| 530 | 383 px = Oprev.oprev op |
| 547 | 384 zc0 : ZChain A ay f mf supO (Oprev.oprev op) |
| 385 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay | |
| 386 zc4 : ZChain A ay f mf supO x | |
| 565 | 387 zc4 with ODC.∋-p O A (* x) |
| 388 ... | no noapx = -- ¬ A ∋ p, just skip | |
| 560 | 389 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 |
| 554 | 390 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 |
| 551 | 391 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 } where -- no extention |
| 565 | 392 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ba : odef A b) → |
| 567 | 393 HasPrev A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (ZChain.chain zc0) (ZChain.chain⊆A zc0) (ZChain.f-total zc0))) ≡ b) → |
| 551 | 394 * a < * b → odef (ZChain.chain zc0) b |
| 565 | 395 zc11 {a} {b} za b<ox ba P a<b with osuc-≡< b<ox |
| 396 ... | case1 eq = ⊥-elim ( noapx (subst (λ k → odef A k) (trans eq (sym &iso)) ba ) ) | |
| 397 ... | case2 lt = ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt ) ba P a<b | |
| 567 | 398 ... | yes apx with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc0) apx f ) -- we have to check adding x preserve is-max ZChain A ay f mf supO x |
| 549 | 399 ... | case1 pr = zc9 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 551 | 400 chain = ZChain.chain zc0 |
| 565 | 401 zc17 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ba : odef A b) → |
| 567 | 402 HasPrev A (ZChain.chain zc0) ba f ∨ (supO (ZChain.chain zc0) (ZChain.chain⊆A zc0) (ZChain.f-total zc0) ≡ b) → |
| 551 | 403 * a < * b → odef (ZChain.chain zc0) b |
| 565 | 404 zc17 {a} {b} za b<ox ba P a<b with osuc-≡< b<ox |
| 405 ... | case2 lt = ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt) ba P a<b | |
| 567 | 406 ... | case1 b=x = subst (λ k → odef chain k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr)) |
| 549 | 407 zc9 : ZChain A ay f mf supO x |
| 408 zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 554 | 409 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 } -- no extention |
| 551 | 410 ... | case2 ¬fy<x with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) )) |
| 560 | 411 ... | case1 x=sup = -- previous ordinal is a sup of a smaller ZChain |
| 561 | 412 record { chain = schain ; chain⊆A = record { incl = A∋schain } ; f-total = scmp ; f-next = scnext |
| 564 | 413 ; initial = scinit ; f-immediate = simm ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax } where -- x is sup |
| 561 | 414 sup0 = supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) |
| 415 sp = SUP.sup sup0 | |
| 551 | 416 chain = ZChain.chain zc0 |
| 561 | 417 sc<A : {y : Ordinal} → odef chain y ∨ FClosure A f (& sp) y → y o< & A |
| 418 sc<A {y} (case1 zx) = subst (λ k → k o< (& A)) &iso ( c<→o< ( incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k) (sym &iso) zx ))) | |
| 419 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 | 420 schain : HOD |
| 561 | 421 schain = record { od = record { def = λ x → odef chain x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = λ {y} sy → sc<A {y} sy } |
| 422 A∋schain : {x : HOD } → schain ∋ x → A ∋ x | |
| 423 A∋schain (case1 zx ) = subst (λ k → odef A k ) &iso (incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k) (sym &iso) zx )) | |
| 424 A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx | |
| 425 cmp : {a b : HOD} (za : odef chain (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) | |
| 426 cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb | |
| 427 ... | case1 sp=a | case1 sp=b = tri≈ (λ lt → <-irr (case1 (sym eq)) lt ) eq (λ lt → <-irr (case1 eq) lt ) where | |
| 428 eq : a ≡ b | |
| 429 eq = trans sp=a (subst₂ (λ j k → j ≡ k ) *iso *iso sp=b ) | |
| 430 ... | 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 | |
| 431 a<b : a < b | |
| 432 a<b = subst (λ k → k < b ) (sym sp=a) (subst₂ (λ j k → j < k ) *iso *iso sp<b ) | |
| 433 ... | 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 | |
| 434 a<b : a < b | |
| 435 a<b = subst (λ k → a < k ) (trans sp=b *iso ) (subst (λ k → a < k ) (sym *iso) a<sp ) | |
| 436 ... | 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 | |
| 437 a<b : a < b | |
| 438 a<b = ptrans (subst (λ k → a < k ) (sym *iso) a<sp ) ( subst₂ (λ j k → j < k ) refl *iso sp<b ) | |
| 439 scmp : {a b : HOD} → odef schain (& a) → odef schain (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 440 scmp (case1 za) (case1 zb) = ZChain.f-total zc0 za zb | |
| 441 scmp {a} {b} (case1 za) (case2 fb) = cmp za fb | |
| 442 scmp (case2 fa) (case1 zb) with cmp zb fa | |
| 443 ... | tri< a ¬b ¬c = tri> ¬c (λ eq → ¬b (sym eq)) a | |
| 444 ... | tri≈ ¬a b ¬c = tri≈ ¬c (sym b) ¬a | |
| 445 ... | tri> ¬a ¬b c = tri< c (λ eq → ¬b (sym eq)) ¬a | |
| 446 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) | |
| 447 scnext : {a : Ordinal} → odef schain a → odef schain (f a) | |
| 448 scnext {x} (case1 zx) = case1 (ZChain.f-next zc0 zx) | |
| 449 scnext {x} (case2 sx) = case2 ( fsuc x sx ) | |
| 450 scinit : {x : Ordinal} → odef schain x → * y ≤ * x | |
| 451 scinit {x} (case1 zx) = ZChain.initial zc0 zx | |
| 452 scinit {x} (case2 sx) with (s≤fc (& sp) f mf sx ) | SUP.x<sup sup0 (subst (λ k → odef chain k ) (sym &iso) ( ZChain.chain∋x zc0 ) ) | |
| 562 | 453 ... | case1 sp=x | case1 y=sp = case1 (trans y=sp (subst (λ k → k ≡ * x ) *iso sp=x ) ) |
| 454 ... | case1 sp=x | case2 y<sp = case2 (subst (λ k → * y < k ) (trans (sym *iso) sp=x) y<sp ) | |
| 455 ... | case2 sp<x | case1 y=sp = case2 (subst (λ k → k < * x ) (trans *iso (sym y=sp )) sp<x ) | |
| 456 ... | case2 sp<x | case2 y<sp = case2 (ptrans y<sp (subst (λ k → k < * x ) *iso sp<x) ) | |
| 457 A∋za : {a : Ordinal } → odef chain a → odef A a | |
| 458 A∋za za = (subst (λ k → odef A k ) &iso (incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k ) (sym &iso) za) ) ) | |
| 459 za<sup : {a : Ordinal } → odef chain a → ( * a ≡ sp ) ∨ ( * a < sp ) | |
| 460 za<sup za = SUP.x<sup sup0 (subst (λ k → odef chain k ) (sym &iso) za ) | |
| 461 simm : {a b : Ordinal} → odef schain a → odef schain b → ¬ (* a < * b) ∧ (* b < * (f a)) | |
| 462 simm {a} {b} (case1 za) (case1 zb) = ZChain.f-immediate zc0 za zb | |
| 463 simm {a} {b} (case1 za) (case2 sb) p with proj1 (mf a (A∋za za) ) | |
| 464 ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) | |
| 465 ... | case2 a<fa with za<sup ( ZChain.f-next zc0 za ) | s≤fc (& sp) f mf sb | |
| 466 ... | case1 fa=sp | case1 sp=b = <-irr (case1 (trans fa=sp (trans (sym *iso) sp=b )) ) ( proj2 p ) | |
| 467 ... | case2 fa<sp | case1 sp=b = <-irr (case2 fa<sp) (subst (λ k → k < * (f a) ) (trans (sym sp=b) *iso) (proj2 p ) ) | |
| 468 ... | 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 ) ) | |
| 469 ... | case2 fa<sp | case2 sp<b = <-irr (case2 (proj2 p )) (ptrans fa<sp (subst (λ k → k < * b ) *iso sp<b ) ) | |
| 470 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) )) ) ) | |
| 471 ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) | |
| 472 ... | case2 b<fb with s≤fc (& sp) f mf sa | za<sup zb | |
| 473 ... | case1 sp=a | case1 b=sp = <-irr (case1 (trans b=sp (trans (sym *iso) sp=a )) ) (proj1 p ) | |
| 474 ... | case1 sp=a | case2 b<sp = <-irr (case2 (subst (λ k → * b < k ) (trans (sym *iso) sp=a) b<sp ) ) (proj1 p ) | |
| 475 ... | case2 sp<a | case1 b=sp = <-irr (case2 (subst ( λ k → k < * a ) (trans *iso (sym b=sp)) sp<a )) (proj1 p ) | |
| 476 ... | case2 sp<a | case2 b<sp = <-irr (case2 (ptrans b<sp (subst (λ k → k < * a) *iso sp<a ))) (proj1 p ) | |
| 564 | 477 simm {a} {b} (case2 sa) (case2 sb) p = fcn-imm {A} (& sp) {a} {b} f mf sa sb p |
| 565 | 478 s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ba : odef A b) → |
| 567 | 479 HasPrev A schain ba f ∨ (& (SUP.sup (supP schain record { incl = A∋schain } scmp)) ≡ b) |
| 565 | 480 → * a < * b → odef schain b |
| 567 | 481 s-ismax {a} {b} (case1 za) b<x ba (case1 p) a<b with osuc-≡< b<x |
| 482 ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) | |
| 483 ... | case2 b<x = z21 p where | |
| 484 z21 : HasPrev A schain ba f → odef schain b | |
| 485 z21 record { y = y ; ay = (case1 zy) ; x=fy = x=fy } = | |
| 486 case1 (ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym ( Oprev.oprev=x op)) b<x ) ba (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) | |
| 487 z21 record { y = y ; ay = (case2 sy) ; x=fy = x=fy } = subst (λ k → odef schain k) (sym x=fy) (case2 (fsuc y sy) ) | |
| 488 s-ismax {a} {b} (case1 za) b<x ba (case2 p) a<b with osuc-≡< b<x | |
| 489 ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) | |
| 490 ... | case2 b<x = z22 p where | |
| 491 z22 : & (SUP.sup (supP schain record { incl = A∋schain } scmp)) ≡ b → odef schain b | |
| 492 z22 p = {!!} | |
| 493 -- case1 (ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym ( Oprev.oprev=x op)) b<x ) ba {!!} a<b ) | |
| 494 s-ismax {a} {b} (case2 sa) b<x ba p a<b = {!!} | |
| 560 | 495 ... | case2 ¬x=sup = -- x is not f y' nor sup of former ZChain from y |
| 496 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 565 | 497 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = z18 } where -- no extention |
| 498 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ba : odef A b) → | |
| 567 | 499 HasPrev A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (ZChain.chain zc0) (ZChain.chain⊆A zc0) (ZChain.f-total zc0))) ≡ b) → |
| 552 | 500 * a < * b → odef (ZChain.chain zc0) b |
| 565 | 501 z18 {a} {b} za b<x ba p a<b with osuc-≡< b<x |
| 502 ... | case2 lt = ZChain.is-max zc0 za (subst (λ k → b o< k ) (sym ( Oprev.oprev=x op)) lt ) ba p a<b | |
| 503 ... | case1 b=x with p | |
| 567 | 504 ... | 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 ) } ) |
| 566 | 505 ... | case2 b=sup = ⊥-elim ( ¬x=sup (sym (trans b=sup b=x )) ) |
| 553 | 506 ... | no ¬ox = {!!} where --- limit ordinal case |
| 554 | 507 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x |
| 553 | 508 field |
| 509 u : Ordinal | |
| 510 u<x : u o< x | |
| 554 | 511 zuy : odef (ZChain.chain (prev u u<x {y} ay )) z |
| 512 Uz : HOD | |
| 513 Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A ; <odmax = {!!} } | |
| 514 u-total : IsTotalOrderSet Uz | |
| 553 | 515 u-total {x} {y} ux uy = {!!} |
| 560 | 516 --- ux ⊆ uy ∨ uy ⊆ ux |
| 553 | 517 |
| 551 | 518 zorn00 : Maximal A |
| 519 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 520 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 521 -- yes we have the maximal | |
| 522 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 523 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 524 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 525 zorn01 = proj1 zorn03 | |
| 526 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 527 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 528 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where | |
| 529 -- if we have no maximal, make ZChain, which contradict SUP condition | |
| 530 nmx : ¬ Maximal A | |
| 531 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 532 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 533 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) ) ⟫ | |
| 534 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO (& A) | |
| 535 zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain A ya f mf supO z } (ind f mf) (& A) | |
| 536 zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) | |
| 537 zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa ) | |
| 538 | |
| 516 | 539 -- usage (see filter.agda ) |
| 540 -- | |
| 497 | 541 -- _⊆'_ : ( A B : HOD ) → Set n |
| 542 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 543 |
| 497 | 544 -- MaximumSubset : {L P : HOD} |
| 545 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 546 -- → IsPartialOrderSet P _⊆'_ | |
| 547 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 548 -- → Maximal P (_⊆'_) | |
| 549 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |