Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 661:9142e834c4c6
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 03 Jul 2022 06:10:51 +0900 |
| parents | db9477c80dce |
| children | a45ec34b9fa7 |
| 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 | |
|
528
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
66 _≤_ : (x y : HOD) → Set (Level.suc n) |
|
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
67 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
|
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
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 | |
| 661 | 242 record UChain (x : Ordinal) (chain : (z : Ordinal ) → z o< x → HOD) (z : Ordinal) : Set n where |
| 655 | 243 -- Union of supf z which o< x |
| 626 | 244 field |
| 655 | 245 u : Ordinal |
| 246 u<x : u o< x | |
| 661 | 247 chain∋z : odef (chain u u<x) z |
| 653 | 248 |
| 656 | 249 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 250 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 251 | |
| 252 data Chain (A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal} (ay : odef A y) : Ordinal → HOD → Set (Level.suc n) where | |
| 253 ch-noax : {x : Ordinal } { chain : HOD } ( op : Oprev Ordinal osuc x ) (noax : ¬ odef A x ) (c : Chain A f ay (Oprev.oprev op) chain) → Chain A f ay x chain | |
| 254 ch-hasprev : {x : Ordinal } { chain : HOD } ( op : Oprev Ordinal osuc x ) (ax : odef A x ) | |
| 255 ( c : Chain A f ay (Oprev.oprev op) chain) ( h : HasPrev A chain ax f ) → Chain A f ay x chain | |
| 256 ch-is-sup : {x : Ordinal } { chain : HOD } ( op : Oprev Ordinal osuc x ) ( ax : odef A x ) | |
| 257 ( c : Chain A f ay (Oprev.oprev op) chain) ( nh : ¬ HasPrev A chain ax f ) ( sup : IsSup A chain ax ) → Chain A f ay x | |
| 258 record { od = record { def = λ x → odef A x ∧ (odef chain x ∨ (FClosure A f y x)) } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } | |
| 259 ch-skip : {x : Ordinal } { chain : HOD } ( op : Oprev Ordinal osuc x ) ( ax : odef A x ) | |
| 260 ( c : Chain A f ay (Oprev.oprev op) chain) ( nh : ¬ HasPrev A chain ax f ) ( nsup : ¬ IsSup A chain ax ) → Chain A f ay x chain | |
| 261 ch-union : {x : Ordinal } { chain : HOD } ( nop : ¬ Oprev Ordinal osuc x ) ( ax : odef A x ) | |
| 661 | 262 → ( chainf : ( z : Ordinal ) → z o< x → HOD ) → ( lt : ( z : Ordinal ) → (z<x : z o< x ) → Chain A f ay z ( chainf z z<x )) |
| 656 | 263 → Chain A f ay x |
| 661 | 264 record { od = record { def = λ z → odef A z ∧ (UChain x chainf z ∨ FClosure A f y z ) } |
| 265 ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } | |
| 653 | 266 |
| 655 | 267 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) ( z : Ordinal ) : Set (Level.suc n) where |
| 268 field | |
| 661 | 269 chain : HOD |
| 270 chain-uniq : Chain A f ay z chain | |
| 655 | 271 |
| 661 | 272 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) {init : Ordinal} (ay : odef A init) (zc0 : (x : Ordinal) → ZChain1 A f ay x ) ( z : Ordinal ) : Set (Level.suc n) where |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
273 chain : HOD |
| 661 | 274 chain = ZChain1.chain (zc0 z) |
| 568 | 275 field |
| 276 chain⊆A : chain ⊆' A | |
| 653 | 277 chain∋init : odef chain init |
| 278 initial : {y : Ordinal } → odef chain y → * init ≤ * y | |
| 568 | 279 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 654 | 280 f-total : IsTotalOrderSet chain |
| 568 | 281 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc z → (ab : odef A b) |
| 574 | 282 → HasPrev A chain ab f ∨ IsSup A chain ab |
| 568 | 283 → * a < * b → odef chain b |
| 653 | 284 |
| 568 | 285 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 286 field | |
| 287 maximal : HOD | |
| 288 A∋maximal : A ∋ maximal | |
| 289 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 567 | 290 |
| 533 | 291 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
| 292 SupCond A B _ _ = SUP A B | |
| 293 | |
| 497 | 294 Zorn-lemma : { A : HOD } |
| 464 | 295 → o∅ o< & A |
| 568 | 296 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 297 → Maximal A |
| 552 | 298 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 299 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 300 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 537 | 301 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 302 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 303 s : HOD |
| 304 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 305 as : A ∋ * ( & s ) |
| 306 as = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) | |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
307 as0 : odef A (& s ) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
308 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 309 s<A : & s o< & A |
| 568 | 310 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 311 HasMaximal : HOD |
| 537 | 312 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 313 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 314 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 315 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 316 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 317 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 318 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 319 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 320 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)) | |
| 321 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 | |
| 322 ¬x<m : ¬ (* x < * m) | |
| 323 ¬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 | 324 |
| 560 | 325 -- Uncountable ascending chain by axiom of choice |
| 530 | 326 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 327 cf nmx x with ODC.∋-p O A (* x) |
| 328 ... | no _ = o∅ | |
| 329 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 330 ... | yes nogt = -- no larger element, so it is maximal |
| 331 ⊥-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 | 332 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 333 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 334 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 335 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 336 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 337 ... | 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 ⟫ ) | |
| 338 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 339 |
| 340 --- | |
| 341 --- infintie ascention sequence of f | |
| 342 --- | |
| 530 | 343 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 344 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 345 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 346 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 347 |
| 661 | 348 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 : (x : Ordinal) → ZChain1 A f as0 x ) (zc : ZChain A f as0 zc0 (& A) ) |
| 653 | 349 (total : IsTotalOrderSet (ZChain.chain zc) ) → SUP A (ZChain.chain zc) |
| 350 sp0 f mf zc0 zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total | |
| 543 | 351 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 352 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 353 | |
| 354 --- | |
| 560 | 355 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 356 --- |
| 661 | 357 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 : (x : Ordinal) → ZChain1 A f as0 x) (zc : ZChain A f as0 zc0 (& A) ) |
|
633
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
358 → (total : IsTotalOrderSet (ZChain.chain zc) ) |
| 653 | 359 → f (& (SUP.sup (sp0 f mf zc0 zc total ))) ≡ & (SUP.sup (sp0 f mf zc0 zc total)) |
| 360 fixpoint f mf zc0 zc total = z14 where | |
| 538 | 361 chain = ZChain.chain zc |
| 653 | 362 sp1 = sp0 f mf zc0 zc total |
| 565 | 363 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc (& A) → (ab : odef A b ) |
| 570 | 364 → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 365 → * a < * b → odef chain b |
| 366 z10 = ZChain.is-max zc | |
| 543 | 367 z11 : & (SUP.sup sp1) o< & A |
| 368 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 369 z12 : odef chain (& (SUP.sup sp1)) |
| 370 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 653 | 371 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 372 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) (ordtrans z11 <-osuc ) (SUP.A∋maximal sp1) | |
| 570 | 373 (case2 z19 ) z13 where |
| 538 | 374 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 653 | 375 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc ) |
| 538 | 376 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 377 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 570 | 378 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) |
| 571 | 379 z19 = record { x<sup = z20 } where |
| 380 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
| 381 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
| 570 | 382 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
| 383 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
| 384 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
| 653 | 385 z14 : f (& (SUP.sup (sp0 f mf zc0 zc total ))) ≡ & (SUP.sup (sp0 f mf zc0 zc total )) |
|
633
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
386 z14 with total (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 631 | 387 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 388 z16 : ⊥ | |
| 389 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 390 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 391 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 392 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 393 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 394 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 395 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) | |
| 396 z17 : ⊥ | |
| 397 z17 with z15 | |
| 398 ... | case1 eq = ¬b eq | |
| 399 ... | case2 lt = ¬a lt | |
| 560 | 400 |
| 401 -- ZChain contradicts ¬ Maximal | |
| 402 -- | |
| 571 | 403 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
| 560 | 404 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
| 405 -- | |
| 661 | 406 z04 : (nmx : ¬ Maximal A ) → (zc0 : (x : Ordinal) → ZChain1 A (cf nmx) as0 x) (zc : ZChain A (cf nmx) as0 zc0 (& A)) → IsTotalOrderSet (ZChain.chain zc) → ⊥ |
| 653 | 407 z04 nmx zc0 zc total = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal sp1 )))) |
| 571 | 408 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) |
| 653 | 409 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc0 zc total ))) -- x ≡ f x ̄ |
|
633
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
410 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1 ))) where -- x < f x |
|
6cd4a483122c
ZChain1 is not strictly positive
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
631
diff
changeset
|
411 sp1 : SUP A (ZChain.chain zc) |
| 653 | 412 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc0 zc total |
| 538 | 413 c = & (SUP.sup sp1) |
| 548 | 414 |
| 560 | 415 -- |
| 547 | 416 -- create all ZChains under o< x |
| 560 | 417 -- |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
418 |
| 630 | 419 sind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 655 | 420 → ((z : Ordinal) → z o< x → ZChain1 A f ay z ) → ZChain1 A f ay x |
| 630 | 421 sind f mf {y} ay x prev with Oprev-p x |
| 422 ... | yes op = sc4 where | |
| 654 | 423 open ZChain1 |
| 630 | 424 px = Oprev.oprev op |
| 656 | 425 px<x : px o< x |
| 426 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 655 | 427 sc : ZChain1 A f ay px |
| 656 | 428 sc = prev px px<x |
| 655 | 429 sc4 : ZChain1 A f ay x |
| 630 | 430 sc4 with ODC.∋-p O A (* x) |
| 661 | 431 ... | no noax = record { chain = ? ; chain-uniq = ? } |
| 432 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain1.chain sc ) ax f ) | |
| 654 | 433 ... | case1 pr = {!!} |
| 661 | 434 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain1.chain sc ) ax ) |
| 654 | 435 ... | case1 is-sup = {!!} where |
| 630 | 436 -- A∋sc -- x is a sup of zc |
| 661 | 437 sup0 : SUP A (ZChain1.chain sc ) |
| 630 | 438 sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where |
| 661 | 439 x21 : {y : HOD} → (ZChain1.chain sc ) ∋ y → (y ≡ * x) ∨ (y < * x) |
| 630 | 440 x21 {y} zy with IsSup.x<sup is-sup zy |
| 634 | 441 ... | case1 y=x = case1 (subst₂ (λ j k → j ≡ * k ) *iso &iso ( cong (*) y=x) ) |
| 630 | 442 ... | case2 y<x = case2 (subst₂ (λ j k → j < * k ) *iso &iso y<x ) |
| 443 sp : HOD | |
| 444 sp = SUP.sup sup0 | |
| 445 schain : HOD | |
| 661 | 446 schain = record { od = record { def = λ x → odef A x ∧ ( odef (ZChain1.chain sc ) x ∨ (FClosure A f (& sp) x)) } |
| 447 ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } | |
| 654 | 448 ... | case2 ¬x=sup = {!!} |
| 655 | 449 ... | no ¬ox = ? where |
| 661 | 450 supf : (z : Ordinal) → z o< x → HOD |
| 451 supf = ? | |
| 655 | 452 sc5 : HOD |
| 661 | 453 sc5 = record { od = record { def = λ z → odef A z ∧ (UChain x supf z ∨ FClosure A f y z)} ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } |
| 630 | 454 |
| 661 | 455 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) → (zc0 : (x : Ordinal) → ZChain1 A f ay x) |
| 655 | 456 → ((z : Ordinal) → z o< x → ZChain A f ay zc0 z) → ZChain A f ay zc0 x |
| 457 ind f mf {y} ay x zc0 prev with Oprev-p x | |
| 548 | 458 ... | yes op = zc4 where |
| 560 | 459 -- |
| 460 -- we have previous ordinal to use induction | |
| 461 -- | |
| 530 | 462 px = Oprev.oprev op |
| 624 | 463 supf : Ordinal → HOD |
| 661 | 464 supf x = ZChain1.chain (zc0 x) |
| 655 | 465 zc : ZChain A f ay zc0 (Oprev.oprev op) |
| 628 | 466 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 626 | 467 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 468 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 604 | 469 px<x : px o< x |
| 470 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 569 | 471 |
| 611 | 472 -- if previous chain satisfies maximality, we caan reuse it |
| 473 -- | |
| 626 | 474 no-extenion : ( {a b : Ordinal} → odef (ZChain.chain zc) a → b o< osuc x → (ab : odef A b) → |
| 475 HasPrev A (ZChain.chain zc) ab f ∨ IsSup A (ZChain.chain zc) ab → | |
| 655 | 476 * a < * b → odef (ZChain.chain zc) b ) → ZChain A f ay ? x |
| 654 | 477 no-extenion is-max = record { chain⊆A = ? -- subst (λ k → k ⊆' A ) {!!} (ZChain.chain⊆A zc) |
| 478 ; initial = subst (λ k → {y₁ : Ordinal} → odef k y₁ → * y ≤ * y₁ ) {!!} (ZChain.initial zc) | |
| 479 ; f-next = subst (λ k → {a : Ordinal} → odef k a → odef k (f a) ) {!!} (ZChain.f-next zc) | |
| 480 ; f-total = ? | |
| 481 ; chain∋init = subst (λ k → odef k y ) {!!} (ZChain.chain∋init zc) | |
| 482 ; is-max = subst (λ k → {a b : Ordinal} → odef k a → b o< osuc x → (ab : odef A b) → | |
| 483 HasPrev A k ab f ∨ IsSup A k ab → * a < * b → odef k b ) {!!} is-max } where | |
| 624 | 484 supf0 : Ordinal → HOD |
| 610 | 485 supf0 z with trio< z x |
| 486 ... | tri< a ¬b ¬c = supf z | |
| 626 | 487 ... | tri≈ ¬a b ¬c = ZChain.chain zc |
| 488 ... | tri> ¬a ¬b c = ZChain.chain zc | |
| 489 seq : ZChain.chain zc ≡ supf0 x | |
| 610 | 490 seq with trio< x x |
| 491 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) | |
| 624 | 492 ... | tri≈ ¬a b ¬c = refl |
| 493 ... | tri> ¬a ¬b c = refl | |
| 494 seq<x : {b : Ordinal } → b o< x → supf b ≡ supf0 b | |
| 611 | 495 seq<x {b} b<x with trio< b x |
| 496 ... | tri< a ¬b ¬c = refl | |
| 497 ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) | |
| 498 ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) | |
| 610 | 499 |
| 655 | 500 zc4 : ZChain A f ay zc0 x |
| 565 | 501 zc4 with ODC.∋-p O A (* x) |
| 626 | 502 ... | no noax = no-extenion zc1 where -- ¬ A ∋ p, just skip |
| 503 zc1 : {a b : Ordinal} → odef (ZChain.chain zc) a → b o< osuc x → (ab : odef A b) → | |
| 504 HasPrev A (ZChain.chain zc) ab f ∨ IsSup A (ZChain.chain zc) ab → | |
| 505 * a < * b → odef (ZChain.chain zc) b | |
| 506 zc1 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox | |
| 568 | 507 ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) ab ) ) |
| 626 | 508 ... | case2 lt = ZChain.is-max zc za (zc-b<x b lt) ab P a<b |
| 653 | 509 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc) ax f ) -- we have to check adding x preserve is-max ZChain A y f mf zc0 x |
| 626 | 510 ... | case1 pr = no-extenion zc7 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 511 chain0 = ZChain.chain zc | |
| 512 zc7 : {a b : Ordinal} → odef (ZChain.chain zc) a → b o< osuc x → (ab : odef A b) → | |
| 513 HasPrev A (ZChain.chain zc) ab f ∨ IsSup A (ZChain.chain zc) ab → | |
| 514 * a < * b → odef (ZChain.chain zc) b | |
| 515 zc7 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox | |
| 516 ... | case2 lt = ZChain.is-max zc za (zc-b<x b lt) ab P a<b | |
| 517 ... | 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)) | |
| 518 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc) ax ) | |
| 519 ... | case1 is-sup = -- x is a sup of zc | |
| 654 | 520 record { chain⊆A = {!!} ; f-next = {!!} ; f-total = {!!} |
| 521 ; initial = {!!} ; chain∋init = {!!} ; is-max = {!!} } where | |
| 626 | 522 sup0 : SUP A (ZChain.chain zc) |
| 571 | 523 sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where |
| 626 | 524 x21 : {y : HOD} → ZChain.chain zc ∋ y → (y ≡ * x) ∨ (y < * x) |
| 571 | 525 x21 {y} zy with IsSup.x<sup is-sup zy |
| 526 ... | case1 y=x = case1 ( subst₂ (λ j k → j ≡ * k ) *iso &iso ( cong (*) y=x) ) | |
| 527 ... | case2 y<x = case2 (subst₂ (λ j k → j < * k ) *iso &iso y<x ) | |
| 570 | 528 sp : HOD |
| 561 | 529 sp = SUP.sup sup0 |
| 570 | 530 x=sup : x ≡ & sp |
| 531 x=sup = sym &iso | |
| 626 | 532 chain0 = ZChain.chain zc |
| 604 | 533 sc<A : {y : Ordinal} → odef chain0 y ∨ FClosure A f (& sp) y → y o< & A |
| 626 | 534 sc<A {y} (case1 zx) = subst (λ k → k o< (& A)) &iso ( c<→o< (ZChain.chain⊆A zc (subst (λ k → odef chain0 k) (sym &iso) zx ))) |
| 561 | 535 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 | 536 schain : HOD |
| 604 | 537 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 } |
| 631 | 538 supf0 : Ordinal → HOD |
| 539 supf0 z with trio< z x | |
| 540 ... | tri< a ¬b ¬c = supf z | |
| 541 ... | tri≈ ¬a b ¬c = schain | |
| 542 ... | tri> ¬a ¬b c = schain | |
| 561 | 543 A∋schain : {x : HOD } → schain ∋ x → A ∋ x |
| 626 | 544 A∋schain (case1 zx ) = ZChain.chain⊆A zc zx |
| 561 | 545 A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx |
| 569 | 546 s⊆A : schain ⊆' A |
| 626 | 547 s⊆A {x} (case1 zx) = ZChain.chain⊆A zc zx |
| 569 | 548 s⊆A {x} (case2 fx) = A∋fc (& sp) f mf fx |
| 604 | 549 cmp : {a b : HOD} (za : odef chain0 (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) |
| 561 | 550 cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb |
| 551 ... | case1 sp=a | case1 sp=b = tri≈ (λ lt → <-irr (case1 (sym eq)) lt ) eq (λ lt → <-irr (case1 eq) lt ) where | |
| 552 eq : a ≡ b | |
| 553 eq = trans sp=a (subst₂ (λ j k → j ≡ k ) *iso *iso sp=b ) | |
| 554 ... | 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 | |
| 555 a<b : a < b | |
| 556 a<b = subst (λ k → k < b ) (sym sp=a) (subst₂ (λ j k → j < k ) *iso *iso sp<b ) | |
| 557 ... | 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 | |
| 558 a<b : a < b | |
| 559 a<b = subst (λ k → a < k ) (trans sp=b *iso ) (subst (λ k → a < k ) (sym *iso) a<sp ) | |
| 560 ... | 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 | |
| 561 a<b : a < b | |
| 562 a<b = ptrans (subst (λ k → a < k ) (sym *iso) a<sp ) ( subst₂ (λ j k → j < k ) refl *iso sp<b ) | |
| 563 scmp : {a b : HOD} → odef schain (& a) → odef schain (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 628 | 564 scmp {a} {b} (case1 za) (case1 zb) = {!!} -- ZChain.f-total zc {px} {px} o≤-refl za zb |
| 561 | 565 scmp {a} {b} (case1 za) (case2 fb) = cmp za fb |
| 566 scmp (case2 fa) (case1 zb) with cmp zb fa | |
| 567 ... | tri< a ¬b ¬c = tri> ¬c (λ eq → ¬b (sym eq)) a | |
| 568 ... | tri≈ ¬a b ¬c = tri≈ ¬c (sym b) ¬a | |
| 569 ... | tri> ¬a ¬b c = tri< c (λ eq → ¬b (sym eq)) ¬a | |
| 570 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) | |
| 571 scnext : {a : Ordinal} → odef schain a → odef schain (f a) | |
| 626 | 572 scnext {x} (case1 zx) = case1 (ZChain.f-next zc zx) |
| 561 | 573 scnext {x} (case2 sx) = case2 ( fsuc x sx ) |
| 574 scinit : {x : Ordinal} → odef schain x → * y ≤ * x | |
| 626 | 575 scinit {x} (case1 zx) = ZChain.initial zc zx |
| 653 | 576 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∋init zc ) ) |
| 562 | 577 ... | case1 sp=x | case1 y=sp = case1 (trans y=sp (subst (λ k → k ≡ * x ) *iso sp=x ) ) |
| 578 ... | case1 sp=x | case2 y<sp = case2 (subst (λ k → * y < k ) (trans (sym *iso) sp=x) y<sp ) | |
| 579 ... | case2 sp<x | case1 y=sp = case2 (subst (λ k → k < * x ) (trans *iso (sym y=sp )) sp<x ) | |
| 580 ... | case2 sp<x | case2 y<sp = case2 (ptrans y<sp (subst (λ k → k < * x ) *iso sp<x) ) | |
| 604 | 581 A∋za : {a : Ordinal } → odef chain0 a → odef A a |
| 626 | 582 A∋za za = ZChain.chain⊆A zc za |
| 604 | 583 za<sup : {a : Ordinal } → odef chain0 a → ( * a ≡ sp ) ∨ ( * a < sp ) |
| 584 za<sup za = SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) za ) | |
| 571 | 585 s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) |
| 586 → HasPrev A schain ab f ∨ IsSup A schain ab | |
| 569 | 587 → * a < * b → odef schain b |
| 571 | 588 s-ismax {a} {b} sa b<ox ab p a<b with osuc-≡< b<ox -- b is x? |
| 600 | 589 ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) |
| 571 | 590 s-ismax {a} {b} (case1 za) b<ox ab (case1 p) a<b | case2 b<x = z21 p where -- has previous |
| 568 | 591 z21 : HasPrev A schain ab f → odef schain b |
| 567 | 592 z21 record { y = y ; ay = (case1 zy) ; x=fy = x=fy } = |
| 626 | 593 case1 (ZChain.is-max zc za (zc-b<x b b<x) ab (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) |
| 567 | 594 z21 record { y = y ; ay = (case2 sy) ; x=fy = x=fy } = subst (λ k → odef schain k) (sym x=fy) (case2 (fsuc y sy) ) |
| 626 | 595 s-ismax {a} {b} (case1 za) b<ox ab (case2 p) a<b | case2 b<x = case1 (ZChain.is-max zc za (zc-b<x b b<x) ab (case2 z22) a<b ) where -- previous sup |
| 596 z22 : IsSup A (ZChain.chain zc) ab | |
| 571 | 597 z22 = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } |
| 598 s-ismax {a} {b} (case2 sa) b<ox ab (case1 p) a<b | case2 b<x with HasPrev.ay p | |
| 626 | 599 ... | case1 zy = case1 (subst (λ k → odef chain0 k ) (sym (HasPrev.x=fy p)) (ZChain.f-next zc zy )) -- in previous closure of f |
| 571 | 600 ... | 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 |
| 626 | 601 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 zc |
| 602 z24 : IsSup A schain ab → IsSup A (ZChain.chain zc) ab | |
| 571 | 603 z24 p = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } |
| 604 | 604 z23 : odef chain0 b |
| 653 | 605 z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋init zc ) |
| 606 ... | case1 y=b = subst (λ k → odef chain0 k ) y=b ( ZChain.chain∋init zc ) | |
| 607 ... | case2 y<b = ZChain.is-max zc (ZChain.chain∋init zc ) (zc-b<x b b<x) ab (case2 (z24 p)) y<b | |
| 624 | 608 seq : schain ≡ supf0 x |
| 611 | 609 seq with trio< x x |
| 610 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) | |
| 624 | 611 ... | tri≈ ¬a b ¬c = refl |
| 612 ... | tri> ¬a ¬b c = refl | |
| 613 seq<x : {b : Ordinal } → b o< x → supf b ≡ supf0 b | |
| 611 | 614 seq<x {b} b<x with trio< b x |
| 615 ... | tri< a ¬b ¬c = refl | |
| 616 ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) | |
| 617 ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) | |
| 618 | |
| 619 ... | case2 ¬x=sup = no-extenion z18 where -- x is not f y' nor sup of former ZChain from y -- no extention | |
| 626 | 620 z18 : {a b : Ordinal} → odef (ZChain.chain zc) a → b o< osuc x → (ab : odef A b) → |
| 621 HasPrev A (ZChain.chain zc) ab f ∨ IsSup A (ZChain.chain zc) ab → | |
| 622 * a < * b → odef (ZChain.chain zc) b | |
| 568 | 623 z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x |
| 626 | 624 ... | case2 lt = ZChain.is-max zc za (zc-b<x b lt) ab p a<b |
| 565 | 625 ... | case1 b=x with p |
| 567 | 626 ... | 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 | 627 ... | case2 b=sup = ⊥-elim ( ¬x=sup record { |
| 628 x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy) } ) | |
| 654 | 629 ... | no ¬ox = record { chain⊆A = {!!} ; f-next = {!!} ; f-total = ? |
| 630 ; initial = {!!} ; chain∋init = {!!} ; is-max = {!!} } where --- limit ordinal case | |
| 653 | 631 supf : Ordinal → HOD |
| 661 | 632 supf x = ZChain1.chain (zc0 x) |
| 633 uzc : {z : Ordinal} → (u : UChain x ? z) → ZChain A f ay zc0 (UChain.u u) | |
| 655 | 634 uzc {z} u = prev (UChain.u u) (UChain.u<x u) |
| 554 | 635 Uz : HOD |
| 661 | 636 Uz = record { od = record { def = λ z → odef A z ∧ ( UChain z ? x ∨ FClosure A f y z ) } ; odmax = & A ; <odmax = ? } |
|
572
427e36467a18
ZChain is monotonic on x, should be in record ZFChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
571
diff
changeset
|
637 u-next : {z : Ordinal} → odef Uz z → odef Uz (f z) |
| 655 | 638 u-next {z} = ? |
| 639 -- (case1 u) = case1 record { u = UChain.u u ; u<x = UChain.u<x u ; chain∋z = ZChain.f-next ( uzc u ) (UChain.chain∋z u) } | |
| 640 -- u-next {z} (case2 u) = case2 ( fsuc _ u ) | |
|
572
427e36467a18
ZChain is monotonic on x, should be in record ZFChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
571
diff
changeset
|
641 u-initial : {z : Ordinal} → odef Uz z → * y ≤ * z |
| 655 | 642 u-initial {z} = ? |
| 643 -- (case1 u) = ZChain.initial ( uzc u ) (UChain.chain∋z u) | |
| 644 -- u-initial {z} (case2 u) = s≤fc _ f mf u | |
| 653 | 645 u-chain∋init : odef Uz y |
| 655 | 646 u-chain∋init = ? -- case2 ( init ay ) |
| 624 | 647 supf0 : Ordinal → HOD |
| 611 | 648 supf0 z with trio< z x |
| 661 | 649 ... | tri< a ¬b ¬c = ZChain1.chain (zc0 z) |
| 624 | 650 ... | tri≈ ¬a b ¬c = Uz |
| 651 ... | tri> ¬a ¬b c = Uz | |
| 654 | 652 u-mono : {z : Ordinal} {w : Ordinal} → z o≤ w → w o≤ x → supf0 z ⊆' supf0 w |
| 653 u-mono {z} {w} z≤w w≤x {i} with trio< z x | trio< w x | |
| 654 ... | s | t = ? | |
| 655 | |
| 624 | 656 seq : Uz ≡ supf0 x |
| 611 | 657 seq with trio< x x |
| 658 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) | |
| 624 | 659 ... | tri≈ ¬a b ¬c = refl |
| 660 ... | tri> ¬a ¬b c = refl | |
| 661 | 661 seq<x : {b : Ordinal } → (b<x : b o< x ) → ZChain1.chain (zc0 b) ≡ supf0 b |
| 611 | 662 seq<x {b} b<x with trio< b x |
| 661 | 663 ... | tri< a ¬b ¬c = cong (λ k → ZChain1.chain (zc0 b)) o<-irr -- b<x ≡ a |
| 611 | 664 ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) |
| 665 ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) | |
| 626 | 666 ord≤< : {x y z : Ordinal} → x o< z → z o≤ y → x o< y |
| 667 ord≤< {x} {y} {z} x<z z≤y with osuc-≡< z≤y | |
| 668 ... | case1 z=y = subst (λ k → x o< k ) z=y x<z | |
| 669 ... | case2 z<y = ordtrans x<z z<y | |
| 553 | 670 |
| 661 | 671 SZ0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain1 A f ay x |
| 672 SZ0 f mf ay x = TransFinite {λ z → ZChain1 A f ay z} (sind f mf ay ) x | |
| 629 | 673 |
| 655 | 674 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f ay (SZ0 f mf ay) (& A) |
| 675 SZ f mf {y} ay = TransFinite {λ z → ZChain A f ay (SZ0 f mf ay) z } (λ x → ind f mf ay x (SZ0 f mf ay) ) (& A) | |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
676 |
| 551 | 677 zorn00 : Maximal A |
| 678 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 679 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 680 -- yes we have the maximal | |
| 681 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 682 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 683 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 684 zorn01 = proj1 zorn03 | |
| 685 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 686 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 653 | 687 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zc0 zorn04 total ) where |
| 551 | 688 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 689 nmx : ¬ Maximal A | |
| 690 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 691 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 692 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) ) ⟫ | |
| 661 | 693 zc0 : (x : Ordinal) → ZChain1 A (cf nmx) as0 x |
| 694 zc0 x = TransFinite {λ z → ZChain1 A (cf nmx) as0 z} (sind (cf nmx) (cf-is-≤-monotonic nmx) as0) x | |
| 655 | 695 zorn04 : ZChain A (cf nmx) as0 zc0 (& A) |
| 653 | 696 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
| 634 | 697 total : IsTotalOrderSet (ZChain.chain zorn04) |
| 654 | 698 total {a} {b} = zorn06 where |
| 699 zorn06 : odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a) | |
| 700 zorn06 = ZChain.f-total (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 |