Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 605:b42f0e50a831
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 14 Jun 2022 16:05:25 +0900 |
| parents | 021fcb324990 |
| children | 9bdb671cbffd |
| 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
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changeset
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66 _≤_ : (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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67 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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68 |
| 554 | 69 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 70 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 71 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 72 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 73 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 74 | |
| 556 | 75 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 76 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 77 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 78 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 79 |
| 561 | 80 ptrans = IsStrictPartialOrder.trans PO |
| 81 | |
| 492 | 82 open _==_ |
| 83 open _⊆_ | |
| 84 | |
| 530 | 85 -- |
| 560 | 86 -- Closure of ≤-monotonic function f has total order |
| 530 | 87 -- |
| 88 | |
| 89 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 90 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 91 | |
| 551 | 92 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 600 | 93 init : odef A s → FClosure A f s s |
| 555 | 94 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 95 |
| 556 | 96 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y |
| 600 | 97 A∋fc {A} s f mf (init as) = as |
| 556 | 98 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 99 |
| 556 | 100 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 600 | 101 s≤fc {A} s {.s} f mf (init x) = case1 refl |
| 556 | 102 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 103 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 104 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 105 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 106 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 107 |
| 557 | 108 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 600 | 109 fcn s mf (init as) = zero |
| 558 | 110 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) |
| 111 ... | case1 eq = fcn s mf p | |
| 112 ... | case2 y<fy = suc (fcn s mf p ) | |
| 557 | 113 |
| 558 | 114 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 115 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 559 | 116 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where |
| 117 fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y | |
| 600 | 118 fc00 zero zero refl (init _) (init x₁) i=x i=y = refl |
| 119 fc00 zero zero refl (init as) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 120 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as) cy i=x i=y ) | |
| 121 fc00 zero zero refl (fsuc x cx) (init as) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 122 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as) i=x i=y ) | |
| 559 | 123 fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) |
| 124 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) | |
| 125 fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 126 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) | |
| 127 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 128 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 129 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 560 | 130 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x |
| 559 | 131 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where |
| 132 fc04 : * x1 ≡ * y | |
| 133 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 134 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 135 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 136 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 137 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 138 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 139 fc05 : * x ≡ * y1 | |
| 140 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 141 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) | |
| 557 | 142 |
| 600 | 143 |
| 557 | 144 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 145 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 558 | 146 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where |
| 147 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y | |
| 148 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 149 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 150 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 151 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 152 ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy | |
| 153 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 154 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 155 fc03 eq = cong pred eq | |
| 156 fc02 : * x < * y1 | |
| 157 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 557 | 158 |
| 559 | 159 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 160 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 559 | 161 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 162 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 163 fc11 : * x < * y | |
| 164 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 165 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 166 fc10 : * x ≡ * y | |
| 167 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 168 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 169 fc12 : * y < * x | |
| 170 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 171 | |
| 600 | 172 |
| 562 | 173 fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 174 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
| 563 | 175 fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where |
| 176 fc20 : fcn s mf cy Data.Nat.< suc (fcn s mf cx) → (fcn s mf cy ≡ fcn s mf cx) ∨ ( fcn s mf cy Data.Nat.< fcn s mf cx ) | |
| 177 fc20 y<sx with <-cmp ( fcn s mf cy ) (fcn s mf cx ) | |
| 178 ... | tri< a ¬b ¬c = case2 a | |
| 179 ... | tri≈ ¬a b ¬c = case1 b | |
| 180 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> y<sx (s≤s c)) | |
| 181 fc17 : {x y : Ordinal } → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → suc (fcn s mf cx) ≡ fcn s mf cy → * (f x ) ≡ * y | |
| 182 fc17 {x} {y} cx cy sx=y = fc18 (fcn s mf cy) cx cy refl sx=y where | |
| 183 fc18 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → suc (fcn s mf cx) ≡ i → * (f x) ≡ * y | |
| 184 fc18 (suc i) {y} cx (fsuc y1 cy) i=y sx=i with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 185 ... | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy ( fc18 (suc i) {y1} cx cy i=y sx=i) -- dereference | |
| 186 ... | case2 y<fy = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k) ) ) fc19) where | |
| 187 fc19 : * x ≡ * y1 | |
| 188 fc19 = fcn-inject s mf cx cy (cong pred ( trans sx=i i=y )) | |
| 189 fc21 : ⊥ | |
| 190 fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy ) | |
| 191 ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy | |
| 192 cxx : FClosure A f s (f x) | |
| 193 cxx = fsuc x cx | |
| 194 fc16 : (x : Ordinal ) → (cx : FClosure A f s x) → (fcn s mf cx ≡ fcn s mf (fsuc x cx)) ∨ ( suc (fcn s mf cx ) ≡ fcn s mf (fsuc x cx)) | |
| 600 | 195 fc16 x (init as) with proj1 (mf s as ) |
| 563 | 196 ... | case1 _ = case1 refl |
| 197 ... | case2 _ = case2 refl | |
| 198 fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) | |
| 199 ... | case1 _ = case1 refl | |
| 200 ... | case2 _ = case2 refl | |
| 201 fc22 : (suc ( fcn s mf cx )) Data.Nat.< (fcn s mf cy ) → * (f x) < * y | |
| 202 fc22 a with fc16 x cx | |
| 203 ... | case1 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq (<-trans a<sa a)) | |
| 204 ... | case2 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq a ) | |
| 205 ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx | |
| 206 ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx | |
| 207 ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y | |
| 208 ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) | |
| 209 | |
| 560 | 210 -- open import Relation.Binary.Properties.Poset as Poset |
| 211 | |
| 212 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 213 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 214 | |
| 567 | 215 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 216 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 217 |
| 568 | 218 _⊆'_ : ( A B : HOD ) → Set n |
| 219 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 220 |
| 221 -- | |
| 222 -- inductive maxmum tree from x | |
| 223 -- tree structure | |
| 224 -- | |
| 554 | 225 |
| 567 | 226 record HasPrev (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 227 field |
| 534 | 228 y : Ordinal |
| 541 | 229 ay : odef B y |
| 534 | 230 x=fy : x ≡ f y |
| 529 | 231 |
| 570 | 232 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 567 | 233 field |
| 571 | 234 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 568 | 235 |
| 603 | 236 record FChain ( A : HOD ) ( f : Ordinal → Ordinal ) (p c z : Ordinal) ( x : Ordinal ) : Set n where |
| 590 | 237 field |
| 598 | 238 fc∨sup : FClosure A f p x |
| 239 chain∋p : odef (* c) p | |
| 592 | 240 |
| 603 | 241 record FSup ( A : HOD ) ( f : Ordinal → Ordinal ) (p c z : Ordinal) ( x : Ordinal ) : Set n where |
| 600 | 242 field |
| 243 sup : (z : Ordinal) → FClosure A f p z → * z < * x | |
| 244 chain∋p : odef (* c) p | |
| 245 | |
| 603 | 246 data Fc∨sup (A : HOD) {y : Ordinal} (ay : odef A y) ( f : Ordinal → Ordinal ) (c z : Ordinal) : (x : Ordinal) → Set n where |
| 247 Finit : {i : Ordinal} → i ≡ y → Fc∨sup A ay f c z i | |
| 248 Fsup : {p x : Ordinal} → p o< x → Fc∨sup A ay f c z p → FSup A f p c z x → x o< osuc z → Fc∨sup A ay f c z x | |
| 249 Fc : {p x : Ordinal} → p o< x → Fc∨sup A ay f c z p → FChain A f p c z x → Fc∨sup A ay f c z x | |
| 592 | 250 |
| 251 record ZChain ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) ( z : Ordinal ) : Set (Level.suc n) where | |
| 568 | 252 field |
| 253 chain : HOD | |
| 254 chain⊆A : chain ⊆' A | |
| 255 chain∋x : odef chain x | |
| 256 initial : {y : Ordinal } → odef chain y → * x ≤ * y | |
| 257 f-total : IsTotalOrderSet chain | |
| 258 f-next : {a : Ordinal } → odef chain a → odef chain (f a) | |
| 259 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
| 260 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc z → (ab : odef A b) | |
| 574 | 261 → HasPrev A chain ab f ∨ IsSup A chain ab |
| 568 | 262 → * a < * b → odef chain b |
| 603 | 263 chain∋sup : (s : HOD) → s ⊆' chain → {b : Ordinal} (ab : odef A b) → b o< osuc z → IsSup A s ab → odef chain b |
| 264 fc∨sup : {c : Ordinal } → ( ca : odef chain c ) → Fc∨sup A (chain⊆A chain∋x) f (& chain) z c | |
| 582 | 265 |
| 595 | 266 |
| 568 | 267 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 268 field | |
| 269 maximal : HOD | |
| 270 A∋maximal : A ∋ maximal | |
| 271 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 567 | 272 |
| 508 | 273 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 503 | 274 field |
| 275 sup : HOD | |
| 276 A∋maximal : A ∋ sup | |
| 277 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 278 | |
| 533 | 279 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
| 280 SupCond A B _ _ = SUP A B | |
| 281 | |
| 497 | 282 Zorn-lemma : { A : HOD } |
| 464 | 283 → o∅ o< & A |
| 568 | 284 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 285 → Maximal A |
| 552 | 286 Zorn-lemma {A} 0<A supP = zorn00 where |
| 568 | 287 supO : (C : HOD ) → C ⊆' A → IsTotalOrderSet C → Ordinal |
| 566 | 288 supO C C⊆A TC = & ( SUP.sup ( supP C C⊆A TC )) |
| 569 | 289 postulate |
| 290 --- irrelevance of ⊆' and compare | |
| 291 sup== : {C C1 : HOD } → C ≡ C1 → {c : C ⊆' A } {c1 : C1 ⊆' A } → {t : IsTotalOrderSet C } {t1 : IsTotalOrderSet C1 } | |
| 292 → SUP.sup ( supP C c t ) ≡ SUP.sup ( supP C1 c1 t1 ) | |
| 571 | 293 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 294 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 537 | 295 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 296 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 297 s : HOD |
| 298 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 299 as : A ∋ * ( & s ) |
| 300 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 )) ) | |
| 547 | 301 s<A : & s o< & A |
| 568 | 302 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 303 HasMaximal : HOD |
| 537 | 304 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 305 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 306 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 307 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 308 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 309 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 310 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 311 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 312 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)) | |
| 313 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 | |
| 314 ¬x<m : ¬ (* x < * m) | |
| 315 ¬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 | 316 |
| 560 | 317 -- Uncountable ascending chain by axiom of choice |
| 530 | 318 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 319 cf nmx x with ODC.∋-p O A (* x) |
| 320 ... | no _ = o∅ | |
| 321 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 322 ... | yes nogt = -- no larger element, so it is maximal |
| 323 ⊥-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 | 324 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 325 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 326 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 327 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 328 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 329 ... | 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 ⟫ ) | |
| 330 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 530 | 331 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 332 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 333 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 334 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 335 |
| 582 | 336 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A (& s) f (& A) ) → SUP A (ZChain.chain zc) |
| 533 | 337 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
| 582 | 338 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A (& s) (cf nmx) (& A) ) |
| 533 | 339 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
| 340 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
| 582 | 341 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (& s) f (& A) ) → SUP A (ZChain.chain zc) |
| 566 | 342 sp0 f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain.f-total zc) |
| 543 | 343 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 344 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 345 | |
| 346 --- | |
| 560 | 347 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 348 --- |
| 582 | 349 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (& s) f (& A) ) |
| 546 | 350 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) |
| 571 | 351 fixpoint f mf zc = z14 where |
| 538 | 352 chain = ZChain.chain zc |
| 353 sp1 = sp0 f mf zc | |
| 565 | 354 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc (& A) → (ab : odef A b ) |
| 570 | 355 → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 356 → * a < * b → odef chain b |
| 357 z10 = ZChain.is-max zc | |
| 543 | 358 z11 : & (SUP.sup sp1) o< & A |
| 359 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 360 z12 : odef chain (& (SUP.sup sp1)) |
| 361 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 362 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
| 569 | 363 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) (ordtrans z11 <-osuc ) (SUP.A∋maximal sp1) |
| 570 | 364 (case2 z19 ) z13 where |
| 538 | 365 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 566 | 366 z13 with SUP.x<sup sp1 ( ZChain.chain∋x zc ) |
| 538 | 367 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 368 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 570 | 369 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) |
| 571 | 370 z19 = record { x<sup = z20 } where |
| 371 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
| 372 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
| 570 | 373 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
| 374 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
| 375 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
| 538 | 376 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) |
| 552 | 377 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 538 | 378 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 379 z16 : ⊥ | |
| 380 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 381 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 382 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 383 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 384 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 385 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 566 | 386 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) |
| 538 | 387 z17 : ⊥ |
| 388 z17 with z15 | |
| 389 ... | case1 eq = ¬b eq | |
| 390 ... | case2 lt = ¬a lt | |
| 560 | 391 |
| 392 -- ZChain contradicts ¬ Maximal | |
| 393 -- | |
| 571 | 394 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
| 560 | 395 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
| 396 -- | |
| 582 | 397 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (& s) (cf nmx) (& A)) → ⊥ |
| 571 | 398 z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
| 399 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) | |
| 400 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) -- x ≡ f x ̄ | |
| 401 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where -- x < f x | |
| 546 | 402 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc |
| 538 | 403 c = & (SUP.sup sp1) |
| 548 | 404 |
| 605 | 405 ys : {y : Ordinal} → (ay : odef A y) (f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → HOD |
| 406 ys {y} ay f mf = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = {!!} } | |
| 407 init-chain : {y x : Ordinal} → (ay : odef A y) (f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → x o< osuc y → ZChain A y f x | |
| 408 init-chain {y} {x} ay f mf x≤y = record { chain = ys ay f mf ; chain⊆A = λ fx → A∋fc y f mf fx | |
| 409 ; f-total = i-total ; f-next = λ {x} sx → fsuc x sx ; chain∋sup = {!!} | |
| 410 ; initial = {!!} ; f-immediate = {!!} ; chain∋x = init ay ; is-max = {!!} ; fc∨sup = {!!} } where | |
| 411 i-total : IsTotalOrderSet (ys ay f mf ) | |
| 412 i-total fa fb = subst₂ (λ a b → Tri (a < b) (a ≡ b) (b < a ) ) *iso *iso (fcn-cmp y f mf fa fb) | |
| 413 | |
| 560 | 414 -- |
| 547 | 415 -- create all ZChains under o< x |
| 560 | 416 -- |
| 546 | 417 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → |
| 582 | 418 ((z : Ordinal) → z o< x → {y : Ordinal} → (ya : odef A y) → ZChain A y f z ) → { y : Ordinal } → (ya : odef A y) → ZChain A y f x |
| 547 | 419 ind f mf x prev {y} ay with Oprev-p x |
| 548 | 420 ... | yes op = zc4 where |
| 560 | 421 -- |
| 422 -- we have previous ordinal to use induction | |
| 423 -- | |
| 604 | 424 open ZChain |
| 425 | |
| 530 | 426 px = Oprev.oprev op |
| 582 | 427 zc0 : ZChain A y f (Oprev.oprev op) |
| 547 | 428 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay |
| 569 | 429 zc0-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 430 zc0-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 604 | 431 px<x : px o< x |
| 432 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 569 | 433 |
| 604 | 434 fcs< : (A : HOD) {w y : Ordinal} (ay : odef A y) (c z : Ordinal) (x : Ordinal) |
| 435 → z o< w → Fc∨sup A ay f c z x → Fc∨sup A ay f c w x | |
| 436 fcs< A ay c z x z<w (Finit x₁) = Finit x₁ | |
| 437 fcs< A {w} ay c z x z<w (Fsup {p} x₁ FC x₂ x₃) = Fsup x₁ (fcs< A ay c z p z<w FC) record { sup = FSup.sup x₂ ; chain∋p = FSup.chain∋p x₂ } | |
| 603 | 438 (x<ow x₃ z<w ) where |
| 439 x<ow : x o< osuc z → z o< w → x o< osuc w | |
| 440 x<ow x<z z<w = ordtrans x<z (osucc z<w) | |
| 604 | 441 fcs< A {w} ay c z x z<w (Fc {p} x₁ FC x₂) = Fc x₁ (fcs< A ay c z p z<w FC) record { fc∨sup = FChain.fc∨sup x₂; chain∋p = FChain.chain∋p x₂} |
| 603 | 442 |
| 582 | 443 zc4 : ZChain A y f x |
| 565 | 444 zc4 with ODC.∋-p O A (* x) |
| 568 | 445 ... | no noax = -- ¬ A ∋ p, just skip |
| 560 | 446 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 |
| 601 | 447 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 ; chain∋sup = {!!} |
| 604 | 448 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 ; fc∨sup = {!!} } where -- no extention |
| 449 zc12 : {c : Ordinal} → odef (ZChain.chain zc0) c → Fc∨sup A (ZChain.chain⊆A zc0 (ZChain.chain∋x zc0)) f (& (ZChain.chain zc0)) x c | |
| 450 zc12 {c} cc = fcs< A (ZChain.chain⊆A zc0 (ZChain.chain∋x zc0)) (& (ZChain.chain zc0)) px c px<x (ZChain.fc∨sup zc0 cc) | |
| 568 | 451 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → |
| 570 | 452 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → |
| 551 | 453 * a < * b → odef (ZChain.chain zc0) b |
| 568 | 454 zc11 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox |
| 455 ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) ab ) ) | |
| 569 | 456 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b |
| 582 | 457 ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc0) ax f ) -- we have to check adding x preserve is-max ZChain A y f mf supO x |
| 549 | 458 ... | case1 pr = zc9 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 604 | 459 chain0 = ZChain.chain zc0 |
| 568 | 460 zc17 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → |
| 570 | 461 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → |
| 551 | 462 * a < * b → odef (ZChain.chain zc0) b |
| 568 | 463 zc17 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox |
| 569 | 464 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b |
| 604 | 465 ... | case1 b=x = subst (λ k → odef chain0 k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr)) |
| 582 | 466 zc9 : ZChain A y f x |
| 574 | 467 zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 -- no extention |
| 601 | 468 ; chain∋sup = {!!} |
| 603 | 469 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 ; fc∨sup = {!!} } |
| 570 | 470 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax ) |
| 571 | 471 ... | case1 is-sup = -- x is a sup of zc0 |
| 601 | 472 record { chain = schain ; chain⊆A = s⊆A ; f-total = scmp ; f-next = scnext ; chain∋sup = {!!} |
| 600 | 473 ; initial = scinit ; f-immediate = simm ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax ; fc∨sup = s-fc∨sup} where |
| 570 | 474 sup0 : SUP A (ZChain.chain zc0) |
| 571 | 475 sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where |
| 476 x21 : {y : HOD} → ZChain.chain zc0 ∋ y → (y ≡ * x) ∨ (y < * x) | |
| 477 x21 {y} zy with IsSup.x<sup is-sup zy | |
| 478 ... | case1 y=x = case1 ( subst₂ (λ j k → j ≡ * k ) *iso &iso ( cong (*) y=x) ) | |
| 479 ... | case2 y<x = case2 (subst₂ (λ j k → j < * k ) *iso &iso y<x ) | |
| 570 | 480 sp : HOD |
| 561 | 481 sp = SUP.sup sup0 |
| 570 | 482 x=sup : x ≡ & sp |
| 483 x=sup = sym &iso | |
| 604 | 484 chain0 = ZChain.chain zc0 |
| 485 sc<A : {y : Ordinal} → odef chain0 y ∨ FClosure A f (& sp) y → y o< & A | |
| 486 sc<A {y} (case1 zx) = subst (λ k → k o< (& A)) &iso ( c<→o< (ZChain.chain⊆A zc0 (subst (λ k → odef chain0 k) (sym &iso) zx ))) | |
| 561 | 487 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 | 488 schain : HOD |
| 604 | 489 schain = record { od = record { def = λ x → odef chain0 x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = λ {y} sy → sc<A {y} sy } |
| 561 | 490 A∋schain : {x : HOD } → schain ∋ x → A ∋ x |
| 569 | 491 A∋schain (case1 zx ) = ZChain.chain⊆A zc0 zx |
| 561 | 492 A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx |
| 569 | 493 s⊆A : schain ⊆' A |
| 494 s⊆A {x} (case1 zx) = ZChain.chain⊆A zc0 zx | |
| 495 s⊆A {x} (case2 fx) = A∋fc (& sp) f mf fx | |
| 603 | 496 s-fc∨sup : {c : Ordinal} → odef schain c → Fc∨sup A (s⊆A (case1 (ZChain.chain∋x zc0))) f (& schain) x c |
| 600 | 497 s-fc∨sup {c} (case1 cx) = {!!} |
| 498 s-fc∨sup {c} (case2 fc) = {!!} | |
| 604 | 499 cmp : {a b : HOD} (za : odef chain0 (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) |
| 561 | 500 cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb |
| 501 ... | case1 sp=a | case1 sp=b = tri≈ (λ lt → <-irr (case1 (sym eq)) lt ) eq (λ lt → <-irr (case1 eq) lt ) where | |
| 502 eq : a ≡ b | |
| 503 eq = trans sp=a (subst₂ (λ j k → j ≡ k ) *iso *iso sp=b ) | |
| 504 ... | 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 | |
| 505 a<b : a < b | |
| 506 a<b = subst (λ k → k < b ) (sym sp=a) (subst₂ (λ j k → j < k ) *iso *iso sp<b ) | |
| 507 ... | 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 | |
| 508 a<b : a < b | |
| 509 a<b = subst (λ k → a < k ) (trans sp=b *iso ) (subst (λ k → a < k ) (sym *iso) a<sp ) | |
| 510 ... | 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 | |
| 511 a<b : a < b | |
| 512 a<b = ptrans (subst (λ k → a < k ) (sym *iso) a<sp ) ( subst₂ (λ j k → j < k ) refl *iso sp<b ) | |
| 513 scmp : {a b : HOD} → odef schain (& a) → odef schain (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 514 scmp (case1 za) (case1 zb) = ZChain.f-total zc0 za zb | |
| 515 scmp {a} {b} (case1 za) (case2 fb) = cmp za fb | |
| 516 scmp (case2 fa) (case1 zb) with cmp zb fa | |
| 517 ... | tri< a ¬b ¬c = tri> ¬c (λ eq → ¬b (sym eq)) a | |
| 518 ... | tri≈ ¬a b ¬c = tri≈ ¬c (sym b) ¬a | |
| 519 ... | tri> ¬a ¬b c = tri< c (λ eq → ¬b (sym eq)) ¬a | |
| 520 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) | |
| 521 scnext : {a : Ordinal} → odef schain a → odef schain (f a) | |
| 522 scnext {x} (case1 zx) = case1 (ZChain.f-next zc0 zx) | |
| 523 scnext {x} (case2 sx) = case2 ( fsuc x sx ) | |
| 524 scinit : {x : Ordinal} → odef schain x → * y ≤ * x | |
| 525 scinit {x} (case1 zx) = ZChain.initial zc0 zx | |
| 604 | 526 scinit {x} (case2 sx) with (s≤fc (& sp) f mf sx ) | SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) ( ZChain.chain∋x zc0 ) ) |
| 562 | 527 ... | case1 sp=x | case1 y=sp = case1 (trans y=sp (subst (λ k → k ≡ * x ) *iso sp=x ) ) |
| 528 ... | case1 sp=x | case2 y<sp = case2 (subst (λ k → * y < k ) (trans (sym *iso) sp=x) y<sp ) | |
| 529 ... | case2 sp<x | case1 y=sp = case2 (subst (λ k → k < * x ) (trans *iso (sym y=sp )) sp<x ) | |
| 530 ... | case2 sp<x | case2 y<sp = case2 (ptrans y<sp (subst (λ k → k < * x ) *iso sp<x) ) | |
| 604 | 531 A∋za : {a : Ordinal } → odef chain0 a → odef A a |
| 569 | 532 A∋za za = ZChain.chain⊆A zc0 za |
| 604 | 533 za<sup : {a : Ordinal } → odef chain0 a → ( * a ≡ sp ) ∨ ( * a < sp ) |
| 534 za<sup za = SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) za ) | |
| 562 | 535 simm : {a b : Ordinal} → odef schain a → odef schain b → ¬ (* a < * b) ∧ (* b < * (f a)) |
| 536 simm {a} {b} (case1 za) (case1 zb) = ZChain.f-immediate zc0 za zb | |
| 537 simm {a} {b} (case1 za) (case2 sb) p with proj1 (mf a (A∋za za) ) | |
| 538 ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) | |
| 539 ... | case2 a<fa with za<sup ( ZChain.f-next zc0 za ) | s≤fc (& sp) f mf sb | |
| 540 ... | case1 fa=sp | case1 sp=b = <-irr (case1 (trans fa=sp (trans (sym *iso) sp=b )) ) ( proj2 p ) | |
| 541 ... | case2 fa<sp | case1 sp=b = <-irr (case2 fa<sp) (subst (λ k → k < * (f a) ) (trans (sym sp=b) *iso) (proj2 p ) ) | |
| 542 ... | 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 ) ) | |
| 543 ... | case2 fa<sp | case2 sp<b = <-irr (case2 (proj2 p )) (ptrans fa<sp (subst (λ k → k < * b ) *iso sp<b ) ) | |
| 544 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) )) ) ) | |
| 545 ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) | |
| 546 ... | case2 b<fb with s≤fc (& sp) f mf sa | za<sup zb | |
| 547 ... | case1 sp=a | case1 b=sp = <-irr (case1 (trans b=sp (trans (sym *iso) sp=a )) ) (proj1 p ) | |
| 548 ... | case1 sp=a | case2 b<sp = <-irr (case2 (subst (λ k → * b < k ) (trans (sym *iso) sp=a) b<sp ) ) (proj1 p ) | |
| 549 ... | case2 sp<a | case1 b=sp = <-irr (case2 (subst ( λ k → k < * a ) (trans *iso (sym b=sp)) sp<a )) (proj1 p ) | |
| 550 ... | case2 sp<a | case2 b<sp = <-irr (case2 (ptrans b<sp (subst (λ k → k < * a) *iso sp<a ))) (proj1 p ) | |
| 564 | 551 simm {a} {b} (case2 sa) (case2 sb) p = fcn-imm {A} (& sp) {a} {b} f mf sa sb p |
| 571 | 552 s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) |
| 553 → HasPrev A schain ab f ∨ IsSup A schain ab | |
| 569 | 554 → * a < * b → odef schain b |
| 571 | 555 s-ismax {a} {b} sa b<ox ab p a<b with osuc-≡< b<ox -- b is x? |
| 600 | 556 ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) |
| 571 | 557 s-ismax {a} {b} (case1 za) b<ox ab (case1 p) a<b | case2 b<x = z21 p where -- has previous |
| 568 | 558 z21 : HasPrev A schain ab f → odef schain b |
| 567 | 559 z21 record { y = y ; ay = (case1 zy) ; x=fy = x=fy } = |
| 569 | 560 case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) |
| 567 | 561 z21 record { y = y ; ay = (case2 sy) ; x=fy = x=fy } = subst (λ k → odef schain k) (sym x=fy) (case2 (fsuc y sy) ) |
| 571 | 562 s-ismax {a} {b} (case1 za) b<ox ab (case2 p) a<b | case2 b<x = case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case2 z22) a<b ) where -- previous sup |
| 563 z22 : IsSup A (ZChain.chain zc0) ab | |
| 564 z22 = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } | |
| 565 s-ismax {a} {b} (case2 sa) b<ox ab (case1 p) a<b | case2 b<x with HasPrev.ay p | |
| 604 | 566 ... | case1 zy = case1 (subst (λ k → odef chain0 k ) (sym (HasPrev.x=fy p)) (ZChain.f-next zc0 zy )) -- in previous closure of f |
| 571 | 567 ... | case2 sy = case2 (subst (λ k → FClosure A f (& (* x)) k ) (sym (HasPrev.x=fy p)) (fsuc (HasPrev.y p) sy )) -- in current closure of f |
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568 s-ismax {a} {b} (case2 sa) b<ox ab (case2 p) a<b | case2 b<x = case1 z23 where -- sup o< x is already in zc0 |
| 571 | 569 z24 : IsSup A schain ab → IsSup A (ZChain.chain zc0) ab |
| 570 z24 p = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } | |
| 604 | 571 z23 : odef chain0 b |
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572 z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋x zc0 ) |
| 604 | 573 ... | case1 y=b = subst (λ k → odef chain0 k ) y=b ( ZChain.chain∋x zc0 ) |
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574 ... | case2 y<b = ZChain.is-max zc0 (ZChain.chain∋x zc0 ) (zc0-b<x b b<x) ab (case2 (z24 p)) y<b |
| 560 | 575 ... | case2 ¬x=sup = -- x is not f y' nor sup of former ZChain from y |
| 601 | 576 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 ; chain∋sup = {!!} |
| 603 | 577 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = z18 ; fc∨sup = {!!} } where |
| 574 | 578 -- no extention |
| 568 | 579 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → |
| 570 | 580 HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → |
| 552 | 581 * a < * b → odef (ZChain.chain zc0) b |
| 568 | 582 z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x |
| 569 | 583 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab p a<b |
| 565 | 584 ... | case1 b=x with p |
| 567 | 585 ... | 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 | 586 ... | case2 b=sup = ⊥-elim ( ¬x=sup record { |
| 587 x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy) } ) | |
| 580 | 588 ... | no ¬ox with trio< x y |
| 605 | 589 ... | tri< a ¬b ¬c = init-chain ay f mf {!!} |
| 590 ... | tri≈ ¬a b ¬c = init-chain ay f mf {!!} | |
| 600 | 591 ... | tri> ¬a ¬b y<x = UnionZ where |
| 582 | 592 UnionZ : ZChain A y f x |
| 601 | 593 UnionZ = record { chain = Uz ; chain⊆A = Uz⊆A ; f-total = u-total ; f-next = u-next ; chain∋sup = {!!} |
| 574 | 594 ; initial = u-initial ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} ; fc∨sup = {!!} } where --- limit ordinal case |
| 554 | 595 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x |
| 553 | 596 field |
| 597 u : Ordinal | |
| 598 u<x : u o< x | |
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599 chain∋z : odef (ZChain.chain (prev u u<x {y} ay )) z |
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600 Uz⊆A : {z : Ordinal} → UZFChain z → odef A z |
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601 Uz⊆A {z} u = ZChain.chain⊆A ( prev (UZFChain.u u) (UZFChain.u<x u) {y} ay ) (UZFChain.chain∋z u) |
| 582 | 602 uzc : {z : Ordinal} → (u : UZFChain z) → ZChain A y f (UZFChain.u u) |
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603 uzc {z} u = prev (UZFChain.u u) (UZFChain.u<x u) {y} ay |
| 554 | 604 Uz : HOD |
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605 Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A |
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606 ; <odmax = λ lt → subst (λ k → k o< & A ) &iso (c<→o< (subst (λ k → odef A k ) (sym &iso) (Uz⊆A lt))) } |
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607 u-next : {z : Ordinal} → odef Uz z → odef Uz (f z) |
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608 u-next {z} u = record { u = UZFChain.u u ; u<x = UZFChain.u<x u ; chain∋z = ZChain.f-next ( uzc u ) (UZFChain.chain∋z u) } |
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609 u-initial : {z : Ordinal} → odef Uz z → * y ≤ * z |
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610 u-initial {z} u = ZChain.initial ( uzc u ) (UZFChain.chain∋z u) |
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611 u-chain∋x : odef Uz y |
| 580 | 612 u-chain∋x = record { u = y ; u<x = y<x ; chain∋z = ZChain.chain∋x (prev y y<x ay ) } |
| 596 | 613 u-mono : ( a b : Ordinal ) → b o< x → a o< osuc b → (za : ZChain A y f a) (zb : ZChain A y f b) → ZChain.chain za ⊆' ZChain.chain zb |
| 597 | 614 u-mono a b b<x a≤b za zb {i} zai = TransFinite0 {λ i → odef (chain za) i → odef (chain zb) i } uind i zai where |
| 590 | 615 open ZChain |
| 597 | 616 uind : (i : Ordinal) |
| 617 → ((j : Ordinal) → j o< i → odef (chain za) j → odef (chain zb) j) | |
| 618 → odef (chain za) i → odef (chain zb) i | |
| 619 uind i previ zai = um01 where | |
| 603 | 620 FC : Fc∨sup A (chain⊆A za (chain∋x za)) f (& (chain za)) a i |
| 621 FC = fc∨sup za zai | |
| 592 | 622 um01 : odef (chain zb) i |
| 623 um01 with FC | |
| 593 | 624 ... | Finit i=y = subst (λ k → odef (chain zb) k ) (sym i=y) ( chain∋x zb ) |
| 603 | 625 ... | Fsup {p} {i} p<i pFC sup i≤b = cb∋i where |
| 601 | 626 i-asup : (z : Ordinal) → FClosure A f p z → * z < * i |
| 627 i-asup = FSup.sup sup | |
| 628 um06 : odef (chain za) p | |
| 629 um06 = subst (λ k → odef k p ) *iso ( FSup.chain∋p sup ) | |
| 630 -- FClosure A f p is a subset of chain zb | |
| 631 um07 : (i2 : Ordinal) → FClosure A f p i2 → odef (chain zb) i2 | |
| 632 um07 i2 (init ap ) = previ p p<i um06 | |
| 633 um07 i (fsuc j fc) = f-next zb ( um07 j fc ) | |
| 634 um08 : odef (chain zb) p | |
| 635 um08 = previ p p<i um06 | |
| 636 cl : HOD | |
| 602 | 637 cl = record { od = record { def = λ x → FClosure A f p x } ; odmax = & A ; <odmax = cl<A } where |
| 638 cl<A : {i2 : Ordinal} → FClosure A f p i2 → i2 o< & A | |
| 639 cl<A {i2} cl = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k) (sym &iso) (A∋fc p f mf cl) ) ) | |
| 601 | 640 cb∋i : odef (chain zb) i |
| 602 | 641 cb∋i = ZChain.chain∋sup zb cl (λ {i2} lt → um07 i2 lt) (chain⊆A za zai) {!!} record { x<sup = λ {i2} cl → case2 (i-asup i2 cl) } |
| 598 | 642 ... | Fc {p} {i} p<i pFC FC with (FChain.fc∨sup FC) |
| 643 ... | fc = um02 i fc where | |
| 597 | 644 um02 : (i2 : Ordinal) → FClosure A f p i2 → odef (chain zb) i2 |
| 600 | 645 um02 i2 (init ap ) = previ p p<i um04 where |
| 597 | 646 um04 : odef (chain za) p |
| 598 | 647 um04 = subst (λ k → odef k p ) *iso ( FChain.chain∋p FC ) |
| 595 | 648 um02 i (fsuc j fc) = f-next zb ( um02 j fc ) |
| 554 | 649 u-total : IsTotalOrderSet Uz |
| 598 | 650 u-total {x} {y} ux uy with trio< (UZFChain.u ux) (UZFChain.u uy) |
| 599 | 651 ... | tri< a ¬b ¬c = ZChain.f-total (uzc uy) (u-mono (UZFChain.u ux) (UZFChain.u uy) |
| 652 (UZFChain.u<x uy) (ordtrans a <-osuc ) (uzc ux) (uzc uy) (UZFChain.chain∋z ux)) (UZFChain.chain∋z uy) | |
| 653 ... | tri≈ ¬a b ¬c = ZChain.f-total (uzc ux) (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) | |
| 654 (UZFChain.u<x ux) (subst (λ k → k o< osuc (UZFChain.u ux)) b <-osuc) (uzc uy) (uzc ux) (UZFChain.chain∋z uy)) | |
| 655 ... | tri> ¬a ¬b c = ZChain.f-total (uzc ux) (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) | |
| 656 (UZFChain.u<x ux) (ordtrans c <-osuc) (uzc uy) (uzc ux) (UZFChain.chain∋z uy)) | |
| 553 | 657 |
| 551 | 658 zorn00 : Maximal A |
| 659 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 660 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 661 -- yes we have the maximal | |
| 662 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 663 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 664 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 665 zorn01 = proj1 zorn03 | |
| 666 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 667 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 668 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where | |
| 669 -- if we have no maximal, make ZChain, which contradict SUP condition | |
| 670 nmx : ¬ Maximal A | |
| 671 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 672 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 673 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) ) ⟫ | |
| 582 | 674 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A (& s) f (& A) |
| 675 zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain A y f z } (ind f mf) (& A) | |
| 676 zorn04 : ZChain A (& s) (cf nmx) (& A) | |
| 568 | 677 zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
| 551 | 678 |
| 516 | 679 -- usage (see filter.agda ) |
| 680 -- | |
| 497 | 681 -- _⊆'_ : ( A B : HOD ) → Set n |
| 682 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 683 |
| 497 | 684 -- MaximumSubset : {L P : HOD} |
| 685 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 686 -- → IsPartialOrderSet P _⊆'_ | |
| 687 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 688 -- → Maximal P (_⊆'_) | |
| 689 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |