Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 560:d09f9a1d6c2f
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 30 Apr 2022 05:11:53 +0900 |
| parents | 9ba98ecfbe62 |
| children | e0cd3ac0087d |
| 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 | |
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528
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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57 _≤_ : (x y : HOD) → Set (Level.suc n) |
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8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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58 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
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59 |
| 554 | 60 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 61 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 62 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 63 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 64 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 65 | |
| 556 | 66 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 67 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 68 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 69 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 70 |
| 492 | 71 open _==_ |
| 72 open _⊆_ | |
| 73 | |
| 530 | 74 -- |
| 560 | 75 -- Closure of ≤-monotonic function f has total order |
| 530 | 76 -- |
| 77 | |
| 78 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 79 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 80 | |
| 560 | 81 -- immieate-f : (A : HOD) → ( f : Ordinal → Ordinal ) → Set n |
| 82 -- immieate-f A f = { x y : Ordinal } → odef A x → odef A y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
| 556 | 83 |
| 551 | 84 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 554 | 85 init : odef A s → FClosure A f s s |
| 555 | 86 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 87 |
| 556 | 88 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 |
| 89 A∋fc {A} s f mf (init as) = as | |
| 90 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) | |
| 555 | 91 |
| 556 | 92 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 93 s≤fc {A} s {.s} f mf (init x) = case1 refl | |
| 94 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) | |
| 95 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 96 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 97 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 98 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 99 |
| 557 | 100 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 101 fcn s mf (init as) = zero | |
| 558 | 102 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) |
| 103 ... | case1 eq = fcn s mf p | |
| 104 ... | case2 y<fy = suc (fcn s mf p ) | |
| 557 | 105 |
| 558 | 106 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 107 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 559 | 108 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 |
| 109 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 | |
| 110 fc00 zero zero refl (init _) (init x₁) i=x i=y = refl | |
| 111 fc00 zero zero refl (init sa) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 112 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init sa) cy i=x i=y ) | |
| 113 fc00 zero zero refl (fsuc x cx) (init sa) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 114 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init sa) i=x i=y ) | |
| 115 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 ) ) | |
| 116 ... | 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 ) | |
| 117 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 ) ) | |
| 118 ... | 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 ) | |
| 119 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 120 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 121 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 560 | 122 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x |
| 559 | 123 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where |
| 124 fc04 : * x1 ≡ * y | |
| 125 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 126 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 127 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 128 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 129 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 130 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 131 fc05 : * x ≡ * y1 | |
| 132 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 133 ... | 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 | 134 |
| 135 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 136 → (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 | 137 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where |
| 138 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 | |
| 139 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 140 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 141 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 142 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 143 ... | 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 | |
| 144 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 145 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 146 fc03 eq = cong pred eq | |
| 147 fc02 : * x < * y1 | |
| 148 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 557 | 149 |
| 559 | 150 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 151 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 559 | 152 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 153 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 154 fc11 : * x < * y | |
| 155 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 156 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 157 fc10 : * x ≡ * y | |
| 158 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 159 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 160 fc12 : * y < * x | |
| 161 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 162 | |
| 560 | 163 -- open import Relation.Binary.Properties.Poset as Poset |
| 164 | |
| 165 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 166 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 167 | |
| 168 | |
| 169 record Maximal ( A : HOD ) : Set (Level.suc n) where | |
| 170 field | |
| 171 maximal : HOD | |
| 172 A∋maximal : A ∋ maximal | |
| 173 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 174 | |
| 175 -- | |
| 176 -- inductive maxmum tree from x | |
| 177 -- tree structure | |
| 178 -- | |
| 554 | 179 |
| 541 | 180 record Prev< (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 181 field |
| 534 | 182 y : Ordinal |
| 541 | 183 ay : odef B y |
| 534 | 184 x=fy : x ≡ f y |
| 529 | 185 |
| 508 | 186 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 503 | 187 field |
| 188 sup : HOD | |
| 189 A∋maximal : A ∋ sup | |
| 190 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 191 | |
| 533 | 192 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
| 193 SupCond A B _ _ = SUP A B | |
| 194 | |
| 546 | 195 record ZChain ( A : HOD ) {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) |
| 547 | 196 (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where |
| 533 | 197 field |
| 198 chain : HOD | |
| 199 chain⊆A : chain ⊆ A | |
| 538 | 200 chain∋x : odef chain x |
| 554 | 201 initial : {y : Ordinal } → odef chain y → * x < * y |
| 533 | 202 f-total : IsTotalOrderSet chain |
| 546 | 203 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 541 | 204 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) |
| 548 | 205 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< z → (ba : odef A b) |
| 541 | 206 → Prev< A chain ba f |
| 207 ∨ (sup (& chain) (subst (λ k → k ⊆ A) (sym *iso) chain⊆A) (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b ) | |
| 534 | 208 → * a < * b → odef chain b |
| 533 | 209 |
| 497 | 210 Zorn-lemma : { A : HOD } |
| 464 | 211 → o∅ o< & A |
| 497 | 212 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 213 → Maximal A | |
| 552 | 214 Zorn-lemma {A} 0<A supP = zorn00 where |
| 535 | 215 supO : (C : Ordinal ) → (* C) ⊆ A → IsTotalOrderSet (* C) → Ordinal |
| 216 supO C C⊆A TC = & ( SUP.sup ( supP (* C) C⊆A TC )) | |
| 493 | 217 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 556 | 218 z01 {a} {b} A∋a A∋b = <-irr |
| 537 | 219 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 220 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 221 s : HOD |
| 222 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 223 sa : A ∋ * ( & s ) | |
| 224 sa = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) | |
| 547 | 225 s<A : & s o< & A |
| 226 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso sa ) | |
| 530 | 227 HasMaximal : HOD |
| 537 | 228 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 229 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 230 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 231 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 232 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 233 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 234 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 235 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 236 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)) | |
| 237 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 | |
| 238 ¬x<m : ¬ (* x < * m) | |
| 239 ¬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 | 240 |
| 560 | 241 -- Uncountable ascending chain by axiom of choice |
| 530 | 242 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 243 cf nmx x with ODC.∋-p O A (* x) |
| 244 ... | no _ = o∅ | |
| 245 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 246 ... | yes nogt = -- no larger element, so it is maximal |
| 247 ⊥-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 | 248 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 249 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 250 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 251 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 252 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 253 ... | 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 ⟫ ) | |
| 254 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 530 | 255 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 256 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 257 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 258 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 259 |
| 547 | 260 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A sa f mf supO (& A) ) → SUP A (ZChain.chain zc) |
| 533 | 261 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
| 547 | 262 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) ) |
| 533 | 263 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
| 264 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
| 547 | 265 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) → SUP A (* (& (ZChain.chain zc))) |
| 538 | 266 sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) |
| 267 (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) ) | |
| 543 | 268 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 269 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 270 | |
| 271 --- | |
| 560 | 272 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 273 --- |
| 547 | 274 z03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) |
| 546 | 275 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) |
| 538 | 276 z03 f mf zc = z14 where |
| 277 chain = ZChain.chain zc | |
| 278 sp1 = sp0 f mf zc | |
| 548 | 279 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
| 541 | 280 → Prev< A chain ab f |
| 281 ∨ (supO (& chain) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc)) ≡ b ) | |
| 538 | 282 → * a < * b → odef chain b |
| 283 z10 = ZChain.is-max zc | |
| 543 | 284 z11 : & (SUP.sup sp1) o< & A |
| 285 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 286 z12 : odef chain (& (SUP.sup sp1)) |
| 287 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 288 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
| 548 | 289 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) z11 (SUP.A∋maximal sp1) (case2 refl ) z13 where |
| 538 | 290 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 291 z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc )) | |
| 292 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) | |
| 293 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 294 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) | |
| 552 | 295 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 538 | 296 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 297 z16 : ⊥ | |
| 298 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 299 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 300 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 301 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 302 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 303 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 546 | 304 z15 = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (ZChain.f-next zc z12 )) |
| 538 | 305 z17 : ⊥ |
| 306 z17 with z15 | |
| 307 ... | case1 eq = ¬b eq | |
| 308 ... | case2 lt = ¬a lt | |
| 560 | 309 |
| 310 -- ZChain contradicts ¬ Maximal | |
| 311 -- | |
| 312 -- ZChain forces fix point on any ≤-monotonic function (z03) | |
| 313 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
| 314 -- | |
| 547 | 315 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥ |
| 537 | 316 z04 nmx zc = z01 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) |
| 538 | 317 (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
| 318 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) | |
| 319 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where | |
| 546 | 320 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc |
| 538 | 321 c = & (SUP.sup sp1) |
| 548 | 322 |
| 560 | 323 -- |
| 547 | 324 -- create all ZChains under o< x |
| 560 | 325 -- |
| 546 | 326 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → |
| 547 | 327 ((z : Ordinal) → z o< x → {y : Ordinal} → (ya : odef A y) → ZChain A ya f mf supO z ) → { y : Ordinal } → (ya : odef A y) → ZChain A ya f mf supO x |
| 328 ind f mf x prev {y} ay with Oprev-p x | |
| 548 | 329 ... | yes op = zc4 where |
| 560 | 330 -- |
| 331 -- we have previous ordinal to use induction | |
| 332 -- | |
| 530 | 333 px = Oprev.oprev op |
| 547 | 334 zc0 : ZChain A ay f mf supO (Oprev.oprev op) |
| 335 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay | |
| 336 zc4 : ZChain A ay f mf supO x | |
| 551 | 337 zc4 with ODC.∋-p O A (* px) |
| 560 | 338 ... | no noapx = -- ¬ A ∋ px, just skip |
| 339 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 | |
| 554 | 340 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 |
| 551 | 341 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 } where -- no extention |
| 342 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
| 343 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
| 344 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 345 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) ≡ b) → | |
| 346 * a < * b → odef (ZChain.chain zc0) b | |
| 347 zc11 {a} {b} za b<x ba P a<b with osuc-≡< (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x) | |
| 348 ... | case1 eq = ⊥-elim ( noapx (subst (λ k → odef A k) (trans eq (sym &iso) ) ba )) | |
| 349 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
| 350 ... | yes apx with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) apx f ) -- we have to check adding x preserve is-max ZChain A ay f mf supO px | |
| 549 | 351 ... | case1 pr = zc9 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 551 | 352 chain = ZChain.chain zc0 |
| 353 zc17 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
| 354 Prev< A (ZChain.chain zc0) ba f ∨ (supO (& (ZChain.chain zc0)) | |
| 355 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 356 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b) → | |
| 357 * a < * b → odef (ZChain.chain zc0) b | |
| 358 zc17 {a} {b} za b<x ba P a<b with osuc-≡< (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x) | |
| 359 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
| 360 ... | case1 b=px = subst (λ k → odef chain k ) (trans (sym (Prev<.x=fy pr )) (trans &iso (sym b=px))) ( ZChain.f-next zc0 (Prev<.ay pr)) | |
| 549 | 361 zc9 : ZChain A ay f mf supO x |
| 362 zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 554 | 363 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 } -- no extention |
| 551 | 364 ... | case2 ¬fy<x with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) )) |
| 560 | 365 ... | case1 x=sup = -- previous ordinal is a sup of a smaller ZChain |
| 366 record { chain = schain ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} | |
| 554 | 367 ; initial = {!!} ; f-immediate = {!!} ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup |
| 551 | 368 sp = SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) |
| 369 chain = ZChain.chain zc0 | |
| 552 | 370 schain : HOD |
| 371 schain = record { od = record { def = λ x → odef chain x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = {!!} } | |
| 560 | 372 ... | case2 ¬x=sup = -- x is not f y' nor sup of former ZChain from y |
| 373 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 554 | 374 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = {!!} } where -- no extention |
| 552 | 375 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → |
| 376 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
| 377 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 378 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) | |
| 379 ≡ b) → | |
| 380 * a < * b → odef (ZChain.chain zc0) b | |
| 381 z18 {a} {b} za b<x ba (case1 p) a<b = {!!} | |
| 382 z18 {a} {b} za b<x ba (case2 p) a<b = {!!} | |
| 553 | 383 ... | no ¬ox = {!!} where --- limit ordinal case |
| 554 | 384 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x |
| 553 | 385 field |
| 386 u : Ordinal | |
| 387 u<x : u o< x | |
| 554 | 388 zuy : odef (ZChain.chain (prev u u<x {y} ay )) z |
| 389 Uz : HOD | |
| 390 Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A ; <odmax = {!!} } | |
| 391 u-total : IsTotalOrderSet Uz | |
| 553 | 392 u-total {x} {y} ux uy = {!!} |
| 560 | 393 --- ux ⊆ uy ∨ uy ⊆ ux |
| 553 | 394 |
| 551 | 395 zorn00 : Maximal A |
| 396 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 397 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 398 -- yes we have the maximal | |
| 399 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 400 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 401 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 402 zorn01 = proj1 zorn03 | |
| 403 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 404 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 405 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where | |
| 406 -- if we have no maximal, make ZChain, which contradict SUP condition | |
| 407 nmx : ¬ Maximal A | |
| 408 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 409 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 410 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) ) ⟫ | |
| 411 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO (& A) | |
| 412 zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain A ya f mf supO z } (ind f mf) (& A) | |
| 413 zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) | |
| 414 zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa ) | |
| 415 | |
| 516 | 416 -- usage (see filter.agda ) |
| 417 -- | |
| 497 | 418 -- _⊆'_ : ( A B : HOD ) → Set n |
| 419 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 420 |
| 497 | 421 -- MaximumSubset : {L P : HOD} |
| 422 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 423 -- → IsPartialOrderSet P _⊆'_ | |
| 424 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 425 -- → Maximal P (_⊆'_) | |
| 426 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |