Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 557:f1e899cbe845
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 29 Apr 2022 18:23:49 +0900 |
| parents | ba889c711524 |
| children | fed1c67b9a65 |
| 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 |
| 10 open import zf | |
| 477 | 11 open import logic |
| 12 -- open import partfunc {n} O | |
| 13 | |
| 14 open import Relation.Nullary | |
| 15 open import Data.Empty | |
| 16 import BAlgbra | |
| 431 | 17 |
| 555 | 18 open import Data.Nat hiding ( _<_ ; _≤_ ) |
| 19 open import Data.Nat.Properties | |
| 20 open import nat | |
| 21 | |
| 431 | 22 |
| 23 open inOrdinal O | |
| 24 open OD O | |
| 25 open OD.OD | |
| 26 open ODAxiom odAxiom | |
| 477 | 27 import OrdUtil |
| 28 import ODUtil | |
| 431 | 29 open Ordinals.Ordinals O |
| 30 open Ordinals.IsOrdinals isOrdinal | |
| 31 open Ordinals.IsNext isNext | |
| 32 open OrdUtil O | |
| 477 | 33 open ODUtil O |
| 34 | |
| 35 | |
| 36 import ODC | |
| 37 | |
| 38 | |
| 39 open _∧_ | |
| 40 open _∨_ | |
| 41 open Bool | |
| 431 | 42 |
| 43 | |
| 44 open HOD | |
| 45 | |
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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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46 _≤_ : (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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47 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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48 |
| 554 | 49 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 50 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 51 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 52 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 53 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 54 | |
| 556 | 55 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 56 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 57 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 58 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 59 |
| 492 | 60 open _==_ |
| 61 open _⊆_ | |
| 62 | |
| 497 | 63 -- open import Relation.Binary.Properties.Poset as Poset |
| 496 | 64 |
| 552 | 65 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) |
| 66 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 498 | 67 |
| 497 | 68 |
| 508 | 69 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 503 | 70 field |
| 71 maximal : HOD | |
| 72 A∋maximal : A ∋ maximal | |
| 73 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 74 | |
| 530 | 75 -- |
| 76 -- inductive maxmum tree from x | |
| 77 -- tree structure | |
| 78 -- | |
| 79 | |
| 80 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 81 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 82 | |
| 556 | 83 immieate-f : (A : HOD) → ( f : Ordinal → Ordinal ) → Set n |
| 84 immieate-f A f = { x y : Ordinal } → odef A x → odef A y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
| 85 | |
| 551 | 86 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 554 | 87 init : odef A s → FClosure A f s s |
| 555 | 88 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 89 |
| 556 | 90 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y |
| 91 A∋fc {A} s f mf (init as) = as | |
| 92 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) | |
| 555 | 93 |
| 556 | 94 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 95 s≤fc {A} s {.s} f mf (init x) = case1 refl | |
| 96 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) | |
| 97 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 98 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 99 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 100 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 101 |
| 557 | 102 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 103 fcn s mf (init as) = zero | |
| 104 fcn {A} s {x} {f} mf (fsuc y p) with mf y (A∋fc f mf p) | |
| 105 ... | ⟪ case1 eq , _ ⟫ = fcn s mf p | |
| 106 ... | ⟪ case2 y<fy , _ ⟫ = suc (fcn s mf p ) | |
| 107 | |
| 108 fcn-suc : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 109 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → suc (fcn s mf cx ) ≡ fcn s mf cy → * x < * y | |
| 110 fcn-suc = ? | |
| 111 | |
| 112 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 113 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 114 fcn-< = ? | |
| 115 | |
| 116 fcn-fsuc : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 117 → (cx : FClosure A f s x ) (cy : FClosure A f s (f x) ) → * x < * (f x) → suc (fcn s mf cx ) ≡ fcn s mf cy | |
| 118 fcn-fsuc = ? | |
| 119 | |
| 556 | 120 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) (imm : immieate-f A f ) |
| 554 | 121 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 556 | 122 fcn-cmp {A} s {.s} {.s} f mf imm (init x) (init x₁) = tri≈ (λ lt → <-irr (case1 refl) lt ) refl (λ lt → <-irr (case1 refl) lt ) |
| 123 fcn-cmp {A} s f mf imm (init x) (fsuc y cy) with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 124 ... | case1 fy=y = subst (λ k → Tri (* s < * k) (* s ≡ * k) (* k < * s ) ) (*≡*→≡ fy=y) ( fcn-cmp {A} s f mf imm (init x) cy ) | |
| 125 ... | case2 fy>y = tri< lt (λ eq → <-irr (case1 (sym eq)) lt ) (λ lt1 → <-irr (case2 lt1) lt ) where | |
| 126 lt : * s < * (f y) | |
| 127 lt with s≤fc s f mf cy | |
| 128 ... | case1 s=y = subst (λ k → * k < * (f y) ) (sym (*≡*→≡ s=y)) fy>y | |
| 129 ... | case2 s<y = IsStrictPartialOrder.trans PO s<y fy>y | |
| 130 fcn-cmp {A} s {x} f mf imm cx (init x₁) with s≤fc s f mf cx | |
| 131 ... | case1 eq = tri≈ (λ lt → <-irr (case1 eq) lt) (sym eq) (λ lt → <-irr (case1 (sym eq)) lt) | |
| 132 ... | case2 s<x = tri> (λ lt → <-irr (case2 s<x) lt) (λ eq → <-irr (case1 eq) s<x ) s<x | |
| 133 fcn-cmp {A} s f mf imm (fsuc x cx) (fsuc y cy) with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 134 ... | case1 x=fx | case1 y=fy = {!!} | |
| 135 ... | case1 x=fx | case2 y<fy = {!!} | |
| 136 ... | case2 x<fx | case1 y=fy = {!!} | |
| 137 ... | case2 x<fx | case2 y<fy = {!!} where | |
| 138 fc-mono : {x y : Ordinal } → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → FClosure A f x y ∨ FClosure A f y x | |
| 139 fc-mono = ? | |
| 140 fc1 : Tri (* (f x) < * (f y)) (* (f x) ≡ * (f y)) (* (f y) < * (f x)) | |
| 141 fc1 with fcn-cmp s f mf imm cx cy | |
| 142 ... | tri< a ¬b ¬c = {!!} | |
| 143 ... | tri≈ ¬a b ¬c = {!!} | |
| 144 ... | tri> ¬a ¬b c = {!!} | |
| 554 | 145 |
| 541 | 146 record Prev< (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 147 field |
| 534 | 148 y : Ordinal |
| 541 | 149 ay : odef B y |
| 534 | 150 x=fy : x ≡ f y |
| 529 | 151 |
| 508 | 152 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 503 | 153 field |
| 154 sup : HOD | |
| 155 A∋maximal : A ∋ sup | |
| 156 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 157 | |
| 533 | 158 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
| 159 SupCond A B _ _ = SUP A B | |
| 160 | |
| 546 | 161 record ZChain ( A : HOD ) {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) |
| 547 | 162 (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where |
| 533 | 163 field |
| 164 chain : HOD | |
| 165 chain⊆A : chain ⊆ A | |
| 538 | 166 chain∋x : odef chain x |
| 554 | 167 initial : {y : Ordinal } → odef chain y → * x < * y |
| 533 | 168 f-total : IsTotalOrderSet chain |
| 546 | 169 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 541 | 170 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) |
| 548 | 171 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< z → (ba : odef A b) |
| 541 | 172 → Prev< A chain ba f |
| 173 ∨ (sup (& chain) (subst (λ k → k ⊆ A) (sym *iso) chain⊆A) (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b ) | |
| 534 | 174 → * a < * b → odef chain b |
| 533 | 175 |
| 497 | 176 Zorn-lemma : { A : HOD } |
| 464 | 177 → o∅ o< & A |
| 497 | 178 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 179 → Maximal A | |
| 552 | 180 Zorn-lemma {A} 0<A supP = zorn00 where |
| 535 | 181 supO : (C : Ordinal ) → (* C) ⊆ A → IsTotalOrderSet (* C) → Ordinal |
| 182 supO C C⊆A TC = & ( SUP.sup ( supP (* C) C⊆A TC )) | |
| 493 | 183 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 556 | 184 z01 {a} {b} A∋a A∋b = <-irr |
| 537 | 185 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 186 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 187 s : HOD |
| 188 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 189 sa : A ∋ * ( & s ) | |
| 190 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 | 191 s<A : & s o< & A |
| 192 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso sa ) | |
| 530 | 193 HasMaximal : HOD |
| 537 | 194 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 195 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 196 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 197 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 198 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 199 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 200 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 201 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 202 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)) | |
| 203 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 | |
| 204 ¬x<m : ¬ (* x < * m) | |
| 205 ¬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 | 206 |
| 207 -- Uncountable acending chain by axiom of choice | |
| 530 | 208 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 209 cf nmx x with ODC.∋-p O A (* x) |
| 210 ... | no _ = o∅ | |
| 211 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 212 ... | yes nogt = -- no larger element, so it is maximal |
| 213 ⊥-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 | 214 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 215 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 216 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 217 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 218 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 219 ... | 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 ⟫ ) | |
| 220 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 530 | 221 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 222 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 223 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 224 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 225 |
| 547 | 226 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A sa f mf supO (& A) ) → SUP A (ZChain.chain zc) |
| 533 | 227 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
| 547 | 228 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) ) |
| 533 | 229 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
| 230 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
| 547 | 231 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 | 232 sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) |
| 233 (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) ) | |
| 543 | 234 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 235 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 236 | |
| 237 --- | |
| 238 --- sup is fix point in maximum chain | |
| 239 --- | |
| 547 | 240 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 | 241 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) |
| 538 | 242 z03 f mf zc = z14 where |
| 243 chain = ZChain.chain zc | |
| 244 sp1 = sp0 f mf zc | |
| 548 | 245 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
| 541 | 246 → Prev< A chain ab f |
| 247 ∨ (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 | 248 → * a < * b → odef chain b |
| 249 z10 = ZChain.is-max zc | |
| 543 | 250 z11 : & (SUP.sup sp1) o< & A |
| 251 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 252 z12 : odef chain (& (SUP.sup sp1)) |
| 253 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 254 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
| 548 | 255 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) z11 (SUP.A∋maximal sp1) (case2 refl ) z13 where |
| 538 | 256 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 257 z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc )) | |
| 258 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) | |
| 259 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 260 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) | |
| 552 | 261 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 538 | 262 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 263 z16 : ⊥ | |
| 264 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 265 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 266 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 267 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 268 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 269 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 546 | 270 z15 = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (ZChain.f-next zc z12 )) |
| 538 | 271 z17 : ⊥ |
| 272 z17 with z15 | |
| 273 ... | case1 eq = ¬b eq | |
| 274 ... | case2 lt = ¬a lt | |
| 547 | 275 -- ZChain requires the Maximal |
| 276 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥ | |
| 537 | 277 z04 nmx zc = z01 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) |
| 538 | 278 (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
| 279 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) | |
| 280 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where | |
| 546 | 281 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc |
| 538 | 282 c = & (SUP.sup sp1) |
| 548 | 283 |
| 550 | 284 -- 3cases : {x y : Ordinal} → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) |
| 285 -- → (ax : odef A x )→ (ay : odef A y ) | |
| 286 -- → (zc0 : ZChain A ay f mf supO x ) | |
| 287 -- → Prev< A (ZChain.chain zc0) ax f | |
| 288 -- ∨ (supO (& (ZChain.chain zc0)) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ x) | |
| 289 -- ∨ ( ( z : Ordinal) → odef (ZChain.chain zc0) z → ¬ ( * z < * x )) | |
| 290 -- 3cases {x} {y} f mf ax ay zc0 = {!!} | |
| 547 | 291 -- create all ZChains under o< x |
| 546 | 292 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → |
| 547 | 293 ((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 |
| 294 ind f mf x prev {y} ay with Oprev-p x | |
| 548 | 295 ... | yes op = zc4 where |
| 530 | 296 px = Oprev.oprev op |
| 547 | 297 zc0 : ZChain A ay f mf supO (Oprev.oprev op) |
| 298 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay | |
| 299 zc4 : ZChain A ay f mf supO x | |
| 551 | 300 zc4 with ODC.∋-p O A (* px) |
| 554 | 301 ... | no noapx = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 |
| 302 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 551 | 303 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 } where -- no extention |
| 304 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
| 305 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
| 306 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 307 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) ≡ b) → | |
| 308 * a < * b → odef (ZChain.chain zc0) b | |
| 309 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) | |
| 310 ... | case1 eq = ⊥-elim ( noapx (subst (λ k → odef A k) (trans eq (sym &iso) ) ba )) | |
| 311 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
| 312 ... | 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 | 313 ... | case1 pr = zc9 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 551 | 314 chain = ZChain.chain zc0 |
| 315 zc17 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
| 316 Prev< A (ZChain.chain zc0) ba f ∨ (supO (& (ZChain.chain zc0)) | |
| 317 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 318 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b) → | |
| 319 * a < * b → odef (ZChain.chain zc0) b | |
| 320 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) | |
| 321 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
| 322 ... | 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 | 323 zc9 : ZChain A ay f mf supO x |
| 324 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 | 325 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 } -- no extention |
| 551 | 326 ... | case2 ¬fy<x with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) )) |
| 552 | 327 ... | case1 x=sup = record { chain = schain ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} |
| 554 | 328 ; initial = {!!} ; f-immediate = {!!} ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup |
| 551 | 329 sp = SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) |
| 330 chain = ZChain.chain zc0 | |
| 552 | 331 schain : HOD |
| 332 schain = record { od = record { def = λ x → odef chain x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = {!!} } | |
| 333 ... | case2 ¬x=sup = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
| 554 | 334 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = {!!} } where -- no extention |
| 552 | 335 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → |
| 336 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
| 337 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
| 338 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) | |
| 339 ≡ b) → | |
| 340 * a < * b → odef (ZChain.chain zc0) b | |
| 341 z18 {a} {b} za b<x ba (case1 p) a<b = {!!} | |
| 342 z18 {a} {b} za b<x ba (case2 p) a<b = {!!} | |
| 553 | 343 ... | no ¬ox = {!!} where --- limit ordinal case |
| 554 | 344 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x |
| 553 | 345 field |
| 346 u : Ordinal | |
| 347 u<x : u o< x | |
| 554 | 348 zuy : odef (ZChain.chain (prev u u<x {y} ay )) z |
| 349 Uz : HOD | |
| 350 Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A ; <odmax = {!!} } | |
| 351 u-total : IsTotalOrderSet Uz | |
| 553 | 352 u-total {x} {y} ux uy = {!!} |
| 353 | |
| 551 | 354 zorn00 : Maximal A |
| 355 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 356 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 357 -- yes we have the maximal | |
| 358 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 359 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 360 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
| 361 zorn01 = proj1 zorn03 | |
| 362 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 363 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 364 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where | |
| 365 -- if we have no maximal, make ZChain, which contradict SUP condition | |
| 366 nmx : ¬ Maximal A | |
| 367 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 368 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 369 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) ) ⟫ | |
| 370 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO (& A) | |
| 371 zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain A ya f mf supO z } (ind f mf) (& A) | |
| 372 zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) | |
| 373 zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa ) | |
| 374 | |
| 516 | 375 -- usage (see filter.agda ) |
| 376 -- | |
| 497 | 377 -- _⊆'_ : ( A B : HOD ) → Set n |
| 378 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 379 |
| 497 | 380 -- MaximumSubset : {L P : HOD} |
| 381 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 382 -- → IsPartialOrderSet P _⊆'_ | |
| 383 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 384 -- → Maximal P (_⊆'_) | |
| 385 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |