Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 754:db177d911091
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 23 Jul 2022 18:40:35 +0900 |
| parents | b96491f7c775 |
| children | b22327e78b03 |
| 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 open _∧_ | |
| 46 open _∨_ | |
| 47 open Bool | |
| 431 | 48 |
| 49 open HOD | |
| 50 | |
| 560 | 51 -- |
| 52 -- Partial Order on HOD ( possibly limited in A ) | |
| 53 -- | |
| 54 | |
| 571 | 55 _<<_ : (x y : Ordinal ) → Set n -- Set n order |
| 570 | 56 x << y = * x < * y |
| 57 | |
| 58 POO : IsStrictPartialOrder _≡_ _<<_ | |
| 59 POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 60 ; trans = IsStrictPartialOrder.trans PO | |
| 61 ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y | |
| 62 ; <-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 } } | |
| 63 | |
|
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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64 _≤_ : (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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65 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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66 |
| 554 | 67 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 68 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
| 69 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 70 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 71 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 72 | |
| 556 | 73 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 74 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 75 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 76 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 77 |
| 561 | 78 ptrans = IsStrictPartialOrder.trans PO |
| 79 | |
| 492 | 80 open _==_ |
| 81 open _⊆_ | |
| 82 | |
| 530 | 83 -- |
| 560 | 84 -- Closure of ≤-monotonic function f has total order |
| 530 | 85 -- |
| 86 | |
| 87 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 88 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 89 | |
| 551 | 90 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 600 | 91 init : odef A s → FClosure A f s s |
| 555 | 92 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 93 |
| 556 | 94 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 | 95 A∋fc {A} s f mf (init as) = as |
| 556 | 96 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 97 |
| 714 | 98 A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s |
| 99 A∋fcs {A} s f mf (init as) = as | |
| 100 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy | |
| 101 | |
| 556 | 102 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 | 103 s≤fc {A} s {.s} f mf (init x) = case1 refl |
| 556 | 104 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 105 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 106 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 107 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 108 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 109 |
| 557 | 110 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 600 | 111 fcn s mf (init as) = zero |
| 558 | 112 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) |
| 113 ... | case1 eq = fcn s mf p | |
| 114 ... | case2 y<fy = suc (fcn s mf p ) | |
| 557 | 115 |
| 558 | 116 fcn-inject : {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 y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 559 | 118 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 |
| 119 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 | 120 fc00 zero zero refl (init _) (init x₁) i=x i=y = refl |
| 121 fc00 zero zero refl (init as) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 122 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as) cy i=x i=y ) | |
| 123 fc00 zero zero refl (fsuc x cx) (init as) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 124 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as) i=x i=y ) | |
| 559 | 125 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 ) ) |
| 126 ... | 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 ) | |
| 127 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 ) ) | |
| 128 ... | 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 ) | |
| 129 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 130 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 131 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 560 | 132 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x |
| 559 | 133 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where |
| 134 fc04 : * x1 ≡ * y | |
| 135 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 136 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 137 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 138 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 139 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 140 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 141 fc05 : * x ≡ * y1 | |
| 142 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 143 ... | 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 | 144 |
| 600 | 145 |
| 557 | 146 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 147 → (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 | 148 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where |
| 149 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 | |
| 150 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 151 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 152 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 153 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 154 ... | 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 | |
| 155 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 156 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 157 fc03 eq = cong pred eq | |
| 158 fc02 : * x < * y1 | |
| 159 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 557 | 160 |
| 559 | 161 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 162 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 559 | 163 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 164 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 165 fc11 : * x < * y | |
| 166 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 167 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 168 fc10 : * x ≡ * y | |
| 169 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 170 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 171 fc12 : * y < * x | |
| 172 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 173 | |
| 600 | 174 |
| 562 | 175 fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 176 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
| 563 | 177 fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where |
| 178 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 ) | |
| 179 fc20 y<sx with <-cmp ( fcn s mf cy ) (fcn s mf cx ) | |
| 180 ... | tri< a ¬b ¬c = case2 a | |
| 181 ... | tri≈ ¬a b ¬c = case1 b | |
| 182 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> y<sx (s≤s c)) | |
| 183 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 | |
| 184 fc17 {x} {y} cx cy sx=y = fc18 (fcn s mf cy) cx cy refl sx=y where | |
| 185 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 | |
| 186 fc18 (suc i) {y} cx (fsuc y1 cy) i=y sx=i with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 187 ... | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy ( fc18 (suc i) {y1} cx cy i=y sx=i) -- dereference | |
| 188 ... | case2 y<fy = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k) ) ) fc19) where | |
| 189 fc19 : * x ≡ * y1 | |
| 190 fc19 = fcn-inject s mf cx cy (cong pred ( trans sx=i i=y )) | |
| 191 fc21 : ⊥ | |
| 192 fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy ) | |
| 193 ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy | |
| 194 cxx : FClosure A f s (f x) | |
| 195 cxx = fsuc x cx | |
| 196 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 | 197 fc16 x (init as) with proj1 (mf s as ) |
| 563 | 198 ... | case1 _ = case1 refl |
| 199 ... | case2 _ = case2 refl | |
| 200 fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) | |
| 201 ... | case1 _ = case1 refl | |
| 202 ... | case2 _ = case2 refl | |
| 203 fc22 : (suc ( fcn s mf cx )) Data.Nat.< (fcn s mf cy ) → * (f x) < * y | |
| 204 fc22 a with fc16 x cx | |
| 205 ... | case1 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq (<-trans a<sa a)) | |
| 206 ... | case2 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq a ) | |
| 207 ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx | |
| 208 ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx | |
| 209 ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y | |
| 210 ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) | |
| 211 | |
| 729 | 212 fc-conv : (A : HOD ) (f : Ordinal → Ordinal) {b u : Ordinal } |
| 213 → {p0 p1 : Ordinal → Ordinal} | |
| 214 → p0 u ≡ p1 u | |
| 215 → FClosure A f (p0 u) b → FClosure A f (p1 u) b | |
| 216 fc-conv A f {.(p0 u)} {u} {p0} {p1} p0u=p1u (init ap0u) = subst (λ k → FClosure A f (p1 u) k) (sym p0u=p1u) | |
| 217 ( init (subst (λ k → odef A k) p0u=p1u ap0u )) | |
| 218 fc-conv A f {_} {u} {p0} {p1} p0u=p1u (fsuc z fc) = fsuc z (fc-conv A f {_} {u} {p0} {p1} p0u=p1u fc) | |
| 219 | |
| 560 | 220 -- open import Relation.Binary.Properties.Poset as Poset |
| 221 | |
| 222 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 223 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 224 | |
| 567 | 225 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 226 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 227 |
| 568 | 228 _⊆'_ : ( A B : HOD ) → Set n |
| 229 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 230 |
| 231 -- | |
| 232 -- inductive maxmum tree from x | |
| 233 -- tree structure | |
| 234 -- | |
| 554 | 235 |
| 567 | 236 record HasPrev (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
| 533 | 237 field |
| 534 | 238 y : Ordinal |
| 541 | 239 ay : odef B y |
| 534 | 240 x=fy : x ≡ f y |
| 529 | 241 |
| 570 | 242 record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 654 | 243 field |
| 571 | 244 x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 568 | 245 |
| 656 | 246 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 247 field | |
| 248 sup : HOD | |
| 249 A∋maximal : A ∋ sup | |
| 250 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
| 251 | |
| 690 | 252 -- |
| 253 -- sup and its fclosure is in a chain HOD | |
| 254 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 255 -- whole chain is a union of separated Chain | |
| 256 -- minimum index is y not ϕ | |
| 257 -- | |
| 258 | |
| 714 | 259 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u z : Ordinal) : Set n where |
| 690 | 260 field |
| 714 | 261 fcy<sup : {z : Ordinal } → FClosure A f y z → z << supf u |
| 739 | 262 csupz : FClosure A f (supf u) z |
| 714 | 263 order : {sup1 z1 : Ordinal} → (lt : sup1 o< u ) → FClosure A f (supf sup1 ) z1 → z1 << supf u |
| 743 | 264 y<s : y << supf u -- not a initial chain |
| 265 supfu=u : supf u ≡ u | |
| 694 | 266 |
| 267 -- Union of supf z which o< x | |
| 268 -- | |
| 690 | 269 |
| 748 | 270 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 271 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where | |
| 272 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 273 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x ) ( is-sup : ChainP A f mf ay supf u z) | |
| 274 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z | |
| 694 | 275 |
| 276 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A | |
| 277 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 278 | |
| 279 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) | |
| 280 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
| 281 UnionCF A f mf ay supf x | |
| 282 = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x z } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } | |
| 662 | 283 |
| 703 | 284 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 285 {init : Ordinal} (ay : odef A init) ( z : Ordinal ) : Set (Level.suc n) where | |
| 655 | 286 field |
| 694 | 287 supf : Ordinal → Ordinal |
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288 chain : HOD |
| 703 | 289 chain = UnionCF A f mf ay supf z |
| 568 | 290 field |
| 291 chain⊆A : chain ⊆' A | |
| 653 | 292 chain∋init : odef chain init |
| 293 initial : {y : Ordinal } → odef chain y → * init ≤ * y | |
| 568 | 294 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
| 654 | 295 f-total : IsTotalOrderSet chain |
| 754 | 296 csupf : {z : Ordinal } → odef chain (supf z) |
| 653 | 297 |
| 728 | 298 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 299 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where | |
| 300 field | |
| 301 chain-mono2 : { a b c : Ordinal } → | |
| 302 a o≤ b → b o≤ z → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c | |
| 303 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay (ZChain.supf zc) z) a ) → b o< z → (ab : odef A b) | |
| 304 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) z) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) z) ab | |
| 305 → * a < * b → odef ((UnionCF A f mf ay (ZChain.supf zc) z)) b | |
| 741 | 306 fcy<sup : {u w : Ordinal } → u o< z → FClosure A f init w → w << (ZChain.supf zc) u |
| 742 | 307 sup=u : {b : Ordinal} → {ab : odef A b} → b o< z → IsSup A (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) ab → (ZChain.supf zc) b ≡ b |
| 308 order : {b sup1 z1 : Ordinal} → b o< z → sup1 o< b → FClosure A f ((ZChain.supf zc) sup1) z1 → z1 << (ZChain.supf zc) b | |
| 728 | 309 |
| 568 | 310 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 311 field | |
| 312 maximal : HOD | |
| 313 A∋maximal : A ∋ maximal | |
| 314 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 567 | 315 |
| 748 | 316 -- data UChain is total |
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317 |
| 694 | 318 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 748 | 319 {s s1 a b : Ordinal } ( ca : UChain A f mf ay supf s a ) ( cb : UChain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) |
| 694 | 320 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where |
| 748 | 321 ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) |
| 322 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb | |
| 323 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub ub<x supb fcb) = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where | |
| 324 ct00 : * a < * (supf ub) | |
| 695 | 325 ct00 = ChainP.fcy<sup supb fca |
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326 ct01 : * a < * b |
| 748 | 327 ct01 with s≤fc (supf ub) f mf fcb |
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328 ... | case1 eq = subst (λ k → * a < k ) eq ct00 |
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329 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 748 | 330 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-init fcb)= tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where |
| 749 | 331 ct00 : * b < * (supf ua) |
| 332 ct00 = ChainP.fcy<sup supa fcb | |
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333 ct01 : * b < * a |
| 749 | 334 ct01 with s≤fc (supf ua) f mf fca |
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parents:
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335 ... | case1 eq = subst (λ k → * b < k ) eq ct00 |
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336 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt |
| 748 | 337 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub |
| 685 | 338 ... | tri< a₁ ¬b ¬c = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where |
| 748 | 339 ct03 : * a < * (supf ub) |
| 340 ct03 = ChainP.order supb a₁ (ChainP.csupz supa) | |
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341 ct02 : * a < * b |
| 748 | 342 ct02 with s≤fc (supf ub) f mf fcb |
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parents:
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diff
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343 ... | case1 eq = subst (λ k → * a < k ) eq ct03 |
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parents:
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344 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt |
| 749 | 345 ... | tri≈ ¬a refl ¬c = fcn-cmp (supf ua) f mf fca fcb |
| 685 | 346 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where |
| 749 | 347 ct05 : * b < * (supf ua) |
| 348 ct05 = ChainP.order supa c (ChainP.csupz supb) | |
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349 ct04 : * b < * a |
| 749 | 350 ct04 with s≤fc (supf ua) f mf fca |
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parents:
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351 ... | case1 eq = subst (λ k → * b < k ) eq ct05 |
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352 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt |
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parents:
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changeset
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353 |
| 743 | 354 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 355 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 748 | 356 init-uchain A f mf ay = ⟪ ay , ch-init (init ay) ⟫ |
| 743 | 357 |
| 698 | 358 ChainP-next : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 359 → {x z : Ordinal } → ChainP A f mf ay supf x z → ChainP A f mf ay supf x (f z ) | |
| 746 | 360 ChainP-next A f mf {y} ay supf {x} {z} cp = record { y<s = ChainP.y<s cp ; supfu=u = ChainP.supfu=u cp |
| 361 ; fcy<sup = ChainP.fcy<sup cp ; csupz = fsuc _ (ChainP.csupz cp) ; order = ChainP.order cp } | |
| 698 | 362 |
| 497 | 363 Zorn-lemma : { A : HOD } |
| 464 | 364 → o∅ o< & A |
| 568 | 365 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 366 → Maximal A |
| 552 | 367 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 368 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 369 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 537 | 370 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 371 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 372 s : HOD |
| 373 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 374 as : A ∋ * ( & s ) |
| 375 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 )) ) | |
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parents:
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376 as0 : odef A (& s ) |
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377 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 378 s<A : & s o< & A |
| 568 | 379 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 380 HasMaximal : HOD |
| 537 | 381 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 382 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 383 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 384 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 385 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 386 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 387 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
| 388 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
| 389 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)) | |
| 390 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 | |
| 391 ¬x<m : ¬ (* x < * m) | |
| 392 ¬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 | 393 |
| 560 | 394 -- Uncountable ascending chain by axiom of choice |
| 530 | 395 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 396 cf nmx x with ODC.∋-p O A (* x) |
| 397 ... | no _ = o∅ | |
| 398 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 399 ... | yes nogt = -- no larger element, so it is maximal |
| 400 ⊥-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 | 401 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 402 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 403 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 404 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 405 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 406 ... | 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 ⟫ ) | |
| 407 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 408 |
| 409 --- | |
| 410 --- infintie ascention sequence of f | |
| 411 --- | |
| 530 | 412 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 413 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 414 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 415 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 416 |
| 703 | 417 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 653 | 418 (total : IsTotalOrderSet (ZChain.chain zc) ) → SUP A (ZChain.chain zc) |
| 703 | 419 sp0 f mf zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total |
| 543 | 420 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
| 421 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
| 422 | |
| 728 | 423 SZ1 :( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 424 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x | |
| 425 SZ1 A f mf {y} ay zc x = TransFinite { λ x → ZChain1 A f mf ay zc x } zc1 x where | |
| 734 | 426 chain-mono2 : (x : Ordinal) {a b c : Ordinal} → a o≤ b → b o≤ x → |
| 427 odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c | |
| 748 | 428 chain-mono2 x {a} {b} {c} a≤b b≤x ⟪ ua , ch-init fc ⟫ = |
| 429 ⟪ ua , ch-init fc ⟫ | |
| 430 chain-mono2 x {a} {b} {c} a≤b b≤x ⟪ uaa , ch-is-sup ua u<x is-sup fc ⟫ = | |
| 749 | 431 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ u<x a≤b ) is-sup fc ⟫ |
| 743 | 432 chain<ZA : {x : Ordinal } → UnionCF A f mf ay (ZChain.supf zc) x ⊆' UnionCF A f mf ay (ZChain.supf zc) (& A) |
| 748 | 433 chain<ZA {x} ux with proj2 ux |
| 434 ... | ch-init fc = ⟪ proj1 ux , ch-init fc ⟫ | |
| 749 | 435 ... | ch-is-sup u pu<x is-sup fc = ⟪ proj1 ux , ch-is-sup u u<x is-sup fc ⟫ where |
| 436 u<A : (& ( * ( ZChain.supf zc u))) o< & A | |
| 437 u<A = c<→o< (subst (λ k → odef A k ) (sym &iso) (A∋fcs _ f mf fc) ) | |
| 438 u<x : u o< & A | |
| 439 u<x = subst (λ k → k o< & A ) (trans &iso (ChainP.supfu=u is-sup)) u<A | |
| 735 | 440 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 441 b o< x → (ab : odef A b) → | |
| 442 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f → | |
| 443 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b | |
| 749 | 444 is-max-hp x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
| 445 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ | |
| 446 ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ⟪ ab , | |
| 447 subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) | |
| 448 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x (ChainP-next A f mf ay _ is-sup) (fsuc _ fc)) ⟫ | |
| 728 | 449 zc1 : (x : Ordinal) → ((y₁ : Ordinal) → y₁ o< x → ZChain1 A f mf ay zc y₁) → ZChain1 A f mf ay zc x |
|
732
ddeb107b6f71
bchain can be reached from upwords by f. so it is worng.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
729
diff
changeset
|
450 zc1 x prev with Oprev-p x |
| 745 | 451 ... | yes op = record { is-max = is-max ; chain-mono2 = chain-mono2 x ; fcy<sup = ? ; sup=u = ? ; order = ? } where |
|
732
ddeb107b6f71
bchain can be reached from upwords by f. so it is worng.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
729
diff
changeset
|
452 px = Oprev.oprev op |
| 735 | 453 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 454 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 754 | 455 fcy<sup : {u w : Ordinal} → u o< x → FClosure A f y w → w << ZChain.supf zc u |
| 456 fcy<sup {u} {w} u<x fc = ? | |
| 728 | 457 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 458 b o< x → (ab : odef A b) → | |
| 459 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → | |
| 460 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b | |
| 735 | 461 is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b |
| 733 | 462 is-max {a} {b} ua b<x ab (case2 is-sup) a<b with ODC.p∨¬p O ( HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ) |
| 735 | 463 ... | case1 has-prev = is-max-hp x {a} {b} ua b<x ab has-prev a<b |
| 734 | 464 ... | case2 ¬fy<x = m01 where |
| 735 | 465 px<x : px o< x |
| 466 px<x = subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc | |
| 728 | 467 m01 : odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
| 736 | 468 m01 with trio< b px --- px < b < x |
| 469 ... | tri> ¬a ¬b c = ⊥-elim (¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫) | |
| 735 | 470 ... | tri< b<px ¬b ¬c = chain-mono2 x ( o<→≤ (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc )) o≤-refl m04 where |
| 471 m03 : odef (UnionCF A f mf ay (ZChain.supf zc) px) a | |
| 749 | 472 m03 with proj2 ua |
| 473 ... | ch-init fc = ⟪ proj1 ua , ch-init fc ⟫ | |
| 474 ... | ch-is-sup u u<x is-sup fc with trio< u px | |
| 475 ... | tri< a ¬b ¬c = ⟪ proj1 ua , ch-is-sup u a is-sup fc ⟫ | |
| 750 | 476 ... | tri≈ ¬a u=px ¬c = ? --- supf u < a < b , |
| 749 | 477 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x ⟫ ) |
| 478 -- = ⟪ proj1 ua , ? ⟫ | |
| 728 | 479 m04 : odef (UnionCF A f mf ay (ZChain.supf zc) px) b |
| 735 | 480 m04 = ZChain1.is-max (prev px px<x) m03 b<px ab |
| 481 (case2 record {x<sup = λ {z} lt → IsSup.x<sup is-sup (chain-mono2 x ( o<→≤ (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc )) o≤-refl lt) } ) a<b | |
| 750 | 482 ... | tri≈ ¬a b=px ¬c = ? -- b = px case, u = px u<x |
| 744 | 483 ... | no lim = record { is-max = is-max ; chain-mono2 = chain-mono2 x ; fcy<sup = fcy<sup ; sup=u = sup=u ; order = order } where |
| 743 | 484 fcy<sup : {u w : Ordinal} → u o< x → FClosure A f y w → w << ZChain.supf zc u |
| 744 | 485 fcy<sup {u} {w} u<x fc = ZChain1.fcy<sup (prev (osuc u) (ob<x lim u<x)) <-osuc fc |
| 486 sup=u : {b : Ordinal} {ab : odef A b} → b o< x → | |
| 487 IsSup A (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) ab → | |
| 488 ZChain.supf zc b ≡ b | |
| 489 sup=u {b} {ab} b<x is-sup = ZChain1.sup=u (prev (osuc b) (ob<x lim b<x)) <-osuc is-sup | |
| 490 order : {b sup1 z1 : Ordinal} → b o< x → sup1 o< b → | |
| 491 FClosure A f (ZChain.supf zc sup1) z1 → z1 << ZChain.supf zc b | |
| 492 order {b} b<x s<b fc = ZChain1.order (prev (osuc b) (ob<x lim b<x)) <-osuc s<b fc | |
| 734 | 493 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → |
| 494 b o< x → (ab : odef A b) → | |
| 495 HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab → | |
| 496 * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b | |
| 735 | 497 is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b |
| 743 | 498 is-max {a} {b} ua b<x ab (case2 is-sup) a<b with IsSup.x<sup is-sup (init-uchain A f mf ay ) |
| 499 ... | case1 b=y = ⊥-elim ( <-irr ( ZChain.initial zc (chain<ZA (chain-mono2 (osuc x) (o<→≤ <-osuc ) o≤-refl ua )) ) | |
| 500 (subst (λ k → * a < * k ) (sym b=y) a<b ) ) | |
| 744 | 501 ... | case2 y<b = chain-mono2 x (o<→≤ (ob<x lim b<x) ) o≤-refl m04 where |
| 743 | 502 y<s : y << ZChain.supf zc b |
| 503 y<s = y<s | |
| 740 | 504 m07 : {z : Ordinal} → FClosure A f y z → z << ZChain.supf zc b |
| 744 | 505 m07 {z} fc = ZChain1.fcy<sup (prev (osuc b) (ob<x lim b<x)) <-osuc fc |
| 741 | 506 m08 : {sup1 z1 : Ordinal} → sup1 o< b → FClosure A f (ZChain.supf zc sup1) z1 → z1 << ZChain.supf zc b |
| 744 | 507 m08 {sup1} {z1} s<b fc = ZChain1.order (prev (osuc b) (ob<x lim b<x) ) <-osuc s<b fc |
| 735 | 508 m05 : b ≡ ZChain.supf zc b |
| 744 | 509 m05 = sym (ZChain1.sup=u (prev (osuc b) (ob<x lim b<x)) {_} {ab} <-osuc |
| 510 record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono2 x (o<→≤ (ob<x lim b<x)) o≤-refl lt )} ) -- ZChain on x | |
| 739 | 511 m06 : ChainP A f mf ay (ZChain.supf zc) b b |
| 744 | 512 m06 = record { fcy<sup = m07 ; csupz = subst (λ k → FClosure A f k b ) m05 (init ab) ; order = m08 ; y<s = y<s |
| 513 ; supfu=u = sym m05 } | |
| 735 | 514 m04 : odef (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) b |
| 750 | 515 m04 = ⟪ ab , ch-is-sup b <-osuc m06 (subst (λ k → FClosure A f k b) m05 (init ab)) ⟫ |
| 727 | 516 |
| 543 | 517 --- |
| 560 | 518 --- the maximum chain has fix point of any ≤-monotonic function |
| 543 | 519 --- |
| 703 | 520 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
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521 → (total : IsTotalOrderSet (ZChain.chain zc) ) |
| 703 | 522 → f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total)) |
| 523 fixpoint f mf zc total = z14 where | |
| 538 | 524 chain = ZChain.chain zc |
| 703 | 525 sp1 = sp0 f mf zc total |
| 712 | 526 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
| 570 | 527 → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) |
| 538 | 528 → * a < * b → odef chain b |
| 728 | 529 z10 = ZChain1.is-max (SZ1 A f mf as0 zc (& A) ) |
| 543 | 530 z11 : & (SUP.sup sp1) o< & A |
| 531 z11 = c<→o< ( SUP.A∋maximal sp1) | |
| 538 | 532 z12 : odef chain (& (SUP.sup sp1)) |
| 533 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
| 653 | 534 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 712 | 535 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z11 (SUP.A∋maximal sp1) |
| 570 | 536 (case2 z19 ) z13 where |
| 538 | 537 z13 : * (& s) < * (& (SUP.sup sp1)) |
| 653 | 538 z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc ) |
| 538 | 539 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
| 540 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 570 | 541 z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) |
| 571 | 542 z19 = record { x<sup = z20 } where |
| 543 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) | |
| 544 z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) | |
| 570 | 545 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
| 546 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
| 547 -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } | |
| 703 | 548 z14 : f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total )) |
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549 z14 with total (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
| 631 | 550 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
| 551 z16 : ⊥ | |
| 552 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
| 553 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
| 554 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
| 555 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
| 556 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
| 557 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
| 558 z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) | |
| 559 z17 : ⊥ | |
| 560 z17 with z15 | |
| 561 ... | case1 eq = ¬b eq | |
| 562 ... | case2 lt = ¬a lt | |
| 560 | 563 |
| 564 -- ZChain contradicts ¬ Maximal | |
| 565 -- | |
| 571 | 566 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) |
| 560 | 567 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain |
| 568 -- | |
| 697 | 569 z04 : (nmx : ¬ Maximal A ) |
| 703 | 570 → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) |
| 664 | 571 → IsTotalOrderSet (ZChain.chain zc) → ⊥ |
| 703 | 572 z04 nmx zc total = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal sp1 )))) |
| 571 | 573 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) |
| 703 | 574 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc total ))) -- x ≡ f x ̄ |
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575 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1 ))) where -- x < f x |
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576 sp1 : SUP A (ZChain.chain zc) |
| 703 | 577 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc total |
| 538 | 578 c = & (SUP.sup sp1) |
| 548 | 579 |
| 711 | 580 inititalChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → ZChain A f mf ay o∅ |
| 754 | 581 inititalChain f mf {y} ay = record { supf = isupf ; chain⊆A = λ lt → proj1 lt ; chain∋init = cy ; csupf = ? |
| 745 | 582 ; initial = isy ; f-next = inext ; f-total = itotal ; sup=u = λ b<0 → ⊥-elim (¬x<0 b<0) ; order = λ b<0 → ⊥-elim (¬x<0 b<0) } where |
| 711 | 583 isupf : Ordinal → Ordinal |
| 584 isupf z = y | |
| 585 cy : odef (UnionCF A f mf ay isupf o∅) y | |
| 750 | 586 cy = ⟪ ay , ch-init (init ay) ⟫ |
| 711 | 587 isy : {z : Ordinal } → odef (UnionCF A f mf ay isupf o∅) z → * y ≤ * z |
| 748 | 588 isy {z} ⟪ az , uz ⟫ with uz |
| 589 ... | ch-init fc = s≤fc y f mf fc | |
| 590 ... | ch-is-sup u u<x is-sup fc = ⊥-elim ( <-irr (case1 refl) ( ChainP.fcy<sup is-sup (init ay) ) ) | |
| 711 | 591 inext : {a : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a → odef (UnionCF A f mf ay isupf o∅) (f a) |
| 748 | 592 inext {a} ua with (proj2 ua) |
| 593 ... | ch-init fc = ⟪ proj2 (mf _ (proj1 ua)) , ch-init (fsuc _ fc ) ⟫ | |
| 594 ... | ch-is-sup u u<x is-sup fc = ⊥-elim ( <-irr (case1 refl) ( ChainP.fcy<sup is-sup (init ay) ) ) | |
| 711 | 595 itotal : IsTotalOrderSet (UnionCF A f mf ay isupf o∅) |
| 596 itotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 597 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 748 | 598 uz01 = chain-total A f mf ay isupf (proj2 ca) (proj2 cb) |
| 711 | 599 imax : {a b : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a → |
| 712 | 600 b o< o∅ → (ab : odef A b) → |
| 711 | 601 HasPrev A (UnionCF A f mf ay isupf o∅) ab f ∨ IsSup A (UnionCF A f mf ay isupf o∅) ab → |
| 602 * a < * b → odef (UnionCF A f mf ay isupf o∅) b | |
| 714 | 603 imax {a} {b} ua b<x ab (case1 hasp) a<b = subst (λ k → odef (UnionCF A f mf ay isupf o∅) k ) (sym (HasPrev.x=fy hasp)) ( inext (HasPrev.ay hasp) ) |
| 604 imax {a} {b} ua b<x ab (case2 sup) a<b = ⊥-elim ( ¬x<0 b<x ) | |
| 711 | 605 |
| 560 | 606 -- |
| 547 | 607 -- create all ZChains under o< x |
| 560 | 608 -- |
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608
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609 |
| 674 | 610 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 611 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 707 | 612 ind f mf {y} ay x prev with Oprev-p x |
| 697 | 613 ... | yes op = zc4 where |
| 682 | 614 -- |
| 615 -- we have previous ordinal to use induction | |
| 616 -- | |
| 617 px = Oprev.oprev op | |
| 703 | 618 zc : ZChain A f mf ay (Oprev.oprev op) |
| 682 | 619 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 620 px<x : px o< x | |
| 621 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
| 709 | 622 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 623 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 697 | 624 |
| 703 | 625 pchain : HOD |
| 626 pchain = UnionCF A f mf ay (ZChain.supf zc) x | |
| 627 ptotal : IsTotalOrderSet pchain | |
| 628 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 629 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 748 | 630 uz01 = chain-total A f mf ay (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb)) |
| 704 | 631 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y |
| 632 pchain⊆A {y} ny = proj1 ny | |
| 633 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
| 749 | 634 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
| 635 pnext {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u<x (ChainP-next A f mf ay _ is-sup ) (fsuc _ fc ) ⟫ | |
| 704 | 636 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
| 748 | 637 pinit {a} ⟪ aa , ua ⟫ with ua |
| 638 ... | ch-init fc = s≤fc y f mf fc | |
| 639 ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (case2 zc7) (s≤fc _ f mf fc) where | |
| 749 | 640 zc7 : y << (ZChain.supf zc) u |
| 707 | 641 zc7 = ChainP.fcy<sup is-sup (init ay) |
| 704 | 642 pcy : odef pchain y |
| 748 | 643 pcy = ⟪ ay , ch-init (init ay) ⟫ |
| 703 | 644 |
| 754 | 645 supf0 = ZChain.supf zc |
| 646 | |
| 647 csupf : {z : Ordinal} → odef (UnionCF A f mf ay supf0 x) (supf0 z) | |
| 648 csupf {z} with ZChain.csupf zc {z} | |
| 649 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 650 ... | ⟪ az , ch-is-sup u u<px is-sup fc ⟫ = ⟪ az , ch-is-sup u (ordtrans u<px px<x ) is-sup fc ⟫ | |
| 745 | 651 |
| 611 | 652 -- if previous chain satisfies maximality, we caan reuse it |
| 653 -- | |
| 727 | 654 no-extension : ZChain A f mf ay x |
| 745 | 655 no-extension = record { supf = supf0 |
| 754 | 656 ; initial = pinit ; chain∋init = pcy ; csupf = csupf |
| 657 ; chain⊆A = pchain⊆A ; f-next = pnext ; f-total = ptotal } | |
| 709 | 658 |
| 703 | 659 zc4 : ZChain A f mf ay x |
| 713 | 660 zc4 with ODC.∋-p O A (* px) |
| 727 | 661 ... | no noapx = no-extension -- ¬ A ∋ p, just skip |
| 713 | 662 ... | yes apx with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc ) apx f ) |
| 703 | 663 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 727 | 664 ... | case1 pr = no-extension -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
| 713 | 665 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc ) apx ) |
| 682 | 666 ... | case1 is-sup = -- x is a sup of zc |
| 754 | 667 record { supf = psupf1 ; chain⊆A = ? ; f-next = ? ; f-total = ? ; csupf = ? |
| 668 ; initial = ? ; chain∋init = ? } where | |
| 750 | 669 psupf1 : Ordinal → Ordinal |
| 670 psupf1 z with trio< z x | |
| 671 ... | tri< a ¬b ¬c = ZChain.supf zc z | |
| 672 ... | tri≈ ¬a b ¬c = x | |
| 673 ... | tri> ¬a ¬b c = x | |
| 727 | 674 ... | case2 ¬x=sup = no-extension -- px is not f y' nor sup of former ZChain from y -- no extention |
| 728 | 675 ... | no lim = zc5 where |
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676 |
| 703 | 677 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
| 678 pzc z z<x = prev z z<x | |
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679 |
| 703 | 680 psupf0 : (z : Ordinal) → Ordinal |
| 681 psupf0 z with trio< z x | |
| 682 ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z | |
| 733 | 683 ... | tri≈ ¬a b ¬c = & A |
| 684 ... | tri> ¬a ¬b c = & A | |
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685 |
| 704 | 686 pchain : HOD |
| 687 pchain = UnionCF A f mf ay psupf0 x | |
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688 |
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689 psupf0=pzc : {z : Ordinal} → (z<x : z o< x) → psupf0 z ≡ ZChain.supf (pzc z z<x) z |
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690 psupf0=pzc {z} z<x with trio< z x |
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|
691 ... | tri≈ ¬a b ¬c = ⊥-elim (¬a z<x) |
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692 ... | tri> ¬a ¬b c = ⊥-elim (¬a z<x) |
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693 ... | tri< a ¬b ¬c with o<-irr z<x a |
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694 ... | refl = refl |
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|
695 |
| 704 | 696 ptotal : IsTotalOrderSet pchain |
| 697 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 703 | 698 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 748 | 699 uz01 = chain-total A f mf ay psupf0 ( (proj2 ca)) ( (proj2 cb)) |
| 704 | 700 |
| 701 pchain⊆A : {y : Ordinal} → odef pchain y → odef A y | |
| 702 pchain⊆A {y} ny = proj1 ny | |
| 703 pnext : {a : Ordinal} → odef pchain a → odef pchain (f a) | |
| 750 | 704 pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-init (fsuc _ fc) ⟫ |
| 705 pnext {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-is-sup u u<x (ChainP-next A f mf ay _ is-sup ) (fsuc _ fc) ⟫ | |
| 704 | 706 pinit : {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁ |
| 748 | 707 pinit {a} ⟪ aa , ua ⟫ with ua |
| 708 ... | ch-init fc = s≤fc y f mf fc | |
| 709 ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (case2 zc7) (s≤fc _ f mf fc) where | |
| 710 zc7 : y << psupf0 ? | |
| 707 | 711 zc7 = ChainP.fcy<sup is-sup (init ay) |
| 704 | 712 pcy : odef pchain y |
| 748 | 713 pcy = ⟪ ay , ch-init (init ay) ⟫ |
| 704 | 714 |
| 715 usup : SUP A pchain | |
| 716 usup = supP pchain (λ lt → proj1 lt) ptotal | |
| 703 | 717 spu = & (SUP.sup usup) |
| 718 psupf : Ordinal → Ordinal | |
| 719 psupf z with trio< z x | |
| 720 ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z | |
| 721 ... | tri≈ ¬a b ¬c = spu | |
| 722 ... | tri> ¬a ¬b c = spu | |
| 704 | 723 |
| 754 | 724 |
| 725 csupf :{z : Ordinal} → odef (UnionCF A f mf ay psupf x) (psupf z) | |
| 726 csupf {z} with trio< z x | |
| 727 ... | tri< a ¬b ¬c = zc11 where | |
| 728 zc11 : odef A (ZChain.supf (pzc z a) z) ∧ UChain A f mf ay psupf x (ZChain.supf (pzc z a) z) | |
| 729 zc11 with ZChain.csupf (pzc z a) {z} | |
| 730 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 731 ... | ⟪ az , ch-is-sup u u<z is-sup fc ⟫ = ⟪ az , ch-is-sup u (ordtrans u<z a) ? (pfc a fc) ⟫ | |
| 732 ... | tri≈ ¬a b ¬c = ? | |
| 733 ... | tri> ¬a ¬b c = ? | |
| 734 | |
| 735 no-extension : ZChain A f mf ay x | |
| 736 no-extension = record { initial = ? ; chain∋init = ? ; supf = psupf ; csupf = ? | |
| 737 ; chain⊆A = ? ; f-next = ? ; f-total = ? } | |
| 738 | |
| 739 | |
| 703 | 740 zc5 : ZChain A f mf ay x |
| 697 | 741 zc5 with ODC.∋-p O A (* x) |
|
732
ddeb107b6f71
bchain can be reached from upwords by f. so it is worng.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
729
diff
changeset
|
742 ... | no noax = no-extension -- ¬ A ∋ p, just skip |
| 704 | 743 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain ax f ) |
| 703 | 744 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 727 | 745 ... | case1 pr = no-extension |
| 704 | 746 ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A pchain ax ) |
| 754 | 747 ... | case1 is-sup = record { initial = {!!} ; chain∋init = {!!} ; supf = psupf1 ; csupf = ? |
| 739 | 748 ; chain⊆A = {!!} ; f-next = {!!} ; f-total = ? } where -- x is a sup of (zc ?) |
| 728 | 749 psupf1 : Ordinal → Ordinal |
| 750 psupf1 z with trio< z x | |
| 751 ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z | |
| 752 ... | tri≈ ¬a b ¬c = x | |
| 753 ... | tri> ¬a ¬b c = x | |
| 727 | 754 ... | case2 ¬x=sup = no-extension -- x is not f y' nor sup of former ZChain from y -- no extention |
| 553 | 755 |
| 703 | 756 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay (& A) |
| 757 SZ f mf {y} ay = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) (& A) | |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
758 |
| 551 | 759 zorn00 : Maximal A |
| 760 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 761 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
| 762 -- yes we have the maximal | |
| 763 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 764 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 765 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 766 zorn01 = proj1 zorn03 | |
| 767 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 768 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 703 | 769 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04 total ) where |
| 551 | 770 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 771 nmx : ¬ Maximal A | |
| 772 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 773 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 774 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) ) ⟫ | |
| 703 | 775 zorn04 : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) |
| 653 | 776 zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) |
| 634 | 777 total : IsTotalOrderSet (ZChain.chain zorn04) |
| 654 | 778 total {a} {b} = zorn06 where |
| 779 zorn06 : odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a) | |
| 780 zorn06 = ZChain.f-total (SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as) ) | |
| 551 | 781 |
| 516 | 782 -- usage (see filter.agda ) |
| 783 -- | |
| 497 | 784 -- _⊆'_ : ( A B : HOD ) → Set n |
| 785 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 786 |
| 497 | 787 -- MaximumSubset : {L P : HOD} |
| 788 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 789 -- → IsPartialOrderSet P _⊆'_ | |
| 790 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 791 -- → Maximal P (_⊆'_) | |
| 792 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |