Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 986:557f8145d3c1
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 12 Nov 2022 18:11:14 +0900 |
| parents | 0d8dafbecb0d |
| children | c8c60a05b39b |
| rev | line source |
|---|---|
| 478 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 508 | 2 open import Level hiding ( suc ; zero ) |
| 431 | 3 open import Ordinals |
| 966 | 4 open import Relation.Binary |
| 552 | 5 open import Relation.Binary.Core |
| 6 open import Relation.Binary.PropositionalEquality | |
| 966 | 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 -- |
| 966 | 11 -- Zorn-lemma : { A : HOD } |
| 12 -- → o∅ o< & A | |
| 560 | 13 -- → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 966 | 14 -- → Maximal A |
| 560 | 15 -- |
| 16 | |
| 431 | 17 open import zf |
| 477 | 18 open import logic |
| 19 -- open import partfunc {n} O | |
| 20 | |
| 966 | 21 open import Relation.Nullary |
| 22 open import Data.Empty | |
| 23 import BAlgbra | |
| 431 | 24 |
| 555 | 25 open import Data.Nat hiding ( _<_ ; _≤_ ) |
| 966 | 26 open import Data.Nat.Properties |
| 27 open import nat | |
| 555 | 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 | |
| 966 | 55 _<<_ : (x y : Ordinal ) → Set n |
| 570 | 56 x << y = * x < * y |
| 57 | |
| 872 | 58 _<=_ : (x y : Ordinal ) → Set n -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain |
| 765 | 59 x <= y = (x ≡ y ) ∨ ( * x < * y ) |
| 60 | |
| 966 | 61 POO : IsStrictPartialOrder _≡_ _<<_ |
| 62 POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 63 ; trans = IsStrictPartialOrder.trans PO | |
| 570 | 64 ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y |
| 966 | 65 ; <-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 } } |
| 66 | |
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67 _≤_ : (x y : HOD) → Set (Level.suc n) |
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68 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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69 |
| 966 | 70 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 554 | 71 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl |
| 72 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 73 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 74 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 75 | |
| 966 | 76 <=-trans : {x y z : Ordinal } → x <= y → y <= z → x <= z |
| 955 | 77 <=-trans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl |
| 78 <=-trans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 79 <=-trans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 80 <=-trans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 779 | 81 |
| 966 | 82 ftrans<=-< : {x y z : Ordinal } → x <= y → y << z → x << z |
| 953 | 83 ftrans<=-< {x} {y} {z} (case1 eq) y<z = subst (λ k → k < * z) (sym (cong (*) eq)) y<z |
| 966 | 84 ftrans<=-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z |
| 951 | 85 |
| 966 | 86 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y |
| 770 | 87 <=to≤ (case1 eq) = case1 (cong (*) eq) |
| 88 <=to≤ (case2 lt) = case2 lt | |
| 89 | |
| 966 | 90 ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y |
| 779 | 91 ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) |
| 92 ≤to<= (case2 lt) = case2 lt | |
| 93 | |
| 556 | 94 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 95 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 96 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 97 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 98 |
| 561 | 99 ptrans = IsStrictPartialOrder.trans PO |
| 100 | |
| 492 | 101 open _==_ |
| 102 open _⊆_ | |
| 103 | |
| 966 | 104 -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A |
| 879 | 105 -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x |
| 106 -- <-TransFinite = ? | |
| 107 | |
| 530 | 108 -- |
| 560 | 109 -- Closure of ≤-monotonic function f has total order |
| 530 | 110 -- |
| 111 | |
| 112 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 113 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 114 | |
| 551 | 115 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 783 | 116 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
| 555 | 117 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 118 |
| 556 | 119 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 |
| 783 | 120 A∋fc {A} s f mf (init as refl ) = as |
| 556 | 121 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 122 |
| 714 | 123 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 |
| 783 | 124 A∋fcs {A} s f mf (init as refl) = as |
| 966 | 125 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
| 714 | 126 |
| 556 | 127 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 783 | 128 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
| 556 | 129 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 130 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 966 | 131 ... | case2 x<fx with s≤fc {A} s f mf fcy |
| 556 | 132 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) |
| 133 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 134 |
| 800 | 135 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 136 fcn s mf (init as refl) = zero | |
| 137 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
| 138 ... | case1 eq = fcn s mf p | |
| 139 ... | case2 y<fy = suc (fcn s mf p ) | |
| 140 | |
| 966 | 141 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 800 | 142 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y |
| 143 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 | |
| 144 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 145 fc06 {x} {y} refl {j} not = fc08 not where | |
| 966 | 146 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
| 800 | 147 fc08 () |
| 148 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
| 149 fc07 {x} (init as refl) eq = refl | |
| 150 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 151 ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) | |
| 152 -- ... | case2 x<fx = ? | |
| 153 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 | |
| 154 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
| 155 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
| 156 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
| 157 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 158 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y ) | |
| 159 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 160 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) i=x i=y ) | |
| 161 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 ) ) | |
| 162 ... | 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 ) | |
| 163 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 ) ) | |
| 164 ... | 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 ) | |
| 165 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 166 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 167 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 168 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 169 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x | |
| 170 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
| 171 fc04 : * x1 ≡ * y | |
| 172 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 173 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 174 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 175 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 176 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 966 | 177 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq |
| 800 | 178 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where |
| 179 fc05 : * x ≡ * y1 | |
| 180 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 181 ... | 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))) | |
| 182 | |
| 183 | |
| 184 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 185 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 186 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
| 187 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 188 fc06 {x} {y} refl {j} not = fc08 not where | |
| 966 | 189 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
| 800 | 190 fc08 () |
| 191 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 | |
| 192 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
| 193 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 966 | 194 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) |
| 800 | 195 ... | case2 y<fy with <-cmp (fcn s mf cx ) i |
| 196 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 966 | 197 ... | 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 |
| 800 | 198 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where |
| 199 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 966 | 200 fc03 eq = cong pred eq |
| 201 fc02 : * x < * y1 | |
| 800 | 202 fc02 = fc01 i cx cy (fc03 i=y ) a |
| 203 | |
| 557 | 204 |
| 966 | 205 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 206 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 800 | 207 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 208 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 209 fc11 : * x < * y | |
| 210 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 211 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 212 fc10 : * x ≡ * y | |
| 213 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 966 | 214 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where |
| 800 | 215 fc12 : * y < * x |
| 216 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 600 | 217 |
| 563 | 218 |
| 729 | 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 |
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236 record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where |
| 533 | 237 field |
| 836 | 238 ax : odef A x |
| 534 | 239 y : Ordinal |
| 541 | 240 ay : odef B y |
| 966 | 241 x=fy : x ≡ f y |
| 529 | 242 |
| 962 | 243 record IsSUP (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 244 field | |
| 245 x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) | |
| 957 | 246 |
| 960 | 247 record IsMinSUP (A B : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (xa : odef A x) : Set n where |
| 654 | 248 field |
| 950 | 249 x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 966 | 250 minsup : { sup1 : Ordinal } → odef A sup1 |
| 954 | 251 → ( {z : Ordinal } → odef B z → (z ≡ sup1 ) ∨ (z << sup1 )) → x o≤ sup1 |
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252 not-hp : ¬ ( HasPrev A B f x ) |
| 568 | 253 |
| 656 | 254 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 255 field | |
| 256 sup : HOD | |
| 804 | 257 as : A ∋ sup |
| 950 | 258 x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive |
| 656 | 259 |
| 690 | 260 -- |
| 261 -- sup and its fclosure is in a chain HOD | |
| 262 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 263 -- whole chain is a union of separated Chain | |
| 966 | 264 -- minimum index is sup of y not ϕ |
| 690 | 265 -- |
| 266 | |
| 787 | 267 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where |
| 690 | 268 field |
| 966 | 269 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
| 828 | 270 order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
| 271 supu=u : supf u ≡ u | |
| 694 | 272 |
| 748 | 273 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 966 | 274 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where |
| 275 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 276 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : supf u o< supf x) ( is-sup : ChainP A f mf ay supf u ) | |
| 748 | 277 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z |
| 694 | 278 |
| 878 | 279 -- |
| 280 -- f (f ( ... (sup y))) f (f ( ... (sup z1))) | |
| 281 -- / | / | | |
| 282 -- / | / | | |
| 283 -- sup y < sup z1 < sup z2 | |
| 284 -- o< o< | |
| 861 | 285 -- data UChain is total |
| 286 | |
| 287 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) | |
| 288 {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 ) | |
| 289 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where | |
| 290 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) | |
| 966 | 291 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb |
| 292 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) with ChainP.fcy<sup supb fca | |
| 293 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 294 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 295 ct00 : * a ≡ * b | |
| 296 ct00 = trans (cong (*) eq) eq1 | |
| 297 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where | |
| 298 ct01 : * a < * b | |
| 299 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
| 300 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where | |
| 301 ct00 : * a < * (supf ub) | |
| 302 ct00 = lt | |
| 303 ct01 : * a < * b | |
| 304 ct01 with s≤fc (supf ub) f mf fcb | |
| 305 ... | case1 eq = subst (λ k → * a < k ) eq ct00 | |
| 306 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 307 ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb | |
| 308 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 309 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 310 ct00 : * a ≡ * b | |
| 311 ct00 = sym (trans (cong (*) eq) eq1 ) | |
| 312 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where | |
| 313 ct01 : * b < * a | |
| 314 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
| 315 ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where | |
| 316 ct00 : * b < * (supf ua) | |
| 317 ct00 = lt | |
| 318 ct01 : * b < * a | |
| 319 ct01 with s≤fc (supf ua) f mf fca | |
| 320 ... | case1 eq = subst (λ k → * b < k ) eq ct00 | |
| 321 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 861 | 322 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 |
| 966 | 323 ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁) fca |
| 324 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 861 | 325 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
| 326 ct00 : * a ≡ * b | |
| 327 ct00 = trans (cong (*) eq) eq1 | |
| 328 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 966 | 329 ct02 : * a < * b |
| 861 | 330 ct02 = subst (λ k → * k < * b ) (sym eq) lt |
| 331 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 332 ct03 : * a < * (supf ub) | |
| 333 ct03 = lt | |
| 966 | 334 ct02 : * a < * b |
| 861 | 335 ct02 with s≤fc (supf ub) f mf fcb |
| 336 ... | case1 eq = subst (λ k → * a < k ) eq ct03 | |
| 337 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt | |
| 966 | 338 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri≈ ¬a eq ¬c |
| 861 | 339 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) |
| 966 | 340 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) fcb |
| 341 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 861 | 342 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
| 343 ct00 : * a ≡ * b | |
| 344 ct00 = sym (trans (cong (*) eq) eq1) | |
| 345 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
| 966 | 346 ct02 : * b < * a |
| 861 | 347 ct02 = subst (λ k → * k < * a ) (sym eq) lt |
| 348 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where | |
| 349 ct05 : * b < * (supf ua) | |
| 350 ct05 = lt | |
| 966 | 351 ct04 : * b < * a |
| 861 | 352 ct04 with s≤fc (supf ua) f mf fca |
| 353 ... | case1 eq = subst (λ k → * b < k ) eq ct05 | |
| 354 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt | |
| 355 | |
| 694 | 356 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 357 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 358 | |
| 803 | 359 -- Union of supf z which o< x |
| 360 -- | |
| 966 | 361 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 694 | 362 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD |
| 363 UnionCF A f mf ay supf x | |
| 894 | 364 = 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 | 365 |
| 966 | 366 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) |
| 367 → supf x o< supf y → x o< y | |
| 842 | 368 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y |
| 369 ... | tri< a ¬b ¬c = a | |
| 370 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 371 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 372 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 373 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 374 | |
| 879 | 375 record MinSUP ( A B : HOD ) : Set n where |
| 376 field | |
| 377 sup : Ordinal | |
| 378 asm : odef A sup | |
| 966 | 379 x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) |
| 380 minsup : { sup1 : Ordinal } → odef A sup1 | |
| 879 | 381 → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 |
| 382 | |
| 383 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A | |
| 384 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 385 | |
| 880 | 386 M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } |
| 387 → (supf : Ordinal → Ordinal ) | |
| 966 | 388 → MinSUP A (UnionCF A f mf ay supf x) |
| 389 → SUP A (UnionCF A f mf ay supf x) | |
| 390 M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) | |
| 950 | 391 ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x≤sup = ms00 } where |
| 880 | 392 msup = MinSUP.sup ms |
| 393 ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup) | |
| 966 | 394 ms00 {z} uz with MinSUP.x≤sup ms uz |
| 880 | 395 ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq)) |
| 396 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso refl lt ) | |
| 397 | |
| 867 | 398 |
| 966 | 399 chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 919 | 400 (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b |
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401 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c |
| 966 | 402 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = |
| 403 ⟪ ua , ch-init fc ⟫ | |
| 919 | 404 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = |
| 966 | 405 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (supf-mono a≤b ) ) is-sup fc ⟫ |
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406 |
| 966 | 407 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 408 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 409 field |
| 966 | 410 supf : Ordinal → Ordinal |
| 411 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z | |
| 412 → IsSUP A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) f b ) → supf b ≡ b | |
| 880 | 413 |
| 868 | 414 asupf : {x : Ordinal } → odef A (supf x) |
| 880 | 415 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y |
| 416 supf-< : {x y : Ordinal } → supf x o< supf y → supf x << supf y | |
| 891 | 417 supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z |
| 880 | 418 |
| 966 | 419 minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) |
| 891 | 420 supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z ) |
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zc10 : supf c ≡ supf (& A) → {x : Ordinal } → odef A x → ¬ ( c << x ) ?
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421 csupf : {b : Ordinal } → supf b o< supf z → odef (UnionCF A f mf ay supf z) (supf b) -- supf z is not an element of this chain |
| 880 | 422 |
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mutual tranfinite in zorn
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423 chain : HOD |
| 703 | 424 chain = UnionCF A f mf ay supf z |
| 861 | 425 chain⊆A : chain ⊆' A |
| 426 chain⊆A = λ lt → proj1 lt | |
| 934 | 427 |
| 966 | 428 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
| 429 sup {x} x≤z = M→S supf (minsup x≤z) | |
| 934 | 430 |
| 431 s=ms : {x : Ordinal } → (x≤z : x o≤ z ) → & (SUP.sup (sup x≤z)) ≡ MinSUP.sup (minsup x≤z) | |
| 432 s=ms {x} x≤z = &iso | |
| 878 | 433 |
| 966 | 434 chain∋init : odef chain y |
| 435 chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ | |
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436 f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a) |
| 966 | 437 f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
| 938 | 438 f-next {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u<x is-sup (fsuc _ fc ) ⟫ |
| 861 | 439 initial : {z : Ordinal } → odef chain z → * y ≤ * z |
| 440 initial {a} ⟪ aa , ua ⟫ with ua | |
| 966 | 441 ... | ch-init fc = s≤fc y f mf fc |
| 938 | 442 ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 966 | 443 zc7 : y <= supf u |
| 861 | 444 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
| 445 f-total : IsTotalOrderSet chain | |
| 966 | 446 f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 861 | 447 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 966 | 448 uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) |
| 861 | 449 |
| 871 | 450 supf-<= : {x y : Ordinal } → supf x <= supf y → supf x o≤ supf y |
| 451 supf-<= {x} {y} (case1 sx=sy) = o≤-refl0 sx=sy | |
| 452 supf-<= {x} {y} (case2 sx<sy) with trio< (supf x) (supf y) | |
| 453 ... | tri< a ¬b ¬c = o<→≤ a | |
| 454 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 455 ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) ) | |
| 456 | |
| 966 | 457 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
| 825 | 458 supf-inject {x} {y} sx<sy with trio< x y |
| 459 ... | tri< a ¬b ¬c = a | |
| 460 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 461 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 462 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 463 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 798 | 464 |
| 966 | 465 fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf |
| 466 fcy<sup {u} {w} u≤z fc with MinSUP.x≤sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) | |
| 467 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ | |
| 468 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) | |
| 469 ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) | |
|
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470 |
| 966 | 471 -- ordering is not proved here but in ZChain1 |
| 825 | 472 |
| 966 | 473 IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp |
| 960 | 474 → ({y : Ordinal} → odef (UnionCF A f mf ay supf x) y → (y ≡ sp ) ∨ (y << sp )) |
| 475 → ( {a : Ordinal } → a << f a ) | |
| 476 → ¬ ( HasPrev A (UnionCF A f mf ay supf x) f sp ) | |
| 477 IsMinSUP→NotHasPrev {x} {sp} asp is-sup <-mono-f hp = ⊥-elim (<-irr ( <=to≤ fsp≤sp) sp<fsp ) where | |
| 478 sp<fsp : sp << f sp | |
| 966 | 479 sp<fsp = <-mono-f |
| 480 pr = HasPrev.y hp | |
| 960 | 481 im00 : f (f pr) <= sp |
| 482 im00 = is-sup ( f-next (f-next (HasPrev.ay hp))) | |
| 483 fsp≤sp : f sp <= sp | |
| 484 fsp≤sp = subst (λ k → f k <= sp ) (sym (HasPrev.x=fy hp)) im00 | |
| 485 | |
| 969 | 486 UChain⊆ : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 487 {z y : Ordinal} (ay : odef A y) { supf supf1 : Ordinal → Ordinal } | |
| 488 → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) | |
| 966 | 489 → ( { x : Ordinal } → x o< z → supf x ≡ supf1 x) |
| 490 → ( { x : Ordinal } → z o≤ x → supf z o≤ supf1 x) | |
| 491 → UnionCF A f mf ay supf z ⊆' UnionCF A f mf ay supf1 z | |
| 969 | 492 UChain⊆ A f mf {z} {y} ay {supf} {supf1} supf-mono eq<x lex ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 978 | 493 UChain⊆ A f mf {z} {y} ay {supf} {supf1} supf-mono eq<x lex ⟪ az , ch-is-sup u {x} u<x is-sup fc ⟫ = ⟪ az , ch-is-sup u u<x1 cp1 fc1 ⟫ where |
| 966 | 494 u<x0 : u o< z |
| 978 | 495 u<x0 = supf-inject0 supf-mono u<x |
| 966 | 496 u<x1 : supf1 u o< supf1 z |
| 497 u<x1 = subst (λ k → k o< supf1 z ) (eq<x u<x0) (ordtrans<-≤ u<x (lex o≤-refl ) ) | |
| 498 fc1 : FClosure A f (supf1 u) x | |
| 499 fc1 = subst (λ k → FClosure A f k x ) (eq<x u<x0) fc | |
| 500 uc01 : {s : Ordinal } → supf1 s o< supf1 u → s o< z | |
| 501 uc01 {s} s<u with trio< s z | |
| 502 ... | tri< a ¬b ¬c = a | |
| 503 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> uc02 s<u ) where -- (supf-mono (o<→≤ u<x0)) | |
| 504 uc02 : supf1 u o≤ supf1 s | |
| 505 uc02 = begin | |
| 506 supf1 u <⟨ u<x1 ⟩ | |
| 507 supf1 z ≡⟨ cong supf1 (sym b) ⟩ | |
| 508 supf1 s ∎ where open o≤-Reasoning O | |
| 509 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> uc03 s<u ) where | |
| 510 uc03 : supf1 u o≤ supf1 s | |
| 511 uc03 = begin | |
| 512 supf1 u ≡⟨ sym (eq<x u<x0) ⟩ | |
| 513 supf u <⟨ u<x ⟩ | |
| 514 supf z ≤⟨ lex (o<→≤ c) ⟩ | |
| 515 supf1 s ∎ where open o≤-Reasoning O | |
| 516 cp1 : ChainP A f mf ay supf1 u | |
| 517 cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x0) (ChainP.fcy<sup is-sup fc ) | |
| 518 ; order = λ {s} {z} s<u fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x0) | |
| 519 (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (eq<x (uc01 s<u) )) (sym (eq<x u<x0)) s<u) | |
| 520 (subst (λ k → FClosure A f k z ) (sym (eq<x (uc01 s<u) )) fc )) | |
| 521 ; supu=u = trans (sym (eq<x u<x0)) (ChainP.supu=u is-sup) } | |
| 522 | |
| 523 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) | |
| 783 | 524 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
| 869 | 525 supf = ZChain.supf zc |
| 728 | 526 field |
| 966 | 527 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → supf b o< supf z → (ab : odef A b) |
| 528 → HasPrev A (UnionCF A f mf ay supf z) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab | |
| 869 | 529 → * a < * b → odef ((UnionCF A f mf ay supf z)) b |
| 949 | 530 order : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b) |
| 531 | |
| 568 | 532 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 533 field | |
| 534 maximal : HOD | |
| 804 | 535 as : A ∋ maximal |
| 568 | 536 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative |
| 567 | 537 |
| 966 | 538 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 539 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 540 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ | |
| 541 | |
| 542 Zorn-lemma : { A : HOD } | |
| 543 → o∅ o< & A | |
| 568 | 544 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 966 | 545 → Maximal A |
| 552 | 546 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 547 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 548 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 549 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 550 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 551 s : HOD |
| 966 | 552 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) |
| 568 | 553 as : A ∋ * ( & s ) |
| 554 as = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) | |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
555 as0 : odef A (& s ) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
556 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 557 s<A : & s o< & A |
| 568 | 558 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 559 HasMaximal : HOD |
| 966 | 560 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 537 | 561 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ |
| 966 | 562 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) |
| 532 | 563 Gtx : { x : HOD} → A ∋ x → HOD |
| 966 | 564 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 537 | 565 z08 : ¬ Maximal A → HasMaximal =h= od∅ |
| 804 | 566 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
| 537 | 567 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
| 568 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)) | |
| 569 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 | |
| 570 ¬x<m : ¬ (* x < * m) | |
| 966 | 571 ¬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 | 572 |
| 966 | 573 minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B |
| 879 | 574 minsupP B B⊆A total = m02 where |
| 575 xsup : (sup : Ordinal ) → Set n | |
| 576 xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) | |
| 577 ∀-imply-or : {A : Ordinal → Set n } {B : Set n } | |
| 578 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B | |
| 579 ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM | |
| 580 ∀-imply-or {A} {B} ∀AB | case1 t = case1 t | |
| 581 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where | |
| 582 lemma : ¬ ((x : Ordinal ) → A x) → B | |
| 583 lemma not with ODC.p∨¬p O B | |
| 584 lemma not | case1 b = b | |
| 585 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
| 586 m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B | |
| 587 m00 x = TransFinite0 ind x where | |
| 588 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) | |
| 589 → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B | |
| 590 ind x prev = ∀-imply-or m01 where | |
| 591 m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B | |
| 592 m01 z with trio< z x | |
| 593 ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 594 ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 595 ... | tri< a ¬b ¬c with prev z a | |
| 596 ... | case2 mins = case2 mins | |
| 597 ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) | |
| 950 | 598 ... | case1 mins = case2 record { sup = z ; asm = proj1 mins ; x≤sup = proj2 mins ; minsup = m04 } where |
| 879 | 599 m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 |
| 600 m04 {s} as lt with trio< z s | |
| 601 ... | tri< a ¬b ¬c = o<→≤ a | |
| 602 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 603 ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) | |
| 604 ... | case2 notz = case1 (λ _ → notz ) | |
| 605 m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) | |
| 606 m03 not = ⊥-elim ( not s1 (z09 (SUP.as S)) ⟪ SUP.as S , m05 ⟫ ) where | |
| 607 S : SUP A B | |
| 608 S = supP B B⊆A total | |
| 609 s1 = & (SUP.sup S) | |
| 610 m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) | |
| 950 | 611 m05 {w} bw with SUP.x≤sup S {* w} (subst (λ k → odef B k) (sym &iso) bw ) |
| 879 | 612 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) |
| 613 ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) | |
| 966 | 614 m02 : MinSUP A B |
| 879 | 615 m02 = dont-or (m00 (& A)) m03 |
| 616 | |
| 560 | 617 -- Uncountable ascending chain by axiom of choice |
| 530 | 618 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 619 cf nmx x with ODC.∋-p O A (* x) |
| 620 ... | no _ = o∅ | |
| 621 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 622 ... | yes nogt = -- no larger element, so it is maximal |
| 623 ⊥-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 | 624 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 625 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 626 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 627 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 628 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 629 ... | 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 ⟫ ) | |
| 630 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 631 |
| 632 --- | |
| 633 --- infintie ascention sequence of f | |
| 634 --- | |
| 530 | 635 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 636 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 637 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 638 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 639 |
| 803 | 640 -- |
| 953 | 641 -- maximality of chain |
| 642 -- | |
| 643 -- supf is fixed for z ≡ & A , we can prove order and is-max | |
| 803 | 644 -- |
| 645 | |
| 966 | 646 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 728 | 647 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
| 953 | 648 SZ1 f mf {y} ay zc x = zc1 x where |
| 900 | 649 chain-mono1 : {a b c : Ordinal} → a o≤ b |
| 788 | 650 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
| 919 | 651 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b |
| 966 | 652 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → (ab : odef A b) |
| 653 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) f b | |
| 920 | 654 → * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
| 655 is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev | |
| 966 | 656 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 938 | 657 ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) |
| 966 | 658 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ |
| 868 | 659 |
| 869 | 660 supf = ZChain.supf zc |
| 661 | |
| 920 | 662 csupf-fc : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1 |
| 663 csupf-fc {b} {s} {z1} b<z ss<sb (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc05 where | |
| 869 | 664 s<b : s o< b |
| 665 s<b = ZChain.supf-inject zc ss<sb | |
| 920 | 666 s<z : s o< & A |
| 667 s<z = ordtrans s<b b<z | |
| 870 | 668 zc04 : odef (UnionCF A f mf ay supf (& A)) (supf s) |
| 986 | 669 zc04 = ZChain.csupf zc (ordtrans<-≤ ss<sb (ZChain.supf-mono zc (o<→≤ b<z))) |
| 869 | 670 zc05 : odef (UnionCF A f mf ay supf b) (supf s) |
| 671 zc05 with zc04 | |
| 966 | 672 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ |
| 938 | 673 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u zc08 is-sup fc ⟫ where |
| 870 | 674 zc07 : FClosure A f (supf u) (supf s) -- supf u ≤ supf s → supf u o≤ supf s |
| 675 zc07 = fc | |
| 869 | 676 zc06 : supf u ≡ u |
| 677 zc06 = ChainP.supu=u is-sup | |
| 966 | 678 zc08 : supf u o< supf b |
| 894 | 679 zc08 = ordtrans≤-< (ZChain.supf-<= zc (≤to<= ( s≤fc _ f mf fc ))) ss<sb |
| 869 | 680 csupf-fc {b} {s} {z1} b<z ss≤sb (fsuc x fc) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc04 where |
| 681 zc04 : odef (UnionCF A f mf ay supf b) (f x) | |
| 682 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb fc ) | |
| 966 | 683 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ |
| 684 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u<x is-sup (fsuc _ fc) ⟫ | |
| 869 | 685 order : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b) |
| 686 order {b} {s} {z1} b<z ss<sb fc = zc04 where | |
| 891 | 687 zc00 : ( z1 ≡ MinSUP.sup (ZChain.minsup zc (o<→≤ b<z) )) ∨ ( z1 << MinSUP.sup ( ZChain.minsup zc (o<→≤ b<z) ) ) |
| 950 | 688 zc00 = MinSUP.x≤sup (ZChain.minsup zc (o<→≤ b<z) ) (subst (λ k → odef (UnionCF A f mf ay (ZChain.supf zc) b) k ) &iso (csupf-fc b<z ss<sb fc )) |
| 870 | 689 -- supf (supf b) ≡ supf b |
| 869 | 690 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) |
| 691 zc04 with zc00 | |
| 892 | 692 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z)) ) eq ) |
| 693 ... | case2 lt = case2 (subst₂ (λ j k → j < * k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z) )) lt ) | |
| 868 | 694 |
| 953 | 695 zc1 : (x : Ordinal) → ZChain1 A f mf ay zc x |
| 696 zc1 x with Oprev-p x -- prev is not used now.... | |
| 949 | 697 ... | yes op = record { is-max = is-max ; order = 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
|
698 px = Oprev.oprev op |
| 919 | 699 zc-b<x : {b : Ordinal } → ZChain.supf zc b o< ZChain.supf zc x → b o< osuc px |
| 700 zc-b<x {b} lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) (ZChain.supf-inject zc lt ) | |
| 894 | 701 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
| 919 | 702 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) → |
| 964 | 703 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → |
| 869 | 704 * a < * b → odef (UnionCF A f mf ay supf x) b |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
705 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P |
| 966 | 706 is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b |
| 707 is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup | |
| 919 | 708 = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
709 b<A : b o< & A |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
710 b<A = z09 ab |
| 919 | 711 b<x : b o< x |
| 712 b<x = ZChain.supf-inject zc sb<sx | |
| 966 | 713 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
| 714 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = | |
| 900 | 715 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) |
| 859 | 716 m05 : ZChain.supf zc b ≡ b |
| 966 | 717 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
718 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
| 872 | 719 m08 {z} fcz = ZChain.fcy<sup zc (o<→≤ b<A) fcz |
| 966 | 720 m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 721 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
| 966 | 722 m09 {s} {z} s<b fcz = order b<A s<b fcz |
| 723 m06 : ChainP A f mf ay supf b | |
| 859 | 724 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = m05 } |
| 949 | 725 ... | no lim = record { is-max = is-max ; order = order } where |
| 869 | 726 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
| 919 | 727 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) → |
| 964 | 728 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → |
| 869 | 729 * a < * b → odef (UnionCF A f mf ay supf x) b |
| 919 | 730 is-max {a} {b} ua sb<sx ab P a<b with ODC.or-exclude O P |
| 966 | 731 is-max {a} {b} ua sb<sx ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b |
| 732 is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (init-uchain A f mf ay ) | |
| 733 ... | case1 b=y = ⟪ subst (λ k → odef A k ) b=y ay , ch-init (subst (λ k → FClosure A f y k ) b=y (init ay refl )) ⟫ | |
| 734 ... | case2 y<b = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where | |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
735 m09 : b o< & A |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
736 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) |
| 966 | 737 b<x : b o< x |
| 919 | 738 b<x = ZChain.supf-inject zc sb<sx |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
739 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
| 872 | 740 m07 {z} fc = ZChain.fcy<sup zc (o<→≤ m09) fc |
| 966 | 741 m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 742 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
| 869 | 743 m08 {s} {z1} s<b fc = order m09 s<b fc |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
744 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
| 966 | 745 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
| 746 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) | |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
747 ; x=fy = HasPrev.x=fy nhp } ) |
| 859 | 748 m05 : ZChain.supf zc b ≡ b |
| 964 | 749 m05 = ZChain.sup=u zc ab (o<→≤ m09) ⟪ record { x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } , m04 ⟫ -- ZChain on x |
| 966 | 750 m06 : ChainP A f mf ay supf b |
| 859 | 751 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = m05 } |
| 727 | 752 |
| 757 | 753 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 966 | 754 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = |
| 757 | 755 λ {z} cz → subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (A∋fc y f mf cz ))) } |
| 756 | |
| 966 | 757 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
| 757 | 758 → IsTotalOrderSet (uchain f mf ay) |
| 966 | 759 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 757 | 760 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 761 uz01 = fcn-cmp y f mf ca cb | |
| 762 | |
| 966 | 763 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
| 928 | 764 → MinSUP A (uchain f mf ay) |
| 966 | 765 ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) |
| 757 | 766 |
|
965
1c1c6a6ed4fa
removing ch-init is no good because of initialization
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
964
diff
changeset
|
767 |
| 793 | 768 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 950 | 769 SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x≤sup = λ lt → SUP.x≤sup sup (B⊆C lt) } |
| 711 | 770 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
771 record xSUP (B : HOD) (f : Ordinal → Ordinal ) (x : Ordinal) : Set n where |
| 833 | 772 field |
| 773 ax : odef A x | |
| 960 | 774 is-sup : IsMinSUP A B f ax |
| 833 | 775 |
| 560 | 776 -- |
| 547 | 777 -- create all ZChains under o< x |
| 560 | 778 -- |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
779 |
| 966 | 780 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 781 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 707 | 782 ind f mf {y} ay x prev with Oprev-p x |
| 954 | 783 ... | yes op = zc41 where |
| 682 | 784 -- |
| 785 -- we have previous ordinal to use induction | |
| 786 -- | |
| 787 px = Oprev.oprev op | |
| 703 | 788 zc : ZChain A f mf ay (Oprev.oprev op) |
| 966 | 789 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 682 | 790 px<x : px o< x |
| 791 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc | |
|
918
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
changeset
|
792 opx=x : osuc px ≡ x |
|
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
changeset
|
793 opx=x = Oprev.oprev=x op |
|
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
changeset
|
794 |
| 709 | 795 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 966 | 796 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt |
| 697 | 797 |
| 754 | 798 supf0 = ZChain.supf zc |
| 869 | 799 pchain : HOD |
| 800 pchain = UnionCF A f mf ay supf0 px | |
| 835 | 801 |
| 966 | 802 supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b |
| 857 | 803 supf-mono = ZChain.supf-mono zc |
| 844 | 804 |
| 861 | 805 zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) |
| 966 | 806 zc04 {b} b≤x with trio< b px |
| 861 | 807 ... | tri< a ¬b ¬c = case1 (o<→≤ a) |
| 808 ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) | |
| 809 ... | tri> ¬a ¬b px<b with osuc-≡< b≤x | |
| 810 ... | case1 eq = case2 eq | |
| 966 | 811 ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) |
| 840 | 812 |
| 954 | 813 -- |
| 814 -- find the next value of supf | |
| 815 -- | |
| 816 | |
| 817 pchainpx : HOD | |
| 966 | 818 pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain A f mf ay supf0 px z ) |
| 954 | 819 ∨ FClosure A f (supf0 px) z } ; odmax = & A ; <odmax = zc00 } where |
| 820 zc00 : {z : Ordinal } → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z → z o< & A | |
| 966 | 821 zc00 {z} (case1 lt) = z07 lt |
| 954 | 822 zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf fc ) |
| 823 | |
| 824 zc02 : { a b : Ordinal } → odef A a ∧ UChain A f mf ay supf0 px a → FClosure A f (supf0 px) b → a <= b | |
| 825 zc02 {a} {b} ca fb = zc05 fb where | |
| 826 zc06 : MinSUP.sup (ZChain.minsup zc o≤-refl) ≡ supf0 px | |
| 827 zc06 = trans (sym ( ZChain.supf-is-minsup zc o≤-refl )) refl | |
| 828 zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a <= b | |
| 829 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb )) | |
| 830 ... | case1 eq = subst (λ k → a <= k ) (subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) eq)) (zc05 fb) | |
| 966 | 831 ... | case2 lt = <=-trans (zc05 fb) (case2 lt) |
| 832 zc05 (init b1 refl) with MinSUP.x≤sup (ZChain.minsup zc o≤-refl) | |
| 954 | 833 (subst (λ k → odef A k ∧ UChain A f mf ay supf0 px k) (sym &iso) ca ) |
| 834 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso zc06 eq ) | |
| 966 | 835 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso (cong (*) zc06) lt ) |
| 836 | |
| 954 | 837 ptotal : IsTotalOrderSet pchainpx |
| 966 | 838 ptotal (case1 a) (case1 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso |
| 839 (chain-total A f mf ay supf0 (proj2 a) (proj2 b)) | |
| 954 | 840 ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b |
| 841 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where | |
| 842 eq1 : a0 ≡ b0 | |
| 843 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 844 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where | |
| 845 lt1 : a0 < b0 | |
| 846 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 847 ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b | |
| 848 ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where | |
| 849 eq1 : a0 ≡ b0 | |
| 850 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 851 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where | |
| 852 lt1 : a0 < b0 | |
| 853 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 854 ptotal (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp (supf0 px) f mf a b) | |
| 966 | 855 |
| 954 | 856 pcha : pchainpx ⊆' A |
| 857 pcha (case1 lt) = proj1 lt | |
| 858 pcha (case2 fc) = A∋fc _ f mf fc | |
| 966 | 859 |
| 860 sup1 : MinSUP A pchainpx | |
| 954 | 861 sup1 = minsupP pchainpx pcha ptotal |
| 862 sp1 = MinSUP.sup sup1 | |
| 863 | |
| 972 | 864 sfpx<=sp1 : supf0 px <= sp1 |
| 865 sfpx<=sp1 = MinSUP.x≤sup sup1 (case2 (init (ZChain.asupf zc {px}) refl )) | |
| 866 | |
| 867 sfpx≤sp1 : supf0 px o≤ sp1 | |
| 868 sfpx≤sp1 = subst ( λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc o≤-refl )) | |
| 869 ( MinSUP.minsup (ZChain.minsup zc o≤-refl) (MinSUP.asm sup1) | |
| 870 (λ {x} ux → MinSUP.x≤sup sup1 (case1 ux)) ) | |
| 967 | 871 |
| 954 | 872 -- |
| 873 -- supf0 px o≤ sp1 | |
| 966 | 874 -- |
| 875 | |
| 967 | 876 --- x ≦ supf px ≦ x ≦ sp ≦ x |
| 877 -- x may apper any place | |
| 878 | |
| 879 -- x < sp → supf x = supf px | |
| 880 -- x ≡ sp → supf x = sp | |
| 881 -- sp < x → supf x = sp ≡ supf px | |
| 882 -- UnionCF A f mf ay supf px ⊆ UnionCF A f mf ay supf x | |
| 883 | |
| 884 -- supf x does not affect UnionCF A f mf ay supf x | |
| 885 | |
| 886 -- supf px < px → UnionCF A f mf ay supf px ≡ UnionCF A f mf ay supf x | |
| 887 -- supf px ≡ px → UnionCF A f mf ay supf px ⊂ UnionCF A f mf ay supf x ≡ pchainx | |
| 888 -- x < supf px → UnionCF A f mf ay supf px ≡ UnionCF A f mf ay supf x | |
| 889 | |
| 972 | 890 zc43 : (x : Ordinal ) → ( x o< sp1 ) ∨ ( sp1 o≤ x ) |
| 891 zc43 x with trio< x sp1 | |
| 971 | 892 ... | tri< a ¬b ¬c = case1 a |
| 893 ... | tri≈ ¬a b ¬c = case2 (o≤-refl0 (sym b)) | |
| 894 ... | tri> ¬a ¬b c = case2 (o<→≤ c) | |
| 967 | 895 |
| 966 | 896 zc41 : ZChain A f mf ay x |
| 972 | 897 zc41 with zc43 x |
| 968 | 898 zc41 | (case2 sp≤x ) = record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf1-mono ; supf-< = ? |
| 901 | 899 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = csupf1 } where |
| 969 | 900 -- supf0 px is included in the chain of sp1 |
| 901 -- supf0 px ≡ px ∧ supf0 px o< sp1 → ( UnionCF A f mf ay supf0 px ∪ FClosure (supf0 px) ) ≡ UnionCF supf1 x | |
| 902 -- else UnionCF A f mf ay supf0 px ≡ UnionCF supf1 x | |
| 901 | 903 -- supf1 x ≡ sp1, which is not included now |
| 883 | 904 |
| 871 | 905 supf1 : Ordinal → Ordinal |
| 966 | 906 supf1 z with trio< z px |
| 871 | 907 ... | tri< a ¬b ¬c = supf0 z |
| 966 | 908 ... | tri≈ ¬a b ¬c = supf0 z |
| 901 | 909 ... | tri> ¬a ¬b c = sp1 |
| 871 | 910 |
| 886 | 911 sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z |
| 901 | 912 sf1=sf0 {z} z≤px with trio< z px |
| 874 | 913 ... | tri< a ¬b ¬c = refl |
| 901 | 914 ... | tri≈ ¬a b ¬c = refl |
| 915 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c ) | |
| 883 | 916 |
| 901 | 917 sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1 |
| 918 sf1=sp1 {z} px<z with trio< z px | |
| 919 ... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a ) | |
| 920 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z ) | |
| 921 ... | tri> ¬a ¬b c = refl | |
| 873 | 922 |
| 968 | 923 sf=eq : { z : Ordinal } → z o< x → supf0 z ≡ supf1 z |
| 924 sf=eq {z} z<x = sym (sf1=sf0 (subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x )) | |
| 925 | |
| 903 | 926 asupf1 : {z : Ordinal } → odef A (supf1 z) |
| 927 asupf1 {z} with trio< z px | |
| 966 | 928 ... | tri< a ¬b ¬c = ZChain.asupf zc |
| 929 ... | tri≈ ¬a b ¬c = ZChain.asupf zc | |
| 903 | 930 ... | tri> ¬a ¬b c = MinSUP.asm sup1 |
| 931 | |
| 966 | 932 supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b |
| 933 supf1-mono {a} {b} a≤b with trio< b px | |
| 901 | 934 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b ) |
| 935 ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (subst (λ k → a o≤ k) b a≤b))) refl ( supf-mono a≤b ) | |
| 936 supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px | |
| 937 ... | tri< a<px ¬b ¬c = zc19 where | |
| 938 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
| 939 zc21 = ZChain.minsup zc (o<→≤ a<px) | |
| 940 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) | |
| 950 | 941 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) ) |
| 966 | 942 zc19 : supf0 a o≤ sp1 |
| 901 | 943 zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) |
| 944 ... | tri≈ ¬a b ¬c = zc18 where | |
| 945 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
| 946 zc21 = ZChain.minsup zc (o≤-refl0 b) | |
| 947 zc20 : MinSUP.sup zc21 ≡ supf0 a | |
| 966 | 948 zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b)) |
| 901 | 949 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) |
| 950 | 950 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) ) |
| 966 | 951 zc18 : supf0 a o≤ sp1 |
| 901 | 952 zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) |
| 953 ... | tri> ¬a ¬b c = o≤-refl | |
| 885 | 954 |
| 968 | 955 sf≤ : { z : Ordinal } → x o≤ z → supf0 x o≤ supf1 z |
| 956 sf≤ {z} x≤z with trio< z px | |
| 957 ... | tri< a ¬b ¬c = ⊥-elim ( o<> (osucc a) (subst (λ k → k o≤ z) (sym (Oprev.oprev=x op)) x≤z ) ) | |
| 958 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) px<x )) | |
| 959 ... | tri> ¬a ¬b c = subst₂ (λ j k → j o≤ k ) (trans (sf1=sf0 o≤-refl ) (sym (ZChain.supfmax zc px<x))) (sf1=sp1 c) | |
| 960 (supf1-mono (o<→≤ c )) | |
| 978 | 961 -- px o<z → supf x ≡ supf0 px ≡ supf1 px o≤ supf1 z |
| 903 | 962 |
| 966 | 963 fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z |
| 903 | 964 fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc |
| 966 | 965 fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z |
| 903 | 966 fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc |
| 967 | 967 |
| 903 | 968 zc11 : {z : Ordinal} → odef (UnionCF A f mf ay supf1 x) z → odef pchainpx z |
| 966 | 969 zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
| 919 | 970 zc11 {z} ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ = zc21 fc where |
| 971 u<x : u o< x | |
| 953 | 972 u<x = supf-inject0 supf1-mono su<sx |
| 973 u≤px : u o≤ px | |
| 974 u≤px = zc-b<x _ u<x | |
| 903 | 975 zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1 |
| 976 zc21 {z1} (fsuc z2 fc ) with zc21 fc | |
| 966 | 977 ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ |
| 978 ... | case1 ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ | |
| 979 ... | case2 fc = case2 (fsuc _ fc) | |
| 953 | 980 zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) | inspect supf1 u |
| 981 ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u a record {fcy<sup = zc13 ; order = zc17 | |
| 982 ; supu=u = trans (sym (sf1=sf0 (o<→≤ u<px))) (ChainP.supu=u is-sup) } (init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where | |
| 983 u<px : u o< px | |
| 984 u<px = ZChain.supf-inject zc a | |
| 985 asp0 : odef A (supf0 u) | |
| 966 | 986 asp0 = ZChain.asupf zc |
| 903 | 987 zc17 : {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 u → |
| 988 FClosure A f (supf0 s) z1 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u) | |
| 966 | 989 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) ((sf1=sf0 u≤px)) ( ChainP.order is-sup |
| 953 | 990 (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 zc18)) (sym (sf1=sf0 u≤px)) ss<spx) (fcpu fc zc18) ) where |
| 903 | 991 zc18 : s o≤ px |
| 953 | 992 zc18 = ordtrans (ZChain.supf-inject zc ss<spx) u≤px |
| 903 | 993 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf0 u) ∨ ( z << supf0 u ) |
| 953 | 994 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sf1=sf0 (o<→≤ u<px)) ( ChainP.fcy<sup is-sup fc ) |
| 995 ... | tri≈ ¬a b ¬c | _ = case2 (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) (sym (trans (sf1=sf0 u≤px) b ))) | |
| 996 ... | tri> ¬a ¬b c | _ = ⊥-elim ( ¬p<x<op ⟪ ZChain.supf-inject zc c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x ⟫ ) | |
| 967 | 997 |
| 885 | 998 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
| 999 field | |
| 907 | 1000 tsup : MinSUP A (UnionCF A f mf ay supf1 z) |
| 966 | 1001 tsup=sup : supf1 z ≡ MinSUP.sup tsup |
| 885 | 1002 |
| 1003 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x | |
| 966 | 1004 sup {z} z≤x with trio< z px |
| 1005 ... | tri< a ¬b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m | |
| 950 | 1006 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o<→≤ a) ) (ZChain.supf-is-minsup zc (o<→≤ a)) } where |
| 885 | 1007 m = ZChain.minsup zc (o<→≤ a) |
| 907 | 1008 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
| 950 | 1009 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1010 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1011 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 1012 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
| 966 | 1013 ... | tri≈ ¬a b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m |
| 950 | 1014 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o≤-refl0 b) ) (ZChain.supf-is-minsup zc (o≤-refl0 b)) } where |
| 885 | 1015 m = ZChain.minsup zc (o≤-refl0 b) |
| 907 | 1016 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
| 950 | 1017 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1018 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1019 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 1020 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
| 901 | 1021 ... | tri> ¬a ¬b px<z = record { tsup = record { sup = sp1 ; asm = MinSUP.asm sup1 |
| 950 | 1022 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = sf1=sp1 px<z } where |
| 907 | 1023 m = sup1 |
| 1024 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) | |
| 950 | 1025 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1026 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1027 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 1028 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
| 885 | 1029 |
| 986 | 1030 csupf0 : {z1 : Ordinal } → supf1 z1 o< supf1 px → z1 o≤ px → odef (UnionCF A f mf ay supf1 x) (supf1 z1) |
| 978 | 1031 csupf0 {z1} s0z<px z≤px = subst (λ k → odef (UnionCF A f mf ay supf1 x) k ) (sym (sf1=sf0 z≤px)) ( |
| 1032 UChain⊆ A f mf {x} {y} ay {supf0} {supf1} (ZChain.supf-mono zc) sf=eq sf≤ | |
| 1033 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ px<x) | |
|
985
0d8dafbecb0d
zc10 : supf c ≡ supf (& A) → {x : Ordinal } → odef A x → ¬ ( c << x ) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
978
diff
changeset
|
1034 (ZChain.csupf zc ? ))) -- (subst (λ k → k o< px) (sf1=sf0 z≤px) s0z<px)))) |
| 978 | 1035 -- px o< z1 , px o≤ supf1 z1 --> px o≤ sp1 o< x -- sp1 ≡ px--> odef (UnionCF A f mf ay supf1 x) sp1 |
| 1036 | |
|
985
0d8dafbecb0d
zc10 : supf c ≡ supf (& A) → {x : Ordinal } → odef A x → ¬ ( c << x ) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
978
diff
changeset
|
1037 csupf1 : {z1 : Ordinal } → supf1 z1 o< supf1 x → odef (UnionCF A f mf ay supf1 x) (supf1 z1) |
| 986 | 1038 csupf1 {z1} sz<sx = ⟪ asupf1 , ch-is-sup (supf1 z1) (subst (λ k → k o< supf1 x) (sym cs00) sz<sx) cp (init asupf1 cs00 ) ⟫ where |
| 1039 z<x : z1 o< x | |
| 1040 z<x = supf-inject0 supf1-mono sz<sx | |
| 1041 cs00 : supf1 (supf1 z1) ≡ supf1 z1 | |
| 1042 cs00 = ? | |
| 1043 cp : ChainP A f mf ay supf1 (supf1 z1) | |
| 1044 cp = ? | |
|
918
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
changeset
|
1045 |
| 877 | 1046 |
| 968 | 1047 zc41 | (case1 x<sp ) = record { supf = supf0 ; sup=u = ? ; asupf = ? ; supf-mono = ? ; supf-< = ? |
| 901 | 1048 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = ? } where |
| 883 | 1049 |
| 901 | 1050 -- supf0 px not is included by the chain |
| 1051 -- supf1 x ≡ supf0 px because of supfmax | |
| 883 | 1052 |
| 872 | 1053 supf1 : Ordinal → Ordinal |
| 966 | 1054 supf1 z with trio< z px |
| 871 | 1055 ... | tri< a ¬b ¬c = supf0 z |
| 968 | 1056 ... | tri≈ ¬a b ¬c = supf0 z |
| 871 | 1057 ... | tri> ¬a ¬b c = supf0 px |
| 1058 | |
| 886 | 1059 sf1=sf0 : {z : Ordinal } → z o< px → supf1 z ≡ supf0 z |
| 1060 sf1=sf0 {z} z<px with trio< z px | |
| 874 | 1061 ... | tri< a ¬b ¬c = refl |
| 1062 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a z<px ) | |
| 1063 ... | tri> ¬a ¬b c = ⊥-elim ( ¬a z<px ) | |
| 1064 | |
| 968 | 1065 sf=eq : { z : Ordinal } → z o< x → supf0 z ≡ supf1 z |
| 1066 sf=eq {z} z<x with trio< z px | |
| 1067 ... | tri< a ¬b ¬c = refl | |
| 1068 ... | tri≈ ¬a b ¬c = refl | |
| 1069 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → z o< k) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 1070 sf≤ : { z : Ordinal } → x o≤ z → supf0 x o≤ supf1 z | |
| 1071 sf≤ {z} x≤z with trio< z px | |
| 1072 ... | tri< a ¬b ¬c = ⊥-elim ( o<> (osucc a) (subst (λ k → k o≤ z) (sym (Oprev.oprev=x op)) x≤z ) ) | |
| 1073 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) px<x )) | |
| 1074 ... | tri> ¬a ¬b c = o≤-refl0 ( ZChain.supfmax zc px<x ) | |
| 1075 | |
| 969 | 1076 sf=eq0 : { z : Ordinal } → z o< x → supf1 z ≡ supf0 z |
| 1077 sf=eq0 {z} z<x with trio< z px | |
| 1078 ... | tri< a ¬b ¬c = refl | |
| 1079 ... | tri≈ ¬a b ¬c = refl | |
| 1080 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → z o< k) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 1081 sf≤0 : { z : Ordinal } → x o≤ z → supf1 x o≤ supf0 z | |
| 1082 sf≤0 {z} x≤z with trio< z px | |
| 1083 ... | tri< a ¬b ¬c = ⊥-elim ( o<> (osucc a) (subst (λ k → k o≤ z) (sym (Oprev.oprev=x op)) x≤z ) ) | |
| 1084 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) px<x )) | |
| 1085 ... | tri> ¬a ¬b c = o≤-refl0 ? -- (sym ( ZChain.supfmax zc px<x )) | |
| 1086 | |
| 874 | 1087 zc17 : {z : Ordinal } → supf0 z o≤ supf0 px |
| 1088 zc17 = ? -- px o< z, px o< supf0 px | |
| 1089 | |
| 1090 supf-mono1 : {z w : Ordinal } → z o≤ w → supf1 z o≤ supf1 w | |
| 966 | 1091 supf-mono1 {z} {w} z≤w with trio< w px |
| 886 | 1092 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (ordtrans≤-< z≤w a))) refl ( supf-mono z≤w ) |
| 874 | 1093 ... | tri≈ ¬a refl ¬c with trio< z px |
| 1094 ... | tri< a ¬b ¬c = zc17 | |
| 1095 ... | tri≈ ¬a refl ¬c = o≤-refl | |
| 1096 ... | tri> ¬a ¬b c = o≤-refl | |
| 1097 supf-mono1 {z} {w} z≤w | tri> ¬a ¬b c with trio< z px | |
| 1098 ... | tri< a ¬b ¬c = zc17 | |
| 968 | 1099 ... | tri≈ ¬a b ¬c = o≤-refl0 ? |
| 874 | 1100 ... | tri> ¬a ¬b c = o≤-refl |
| 1101 | |
| 872 | 1102 pchain1 : HOD |
| 1103 pchain1 = UnionCF A f mf ay supf1 x | |
| 871 | 1104 |
| 863 | 1105 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z |
| 966 | 1106 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 894 | 1107 zc10 {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ |
| 873 | 1108 |
| 966 | 1109 zc111 : {z : Ordinal} → z o< px → OD.def (od pchain1) z → OD.def (od pchain) z |
| 1110 zc111 {z} z<px ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 894 | 1111 zc111 {z} z<px ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ |
| 873 | 1112 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1113 zc11 : (¬ xSUP (UnionCF A f mf ay supf0 px) f x ) ∨ (HasPrev A pchain f x ) |
| 864 | 1114 → {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z ∨ ( (supf0 px ≡ px) ∧ FClosure A f px z ) |
| 966 | 1115 zc11 P {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
| 1116 zc11 P {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ with trio< u1 px | |
| 1117 ... | tri< u1<px ¬b ¬c = case1 ⟪ az , ch-is-sup u1 ? ? fc ⟫ | |
| 872 | 1118 ... | tri≈ ¬a eq ¬c = case2 ⟪ subst (λ k → supf0 k ≡ k) eq s1u=u , subst (λ k → FClosure A f k z) zc12 ? ⟫ where |
| 863 | 1119 s1u=u : supf0 u1 ≡ u1 |
| 872 | 1120 s1u=u = ? -- ChainP.supu=u u1-is-sup |
| 966 | 1121 zc12 : supf0 u1 ≡ px |
| 872 | 1122 zc12 = trans s1u=u eq |
| 863 | 1123 zc11 (case1 ¬sp=x) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = ⊥-elim (¬sp=x zcsup) where |
| 966 | 1124 eq : u1 ≡ x |
| 863 | 1125 eq with trio< u1 x |
| 1126 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) | |
| 1127 ... | tri≈ ¬a b ¬c = b | |
| 890 | 1128 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x ? ) |
| 863 | 1129 s1u=x : supf0 u1 ≡ x |
| 872 | 1130 s1u=x = trans ? eq |
| 863 | 1131 zc13 : osuc px o< osuc u1 |
| 966 | 1132 zc13 = o≤-refl0 ( trans (Oprev.oprev=x op) (sym eq ) ) |
| 950 | 1133 x≤sup : {w : Ordinal} → odef (UnionCF A f mf ay supf0 px) w → (w ≡ x) ∨ (w << x) |
| 966 | 1134 x≤sup {w} ⟪ az , ch-init {w} fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? |
| 950 | 1135 x≤sup {w} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u<x) ? )) |
| 890 | 1136 ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 ? ) where |
| 851 | 1137 zc14 : u ≡ osuc px |
| 1138 zc14 = begin | |
| 966 | 1139 u ≡⟨ sym ( ChainP.supu=u is-sup) ⟩ |
| 1140 supf0 u ≡⟨ ? ⟩ | |
| 1141 supf0 u1 ≡⟨ s1u=x ⟩ | |
| 1142 x ≡⟨ sym (Oprev.oprev=x op) ⟩ | |
| 851 | 1143 osuc px ∎ where open ≡-Reasoning |
| 872 | 1144 ... | case2 lt = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? |
| 863 | 1145 zc12 : supf0 x ≡ u1 |
| 872 | 1146 zc12 = subst (λ k → supf0 k ≡ u1) eq ? |
| 966 | 1147 zcsup : xSUP (UnionCF A f mf ay supf0 px) f x |
| 868 | 1148 zcsup = record { ax = subst (λ k → odef A k) (trans zc12 eq) (ZChain.asupf zc) |
| 954 | 1149 ; is-sup = record { x≤sup = x≤sup ; minsup = ? } } |
| 872 | 1150 zc11 (case2 hp) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = case1 ? where |
| 966 | 1151 eq : u1 ≡ x |
| 864 | 1152 eq with trio< u1 x |
| 1153 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) | |
| 1154 ... | tri≈ ¬a b ¬c = b | |
| 890 | 1155 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x ? ) |
| 858 | 1156 zc20 : {z : Ordinal} → FClosure A f (supf0 u1) z → OD.def (od pchain) z |
| 1157 zc20 {z} (init asu su=z ) = zc13 where | |
| 1158 zc14 : x ≡ z | |
| 1159 zc14 = begin | |
| 1160 x ≡⟨ sym eq ⟩ | |
| 872 | 1161 u1 ≡⟨ sym ? ⟩ |
| 858 | 1162 supf0 u1 ≡⟨ su=z ⟩ |
| 1163 z ∎ where open ≡-Reasoning | |
| 1164 zc13 : odef pchain z | |
| 1165 zc13 = subst (λ k → odef pchain k) (trans (sym (HasPrev.x=fy hp)) zc14) ( ZChain.f-next zc (HasPrev.ay hp) ) | |
| 1166 zc20 {.(f w)} (fsuc w fc) = ZChain.f-next zc (zc20 fc) | |
| 891 | 1167 |
| 857 | 1168 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
| 1169 field | |
| 891 | 1170 tsup : MinSUP A (UnionCF A f mf ay supf1 z) |
| 966 | 1171 tsup=sup : supf1 z ≡ MinSUP.sup tsup |
| 891 | 1172 |
| 857 | 1173 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x |
| 966 | 1174 sup {z} z≤x with trio< z px |
| 891 | 1175 ... | tri< a ¬b ¬c = ? -- jrecord { tsup = ZChain.minsup zc (o<→≤ a) ; tsup=sup = ZChain.supf-is-minsup zc (o<→≤ a) } |
| 1176 ... | tri≈ ¬a b ¬c = ? -- record { tsup = ZChain.minsup zc (o≤-refl0 b) ; tsup=sup = ZChain.supf-is-minsup zc (o≤-refl0 b) } | |
| 865 | 1177 ... | tri> ¬a ¬b px<z = zc35 where |
| 840 | 1178 zc30 : z ≡ x |
| 1179 zc30 with osuc-≡< z≤x | |
| 1180 ... | case1 eq = eq | |
| 1181 ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 966 | 1182 zc32 = ZChain.sup zc o≤-refl |
| 865 | 1183 zc34 : ¬ (supf0 px ≡ px) → {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup zc32) ∨ (w < SUP.sup zc32) |
| 882 | 1184 zc34 ne {w} lt with zc11 ? ⟪ proj1 lt , ? ⟫ |
| 966 | 1185 ... | case1 lt = SUP.x≤sup zc32 lt |
| 865 | 1186 ... | case2 ⟪ spx=px , fc ⟫ = ⊥-elim ( ne spx=px ) |
| 857 | 1187 zc33 : supf0 z ≡ & (SUP.sup zc32) |
| 891 | 1188 zc33 = ? -- trans (sym (supfx (o≤-refl0 (sym zc30)))) ( ZChain.supf-is-minsup zc o≤-refl ) |
| 865 | 1189 zc36 : ¬ (supf0 px ≡ px) → STMP z≤x |
| 966 | 1190 zc36 ne = ? -- record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x≤sup = zc34 ne } ; tsup=sup = zc33 } |
| 865 | 1191 zc35 : STMP z≤x |
| 1192 zc35 with trio< (supf0 px) px | |
| 1193 ... | tri< a ¬b ¬c = zc36 ¬b | |
| 1194 ... | tri> ¬a ¬b c = zc36 ¬b | |
| 891 | 1195 ... | tri≈ ¬a b ¬c = record { tsup = ? ; tsup=sup = ? } where |
| 1196 zc37 : MinSUP A (UnionCF A f mf ay supf0 z) | |
| 950 | 1197 zc37 = record { sup = ? ; asm = ? ; x≤sup = ? } |
| 803 | 1198 sup=u : {b : Ordinal} (ab : odef A b) → |
| 960 | 1199 b o≤ x → IsMinSUP A (UnionCF A f mf ay supf0 b) supf0 ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf0 b) f b ) → supf0 b ≡ b |
| 814 | 1200 sup=u {b} ab b≤x is-sup with trio< b px |
| 966 | 1201 ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ |
| 1202 ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (o≤-refl0 b) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ | |
| 882 | 1203 ... | tri> ¬a ¬b px<b = zc31 ? where |
| 815 | 1204 zc30 : x ≡ b |
| 1205 zc30 with osuc-≡< b≤x | |
| 1206 ... | case1 eq = sym (eq) | |
| 1207 ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
| 966 | 1208 zcsup : xSUP (UnionCF A f mf ay supf0 px) supf0 x |
| 859 | 1209 zcsup with zc30 |
| 966 | 1210 ... | refl = record { ax = ab ; is-sup = record { x≤sup = λ {w} lt → |
| 1211 IsMinSUP.x≤sup (proj1 is-sup) ? ; minsup = ? } } | |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1212 zc31 : ( (¬ xSUP (UnionCF A f mf ay supf0 px) supf0 x ) ∨ HasPrev A (UnionCF A f mf ay supf0 px) f x ) → supf0 b ≡ b |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1213 zc31 (case1 ¬sp=x) with zc30 |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1214 ... | refl = ⊥-elim (¬sp=x zcsup ) |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1215 zc31 (case2 hasPrev ) with zc30 |
| 966 | 1216 ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev |
| 1217 ; ay = ? ; x=fy = HasPrev.x=fy hasPrev } ) | |
| 833 | 1218 |
| 728 | 1219 ... | no lim = zc5 where |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1220 |
| 703 | 1221 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
| 1222 pzc z z<x = prev z z<x | |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1223 |
| 928 | 1224 ysp = MinSUP.sup (ysup f mf ay) |
| 755 | 1225 |
| 835 | 1226 supf0 : Ordinal → Ordinal |
| 1227 supf0 z with trio< z x | |
| 1228 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 1229 ... | tri≈ ¬a b ¬c = ysp |
| 1230 ... | tri> ¬a ¬b c = ysp | |
| 835 | 1231 |
| 838 | 1232 pchain : HOD |
| 1233 pchain = UnionCF A f mf ay supf0 x | |
| 835 | 1234 |
| 838 | 1235 ptotal0 : IsTotalOrderSet pchain |
| 966 | 1236 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 835 | 1237 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 966 | 1238 uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb)) |
| 1239 | |
| 880 | 1240 usup : MinSUP A pchain |
| 1241 usup = minsupP pchain (λ lt → proj1 lt) ptotal0 | |
| 1242 spu = MinSUP.sup usup | |
| 834 | 1243 |
| 794 | 1244 supf1 : Ordinal → Ordinal |
| 835 | 1245 supf1 z with trio< z x |
| 1246 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 1247 ... | tri≈ ¬a b ¬c = spu |
| 1248 ... | tri> ¬a ¬b c = spu | |
| 755 | 1249 |
| 838 | 1250 pchain1 : HOD |
| 1251 pchain1 = UnionCF A f mf ay supf1 x | |
| 704 | 1252 |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1253 is-max-hp : (supf : Ordinal → Ordinal) (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
|
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1254 b o< x → (ab : odef A b) → |
| 966 | 1255 HasPrev A (UnionCF A f mf ay supf x) f b → |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1256 * a < * b → odef (UnionCF A f mf ay supf x) b |
|
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1257 is-max-hp supf x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
| 966 | 1258 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 1259 ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ? -- ⟪ ab , | |
| 890 | 1260 -- subst (λ k → UChain A f mf ay supf x k ) |
| 966 | 1261 -- (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1262 |
| 966 | 1263 zc70 : HasPrev A pchain f x → ¬ xSUP pchain f x |
| 844 | 1264 zc70 pr xsup = ? |
| 1265 | |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1266 no-extension : ¬ ( xSUP (UnionCF A f mf ay supf0 x) supf0 x ) → ZChain A f mf ay x |
| 966 | 1267 no-extension ¬sp=x = ? where -- record { supf = supf1 ; sup=u = sup=u |
| 879 | 1268 -- ; sup = sup ; supf-is-sup = sis ; supf-mono = {!!} ; asupf = ? } where |
|
795
408e7e8a3797
csupf depends on order cyclicly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
794
diff
changeset
|
1269 supfu : {u : Ordinal } → ( a : u o< x ) → (z : Ordinal) → Ordinal |
|
408e7e8a3797
csupf depends on order cyclicly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
794
diff
changeset
|
1270 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z |
| 838 | 1271 pchain0=1 : pchain ≡ pchain1 |
| 1272 pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
| 1273 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z | |
| 966 | 1274 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 938 | 1275 zc10 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc12 fc where |
| 838 | 1276 zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z |
| 1277 zc12 (fsuc x fc) with zc12 fc | |
| 966 | 1278 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ |
| 1279 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ | |
| 1280 zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u ? ? (init ? ? ) ⟫ | |
| 838 | 1281 zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z |
| 966 | 1282 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 938 | 1283 zc11 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc13 fc where |
| 838 | 1284 zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z |
| 1285 zc13 (fsuc x fc) with zc13 fc | |
| 966 | 1286 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ |
| 1287 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ | |
| 838 | 1288 zc13 (init asu su=z ) with trio< u x |
| 966 | 1289 ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u ? ? (init ? ? ) ⟫ |
| 838 | 1290 ... | tri≈ ¬a b ¬c = ? |
| 938 | 1291 ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u<x c ) |
| 832 | 1292 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 797 | 1293 sup {z} z≤x with trio< z x |
| 838 | 1294 ... | tri< a ¬b ¬c = SUP⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} ) |
| 815 | 1295 ... | tri≈ ¬a b ¬c = {!!} |
| 966 | 1296 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 832 | 1297 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup {!!})) |
| 843 | 1298 sis {z} z≤x with trio< z x |
| 800 | 1299 ... | tri< a ¬b ¬c = {!!} where |
| 891 | 1300 zc8 = ZChain.supf-is-minsup (pzc z a) {!!} |
| 815 | 1301 ... | tri≈ ¬a b ¬c = {!!} |
| 966 | 1302 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 960 | 1303 sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsMinSUP A (UnionCF A f mf ay supf1 b) f ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf1 b) f b ) → supf1 b ≡ b |
| 843 | 1304 sup=u {z} ab z≤x is-sup with trio< z x |
| 950 | 1305 ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x≤sup = {!!} } |
| 1306 ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?) ?) ab {!!} record { x≤sup = {!!} } | |
| 966 | 1307 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 797 | 1308 |
| 966 | 1309 zc5 : ZChain A f mf ay x |
| 697 | 1310 zc5 with ODC.∋-p O A (* x) |
| 796 | 1311 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
| 966 | 1312 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain f x ) |
| 703 | 1313 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 966 | 1314 ... | case1 pr = no-extension {!!} |
| 960 | 1315 ... | case2 ¬fy<x with ODC.p∨¬p O (IsMinSUP A pchain f ax ) |
| 966 | 1316 ... | case1 is-sup = ? -- record { supf = supf1 ; sup=u = {!!} |
| 1317 -- ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!}; asupf = {!!} } -- where -- x is a sup of (zc ?) | |
| 796 | 1318 ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention |
| 966 | 1319 |
| 921 | 1320 --- |
| 1321 --- the maximum chain has fix point of any ≤-monotonic function | |
| 1322 --- | |
| 1323 | |
| 1324 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf ay x | |
| 1325 SZ f mf {y} ay x = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) x | |
| 1326 | |
| 966 | 1327 msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
| 1328 → (zc : ZChain A f mf ay x ) | |
| 934 | 1329 → MinSUP A (UnionCF A f mf ay (ZChain.supf zc) x) |
| 960 | 1330 msp0 f mf {x} ay zc = minsupP (UnionCF A f mf ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc) |
| 922 | 1331 |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1332 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 966 | 1333 → (sp1 : MinSUP A (ZChain.chain zc)) |
| 959 | 1334 → (ssp<as : ZChain.supf zc (MinSUP.sup sp1) o< ZChain.supf zc (& A)) |
| 1335 → f (MinSUP.sup sp1) ≡ MinSUP.sup sp1 | |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1336 fixpoint f mf zc sp1 ssp<as = z14 where |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1337 chain = ZChain.chain zc |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1338 supf = ZChain.supf zc |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1339 sp : Ordinal |
| 959 | 1340 sp = MinSUP.sup sp1 |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1341 asp : odef A sp |
| 959 | 1342 asp = MinSUP.asm sp1 |
| 966 | 1343 z10 : {a b : Ordinal } → (ca : odef chain a ) → supf b o< supf (& A) → (ab : odef A b ) |
| 964 | 1344 → HasPrev A chain f b ∨ IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) b) ab |
| 921 | 1345 → * a < * b → odef chain b |
| 960 | 1346 z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) ) |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1347 z22 : sp o< & A |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1348 z22 = z09 asp |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1349 z12 : odef chain sp |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1350 z12 with o≡? (& s) sp |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1351 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 961 | 1352 ... | no ne = ZChain1.is-max (SZ1 f mf as0 zc (& A)) {& s} {sp} ( ZChain.chain∋init zc ) ssp<as asp (case2 z19 ) z13 where |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1353 z13 : * (& s) < * sp |
| 960 | 1354 z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc ) |
| 1355 ... | case1 eq = ⊥-elim ( ne eq ) | |
| 966 | 1356 ... | case2 lt = lt |
| 964 | 1357 z19 : IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) sp) asp |
| 1358 z19 = record { x≤sup = z20 } where | |
| 959 | 1359 z20 : {y : Ordinal} → odef (UnionCF A f mf as0 (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp) |
| 966 | 1360 z20 {y} zy with MinSUP.x≤sup sp1 |
| 961 | 1361 (subst (λ k → odef chain k ) (sym &iso) (chain-mono f mf as0 supf (ZChain.supf-mono zc) (o<→≤ z22) zy )) |
| 966 | 1362 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p ) |
| 960 | 1363 ... | case2 y<p = case2 (subst (λ k → * k < _ ) &iso y<p ) |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1364 z14 : f sp ≡ sp |
| 960 | 1365 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) (subst (λ k → odef chain k) (sym &iso) z12 ) |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1366 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1367 z16 : ⊥ |
| 959 | 1368 z16 with proj1 (mf (( MinSUP.sup sp1)) ( MinSUP.asm sp1 )) |
| 966 | 1369 ... | case1 eq = ⊥-elim (¬b (sym eq) ) |
| 1370 ... | case2 lt = ⊥-elim (¬c lt ) | |
| 1371 ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) b ) | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1372 ... | tri> ¬a ¬b c = ⊥-elim z17 where |
| 959 | 1373 z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) ) |
| 960 | 1374 z15 = MinSUP.x≤sup sp1 (ZChain.f-next zc z12 ) |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1375 z17 : ⊥ |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1376 z17 with z15 |
| 960 | 1377 ... | case1 eq = ¬b (cong (*) eq) |
| 1378 ... | case2 lt = ¬a lt | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1379 |
| 952 | 1380 tri : {n : Level} (u w : Ordinal ) { R : Set n } → ( u o< w → R ) → ( u ≡ w → R ) → ( w o< u → R ) → R |
| 1381 tri {_} u w p q r with trio< u w | |
| 1382 ... | tri< a ¬b ¬c = p a | |
| 1383 ... | tri≈ ¬a b ¬c = q b | |
| 1384 ... | tri> ¬a ¬b c = r c | |
| 1385 | |
| 1386 or : {n m r : Level } {P : Set n } {Q : Set m} {R : Set r} → P ∨ Q → ( P → R ) → (Q → R ) → R | |
| 1387 or (case1 p) p→r q→r = p→r p | |
| 1388 or (case2 q) p→r q→r = q→r q | |
| 1389 | |
| 921 | 1390 |
| 1391 -- ZChain contradicts ¬ Maximal | |
| 1392 -- | |
| 1393 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) | |
| 1394 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
| 1395 -- | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1396 |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1397 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥ |
| 966 | 1398 z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} |
| 1399 (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.asm msp1 )))) | |
|
965
1c1c6a6ed4fa
removing ch-init is no good because of initialization
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
964
diff
changeset
|
1400 (subst (λ k → odef A k) (sym &iso) (MinSUP.asm msp1) ) |
| 959 | 1401 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc msp1 ss<sa ))) -- x ≡ f x ̄ |
| 1402 (proj1 (cf-is-<-monotonic nmx c (MinSUP.asm msp1 ))) where -- x < f x | |
| 937 | 1403 |
| 927 | 1404 supf = ZChain.supf zc |
| 934 | 1405 msp1 : MinSUP A (ZChain.chain zc) |
| 966 | 1406 msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc |
| 1407 c : Ordinal | |
| 1408 c = MinSUP.sup msp1 | |
|
985
0d8dafbecb0d
zc10 : supf c ≡ supf (& A) → {x : Ordinal } → odef A x → ¬ ( c << x ) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
978
diff
changeset
|
1409 c<A : c o< & A |
|
0d8dafbecb0d
zc10 : supf c ≡ supf (& A) → {x : Ordinal } → odef A x → ¬ ( c << x ) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
978
diff
changeset
|
1410 c<A = ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫ |
|
0d8dafbecb0d
zc10 : supf c ≡ supf (& A) → {x : Ordinal } → odef A x → ¬ ( c << x ) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
978
diff
changeset
|
1411 asc : odef A (supf c) |
| 928 | 1412 asc = ZChain.asupf zc |
|
985
0d8dafbecb0d
zc10 : supf c ≡ supf (& A) → {x : Ordinal } → odef A x → ¬ ( c << x ) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
978
diff
changeset
|
1413 |
| 966 | 1414 spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) |
| 928 | 1415 spd = ysup (cf nmx) (cf-is-≤-monotonic nmx) asc |
| 1416 d = MinSUP.sup spd | |
| 1417 d<A : d o< & A | |
| 1418 d<A = ∈∧P→o< ⟪ MinSUP.asm spd , lift true ⟫ | |
| 929 | 1419 msup : MinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) |
| 966 | 1420 msup = ZChain.minsup zc (o<→≤ d<A) |
| 928 | 1421 sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) ) |
| 1422 sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A) | |
| 937 | 1423 |
| 966 | 1424 sc<<d : {mc : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) |
| 934 | 1425 → supf mc << MinSUP.sup spd |
| 986 | 1426 sc<<d {mc} asc spd with MinSUP.x≤sup spd (init asc refl) |
| 1427 ... | case1 eq = ? | |
| 1428 ... | case2 lt = ? | |
| 928 | 1429 |
| 927 | 1430 ss<sa : supf c o< supf (& A) |
| 986 | 1431 ss<sa with osuc-≡< ( ZChain.supf-mono zc (o<→≤ c<A)) |
| 1432 ... | case2 sc<sa = sc<sa | |
| 1433 ... | case1 sc=sa = ⊥-elim ( nmx record { maximal = * d ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm spd) | |
| 1434 ; ¬maximal<x = λ {x} ax → subst₂ (λ j k → ¬ ( j < k)) refl *iso (zc10 sc=sa ax) } ) where | |
| 1435 zc10 : supf c ≡ supf (& A) → {x : Ordinal } → odef A x → ¬ ( d << x ) | |
| 1436 zc10 = ? -- supf x o≤ supf c → supf x ≡ supf c ∨ supf x o< supf c | |
| 1437 -- c << x → x is sup of chain or x = f ( .. ( f c )) | |
| 1438 -- c << x → x is not in chain | |
| 1439 -- supf c o≤ x (minimulity) | |
| 1440 -- odef chain z → supf z o< supf (& A) ≡ supf c → minimulity c o≤ supf c | |
| 1441 -- supf c o≤ supf (supf c) o≤ supf x o≤ supf (& A) | |
| 1442 -- supf c ≡ supf (supf c) ≡ supf x ≡ supf (& A) means supf of FClosure of (supf c) is Maximal | |
| 934 | 1443 |
| 966 | 1444 zorn00 : Maximal A |
| 1445 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 804 | 1446 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
| 551 | 1447 -- yes we have the maximal |
| 1448 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 1449 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 1450 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 966 | 1451 zorn01 = proj1 zorn03 |
| 551 | 1452 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) |
| 1453 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 927 | 1454 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where |
| 551 | 1455 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 966 | 1456 nmx : ¬ Maximal A |
| 551 | 1457 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where |
| 966 | 1458 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) |
| 804 | 1459 zc5 = ⟪ Maximal.as 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) ) ⟫ |
| 551 | 1460 |
| 516 | 1461 -- usage (see filter.agda ) |
| 1462 -- | |
| 497 | 1463 -- _⊆'_ : ( A B : HOD ) → Set n |
| 1464 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 1465 |
| 966 | 1466 -- MaximumSubset : {L P : HOD} |
| 497 | 1467 -- → o∅ o< & L → o∅ o< & P → P ⊆ L |
| 1468 -- → IsPartialOrderSet P _⊆'_ | |
| 1469 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 1470 -- → Maximal P (_⊆'_) | |
| 1471 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |