Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 991:c190f708862a
f (supf x) = supf x
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 17 Nov 2022 00:22:58 +0900 |
| parents | 9672214d4e13 |
| children | 4aaecae58da5 |
| 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 | |
|
528
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
67 _≤_ : (x y : HOD) → Set (Level.suc n) |
|
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
68 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
|
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
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 |
| 990 | 86 ftrans<-<= : {x y z : Ordinal } → x << y → y <= z → x << z |
| 87 ftrans<-<= {x} {y} {z} x<y (case1 eq) = subst (λ k → * x < k ) ((cong (*) eq)) x<y | |
| 88 ftrans<-<= {x} {y} {z} x<y (case2 lt) = IsStrictPartialOrder.trans PO x<y lt | |
| 89 | |
| 966 | 90 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y |
| 770 | 91 <=to≤ (case1 eq) = case1 (cong (*) eq) |
| 92 <=to≤ (case2 lt) = case2 lt | |
| 93 | |
| 966 | 94 ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y |
| 779 | 95 ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) |
| 96 ≤to<= (case2 lt) = case2 lt | |
| 97 | |
| 556 | 98 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 99 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 100 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 101 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 102 |
| 561 | 103 ptrans = IsStrictPartialOrder.trans PO |
| 104 | |
| 492 | 105 open _==_ |
| 106 open _⊆_ | |
| 107 | |
| 966 | 108 -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A |
| 879 | 109 -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x |
| 110 -- <-TransFinite = ? | |
| 111 | |
| 530 | 112 -- |
| 560 | 113 -- Closure of ≤-monotonic function f has total order |
| 530 | 114 -- |
| 115 | |
| 116 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 117 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 118 | |
| 551 | 119 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 783 | 120 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
| 555 | 121 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 122 |
| 556 | 123 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 | 124 A∋fc {A} s f mf (init as refl ) = as |
| 556 | 125 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 126 |
| 714 | 127 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 | 128 A∋fcs {A} s f mf (init as refl) = as |
| 966 | 129 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
| 714 | 130 |
| 556 | 131 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 | 132 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
| 556 | 133 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 134 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 966 | 135 ... | case2 x<fx with s≤fc {A} s f mf fcy |
| 556 | 136 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) |
| 137 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 138 |
| 800 | 139 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 140 fcn s mf (init as refl) = zero | |
| 141 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
| 142 ... | case1 eq = fcn s mf p | |
| 143 ... | case2 y<fy = suc (fcn s mf p ) | |
| 144 | |
| 966 | 145 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 800 | 146 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y |
| 147 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 | |
| 148 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 149 fc06 {x} {y} refl {j} not = fc08 not where | |
| 966 | 150 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
| 800 | 151 fc08 () |
| 152 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
| 153 fc07 {x} (init as refl) eq = refl | |
| 154 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 155 ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) | |
| 156 -- ... | case2 x<fx = ? | |
| 157 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 | |
| 158 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
| 159 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
| 160 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
| 161 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 162 ... | 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 ) | |
| 163 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 164 ... | 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 ) | |
| 165 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 ) ) | |
| 166 ... | 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 ) | |
| 167 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 ) ) | |
| 168 ... | 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 ) | |
| 169 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 170 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 171 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 172 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 173 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x | |
| 174 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
| 175 fc04 : * x1 ≡ * y | |
| 176 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 177 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 178 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 179 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 180 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 966 | 181 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq |
| 800 | 182 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where |
| 183 fc05 : * x ≡ * y1 | |
| 184 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 185 ... | 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))) | |
| 186 | |
| 187 | |
| 188 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 189 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 190 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
| 191 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 192 fc06 {x} {y} refl {j} not = fc08 not where | |
| 966 | 193 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
| 800 | 194 fc08 () |
| 195 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 | |
| 196 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
| 197 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 | 198 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) |
| 800 | 199 ... | case2 y<fy with <-cmp (fcn s mf cx ) i |
| 200 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 966 | 201 ... | 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 | 202 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where |
| 203 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 966 | 204 fc03 eq = cong pred eq |
| 205 fc02 : * x < * y1 | |
| 800 | 206 fc02 = fc01 i cx cy (fc03 i=y ) a |
| 207 | |
| 557 | 208 |
| 966 | 209 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 210 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 800 | 211 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 212 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 213 fc11 : * x < * y | |
| 214 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 215 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 216 fc10 : * x ≡ * y | |
| 217 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 966 | 218 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where |
| 800 | 219 fc12 : * y < * x |
| 220 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 600 | 221 |
| 563 | 222 |
| 729 | 223 |
| 560 | 224 -- open import Relation.Binary.Properties.Poset as Poset |
| 225 | |
| 226 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 227 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 228 | |
| 567 | 229 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 230 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 231 |
| 568 | 232 _⊆'_ : ( A B : HOD ) → Set n |
| 233 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 234 |
| 235 -- | |
| 236 -- inductive maxmum tree from x | |
| 237 -- tree structure | |
| 238 -- | |
| 554 | 239 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
240 record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where |
| 533 | 241 field |
| 836 | 242 ax : odef A x |
| 534 | 243 y : Ordinal |
| 541 | 244 ay : odef B y |
| 966 | 245 x=fy : x ≡ f y |
| 529 | 246 |
| 962 | 247 record IsSUP (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 248 field | |
| 249 x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) | |
| 957 | 250 |
| 960 | 251 record IsMinSUP (A B : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (xa : odef A x) : Set n where |
| 654 | 252 field |
| 950 | 253 x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 966 | 254 minsup : { sup1 : Ordinal } → odef A sup1 |
| 954 | 255 → ( {z : Ordinal } → odef B z → (z ≡ sup1 ) ∨ (z << sup1 )) → x o≤ sup1 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
256 not-hp : ¬ ( HasPrev A B f x ) |
| 568 | 257 |
| 656 | 258 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 259 field | |
| 260 sup : HOD | |
| 804 | 261 as : A ∋ sup |
| 950 | 262 x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive |
| 656 | 263 |
| 690 | 264 -- |
| 265 -- sup and its fclosure is in a chain HOD | |
| 266 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 267 -- whole chain is a union of separated Chain | |
| 966 | 268 -- minimum index is sup of y not ϕ |
| 690 | 269 -- |
| 270 | |
| 787 | 271 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 | 272 field |
| 966 | 273 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
| 828 | 274 order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
| 275 supu=u : supf u ≡ u | |
| 694 | 276 |
| 748 | 277 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 966 | 278 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where |
| 279 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 280 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : supf u o< supf x) ( is-sup : ChainP A f mf ay supf u ) | |
| 748 | 281 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z |
| 694 | 282 |
| 878 | 283 -- |
| 990 | 284 -- f (f ( ... (supf y))) f (f ( ... (supf z1))) |
| 878 | 285 -- / | / | |
| 286 -- / | / | | |
| 990 | 287 -- supf y < supf z1 < supf z2 |
| 878 | 288 -- o< o< |
| 990 | 289 -- |
| 290 -- if sup z1 ≡ sup z2, the chain is stopped at sup z1, then f (sup z1) ≡ sup z1 | |
| 291 | |
| 292 | |
| 293 fc-stop : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) { a b : Ordinal } | |
| 294 → (aa : odef A a ) →( {y : Ordinal} → FClosure A f a y → (y ≡ b ) ∨ (y << b )) → a ≡ b → f a ≡ a | |
| 295 fc-stop A f mf {a} {b} aa x≤sup a=b with x≤sup (fsuc a (init aa refl )) | |
| 296 ... | case1 eq = trans eq (sym a=b) | |
| 297 ... | case2 lt = ⊥-elim (<-irr (case1 (cong (λ k → * (f k) ) (sym a=b))) (ftrans<-<= lt (≤to<= fc00 )) ) where | |
| 298 fc00 : * b ≤ * (f b) | |
| 299 fc00 = proj1 (mf _ (subst (λ k → odef A k) a=b aa )) | |
| 300 | |
| 301 -- | |
| 861 | 302 -- data UChain is total |
| 303 | |
| 304 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) | |
| 305 {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 ) | |
| 306 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where | |
| 307 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 | 308 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb |
| 309 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) with ChainP.fcy<sup supb fca | |
| 310 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 311 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 312 ct00 : * a ≡ * b | |
| 313 ct00 = trans (cong (*) eq) eq1 | |
| 314 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where | |
| 315 ct01 : * a < * b | |
| 316 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
| 317 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 | |
| 318 ct00 : * a < * (supf ub) | |
| 319 ct00 = lt | |
| 320 ct01 : * a < * b | |
| 321 ct01 with s≤fc (supf ub) f mf fcb | |
| 322 ... | case1 eq = subst (λ k → * a < k ) eq ct00 | |
| 323 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 324 ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb | |
| 325 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 326 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 327 ct00 : * a ≡ * b | |
| 328 ct00 = sym (trans (cong (*) eq) eq1 ) | |
| 329 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where | |
| 330 ct01 : * b < * a | |
| 331 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
| 332 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 | |
| 333 ct00 : * b < * (supf ua) | |
| 334 ct00 = lt | |
| 335 ct01 : * b < * a | |
| 336 ct01 with s≤fc (supf ua) f mf fca | |
| 337 ... | case1 eq = subst (λ k → * b < k ) eq ct00 | |
| 338 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 861 | 339 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 | 340 ... | 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 |
| 341 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 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 = trans (cong (*) eq) eq1 | |
| 345 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 966 | 346 ct02 : * a < * b |
| 861 | 347 ct02 = subst (λ k → * k < * b ) (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< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 349 ct03 : * a < * (supf ub) | |
| 350 ct03 = lt | |
| 966 | 351 ct02 : * a < * b |
| 861 | 352 ct02 with s≤fc (supf ub) f mf fcb |
| 353 ... | case1 eq = subst (λ k → * a < k ) eq ct03 | |
| 354 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt | |
| 966 | 355 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 | 356 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) |
| 966 | 357 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 |
| 358 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 861 | 359 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
| 360 ct00 : * a ≡ * b | |
| 361 ct00 = sym (trans (cong (*) eq) eq1) | |
| 362 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
| 966 | 363 ct02 : * b < * a |
| 861 | 364 ct02 = subst (λ k → * k < * a ) (sym eq) lt |
| 365 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 | |
| 366 ct05 : * b < * (supf ua) | |
| 367 ct05 = lt | |
| 966 | 368 ct04 : * b < * a |
| 861 | 369 ct04 with s≤fc (supf ua) f mf fca |
| 370 ... | case1 eq = subst (λ k → * b < k ) eq ct05 | |
| 371 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt | |
| 372 | |
| 694 | 373 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 374 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 375 | |
| 803 | 376 -- Union of supf z which o< x |
| 377 -- | |
| 966 | 378 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 694 | 379 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD |
| 380 UnionCF A f mf ay supf x | |
| 894 | 381 = 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 | 382 |
| 966 | 383 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) |
| 384 → supf x o< supf y → x o< y | |
| 842 | 385 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y |
| 386 ... | tri< a ¬b ¬c = a | |
| 387 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 388 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 389 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 390 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 391 | |
| 879 | 392 record MinSUP ( A B : HOD ) : Set n where |
| 393 field | |
| 394 sup : Ordinal | |
| 395 asm : odef A sup | |
| 966 | 396 x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) |
| 397 minsup : { sup1 : Ordinal } → odef A sup1 | |
| 879 | 398 → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 |
| 399 | |
| 400 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A | |
| 401 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 402 | |
| 880 | 403 M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } |
| 404 → (supf : Ordinal → Ordinal ) | |
| 966 | 405 → MinSUP A (UnionCF A f mf ay supf x) |
| 406 → SUP A (UnionCF A f mf ay supf x) | |
| 407 M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) | |
| 950 | 408 ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x≤sup = ms00 } where |
| 880 | 409 msup = MinSUP.sup ms |
| 410 ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup) | |
| 966 | 411 ms00 {z} uz with MinSUP.x≤sup ms uz |
| 880 | 412 ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq)) |
| 413 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso refl lt ) | |
| 414 | |
| 867 | 415 |
| 966 | 416 chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 919 | 417 (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b |
|
908
d917831fb607
supf (supf x) ≡ supf x is bad
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
907
diff
changeset
|
418 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c |
| 966 | 419 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = |
| 420 ⟪ ua , ch-init fc ⟫ | |
| 919 | 421 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = |
| 966 | 422 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (supf-mono a≤b ) ) is-sup fc ⟫ |
|
908
d917831fb607
supf (supf x) ≡ supf x is bad
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
907
diff
changeset
|
423 |
| 966 | 424 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 425 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 426 field |
| 966 | 427 supf : Ordinal → Ordinal |
| 428 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z | |
| 429 → IsSUP A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) f b ) → supf b ≡ b | |
| 990 | 430 fcs<sup : {a b w : Ordinal } → a o< b → b o≤ z → FClosure A f (supf a) w → odef (UnionCF A f mf ay supf b) w |
| 868 | 431 asupf : {x : Ordinal } → odef A (supf x) |
| 880 | 432 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y |
| 433 supf-< : {x y : Ordinal } → supf x o< supf y → supf x << supf y | |
| 891 | 434 supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z |
| 880 | 435 |
| 966 | 436 minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) |
| 891 | 437 supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z ) |
|
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
|
438 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 | 439 |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
440 chain : HOD |
| 703 | 441 chain = UnionCF A f mf ay supf z |
| 861 | 442 chain⊆A : chain ⊆' A |
| 443 chain⊆A = λ lt → proj1 lt | |
| 934 | 444 |
| 966 | 445 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
| 446 sup {x} x≤z = M→S supf (minsup x≤z) | |
| 934 | 447 |
| 448 s=ms : {x : Ordinal } → (x≤z : x o≤ z ) → & (SUP.sup (sup x≤z)) ≡ MinSUP.sup (minsup x≤z) | |
| 449 s=ms {x} x≤z = &iso | |
| 878 | 450 |
| 966 | 451 chain∋init : odef chain y |
| 452 chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ | |
|
908
d917831fb607
supf (supf x) ≡ supf x is bad
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
907
diff
changeset
|
453 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 | 454 f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
| 938 | 455 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 | 456 initial : {z : Ordinal } → odef chain z → * y ≤ * z |
| 457 initial {a} ⟪ aa , ua ⟫ with ua | |
| 966 | 458 ... | ch-init fc = s≤fc y f mf fc |
| 938 | 459 ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 966 | 460 zc7 : y <= supf u |
| 861 | 461 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
| 462 f-total : IsTotalOrderSet chain | |
| 966 | 463 f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 861 | 464 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 966 | 465 uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) |
| 861 | 466 |
| 871 | 467 supf-<= : {x y : Ordinal } → supf x <= supf y → supf x o≤ supf y |
| 468 supf-<= {x} {y} (case1 sx=sy) = o≤-refl0 sx=sy | |
| 469 supf-<= {x} {y} (case2 sx<sy) with trio< (supf x) (supf y) | |
| 470 ... | tri< a ¬b ¬c = o<→≤ a | |
| 471 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 472 ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) ) | |
| 473 | |
| 966 | 474 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
| 825 | 475 supf-inject {x} {y} sx<sy with trio< x y |
| 476 ... | tri< a ¬b ¬c = a | |
| 477 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 478 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 479 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 480 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 798 | 481 |
| 966 | 482 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 |
| 483 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) | |
| 484 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ | |
| 485 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) | |
| 486 ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) | |
|
965
1c1c6a6ed4fa
removing ch-init is no good because of initialization
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
964
diff
changeset
|
487 |
| 966 | 488 -- ordering is not proved here but in ZChain1 |
| 825 | 489 |
| 966 | 490 IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp |
| 960 | 491 → ({y : Ordinal} → odef (UnionCF A f mf ay supf x) y → (y ≡ sp ) ∨ (y << sp )) |
| 492 → ( {a : Ordinal } → a << f a ) | |
| 493 → ¬ ( HasPrev A (UnionCF A f mf ay supf x) f sp ) | |
| 494 IsMinSUP→NotHasPrev {x} {sp} asp is-sup <-mono-f hp = ⊥-elim (<-irr ( <=to≤ fsp≤sp) sp<fsp ) where | |
| 495 sp<fsp : sp << f sp | |
| 966 | 496 sp<fsp = <-mono-f |
| 497 pr = HasPrev.y hp | |
| 960 | 498 im00 : f (f pr) <= sp |
| 499 im00 = is-sup ( f-next (f-next (HasPrev.ay hp))) | |
| 500 fsp≤sp : f sp <= sp | |
| 501 fsp≤sp = subst (λ k → f k <= sp ) (sym (HasPrev.x=fy hp)) im00 | |
| 502 | |
| 969 | 503 UChain⊆ : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 504 {z y : Ordinal} (ay : odef A y) { supf supf1 : Ordinal → Ordinal } | |
| 505 → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) | |
| 966 | 506 → ( { x : Ordinal } → x o< z → supf x ≡ supf1 x) |
| 507 → ( { x : Ordinal } → z o≤ x → supf z o≤ supf1 x) | |
| 508 → UnionCF A f mf ay supf z ⊆' UnionCF A f mf ay supf1 z | |
| 969 | 509 UChain⊆ A f mf {z} {y} ay {supf} {supf1} supf-mono eq<x lex ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 978 | 510 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 | 511 u<x0 : u o< z |
| 978 | 512 u<x0 = supf-inject0 supf-mono u<x |
| 966 | 513 u<x1 : supf1 u o< supf1 z |
| 514 u<x1 = subst (λ k → k o< supf1 z ) (eq<x u<x0) (ordtrans<-≤ u<x (lex o≤-refl ) ) | |
| 515 fc1 : FClosure A f (supf1 u) x | |
| 516 fc1 = subst (λ k → FClosure A f k x ) (eq<x u<x0) fc | |
| 517 uc01 : {s : Ordinal } → supf1 s o< supf1 u → s o< z | |
| 518 uc01 {s} s<u with trio< s z | |
| 519 ... | tri< a ¬b ¬c = a | |
| 520 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> uc02 s<u ) where -- (supf-mono (o<→≤ u<x0)) | |
| 521 uc02 : supf1 u o≤ supf1 s | |
| 522 uc02 = begin | |
| 523 supf1 u <⟨ u<x1 ⟩ | |
| 524 supf1 z ≡⟨ cong supf1 (sym b) ⟩ | |
| 525 supf1 s ∎ where open o≤-Reasoning O | |
| 526 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> uc03 s<u ) where | |
| 527 uc03 : supf1 u o≤ supf1 s | |
| 528 uc03 = begin | |
| 529 supf1 u ≡⟨ sym (eq<x u<x0) ⟩ | |
| 530 supf u <⟨ u<x ⟩ | |
| 531 supf z ≤⟨ lex (o<→≤ c) ⟩ | |
| 532 supf1 s ∎ where open o≤-Reasoning O | |
| 533 cp1 : ChainP A f mf ay supf1 u | |
| 534 cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x0) (ChainP.fcy<sup is-sup fc ) | |
| 535 ; order = λ {s} {z} s<u fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x0) | |
| 536 (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (eq<x (uc01 s<u) )) (sym (eq<x u<x0)) s<u) | |
| 537 (subst (λ k → FClosure A f k z ) (sym (eq<x (uc01 s<u) )) fc )) | |
| 538 ; supu=u = trans (sym (eq<x u<x0)) (ChainP.supu=u is-sup) } | |
| 539 | |
| 540 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) | |
| 783 | 541 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
| 869 | 542 supf = ZChain.supf zc |
| 728 | 543 field |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
544 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → b o< z → (ab : odef A b) |
| 966 | 545 → HasPrev A (UnionCF A f mf ay supf z) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab |
| 869 | 546 → * a < * b → odef ((UnionCF A f mf ay supf z)) b |
| 949 | 547 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) |
| 548 | |
| 568 | 549 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 550 field | |
| 551 maximal : HOD | |
| 804 | 552 as : A ∋ maximal |
| 568 | 553 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative |
| 567 | 554 |
| 966 | 555 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 556 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 557 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ | |
| 558 | |
| 559 Zorn-lemma : { A : HOD } | |
| 560 → o∅ o< & A | |
| 568 | 561 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 966 | 562 → Maximal A |
| 552 | 563 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 564 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 565 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 566 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 567 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 568 s : HOD |
| 966 | 569 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) |
| 568 | 570 as : A ∋ * ( & s ) |
| 571 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
|
572 as0 : odef A (& s ) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
573 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 574 s<A : & s o< & A |
| 568 | 575 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 576 HasMaximal : HOD |
| 966 | 577 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 537 | 578 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ |
| 966 | 579 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) |
| 532 | 580 Gtx : { x : HOD} → A ∋ x → HOD |
| 966 | 581 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 537 | 582 z08 : ¬ Maximal A → HasMaximal =h= od∅ |
| 804 | 583 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
| 537 | 584 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
| 585 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)) | |
| 586 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 | |
| 587 ¬x<m : ¬ (* x < * m) | |
| 966 | 588 ¬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 | 589 |
| 966 | 590 minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B |
| 879 | 591 minsupP B B⊆A total = m02 where |
| 592 xsup : (sup : Ordinal ) → Set n | |
| 593 xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) | |
| 594 ∀-imply-or : {A : Ordinal → Set n } {B : Set n } | |
| 595 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B | |
| 596 ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM | |
| 597 ∀-imply-or {A} {B} ∀AB | case1 t = case1 t | |
| 598 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where | |
| 599 lemma : ¬ ((x : Ordinal ) → A x) → B | |
| 600 lemma not with ODC.p∨¬p O B | |
| 601 lemma not | case1 b = b | |
| 602 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
| 603 m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B | |
| 604 m00 x = TransFinite0 ind x where | |
| 605 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) | |
| 606 → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B | |
| 607 ind x prev = ∀-imply-or m01 where | |
| 608 m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B | |
| 609 m01 z with trio< z x | |
| 610 ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 611 ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 612 ... | tri< a ¬b ¬c with prev z a | |
| 613 ... | case2 mins = case2 mins | |
| 614 ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) | |
| 950 | 615 ... | case1 mins = case2 record { sup = z ; asm = proj1 mins ; x≤sup = proj2 mins ; minsup = m04 } where |
| 879 | 616 m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 |
| 617 m04 {s} as lt with trio< z s | |
| 618 ... | tri< a ¬b ¬c = o<→≤ a | |
| 619 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 620 ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) | |
| 621 ... | case2 notz = case1 (λ _ → notz ) | |
| 622 m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) | |
| 623 m03 not = ⊥-elim ( not s1 (z09 (SUP.as S)) ⟪ SUP.as S , m05 ⟫ ) where | |
| 624 S : SUP A B | |
| 625 S = supP B B⊆A total | |
| 626 s1 = & (SUP.sup S) | |
| 627 m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) | |
| 950 | 628 m05 {w} bw with SUP.x≤sup S {* w} (subst (λ k → odef B k) (sym &iso) bw ) |
| 879 | 629 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) |
| 630 ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) | |
| 966 | 631 m02 : MinSUP A B |
| 879 | 632 m02 = dont-or (m00 (& A)) m03 |
| 633 | |
| 560 | 634 -- Uncountable ascending chain by axiom of choice |
| 530 | 635 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 636 cf nmx x with ODC.∋-p O A (* x) |
| 637 ... | no _ = o∅ | |
| 638 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 639 ... | yes nogt = -- no larger element, so it is maximal |
| 640 ⊥-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 | 641 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 642 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 643 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 644 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 645 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 646 ... | 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 ⟫ ) | |
| 647 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 648 |
| 649 --- | |
| 650 --- infintie ascention sequence of f | |
| 651 --- | |
| 530 | 652 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 653 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 654 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 655 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 656 |
| 803 | 657 -- |
| 953 | 658 -- maximality of chain |
| 659 -- | |
| 660 -- supf is fixed for z ≡ & A , we can prove order and is-max | |
| 803 | 661 -- |
| 662 | |
| 966 | 663 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 728 | 664 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
665 SZ1 f mf {y} ay zc x = TransFinite { λ x → ZChain1 A f mf ay zc x } zc1 x where |
| 900 | 666 chain-mono1 : {a b c : Ordinal} → a o≤ b |
| 788 | 667 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
| 919 | 668 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b |
| 966 | 669 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → (ab : odef A b) |
| 670 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) f b | |
| 920 | 671 → * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
| 672 is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev | |
| 966 | 673 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 938 | 674 ... | ⟪ 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 | 675 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ |
| 868 | 676 |
| 869 | 677 supf = ZChain.supf zc |
| 678 | |
| 920 | 679 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 |
| 680 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 | 681 s<b : s o< b |
| 682 s<b = ZChain.supf-inject zc ss<sb | |
| 920 | 683 s<z : s o< & A |
| 684 s<z = ordtrans s<b b<z | |
| 870 | 685 zc04 : odef (UnionCF A f mf ay supf (& A)) (supf s) |
| 986 | 686 zc04 = ZChain.csupf zc (ordtrans<-≤ ss<sb (ZChain.supf-mono zc (o<→≤ b<z))) |
| 869 | 687 zc05 : odef (UnionCF A f mf ay supf b) (supf s) |
| 688 zc05 with zc04 | |
| 966 | 689 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ |
| 938 | 690 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u zc08 is-sup fc ⟫ where |
| 870 | 691 zc07 : FClosure A f (supf u) (supf s) -- supf u ≤ supf s → supf u o≤ supf s |
| 692 zc07 = fc | |
| 869 | 693 zc06 : supf u ≡ u |
| 694 zc06 = ChainP.supu=u is-sup | |
| 966 | 695 zc08 : supf u o< supf b |
| 894 | 696 zc08 = ordtrans≤-< (ZChain.supf-<= zc (≤to<= ( s≤fc _ f mf fc ))) ss<sb |
| 869 | 697 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 |
| 698 zc04 : odef (UnionCF A f mf ay supf b) (f x) | |
| 699 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb fc ) | |
| 966 | 700 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ |
| 701 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u<x is-sup (fsuc _ fc) ⟫ | |
| 869 | 702 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) |
| 703 order {b} {s} {z1} b<z ss<sb fc = zc04 where | |
| 891 | 704 zc00 : ( z1 ≡ MinSUP.sup (ZChain.minsup zc (o<→≤ b<z) )) ∨ ( z1 << MinSUP.sup ( ZChain.minsup zc (o<→≤ b<z) ) ) |
| 950 | 705 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 | 706 -- supf (supf b) ≡ supf b |
| 869 | 707 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) |
| 708 zc04 with zc00 | |
| 892 | 709 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z)) ) eq ) |
| 710 ... | case2 lt = case2 (subst₂ (λ j k → j < * k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z) )) lt ) | |
| 868 | 711 |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
712 zc1 : (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain1 A f mf ay zc y) → ZChain1 A f mf ay zc x |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
713 zc1 x prev with Oprev-p x |
| 949 | 714 ... | yes op = record { is-max = is-max ; order = order } where |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
715 px = Oprev.oprev op |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
716 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
717 b o< x → (ab : odef A b) → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
718 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
719 * a < * b → odef (UnionCF A f mf ay supf x) b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
720 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
721 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 |
| 989 | 722 is-max {a} {b} ua b<x ab P a<b | case2 is-sup with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) |
| 723 ... | case2 sb<sx = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where | |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
724 b<A : b o< & A |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
725 b<A = z09 ab |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
726 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
727 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
728 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
729 m05 : ZChain.supf zc b ≡ b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
730 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
731 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
732 m08 {z} fcz = ZChain.fcy<sup zc (o<→≤ b<A) fcz |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
733 m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
734 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
735 m09 {s} {z} s<b fcz = order b<A s<b fcz |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
736 m06 : ChainP A f mf ay supf b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
737 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = m05 } |
| 989 | 738 ... | case1 sb=sx = ? where |
| 990 | 739 b<A : b o< & A |
| 740 b<A = z09 ab | |
| 989 | 741 m09 : IsSUP A (UnionCF A f mf ay (ZChain.supf zc) b) ab |
| 742 m09 = proj2 is-sup | |
| 743 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b | |
| 744 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = | |
| 745 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) | |
| 746 m05 : ZChain.supf zc b ≡ b | |
| 747 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ | |
| 748 m07 : MinSUP A (UnionCF A f mf ay supf (supf x)) -- supf z o< supf ( supf x ) | |
| 749 m07 = ZChain.minsup zc (o<→≤ (z09 (ZChain.asupf zc))) | |
| 990 | 750 m29 : x o≤ & A |
| 751 m29 = ? | |
| 752 m15 : supf (supf x) ≡ MinSUP.sup m07 | |
| 753 m15 = ZChain.supf-is-minsup zc (o<→≤ (z09 (ZChain.asupf zc))) | |
| 754 m17 : MinSUP A (UnionCF A f mf ay supf x) -- supf z o< supf ( supf x ) | |
| 755 m17 = ZChain.minsup zc m29 | |
| 756 m18 : supf x ≡ MinSUP.sup m17 | |
| 757 m18 = ZChain.supf-is-minsup zc m29 | |
| 758 m10 : f (supf b) ≡ supf b | |
| 759 m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where | |
| 760 m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) | |
| 761 m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where | |
| 762 m13 : odef (UnionCF A f mf ay supf x) z | |
| 763 m13 = ZChain.fcs<sup zc b<x m29 fc | |
| 764 m12 : odef (UnionCF A f mf ay supf (& A)) z | |
| 765 m12 = ZChain.fcs<sup zc ? ? fc | |
| 766 -- m14 : odef (UnionCF A f mf ay supf x) z | |
| 767 -- m14 = ZChain.fcs<sup zc ? ? | |
| 989 | 768 m08 : MinSUP A (UnionCF A f mf ay supf b) |
| 769 m08 = ZChain.minsup zc (o<→≤ (z09 ab)) -- supf z o< supf b | |
| 770 | |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
771 ... | no lim = record { is-max = is-max ; order = order } where |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
772 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
773 b o< x → (ab : odef A b) → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
774 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
775 * a < * b → odef (UnionCF A f mf ay supf x) b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
776 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
777 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 |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
778 is-max {a} {b} ua b<x ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (init-uchain A f mf ay ) |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
779 ... | 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 )) ⟫ |
| 990 | 780 ... | case2 y<b with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) |
| 781 ... | case2 sb<sx = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where | |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
782 m09 : b o< & A |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
783 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
784 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
785 m07 {z} fc = ZChain.fcy<sup zc (o<→≤ m09) fc |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
786 m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 787 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
788 m08 {s} {z1} s<b fc = order m09 s<b fc |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
789 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
790 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
791 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
792 ; x=fy = HasPrev.x=fy nhp } ) |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
793 m05 : ZChain.supf zc b ≡ b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
794 m05 = ZChain.sup=u zc ab (o<→≤ m09) ⟪ record { x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } , m04 ⟫ -- ZChain on x |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
795 m06 : ChainP A f mf ay supf b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
796 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = m05 } |
| 990 | 797 ... | case1 sb=sx = ? where |
| 798 m17 : MinSUP A (UnionCF A f mf ay supf x) -- supf z o< supf ( supf x ) | |
| 799 m17 = ZChain.minsup zc ? | |
| 800 m18 : supf x ≡ MinSUP.sup m17 | |
| 801 m18 = ZChain.supf-is-minsup zc ? | |
| 802 m10 : f (supf b) ≡ supf b | |
| 803 m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where | |
| 804 m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) | |
| 805 m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where | |
| 806 m13 : odef (UnionCF A f mf ay supf x) z | |
| 807 m13 = ZChain.fcs<sup zc b<x ? fc | |
| 727 | 808 |
| 757 | 809 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 966 | 810 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = |
| 757 | 811 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 ))) } |
| 812 | |
| 966 | 813 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
| 757 | 814 → IsTotalOrderSet (uchain f mf ay) |
| 966 | 815 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 | 816 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 817 uz01 = fcn-cmp y f mf ca cb | |
| 818 | |
| 966 | 819 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
| 928 | 820 → MinSUP A (uchain f mf ay) |
| 966 | 821 ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) |
| 757 | 822 |
|
965
1c1c6a6ed4fa
removing ch-init is no good because of initialization
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
964
diff
changeset
|
823 |
| 793 | 824 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 950 | 825 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 | 826 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
827 record xSUP (B : HOD) (f : Ordinal → Ordinal ) (x : Ordinal) : Set n where |
| 833 | 828 field |
| 829 ax : odef A x | |
| 960 | 830 is-sup : IsMinSUP A B f ax |
| 833 | 831 |
| 991 | 832 supf-fc : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → Ordinal → Ordinal |
| 833 supf-fc f mf {y} ay x = TransFinite0 ind x where | |
| 834 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → Ordinal ) → Ordinal | |
| 835 ind x prev with trio< x o∅ | |
| 836 ... | tri< a ¬b ¬c = ? | |
| 837 ... | tri≈ ¬a b ¬c = MinSUP.sup (ysup f mf ay) | |
| 838 ... | tri> ¬a ¬b 0<x with Oprev-p x | |
| 839 ... | yes op = ? | |
| 840 ... | no lim = ? | |
| 841 | |
| 560 | 842 -- |
| 547 | 843 -- create all ZChains under o< x |
| 560 | 844 -- |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
845 |
| 966 | 846 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 847 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 707 | 848 ind f mf {y} ay x prev with Oprev-p x |
| 954 | 849 ... | yes op = zc41 where |
| 682 | 850 -- |
| 851 -- we have previous ordinal to use induction | |
| 852 -- | |
| 853 px = Oprev.oprev op | |
| 703 | 854 zc : ZChain A f mf ay (Oprev.oprev op) |
| 966 | 855 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 682 | 856 px<x : px o< x |
| 857 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
|
858 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
|
859 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
|
860 |
| 709 | 861 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 966 | 862 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt |
| 697 | 863 |
| 754 | 864 supf0 = ZChain.supf zc |
| 869 | 865 pchain : HOD |
| 866 pchain = UnionCF A f mf ay supf0 px | |
| 835 | 867 |
| 966 | 868 supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b |
| 857 | 869 supf-mono = ZChain.supf-mono zc |
| 844 | 870 |
| 861 | 871 zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) |
| 966 | 872 zc04 {b} b≤x with trio< b px |
| 861 | 873 ... | tri< a ¬b ¬c = case1 (o<→≤ a) |
| 874 ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) | |
| 875 ... | tri> ¬a ¬b px<b with osuc-≡< b≤x | |
| 876 ... | case1 eq = case2 eq | |
| 966 | 877 ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) |
| 840 | 878 |
| 954 | 879 -- |
| 880 -- find the next value of supf | |
| 881 -- | |
| 882 | |
| 883 pchainpx : HOD | |
| 966 | 884 pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain A f mf ay supf0 px z ) |
| 954 | 885 ∨ FClosure A f (supf0 px) z } ; odmax = & A ; <odmax = zc00 } where |
| 886 zc00 : {z : Ordinal } → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z → z o< & A | |
| 966 | 887 zc00 {z} (case1 lt) = z07 lt |
| 954 | 888 zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf fc ) |
| 889 | |
| 890 zc02 : { a b : Ordinal } → odef A a ∧ UChain A f mf ay supf0 px a → FClosure A f (supf0 px) b → a <= b | |
| 891 zc02 {a} {b} ca fb = zc05 fb where | |
| 892 zc06 : MinSUP.sup (ZChain.minsup zc o≤-refl) ≡ supf0 px | |
| 893 zc06 = trans (sym ( ZChain.supf-is-minsup zc o≤-refl )) refl | |
| 894 zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a <= b | |
| 895 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb )) | |
| 896 ... | case1 eq = subst (λ k → a <= k ) (subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) eq)) (zc05 fb) | |
| 966 | 897 ... | case2 lt = <=-trans (zc05 fb) (case2 lt) |
| 898 zc05 (init b1 refl) with MinSUP.x≤sup (ZChain.minsup zc o≤-refl) | |
| 954 | 899 (subst (λ k → odef A k ∧ UChain A f mf ay supf0 px k) (sym &iso) ca ) |
| 900 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso zc06 eq ) | |
| 966 | 901 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso (cong (*) zc06) lt ) |
| 902 | |
| 954 | 903 ptotal : IsTotalOrderSet pchainpx |
| 966 | 904 ptotal (case1 a) (case1 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso |
| 905 (chain-total A f mf ay supf0 (proj2 a) (proj2 b)) | |
| 954 | 906 ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b |
| 907 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where | |
| 908 eq1 : a0 ≡ b0 | |
| 909 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 910 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where | |
| 911 lt1 : a0 < b0 | |
| 912 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 913 ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b | |
| 914 ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where | |
| 915 eq1 : a0 ≡ b0 | |
| 916 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 917 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where | |
| 918 lt1 : a0 < b0 | |
| 919 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 920 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 | 921 |
| 954 | 922 pcha : pchainpx ⊆' A |
| 923 pcha (case1 lt) = proj1 lt | |
| 924 pcha (case2 fc) = A∋fc _ f mf fc | |
| 966 | 925 |
| 926 sup1 : MinSUP A pchainpx | |
| 954 | 927 sup1 = minsupP pchainpx pcha ptotal |
| 928 sp1 = MinSUP.sup sup1 | |
| 929 | |
| 972 | 930 sfpx<=sp1 : supf0 px <= sp1 |
| 931 sfpx<=sp1 = MinSUP.x≤sup sup1 (case2 (init (ZChain.asupf zc {px}) refl )) | |
| 932 | |
| 933 sfpx≤sp1 : supf0 px o≤ sp1 | |
| 934 sfpx≤sp1 = subst ( λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc o≤-refl )) | |
| 935 ( MinSUP.minsup (ZChain.minsup zc o≤-refl) (MinSUP.asm sup1) | |
| 936 (λ {x} ux → MinSUP.x≤sup sup1 (case1 ux)) ) | |
| 967 | 937 |
| 954 | 938 -- |
| 939 -- supf0 px o≤ sp1 | |
| 966 | 940 -- |
| 941 | |
| 967 | 942 --- x ≦ supf px ≦ x ≦ sp ≦ x |
| 943 -- x may apper any place | |
| 944 | |
| 945 -- x < sp → supf x = supf px | |
| 946 -- x ≡ sp → supf x = sp | |
| 947 -- sp < x → supf x = sp ≡ supf px | |
| 948 -- UnionCF A f mf ay supf px ⊆ UnionCF A f mf ay supf x | |
| 949 | |
| 950 -- supf x does not affect UnionCF A f mf ay supf x | |
| 951 | |
| 952 -- supf px < px → UnionCF A f mf ay supf px ≡ UnionCF A f mf ay supf x | |
| 953 -- supf px ≡ px → UnionCF A f mf ay supf px ⊂ UnionCF A f mf ay supf x ≡ pchainx | |
| 954 -- x < supf px → UnionCF A f mf ay supf px ≡ UnionCF A f mf ay supf x | |
| 955 | |
| 972 | 956 zc43 : (x : Ordinal ) → ( x o< sp1 ) ∨ ( sp1 o≤ x ) |
| 957 zc43 x with trio< x sp1 | |
| 971 | 958 ... | tri< a ¬b ¬c = case1 a |
| 959 ... | tri≈ ¬a b ¬c = case2 (o≤-refl0 (sym b)) | |
| 960 ... | tri> ¬a ¬b c = case2 (o<→≤ c) | |
| 967 | 961 |
| 966 | 962 zc41 : ZChain A f mf ay x |
| 972 | 963 zc41 with zc43 x |
| 968 | 964 zc41 | (case2 sp≤x ) = record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf1-mono ; supf-< = ? |
| 901 | 965 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = csupf1 } where |
| 969 | 966 -- supf0 px is included in the chain of sp1 |
| 967 -- supf0 px ≡ px ∧ supf0 px o< sp1 → ( UnionCF A f mf ay supf0 px ∪ FClosure (supf0 px) ) ≡ UnionCF supf1 x | |
| 968 -- else UnionCF A f mf ay supf0 px ≡ UnionCF supf1 x | |
| 901 | 969 -- supf1 x ≡ sp1, which is not included now |
| 883 | 970 |
| 871 | 971 supf1 : Ordinal → Ordinal |
| 966 | 972 supf1 z with trio< z px |
| 871 | 973 ... | tri< a ¬b ¬c = supf0 z |
| 966 | 974 ... | tri≈ ¬a b ¬c = supf0 z |
| 901 | 975 ... | tri> ¬a ¬b c = sp1 |
| 871 | 976 |
| 886 | 977 sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z |
| 901 | 978 sf1=sf0 {z} z≤px with trio< z px |
| 874 | 979 ... | tri< a ¬b ¬c = refl |
| 901 | 980 ... | tri≈ ¬a b ¬c = refl |
| 981 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c ) | |
| 883 | 982 |
| 901 | 983 sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1 |
| 984 sf1=sp1 {z} px<z with trio< z px | |
| 985 ... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a ) | |
| 986 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z ) | |
| 987 ... | tri> ¬a ¬b c = refl | |
| 873 | 988 |
| 968 | 989 sf=eq : { z : Ordinal } → z o< x → supf0 z ≡ supf1 z |
| 990 sf=eq {z} z<x = sym (sf1=sf0 (subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x )) | |
| 991 | |
| 903 | 992 asupf1 : {z : Ordinal } → odef A (supf1 z) |
| 993 asupf1 {z} with trio< z px | |
| 966 | 994 ... | tri< a ¬b ¬c = ZChain.asupf zc |
| 995 ... | tri≈ ¬a b ¬c = ZChain.asupf zc | |
| 903 | 996 ... | tri> ¬a ¬b c = MinSUP.asm sup1 |
| 997 | |
| 966 | 998 supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b |
| 999 supf1-mono {a} {b} a≤b with trio< b px | |
| 901 | 1000 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b ) |
| 1001 ... | 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 ) | |
| 1002 supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px | |
| 1003 ... | tri< a<px ¬b ¬c = zc19 where | |
| 1004 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
| 1005 zc21 = ZChain.minsup zc (o<→≤ a<px) | |
| 1006 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) | |
| 950 | 1007 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) ) |
| 966 | 1008 zc19 : supf0 a o≤ sp1 |
| 901 | 1009 zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) |
| 1010 ... | tri≈ ¬a b ¬c = zc18 where | |
| 1011 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
| 1012 zc21 = ZChain.minsup zc (o≤-refl0 b) | |
| 1013 zc20 : MinSUP.sup zc21 ≡ supf0 a | |
| 966 | 1014 zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b)) |
| 901 | 1015 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) |
| 950 | 1016 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) ) |
| 966 | 1017 zc18 : supf0 a o≤ sp1 |
| 901 | 1018 zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) |
| 1019 ... | tri> ¬a ¬b c = o≤-refl | |
| 885 | 1020 |
| 968 | 1021 sf≤ : { z : Ordinal } → x o≤ z → supf0 x o≤ supf1 z |
| 1022 sf≤ {z} x≤z with trio< z px | |
| 1023 ... | tri< a ¬b ¬c = ⊥-elim ( o<> (osucc a) (subst (λ k → k o≤ z) (sym (Oprev.oprev=x op)) x≤z ) ) | |
| 1024 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) px<x )) | |
| 1025 ... | 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) | |
| 1026 (supf1-mono (o<→≤ c )) | |
| 978 | 1027 -- px o<z → supf x ≡ supf0 px ≡ supf1 px o≤ supf1 z |
| 903 | 1028 |
| 966 | 1029 fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z |
| 903 | 1030 fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc |
| 966 | 1031 fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z |
| 903 | 1032 fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc |
| 967 | 1033 |
| 903 | 1034 zc11 : {z : Ordinal} → odef (UnionCF A f mf ay supf1 x) z → odef pchainpx z |
| 966 | 1035 zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
| 919 | 1036 zc11 {z} ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ = zc21 fc where |
| 1037 u<x : u o< x | |
| 953 | 1038 u<x = supf-inject0 supf1-mono su<sx |
| 1039 u≤px : u o≤ px | |
| 1040 u≤px = zc-b<x _ u<x | |
| 903 | 1041 zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1 |
| 1042 zc21 {z1} (fsuc z2 fc ) with zc21 fc | |
| 966 | 1043 ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ |
| 1044 ... | 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) ⟫ | |
| 1045 ... | case2 fc = case2 (fsuc _ fc) | |
| 953 | 1046 zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) | inspect supf1 u |
| 1047 ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u a record {fcy<sup = zc13 ; order = zc17 | |
| 1048 ; supu=u = trans (sym (sf1=sf0 (o<→≤ u<px))) (ChainP.supu=u is-sup) } (init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where | |
| 1049 u<px : u o< px | |
| 1050 u<px = ZChain.supf-inject zc a | |
| 1051 asp0 : odef A (supf0 u) | |
| 966 | 1052 asp0 = ZChain.asupf zc |
| 903 | 1053 zc17 : {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 u → |
| 1054 FClosure A f (supf0 s) z1 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u) | |
| 966 | 1055 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) ((sf1=sf0 u≤px)) ( ChainP.order is-sup |
| 953 | 1056 (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 zc18)) (sym (sf1=sf0 u≤px)) ss<spx) (fcpu fc zc18) ) where |
| 903 | 1057 zc18 : s o≤ px |
| 953 | 1058 zc18 = ordtrans (ZChain.supf-inject zc ss<spx) u≤px |
| 903 | 1059 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf0 u) ∨ ( z << supf0 u ) |
| 953 | 1060 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sf1=sf0 (o<→≤ u<px)) ( ChainP.fcy<sup is-sup fc ) |
| 1061 ... | tri≈ ¬a b ¬c | _ = case2 (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) (sym (trans (sf1=sf0 u≤px) b ))) | |
| 1062 ... | 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 | 1063 |
| 885 | 1064 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
| 1065 field | |
| 907 | 1066 tsup : MinSUP A (UnionCF A f mf ay supf1 z) |
| 966 | 1067 tsup=sup : supf1 z ≡ MinSUP.sup tsup |
| 885 | 1068 |
| 1069 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x | |
| 966 | 1070 sup {z} z≤x with trio< z px |
| 1071 ... | tri< a ¬b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m | |
| 950 | 1072 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o<→≤ a) ) (ZChain.supf-is-minsup zc (o<→≤ a)) } where |
| 885 | 1073 m = ZChain.minsup zc (o<→≤ a) |
| 907 | 1074 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
| 950 | 1075 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1076 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1077 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 1078 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
| 966 | 1079 ... | tri≈ ¬a b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m |
| 950 | 1080 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o≤-refl0 b) ) (ZChain.supf-is-minsup zc (o≤-refl0 b)) } where |
| 885 | 1081 m = ZChain.minsup zc (o≤-refl0 b) |
| 907 | 1082 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
| 950 | 1083 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1084 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1085 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 1086 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
| 901 | 1087 ... | tri> ¬a ¬b px<z = record { tsup = record { sup = sp1 ; asm = MinSUP.asm sup1 |
| 950 | 1088 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = sf1=sp1 px<z } where |
| 907 | 1089 m = sup1 |
| 1090 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) | |
| 950 | 1091 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1092 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1093 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 1094 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
| 885 | 1095 |
| 986 | 1096 csupf0 : {z1 : Ordinal } → supf1 z1 o< supf1 px → z1 o≤ px → odef (UnionCF A f mf ay supf1 x) (supf1 z1) |
| 978 | 1097 csupf0 {z1} s0z<px z≤px = subst (λ k → odef (UnionCF A f mf ay supf1 x) k ) (sym (sf1=sf0 z≤px)) ( |
| 1098 UChain⊆ A f mf {x} {y} ay {supf0} {supf1} (ZChain.supf-mono zc) sf=eq sf≤ | |
| 1099 (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
|
1100 (ZChain.csupf zc ? ))) -- (subst (λ k → k o< px) (sf1=sf0 z≤px) s0z<px)))) |
| 978 | 1101 -- px o< z1 , px o≤ supf1 z1 --> px o≤ sp1 o< x -- sp1 ≡ px--> odef (UnionCF A f mf ay supf1 x) sp1 |
| 1102 | |
|
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
|
1103 csupf1 : {z1 : Ordinal } → supf1 z1 o< supf1 x → odef (UnionCF A f mf ay supf1 x) (supf1 z1) |
| 986 | 1104 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 |
| 1105 z<x : z1 o< x | |
| 1106 z<x = supf-inject0 supf1-mono sz<sx | |
| 1107 cs00 : supf1 (supf1 z1) ≡ supf1 z1 | |
| 1108 cs00 = ? | |
| 1109 cp : ChainP A f mf ay supf1 (supf1 z1) | |
| 1110 cp = ? | |
|
918
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
changeset
|
1111 |
| 877 | 1112 |
| 968 | 1113 zc41 | (case1 x<sp ) = record { supf = supf0 ; sup=u = ? ; asupf = ? ; supf-mono = ? ; supf-< = ? |
| 901 | 1114 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = ? } where |
| 883 | 1115 |
| 901 | 1116 -- supf0 px not is included by the chain |
| 1117 -- supf1 x ≡ supf0 px because of supfmax | |
| 883 | 1118 |
| 872 | 1119 supf1 : Ordinal → Ordinal |
| 966 | 1120 supf1 z with trio< z px |
| 871 | 1121 ... | tri< a ¬b ¬c = supf0 z |
| 968 | 1122 ... | tri≈ ¬a b ¬c = supf0 z |
| 871 | 1123 ... | tri> ¬a ¬b c = supf0 px |
| 1124 | |
| 886 | 1125 sf1=sf0 : {z : Ordinal } → z o< px → supf1 z ≡ supf0 z |
| 1126 sf1=sf0 {z} z<px with trio< z px | |
| 874 | 1127 ... | tri< a ¬b ¬c = refl |
| 1128 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a z<px ) | |
| 1129 ... | tri> ¬a ¬b c = ⊥-elim ( ¬a z<px ) | |
| 1130 | |
| 968 | 1131 sf=eq : { z : Ordinal } → z o< x → supf0 z ≡ supf1 z |
| 1132 sf=eq {z} z<x with trio< z px | |
| 1133 ... | tri< a ¬b ¬c = refl | |
| 1134 ... | tri≈ ¬a b ¬c = refl | |
| 1135 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → z o< k) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 1136 sf≤ : { z : Ordinal } → x o≤ z → supf0 x o≤ supf1 z | |
| 1137 sf≤ {z} x≤z with trio< z px | |
| 1138 ... | tri< a ¬b ¬c = ⊥-elim ( o<> (osucc a) (subst (λ k → k o≤ z) (sym (Oprev.oprev=x op)) x≤z ) ) | |
| 1139 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) px<x )) | |
| 1140 ... | tri> ¬a ¬b c = o≤-refl0 ( ZChain.supfmax zc px<x ) | |
| 1141 | |
| 969 | 1142 sf=eq0 : { z : Ordinal } → z o< x → supf1 z ≡ supf0 z |
| 1143 sf=eq0 {z} z<x with trio< z px | |
| 1144 ... | tri< a ¬b ¬c = refl | |
| 1145 ... | tri≈ ¬a b ¬c = refl | |
| 1146 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → z o< k) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 1147 sf≤0 : { z : Ordinal } → x o≤ z → supf1 x o≤ supf0 z | |
| 1148 sf≤0 {z} x≤z with trio< z px | |
| 1149 ... | tri< a ¬b ¬c = ⊥-elim ( o<> (osucc a) (subst (λ k → k o≤ z) (sym (Oprev.oprev=x op)) x≤z ) ) | |
| 1150 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) px<x )) | |
| 1151 ... | tri> ¬a ¬b c = o≤-refl0 ? -- (sym ( ZChain.supfmax zc px<x )) | |
| 1152 | |
| 874 | 1153 zc17 : {z : Ordinal } → supf0 z o≤ supf0 px |
| 1154 zc17 = ? -- px o< z, px o< supf0 px | |
| 1155 | |
| 1156 supf-mono1 : {z w : Ordinal } → z o≤ w → supf1 z o≤ supf1 w | |
| 966 | 1157 supf-mono1 {z} {w} z≤w with trio< w px |
| 886 | 1158 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (ordtrans≤-< z≤w a))) refl ( supf-mono z≤w ) |
| 874 | 1159 ... | tri≈ ¬a refl ¬c with trio< z px |
| 1160 ... | tri< a ¬b ¬c = zc17 | |
| 1161 ... | tri≈ ¬a refl ¬c = o≤-refl | |
| 1162 ... | tri> ¬a ¬b c = o≤-refl | |
| 1163 supf-mono1 {z} {w} z≤w | tri> ¬a ¬b c with trio< z px | |
| 1164 ... | tri< a ¬b ¬c = zc17 | |
| 968 | 1165 ... | tri≈ ¬a b ¬c = o≤-refl0 ? |
| 874 | 1166 ... | tri> ¬a ¬b c = o≤-refl |
| 1167 | |
| 872 | 1168 pchain1 : HOD |
| 1169 pchain1 = UnionCF A f mf ay supf1 x | |
| 871 | 1170 |
| 863 | 1171 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z |
| 966 | 1172 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 894 | 1173 zc10 {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ |
| 873 | 1174 |
| 966 | 1175 zc111 : {z : Ordinal} → z o< px → OD.def (od pchain1) z → OD.def (od pchain) z |
| 1176 zc111 {z} z<px ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 894 | 1177 zc111 {z} z<px ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ |
| 873 | 1178 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1179 zc11 : (¬ xSUP (UnionCF A f mf ay supf0 px) f x ) ∨ (HasPrev A pchain f x ) |
| 864 | 1180 → {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z ∨ ( (supf0 px ≡ px) ∧ FClosure A f px z ) |
| 966 | 1181 zc11 P {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
| 1182 zc11 P {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ with trio< u1 px | |
| 1183 ... | tri< u1<px ¬b ¬c = case1 ⟪ az , ch-is-sup u1 ? ? fc ⟫ | |
| 872 | 1184 ... | 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 | 1185 s1u=u : supf0 u1 ≡ u1 |
| 872 | 1186 s1u=u = ? -- ChainP.supu=u u1-is-sup |
| 966 | 1187 zc12 : supf0 u1 ≡ px |
| 872 | 1188 zc12 = trans s1u=u eq |
| 863 | 1189 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 | 1190 eq : u1 ≡ x |
| 863 | 1191 eq with trio< u1 x |
| 1192 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) | |
| 1193 ... | tri≈ ¬a b ¬c = b | |
| 890 | 1194 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x ? ) |
| 863 | 1195 s1u=x : supf0 u1 ≡ x |
| 872 | 1196 s1u=x = trans ? eq |
| 863 | 1197 zc13 : osuc px o< osuc u1 |
| 966 | 1198 zc13 = o≤-refl0 ( trans (Oprev.oprev=x op) (sym eq ) ) |
| 950 | 1199 x≤sup : {w : Ordinal} → odef (UnionCF A f mf ay supf0 px) w → (w ≡ x) ∨ (w << x) |
| 966 | 1200 x≤sup {w} ⟪ az , ch-init {w} fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? |
| 950 | 1201 x≤sup {w} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u<x) ? )) |
| 890 | 1202 ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 ? ) where |
| 851 | 1203 zc14 : u ≡ osuc px |
| 1204 zc14 = begin | |
| 966 | 1205 u ≡⟨ sym ( ChainP.supu=u is-sup) ⟩ |
| 1206 supf0 u ≡⟨ ? ⟩ | |
| 1207 supf0 u1 ≡⟨ s1u=x ⟩ | |
| 1208 x ≡⟨ sym (Oprev.oprev=x op) ⟩ | |
| 851 | 1209 osuc px ∎ where open ≡-Reasoning |
| 872 | 1210 ... | case2 lt = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? |
| 863 | 1211 zc12 : supf0 x ≡ u1 |
| 872 | 1212 zc12 = subst (λ k → supf0 k ≡ u1) eq ? |
| 966 | 1213 zcsup : xSUP (UnionCF A f mf ay supf0 px) f x |
| 868 | 1214 zcsup = record { ax = subst (λ k → odef A k) (trans zc12 eq) (ZChain.asupf zc) |
| 954 | 1215 ; is-sup = record { x≤sup = x≤sup ; minsup = ? } } |
| 872 | 1216 zc11 (case2 hp) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = case1 ? where |
| 966 | 1217 eq : u1 ≡ x |
| 864 | 1218 eq with trio< u1 x |
| 1219 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) | |
| 1220 ... | tri≈ ¬a b ¬c = b | |
| 890 | 1221 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x ? ) |
| 858 | 1222 zc20 : {z : Ordinal} → FClosure A f (supf0 u1) z → OD.def (od pchain) z |
| 1223 zc20 {z} (init asu su=z ) = zc13 where | |
| 1224 zc14 : x ≡ z | |
| 1225 zc14 = begin | |
| 1226 x ≡⟨ sym eq ⟩ | |
| 872 | 1227 u1 ≡⟨ sym ? ⟩ |
| 858 | 1228 supf0 u1 ≡⟨ su=z ⟩ |
| 1229 z ∎ where open ≡-Reasoning | |
| 1230 zc13 : odef pchain z | |
| 1231 zc13 = subst (λ k → odef pchain k) (trans (sym (HasPrev.x=fy hp)) zc14) ( ZChain.f-next zc (HasPrev.ay hp) ) | |
| 1232 zc20 {.(f w)} (fsuc w fc) = ZChain.f-next zc (zc20 fc) | |
| 891 | 1233 |
| 857 | 1234 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
| 1235 field | |
| 891 | 1236 tsup : MinSUP A (UnionCF A f mf ay supf1 z) |
| 966 | 1237 tsup=sup : supf1 z ≡ MinSUP.sup tsup |
| 891 | 1238 |
| 857 | 1239 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x |
| 966 | 1240 sup {z} z≤x with trio< z px |
| 891 | 1241 ... | tri< a ¬b ¬c = ? -- jrecord { tsup = ZChain.minsup zc (o<→≤ a) ; tsup=sup = ZChain.supf-is-minsup zc (o<→≤ a) } |
| 1242 ... | tri≈ ¬a b ¬c = ? -- record { tsup = ZChain.minsup zc (o≤-refl0 b) ; tsup=sup = ZChain.supf-is-minsup zc (o≤-refl0 b) } | |
| 865 | 1243 ... | tri> ¬a ¬b px<z = zc35 where |
| 840 | 1244 zc30 : z ≡ x |
| 1245 zc30 with osuc-≡< z≤x | |
| 1246 ... | case1 eq = eq | |
| 1247 ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 966 | 1248 zc32 = ZChain.sup zc o≤-refl |
| 865 | 1249 zc34 : ¬ (supf0 px ≡ px) → {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup zc32) ∨ (w < SUP.sup zc32) |
| 882 | 1250 zc34 ne {w} lt with zc11 ? ⟪ proj1 lt , ? ⟫ |
| 966 | 1251 ... | case1 lt = SUP.x≤sup zc32 lt |
| 865 | 1252 ... | case2 ⟪ spx=px , fc ⟫ = ⊥-elim ( ne spx=px ) |
| 857 | 1253 zc33 : supf0 z ≡ & (SUP.sup zc32) |
| 891 | 1254 zc33 = ? -- trans (sym (supfx (o≤-refl0 (sym zc30)))) ( ZChain.supf-is-minsup zc o≤-refl ) |
| 865 | 1255 zc36 : ¬ (supf0 px ≡ px) → STMP z≤x |
| 966 | 1256 zc36 ne = ? -- record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x≤sup = zc34 ne } ; tsup=sup = zc33 } |
| 865 | 1257 zc35 : STMP z≤x |
| 1258 zc35 with trio< (supf0 px) px | |
| 1259 ... | tri< a ¬b ¬c = zc36 ¬b | |
| 1260 ... | tri> ¬a ¬b c = zc36 ¬b | |
| 891 | 1261 ... | tri≈ ¬a b ¬c = record { tsup = ? ; tsup=sup = ? } where |
| 1262 zc37 : MinSUP A (UnionCF A f mf ay supf0 z) | |
| 950 | 1263 zc37 = record { sup = ? ; asm = ? ; x≤sup = ? } |
| 803 | 1264 sup=u : {b : Ordinal} (ab : odef A b) → |
| 960 | 1265 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 | 1266 sup=u {b} ab b≤x is-sup with trio< b px |
| 966 | 1267 ... | 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 ⟫ |
| 1268 ... | 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 | 1269 ... | tri> ¬a ¬b px<b = zc31 ? where |
| 815 | 1270 zc30 : x ≡ b |
| 1271 zc30 with osuc-≡< b≤x | |
| 1272 ... | case1 eq = sym (eq) | |
| 1273 ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
| 966 | 1274 zcsup : xSUP (UnionCF A f mf ay supf0 px) supf0 x |
| 859 | 1275 zcsup with zc30 |
| 966 | 1276 ... | refl = record { ax = ab ; is-sup = record { x≤sup = λ {w} lt → |
| 1277 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
|
1278 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
|
1279 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
|
1280 ... | refl = ⊥-elim (¬sp=x zcsup ) |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1281 zc31 (case2 hasPrev ) with zc30 |
| 966 | 1282 ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev |
| 1283 ; ay = ? ; x=fy = HasPrev.x=fy hasPrev } ) | |
| 833 | 1284 |
| 728 | 1285 ... | no lim = zc5 where |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1286 |
| 703 | 1287 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
| 1288 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
|
1289 |
| 928 | 1290 ysp = MinSUP.sup (ysup f mf ay) |
| 755 | 1291 |
| 835 | 1292 supf0 : Ordinal → Ordinal |
| 1293 supf0 z with trio< z x | |
| 1294 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 1295 ... | tri≈ ¬a b ¬c = ysp |
| 1296 ... | tri> ¬a ¬b c = ysp | |
| 835 | 1297 |
| 838 | 1298 pchain : HOD |
| 1299 pchain = UnionCF A f mf ay supf0 x | |
| 835 | 1300 |
| 838 | 1301 ptotal0 : IsTotalOrderSet pchain |
| 966 | 1302 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 835 | 1303 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 966 | 1304 uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb)) |
| 1305 | |
| 880 | 1306 usup : MinSUP A pchain |
| 1307 usup = minsupP pchain (λ lt → proj1 lt) ptotal0 | |
| 1308 spu = MinSUP.sup usup | |
| 834 | 1309 |
| 794 | 1310 supf1 : Ordinal → Ordinal |
| 835 | 1311 supf1 z with trio< z x |
| 1312 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 1313 ... | tri≈ ¬a b ¬c = spu |
| 1314 ... | tri> ¬a ¬b c = spu | |
| 755 | 1315 |
| 838 | 1316 pchain1 : HOD |
| 1317 pchain1 = UnionCF A f mf ay supf1 x | |
| 704 | 1318 |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1319 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
|
1320 b o< x → (ab : odef A b) → |
| 966 | 1321 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
|
1322 * 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
|
1323 is-max-hp supf x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
| 966 | 1324 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 1325 ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ? -- ⟪ ab , | |
| 890 | 1326 -- subst (λ k → UChain A f mf ay supf x k ) |
| 966 | 1327 -- (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
|
1328 |
| 966 | 1329 zc70 : HasPrev A pchain f x → ¬ xSUP pchain f x |
| 844 | 1330 zc70 pr xsup = ? |
| 1331 | |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1332 no-extension : ¬ ( xSUP (UnionCF A f mf ay supf0 x) supf0 x ) → ZChain A f mf ay x |
| 966 | 1333 no-extension ¬sp=x = ? where -- record { supf = supf1 ; sup=u = sup=u |
| 879 | 1334 -- ; 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
|
1335 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
|
1336 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z |
| 838 | 1337 pchain0=1 : pchain ≡ pchain1 |
| 1338 pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
| 1339 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z | |
| 966 | 1340 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 938 | 1341 zc10 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc12 fc where |
| 838 | 1342 zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z |
| 1343 zc12 (fsuc x fc) with zc12 fc | |
| 966 | 1344 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ |
| 1345 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ | |
| 1346 zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u ? ? (init ? ? ) ⟫ | |
| 838 | 1347 zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z |
| 966 | 1348 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 938 | 1349 zc11 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc13 fc where |
| 838 | 1350 zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z |
| 1351 zc13 (fsuc x fc) with zc13 fc | |
| 966 | 1352 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ |
| 1353 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ | |
| 838 | 1354 zc13 (init asu su=z ) with trio< u x |
| 966 | 1355 ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u ? ? (init ? ? ) ⟫ |
| 838 | 1356 ... | tri≈ ¬a b ¬c = ? |
| 938 | 1357 ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u<x c ) |
| 832 | 1358 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 797 | 1359 sup {z} z≤x with trio< z x |
| 838 | 1360 ... | tri< a ¬b ¬c = SUP⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} ) |
| 815 | 1361 ... | tri≈ ¬a b ¬c = {!!} |
| 966 | 1362 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 832 | 1363 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup {!!})) |
| 843 | 1364 sis {z} z≤x with trio< z x |
| 800 | 1365 ... | tri< a ¬b ¬c = {!!} where |
| 891 | 1366 zc8 = ZChain.supf-is-minsup (pzc z a) {!!} |
| 815 | 1367 ... | tri≈ ¬a b ¬c = {!!} |
| 966 | 1368 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 960 | 1369 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 | 1370 sup=u {z} ab z≤x is-sup with trio< z x |
| 950 | 1371 ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x≤sup = {!!} } |
| 1372 ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?) ?) ab {!!} record { x≤sup = {!!} } | |
| 966 | 1373 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 797 | 1374 |
| 966 | 1375 zc5 : ZChain A f mf ay x |
| 697 | 1376 zc5 with ODC.∋-p O A (* x) |
| 796 | 1377 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
| 966 | 1378 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain f x ) |
| 703 | 1379 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 966 | 1380 ... | case1 pr = no-extension {!!} |
| 960 | 1381 ... | case2 ¬fy<x with ODC.p∨¬p O (IsMinSUP A pchain f ax ) |
| 966 | 1382 ... | case1 is-sup = ? -- record { supf = supf1 ; sup=u = {!!} |
| 1383 -- ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!}; asupf = {!!} } -- where -- x is a sup of (zc ?) | |
| 796 | 1384 ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention |
| 966 | 1385 |
| 921 | 1386 --- |
| 1387 --- the maximum chain has fix point of any ≤-monotonic function | |
| 1388 --- | |
| 1389 | |
| 1390 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf ay x | |
| 1391 SZ f mf {y} ay x = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) x | |
| 1392 | |
| 966 | 1393 msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
| 1394 → (zc : ZChain A f mf ay x ) | |
| 934 | 1395 → MinSUP A (UnionCF A f mf ay (ZChain.supf zc) x) |
| 960 | 1396 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 | 1397 |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1398 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 966 | 1399 → (sp1 : MinSUP A (ZChain.chain zc)) |
| 959 | 1400 → f (MinSUP.sup sp1) ≡ MinSUP.sup sp1 |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
1401 fixpoint f mf zc sp1 = z14 where |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1402 chain = ZChain.chain zc |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1403 supf = ZChain.supf zc |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1404 sp : Ordinal |
| 959 | 1405 sp = MinSUP.sup sp1 |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1406 asp : odef A sp |
| 959 | 1407 asp = MinSUP.asm sp1 |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
1408 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< (& A) → (ab : odef A b ) |
| 964 | 1409 → HasPrev A chain f b ∨ IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) b) ab |
| 921 | 1410 → * a < * b → odef chain b |
| 960 | 1411 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
|
1412 z22 : sp o< & A |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1413 z22 = z09 asp |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1414 z12 : odef chain sp |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1415 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
|
1416 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
1417 ... | no ne = ZChain1.is-max (SZ1 f mf as0 zc (& A)) {& s} {sp} ( ZChain.chain∋init zc ) (z09 asp) asp (case2 z19 ) z13 where |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1418 z13 : * (& s) < * sp |
| 960 | 1419 z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc ) |
| 1420 ... | case1 eq = ⊥-elim ( ne eq ) | |
| 966 | 1421 ... | case2 lt = lt |
| 964 | 1422 z19 : IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) sp) asp |
| 1423 z19 = record { x≤sup = z20 } where | |
| 959 | 1424 z20 : {y : Ordinal} → odef (UnionCF A f mf as0 (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp) |
| 966 | 1425 z20 {y} zy with MinSUP.x≤sup sp1 |
| 961 | 1426 (subst (λ k → odef chain k ) (sym &iso) (chain-mono f mf as0 supf (ZChain.supf-mono zc) (o<→≤ z22) zy )) |
| 966 | 1427 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p ) |
| 960 | 1428 ... | 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
|
1429 z14 : f sp ≡ sp |
| 960 | 1430 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
|
1431 ... | 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
|
1432 z16 : ⊥ |
| 959 | 1433 z16 with proj1 (mf (( MinSUP.sup sp1)) ( MinSUP.asm sp1 )) |
| 966 | 1434 ... | case1 eq = ⊥-elim (¬b (sym eq) ) |
| 1435 ... | case2 lt = ⊥-elim (¬c lt ) | |
| 1436 ... | 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
|
1437 ... | tri> ¬a ¬b c = ⊥-elim z17 where |
| 959 | 1438 z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) ) |
| 960 | 1439 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
|
1440 z17 : ⊥ |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1441 z17 with z15 |
| 960 | 1442 ... | case1 eq = ¬b (cong (*) eq) |
| 1443 ... | case2 lt = ¬a lt | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1444 |
| 952 | 1445 tri : {n : Level} (u w : Ordinal ) { R : Set n } → ( u o< w → R ) → ( u ≡ w → R ) → ( w o< u → R ) → R |
| 1446 tri {_} u w p q r with trio< u w | |
| 1447 ... | tri< a ¬b ¬c = p a | |
| 1448 ... | tri≈ ¬a b ¬c = q b | |
| 1449 ... | tri> ¬a ¬b c = r c | |
| 1450 | |
| 1451 or : {n m r : Level } {P : Set n } {Q : Set m} {R : Set r} → P ∨ Q → ( P → R ) → (Q → R ) → R | |
| 1452 or (case1 p) p→r q→r = p→r p | |
| 1453 or (case2 q) p→r q→r = q→r q | |
| 1454 | |
| 921 | 1455 |
| 1456 -- ZChain contradicts ¬ Maximal | |
| 1457 -- | |
| 1458 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) | |
| 1459 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
| 1460 -- | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1461 |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1462 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥ |
| 966 | 1463 z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} |
| 1464 (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
|
1465 (subst (λ k → odef A k) (sym &iso) (MinSUP.asm msp1) ) |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
1466 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc msp1 ))) -- x ≡ f x ̄ |
| 959 | 1467 (proj1 (cf-is-<-monotonic nmx c (MinSUP.asm msp1 ))) where -- x < f x |
| 937 | 1468 |
| 927 | 1469 supf = ZChain.supf zc |
| 934 | 1470 msp1 : MinSUP A (ZChain.chain zc) |
| 966 | 1471 msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc |
| 1472 c : Ordinal | |
| 1473 c = MinSUP.sup msp1 | |
| 934 | 1474 |
| 966 | 1475 zorn00 : Maximal A |
| 1476 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 804 | 1477 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
| 551 | 1478 -- yes we have the maximal |
| 1479 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 1480 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 1481 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 966 | 1482 zorn01 = proj1 zorn03 |
| 551 | 1483 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) |
| 1484 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 927 | 1485 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where |
| 551 | 1486 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 966 | 1487 nmx : ¬ Maximal A |
| 551 | 1488 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where |
| 966 | 1489 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) |
| 804 | 1490 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 | 1491 |
| 516 | 1492 -- usage (see filter.agda ) |
| 1493 -- | |
| 497 | 1494 -- _⊆'_ : ( A B : HOD ) → Set n |
| 1495 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 1496 |
| 966 | 1497 -- MaximumSubset : {L P : HOD} |
| 497 | 1498 -- → o∅ o< & L → o∅ o< & P → P ⊆ L |
| 1499 -- → IsPartialOrderSet P _⊆'_ | |
| 1500 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 1501 -- → Maximal P (_⊆'_) | |
| 1502 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |