Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 1077:faa512b2425c
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 16 Dec 2022 10:08:05 +0900 |
| parents | 7e047b46c3b2 |
| children | 2624f8a9f6ed |
| 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 ( _<_ ; _≤_ ) |
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26 open import Data.Nat.Properties |
| 966 | 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 | |
| 1031 | 58 _≤_ : (x y : Ordinal ) → Set n -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain |
| 59 x ≤ y = (x ≡ y ) ∨ ( * x < * y ) | |
| 765 | 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 | |
| 1031 | 67 ≤-ftrans : {x y z : Ordinal } → x ≤ y → y ≤ z → x ≤ z |
| 554 | 68 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl |
| 69 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 70 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 71 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 72 | |
| 1031 | 73 ftrans≤-< : {x y z : Ordinal } → x ≤ y → y << z → x << z |
| 74 ftrans≤-< {x} {y} {z} (case1 eq) y<z = subst (λ k → k < * z) (sym (cong (*) eq)) y<z | |
| 75 ftrans≤-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z | |
| 951 | 76 |
| 1031 | 77 ftrans<-≤ : {x y z : Ordinal } → x << y → y ≤ z → x << z |
| 78 ftrans<-≤ {x} {y} {z} x<y (case1 eq) = subst (λ k → * x < k ) ((cong (*) eq)) x<y | |
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79 ftrans<-≤ {x} {y} {z} x<y (case2 lt) = IsStrictPartialOrder.trans PO x<y lt |
| 779 | 80 |
| 556 | 81 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 82 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 83 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 84 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 85 |
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86 <<-irr : {a b : Ordinal } → a ≤ b → b << a → ⊥ |
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382680c3e9be
minsup is not obvious in ZChain
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parents:
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87 <<-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (cong (*) (sym a=b)) b<a |
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382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
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88 <<-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl (IsStrictPartialOrder.trans PO b<a a<b) |
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382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
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89 |
| 561 | 90 ptrans = IsStrictPartialOrder.trans PO |
| 91 | |
| 492 | 92 open _==_ |
| 93 open _⊆_ | |
| 94 | |
| 966 | 95 -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A |
| 879 | 96 -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x |
| 97 -- <-TransFinite = ? | |
| 98 | |
| 530 | 99 -- |
| 560 | 100 -- Closure of ≤-monotonic function f has total order |
| 530 | 101 -- |
| 102 | |
| 1031 | 103 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set n |
| 104 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( x ≤ (f x) ) ∧ odef A (f x ) | |
| 530 | 105 |
| 992 | 106 <-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set n |
| 107 <-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x < * (f x) ) ∧ odef A (f x ) | |
| 108 | |
| 551 | 109 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 783 | 110 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
| 555 | 111 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 112 |
| 556 | 113 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 | 114 A∋fc {A} s f mf (init as refl ) = as |
| 556 | 115 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 116 |
| 714 | 117 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 | 118 A∋fcs {A} s f mf (init as refl) = as |
| 966 | 119 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
| 714 | 120 |
| 1031 | 121 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 | 122 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
| 556 | 123 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 1031 | 124 ... | case1 x=fx = subst₂ (λ j k → j ≤ k ) refl x=fx (s≤fc s f mf fcy) |
| 966 | 125 ... | case2 x<fx with s≤fc {A} s f mf fcy |
| 1031 | 126 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym (cong (*) s≡x )) refl x<fx ) |
| 556 | 127 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) |
| 555 | 128 |
| 800 | 129 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 130 fcn s mf (init as refl) = zero | |
| 131 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
| 132 ... | case1 eq = fcn s mf p | |
| 133 ... | case2 y<fy = suc (fcn s mf p ) | |
| 134 | |
| 966 | 135 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 800 | 136 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y |
| 137 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 | |
| 138 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 139 fc06 {x} {y} refl {j} not = fc08 not where | |
| 966 | 140 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
| 800 | 141 fc08 () |
| 142 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
| 143 fc07 {x} (init as refl) eq = refl | |
| 144 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 1031 | 145 ... | case1 x=fx = subst (λ k → * s ≡ k ) (cong (*) x=fx) ( fc07 cx eq ) |
| 800 | 146 -- ... | case2 x<fx = ? |
| 147 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 | |
| 148 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
| 149 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
| 150 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
| 151 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 1031 | 152 ... | 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 ) |
| 800 | 153 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) |
| 1031 | 154 ... | 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 ) |
| 800 | 155 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 ) ) |
| 1031 | 156 ... | 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 ) |
| 800 | 157 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 ) ) |
| 1031 | 158 ... | 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 ) |
| 159 ... | case1 x=fx | case2 y<fy = subst (λ k → * k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 800 | 160 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) |
| 161 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 162 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 1031 | 163 ... | case1 eq = trans (sym (cong (*) eq )) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x |
| 800 | 164 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where |
| 165 fc04 : * x1 ≡ * y | |
| 166 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 1031 | 167 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ * k ) y=fy (fc03 y cy j=y) where |
| 800 | 168 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 |
| 169 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 170 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
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parents:
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171 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) (cong (*) eq) |
| 800 | 172 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where |
| 173 fc05 : * x ≡ * y1 | |
| 174 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 175 ... | 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))) | |
| 176 | |
| 177 | |
| 178 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 179 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 180 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
| 181 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 182 fc06 {x} {y} refl {j} not = fc08 not where | |
| 966 | 183 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
| 800 | 184 fc08 () |
| 185 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 | |
| 186 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
| 187 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 1031 | 188 ... | case1 y=fy = subst (λ k → * x < k ) (cong (*) y=fy) ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) |
| 800 | 189 ... | case2 y<fy with <-cmp (fcn s mf cx ) i |
| 190 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 966 | 191 ... | 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 | 192 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where |
| 193 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 966 | 194 fc03 eq = cong pred eq |
| 195 fc02 : * x < * y1 | |
| 800 | 196 fc02 = fc01 i cx cy (fc03 i=y ) a |
| 197 | |
| 557 | 198 |
| 966 | 199 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 200 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 800 | 201 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 202 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 203 fc11 : * x < * y | |
| 204 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 205 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 206 fc10 : * x ≡ * y | |
| 207 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 966 | 208 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where |
| 800 | 209 fc12 : * y < * x |
| 210 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 600 | 211 |
| 563 | 212 |
| 729 | 213 |
| 560 | 214 -- open import Relation.Binary.Properties.Poset as Poset |
| 215 | |
| 216 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 217 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 218 | |
| 567 | 219 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 220 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 221 |
| 568 | 222 _⊆'_ : ( A B : HOD ) → Set n |
| 223 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 224 |
| 225 -- | |
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226 -- inductive masum tree from x |
| 560 | 227 -- tree structure |
| 228 -- | |
| 554 | 229 |
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230 record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where |
| 533 | 231 field |
| 836 | 232 ax : odef A x |
| 534 | 233 y : Ordinal |
| 541 | 234 ay : odef B y |
| 966 | 235 x=fy : x ≡ f y |
| 529 | 236 |
| 1031 | 237 record IsSUP (A B : HOD) (x : Ordinal ) : Set n where |
| 962 | 238 field |
| 1031 | 239 ax : odef A x |
| 240 x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) -- B is Total, use positive | |
| 568 | 241 |
| 656 | 242 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 243 field | |
| 244 sup : HOD | |
| 1031 | 245 isSUP : IsSUP A B (& sup) |
| 246 ax = IsSUP.ax isSUP | |
| 247 x≤sup = IsSUP.x≤sup isSUP | |
| 656 | 248 |
| 690 | 249 -- |
| 878 | 250 -- |
| 990 | 251 -- f (f ( ... (supf y))) f (f ( ... (supf z1))) |
| 878 | 252 -- / | / | |
| 253 -- / | / | | |
| 990 | 254 -- supf y < supf z1 < supf z2 |
| 878 | 255 -- o< o< |
| 990 | 256 -- |
| 257 -- if sup z1 ≡ sup z2, the chain is stopped at sup z1, then f (sup z1) ≡ sup z1 | |
| 1039 | 258 -- this means sup z1 is the Maximal, so f is <-monotonic if we have no Maximal. |
| 259 -- | |
| 990 | 260 |
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pxhainx conditon is requied?
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parents:
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261 fc-stop : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) { a b : Ordinal } |
| 990 | 262 → (aa : odef A a ) →( {y : Ordinal} → FClosure A f a y → (y ≡ b ) ∨ (y << b )) → a ≡ b → f a ≡ a |
| 263 fc-stop A f mf {a} {b} aa x≤sup a=b with x≤sup (fsuc a (init aa refl )) | |
| 264 ... | case1 eq = trans eq (sym a=b) | |
| 1031 | 265 ... | case2 lt = ⊥-elim (<-irr (case1 (cong (λ k → * (f k) ) (sym a=b))) (ftrans<-≤ lt fc00 ) ) where |
| 266 fc00 : b ≤ (f b) | |
| 990 | 267 fc00 = proj1 (mf _ (subst (λ k → odef A k) a=b aa )) |
| 268 | |
| 694 | 269 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 270 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 271 | |
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272 -- Union of supf z and FClosure A f y |
| 662 | 273 |
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274 data UChain { A : HOD } { f : Ordinal → Ordinal } {supf : Ordinal → Ordinal} {y : Ordinal } (ay : odef A y ) |
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275 (x : Ordinal) : (z : Ordinal) → Set n where |
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276 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain ay x z |
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277 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) (supu=u : supf u ≡ u) ( fc : FClosure A f (supf u) z ) → UChain ay x z |
| 1030 | 278 |
| 1060 | 279 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD |
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280 UnionCF A f ay supf x |
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281 = record { od = record { def = λ z → odef A z ∧ UChain {A} {f} {supf} ay x z } ; |
| 1028 | 282 odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } |
| 283 | |
| 1061 | 284 -- Union of chain lower than x |
| 1060 | 285 |
| 1063 | 286 data IChain {A : HOD} { f : Ordinal → Ordinal } {y : Ordinal } (ay : odef A y ) |
| 287 {x : Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) : (z : Ordinal ) → Set n where | |
| 288 ic-init : {z : Ordinal } (fc : FClosure A f y z) → IChain ay supfz z | |
| 289 ic-isup : {z : Ordinal} (i : Ordinal) (i<x : i o< x) (s<x : supfz i<x o≤ i ) (fc : FClosure A f (supfz i<x) z) → IChain ay supfz z | |
| 1060 | 290 |
| 1063 | 291 UnionIC : ( A : HOD ) ( f : Ordinal → Ordinal ) { x : Ordinal } {y : Ordinal } (ay : odef A y ) (supfz : {z : Ordinal } → z o< x → Ordinal) → HOD |
| 292 UnionIC A f ay supfz | |
| 293 = record { od = record { def = λ z → odef A z ∧ IChain {A} {f} ay supfz z } ; | |
| 1060 | 294 odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } |
| 1028 | 295 |
| 966 | 296 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) |
| 297 → supf x o< supf y → x o< y | |
| 842 | 298 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y |
| 299 ... | tri< a ¬b ¬c = a | |
| 300 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 301 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 302 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 303 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 304 | |
| 1031 | 305 record IsMinSUP ( A B : HOD ) (sup : Ordinal) : Set n where |
| 306 field | |
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307 as : odef A sup |
| 1031 | 308 x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) |
| 309 minsup : { sup1 : Ordinal } → odef A sup1 | |
| 310 → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 | |
| 311 | |
| 879 | 312 record MinSUP ( A B : HOD ) : Set n where |
| 313 field | |
| 314 sup : Ordinal | |
| 1031 | 315 isMinSUP : IsMinSUP A B sup |
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316 as = IsMinSUP.as isMinSUP |
| 1031 | 317 x≤sup = IsMinSUP.x≤sup isMinSUP |
| 318 minsup = IsMinSUP.minsup isMinSUP | |
| 879 | 319 |
| 320 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A | |
| 321 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 322 | |
| 880 | 323 M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } |
| 324 → (supf : Ordinal → Ordinal ) | |
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325 → MinSUP A (UnionCF A f ay supf x) |
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326 → SUP A (UnionCF A f ay supf x) |
| 966 | 327 M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) |
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328 ; isSUP = record { ax = subst (λ k → odef A k) (sym &iso) (MinSUP.as ms) ; x≤sup = ms00 } } where |
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329 ms00 : {z : Ordinal} → odef (UnionCF A f ay supf x) z → (z ≡ & (* (MinSUP.sup ms))) ∨ (z << & (* (MinSUP.sup ms))) |
| 966 | 330 ms00 {z} uz with MinSUP.x≤sup ms uz |
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331 ... | case1 eq = case1 (subst (λ k → z ≡ k) (sym &iso) eq) |
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332 ... | case2 lt = case2 (subst (λ k → * z < k ) (sym *iso) lt ) |
| 880 | 333 |
| 867 | 334 |
| 966 | 335 chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 919 | 336 (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b |
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337 → odef (UnionCF A f ay supf a) c → odef (UnionCF A f ay supf b) c |
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338 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = ⟪ ua , ch-init fc ⟫ |
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339 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-is-sup u u<x supu=u fc ⟫ = ⟪ ua , ch-is-sup u (ordtrans<-≤ u<x a≤b) supu=u fc ⟫ |
|
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340 |
| 1038 | 341 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f) |
| 783 | 342 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 343 field |
| 966 | 344 supf : Ordinal → Ordinal |
| 1028 | 345 |
| 1046 | 346 cfcs : {a b w : Ordinal } → a o< b → b o≤ z → supf a o< b → FClosure A f (supf a) w → odef (UnionCF A f ay supf b) w |
| 347 order : {a b w : Ordinal } → b o≤ z → supf a o< supf b → FClosure A f (supf a) w → w ≤ supf b | |
| 994 | 348 |
| 868 | 349 asupf : {x : Ordinal } → odef A (supf x) |
| 880 | 350 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y |
| 1059 | 351 zo≤sz : {x : Ordinal } → x o≤ z → x o≤ supf x |
| 1068 | 352 0<supfz : {x : Ordinal } → o∅ o< supf x |
| 880 | 353 |
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354 is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → IsMinSUP A (UnionCF A f ay supf x) (supf x) |
| 994 | 355 |
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356 chain : HOD |
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357 chain = UnionCF A f ay supf z |
| 861 | 358 chain⊆A : chain ⊆' A |
| 359 chain⊆A = λ lt → proj1 lt | |
| 934 | 360 |
| 1058 | 361 chain∋init : {x : Ordinal } → odef (UnionCF A f ay supf x) y |
| 362 chain∋init {x} = ⟪ ay , ch-init (init ay refl) ⟫ | |
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363 |
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364 mf : ≤-monotonic-f A f |
| 1040 | 365 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where |
| 366 mf00 : * x < * (f x) | |
| 367 mf00 = proj1 ( mf< x ax ) | |
| 1038 | 368 |
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369 f-next : {a z : Ordinal} → odef (UnionCF A f ay supf z) a → odef (UnionCF A f ay supf z) (f a) |
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370 f-next {a} ⟪ ua , ch-init fc ⟫ = ⟪ proj2 ( mf _ ua) , ch-init (fsuc _ fc) ⟫ |
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371 f-next {a} ⟪ ua , ch-is-sup u su<x su=u fc ⟫ = ⟪ proj2 ( mf _ ua) , ch-is-sup u su<x su=u (fsuc _ fc) ⟫ |
| 861 | 372 |
| 966 | 373 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
| 825 | 374 supf-inject {x} {y} sx<sy with trio< x y |
| 375 ... | tri< a ¬b ¬c = a | |
| 376 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 377 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 378 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 379 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 798 | 380 |
| 1005 | 381 supf<A : {x : Ordinal } → supf x o< & A |
| 382 supf<A = z09 asupf | |
| 383 | |
| 1038 | 384 csupf : {b : Ordinal } → supf b o< supf z → supf b o< z → odef (UnionCF A f ay supf z) (supf b) -- supf z is not an element of this chain |
| 385 csupf {b} sb<sz sb<z = cfcs (supf-inject sb<sz) o≤-refl sb<z (init asupf refl) | |
| 994 | 386 |
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387 minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f ay supf x) |
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388 minsup {x} x≤z = record { sup = supf x ; isMinSUP = is-minsup x≤z } |
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389 |
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390 supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup (minsup x≤z) |
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391 supf-is-minsup _ = refl |
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392 |
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393 -- different from order because y o< supf |
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394 fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) |
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395 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) |
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396 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ |
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397 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) |
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398 ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) |
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399 |
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400 initial : {x : Ordinal } → x o≤ z → odef (UnionCF A f ay supf x) x → y ≤ x |
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401 initial {x} x≤z ⟪ aa , ch-init fc ⟫ = s≤fc y f mf fc |
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402 initial {x} x≤z ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ≤-ftrans (fcy<sup (ordtrans u<x x≤z) (init ay refl)) (s≤fc _ f mf fc) |
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403 |
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404 f-total : IsTotalOrderSet chain |
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405 f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = |
| 1035 | 406 subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso fc-total where |
| 407 fc-total : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 408 fc-total with trio< ua ub | |
| 1046 | 409 ... | tri< a₁ ¬b ¬c with ≤-ftrans (order (o<→≤ sub<x) (subst₂ (λ j k → j o< k) (sym sua=ua) (sym sub=ub) a₁) fca ) (s≤fc (supf ub) f mf fcb ) |
| 1035 | 410 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
| 411 ct00 : * (& a) ≡ * (& b) | |
|
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412 ct00 = cong (*) eq1 |
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413 ... | case2 a<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt) |
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|
414 fc-total | tri≈ _ refl _ = fcn-cmp _ f mf fca fcb |
| 1046 | 415 fc-total | tri> ¬a ¬b c with ≤-ftrans (order (o<→≤ sua<x) (subst₂ (λ j k → j o< k) (sym sub=ub) (sym sua=ua) c) fcb ) (s≤fc (supf ua) f mf fca ) |
| 1035 | 416 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
| 417 ct00 : * (& a) ≡ * (& b) | |
|
1042
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parents:
1041
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418 ct00 = cong (*) (sym eq1) |
| 1035 | 419 ... | case2 b<a = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a |
|
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ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
420 f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = ft00 where |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
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421 ft01 : (& a) ≤ (& b) → Tri ( a < b) ( a ≡ b) ( b < a ) |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
422 ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b (λ lt → ⊥-elim (<-irr (case1 a=b) lt)) where |
|
1042
912ca4abe806
pxhainx conditon is requied?
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parents:
1041
diff
changeset
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423 a=b : a ≡ b |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
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424 a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq) |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
425 ft01 (case2 lt) = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt) where |
|
1042
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pxhainx conditon is requied?
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parents:
1041
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changeset
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426 a<b : a < b |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
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427 a<b = subst₂ (λ j k → j < k ) *iso *iso lt |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
428 ft00 : Tri ( a < b) ( a ≡ b) ( b < a ) |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
429 ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sub<x) fca) (s≤fc {A} _ f mf fcb)) |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
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430 f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-init fcb ⟫ = ft00 where |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
431 ft01 : (& b) ≤ (& a) → Tri ( a < b) ( a ≡ b) ( b < a ) |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
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432 ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b (λ lt → ⊥-elim (<-irr (case1 a=b) lt)) where |
|
1042
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pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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433 a=b : a ≡ b |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
434 a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (sym eq)) |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
435 ft01 (case2 lt) = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a where |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
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436 b<a : b < a |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
437 b<a = subst₂ (λ j k → j < k ) *iso *iso lt |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
438 ft00 : Tri ( a < b) ( a ≡ b) ( b < a ) |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
439 ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sua<x) fcb) (s≤fc {A} _ f mf fca)) |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
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440 f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-init fcb ⟫ = |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
441 subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp y f mf fca fcb ) |
| 825 | 442 |
| 1048 | 443 supf-mono< : {a b : Ordinal } → b o≤ z → supf a o< supf b → supf a << supf b |
| 444 supf-mono< {a} {b} b≤z sa<sb with order {a} {b} b≤z sa<sb (init asupf refl) | |
| 445 ... | case2 lt = lt | |
| 446 ... | case1 eq = ⊥-elim ( o<¬≡ eq sa<sb ) | |
| 447 | |
| 1053 | 448 supfeq : {a b : Ordinal } → a o≤ z → b o≤ z → UnionCF A f ay supf a ≡ UnionCF A f ay supf b → supf a ≡ supf b |
| 449 supfeq {a} {b} a≤z b≤z ua=ub with trio< (supf a) (supf b) | |
| 450 ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> ( | |
| 451 IsMinSUP.minsup (is-minsup b≤z) asupf (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (subst (λ k → odef k z) (sym ua=ub) uzb)) ) sa<sb ) | |
| 452 ... | tri≈ ¬a b ¬c = b | |
| 453 ... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> ( | |
| 454 IsMinSUP.minsup (is-minsup a≤z) asupf (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (subst (λ k → odef k z) ua=ub uza)) ) sb<sa ) | |
| 1052 | 455 |
| 1053 | 456 union-max : {a b : Ordinal } → z o≤ supf a → b o≤ z → supf a o< supf b → UnionCF A f ay supf a ≡ UnionCF A f ay supf b |
| 457 union-max {a} {b} z≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where | |
| 458 z47 : {w : Ordinal } → odef (UnionCF A f ay supf a ) w → odef ( UnionCF A f ay supf b ) w | |
| 459 z47 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫ | |
| 460 z47 {w} ⟪ aw , ch-is-sup u u<a su=u fc ⟫ = ⟪ aw , ch-is-sup u u<b su=u fc ⟫ where | |
| 461 u<b : u o< b | |
| 462 u<b = ordtrans u<a (supf-inject sa<sb ) | |
| 463 z48 : {w : Ordinal } → odef (UnionCF A f ay supf b ) w → odef ( UnionCF A f ay supf a ) w | |
| 464 z48 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫ | |
| 465 z48 {w} ⟪ aw , ch-is-sup u u<b su=u fc ⟫ = ⟪ aw , ch-is-sup u u<a su=u fc ⟫ where | |
| 466 u<a : u o< a | |
| 467 u<a = supf-inject ( osucprev (begin | |
| 468 osuc (supf u) ≡⟨ cong osuc su=u ⟩ | |
| 469 osuc u ≤⟨ osucc (ordtrans<-≤ u<b b≤z ) ⟩ | |
| 470 z ≤⟨ z≤sa ⟩ | |
| 471 supf a ∎ )) where open o≤-Reasoning O | |
| 1023 | 472 |
| 1059 | 473 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z |
| 474 → IsSUP A (UnionCF A f ay supf b) b → supf b ≡ b | |
| 475 sup=u {b} ab b≤z is-sup = z50 where | |
| 476 z48 : supf b o≤ b | |
| 477 z48 = IsMinSUP.minsup (is-minsup b≤z) ab (λ ux → IsSUP.x≤sup is-sup ux ) | |
| 478 z50 : supf b ≡ b | |
| 479 z50 with trio< (supf b) b | |
| 480 ... | tri< sb<b ¬b ¬c = ⊥-elim ( o≤> z47 sb<b ) where | |
| 481 z47 : b o≤ supf b | |
| 482 z47 = zo≤sz b≤z | |
| 483 ... | tri≈ ¬a b ¬c = b | |
| 484 ... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb ) | |
| 485 | |
| 486 supf-idem : {b : Ordinal } → b o≤ z → supf b o≤ z → supf (supf b) ≡ supf b | |
| 487 supf-idem {b} b≤z sfb≤x = z52 where | |
| 488 z54 : {w : Ordinal} → odef (UnionCF A f ay supf (supf b)) w → (w ≡ supf b) ∨ (w << supf b) | |
| 489 z54 {w} ⟪ aw , ch-init fc ⟫ = fcy<sup b≤z fc | |
| 490 z54 {w} ⟪ aw , ch-is-sup u u<x su=u fc ⟫ = order b≤z (subst (λ k → k o< supf b) (sym su=u) u<x) fc where | |
| 491 u<b : u o< b | |
| 492 u<b = supf-inject (subst (λ k → k o< supf b ) (sym (su=u)) u<x ) | |
| 493 z52 : supf (supf b) ≡ supf b | |
| 494 z52 = sup=u asupf sfb≤x record { ax = asupf ; x≤sup = z54 } | |
| 495 | |
| 496 x≤supfx : {x : Ordinal } → x o≤ z → supf x o≤ z → x o≤ supf x | |
| 497 x≤supfx {x} x≤z sx≤z with x<y∨y≤x (supf x) x | |
| 498 ... | case2 le = le | |
| 499 ... | case1 spx<px = ⊥-elim ( <<-irr z45 (proj1 ( mf< (supf x) asupf ))) where | |
| 500 z46 : odef (UnionCF A f ay supf x) (f (supf x)) | |
| 501 z46 = ⟪ proj2 ( mf (supf x) asupf ) , ch-is-sup (supf x) spx<px z47 (fsuc _ (init asupf z47 )) ⟫ where | |
| 502 z47 : supf (supf x) ≡ supf x | |
| 503 z47 = supf-idem x≤z sx≤z | |
| 504 z45 : f (supf x) ≤ supf x | |
| 505 z45 = IsMinSUP.x≤sup (is-minsup x≤z ) z46 | |
| 506 | |
| 1077 | 507 order0 : {a b w : Ordinal } → b o≤ z → supf a o< supf b → supf a o≤ z → FClosure A f (supf a) w → w ≤ supf b |
| 508 order0 {a} {b} {w} b≤z sa<sb sa≤z fc = IsMinSUP.x≤sup (is-minsup b≤z) (cfcs (supf-inject sa<sb) b≤z sa<b fc) where | |
| 509 sa<b : supf a o< b | |
| 510 sa<b with x<y∨y≤x (supf a) b | |
| 511 ... | case1 lt = lt | |
| 512 ... | case2 b≤sa = ⊥-elim ( o≤> b≤sa ( supf-inject ( osucprev ( begin | |
| 513 osuc (supf (supf a)) ≡⟨ cong osuc (supf-idem (ordtrans (supf-inject sa<sb) b≤z) sa≤z) ⟩ | |
| 514 osuc (supf a) ≤⟨ osucc sa<sb ⟩ | |
| 515 supf b ∎ )))) where open o≤-Reasoning O | |
| 516 | |
| 1038 | 517 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f) |
| 518 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf< ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where | |
| 1027 | 519 supf = ZChain.supf zc |
| 520 field | |
|
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ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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parents:
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521 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f ay supf z) a ) → b o< z → (ab : odef A b) |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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parents:
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changeset
|
522 → HasPrev A (UnionCF A f ay supf z) f b ∨ IsSUP A (UnionCF A f ay supf b) b |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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parents:
1032
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changeset
|
523 → * a < * b → odef ((UnionCF A f ay supf z)) b |
| 1027 | 524 |
| 525 record Maximal ( A : HOD ) : Set (Level.suc n) where | |
| 526 field | |
| 527 maximal : HOD | |
| 528 as : A ∋ maximal | |
| 529 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
| 530 | |
| 531 -- | |
| 532 -- supf in TransFinite indution may differ each other, but it is the same because of the minimul sup | |
| 533 -- | |
| 1038 | 534 supf-unique : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f) |
|
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parents:
1041
diff
changeset
|
535 {y xa xb : Ordinal} → (ay : odef A y) → (xa o≤ xb ) → (za : ZChain A f mf< ay xa ) (zb : ZChain A f mf< ay xb ) |
| 1007 | 536 → {z : Ordinal } → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z |
| 1038 | 537 supf-unique A f mf< {y} {xa} {xb} ay xa≤xb za zb {z} z≤xa = TransFinite0 {λ z → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z } ind z z≤xa where |
| 1007 | 538 supfa = ZChain.supf za |
| 539 supfb = ZChain.supf zb | |
| 540 ind : (x : Ordinal) → ((w : Ordinal) → w o< x → w o≤ xa → supfa w ≡ supfb w) → x o≤ xa → supfa x ≡ supfb x | |
| 1008 | 541 ind x prev x≤xa = sxa=sxb where |
|
1042
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pxhainx conditon is requied?
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parents:
1041
diff
changeset
|
542 ma = ZChain.minsup za x≤xa |
| 1008 | 543 mb = ZChain.minsup zb (OrdTrans x≤xa xa≤xb ) |
| 544 spa = MinSUP.sup ma | |
| 545 spb = MinSUP.sup mb | |
| 546 sax=spa : supfa x ≡ spa | |
|
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parents:
1041
diff
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|
547 sax=spa = ZChain.supf-is-minsup za x≤xa |
| 1008 | 548 sbx=spb : supfb x ≡ spb |
| 549 sbx=spb = ZChain.supf-is-minsup zb (OrdTrans x≤xa xa≤xb ) | |
| 1007 | 550 sxa=sxb : supfa x ≡ supfb x |
| 551 sxa=sxb with trio< (supfa x) (supfb x) | |
| 552 ... | tri≈ ¬a b ¬c = b | |
| 553 ... | tri< a ¬b ¬c = ⊥-elim ( o≤> ( | |
| 554 begin | |
| 1008 | 555 supfb x ≡⟨ sbx=spb ⟩ |
|
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parents:
1031
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|
556 spb ≤⟨ MinSUP.minsup mb (MinSUP.as ma) (λ {z} uzb → MinSUP.x≤sup ma (z53 uzb)) ⟩ |
|
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parents:
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557 spa ≡⟨ sym sax=spa ⟩ |
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558 supfa x ∎ ) a ) where |
| 1008 | 559 open o≤-Reasoning O |
|
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ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
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|
560 z53 : {z : Ordinal } → odef (UnionCF A f ay (ZChain.supf zb) x) z → odef (UnionCF A f ay (ZChain.supf za) x) z |
|
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parents:
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diff
changeset
|
561 z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ |
| 1034 | 562 z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ = ⟪ as , ch-is-sup u u<x (trans ua=ub su=u) z55 ⟫ where |
| 563 ua=ub : supfa u ≡ supfb u | |
| 564 ua=ub = prev u u<x (ordtrans u<x x≤xa ) | |
| 565 z55 : FClosure A f (ZChain.supf za u) z | |
| 566 z55 = subst (λ k → FClosure A f k z ) (sym ua=ub) fc | |
| 1007 | 567 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> ( |
| 568 begin | |
| 1008 | 569 supfa x ≡⟨ sax=spa ⟩ |
|
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minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
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|
570 spa ≤⟨ MinSUP.minsup ma (MinSUP.as mb) (λ uza → MinSUP.x≤sup mb (z53 uza)) ⟩ |
| 1008 | 571 spb ≡⟨ sym sbx=spb ⟩ |
|
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parents:
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diff
changeset
|
572 supfb x ∎ ) c ) where |
| 1009 | 573 open o≤-Reasoning O |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
574 z53 : {z : Ordinal } → odef (UnionCF A f ay (ZChain.supf za) x) z → odef (UnionCF A f ay (ZChain.supf zb) x) z |
|
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parents:
1041
diff
changeset
|
575 z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ |
| 1034 | 576 z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ = ⟪ as , ch-is-sup u u<x (trans ub=ua su=u) z55 ⟫ where |
| 577 ub=ua : supfb u ≡ supfa u | |
| 578 ub=ua = sym ( prev u u<x (ordtrans u<x x≤xa )) | |
| 579 z55 : FClosure A f (ZChain.supf zb u) z | |
| 580 z55 = subst (λ k → FClosure A f k z ) (sym ub=ua) fc | |
| 1028 | 581 |
| 966 | 582 Zorn-lemma : { A : HOD } |
| 583 → o∅ o< & A | |
| 568 | 584 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 966 | 585 → Maximal A |
| 552 | 586 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 587 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 588 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 589 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 590 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 591 s : HOD |
| 966 | 592 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) |
| 568 | 593 as : A ∋ * ( & s ) |
| 594 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
|
595 as0 : odef A (& s ) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
596 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 597 s<A : & s o< & A |
| 568 | 598 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 599 HasMaximal : HOD |
| 966 | 600 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 537 | 601 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ |
| 966 | 602 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) |
| 532 | 603 Gtx : { x : HOD} → A ∋ x → HOD |
| 966 | 604 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 537 | 605 z08 : ¬ Maximal A → HasMaximal =h= od∅ |
| 804 | 606 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
| 537 | 607 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
| 608 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)) | |
| 609 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 | |
| 610 ¬x<m : ¬ (* x < * m) | |
| 966 | 611 ¬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 | 612 |
| 1027 | 613 -- |
| 614 -- we have minsup using LEM, this is similar to the proof of the axiom of choice | |
| 615 -- | |
| 966 | 616 minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B |
| 879 | 617 minsupP B B⊆A total = m02 where |
| 618 xsup : (sup : Ordinal ) → Set n | |
| 619 xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) | |
| 620 ∀-imply-or : {A : Ordinal → Set n } {B : Set n } | |
| 621 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B | |
| 622 ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM | |
| 623 ∀-imply-or {A} {B} ∀AB | case1 t = case1 t | |
| 624 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where | |
| 625 lemma : ¬ ((x : Ordinal ) → A x) → B | |
| 626 lemma not with ODC.p∨¬p O B | |
| 627 lemma not | case1 b = b | |
| 628 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
| 629 m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B | |
| 630 m00 x = TransFinite0 ind x where | |
| 631 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) | |
| 632 → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B | |
| 633 ind x prev = ∀-imply-or m01 where | |
| 634 m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B | |
| 635 m01 z with trio< z x | |
| 636 ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 637 ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 638 ... | tri< a ¬b ¬c with prev z a | |
| 639 ... | case2 mins = case2 mins | |
| 640 ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) | |
|
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
641 ... | case1 mins = case2 record { sup = z ; isMinSUP = record { as = proj1 mins ; x≤sup = proj2 mins ; minsup = m04 } } where |
| 879 | 642 m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 |
| 643 m04 {s} as lt with trio< z s | |
| 644 ... | tri< a ¬b ¬c = o<→≤ a | |
| 645 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 646 ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) | |
| 647 ... | case2 notz = case1 (λ _ → notz ) | |
| 648 m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) | |
| 1031 | 649 m03 not = ⊥-elim ( not s1 (z09 (SUP.ax S)) ⟪ SUP.ax S , m05 ⟫ ) where |
| 879 | 650 S : SUP A B |
| 651 S = supP B B⊆A total | |
| 652 s1 = & (SUP.sup S) | |
| 653 m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) | |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
654 m05 {w} bw with SUP.x≤sup S bw |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
655 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (trans &iso eq)) |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
656 ... | case2 lt = case2 lt |
| 966 | 657 m02 : MinSUP A B |
| 879 | 658 m02 = dont-or (m00 (& A)) m03 |
| 659 | |
| 560 | 660 -- Uncountable ascending chain by axiom of choice |
| 530 | 661 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 662 cf nmx x with ODC.∋-p O A (* x) |
| 663 ... | no _ = o∅ | |
| 664 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 665 ... | yes nogt = -- no larger element, so it is maximal |
| 666 ⊥-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 | 667 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 668 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 669 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 670 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 671 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 672 ... | 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 ⟫ ) | |
| 673 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 674 |
| 675 --- | |
| 676 --- infintie ascention sequence of f | |
| 677 --- | |
| 530 | 678 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 679 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 680 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 681 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 682 |
| 803 | 683 -- |
| 953 | 684 -- maximality of chain |
| 685 -- | |
| 686 -- supf is fixed for z ≡ & A , we can prove order and is-max | |
| 1016 | 687 -- we have supf-unique now, it is provable in the first Tranfinte induction |
| 803 | 688 |
| 992 | 689 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) (mf< : <-monotonic-f A f) |
| 1038 | 690 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf< ay (& A)) (x : Ordinal) → x o≤ & A → ZChain1 A f mf< ay zc x |
| 993 | 691 SZ1 f mf mf< {y} ay zc x x≤A = zc1 x x≤A where |
| 900 | 692 chain-mono1 : {a b c : Ordinal} → a o≤ b |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
693 → odef (UnionCF A f ay (ZChain.supf zc) a) c → odef (UnionCF A f ay (ZChain.supf zc) b) c |
| 919 | 694 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
695 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) x) a → (ab : odef A b) |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
696 → HasPrev A (UnionCF A f ay (ZChain.supf zc) x) f b |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
697 → * a < * b → odef (UnionCF A f ay (ZChain.supf zc) x) b |
| 920 | 698 is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev |
| 1034 | 699 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 700 ... | ⟪ ab0 , ch-is-sup u u<x su=u fc ⟫ = ⟪ ab , subst (λ k → UChain ay x k ) | |
| 701 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x su=u (fsuc _ fc)) ⟫ | |
| 702 | |
| 869 | 703 supf = ZChain.supf zc |
| 704 | |
| 1038 | 705 zc1 : (x : Ordinal ) → x o≤ & A → ZChain1 A f mf< ay zc x |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
706 zc1 x x≤A with Oprev-p x |
|
1024
ab72526316bd
supf-< and ZChain1.order is removed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1023
diff
changeset
|
707 ... | yes op = record { is-max = is-max } where |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
708 px = Oprev.oprev op |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
709 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a → |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
710 b o< x → (ab : odef A b) → |
| 1034 | 711 HasPrev A (UnionCF A f ay supf x) f b ∨ IsSUP A (UnionCF A f ay supf b) b → |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
712 * a < * b → odef (UnionCF A f ay supf x) b |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
713 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
|
714 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 | 715 is-max {a} {b} ua b<x ab P a<b | case2 is-sup with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) |
|
1024
ab72526316bd
supf-< and ZChain1.order is removed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1023
diff
changeset
|
716 ... | case2 sb<sx = m10 where |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
717 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
|
718 b<A = z09 ab |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
719 m05 : ZChain.supf zc b ≡ b |
| 1058 | 720 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
721 m10 : odef (UnionCF A f ay supf x) b |
| 1039 | 722 m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05) |
| 992 | 723 ... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
724 m17 : MinSUP A (UnionCF A f ay supf x) -- supf z o< supf ( supf x ) |
| 992 | 725 m17 = ZChain.minsup zc x≤A |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
726 m18 : supf x ≡ MinSUP.sup m17 |
| 992 | 727 m18 = ZChain.supf-is-minsup zc x≤A |
| 990 | 728 m10 : f (supf b) ≡ supf b |
| 729 m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where | |
| 730 m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) | |
| 731 m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where | |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
732 m04 : ¬ HasPrev A (UnionCF A f ay supf b) f b |
| 1019 | 733 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
| 734 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) | |
| 735 m05 : ZChain.supf zc b ≡ b | |
| 1058 | 736 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } |
| 1019 | 737 m14 : ZChain.supf zc b o< x |
| 738 m14 = subst (λ k → k o< x ) (sym m05) b<x | |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
739 m13 : odef (UnionCF A f ay supf x) z |
| 1039 | 740 m13 = ZChain.cfcs zc b<x x≤A m14 fc |
| 989 | 741 |
|
1024
ab72526316bd
supf-< and ZChain1.order is removed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1023
diff
changeset
|
742 ... | no lim = record { is-max = is-max } where |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
743 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a → |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
744 b o< x → (ab : odef A b) → |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
745 HasPrev A (UnionCF A f ay supf x) f b ∨ IsSUP A (UnionCF A f ay supf b) b → |
|
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
746 * a < * b → odef (UnionCF A f ay supf x) b |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
747 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
|
748 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 |
| 1058 | 749 is-max {a} {b} ua b<x ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (ZChain.chain∋init zc ) |
| 1034 | 750 ... | 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 | 751 ... | case2 y<b with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) |
|
1024
ab72526316bd
supf-< and ZChain1.order is removed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1023
diff
changeset
|
752 ... | case2 sb<sx = m10 where |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
753 m09 : b o< & A |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
754 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
|
755 m05 : ZChain.supf zc b ≡ b |
| 1058 | 756 m05 = ZChain.sup=u zc ab (o<→≤ m09) record { ax = ab ; x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
757 m10 : odef (UnionCF A f ay supf x) b |
| 1039 | 758 m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05) |
| 992 | 759 ... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
760 m17 : MinSUP A (UnionCF A f ay supf x) -- supf z o< supf ( supf x ) |
| 992 | 761 m17 = ZChain.minsup zc x≤A |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
762 m18 : supf x ≡ MinSUP.sup m17 |
| 992 | 763 m18 = ZChain.supf-is-minsup zc x≤A |
| 990 | 764 m10 : f (supf b) ≡ supf b |
| 765 m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where | |
| 766 m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) | |
| 767 m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where | |
| 1058 | 768 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } |
| 1019 | 769 m14 : ZChain.supf zc b o< x |
| 770 m14 = subst (λ k → k o< x ) (sym m05) b<x | |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
771 m13 : odef (UnionCF A f ay supf x) z |
| 1039 | 772 m13 = ZChain.cfcs zc b<x x≤A m14 fc |
| 727 | 773 |
| 757 | 774 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 966 | 775 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = |
| 757 | 776 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 ))) } |
| 777 | |
| 966 | 778 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
| 757 | 779 → IsTotalOrderSet (uchain f mf ay) |
| 966 | 780 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 | 781 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 782 uz01 = fcn-cmp y f mf ca cb | |
| 783 | |
| 966 | 784 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
| 928 | 785 → MinSUP A (uchain f mf ay) |
| 966 | 786 ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) |
| 757 | 787 |
|
965
1c1c6a6ed4fa
removing ch-init is no good because of initialization
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
964
diff
changeset
|
788 |
| 793 | 789 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 1031 | 790 SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; isSUP = record { ax = SUP.ax sup ; x≤sup = λ lt → SUP.x≤sup sup (B⊆C lt) } } |
| 711 | 791 |
| 1007 | 792 zc43 : (x sp1 : Ordinal ) → ( x o< sp1 ) ∨ ( sp1 o≤ x ) |
| 793 zc43 x sp1 with trio< x sp1 | |
| 794 ... | tri< a ¬b ¬c = case1 a | |
| 795 ... | tri≈ ¬a b ¬c = case2 (o≤-refl0 (sym b)) | |
| 796 ... | tri> ¬a ¬b c = case2 (o<→≤ c) | |
| 797 | |
| 560 | 798 -- |
| 547 | 799 -- create all ZChains under o< x |
| 560 | 800 -- |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
801 |
| 1038 | 802 ind : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 803 → ((z : Ordinal) → z o< x → ZChain A f mf< ay z) → ZChain A f mf< ay x | |
| 804 ind f mf< {y} ay x prev with Oprev-p x | |
| 954 | 805 ... | yes op = zc41 where |
| 682 | 806 -- |
| 807 -- we have previous ordinal to use induction | |
| 808 -- | |
| 809 px = Oprev.oprev op | |
| 1038 | 810 zc : ZChain A f mf< ay (Oprev.oprev op) |
| 966 | 811 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 682 | 812 px<x : px o< x |
| 813 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
|
814 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
|
815 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
|
816 |
| 709 | 817 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 966 | 818 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt |
| 697 | 819 |
| 754 | 820 supf0 = ZChain.supf zc |
| 869 | 821 pchain : HOD |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
822 pchain = UnionCF A f ay supf0 px |
| 835 | 823 |
| 966 | 824 supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b |
| 857 | 825 supf-mono = ZChain.supf-mono zc |
| 844 | 826 |
| 861 | 827 zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) |
| 966 | 828 zc04 {b} b≤x with trio< b px |
| 861 | 829 ... | tri< a ¬b ¬c = case1 (o<→≤ a) |
| 830 ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) | |
| 831 ... | tri> ¬a ¬b px<b with osuc-≡< b≤x | |
| 832 ... | case1 eq = case2 eq | |
| 966 | 833 ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) |
| 840 | 834 |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
835 mf : ≤-monotonic-f A f |
| 1039 | 836 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where |
| 837 mf00 : * x < * (f x) | |
| 838 mf00 = proj1 ( mf< x ax ) | |
| 1038 | 839 |
| 954 | 840 -- |
| 841 -- find the next value of supf | |
| 842 -- | |
| 843 | |
| 844 pchainpx : HOD | |
| 1034 | 845 pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain ay px z ) |
| 1043 | 846 ∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x)) } ; odmax = & A ; <odmax = zc00 } where |
| 847 zc00 : {z : Ordinal } → (odef A z ∧ UChain ay px z ) ∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x) )→ z o< & A | |
| 966 | 848 zc00 {z} (case1 lt) = z07 lt |
| 1043 | 849 zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf (proj1 fc) ) |
| 954 | 850 |
| 1043 | 851 zc02 : { a b : Ordinal } → odef A a ∧ UChain ay px a → FClosure A f (supf0 px) b ∧ ( supf0 px o< x) → a ≤ b |
| 852 zc02 {a} {b} ca fb = zc05 (proj1 fb) where | |
| 1031 | 853 zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a ≤ b |
| 954 | 854 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb )) |
| 1034 | 855 ... | case1 eq = subst (λ k → a ≤ k ) eq (zc05 fb) |
| 1031 | 856 ... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt) |
| 1034 | 857 zc05 (init b1 refl) = MinSUP.x≤sup (ZChain.minsup zc o≤-refl) ca |
| 966 | 858 |
| 954 | 859 ptotal : IsTotalOrderSet pchainpx |
| 1034 | 860 ptotal (case1 a) (case1 b) = ZChain.f-total zc a b |
| 954 | 861 ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b |
| 862 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where | |
| 863 eq1 : a0 ≡ b0 | |
| 864 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 865 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where | |
| 866 lt1 : a0 < b0 | |
| 867 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 868 ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b | |
| 869 ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where | |
| 870 eq1 : a0 ≡ b0 | |
| 871 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 872 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where | |
| 873 lt1 : a0 < b0 | |
| 874 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 1043 | 875 ptotal (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp (supf0 px) f mf (proj1 a) (proj1 b)) |
| 966 | 876 |
| 954 | 877 pcha : pchainpx ⊆' A |
| 878 pcha (case1 lt) = proj1 lt | |
| 1043 | 879 pcha (case2 fc) = A∋fc _ f mf (proj1 fc) |
| 966 | 880 |
| 881 sup1 : MinSUP A pchainpx | |
| 954 | 882 sup1 = minsupP pchainpx pcha ptotal |
| 883 sp1 = MinSUP.sup sup1 | |
| 884 | |
| 885 -- | |
| 886 -- supf0 px o≤ sp1 | |
| 966 | 887 -- |
| 888 | |
| 1043 | 889 sfpx≤sp1 : supf0 px o< x → supf0 px ≤ sp1 |
| 890 sfpx≤sp1 spx<x = MinSUP.x≤sup sup1 (case2 ⟪ init (ZChain.asupf zc {px}) refl , spx<x ⟫ ) | |
| 1034 | 891 |
| 1043 | 892 m13 : supf0 px o< x → supf0 px o≤ sp1 |
| 893 m13 spx<x = IsMinSUP.minsup (ZChain.is-minsup zc o≤-refl ) (MinSUP.as sup1) m14 where | |
| 1034 | 894 m14 : {z : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) px) z → (z ≡ sp1) ∨ (z << sp1) |
| 1046 | 895 m14 {z} uz = MinSUP.x≤sup sup1 (case1 uz) |
| 1034 | 896 |
| 1038 | 897 zc41 : ZChain A f mf< ay x |
| 1070 | 898 zc41 = record { supf = supf1 ; asupf = asupf1 ; supf-mono = supf1-mono ; order = order |
| 899 ; 0<supfz = 0<supfz | |
| 1071 | 900 ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = cfcs } where |
| 883 | 901 |
| 871 | 902 supf1 : Ordinal → Ordinal |
| 966 | 903 supf1 z with trio< z px |
| 871 | 904 ... | tri< a ¬b ¬c = supf0 z |
| 966 | 905 ... | tri≈ ¬a b ¬c = supf0 z |
| 901 | 906 ... | tri> ¬a ¬b c = sp1 |
| 871 | 907 |
| 886 | 908 sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z |
| 901 | 909 sf1=sf0 {z} z≤px with trio< z px |
| 874 | 910 ... | tri< a ¬b ¬c = refl |
| 901 | 911 ... | tri≈ ¬a b ¬c = refl |
| 912 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c ) | |
| 883 | 913 |
| 901 | 914 sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1 |
| 915 sf1=sp1 {z} px<z with trio< z px | |
| 916 ... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a ) | |
| 917 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z ) | |
| 918 ... | tri> ¬a ¬b c = refl | |
| 873 | 919 |
| 968 | 920 sf=eq : { z : Ordinal } → z o< x → supf0 z ≡ supf1 z |
| 921 sf=eq {z} z<x = sym (sf1=sf0 (subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x )) | |
| 922 | |
| 903 | 923 asupf1 : {z : Ordinal } → odef A (supf1 z) |
| 924 asupf1 {z} with trio< z px | |
| 966 | 925 ... | tri< a ¬b ¬c = ZChain.asupf zc |
| 926 ... | tri≈ ¬a b ¬c = ZChain.asupf zc | |
|
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
927 ... | tri> ¬a ¬b c = MinSUP.as sup1 |
| 903 | 928 |
| 966 | 929 supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b |
| 930 supf1-mono {a} {b} a≤b with trio< b px | |
| 901 | 931 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b ) |
| 932 ... | 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 ) | |
| 933 supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px | |
| 934 ... | tri< a<px ¬b ¬c = zc19 where | |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
935 zc21 : MinSUP A (UnionCF A f ay supf0 a) |
| 901 | 936 zc21 = ZChain.minsup zc (o<→≤ a<px) |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
937 zc24 : {x₁ : Ordinal} → odef (UnionCF A f ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) |
| 950 | 938 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) ) |
| 966 | 939 zc19 : supf0 a o≤ sp1 |
|
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
940 zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.as sup1) zc24 ) |
| 901 | 941 ... | tri≈ ¬a b ¬c = zc18 where |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
942 zc21 : MinSUP A (UnionCF A f ay supf0 a) |
| 901 | 943 zc21 = ZChain.minsup zc (o≤-refl0 b) |
| 944 zc20 : MinSUP.sup zc21 ≡ supf0 a | |
| 966 | 945 zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b)) |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
946 zc24 : {x₁ : Ordinal} → odef (UnionCF A f ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) |
| 950 | 947 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) ) |
| 966 | 948 zc18 : supf0 a o≤ sp1 |
|
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
949 zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.as sup1) zc24 ) |
| 901 | 950 ... | tri> ¬a ¬b c = o≤-refl |
| 885 | 951 |
| 1070 | 952 0<supfz : {x : Ordinal } → o∅ o< supf1 x |
| 953 0<supfz = ordtrans<-≤ (ZChain.0<supfz zc) (OrdTrans (o≤-refl0 (sym (sf1=sf0 o∅≤z))) (supf1-mono o∅≤z)) | |
| 954 | |
| 966 | 955 fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z |
| 903 | 956 fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc |
| 966 | 957 fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z |
| 903 | 958 fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc |
| 967 | 959 |
|
999
3ffbdd53d1ea
fcs<sup requires <-monotonicity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
998
diff
changeset
|
960 -- this is a kind of maximality, so we cannot prove this without <-monotonicity |
|
3ffbdd53d1ea
fcs<sup requires <-monotonicity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
998
diff
changeset
|
961 -- |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
962 cfcs : {a b w : Ordinal } |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
963 → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w |
| 1039 | 964 cfcs {a} {b} {w} a<b b≤x sa<b fc with zc43 (supf0 a) px |
| 1012 | 965 ... | case2 px≤sa = z50 where |
| 1023 | 966 a<x : a o< x |
| 967 a<x = ordtrans<-≤ a<b b≤x | |
| 1012 | 968 a≤px : a o≤ px |
| 969 a≤px = subst (λ k → a o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ a<b b≤x) | |
| 970 -- supf0 a ≡ px we cannot use previous cfcs, it is in the chain because | |
| 971 -- supf0 a ≡ supf0 (supf0 a) ≡ supf0 px o< x | |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
972 z50 : odef (UnionCF A f ay supf1 b) w |
| 1012 | 973 z50 with osuc-≡< px≤sa |
| 1034 | 974 ... | case1 px=sa = ⟪ A∋fc {A} _ f mf fc , cp ⟫ where |
| 1023 | 975 sa≤px : supf0 a o≤ px |
| 976 sa≤px = subst₂ (λ j k → j o< k) px=sa (sym (Oprev.oprev=x op)) px<x | |
| 1026 | 977 spx=sa : supf0 px ≡ supf0 a |
| 978 spx=sa = begin | |
|
1042
912ca4abe806
pxhainx conditon is requied?
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parents:
1041
diff
changeset
|
979 supf0 px ≡⟨ cong supf0 px=sa ⟩ |
| 1039 | 980 supf0 (supf0 a) ≡⟨ ZChain.supf-idem zc a≤px sa≤px ⟩ |
| 1026 | 981 supf0 a ∎ where open ≡-Reasoning |
| 1020 | 982 z51 : supf0 px o< b |
|
1042
912ca4abe806
pxhainx conditon is requied?
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parents:
1041
diff
changeset
|
983 z51 = subst (λ k → k o< b ) (sym ( begin supf0 px ≡⟨ spx=sa ⟩ |
|
912ca4abe806
pxhainx conditon is requied?
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parents:
1041
diff
changeset
|
984 supf0 a ≡⟨ sym (sf1=sf0 a≤px) ⟩ |
| 1025 | 985 supf1 a ∎ )) sa<b where open ≡-Reasoning |
| 1020 | 986 z52 : supf1 a ≡ supf1 (supf0 px) |
|
1042
912ca4abe806
pxhainx conditon is requied?
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parents:
1041
diff
changeset
|
987 z52 = begin supf1 a ≡⟨ sf1=sf0 a≤px ⟩ |
|
912ca4abe806
pxhainx conditon is requied?
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parents:
1041
diff
changeset
|
988 supf0 a ≡⟨ sym (ZChain.supf-idem zc a≤px sa≤px ) ⟩ |
|
912ca4abe806
pxhainx conditon is requied?
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parents:
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diff
changeset
|
989 supf0 (supf0 a) ≡⟨ sym (sf1=sf0 sa≤px) ⟩ |
|
912ca4abe806
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parents:
1041
diff
changeset
|
990 supf1 (supf0 a) ≡⟨ cong supf1 (sym spx=sa) ⟩ |
| 1025 | 991 supf1 (supf0 px) ∎ where open ≡-Reasoning |
| 992 z53 : supf1 (supf0 px) ≡ supf0 px | |
| 993 z53 = begin | |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
994 supf1 (supf0 px) ≡⟨ cong supf1 spx=sa ⟩ |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
995 supf1 (supf0 a) ≡⟨ sf1=sf0 sa≤px ⟩ |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
996 supf0 (supf0 a) ≡⟨ sym ( cong supf0 px=sa ) ⟩ |
|
912ca4abe806
pxhainx conditon is requied?
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parents:
1041
diff
changeset
|
997 supf0 px ∎ where open ≡-Reasoning |
| 1034 | 998 cp : UChain ay b w |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
999 cp = ch-is-sup (supf0 px) z51 z53 (subst (λ k → FClosure A f k w) z52 fc) |
| 1020 | 1000 ... | case2 px<sa = ⊥-elim ( ¬p<x<op ⟪ px<sa , subst₂ (λ j k → j o< k ) (sf1=sf0 a≤px) (sym (Oprev.oprev=x op)) z53 ⟫ ) where |
| 1001 z53 : supf1 a o< x | |
| 1002 z53 = ordtrans<-≤ sa<b b≤x | |
| 1012 | 1003 ... | case1 sa<px with trio< a px |
| 996 | 1004 ... | tri< a<px ¬b ¬c = z50 where |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1005 z50 : odef (UnionCF A f ay supf1 b) w |
| 997 | 1006 z50 with osuc-≡< b≤x |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1007 ... | case2 lt with ZChain.cfcs zc a<b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) lt) sa<b fc |
|
912ca4abe806
pxhainx conditon is requied?
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parents:
1041
diff
changeset
|
1008 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 1034 | 1009 ... | ⟪ az , ch-is-sup u u<b su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc u≤px ) ⟫ where |
| 1010 u≤px : u o≤ px | |
| 1011 u≤px = subst (λ k → u o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ u<b b≤x ) | |
| 1012 u<x : u o< x | |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1013 u<x = ordtrans<-≤ u<b b≤x |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1014 z50 | case1 eq with ZChain.cfcs zc a<px o≤-refl sa<px fc |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1015 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 1034 | 1016 ... | ⟪ az , ch-is-sup u u<px su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc (o<→≤ u<px)) ⟫ where -- u o< px → u o< b ? |
| 1017 u<b : u o< b | |
| 1018 u<b = subst (λ k → u o< k ) (trans (Oprev.oprev=x op) (sym eq) ) (ordtrans u<px <-osuc ) | |
| 1019 u<x : u o< x | |
| 1020 u<x = subst (λ k → u o< k ) (Oprev.oprev=x op) ( ordtrans u<px <-osuc ) | |
| 1000 | 1021 ... | tri≈ ¬a a=px ¬c = csupf1 where |
|
1042
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pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1022 -- a ≡ px , b ≡ x, sp o≤ x |
| 995 | 1023 px<b : px o< b |
| 1024 px<b = subst₂ (λ j k → j o< k) a=px refl a<b | |
| 1025 b=x : b ≡ x | |
| 1026 b=x with trio< b x | |
| 996 | 1027 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) a ⟫ ) -- px o< b o< x |
| 995 | 1028 ... | tri≈ ¬a b ¬c = b |
| 996 | 1029 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> b≤x c ) -- x o< b |
| 997 | 1030 z51 : FClosure A f (supf1 px) w |
| 1031 z51 = subst (λ k → FClosure A f k w) (sym (trans (cong supf1 (sym a=px)) (sf1=sf0 (o≤-refl0 a=px) ))) fc | |
| 1001 | 1032 z53 : odef A w |
| 1033 z53 = A∋fc {A} _ f mf fc | |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1034 csupf1 : odef (UnionCF A f ay supf1 b) w |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1035 csupf1 with zc43 px (supf0 px) |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1036 ... | case2 spx≤px = ⟪ z53 , ch-is-sup (supf0 px) z54 z52 fc1 ⟫ where |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1037 z54 : supf0 px o< b |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1038 z54 = subst (λ k → supf0 px o< k ) (trans (Oprev.oprev=x op) (sym b=x) ) spx≤px |
| 1003 | 1039 z52 : supf1 (supf0 px) ≡ supf0 px |
|
1042
912ca4abe806
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parents:
1041
diff
changeset
|
1040 z52 = trans (sf1=sf0 spx≤px ) ( ZChain.supf-idem zc o≤-refl spx≤px ) |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1041 fc1 : FClosure A f (supf1 (supf0 px)) w |
| 1004 | 1042 fc1 = subst (λ k → FClosure A f k w ) (trans (cong supf0 a=px) (sym z52) ) fc |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1043 ... | case1 px<spx = ⊥-elim (¬p<x<op ⟪ px<spx , z54 ⟫ ) where -- supf1 px o≤ spuf1 x → supf1 px ≡ x o< x |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1044 z54 : supf0 px o≤ px |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1045 z54 = subst₂ (λ j k → supf0 j o< k ) a=px (trans b=x (sym (Oprev.oprev=x op))) sa<b |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1046 |
| 996 | 1047 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → a o< k ) (sym (Oprev.oprev=x op)) ( ordtrans<-≤ a<b b≤x) ⟫ ) -- px o< a o< b o≤ x |
| 994 | 1048 |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1049 zc11 : {z : Ordinal} → odef (UnionCF A f ay supf1 x) z → odef pchainpx z |
| 1034 | 1050 zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
| 1051 zc11 {z} ⟪ az , ch-is-sup u u<x su=u fc ⟫ = zc21 fc where | |
| 1052 zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1 | |
| 903 | 1053 zc21 {z1} (fsuc z2 fc ) with zc21 fc |
| 1034 | 1054 ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ |
| 1055 ... | case1 ⟪ ua1 , ch-is-sup u u<x su=u fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x su=u (fsuc _ fc₁) ⟫ | |
| 1043 | 1056 ... | case2 fc = case2 ⟪ fsuc _ (proj1 fc) , proj2 fc ⟫ |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1057 zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) |
| 1036 | 1058 ... | tri< a ¬b ¬c = case1 ⟪ asp , ch-is-sup u u<px (trans (sym (sf1=sf0 (o<→≤ u<px))) su=u )(init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where |
| 1059 u<px : u o< px | |
| 1060 u<px = ZChain.supf-inject zc a | |
| 1061 asp0 : odef A (supf0 u) | |
| 1062 asp0 = ZChain.asupf zc | |
| 1043 | 1063 ... | tri≈ ¬a b ¬c = case2 ⟪ (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) |
| 1044 | 1064 (sym (trans (sf1=sf0 (zc-b<x _ u<x)) b ))) , spx<x ⟫ where |
| 1065 spx<x : supf0 px o< x | |
| 1066 spx<x = osucprev ( begin | |
| 1067 osuc (supf0 px) ≡⟨ cong osuc (sym b) ⟩ | |
| 1068 osuc (supf0 u) ≡⟨ cong osuc (sym (sf1=sf0 (zc-b<x _ u<x) )) ⟩ | |
| 1069 osuc (supf1 u) ≡⟨ cong osuc su=u ⟩ | |
| 1070 osuc u ≤⟨ osucc u<x ⟩ | |
| 1071 x ∎ ) where open o≤-Reasoning O | |
| 1036 | 1072 ... | 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 | 1073 |
| 1035 | 1074 is-minsup : {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z) |
| 1075 is-minsup {z} z≤x with osuc-≡< z≤x | |
| 1036 | 1076 ... | case1 z=x = record { as = zc22 ; x≤sup = z23 ; minsup = z24 } where |
| 1077 px<z : px o< z | |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1078 px<z = subst (λ k → px o< k) (sym z=x) px<x |
| 1036 | 1079 zc22 : odef A (supf1 z) |
| 1080 zc22 = subst (λ k → odef A k ) (sym (sf1=sp1 px<z )) ( MinSUP.as sup1 ) | |
| 1081 z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z | |
|
1045
022d2ef3f20b
is-minsup in px case done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1044
diff
changeset
|
1082 z23 {w} uz = subst (λ k → w ≤ k ) (sym (sf1=sp1 px<z)) ( MinSUP.x≤sup sup1 ( |
|
022d2ef3f20b
is-minsup in px case done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1044
diff
changeset
|
1083 zc11 (subst (λ k → odef (UnionCF A f ay supf1 k) w) z=x uz ))) |
|
1042
912ca4abe806
pxhainx conditon is requied?
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parents:
1041
diff
changeset
|
1084 z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s ) |
| 1036 | 1085 → supf1 z o≤ s |
| 1086 z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sp1 px<z)) ( MinSUP.minsup sup1 as z25 ) where | |
|
1042
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parents:
1041
diff
changeset
|
1087 z25 : {w : Ordinal } → odef pchainpx w → w ≤ s |
| 1043 | 1088 z25 {w} (case2 fc) = sup ⟪ A∋fc _ f mf (proj1 fc) , ch-is-sup (supf0 px) z28 z27 fc1 ⟫ where |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1089 -- z=x , supf0 px o< x |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1090 z28 : supf0 px o< z -- supf0 px ≡ supf1 px o≤ supf1 x ≡ sp1 o≤ x ≡ z |
| 1044 | 1091 z28 = subst (λ k → supf0 px o< k) (sym z=x) (proj2 fc) |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1092 z29 : supf0 px o≤ px |
| 1044 | 1093 z29 = zc-b<x _ (proj2 fc) |
| 1040 | 1094 z27 : supf1 (supf0 px) ≡ supf0 px |
| 1095 z27 = trans (sf1=sf0 z29) ( ZChain.supf-idem zc o≤-refl z29 ) | |
| 1096 fc1 : FClosure A f (supf1 (supf0 px)) w | |
| 1043 | 1097 fc1 = subst (λ k → FClosure A f k w) (sym z27) (proj1 fc) |
| 1036 | 1098 z25 {w} (case1 ⟪ ua , ch-init fc ⟫) = sup ⟪ ua , ch-init fc ⟫ |
| 1099 z25 {w} (case1 ⟪ ua , ch-is-sup u u<x su=u fc ⟫) = sup ⟪ ua , ch-is-sup u u<z | |
| 1100 (trans (sf1=sf0 u≤px) su=u) (fcpu fc u≤px) ⟫ where | |
|
1042
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1101 u≤px : u o< osuc px |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1102 u≤px = ordtrans u<x <-osuc |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1103 u<z : u o< z |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1104 u<z = ordtrans u<x (subst (λ k → px o< k ) (sym z=x) px<x ) |
| 1036 | 1105 ... | case2 z<x = record { as = zc22 ; x≤sup = z23 ; minsup = z24 } where |
| 1106 z≤px = zc-b<x _ z<x | |
| 1107 m = ZChain.is-minsup zc z≤px | |
| 1108 zc22 : odef A (supf1 z) | |
| 1109 zc22 = subst (λ k → odef A k ) (sym (sf1=sf0 z≤px)) ( IsMinSUP.as m ) | |
| 1110 z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z | |
| 1111 z23 {w} ⟪ ua , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px)) ( ZChain.fcy<sup zc z≤px fc ) | |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1112 z23 {w} ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px)) |
| 1036 | 1113 (IsMinSUP.x≤sup m ⟪ ua , ch-is-sup u u<x (trans (sym (sf1=sf0 u≤px )) su=u) (fcup fc u≤px ) ⟫ ) where |
| 1114 u≤px : u o≤ px | |
| 1115 u≤px = ordtrans u<x z≤px | |
| 1116 z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s ) | |
| 1117 → supf1 z o≤ s | |
| 1118 z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf0 z≤px)) ( IsMinSUP.minsup m as z25 ) where | |
|
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parents:
1041
diff
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|
1119 z25 : {w : Ordinal } → odef ( UnionCF A f ay supf0 z ) w → w ≤ s |
| 1036 | 1120 z25 {w} ⟪ ua , ch-init fc ⟫ = sup ⟪ ua , ch-init fc ⟫ |
|
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pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1121 z25 {w} ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = sup ⟪ ua , ch-is-sup u u<x |
| 1036 | 1122 (trans (sf1=sf0 u≤px) su=u) (fcpu fc u≤px) ⟫ where |
| 1123 u≤px : u o≤ px | |
| 1124 u≤px = ordtrans u<x z≤px | |
| 885 | 1125 |
| 1053 | 1126 supfeq1 : {a b : Ordinal } → a o≤ x → b o≤ x → UnionCF A f ay supf1 a ≡ UnionCF A f ay supf1 b → supf1 a ≡ supf1 b |
| 1127 supfeq1 {a} {b} a≤z b≤z ua=ub with trio< (supf1 a) (supf1 b) | |
| 1128 ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> ( | |
| 1129 IsMinSUP.minsup (is-minsup b≤z) asupf1 (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (subst (λ k → odef k z) (sym ua=ub) uzb)) ) sa<sb ) | |
| 1130 ... | tri≈ ¬a b ¬c = b | |
| 1131 ... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> ( | |
| 1132 IsMinSUP.minsup (is-minsup a≤z) asupf1 (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (subst (λ k → odef k z) ua=ub uza)) ) sb<sa ) | |
| 1133 | |
| 1134 union-max1 : {a b : Ordinal } → x o≤ supf1 a → b o≤ x → supf1 a o< supf1 b → UnionCF A f ay supf1 a ≡ UnionCF A f ay supf1 b | |
| 1135 union-max1 {a} {b} z≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where | |
| 1136 z47 : {w : Ordinal } → odef (UnionCF A f ay supf1 a ) w → odef ( UnionCF A f ay supf1 b ) w | |
| 1137 z47 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫ | |
| 1138 z47 {w} ⟪ aw , ch-is-sup u u<a su=u fc ⟫ = ⟪ aw , ch-is-sup u u<b su=u fc ⟫ where | |
| 1139 u<b : u o< b | |
| 1140 u<b = ordtrans u<a (supf-inject0 supf1-mono sa<sb ) | |
| 1141 z48 : {w : Ordinal } → odef (UnionCF A f ay supf1 b ) w → odef ( UnionCF A f ay supf1 a ) w | |
| 1142 z48 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫ | |
| 1143 z48 {w} ⟪ aw , ch-is-sup u u<b su=u fc ⟫ = ⟪ aw , ch-is-sup u u<a su=u fc ⟫ where | |
| 1144 u<a : u o< a | |
| 1145 u<a = supf-inject0 supf1-mono ( osucprev (begin | |
| 1146 osuc (supf1 u) ≡⟨ cong osuc su=u ⟩ | |
| 1147 osuc u ≤⟨ osucc (ordtrans<-≤ u<b b≤z ) ⟩ | |
| 1148 x ≤⟨ z≤sa ⟩ | |
| 1149 supf1 a ∎ )) where open o≤-Reasoning O | |
| 1150 | |
| 1059 | 1151 |
| 1152 zo≤sz : {z : Ordinal} → z o≤ x → z o≤ supf1 z | |
| 1153 zo≤sz {z} z≤x with osuc-≡< z≤x | |
| 1154 ... | case2 z<x = subst (λ k → z o≤ k) (sym (sf1=sf0 (zc-b<x _ z<x ))) (ZChain.zo≤sz zc (zc-b<x _ z<x )) | |
| 1155 ... | case1 refl with osuc-≡< (supf1-mono (o<→≤ (px<x))) -- px o≤ supf1 px o≤ supf1 x ≡ sp1 → x o≤ sp1 | |
| 1156 ... | case2 lt = begin | |
| 1157 x ≡⟨ sym (Oprev.oprev=x op) ⟩ | |
| 1158 osuc px ≤⟨ osucc (ZChain.zo≤sz zc o≤-refl) ⟩ | |
| 1159 osuc (supf0 px) ≡⟨ sym (cong osuc (sf1=sf0 o≤-refl )) ⟩ | |
| 1160 osuc (supf1 px) ≤⟨ osucc lt ⟩ | |
| 1161 supf1 x ∎ where open o≤-Reasoning O | |
| 1162 ... | case1 spx=sx with osuc-≡< ( ZChain.zo≤sz zc o≤-refl ) | |
| 1163 ... | case2 lt = begin | |
| 1164 x ≡⟨ sym (Oprev.oprev=x op) ⟩ | |
| 1165 osuc px ≤⟨ osucc lt ⟩ | |
| 1166 supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ | |
| 1167 supf1 px ≤⟨ supf1-mono (o<→≤ px<x) ⟩ | |
| 1168 supf1 x ∎ where open o≤-Reasoning O | |
| 1169 ... | case1 px=spx = ⊥-elim ( <<-irr zc40 (proj1 ( mf< (supf0 px) (ZChain.asupf zc))) ) where | |
| 1170 zc37 : supf0 px ≡ px | |
| 1171 zc37 = sym px=spx | |
| 1172 zc39 : supf0 px ≡ sp1 | |
| 1173 zc39 = begin | |
| 1174 supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ | |
| 1175 supf1 px ≡⟨ spx=sx ⟩ | |
| 1176 supf1 x ≡⟨ sf1=sp1 px<x ⟩ | |
| 1177 sp1 ∎ where open ≡-Reasoning | |
| 1178 zc40 : f (supf0 px) ≤ supf0 px | |
| 1179 zc40 = subst (λ k → f (supf0 px) ≤ k ) (sym zc39) | |
| 1180 ( MinSUP.x≤sup sup1 (case2 ⟪ fsuc _ (init (ZChain.asupf zc) refl) , subst (λ k → k o< x) (sym zc37) px<x ⟫ )) | |
| 1181 | |
| 1051 | 1182 order : {a b : Ordinal} {w : Ordinal} → |
| 1183 b o≤ x → supf1 a o< supf1 b → FClosure A f (supf1 a) w → w ≤ supf1 b | |
| 1184 order {a} {b} {w} b≤x sa<sb fc = z20 where | |
| 1185 a<b : a o< b | |
| 1186 a<b = supf-inject0 supf1-mono sa<sb | |
| 1187 a≤px : a o≤ px | |
| 1188 a≤px with trio< a px | |
| 1189 ... | tri< a ¬b ¬c = o<→≤ a | |
| 1190 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 1191 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → a o< k) (sym (Oprev.oprev=x op)) | |
| 1192 ( ordtrans<-≤ a<b b≤x) ⟫ ) -- px o< a o< b o≤ x | |
| 1193 z20 : w ≤ supf1 b | |
| 1194 z20 with trio< b px | |
| 1195 ... | tri< b<px ¬b ¬c = ZChain.order zc (o<→≤ b<px) (subst (λ k → k o< supf0 b) (sf1=sf0 (o<→≤ (ordtrans a<b b<px))) sa<sb) | |
| 1196 (fcup fc (o<→≤ (ordtrans a<b b<px))) | |
| 1197 ... | tri≈ ¬a b=px ¬c = IsMinSUP.x≤sup (ZChain.is-minsup zc (o≤-refl0 b=px)) z26 where | |
| 1054 | 1198 a≤x : a o≤ x |
| 1199 a≤x = o<→≤ (ordtrans ( subst (λ k → a o< k ) b=px a<b ) px<x ) | |
| 1051 | 1200 z26 : odef ( UnionCF A f ay supf0 b ) w |
| 1052 | 1201 z26 with x<y∨y≤x (supf1 a) b |
| 1054 | 1202 ... | case2 b≤sa = z27 where |
| 1203 z27 : odef (UnionCF A f ay supf0 b) w | |
| 1204 z27 with osuc-≡< b≤sa | |
| 1205 ... | case2 b<sa = ⊥-elim ( o<¬≡ ( supfeq1 a≤x b≤x | |
| 1206 ( union-max1 x≤sa b≤x (subst (λ k → supf1 a o< k) (sym (sf1=sf0 (o≤-refl0 b=px))) sa<sb) )) | |
| 1207 (ordtrans<-≤ sa<sb (o≤-refl0 (sym (sf1=sf0 (o≤-refl0 b=px))) ))) where | |
| 1208 x≤sa : x o≤ supf1 a | |
| 1209 x≤sa = subst (λ k → k o≤ supf1 a ) (trans (cong osuc b=px) (Oprev.oprev=x op)) (osucc b<sa ) | |
| 1055 | 1210 ... | case1 b=sa = ⊥-elim (o<¬≡ sa=sb sa<sb) where |
| 1211 sa=sb : supf1 a ≡ supf0 b | |
| 1212 sa=sb = begin | |
| 1213 supf1 a ≡⟨ sf1=sf0 a≤px ⟩ | |
| 1214 supf0 a ≡⟨ sym (ZChain.supf-idem zc a≤px (o≤-refl0 (sym (trans (sym b=px) (trans b=sa (sf1=sf0 a≤px) ))))) ⟩ | |
| 1215 supf0 (supf0 a) ≡⟨ cong supf0 (sym (sf1=sf0 a≤px )) ⟩ | |
| 1216 supf0 (supf1 a) ≡⟨ cong supf0 (sym b=sa) ⟩ | |
| 1217 supf0 b ∎ where open ≡-Reasoning | |
| 1052 | 1218 ... | case1 sa<b with cfcs a<b b≤x sa<b fc |
| 1051 | 1219 ... | ⟪ ua , ch-init fc ⟫ = ⟪ ua , ch-init fc ⟫ |
| 1054 | 1220 ... | ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = ⟪ ua , ch-is-sup u u<x (trans (sym (sf1=sf0 u≤px)) su=u) (fcup fc u≤px) ⟫ where |
| 1221 u≤px : u o≤ px | |
| 1222 u≤px = o<→≤ ( subst (λ k → u o< k ) b=px u<x ) | |
| 1052 | 1223 ... | tri> ¬a ¬b px<b with x<y∨y≤x (supf1 a) b |
| 1224 ... | case1 sa<b = MinSUP.x≤sup sup1 (zc11 ( chain-mono f mf ay supf1 supf1-mono b≤x (cfcs a<b b≤x sa<b fc))) | |
| 1054 | 1225 ... | case2 b≤sa = ⊥-elim ( o<¬≡ z27 sa<sb ) where -- x=b x o≤ sa UnionCF a ≡ UnionCF b → supf1 a ≡ supfb b → ⊥ |
| 1053 | 1226 b=x : b ≡ x |
| 1227 b=x with trio< b x | |
| 1228 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<b , zc-b<x _ a ⟫ ) -- px o< b o< x | |
| 1229 ... | tri≈ ¬a b ¬c = b | |
| 1230 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> b≤x c ) -- x o< b ≤ x | |
| 1054 | 1231 a≤x : a o≤ x |
| 1232 a≤x = o<→≤ ( subst (λ k → a o< k ) b=x a<b ) | |
| 1233 sf1b=sp1 : supf1 b ≡ sp1 | |
| 1234 sf1b=sp1 = sf1=sp1 (subst (λ k → px o< k) (trans (Oprev.oprev=x op) (sym b=x)) <-osuc) | |
| 1235 z27 : supf1 a ≡ sp1 | |
| 1236 z27 = trans ( supfeq1 a≤x b≤x ( union-max1 (subst (λ k → k o≤ supf1 a) b=x b≤sa) | |
| 1237 b≤x (subst (λ k → supf1 a o< k ) (sym sf1b=sp1) sa<sb ) ) ) sf1b=sp1 | |
| 1051 | 1238 |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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parents:
1032
diff
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|
1239 ... | no lim with trio< x o∅ |
|
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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|
1240 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1068 | 1241 ... | tri≈ ¬a b ¬c = record { supf = λ _ → MinSUP.sup (ysup f mf ay) ; asupf = MinSUP.as (ysup f mf ay) ; 0<supfz = ? |
| 1058 | 1242 ; supf-mono = λ _ → o≤-refl ; order = λ _ s<s → ⊥-elim ( o<¬≡ refl s<s ) |
| 1071 | 1243 ; zo≤sz = ? ; is-minsup = ? ; cfcs = ? } where |
| 1058 | 1244 mf : ≤-monotonic-f A f |
| 1245 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where | |
| 1246 mf00 : * x < * (f x) | |
| 1247 mf00 = proj1 ( mf< x ax ) | |
| 1077 | 1248 ... | tri> ¬a ¬b 0<x = record { supf = supf1 ; asupf = asupf ; supf-mono = supf-mono ; order = order ; 0<supfz = 0<sufz |
| 1072 | 1249 ; zo≤sz = ? ; is-minsup = is-minsup ; cfcs = cfcs } where |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1250 |
|
1042
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1251 mf : ≤-monotonic-f A f |
| 1039 | 1252 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where |
| 1253 mf00 : * x < * (f x) | |
| 1254 mf00 = proj1 ( mf< x ax ) | |
| 1038 | 1255 |
| 1039 | 1256 pzc : {z : Ordinal} → z o< x → ZChain A f mf< ay z |
| 1009 | 1257 pzc {z} z<x = prev z z<x |
|
726
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psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1258 |
| 928 | 1259 ysp = MinSUP.sup (ysup f mf ay) |
| 755 | 1260 |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1261 supfz : {z : Ordinal } → z o< x → Ordinal |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1262 supfz {z} z<x = ZChain.supf (pzc (ob<x lim z<x)) z |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1263 |
| 1069 | 1264 pchainU : HOD |
| 1265 pchainU = UnionIC A f ay supfz | |
| 1012 | 1266 |
|
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parents:
1041
diff
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|
1267 zeq : {xa xb z : Ordinal } |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
changeset
|
1268 → (xa<x : xa o< x) → (xb<x : xb o< x) → xa o≤ xb → z o≤ xa |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1269 → ZChain.supf (pzc xa<x) z ≡ ZChain.supf (pzc xb<x) z |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1270 zeq {xa} {xb} {z} xa<x xb<x xa≤xb z≤xa = supf-unique A f mf< ay xa≤xb |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1271 (pzc xa<x) (pzc xb<x) z≤xa |
| 835 | 1272 |
|
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pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1273 iceq : {ix iy : Ordinal } → ix ≡ iy → {i<x : ix o< x } {i<y : iy o< x } → supfz i<x ≡ supfz i<y |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1274 iceq refl = cong supfz o<-irr |
| 1027 | 1275 |
| 1063 | 1276 IChain-i : {z : Ordinal } → IChain ay supfz z → Ordinal |
| 1277 IChain-i (ic-init fc) = o∅ | |
| 1278 IChain-i (ic-isup ia ia<x sa<x fca) = ia | |
| 1027 | 1279 |
| 1063 | 1280 pic<x : {z : Ordinal } → (ic : IChain ay supfz z ) → osuc (IChain-i ic) o< x |
| 1281 pic<x {z} (ic-init fc) = ob<x lim 0<x -- 0<x ∧ lim x → osuc o∅ o< x | |
| 1282 pic<x {z} (ic-isup ia ia<x sa<x fca) = ob<x lim ia<x | |
| 1283 | |
| 1069 | 1284 pchainU⊆chain : {z : Ordinal } → (pz : odef pchainU z) → odef (ZChain.chain (pzc (pic<x (proj2 pz)))) z |
| 1285 pchainU⊆chain {z} ⟪ aw , ic-init fc ⟫ = ⟪ aw , ch-init fc ⟫ | |
| 1286 pchainU⊆chain {z} ⟪ aw , (ic-isup ia ia<x sa<x fca) ⟫ = ZChain.cfcs (pzc (ob<x lim ia<x) ) <-osuc o≤-refl uz03 fca where | |
| 1063 | 1287 uz02 : FClosure A f (ZChain.supf (pzc (ob<x lim ia<x)) ia ) z |
| 1288 uz02 = fca | |
| 1289 uz03 : ZChain.supf (pzc (ob<x lim ia<x)) ia o≤ ia | |
| 1290 uz03 = sa<x | |
| 1062 | 1291 |
| 1069 | 1292 chain⊆pchainU : {z w : Ordinal } → (oz<x : osuc z o< x) → odef (ZChain.chain (pzc oz<x)) w → odef pchainU w |
| 1293 chain⊆pchainU {z} {w} oz<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫ | |
| 1294 chain⊆pchainU {z} {w} oz<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫ | |
| 1064 | 1295 = ⟪ aw , ic-isup u u<x (o≤-refl0 su≡u) (subst (λ k → FClosure A f k w ) su=su fc) ⟫ where |
| 1296 u<x : u o< x | |
| 1297 u<x = ordtrans u<oz oz<x | |
| 1298 su=su : ZChain.supf (pzc oz<x) u ≡ supfz u<x | |
| 1299 su=su = sym ( zeq _ _ (osucc u<oz) (o<→≤ <-osuc) ) | |
| 1300 su≡u : supfz u<x ≡ u | |
| 1301 su≡u = begin | |
| 1302 ZChain.supf (pzc (ob<x lim u<x )) u ≡⟨ sym su=su ⟩ | |
| 1303 ZChain.supf (pzc oz<x) u ≡⟨ su=u ⟩ | |
| 1304 u ∎ where open ≡-Reasoning | |
| 1305 | |
| 1072 | 1306 chain⊆pchainU1 : {z w : Ordinal } → (z<x : z o< x) → odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim z<x))) z) w → odef pchainU w |
| 1307 chain⊆pchainU1 {z} {w} z<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫ | |
| 1308 chain⊆pchainU1 {z} {w} z<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫ | |
| 1309 = ⟪ aw , ic-isup u u<x (o≤-refl0 su≡u) (subst (λ k → FClosure A f k w ) su=su fc) ⟫ where | |
| 1310 u<x : u o< x | |
| 1311 u<x = ordtrans u<oz z<x | |
| 1312 su=su : ZChain.supf (pzc (ob<x lim z<x)) u ≡ supfz u<x | |
| 1313 su=su = sym ( zeq _ _ (o<→≤ (osucc u<oz)) (o<→≤ <-osuc) ) | |
| 1314 su≡u : supfz u<x ≡ u | |
| 1315 su≡u = begin | |
| 1316 ZChain.supf (pzc (ob<x lim u<x )) u ≡⟨ sym su=su ⟩ | |
| 1317 ZChain.supf (pzc (ob<x lim z<x)) u ≡⟨ su=u ⟩ | |
| 1318 u ∎ where open ≡-Reasoning | |
| 1319 | |
| 1064 | 1320 ichain-inject : {a b : Ordinal } {ia : IChain ay supfz a } {ib : IChain ay supfz b } |
| 1321 → ZChain.supf (pzc (pic<x ia)) (IChain-i ia) o< ZChain.supf (pzc (pic<x ib)) (IChain-i ib) | |
| 1322 → IChain-i ia o< IChain-i ib | |
| 1065 | 1323 ichain-inject {a} {b} {ia} {ib} sa<sb = uz11 where |
| 1324 uz11 : IChain-i ia o< IChain-i ib | |
| 1325 uz11 with trio< (IChain-i ia ) (IChain-i ib) | |
| 1326 ... | tri< a ¬b ¬c = a | |
| 1327 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (trans (zeq _ _ (o≤-refl0 (cong osuc b)) (o<→≤ <-osuc) ) | |
| 1328 ( cong (ZChain.supf (pzc (pic<x ib))) b )) sa<sb ) | |
| 1329 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> ( begin | |
| 1330 ZChain.supf (pzc (pic<x ib)) (IChain-i ib) ≡⟨ zeq _ _ (o<→≤ (osucc c)) (o<→≤ <-osuc) ⟩ | |
| 1331 ZChain.supf (pzc (pic<x ia)) (IChain-i ib) ≤⟨ ZChain.supf-mono (pzc (pic<x ia)) (o<→≤ c) ⟩ | |
| 1332 ZChain.supf (pzc (pic<x ia)) (IChain-i ia) ∎ ) sa<sb ) where open o≤-Reasoning O | |
| 1062 | 1333 |
| 1068 | 1334 IC⊆ : {a b : Ordinal } (ia : IChain ay supfz a ) (ib : IChain ay supfz b ) |
| 1335 → IChain-i ia o≤ IChain-i ib → odef (ZChain.chain (pzc (pic<x ib))) a | |
| 1336 IC⊆ {a} {b} (ic-init fc ) ib ia≤ib = ⟪ A∋fc _ f mf fc , ch-init fc ⟫ | |
| 1337 IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-init fcb ) ia≤ib = ⊥-elim ( o≤> ia≤ib ic01 ) where | |
| 1338 ic02 : o∅ o< supfz i<x | |
| 1339 ic02 = ZChain.0<supfz (pzc (ob<x lim i<x)) | |
| 1340 ic01 : o∅ o< i | |
| 1341 ic01 = ordtrans<-≤ ic02 sa<x | |
| 1342 IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-isup j j<x sb<x fcb ) ia≤ib | |
| 1343 = ZChain.cfcs (pzc (ob<x lim j<x) ) ia≤ib o≤-refl (OrdTrans (ZChain.supf-mono (pzc (ob<x lim j<x)) ia≤ib) sb<x) | |
| 1344 (subst (λ k → FClosure A f k a) (zeq _ _ (osucc ia≤ib) (o<→≤ <-osuc)) fc ) | |
| 1345 | |
| 1069 | 1346 ptotalU : IsTotalOrderSet pchainU |
| 1347 ptotalU {a} {b} ia ib with trio< (IChain-i (proj2 ia)) (IChain-i (proj2 ib)) | |
| 1348 ... | tri< ia<ib ¬b ¬c = ZChain.f-total (pzc (pic<x (proj2 ib))) (IC⊆ (proj2 ia) (proj2 ib) (o<→≤ ia<ib)) (pchainU⊆chain ib) | |
| 1349 ... | tri≈ ¬a ia=ib ¬c = ZChain.f-total (pzc (pic<x (proj2 ib))) (IC⊆ (proj2 ia) (proj2 ib) (o≤-refl0 ia=ib)) (pchainU⊆chain ib) | |
| 1350 ... | tri> ¬a ¬b ib<ia = ZChain.f-total (pzc (pic<x (proj2 ia))) (pchainU⊆chain ia) (IC⊆ (proj2 ib) (proj2 ia) (o<→≤ ib<ia)) | |
| 1351 | |
| 1352 usup : MinSUP A pchainU | |
| 1353 usup = minsupP pchainU (λ ic → proj1 ic ) ptotalU | |
| 1354 spu = MinSUP.sup usup | |
| 1355 | |
| 1356 | |
| 1357 pchainS : HOD | |
| 1358 pchainS = record { od = record { def = λ z → (odef A z ∧ IChain ay supfz z ) | |
| 1359 ∨ (FClosure A f spu z ∧ (spu o< x)) } ; odmax = & A ; <odmax = zc00 } where | |
| 1360 zc00 : {z : Ordinal } → (odef A z ∧ IChain ay supfz z ) ∨ (FClosure A f spu z ∧ (spu o< x) )→ z o< & A | |
| 1361 zc00 {z} (case1 lt) = z07 lt | |
| 1362 zc00 {z} (case2 fc) = z09 ( A∋fc spu f mf (proj1 fc) ) | |
| 1363 | |
| 1364 zc02 : { a b : Ordinal } → odef A a ∧ IChain ay supfz a → FClosure A f spu b ∧ ( spu o< x) → a ≤ b | |
| 1365 zc02 {a} {b} ca fb = zc05 (proj1 fb) where | |
| 1366 zc05 : {b : Ordinal } → FClosure A f spu b → a ≤ b | |
| 1367 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc spu f mf fb )) | |
| 1368 ... | case1 eq = subst (λ k → a ≤ k ) eq (zc05 fb) | |
| 1369 ... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt) | |
| 1370 zc05 (init b1 refl) = MinSUP.x≤sup usup ca | |
| 966 | 1371 |
| 1069 | 1372 ptotalS : IsTotalOrderSet pchainS |
| 1373 ptotalS (case1 a) (case1 b) = ptotalU a b | |
| 1374 ptotalS {a0} {b0} (case1 a) (case2 b) with zc02 a b | |
| 1375 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where | |
| 1376 eq1 : a0 ≡ b0 | |
| 1377 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 1378 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where | |
| 1379 lt1 : a0 < b0 | |
| 1380 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 1381 ptotalS {b0} {a0} (case2 b) (case1 a) with zc02 a b | |
| 1382 ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where | |
| 1383 eq1 : a0 ≡ b0 | |
| 1384 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 1385 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where | |
| 1386 lt1 : a0 < b0 | |
| 1387 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 1388 ptotalS (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp spu f mf (proj1 a) (proj1 b)) | |
| 1389 | |
| 1390 S⊆A : pchainS ⊆' A | |
| 1391 S⊆A (case1 lt) = proj1 lt | |
| 1392 S⊆A (case2 fc) = A∋fc _ f mf (proj1 fc) | |
| 1393 | |
| 1394 ssup : MinSUP A pchainS | |
| 1395 ssup = minsupP pchainS S⊆A ptotalS | |
| 1396 | |
| 1397 sps = MinSUP.sup ssup | |
| 834 | 1398 |
| 794 | 1399 supf1 : Ordinal → Ordinal |
| 835 | 1400 supf1 z with trio< z x |
| 1069 | 1401 ... | tri< a ¬b ¬c = ZChain.supf (pzc (ob<x lim a)) z -- each sup o< x |
| 1402 ... | tri≈ ¬a b ¬c = spu -- sup of all sup o< x | |
| 1403 ... | tri> ¬a ¬b c = sps -- sup of spu which o< x | |
| 1404 -- if x o< spu, spu is not included in UnionCF x | |
| 1405 -- the chain | |
| 755 | 1406 |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1407 pchain : HOD |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1408 pchain = UnionCF A f ay supf1 x |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1409 |
| 1069 | 1410 -- pchain ⊆ pchainU ⊆ pchianS |
| 1009 | 1411 |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1412 sf1=sf : {z : Ordinal } → (a : z o< x ) → supf1 z ≡ ZChain.supf (pzc (ob<x lim a)) z |
| 1009 | 1413 sf1=sf {z} z<x with trio< z x |
| 1414 ... | tri< a ¬b ¬c = cong ( λ k → ZChain.supf (pzc (ob<x lim k)) z) o<-irr | |
| 1415 ... | tri≈ ¬a b ¬c = ⊥-elim (¬a z<x) | |
| 1416 ... | tri> ¬a ¬b c = ⊥-elim (¬a z<x) | |
| 1417 | |
| 1071 | 1418 sf1=spu : {z : Ordinal } → x ≡ z → supf1 z ≡ spu |
| 1419 sf1=spu {z} eq with trio< z x | |
| 1420 ... | tri< a ¬b ¬c = ⊥-elim (¬b (sym eq)) | |
| 1009 | 1421 ... | tri≈ ¬a b ¬c = refl |
| 1071 | 1422 ... | tri> ¬a ¬b c = ⊥-elim (¬b (sym eq)) |
| 1009 | 1423 |
| 1071 | 1424 sf1=sps : {z : Ordinal } → (a : x o< z ) → supf1 z ≡ sps |
| 1425 sf1=sps {z} x<z with trio< z x | |
| 1426 ... | tri< a ¬b ¬c = ⊥-elim (o<> x<z a) | |
| 1427 ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x<z ) | |
| 1428 ... | tri> ¬a ¬b c = refl | |
| 1069 | 1429 |
| 1430 zc11 : {z : Ordinal } → odef pchain z → odef pchainS z | |
| 1027 | 1431 zc11 {z} lt = ? |
| 1009 | 1432 |
| 1069 | 1433 sfpx≤spu : {z : Ordinal } → supf1 z ≤ sps |
| 1031 | 1434 sfpx≤spu {z} with trio< z x |
| 1069 | 1435 ... | tri< a ¬b ¬c = ? -- MinSUP.x≤sup usup ? -- (init (ZChain.asupf (pzc (ob<x lim a)) ) refl ) |
| 1436 ... | tri≈ ¬a b ¬c = case1 ? | |
| 1437 ... | tri> ¬a ¬b c = case1 ? | |
| 844 | 1438 |
| 1069 | 1439 sfpxo≤spu : {z : Ordinal } → supf1 z o≤ sps |
| 1057 | 1440 sfpxo≤spu {z} with trio< z x |
| 1441 ... | tri< a ¬b ¬c = ? -- MinSUP.x≤sup usup ? -- (init (ZChain.asupf (pzc (ob<x lim a)) ) refl ) | |
| 1442 ... | tri≈ ¬a b ¬c = o≤-refl0 ? | |
| 1443 ... | tri> ¬a ¬b c = o≤-refl0 ? | |
| 1444 | |
| 1445 asupf : {z : Ordinal } → odef A (supf1 z) | |
| 1446 asupf {z} with trio< z x | |
| 1447 ... | tri< a ¬b ¬c = ZChain.asupf (pzc (ob<x lim a)) | |
| 1448 ... | tri≈ ¬a b ¬c = MinSUP.as usup | |
| 1069 | 1449 ... | tri> ¬a ¬b c = MinSUP.as ssup |
| 1057 | 1450 |
| 1451 supf-mono : {z y : Ordinal } → z o≤ y → supf1 z o≤ supf1 y | |
| 1452 supf-mono {z} {y} z≤y with trio< y x | |
| 1071 | 1453 ... | tri< y<x ¬b ¬c = zc01 where |
| 1057 | 1454 open o≤-Reasoning O |
| 1071 | 1455 zc01 : supf1 z o≤ ZChain.supf (pzc (ob<x lim y<x)) y |
| 1072 | 1456 zc01 = begin |
| 1457 supf1 z ≡⟨ sf1=sf (ordtrans≤-< z≤y y<x) ⟩ | |
| 1458 ZChain.supf (pzc (ob<x lim (ordtrans≤-< z≤y y<x))) z ≡⟨ zeq _ _ (osucc z≤y) (o<→≤ <-osuc) ⟩ | |
| 1070 | 1459 ZChain.supf (pzc (ob<x lim y<x)) z ≤⟨ ZChain.supf-mono (pzc (ob<x lim y<x)) z≤y ⟩ |
| 1071 | 1460 ZChain.supf (pzc (ob<x lim y<x)) y ∎ |
| 1072 | 1461 ... | tri≈ ¬a b ¬c = zc01 where -- supf1 z o≤ spu |
| 1462 open o≤-Reasoning O | |
| 1463 zc01 : supf1 z o≤ spu | |
| 1464 zc01 with osuc-≡< (subst (λ k → z o≤ k) b z≤y) | |
| 1465 ... | case1 z=x = o≤-refl0 (sf1=spu (sym z=x)) | |
| 1466 ... | case2 z<x = subst (λ k → k o≤ spu ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) ) | |
| 1467 (MinSUP.as usup) (λ uw → MinSUP.x≤sup usup (chain⊆pchainU1 z<x uw)) ) | |
| 1468 ... | tri> ¬a ¬b c = zc01 where -- supf1 z o≤ sps | |
| 1469 zc01 : supf1 z o≤ sps | |
| 1470 zc01 with trio< z x | |
| 1471 ... | tri< z<x ¬b ¬c = IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) ) | |
| 1472 (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 (chain⊆pchainU1 z<x uw)) ) | |
| 1473 ... | tri≈ ¬a z=x ¬c = MinSUP.minsup usup (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 uw) ) | |
| 1474 ... | tri> ¬a ¬b c = o≤-refl -- (sf1=sps c) | |
| 797 | 1475 |
| 1070 | 1476 0<sufz : {x : Ordinal } → o∅ o< supf1 x |
| 1477 0<sufz = ordtrans<-≤ (ZChain.0<supfz (pzc (ob<x lim 0<x ))) (OrdTrans (o≤-refl0 (sym (sf1=sf 0<x ))) (supf-mono o∅≤z)) | |
| 1478 | |
| 1035 | 1479 is-minsup : {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z) |
| 1073 | 1480 is-minsup {z} z≤x with osuc-≡< z≤x |
| 1075 | 1481 ... | case1 z=x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where |
| 1482 zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z | |
| 1076 | 1483 zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x))) ( MinSUP.x≤sup usup ⟪ az , ic-init fc ⟫ ) |
| 1484 zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x))) | |
| 1485 ( MinSUP.x≤sup usup ⟪ az , ic-isup u u<x (o≤-refl0 zm05) (subst (λ k → FClosure A f k w) (sym zm06) fc) ⟫ ) where | |
| 1486 u<x : u o< x | |
| 1487 u<x = subst (λ k → u o< k) z=x u<b | |
| 1488 zm06 : supfz (subst (λ k → u o< k) z=x u<b) ≡ supf1 u | |
| 1489 zm06 = trans (zeq _ _ o≤-refl (o<→≤ <-osuc) ) (sym (sf1=sf u<x )) | |
| 1490 zm05 : supfz (subst (λ k → u o< k) z=x u<b) ≡ u | |
| 1491 zm05 = trans zm06 su=u | |
| 1075 | 1492 zm01 : { s : Ordinal } → odef A s → ( {x : Ordinal } → odef (UnionCF A f ay supf1 z) x → x ≤ s ) → supf1 z o≤ s |
| 1493 zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=spu (sym z=x))) ( MinSUP.minsup usup as zm02 ) where | |
| 1494 zm02 : {w : Ordinal } → odef pchainU w → w ≤ s | |
| 1076 | 1495 zm02 {w} uw with pchainU⊆chain uw |
| 1496 ... | ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫ | |
| 1497 ... | ⟪ az , ch-is-sup u1 u<b su=u fc ⟫ = sup ⟪ az , ch-is-sup u1 (ordtrans u<b zm05) (trans zm03 su=u) zm04 ⟫ where | |
| 1498 zm05 : osuc (IChain-i (proj2 uw)) o< z | |
| 1499 zm05 = subst (λ k → osuc (IChain-i (proj2 uw)) o< k) (sym z=x) ( pic<x (proj2 uw) ) | |
| 1500 u<x : u1 o< x | |
| 1501 u<x = subst (λ k → u1 o< k) z=x ( ordtrans u<b zm05 ) | |
| 1502 zm03 : supf1 u1 ≡ ZChain.supf (prev (osuc (IChain-i (proj2 uw))) (pic<x (proj2 uw))) u1 | |
| 1503 zm03 = trans (sf1=sf u<x) (zeq _ _ (osucc u<b) (o<→≤ <-osuc) ) | |
| 1504 zm04 : FClosure A f (supf1 u1) w | |
| 1505 zm04 = subst (λ k → FClosure A f k w) (sym zm03) fc | |
| 1073 | 1506 ... | case2 z<x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where |
| 1507 supf0 = ZChain.supf (pzc (ob<x lim z<x)) | |
| 1508 msup : IsMinSUP A (UnionCF A f ay supf0 z) (supf0 z) | |
| 1509 msup = ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) | |
| 1074 | 1510 s1=0 : {u : Ordinal } → u o< z → supf1 u ≡ supf0 u |
| 1511 s1=0 {u} u<z = trans (sf1=sf (ordtrans u<z z<x)) (zeq _ _ (o<→≤ (osucc u<z)) (o<→≤ <-osuc) ) | |
| 1073 | 1512 zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z |
| 1074 | 1513 zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x)) ( IsMinSUP.x≤sup msup ⟪ az , ch-init fc ⟫ ) |
| 1073 | 1514 zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x)) |
| 1074 | 1515 ( IsMinSUP.x≤sup msup ⟪ az , ch-is-sup u u<b (trans (sym (s1=0 u<b)) su=u) (subst (λ k → FClosure A f k w) (s1=0 u<b) fc) ⟫ ) |
| 1073 | 1516 zm01 : { s : Ordinal } → odef A s → ( {x : Ordinal } → odef (UnionCF A f ay supf1 z) x → x ≤ s ) → supf1 z o≤ s |
| 1074 | 1517 zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup msup as zm02 ) where |
| 1073 | 1518 zm02 : {w : Ordinal } → odef (UnionCF A f ay supf0 z) w → w ≤ s |
| 1519 zm02 {w} ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫ | |
| 1074 | 1520 zm02 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = sup |
| 1521 ⟪ az , ch-is-sup u u<b (trans (s1=0 u<b) su=u) (subst (λ k → FClosure A f k w) (sym (s1=0 u<b)) fc) ⟫ | |
| 1073 | 1522 |
| 1523 | |
| 1035 | 1524 |
| 1039 | 1525 cfcs : {a b w : Ordinal } → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w |
| 1526 cfcs {a} {b} {w} a<b b≤x sa<b fc with osuc-≡< b≤x | |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1527 ... | case1 b=x with trio< a x |
| 1016 | 1528 ... | tri< a<x ¬b ¬c = zc40 where |
| 1529 sa = ZChain.supf (pzc (ob<x lim a<x)) a | |
| 1020 | 1530 m = omax a sa -- x is limit ordinal, so we have sa o< m o< x |
| 1016 | 1531 m<x : m o< x |
| 1532 m<x with trio< a sa | inspect (omax a) sa | |
| 1020 | 1533 ... | tri< a<sa ¬b ¬c | record { eq = eq } = ob<x lim (ordtrans<-≤ sa<b b≤x ) |
| 1016 | 1534 ... | tri≈ ¬a a=sa ¬c | record { eq = eq } = subst (λ k → k o< x) eq zc41 where |
| 1535 zc41 : omax a sa o< x | |
| 1536 zc41 = osucprev ( begin | |
| 1537 osuc ( omax a sa ) ≡⟨ cong (λ k → osuc (omax a k)) (sym a=sa) ⟩ | |
| 1538 osuc ( omax a a ) ≡⟨ cong osuc (omxx _) ⟩ | |
| 1539 osuc ( osuc a ) ≤⟨ o<→≤ (ob<x lim (ob<x lim a<x)) ⟩ | |
| 1540 x ∎ ) where open o≤-Reasoning O | |
| 1541 ... | tri> ¬a ¬b c | record { eq = eq } = ob<x lim a<x | |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1542 sam = ZChain.supf (pzc (ob<x lim m<x)) a |
| 1016 | 1543 zc42 : osuc a o≤ osuc m |
| 1544 zc42 = osucc (o<→≤ ( omax-x _ _ ) ) | |
| 1545 sam<m : sam o< m | |
| 1039 | 1546 sam<m = subst (λ k → k o< m ) (supf-unique A f mf< ay zc42 (pzc (ob<x lim a<x)) (pzc (ob<x lim m<x)) (o<→≤ <-osuc)) ( omax-y _ _ ) |
| 1016 | 1547 fcm : FClosure A f (ZChain.supf (pzc (ob<x lim m<x)) a) w |
| 1548 fcm = subst (λ k → FClosure A f k w ) (zeq (ob<x lim a<x) (ob<x lim m<x) zc42 (o<→≤ <-osuc) ) fc | |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1549 zcm : odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim m<x))) (osuc (omax a sa))) w |
| 1039 | 1550 zcm = ZChain.cfcs (pzc (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1551 zc40 : odef (UnionCF A f ay supf1 b) w |
| 1039 | 1552 zc40 with ZChain.cfcs (pzc (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm |
| 1058 | 1553 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 1070 | 1554 ... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans zc45 su=u) zc44 ⟫ where |
| 1555 u<b : u o< b | |
| 1556 u<b = osucprev ( begin | |
| 1557 osuc u ≤⟨ osucc u<x ⟩ | |
| 1558 osuc m ≤⟨ osucc m<x ⟩ | |
| 1559 x ≡⟨ sym b=x ⟩ | |
| 1560 b ∎ ) where open o≤-Reasoning O | |
| 1561 zc45 : supf1 u ≡ ZChain.supf (pzc (ob<x lim m<x)) u | |
| 1562 zc45 = begin | |
| 1563 supf1 u ≡⟨ sf1=sf (subst (λ k → u o< k) b=x u<b ) ⟩ | |
| 1564 ZChain.supf (pzc (ob<x lim (subst (λ k → u o< k) b=x u<b ))) u ≡⟨ zeq _ _ (osucc u<x) (o<→≤ <-osuc) ⟩ | |
| 1565 ZChain.supf (pzc (ob<x lim m<x)) u ∎ where open ≡-Reasoning | |
| 1566 zc44 : FClosure A f (supf1 u) w | |
| 1567 zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc | |
| 1016 | 1568 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) |
| 1569 ... | tri> ¬a ¬b c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) | |
| 1039 | 1570 cfcs {a} {b} {w} a<b b≤x sa<b fc | case2 b<x = zc40 where |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1571 supfb = ZChain.supf (pzc (ob<x lim b<x)) |
|
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1572 sb=sa : {a : Ordinal } → a o< b → supf1 a ≡ ZChain.supf (pzc (ob<x lim b<x)) a |
| 1020 | 1573 sb=sa {a} a<b = trans (sf1=sf (ordtrans<-≤ a<b b≤x)) (zeq _ _ (o<→≤ (osucc a<b)) (o<→≤ <-osuc) ) |
|
1017
ffdfd8d1303a
trying cscf as odef (UnionCF A f mf ay supf z) w
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1016
diff
changeset
|
1574 fcb : FClosure A f (supfb a) w |
|
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1575 fcb = subst (λ k → FClosure A f k w) (sb=sa a<b) fc |
| 1020 | 1576 -- supfb a o< b assures it is in Union b |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1577 zcb : odef (UnionCF A f ay supfb b) w |
| 1039 | 1578 zcb = ZChain.cfcs (pzc (ob<x lim b<x)) a<b (o<→≤ <-osuc) (subst (λ k → k o< b) (sb=sa a<b) sa<b) fcb |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1579 zc40 : odef (UnionCF A f ay supf1 b) w |
| 1016 | 1580 zc40 with zcb |
| 1058 | 1581 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 1070 | 1582 ... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<x (trans zc45 su=u) zc44 ⟫ where |
| 1583 zc45 : supf1 u ≡ ZChain.supf (pzc (ob<x lim b<x)) u | |
| 1584 zc45 = begin | |
| 1585 supf1 u ≡⟨ sf1=sf (ordtrans u<x b<x) ⟩ | |
| 1586 ZChain.supf (pzc (ob<x lim (ordtrans u<x b<x) )) u ≡⟨ zeq _ _ (o<→≤ (osucc u<x)) (o<→≤ <-osuc) ⟩ | |
| 1587 ZChain.supf (pzc (ob<x lim b<x )) u ∎ where open ≡-Reasoning | |
| 1588 zc44 : FClosure A f (supf1 u) w | |
| 1589 zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc | |
| 1028 | 1590 |
| 1077 | 1591 order : {a b w : Ordinal } → b o≤ x → supf1 a o< supf1 b → FClosure A f (supf1 a) w → w ≤ supf1 b |
| 1592 order = ? | |
| 1593 | |
| 1594 | |
| 921 | 1595 --- |
| 1596 --- the maximum chain has fix point of any ≤-monotonic function | |
| 1597 --- | |
| 1598 | |
| 1039 | 1599 SZ : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf< ay x |
| 1600 SZ f mf< {y} ay x = TransFinite {λ z → ZChain A f mf< ay z } (λ x → ind f mf< ay x ) x | |
| 921 | 1601 |
| 1039 | 1602 msp0 : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
| 1603 → (zc : ZChain A f mf< ay x ) | |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1604 → MinSUP A (UnionCF A f ay (ZChain.supf zc) x) |
| 1039 | 1605 msp0 f mf< {x} ay zc = minsupP (UnionCF A f ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc) |
| 922 | 1606 |
| 1039 | 1607 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (mf< : <-monotonic-f A f ) (zc : ZChain A f mf< as0 (& A) ) |
| 966 | 1608 → (sp1 : MinSUP A (ZChain.chain zc)) |
| 959 | 1609 → f (MinSUP.sup sp1) ≡ MinSUP.sup sp1 |
| 992 | 1610 fixpoint f mf 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
|
1611 chain = ZChain.chain zc |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1612 supf = ZChain.supf zc |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1613 sp : Ordinal |
| 959 | 1614 sp = MinSUP.sup sp1 |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1615 asp : odef A sp |
|
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1616 asp = MinSUP.as sp1 |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1617 ay = as0 |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
1618 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< (& A) → (ab : odef A b ) |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1619 → HasPrev A chain f b ∨ IsSUP A (UnionCF A f ay (ZChain.supf zc) b) b |
| 921 | 1620 → * a < * b → odef chain b |
| 993 | 1621 z10 = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl ) |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1622 z22 : sp o< & A |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1623 z22 = z09 asp |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1624 z12 : odef chain sp |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1625 z12 with o≡? (& s) sp |
| 1058 | 1626 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 1627 ... | no ne = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl) {& 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
|
1628 z13 : * (& s) < * sp |
| 1058 | 1629 z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc ) |
| 960 | 1630 ... | case1 eq = ⊥-elim ( ne eq ) |
| 966 | 1631 ... | case2 lt = lt |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1632 z19 : IsSUP A (UnionCF A f ay (ZChain.supf zc) sp) sp |
| 1069 | 1633 z19 = record { ax = asp ; x≤sup = z20 } where |
|
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1634 z20 : {y : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp) |
| 966 | 1635 z20 {y} zy with MinSUP.x≤sup sp1 |
| 961 | 1636 (subst (λ k → odef chain k ) (sym &iso) (chain-mono f mf as0 supf (ZChain.supf-mono zc) (o<→≤ z22) zy )) |
| 966 | 1637 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p ) |
| 960 | 1638 ... | 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
|
1639 z14 : f sp ≡ sp |
| 960 | 1640 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
|
1641 ... | 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
|
1642 z16 : ⊥ |
|
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1643 z16 with proj1 (mf (( MinSUP.sup sp1)) ( MinSUP.as sp1 )) |
| 1031 | 1644 ... | case1 eq = ⊥-elim (¬b (sym (cong (*) eq ) )) |
| 966 | 1645 ... | case2 lt = ⊥-elim (¬c lt ) |
| 1646 ... | 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
|
1647 ... | tri> ¬a ¬b c = ⊥-elim z17 where |
| 959 | 1648 z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) ) |
| 960 | 1649 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
|
1650 z17 : ⊥ |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1651 z17 with z15 |
| 960 | 1652 ... | case1 eq = ¬b (cong (*) eq) |
| 1653 ... | case2 lt = ¬a lt | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1654 |
| 921 | 1655 -- ZChain contradicts ¬ Maximal |
| 1656 -- | |
| 1657 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) | |
| 1658 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
| 1659 -- | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1660 |
| 1039 | 1661 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-<-monotonic nmx) as0 (& A)) → ⊥ |
| 966 | 1662 z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} |
|
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1663 (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.as msp1 )))) |
|
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1664 (subst (λ k → odef A k) (sym &iso) (MinSUP.as msp1) ) |
| 992 | 1665 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) (cf-is-<-monotonic nmx ) zc msp1 ))) -- x ≡ f x ̄ |
|
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1666 (proj1 (cf-is-<-monotonic nmx c (MinSUP.as msp1 ))) where -- x < f x |
| 937 | 1667 |
| 927 | 1668 supf = ZChain.supf zc |
| 934 | 1669 msp1 : MinSUP A (ZChain.chain zc) |
| 1039 | 1670 msp1 = msp0 (cf nmx) (cf-is-<-monotonic nmx) as0 zc |
| 966 | 1671 c : Ordinal |
| 1672 c = MinSUP.sup msp1 | |
| 934 | 1673 |
| 966 | 1674 zorn00 : Maximal A |
| 1675 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 804 | 1676 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
| 551 | 1677 -- yes we have the maximal |
| 1678 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 1679 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 1680 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 966 | 1681 zorn01 = proj1 zorn03 |
| 551 | 1682 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) |
| 1683 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 1039 | 1684 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-<-monotonic nmx) as0 (& A) )) where |
| 551 | 1685 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 966 | 1686 nmx : ¬ Maximal A |
| 551 | 1687 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where |
| 966 | 1688 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) |
| 804 | 1689 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 | 1690 |
| 516 | 1691 -- usage (see filter.agda ) |
| 1692 -- | |
| 497 | 1693 -- _⊆'_ : ( A B : HOD ) → Set n |
| 1694 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 1695 |
| 966 | 1696 -- MaximumSubset : {L P : HOD} |
| 497 | 1697 -- → o∅ o< & L → o∅ o< & P → P ⊆ L |
| 1698 -- → IsPartialOrderSet P _⊆'_ | |
| 1699 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 1700 -- → Maximal P (_⊆'_) | |
| 1701 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |