Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 1023:52272b5c9d58
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 25 Nov 2022 16:57:33 +0900 |
| parents | 1b87669d9b11 |
| children | ab72526316bd |
| rev | line source |
|---|---|
| 478 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 508 | 2 open import Level hiding ( suc ; zero ) |
| 431 | 3 open import Ordinals |
| 966 | 4 open import Relation.Binary |
| 552 | 5 open import Relation.Binary.Core |
| 6 open import Relation.Binary.PropositionalEquality | |
| 966 | 7 import OD |
| 552 | 8 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where |
| 431 | 9 |
| 560 | 10 -- |
| 966 | 11 -- Zorn-lemma : { A : HOD } |
| 12 -- → o∅ o< & A | |
| 560 | 13 -- → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 966 | 14 -- → Maximal A |
| 560 | 15 -- |
| 16 | |
| 431 | 17 open import zf |
| 477 | 18 open import logic |
| 19 -- open import partfunc {n} O | |
| 20 | |
| 966 | 21 open import Relation.Nullary |
| 22 open import Data.Empty | |
| 23 import BAlgbra | |
| 431 | 24 |
| 555 | 25 open import Data.Nat hiding ( _<_ ; _≤_ ) |
| 966 | 26 open import Data.Nat.Properties |
| 27 open import nat | |
| 555 | 28 |
| 431 | 29 |
| 30 open inOrdinal O | |
| 31 open OD O | |
| 32 open OD.OD | |
| 33 open ODAxiom odAxiom | |
| 477 | 34 import OrdUtil |
| 35 import ODUtil | |
| 431 | 36 open Ordinals.Ordinals O |
| 37 open Ordinals.IsOrdinals isOrdinal | |
| 38 open Ordinals.IsNext isNext | |
| 39 open OrdUtil O | |
| 477 | 40 open ODUtil O |
| 41 | |
| 42 | |
| 43 import ODC | |
| 44 | |
| 45 open _∧_ | |
| 46 open _∨_ | |
| 47 open Bool | |
| 431 | 48 |
| 49 open HOD | |
| 50 | |
| 560 | 51 -- |
| 52 -- Partial Order on HOD ( possibly limited in A ) | |
| 53 -- | |
| 54 | |
| 966 | 55 _<<_ : (x y : Ordinal ) → Set n |
| 570 | 56 x << y = * x < * y |
| 57 | |
| 872 | 58 _<=_ : (x y : Ordinal ) → Set n -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain |
| 765 | 59 x <= y = (x ≡ y ) ∨ ( * x < * y ) |
| 60 | |
| 966 | 61 POO : IsStrictPartialOrder _≡_ _<<_ |
| 62 POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 63 ; trans = IsStrictPartialOrder.trans PO | |
| 570 | 64 ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y |
| 966 | 65 ; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } } |
| 66 | |
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67 _≤_ : (x y : HOD) → Set (Level.suc n) |
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68 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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69 |
| 966 | 70 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 554 | 71 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl |
| 72 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 73 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 74 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 75 | |
| 966 | 76 <=-trans : {x y z : Ordinal } → x <= y → y <= z → x <= z |
| 955 | 77 <=-trans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl |
| 78 <=-trans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
| 79 <=-trans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
| 80 <=-trans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
| 779 | 81 |
| 966 | 82 ftrans<=-< : {x y z : Ordinal } → x <= y → y << z → x << z |
| 953 | 83 ftrans<=-< {x} {y} {z} (case1 eq) y<z = subst (λ k → k < * z) (sym (cong (*) eq)) y<z |
| 966 | 84 ftrans<=-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z |
| 951 | 85 |
| 990 | 86 ftrans<-<= : {x y z : Ordinal } → x << y → y <= z → x << z |
| 87 ftrans<-<= {x} {y} {z} x<y (case1 eq) = subst (λ k → * x < k ) ((cong (*) eq)) x<y | |
| 88 ftrans<-<= {x} {y} {z} x<y (case2 lt) = IsStrictPartialOrder.trans PO x<y lt | |
| 89 | |
| 966 | 90 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y |
| 770 | 91 <=to≤ (case1 eq) = case1 (cong (*) eq) |
| 92 <=to≤ (case2 lt) = case2 lt | |
| 93 | |
| 966 | 94 ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y |
| 779 | 95 ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) |
| 96 ≤to<= (case2 lt) = case2 lt | |
| 97 | |
| 556 | 98 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 99 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 100 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 101 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 102 |
| 561 | 103 ptrans = IsStrictPartialOrder.trans PO |
| 104 | |
| 492 | 105 open _==_ |
| 106 open _⊆_ | |
| 107 | |
| 966 | 108 -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A |
| 879 | 109 -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x |
| 110 -- <-TransFinite = ? | |
| 111 | |
| 530 | 112 -- |
| 560 | 113 -- Closure of ≤-monotonic function f has total order |
| 530 | 114 -- |
| 115 | |
| 116 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 117 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 118 | |
| 992 | 119 <-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set n |
| 120 <-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x < * (f x) ) ∧ odef A (f x ) | |
| 121 | |
| 551 | 122 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 783 | 123 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
| 555 | 124 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 125 |
| 556 | 126 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 | 127 A∋fc {A} s f mf (init as refl ) = as |
| 556 | 128 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 129 |
| 714 | 130 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 | 131 A∋fcs {A} s f mf (init as refl) = as |
| 966 | 132 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
| 714 | 133 |
| 556 | 134 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 | 135 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
| 556 | 136 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 137 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 966 | 138 ... | case2 x<fx with s≤fc {A} s f mf fcy |
| 556 | 139 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) |
| 140 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 141 |
| 800 | 142 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 143 fcn s mf (init as refl) = zero | |
| 144 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
| 145 ... | case1 eq = fcn s mf p | |
| 146 ... | case2 y<fy = suc (fcn s mf p ) | |
| 147 | |
| 966 | 148 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
| 800 | 149 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y |
| 150 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 | |
| 151 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 152 fc06 {x} {y} refl {j} not = fc08 not where | |
| 966 | 153 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
| 800 | 154 fc08 () |
| 155 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
| 156 fc07 {x} (init as refl) eq = refl | |
| 157 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 158 ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) | |
| 159 -- ... | case2 x<fx = ? | |
| 160 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 | |
| 161 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
| 162 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
| 163 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
| 164 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 165 ... | 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 ) | |
| 166 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 167 ... | 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 ) | |
| 168 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 ) ) | |
| 169 ... | 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 ) | |
| 170 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 ) ) | |
| 171 ... | 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 ) | |
| 172 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 173 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 174 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 175 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 176 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x | |
| 177 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
| 178 fc04 : * x1 ≡ * y | |
| 179 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 180 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 181 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 182 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 183 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 966 | 184 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq |
| 800 | 185 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where |
| 186 fc05 : * x ≡ * y1 | |
| 187 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 188 ... | 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))) | |
| 189 | |
| 190 | |
| 191 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 192 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 193 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
| 194 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 195 fc06 {x} {y} refl {j} not = fc08 not where | |
| 966 | 196 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
| 800 | 197 fc08 () |
| 198 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 | |
| 199 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
| 200 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 966 | 201 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) |
| 800 | 202 ... | case2 y<fy with <-cmp (fcn s mf cx ) i |
| 203 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 966 | 204 ... | 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 | 205 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where |
| 206 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 966 | 207 fc03 eq = cong pred eq |
| 208 fc02 : * x < * y1 | |
| 800 | 209 fc02 = fc01 i cx cy (fc03 i=y ) a |
| 210 | |
| 557 | 211 |
| 966 | 212 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 213 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 800 | 214 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 215 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 216 fc11 : * x < * y | |
| 217 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 218 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 219 fc10 : * x ≡ * y | |
| 220 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 966 | 221 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where |
| 800 | 222 fc12 : * y < * x |
| 223 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 600 | 224 |
| 563 | 225 |
| 729 | 226 |
| 560 | 227 -- open import Relation.Binary.Properties.Poset as Poset |
| 228 | |
| 229 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 230 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 231 | |
| 567 | 232 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 233 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 234 |
| 568 | 235 _⊆'_ : ( A B : HOD ) → Set n |
| 236 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 237 |
| 238 -- | |
| 239 -- inductive maxmum tree from x | |
| 240 -- tree structure | |
| 241 -- | |
| 554 | 242 |
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243 record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where |
| 533 | 244 field |
| 836 | 245 ax : odef A x |
| 534 | 246 y : Ordinal |
| 541 | 247 ay : odef B y |
| 966 | 248 x=fy : x ≡ f y |
| 529 | 249 |
| 962 | 250 record IsSUP (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 251 field | |
| 252 x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) | |
| 957 | 253 |
| 960 | 254 record IsMinSUP (A B : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (xa : odef A x) : Set n where |
| 654 | 255 field |
| 950 | 256 x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 966 | 257 minsup : { sup1 : Ordinal } → odef A sup1 |
| 954 | 258 → ( {z : Ordinal } → odef B z → (z ≡ sup1 ) ∨ (z << sup1 )) → x o≤ sup1 |
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259 not-hp : ¬ ( HasPrev A B f x ) |
| 568 | 260 |
| 656 | 261 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 262 field | |
| 263 sup : HOD | |
| 804 | 264 as : A ∋ sup |
| 950 | 265 x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive |
| 656 | 266 |
| 690 | 267 -- |
| 268 -- sup and its fclosure is in a chain HOD | |
| 269 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 270 -- whole chain is a union of separated Chain | |
| 966 | 271 -- minimum index is sup of y not ϕ |
| 690 | 272 -- |
| 273 | |
| 787 | 274 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where |
| 690 | 275 field |
| 966 | 276 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
| 828 | 277 order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
| 278 supu=u : supf u ≡ u | |
| 694 | 279 |
| 748 | 280 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 966 | 281 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where |
| 282 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 995 | 283 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) ( is-sup : ChainP A f mf ay supf u ) |
| 748 | 284 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z |
| 694 | 285 |
| 878 | 286 -- |
| 990 | 287 -- f (f ( ... (supf y))) f (f ( ... (supf z1))) |
| 878 | 288 -- / | / | |
| 289 -- / | / | | |
| 990 | 290 -- supf y < supf z1 < supf z2 |
| 878 | 291 -- o< o< |
| 990 | 292 -- |
| 293 -- if sup z1 ≡ sup z2, the chain is stopped at sup z1, then f (sup z1) ≡ sup z1 | |
| 294 | |
| 295 | |
| 296 fc-stop : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) { a b : Ordinal } | |
| 297 → (aa : odef A a ) →( {y : Ordinal} → FClosure A f a y → (y ≡ b ) ∨ (y << b )) → a ≡ b → f a ≡ a | |
| 298 fc-stop A f mf {a} {b} aa x≤sup a=b with x≤sup (fsuc a (init aa refl )) | |
| 299 ... | case1 eq = trans eq (sym a=b) | |
| 300 ... | case2 lt = ⊥-elim (<-irr (case1 (cong (λ k → * (f k) ) (sym a=b))) (ftrans<-<= lt (≤to<= fc00 )) ) where | |
| 301 fc00 : * b ≤ * (f b) | |
| 302 fc00 = proj1 (mf _ (subst (λ k → odef A k) a=b aa )) | |
| 303 | |
| 304 -- | |
| 861 | 305 -- data UChain is total |
| 306 | |
| 307 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) | |
| 308 {s s1 a b : Ordinal } ( ca : UChain A f mf ay supf s a ) ( cb : UChain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) | |
| 309 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where | |
| 310 ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) | |
| 966 | 311 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb |
| 312 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) with ChainP.fcy<sup supb fca | |
| 313 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 314 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 315 ct00 : * a ≡ * b | |
| 316 ct00 = trans (cong (*) eq) eq1 | |
| 317 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where | |
| 318 ct01 : * a < * b | |
| 319 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
| 320 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where | |
| 321 ct00 : * a < * (supf ub) | |
| 322 ct00 = lt | |
| 323 ct01 : * a < * b | |
| 324 ct01 with s≤fc (supf ub) f mf fcb | |
| 325 ... | case1 eq = subst (λ k → * a < k ) eq ct00 | |
| 326 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 327 ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb | |
| 328 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 329 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 330 ct00 : * a ≡ * b | |
| 331 ct00 = sym (trans (cong (*) eq) eq1 ) | |
| 332 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where | |
| 333 ct01 : * b < * a | |
| 334 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
| 335 ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where | |
| 336 ct00 : * b < * (supf ua) | |
| 337 ct00 = lt | |
| 338 ct01 : * b < * a | |
| 339 ct01 with s≤fc (supf ua) f mf fca | |
| 340 ... | case1 eq = subst (λ k → * b < k ) eq ct00 | |
| 341 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 861 | 342 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub |
| 966 | 343 ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁) fca |
| 344 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 861 | 345 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
| 346 ct00 : * a ≡ * b | |
| 347 ct00 = trans (cong (*) eq) eq1 | |
| 348 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 966 | 349 ct02 : * a < * b |
| 861 | 350 ct02 = subst (λ k → * k < * b ) (sym eq) lt |
| 351 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 352 ct03 : * a < * (supf ub) | |
| 353 ct03 = lt | |
| 966 | 354 ct02 : * a < * b |
| 861 | 355 ct02 with s≤fc (supf ub) f mf fcb |
| 356 ... | case1 eq = subst (λ k → * a < k ) eq ct03 | |
| 357 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt | |
| 966 | 358 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri≈ ¬a eq ¬c |
| 861 | 359 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) |
| 966 | 360 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) fcb |
| 361 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 861 | 362 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
| 363 ct00 : * a ≡ * b | |
| 364 ct00 = sym (trans (cong (*) eq) eq1) | |
| 365 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
| 966 | 366 ct02 : * b < * a |
| 861 | 367 ct02 = subst (λ k → * k < * a ) (sym eq) lt |
| 368 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where | |
| 369 ct05 : * b < * (supf ua) | |
| 370 ct05 = lt | |
| 966 | 371 ct04 : * b < * a |
| 861 | 372 ct04 with s≤fc (supf ua) f mf fca |
| 373 ... | case1 eq = subst (λ k → * b < k ) eq ct05 | |
| 374 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt | |
| 375 | |
| 694 | 376 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 377 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 378 | |
| 803 | 379 -- Union of supf z which o< x |
| 380 -- | |
| 966 | 381 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 694 | 382 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD |
| 383 UnionCF A f mf ay supf x | |
| 894 | 384 = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x z } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } |
| 662 | 385 |
| 966 | 386 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) |
| 387 → supf x o< supf y → x o< y | |
| 842 | 388 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y |
| 389 ... | tri< a ¬b ¬c = a | |
| 390 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 391 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 392 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 393 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 394 | |
| 879 | 395 record MinSUP ( A B : HOD ) : Set n where |
| 396 field | |
| 397 sup : Ordinal | |
| 398 asm : odef A sup | |
| 966 | 399 x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) |
| 400 minsup : { sup1 : Ordinal } → odef A sup1 | |
| 879 | 401 → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 |
| 402 | |
| 403 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A | |
| 404 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 405 | |
| 880 | 406 M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } |
| 407 → (supf : Ordinal → Ordinal ) | |
| 966 | 408 → MinSUP A (UnionCF A f mf ay supf x) |
| 409 → SUP A (UnionCF A f mf ay supf x) | |
| 410 M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) | |
| 950 | 411 ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x≤sup = ms00 } where |
| 880 | 412 msup = MinSUP.sup ms |
| 413 ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup) | |
| 966 | 414 ms00 {z} uz with MinSUP.x≤sup ms uz |
| 880 | 415 ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq)) |
| 416 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso refl lt ) | |
| 417 | |
| 867 | 418 |
| 966 | 419 chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 919 | 420 (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b |
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421 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c |
| 966 | 422 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = |
| 423 ⟪ ua , ch-init fc ⟫ | |
| 919 | 424 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = |
| 995 | 425 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x a≤b) is-sup fc ⟫ |
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426 |
| 966 | 427 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 428 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 429 field |
| 966 | 430 supf : Ordinal → Ordinal |
| 431 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z | |
| 432 → IsSUP A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) f b ) → supf b ≡ b | |
| 1001 | 433 cfcs : (mf< : <-monotonic-f A f) |
| 1018 | 434 {a b w : Ordinal } → a o< b → b o≤ z → supf a o< b → FClosure A f (supf a) w → odef (UnionCF A f mf ay supf b) w |
| 994 | 435 |
| 868 | 436 asupf : {x : Ordinal } → odef A (supf x) |
| 880 | 437 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y |
| 438 supf-< : {x y : Ordinal } → supf x o< supf y → supf x << supf y | |
| 891 | 439 supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z |
| 880 | 440 |
| 966 | 441 minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) |
| 891 | 442 supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z ) |
| 994 | 443 |
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444 chain : HOD |
| 703 | 445 chain = UnionCF A f mf ay supf z |
| 861 | 446 chain⊆A : chain ⊆' A |
| 447 chain⊆A = λ lt → proj1 lt | |
| 934 | 448 |
| 966 | 449 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
| 450 sup {x} x≤z = M→S supf (minsup x≤z) | |
| 934 | 451 |
| 452 s=ms : {x : Ordinal } → (x≤z : x o≤ z ) → & (SUP.sup (sup x≤z)) ≡ MinSUP.sup (minsup x≤z) | |
| 453 s=ms {x} x≤z = &iso | |
| 878 | 454 |
| 966 | 455 chain∋init : odef chain y |
| 456 chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ | |
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457 f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a) |
| 966 | 458 f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
| 938 | 459 f-next {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u<x is-sup (fsuc _ fc ) ⟫ |
| 861 | 460 initial : {z : Ordinal } → odef chain z → * y ≤ * z |
| 461 initial {a} ⟪ aa , ua ⟫ with ua | |
| 966 | 462 ... | ch-init fc = s≤fc y f mf fc |
| 938 | 463 ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 966 | 464 zc7 : y <= supf u |
| 861 | 465 zc7 = ChainP.fcy<sup is-sup (init ay refl) |
| 466 f-total : IsTotalOrderSet chain | |
| 966 | 467 f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 861 | 468 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 966 | 469 uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) |
| 861 | 470 |
| 871 | 471 supf-<= : {x y : Ordinal } → supf x <= supf y → supf x o≤ supf y |
| 472 supf-<= {x} {y} (case1 sx=sy) = o≤-refl0 sx=sy | |
| 473 supf-<= {x} {y} (case2 sx<sy) with trio< (supf x) (supf y) | |
| 474 ... | tri< a ¬b ¬c = o<→≤ a | |
| 475 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 476 ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) ) | |
| 477 | |
| 1014 | 478 supf-<inject : {x y : Ordinal } → supf x << supf y → supf x o< supf y |
| 479 supf-<inject {x} {y} sx<sy with trio< (supf x) (supf y) | |
| 480 ... | tri< a ¬b ¬c = a | |
| 481 ... | tri≈ ¬a b ¬c = ⊥-elim (<-irr (case1 (cong (*) (sym b)) ) sx<sy ) | |
| 482 ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) ) | |
| 483 | |
| 966 | 484 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
| 825 | 485 supf-inject {x} {y} sx<sy with trio< x y |
| 486 ... | tri< a ¬b ¬c = a | |
| 487 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 488 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 489 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 490 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 798 | 491 |
| 1005 | 492 supf<A : {x : Ordinal } → supf x o< & A |
| 493 supf<A = z09 asupf | |
| 494 | |
| 1000 | 495 csupf : (mf< : <-monotonic-f A f) {b : Ordinal } |
| 1005 | 496 → supf b o< supf z → supf b o< z → odef (UnionCF A f mf ay supf z) (supf b) -- supf z is not an element of this chain |
| 497 csupf mf< {b} sb<sz sb<z = cfcs mf< (supf-inject sb<sz) o≤-refl sb<z (init asupf refl) | |
| 994 | 498 |
| 966 | 499 fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf |
| 500 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) | |
| 501 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ | |
| 502 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) | |
| 503 ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) | |
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504 |
| 966 | 505 -- ordering is not proved here but in ZChain1 |
| 825 | 506 |
| 966 | 507 IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp |
| 960 | 508 → ({y : Ordinal} → odef (UnionCF A f mf ay supf x) y → (y ≡ sp ) ∨ (y << sp )) |
| 1000 | 509 → ( {a : Ordinal } → odef A a → a << f a ) |
| 960 | 510 → ¬ ( HasPrev A (UnionCF A f mf ay supf x) f sp ) |
| 511 IsMinSUP→NotHasPrev {x} {sp} asp is-sup <-mono-f hp = ⊥-elim (<-irr ( <=to≤ fsp≤sp) sp<fsp ) where | |
| 512 sp<fsp : sp << f sp | |
| 1000 | 513 sp<fsp = <-mono-f asp |
| 966 | 514 pr = HasPrev.y hp |
| 960 | 515 im00 : f (f pr) <= sp |
| 516 im00 = is-sup ( f-next (f-next (HasPrev.ay hp))) | |
| 517 fsp≤sp : f sp <= sp | |
| 518 fsp≤sp = subst (λ k → f k <= sp ) (sym (HasPrev.x=fy hp)) im00 | |
| 519 | |
| 1013 | 520 supf-¬hp : {x : Ordinal } → x o≤ z |
| 521 → ( {a : Ordinal } → odef A a → a << f a ) | |
| 522 → ¬ ( HasPrev A (UnionCF A f mf ay supf x) f (supf x) ) | |
| 523 supf-¬hp {x} x≤z <-mono hp = IsMinSUP→NotHasPrev asupf (λ {w} uw → | |
| 524 (subst (λ k → w <= k) (sym (supf-is-minsup x≤z)) ( MinSUP.x≤sup (minsup x≤z) uw) )) <-mono hp | |
| 525 | |
| 1006 | 526 supf-idem : (mf< : <-monotonic-f A f) {b : Ordinal } → b o≤ z → supf b o≤ z → supf (supf b) ≡ supf b |
| 527 supf-idem mf< {b} b≤z sfb≤x = z52 where | |
| 1005 | 528 z54 : {w : Ordinal} → odef (UnionCF A f mf ay supf (supf b)) w → (w ≡ supf b) ∨ (w << supf b) |
| 529 z54 {w} ⟪ aw , ch-init fc ⟫ = fcy<sup b≤z fc | |
| 530 z54 {w} ⟪ aw , ch-is-sup u u<x is-sup fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k )) | |
| 531 (sym (supf-is-minsup b≤z)) | |
| 1019 | 532 (MinSUP.x≤sup (minsup b≤z) (cfcs mf< u<b b≤z (subst (λ k → k o< b) (sym (ChainP.supu=u is-sup)) u<b) fc )) where |
| 1005 | 533 u<b : u o< b |
| 534 u<b = supf-inject ( subst (λ k → k o< supf b) (sym (ChainP.supu=u is-sup)) u<x ) | |
| 535 z52 : supf (supf b) ≡ supf b | |
| 1006 | 536 z52 = sup=u asupf sfb≤x ⟪ record { x≤sup = z54 } , IsMinSUP→NotHasPrev asupf z54 ( λ ax → proj1 (mf< _ ax)) ⟫ |
| 1005 | 537 |
| 1023 | 538 -- cp : (mf< : <-monotonic-f A f) {b : Ordinal } → b o≤ z → supf b o≤ z → ChainP A f mf ay supf (supf b) |
| 539 -- the condition of cfcs is satisfied, this is obvious | |
| 540 | |
| 1013 | 541 supf-unique : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 1007 | 542 {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 ) |
| 543 → {z : Ordinal } → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z | |
| 1013 | 544 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 | 545 supfa = ZChain.supf za |
| 546 supfb = ZChain.supf zb | |
| 547 ind : (x : Ordinal) → ((w : Ordinal) → w o< x → w o≤ xa → supfa w ≡ supfb w) → x o≤ xa → supfa x ≡ supfb x | |
| 1008 | 548 ind x prev x≤xa = sxa=sxb where |
| 549 ma = ZChain.minsup za x≤xa | |
| 550 mb = ZChain.minsup zb (OrdTrans x≤xa xa≤xb ) | |
| 551 spa = MinSUP.sup ma | |
| 552 spb = MinSUP.sup mb | |
| 553 sax=spa : supfa x ≡ spa | |
| 554 sax=spa = ZChain.supf-is-minsup za x≤xa | |
| 555 sbx=spb : supfb x ≡ spb | |
| 556 sbx=spb = ZChain.supf-is-minsup zb (OrdTrans x≤xa xa≤xb ) | |
| 1007 | 557 sxa=sxb : supfa x ≡ supfb x |
| 558 sxa=sxb with trio< (supfa x) (supfb x) | |
| 559 ... | tri≈ ¬a b ¬c = b | |
| 560 ... | tri< a ¬b ¬c = ⊥-elim ( o≤> ( | |
| 561 begin | |
| 1008 | 562 supfb x ≡⟨ sbx=spb ⟩ |
| 563 spb ≤⟨ MinSUP.minsup mb (MinSUP.asm ma) (λ {z} uzb → MinSUP.x≤sup ma (z53 uzb)) ⟩ | |
| 564 spa ≡⟨ sym sax=spa ⟩ | |
| 565 supfa x ∎ ) a ) where | |
| 566 open o≤-Reasoning O | |
| 567 z53 : {z : Ordinal } → odef (UnionCF A f mf ay (ZChain.supf zb) x) z → odef (UnionCF A f mf ay (ZChain.supf za) x) z | |
| 568 z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ | |
| 1009 | 569 z53 {z} ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u u<x z54 z55 ⟫ where |
| 570 ua=ub : supfa u ≡ supfb u | |
| 571 ua=ub = prev u u<x (ordtrans u<x x≤xa ) | |
| 572 order : {s z1 : Ordinal} → ZChain.supf za s o< ZChain.supf za u → FClosure A f (ZChain.supf za s) z1 → | |
| 573 (z1 ≡ ZChain.supf za u) ∨ (z1 << ZChain.supf za u) | |
| 574 order {s} {z1} lt fc = subst (λ k → z1 <= k) (sym ua=ub) | |
| 575 (ChainP.order is-sup (subst₂ ( λ j k → j o< k ) z56 ua=ub lt ) (subst (λ k → FClosure A f k z1 ) z56 fc )) where | |
| 576 s<x : s o< x | |
| 577 s<x = ordtrans (ZChain.supf-inject za lt) u<x | |
| 578 z56 : supfa s ≡ supfb s | |
| 579 z56 = prev s s<x (ordtrans s<x x≤xa) | |
| 580 z54 : ChainP A f mf ay (ZChain.supf za) u | |
| 581 z54 = record { fcy<sup = λ {w} fc → subst (λ k → w <= k ) (sym ua=ub) (ChainP.fcy<sup is-sup fc ) | |
| 582 ; order = order | |
| 583 ; supu=u = trans ua=ub (ChainP.supu=u is-sup) } | |
| 584 z55 : FClosure A f (ZChain.supf za u) z | |
| 585 z55 = subst (λ k → FClosure A f k z ) (sym ua=ub) fc | |
| 1007 | 586 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> ( |
| 587 begin | |
| 1008 | 588 supfa x ≡⟨ sax=spa ⟩ |
| 1009 | 589 spa ≤⟨ MinSUP.minsup ma (MinSUP.asm mb) (λ uza → MinSUP.x≤sup mb (z53 uza)) ⟩ |
| 1008 | 590 spb ≡⟨ sym sbx=spb ⟩ |
| 1009 | 591 supfb x ∎ ) c ) where |
| 592 open o≤-Reasoning O | |
| 593 z53 : {z : Ordinal } → odef (UnionCF A f mf ay (ZChain.supf za) x) z → odef (UnionCF A f mf ay (ZChain.supf zb) x) z | |
| 594 z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ | |
| 595 z53 {z} ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u u<x z54 z55 ⟫ where | |
| 596 ub=ua : supfb u ≡ supfa u | |
| 597 ub=ua = sym ( prev u u<x (ordtrans u<x x≤xa )) | |
| 598 order : {s z1 : Ordinal} → ZChain.supf zb s o< ZChain.supf zb u → FClosure A f (ZChain.supf zb s) z1 → | |
| 599 (z1 ≡ ZChain.supf zb u) ∨ (z1 << ZChain.supf zb u) | |
| 600 order {s} {z1} lt fc = subst (λ k → z1 <= k) (sym ub=ua) | |
| 601 (ChainP.order is-sup (subst₂ ( λ j k → j o< k ) z56 ub=ua lt ) (subst (λ k → FClosure A f k z1 ) z56 fc )) where | |
| 602 s<x : s o< x | |
| 603 s<x = ordtrans (ZChain.supf-inject zb lt) u<x | |
| 604 z56 : supfb s ≡ supfa s | |
| 605 z56 = sym (prev s s<x (ordtrans s<x x≤xa)) | |
| 606 z54 : ChainP A f mf ay (ZChain.supf zb) u | |
| 607 z54 = record { fcy<sup = λ {w} fc → subst (λ k → w <= k ) (sym ub=ua) (ChainP.fcy<sup is-sup fc ) | |
| 608 ; order = order | |
| 609 ; supu=u = trans ub=ua (ChainP.supu=u is-sup) } | |
| 610 z55 : FClosure A f (ZChain.supf zb u) z | |
| 611 z55 = subst (λ k → FClosure A f k z ) (sym ub=ua) fc | |
| 1007 | 612 |
| 1013 | 613 |
| 969 | 614 UChain⊆ : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 615 {z y : Ordinal} (ay : odef A y) { supf supf1 : Ordinal → Ordinal } | |
| 616 → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) | |
| 966 | 617 → ( { x : Ordinal } → x o< z → supf x ≡ supf1 x) |
| 618 → ( { x : Ordinal } → z o≤ x → supf z o≤ supf1 x) | |
| 619 → UnionCF A f mf ay supf z ⊆' UnionCF A f mf ay supf1 z | |
| 969 | 620 UChain⊆ A f mf {z} {y} ay {supf} {supf1} supf-mono eq<x lex ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 995 | 621 UChain⊆ A f mf {z} {y} ay {supf} {supf1} supf-mono eq<x lex ⟪ az , ch-is-sup u {x} u<x is-sup fc ⟫ = ⟪ az , ch-is-sup u u<x cp1 fc1 ⟫ where |
| 966 | 622 fc1 : FClosure A f (supf1 u) x |
| 995 | 623 fc1 = subst (λ k → FClosure A f k x ) (eq<x u<x) fc |
| 624 supf1-mono : {x y : Ordinal } → x o≤ y → supf1 x o≤ supf1 y | |
| 625 supf1-mono = ? | |
| 966 | 626 uc01 : {s : Ordinal } → supf1 s o< supf1 u → s o< z |
| 627 uc01 {s} s<u with trio< s z | |
| 628 ... | tri< a ¬b ¬c = a | |
| 629 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> uc02 s<u ) where -- (supf-mono (o<→≤ u<x0)) | |
| 630 uc02 : supf1 u o≤ supf1 s | |
| 631 uc02 = begin | |
| 995 | 632 supf1 u ≤⟨ supf1-mono (o<→≤ u<x) ⟩ |
| 966 | 633 supf1 z ≡⟨ cong supf1 (sym b) ⟩ |
| 634 supf1 s ∎ where open o≤-Reasoning O | |
| 635 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> uc03 s<u ) where | |
| 636 uc03 : supf1 u o≤ supf1 s | |
| 637 uc03 = begin | |
| 995 | 638 supf1 u ≡⟨ sym (eq<x u<x) ⟩ |
| 639 supf u ≤⟨ supf-mono (o<→≤ u<x) ⟩ | |
| 966 | 640 supf z ≤⟨ lex (o<→≤ c) ⟩ |
| 641 supf1 s ∎ where open o≤-Reasoning O | |
| 642 cp1 : ChainP A f mf ay supf1 u | |
| 995 | 643 cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x) (ChainP.fcy<sup is-sup fc ) |
| 644 ; order = λ {s} {z} s<u fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x) | |
| 645 (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (eq<x (uc01 s<u) )) (sym (eq<x u<x)) s<u) | |
| 966 | 646 (subst (λ k → FClosure A f k z ) (sym (eq<x (uc01 s<u) )) fc )) |
| 995 | 647 ; supu=u = trans (sym (eq<x u<x)) (ChainP.supu=u is-sup) } |
| 966 | 648 |
| 649 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) | |
| 783 | 650 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
| 869 | 651 supf = ZChain.supf zc |
| 728 | 652 field |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
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653 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → b o< z → (ab : odef A b) |
| 966 | 654 → HasPrev A (UnionCF A f mf ay supf z) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab |
| 869 | 655 → * a < * b → odef ((UnionCF A f mf ay supf z)) b |
| 949 | 656 order : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b) |
| 657 | |
| 568 | 658 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 659 field | |
| 660 maximal : HOD | |
| 804 | 661 as : A ∋ maximal |
| 568 | 662 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative |
| 567 | 663 |
| 966 | 664 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 665 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 666 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ | |
| 667 | |
|
1011
66154af40f89
IChain recursive record avoided
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parents:
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668 record IChain (A : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) (z : Ordinal ) : Set n where |
|
66154af40f89
IChain recursive record avoided
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parents:
1010
diff
changeset
|
669 field |
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66154af40f89
IChain recursive record avoided
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
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670 i : Ordinal |
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66154af40f89
IChain recursive record avoided
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
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671 i<x : i o< x |
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66154af40f89
IChain recursive record avoided
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
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672 fc : FClosure A f (supfz i<x) z |
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66154af40f89
IChain recursive record avoided
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parents:
1010
diff
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|
673 |
| 966 | 674 Zorn-lemma : { A : HOD } |
| 675 → o∅ o< & A | |
| 568 | 676 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 966 | 677 → Maximal A |
| 552 | 678 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 679 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 680 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 681 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 682 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 683 s : HOD |
| 966 | 684 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) |
| 568 | 685 as : A ∋ * ( & s ) |
| 686 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
|
687 as0 : odef A (& s ) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
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|
688 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 689 s<A : & s o< & A |
| 568 | 690 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 691 HasMaximal : HOD |
| 966 | 692 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 537 | 693 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ |
| 966 | 694 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) |
| 532 | 695 Gtx : { x : HOD} → A ∋ x → HOD |
| 966 | 696 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 537 | 697 z08 : ¬ Maximal A → HasMaximal =h= od∅ |
| 804 | 698 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
| 537 | 699 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
| 700 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)) | |
| 701 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 | |
| 702 ¬x<m : ¬ (* x < * m) | |
| 966 | 703 ¬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 | 704 |
| 966 | 705 minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B |
| 879 | 706 minsupP B B⊆A total = m02 where |
| 707 xsup : (sup : Ordinal ) → Set n | |
| 708 xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) | |
| 709 ∀-imply-or : {A : Ordinal → Set n } {B : Set n } | |
| 710 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B | |
| 711 ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM | |
| 712 ∀-imply-or {A} {B} ∀AB | case1 t = case1 t | |
| 713 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where | |
| 714 lemma : ¬ ((x : Ordinal ) → A x) → B | |
| 715 lemma not with ODC.p∨¬p O B | |
| 716 lemma not | case1 b = b | |
| 717 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
| 718 m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B | |
| 719 m00 x = TransFinite0 ind x where | |
| 720 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) | |
| 721 → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B | |
| 722 ind x prev = ∀-imply-or m01 where | |
| 723 m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B | |
| 724 m01 z with trio< z x | |
| 725 ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 726 ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 727 ... | tri< a ¬b ¬c with prev z a | |
| 728 ... | case2 mins = case2 mins | |
| 729 ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) | |
| 950 | 730 ... | case1 mins = case2 record { sup = z ; asm = proj1 mins ; x≤sup = proj2 mins ; minsup = m04 } where |
| 879 | 731 m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 |
| 732 m04 {s} as lt with trio< z s | |
| 733 ... | tri< a ¬b ¬c = o<→≤ a | |
| 734 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 735 ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) | |
| 736 ... | case2 notz = case1 (λ _ → notz ) | |
| 737 m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) | |
| 738 m03 not = ⊥-elim ( not s1 (z09 (SUP.as S)) ⟪ SUP.as S , m05 ⟫ ) where | |
| 739 S : SUP A B | |
| 740 S = supP B B⊆A total | |
| 741 s1 = & (SUP.sup S) | |
| 742 m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) | |
| 950 | 743 m05 {w} bw with SUP.x≤sup S {* w} (subst (λ k → odef B k) (sym &iso) bw ) |
| 879 | 744 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) |
| 745 ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) | |
| 966 | 746 m02 : MinSUP A B |
| 879 | 747 m02 = dont-or (m00 (& A)) m03 |
| 748 | |
| 560 | 749 -- Uncountable ascending chain by axiom of choice |
| 530 | 750 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 751 cf nmx x with ODC.∋-p O A (* x) |
| 752 ... | no _ = o∅ | |
| 753 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 754 ... | yes nogt = -- no larger element, so it is maximal |
| 755 ⊥-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 | 756 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 757 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 758 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 759 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 760 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 761 ... | 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 ⟫ ) | |
| 762 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 763 |
| 764 --- | |
| 765 --- infintie ascention sequence of f | |
| 766 --- | |
| 530 | 767 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 768 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 769 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 770 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 771 |
| 803 | 772 -- |
| 953 | 773 -- maximality of chain |
| 774 -- | |
| 775 -- supf is fixed for z ≡ & A , we can prove order and is-max | |
| 1016 | 776 -- we have supf-unique now, it is provable in the first Tranfinte induction |
| 803 | 777 |
| 992 | 778 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) (mf< : <-monotonic-f A f) |
| 993 | 779 {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 |
| 780 SZ1 f mf mf< {y} ay zc x x≤A = zc1 x x≤A where | |
| 900 | 781 chain-mono1 : {a b c : Ordinal} → a o≤ b |
| 788 | 782 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
| 919 | 783 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b |
| 966 | 784 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → (ab : odef A b) |
| 785 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) f b | |
| 920 | 786 → * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
| 787 is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev | |
| 966 | 788 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 938 | 789 ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) |
| 966 | 790 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ |
| 868 | 791 |
| 869 | 792 supf = ZChain.supf zc |
| 793 | |
| 920 | 794 csupf-fc : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1 |
| 795 csupf-fc {b} {s} {z1} b<z ss<sb (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc05 where | |
| 869 | 796 s<b : s o< b |
| 797 s<b = ZChain.supf-inject zc ss<sb | |
| 920 | 798 s<z : s o< & A |
| 799 s<z = ordtrans s<b b<z | |
| 870 | 800 zc04 : odef (UnionCF A f mf ay supf (& A)) (supf s) |
| 1005 | 801 zc04 = ZChain.csupf zc mf< (ordtrans<-≤ ss<sb (ZChain.supf-mono zc (o<→≤ b<z))) (ZChain.supf<A zc) |
| 869 | 802 zc05 : odef (UnionCF A f mf ay supf b) (supf s) |
| 803 zc05 with zc04 | |
| 966 | 804 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ |
| 995 | 805 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u (ZChain.supf-inject zc zc08) is-sup fc ⟫ where |
| 870 | 806 zc07 : FClosure A f (supf u) (supf s) -- supf u ≤ supf s → supf u o≤ supf s |
| 807 zc07 = fc | |
| 869 | 808 zc06 : supf u ≡ u |
| 809 zc06 = ChainP.supu=u is-sup | |
| 966 | 810 zc08 : supf u o< supf b |
| 894 | 811 zc08 = ordtrans≤-< (ZChain.supf-<= zc (≤to<= ( s≤fc _ f mf fc ))) ss<sb |
| 869 | 812 csupf-fc {b} {s} {z1} b<z ss≤sb (fsuc x fc) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc04 where |
| 813 zc04 : odef (UnionCF A f mf ay supf b) (f x) | |
| 814 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb fc ) | |
| 966 | 815 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ |
| 816 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u<x is-sup (fsuc _ fc) ⟫ | |
| 869 | 817 order : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b) |
| 818 order {b} {s} {z1} b<z ss<sb fc = zc04 where | |
| 891 | 819 zc00 : ( z1 ≡ MinSUP.sup (ZChain.minsup zc (o<→≤ b<z) )) ∨ ( z1 << MinSUP.sup ( ZChain.minsup zc (o<→≤ b<z) ) ) |
| 950 | 820 zc00 = MinSUP.x≤sup (ZChain.minsup zc (o<→≤ b<z) ) (subst (λ k → odef (UnionCF A f mf ay (ZChain.supf zc) b) k ) &iso (csupf-fc b<z ss<sb fc )) |
| 870 | 821 -- supf (supf b) ≡ supf b |
| 869 | 822 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) |
| 823 zc04 with zc00 | |
| 892 | 824 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z)) ) eq ) |
| 825 ... | case2 lt = case2 (subst₂ (λ j k → j < * k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z) )) lt ) | |
| 868 | 826 |
| 993 | 827 zc1 : (x : Ordinal ) → x o≤ & A → ZChain1 A f mf ay zc x |
| 828 zc1 x x≤A with Oprev-p x | |
| 949 | 829 ... | yes op = record { is-max = is-max ; order = order } where |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
830 px = Oprev.oprev op |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
831 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
832 b o< x → (ab : odef A b) → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
833 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
834 * a < * b → odef (UnionCF A f mf ay supf x) b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
835 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
|
836 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 | 837 is-max {a} {b} ua b<x ab P a<b | case2 is-sup with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) |
| 995 | 838 ... | case2 sb<sx = ⟪ ab , ch-is-sup b b<x m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
839 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
|
840 b<A = z09 ab |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
841 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
842 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
843 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
844 m05 : ZChain.supf zc b ≡ b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
845 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
846 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
847 m08 {z} fcz = ZChain.fcy<sup zc (o<→≤ b<A) fcz |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
848 m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
849 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
850 m09 {s} {z} s<b fcz = order b<A s<b fcz |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
851 m06 : ChainP A f mf ay supf b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
852 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = m05 } |
| 992 | 853 ... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where |
| 990 | 854 m17 : MinSUP A (UnionCF A f mf ay supf x) -- supf z o< supf ( supf x ) |
| 992 | 855 m17 = ZChain.minsup zc x≤A |
| 990 | 856 m18 : supf x ≡ MinSUP.sup m17 |
| 992 | 857 m18 = ZChain.supf-is-minsup zc x≤A |
| 990 | 858 m10 : f (supf b) ≡ supf b |
| 859 m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where | |
| 860 m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) | |
| 861 m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where | |
| 1019 | 862 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
| 863 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = | |
| 864 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) | |
| 865 m05 : ZChain.supf zc b ≡ b | |
| 866 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ | |
| 867 m14 : ZChain.supf zc b o< x | |
| 868 m14 = subst (λ k → k o< x ) (sym m05) b<x | |
| 990 | 869 m13 : odef (UnionCF A f mf ay supf x) z |
| 1019 | 870 m13 = ZChain.cfcs zc mf< b<x x≤A m14 fc |
| 989 | 871 |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
872 ... | no lim = record { is-max = is-max ; order = order } where |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
873 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
874 b o< x → (ab : odef A b) → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
875 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
876 * a < * b → odef (UnionCF A f mf ay supf x) b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
877 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
|
878 is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
879 is-max {a} {b} ua b<x ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (init-uchain A f mf ay ) |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
880 ... | 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 | 881 ... | case2 y<b with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) |
| 995 | 882 ... | case2 sb<sx = ⟪ ab , ch-is-sup b b<x m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
883 m09 : b o< & A |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
884 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
|
885 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
886 m07 {z} fc = ZChain.fcy<sup zc (o<→≤ m09) fc |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
887 m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 888 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
|
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
889 m08 {s} {z1} s<b fc = order m09 s<b fc |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
890 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
891 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
892 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
893 ; x=fy = HasPrev.x=fy nhp } ) |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
894 m05 : ZChain.supf zc b ≡ b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
895 m05 = ZChain.sup=u zc ab (o<→≤ m09) ⟪ record { x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } , m04 ⟫ -- ZChain on x |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
896 m06 : ChainP A f mf ay supf b |
|
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
897 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = m05 } |
| 992 | 898 ... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where |
| 990 | 899 m17 : MinSUP A (UnionCF A f mf ay supf x) -- supf z o< supf ( supf x ) |
| 992 | 900 m17 = ZChain.minsup zc x≤A |
| 990 | 901 m18 : supf x ≡ MinSUP.sup m17 |
| 992 | 902 m18 = ZChain.supf-is-minsup zc x≤A |
| 990 | 903 m10 : f (supf b) ≡ supf b |
| 904 m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where | |
| 905 m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) | |
| 906 m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where | |
| 1019 | 907 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
| 908 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = | |
| 909 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) | |
| 910 m05 : ZChain.supf zc b ≡ b | |
| 911 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ | |
| 912 m14 : ZChain.supf zc b o< x | |
| 913 m14 = subst (λ k → k o< x ) (sym m05) b<x | |
| 990 | 914 m13 : odef (UnionCF A f mf ay supf x) z |
| 1019 | 915 m13 = ZChain.cfcs zc mf< b<x x≤A m14 fc |
| 727 | 916 |
| 757 | 917 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 966 | 918 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = |
| 757 | 919 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 ))) } |
| 920 | |
| 966 | 921 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
| 757 | 922 → IsTotalOrderSet (uchain f mf ay) |
| 966 | 923 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 | 924 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
| 925 uz01 = fcn-cmp y f mf ca cb | |
| 926 | |
| 966 | 927 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
| 928 | 928 → MinSUP A (uchain f mf ay) |
| 966 | 929 ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) |
| 757 | 930 |
|
965
1c1c6a6ed4fa
removing ch-init is no good because of initialization
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
964
diff
changeset
|
931 |
| 793 | 932 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 950 | 933 SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x≤sup = λ lt → SUP.x≤sup sup (B⊆C lt) } |
| 711 | 934 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
935 record xSUP (B : HOD) (f : Ordinal → Ordinal ) (x : Ordinal) : Set n where |
| 833 | 936 field |
| 937 ax : odef A x | |
| 960 | 938 is-sup : IsMinSUP A B f ax |
| 833 | 939 |
| 1007 | 940 zc43 : (x sp1 : Ordinal ) → ( x o< sp1 ) ∨ ( sp1 o≤ x ) |
| 941 zc43 x sp1 with trio< x sp1 | |
| 942 ... | tri< a ¬b ¬c = case1 a | |
| 943 ... | tri≈ ¬a b ¬c = case2 (o≤-refl0 (sym b)) | |
| 944 ... | tri> ¬a ¬b c = case2 (o<→≤ c) | |
| 945 | |
| 560 | 946 -- |
| 547 | 947 -- create all ZChains under o< x |
| 560 | 948 -- |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
949 |
| 966 | 950 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 951 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 707 | 952 ind f mf {y} ay x prev with Oprev-p x |
| 954 | 953 ... | yes op = zc41 where |
| 682 | 954 -- |
| 955 -- we have previous ordinal to use induction | |
| 956 -- | |
| 957 px = Oprev.oprev op | |
| 703 | 958 zc : ZChain A f mf ay (Oprev.oprev op) |
| 966 | 959 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 682 | 960 px<x : px o< x |
| 961 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
|
962 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
|
963 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
|
964 |
| 709 | 965 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 966 | 966 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt |
| 697 | 967 |
| 754 | 968 supf0 = ZChain.supf zc |
| 869 | 969 pchain : HOD |
| 970 pchain = UnionCF A f mf ay supf0 px | |
| 835 | 971 |
| 966 | 972 supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b |
| 857 | 973 supf-mono = ZChain.supf-mono zc |
| 844 | 974 |
| 861 | 975 zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) |
| 966 | 976 zc04 {b} b≤x with trio< b px |
| 861 | 977 ... | tri< a ¬b ¬c = case1 (o<→≤ a) |
| 978 ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) | |
| 979 ... | tri> ¬a ¬b px<b with osuc-≡< b≤x | |
| 980 ... | case1 eq = case2 eq | |
| 966 | 981 ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) |
| 840 | 982 |
| 954 | 983 -- |
| 984 -- find the next value of supf | |
| 985 -- | |
| 986 | |
| 987 pchainpx : HOD | |
| 966 | 988 pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain A f mf ay supf0 px z ) |
| 954 | 989 ∨ FClosure A f (supf0 px) z } ; odmax = & A ; <odmax = zc00 } where |
| 990 zc00 : {z : Ordinal } → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z → z o< & A | |
| 966 | 991 zc00 {z} (case1 lt) = z07 lt |
| 954 | 992 zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf fc ) |
| 993 | |
| 994 zc02 : { a b : Ordinal } → odef A a ∧ UChain A f mf ay supf0 px a → FClosure A f (supf0 px) b → a <= b | |
| 995 zc02 {a} {b} ca fb = zc05 fb where | |
| 996 zc06 : MinSUP.sup (ZChain.minsup zc o≤-refl) ≡ supf0 px | |
| 997 zc06 = trans (sym ( ZChain.supf-is-minsup zc o≤-refl )) refl | |
| 998 zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a <= b | |
| 999 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb )) | |
| 1000 ... | case1 eq = subst (λ k → a <= k ) (subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) eq)) (zc05 fb) | |
| 966 | 1001 ... | case2 lt = <=-trans (zc05 fb) (case2 lt) |
| 1002 zc05 (init b1 refl) with MinSUP.x≤sup (ZChain.minsup zc o≤-refl) | |
| 954 | 1003 (subst (λ k → odef A k ∧ UChain A f mf ay supf0 px k) (sym &iso) ca ) |
| 1004 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso zc06 eq ) | |
| 966 | 1005 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso (cong (*) zc06) lt ) |
| 1006 | |
| 954 | 1007 ptotal : IsTotalOrderSet pchainpx |
| 966 | 1008 ptotal (case1 a) (case1 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso |
| 1009 (chain-total A f mf ay supf0 (proj2 a) (proj2 b)) | |
| 954 | 1010 ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b |
| 1011 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where | |
| 1012 eq1 : a0 ≡ b0 | |
| 1013 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 1014 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where | |
| 1015 lt1 : a0 < b0 | |
| 1016 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 1017 ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b | |
| 1018 ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where | |
| 1019 eq1 : a0 ≡ b0 | |
| 1020 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 1021 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where | |
| 1022 lt1 : a0 < b0 | |
| 1023 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 1024 ptotal (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp (supf0 px) f mf a b) | |
| 966 | 1025 |
| 954 | 1026 pcha : pchainpx ⊆' A |
| 1027 pcha (case1 lt) = proj1 lt | |
| 1028 pcha (case2 fc) = A∋fc _ f mf fc | |
| 966 | 1029 |
| 1030 sup1 : MinSUP A pchainpx | |
| 954 | 1031 sup1 = minsupP pchainpx pcha ptotal |
| 1032 sp1 = MinSUP.sup sup1 | |
| 1033 | |
| 972 | 1034 sfpx<=sp1 : supf0 px <= sp1 |
| 1035 sfpx<=sp1 = MinSUP.x≤sup sup1 (case2 (init (ZChain.asupf zc {px}) refl )) | |
| 1036 | |
| 1037 sfpx≤sp1 : supf0 px o≤ sp1 | |
| 1038 sfpx≤sp1 = subst ( λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc o≤-refl )) | |
| 1039 ( MinSUP.minsup (ZChain.minsup zc o≤-refl) (MinSUP.asm sup1) | |
| 1040 (λ {x} ux → MinSUP.x≤sup sup1 (case1 ux)) ) | |
| 967 | 1041 |
| 954 | 1042 -- |
| 1043 -- supf0 px o≤ sp1 | |
| 966 | 1044 -- |
| 1045 | |
| 1046 zc41 : ZChain A f mf ay x | |
| 1007 | 1047 zc41 with zc43 x sp1 |
| 968 | 1048 zc41 | (case2 sp≤x ) = record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf1-mono ; supf-< = ? |
| 1001 | 1049 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; cfcs = ? } where |
| 883 | 1050 |
| 871 | 1051 supf1 : Ordinal → Ordinal |
| 966 | 1052 supf1 z with trio< z px |
| 871 | 1053 ... | tri< a ¬b ¬c = supf0 z |
| 966 | 1054 ... | tri≈ ¬a b ¬c = supf0 z |
| 901 | 1055 ... | tri> ¬a ¬b c = sp1 |
| 871 | 1056 |
| 886 | 1057 sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z |
| 901 | 1058 sf1=sf0 {z} z≤px with trio< z px |
| 874 | 1059 ... | tri< a ¬b ¬c = refl |
| 901 | 1060 ... | tri≈ ¬a b ¬c = refl |
| 1061 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c ) | |
| 883 | 1062 |
| 901 | 1063 sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1 |
| 1064 sf1=sp1 {z} px<z with trio< z px | |
| 1065 ... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a ) | |
| 1066 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z ) | |
| 1067 ... | tri> ¬a ¬b c = refl | |
| 873 | 1068 |
| 968 | 1069 sf=eq : { z : Ordinal } → z o< x → supf0 z ≡ supf1 z |
| 1070 sf=eq {z} z<x = sym (sf1=sf0 (subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x )) | |
| 1071 | |
| 903 | 1072 asupf1 : {z : Ordinal } → odef A (supf1 z) |
| 1073 asupf1 {z} with trio< z px | |
| 966 | 1074 ... | tri< a ¬b ¬c = ZChain.asupf zc |
| 1075 ... | tri≈ ¬a b ¬c = ZChain.asupf zc | |
| 903 | 1076 ... | tri> ¬a ¬b c = MinSUP.asm sup1 |
| 1077 | |
| 966 | 1078 supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b |
| 1079 supf1-mono {a} {b} a≤b with trio< b px | |
| 901 | 1080 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b ) |
| 1081 ... | 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 ) | |
| 1082 supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px | |
| 1083 ... | tri< a<px ¬b ¬c = zc19 where | |
| 1084 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
| 1085 zc21 = ZChain.minsup zc (o<→≤ a<px) | |
| 1086 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) | |
| 950 | 1087 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) ) |
| 966 | 1088 zc19 : supf0 a o≤ sp1 |
| 901 | 1089 zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) |
| 1090 ... | tri≈ ¬a b ¬c = zc18 where | |
| 1091 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
| 1092 zc21 = ZChain.minsup zc (o≤-refl0 b) | |
| 1093 zc20 : MinSUP.sup zc21 ≡ supf0 a | |
| 966 | 1094 zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b)) |
| 901 | 1095 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) |
| 950 | 1096 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) ) |
| 966 | 1097 zc18 : supf0 a o≤ sp1 |
| 901 | 1098 zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) |
| 1099 ... | tri> ¬a ¬b c = o≤-refl | |
| 885 | 1100 |
| 968 | 1101 sf≤ : { z : Ordinal } → x o≤ z → supf0 x o≤ supf1 z |
| 1102 sf≤ {z} x≤z with trio< z px | |
| 1103 ... | tri< a ¬b ¬c = ⊥-elim ( o<> (osucc a) (subst (λ k → k o≤ z) (sym (Oprev.oprev=x op)) x≤z ) ) | |
| 1104 ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) px<x )) | |
| 1105 ... | tri> ¬a ¬b c = subst₂ (λ j k → j o≤ k ) (trans (sf1=sf0 o≤-refl ) (sym (ZChain.supfmax zc px<x))) (sf1=sp1 c) | |
| 1106 (supf1-mono (o<→≤ c )) | |
| 978 | 1107 -- px o<z → supf x ≡ supf0 px ≡ supf1 px o≤ supf1 z |
| 903 | 1108 |
| 966 | 1109 fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z |
| 903 | 1110 fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc |
| 966 | 1111 fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z |
| 903 | 1112 fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc |
| 967 | 1113 |
|
999
3ffbdd53d1ea
fcs<sup requires <-monotonicity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
998
diff
changeset
|
1114 -- 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
|
1115 -- |
| 1001 | 1116 cfcs : (mf< : <-monotonic-f A f) {a b w : Ordinal } |
| 1018 | 1117 → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f mf ay supf1 b) w |
| 1118 cfcs mf< {a} {b} {w} a<b b≤x sa<b fc with zc43 (supf0 a) px | |
| 1012 | 1119 ... | case2 px≤sa = z50 where |
| 1023 | 1120 a<x : a o< x |
| 1121 a<x = ordtrans<-≤ a<b b≤x | |
| 1012 | 1122 a≤px : a o≤ px |
| 1123 a≤px = subst (λ k → a o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ a<b b≤x) | |
| 1124 -- supf0 a ≡ px we cannot use previous cfcs, it is in the chain because | |
| 1125 -- supf0 a ≡ supf0 (supf0 a) ≡ supf0 px o< x | |
| 1018 | 1126 z50 : odef (UnionCF A f mf ay supf1 b) w |
| 1012 | 1127 z50 with osuc-≡< px≤sa |
| 1020 | 1128 ... | case1 px=sa = ⟪ A∋fc {A} _ f mf fc , ch-is-sup (supf0 px) z51 ? (subst (λ k → FClosure A f k w) z52 fc) ⟫ where |
| 1023 | 1129 sa≤px : supf0 a o≤ px |
| 1130 sa≤px = subst₂ (λ j k → j o< k) px=sa (sym (Oprev.oprev=x op)) px<x | |
| 1020 | 1131 z51 : supf0 px o< b |
| 1023 | 1132 z51 = subst (λ k → k o< b ) (sym ( begin supf0 px ≡⟨ cong supf0 px=sa ⟩ |
| 1133 supf0 (supf0 a ) ≡⟨ ZChain.supf-idem zc mf< a≤px sa≤px ⟩ | |
| 1134 supf0 a ≡⟨ sym (sf1=sf0 a≤px) ⟩ | |
| 1135 supf1 a ∎ )) sa<b where open ≡-Reasoning | |
| 1020 | 1136 z52 : supf1 a ≡ supf1 (supf0 px) |
| 1023 | 1137 z52 = begin supf1 a ≡⟨ sf1=sf0 a≤px ⟩ |
| 1138 supf0 a ≡⟨ sym (ZChain.supf-idem zc mf< a≤px sa≤px ) ⟩ | |
| 1139 supf0 (supf0 a) ≡⟨ sym (sf1=sf0 sa≤px) ⟩ | |
| 1140 supf1 (supf0 a) ≡⟨ cong supf1 (sym (ZChain.supf-idem zc mf< a≤px sa≤px )) ⟩ | |
| 1141 supf1 (supf0 (supf0 a)) ≡⟨ cong (λ k → supf1 (supf0 k)) (sym px=sa) ⟩ | |
| 1142 supf1 (supf0 px) ∎ where open ≡-Reasoning | |
| 1020 | 1143 ... | 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 |
| 1144 z53 : supf1 a o< x | |
| 1145 z53 = ordtrans<-≤ sa<b b≤x | |
| 1012 | 1146 ... | case1 sa<px with trio< a px |
| 996 | 1147 ... | tri< a<px ¬b ¬c = z50 where |
| 1018 | 1148 z50 : odef (UnionCF A f mf ay supf1 b) w |
| 997 | 1149 z50 with osuc-≡< b≤x |
| 1018 | 1150 ... | case2 lt with ZChain.cfcs zc mf< a<b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) lt) ? fc |
| 996 | 1151 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 1018 | 1152 ... | ⟪ az , ch-is-sup u u<b is-sup fc ⟫ = ⟪ az , ch-is-sup u u<b cp1 (fcpu fc u≤px ) ⟫ where -- u o< px → u o< b ? |
| 997 | 1153 u≤px : u o≤ px |
| 1018 | 1154 u≤px = subst (λ k → u o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ u<b b≤x ) |
| 997 | 1155 u<x : u o< x |
| 1018 | 1156 u<x = ordtrans<-≤ u<b b≤x |
| 997 | 1157 cp1 : ChainP A f mf ay supf1 u |
| 1158 cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sf=eq u<x) (ChainP.fcy<sup is-sup fc ) | |
| 1159 ; order = λ {s} {z} s<u fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sf=eq u<x) | |
| 1160 (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (sf=eq (ordtrans (supf-inject0 supf1-mono s<u) u<x) )) | |
| 1161 (sym (sf=eq u<x)) s<u) | |
| 1162 (subst (λ k → FClosure A f k z ) (sym (sf=eq (ordtrans (supf-inject0 supf1-mono s<u) u<x) )) fc )) | |
| 1163 ; supu=u = trans (sym (sf=eq u<x)) (ChainP.supu=u is-sup) } | |
| 1012 | 1164 z50 | case1 eq with ZChain.cfcs zc mf< a<px o≤-refl sa<px fc |
| 997 | 1165 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
| 1018 | 1166 ... | ⟪ az , ch-is-sup u u<px is-sup fc ⟫ = ⟪ az , ch-is-sup u u<b cp1 (fcpu fc (o<→≤ u<px)) ⟫ where -- u o< px → u o< b ? |
| 997 | 1167 u<b : u o< b |
| 1168 u<b = subst (λ k → u o< k ) (trans (Oprev.oprev=x op) (sym eq) ) (ordtrans u<px <-osuc ) | |
| 1169 u<x : u o< x | |
| 1170 u<x = subst (λ k → u o< k ) (Oprev.oprev=x op) ( ordtrans u<px <-osuc ) | |
| 1171 cp1 : ChainP A f mf ay supf1 u | |
| 1172 cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sf=eq u<x) (ChainP.fcy<sup is-sup fc ) | |
| 1173 ; order = λ {s} {z} s<u fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sf=eq u<x) | |
| 1174 (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (sf=eq (ordtrans (supf-inject0 supf1-mono s<u) u<x) )) | |
| 1175 (sym (sf=eq u<x)) s<u) | |
| 1176 (subst (λ k → FClosure A f k z ) (sym (sf=eq (ordtrans (supf-inject0 supf1-mono s<u) u<x) )) fc )) | |
| 1177 ; supu=u = trans (sym (sf=eq u<x)) (ChainP.supu=u is-sup) } | |
| 1000 | 1178 ... | tri≈ ¬a a=px ¬c = csupf1 where |
| 1179 -- a ≡ px , b ≡ x, sp o≤ x | |
| 995 | 1180 px<b : px o< b |
| 1181 px<b = subst₂ (λ j k → j o< k) a=px refl a<b | |
| 1182 b=x : b ≡ x | |
| 1183 b=x with trio< b x | |
| 996 | 1184 ... | 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 | 1185 ... | tri≈ ¬a b ¬c = b |
| 996 | 1186 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> b≤x c ) -- x o< b |
| 997 | 1187 z51 : FClosure A f (supf1 px) w |
| 1188 z51 = subst (λ k → FClosure A f k w) (sym (trans (cong supf1 (sym a=px)) (sf1=sf0 (o≤-refl0 a=px) ))) fc | |
| 1001 | 1189 z53 : odef A w |
| 1190 z53 = A∋fc {A} _ f mf fc | |
| 1018 | 1191 csupf1 : odef (UnionCF A f mf ay supf1 b) w |
| 1000 | 1192 csupf1 with trio< (supf0 px) x |
| 1018 | 1193 ... | tri< sfpx<x ¬b ¬c = ⟪ z53 , ch-is-sup spx (subst (λ k → spx o< k) (sym b=x) sfpx<x) cp1 fc1 ⟫ where |
| 1003 | 1194 spx = supf0 px |
| 1004 | 1195 spx≤px : supf0 px o≤ px |
| 1196 spx≤px = zc-b<x _ sfpx<x | |
| 1003 | 1197 z52 : supf1 (supf0 px) ≡ supf0 px |
| 1012 | 1198 z52 = trans (sf1=sf0 (zc-b<x _ sfpx<x)) ( ZChain.supf-idem zc mf< o≤-refl (zc-b<x _ sfpx<x ) ) |
| 1004 | 1199 fc1 : FClosure A f (supf1 spx) w |
| 1200 fc1 = subst (λ k → FClosure A f k w ) (trans (cong supf0 a=px) (sym z52) ) fc | |
| 1003 | 1201 order : {s z1 : Ordinal} → supf1 s o< supf1 spx → FClosure A f (supf1 s) z1 → (z1 ≡ supf1 spx) ∨ (z1 << supf1 spx) |
| 1202 order {s} {z1} ss<spx fcs = subst (λ k → (z1 ≡ k) ∨ (z1 << k )) | |
| 1004 | 1203 (trans (sym (ZChain.supf-is-minsup zc spx≤px )) (sym (sf1=sf0 spx≤px) ) ) |
| 1018 | 1204 (MinSUP.x≤sup (ZChain.minsup zc spx≤px) (ZChain.cfcs zc mf< (supf-inject0 supf1-mono ss<spx) |
| 1019 | 1205 spx≤px ss0<spx (fcup fcs (ordtrans (supf-inject0 supf1-mono ss<spx) spx≤px ) ))) where |
| 1020 | 1206 ss0<spx : supf0 s o< spx |
| 1207 ss0<spx = osucprev ( begin | |
| 1012 | 1208 osuc (supf0 s) ≡⟨ cong osuc (sym (sf1=sf0 ( begin |
| 1209 s <⟨ supf-inject0 supf1-mono ss<spx ⟩ | |
| 1210 supf0 px ≤⟨ spx≤px ⟩ | |
| 1211 px ∎ ) )) ⟩ | |
| 1212 osuc (supf1 s) ≤⟨ osucc ss<spx ⟩ | |
| 1213 supf1 spx ≡⟨ sf1=sf0 spx≤px ⟩ | |
| 1214 supf0 spx ≤⟨ ZChain.supf-mono zc spx≤px ⟩ | |
| 1020 | 1215 supf0 px ∎ ) where open o≤-Reasoning O |
| 1003 | 1216 cp1 : ChainP A f mf ay supf1 spx |
| 1004 | 1217 cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ (z << k )) (sym (sf1=sf0 spx≤px )) |
| 1218 ( ZChain.fcy<sup zc spx≤px fc ) | |
| 1002 | 1219 ; order = order |
| 1004 | 1220 ; supu=u = z52 } |
| 1000 | 1221 ... | tri≈ ¬a spx=x ¬c = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf0 px) (ZChain.asupf zc)))) where |
| 1222 -- supf px ≡ x then the chain is stopped, which cannot happen when <-monotonic case | |
| 1223 m12 : supf0 px ≡ sp1 | |
| 1224 m12 with osuc-≡< sfpx≤sp1 | |
| 1225 ... | case1 eq = eq | |
| 1226 ... | case2 lt = ⊥-elim ( o≤> sp≤x (subst (λ k → k o< sp1) spx=x lt )) -- supf0 px o< sp1 , x o< sp1 | |
| 1227 m10 : f (supf0 px) ≡ supf0 px | |
| 1228 m10 = fc-stop A f mf (ZChain.asupf zc) m11 m12 where | |
| 1229 m11 : {z : Ordinal} → FClosure A f (supf0 px) z → (z ≡ sp1) ∨ (z << sp1) | |
| 1230 m11 {z} fc = MinSUP.x≤sup sup1 (case2 fc) | |
| 1231 ... | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (ordtrans<-≤ c (OrdTrans sfpx≤sp1 sp≤x))) -- x o< supf0 px o≤ sp1 ≤ x | |
| 996 | 1232 ... | 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 | 1233 |
| 903 | 1234 zc11 : {z : Ordinal} → odef (UnionCF A f mf ay supf1 x) z → odef pchainpx z |
| 966 | 1235 zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
| 995 | 1236 zc11 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc21 fc where |
| 953 | 1237 u≤px : u o≤ px |
| 1238 u≤px = zc-b<x _ u<x | |
| 903 | 1239 zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1 |
| 1240 zc21 {z1} (fsuc z2 fc ) with zc21 fc | |
| 966 | 1241 ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ |
| 1242 ... | case1 ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ | |
| 1243 ... | case2 fc = case2 (fsuc _ fc) | |
| 953 | 1244 zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) | inspect supf1 u |
| 995 | 1245 ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u u<px record {fcy<sup = zc13 ; order = zc17 |
| 953 | 1246 ; supu=u = trans (sym (sf1=sf0 (o<→≤ u<px))) (ChainP.supu=u is-sup) } (init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where |
| 1247 u<px : u o< px | |
| 1248 u<px = ZChain.supf-inject zc a | |
| 1249 asp0 : odef A (supf0 u) | |
| 966 | 1250 asp0 = ZChain.asupf zc |
| 903 | 1251 zc17 : {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 u → |
| 1252 FClosure A f (supf0 s) z1 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u) | |
| 966 | 1253 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) ((sf1=sf0 u≤px)) ( ChainP.order is-sup |
| 953 | 1254 (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 zc18)) (sym (sf1=sf0 u≤px)) ss<spx) (fcpu fc zc18) ) where |
| 903 | 1255 zc18 : s o≤ px |
| 953 | 1256 zc18 = ordtrans (ZChain.supf-inject zc ss<spx) u≤px |
| 903 | 1257 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf0 u) ∨ ( z << supf0 u ) |
| 953 | 1258 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sf1=sf0 (o<→≤ u<px)) ( ChainP.fcy<sup is-sup fc ) |
| 1259 ... | tri≈ ¬a b ¬c | _ = case2 (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) (sym (trans (sf1=sf0 u≤px) b ))) | |
| 1260 ... | 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 | 1261 |
| 885 | 1262 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
| 1263 field | |
| 907 | 1264 tsup : MinSUP A (UnionCF A f mf ay supf1 z) |
| 966 | 1265 tsup=sup : supf1 z ≡ MinSUP.sup tsup |
| 885 | 1266 |
| 1267 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x | |
| 966 | 1268 sup {z} z≤x with trio< z px |
| 1269 ... | tri< a ¬b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m | |
| 950 | 1270 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o<→≤ a) ) (ZChain.supf-is-minsup zc (o<→≤ a)) } where |
| 885 | 1271 m = ZChain.minsup zc (o<→≤ a) |
| 907 | 1272 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
| 950 | 1273 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1274 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1275 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 997 | 1276 ms01 {sup2} us P = MinSUP.minsup m us ? |
| 966 | 1277 ... | tri≈ ¬a b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m |
| 950 | 1278 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o≤-refl0 b) ) (ZChain.supf-is-minsup zc (o≤-refl0 b)) } where |
| 885 | 1279 m = ZChain.minsup zc (o≤-refl0 b) |
| 907 | 1280 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
| 950 | 1281 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1282 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1283 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 997 | 1284 ms01 {sup2} us P = MinSUP.minsup m us ? |
| 901 | 1285 ... | tri> ¬a ¬b px<z = record { tsup = record { sup = sp1 ; asm = MinSUP.asm sup1 |
| 950 | 1286 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = sf1=sp1 px<z } where |
| 907 | 1287 m = sup1 |
| 1288 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) | |
| 950 | 1289 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1290 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1291 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 997 | 1292 ms01 {sup2} us P = MinSUP.minsup m us ? |
| 885 | 1293 |
| 877 | 1294 |
| 968 | 1295 zc41 | (case1 x<sp ) = record { supf = supf0 ; sup=u = ? ; asupf = ? ; supf-mono = ? ; supf-< = ? |
| 1005 | 1296 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; cfcs = cfcs } where |
| 883 | 1297 |
| 901 | 1298 -- supf0 px not is included by the chain |
| 1299 -- supf1 x ≡ supf0 px because of supfmax | |
| 883 | 1300 |
| 1005 | 1301 cfcs : (mf< : <-monotonic-f A f) {a b w : Ordinal } |
| 1018 | 1302 → a o< b → b o≤ x → supf0 a o< b → FClosure A f (supf0 a) w → odef (UnionCF A f mf ay supf0 b) w |
| 1303 cfcs mf< {a} {b} {w} a<b b≤x sa<b fc with trio< b px | |
| 1020 | 1304 ... | tri< a ¬b ¬c = ZChain.cfcs zc mf< a<b (o<→≤ a) sa<b fc |
| 1305 ... | tri≈ ¬a refl ¬c = ZChain.cfcs zc mf< a<b o≤-refl sa<b fc | |
| 1022 | 1306 ... | tri> ¬a ¬b px<b = cfcs1 where |
| 1020 | 1307 x=b : x ≡ b |
| 1022 | 1308 x=b with trio< x b |
| 1309 ... | tri< a ¬b ¬c = ⊥-elim ( o≤> b≤x a ) | |
| 1310 ... | tri≈ ¬a b ¬c = b | |
| 1311 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ px<b , zc-b<x _ c ⟫ ) -- px o< b o< x | |
| 1020 | 1312 -- a o< x, supf a o< x |
| 1313 -- a o< px , supf a o< px → odef U w | |
| 1314 -- a ≡ px -- supf0 px o< x → odef U w | |
| 1315 -- supf a ≡ px -- a o< px → odef U w | |
| 1316 -- a ≡ px → supf px ≡ px → odef U w | |
| 1317 | |
| 1022 | 1318 cfcs0 : a ≡ px → odef (UnionCF A f mf ay supf0 b) w |
| 1023 | 1319 cfcs0 a=px = ⟪ A∋fc {A} _ f mf fc , ch-is-sup (supf0 px) spx<b ? fc1 ⟫ where |
| 1022 | 1320 spx<b : supf0 px o< b |
| 1023 | 1321 spx<b = subst (λ k → supf0 k o< b) a=px sa<b |
| 1322 cs01 : supf0 a ≡ supf0 (supf0 px) | |
| 1323 cs01 = trans (cong supf0 a=px) ( sym ( ZChain.supf-idem zc mf< o≤-refl | |
| 1324 (subst (λ k → supf0 px o< k ) (sym (Oprev.oprev=x op)) (ordtrans<-≤ spx<b b≤x)))) | |
| 1325 fc1 : FClosure A f (supf0 (supf0 px)) w | |
| 1326 fc1 = subst (λ k → FClosure A f k w) cs01 fc | |
| 1327 | |
| 1022 | 1328 |
| 1020 | 1329 cfcs1 : odef (UnionCF A f mf ay supf0 b) w |
| 1330 cfcs1 with trio< a px | |
| 1022 | 1331 ... | tri< a<px ¬b ¬c = cfcs2 where |
| 1332 sa<x : supf0 a o< x | |
| 1333 sa<x = ordtrans<-≤ sa<b b≤x | |
| 1334 cfcs2 : odef (UnionCF A f mf ay supf0 b) w | |
| 1335 cfcs2 with trio< (supf0 a) px | |
| 1336 ... | tri< sa<x ¬b ¬c = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) (o<→≤ px<b) | |
| 1337 ( ZChain.cfcs zc mf< a<px o≤-refl sa<x fc ) | |
| 1338 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , (zc-b<x _ sa<x) ⟫ ) | |
| 1339 ... | tri≈ ¬a sa=px ¬c with trio< a px | |
| 1023 | 1340 ... | tri< a<px ¬b ¬c = ⟪ A∋fc {A} _ f mf fc , ch-is-sup (supf0 a) sa<b ? fc1 ⟫ where |
| 1341 cs01 : supf0 a ≡ supf0 (supf0 a) | |
| 1342 cs01 = sym ( ZChain.supf-idem zc mf< (zc-b<x _ (ordtrans<-≤ a<b b≤x)) (zc-b<x _ (ordtrans<-≤ sa<b b≤x))) | |
| 1343 fc1 : FClosure A f (supf0 (supf0 a)) w | |
| 1344 fc1 = subst (λ k → FClosure A f k w) cs01 fc | |
| 1022 | 1345 ... | tri≈ ¬a a=px ¬c = cfcs0 a=px |
| 1346 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , (zc-b<x _ (ordtrans<-≤ a<b b≤x) ) ⟫ ) | |
| 1347 ... | tri≈ ¬a a=px ¬c = cfcs0 a=px | |
| 1348 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> (zc-b<x _ (ordtrans<-≤ a<b b≤x)) c ) | |
| 969 | 1349 |
| 874 | 1350 zc17 : {z : Ordinal } → supf0 z o≤ supf0 px |
| 995 | 1351 zc17 {z} with trio< z px |
| 1352 ... | tri< a ¬b ¬c = ZChain.supf-mono zc (o<→≤ a) | |
| 1353 ... | tri≈ ¬a b ¬c = o≤-refl0 (cong supf0 b) | |
| 1354 ... | tri> ¬a ¬b px<z = o≤-refl0 zc177 where | |
| 1355 zc177 : supf0 z ≡ supf0 px | |
| 1356 zc177 = ZChain.supfmax zc px<z -- px o< z, px o< supf0 px | |
| 874 | 1357 |
| 857 | 1358 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
| 1359 field | |
| 1005 | 1360 tsup : MinSUP A (UnionCF A f mf ay supf0 z) |
| 1361 tsup=sup : supf0 z ≡ MinSUP.sup tsup | |
| 891 | 1362 |
| 857 | 1363 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x |
| 966 | 1364 sup {z} z≤x with trio< z px |
| 891 | 1365 ... | tri< a ¬b ¬c = ? -- jrecord { tsup = ZChain.minsup zc (o<→≤ a) ; tsup=sup = ZChain.supf-is-minsup zc (o<→≤ a) } |
| 1366 ... | tri≈ ¬a b ¬c = ? -- record { tsup = ZChain.minsup zc (o≤-refl0 b) ; tsup=sup = ZChain.supf-is-minsup zc (o≤-refl0 b) } | |
| 865 | 1367 ... | tri> ¬a ¬b px<z = zc35 where |
| 840 | 1368 zc30 : z ≡ x |
| 1369 zc30 with osuc-≡< z≤x | |
| 1370 ... | case1 eq = eq | |
| 1371 ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 966 | 1372 zc32 = ZChain.sup zc o≤-refl |
| 865 | 1373 zc34 : ¬ (supf0 px ≡ px) → {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup zc32) ∨ (w < SUP.sup zc32) |
| 1005 | 1374 zc34 ne {w} lt = ? |
| 857 | 1375 zc33 : supf0 z ≡ & (SUP.sup zc32) |
| 891 | 1376 zc33 = ? -- trans (sym (supfx (o≤-refl0 (sym zc30)))) ( ZChain.supf-is-minsup zc o≤-refl ) |
| 865 | 1377 zc36 : ¬ (supf0 px ≡ px) → STMP z≤x |
| 966 | 1378 zc36 ne = ? -- record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x≤sup = zc34 ne } ; tsup=sup = zc33 } |
| 865 | 1379 zc35 : STMP z≤x |
| 1380 zc35 with trio< (supf0 px) px | |
| 1381 ... | tri< a ¬b ¬c = zc36 ¬b | |
| 1382 ... | tri> ¬a ¬b c = zc36 ¬b | |
| 891 | 1383 ... | tri≈ ¬a b ¬c = record { tsup = ? ; tsup=sup = ? } where |
| 1384 zc37 : MinSUP A (UnionCF A f mf ay supf0 z) | |
| 950 | 1385 zc37 = record { sup = ? ; asm = ? ; x≤sup = ? } |
| 803 | 1386 sup=u : {b : Ordinal} (ab : odef A b) → |
| 960 | 1387 b o≤ x → IsMinSUP A (UnionCF A f mf ay supf0 b) supf0 ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf0 b) f b ) → supf0 b ≡ b |
| 814 | 1388 sup=u {b} ab b≤x is-sup with trio< b px |
| 966 | 1389 ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ |
| 1390 ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (o≤-refl0 b) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ | |
| 882 | 1391 ... | tri> ¬a ¬b px<b = zc31 ? where |
| 815 | 1392 zc30 : x ≡ b |
| 1393 zc30 with osuc-≡< b≤x | |
| 1394 ... | case1 eq = sym (eq) | |
| 1395 ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
| 966 | 1396 zcsup : xSUP (UnionCF A f mf ay supf0 px) supf0 x |
| 859 | 1397 zcsup with zc30 |
| 966 | 1398 ... | refl = record { ax = ab ; is-sup = record { x≤sup = λ {w} lt → |
| 1399 IsMinSUP.x≤sup (proj1 is-sup) ? ; minsup = ? } } | |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1400 zc31 : ( (¬ xSUP (UnionCF A f mf ay supf0 px) supf0 x ) ∨ HasPrev A (UnionCF A f mf ay supf0 px) f x ) → supf0 b ≡ b |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1401 zc31 (case1 ¬sp=x) with zc30 |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1402 ... | refl = ⊥-elim (¬sp=x zcsup ) |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1403 zc31 (case2 hasPrev ) with zc30 |
| 966 | 1404 ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev |
| 1405 ; ay = ? ; x=fy = HasPrev.x=fy hasPrev } ) | |
| 833 | 1406 |
| 1016 | 1407 ... | no lim = ? where |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1408 |
| 1009 | 1409 pzc : {z : Ordinal} → z o< x → ZChain A f mf ay z |
| 1410 pzc {z} z<x = prev z z<x | |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1411 |
| 928 | 1412 ysp = MinSUP.sup (ysup f mf ay) |
| 755 | 1413 |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1414 supfz : {z : Ordinal } → z o< x → Ordinal |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1415 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
|
1416 |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1417 pchainx : HOD |
|
1011
66154af40f89
IChain recursive record avoided
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1010
diff
changeset
|
1418 pchainx = record { od = record { def = λ z → IChain A f supfz z } ; odmax = & A ; <odmax = zc00 } where |
|
66154af40f89
IChain recursive record avoided
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1010
diff
changeset
|
1419 zc00 : {z : Ordinal } → IChain A f supfz z → z o< & A |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1420 zc00 {z} ic = z09 ( A∋fc (supfz (IChain.i<x ic)) f mf (IChain.fc ic) ) |
| 835 | 1421 |
| 1012 | 1422 aic : {z : Ordinal } → IChain A f supfz z → odef A z |
| 1423 aic {z} ic = ? | |
| 1424 | |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1425 zeq : {xa xb z : Ordinal } |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1426 → (xa<x : xa o< x) → (xb<x : xb o< x) → xa o≤ xb → z o≤ xa |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1427 → ZChain.supf (pzc xa<x) z ≡ ZChain.supf (pzc xb<x) z |
| 1013 | 1428 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
|
1429 (pzc xa<x) (pzc xb<x) z≤xa |
| 835 | 1430 |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1431 ptotalx : IsTotalOrderSet pchainx |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1432 ptotalx {a} {b} ia ib = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 835 | 1433 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1434 uz01 with trio< (IChain.i ia) (IChain.i ib) |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1435 ... | tri< a ¬b ¬c = ? |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1436 ... | tri≈ ¬a b ¬c = ? |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1437 ... | tri> ¬a ¬b c = ? |
| 966 | 1438 |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1439 usup : MinSUP A pchainx |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1440 usup = minsupP pchainx (λ lt → ? ) ptotalx |
| 880 | 1441 spu = MinSUP.sup usup |
| 834 | 1442 |
| 794 | 1443 supf1 : Ordinal → Ordinal |
| 835 | 1444 supf1 z with trio< z x |
| 1009 | 1445 ... | tri< a ¬b ¬c = ZChain.supf (pzc (ob<x lim a)) z |
| 836 | 1446 ... | tri≈ ¬a b ¬c = spu |
| 1447 ... | tri> ¬a ¬b c = spu | |
| 755 | 1448 |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1449 pchain : HOD |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1450 pchain = UnionCF A f mf ay supf1 x |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1451 |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1452 -- pchain ⊆ pchainx |
| 704 | 1453 |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1454 ptotal : IsTotalOrderSet pchain |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1455 ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1456 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1457 uz01 = chain-total A f mf ay supf1 ( (proj2 ca)) ( (proj2 cb)) |
| 1009 | 1458 |
| 1459 sf1=sf : {z : Ordinal } → (a : z o< x ) → supf1 z ≡ ZChain.supf (pzc (ob<x lim a)) z | |
| 1460 sf1=sf {z} z<x with trio< z x | |
| 1461 ... | tri< a ¬b ¬c = cong ( λ k → ZChain.supf (pzc (ob<x lim k)) z) o<-irr | |
| 1462 ... | tri≈ ¬a b ¬c = ⊥-elim (¬a z<x) | |
| 1463 ... | tri> ¬a ¬b c = ⊥-elim (¬a z<x) | |
| 1464 | |
| 1465 sf1=spu : {z : Ordinal } → (a : x o≤ z ) → supf1 z ≡ spu | |
| 1466 sf1=spu {z} x≤z with trio< z x | |
| 1467 ... | tri< a ¬b ¬c = ⊥-elim (o≤> x≤z a) | |
| 1468 ... | tri≈ ¬a b ¬c = refl | |
| 1469 ... | tri> ¬a ¬b c = refl | |
| 1470 | |
|
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1471 -- zc11 : {z : Ordinal } → (a : z o< x ) → odef pchain (ZChain.supf (pzc (ob<x lim a)) z) |
|
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1472 -- zc11 {z} z<x = ⟪ ZChain.asupf (pzc (ob<x lim z<x)) , ch-is-sup (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
|
1473 -- ? ? (init ? ?) ⟫ |
| 1009 | 1474 |
| 1007 | 1475 sfpx<=spu : {z : Ordinal } → supf1 z <= spu |
| 1009 | 1476 sfpx<=spu {z} with trio< z x |
| 1477 ... | tri< a ¬b ¬c = MinSUP.x≤sup usup ? -- (init (ZChain.asupf (pzc (ob<x lim a)) ) refl ) | |
| 1478 ... | tri≈ ¬a b ¬c = case1 refl | |
| 1479 ... | tri> ¬a ¬b c = case1 refl | |
| 844 | 1480 |
| 1007 | 1481 sfpx≤spu : {z : Ordinal } → supf1 z o≤ spu |
| 1009 | 1482 sfpx≤spu {z} with trio< z x |
| 1483 ... | tri< a ¬b ¬c = subst ( λ k → k o≤ spu) ? | |
| 1484 ( MinSUP.minsup (ZChain.minsup ? o≤-refl) ? (λ {x} ux → MinSUP.x≤sup ? ?) ) | |
| 1485 ... | tri≈ ¬a b ¬c = ? | |
| 1486 ... | tri> ¬a ¬b c = ? | |
| 1007 | 1487 |
| 1488 supf-mono : {x y : Ordinal } → x o≤ y → supf1 x o≤ supf1 y | |
| 1009 | 1489 supf-mono {x} {y} x≤y with trio< y x |
| 1490 ... | tri< a ¬b ¬c = ? | |
| 1491 ... | tri≈ ¬a b ¬c = ? | |
| 1492 ... | tri> ¬a ¬b c = ? | |
| 797 | 1493 |
| 1016 | 1494 cfcs : (mf< : <-monotonic-f A f) {a b w : Ordinal } |
| 1018 | 1495 → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f mf ay supf1 b) w |
| 1496 cfcs mf< {a} {b} {w} a<b b≤x sa<b fc with osuc-≡< b≤x | |
| 1016 | 1497 ... | case1 b=x with trio< a x |
| 1498 ... | tri< a<x ¬b ¬c = zc40 where | |
| 1499 sa = ZChain.supf (pzc (ob<x lim a<x)) a | |
| 1020 | 1500 m = omax a sa -- x is limit ordinal, so we have sa o< m o< x |
| 1016 | 1501 m<x : m o< x |
| 1502 m<x with trio< a sa | inspect (omax a) sa | |
| 1020 | 1503 ... | tri< a<sa ¬b ¬c | record { eq = eq } = ob<x lim (ordtrans<-≤ sa<b b≤x ) |
| 1016 | 1504 ... | tri≈ ¬a a=sa ¬c | record { eq = eq } = subst (λ k → k o< x) eq zc41 where |
| 1505 zc41 : omax a sa o< x | |
| 1506 zc41 = osucprev ( begin | |
| 1507 osuc ( omax a sa ) ≡⟨ cong (λ k → osuc (omax a k)) (sym a=sa) ⟩ | |
| 1508 osuc ( omax a a ) ≡⟨ cong osuc (omxx _) ⟩ | |
| 1509 osuc ( osuc a ) ≤⟨ o<→≤ (ob<x lim (ob<x lim a<x)) ⟩ | |
| 1510 x ∎ ) where open o≤-Reasoning O | |
| 1511 ... | tri> ¬a ¬b c | record { eq = eq } = ob<x lim a<x | |
| 1512 sam = ZChain.supf (pzc (ob<x lim m<x)) a | |
| 1513 zc42 : osuc a o≤ osuc m | |
| 1514 zc42 = osucc (o<→≤ ( omax-x _ _ ) ) | |
| 1515 sam<m : sam o< m | |
| 1516 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 _ _ ) | |
| 1517 fcm : FClosure A f (ZChain.supf (pzc (ob<x lim m<x)) a) w | |
| 1518 fcm = subst (λ k → FClosure A f k w ) (zeq (ob<x lim a<x) (ob<x lim m<x) zc42 (o<→≤ <-osuc) ) fc | |
| 1519 zcm : odef (UnionCF A f mf ay (ZChain.supf (pzc (ob<x lim m<x))) (osuc (omax a sa))) w | |
| 1520 zcm = ZChain.cfcs (pzc (ob<x lim m<x)) mf< (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm | |
| 1018 | 1521 zc40 : odef (UnionCF A f mf ay supf1 b) w |
| 1016 | 1522 zc40 with ZChain.cfcs (pzc (ob<x lim m<x)) mf< (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm |
| 1523 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 1021 | 1524 ... | ⟪ az , ch-is-sup u u<x is-sup fc1 ⟫ = ⟪ az , ch-is-sup u u<b cp fc2 ⟫ where |
| 1016 | 1525 zc55 : u o< osuc m |
| 1526 zc55 = u<x | |
| 1020 | 1527 u<b : u o< b |
|
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
|
1528 u<b = subst (λ k → u o< k ) (sym b=x) ( ordtrans u<x (ob<x lim m<x)) |
| 1016 | 1529 fc1m : FClosure A f (ZChain.supf (pzc (ob<x lim m<x)) u) w |
| 1530 fc1m = fc1 | |
| 1531 fc1a : FClosure A f (ZChain.supf (pzc (ob<x lim a<x)) a) w | |
| 1532 fc1a = fc | |
| 1533 fc2 : FClosure A f (supf1 u) w | |
| 1534 fc2 = subst (λ k → FClosure A f k w) (trans (sym (zeq _ _ zc57 (o<→≤ <-osuc))) (sym (sf1=sf (ordtrans≤-< u<x m<x))) ) fc1 where | |
| 1535 zc57 : osuc u o≤ osuc m | |
| 1536 zc57 = osucc u<x | |
| 1021 | 1537 sb=sa : {a : Ordinal } → a o≤ m → supf1 a ≡ ZChain.supf (pzc (ob<x lim m<x)) a |
| 1538 sb=sa {a} a≤m = trans (sf1=sf (ordtrans≤-< a≤m m<x)) (zeq _ _ (osucc a≤m) (o<→≤ <-osuc)) | |
| 1539 cp : ChainP A f mf ay supf1 u | |
| 1540 cp = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sym (sb=sa u<x)) (ChainP.fcy<sup is-sup fc ) | |
| 1541 ; order = order | |
| 1542 ; supu=u = trans (sb=sa u<x ) (ChainP.supu=u is-sup) } where | |
| 1543 order : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 u → | |
| 1544 FClosure A f (supf1 s) z1 → (z1 ≡ supf1 u) ∨ (z1 << supf1 u) | |
| 1545 order {s} {z} s<u fc = subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sym (sb=sa u<x)) | |
| 1546 (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sb=sa s≤m) (sb=sa u<x) s<u) | |
| 1547 (subst (λ k → FClosure A f k z) (sb=sa s≤m ) fc )) where | |
| 1548 s≤m : s o≤ m | |
| 1549 s≤m = ordtrans (supf-inject0 supf-mono s<u ) u<x | |
| 1016 | 1550 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) |
| 1551 ... | tri> ¬a ¬b c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) | |
| 1018 | 1552 cfcs mf< {a} {b} {w} a<b b≤x sa<b fc | case2 b<x = zc40 where |
|
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
|
1553 supfb = ZChain.supf (pzc (ob<x lim b<x)) |
| 1020 | 1554 sb=sa : {a : Ordinal } → a o< b → supf1 a ≡ ZChain.supf (pzc (ob<x lim b<x)) a |
| 1555 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
|
1556 fcb : FClosure A f (supfb a) w |
| 1020 | 1557 fcb = subst (λ k → FClosure A f k w) (sb=sa a<b) fc |
| 1558 -- supfb a o< b assures it is in Union b | |
| 1018 | 1559 zcb : odef (UnionCF A f mf ay supfb b) w |
| 1020 | 1560 zcb = ZChain.cfcs (pzc (ob<x lim b<x)) mf< a<b (o<→≤ <-osuc) (subst (λ k → k o< b) (sb=sa a<b) sa<b) fcb |
| 1018 | 1561 zc40 : odef (UnionCF A f mf ay supf1 b) w |
| 1016 | 1562 zc40 with zcb |
| 1563 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 1020 | 1564 ... | ⟪ az , ch-is-sup u u<x is-sup fc1 ⟫ = ⟪ az , ch-is-sup u u<x cp (subst (λ k → FClosure A f k w) (sym (sb=sa u<x)) fc1 ) ⟫ where |
| 1565 cp : ChainP A f mf ay supf1 u | |
| 1021 | 1566 cp = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sym (sb=sa u<x)) (ChainP.fcy<sup is-sup fc ) |
| 1567 ; order = order | |
| 1568 ; supu=u = trans (sb=sa u<x) (ChainP.supu=u is-sup) } where | |
| 1569 order : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 u → | |
| 1570 FClosure A f (supf1 s) z1 → (z1 ≡ supf1 u) ∨ (z1 << supf1 u) | |
| 1571 order {s} {z} s<u fc = subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sym (sb=sa u<x)) | |
| 1572 (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sb=sa s<b) (sb=sa u<x) s<u) | |
| 1573 (subst (λ k → FClosure A f k z) (sb=sa s<b ) fc )) where | |
| 1574 s<b : s o< b | |
| 1575 s<b = ordtrans (supf-inject0 supf-mono s<u ) u<x | |
| 921 | 1576 --- |
| 1577 --- the maximum chain has fix point of any ≤-monotonic function | |
| 1578 --- | |
| 1579 | |
| 1580 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf ay x | |
| 1581 SZ f mf {y} ay x = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) x | |
| 1582 | |
| 966 | 1583 msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
| 1584 → (zc : ZChain A f mf ay x ) | |
| 934 | 1585 → MinSUP A (UnionCF A f mf ay (ZChain.supf zc) x) |
| 960 | 1586 msp0 f mf {x} ay zc = minsupP (UnionCF A f mf ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc) |
| 922 | 1587 |
| 992 | 1588 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (mf< : <-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 966 | 1589 → (sp1 : MinSUP A (ZChain.chain zc)) |
| 959 | 1590 → f (MinSUP.sup sp1) ≡ MinSUP.sup sp1 |
| 992 | 1591 fixpoint f mf mf< zc sp1 = z14 where |
|
924
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|
1592 chain = ZChain.chain zc |
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1593 supf = ZChain.supf zc |
|
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1594 sp : Ordinal |
| 959 | 1595 sp = MinSUP.sup sp1 |
|
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1596 asp : odef A sp |
| 959 | 1597 asp = MinSUP.asm sp1 |
|
988
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is-max and supf b = supf x
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987
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|
1598 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< (& A) → (ab : odef A b ) |
| 964 | 1599 → HasPrev A chain f b ∨ IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) b) ab |
| 921 | 1600 → * a < * b → odef chain b |
| 993 | 1601 z10 = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl ) |
|
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1602 z22 : sp o< & A |
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1603 z22 = z09 asp |
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1604 z12 : odef chain sp |
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|
1605 z12 with o≡? (& s) sp |
|
924
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parents:
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1606 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
| 993 | 1607 ... | 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 |
|
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1608 z13 : * (& s) < * sp |
| 960 | 1609 z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc ) |
| 1610 ... | case1 eq = ⊥-elim ( ne eq ) | |
| 966 | 1611 ... | case2 lt = lt |
| 964 | 1612 z19 : IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) sp) asp |
| 1613 z19 = record { x≤sup = z20 } where | |
| 959 | 1614 z20 : {y : Ordinal} → odef (UnionCF A f mf as0 (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp) |
| 966 | 1615 z20 {y} zy with MinSUP.x≤sup sp1 |
| 961 | 1616 (subst (λ k → odef chain k ) (sym &iso) (chain-mono f mf as0 supf (ZChain.supf-mono zc) (o<→≤ z22) zy )) |
| 966 | 1617 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p ) |
| 960 | 1618 ... | case2 y<p = case2 (subst (λ k → * k < _ ) &iso y<p ) |
|
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1619 z14 : f sp ≡ sp |
| 960 | 1620 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
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parents:
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|
1621 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
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1622 z16 : ⊥ |
| 959 | 1623 z16 with proj1 (mf (( MinSUP.sup sp1)) ( MinSUP.asm sp1 )) |
| 966 | 1624 ... | case1 eq = ⊥-elim (¬b (sym eq) ) |
| 1625 ... | case2 lt = ⊥-elim (¬c lt ) | |
| 1626 ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) b ) | |
|
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parents:
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1627 ... | tri> ¬a ¬b c = ⊥-elim z17 where |
| 959 | 1628 z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) ) |
| 960 | 1629 z15 = MinSUP.x≤sup sp1 (ZChain.f-next zc z12 ) |
|
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parents:
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1630 z17 : ⊥ |
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parents:
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|
1631 z17 with z15 |
| 960 | 1632 ... | case1 eq = ¬b (cong (*) eq) |
| 1633 ... | case2 lt = ¬a lt | |
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1634 |
| 952 | 1635 tri : {n : Level} (u w : Ordinal ) { R : Set n } → ( u o< w → R ) → ( u ≡ w → R ) → ( w o< u → R ) → R |
| 1636 tri {_} u w p q r with trio< u w | |
| 1637 ... | tri< a ¬b ¬c = p a | |
| 1638 ... | tri≈ ¬a b ¬c = q b | |
| 1639 ... | tri> ¬a ¬b c = r c | |
| 1640 | |
| 1641 or : {n m r : Level } {P : Set n } {Q : Set m} {R : Set r} → P ∨ Q → ( P → R ) → (Q → R ) → R | |
| 1642 or (case1 p) p→r q→r = p→r p | |
| 1643 or (case2 q) p→r q→r = q→r q | |
| 1644 | |
| 921 | 1645 |
| 1646 -- ZChain contradicts ¬ Maximal | |
| 1647 -- | |
| 1648 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) | |
| 1649 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
| 1650 -- | |
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|
1651 |
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a48dc906796c
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parents:
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1652 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥ |
| 966 | 1653 z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} |
| 1654 (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.asm msp1 )))) | |
|
965
1c1c6a6ed4fa
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parents:
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|
1655 (subst (λ k → odef A k) (sym &iso) (MinSUP.asm msp1) ) |
| 992 | 1656 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) (cf-is-<-monotonic nmx ) zc msp1 ))) -- x ≡ f x ̄ |
| 959 | 1657 (proj1 (cf-is-<-monotonic nmx c (MinSUP.asm msp1 ))) where -- x < f x |
| 937 | 1658 |
| 927 | 1659 supf = ZChain.supf zc |
| 934 | 1660 msp1 : MinSUP A (ZChain.chain zc) |
| 966 | 1661 msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc |
| 1662 c : Ordinal | |
| 1663 c = MinSUP.sup msp1 | |
| 934 | 1664 |
| 966 | 1665 zorn00 : Maximal A |
| 1666 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 804 | 1667 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
| 551 | 1668 -- yes we have the maximal |
| 1669 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 1670 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 1671 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 966 | 1672 zorn01 = proj1 zorn03 |
| 551 | 1673 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) |
| 1674 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 927 | 1675 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where |
| 551 | 1676 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 966 | 1677 nmx : ¬ Maximal A |
| 551 | 1678 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where |
| 966 | 1679 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) |
| 804 | 1680 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 | 1681 |
| 516 | 1682 -- usage (see filter.agda ) |
| 1683 -- | |
| 497 | 1684 -- _⊆'_ : ( A B : HOD ) → Set n |
| 1685 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 1686 |
| 966 | 1687 -- MaximumSubset : {L P : HOD} |
| 497 | 1688 -- → o∅ o< & L → o∅ o< & P → P ⊆ L |
| 1689 -- → IsPartialOrderSet P _⊆'_ | |
| 1690 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 1691 -- → Maximal P (_⊆'_) | |
| 1692 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |