Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 960:b7370c39769e
IsMinSUP< is wrong
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 04 Nov 2022 17:27:12 +0900 |
| parents | 1ef03eedd148 |
| children | 811135ad1904 |
| rev | line source |
|---|---|
| 478 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 508 | 2 open import Level hiding ( suc ; zero ) |
| 431 | 3 open import Ordinals |
| 552 | 4 open import Relation.Binary |
| 5 open import Relation.Binary.Core | |
| 6 open import Relation.Binary.PropositionalEquality | |
| 497 | 7 import OD |
| 552 | 8 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where |
| 431 | 9 |
| 560 | 10 -- |
| 11 -- Zorn-lemma : { A : HOD } | |
| 12 -- → o∅ o< & A | |
| 13 -- → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition | |
| 14 -- → Maximal A | |
| 15 -- | |
| 16 | |
| 431 | 17 open import zf |
| 477 | 18 open import logic |
| 19 -- open import partfunc {n} O | |
| 20 | |
| 21 open import Relation.Nullary | |
| 22 open import Data.Empty | |
| 23 import BAlgbra | |
| 431 | 24 |
| 555 | 25 open import Data.Nat hiding ( _<_ ; _≤_ ) |
| 26 open import Data.Nat.Properties | |
| 27 open import nat | |
| 28 | |
| 431 | 29 |
| 30 open inOrdinal O | |
| 31 open OD O | |
| 32 open OD.OD | |
| 33 open ODAxiom odAxiom | |
| 477 | 34 import OrdUtil |
| 35 import ODUtil | |
| 431 | 36 open Ordinals.Ordinals O |
| 37 open Ordinals.IsOrdinals isOrdinal | |
| 38 open Ordinals.IsNext isNext | |
| 39 open OrdUtil O | |
| 477 | 40 open ODUtil O |
| 41 | |
| 42 | |
| 43 import ODC | |
| 44 | |
| 45 open _∧_ | |
| 46 open _∨_ | |
| 47 open Bool | |
| 431 | 48 |
| 49 open HOD | |
| 50 | |
| 560 | 51 -- |
| 52 -- Partial Order on HOD ( possibly limited in A ) | |
| 53 -- | |
| 54 | |
| 872 | 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 | |
| 570 | 61 POO : IsStrictPartialOrder _≡_ _<<_ |
| 62 POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
| 63 ; trans = IsStrictPartialOrder.trans PO | |
| 64 ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y | |
| 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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TransitiveClosure with x <= f x is possible
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67 _≤_ : (x y : HOD) → Set (Level.suc n) |
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TransitiveClosure with x <= f x is possible
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68 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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TransitiveClosure with x <= f x is possible
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69 |
| 554 | 70 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
| 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 | |
| 955 | 76 <=-trans : {x y z : Ordinal } → x <= y → y <= z → x <= z |
| 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 |
| 951 | 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 |
| 951 | 84 ftrans<=-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z |
| 85 | |
| 770 | 86 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y |
| 87 <=to≤ (case1 eq) = case1 (cong (*) eq) | |
| 88 <=to≤ (case2 lt) = case2 lt | |
| 89 | |
| 779 | 90 ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y |
| 91 ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) | |
| 92 ≤to<= (case2 lt) = case2 lt | |
| 93 | |
| 556 | 94 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 95 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
| 96 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
| 97 (IsStrictPartialOrder.trans PO b<a a<b) | |
| 490 | 98 |
| 561 | 99 ptrans = IsStrictPartialOrder.trans PO |
| 100 | |
| 492 | 101 open _==_ |
| 102 open _⊆_ | |
| 103 | |
| 879 | 104 -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A |
| 105 -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x | |
| 106 -- <-TransFinite = ? | |
| 107 | |
| 530 | 108 -- |
| 560 | 109 -- Closure of ≤-monotonic function f has total order |
| 530 | 110 -- |
| 111 | |
| 112 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
| 113 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
| 114 | |
| 551 | 115 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
| 783 | 116 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
| 555 | 117 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
| 554 | 118 |
| 556 | 119 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y |
| 783 | 120 A∋fc {A} s f mf (init as refl ) = as |
| 556 | 121 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
| 555 | 122 |
| 714 | 123 A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s |
| 783 | 124 A∋fcs {A} s f mf (init as refl) = as |
| 714 | 125 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
| 126 | |
| 556 | 127 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
| 783 | 128 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
| 556 | 129 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
| 130 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
| 131 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
| 132 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
| 133 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
| 555 | 134 |
| 800 | 135 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
| 136 fcn s mf (init as refl) = zero | |
| 137 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
| 138 ... | case1 eq = fcn s mf p | |
| 139 ... | case2 y<fy = suc (fcn s mf p ) | |
| 140 | |
| 141 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 142 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
| 143 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where | |
| 144 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 145 fc06 {x} {y} refl {j} not = fc08 not where | |
| 146 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 147 fc08 () | |
| 148 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
| 149 fc07 {x} (init as refl) eq = refl | |
| 150 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 151 ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) | |
| 152 -- ... | case2 x<fx = ? | |
| 153 fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y | |
| 154 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
| 155 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
| 156 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
| 157 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
| 158 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y ) | |
| 159 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
| 160 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) i=x i=y ) | |
| 161 fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 162 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) | |
| 163 fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
| 164 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) | |
| 165 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
| 166 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
| 167 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 168 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
| 169 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x | |
| 170 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
| 171 fc04 : * x1 ≡ * y | |
| 172 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
| 173 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
| 174 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
| 175 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
| 176 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
| 177 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
| 178 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
| 179 fc05 : * x ≡ * y1 | |
| 180 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
| 181 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) | |
| 182 | |
| 183 | |
| 184 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
| 185 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
| 186 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
| 187 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
| 188 fc06 {x} {y} refl {j} not = fc08 not where | |
| 189 fc08 : {j : ℕ} → ¬ suc j ≡ 0 | |
| 190 fc08 () | |
| 191 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y | |
| 192 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
| 193 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
| 194 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
| 195 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
| 196 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
| 197 ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy | |
| 198 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
| 199 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
| 200 fc03 eq = cong pred eq | |
| 201 fc02 : * x < * y1 | |
| 202 fc02 = fc01 i cx cy (fc03 i=y ) a | |
| 203 | |
| 557 | 204 |
| 559 | 205 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
| 554 | 206 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
| 800 | 207 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
| 208 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
| 209 fc11 : * x < * y | |
| 210 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
| 211 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
| 212 fc10 : * x ≡ * y | |
| 213 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
| 214 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
| 215 fc12 : * y < * x | |
| 216 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
| 600 | 217 |
| 563 | 218 |
| 729 | 219 |
| 560 | 220 -- open import Relation.Binary.Properties.Poset as Poset |
| 221 | |
| 222 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
| 223 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
| 224 | |
| 567 | 225 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
| 568 | 226 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
| 567 | 227 |
| 568 | 228 _⊆'_ : ( A B : HOD ) → Set n |
| 229 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
| 560 | 230 |
| 231 -- | |
| 232 -- inductive maxmum tree from x | |
| 233 -- tree structure | |
| 234 -- | |
| 554 | 235 |
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236 record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where |
| 533 | 237 field |
| 836 | 238 ax : odef A x |
| 534 | 239 y : Ordinal |
| 541 | 240 ay : odef B y |
| 534 | 241 x=fy : x ≡ f y |
| 529 | 242 |
| 957 | 243 -- record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where |
| 244 -- field | |
| 245 -- x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) | |
| 246 | |
| 960 | 247 record IsMinSUP (A B : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (xa : odef A x) : Set n where |
| 654 | 248 field |
| 950 | 249 x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) |
| 954 | 250 minsup : { sup1 : Ordinal } → odef A sup1 |
| 251 → ( {z : Ordinal } → odef B z → (z ≡ sup1 ) ∨ (z << sup1 )) → x o≤ sup1 | |
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252 not-hp : ¬ ( HasPrev A B f x ) |
| 568 | 253 |
| 656 | 254 record SUP ( A B : HOD ) : Set (Level.suc n) where |
| 255 field | |
| 256 sup : HOD | |
| 804 | 257 as : A ∋ sup |
| 950 | 258 x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive |
| 656 | 259 |
| 690 | 260 -- |
| 261 -- sup and its fclosure is in a chain HOD | |
| 262 -- chain HOD is sorted by sup as Ordinal and <-ordered | |
| 263 -- whole chain is a union of separated Chain | |
| 803 | 264 -- minimum index is sup of y not ϕ |
| 690 | 265 -- |
| 266 | |
| 787 | 267 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where |
| 690 | 268 field |
| 765 | 269 fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) |
| 828 | 270 order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) |
| 271 supu=u : supf u ≡ u | |
| 694 | 272 |
| 748 | 273 data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 274 (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where | |
| 275 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z | |
| 919 | 276 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : supf u o< supf x) ( is-sup : ChainP A f mf ay supf u ) |
| 748 | 277 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z |
| 694 | 278 |
| 878 | 279 -- |
| 280 -- f (f ( ... (sup y))) f (f ( ... (sup z1))) | |
| 281 -- / | / | | |
| 282 -- / | / | | |
| 283 -- sup y < sup z1 < sup z2 | |
| 284 -- o< o< | |
| 861 | 285 -- data UChain is total |
| 286 | |
| 287 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) | |
| 288 {s s1 a b : Ordinal } ( ca : UChain A f mf ay supf s a ) ( cb : UChain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) | |
| 289 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where | |
| 290 ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) | |
| 291 ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb | |
| 938 | 292 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) with ChainP.fcy<sup supb fca |
| 861 | 293 ... | case1 eq with s≤fc (supf ub) f mf fcb |
| 294 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 295 ct00 : * a ≡ * b | |
| 296 ct00 = trans (cong (*) eq) eq1 | |
| 297 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where | |
| 298 ct01 : * a < * b | |
| 299 ct01 = subst (λ k → * k < * b ) (sym eq) lt | |
| 938 | 300 ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where |
| 861 | 301 ct00 : * a < * (supf ub) |
| 302 ct00 = lt | |
| 303 ct01 : * a < * b | |
| 304 ct01 with s≤fc (supf ub) f mf fcb | |
| 305 ... | case1 eq = subst (λ k → * a < k ) eq ct00 | |
| 306 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 938 | 307 ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb |
| 861 | 308 ... | case1 eq with s≤fc (supf ua) f mf fca |
| 309 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 310 ct00 : * a ≡ * b | |
| 311 ct00 = sym (trans (cong (*) eq) eq1 ) | |
| 312 ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where | |
| 313 ct01 : * b < * a | |
| 314 ct01 = subst (λ k → * k < * a ) (sym eq) lt | |
| 938 | 315 ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where |
| 861 | 316 ct00 : * b < * (supf ua) |
| 317 ct00 = lt | |
| 318 ct01 : * b < * a | |
| 319 ct01 with s≤fc (supf ua) f mf fca | |
| 320 ... | case1 eq = subst (λ k → * b < k ) eq ct00 | |
| 321 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt | |
| 322 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub | |
| 323 ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁) fca | |
| 324 ... | case1 eq with s≤fc (supf ub) f mf fcb | |
| 325 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 326 ct00 : * a ≡ * b | |
| 327 ct00 = trans (cong (*) eq) eq1 | |
| 328 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 329 ct02 : * a < * b | |
| 330 ct02 = subst (λ k → * k < * b ) (sym eq) lt | |
| 331 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where | |
| 332 ct03 : * a < * (supf ub) | |
| 333 ct03 = lt | |
| 334 ct02 : * a < * b | |
| 335 ct02 with s≤fc (supf ub) f mf fcb | |
| 336 ... | case1 eq = subst (λ k → * a < k ) eq ct03 | |
| 337 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt | |
| 338 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri≈ ¬a eq ¬c | |
| 339 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) | |
| 340 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) fcb | |
| 341 ... | case1 eq with s≤fc (supf ua) f mf fca | |
| 342 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where | |
| 343 ct00 : * a ≡ * b | |
| 344 ct00 = sym (trans (cong (*) eq) eq1) | |
| 345 ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where | |
| 346 ct02 : * b < * a | |
| 347 ct02 = subst (λ k → * k < * a ) (sym eq) lt | |
| 348 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where | |
| 349 ct05 : * b < * (supf ua) | |
| 350 ct05 = lt | |
| 351 ct04 : * b < * a | |
| 352 ct04 with s≤fc (supf ua) f mf fca | |
| 353 ... | case1 eq = subst (λ k → * b < k ) eq ct05 | |
| 354 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt | |
| 355 | |
| 694 | 356 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
| 357 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 358 | |
| 803 | 359 -- Union of supf z which o< x |
| 360 -- | |
| 694 | 361 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) |
| 362 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD | |
| 363 UnionCF A f mf ay supf x | |
| 894 | 364 = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x z } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } |
| 662 | 365 |
| 842 | 366 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) |
| 367 → supf x o< supf y → x o< y | |
| 368 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y | |
| 369 ... | tri< a ¬b ¬c = a | |
| 370 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 371 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 372 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 373 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 374 | |
| 879 | 375 record MinSUP ( A B : HOD ) : Set n where |
| 376 field | |
| 377 sup : Ordinal | |
| 378 asm : odef A sup | |
| 950 | 379 x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) |
| 879 | 380 minsup : { sup1 : Ordinal } → odef A sup1 |
| 381 → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 | |
| 382 | |
| 383 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A | |
| 384 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
| 385 | |
| 880 | 386 M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } |
| 387 → (supf : Ordinal → Ordinal ) | |
| 388 → MinSUP A (UnionCF A f mf ay supf x) | |
| 389 → SUP A (UnionCF A f mf ay supf x) | |
| 390 M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) | |
| 950 | 391 ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x≤sup = ms00 } where |
| 880 | 392 msup = MinSUP.sup ms |
| 393 ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup) | |
| 950 | 394 ms00 {z} uz with MinSUP.x≤sup ms uz |
| 880 | 395 ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq)) |
| 396 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso refl lt ) | |
| 397 | |
| 867 | 398 |
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399 chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
| 919 | 400 (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b |
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401 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c |
| 919 | 402 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = |
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403 ⟪ ua , ch-init fc ⟫ |
| 919 | 404 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = |
| 405 ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (supf-mono a≤b ) ) is-sup fc ⟫ | |
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406 |
| 703 | 407 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 408 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
| 655 | 409 field |
| 694 | 410 supf : Ordinal → Ordinal |
| 880 | 411 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z |
| 960 | 412 → IsMinSUP A (UnionCF A f mf ay supf b) f ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) f b ) → supf b ≡ b |
| 880 | 413 |
| 868 | 414 asupf : {x : Ordinal } → odef A (supf x) |
| 880 | 415 supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y |
| 416 supf-< : {x y : Ordinal } → supf x o< supf y → supf x << supf y | |
| 891 | 417 supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z |
| 880 | 418 |
| 419 minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) | |
| 891 | 420 supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z ) |
| 880 | 421 csupf : {b : Ordinal } → supf b o< z → odef (UnionCF A f mf ay supf z) (supf b) -- supf z is not an element of this chain |
| 422 | |
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423 chain : HOD |
| 703 | 424 chain = UnionCF A f mf ay supf z |
| 861 | 425 chain⊆A : chain ⊆' A |
| 426 chain⊆A = λ lt → proj1 lt | |
| 934 | 427 |
| 879 | 428 sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) |
| 880 | 429 sup {x} x≤z = M→S supf (minsup x≤z) |
| 934 | 430 |
| 431 s=ms : {x : Ordinal } → (x≤z : x o≤ z ) → & (SUP.sup (sup x≤z)) ≡ MinSUP.sup (minsup x≤z) | |
| 432 s=ms {x} x≤z = &iso | |
| 878 | 433 |
| 861 | 434 chain∋init : odef chain y |
| 435 chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ | |
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436 f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a) |
| 861 | 437 f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ |
| 938 | 438 f-next {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u<x is-sup (fsuc _ fc ) ⟫ |
| 861 | 439 initial : {z : Ordinal } → odef chain z → * y ≤ * z |
| 440 initial {a} ⟪ aa , ua ⟫ with ua | |
| 441 ... | ch-init fc = s≤fc y f mf fc | |
| 938 | 442 ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where |
| 861 | 443 zc7 : y <= supf u |
| 444 zc7 = ChainP.fcy<sup is-sup (init ay refl) | |
| 445 f-total : IsTotalOrderSet chain | |
| 446 f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 447 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 448 uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) | |
| 449 | |
| 871 | 450 supf-<= : {x y : Ordinal } → supf x <= supf y → supf x o≤ supf y |
| 451 supf-<= {x} {y} (case1 sx=sy) = o≤-refl0 sx=sy | |
| 452 supf-<= {x} {y} (case2 sx<sy) with trio< (supf x) (supf y) | |
| 453 ... | tri< a ¬b ¬c = o<→≤ a | |
| 454 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 455 ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) ) | |
| 456 | |
| 825 | 457 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
| 458 supf-inject {x} {y} sx<sy with trio< x y | |
| 459 ... | tri< a ¬b ¬c = a | |
| 460 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
| 461 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
| 462 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
| 463 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
| 798 | 464 |
| 872 | 465 fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf |
| 950 | 466 fcy<sup {u} {w} u≤z fc with MinSUP.x≤sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) |
| 798 | 467 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ |
| 892 | 468 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) |
| 469 ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) | |
| 825 | 470 |
| 871 | 471 -- ordering is not proved here but in ZChain1 |
| 756 | 472 |
| 960 | 473 IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp |
| 474 → ({y : Ordinal} → odef (UnionCF A f mf ay supf x) y → (y ≡ sp ) ∨ (y << sp )) | |
| 475 → ( {a : Ordinal } → a << f a ) | |
| 476 → ¬ ( HasPrev A (UnionCF A f mf ay supf x) f sp ) | |
| 477 IsMinSUP→NotHasPrev {x} {sp} asp is-sup <-mono-f hp = ⊥-elim (<-irr ( <=to≤ fsp≤sp) sp<fsp ) where | |
| 478 sp<fsp : sp << f sp | |
| 479 sp<fsp = <-mono-f | |
| 480 pr = HasPrev.y hp | |
| 481 im00 : f (f pr) <= sp | |
| 482 im00 = is-sup ( f-next (f-next (HasPrev.ay hp))) | |
| 483 fsp≤sp : f sp <= sp | |
| 484 fsp≤sp = subst (λ k → f k <= sp ) (sym (HasPrev.x=fy hp)) im00 | |
| 485 | |
| 486 IsMinSUP< : ( {a : Ordinal } → a << f a ) | |
| 487 → {b x : Ordinal } → {ab : odef A b} → x o≤ z → b o< x | |
| 488 → IsMinSUP A (UnionCF A f mf ay supf x) f ab → IsMinSUP A (UnionCF A f mf ay supf b) f ab | |
| 489 IsMinSUP< <-mono-f {b} {x} {ab} x≤z b<x record { x≤sup = x≤sup ; minsup = minsup ; not-hp = nhp } | |
| 490 = record { x≤sup = m02 ; minsup = m07 ; not-hp = IsMinSUP→NotHasPrev ab m02 <-mono-f } where | |
| 491 m02 : {z : Ordinal} → odef (UnionCF A f mf ay supf b) z → (z ≡ b) ∨ (z << b) | |
| 492 m02 {z} uz = x≤sup (chain-mono f mf ay supf supf-mono (o<→≤ b<x) uz) | |
| 493 m10 : {s : Ordinal } → (odef A s ) | |
| 494 → ( {w : Ordinal} → odef (UnionCF A f mf ay supf b) w → (w ≡ s) ∨ (w << s) ) | |
| 495 → {w : Ordinal} → odef (UnionCF A f mf ay supf x) w → (w ≡ s) ∨ (w << s) | |
| 496 m10 {s} as b-is-sup ⟪ aa , ch-init fc ⟫ = ? | |
| 497 m10 {s} as b-is-sup ⟪ aa , ch-is-sup u {w} u<x is-sup-z fc ⟫ = m01 where | |
| 498 m01 : w <= s | |
| 499 m01 with trio< (supf u) (supf b) | |
| 500 ... | tri< a ¬b ¬c = b-is-sup ⟪ aa , ch-is-sup u {w} a is-sup-z fc ⟫ | |
| 501 ... | tri≈ ¬a b ¬c = ? | |
| 502 ... | tri> ¬a ¬b c = ? | |
| 503 m07 : {s : Ordinal} → odef A s → ({z : Ordinal} → | |
| 504 odef (UnionCF A f mf ay supf b) z → (z ≡ s) ∨ (z << s)) → b o≤ s | |
| 505 m07 {s} as b-is-sup = minsup as (m10 as b-is-sup ) | |
| 955 | 506 |
| 728 | 507 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 783 | 508 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
| 869 | 509 supf = ZChain.supf zc |
| 728 | 510 field |
| 919 | 511 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → supf b o< supf z → (ab : odef A b) |
| 960 | 512 → HasPrev A (UnionCF A f mf ay supf z) f b ∨ IsMinSUP A (UnionCF A f mf ay supf b) f ab |
| 869 | 513 → * a < * b → odef ((UnionCF A f mf ay supf z)) b |
| 949 | 514 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) |
| 515 | |
| 568 | 516 record Maximal ( A : HOD ) : Set (Level.suc n) where |
| 517 field | |
| 518 maximal : HOD | |
| 804 | 519 as : A ∋ maximal |
| 568 | 520 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative |
| 567 | 521 |
| 743 | 522 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
| 523 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
| 783 | 524 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ |
| 743 | 525 |
| 497 | 526 Zorn-lemma : { A : HOD } |
| 464 | 527 → o∅ o< & A |
| 568 | 528 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
| 497 | 529 → Maximal A |
| 552 | 530 Zorn-lemma {A} 0<A supP = zorn00 where |
| 571 | 531 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
| 532 <-irr0 {a} {b} A∋a A∋b = <-irr | |
| 788 | 533 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
| 534 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
| 530 | 535 s : HOD |
| 536 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
| 568 | 537 as : A ∋ * ( & s ) |
| 538 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
|
539 as0 : odef A (& s ) |
|
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
540 as0 = subst (λ k → odef A k ) &iso as |
| 547 | 541 s<A : & s o< & A |
| 568 | 542 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
| 530 | 543 HasMaximal : HOD |
| 537 | 544 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
| 545 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
| 546 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
| 532 | 547 Gtx : { x : HOD} → A ∋ x → HOD |
| 537 | 548 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
| 549 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
| 804 | 550 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
| 537 | 551 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
| 552 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)) | |
| 553 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 | |
| 554 ¬x<m : ¬ (* x < * m) | |
| 555 ¬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 | 556 |
| 879 | 557 minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B |
| 558 minsupP B B⊆A total = m02 where | |
| 559 xsup : (sup : Ordinal ) → Set n | |
| 560 xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) | |
| 561 ∀-imply-or : {A : Ordinal → Set n } {B : Set n } | |
| 562 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B | |
| 563 ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM | |
| 564 ∀-imply-or {A} {B} ∀AB | case1 t = case1 t | |
| 565 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where | |
| 566 lemma : ¬ ((x : Ordinal ) → A x) → B | |
| 567 lemma not with ODC.p∨¬p O B | |
| 568 lemma not | case1 b = b | |
| 569 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
| 570 m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B | |
| 571 m00 x = TransFinite0 ind x where | |
| 572 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) | |
| 573 → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B | |
| 574 ind x prev = ∀-imply-or m01 where | |
| 575 m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B | |
| 576 m01 z with trio< z x | |
| 577 ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 578 ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
| 579 ... | tri< a ¬b ¬c with prev z a | |
| 580 ... | case2 mins = case2 mins | |
| 581 ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) | |
| 950 | 582 ... | case1 mins = case2 record { sup = z ; asm = proj1 mins ; x≤sup = proj2 mins ; minsup = m04 } where |
| 879 | 583 m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 |
| 584 m04 {s} as lt with trio< z s | |
| 585 ... | tri< a ¬b ¬c = o<→≤ a | |
| 586 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
| 587 ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) | |
| 588 ... | case2 notz = case1 (λ _ → notz ) | |
| 589 m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) | |
| 590 m03 not = ⊥-elim ( not s1 (z09 (SUP.as S)) ⟪ SUP.as S , m05 ⟫ ) where | |
| 591 S : SUP A B | |
| 592 S = supP B B⊆A total | |
| 593 s1 = & (SUP.sup S) | |
| 594 m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) | |
| 950 | 595 m05 {w} bw with SUP.x≤sup S {* w} (subst (λ k → odef B k) (sym &iso) bw ) |
| 879 | 596 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) |
| 597 ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) | |
| 598 m02 : MinSUP A B | |
| 599 m02 = dont-or (m00 (& A)) m03 | |
| 600 | |
| 560 | 601 -- Uncountable ascending chain by axiom of choice |
| 530 | 602 cf : ¬ Maximal A → Ordinal → Ordinal |
| 532 | 603 cf nmx x with ODC.∋-p O A (* x) |
| 604 ... | no _ = o∅ | |
| 605 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 538 | 606 ... | yes nogt = -- no larger element, so it is maximal |
| 607 ⊥-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 | 608 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
| 537 | 609 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
| 610 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
| 611 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
| 612 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
| 613 ... | 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 ⟫ ) | |
| 614 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
| 606 | 615 |
| 616 --- | |
| 617 --- infintie ascention sequence of f | |
| 618 --- | |
| 530 | 619 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
| 537 | 620 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
| 530 | 621 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
| 532 | 622 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
| 543 | 623 |
| 803 | 624 -- |
| 953 | 625 -- maximality of chain |
| 626 -- | |
| 627 -- supf is fixed for z ≡ & A , we can prove order and is-max | |
| 803 | 628 -- |
| 629 | |
| 793 | 630 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
| 728 | 631 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
| 953 | 632 SZ1 f mf {y} ay zc x = zc1 x where |
| 900 | 633 chain-mono1 : {a b c : Ordinal} → a o≤ b |
| 788 | 634 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
| 919 | 635 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b |
| 920 | 636 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → (ab : odef A b) |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
637 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) f b |
| 920 | 638 → * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
| 639 is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev | |
| 749 | 640 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 938 | 641 ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) |
| 642 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ | |
| 868 | 643 |
| 869 | 644 supf = ZChain.supf zc |
| 645 | |
| 920 | 646 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 |
| 647 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 | 648 s<b : s o< b |
| 649 s<b = ZChain.supf-inject zc ss<sb | |
| 920 | 650 s<z : s o< & A |
| 651 s<z = ordtrans s<b b<z | |
| 870 | 652 zc04 : odef (UnionCF A f mf ay supf (& A)) (supf s) |
| 874 | 653 zc04 = ZChain.csupf zc (z09 (ZChain.asupf zc)) |
| 869 | 654 zc05 : odef (UnionCF A f mf ay supf b) (supf s) |
| 655 zc05 with zc04 | |
| 656 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ | |
| 938 | 657 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u zc08 is-sup fc ⟫ where |
| 870 | 658 zc07 : FClosure A f (supf u) (supf s) -- supf u ≤ supf s → supf u o≤ supf s |
| 659 zc07 = fc | |
| 869 | 660 zc06 : supf u ≡ u |
| 661 zc06 = ChainP.supu=u is-sup | |
| 894 | 662 zc08 : supf u o< supf b |
| 663 zc08 = ordtrans≤-< (ZChain.supf-<= zc (≤to<= ( s≤fc _ f mf fc ))) ss<sb | |
| 869 | 664 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 |
| 665 zc04 : odef (UnionCF A f mf ay supf b) (f x) | |
| 666 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb fc ) | |
| 667 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ | |
| 938 | 668 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u<x is-sup (fsuc _ fc) ⟫ |
| 869 | 669 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) |
| 670 order {b} {s} {z1} b<z ss<sb fc = zc04 where | |
| 891 | 671 zc00 : ( z1 ≡ MinSUP.sup (ZChain.minsup zc (o<→≤ b<z) )) ∨ ( z1 << MinSUP.sup ( ZChain.minsup zc (o<→≤ b<z) ) ) |
| 950 | 672 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 | 673 -- supf (supf b) ≡ supf b |
| 869 | 674 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) |
| 675 zc04 with zc00 | |
| 892 | 676 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z)) ) eq ) |
| 677 ... | case2 lt = case2 (subst₂ (λ j k → j < * k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z) )) lt ) | |
| 868 | 678 |
| 953 | 679 zc1 : (x : Ordinal) → ZChain1 A f mf ay zc x |
| 680 zc1 x with Oprev-p x -- prev is not used now.... | |
| 949 | 681 ... | yes op = record { is-max = is-max ; order = order } where |
|
732
ddeb107b6f71
bchain can be reached from upwords by f. so it is worng.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
729
diff
changeset
|
682 px = Oprev.oprev op |
| 919 | 683 zc-b<x : {b : Ordinal } → ZChain.supf zc b o< ZChain.supf zc x → b o< osuc px |
| 684 zc-b<x {b} lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) (ZChain.supf-inject zc lt ) | |
| 894 | 685 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
| 919 | 686 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) → |
| 960 | 687 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsMinSUP A (UnionCF A f mf ay supf b) f ab → |
| 869 | 688 * a < * b → odef (UnionCF A f mf ay supf x) b |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
689 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P |
| 920 | 690 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 |
| 919 | 691 is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup |
| 692 = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where | |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
693 b<A : b o< & A |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
694 b<A = z09 ab |
| 919 | 695 b<x : b o< x |
| 696 b<x = ZChain.supf-inject zc sb<sx | |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
697 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
| 894 | 698 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
| 900 | 699 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) |
| 859 | 700 m05 : ZChain.supf zc b ≡ b |
| 701 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) | |
| 960 | 702 ⟪ record { x≤sup = λ {z} uz → IsMinSUP.x≤sup (proj2 is-sup) uz |
| 959 | 703 ; minsup = m07 ; not-hp = m04 } , m04 ⟫ where |
| 956 | 704 m07 : {s : Ordinal} → odef A s → ({z : Ordinal} → |
| 705 odef (UnionCF A f mf ay (ZChain.supf zc) b) z → (z ≡ s) ∨ (z << s)) → b o≤ s | |
| 960 | 706 m07 {s} as s-is-sup = IsMinSUP.minsup (proj2 is-sup) as s-is-sup |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
707 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
| 872 | 708 m08 {z} fcz = ZChain.fcy<sup zc (o<→≤ b<A) fcz |
| 828 | 709 m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 710 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
| 869 | 711 m09 {s} {z} s<b fcz = order b<A s<b fcz |
| 712 m06 : ChainP A f mf ay supf b | |
| 859 | 713 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = m05 } |
| 949 | 714 ... | no lim = record { is-max = is-max ; order = order } where |
| 869 | 715 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
| 919 | 716 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) → |
| 960 | 717 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsMinSUP A (UnionCF A f mf ay supf b) f ab → |
| 869 | 718 * a < * b → odef (UnionCF A f mf ay supf x) b |
| 919 | 719 is-max {a} {b} ua sb<sx ab P a<b with ODC.or-exclude O P |
| 920 | 720 is-max {a} {b} ua sb<sx ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b |
| 960 | 721 is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup with IsMinSUP.x≤sup (proj2 is-sup) (init-uchain A f mf ay ) |
| 789 | 722 ... | 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 )) ⟫ |
| 919 | 723 ... | case2 y<b = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
724 m09 : b o< & A |
|
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
725 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) |
| 919 | 726 b<x : b o< x |
| 727 b<x = ZChain.supf-inject zc sb<sx | |
|
790
201b66da4e69
remove unnesesary part in SZ1 the second TransFinite induction for is-max
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
789
diff
changeset
|
728 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
| 872 | 729 m07 {z} fc = ZChain.fcy<sup zc (o<→≤ m09) fc |
| 828 | 730 m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
| 825 | 731 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
| 869 | 732 m08 {s} {z1} s<b fc = order m09 s<b fc |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
733 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
| 894 | 734 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
| 900 | 735 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) |
|
860
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
736 ; x=fy = HasPrev.x=fy nhp } ) |
| 859 | 737 m05 : ZChain.supf zc b ≡ b |
| 738 m05 = ZChain.sup=u zc ab (o<→≤ m09) | |
| 960 | 739 ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj2 is-sup) lt |
| 740 ; minsup = IsMinSUP.minsup (proj2 is-sup) ; not-hp = m04 } , m04 ⟫ -- ZChain on x | |
| 869 | 741 m06 : ChainP A f mf ay supf b |
| 859 | 742 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = m05 } |
| 727 | 743 |
| 757 | 744 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
| 745 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = | |
| 746 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 ))) } | |
| 747 | |
| 748 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 749 → IsTotalOrderSet (uchain f mf ay) | |
| 750 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
| 751 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 752 uz01 = fcn-cmp y f mf ca cb | |
| 753 | |
| 754 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
| 928 | 755 → MinSUP A (uchain f mf ay) |
| 756 ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) | |
| 757 | 757 |
| 793 | 758 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
| 950 | 759 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 | 760 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
761 record xSUP (B : HOD) (f : Ordinal → Ordinal ) (x : Ordinal) : Set n where |
| 833 | 762 field |
| 763 ax : odef A x | |
| 960 | 764 is-sup : IsMinSUP A B f ax |
| 833 | 765 |
| 560 | 766 -- |
| 547 | 767 -- create all ZChains under o< x |
| 560 | 768 -- |
|
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
769 |
| 674 | 770 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
| 703 | 771 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
| 707 | 772 ind f mf {y} ay x prev with Oprev-p x |
| 954 | 773 ... | yes op = zc41 where |
| 682 | 774 -- |
| 775 -- we have previous ordinal to use induction | |
| 776 -- | |
| 777 px = Oprev.oprev op | |
| 703 | 778 zc : ZChain A f mf ay (Oprev.oprev op) |
| 682 | 779 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
| 780 px<x : px o< x | |
| 781 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
|
782 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
|
783 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
|
784 |
| 709 | 785 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
| 786 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
| 697 | 787 |
| 754 | 788 supf0 = ZChain.supf zc |
| 869 | 789 pchain : HOD |
| 790 pchain = UnionCF A f mf ay supf0 px | |
| 835 | 791 |
| 857 | 792 supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b |
| 793 supf-mono = ZChain.supf-mono zc | |
| 844 | 794 |
| 861 | 795 zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) |
| 796 zc04 {b} b≤x with trio< b px | |
| 797 ... | tri< a ¬b ¬c = case1 (o<→≤ a) | |
| 798 ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) | |
| 799 ... | tri> ¬a ¬b px<b with osuc-≡< b≤x | |
| 800 ... | case1 eq = case2 eq | |
| 801 ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
| 840 | 802 |
| 954 | 803 -- |
| 804 -- find the next value of supf | |
| 805 -- | |
| 806 | |
| 807 pchainpx : HOD | |
| 808 pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain A f mf ay supf0 px z ) | |
| 809 ∨ FClosure A f (supf0 px) z } ; odmax = & A ; <odmax = zc00 } where | |
| 810 zc00 : {z : Ordinal } → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z → z o< & A | |
| 811 zc00 {z} (case1 lt) = z07 lt | |
| 812 zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf fc ) | |
| 813 | |
| 814 zc02 : { a b : Ordinal } → odef A a ∧ UChain A f mf ay supf0 px a → FClosure A f (supf0 px) b → a <= b | |
| 815 zc02 {a} {b} ca fb = zc05 fb where | |
| 816 zc06 : MinSUP.sup (ZChain.minsup zc o≤-refl) ≡ supf0 px | |
| 817 zc06 = trans (sym ( ZChain.supf-is-minsup zc o≤-refl )) refl | |
| 818 zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a <= b | |
| 819 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb )) | |
| 820 ... | case1 eq = subst (λ k → a <= k ) (subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) eq)) (zc05 fb) | |
| 955 | 821 ... | case2 lt = <=-trans (zc05 fb) (case2 lt) |
| 954 | 822 zc05 (init b1 refl) with MinSUP.x≤sup (ZChain.minsup zc o≤-refl) |
| 823 (subst (λ k → odef A k ∧ UChain A f mf ay supf0 px k) (sym &iso) ca ) | |
| 824 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso zc06 eq ) | |
| 825 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso (cong (*) zc06) lt ) | |
| 826 | |
| 827 ptotal : IsTotalOrderSet pchainpx | |
| 828 ptotal (case1 a) (case1 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso | |
| 829 (chain-total A f mf ay supf0 (proj2 a) (proj2 b)) | |
| 830 ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b | |
| 831 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where | |
| 832 eq1 : a0 ≡ b0 | |
| 833 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 834 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where | |
| 835 lt1 : a0 < b0 | |
| 836 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 837 ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b | |
| 838 ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where | |
| 839 eq1 : a0 ≡ b0 | |
| 840 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
| 841 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where | |
| 842 lt1 : a0 < b0 | |
| 843 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
| 844 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) | |
| 845 | |
| 846 pcha : pchainpx ⊆' A | |
| 847 pcha (case1 lt) = proj1 lt | |
| 848 pcha (case2 fc) = A∋fc _ f mf fc | |
| 849 | |
| 850 sup1 : MinSUP A pchainpx | |
| 851 sup1 = minsupP pchainpx pcha ptotal | |
| 852 sp1 = MinSUP.sup sup1 | |
| 853 | |
| 854 -- | |
| 855 -- supf0 px o≤ sp1 | |
| 856 -- | |
| 857 | |
| 858 zc41 : ZChain A f mf ay x | |
| 859 zc41 with MinSUP.x≤sup sup1 (case2 (init (ZChain.asupf zc {px}) refl )) | |
| 860 zc41 | (case2 sfpx<sp1) = record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf1-mono ; supf-< = ? | |
| 901 | 861 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = csupf1 } where |
| 954 | 862 -- supf0 px is included by the chain of sp1 |
| 901 | 863 -- ( UnionCF A f mf ay supf0 px ∪ FClosure (supf0 px) ) ≡ UnionCF supf1 x |
| 864 -- supf1 x ≡ sp1, which is not included now | |
| 883 | 865 |
| 871 | 866 supf1 : Ordinal → Ordinal |
| 867 supf1 z with trio< z px | |
| 868 ... | tri< a ¬b ¬c = supf0 z | |
| 901 | 869 ... | tri≈ ¬a b ¬c = supf0 z |
| 870 ... | tri> ¬a ¬b c = sp1 | |
| 871 | 871 |
| 886 | 872 sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z |
| 901 | 873 sf1=sf0 {z} z≤px with trio< z px |
| 874 | 874 ... | tri< a ¬b ¬c = refl |
| 901 | 875 ... | tri≈ ¬a b ¬c = refl |
| 876 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c ) | |
| 883 | 877 |
| 901 | 878 sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1 |
| 879 sf1=sp1 {z} px<z with trio< z px | |
| 880 ... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a ) | |
| 881 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z ) | |
| 882 ... | tri> ¬a ¬b c = refl | |
| 873 | 883 |
| 903 | 884 asupf1 : {z : Ordinal } → odef A (supf1 z) |
| 885 asupf1 {z} with trio< z px | |
| 886 ... | tri< a ¬b ¬c = ZChain.asupf zc | |
| 887 ... | tri≈ ¬a b ¬c = ZChain.asupf zc | |
| 888 ... | tri> ¬a ¬b c = MinSUP.asm sup1 | |
| 889 | |
| 901 | 890 supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b |
| 891 supf1-mono {a} {b} a≤b with trio< b px | |
| 892 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b ) | |
| 893 ... | 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 ) | |
| 894 supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px | |
| 895 ... | tri< a<px ¬b ¬c = zc19 where | |
| 896 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
| 897 zc21 = ZChain.minsup zc (o<→≤ a<px) | |
| 898 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) | |
| 950 | 899 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) ) |
| 901 | 900 zc19 : supf0 a o≤ sp1 |
| 901 zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) | |
| 902 ... | tri≈ ¬a b ¬c = zc18 where | |
| 903 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
| 904 zc21 = ZChain.minsup zc (o≤-refl0 b) | |
| 905 zc20 : MinSUP.sup zc21 ≡ supf0 a | |
| 906 zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b)) | |
| 907 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) | |
| 950 | 908 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) ) |
| 901 | 909 zc18 : supf0 a o≤ sp1 |
| 910 zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) | |
| 911 ... | tri> ¬a ¬b c = o≤-refl | |
| 885 | 912 |
| 903 | 913 |
| 914 fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z | |
| 915 fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc | |
| 916 fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z | |
| 917 fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc | |
| 918 zc11 : {z : Ordinal} → odef (UnionCF A f mf ay supf1 x) z → odef pchainpx z | |
| 919 zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ | |
| 919 | 920 zc11 {z} ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ = zc21 fc where |
| 921 u<x : u o< x | |
| 953 | 922 u<x = supf-inject0 supf1-mono su<sx |
| 923 u≤px : u o≤ px | |
| 924 u≤px = zc-b<x _ u<x | |
| 903 | 925 zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1 |
| 926 zc21 {z1} (fsuc z2 fc ) with zc21 fc | |
| 927 ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 928 ... | 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) ⟫ | |
| 929 ... | case2 fc = case2 (fsuc _ fc) | |
| 953 | 930 zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) | inspect supf1 u |
| 931 ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u a record {fcy<sup = zc13 ; order = zc17 | |
| 932 ; supu=u = trans (sym (sf1=sf0 (o<→≤ u<px))) (ChainP.supu=u is-sup) } (init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where | |
| 933 u<px : u o< px | |
| 934 u<px = ZChain.supf-inject zc a | |
| 935 asp0 : odef A (supf0 u) | |
| 936 asp0 = ZChain.asupf zc | |
| 903 | 937 zc17 : {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 u → |
| 938 FClosure A f (supf0 s) z1 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u) | |
| 953 | 939 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) ((sf1=sf0 u≤px)) ( ChainP.order is-sup |
| 940 (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 zc18)) (sym (sf1=sf0 u≤px)) ss<spx) (fcpu fc zc18) ) where | |
| 903 | 941 zc18 : s o≤ px |
| 953 | 942 zc18 = ordtrans (ZChain.supf-inject zc ss<spx) u≤px |
| 903 | 943 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf0 u) ∨ ( z << supf0 u ) |
| 953 | 944 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sf1=sf0 (o<→≤ u<px)) ( ChainP.fcy<sup is-sup fc ) |
| 945 ... | tri≈ ¬a b ¬c | _ = case2 (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) (sym (trans (sf1=sf0 u≤px) b ))) | |
| 946 ... | 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 ⟫ ) | |
| 903 | 947 zc12 : {z : Ordinal} → odef pchainpx z → odef (UnionCF A f mf ay supf1 x) z |
| 948 zc12 {z} (case1 ⟪ az , ch-init fc ⟫ ) = ⟪ az , ch-init fc ⟫ | |
| 919 | 949 zc12 {z} (case1 ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ ) = zc21 fc where |
| 953 | 950 u<px : u o< px |
| 951 u<px = ZChain.supf-inject zc su<sx | |
| 952 u<x : u o< x | |
| 953 u<x = ordtrans u<px px<x | |
| 954 u≤px : u o≤ px | |
| 955 u≤px = o<→≤ u<px | |
| 956 s1u<s1x : supf1 u o< supf1 x | |
| 957 s1u<s1x = ordtrans<-≤ (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 u≤px )) (sym (sf1=sf0 o≤-refl)) su<sx) (supf1-mono (o<→≤ px<x) ) | |
| 903 | 958 zc21 : {z1 : Ordinal } → FClosure A f (supf0 u) z1 → odef (UnionCF A f mf ay supf1 x) z1 |
| 959 zc21 {z1} (fsuc z2 fc ) with zc21 fc | |
| 960 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 938 | 961 ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ |
| 903 | 962 zc21 (init asp refl ) with trio< u px | inspect supf1 u |
| 963 ... | tri< a ¬b ¬c | _ = ⟪ asp , ch-is-sup u | |
| 953 | 964 s1u<s1x |
| 965 record {fcy<sup = zc13 ; order = zc17 ; supu=u = trans (sf1=sf0 u≤px ) (ChainP.supu=u is-sup) } zc14 ⟫ where | |
| 903 | 966 zc17 : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 u → |
| 967 FClosure A f (supf1 s) z1 → (z1 ≡ supf1 u) ∨ (z1 << supf1 u) | |
| 953 | 968 zc17 {s} {z1} ss<su fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) (sym (sf1=sf0 (o<→≤ u<px))) ( ChainP.order is-sup |
| 969 (subst₂ (λ j k → j o< k ) (sf1=sf0 s≤px) (sf1=sf0 (o<→≤ u<px)) ss<su) (fcup fc s≤px) ) where | |
| 970 s≤px : s o≤ px -- ss<su : supf1 s o< supf1 u | |
| 971 s≤px = ordtrans ( supf-inject0 supf1-mono ss<su ) (o<→≤ u<px) | |
| 903 | 972 zc14 : FClosure A f (supf1 u) (supf0 u) |
| 953 | 973 zc14 = init (subst (λ k → odef A k ) (sym (sf1=sf0 u≤px)) asp) (sf1=sf0 u≤px) |
| 903 | 974 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u ) |
| 953 | 975 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 u≤px )) ( ChainP.fcy<sup is-sup fc ) |
| 976 ... | tri≈ ¬a b ¬c | _ = ⟪ asp , ch-is-sup px (subst (λ k → supf1 k o< supf1 x) b s1u<s1x) record { fcy<sup = zc13 | |
| 903 | 977 ; order = zc17 ; supu=u = zc18 } (init asupf1 (trans (sf1=sf0 o≤-refl ) (cong supf0 (sym b))) ) ⟫ where |
| 978 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 px) ∨ ( z << supf1 px ) | |
| 979 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (trans (cong supf0 b) (sym (sf1=sf0 o≤-refl))) (ChainP.fcy<sup is-sup fc ) | |
| 980 zc18 : supf1 px ≡ px | |
| 981 zc18 = begin | |
| 982 supf1 px ≡⟨ sf1=sf0 o≤-refl ⟩ | |
| 983 supf0 px ≡⟨ cong supf0 (sym b) ⟩ | |
| 984 supf0 u ≡⟨ ChainP.supu=u is-sup ⟩ | |
| 985 u ≡⟨ b ⟩ | |
| 986 px ∎ where open ≡-Reasoning | |
| 987 zc17 : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 px → | |
| 988 FClosure A f (supf1 s) z1 → (z1 ≡ supf1 px) ∨ (z1 << supf1 px) | |
| 989 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) (trans (cong supf0 b) (sym (sf1=sf0 o≤-refl))) | |
| 990 ( ChainP.order is-sup (subst₂ (λ j k → j o< k) (sf1=sf0 s≤px) zc19 ss<spx) (fcup fc s≤px) ) where | |
| 991 zc19 : supf1 px ≡ supf0 u | |
| 992 zc19 = trans (sf1=sf0 o≤-refl) (cong supf0 (sym b)) | |
| 993 s≤px : s o≤ px | |
| 994 s≤px = o<→≤ (supf-inject0 supf1-mono ss<spx) | |
| 953 | 995 ... | tri> ¬a ¬b c | _ = ⊥-elim ( ¬p<x<op ⟪ c , u≤px ⟫ ) |
| 903 | 996 zc12 {z} (case2 fc) = zc21 fc where |
| 954 | 997 zc20 : (supf0 px ≡ px ) ∨ ( supf0 px o< px ) |
| 998 zc20 = ? | |
| 903 | 999 zc21 : {z1 : Ordinal } → FClosure A f (supf0 px) z1 → odef (UnionCF A f mf ay supf1 x) z1 |
| 1000 zc21 {z1} (fsuc z2 fc ) with zc21 fc | |
| 1001 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 938 | 1002 ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ |
| 954 | 1003 zc21 (init asp refl ) with zc20 |
| 953 | 1004 ... | case1 sfpx=px = ⟪ asp , ch-is-sup px zc18 |
| 905 | 1005 record {fcy<sup = zc13 ; order = zc17 ; supu=u = zc15 } zc14 ⟫ where |
| 1006 zc15 : supf1 px ≡ px | |
| 1007 zc15 = trans (sf1=sf0 o≤-refl ) (sfpx=px) | |
| 953 | 1008 zc18 : supf1 px o< supf1 x |
| 1009 zc18 = ? | |
| 905 | 1010 zc14 : FClosure A f (supf1 px) (supf0 px) |
| 1011 zc14 = init (subst (λ k → odef A k) (sym (sf1=sf0 o≤-refl)) asp) (sf1=sf0 o≤-refl) | |
| 1012 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 px ) ∨ ( z << supf1 px ) | |
| 1013 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 o≤-refl)) ( ZChain.fcy<sup zc o≤-refl fc ) | |
| 1014 zc17 : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 px → | |
| 1015 FClosure A f (supf1 s) z1 → (z1 ≡ supf1 px) ∨ (z1 << supf1 px) | |
| 1016 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k )) mins-is-spx | |
| 950 | 1017 (MinSUP.x≤sup mins (csupf17 (fcup fc (o<→≤ s<px) )) ) where |
| 905 | 1018 mins : MinSUP A (UnionCF A f mf ay supf0 px) |
| 1019 mins = ZChain.minsup zc o≤-refl | |
| 1020 mins-is-spx : MinSUP.sup mins ≡ supf1 px | |
| 1021 mins-is-spx = trans (sym ( ZChain.supf-is-minsup zc o≤-refl ) ) (sym (sf1=sf0 o≤-refl )) | |
| 1022 s<px : s o< px | |
| 1023 s<px = supf-inject0 supf1-mono ss<spx | |
| 1024 sf<px : supf0 s o< px | |
| 1025 sf<px = subst₂ (λ j k → j o< k ) (sf1=sf0 (o<→≤ s<px)) (trans (sf1=sf0 o≤-refl) (sfpx=px)) ss<spx | |
| 1026 csupf17 : {z1 : Ordinal } → FClosure A f (supf0 s) z1 → odef (UnionCF A f mf ay supf0 px) z1 | |
| 1027 csupf17 (init as refl ) = ZChain.csupf zc sf<px | |
| 1028 csupf17 (fsuc x fc) = ZChain.f-next zc (csupf17 fc) | |
| 1029 | |
| 1030 ... | case2 sfp<px with ZChain.csupf zc sfp<px -- odef (UnionCF A f mf ay supf0 px) (supf0 px) | |
| 1031 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ ua1 , ch-init fc₁ ⟫ | |
| 954 | 1032 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ ua1 , ch-is-sup u zc18 |
| 919 | 1033 record { fcy<sup = z10 ; order = z11 ; supu=u = z12 } (fcpu fc₁ ? ) ⟫ where |
| 905 | 1034 z10 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u ) |
| 919 | 1035 z10 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? )) (ChainP.fcy<sup is-sup fc) |
| 905 | 1036 z11 : {s z1 : Ordinal} → (lt : supf1 s o< supf1 u ) → FClosure A f (supf1 s ) z1 |
| 1037 → (z1 ≡ supf1 u ) ∨ ( z1 << supf1 u ) | |
| 919 | 1038 z11 {s} {z} lt fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? )) |
| 905 | 1039 (ChainP.order is-sup lt0 (fcup fc s≤px )) where |
| 1040 s<u : s o< u | |
| 1041 s<u = supf-inject0 supf1-mono lt | |
| 1042 s≤px : s o≤ px | |
| 938 | 1043 s≤px = ordtrans s<u ? -- (o<→≤ u<x) |
| 905 | 1044 lt0 : supf0 s o< supf0 u |
| 919 | 1045 lt0 = subst₂ (λ j k → j o< k ) (sf1=sf0 s≤px) (sf1=sf0 ? ) lt |
| 905 | 1046 z12 : supf1 u ≡ u |
| 919 | 1047 z12 = trans (sf1=sf0 ? ) (ChainP.supu=u is-sup) |
| 954 | 1048 zc18 : supf1 u o< supf1 x |
| 1049 zc18 = ? | |
| 905 | 1050 |
| 903 | 1051 |
| 1052 | |
| 885 | 1053 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
| 1054 field | |
| 907 | 1055 tsup : MinSUP A (UnionCF A f mf ay supf1 z) |
| 885 | 1056 tsup=sup : supf1 z ≡ MinSUP.sup tsup |
| 1057 | |
| 1058 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x | |
| 1059 sup {z} z≤x with trio< z px | |
| 1060 ... | tri< a ¬b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m | |
| 950 | 1061 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o<→≤ a) ) (ZChain.supf-is-minsup zc (o<→≤ a)) } where |
| 885 | 1062 m = ZChain.minsup zc (o<→≤ a) |
| 907 | 1063 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
| 950 | 1064 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1065 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1066 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 1067 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
| 885 | 1068 ... | tri≈ ¬a b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m |
| 950 | 1069 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o≤-refl0 b) ) (ZChain.supf-is-minsup zc (o≤-refl0 b)) } where |
| 885 | 1070 m = ZChain.minsup zc (o≤-refl0 b) |
| 907 | 1071 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
| 950 | 1072 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1073 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1074 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 1075 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
| 901 | 1076 ... | tri> ¬a ¬b px<z = record { tsup = record { sup = sp1 ; asm = MinSUP.asm sup1 |
| 950 | 1077 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = sf1=sp1 px<z } where |
| 907 | 1078 m = sup1 |
| 1079 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) | |
| 950 | 1080 ms00 {x} ux = MinSUP.x≤sup m ? |
| 907 | 1081 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
| 1082 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
| 1083 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
| 885 | 1084 |
| 887 | 1085 csupf1 : {z1 : Ordinal } → supf1 z1 o< x → odef (UnionCF A f mf ay supf1 x) (supf1 z1) |
| 906 | 1086 csupf1 {z1} sz1<x = csupf2 where |
| 1087 -- z1 o< px → supf1 z1 ≡ supf0 z1 | |
| 1088 -- z1 ≡ px , supf0 px o< px .. px o< z1, x o≤ z1 ... supf1 z1 ≡ sp1 | |
| 1089 -- z1 ≡ px , supf0 px ≡ px | |
| 1090 psz1≤px : supf1 z1 o≤ px | |
|
918
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
changeset
|
1091 psz1≤px = subst (λ k → supf1 z1 o< k ) (sym opx=x) sz1<x |
| 906 | 1092 csupf2 : odef (UnionCF A f mf ay supf1 x) (supf1 z1) |
| 907 | 1093 csupf2 with trio< z1 px | inspect supf1 z1 |
| 1094 csupf2 | tri< a ¬b ¬c | record { eq = eq1 } with osuc-≡< psz1≤px | |
| 909 | 1095 ... | case2 lt = zc12 (case1 cs03) where |
| 1096 cs03 : odef (UnionCF A f mf ay supf0 px) (supf0 z1) | |
| 1097 cs03 = ZChain.csupf zc (subst (λ k → k o< px) (sf1=sf0 (o<→≤ a)) lt ) | |
| 910 | 1098 ... | case1 sfz=px with osuc-≡< ( supf1-mono (o<→≤ a) ) |
| 1099 ... | case1 sfz=sfpx = zc12 (case2 (init (ZChain.asupf zc) cs04 )) where | |
| 1100 supu=u : supf1 (supf1 z1) ≡ supf1 z1 | |
| 1101 supu=u = trans (cong supf1 sfz=px) (sym sfz=sfpx) | |
| 1102 cs04 : supf0 px ≡ supf0 z1 | |
| 1103 cs04 = begin | |
| 911 | 1104 supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ |
| 1105 supf1 px ≡⟨ sym sfz=sfpx ⟩ | |
| 1106 supf1 z1 ≡⟨ sf1=sf0 (o<→≤ a) ⟩ | |
| 1107 supf0 z1 ∎ where open ≡-Reasoning | |
|
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supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
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|
1108 ... | case2 sfz<sfpx = ⊥-elim ( ¬p<x<op ⟪ cs05 , cs06 ⟫ ) where |
|
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
changeset
|
1109 --- supf1 z1 ≡ px , z1 o< px , px ≡ supf0 z1 o< supf0 px o< x |
|
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
changeset
|
1110 cs05 : px o< supf0 px |
|
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
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|
1111 cs05 = subst₂ ( λ j k → j o< k ) sfz=px (sf1=sf0 o≤-refl ) sfz<sfpx |
|
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
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|
1112 cs06 : supf0 px o< osuc px |
| 954 | 1113 cs06 = subst (λ k → supf0 px o< k ) (sym opx=x) ? |
| 909 | 1114 csupf2 | tri≈ ¬a b ¬c | record { eq = eq1 } = zc12 (case2 (init (ZChain.asupf zc) (cong supf0 (sym b)))) |
| 919 | 1115 csupf2 | tri> ¬a ¬b px<z1 | record { eq = eq1 } = ? |
| 1116 -- ⊥-elim ( ¬p<x<op ⟪ px<z1 , subst (λ k → z1 o< k) (sym opx=x) z1<x ⟫ ) | |
|
918
4c33f8383d7d
supf px o< px is in csupf
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
911
diff
changeset
|
1117 |
| 877 | 1118 |
| 954 | 1119 zc41 | (case1 sfp=sp1 ) = record { supf = supf0 ; sup=u = ? ; asupf = ? ; supf-mono = ? ; supf-< = ? |
| 901 | 1120 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = ? } where |
| 883 | 1121 |
| 901 | 1122 -- supf0 px not is included by the chain |
| 1123 -- supf1 x ≡ supf0 px because of supfmax | |
| 883 | 1124 |
| 872 | 1125 supf1 : Ordinal → Ordinal |
| 1126 supf1 z with trio< z px | |
| 871 | 1127 ... | tri< a ¬b ¬c = supf0 z |
| 872 | 1128 ... | tri≈ ¬a b ¬c = supf0 px |
| 871 | 1129 ... | tri> ¬a ¬b c = supf0 px |
| 1130 | |
| 886 | 1131 sf1=sf0 : {z : Ordinal } → z o< px → supf1 z ≡ supf0 z |
| 1132 sf1=sf0 {z} z<px with trio< z px | |
| 874 | 1133 ... | tri< a ¬b ¬c = refl |
| 1134 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a z<px ) | |
| 1135 ... | tri> ¬a ¬b c = ⊥-elim ( ¬a z<px ) | |
| 1136 | |
| 1137 zc17 : {z : Ordinal } → supf0 z o≤ supf0 px | |
| 1138 zc17 = ? -- px o< z, px o< supf0 px | |
| 1139 | |
| 1140 supf-mono1 : {z w : Ordinal } → z o≤ w → supf1 z o≤ supf1 w | |
| 1141 supf-mono1 {z} {w} z≤w with trio< w px | |
| 886 | 1142 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (ordtrans≤-< z≤w a))) refl ( supf-mono z≤w ) |
| 874 | 1143 ... | tri≈ ¬a refl ¬c with trio< z px |
| 1144 ... | tri< a ¬b ¬c = zc17 | |
| 1145 ... | tri≈ ¬a refl ¬c = o≤-refl | |
| 1146 ... | tri> ¬a ¬b c = o≤-refl | |
| 1147 supf-mono1 {z} {w} z≤w | tri> ¬a ¬b c with trio< z px | |
| 1148 ... | tri< a ¬b ¬c = zc17 | |
| 1149 ... | tri≈ ¬a b ¬c = o≤-refl | |
| 1150 ... | tri> ¬a ¬b c = o≤-refl | |
| 1151 | |
| 872 | 1152 pchain1 : HOD |
| 1153 pchain1 = UnionCF A f mf ay supf1 x | |
| 871 | 1154 |
| 863 | 1155 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z |
| 1156 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 894 | 1157 zc10 {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ |
| 873 | 1158 |
| 1159 zc111 : {z : Ordinal} → z o< px → OD.def (od pchain1) z → OD.def (od pchain) z | |
| 1160 zc111 {z} z<px ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 894 | 1161 zc111 {z} z<px ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ |
| 873 | 1162 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1163 zc11 : (¬ xSUP (UnionCF A f mf ay supf0 px) f x ) ∨ (HasPrev A pchain f x ) |
| 864 | 1164 → {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z ∨ ( (supf0 px ≡ px) ∧ FClosure A f px z ) |
| 863 | 1165 zc11 P {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
| 1166 zc11 P {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ with trio< u1 px | |
| 890 | 1167 ... | tri< u1<px ¬b ¬c = case1 ⟪ az , ch-is-sup u1 ? ? fc ⟫ |
| 872 | 1168 ... | tri≈ ¬a eq ¬c = case2 ⟪ subst (λ k → supf0 k ≡ k) eq s1u=u , subst (λ k → FClosure A f k z) zc12 ? ⟫ where |
| 863 | 1169 s1u=u : supf0 u1 ≡ u1 |
| 872 | 1170 s1u=u = ? -- ChainP.supu=u u1-is-sup |
| 864 | 1171 zc12 : supf0 u1 ≡ px |
| 872 | 1172 zc12 = trans s1u=u eq |
| 863 | 1173 zc11 (case1 ¬sp=x) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = ⊥-elim (¬sp=x zcsup) where |
| 1174 eq : u1 ≡ x | |
| 1175 eq with trio< u1 x | |
| 1176 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) | |
| 1177 ... | tri≈ ¬a b ¬c = b | |
| 890 | 1178 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x ? ) |
| 863 | 1179 s1u=x : supf0 u1 ≡ x |
| 872 | 1180 s1u=x = trans ? eq |
| 863 | 1181 zc13 : osuc px o< osuc u1 |
| 1182 zc13 = o≤-refl0 ( trans (Oprev.oprev=x op) (sym eq ) ) | |
| 950 | 1183 x≤sup : {w : Ordinal} → odef (UnionCF A f mf ay supf0 px) w → (w ≡ x) ∨ (w << x) |
| 1184 x≤sup {w} ⟪ az , ch-init {w} fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? | |
| 1185 x≤sup {w} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u<x) ? )) | |
| 890 | 1186 ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 ? ) where |
| 851 | 1187 zc14 : u ≡ osuc px |
| 1188 zc14 = begin | |
| 1189 u ≡⟨ sym ( ChainP.supu=u is-sup) ⟩ | |
| 894 | 1190 supf0 u ≡⟨ ? ⟩ |
| 857 | 1191 supf0 u1 ≡⟨ s1u=x ⟩ |
| 851 | 1192 x ≡⟨ sym (Oprev.oprev=x op) ⟩ |
| 1193 osuc px ∎ where open ≡-Reasoning | |
| 872 | 1194 ... | case2 lt = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? |
| 863 | 1195 zc12 : supf0 x ≡ u1 |
| 872 | 1196 zc12 = subst (λ k → supf0 k ≡ u1) eq ? |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1197 zcsup : xSUP (UnionCF A f mf ay supf0 px) f x |
| 868 | 1198 zcsup = record { ax = subst (λ k → odef A k) (trans zc12 eq) (ZChain.asupf zc) |
| 954 | 1199 ; is-sup = record { x≤sup = x≤sup ; minsup = ? } } |
| 872 | 1200 zc11 (case2 hp) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = case1 ? where |
| 863 | 1201 eq : u1 ≡ x |
| 864 | 1202 eq with trio< u1 x |
| 1203 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) | |
| 1204 ... | tri≈ ¬a b ¬c = b | |
| 890 | 1205 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x ? ) |
| 858 | 1206 zc20 : {z : Ordinal} → FClosure A f (supf0 u1) z → OD.def (od pchain) z |
| 1207 zc20 {z} (init asu su=z ) = zc13 where | |
| 1208 zc14 : x ≡ z | |
| 1209 zc14 = begin | |
| 1210 x ≡⟨ sym eq ⟩ | |
| 872 | 1211 u1 ≡⟨ sym ? ⟩ |
| 858 | 1212 supf0 u1 ≡⟨ su=z ⟩ |
| 1213 z ∎ where open ≡-Reasoning | |
| 1214 zc13 : odef pchain z | |
| 1215 zc13 = subst (λ k → odef pchain k) (trans (sym (HasPrev.x=fy hp)) zc14) ( ZChain.f-next zc (HasPrev.ay hp) ) | |
| 1216 zc20 {.(f w)} (fsuc w fc) = ZChain.f-next zc (zc20 fc) | |
| 891 | 1217 |
| 857 | 1218 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
| 1219 field | |
| 891 | 1220 tsup : MinSUP A (UnionCF A f mf ay supf1 z) |
| 1221 tsup=sup : supf1 z ≡ MinSUP.sup tsup | |
| 1222 | |
| 857 | 1223 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x |
| 1224 sup {z} z≤x with trio< z px | |
| 891 | 1225 ... | tri< a ¬b ¬c = ? -- jrecord { tsup = ZChain.minsup zc (o<→≤ a) ; tsup=sup = ZChain.supf-is-minsup zc (o<→≤ a) } |
| 1226 ... | tri≈ ¬a b ¬c = ? -- record { tsup = ZChain.minsup zc (o≤-refl0 b) ; tsup=sup = ZChain.supf-is-minsup zc (o≤-refl0 b) } | |
| 865 | 1227 ... | tri> ¬a ¬b px<z = zc35 where |
| 840 | 1228 zc30 : z ≡ x |
| 1229 zc30 with osuc-≡< z≤x | |
| 1230 ... | case1 eq = eq | |
| 1231 ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
| 865 | 1232 zc32 = ZChain.sup zc o≤-refl |
| 1233 zc34 : ¬ (supf0 px ≡ px) → {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup zc32) ∨ (w < SUP.sup zc32) | |
| 882 | 1234 zc34 ne {w} lt with zc11 ? ⟪ proj1 lt , ? ⟫ |
| 950 | 1235 ... | case1 lt = SUP.x≤sup zc32 lt |
| 865 | 1236 ... | case2 ⟪ spx=px , fc ⟫ = ⊥-elim ( ne spx=px ) |
| 857 | 1237 zc33 : supf0 z ≡ & (SUP.sup zc32) |
| 891 | 1238 zc33 = ? -- trans (sym (supfx (o≤-refl0 (sym zc30)))) ( ZChain.supf-is-minsup zc o≤-refl ) |
| 865 | 1239 zc36 : ¬ (supf0 px ≡ px) → STMP z≤x |
| 950 | 1240 zc36 ne = ? -- record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x≤sup = zc34 ne } ; tsup=sup = zc33 } |
| 865 | 1241 zc35 : STMP z≤x |
| 1242 zc35 with trio< (supf0 px) px | |
| 1243 ... | tri< a ¬b ¬c = zc36 ¬b | |
| 1244 ... | tri> ¬a ¬b c = zc36 ¬b | |
| 891 | 1245 ... | tri≈ ¬a b ¬c = record { tsup = ? ; tsup=sup = ? } where |
| 1246 zc37 : MinSUP A (UnionCF A f mf ay supf0 z) | |
| 950 | 1247 zc37 = record { sup = ? ; asm = ? ; x≤sup = ? } |
| 803 | 1248 sup=u : {b : Ordinal} (ab : odef A b) → |
| 960 | 1249 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 | 1250 sup=u {b} ab b≤x is-sup with trio< b px |
| 960 | 1251 ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt ; minsup = ? } , proj2 is-sup ⟫ |
| 1252 ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (o≤-refl0 b) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt ; minsup = ? } , proj2 is-sup ⟫ | |
| 882 | 1253 ... | tri> ¬a ¬b px<b = zc31 ? where |
| 815 | 1254 zc30 : x ≡ b |
| 1255 zc30 with osuc-≡< b≤x | |
| 1256 ... | case1 eq = sym (eq) | |
| 1257 ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1258 zcsup : xSUP (UnionCF A f mf ay supf0 px) supf0 x |
| 859 | 1259 zcsup with zc30 |
| 950 | 1260 ... | refl = record { ax = ab ; is-sup = record { x≤sup = λ {w} lt → |
| 960 | 1261 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
|
1262 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
|
1263 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
|
1264 ... | refl = ⊥-elim (¬sp=x zcsup ) |
|
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1265 zc31 (case2 hasPrev ) with zc30 |
| 863 | 1266 ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev |
| 872 | 1267 ; ay = ? ; x=fy = HasPrev.x=fy hasPrev } ) |
| 833 | 1268 |
| 728 | 1269 ... | no lim = zc5 where |
|
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1270 |
| 703 | 1271 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
| 1272 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
|
1273 |
| 928 | 1274 ysp = MinSUP.sup (ysup f mf ay) |
| 755 | 1275 |
| 835 | 1276 supf0 : Ordinal → Ordinal |
| 1277 supf0 z with trio< z x | |
| 1278 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 1279 ... | tri≈ ¬a b ¬c = ysp |
| 1280 ... | tri> ¬a ¬b c = ysp | |
| 835 | 1281 |
| 838 | 1282 pchain : HOD |
| 1283 pchain = UnionCF A f mf ay supf0 x | |
| 835 | 1284 |
| 838 | 1285 ptotal0 : IsTotalOrderSet pchain |
| 835 | 1286 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
| 1287 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
| 1288 uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb)) | |
| 844 | 1289 |
| 880 | 1290 usup : MinSUP A pchain |
| 1291 usup = minsupP pchain (λ lt → proj1 lt) ptotal0 | |
| 1292 spu = MinSUP.sup usup | |
| 834 | 1293 |
| 794 | 1294 supf1 : Ordinal → Ordinal |
| 835 | 1295 supf1 z with trio< z x |
| 1296 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
| 836 | 1297 ... | tri≈ ¬a b ¬c = spu |
| 1298 ... | tri> ¬a ¬b c = spu | |
| 755 | 1299 |
| 838 | 1300 pchain1 : HOD |
| 1301 pchain1 = UnionCF A f mf ay supf1 x | |
| 704 | 1302 |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1303 is-max-hp : (supf : Ordinal → Ordinal) (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
|
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1304 b o< x → (ab : odef A b) → |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1305 HasPrev A (UnionCF A f mf ay supf x) f b → |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1306 * a < * b → odef (UnionCF A f mf ay supf x) b |
|
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1307 is-max-hp supf x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev |
|
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1308 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
| 938 | 1309 ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ? -- ⟪ ab , |
| 890 | 1310 -- subst (λ k → UChain A f mf ay supf x k ) |
| 938 | 1311 -- (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ |
|
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1312 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1313 zc70 : HasPrev A pchain f x → ¬ xSUP pchain f x |
| 844 | 1314 zc70 pr xsup = ? |
| 1315 | |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1316 no-extension : ¬ ( xSUP (UnionCF A f mf ay supf0 x) supf0 x ) → ZChain A f mf ay x |
| 879 | 1317 no-extension ¬sp=x = ? where -- record { supf = supf1 ; sup=u = sup=u |
| 1318 -- ; sup = sup ; supf-is-sup = sis ; supf-mono = {!!} ; asupf = ? } where | |
|
795
408e7e8a3797
csupf depends on order cyclicly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
794
diff
changeset
|
1319 supfu : {u : Ordinal } → ( a : u o< x ) → (z : Ordinal) → Ordinal |
|
408e7e8a3797
csupf depends on order cyclicly
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
794
diff
changeset
|
1320 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z |
| 838 | 1321 pchain0=1 : pchain ≡ pchain1 |
| 1322 pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
| 1323 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z | |
| 1324 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 938 | 1325 zc10 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc12 fc where |
| 838 | 1326 zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z |
| 1327 zc12 (fsuc x fc) with zc12 fc | |
| 1328 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 938 | 1329 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ |
| 890 | 1330 zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u ? ? (init ? ? ) ⟫ |
| 838 | 1331 zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z |
| 1332 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
| 938 | 1333 zc11 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc13 fc where |
| 838 | 1334 zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z |
| 1335 zc13 (fsuc x fc) with zc13 fc | |
| 1336 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
| 938 | 1337 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ |
| 838 | 1338 zc13 (init asu su=z ) with trio< u x |
| 890 | 1339 ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u ? ? (init ? ? ) ⟫ |
| 838 | 1340 ... | tri≈ ¬a b ¬c = ? |
| 938 | 1341 ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u<x c ) |
| 832 | 1342 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
| 797 | 1343 sup {z} z≤x with trio< z x |
| 838 | 1344 ... | tri< a ¬b ¬c = SUP⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} ) |
| 815 | 1345 ... | tri≈ ¬a b ¬c = {!!} |
| 843 | 1346 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 832 | 1347 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup {!!})) |
| 843 | 1348 sis {z} z≤x with trio< z x |
| 800 | 1349 ... | tri< a ¬b ¬c = {!!} where |
| 891 | 1350 zc8 = ZChain.supf-is-minsup (pzc z a) {!!} |
| 815 | 1351 ... | tri≈ ¬a b ¬c = {!!} |
| 843 | 1352 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 960 | 1353 sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsMinSUP A (UnionCF A f mf ay supf1 b) f ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf1 b) f b ) → supf1 b ≡ b |
| 843 | 1354 sup=u {z} ab z≤x is-sup with trio< z x |
| 950 | 1355 ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x≤sup = {!!} } |
| 1356 ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?) ?) ab {!!} record { x≤sup = {!!} } | |
| 843 | 1357 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
| 797 | 1358 |
| 703 | 1359 zc5 : ZChain A f mf ay x |
| 697 | 1360 zc5 with ODC.∋-p O A (* x) |
| 796 | 1361 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1362 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain f x ) |
| 703 | 1363 -- we have to check adding x preserve is-max ZChain A y f mf x |
| 796 | 1364 ... | case1 pr = no-extension {!!} |
| 960 | 1365 ... | case2 ¬fy<x with ODC.p∨¬p O (IsMinSUP A pchain f ax ) |
| 879 | 1366 ... | case1 is-sup = ? -- record { supf = supf1 ; sup=u = {!!} |
| 1367 -- ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!}; asupf = {!!} } -- where -- x is a sup of (zc ?) | |
| 796 | 1368 ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention |
| 553 | 1369 |
| 921 | 1370 --- |
| 1371 --- the maximum chain has fix point of any ≤-monotonic function | |
| 1372 --- | |
| 1373 | |
| 1374 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf ay x | |
| 1375 SZ f mf {y} ay x = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) x | |
| 1376 | |
| 934 | 1377 msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
| 1378 → (zc : ZChain A f mf ay x ) | |
| 1379 → MinSUP A (UnionCF A f mf ay (ZChain.supf zc) x) | |
| 960 | 1380 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 | 1381 |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1382 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
| 960 | 1383 → (sp1 : MinSUP A (ZChain.chain zc)) |
| 959 | 1384 → (ssp<as : ZChain.supf zc (MinSUP.sup sp1) o< ZChain.supf zc (& A)) |
| 1385 → f (MinSUP.sup sp1) ≡ MinSUP.sup sp1 | |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1386 fixpoint f mf zc sp1 ssp<as = z14 where |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1387 chain = ZChain.chain zc |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1388 supf = ZChain.supf zc |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1389 sp : Ordinal |
| 959 | 1390 sp = MinSUP.sup sp1 |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1391 asp : odef A sp |
| 959 | 1392 asp = MinSUP.asm sp1 |
| 921 | 1393 z10 : {a b : Ordinal } → (ca : odef chain a ) → supf b o< supf (& A) → (ab : odef A b ) |
| 960 | 1394 → HasPrev A chain f b ∨ IsMinSUP A (UnionCF A f mf as0 (ZChain.supf zc) b) f ab |
| 921 | 1395 → * a < * b → odef chain b |
| 960 | 1396 z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) ) |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1397 z22 : sp o< & A |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1398 z22 = z09 asp |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1399 z12 : odef chain sp |
|
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1400 z12 with o≡? (& s) sp |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1401 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1402 ... | no ne = ZChain1.is-max (SZ1 f mf as0 zc (& A)) {& s} {sp} ( ZChain.chain∋init zc ) ssp<as asp |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1403 (case2 z19 ) z13 where |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1404 z13 : * (& s) < * sp |
| 960 | 1405 z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc ) |
| 1406 ... | case1 eq = ⊥-elim ( ne eq ) | |
| 1407 ... | case2 lt = lt -- subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
| 1408 z19 : IsMinSUP A (UnionCF A f mf as0 (ZChain.supf zc) sp) f asp | |
| 1409 z19 = record { x≤sup = z20 ; minsup = ? ; not-hp = ZChain.IsMinSUP→NotHasPrev zc ? z20 ? } where | |
| 959 | 1410 z20 : {y : Ordinal} → odef (UnionCF A f mf as0 (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp) |
| 960 | 1411 z20 {y} zy with MinSUP.x≤sup sp1 (subst (λ k → odef chain k ) (sym &iso) (chain-mono f mf as0 supf (ZChain.supf-mono zc) (o<→≤ z22) zy )) |
| 1412 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p ) | |
| 1413 ... | case2 y<p = case2 (subst (λ k → * k < _ ) &iso y<p ) | |
|
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1414 z14 : f sp ≡ sp |
| 960 | 1415 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) (subst (λ k → odef chain k) (sym &iso) z12 ) |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1416 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1417 z16 : ⊥ |
| 959 | 1418 z16 with proj1 (mf (( MinSUP.sup sp1)) ( MinSUP.asm sp1 )) |
| 960 | 1419 ... | case1 eq = ⊥-elim (¬b (sym eq) ) |
| 1420 ... | case2 lt = ⊥-elim (¬c lt ) | |
| 1421 ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) b ) | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1422 ... | tri> ¬a ¬b c = ⊥-elim z17 where |
| 959 | 1423 z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) ) |
| 960 | 1424 z15 = MinSUP.x≤sup sp1 (ZChain.f-next zc z12 ) |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1425 z17 : ⊥ |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1426 z17 with z15 |
| 960 | 1427 ... | case1 eq = ¬b (cong (*) eq) |
| 1428 ... | case2 lt = ¬a lt | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1429 |
| 952 | 1430 tri : {n : Level} (u w : Ordinal ) { R : Set n } → ( u o< w → R ) → ( u ≡ w → R ) → ( w o< u → R ) → R |
| 1431 tri {_} u w p q r with trio< u w | |
| 1432 ... | tri< a ¬b ¬c = p a | |
| 1433 ... | tri≈ ¬a b ¬c = q b | |
| 1434 ... | tri> ¬a ¬b c = r c | |
| 1435 | |
| 1436 or : {n m r : Level } {P : Set n } {Q : Set m} {R : Set r} → P ∨ Q → ( P → R ) → (Q → R ) → R | |
| 1437 or (case1 p) p→r q→r = p→r p | |
| 1438 or (case2 q) p→r q→r = q→r q | |
| 1439 | |
| 921 | 1440 |
| 1441 -- ZChain contradicts ¬ Maximal | |
| 1442 -- | |
| 1443 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) | |
| 1444 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
| 1445 -- | |
|
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1446 |
|
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1447 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥ |
| 959 | 1448 z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} |
| 1449 (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.asm msp1 )))) | |
| 1450 (subst (λ k → odef A k) ? (MinSUP.asm msp1) ) | |
| 1451 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc msp1 ss<sa ))) -- x ≡ f x ̄ | |
| 1452 (proj1 (cf-is-<-monotonic nmx c (MinSUP.asm msp1 ))) where -- x < f x | |
| 937 | 1453 |
| 927 | 1454 supf = ZChain.supf zc |
| 934 | 1455 msp1 : MinSUP A (ZChain.chain zc) |
| 1456 msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc | |
| 1457 c : Ordinal | |
| 959 | 1458 c = MinSUP.sup msp1 |
| 1459 mc = c | |
| 943 | 1460 mc<A : mc o< & A |
| 1461 mc<A = ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫ | |
| 934 | 1462 c=mc : c ≡ mc |
| 959 | 1463 c=mc = refl |
| 934 | 1464 z20 : mc << cf nmx mc |
| 1465 z20 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1) ) | |
| 1466 asc : odef A (supf mc) | |
| 928 | 1467 asc = ZChain.asupf zc |
| 1468 spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) | |
| 1469 spd = ysup (cf nmx) (cf-is-≤-monotonic nmx) asc | |
| 1470 d = MinSUP.sup spd | |
| 1471 d<A : d o< & A | |
| 1472 d<A = ∈∧P→o< ⟪ MinSUP.asm spd , lift true ⟫ | |
| 929 | 1473 msup : MinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) |
| 1474 msup = ZChain.minsup zc (o<→≤ d<A) | |
| 928 | 1475 sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) ) |
| 1476 sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A) | |
| 937 | 1477 |
| 943 | 1478 sc<<d : {mc : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) |
| 934 | 1479 → supf mc << MinSUP.sup spd |
| 943 | 1480 sc<<d {mc} asc spd = z25 where |
| 934 | 1481 d1 : Ordinal |
| 938 | 1482 d1 = MinSUP.sup spd -- supf d1 ≡ d |
| 934 | 1483 z24 : (supf mc ≡ d1) ∨ ( supf mc << d1 ) |
| 950 | 1484 z24 = MinSUP.x≤sup spd (init asc refl) |
| 939 | 1485 -- |
| 1486 -- f ( f .. ( supf mc ) <= d1 | |
| 1487 -- f d1 <= d1 | |
| 1488 -- | |
| 934 | 1489 z25 : supf mc << d1 |
| 1490 z25 with z24 | |
| 1491 ... | case2 lt = lt | |
| 938 | 1492 ... | case1 eq = ⊥-elim ( <-irr z29 (proj1 (cf-is-<-monotonic nmx d1 (MinSUP.asm spd)) ) ) where |
| 1493 -- supf mc ≡ d1 | |
| 939 | 1494 z32 : ((cf nmx (supf mc)) ≡ d1) ∨ ( (cf nmx (supf mc)) << d1 ) |
| 950 | 1495 z32 = MinSUP.x≤sup spd (fsuc _ (init asc refl)) |
| 938 | 1496 z29 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1) |
| 939 | 1497 z29 with z32 |
| 1498 ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) ) | |
| 1499 ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt) | |
| 1500 | |
| 946 | 1501 fsc<<d : {mc z : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) |
| 1502 → (fc : FClosure A (cf nmx) (supf mc) z) → z << MinSUP.sup spd | |
| 1503 fsc<<d {mc} {z} asc spd fc = z25 where | |
| 1504 d1 : Ordinal | |
| 1505 d1 = MinSUP.sup spd -- supf d1 ≡ d | |
| 1506 z24 : (z ≡ d1) ∨ ( z << d1 ) | |
| 950 | 1507 z24 = MinSUP.x≤sup spd fc |
| 946 | 1508 -- |
| 1509 -- f ( f .. ( supf mc ) <= d1 | |
| 1510 -- f d1 <= d1 | |
| 1511 -- | |
| 1512 z25 : z << d1 | |
| 1513 z25 with z24 | |
| 1514 ... | case2 lt = lt | |
| 1515 ... | case1 eq = ⊥-elim ( <-irr z29 (proj1 (cf-is-<-monotonic nmx d1 (MinSUP.asm spd)) ) ) where | |
| 1516 -- supf mc ≡ d1 | |
| 1517 z32 : ((cf nmx z) ≡ d1) ∨ ( (cf nmx z) << d1 ) | |
| 950 | 1518 z32 = MinSUP.x≤sup spd (fsuc _ fc) |
| 946 | 1519 z29 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1) |
| 1520 z29 with z32 | |
| 1521 ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) ) | |
| 1522 ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt) | |
| 1523 | |
| 943 | 1524 smc<<d : supf mc << d |
| 1525 smc<<d = sc<<d asc spd | |
| 1526 | |
| 1527 sz<<c : {z : Ordinal } → z o< & A → supf z <= mc | |
| 950 | 1528 sz<<c z<A = MinSUP.x≤sup msp1 (ZChain.csupf zc (z09 (ZChain.asupf zc) )) |
| 943 | 1529 |
| 1530 sc=c : supf mc ≡ mc | |
| 1531 sc=c = ZChain.sup=u zc (MinSUP.asm msp1) (o<→≤ (∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫ )) ⟪ is-sup , not-hasprev ⟫ where | |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1532 not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc) (cf nmx) mc |
| 952 | 1533 not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-init fc ⟫ ; x=fy = x=fy } = ⊥-elim ( <-irr z31 z30 ) where |
| 1534 z30 : * mc < * (cf nmx mc) | |
| 1535 z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1)) | |
| 1536 z31 : ( * (cf nmx mc) ≡ * mc ) ∨ ( * (cf nmx mc) < * mc ) | |
| 1537 z31 = <=to≤ ( MinSUP.x≤sup msp1 (subst (λ k → odef (ZChain.chain zc) (cf nmx k)) (sym x=fy) | |
| 1538 ⟪ proj2 (cf-is-≤-monotonic nmx _ (proj2 (cf-is-≤-monotonic nmx _ ua1 ) )) , ch-init (fsuc _ (fsuc _ fc)) ⟫ )) | |
| 951 | 1539 not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-is-sup u u<x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z48 z32 ) where |
| 943 | 1540 z30 : * mc < * (cf nmx mc) |
| 1541 z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1)) | |
| 951 | 1542 z31 : ( supf mc ≡ mc ) ∨ ( * (supf mc) < * mc ) |
| 1543 z31 = MinSUP.x≤sup msp1 (ZChain.csupf zc (z09 (ZChain.asupf zc) )) | |
| 1544 z32 : * (supf mc) < * (cf nmx (cf nmx y)) | |
| 1545 z32 = ftrans<=-< z31 (subst (λ k → * mc < * k ) (cong (cf nmx) x=fy) z30 ) | |
| 1546 z48 : ( * (cf nmx (cf nmx y)) ≡ * (supf mc)) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) | |
| 1547 z48 = <=to≤ (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A u<x (fsuc _ ( fsuc _ fc ))) | |
| 960 | 1548 is-sup : IsMinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc) (cf nmx) (MinSUP.asm msp1) |
| 1549 is-sup = record { x≤sup = λ zy → MinSUP.x≤sup msp1 (chain-mono (cf nmx) (cf-is-≤-monotonic nmx) as0 supf (ZChain.supf-mono zc) (o<→≤ mc<A) zy ) | |
| 1550 ; minsup = ? ; not-hp = not-hasprev } | |
| 943 | 1551 |
|
940
aee83a7f9f57
not-hasprev z29 and z31 cause memory exhaust
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
939
diff
changeset
|
1552 |
|
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1553 not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) (cf nmx) d |
| 952 | 1554 not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-init fc ⟫ ; x=fy = x=fy } = ⊥-elim ( <-irr z31 z30 ) where |
| 1555 z30 : * d < * (cf nmx d) | |
| 1556 z30 = proj1 (cf-is-<-monotonic nmx d (MinSUP.asm spd)) | |
| 1557 z32 : ( cf nmx (cf nmx y) ≡ supf mc ) ∨ ( * (cf nmx (cf nmx y)) < * (supf mc) ) | |
| 1558 z32 = ZChain.fcy<sup zc (o<→≤ mc<A) (fsuc _ (fsuc _ fc)) | |
| 1559 z31 : ( * (cf nmx d) ≡ * d ) ∨ ( * (cf nmx d) < * d ) | |
| 1560 z31 = case2 ( subst (λ k → * (cf nmx k) < * d ) (sym x=fy) ( ftrans<=-< z32 ( sc<<d {mc} asc spd ) )) | |
| 948 | 1561 not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-is-sup u u<x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z46 z30 ) where |
| 1562 z45 : (* (cf nmx (cf nmx y)) ≡ * d) ∨ (* (cf nmx (cf nmx y)) < * d) → (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d) | |
| 1563 z45 p = subst (λ k → (* (cf nmx k) ≡ * d) ∨ (* (cf nmx k) < * d)) (sym x=fy) p | |
| 1564 z48 : supf mc << d | |
| 1565 z48 = sc<<d {mc} asc spd | |
| 949 | 1566 z53 : supf u o< supf (& A) |
| 1567 z53 = ordtrans<-≤ u<x (ZChain.supf-mono zc (o<→≤ d<A) ) | |
| 1568 z52 : ( u ≡ mc ) ∨ ( u << mc ) | |
| 950 | 1569 z52 = MinSUP.x≤sup msp1 ⟪ subst (λ k → odef A k ) (ChainP.supu=u is-sup1) (A∋fcs _ _ (cf-is-≤-monotonic nmx) fc) |
| 949 | 1570 , ch-is-sup u z53 is-sup1 (init (ZChain.asupf zc) (ChainP.supu=u is-sup1)) ⟫ |
| 1571 z51 : supf u o≤ supf mc | |
| 952 | 1572 z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57 where |
| 950 | 1573 z56 : u ≡ mc → supf u ≡ supf mc |
| 1574 z56 eq = cong supf eq | |
| 1575 z57 : u << mc → supf u o≤ supf mc | |
| 1576 z57 lt = ZChain.supf-<= zc (case2 z58) where | |
| 951 | 1577 z58 : supf u << supf mc -- supf u o< supf d -- supf u << supf d |
| 1578 z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c) lt | |
| 949 | 1579 z49 : supf u o< supf mc → (cf nmx (cf nmx y) ≡ supf mc) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) |
| 1580 z49 su<smc = ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A su<smc (fsuc _ ( fsuc _ fc )) | |
| 1581 z50 : (cf nmx (cf nmx y) ≡ supf d) ∨ (* (cf nmx (cf nmx y)) < * (supf d) ) | |
| 1582 z50 = ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) d<A u<x (fsuc _ ( fsuc _ fc )) | |
| 948 | 1583 z47 : {mc d1 : Ordinal } {asc : odef A (supf mc)} (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) |
| 951 | 1584 → (cf nmx (cf nmx y) ≡ supf mc) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) → supf mc << d1 |
| 948 | 1585 → * (cf nmx (cf nmx y)) < * d1 |
| 950 | 1586 z47 {mc} {d1} {asc} spd (case1 eq) smc<d = subst (λ k → k < * d1 ) (sym (cong (*) eq)) smc<d |
| 1587 z47 {mc} {d1} {asc} spd (case2 lt) smc<d = IsStrictPartialOrder.trans PO lt smc<d | |
| 948 | 1588 z30 : * d < * (cf nmx d) |
| 1589 z30 = proj1 (cf-is-<-monotonic nmx d (MinSUP.asm spd)) | |
| 1590 z46 : (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d) | |
| 952 | 1591 z46 = or (osuc-≡< z51) z55 z54 where |
| 950 | 1592 z55 : supf u ≡ supf mc → (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d) |
| 1593 z55 eq = <=to≤ (MinSUP.x≤sup spd ( subst₂ (λ j k → FClosure A (cf nmx) j (cf nmx k) ) eq (sym x=fy ) (fsuc _ (fsuc _ fc)) ) ) | |
| 1594 z54 : supf u o< supf mc → (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d) | |
| 1595 z54 lt = z45 (case2 (z47 {mc} {d} {asc} spd (z49 lt) z48 )) | |
| 1596 -- z46 with osuc-≡< z51 | |
| 1597 -- ... | case1 eq = MinSUP.x≤sup spd ( subst₂ (λ j k → FClosure A (cf nmx) j k ) (trans (ChainP.supu=u is-sup1) eq) refl fc ) | |
| 1598 -- ... | case2 lt = z45 (case2 (z47 {mc} {d} {asc} spd (z49 lt) z48 )) | |
| 948 | 1599 |
| 960 | 1600 is-sup : IsMinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) (cf nmx) (MinSUP.asm spd) |
| 1601 is-sup = record { x≤sup = z22 ; minsup = ? ; not-hp = not-hasprev } where | |
| 1602 z23 : {z : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z → (z ≡ MinSUP.sup spd) ∨ (z << MinSUP.sup spd) | |
| 1603 z23 lt = MinSUP.x≤sup spd lt | |
| 1604 z22 : {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y → | |
| 1605 (y ≡ MinSUP.sup spd) ∨ (y << MinSUP.sup spd) | |
| 1606 z22 {a} ⟪ aa , ch-init fc ⟫ = case2 ( ( ftrans<=-< z32 ( sc<<d {mc} asc spd ) )) where | |
| 1607 z32 : ( a ≡ supf mc ) ∨ ( * a < * (supf mc) ) | |
| 1608 z32 = ZChain.fcy<sup zc (o<→≤ mc<A) fc | |
| 1609 z22 {a} ⟪ aa , ch-is-sup u u<x is-sup1 fc ⟫ = tri u (supf mc) | |
| 1610 z60 z61 ( λ sc<u → ⊥-elim ( o≤> ( subst (λ k → k o≤ supf mc) (ChainP.supu=u is-sup1) z51) sc<u )) where | |
| 1611 z53 : supf u o< supf (& A) | |
| 1612 z53 = ordtrans<-≤ u<x (ZChain.supf-mono zc (o<→≤ d<A) ) | |
| 1613 z52 : ( u ≡ mc ) ∨ ( u << mc ) | |
| 1614 z52 = MinSUP.x≤sup msp1 ⟪ subst (λ k → odef A k ) (ChainP.supu=u is-sup1) (A∋fcs _ _ (cf-is-≤-monotonic nmx) fc) | |
| 1615 , ch-is-sup u z53 is-sup1 (init (ZChain.asupf zc) (ChainP.supu=u is-sup1)) ⟫ | |
| 1616 z56 : u ≡ mc → supf u ≡ supf mc | |
| 1617 z56 eq = cong supf eq | |
| 1618 z57 : u << mc → supf u o≤ supf mc | |
| 1619 z57 lt = ZChain.supf-<= zc (case2 z58) where | |
| 1620 z58 : supf u << supf mc -- supf u o< supf d -- supf u << supf d | |
| 1621 z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c) lt | |
| 1622 z51 : supf u o≤ supf mc | |
| 1623 z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57 | |
| 1624 z60 : u o< supf mc → (a ≡ d ) ∨ ( * a < * d ) | |
| 1625 z60 u<smc = case2 ( ftrans<=-< (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A | |
| 1626 (subst (λ k → k o< supf mc) (sym (ChainP.supu=u is-sup1)) u<smc) fc ) smc<<d ) | |
| 1627 z61 : u ≡ supf mc → (a ≡ d ) ∨ ( * a < * d ) | |
| 1628 z61 u=sc = case2 (fsc<<d {mc} asc spd (subst (λ k → FClosure A (cf nmx) k a) (trans (ChainP.supu=u is-sup1) u=sc) fc ) ) | |
| 1629 -- u<x : ZChain.supf zc u o< ZChain.supf zc d | |
| 1630 -- supf u o< spuf c → order | |
| 1631 | |
|
940
aee83a7f9f57
not-hasprev z29 and z31 cause memory exhaust
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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1632 sd=d : supf d ≡ d |
|
aee83a7f9f57
not-hasprev z29 and z31 cause memory exhaust
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
939
diff
changeset
|
1633 sd=d = ZChain.sup=u zc (MinSUP.asm spd) (o<→≤ d<A) ⟪ is-sup , not-hasprev ⟫ |
|
aee83a7f9f57
not-hasprev z29 and z31 cause memory exhaust
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
939
diff
changeset
|
1634 |
| 934 | 1635 sc<sd : {mc d : Ordinal } → supf mc << supf d → supf mc o< supf d |
| 1636 sc<sd {mc} {d} sc<<sd with osuc-≡< ( ZChain.supf-<= zc (case2 sc<<sd ) ) | |
| 1637 ... | case1 eq = ⊥-elim ( <-irr (case1 (cong (*) (sym eq) )) sc<<sd ) | |
| 1638 ... | case2 lt = lt | |
| 1639 | |
| 1640 sms<sa : supf mc o< supf (& A) | |
| 1641 sms<sa with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫) )) | |
| 1642 ... | case2 lt = lt | |
| 943 | 1643 ... | case1 eq = ⊥-elim ( o<¬≡ eq ( ordtrans<-≤ (sc<sd (subst (λ k → supf mc << k ) (sym sd=d) (sc<<d {mc} asc spd)) ) |
| 934 | 1644 ( ZChain.supf-mono zc (o<→≤ d<A )))) |
| 928 | 1645 |
| 927 | 1646 ss<sa : supf c o< supf (& A) |
| 934 | 1647 ss<sa = subst (λ k → supf k o< supf (& A)) (sym c=mc) sms<sa |
| 1648 | |
| 551 | 1649 zorn00 : Maximal A |
| 1650 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
| 804 | 1651 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
| 551 | 1652 -- yes we have the maximal |
| 1653 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
| 606 | 1654 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
| 551 | 1655 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
| 1656 zorn01 = proj1 zorn03 | |
| 1657 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
| 1658 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
| 927 | 1659 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where |
| 551 | 1660 -- if we have no maximal, make ZChain, which contradict SUP condition |
| 1661 nmx : ¬ Maximal A | |
| 1662 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
| 1663 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
| 804 | 1664 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 | 1665 |
| 516 | 1666 -- usage (see filter.agda ) |
| 1667 -- | |
| 497 | 1668 -- _⊆'_ : ( A B : HOD ) → Set n |
| 1669 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
| 482 | 1670 |
| 497 | 1671 -- MaximumSubset : {L P : HOD} |
| 1672 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 1673 -- → IsPartialOrderSet P _⊆'_ | |
| 1674 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 1675 -- → Maximal P (_⊆'_) | |
| 1676 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |