Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 958:33891adf80ea
IsMinSup contains not HasPrev
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Fri, 04 Nov 2022 06:47:23 +0900 |
parents | ce42b1c5cf42 |
children | 1ef03eedd148 |
rev | line source |
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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 | |
528
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67 _≤_ : (x y : HOD) → Set (Level.suc n) |
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68 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
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69 |
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 | |
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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 | |
958
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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 |
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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 |
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473 -- b o< x → IsMinSup A (UnionCF A f mf ay supf x) ab → IsMinSup A (UnionCF A f mf ay supf b) ab |
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474 -- is not valid without ¬ HasPrev |
955 | 475 |
728 | 476 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
783 | 477 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where |
869 | 478 supf = ZChain.supf zc |
728 | 479 field |
919 | 480 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → supf b o< supf z → (ab : odef A b) |
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481 → HasPrev A (UnionCF A f mf ay supf z) f b ∨ IsMinSup A (UnionCF A f mf ay supf b) f ab |
869 | 482 → * a < * b → odef ((UnionCF A f mf ay supf z)) b |
949 | 483 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) |
484 | |
568 | 485 record Maximal ( A : HOD ) : Set (Level.suc n) where |
486 field | |
487 maximal : HOD | |
804 | 488 as : A ∋ maximal |
568 | 489 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative |
567 | 490 |
743 | 491 init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) |
492 { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y | |
783 | 493 init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ |
743 | 494 |
497 | 495 Zorn-lemma : { A : HOD } |
464 | 496 → o∅ o< & A |
568 | 497 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
497 | 498 → Maximal A |
552 | 499 Zorn-lemma {A} 0<A supP = zorn00 where |
571 | 500 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
501 <-irr0 {a} {b} A∋a A∋b = <-irr | |
788 | 502 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
503 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
530 | 504 s : HOD |
505 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
568 | 506 as : A ∋ * ( & s ) |
507 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 )) ) | |
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508 as0 : odef A (& s ) |
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509 as0 = subst (λ k → odef A k ) &iso as |
547 | 510 s<A : & s o< & A |
568 | 511 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
530 | 512 HasMaximal : HOD |
537 | 513 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
514 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
515 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
532 | 516 Gtx : { x : HOD} → A ∋ x → HOD |
537 | 517 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
518 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
804 | 519 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
537 | 520 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
521 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)) | |
522 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 | |
523 ¬x<m : ¬ (* x < * m) | |
524 ¬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 | 525 |
879 | 526 minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B |
527 minsupP B B⊆A total = m02 where | |
528 xsup : (sup : Ordinal ) → Set n | |
529 xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) | |
530 ∀-imply-or : {A : Ordinal → Set n } {B : Set n } | |
531 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B | |
532 ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM | |
533 ∀-imply-or {A} {B} ∀AB | case1 t = case1 t | |
534 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where | |
535 lemma : ¬ ((x : Ordinal ) → A x) → B | |
536 lemma not with ODC.p∨¬p O B | |
537 lemma not | case1 b = b | |
538 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
539 m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B | |
540 m00 x = TransFinite0 ind x where | |
541 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) | |
542 → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B | |
543 ind x prev = ∀-imply-or m01 where | |
544 m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B | |
545 m01 z with trio< z x | |
546 ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
547 ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
548 ... | tri< a ¬b ¬c with prev z a | |
549 ... | case2 mins = case2 mins | |
550 ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) | |
950 | 551 ... | case1 mins = case2 record { sup = z ; asm = proj1 mins ; x≤sup = proj2 mins ; minsup = m04 } where |
879 | 552 m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 |
553 m04 {s} as lt with trio< z s | |
554 ... | tri< a ¬b ¬c = o<→≤ a | |
555 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
556 ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) | |
557 ... | case2 notz = case1 (λ _ → notz ) | |
558 m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) | |
559 m03 not = ⊥-elim ( not s1 (z09 (SUP.as S)) ⟪ SUP.as S , m05 ⟫ ) where | |
560 S : SUP A B | |
561 S = supP B B⊆A total | |
562 s1 = & (SUP.sup S) | |
563 m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) | |
950 | 564 m05 {w} bw with SUP.x≤sup S {* w} (subst (λ k → odef B k) (sym &iso) bw ) |
879 | 565 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) |
566 ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) | |
567 m02 : MinSUP A B | |
568 m02 = dont-or (m00 (& A)) m03 | |
569 | |
560 | 570 -- Uncountable ascending chain by axiom of choice |
530 | 571 cf : ¬ Maximal A → Ordinal → Ordinal |
532 | 572 cf nmx x with ODC.∋-p O A (* x) |
573 ... | no _ = o∅ | |
574 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
538 | 575 ... | yes nogt = -- no larger element, so it is maximal |
576 ⊥-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 | 577 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
537 | 578 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
579 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
580 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
581 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
582 ... | 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 ⟫ ) | |
583 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
606 | 584 |
585 --- | |
586 --- infintie ascention sequence of f | |
587 --- | |
530 | 588 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
537 | 589 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
530 | 590 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
532 | 591 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
543 | 592 |
803 | 593 -- |
953 | 594 -- maximality of chain |
595 -- | |
596 -- supf is fixed for z ≡ & A , we can prove order and is-max | |
803 | 597 -- |
598 | |
793 | 599 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) |
728 | 600 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x |
953 | 601 SZ1 f mf {y} ay zc x = zc1 x where |
900 | 602 chain-mono1 : {a b c : Ordinal} → a o≤ b |
788 | 603 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c |
919 | 604 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b |
920 | 605 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → (ab : odef A b) |
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606 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) f b |
920 | 607 → * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b |
608 is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev | |
749 | 609 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
938 | 610 ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) |
611 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc)) ⟫ | |
868 | 612 |
869 | 613 supf = ZChain.supf zc |
614 | |
920 | 615 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 |
616 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 | 617 s<b : s o< b |
618 s<b = ZChain.supf-inject zc ss<sb | |
920 | 619 s<z : s o< & A |
620 s<z = ordtrans s<b b<z | |
870 | 621 zc04 : odef (UnionCF A f mf ay supf (& A)) (supf s) |
874 | 622 zc04 = ZChain.csupf zc (z09 (ZChain.asupf zc)) |
869 | 623 zc05 : odef (UnionCF A f mf ay supf b) (supf s) |
624 zc05 with zc04 | |
625 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ | |
938 | 626 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u zc08 is-sup fc ⟫ where |
870 | 627 zc07 : FClosure A f (supf u) (supf s) -- supf u ≤ supf s → supf u o≤ supf s |
628 zc07 = fc | |
869 | 629 zc06 : supf u ≡ u |
630 zc06 = ChainP.supu=u is-sup | |
894 | 631 zc08 : supf u o< supf b |
632 zc08 = ordtrans≤-< (ZChain.supf-<= zc (≤to<= ( s≤fc _ f mf fc ))) ss<sb | |
869 | 633 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 |
634 zc04 : odef (UnionCF A f mf ay supf b) (f x) | |
635 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb fc ) | |
636 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫ | |
938 | 637 ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u<x is-sup (fsuc _ fc) ⟫ |
869 | 638 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) |
639 order {b} {s} {z1} b<z ss<sb fc = zc04 where | |
891 | 640 zc00 : ( z1 ≡ MinSUP.sup (ZChain.minsup zc (o<→≤ b<z) )) ∨ ( z1 << MinSUP.sup ( ZChain.minsup zc (o<→≤ b<z) ) ) |
950 | 641 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 | 642 -- supf (supf b) ≡ supf b |
869 | 643 zc04 : (z1 ≡ supf b) ∨ (z1 << supf b) |
644 zc04 with zc00 | |
892 | 645 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z)) ) eq ) |
646 ... | case2 lt = case2 (subst₂ (λ j k → j < * k ) refl (sym (ZChain.supf-is-minsup zc (o<→≤ b<z) )) lt ) | |
868 | 647 |
953 | 648 zc1 : (x : Ordinal) → ZChain1 A f mf ay zc x |
649 zc1 x with Oprev-p x -- prev is not used now.... | |
949 | 650 ... | yes op = record { is-max = is-max ; order = order } where |
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651 px = Oprev.oprev op |
919 | 652 zc-b<x : {b : Ordinal } → ZChain.supf zc b o< ZChain.supf zc x → b o< osuc px |
653 zc-b<x {b} lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) (ZChain.supf-inject zc lt ) | |
894 | 654 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
919 | 655 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) → |
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656 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsMinSup A (UnionCF A f mf ay supf b) f ab → |
869 | 657 * a < * b → odef (UnionCF A f mf ay supf x) b |
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658 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P |
920 | 659 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 | 660 is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup |
661 = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫ where | |
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662 b<A : b o< & A |
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663 b<A = z09 ab |
919 | 664 b<x : b o< x |
665 b<x = ZChain.supf-inject zc sb<sx | |
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666 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
894 | 667 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
900 | 668 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) |
859 | 669 m05 : ZChain.supf zc b ≡ b |
670 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) | |
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671 ⟪ record { x≤sup = λ {z} uz → IsMinSup.x≤sup (proj2 is-sup) ? -- (chain-mono1 (o<→≤ b<x) uz) |
956 | 672 ; minsup = m07 } , m04 ⟫ where |
673 m10 : {s : Ordinal } → (odef A s ) | |
674 → ( {z : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) b) z → (z ≡ s) ∨ (z << s) ) | |
957 | 675 → {z : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) b) z → (z ≡ s) ∨ (z << s) |
956 | 676 m10 = ? |
677 m07 : {s : Ordinal} → odef A s → ({z : Ordinal} → | |
678 odef (UnionCF A f mf ay (ZChain.supf zc) b) z → (z ≡ s) ∨ (z << s)) → b o≤ s | |
957 | 679 m07 {s} as s-is-sup = ? -- IsSup.minsup (proj2 is-sup) as (m10 as s-is-sup) |
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680 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b |
872 | 681 m08 {z} fcz = ZChain.fcy<sup zc (o<→≤ b<A) fcz |
828 | 682 m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
825 | 683 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
869 | 684 m09 {s} {z} s<b fcz = order b<A s<b fcz |
685 m06 : ChainP A f mf ay supf b | |
859 | 686 m06 = record { fcy<sup = m08 ; order = m09 ; supu=u = m05 } |
949 | 687 ... | no lim = record { is-max = is-max ; order = order } where |
869 | 688 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → |
919 | 689 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) → |
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690 HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsMinSup A (UnionCF A f mf ay supf b) f ab → |
869 | 691 * a < * b → odef (UnionCF A f mf ay supf x) b |
919 | 692 is-max {a} {b} ua sb<sx ab P a<b with ODC.or-exclude O P |
920 | 693 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 |
957 | 694 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 | 695 ... | 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 | 696 ... | 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 |
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697 m09 : b o< & A |
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698 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) |
919 | 699 b<x : b o< x |
700 b<x = ZChain.supf-inject zc sb<sx | |
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701 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b |
872 | 702 m07 {z} fc = ZChain.fcy<sup zc (o<→≤ m09) fc |
828 | 703 m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b |
825 | 704 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b |
869 | 705 m08 {s} {z1} s<b fc = order m09 s<b fc |
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706 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b |
894 | 707 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
900 | 708 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) |
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709 ; x=fy = HasPrev.x=fy nhp } ) |
859 | 710 m05 : ZChain.supf zc b ≡ b |
711 m05 = ZChain.sup=u zc ab (o<→≤ m09) | |
957 | 712 ⟪ record { x≤sup = λ lt → IsMinSup.x≤sup (proj2 is-sup) ? -- (chain-mono1 (o<→≤ b<x) lt ) |
954 | 713 ; minsup = ? } , m04 ⟫ -- ZChain on x |
869 | 714 m06 : ChainP A f mf ay supf b |
859 | 715 m06 = record { fcy<sup = m07 ; order = m08 ; supu=u = m05 } |
727 | 716 |
757 | 717 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
718 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = | |
719 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 ))) } | |
720 | |
721 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
722 → IsTotalOrderSet (uchain f mf ay) | |
723 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
724 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
725 uz01 = fcn-cmp y f mf ca cb | |
726 | |
727 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) | |
928 | 728 → MinSUP A (uchain f mf ay) |
729 ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) | |
757 | 730 |
793 | 731 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B |
950 | 732 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 | 733 |
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734 record xSUP (B : HOD) (f : Ordinal → Ordinal ) (x : Ordinal) : Set n where |
833 | 735 field |
736 ax : odef A x | |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
737 is-sup : IsMinSup A B f ax |
833 | 738 |
560 | 739 -- |
547 | 740 -- create all ZChains under o< x |
560 | 741 -- |
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
742 |
674 | 743 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
703 | 744 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x |
707 | 745 ind f mf {y} ay x prev with Oprev-p x |
954 | 746 ... | yes op = zc41 where |
682 | 747 -- |
748 -- we have previous ordinal to use induction | |
749 -- | |
750 px = Oprev.oprev op | |
703 | 751 zc : ZChain A f mf ay (Oprev.oprev op) |
682 | 752 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
753 px<x : px o< x | |
754 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
|
755 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
|
756 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
|
757 |
709 | 758 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
759 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt | |
697 | 760 |
754 | 761 supf0 = ZChain.supf zc |
869 | 762 pchain : HOD |
763 pchain = UnionCF A f mf ay supf0 px | |
835 | 764 |
857 | 765 supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b |
766 supf-mono = ZChain.supf-mono zc | |
844 | 767 |
861 | 768 zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) |
769 zc04 {b} b≤x with trio< b px | |
770 ... | tri< a ¬b ¬c = case1 (o<→≤ a) | |
771 ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) | |
772 ... | tri> ¬a ¬b px<b with osuc-≡< b≤x | |
773 ... | case1 eq = case2 eq | |
774 ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) | |
840 | 775 |
954 | 776 -- |
777 -- find the next value of supf | |
778 -- | |
779 | |
780 pchainpx : HOD | |
781 pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain A f mf ay supf0 px z ) | |
782 ∨ FClosure A f (supf0 px) z } ; odmax = & A ; <odmax = zc00 } where | |
783 zc00 : {z : Ordinal } → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z → z o< & A | |
784 zc00 {z} (case1 lt) = z07 lt | |
785 zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf fc ) | |
786 | |
787 zc02 : { a b : Ordinal } → odef A a ∧ UChain A f mf ay supf0 px a → FClosure A f (supf0 px) b → a <= b | |
788 zc02 {a} {b} ca fb = zc05 fb where | |
789 zc06 : MinSUP.sup (ZChain.minsup zc o≤-refl) ≡ supf0 px | |
790 zc06 = trans (sym ( ZChain.supf-is-minsup zc o≤-refl )) refl | |
791 zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a <= b | |
792 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb )) | |
793 ... | case1 eq = subst (λ k → a <= k ) (subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) eq)) (zc05 fb) | |
955 | 794 ... | case2 lt = <=-trans (zc05 fb) (case2 lt) |
954 | 795 zc05 (init b1 refl) with MinSUP.x≤sup (ZChain.minsup zc o≤-refl) |
796 (subst (λ k → odef A k ∧ UChain A f mf ay supf0 px k) (sym &iso) ca ) | |
797 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso zc06 eq ) | |
798 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso (cong (*) zc06) lt ) | |
799 | |
800 ptotal : IsTotalOrderSet pchainpx | |
801 ptotal (case1 a) (case1 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso | |
802 (chain-total A f mf ay supf0 (proj2 a) (proj2 b)) | |
803 ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b | |
804 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where | |
805 eq1 : a0 ≡ b0 | |
806 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
807 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where | |
808 lt1 : a0 < b0 | |
809 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
810 ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b | |
811 ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where | |
812 eq1 : a0 ≡ b0 | |
813 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
814 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where | |
815 lt1 : a0 < b0 | |
816 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
817 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) | |
818 | |
819 pcha : pchainpx ⊆' A | |
820 pcha (case1 lt) = proj1 lt | |
821 pcha (case2 fc) = A∋fc _ f mf fc | |
822 | |
823 sup1 : MinSUP A pchainpx | |
824 sup1 = minsupP pchainpx pcha ptotal | |
825 sp1 = MinSUP.sup sup1 | |
826 | |
827 -- | |
828 -- supf0 px o≤ sp1 | |
829 -- | |
830 | |
831 zc41 : ZChain A f mf ay x | |
832 zc41 with MinSUP.x≤sup sup1 (case2 (init (ZChain.asupf zc {px}) refl )) | |
833 zc41 | (case2 sfpx<sp1) = record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf1-mono ; supf-< = ? | |
901 | 834 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = csupf1 } where |
954 | 835 -- supf0 px is included by the chain of sp1 |
901 | 836 -- ( UnionCF A f mf ay supf0 px ∪ FClosure (supf0 px) ) ≡ UnionCF supf1 x |
837 -- supf1 x ≡ sp1, which is not included now | |
883 | 838 |
871 | 839 supf1 : Ordinal → Ordinal |
840 supf1 z with trio< z px | |
841 ... | tri< a ¬b ¬c = supf0 z | |
901 | 842 ... | tri≈ ¬a b ¬c = supf0 z |
843 ... | tri> ¬a ¬b c = sp1 | |
871 | 844 |
886 | 845 sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z |
901 | 846 sf1=sf0 {z} z≤px with trio< z px |
874 | 847 ... | tri< a ¬b ¬c = refl |
901 | 848 ... | tri≈ ¬a b ¬c = refl |
849 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c ) | |
883 | 850 |
901 | 851 sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1 |
852 sf1=sp1 {z} px<z with trio< z px | |
853 ... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a ) | |
854 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z ) | |
855 ... | tri> ¬a ¬b c = refl | |
873 | 856 |
903 | 857 asupf1 : {z : Ordinal } → odef A (supf1 z) |
858 asupf1 {z} with trio< z px | |
859 ... | tri< a ¬b ¬c = ZChain.asupf zc | |
860 ... | tri≈ ¬a b ¬c = ZChain.asupf zc | |
861 ... | tri> ¬a ¬b c = MinSUP.asm sup1 | |
862 | |
901 | 863 supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b |
864 supf1-mono {a} {b} a≤b with trio< b px | |
865 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b ) | |
866 ... | 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 ) | |
867 supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px | |
868 ... | tri< a<px ¬b ¬c = zc19 where | |
869 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
870 zc21 = ZChain.minsup zc (o<→≤ a<px) | |
871 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) | |
950 | 872 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) ) |
901 | 873 zc19 : supf0 a o≤ sp1 |
874 zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) | |
875 ... | tri≈ ¬a b ¬c = zc18 where | |
876 zc21 : MinSUP A (UnionCF A f mf ay supf0 a) | |
877 zc21 = ZChain.minsup zc (o≤-refl0 b) | |
878 zc20 : MinSUP.sup zc21 ≡ supf0 a | |
879 zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b)) | |
880 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) | |
950 | 881 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) ) |
901 | 882 zc18 : supf0 a o≤ sp1 |
883 zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 ) | |
884 ... | tri> ¬a ¬b c = o≤-refl | |
885 | 885 |
903 | 886 |
887 fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z | |
888 fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc | |
889 fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z | |
890 fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc | |
891 zc11 : {z : Ordinal} → odef (UnionCF A f mf ay supf1 x) z → odef pchainpx z | |
892 zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ | |
919 | 893 zc11 {z} ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ = zc21 fc where |
894 u<x : u o< x | |
953 | 895 u<x = supf-inject0 supf1-mono su<sx |
896 u≤px : u o≤ px | |
897 u≤px = zc-b<x _ u<x | |
903 | 898 zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1 |
899 zc21 {z1} (fsuc z2 fc ) with zc21 fc | |
900 ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
901 ... | 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) ⟫ | |
902 ... | case2 fc = case2 (fsuc _ fc) | |
953 | 903 zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) | inspect supf1 u |
904 ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u a record {fcy<sup = zc13 ; order = zc17 | |
905 ; supu=u = trans (sym (sf1=sf0 (o<→≤ u<px))) (ChainP.supu=u is-sup) } (init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where | |
906 u<px : u o< px | |
907 u<px = ZChain.supf-inject zc a | |
908 asp0 : odef A (supf0 u) | |
909 asp0 = ZChain.asupf zc | |
903 | 910 zc17 : {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 u → |
911 FClosure A f (supf0 s) z1 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u) | |
953 | 912 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) ((sf1=sf0 u≤px)) ( ChainP.order is-sup |
913 (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 zc18)) (sym (sf1=sf0 u≤px)) ss<spx) (fcpu fc zc18) ) where | |
903 | 914 zc18 : s o≤ px |
953 | 915 zc18 = ordtrans (ZChain.supf-inject zc ss<spx) u≤px |
903 | 916 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf0 u) ∨ ( z << supf0 u ) |
953 | 917 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sf1=sf0 (o<→≤ u<px)) ( ChainP.fcy<sup is-sup fc ) |
918 ... | tri≈ ¬a b ¬c | _ = case2 (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) (sym (trans (sf1=sf0 u≤px) b ))) | |
919 ... | 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 | 920 zc12 : {z : Ordinal} → odef pchainpx z → odef (UnionCF A f mf ay supf1 x) z |
921 zc12 {z} (case1 ⟪ az , ch-init fc ⟫ ) = ⟪ az , ch-init fc ⟫ | |
919 | 922 zc12 {z} (case1 ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ ) = zc21 fc where |
953 | 923 u<px : u o< px |
924 u<px = ZChain.supf-inject zc su<sx | |
925 u<x : u o< x | |
926 u<x = ordtrans u<px px<x | |
927 u≤px : u o≤ px | |
928 u≤px = o<→≤ u<px | |
929 s1u<s1x : supf1 u o< supf1 x | |
930 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 | 931 zc21 : {z1 : Ordinal } → FClosure A f (supf0 u) z1 → odef (UnionCF A f mf ay supf1 x) z1 |
932 zc21 {z1} (fsuc z2 fc ) with zc21 fc | |
933 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
938 | 934 ... | ⟪ 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 | 935 zc21 (init asp refl ) with trio< u px | inspect supf1 u |
936 ... | tri< a ¬b ¬c | _ = ⟪ asp , ch-is-sup u | |
953 | 937 s1u<s1x |
938 record {fcy<sup = zc13 ; order = zc17 ; supu=u = trans (sf1=sf0 u≤px ) (ChainP.supu=u is-sup) } zc14 ⟫ where | |
903 | 939 zc17 : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 u → |
940 FClosure A f (supf1 s) z1 → (z1 ≡ supf1 u) ∨ (z1 << supf1 u) | |
953 | 941 zc17 {s} {z1} ss<su fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) (sym (sf1=sf0 (o<→≤ u<px))) ( ChainP.order is-sup |
942 (subst₂ (λ j k → j o< k ) (sf1=sf0 s≤px) (sf1=sf0 (o<→≤ u<px)) ss<su) (fcup fc s≤px) ) where | |
943 s≤px : s o≤ px -- ss<su : supf1 s o< supf1 u | |
944 s≤px = ordtrans ( supf-inject0 supf1-mono ss<su ) (o<→≤ u<px) | |
903 | 945 zc14 : FClosure A f (supf1 u) (supf0 u) |
953 | 946 zc14 = init (subst (λ k → odef A k ) (sym (sf1=sf0 u≤px)) asp) (sf1=sf0 u≤px) |
903 | 947 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u ) |
953 | 948 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 u≤px )) ( ChainP.fcy<sup is-sup fc ) |
949 ... | 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 | 950 ; order = zc17 ; supu=u = zc18 } (init asupf1 (trans (sf1=sf0 o≤-refl ) (cong supf0 (sym b))) ) ⟫ where |
951 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 px) ∨ ( z << supf1 px ) | |
952 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 ) | |
953 zc18 : supf1 px ≡ px | |
954 zc18 = begin | |
955 supf1 px ≡⟨ sf1=sf0 o≤-refl ⟩ | |
956 supf0 px ≡⟨ cong supf0 (sym b) ⟩ | |
957 supf0 u ≡⟨ ChainP.supu=u is-sup ⟩ | |
958 u ≡⟨ b ⟩ | |
959 px ∎ where open ≡-Reasoning | |
960 zc17 : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 px → | |
961 FClosure A f (supf1 s) z1 → (z1 ≡ supf1 px) ∨ (z1 << supf1 px) | |
962 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) (trans (cong supf0 b) (sym (sf1=sf0 o≤-refl))) | |
963 ( ChainP.order is-sup (subst₂ (λ j k → j o< k) (sf1=sf0 s≤px) zc19 ss<spx) (fcup fc s≤px) ) where | |
964 zc19 : supf1 px ≡ supf0 u | |
965 zc19 = trans (sf1=sf0 o≤-refl) (cong supf0 (sym b)) | |
966 s≤px : s o≤ px | |
967 s≤px = o<→≤ (supf-inject0 supf1-mono ss<spx) | |
953 | 968 ... | tri> ¬a ¬b c | _ = ⊥-elim ( ¬p<x<op ⟪ c , u≤px ⟫ ) |
903 | 969 zc12 {z} (case2 fc) = zc21 fc where |
954 | 970 zc20 : (supf0 px ≡ px ) ∨ ( supf0 px o< px ) |
971 zc20 = ? | |
903 | 972 zc21 : {z1 : Ordinal } → FClosure A f (supf0 px) z1 → odef (UnionCF A f mf ay supf1 x) z1 |
973 zc21 {z1} (fsuc z2 fc ) with zc21 fc | |
974 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
938 | 975 ... | ⟪ 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 | 976 zc21 (init asp refl ) with zc20 |
953 | 977 ... | case1 sfpx=px = ⟪ asp , ch-is-sup px zc18 |
905 | 978 record {fcy<sup = zc13 ; order = zc17 ; supu=u = zc15 } zc14 ⟫ where |
979 zc15 : supf1 px ≡ px | |
980 zc15 = trans (sf1=sf0 o≤-refl ) (sfpx=px) | |
953 | 981 zc18 : supf1 px o< supf1 x |
982 zc18 = ? | |
905 | 983 zc14 : FClosure A f (supf1 px) (supf0 px) |
984 zc14 = init (subst (λ k → odef A k) (sym (sf1=sf0 o≤-refl)) asp) (sf1=sf0 o≤-refl) | |
985 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 px ) ∨ ( z << supf1 px ) | |
986 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 o≤-refl)) ( ZChain.fcy<sup zc o≤-refl fc ) | |
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 )) mins-is-spx | |
950 | 990 (MinSUP.x≤sup mins (csupf17 (fcup fc (o<→≤ s<px) )) ) where |
905 | 991 mins : MinSUP A (UnionCF A f mf ay supf0 px) |
992 mins = ZChain.minsup zc o≤-refl | |
993 mins-is-spx : MinSUP.sup mins ≡ supf1 px | |
994 mins-is-spx = trans (sym ( ZChain.supf-is-minsup zc o≤-refl ) ) (sym (sf1=sf0 o≤-refl )) | |
995 s<px : s o< px | |
996 s<px = supf-inject0 supf1-mono ss<spx | |
997 sf<px : supf0 s o< px | |
998 sf<px = subst₂ (λ j k → j o< k ) (sf1=sf0 (o<→≤ s<px)) (trans (sf1=sf0 o≤-refl) (sfpx=px)) ss<spx | |
999 csupf17 : {z1 : Ordinal } → FClosure A f (supf0 s) z1 → odef (UnionCF A f mf ay supf0 px) z1 | |
1000 csupf17 (init as refl ) = ZChain.csupf zc sf<px | |
1001 csupf17 (fsuc x fc) = ZChain.f-next zc (csupf17 fc) | |
1002 | |
1003 ... | case2 sfp<px with ZChain.csupf zc sfp<px -- odef (UnionCF A f mf ay supf0 px) (supf0 px) | |
1004 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ ua1 , ch-init fc₁ ⟫ | |
954 | 1005 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ ua1 , ch-is-sup u zc18 |
919 | 1006 record { fcy<sup = z10 ; order = z11 ; supu=u = z12 } (fcpu fc₁ ? ) ⟫ where |
905 | 1007 z10 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u ) |
919 | 1008 z10 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? )) (ChainP.fcy<sup is-sup fc) |
905 | 1009 z11 : {s z1 : Ordinal} → (lt : supf1 s o< supf1 u ) → FClosure A f (supf1 s ) z1 |
1010 → (z1 ≡ supf1 u ) ∨ ( z1 << supf1 u ) | |
919 | 1011 z11 {s} {z} lt fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? )) |
905 | 1012 (ChainP.order is-sup lt0 (fcup fc s≤px )) where |
1013 s<u : s o< u | |
1014 s<u = supf-inject0 supf1-mono lt | |
1015 s≤px : s o≤ px | |
938 | 1016 s≤px = ordtrans s<u ? -- (o<→≤ u<x) |
905 | 1017 lt0 : supf0 s o< supf0 u |
919 | 1018 lt0 = subst₂ (λ j k → j o< k ) (sf1=sf0 s≤px) (sf1=sf0 ? ) lt |
905 | 1019 z12 : supf1 u ≡ u |
919 | 1020 z12 = trans (sf1=sf0 ? ) (ChainP.supu=u is-sup) |
954 | 1021 zc18 : supf1 u o< supf1 x |
1022 zc18 = ? | |
905 | 1023 |
903 | 1024 |
1025 | |
885 | 1026 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
1027 field | |
907 | 1028 tsup : MinSUP A (UnionCF A f mf ay supf1 z) |
885 | 1029 tsup=sup : supf1 z ≡ MinSUP.sup tsup |
1030 | |
1031 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x | |
1032 sup {z} z≤x with trio< z px | |
1033 ... | tri< a ¬b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m | |
950 | 1034 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o<→≤ a) ) (ZChain.supf-is-minsup zc (o<→≤ a)) } where |
885 | 1035 m = ZChain.minsup zc (o<→≤ a) |
907 | 1036 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
950 | 1037 ms00 {x} ux = MinSUP.x≤sup m ? |
907 | 1038 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
1039 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
1040 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
885 | 1041 ... | tri≈ ¬a b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m |
950 | 1042 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o≤-refl0 b) ) (ZChain.supf-is-minsup zc (o≤-refl0 b)) } where |
885 | 1043 m = ZChain.minsup zc (o≤-refl0 b) |
907 | 1044 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) |
950 | 1045 ms00 {x} ux = MinSUP.x≤sup m ? |
907 | 1046 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
1047 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
1048 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
901 | 1049 ... | tri> ¬a ¬b px<z = record { tsup = record { sup = sp1 ; asm = MinSUP.asm sup1 |
950 | 1050 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = sf1=sp1 px<z } where |
907 | 1051 m = sup1 |
1052 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) | |
950 | 1053 ms00 {x} ux = MinSUP.x≤sup m ? |
907 | 1054 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → |
1055 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 | |
1056 ms01 {sup2} us P = MinSUP.minsup m ? ? | |
885 | 1057 |
887 | 1058 csupf1 : {z1 : Ordinal } → supf1 z1 o< x → odef (UnionCF A f mf ay supf1 x) (supf1 z1) |
906 | 1059 csupf1 {z1} sz1<x = csupf2 where |
1060 -- z1 o< px → supf1 z1 ≡ supf0 z1 | |
1061 -- z1 ≡ px , supf0 px o< px .. px o< z1, x o≤ z1 ... supf1 z1 ≡ sp1 | |
1062 -- z1 ≡ px , supf0 px ≡ px | |
1063 psz1≤px : supf1 z1 o≤ px | |
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1064 psz1≤px = subst (λ k → supf1 z1 o< k ) (sym opx=x) sz1<x |
906 | 1065 csupf2 : odef (UnionCF A f mf ay supf1 x) (supf1 z1) |
907 | 1066 csupf2 with trio< z1 px | inspect supf1 z1 |
1067 csupf2 | tri< a ¬b ¬c | record { eq = eq1 } with osuc-≡< psz1≤px | |
909 | 1068 ... | case2 lt = zc12 (case1 cs03) where |
1069 cs03 : odef (UnionCF A f mf ay supf0 px) (supf0 z1) | |
1070 cs03 = ZChain.csupf zc (subst (λ k → k o< px) (sf1=sf0 (o<→≤ a)) lt ) | |
910 | 1071 ... | case1 sfz=px with osuc-≡< ( supf1-mono (o<→≤ a) ) |
1072 ... | case1 sfz=sfpx = zc12 (case2 (init (ZChain.asupf zc) cs04 )) where | |
1073 supu=u : supf1 (supf1 z1) ≡ supf1 z1 | |
1074 supu=u = trans (cong supf1 sfz=px) (sym sfz=sfpx) | |
1075 cs04 : supf0 px ≡ supf0 z1 | |
1076 cs04 = begin | |
911 | 1077 supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ |
1078 supf1 px ≡⟨ sym sfz=sfpx ⟩ | |
1079 supf1 z1 ≡⟨ sf1=sf0 (o<→≤ a) ⟩ | |
1080 supf0 z1 ∎ where open ≡-Reasoning | |
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1081 ... | case2 sfz<sfpx = ⊥-elim ( ¬p<x<op ⟪ cs05 , cs06 ⟫ ) where |
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1082 --- supf1 z1 ≡ px , z1 o< px , px ≡ supf0 z1 o< supf0 px o< x |
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1083 cs05 : px o< supf0 px |
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1084 cs05 = subst₂ ( λ j k → j o< k ) sfz=px (sf1=sf0 o≤-refl ) sfz<sfpx |
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1085 cs06 : supf0 px o< osuc px |
954 | 1086 cs06 = subst (λ k → supf0 px o< k ) (sym opx=x) ? |
909 | 1087 csupf2 | tri≈ ¬a b ¬c | record { eq = eq1 } = zc12 (case2 (init (ZChain.asupf zc) (cong supf0 (sym b)))) |
919 | 1088 csupf2 | tri> ¬a ¬b px<z1 | record { eq = eq1 } = ? |
1089 -- ⊥-elim ( ¬p<x<op ⟪ px<z1 , subst (λ k → z1 o< k) (sym opx=x) z1<x ⟫ ) | |
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1090 |
877 | 1091 |
954 | 1092 zc41 | (case1 sfp=sp1 ) = record { supf = supf0 ; sup=u = ? ; asupf = ? ; supf-mono = ? ; supf-< = ? |
901 | 1093 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = ? } where |
883 | 1094 |
901 | 1095 -- supf0 px not is included by the chain |
1096 -- supf1 x ≡ supf0 px because of supfmax | |
883 | 1097 |
872 | 1098 supf1 : Ordinal → Ordinal |
1099 supf1 z with trio< z px | |
871 | 1100 ... | tri< a ¬b ¬c = supf0 z |
872 | 1101 ... | tri≈ ¬a b ¬c = supf0 px |
871 | 1102 ... | tri> ¬a ¬b c = supf0 px |
1103 | |
886 | 1104 sf1=sf0 : {z : Ordinal } → z o< px → supf1 z ≡ supf0 z |
1105 sf1=sf0 {z} z<px with trio< z px | |
874 | 1106 ... | tri< a ¬b ¬c = refl |
1107 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a z<px ) | |
1108 ... | tri> ¬a ¬b c = ⊥-elim ( ¬a z<px ) | |
1109 | |
1110 zc17 : {z : Ordinal } → supf0 z o≤ supf0 px | |
1111 zc17 = ? -- px o< z, px o< supf0 px | |
1112 | |
1113 supf-mono1 : {z w : Ordinal } → z o≤ w → supf1 z o≤ supf1 w | |
1114 supf-mono1 {z} {w} z≤w with trio< w px | |
886 | 1115 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (ordtrans≤-< z≤w a))) refl ( supf-mono z≤w ) |
874 | 1116 ... | tri≈ ¬a refl ¬c with trio< z px |
1117 ... | tri< a ¬b ¬c = zc17 | |
1118 ... | tri≈ ¬a refl ¬c = o≤-refl | |
1119 ... | tri> ¬a ¬b c = o≤-refl | |
1120 supf-mono1 {z} {w} z≤w | tri> ¬a ¬b c with trio< z px | |
1121 ... | tri< a ¬b ¬c = zc17 | |
1122 ... | tri≈ ¬a b ¬c = o≤-refl | |
1123 ... | tri> ¬a ¬b c = o≤-refl | |
1124 | |
872 | 1125 pchain1 : HOD |
1126 pchain1 = UnionCF A f mf ay supf1 x | |
871 | 1127 |
863 | 1128 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z |
1129 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
894 | 1130 zc10 {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ |
873 | 1131 |
1132 zc111 : {z : Ordinal} → z o< px → OD.def (od pchain1) z → OD.def (od pchain) z | |
1133 zc111 {z} z<px ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
894 | 1134 zc111 {z} z<px ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ? ? ⟫ |
873 | 1135 |
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|
1136 zc11 : (¬ xSUP (UnionCF A f mf ay supf0 px) f x ) ∨ (HasPrev A pchain f x ) |
864 | 1137 → {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z ∨ ( (supf0 px ≡ px) ∧ FClosure A f px z ) |
863 | 1138 zc11 P {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
1139 zc11 P {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ with trio< u1 px | |
890 | 1140 ... | tri< u1<px ¬b ¬c = case1 ⟪ az , ch-is-sup u1 ? ? fc ⟫ |
872 | 1141 ... | 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 | 1142 s1u=u : supf0 u1 ≡ u1 |
872 | 1143 s1u=u = ? -- ChainP.supu=u u1-is-sup |
864 | 1144 zc12 : supf0 u1 ≡ px |
872 | 1145 zc12 = trans s1u=u eq |
863 | 1146 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 |
1147 eq : u1 ≡ x | |
1148 eq with trio< u1 x | |
1149 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ ) | |
1150 ... | tri≈ ¬a b ¬c = b | |
890 | 1151 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x ? ) |
863 | 1152 s1u=x : supf0 u1 ≡ x |
872 | 1153 s1u=x = trans ? eq |
863 | 1154 zc13 : osuc px o< osuc u1 |
1155 zc13 = o≤-refl0 ( trans (Oprev.oprev=x op) (sym eq ) ) | |
950 | 1156 x≤sup : {w : Ordinal} → odef (UnionCF A f mf ay supf0 px) w → (w ≡ x) ∨ (w << x) |
1157 x≤sup {w} ⟪ az , ch-init {w} fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? | |
1158 x≤sup {w} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u<x) ? )) | |
890 | 1159 ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 ? ) where |
851 | 1160 zc14 : u ≡ osuc px |
1161 zc14 = begin | |
1162 u ≡⟨ sym ( ChainP.supu=u is-sup) ⟩ | |
894 | 1163 supf0 u ≡⟨ ? ⟩ |
857 | 1164 supf0 u1 ≡⟨ s1u=x ⟩ |
851 | 1165 x ≡⟨ sym (Oprev.oprev=x op) ⟩ |
1166 osuc px ∎ where open ≡-Reasoning | |
872 | 1167 ... | case2 lt = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ? |
863 | 1168 zc12 : supf0 x ≡ u1 |
872 | 1169 zc12 = subst (λ k → supf0 k ≡ u1) eq ? |
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1170 zcsup : xSUP (UnionCF A f mf ay supf0 px) f x |
868 | 1171 zcsup = record { ax = subst (λ k → odef A k) (trans zc12 eq) (ZChain.asupf zc) |
954 | 1172 ; is-sup = record { x≤sup = x≤sup ; minsup = ? } } |
872 | 1173 zc11 (case2 hp) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = case1 ? where |
863 | 1174 eq : u1 ≡ x |
864 | 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 ? ) |
858 | 1179 zc20 : {z : Ordinal} → FClosure A f (supf0 u1) z → OD.def (od pchain) z |
1180 zc20 {z} (init asu su=z ) = zc13 where | |
1181 zc14 : x ≡ z | |
1182 zc14 = begin | |
1183 x ≡⟨ sym eq ⟩ | |
872 | 1184 u1 ≡⟨ sym ? ⟩ |
858 | 1185 supf0 u1 ≡⟨ su=z ⟩ |
1186 z ∎ where open ≡-Reasoning | |
1187 zc13 : odef pchain z | |
1188 zc13 = subst (λ k → odef pchain k) (trans (sym (HasPrev.x=fy hp)) zc14) ( ZChain.f-next zc (HasPrev.ay hp) ) | |
1189 zc20 {.(f w)} (fsuc w fc) = ZChain.f-next zc (zc20 fc) | |
891 | 1190 |
857 | 1191 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where |
1192 field | |
891 | 1193 tsup : MinSUP A (UnionCF A f mf ay supf1 z) |
1194 tsup=sup : supf1 z ≡ MinSUP.sup tsup | |
1195 | |
857 | 1196 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x |
1197 sup {z} z≤x with trio< z px | |
891 | 1198 ... | tri< a ¬b ¬c = ? -- jrecord { tsup = ZChain.minsup zc (o<→≤ a) ; tsup=sup = ZChain.supf-is-minsup zc (o<→≤ a) } |
1199 ... | tri≈ ¬a b ¬c = ? -- record { tsup = ZChain.minsup zc (o≤-refl0 b) ; tsup=sup = ZChain.supf-is-minsup zc (o≤-refl0 b) } | |
865 | 1200 ... | tri> ¬a ¬b px<z = zc35 where |
840 | 1201 zc30 : z ≡ x |
1202 zc30 with osuc-≡< z≤x | |
1203 ... | case1 eq = eq | |
1204 ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ ) | |
865 | 1205 zc32 = ZChain.sup zc o≤-refl |
1206 zc34 : ¬ (supf0 px ≡ px) → {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup zc32) ∨ (w < SUP.sup zc32) | |
882 | 1207 zc34 ne {w} lt with zc11 ? ⟪ proj1 lt , ? ⟫ |
950 | 1208 ... | case1 lt = SUP.x≤sup zc32 lt |
865 | 1209 ... | case2 ⟪ spx=px , fc ⟫ = ⊥-elim ( ne spx=px ) |
857 | 1210 zc33 : supf0 z ≡ & (SUP.sup zc32) |
891 | 1211 zc33 = ? -- trans (sym (supfx (o≤-refl0 (sym zc30)))) ( ZChain.supf-is-minsup zc o≤-refl ) |
865 | 1212 zc36 : ¬ (supf0 px ≡ px) → STMP z≤x |
950 | 1213 zc36 ne = ? -- record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x≤sup = zc34 ne } ; tsup=sup = zc33 } |
865 | 1214 zc35 : STMP z≤x |
1215 zc35 with trio< (supf0 px) px | |
1216 ... | tri< a ¬b ¬c = zc36 ¬b | |
1217 ... | tri> ¬a ¬b c = zc36 ¬b | |
891 | 1218 ... | tri≈ ¬a b ¬c = record { tsup = ? ; tsup=sup = ? } where |
1219 zc37 : MinSUP A (UnionCF A f mf ay supf0 z) | |
950 | 1220 zc37 = record { sup = ? ; asm = ? ; x≤sup = ? } |
803 | 1221 sup=u : {b : Ordinal} (ab : odef A b) → |
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1222 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 | 1223 sup=u {b} ab b≤x is-sup with trio< b px |
957 | 1224 ... | 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 ⟫ |
1225 ... | 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 | 1226 ... | tri> ¬a ¬b px<b = zc31 ? where |
815 | 1227 zc30 : x ≡ b |
1228 zc30 with osuc-≡< b≤x | |
1229 ... | case1 eq = sym (eq) | |
1230 ... | 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
|
1231 zcsup : xSUP (UnionCF A f mf ay supf0 px) supf0 x |
859 | 1232 zcsup with zc30 |
950 | 1233 ... | refl = record { ax = ab ; is-sup = record { x≤sup = λ {w} lt → |
957 | 1234 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
|
1235 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
|
1236 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
|
1237 ... | refl = ⊥-elim (¬sp=x zcsup ) |
105f8d6c51fb
no-extension on immidate ordinal passed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
859
diff
changeset
|
1238 zc31 (case2 hasPrev ) with zc30 |
863 | 1239 ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev |
872 | 1240 ; ay = ? ; x=fy = HasPrev.x=fy hasPrev } ) |
833 | 1241 |
728 | 1242 ... | no lim = zc5 where |
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1243 |
703 | 1244 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z |
1245 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
|
1246 |
928 | 1247 ysp = MinSUP.sup (ysup f mf ay) |
755 | 1248 |
835 | 1249 supf0 : Ordinal → Ordinal |
1250 supf0 z with trio< z x | |
1251 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
836 | 1252 ... | tri≈ ¬a b ¬c = ysp |
1253 ... | tri> ¬a ¬b c = ysp | |
835 | 1254 |
838 | 1255 pchain : HOD |
1256 pchain = UnionCF A f mf ay supf0 x | |
835 | 1257 |
838 | 1258 ptotal0 : IsTotalOrderSet pchain |
835 | 1259 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
1260 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
1261 uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb)) | |
844 | 1262 |
880 | 1263 usup : MinSUP A pchain |
1264 usup = minsupP pchain (λ lt → proj1 lt) ptotal0 | |
1265 spu = MinSUP.sup usup | |
834 | 1266 |
794 | 1267 supf1 : Ordinal → Ordinal |
835 | 1268 supf1 z with trio< z x |
1269 ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z | |
836 | 1270 ... | tri≈ ¬a b ¬c = spu |
1271 ... | tri> ¬a ¬b c = spu | |
755 | 1272 |
838 | 1273 pchain1 : HOD |
1274 pchain1 = UnionCF A f mf ay supf1 x | |
704 | 1275 |
758
a2947dfff80d
is-max on first transfinite induction is not good
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
757
diff
changeset
|
1276 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
|
1277 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
|
1278 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
|
1279 * 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
|
1280 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
|
1281 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
938 | 1282 ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ? -- ⟪ ab , |
890 | 1283 -- subst (λ k → UChain A f mf ay supf x k ) |
938 | 1284 -- (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
|
1285 |
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1286 zc70 : HasPrev A pchain f x → ¬ xSUP pchain f x |
844 | 1287 zc70 pr xsup = ? |
1288 | |
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1289 no-extension : ¬ ( xSUP (UnionCF A f mf ay supf0 x) supf0 x ) → ZChain A f mf ay x |
879 | 1290 no-extension ¬sp=x = ? where -- record { supf = supf1 ; sup=u = sup=u |
1291 -- ; 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
|
1292 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
|
1293 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z |
838 | 1294 pchain0=1 : pchain ≡ pchain1 |
1295 pchain0=1 = ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where | |
1296 zc10 : {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z | |
1297 zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
938 | 1298 zc10 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc12 fc where |
838 | 1299 zc12 : {z : Ordinal} → FClosure A f (supf0 u) z → odef pchain1 z |
1300 zc12 (fsuc x fc) with zc12 fc | |
1301 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
938 | 1302 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ |
890 | 1303 zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u ? ? (init ? ? ) ⟫ |
838 | 1304 zc11 : {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z |
1305 zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ | |
938 | 1306 zc11 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc13 fc where |
838 | 1307 zc13 : {z : Ordinal} → FClosure A f (supf1 u) z → odef pchain z |
1308 zc13 (fsuc x fc) with zc13 fc | |
1309 ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ | |
938 | 1310 ... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x is-sup (fsuc _ fc₁) ⟫ |
838 | 1311 zc13 (init asu su=z ) with trio< u x |
890 | 1312 ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u ? ? (init ? ? ) ⟫ |
838 | 1313 ... | tri≈ ¬a b ¬c = ? |
938 | 1314 ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u<x c ) |
832 | 1315 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z) |
797 | 1316 sup {z} z≤x with trio< z x |
838 | 1317 ... | tri< a ¬b ¬c = SUP⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} ) |
815 | 1318 ... | tri≈ ¬a b ¬c = {!!} |
843 | 1319 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
832 | 1320 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup {!!})) |
843 | 1321 sis {z} z≤x with trio< z x |
800 | 1322 ... | tri< a ¬b ¬c = {!!} where |
891 | 1323 zc8 = ZChain.supf-is-minsup (pzc z a) {!!} |
815 | 1324 ... | tri≈ ¬a b ¬c = {!!} |
843 | 1325 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1326 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 | 1327 sup=u {z} ab z≤x is-sup with trio< z x |
950 | 1328 ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b) (ob<x lim a)) ab {!!} record { x≤sup = {!!} } |
1329 ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?) ?) ab {!!} record { x≤sup = {!!} } | |
843 | 1330 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x )) |
797 | 1331 |
703 | 1332 zc5 : ZChain A f mf ay x |
697 | 1333 zc5 with ODC.∋-p O A (* x) |
796 | 1334 ... | 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
|
1335 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain f x ) |
703 | 1336 -- we have to check adding x preserve is-max ZChain A y f mf x |
796 | 1337 ... | case1 pr = no-extension {!!} |
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1338 ... | case2 ¬fy<x with ODC.p∨¬p O (IsMinSup A pchain f ax ) |
879 | 1339 ... | case1 is-sup = ? -- record { supf = supf1 ; sup=u = {!!} |
1340 -- ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!}; asupf = {!!} } -- where -- x is a sup of (zc ?) | |
796 | 1341 ... | case2 ¬x=sup = no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention |
553 | 1342 |
921 | 1343 --- |
1344 --- the maximum chain has fix point of any ≤-monotonic function | |
1345 --- | |
1346 | |
1347 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf ay x | |
1348 SZ f mf {y} ay x = TransFinite {λ z → ZChain A f mf ay z } (λ x → ind f mf ay x ) x | |
1349 | |
934 | 1350 msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
1351 → (zc : ZChain A f mf ay x ) | |
1352 → MinSUP A (UnionCF A f mf ay (ZChain.supf zc) x) | |
1353 msp0 f mf {x} ay zc = minsupP (UnionCF A f mf ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) ztotal where | |
1354 ztotal : IsTotalOrderSet (ZChain.chain zc) | |
1355 ztotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where | |
1356 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
1357 uz01 = chain-total A f mf ay (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb)) | |
922 | 1358 |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1359 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
923
85f6238a38db
use supf of zchain for (nmx : ¬ Maximal A ) → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
922
diff
changeset
|
1360 → (zc : ZChain A f mf ay x ) |
85f6238a38db
use supf of zchain for (nmx : ¬ Maximal A ) → ⊥
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
922
diff
changeset
|
1361 → SUP A (UnionCF A f mf ay (ZChain.supf zc) x) |
934 | 1362 sp0 f mf ay zc = M→S (ZChain.supf zc) (msp0 f mf ay zc ) |
921 | 1363 |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1364 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1365 → (sp1 : SUP A (ZChain.chain zc)) -- & (SUP.sup (sp0 f mf as0 zc )) |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1366 → (ssp<as : ZChain.supf zc (& (SUP.sup sp1)) o< ZChain.supf zc (& A)) |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1367 → f (& (SUP.sup sp1)) ≡ & (SUP.sup sp1) |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1368 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
|
1369 chain = ZChain.chain zc |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1370 supf = ZChain.supf zc |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1371 sp : Ordinal |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1372 sp = & (SUP.sup sp1) |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1373 asp : odef A sp |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1374 asp = SUP.as sp1 |
921 | 1375 z10 : {a b : Ordinal } → (ca : odef chain a ) → supf b o< supf (& A) → (ab : odef A b ) |
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1376 → HasPrev A chain f b ∨ IsMinSup A (UnionCF A f mf ab (ZChain.supf zc) b) f ab |
921 | 1377 → * a < * b → odef chain b |
957 | 1378 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
|
1379 z22 : sp o< & A |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1380 z22 = z09 asp |
950 | 1381 x≤sup : {x : HOD} → chain ∋ x → (x ≡ SUP.sup sp1 ) ∨ (x < SUP.sup sp1 ) |
1382 x≤sup bz = SUP.x≤sup sp1 bz | |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1383 z12 : odef chain sp |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1384 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
|
1385 ... | 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
|
1386 ... | 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
|
1387 (case2 z19 ) z13 where |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1388 z13 : * (& s) < * sp |
950 | 1389 z13 with x≤sup ( ZChain.chain∋init zc ) |
921 | 1390 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) |
1391 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1392 z19 : IsMinSup A (UnionCF A f mf as0 (ZChain.supf zc) sp) f asp |
957 | 1393 z19 = record { x≤sup = ? ; minsup = ? } where |
921 | 1394 z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) |
950 | 1395 z20 {y} zy with x≤sup (subst (λ k → odef chain k ) (sym &iso) zy) |
921 | 1396 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) |
1397 ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) | |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1398 ztotal : IsTotalOrderSet (ZChain.chain zc) |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1399 ztotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1400 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1401 uz01 = chain-total A f mf as0 supf ( (proj2 ca)) ( (proj2 cb)) |
935
ed711d7be191
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
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|
1402 z14 : f sp ≡ sp |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1403 z14 with ztotal (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1404 ... | 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
|
1405 z16 : ⊥ |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1406 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.as sp1 )) |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1407 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1408 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1409 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1410 ... | tri> ¬a ¬b c = ⊥-elim z17 where |
936 | 1411 z15 : (* (f sp) ≡ SUP.sup sp1) ∨ (* (f sp) < SUP.sup sp1 ) |
950 | 1412 z15 = x≤sup (subst (λ k → odef chain k ) (sym &iso) (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
|
1413 z17 : ⊥ |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1414 z17 with z15 |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1415 ... | case1 eq = ¬b eq |
936 | 1416 ... | case2 lt = ¬a lt |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1417 |
952 | 1418 tri : {n : Level} (u w : Ordinal ) { R : Set n } → ( u o< w → R ) → ( u ≡ w → R ) → ( w o< u → R ) → R |
1419 tri {_} u w p q r with trio< u w | |
1420 ... | tri< a ¬b ¬c = p a | |
1421 ... | tri≈ ¬a b ¬c = q b | |
1422 ... | tri> ¬a ¬b c = r c | |
1423 | |
1424 or : {n m r : Level } {P : Set n } {Q : Set m} {R : Set r} → P ∨ Q → ( P → R ) → (Q → R ) → R | |
1425 or (case1 p) p→r q→r = p→r p | |
1426 or (case2 q) p→r q→r = q→r q | |
1427 | |
921 | 1428 |
1429 -- ZChain contradicts ¬ Maximal | |
1430 -- | |
1431 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) | |
1432 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
1433 -- | |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1434 |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1435 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥ |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1436 z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.as sp1 )))) |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1437 (subst (λ k → odef A (& k)) (sym *iso) (SUP.as sp1) ) |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1438 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1439 (sp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc ) ss<sa ))) -- x ≡ f x ̄ |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1440 (proj1 (cf-is-<-monotonic nmx c (SUP.as sp1 ))) where -- x < f x |
937 | 1441 |
927 | 1442 supf = ZChain.supf zc |
934 | 1443 msp1 : MinSUP A (ZChain.chain zc) |
1444 msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc | |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1445 sp1 : SUP A (ZChain.chain zc) |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1446 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc |
934 | 1447 c : Ordinal |
1448 c = & ( SUP.sup sp1 ) | |
1449 mc = MinSUP.sup msp1 | |
943 | 1450 mc<A : mc o< & A |
1451 mc<A = ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫ | |
934 | 1452 c=mc : c ≡ mc |
1453 c=mc = &iso | |
1454 z20 : mc << cf nmx mc | |
1455 z20 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1) ) | |
1456 asc : odef A (supf mc) | |
928 | 1457 asc = ZChain.asupf zc |
1458 spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) | |
1459 spd = ysup (cf nmx) (cf-is-≤-monotonic nmx) asc | |
1460 d = MinSUP.sup spd | |
1461 d<A : d o< & A | |
1462 d<A = ∈∧P→o< ⟪ MinSUP.asm spd , lift true ⟫ | |
929 | 1463 msup : MinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) |
1464 msup = ZChain.minsup zc (o<→≤ d<A) | |
928 | 1465 sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) ) |
1466 sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A) | |
937 | 1467 |
943 | 1468 sc<<d : {mc : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) |
934 | 1469 → supf mc << MinSUP.sup spd |
943 | 1470 sc<<d {mc} asc spd = z25 where |
934 | 1471 d1 : Ordinal |
938 | 1472 d1 = MinSUP.sup spd -- supf d1 ≡ d |
934 | 1473 z24 : (supf mc ≡ d1) ∨ ( supf mc << d1 ) |
950 | 1474 z24 = MinSUP.x≤sup spd (init asc refl) |
939 | 1475 -- |
1476 -- f ( f .. ( supf mc ) <= d1 | |
1477 -- f d1 <= d1 | |
1478 -- | |
934 | 1479 z25 : supf mc << d1 |
1480 z25 with z24 | |
1481 ... | case2 lt = lt | |
938 | 1482 ... | case1 eq = ⊥-elim ( <-irr z29 (proj1 (cf-is-<-monotonic nmx d1 (MinSUP.asm spd)) ) ) where |
1483 -- supf mc ≡ d1 | |
939 | 1484 z32 : ((cf nmx (supf mc)) ≡ d1) ∨ ( (cf nmx (supf mc)) << d1 ) |
950 | 1485 z32 = MinSUP.x≤sup spd (fsuc _ (init asc refl)) |
938 | 1486 z29 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1) |
939 | 1487 z29 with z32 |
1488 ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) ) | |
1489 ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt) | |
1490 | |
946 | 1491 fsc<<d : {mc z : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) |
1492 → (fc : FClosure A (cf nmx) (supf mc) z) → z << MinSUP.sup spd | |
1493 fsc<<d {mc} {z} asc spd fc = z25 where | |
1494 d1 : Ordinal | |
1495 d1 = MinSUP.sup spd -- supf d1 ≡ d | |
1496 z24 : (z ≡ d1) ∨ ( z << d1 ) | |
950 | 1497 z24 = MinSUP.x≤sup spd fc |
946 | 1498 -- |
1499 -- f ( f .. ( supf mc ) <= d1 | |
1500 -- f d1 <= d1 | |
1501 -- | |
1502 z25 : z << d1 | |
1503 z25 with z24 | |
1504 ... | case2 lt = lt | |
1505 ... | case1 eq = ⊥-elim ( <-irr z29 (proj1 (cf-is-<-monotonic nmx d1 (MinSUP.asm spd)) ) ) where | |
1506 -- supf mc ≡ d1 | |
1507 z32 : ((cf nmx z) ≡ d1) ∨ ( (cf nmx z) << d1 ) | |
950 | 1508 z32 = MinSUP.x≤sup spd (fsuc _ fc) |
946 | 1509 z29 : (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1) |
1510 z29 with z32 | |
1511 ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1) ) | |
1512 ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt) | |
1513 | |
943 | 1514 smc<<d : supf mc << d |
1515 smc<<d = sc<<d asc spd | |
1516 | |
1517 sz<<c : {z : Ordinal } → z o< & A → supf z <= mc | |
950 | 1518 sz<<c z<A = MinSUP.x≤sup msp1 (ZChain.csupf zc (z09 (ZChain.asupf zc) )) |
943 | 1519 |
1520 sc=c : supf mc ≡ mc | |
1521 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
|
1522 is-sup : IsMinSup A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc) (cf nmx) (MinSUP.asm msp1) |
954 | 1523 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 ) |
1524 ; minsup = ? } | |
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1525 not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc) (cf nmx) mc |
952 | 1526 not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-init fc ⟫ ; x=fy = x=fy } = ⊥-elim ( <-irr z31 z30 ) where |
1527 z30 : * mc < * (cf nmx mc) | |
1528 z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1)) | |
1529 z31 : ( * (cf nmx mc) ≡ * mc ) ∨ ( * (cf nmx mc) < * mc ) | |
1530 z31 = <=to≤ ( MinSUP.x≤sup msp1 (subst (λ k → odef (ZChain.chain zc) (cf nmx k)) (sym x=fy) | |
1531 ⟪ proj2 (cf-is-≤-monotonic nmx _ (proj2 (cf-is-≤-monotonic nmx _ ua1 ) )) , ch-init (fsuc _ (fsuc _ fc)) ⟫ )) | |
951 | 1532 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 | 1533 z30 : * mc < * (cf nmx mc) |
1534 z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1)) | |
951 | 1535 z31 : ( supf mc ≡ mc ) ∨ ( * (supf mc) < * mc ) |
1536 z31 = MinSUP.x≤sup msp1 (ZChain.csupf zc (z09 (ZChain.asupf zc) )) | |
1537 z32 : * (supf mc) < * (cf nmx (cf nmx y)) | |
1538 z32 = ftrans<=-< z31 (subst (λ k → * mc < * k ) (cong (cf nmx) x=fy) z30 ) | |
1539 z48 : ( * (cf nmx (cf nmx y)) ≡ * (supf mc)) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) | |
1540 z48 = <=to≤ (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A u<x (fsuc _ ( fsuc _ fc ))) | |
943 | 1541 |
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1542 is-sup : IsMinSup A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) (cf nmx) (MinSUP.asm spd) |
954 | 1543 is-sup = record { x≤sup = z22 ; minsup = ? } where |
952 | 1544 z23 : {z : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z → (z ≡ MinSUP.sup spd) ∨ (z << MinSUP.sup spd) |
1545 z23 lt = MinSUP.x≤sup spd lt | |
1546 z22 : {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y → | |
940
aee83a7f9f57
not-hasprev z29 and z31 cause memory exhaust
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
939
diff
changeset
|
1547 (y ≡ MinSUP.sup spd) ∨ (y << MinSUP.sup spd) |
952 | 1548 z22 {a} ⟪ aa , ch-init fc ⟫ = case2 ( ( ftrans<=-< z32 ( sc<<d {mc} asc spd ) )) where |
1549 z32 : ( a ≡ supf mc ) ∨ ( * a < * (supf mc) ) | |
1550 z32 = ZChain.fcy<sup zc (o<→≤ mc<A) fc | |
1551 z22 {a} ⟪ aa , ch-is-sup u u<x is-sup1 fc ⟫ = tri u (supf mc) | |
1552 z60 z61 ( λ sc<u → ⊥-elim ( o≤> ( subst (λ k → k o≤ supf mc) (ChainP.supu=u is-sup1) z51) sc<u )) where | |
1553 z53 : supf u o< supf (& A) | |
1554 z53 = ordtrans<-≤ u<x (ZChain.supf-mono zc (o<→≤ d<A) ) | |
1555 z52 : ( u ≡ mc ) ∨ ( u << mc ) | |
1556 z52 = MinSUP.x≤sup msp1 ⟪ subst (λ k → odef A k ) (ChainP.supu=u is-sup1) (A∋fcs _ _ (cf-is-≤-monotonic nmx) fc) | |
1557 , ch-is-sup u z53 is-sup1 (init (ZChain.asupf zc) (ChainP.supu=u is-sup1)) ⟫ | |
1558 z56 : u ≡ mc → supf u ≡ supf mc | |
1559 z56 eq = cong supf eq | |
1560 z57 : u << mc → supf u o≤ supf mc | |
1561 z57 lt = ZChain.supf-<= zc (case2 z58) where | |
1562 z58 : supf u << supf mc -- supf u o< supf d -- supf u << supf d | |
1563 z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c) lt | |
1564 z51 : supf u o≤ supf mc | |
1565 z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57 | |
1566 z60 : u o< supf mc → (a ≡ d ) ∨ ( * a < * d ) | |
1567 z60 u<smc = case2 ( ftrans<=-< (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A | |
1568 (subst (λ k → k o< supf mc) (sym (ChainP.supu=u is-sup1)) u<smc) fc ) smc<<d ) | |
1569 z61 : u ≡ supf mc → (a ≡ d ) ∨ ( * a < * d ) | |
1570 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 ) ) | |
940
aee83a7f9f57
not-hasprev z29 and z31 cause memory exhaust
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
939
diff
changeset
|
1571 -- u<x : ZChain.supf zc u o< ZChain.supf zc d |
aee83a7f9f57
not-hasprev z29 and z31 cause memory exhaust
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
939
diff
changeset
|
1572 -- supf u o< spuf c → order |
aee83a7f9f57
not-hasprev z29 and z31 cause memory exhaust
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
939
diff
changeset
|
1573 |
958
33891adf80ea
IsMinSup contains not HasPrev
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
957
diff
changeset
|
1574 not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) (cf nmx) d |
952 | 1575 not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-init fc ⟫ ; x=fy = x=fy } = ⊥-elim ( <-irr z31 z30 ) where |
1576 z30 : * d < * (cf nmx d) | |
1577 z30 = proj1 (cf-is-<-monotonic nmx d (MinSUP.asm spd)) | |
1578 z32 : ( cf nmx (cf nmx y) ≡ supf mc ) ∨ ( * (cf nmx (cf nmx y)) < * (supf mc) ) | |
1579 z32 = ZChain.fcy<sup zc (o<→≤ mc<A) (fsuc _ (fsuc _ fc)) | |
1580 z31 : ( * (cf nmx d) ≡ * d ) ∨ ( * (cf nmx d) < * d ) | |
1581 z31 = case2 ( subst (λ k → * (cf nmx k) < * d ) (sym x=fy) ( ftrans<=-< z32 ( sc<<d {mc} asc spd ) )) | |
948 | 1582 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 |
1583 z45 : (* (cf nmx (cf nmx y)) ≡ * d) ∨ (* (cf nmx (cf nmx y)) < * d) → (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d) | |
1584 z45 p = subst (λ k → (* (cf nmx k) ≡ * d) ∨ (* (cf nmx k) < * d)) (sym x=fy) p | |
1585 z48 : supf mc << d | |
1586 z48 = sc<<d {mc} asc spd | |
949 | 1587 z53 : supf u o< supf (& A) |
1588 z53 = ordtrans<-≤ u<x (ZChain.supf-mono zc (o<→≤ d<A) ) | |
1589 z52 : ( u ≡ mc ) ∨ ( u << mc ) | |
950 | 1590 z52 = MinSUP.x≤sup msp1 ⟪ subst (λ k → odef A k ) (ChainP.supu=u is-sup1) (A∋fcs _ _ (cf-is-≤-monotonic nmx) fc) |
949 | 1591 , ch-is-sup u z53 is-sup1 (init (ZChain.asupf zc) (ChainP.supu=u is-sup1)) ⟫ |
1592 z51 : supf u o≤ supf mc | |
952 | 1593 z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57 where |
950 | 1594 z56 : u ≡ mc → supf u ≡ supf mc |
1595 z56 eq = cong supf eq | |
1596 z57 : u << mc → supf u o≤ supf mc | |
1597 z57 lt = ZChain.supf-<= zc (case2 z58) where | |
951 | 1598 z58 : supf u << supf mc -- supf u o< supf d -- supf u << supf d |
1599 z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c) lt | |
949 | 1600 z49 : supf u o< supf mc → (cf nmx (cf nmx y) ≡ supf mc) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) |
1601 z49 su<smc = ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A su<smc (fsuc _ ( fsuc _ fc )) | |
1602 z50 : (cf nmx (cf nmx y) ≡ supf d) ∨ (* (cf nmx (cf nmx y)) < * (supf d) ) | |
1603 z50 = ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) d<A u<x (fsuc _ ( fsuc _ fc )) | |
948 | 1604 z47 : {mc d1 : Ordinal } {asc : odef A (supf mc)} (spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc )) |
951 | 1605 → (cf nmx (cf nmx y) ≡ supf mc) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) → supf mc << d1 |
948 | 1606 → * (cf nmx (cf nmx y)) < * d1 |
950 | 1607 z47 {mc} {d1} {asc} spd (case1 eq) smc<d = subst (λ k → k < * d1 ) (sym (cong (*) eq)) smc<d |
1608 z47 {mc} {d1} {asc} spd (case2 lt) smc<d = IsStrictPartialOrder.trans PO lt smc<d | |
948 | 1609 z30 : * d < * (cf nmx d) |
1610 z30 = proj1 (cf-is-<-monotonic nmx d (MinSUP.asm spd)) | |
1611 z46 : (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d) | |
952 | 1612 z46 = or (osuc-≡< z51) z55 z54 where |
950 | 1613 z55 : supf u ≡ supf mc → (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d) |
1614 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)) ) ) | |
1615 z54 : supf u o< supf mc → (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d) | |
1616 z54 lt = z45 (case2 (z47 {mc} {d} {asc} spd (z49 lt) z48 )) | |
1617 -- z46 with osuc-≡< z51 | |
1618 -- ... | 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 ) | |
1619 -- ... | case2 lt = z45 (case2 (z47 {mc} {d} {asc} spd (z49 lt) z48 )) | |
948 | 1620 |
940
aee83a7f9f57
not-hasprev z29 and z31 cause memory exhaust
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
939
diff
changeset
|
1621 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
|
1622 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
|
1623 |
934 | 1624 sc<sd : {mc d : Ordinal } → supf mc << supf d → supf mc o< supf d |
1625 sc<sd {mc} {d} sc<<sd with osuc-≡< ( ZChain.supf-<= zc (case2 sc<<sd ) ) | |
1626 ... | case1 eq = ⊥-elim ( <-irr (case1 (cong (*) (sym eq) )) sc<<sd ) | |
1627 ... | case2 lt = lt | |
1628 | |
1629 sms<sa : supf mc o< supf (& A) | |
1630 sms<sa with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫) )) | |
1631 ... | case2 lt = lt | |
943 | 1632 ... | case1 eq = ⊥-elim ( o<¬≡ eq ( ordtrans<-≤ (sc<sd (subst (λ k → supf mc << k ) (sym sd=d) (sc<<d {mc} asc spd)) ) |
934 | 1633 ( ZChain.supf-mono zc (o<→≤ d<A )))) |
928 | 1634 |
927 | 1635 ss<sa : supf c o< supf (& A) |
934 | 1636 ss<sa = subst (λ k → supf k o< supf (& A)) (sym c=mc) sms<sa |
1637 | |
551 | 1638 zorn00 : Maximal A |
1639 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
804 | 1640 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
551 | 1641 -- yes we have the maximal |
1642 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
606 | 1643 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
551 | 1644 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
1645 zorn01 = proj1 zorn03 | |
1646 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
1647 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
927 | 1648 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where |
551 | 1649 -- if we have no maximal, make ZChain, which contradict SUP condition |
1650 nmx : ¬ Maximal A | |
1651 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
1652 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
804 | 1653 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 | 1654 |
516 | 1655 -- usage (see filter.agda ) |
1656 -- | |
497 | 1657 -- _⊆'_ : ( A B : HOD ) → Set n |
1658 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
482 | 1659 |
497 | 1660 -- MaximumSubset : {L P : HOD} |
1661 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
1662 -- → IsPartialOrderSet P _⊆'_ | |
1663 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
1664 -- → Maximal P (_⊆'_) | |
1665 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |