Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 1090:2cf182f0f583
order removal
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 19 Dec 2022 09:50:51 +0900 |
parents | b627e3ea7266 |
children | 63c1167b2343 |
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 |
966 | 4 open import Relation.Binary |
552 | 5 open import Relation.Binary.Core |
6 open import Relation.Binary.PropositionalEquality | |
966 | 7 import OD |
552 | 8 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where |
431 | 9 |
560 | 10 -- |
966 | 11 -- Zorn-lemma : { A : HOD } |
12 -- → o∅ o< & A | |
560 | 13 -- → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
966 | 14 -- → Maximal A |
560 | 15 -- |
16 | |
431 | 17 open import zf |
477 | 18 open import logic |
19 -- open import partfunc {n} O | |
20 | |
966 | 21 open import Relation.Nullary |
22 open import Data.Empty | |
23 import BAlgbra | |
431 | 24 |
555 | 25 open import Data.Nat hiding ( _<_ ; _≤_ ) |
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26 open import Data.Nat.Properties |
966 | 27 open import nat |
555 | 28 |
431 | 29 |
30 open inOrdinal O | |
31 open OD O | |
32 open OD.OD | |
33 open ODAxiom odAxiom | |
477 | 34 import OrdUtil |
35 import ODUtil | |
431 | 36 open Ordinals.Ordinals O |
37 open Ordinals.IsOrdinals isOrdinal | |
38 open Ordinals.IsNext isNext | |
39 open OrdUtil O | |
477 | 40 open ODUtil O |
41 | |
42 | |
43 import ODC | |
44 | |
45 open _∧_ | |
46 open _∨_ | |
47 open Bool | |
431 | 48 |
49 open HOD | |
50 | |
560 | 51 -- |
52 -- Partial Order on HOD ( possibly limited in A ) | |
53 -- | |
54 | |
966 | 55 _<<_ : (x y : Ordinal ) → Set n |
570 | 56 x << y = * x < * y |
57 | |
1031 | 58 _≤_ : (x y : Ordinal ) → Set n -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain |
59 x ≤ y = (x ≡ y ) ∨ ( * x < * y ) | |
765 | 60 |
966 | 61 POO : IsStrictPartialOrder _≡_ _<<_ |
62 POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } | |
63 ; trans = IsStrictPartialOrder.trans PO | |
570 | 64 ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y |
966 | 65 ; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } } |
66 | |
1031 | 67 ≤-ftrans : {x y z : Ordinal } → x ≤ y → y ≤ z → x ≤ z |
554 | 68 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl |
69 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
70 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
71 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
72 | |
1031 | 73 ftrans≤-< : {x y z : Ordinal } → x ≤ y → y << z → x << z |
74 ftrans≤-< {x} {y} {z} (case1 eq) y<z = subst (λ k → k < * z) (sym (cong (*) eq)) y<z | |
75 ftrans≤-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z | |
951 | 76 |
1031 | 77 ftrans<-≤ : {x y z : Ordinal } → x << y → y ≤ z → x << z |
78 ftrans<-≤ {x} {y} {z} x<y (case1 eq) = subst (λ k → * x < k ) ((cong (*) eq)) x<y | |
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79 ftrans<-≤ {x} {y} {z} x<y (case2 lt) = IsStrictPartialOrder.trans PO x<y lt |
779 | 80 |
556 | 81 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
82 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
83 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
84 (IsStrictPartialOrder.trans PO b<a a<b) | |
490 | 85 |
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86 <<-irr : {a b : Ordinal } → a ≤ b → b << a → ⊥ |
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87 <<-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (cong (*) (sym a=b)) b<a |
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88 <<-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl (IsStrictPartialOrder.trans PO b<a a<b) |
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89 |
561 | 90 ptrans = IsStrictPartialOrder.trans PO |
91 | |
492 | 92 open _==_ |
93 open _⊆_ | |
94 | |
966 | 95 -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A |
879 | 96 -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x |
97 -- <-TransFinite = ? | |
98 | |
530 | 99 -- |
560 | 100 -- Closure of ≤-monotonic function f has total order |
530 | 101 -- |
102 | |
1031 | 103 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set n |
104 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( x ≤ (f x) ) ∧ odef A (f x ) | |
530 | 105 |
992 | 106 <-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set n |
107 <-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x < * (f x) ) ∧ odef A (f x ) | |
108 | |
551 | 109 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
783 | 110 init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1 |
555 | 111 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
554 | 112 |
556 | 113 A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y |
783 | 114 A∋fc {A} s f mf (init as refl ) = as |
556 | 115 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) |
555 | 116 |
714 | 117 A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s |
783 | 118 A∋fcs {A} s f mf (init as refl) = as |
966 | 119 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy |
714 | 120 |
1031 | 121 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → s ≤ y |
783 | 122 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl |
556 | 123 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) |
1031 | 124 ... | case1 x=fx = subst₂ (λ j k → j ≤ k ) refl x=fx (s≤fc s f mf fcy) |
966 | 125 ... | case2 x<fx with s≤fc {A} s f mf fcy |
1031 | 126 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym (cong (*) s≡x )) refl x<fx ) |
556 | 127 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) |
555 | 128 |
800 | 129 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
130 fcn s mf (init as refl) = zero | |
131 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) | |
132 ... | case1 eq = fcn s mf p | |
133 ... | case2 y<fy = suc (fcn s mf p ) | |
134 | |
966 | 135 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
800 | 136 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y |
137 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where | |
138 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
139 fc06 {x} {y} refl {j} not = fc08 not where | |
966 | 140 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
800 | 141 fc08 () |
142 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x | |
143 fc07 {x} (init as refl) eq = refl | |
144 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) | |
1031 | 145 ... | case1 x=fx = subst (λ k → * s ≡ k ) (cong (*) x=fx) ( fc07 cx eq ) |
800 | 146 -- ... | case2 x<fx = ? |
147 fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → * x ≡ * y | |
148 fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ ) | |
149 fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ ) | |
150 fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl | |
151 fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
1031 | 152 ... | case1 y=fy = subst (λ k → * s ≡ * k ) y=fy (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y ) |
800 | 153 fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) |
1031 | 154 ... | case1 x=fx = subst (λ k → * k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) i=x i=y ) |
800 | 155 fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) |
1031 | 156 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → * j ≡ * k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) |
800 | 157 fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) |
1031 | 158 ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → * j ≡ * k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) |
159 ... | case1 x=fx | case2 y<fy = subst (λ k → * k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
800 | 160 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) |
161 fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
162 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
1031 | 163 ... | case1 eq = trans (sym (cong (*) eq )) ( fc02 x1 cx1 i=x1 ) -- derefence while f x ≡ x |
800 | 164 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where |
165 fc04 : * x1 ≡ * y | |
166 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
1031 | 167 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ * k ) y=fy (fc03 y cy j=y) where |
800 | 168 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 |
169 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x) | |
170 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
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171 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) (cong (*) eq) |
800 | 172 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where |
173 fc05 : * x ≡ * y1 | |
174 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
175 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) | |
176 | |
177 | |
178 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
179 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
180 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where | |
181 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) | |
182 fc06 {x} {y} refl {j} not = fc08 not where | |
966 | 183 fc08 : {j : ℕ} → ¬ suc j ≡ 0 |
800 | 184 fc08 () |
185 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y | |
186 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) | |
187 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
1031 | 188 ... | case1 y=fy = subst (λ k → * x < k ) (cong (*) y=fy) ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) |
800 | 189 ... | case2 y<fy with <-cmp (fcn s mf cx ) i |
190 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
966 | 191 ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy |
800 | 192 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where |
193 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
966 | 194 fc03 eq = cong pred eq |
195 fc02 : * x < * y1 | |
800 | 196 fc02 = fc01 i cx cy (fc03 i=y ) a |
197 | |
557 | 198 |
966 | 199 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
554 | 200 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
800 | 201 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
202 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
203 fc11 : * x < * y | |
204 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
205 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
206 fc10 : * x ≡ * y | |
207 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
966 | 208 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where |
800 | 209 fc12 : * y < * x |
210 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
600 | 211 |
563 | 212 |
729 | 213 |
560 | 214 -- open import Relation.Binary.Properties.Poset as Poset |
215 | |
216 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) | |
217 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
218 | |
567 | 219 ⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B |
568 | 220 ⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) |
567 | 221 |
568 | 222 _⊆'_ : ( A B : HOD ) → Set n |
223 _⊆'_ A B = {x : Ordinal } → odef A x → odef B x | |
560 | 224 |
225 -- | |
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226 -- inductive masum tree from x |
560 | 227 -- tree structure |
228 -- | |
554 | 229 |
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230 record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where |
533 | 231 field |
836 | 232 ax : odef A x |
534 | 233 y : Ordinal |
541 | 234 ay : odef B y |
966 | 235 x=fy : x ≡ f y |
529 | 236 |
1031 | 237 record IsSUP (A B : HOD) (x : Ordinal ) : Set n where |
962 | 238 field |
1031 | 239 ax : odef A x |
240 x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) -- B is Total, use positive | |
568 | 241 |
656 | 242 record SUP ( A B : HOD ) : Set (Level.suc n) where |
243 field | |
244 sup : HOD | |
1031 | 245 isSUP : IsSUP A B (& sup) |
246 ax = IsSUP.ax isSUP | |
247 x≤sup = IsSUP.x≤sup isSUP | |
656 | 248 |
690 | 249 -- |
878 | 250 -- |
990 | 251 -- f (f ( ... (supf y))) f (f ( ... (supf z1))) |
878 | 252 -- / | / | |
253 -- / | / | | |
990 | 254 -- supf y < supf z1 < supf z2 |
878 | 255 -- o< o< |
990 | 256 -- |
257 -- if sup z1 ≡ sup z2, the chain is stopped at sup z1, then f (sup z1) ≡ sup z1 | |
1039 | 258 -- this means sup z1 is the Maximal, so f is <-monotonic if we have no Maximal. |
259 -- | |
990 | 260 |
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261 fc-stop : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) { a b : Ordinal } |
990 | 262 → (aa : odef A a ) →( {y : Ordinal} → FClosure A f a y → (y ≡ b ) ∨ (y << b )) → a ≡ b → f a ≡ a |
263 fc-stop A f mf {a} {b} aa x≤sup a=b with x≤sup (fsuc a (init aa refl )) | |
264 ... | case1 eq = trans eq (sym a=b) | |
1031 | 265 ... | case2 lt = ⊥-elim (<-irr (case1 (cong (λ k → * (f k) ) (sym a=b))) (ftrans<-≤ lt fc00 ) ) where |
266 fc00 : b ≤ (f b) | |
990 | 267 fc00 = proj1 (mf _ (subst (λ k → odef A k) a=b aa )) |
268 | |
694 | 269 ∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
270 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
271 | |
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272 -- Union of supf z and FClosure A f y |
662 | 273 |
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274 data UChain { A : HOD } { f : Ordinal → Ordinal } {supf : Ordinal → Ordinal} {y : Ordinal } (ay : odef A y ) |
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275 (x : Ordinal) : (z : Ordinal) → Set n where |
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276 ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain ay x z |
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277 ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) (supu=u : supf u ≡ u) ( fc : FClosure A f (supf u) z ) → UChain ay x z |
1030 | 278 |
1060 | 279 UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD |
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280 UnionCF A f ay supf x |
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281 = record { od = record { def = λ z → odef A z ∧ UChain {A} {f} {supf} ay x z } ; |
1028 | 282 odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } |
283 | |
1061 | 284 -- Union of chain lower than x |
1060 | 285 |
1078 | 286 data IChain {A : HOD} { f : Ordinal → Ordinal } {y : Ordinal } (ay : odef A y ) |
1063 | 287 {x : Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) : (z : Ordinal ) → Set n where |
288 ic-init : {z : Ordinal } (fc : FClosure A f y z) → IChain ay supfz z | |
289 ic-isup : {z : Ordinal} (i : Ordinal) (i<x : i o< x) (s<x : supfz i<x o≤ i ) (fc : FClosure A f (supfz i<x) z) → IChain ay supfz z | |
1060 | 290 |
1063 | 291 UnionIC : ( A : HOD ) ( f : Ordinal → Ordinal ) { x : Ordinal } {y : Ordinal } (ay : odef A y ) (supfz : {z : Ordinal } → z o< x → Ordinal) → HOD |
1078 | 292 UnionIC A f ay supfz |
1063 | 293 = record { od = record { def = λ z → odef A z ∧ IChain {A} {f} ay supfz z } ; |
1060 | 294 odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } |
1028 | 295 |
966 | 296 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) |
297 → supf x o< supf y → x o< y | |
842 | 298 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y |
299 ... | tri< a ¬b ¬c = a | |
300 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
301 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
302 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
303 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
304 | |
1031 | 305 record IsMinSUP ( A B : HOD ) (sup : Ordinal) : Set n where |
306 field | |
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307 as : odef A sup |
1031 | 308 x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) |
309 minsup : { sup1 : Ordinal } → odef A sup1 | |
310 → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 | |
311 | |
879 | 312 record MinSUP ( A B : HOD ) : Set n where |
313 field | |
314 sup : Ordinal | |
1031 | 315 isMinSUP : IsMinSUP A B sup |
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316 as = IsMinSUP.as isMinSUP |
1031 | 317 x≤sup = IsMinSUP.x≤sup isMinSUP |
318 minsup = IsMinSUP.minsup isMinSUP | |
879 | 319 |
320 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A | |
321 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) | |
322 | |
966 | 323 chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) |
919 | 324 (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b |
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325 → odef (UnionCF A f ay supf a) c → odef (UnionCF A f ay supf b) c |
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326 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = ⟪ ua , ch-init fc ⟫ |
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327 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-is-sup u u<x supu=u fc ⟫ = ⟪ ua , ch-is-sup u (ordtrans<-≤ u<x a≤b) supu=u fc ⟫ |
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328 |
1038 | 329 record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f) |
783 | 330 {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where |
655 | 331 field |
966 | 332 supf : Ordinal → Ordinal |
1028 | 333 |
1090 | 334 supf-mono : {a b : Ordinal } → a o≤ b → supf a o≤ supf b |
1046 | 335 cfcs : {a b w : Ordinal } → a o< b → b o≤ z → supf a o< b → FClosure A f (supf a) w → odef (UnionCF A f ay supf b) w |
868 | 336 asupf : {x : Ordinal } → odef A (supf x) |
1059 | 337 zo≤sz : {x : Ordinal } → x o≤ z → x o≤ supf x |
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338 is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → IsMinSUP A (UnionCF A f ay supf x) (supf x) |
994 | 339 |
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340 chain : HOD |
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341 chain = UnionCF A f ay supf z |
861 | 342 chain⊆A : chain ⊆' A |
343 chain⊆A = λ lt → proj1 lt | |
934 | 344 |
1058 | 345 chain∋init : {x : Ordinal } → odef (UnionCF A f ay supf x) y |
346 chain∋init {x} = ⟪ ay , ch-init (init ay refl) ⟫ | |
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347 |
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348 mf : ≤-monotonic-f A f |
1040 | 349 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where |
350 mf00 : * x < * (f x) | |
351 mf00 = proj1 ( mf< x ax ) | |
1038 | 352 |
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353 f-next : {a z : Ordinal} → odef (UnionCF A f ay supf z) a → odef (UnionCF A f ay supf z) (f a) |
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354 f-next {a} ⟪ ua , ch-init fc ⟫ = ⟪ proj2 ( mf _ ua) , ch-init (fsuc _ fc) ⟫ |
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355 f-next {a} ⟪ ua , ch-is-sup u su<x su=u fc ⟫ = ⟪ proj2 ( mf _ ua) , ch-is-sup u su<x su=u (fsuc _ fc) ⟫ |
861 | 356 |
966 | 357 supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y |
825 | 358 supf-inject {x} {y} sx<sy with trio< x y |
359 ... | tri< a ¬b ¬c = a | |
360 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) | |
361 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) | |
362 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) | |
363 ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) | |
798 | 364 |
1038 | 365 csupf : {b : Ordinal } → supf b o< supf z → supf b o< z → odef (UnionCF A f ay supf z) (supf b) -- supf z is not an element of this chain |
366 csupf {b} sb<sz sb<z = cfcs (supf-inject sb<sz) o≤-refl sb<z (init asupf refl) | |
994 | 367 |
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368 minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f ay supf x) |
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369 minsup {x} x≤z = record { sup = supf x ; isMinSUP = is-minsup x≤z } |
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370 |
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371 supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup (minsup x≤z) |
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372 supf-is-minsup _ = refl |
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373 |
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374 -- different from order because y o< supf |
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375 fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) |
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376 fcy<sup {u} {w} u≤z fc with MinSUP.x≤sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc) |
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377 , ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫ |
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378 ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) )) |
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379 ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt ) |
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380 |
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381 initial : {x : Ordinal } → x o≤ z → odef (UnionCF A f ay supf x) x → y ≤ x |
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382 initial {x} x≤z ⟪ aa , ch-init fc ⟫ = s≤fc y f mf fc |
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383 initial {x} x≤z ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ≤-ftrans (fcy<sup (ordtrans u<x x≤z) (init ay refl)) (s≤fc _ f mf fc) |
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384 |
1090 | 385 supfeq : {a b : Ordinal } → a o≤ z → b o≤ z → UnionCF A f ay supf a ≡ UnionCF A f ay supf b → supf a ≡ supf b |
386 supfeq {a} {b} a≤z b≤z ua=ub with trio< (supf a) (supf b) | |
387 ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> ( | |
388 IsMinSUP.minsup (is-minsup b≤z) asupf (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (subst (λ k → odef k z) (sym ua=ub) uzb)) ) sa<sb ) | |
389 ... | tri≈ ¬a b ¬c = b | |
390 ... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> ( | |
391 IsMinSUP.minsup (is-minsup a≤z) asupf (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (subst (λ k → odef k z) ua=ub uza)) ) sb<sa ) | |
392 | |
393 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z | |
394 → IsSUP A (UnionCF A f ay supf b) b → supf b ≡ b | |
395 sup=u {b} ab b≤z is-sup = z50 where | |
396 z48 : supf b o≤ b | |
397 z48 = IsMinSUP.minsup (is-minsup b≤z) ab (λ ux → IsSUP.x≤sup is-sup ux ) | |
398 z50 : supf b ≡ b | |
399 z50 with trio< (supf b) b | |
400 ... | tri< sb<b ¬b ¬c = ⊥-elim ( o≤> z47 sb<b ) where | |
401 z47 : b o≤ supf b | |
402 z47 = zo≤sz b≤z | |
403 ... | tri≈ ¬a b ¬c = b | |
404 ... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb ) | |
405 | |
406 union-max : {a b : Ordinal } → b o≤ supf a → b o≤ z → supf a o< supf b → UnionCF A f ay supf a ≡ UnionCF A f ay supf b | |
407 union-max {a} {b} b≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where | |
408 z47 : {w : Ordinal } → odef (UnionCF A f ay supf a ) w → odef ( UnionCF A f ay supf b ) w | |
409 z47 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫ | |
410 z47 {w} ⟪ aw , ch-is-sup u u<a su=u fc ⟫ = ⟪ aw , ch-is-sup u u<b su=u fc ⟫ where | |
411 u<b : u o< b | |
412 u<b = ordtrans u<a (supf-inject sa<sb ) | |
413 z48 : {w : Ordinal } → odef (UnionCF A f ay supf b ) w → odef ( UnionCF A f ay supf a ) w | |
414 z48 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫ | |
415 z48 {w} ⟪ aw , ch-is-sup u u<b su=u fc ⟫ = ⟪ aw , ch-is-sup u u<a su=u fc ⟫ where | |
416 u<a : u o< a | |
417 u<a = supf-inject ( osucprev (begin | |
418 osuc (supf u) ≡⟨ cong osuc su=u ⟩ | |
419 osuc u ≤⟨ osucc u<b ⟩ | |
420 b ≤⟨ b≤sa ⟩ | |
421 supf a ∎ )) where open o≤-Reasoning O | |
422 | |
423 x≤supfx→¬sa<sa : {a b : Ordinal } → b o≤ z → b o≤ supf a → ¬ (supf a o< supf b ) | |
424 x≤supfx→¬sa<sa {a} {b} b≤z b≤sa sa<sb = ⊥-elim ( o<¬≡ z27 sa<sb ) where -- x o≤ supf a ∧ supf a o< supf b → ⊥, because it defines the same UnionCF | |
425 z27 : supf a ≡ supf b | |
426 z27 = supfeq (ordtrans (supf-inject sa<sb) b≤z) b≤z ( union-max b≤sa b≤z sa<sb) | |
427 | |
428 order : {a b w : Ordinal } → b o≤ z → supf a o< supf b → FClosure A f (supf a) w → w ≤ supf b | |
429 order {a} {b} {w} b≤z sa<sb fc = IsMinSUP.x≤sup (is-minsup b≤z) (cfcs (supf-inject sa<sb) b≤z sa<b fc) where | |
430 sa<b : supf a o< b | |
431 sa<b with x<y∨y≤x (supf a) b | |
432 ... | case1 lt = lt | |
433 ... | case2 b≤sa = ⊥-elim (x≤supfx→¬sa<sa b≤z b≤sa sa<sb) | |
434 | |
435 supf-idem : {b : Ordinal } → b o≤ z → supf b o≤ z → supf (supf b) ≡ supf b | |
436 supf-idem {b} b≤z sfb≤x = z52 where | |
437 z54 : {w : Ordinal} → odef (UnionCF A f ay supf (supf b)) w → (w ≡ supf b) ∨ (w << supf b) | |
438 z54 {w} ⟪ aw , ch-init fc ⟫ = fcy<sup b≤z fc | |
439 z54 {w} ⟪ aw , ch-is-sup u u<x su=u fc ⟫ = order b≤z (subst (λ k → k o< supf b) (sym su=u) u<x) fc where | |
440 u<b : u o< b | |
441 u<b = supf-inject (subst (λ k → k o< supf b ) (sym (su=u)) u<x ) | |
442 z52 : supf (supf b) ≡ supf b | |
443 z52 = sup=u asupf sfb≤x record { ax = asupf ; x≤sup = z54 } | |
444 | |
445 supf-mono< : {a b : Ordinal } → b o≤ z → supf a o< supf b → supf a << supf b | |
446 supf-mono< {a} {b} b≤z sa<sb with order {a} {b} b≤z sa<sb (init asupf refl) | |
447 ... | case2 lt = lt | |
448 ... | case1 eq = ⊥-elim ( o<¬≡ eq sa<sb ) | |
449 | |
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450 f-total : IsTotalOrderSet chain |
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451 f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = |
1035 | 452 subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso fc-total where |
453 fc-total : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) | |
454 fc-total with trio< ua ub | |
1046 | 455 ... | tri< a₁ ¬b ¬c with ≤-ftrans (order (o<→≤ sub<x) (subst₂ (λ j k → j o< k) (sym sua=ua) (sym sub=ub) a₁) fca ) (s≤fc (supf ub) f mf fcb ) |
1035 | 456 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
457 ct00 : * (& a) ≡ * (& b) | |
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458 ct00 = cong (*) eq1 |
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459 ... | case2 a<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt) |
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460 fc-total | tri≈ _ refl _ = fcn-cmp _ f mf fca fcb |
1046 | 461 fc-total | tri> ¬a ¬b c with ≤-ftrans (order (o<→≤ sua<x) (subst₂ (λ j k → j o< k) (sym sub=ub) (sym sua=ua) c) fcb ) (s≤fc (supf ua) f mf fca ) |
1035 | 462 ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
463 ct00 : * (& a) ≡ * (& b) | |
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464 ct00 = cong (*) (sym eq1) |
1035 | 465 ... | case2 b<a = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a |
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parents:
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466 f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = ft00 where |
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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parents:
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467 ft01 : (& a) ≤ (& b) → Tri ( a < b) ( a ≡ b) ( b < a ) |
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468 ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b (λ lt → ⊥-elim (<-irr (case1 a=b) lt)) where |
1042
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469 a=b : a ≡ b |
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ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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parents:
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470 a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq) |
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ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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471 ft01 (case2 lt) = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt) where |
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472 a<b : a < b |
1033
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parents:
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473 a<b = subst₂ (λ j k → j < k ) *iso *iso lt |
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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474 ft00 : Tri ( a < b) ( a ≡ b) ( b < a ) |
2da8dcbb0825
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475 ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sub<x) fca) (s≤fc {A} _ f mf fcb)) |
2da8dcbb0825
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parents:
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476 f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-init fcb ⟫ = ft00 where |
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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changeset
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477 ft01 : (& b) ≤ (& a) → Tri ( a < b) ( a ≡ b) ( b < a ) |
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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parents:
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478 ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b (λ lt → ⊥-elim (<-irr (case1 a=b) lt)) where |
1042
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479 a=b : a ≡ b |
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480 a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (sym eq)) |
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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parents:
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481 ft01 (case2 lt) = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a where |
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482 b<a : b < a |
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parents:
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483 b<a = subst₂ (λ j k → j < k ) *iso *iso lt |
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
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parents:
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484 ft00 : Tri ( a < b) ( a ≡ b) ( b < a ) |
2da8dcbb0825
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485 ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sua<x) fcb) (s≤fc {A} _ f mf fca)) |
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486 f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-init fcb ⟫ = |
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487 subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp y f mf fca fcb ) |
825 | 488 |
1038 | 489 record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f) |
490 {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf< ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where | |
1027 | 491 supf = ZChain.supf zc |
492 field | |
1033
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parents:
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493 is-max : {a b : Ordinal } → (ca : odef (UnionCF A f ay supf z) a ) → b o< z → (ab : odef A b) |
2da8dcbb0825
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494 → HasPrev A (UnionCF A f ay supf z) f b ∨ IsSUP A (UnionCF A f ay supf b) b |
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495 → * a < * b → odef ((UnionCF A f ay supf z)) b |
1027 | 496 |
497 record Maximal ( A : HOD ) : Set (Level.suc n) where | |
498 field | |
499 maximal : HOD | |
500 as : A ∋ maximal | |
501 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
502 | |
503 -- | |
504 -- supf in TransFinite indution may differ each other, but it is the same because of the minimul sup | |
505 -- | |
1038 | 506 supf-unique : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f) |
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507 {y xa xb : Ordinal} → (ay : odef A y) → (xa o≤ xb ) → (za : ZChain A f mf< ay xa ) (zb : ZChain A f mf< ay xb ) |
1007 | 508 → {z : Ordinal } → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z |
1038 | 509 supf-unique A f mf< {y} {xa} {xb} ay xa≤xb za zb {z} z≤xa = TransFinite0 {λ z → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z } ind z z≤xa where |
1007 | 510 supfa = ZChain.supf za |
511 supfb = ZChain.supf zb | |
512 ind : (x : Ordinal) → ((w : Ordinal) → w o< x → w o≤ xa → supfa w ≡ supfb w) → x o≤ xa → supfa x ≡ supfb x | |
1008 | 513 ind x prev x≤xa = sxa=sxb where |
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514 ma = ZChain.minsup za x≤xa |
1008 | 515 mb = ZChain.minsup zb (OrdTrans x≤xa xa≤xb ) |
516 spa = MinSUP.sup ma | |
517 spb = MinSUP.sup mb | |
518 sax=spa : supfa x ≡ spa | |
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519 sax=spa = ZChain.supf-is-minsup za x≤xa |
1008 | 520 sbx=spb : supfb x ≡ spb |
521 sbx=spb = ZChain.supf-is-minsup zb (OrdTrans x≤xa xa≤xb ) | |
1007 | 522 sxa=sxb : supfa x ≡ supfb x |
523 sxa=sxb with trio< (supfa x) (supfb x) | |
524 ... | tri≈ ¬a b ¬c = b | |
525 ... | tri< a ¬b ¬c = ⊥-elim ( o≤> ( | |
526 begin | |
1008 | 527 supfb x ≡⟨ sbx=spb ⟩ |
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528 spb ≤⟨ MinSUP.minsup mb (MinSUP.as ma) (λ {z} uzb → MinSUP.x≤sup ma (z53 uzb)) ⟩ |
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529 spa ≡⟨ sym sax=spa ⟩ |
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530 supfa x ∎ ) a ) where |
1008 | 531 open o≤-Reasoning O |
1033
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532 z53 : {z : Ordinal } → odef (UnionCF A f ay (ZChain.supf zb) x) z → odef (UnionCF A f ay (ZChain.supf za) x) z |
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533 z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ |
1034 | 534 z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ = ⟪ as , ch-is-sup u u<x (trans ua=ub su=u) z55 ⟫ where |
535 ua=ub : supfa u ≡ supfb u | |
536 ua=ub = prev u u<x (ordtrans u<x x≤xa ) | |
537 z55 : FClosure A f (ZChain.supf za u) z | |
538 z55 = subst (λ k → FClosure A f k z ) (sym ua=ub) fc | |
1007 | 539 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> ( |
540 begin | |
1008 | 541 supfa x ≡⟨ sax=spa ⟩ |
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542 spa ≤⟨ MinSUP.minsup ma (MinSUP.as mb) (λ uza → MinSUP.x≤sup mb (z53 uza)) ⟩ |
1008 | 543 spb ≡⟨ sym sbx=spb ⟩ |
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544 supfb x ∎ ) c ) where |
1009 | 545 open o≤-Reasoning O |
1033
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parents:
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546 z53 : {z : Ordinal } → odef (UnionCF A f ay (ZChain.supf za) x) z → odef (UnionCF A f ay (ZChain.supf zb) x) z |
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547 z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ |
1034 | 548 z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ = ⟪ as , ch-is-sup u u<x (trans ub=ua su=u) z55 ⟫ where |
549 ub=ua : supfb u ≡ supfa u | |
550 ub=ua = sym ( prev u u<x (ordtrans u<x x≤xa )) | |
551 z55 : FClosure A f (ZChain.supf zb u) z | |
552 z55 = subst (λ k → FClosure A f k z ) (sym ub=ua) fc | |
1028 | 553 |
966 | 554 Zorn-lemma : { A : HOD } |
555 → o∅ o< & A | |
568 | 556 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
966 | 557 → Maximal A |
552 | 558 Zorn-lemma {A} 0<A supP = zorn00 where |
571 | 559 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
560 <-irr0 {a} {b} A∋a A∋b = <-irr | |
788 | 561 z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A |
562 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
530 | 563 s : HOD |
966 | 564 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) |
568 | 565 as : A ∋ * ( & s ) |
566 as = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) | |
608
6655f03984f9
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607
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567 as0 : odef A (& s ) |
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568 as0 = subst (λ k → odef A k ) &iso as |
547 | 569 s<A : & s o< & A |
568 | 570 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) |
530 | 571 HasMaximal : HOD |
966 | 572 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
537 | 573 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ |
966 | 574 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) |
532 | 575 Gtx : { x : HOD} → A ∋ x → HOD |
966 | 576 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
537 | 577 z08 : ¬ Maximal A → HasMaximal =h= od∅ |
804 | 578 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) |
537 | 579 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} |
580 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)) | |
581 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 | |
582 ¬x<m : ¬ (* x < * m) | |
966 | 583 ¬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 | 584 |
1027 | 585 -- |
586 -- we have minsup using LEM, this is similar to the proof of the axiom of choice | |
587 -- | |
966 | 588 minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B |
879 | 589 minsupP B B⊆A total = m02 where |
590 xsup : (sup : Ordinal ) → Set n | |
591 xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) | |
592 ∀-imply-or : {A : Ordinal → Set n } {B : Set n } | |
593 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B | |
594 ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM | |
595 ∀-imply-or {A} {B} ∀AB | case1 t = case1 t | |
596 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where | |
597 lemma : ¬ ((x : Ordinal ) → A x) → B | |
598 lemma not with ODC.p∨¬p O B | |
599 lemma not | case1 b = b | |
600 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
601 m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B | |
602 m00 x = TransFinite0 ind x where | |
603 ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) | |
604 → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B | |
605 ind x prev = ∀-imply-or m01 where | |
606 m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B | |
607 m01 z with trio< z x | |
608 ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
609 ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) | |
610 ... | tri< a ¬b ¬c with prev z a | |
611 ... | case2 mins = case2 mins | |
612 ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) | |
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
613 ... | case1 mins = case2 record { sup = z ; isMinSUP = record { as = proj1 mins ; x≤sup = proj2 mins ; minsup = m04 } } where |
879 | 614 m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 |
615 m04 {s} as lt with trio< z s | |
616 ... | tri< a ¬b ¬c = o<→≤ a | |
617 ... | tri≈ ¬a b ¬c = o≤-refl0 b | |
618 ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) | |
619 ... | case2 notz = case1 (λ _ → notz ) | |
620 m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) | |
1031 | 621 m03 not = ⊥-elim ( not s1 (z09 (SUP.ax S)) ⟪ SUP.ax S , m05 ⟫ ) where |
879 | 622 S : SUP A B |
623 S = supP B B⊆A total | |
624 s1 = & (SUP.sup S) | |
625 m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) | |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
626 m05 {w} bw with SUP.x≤sup S bw |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
627 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (trans &iso eq)) |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
628 ... | case2 lt = case2 lt |
966 | 629 m02 : MinSUP A B |
879 | 630 m02 = dont-or (m00 (& A)) m03 |
631 | |
560 | 632 -- Uncountable ascending chain by axiom of choice |
530 | 633 cf : ¬ Maximal A → Ordinal → Ordinal |
532 | 634 cf nmx x with ODC.∋-p O A (* x) |
635 ... | no _ = o∅ | |
636 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
538 | 637 ... | yes nogt = -- no larger element, so it is maximal |
638 ⊥-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 | 639 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
537 | 640 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
641 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
642 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
643 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
644 ... | 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 ⟫ ) | |
645 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
606 | 646 |
647 --- | |
648 --- infintie ascention sequence of f | |
649 --- | |
530 | 650 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
537 | 651 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
530 | 652 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
532 | 653 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
543 | 654 |
803 | 655 -- |
953 | 656 -- maximality of chain |
657 -- | |
658 -- supf is fixed for z ≡ & A , we can prove order and is-max | |
1016 | 659 -- we have supf-unique now, it is provable in the first Tranfinte induction |
803 | 660 |
992 | 661 SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) (mf< : <-monotonic-f A f) |
1038 | 662 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf< ay (& A)) (x : Ordinal) → x o≤ & A → ZChain1 A f mf< ay zc x |
993 | 663 SZ1 f mf mf< {y} ay zc x x≤A = zc1 x x≤A where |
900 | 664 chain-mono1 : {a b c : Ordinal} → a o≤ b |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
665 → odef (UnionCF A f ay (ZChain.supf zc) a) c → odef (UnionCF A f ay (ZChain.supf zc) b) c |
919 | 666 chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
667 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) x) a → (ab : odef A b) |
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
668 → HasPrev A (UnionCF A f ay (ZChain.supf zc) x) f b |
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
669 → * a < * b → odef (UnionCF A f ay (ZChain.supf zc) x) b |
920 | 670 is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev |
1034 | 671 ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫ |
672 ... | ⟪ ab0 , ch-is-sup u u<x su=u fc ⟫ = ⟪ ab , subst (λ k → UChain ay x k ) | |
673 (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x su=u (fsuc _ fc)) ⟫ | |
674 | |
869 | 675 supf = ZChain.supf zc |
676 | |
1038 | 677 zc1 : (x : Ordinal ) → x o≤ & A → ZChain1 A f mf< ay zc x |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
678 zc1 x x≤A with Oprev-p x |
1024
ab72526316bd
supf-< and ZChain1.order is removed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1023
diff
changeset
|
679 ... | yes op = record { is-max = is-max } where |
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
680 px = Oprev.oprev op |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
681 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a → |
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
682 b o< x → (ab : odef A b) → |
1034 | 683 HasPrev A (UnionCF A f ay supf x) f b ∨ IsSUP A (UnionCF A f ay supf b) b → |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
684 * a < * b → odef (UnionCF A f ay supf x) b |
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
685 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P |
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
686 is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b |
989 | 687 is-max {a} {b} ua b<x ab P a<b | case2 is-sup with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) |
1024
ab72526316bd
supf-< and ZChain1.order is removed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1023
diff
changeset
|
688 ... | case2 sb<sx = m10 where |
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
689 b<A : b o< & A |
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
690 b<A = z09 ab |
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
691 m05 : ZChain.supf zc b ≡ b |
1078 | 692 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
693 m10 : odef (UnionCF A f ay supf x) b |
1039 | 694 m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05) |
992 | 695 ... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
696 m17 : MinSUP A (UnionCF A f ay supf x) -- supf z o< supf ( supf x ) |
992 | 697 m17 = ZChain.minsup zc x≤A |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
698 m18 : supf x ≡ MinSUP.sup m17 |
992 | 699 m18 = ZChain.supf-is-minsup zc x≤A |
990 | 700 m10 : f (supf b) ≡ supf b |
701 m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where | |
702 m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) | |
703 m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where | |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
704 m04 : ¬ HasPrev A (UnionCF A f ay supf b) f b |
1019 | 705 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay = |
706 chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } ) | |
707 m05 : ZChain.supf zc b ≡ b | |
1078 | 708 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } |
1019 | 709 m14 : ZChain.supf zc b o< x |
710 m14 = subst (λ k → k o< x ) (sym m05) b<x | |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
711 m13 : odef (UnionCF A f ay supf x) z |
1039 | 712 m13 = ZChain.cfcs zc b<x x≤A m14 fc |
989 | 713 |
1024
ab72526316bd
supf-< and ZChain1.order is removed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1023
diff
changeset
|
714 ... | no lim = record { is-max = is-max } where |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
715 is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a → |
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
716 b o< x → (ab : odef A b) → |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
717 HasPrev A (UnionCF A f ay supf x) f b ∨ IsSUP A (UnionCF A f ay supf b) b → |
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
718 * a < * b → odef (UnionCF A f ay supf x) b |
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
719 is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P |
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
720 is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b |
1058 | 721 is-max {a} {b} ua b<x ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (ZChain.chain∋init zc ) |
1034 | 722 ... | case1 b=y = ⟪ subst (λ k → odef A k ) b=y ay , ch-init (subst (λ k → FClosure A f y k ) b=y (init ay refl )) ⟫ |
990 | 723 ... | case2 y<b with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x)) |
1024
ab72526316bd
supf-< and ZChain1.order is removed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1023
diff
changeset
|
724 ... | case2 sb<sx = m10 where |
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
725 m09 : b o< & A |
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
726 m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) |
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
727 m05 : ZChain.supf zc b ≡ b |
1078 | 728 m05 = ZChain.sup=u zc ab (o<→≤ m09) record { ax = ab ; x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
729 m10 : odef (UnionCF A f ay supf x) b |
1039 | 730 m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05) |
992 | 731 ... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
732 m17 : MinSUP A (UnionCF A f ay supf x) -- supf z o< supf ( supf x ) |
992 | 733 m17 = ZChain.minsup zc x≤A |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
734 m18 : supf x ≡ MinSUP.sup m17 |
992 | 735 m18 = ZChain.supf-is-minsup zc x≤A |
990 | 736 m10 : f (supf b) ≡ supf b |
737 m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where | |
738 m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x) | |
739 m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where | |
1078 | 740 m05 = ZChain.sup=u zc ab (o<→≤ (z09 ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } |
1019 | 741 m14 : ZChain.supf zc b o< x |
742 m14 = subst (λ k → k o< x ) (sym m05) b<x | |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
743 m13 : odef (UnionCF A f ay supf x) z |
1039 | 744 m13 = ZChain.cfcs zc b<x x≤A m14 fc |
727 | 745 |
757 | 746 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD |
966 | 747 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax = |
757 | 748 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 ))) } |
749 | |
966 | 750 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
757 | 751 → IsTotalOrderSet (uchain f mf ay) |
966 | 752 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where |
757 | 753 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
754 uz01 = fcn-cmp y f mf ca cb | |
755 | |
966 | 756 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) |
928 | 757 → MinSUP A (uchain f mf ay) |
966 | 758 ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay) |
757 | 759 |
560 | 760 -- |
547 | 761 -- create all ZChains under o< x |
560 | 762 -- |
608
6655f03984f9
mutual tranfinite in zorn
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
607
diff
changeset
|
763 |
1038 | 764 ind : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) |
765 → ((z : Ordinal) → z o< x → ZChain A f mf< ay z) → ZChain A f mf< ay x | |
766 ind f mf< {y} ay x prev with Oprev-p x | |
1089
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
767 ... | yes op = zc41 sup1 where |
682 | 768 -- |
769 -- we have previous ordinal to use induction | |
770 -- | |
771 px = Oprev.oprev op | |
1038 | 772 zc : ZChain A f mf< ay (Oprev.oprev op) |
966 | 773 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) |
682 | 774 px<x : px o< x |
775 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
|
776 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
|
777 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
|
778 |
709 | 779 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px |
966 | 780 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt |
697 | 781 |
754 | 782 supf0 = ZChain.supf zc |
869 | 783 pchain : HOD |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
784 pchain = UnionCF A f ay supf0 px |
835 | 785 |
857 | 786 supf-mono = ZChain.supf-mono zc |
844 | 787 |
861 | 788 zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) |
966 | 789 zc04 {b} b≤x with trio< b px |
861 | 790 ... | tri< a ¬b ¬c = case1 (o<→≤ a) |
791 ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) | |
792 ... | tri> ¬a ¬b px<b with osuc-≡< b≤x | |
793 ... | case1 eq = case2 eq | |
966 | 794 ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) |
840 | 795 |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
796 mf : ≤-monotonic-f A f |
1039 | 797 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where |
798 mf00 : * x < * (f x) | |
799 mf00 = proj1 ( mf< x ax ) | |
1038 | 800 |
954 | 801 -- |
802 -- find the next value of supf | |
803 -- | |
804 | |
805 pchainpx : HOD | |
1034 | 806 pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain ay px z ) |
1043 | 807 ∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x)) } ; odmax = & A ; <odmax = zc00 } where |
808 zc00 : {z : Ordinal } → (odef A z ∧ UChain ay px z ) ∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x) )→ z o< & A | |
966 | 809 zc00 {z} (case1 lt) = z07 lt |
1043 | 810 zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf (proj1 fc) ) |
954 | 811 |
1043 | 812 zc02 : { a b : Ordinal } → odef A a ∧ UChain ay px a → FClosure A f (supf0 px) b ∧ ( supf0 px o< x) → a ≤ b |
813 zc02 {a} {b} ca fb = zc05 (proj1 fb) where | |
1031 | 814 zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a ≤ b |
954 | 815 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb )) |
1034 | 816 ... | case1 eq = subst (λ k → a ≤ k ) eq (zc05 fb) |
1031 | 817 ... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt) |
1034 | 818 zc05 (init b1 refl) = MinSUP.x≤sup (ZChain.minsup zc o≤-refl) ca |
966 | 819 |
954 | 820 ptotal : IsTotalOrderSet pchainpx |
1034 | 821 ptotal (case1 a) (case1 b) = ZChain.f-total zc a b |
954 | 822 ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b |
823 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where | |
824 eq1 : a0 ≡ b0 | |
825 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
826 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where | |
827 lt1 : a0 < b0 | |
828 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
829 ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b | |
830 ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where | |
831 eq1 : a0 ≡ b0 | |
832 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) | |
833 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where | |
834 lt1 : a0 < b0 | |
835 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt | |
1043 | 836 ptotal (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp (supf0 px) f mf (proj1 a) (proj1 b)) |
966 | 837 |
954 | 838 pcha : pchainpx ⊆' A |
839 pcha (case1 lt) = proj1 lt | |
1043 | 840 pcha (case2 fc) = A∋fc _ f mf (proj1 fc) |
966 | 841 |
842 sup1 : MinSUP A pchainpx | |
954 | 843 sup1 = minsupP pchainpx pcha ptotal |
844 | |
845 -- | |
846 -- supf0 px o≤ sp1 | |
966 | 847 -- |
848 | |
1089
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
849 zc41 : MinSUP A pchainpx → ZChain A f mf< ay x |
1090 | 850 zc41 sup1 = record { supf = supf1 ; asupf = asupf1 ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = cfcs ; supf-mono = supf1-mono } where |
1034 | 851 |
1089
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
852 sp1 = MinSUP.sup sup1 |
883 | 853 |
871 | 854 supf1 : Ordinal → Ordinal |
966 | 855 supf1 z with trio< z px |
871 | 856 ... | tri< a ¬b ¬c = supf0 z |
966 | 857 ... | tri≈ ¬a b ¬c = supf0 z |
901 | 858 ... | tri> ¬a ¬b c = sp1 |
871 | 859 |
886 | 860 sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z |
901 | 861 sf1=sf0 {z} z≤px with trio< z px |
874 | 862 ... | tri< a ¬b ¬c = refl |
901 | 863 ... | tri≈ ¬a b ¬c = refl |
864 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c ) | |
883 | 865 |
901 | 866 sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1 |
867 sf1=sp1 {z} px<z with trio< z px | |
868 ... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a ) | |
869 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z ) | |
870 ... | tri> ¬a ¬b c = refl | |
873 | 871 |
968 | 872 sf=eq : { z : Ordinal } → z o< x → supf0 z ≡ supf1 z |
873 sf=eq {z} z<x = sym (sf1=sf0 (subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x )) | |
874 | |
903 | 875 asupf1 : {z : Ordinal } → odef A (supf1 z) |
876 asupf1 {z} with trio< z px | |
966 | 877 ... | tri< a ¬b ¬c = ZChain.asupf zc |
878 ... | tri≈ ¬a b ¬c = ZChain.asupf zc | |
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
879 ... | tri> ¬a ¬b c = MinSUP.as sup1 |
903 | 880 |
966 | 881 supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b |
882 supf1-mono {a} {b} a≤b with trio< b px | |
901 | 883 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b ) |
884 ... | 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 ) | |
885 supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px | |
886 ... | tri< a<px ¬b ¬c = zc19 where | |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
887 zc21 : MinSUP A (UnionCF A f ay supf0 a) |
901 | 888 zc21 = ZChain.minsup zc (o<→≤ a<px) |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
889 zc24 : {x₁ : Ordinal} → odef (UnionCF A f ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) |
950 | 890 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) ) |
966 | 891 zc19 : supf0 a o≤ sp1 |
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
892 zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.as sup1) zc24 ) |
901 | 893 ... | tri≈ ¬a b ¬c = zc18 where |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
894 zc21 : MinSUP A (UnionCF A f ay supf0 a) |
901 | 895 zc21 = ZChain.minsup zc (o≤-refl0 b) |
896 zc20 : MinSUP.sup zc21 ≡ supf0 a | |
966 | 897 zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b)) |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
898 zc24 : {x₁ : Ordinal} → odef (UnionCF A f ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1) |
950 | 899 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) ) |
966 | 900 zc18 : supf0 a o≤ sp1 |
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
901 zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.as sup1) zc24 ) |
901 | 902 ... | tri> ¬a ¬b c = o≤-refl |
885 | 903 |
966 | 904 fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z |
903 | 905 fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc |
966 | 906 fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z |
903 | 907 fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc |
967 | 908 |
999
3ffbdd53d1ea
fcs<sup requires <-monotonicity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
998
diff
changeset
|
909 -- this is a kind of maximality, so we cannot prove this without <-monotonicity |
3ffbdd53d1ea
fcs<sup requires <-monotonicity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
998
diff
changeset
|
910 -- |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
911 cfcs : {a b w : Ordinal } |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
912 → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w |
1090 | 913 cfcs {a} {b} {w} a<b b≤x sa<b fc with x<y∨y≤x (supf0 a) px |
1012 | 914 ... | case2 px≤sa = z50 where |
1023 | 915 a<x : a o< x |
916 a<x = ordtrans<-≤ a<b b≤x | |
1012 | 917 a≤px : a o≤ px |
918 a≤px = subst (λ k → a o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ a<b b≤x) | |
919 -- supf0 a ≡ px we cannot use previous cfcs, it is in the chain because | |
920 -- supf0 a ≡ supf0 (supf0 a) ≡ supf0 px o< x | |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
921 z50 : odef (UnionCF A f ay supf1 b) w |
1012 | 922 z50 with osuc-≡< px≤sa |
1034 | 923 ... | case1 px=sa = ⟪ A∋fc {A} _ f mf fc , cp ⟫ where |
1023 | 924 sa≤px : supf0 a o≤ px |
925 sa≤px = subst₂ (λ j k → j o< k) px=sa (sym (Oprev.oprev=x op)) px<x | |
1026 | 926 spx=sa : supf0 px ≡ supf0 a |
927 spx=sa = begin | |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
928 supf0 px ≡⟨ cong supf0 px=sa ⟩ |
1039 | 929 supf0 (supf0 a) ≡⟨ ZChain.supf-idem zc a≤px sa≤px ⟩ |
1026 | 930 supf0 a ∎ where open ≡-Reasoning |
1020 | 931 z51 : supf0 px o< b |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
932 z51 = subst (λ k → k o< b ) (sym ( begin supf0 px ≡⟨ spx=sa ⟩ |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
933 supf0 a ≡⟨ sym (sf1=sf0 a≤px) ⟩ |
1025 | 934 supf1 a ∎ )) sa<b where open ≡-Reasoning |
1020 | 935 z52 : supf1 a ≡ supf1 (supf0 px) |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
936 z52 = begin supf1 a ≡⟨ sf1=sf0 a≤px ⟩ |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
937 supf0 a ≡⟨ sym (ZChain.supf-idem zc a≤px sa≤px ) ⟩ |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
938 supf0 (supf0 a) ≡⟨ sym (sf1=sf0 sa≤px) ⟩ |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
939 supf1 (supf0 a) ≡⟨ cong supf1 (sym spx=sa) ⟩ |
1025 | 940 supf1 (supf0 px) ∎ where open ≡-Reasoning |
941 z53 : supf1 (supf0 px) ≡ supf0 px | |
942 z53 = begin | |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
943 supf1 (supf0 px) ≡⟨ cong supf1 spx=sa ⟩ |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
944 supf1 (supf0 a) ≡⟨ sf1=sf0 sa≤px ⟩ |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
945 supf0 (supf0 a) ≡⟨ sym ( cong supf0 px=sa ) ⟩ |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
946 supf0 px ∎ where open ≡-Reasoning |
1034 | 947 cp : UChain ay b w |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
948 cp = ch-is-sup (supf0 px) z51 z53 (subst (λ k → FClosure A f k w) z52 fc) |
1020 | 949 ... | case2 px<sa = ⊥-elim ( ¬p<x<op ⟪ px<sa , subst₂ (λ j k → j o< k ) (sf1=sf0 a≤px) (sym (Oprev.oprev=x op)) z53 ⟫ ) where |
950 z53 : supf1 a o< x | |
951 z53 = ordtrans<-≤ sa<b b≤x | |
1012 | 952 ... | case1 sa<px with trio< a px |
996 | 953 ... | tri< a<px ¬b ¬c = z50 where |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
954 z50 : odef (UnionCF A f ay supf1 b) w |
997 | 955 z50 with osuc-≡< b≤x |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
956 ... | case2 lt with ZChain.cfcs zc a<b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) lt) sa<b fc |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
957 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
1034 | 958 ... | ⟪ az , ch-is-sup u u<b su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc u≤px ) ⟫ where |
959 u≤px : u o≤ px | |
960 u≤px = subst (λ k → u o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ u<b b≤x ) | |
961 u<x : u o< x | |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
962 u<x = ordtrans<-≤ u<b b≤x |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
963 z50 | case1 eq with ZChain.cfcs zc a<px o≤-refl sa<px fc |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
964 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
1034 | 965 ... | ⟪ az , ch-is-sup u u<px su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc (o<→≤ u<px)) ⟫ where -- u o< px → u o< b ? |
966 u<b : u o< b | |
967 u<b = subst (λ k → u o< k ) (trans (Oprev.oprev=x op) (sym eq) ) (ordtrans u<px <-osuc ) | |
968 u<x : u o< x | |
969 u<x = subst (λ k → u o< k ) (Oprev.oprev=x op) ( ordtrans u<px <-osuc ) | |
1000 | 970 ... | tri≈ ¬a a=px ¬c = csupf1 where |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
971 -- a ≡ px , b ≡ x, sp o≤ x |
995 | 972 px<b : px o< b |
973 px<b = subst₂ (λ j k → j o< k) a=px refl a<b | |
974 b=x : b ≡ x | |
975 b=x with trio< b x | |
996 | 976 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) a ⟫ ) -- px o< b o< x |
995 | 977 ... | tri≈ ¬a b ¬c = b |
996 | 978 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> b≤x c ) -- x o< b |
997 | 979 z51 : FClosure A f (supf1 px) w |
980 z51 = subst (λ k → FClosure A f k w) (sym (trans (cong supf1 (sym a=px)) (sf1=sf0 (o≤-refl0 a=px) ))) fc | |
1001 | 981 z53 : odef A w |
982 z53 = A∋fc {A} _ f mf fc | |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
983 csupf1 : odef (UnionCF A f ay supf1 b) w |
1090 | 984 csupf1 with x<y∨y≤x px (supf0 px) |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
985 ... | case2 spx≤px = ⟪ z53 , ch-is-sup (supf0 px) z54 z52 fc1 ⟫ where |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
986 z54 : supf0 px o< b |
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
987 z54 = subst (λ k → supf0 px o< k ) (trans (Oprev.oprev=x op) (sym b=x) ) spx≤px |
1003 | 988 z52 : supf1 (supf0 px) ≡ supf0 px |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
989 z52 = trans (sf1=sf0 spx≤px ) ( ZChain.supf-idem zc o≤-refl spx≤px ) |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
990 fc1 : FClosure A f (supf1 (supf0 px)) w |
1004 | 991 fc1 = subst (λ k → FClosure A f k w ) (trans (cong supf0 a=px) (sym z52) ) fc |
1042
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
992 ... | case1 px<spx = ⊥-elim (¬p<x<op ⟪ px<spx , z54 ⟫ ) where -- supf1 px o≤ spuf1 x → supf1 px ≡ x o< x |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
993 z54 : supf0 px o≤ px |
912ca4abe806
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parents:
1041
diff
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|
994 z54 = subst₂ (λ j k → supf0 j o< k ) a=px (trans b=x (sym (Oprev.oprev=x op))) sa<b |
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
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|
995 |
996 | 996 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → a o< k ) (sym (Oprev.oprev=x op)) ( ordtrans<-≤ a<b b≤x) ⟫ ) -- px o< a o< b o≤ x |
994 | 997 |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
998 zc11 : {z : Ordinal} → odef (UnionCF A f ay supf1 x) z → odef pchainpx z |
1034 | 999 zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫ |
1000 zc11 {z} ⟪ az , ch-is-sup u u<x su=u fc ⟫ = zc21 fc where | |
1001 zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1 | |
903 | 1002 zc21 {z1} (fsuc z2 fc ) with zc21 fc |
1034 | 1003 ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫ |
1004 ... | case1 ⟪ ua1 , ch-is-sup u u<x su=u fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x su=u (fsuc _ fc₁) ⟫ | |
1043 | 1005 ... | case2 fc = case2 ⟪ fsuc _ (proj1 fc) , proj2 fc ⟫ |
1042
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
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|
1006 zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) |
1036 | 1007 ... | tri< a ¬b ¬c = case1 ⟪ asp , ch-is-sup u u<px (trans (sym (sf1=sf0 (o<→≤ u<px))) su=u )(init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where |
1008 u<px : u o< px | |
1009 u<px = ZChain.supf-inject zc a | |
1010 asp0 : odef A (supf0 u) | |
1011 asp0 = ZChain.asupf zc | |
1043 | 1012 ... | tri≈ ¬a b ¬c = case2 ⟪ (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) |
1044 | 1013 (sym (trans (sf1=sf0 (zc-b<x _ u<x)) b ))) , spx<x ⟫ where |
1014 spx<x : supf0 px o< x | |
1015 spx<x = osucprev ( begin | |
1016 osuc (supf0 px) ≡⟨ cong osuc (sym b) ⟩ | |
1017 osuc (supf0 u) ≡⟨ cong osuc (sym (sf1=sf0 (zc-b<x _ u<x) )) ⟩ | |
1018 osuc (supf1 u) ≡⟨ cong osuc su=u ⟩ | |
1019 osuc u ≤⟨ osucc u<x ⟩ | |
1020 x ∎ ) where open o≤-Reasoning O | |
1036 | 1021 ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ ZChain.supf-inject zc c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x ⟫ ) |
967 | 1022 |
1035 | 1023 is-minsup : {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z) |
1024 is-minsup {z} z≤x with osuc-≡< z≤x | |
1036 | 1025 ... | case1 z=x = record { as = zc22 ; x≤sup = z23 ; minsup = z24 } where |
1026 px<z : px o< z | |
1042
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diff
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|
1027 px<z = subst (λ k → px o< k) (sym z=x) px<x |
1036 | 1028 zc22 : odef A (supf1 z) |
1029 zc22 = subst (λ k → odef A k ) (sym (sf1=sp1 px<z )) ( MinSUP.as sup1 ) | |
1030 z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z | |
1045
022d2ef3f20b
is-minsup in px case done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1044
diff
changeset
|
1031 z23 {w} uz = subst (λ k → w ≤ k ) (sym (sf1=sp1 px<z)) ( MinSUP.x≤sup sup1 ( |
1078 | 1032 zc11 (subst (λ k → odef (UnionCF A f ay supf1 k) w) z=x uz ))) |
1042
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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diff
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|
1033 z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s ) |
1036 | 1034 → supf1 z o≤ s |
1035 z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sp1 px<z)) ( MinSUP.minsup sup1 as z25 ) where | |
1042
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
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|
1036 z25 : {w : Ordinal } → odef pchainpx w → w ≤ s |
1043 | 1037 z25 {w} (case2 fc) = sup ⟪ A∋fc _ f mf (proj1 fc) , ch-is-sup (supf0 px) z28 z27 fc1 ⟫ where |
1042
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
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|
1038 -- z=x , supf0 px o< x |
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
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|
1039 z28 : supf0 px o< z -- supf0 px ≡ supf1 px o≤ supf1 x ≡ sp1 o≤ x ≡ z |
1044 | 1040 z28 = subst (λ k → supf0 px o< k) (sym z=x) (proj2 fc) |
1042
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1041 z29 : supf0 px o≤ px |
1044 | 1042 z29 = zc-b<x _ (proj2 fc) |
1040 | 1043 z27 : supf1 (supf0 px) ≡ supf0 px |
1044 z27 = trans (sf1=sf0 z29) ( ZChain.supf-idem zc o≤-refl z29 ) | |
1045 fc1 : FClosure A f (supf1 (supf0 px)) w | |
1043 | 1046 fc1 = subst (λ k → FClosure A f k w) (sym z27) (proj1 fc) |
1036 | 1047 z25 {w} (case1 ⟪ ua , ch-init fc ⟫) = sup ⟪ ua , ch-init fc ⟫ |
1048 z25 {w} (case1 ⟪ ua , ch-is-sup u u<x su=u fc ⟫) = sup ⟪ ua , ch-is-sup u u<z | |
1049 (trans (sf1=sf0 u≤px) su=u) (fcpu fc u≤px) ⟫ where | |
1042
912ca4abe806
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
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|
1050 u≤px : u o< osuc px |
912ca4abe806
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parents:
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diff
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|
1051 u≤px = ordtrans u<x <-osuc |
912ca4abe806
pxhainx conditon is requied?
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parents:
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diff
changeset
|
1052 u<z : u o< z |
912ca4abe806
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parents:
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diff
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|
1053 u<z = ordtrans u<x (subst (λ k → px o< k ) (sym z=x) px<x ) |
1036 | 1054 ... | case2 z<x = record { as = zc22 ; x≤sup = z23 ; minsup = z24 } where |
1055 z≤px = zc-b<x _ z<x | |
1056 m = ZChain.is-minsup zc z≤px | |
1057 zc22 : odef A (supf1 z) | |
1058 zc22 = subst (λ k → odef A k ) (sym (sf1=sf0 z≤px)) ( IsMinSUP.as m ) | |
1059 z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z | |
1060 z23 {w} ⟪ ua , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px)) ( ZChain.fcy<sup zc z≤px fc ) | |
1042
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parents:
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diff
changeset
|
1061 z23 {w} ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px)) |
1036 | 1062 (IsMinSUP.x≤sup m ⟪ ua , ch-is-sup u u<x (trans (sym (sf1=sf0 u≤px )) su=u) (fcup fc u≤px ) ⟫ ) where |
1063 u≤px : u o≤ px | |
1064 u≤px = ordtrans u<x z≤px | |
1065 z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s ) | |
1066 → supf1 z o≤ s | |
1067 z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf0 z≤px)) ( IsMinSUP.minsup m as z25 ) where | |
1042
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
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|
1068 z25 : {w : Ordinal } → odef ( UnionCF A f ay supf0 z ) w → w ≤ s |
1036 | 1069 z25 {w} ⟪ ua , ch-init fc ⟫ = sup ⟪ ua , ch-init fc ⟫ |
1042
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parents:
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diff
changeset
|
1070 z25 {w} ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = sup ⟪ ua , ch-is-sup u u<x |
1036 | 1071 (trans (sf1=sf0 u≤px) su=u) (fcpu fc u≤px) ⟫ where |
1072 u≤px : u o≤ px | |
1073 u≤px = ordtrans u<x z≤px | |
885 | 1074 |
1059 | 1075 zo≤sz : {z : Ordinal} → z o≤ x → z o≤ supf1 z |
1076 zo≤sz {z} z≤x with osuc-≡< z≤x | |
1077 ... | case2 z<x = subst (λ k → z o≤ k) (sym (sf1=sf0 (zc-b<x _ z<x ))) (ZChain.zo≤sz zc (zc-b<x _ z<x )) | |
1078 ... | case1 refl with osuc-≡< (supf1-mono (o<→≤ (px<x))) -- px o≤ supf1 px o≤ supf1 x ≡ sp1 → x o≤ sp1 | |
1078 | 1079 ... | case2 lt = begin |
1080 x ≡⟨ sym (Oprev.oprev=x op) ⟩ | |
1081 osuc px ≤⟨ osucc (ZChain.zo≤sz zc o≤-refl) ⟩ | |
1082 osuc (supf0 px) ≡⟨ sym (cong osuc (sf1=sf0 o≤-refl )) ⟩ | |
1083 osuc (supf1 px) ≤⟨ osucc lt ⟩ | |
1059 | 1084 supf1 x ∎ where open o≤-Reasoning O |
1085 ... | case1 spx=sx with osuc-≡< ( ZChain.zo≤sz zc o≤-refl ) | |
1086 ... | case2 lt = begin | |
1078 | 1087 x ≡⟨ sym (Oprev.oprev=x op) ⟩ |
1088 osuc px ≤⟨ osucc lt ⟩ | |
1089 supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ | |
1090 supf1 px ≤⟨ supf1-mono (o<→≤ px<x) ⟩ | |
1059 | 1091 supf1 x ∎ where open o≤-Reasoning O |
1092 ... | case1 px=spx = ⊥-elim ( <<-irr zc40 (proj1 ( mf< (supf0 px) (ZChain.asupf zc))) ) where | |
1093 zc37 : supf0 px ≡ px | |
1094 zc37 = sym px=spx | |
1095 zc39 : supf0 px ≡ sp1 | |
1096 zc39 = begin | |
1097 supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ | |
1098 supf1 px ≡⟨ spx=sx ⟩ | |
1099 supf1 x ≡⟨ sf1=sp1 px<x ⟩ | |
1100 sp1 ∎ where open ≡-Reasoning | |
1078 | 1101 zc40 : f (supf0 px) ≤ supf0 px |
1102 zc40 = subst (λ k → f (supf0 px) ≤ k ) (sym zc39) | |
1059 | 1103 ( MinSUP.x≤sup sup1 (case2 ⟪ fsuc _ (init (ZChain.asupf zc) refl) , subst (λ k → k o< x) (sym zc37) px<x ⟫ )) |
1104 | |
1033
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parents:
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|
1105 ... | no lim with trio< x o∅ |
1042
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1106 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
1090 | 1107 ... | tri≈ ¬a x=0 ¬c = record { supf = λ _ → MinSUP.sup (ysup f mf ay) ; asupf = MinSUP.as (ysup f mf ay) ; supf-mono = λ _ → o≤-refl |
1089
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parents:
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|
1108 ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = λ a<b b≤0 → ⊥-elim ( ¬x<0 (subst (λ k → _ o< k ) x=0 (ordtrans<-≤ a<b b≤0))) } where |
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|
1109 |
1081 | 1110 mf : ≤-monotonic-f A f |
1111 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where | |
1112 mf00 : * x < * (f x) | |
1113 mf00 = proj1 ( mf< x ax ) | |
1114 ym = MinSUP.sup (ysup f mf ay) | |
1083 | 1115 |
1081 | 1116 zo≤sz : {z : Ordinal} → z o≤ x → z o≤ MinSUP.sup (ysup f mf ay) |
1117 zo≤sz {z} z≤x with osuc-≡< z≤x | |
1083 | 1118 ... | case1 refl = subst (λ k → k o≤ _) (sym x=0) o∅≤z |
1081 | 1119 ... | case2 lt = ⊥-elim ( ¬x<0 (subst (λ k → z o< k ) x=0 lt ) ) |
1083 | 1120 |
1081 | 1121 is-minsup : {z : Ordinal} → z o≤ x → |
1122 IsMinSUP A (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) z) (MinSUP.sup (ysup f mf ay)) | |
1123 is-minsup {z} z≤x with osuc-≡< z≤x | |
1124 ... | case1 refl = record { as = MinSUP.as (ysup f mf ay) ; x≤sup = λ {w} uw → is00 uw ; minsup = λ {s} as sup → is01 as sup } where | |
1125 is00 : {w : Ordinal } → odef (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) x ) w → w ≤ MinSUP.sup (ysup f mf ay) | |
1126 is00 {w} ⟪ aw , ch-init fc ⟫ = MinSUP.x≤sup (ysup f mf ay) fc | |
1127 is00 {w} ⟪ aw , ch-is-sup u u<z su=u fc ⟫ = ⊥-elim (¬x<0 (subst (λ k → u o< k ) x=0 u<z )) | |
1128 is01 : { s : Ordinal } → odef A s → ( {w : Ordinal } → odef (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) x ) w → w ≤ s ) | |
1129 → ym o≤ s | |
1130 is01 {s} as sup = MinSUP.minsup (ysup f mf ay) as is02 where | |
1131 is02 : {w : Ordinal } → odef (uchain f mf ay) w → (w ≡ s) ∨ (w << s) | |
1132 is02 fc = sup ⟪ A∋fc _ f mf fc , ch-init fc ⟫ | |
1133 ... | case2 lt = ⊥-elim ( ¬x<0 (subst (λ k → z o< k ) x=0 lt ) ) | |
1089
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1134 |
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1135 ... | tri> ¬a ¬b 0<x = zc400 usup ssup where |
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1136 |
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1137 mf : ≤-monotonic-f A f |
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1138 mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where |
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1139 mf00 : * x < * (f x) |
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1140 mf00 = proj1 ( mf< x ax ) |
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1141 |
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1142 pzc : {z : Ordinal} → z o< x → ZChain A f mf< ay z |
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1143 pzc {z} z<x = prev z z<x |
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1144 |
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1145 ysp = MinSUP.sup (ysup f mf ay) |
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|
1146 |
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|
1147 supfz : {z : Ordinal } → z o< x → Ordinal |
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|
1148 supfz {z} z<x = ZChain.supf (pzc (ob<x lim z<x)) z |
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parents:
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|
1149 |
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|
1150 pchainU : HOD |
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parents:
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|
1151 pchainU = UnionIC A f ay supfz |
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parents:
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changeset
|
1152 |
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parents:
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|
1153 zeq : {xa xb z : Ordinal } |
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parents:
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|
1154 → (xa<x : xa o< x) → (xb<x : xb o< x) → xa o≤ xb → z o≤ xa |
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|
1155 → ZChain.supf (pzc xa<x) z ≡ ZChain.supf (pzc xb<x) z |
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parents:
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|
1156 zeq {xa} {xb} {z} xa<x xb<x xa≤xb z≤xa = supf-unique A f mf< ay xa≤xb |
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parents:
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|
1157 (pzc xa<x) (pzc xb<x) z≤xa |
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parents:
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changeset
|
1158 |
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parents:
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|
1159 iceq : {ix iy : Ordinal } → ix ≡ iy → {i<x : ix o< x } {i<y : iy o< x } → supfz i<x ≡ supfz i<y |
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parents:
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changeset
|
1160 iceq refl = cong supfz o<-irr |
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parents:
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changeset
|
1161 |
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parents:
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changeset
|
1162 IChain-i : {z : Ordinal } → IChain ay supfz z → Ordinal |
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parents:
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|
1163 IChain-i (ic-init fc) = o∅ |
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changeset
|
1164 IChain-i (ic-isup ia ia<x sa<x fca) = ia |
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parents:
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changeset
|
1165 |
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parents:
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changeset
|
1166 pic<x : {z : Ordinal } → (ic : IChain ay supfz z ) → osuc (IChain-i ic) o< x |
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changeset
|
1167 pic<x {z} (ic-init fc) = ob<x lim 0<x -- 0<x ∧ lim x → osuc o∅ o< x |
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changeset
|
1168 pic<x {z} (ic-isup ia ia<x sa<x fca) = ob<x lim ia<x |
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changeset
|
1169 |
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changeset
|
1170 pchainU⊆chain : {z : Ordinal } → (pz : odef pchainU z) → odef (ZChain.chain (pzc (pic<x (proj2 pz)))) z |
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changeset
|
1171 pchainU⊆chain {z} ⟪ aw , ic-init fc ⟫ = ⟪ aw , ch-init fc ⟫ |
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|
1172 pchainU⊆chain {z} ⟪ aw , (ic-isup ia ia<x sa<x fca) ⟫ = ZChain.cfcs (pzc (ob<x lim ia<x) ) <-osuc o≤-refl uz03 fca where |
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|
1173 uz02 : FClosure A f (ZChain.supf (pzc (ob<x lim ia<x)) ia ) z |
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changeset
|
1174 uz02 = fca |
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|
1175 uz03 : ZChain.supf (pzc (ob<x lim ia<x)) ia o≤ ia |
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changeset
|
1176 uz03 = sa<x |
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changeset
|
1177 |
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parents:
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changeset
|
1178 chain⊆pchainU : {z w : Ordinal } → (oz<x : osuc z o< x) → odef (ZChain.chain (pzc oz<x)) w → odef pchainU w |
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changeset
|
1179 chain⊆pchainU {z} {w} oz<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫ |
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changeset
|
1180 chain⊆pchainU {z} {w} oz<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫ |
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parents:
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|
1181 = ⟪ aw , ic-isup u u<x (o≤-refl0 su≡u) (subst (λ k → FClosure A f k w ) su=su fc) ⟫ where |
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parents:
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|
1182 u<x : u o< x |
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|
1183 u<x = ordtrans u<oz oz<x |
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changeset
|
1184 su=su : ZChain.supf (pzc oz<x) u ≡ supfz u<x |
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changeset
|
1185 su=su = sym ( zeq _ _ (osucc u<oz) (o<→≤ <-osuc) ) |
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|
1186 su≡u : supfz u<x ≡ u |
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changeset
|
1187 su≡u = begin |
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changeset
|
1188 ZChain.supf (pzc (ob<x lim u<x )) u ≡⟨ sym su=su ⟩ |
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parents:
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changeset
|
1189 ZChain.supf (pzc oz<x) u ≡⟨ su=u ⟩ |
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parents:
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|
1190 u ∎ where open ≡-Reasoning |
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changeset
|
1191 |
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parents:
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changeset
|
1192 chain⊆pchainU1 : {z w : Ordinal } → (z<x : z o< x) → odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim z<x))) z) w → odef pchainU w |
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changeset
|
1193 chain⊆pchainU1 {z} {w} z<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫ |
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changeset
|
1194 chain⊆pchainU1 {z} {w} z<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫ |
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changeset
|
1195 = ⟪ aw , ic-isup u u<x (o≤-refl0 su≡u) (subst (λ k → FClosure A f k w ) su=su fc) ⟫ where |
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parents:
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changeset
|
1196 u<x : u o< x |
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|
1197 u<x = ordtrans u<oz z<x |
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|
1198 su=su : ZChain.supf (pzc (ob<x lim z<x)) u ≡ supfz u<x |
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changeset
|
1199 su=su = sym ( zeq _ _ (o<→≤ (osucc u<oz)) (o<→≤ <-osuc) ) |
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|
1200 su≡u : supfz u<x ≡ u |
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changeset
|
1201 su≡u = begin |
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parents:
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changeset
|
1202 ZChain.supf (pzc (ob<x lim u<x )) u ≡⟨ sym su=su ⟩ |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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changeset
|
1203 ZChain.supf (pzc (ob<x lim z<x)) u ≡⟨ su=u ⟩ |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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|
1204 u ∎ where open ≡-Reasoning |
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parents:
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changeset
|
1205 |
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parents:
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changeset
|
1206 ichain-inject : {a b : Ordinal } {ia : IChain ay supfz a } {ib : IChain ay supfz b } |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
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|
1207 → ZChain.supf (pzc (pic<x ia)) (IChain-i ia) o< ZChain.supf (pzc (pic<x ib)) (IChain-i ib) |
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parents:
1084
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changeset
|
1208 → IChain-i ia o< IChain-i ib |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1209 ichain-inject {a} {b} {ia} {ib} sa<sb = uz11 where |
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parents:
1084
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|
1210 uz11 : IChain-i ia o< IChain-i ib |
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parents:
1084
diff
changeset
|
1211 uz11 with trio< (IChain-i ia ) (IChain-i ib) |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
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changeset
|
1212 ... | tri< a ¬b ¬c = a |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1213 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (trans (zeq _ _ (o≤-refl0 (cong osuc b)) (o<→≤ <-osuc) ) |
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parents:
1084
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changeset
|
1214 ( cong (ZChain.supf (pzc (pic<x ib))) b )) sa<sb ) |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
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changeset
|
1215 ... | tri> ¬a ¬b c = ⊥-elim ( o≤> ( begin |
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parents:
1084
diff
changeset
|
1216 ZChain.supf (pzc (pic<x ib)) (IChain-i ib) ≡⟨ zeq _ _ (o<→≤ (osucc c)) (o<→≤ <-osuc) ⟩ |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1217 ZChain.supf (pzc (pic<x ia)) (IChain-i ib) ≤⟨ ZChain.supf-mono (pzc (pic<x ia)) (o<→≤ c) ⟩ |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
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changeset
|
1218 ZChain.supf (pzc (pic<x ia)) (IChain-i ia) ∎ ) sa<sb ) where open o≤-Reasoning O |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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changeset
|
1219 |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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changeset
|
1220 IC⊆ : {a b : Ordinal } (ia : IChain ay supfz a ) (ib : IChain ay supfz b ) |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
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changeset
|
1221 → IChain-i ia o< IChain-i ib → odef (ZChain.chain (pzc (pic<x ib))) a |
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parents:
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diff
changeset
|
1222 IC⊆ {a} {b} (ic-init fc ) ib ia<ib = ⟪ A∋fc _ f mf fc , ch-init fc ⟫ |
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parents:
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changeset
|
1223 IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-init fcb ) ia<ib = ⊥-elim ( ¬x<0 ia<ib ) |
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parents:
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changeset
|
1224 IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-isup j j<x sb<x fcb ) ia<ib |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1225 = ZChain.cfcs (pzc (ob<x lim j<x) ) (o<→≤ ia<ib) o≤-refl (OrdTrans (ZChain.supf-mono (pzc (ob<x lim j<x)) (o<→≤ ia<ib)) sb<x) |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1226 (subst (λ k → FClosure A f k a) (zeq _ _ (osucc (o<→≤ ia<ib)) (o<→≤ <-osuc)) fc ) |
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parents:
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changeset
|
1227 |
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parents:
1084
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changeset
|
1228 ptotalU : IsTotalOrderSet pchainU |
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parents:
1084
diff
changeset
|
1229 ptotalU {a} {b} ia ib with trio< (IChain-i (proj2 ia)) (IChain-i (proj2 ib)) |
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parents:
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|
1230 ... | tri< ia<ib ¬b ¬c = ZChain.f-total (pzc (pic<x (proj2 ib))) (IC⊆ (proj2 ia) (proj2 ib) ia<ib) (pchainU⊆chain ib) |
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parents:
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|
1231 ... | tri≈ ¬a ia=ib ¬c = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso ( pcmp (proj2 ia) (proj2 ib) ia=ib ) where |
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parents:
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|
1232 pcmp : (ia : IChain ay supfz (& a)) → (ib : IChain ay supfz (& b)) → IChain-i ia ≡ IChain-i ib |
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parents:
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|
1233 → Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
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|
1234 pcmp (ic-init fca) (ic-init fcb) eq = fcn-cmp _ f mf fca fcb |
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parents:
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changeset
|
1235 pcmp (ic-init fca) (ic-isup i i<x s<x fcb) eq with ZChain.fcy<sup (pzc i<x) o≤-refl fca |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1236 ... | case1 eq1 = ct22 where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1237 ct22 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1238 ct22 with subst (λ k → k ≤ & b) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fcb ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1239 ... | case1 eq2 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1240 ct00 : * (& a) ≡ * (& b) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1241 ct00 = cong (*) (trans eq1 eq2) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1242 ... | case2 lt = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1243 fc11 : * (& a) < * (& b) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1244 fc11 = subst (λ k → k < * (& b) ) (cong (*) (sym eq1)) lt |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1245 ... | case2 lt = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1246 fc11 : * (& a) < * (& b) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1247 fc11 = ftrans<-≤ lt (subst (λ k → k ≤ & b) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fcb ) ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1248 pcmp (ic-isup i i<x s<x fca) (ic-init fcb) eq with ZChain.fcy<sup (pzc i<x) o≤-refl fcb |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1249 ... | case1 eq1 = ct22 where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1250 ct22 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1251 ct22 with subst (λ k → k ≤ & a) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fca ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1252 ... | case1 eq2 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1253 ct00 : * (& a) ≡ * (& b) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1254 ct00 = cong (*) (sym (trans eq1 eq2)) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1255 ... | case2 lt = tri> (λ lt → <-irr (case2 fc11) lt) (λ eq → <-irr (case1 eq) fc11) fc11 where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1256 fc11 : * (& b) < * (& a) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1257 fc11 = subst (λ k → k < * (& a) ) (cong (*) (sym eq1)) lt |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1258 ... | case2 lt = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1259 fc12 : * (& b) < * (& a) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1260 fc12 = ftrans<-≤ lt (subst (λ k → k ≤ & a) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fca ) ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1261 pcmp (ic-isup i i<x s<x fca) (ic-isup i i<y s<y fcb) refl = fcn-cmp _ f mf fca (subst (λ k → FClosure A f k (& b)) pc01 fcb ) where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1262 pc01 : supfz i<y ≡ supfz i<x |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1263 pc01 = cong supfz o<-irr |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1264 ... | tri> ¬a ¬b ib<ia = ZChain.f-total (pzc (pic<x (proj2 ia))) (pchainU⊆chain ia) (IC⊆ (proj2 ib) (proj2 ia) ib<ia) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1265 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1266 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1267 usup : MinSUP A pchainU |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1268 usup = minsupP pchainU (λ ic → proj1 ic ) ptotalU |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1269 spu0 = MinSUP.sup usup |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1270 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1271 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1272 pchainS : HOD |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1273 pchainS = record { od = record { def = λ z → (odef A z ∧ IChain ay supfz z ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1274 ∨ (FClosure A f spu0 z ∧ (spu0 o< x)) } ; odmax = & A ; <odmax = zc00 } where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1275 zc00 : {z : Ordinal } → (odef A z ∧ IChain ay supfz z ) ∨ (FClosure A f spu0 z ∧ (spu0 o< x) )→ z o< & A |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1276 zc00 {z} (case1 lt) = z07 lt |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1277 zc00 {z} (case2 fc) = z09 ( A∋fc spu0 f mf (proj1 fc) ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1278 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1279 zc02 : { a b : Ordinal } → odef A a ∧ IChain ay supfz a → FClosure A f spu0 b ∧ ( spu0 o< x) → a ≤ b |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1280 zc02 {a} {b} ca fb = zc05 (proj1 fb) where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1281 zc05 : {b : Ordinal } → FClosure A f spu0 b → a ≤ b |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1282 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc spu0 f mf fb )) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1283 ... | case1 eq = subst (λ k → a ≤ k ) eq (zc05 fb) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1284 ... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1285 zc05 (init b1 refl) = MinSUP.x≤sup usup ca |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1286 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1287 ptotalS : IsTotalOrderSet pchainS |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1288 ptotalS (case1 a) (case1 b) = ptotalU a b |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1289 ptotalS {a0} {b0} (case1 a) (case2 b) with zc02 a b |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1290 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1291 eq1 : a0 ≡ b0 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1292 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1293 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1294 lt1 : a0 < b0 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1295 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1296 ptotalS {b0} {a0} (case2 b) (case1 a) with zc02 a b |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1297 ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1298 eq1 : a0 ≡ b0 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1299 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1300 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1301 lt1 : a0 < b0 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1302 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1303 ptotalS (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp spu0 f mf (proj1 a) (proj1 b)) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1304 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1305 S⊆A : pchainS ⊆' A |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1306 S⊆A (case1 lt) = proj1 lt |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1307 S⊆A (case2 fc) = A∋fc _ f mf (proj1 fc) |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1308 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1309 ssup : MinSUP A pchainS |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1310 ssup = minsupP pchainS S⊆A ptotalS |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1311 |
b627e3ea7266
try to hide spu from source
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1084
diff
changeset
|
1312 zc400 : MinSUP A pchainU → MinSUP A pchainS → ZChain A f mf< ay x |
1090 | 1313 zc400 usup ssup = record { supf = supf1 ; asupf = asupf ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = cfcs ; supf-mono = supf-mono } where |
726
b2e2cd12e38f
psupf-mono and is-max conflict
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
725
diff
changeset
|
1314 |
1069 | 1315 spu = MinSUP.sup usup |
1316 sps = MinSUP.sup ssup | |
834 | 1317 |
794 | 1318 supf1 : Ordinal → Ordinal |
835 | 1319 supf1 z with trio< z x |
1069 | 1320 ... | tri< a ¬b ¬c = ZChain.supf (pzc (ob<x lim a)) z -- each sup o< x |
1321 ... | tri≈ ¬a b ¬c = spu -- sup of all sup o< x | |
1078 | 1322 ... | tri> ¬a ¬b c = sps -- sup of spu which o< x |
1069 | 1323 -- if x o< spu, spu is not included in UnionCF x |
1324 -- the chain | |
755 | 1325 |
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1326 pchain : HOD |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1327 pchain = UnionCF A f ay supf1 x |
1010
f80d525e6a6b
Recursive record IChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1009
diff
changeset
|
1328 |
1069 | 1329 -- pchain ⊆ pchainU ⊆ pchianS |
1009 | 1330 |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1331 sf1=sf : {z : Ordinal } → (a : z o< x ) → supf1 z ≡ ZChain.supf (pzc (ob<x lim a)) z |
1009 | 1332 sf1=sf {z} z<x with trio< z x |
1333 ... | tri< a ¬b ¬c = cong ( λ k → ZChain.supf (pzc (ob<x lim k)) z) o<-irr | |
1334 ... | tri≈ ¬a b ¬c = ⊥-elim (¬a z<x) | |
1335 ... | tri> ¬a ¬b c = ⊥-elim (¬a z<x) | |
1336 | |
1071 | 1337 sf1=spu : {z : Ordinal } → x ≡ z → supf1 z ≡ spu |
1338 sf1=spu {z} eq with trio< z x | |
1078 | 1339 ... | tri< a ¬b ¬c = ⊥-elim (¬b (sym eq)) |
1009 | 1340 ... | tri≈ ¬a b ¬c = refl |
1078 | 1341 ... | tri> ¬a ¬b c = ⊥-elim (¬b (sym eq)) |
1009 | 1342 |
1071 | 1343 sf1=sps : {z : Ordinal } → (a : x o< z ) → supf1 z ≡ sps |
1344 sf1=sps {z} x<z with trio< z x | |
1345 ... | tri< a ¬b ¬c = ⊥-elim (o<> x<z a) | |
1346 ... | tri≈ ¬a b ¬c = ⊥-elim (¬c x<z ) | |
1347 ... | tri> ¬a ¬b c = refl | |
1069 | 1348 |
1057 | 1349 asupf : {z : Ordinal } → odef A (supf1 z) |
1350 asupf {z} with trio< z x | |
1078 | 1351 ... | tri< a ¬b ¬c = ZChain.asupf (pzc (ob<x lim a)) |
1057 | 1352 ... | tri≈ ¬a b ¬c = MinSUP.as usup |
1069 | 1353 ... | tri> ¬a ¬b c = MinSUP.as ssup |
1057 | 1354 |
1355 supf-mono : {z y : Ordinal } → z o≤ y → supf1 z o≤ supf1 y | |
1356 supf-mono {z} {y} z≤y with trio< y x | |
1071 | 1357 ... | tri< y<x ¬b ¬c = zc01 where |
1057 | 1358 open o≤-Reasoning O |
1071 | 1359 zc01 : supf1 z o≤ ZChain.supf (pzc (ob<x lim y<x)) y |
1072 | 1360 zc01 = begin |
1361 supf1 z ≡⟨ sf1=sf (ordtrans≤-< z≤y y<x) ⟩ | |
1362 ZChain.supf (pzc (ob<x lim (ordtrans≤-< z≤y y<x))) z ≡⟨ zeq _ _ (osucc z≤y) (o<→≤ <-osuc) ⟩ | |
1070 | 1363 ZChain.supf (pzc (ob<x lim y<x)) z ≤⟨ ZChain.supf-mono (pzc (ob<x lim y<x)) z≤y ⟩ |
1078 | 1364 ZChain.supf (pzc (ob<x lim y<x)) y ∎ |
1072 | 1365 ... | tri≈ ¬a b ¬c = zc01 where -- supf1 z o≤ spu |
1366 open o≤-Reasoning O | |
1367 zc01 : supf1 z o≤ spu | |
1368 zc01 with osuc-≡< (subst (λ k → z o≤ k) b z≤y) | |
1369 ... | case1 z=x = o≤-refl0 (sf1=spu (sym z=x)) | |
1078 | 1370 ... | case2 z<x = subst (λ k → k o≤ spu ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) ) |
1072 | 1371 (MinSUP.as usup) (λ uw → MinSUP.x≤sup usup (chain⊆pchainU1 z<x uw)) ) |
1372 ... | tri> ¬a ¬b c = zc01 where -- supf1 z o≤ sps | |
1373 zc01 : supf1 z o≤ sps | |
1374 zc01 with trio< z x | |
1078 | 1375 ... | tri< z<x ¬b ¬c = IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) ) |
1072 | 1376 (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 (chain⊆pchainU1 z<x uw)) ) |
1377 ... | tri≈ ¬a z=x ¬c = MinSUP.minsup usup (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 uw) ) | |
1378 ... | tri> ¬a ¬b c = o≤-refl -- (sf1=sps c) | |
797 | 1379 |
1035 | 1380 is-minsup : {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z) |
1073 | 1381 is-minsup {z} z≤x with osuc-≡< z≤x |
1075 | 1382 ... | case1 z=x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where |
1078 | 1383 zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z |
1384 zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x))) ( MinSUP.x≤sup usup ⟪ az , ic-init fc ⟫ ) | |
1385 zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x))) | |
1076 | 1386 ( MinSUP.x≤sup usup ⟪ az , ic-isup u u<x (o≤-refl0 zm05) (subst (λ k → FClosure A f k w) (sym zm06) fc) ⟫ ) where |
1387 u<x : u o< x | |
1388 u<x = subst (λ k → u o< k) z=x u<b | |
1389 zm06 : supfz (subst (λ k → u o< k) z=x u<b) ≡ supf1 u | |
1390 zm06 = trans (zeq _ _ o≤-refl (o<→≤ <-osuc) ) (sym (sf1=sf u<x )) | |
1078 | 1391 zm05 : supfz (subst (λ k → u o< k) z=x u<b) ≡ u |
1076 | 1392 zm05 = trans zm06 su=u |
1075 | 1393 zm01 : { s : Ordinal } → odef A s → ( {x : Ordinal } → odef (UnionCF A f ay supf1 z) x → x ≤ s ) → supf1 z o≤ s |
1078 | 1394 zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=spu (sym z=x))) ( MinSUP.minsup usup as zm02 ) where |
1075 | 1395 zm02 : {w : Ordinal } → odef pchainU w → w ≤ s |
1076 | 1396 zm02 {w} uw with pchainU⊆chain uw |
1078 | 1397 ... | ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫ |
1076 | 1398 ... | ⟪ az , ch-is-sup u1 u<b su=u fc ⟫ = sup ⟪ az , ch-is-sup u1 (ordtrans u<b zm05) (trans zm03 su=u) zm04 ⟫ where |
1078 | 1399 zm05 : osuc (IChain-i (proj2 uw)) o< z |
1076 | 1400 zm05 = subst (λ k → osuc (IChain-i (proj2 uw)) o< k) (sym z=x) ( pic<x (proj2 uw) ) |
1401 u<x : u1 o< x | |
1402 u<x = subst (λ k → u1 o< k) z=x ( ordtrans u<b zm05 ) | |
1403 zm03 : supf1 u1 ≡ ZChain.supf (prev (osuc (IChain-i (proj2 uw))) (pic<x (proj2 uw))) u1 | |
1404 zm03 = trans (sf1=sf u<x) (zeq _ _ (osucc u<b) (o<→≤ <-osuc) ) | |
1405 zm04 : FClosure A f (supf1 u1) w | |
1406 zm04 = subst (λ k → FClosure A f k w) (sym zm03) fc | |
1073 | 1407 ... | case2 z<x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where |
1078 | 1408 supf0 = ZChain.supf (pzc (ob<x lim z<x)) |
1073 | 1409 msup : IsMinSUP A (UnionCF A f ay supf0 z) (supf0 z) |
1410 msup = ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) | |
1074 | 1411 s1=0 : {u : Ordinal } → u o< z → supf1 u ≡ supf0 u |
1078 | 1412 s1=0 {u} u<z = trans (sf1=sf (ordtrans u<z z<x)) (zeq _ _ (o<→≤ (osucc u<z)) (o<→≤ <-osuc) ) |
1413 zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z | |
1414 zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x)) ( IsMinSUP.x≤sup msup ⟪ az , ch-init fc ⟫ ) | |
1415 zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x)) | |
1074 | 1416 ( IsMinSUP.x≤sup msup ⟪ az , ch-is-sup u u<b (trans (sym (s1=0 u<b)) su=u) (subst (λ k → FClosure A f k w) (s1=0 u<b) fc) ⟫ ) |
1073 | 1417 zm01 : { s : Ordinal } → odef A s → ( {x : Ordinal } → odef (UnionCF A f ay supf1 z) x → x ≤ s ) → supf1 z o≤ s |
1078 | 1418 zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup msup as zm02 ) where |
1073 | 1419 zm02 : {w : Ordinal } → odef (UnionCF A f ay supf0 z) w → w ≤ s |
1420 zm02 {w} ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫ | |
1078 | 1421 zm02 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = sup |
1422 ⟪ az , ch-is-sup u u<b (trans (s1=0 u<b) su=u) (subst (λ k → FClosure A f k w) (sym (s1=0 u<b)) fc) ⟫ | |
1073 | 1423 |
1424 | |
1035 | 1425 |
1039 | 1426 cfcs : {a b w : Ordinal } → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w |
1427 cfcs {a} {b} {w} a<b b≤x sa<b fc with osuc-≡< b≤x | |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1428 ... | case1 b=x with trio< a x |
1016 | 1429 ... | tri< a<x ¬b ¬c = zc40 where |
1430 sa = ZChain.supf (pzc (ob<x lim a<x)) a | |
1020 | 1431 m = omax a sa -- x is limit ordinal, so we have sa o< m o< x |
1016 | 1432 m<x : m o< x |
1433 m<x with trio< a sa | inspect (omax a) sa | |
1020 | 1434 ... | tri< a<sa ¬b ¬c | record { eq = eq } = ob<x lim (ordtrans<-≤ sa<b b≤x ) |
1016 | 1435 ... | tri≈ ¬a a=sa ¬c | record { eq = eq } = subst (λ k → k o< x) eq zc41 where |
1436 zc41 : omax a sa o< x | |
1437 zc41 = osucprev ( begin | |
1438 osuc ( omax a sa ) ≡⟨ cong (λ k → osuc (omax a k)) (sym a=sa) ⟩ | |
1439 osuc ( omax a a ) ≡⟨ cong osuc (omxx _) ⟩ | |
1440 osuc ( osuc a ) ≤⟨ o<→≤ (ob<x lim (ob<x lim a<x)) ⟩ | |
1441 x ∎ ) where open o≤-Reasoning O | |
1442 ... | tri> ¬a ¬b c | record { eq = eq } = ob<x lim a<x | |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1443 sam = ZChain.supf (pzc (ob<x lim m<x)) a |
1016 | 1444 zc42 : osuc a o≤ osuc m |
1445 zc42 = osucc (o<→≤ ( omax-x _ _ ) ) | |
1446 sam<m : sam o< m | |
1039 | 1447 sam<m = subst (λ k → k o< m ) (supf-unique A f mf< ay zc42 (pzc (ob<x lim a<x)) (pzc (ob<x lim m<x)) (o<→≤ <-osuc)) ( omax-y _ _ ) |
1016 | 1448 fcm : FClosure A f (ZChain.supf (pzc (ob<x lim m<x)) a) w |
1449 fcm = subst (λ k → FClosure A f k w ) (zeq (ob<x lim a<x) (ob<x lim m<x) zc42 (o<→≤ <-osuc) ) fc | |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1450 zcm : odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim m<x))) (osuc (omax a sa))) w |
1039 | 1451 zcm = ZChain.cfcs (pzc (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1452 zc40 : odef (UnionCF A f ay supf1 b) w |
1039 | 1453 zc40 with ZChain.cfcs (pzc (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm |
1058 | 1454 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
1070 | 1455 ... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans zc45 su=u) zc44 ⟫ where |
1456 u<b : u o< b | |
1457 u<b = osucprev ( begin | |
1458 osuc u ≤⟨ osucc u<x ⟩ | |
1459 osuc m ≤⟨ osucc m<x ⟩ | |
1460 x ≡⟨ sym b=x ⟩ | |
1461 b ∎ ) where open o≤-Reasoning O | |
1078 | 1462 zc45 : supf1 u ≡ ZChain.supf (pzc (ob<x lim m<x)) u |
1070 | 1463 zc45 = begin |
1464 supf1 u ≡⟨ sf1=sf (subst (λ k → u o< k) b=x u<b ) ⟩ | |
1465 ZChain.supf (pzc (ob<x lim (subst (λ k → u o< k) b=x u<b ))) u ≡⟨ zeq _ _ (osucc u<x) (o<→≤ <-osuc) ⟩ | |
1078 | 1466 ZChain.supf (pzc (ob<x lim m<x)) u ∎ where open ≡-Reasoning |
1070 | 1467 zc44 : FClosure A f (supf1 u) w |
1468 zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc | |
1016 | 1469 ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) |
1470 ... | tri> ¬a ¬b c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) | |
1039 | 1471 cfcs {a} {b} {w} a<b b≤x sa<b fc | case2 b<x = zc40 where |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1472 supfb = ZChain.supf (pzc (ob<x lim b<x)) |
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1473 sb=sa : {a : Ordinal } → a o< b → supf1 a ≡ ZChain.supf (pzc (ob<x lim b<x)) a |
1020 | 1474 sb=sa {a} a<b = trans (sf1=sf (ordtrans<-≤ a<b b≤x)) (zeq _ _ (o<→≤ (osucc a<b)) (o<→≤ <-osuc) ) |
1017
ffdfd8d1303a
trying cscf as odef (UnionCF A f mf ay supf z) w
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1016
diff
changeset
|
1475 fcb : FClosure A f (supfb a) w |
1042
912ca4abe806
pxhainx conditon is requied?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1041
diff
changeset
|
1476 fcb = subst (λ k → FClosure A f k w) (sb=sa a<b) fc |
1020 | 1477 -- supfb a o< b assures it is in Union b |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1478 zcb : odef (UnionCF A f ay supfb b) w |
1039 | 1479 zcb = ZChain.cfcs (pzc (ob<x lim b<x)) a<b (o<→≤ <-osuc) (subst (λ k → k o< b) (sb=sa a<b) sa<b) fcb |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1480 zc40 : odef (UnionCF A f ay supf1 b) w |
1016 | 1481 zc40 with zcb |
1058 | 1482 ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ |
1070 | 1483 ... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<x (trans zc45 su=u) zc44 ⟫ where |
1078 | 1484 zc45 : supf1 u ≡ ZChain.supf (pzc (ob<x lim b<x)) u |
1070 | 1485 zc45 = begin |
1486 supf1 u ≡⟨ sf1=sf (ordtrans u<x b<x) ⟩ | |
1487 ZChain.supf (pzc (ob<x lim (ordtrans u<x b<x) )) u ≡⟨ zeq _ _ (o<→≤ (osucc u<x)) (o<→≤ <-osuc) ⟩ | |
1078 | 1488 ZChain.supf (pzc (ob<x lim b<x )) u ∎ where open ≡-Reasoning |
1070 | 1489 zc44 : FClosure A f (supf1 u) w |
1490 zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc | |
1028 | 1491 |
1079 | 1492 zo≤sz : {z : Ordinal} → z o≤ x → z o≤ supf1 z |
1080 | 1493 zo≤sz {z} z≤x with osuc-≡< z≤x |
1494 ... | case2 z<x = subst (λ k → z o≤ k) (sym (trans (sf1=sf z<x) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl)))) ( ZChain.zo≤sz (pzc z<x) o≤-refl ) | |
1083 | 1495 ... | case1 refl with x<y∨y≤x (supf1 spu) x |
1496 ... | case2 x≤ssp = z40 where | |
1497 z40 : z o≤ supf1 z | |
1498 z40 with x<y∨y≤x z spu | |
1499 ... | case1 z<spu = o<→≤ ( subst (λ k → z o< k ) (sym (sf1=spu refl)) z<spu ) | |
1500 ... | case2 spu≤z = begin -- x ≡ supf1 spu ≡ spu ≡ supf1 x | |
1501 x ≤⟨ x≤ssp ⟩ | |
1502 supf1 spu ≤⟨ supf-mono spu≤z ⟩ | |
1503 supf1 x ∎ where open o≤-Reasoning O | |
1504 ... | case1 ssp<x = subst (λ k → x o≤ k) (sym (sf1=spu refl)) z47 where | |
1080 | 1505 z47 : x o≤ spu |
1506 z47 with x<y∨y≤x spu x | |
1507 ... | case2 lt = lt | |
1508 ... | case1 spu<x = ⊥-elim ( <<-irr (MinSUP.x≤sup usup z48) (proj1 ( mf< spu (MinSUP.as usup)))) where | |
1083 | 1509 z70 : odef (UnionCF A f ay supf1 z) (supf1 spu) |
1510 z70 = cfcs spu<x o≤-refl ssp<x (init asupf refl ) | |
1511 z73 : IsSUP A (UnionCF A f ay (ZChain.supf (pzc (ob<x lim spu<x))) spu) spu | |
1512 z73 = record { ax = MinSUP.as usup ; x≤sup = λ uw → MinSUP.x≤sup usup (chain⊆pchainU1 spu<x uw ) } | |
1080 | 1513 z49 : supfz spu<x ≡ spu |
1083 | 1514 z49 = begin |
1515 supfz spu<x ≡⟨ ZChain.sup=u (pzc (ob<x lim spu<x)) (MinSUP.as usup) (o<→≤ <-osuc) z73 ⟩ | |
1516 spu ∎ where open ≡-Reasoning | |
1082 | 1517 z50 : supfz spu<x o≤ spu |
1518 z50 = o≤-refl0 z49 | |
1080 | 1519 z48 : odef pchainU (f spu) |
1082 | 1520 z48 = ⟪ proj2 (mf _ (MinSUP.as usup) ) , ic-isup _ (subst (λ k → k o< x) refl spu<x) z50 |
1080 | 1521 (fsuc _ (init (ZChain.asupf (pzc (ob<x lim spu<x))) z49)) ⟫ |
1079 | 1522 |
1077 | 1523 |
921 | 1524 --- |
1525 --- the maximum chain has fix point of any ≤-monotonic function | |
1526 --- | |
1527 | |
1039 | 1528 SZ : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf< ay x |
1529 SZ f mf< {y} ay x = TransFinite {λ z → ZChain A f mf< ay z } (λ x → ind f mf< ay x ) x | |
921 | 1530 |
1039 | 1531 msp0 : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) |
1532 → (zc : ZChain A f mf< ay x ) | |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1533 → MinSUP A (UnionCF A f ay (ZChain.supf zc) x) |
1039 | 1534 msp0 f mf< {x} ay zc = minsupP (UnionCF A f ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc) |
922 | 1535 |
1090 | 1536 -- f eventualy stop |
1537 -- we can prove contradict here, it is here for a historical reason | |
1538 -- | |
1039 | 1539 fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (mf< : <-monotonic-f A f ) (zc : ZChain A f mf< as0 (& A) ) |
966 | 1540 → (sp1 : MinSUP A (ZChain.chain zc)) |
959 | 1541 → f (MinSUP.sup sp1) ≡ MinSUP.sup sp1 |
992 | 1542 fixpoint f mf mf< zc sp1 = z14 where |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1543 chain = ZChain.chain zc |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1544 supf = ZChain.supf zc |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1545 sp : Ordinal |
959 | 1546 sp = MinSUP.sup sp1 |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1547 asp : odef A sp |
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1548 asp = MinSUP.as sp1 |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1549 ay = as0 |
988
9a85233384f7
is-max and supf b = supf x
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
987
diff
changeset
|
1550 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< (& A) → (ab : odef A b ) |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1551 → HasPrev A chain f b ∨ IsSUP A (UnionCF A f ay (ZChain.supf zc) b) b |
921 | 1552 → * a < * b → odef chain b |
993 | 1553 z10 = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl ) |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1554 z22 : sp o< & A |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1555 z22 = z09 asp |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1556 z12 : odef chain sp |
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1557 z12 with o≡? (& s) sp |
1058 | 1558 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc ) |
1559 ... | no ne = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl) {& s} {sp} ( ZChain.chain∋init zc ) (z09 asp) asp (case2 z19 ) z13 where | |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1560 z13 : * (& s) < * sp |
1058 | 1561 z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc ) |
960 | 1562 ... | case1 eq = ⊥-elim ( ne eq ) |
966 | 1563 ... | case2 lt = lt |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1564 z19 : IsSUP A (UnionCF A f ay (ZChain.supf zc) sp) sp |
1069 | 1565 z19 = record { ax = asp ; x≤sup = z20 } where |
1033
2da8dcbb0825
ch-init again, because ch-is-sup require u<x which is not valid supf o∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1032
diff
changeset
|
1566 z20 : {y : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp) |
966 | 1567 z20 {y} zy with MinSUP.x≤sup sp1 |
961 | 1568 (subst (λ k → odef chain k ) (sym &iso) (chain-mono f mf as0 supf (ZChain.supf-mono zc) (o<→≤ z22) zy )) |
966 | 1569 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p ) |
960 | 1570 ... | case2 y<p = case2 (subst (λ k → * k < _ ) &iso y<p ) |
935
ed711d7be191
mem exhaust fix on fixpoint
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
934
diff
changeset
|
1571 z14 : f sp ≡ sp |
960 | 1572 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) (subst (λ k → odef chain k) (sym &iso) z12 ) |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1573 ... | 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
|
1574 z16 : ⊥ |
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1575 z16 with proj1 (mf (( MinSUP.sup sp1)) ( MinSUP.as sp1 )) |
1031 | 1576 ... | case1 eq = ⊥-elim (¬b (sym (cong (*) eq ) )) |
966 | 1577 ... | case2 lt = ⊥-elim (¬c lt ) |
1578 ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) b ) | |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1579 ... | tri> ¬a ¬b c = ⊥-elim z17 where |
959 | 1580 z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) ) |
960 | 1581 z15 = MinSUP.x≤sup sp1 (ZChain.f-next zc z12 ) |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1582 z17 : ⊥ |
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1583 z17 with z15 |
960 | 1584 ... | case1 eq = ¬b (cong (*) eq) |
1585 ... | case2 lt = ¬a lt | |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1586 |
921 | 1587 -- ZChain contradicts ¬ Maximal |
1588 -- | |
1589 -- ZChain forces fix point on any ≤-monotonic function (fixpoint) | |
1590 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain | |
1591 -- | |
924
a48dc906796c
supf usp0 instead of supf (& A) ?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
923
diff
changeset
|
1592 |
1090 | 1593 ¬Maximal→¬cf-mono : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-<-monotonic nmx) as0 (& A)) → ⊥ |
1594 ¬Maximal→¬cf-mono nmx zc = <-irr0 {* (cf nmx c)} {* c} | |
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1595 (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.as msp1 )))) |
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1596 (subst (λ k → odef A k) (sym &iso) (MinSUP.as msp1) ) |
992 | 1597 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) (cf-is-<-monotonic nmx ) zc msp1 ))) -- x ≡ f x ̄ |
1032
382680c3e9be
minsup is not obvious in ZChain
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1031
diff
changeset
|
1598 (proj1 (cf-is-<-monotonic nmx c (MinSUP.as msp1 ))) where -- x < f x |
937 | 1599 |
927 | 1600 supf = ZChain.supf zc |
934 | 1601 msp1 : MinSUP A (ZChain.chain zc) |
1039 | 1602 msp1 = msp0 (cf nmx) (cf-is-<-monotonic nmx) as0 zc |
966 | 1603 c : Ordinal |
1604 c = MinSUP.sup msp1 | |
934 | 1605 |
966 | 1606 zorn00 : Maximal A |
1090 | 1607 zorn00 with is-o∅ ( & HasMaximal ) |
804 | 1608 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where |
1090 | 1609 -- yes we have the maximal because of the axiom of choice |
551 | 1610 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) |
606 | 1611 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice |
551 | 1612 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) |
966 | 1613 zorn01 = proj1 zorn03 |
551 | 1614 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) |
1615 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
1090 | 1616 ... | yes ¬Maximal = ⊥-elim ( ¬Maximal→¬cf-mono nmx (SZ (cf nmx) (cf-is-<-monotonic nmx) as0 (& A) )) where |
551 | 1617 -- if we have no maximal, make ZChain, which contradict SUP condition |
966 | 1618 nmx : ¬ Maximal A |
551 | 1619 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where |
966 | 1620 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) |
804 | 1621 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 | 1622 |
516 | 1623 -- usage (see filter.agda ) |
1624 -- | |
497 | 1625 -- _⊆'_ : ( A B : HOD ) → Set n |
1626 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
482 | 1627 |
966 | 1628 -- MaximumSubset : {L P : HOD} |
497 | 1629 -- → o∅ o< & L → o∅ o< & P → P ⊆ L |
1630 -- → IsPartialOrderSet P _⊆'_ | |
1631 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
1632 -- → Maximal P (_⊆'_) | |
1633 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |