Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/zorn.agda @ 559:9ba98ecfbe62
fcn-cmp done
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Sat, 30 Apr 2022 04:41:06 +0900 |
parents | fed1c67b9a65 |
children | d09f9a1d6c2f |
rev | line source |
---|---|
478 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
508 | 2 open import Level hiding ( suc ; zero ) |
431 | 3 open import Ordinals |
552 | 4 open import Relation.Binary |
5 open import Relation.Binary.Core | |
6 open import Relation.Binary.PropositionalEquality | |
497 | 7 import OD |
552 | 8 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where |
431 | 9 |
10 open import zf | |
477 | 11 open import logic |
12 -- open import partfunc {n} O | |
13 | |
14 open import Relation.Nullary | |
15 open import Data.Empty | |
16 import BAlgbra | |
431 | 17 |
555 | 18 open import Data.Nat hiding ( _<_ ; _≤_ ) |
19 open import Data.Nat.Properties | |
20 open import nat | |
21 | |
431 | 22 |
23 open inOrdinal O | |
24 open OD O | |
25 open OD.OD | |
26 open ODAxiom odAxiom | |
477 | 27 import OrdUtil |
28 import ODUtil | |
431 | 29 open Ordinals.Ordinals O |
30 open Ordinals.IsOrdinals isOrdinal | |
31 open Ordinals.IsNext isNext | |
32 open OrdUtil O | |
477 | 33 open ODUtil O |
34 | |
35 | |
36 import ODC | |
37 | |
38 | |
39 open _∧_ | |
40 open _∨_ | |
41 open Bool | |
431 | 42 |
43 | |
44 open HOD | |
45 | |
528
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
46 _≤_ : (x y : HOD) → Set (Level.suc n) |
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
47 x ≤ y = ( x ≡ y ) ∨ ( x < y ) |
8facdd7cc65a
TransitiveClosure with x <= f x is possible
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
527
diff
changeset
|
48 |
554 | 49 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z |
50 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl | |
51 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z | |
52 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y | |
53 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) | |
54 | |
556 | 55 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
56 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a | |
57 <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl | |
58 (IsStrictPartialOrder.trans PO b<a a<b) | |
490 | 59 |
492 | 60 open _==_ |
61 open _⊆_ | |
62 | |
497 | 63 -- open import Relation.Binary.Properties.Poset as Poset |
496 | 64 |
552 | 65 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) |
66 IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) | |
498 | 67 |
497 | 68 |
508 | 69 record Maximal ( A : HOD ) : Set (Level.suc n) where |
503 | 70 field |
71 maximal : HOD | |
72 A∋maximal : A ∋ maximal | |
73 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative | |
74 | |
530 | 75 -- |
76 -- inductive maxmum tree from x | |
77 -- tree structure | |
78 -- | |
79 | |
80 ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n) | |
81 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) | |
82 | |
556 | 83 immieate-f : (A : HOD) → ( f : Ordinal → Ordinal ) → Set n |
84 immieate-f A f = { x y : Ordinal } → odef A x → odef A y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) | |
85 | |
551 | 86 data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where |
554 | 87 init : odef A s → FClosure A f s s |
555 | 88 fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x) |
554 | 89 |
556 | 90 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 |
91 A∋fc {A} s f mf (init as) = as | |
92 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) ) | |
555 | 93 |
556 | 94 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y |
95 s≤fc {A} s {.s} f mf (init x) = case1 refl | |
96 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) ) | |
97 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) | |
98 ... | case2 x<fx with s≤fc {A} s f mf fcy | |
99 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) | |
100 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) | |
555 | 101 |
557 | 102 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ |
103 fcn s mf (init as) = zero | |
558 | 104 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) |
105 ... | case1 eq = fcn s mf p | |
106 ... | case2 y<fy = suc (fcn s mf p ) | |
557 | 107 |
558 | 108 fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) |
109 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y | |
559 | 110 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 |
111 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 | |
112 fc00 zero zero refl (init _) (init x₁) i=x i=y = refl | |
113 fc00 zero zero refl (init sa) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) ) | |
114 ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init sa) cy i=x i=y ) | |
115 fc00 zero zero refl (fsuc x cx) (init sa) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | |
116 ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init sa) i=x i=y ) | |
117 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 ) ) | |
118 ... | 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 ) | |
119 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 ) ) | |
120 ... | 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 ) | |
121 ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where | |
122 fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y) | |
123 fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) ) | |
124 ... | case1 eq = trans (sym eq) ( fc02 x1 cx1 i=x1 ) | |
125 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where | |
126 fc04 : * x1 ≡ * y | |
127 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y) | |
128 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where | |
129 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → * (f x) ≡ * y1 | |
130 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) ) | |
131 ... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq | |
132 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where | |
133 fc05 : * x ≡ * y1 | |
134 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) | |
135 ... | 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))) | |
557 | 136 |
137 fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) | |
138 → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y | |
558 | 139 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where |
140 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 | |
141 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) | |
142 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) | |
143 ... | case2 y<fy with <-cmp (fcn s mf cx ) i | |
144 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) | |
145 ... | 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 | |
146 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where | |
147 fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy | |
148 fc03 eq = cong pred eq | |
149 fc02 : * x < * y1 | |
150 fc02 = fc01 i cx cy (fc03 i=y ) a | |
557 | 151 |
559 | 152 fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) |
554 | 153 → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) |
559 | 154 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) |
155 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where | |
156 fc11 : * x < * y | |
157 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a | |
158 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where | |
159 fc10 : * x ≡ * y | |
160 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b | |
161 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where | |
162 fc12 : * y < * x | |
163 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c | |
164 | |
165 -- fcn-cmp {A} s {.s} {.s} f mf (init x) (init x₁) = tri≈ (λ lt → <-irr (case1 refl) lt ) refl (λ lt → <-irr (case1 refl) lt ) | |
166 -- fcn-cmp {A} s f mf imm (init x) (fsuc y cy) with proj1 (mf y (A∋fc s f mf cy ) ) | |
167 -- ... | case1 fy=y = subst (λ k → Tri (* s < * k) (* s ≡ * k) (* k < * s ) ) (*≡*→≡ fy=y) ( fcn-cmp {A} s f mf imm (init x) cy ) | |
168 -- ... | case2 fy>y = tri< lt (λ eq → <-irr (case1 (sym eq)) lt ) (λ lt1 → <-irr (case2 lt1) lt ) where | |
169 -- lt : * s < * (f y) | |
170 -- lt with s≤fc s f mf cy | |
171 -- ... | case1 s=y = subst (λ k → * k < * (f y) ) (sym (*≡*→≡ s=y)) fy>y | |
172 -- ... | case2 s<y = IsStrictPartialOrder.trans PO s<y fy>y | |
173 -- fcn-cmp {A} s {x} f mf imm cx (init x₁) with s≤fc s f mf cx | |
174 -- ... | case1 eq = tri≈ (λ lt → <-irr (case1 eq) lt) (sym eq) (λ lt → <-irr (case1 (sym eq)) lt) | |
175 -- ... | case2 s<x = tri> (λ lt → <-irr (case2 s<x) lt) (λ eq → <-irr (case1 eq) s<x ) s<x | |
176 -- fcn-cmp {A} s f mf imm (fsuc x cx) (fsuc y cy) with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) ) | |
177 -- ... | case1 x=fx | case1 y=fy = {!!} | |
178 -- ... | case1 x=fx | case2 y<fy = {!!} | |
179 -- ... | case2 x<fx | case1 y=fy = {!!} | |
180 -- ... | case2 x<fx | case2 y<fy = {!!} where | |
181 -- fc-mono : {x y : Ordinal } → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → FClosure A f x y ∨ FClosure A f y x | |
182 -- fc-mono = {!!} | |
183 -- fc1 : Tri (* (f x) < * (f y)) (* (f x) ≡ * (f y)) (* (f y) < * (f x)) | |
184 -- fc1 with fcn-cmp s f mf imm cx cy | |
185 -- ... | tri< a ¬b ¬c = {!!} | |
186 -- ... | tri≈ ¬a b ¬c = {!!} | |
187 -- ... | tri> ¬a ¬b c = {!!} | |
554 | 188 |
541 | 189 record Prev< (A B : HOD) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where |
533 | 190 field |
534 | 191 y : Ordinal |
541 | 192 ay : odef B y |
534 | 193 x=fy : x ≡ f y |
529 | 194 |
508 | 195 record SUP ( A B : HOD ) : Set (Level.suc n) where |
503 | 196 field |
197 sup : HOD | |
198 A∋maximal : A ∋ sup | |
199 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive | |
200 | |
533 | 201 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) |
202 SupCond A B _ _ = SUP A B | |
203 | |
546 | 204 record ZChain ( A : HOD ) {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f ) |
547 | 205 (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) ( z : Ordinal ) : Set (Level.suc n) where |
533 | 206 field |
207 chain : HOD | |
208 chain⊆A : chain ⊆ A | |
538 | 209 chain∋x : odef chain x |
554 | 210 initial : {y : Ordinal } → odef chain y → * x < * y |
533 | 211 f-total : IsTotalOrderSet chain |
546 | 212 f-next : {a : Ordinal } → odef chain a → odef chain (f a) |
541 | 213 f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) |
548 | 214 is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< z → (ba : odef A b) |
541 | 215 → Prev< A chain ba f |
216 ∨ (sup (& chain) (subst (λ k → k ⊆ A) (sym *iso) chain⊆A) (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b ) | |
534 | 217 → * a < * b → odef chain b |
533 | 218 |
497 | 219 Zorn-lemma : { A : HOD } |
464 | 220 → o∅ o< & A |
497 | 221 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition |
222 → Maximal A | |
552 | 223 Zorn-lemma {A} 0<A supP = zorn00 where |
535 | 224 supO : (C : Ordinal ) → (* C) ⊆ A → IsTotalOrderSet (* C) → Ordinal |
225 supO C C⊆A TC = & ( SUP.sup ( supP (* C) C⊆A TC )) | |
493 | 226 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ |
556 | 227 z01 {a} {b} A∋a A∋b = <-irr |
537 | 228 z07 : {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A |
229 z07 {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) | |
530 | 230 s : HOD |
231 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) | |
232 sa : A ∋ * ( & s ) | |
233 sa = 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 )) ) | |
547 | 234 s<A : & s o< & A |
235 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso sa ) | |
530 | 236 HasMaximal : HOD |
537 | 237 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } |
238 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ | |
239 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) | |
532 | 240 Gtx : { x : HOD} → A ∋ x → HOD |
537 | 241 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } |
242 z08 : ¬ Maximal A → HasMaximal =h= od∅ | |
243 z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; A∋maximal = subst (λ k → odef A k) (sym &iso) (proj1 lt) | |
244 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} | |
245 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)) | |
246 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 | |
247 ¬x<m : ¬ (* x < * m) | |
248 ¬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 | 249 |
250 -- Uncountable acending chain by axiom of choice | |
530 | 251 cf : ¬ Maximal A → Ordinal → Ordinal |
532 | 252 cf nmx x with ODC.∋-p O A (* x) |
253 ... | no _ = o∅ | |
254 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
538 | 255 ... | yes nogt = -- no larger element, so it is maximal |
256 ⊥-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 | 257 ... | no not = & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))) |
537 | 258 is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) ) |
259 is-cf nmx {x} ax with ODC.∋-p O A (* x) | |
260 ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax )) | |
261 ... | yes ax with is-o∅ (& ( Gtx ax )) | |
262 ... | 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 ⟫ ) | |
263 ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) | |
530 | 264 cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x ) |
537 | 265 cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫ |
530 | 266 cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx ) |
532 | 267 cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫ |
543 | 268 |
547 | 269 zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc : ZChain A sa f mf supO (& A) ) → SUP A (ZChain.chain zc) |
533 | 270 zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) |
547 | 271 A∋zsup : (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) ) |
533 | 272 → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )) |
273 A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ) | |
547 | 274 sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) → SUP A (* (& (ZChain.chain zc))) |
538 | 275 sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) |
276 (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) ) | |
543 | 277 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P |
278 zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) | |
279 | |
280 --- | |
281 --- sup is fix point in maximum chain | |
282 --- | |
547 | 283 z03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO (& A) ) |
546 | 284 → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc )) |
538 | 285 z03 f mf zc = z14 where |
286 chain = ZChain.chain zc | |
287 sp1 = sp0 f mf zc | |
548 | 288 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) |
541 | 289 → Prev< A chain ab f |
290 ∨ (supO (& chain) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc)) (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc)) ≡ b ) | |
538 | 291 → * a < * b → odef chain b |
292 z10 = ZChain.is-max zc | |
543 | 293 z11 : & (SUP.sup sp1) o< & A |
294 z11 = c<→o< ( SUP.A∋maximal sp1) | |
538 | 295 z12 : odef chain (& (SUP.sup sp1)) |
296 z12 with o≡? (& s) (& (SUP.sup sp1)) | |
297 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) | |
548 | 298 ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) z11 (SUP.A∋maximal sp1) (case2 refl ) z13 where |
538 | 299 z13 : * (& s) < * (& (SUP.sup sp1)) |
300 z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc )) | |
301 ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) | |
302 ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt | |
303 z14 : f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc)) | |
552 | 304 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 |
538 | 305 ... | tri< a ¬b ¬c = ⊥-elim z16 where |
306 z16 : ⊥ | |
307 z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) | |
308 ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) | |
309 ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) | |
310 ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) | |
311 ... | tri> ¬a ¬b c = ⊥-elim z17 where | |
312 z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) | |
546 | 313 z15 = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso) (ZChain.f-next zc z12 )) |
538 | 314 z17 : ⊥ |
315 z17 with z15 | |
316 ... | case1 eq = ¬b eq | |
317 ... | case2 lt = ¬a lt | |
547 | 318 -- ZChain requires the Maximal |
319 z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥ | |
537 | 320 z04 nmx zc = z01 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) |
538 | 321 (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) |
322 (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) | |
323 (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where | |
546 | 324 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc |
538 | 325 c = & (SUP.sup sp1) |
548 | 326 |
550 | 327 -- 3cases : {x y : Ordinal} → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) |
328 -- → (ax : odef A x )→ (ay : odef A y ) | |
329 -- → (zc0 : ZChain A ay f mf supO x ) | |
330 -- → Prev< A (ZChain.chain zc0) ax f | |
331 -- ∨ (supO (& (ZChain.chain zc0)) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ x) | |
332 -- ∨ ( ( z : Ordinal) → odef (ZChain.chain zc0) z → ¬ ( * z < * x )) | |
333 -- 3cases {x} {y} f mf ax ay zc0 = {!!} | |
547 | 334 -- create all ZChains under o< x |
546 | 335 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → |
547 | 336 ((z : Ordinal) → z o< x → {y : Ordinal} → (ya : odef A y) → ZChain A ya f mf supO z ) → { y : Ordinal } → (ya : odef A y) → ZChain A ya f mf supO x |
337 ind f mf x prev {y} ay with Oprev-p x | |
548 | 338 ... | yes op = zc4 where |
530 | 339 px = Oprev.oprev op |
547 | 340 zc0 : ZChain A ay f mf supO (Oprev.oprev op) |
341 zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay | |
342 zc4 : ZChain A ay f mf supO x | |
551 | 343 zc4 with ODC.∋-p O A (* px) |
554 | 344 ... | no noapx = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 |
345 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
551 | 346 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc11 } where -- no extention |
347 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
348 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
349 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
350 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) ≡ b) → | |
351 * a < * b → odef (ZChain.chain zc0) b | |
352 zc11 {a} {b} za b<x ba P a<b with osuc-≡< (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x) | |
353 ... | case1 eq = ⊥-elim ( noapx (subst (λ k → odef A k) (trans eq (sym &iso) ) ba )) | |
354 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
355 ... | yes apx with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) apx f ) -- we have to check adding x preserve is-max ZChain A ay f mf supO px | |
549 | 356 ... | case1 pr = zc9 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next |
551 | 357 chain = ZChain.chain zc0 |
358 zc17 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → | |
359 Prev< A (ZChain.chain zc0) ba f ∨ (supO (& (ZChain.chain zc0)) | |
360 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
361 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b) → | |
362 * a < * b → odef (ZChain.chain zc0) b | |
363 zc17 {a} {b} za b<x ba P a<b with osuc-≡< (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x) | |
364 ... | case2 lt = ZChain.is-max zc0 za lt ba P a<b | |
365 ... | case1 b=px = subst (λ k → odef chain k ) (trans (sym (Prev<.x=fy pr )) (trans &iso (sym b=px))) ( ZChain.f-next zc0 (Prev<.ay pr)) | |
549 | 366 zc9 : ZChain A ay f mf supO x |
367 zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
554 | 368 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = zc17 } -- no extention |
551 | 369 ... | case2 ¬fy<x with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) )) |
552 | 370 ... | case1 x=sup = record { chain = schain ; chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} |
554 | 371 ; initial = {!!} ; f-immediate = {!!} ; chain∋x = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup |
551 | 372 sp = SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) |
373 chain = ZChain.chain zc0 | |
552 | 374 schain : HOD |
375 schain = record { od = record { def = λ x → odef chain x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = {!!} } | |
376 ... | case2 ¬x=sup = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0 | |
554 | 377 ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x = ZChain.chain∋x zc0 ; is-max = {!!} } where -- no extention |
552 | 378 z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< x → (ba : odef A b) → |
379 Prev< A (ZChain.chain zc0) ba f ∨ (& (SUP.sup (supP (* (& (ZChain.chain zc0))) | |
380 (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0)) | |
381 (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)))) | |
382 ≡ b) → | |
383 * a < * b → odef (ZChain.chain zc0) b | |
384 z18 {a} {b} za b<x ba (case1 p) a<b = {!!} | |
385 z18 {a} {b} za b<x ba (case2 p) a<b = {!!} | |
553 | 386 ... | no ¬ox = {!!} where --- limit ordinal case |
554 | 387 record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x |
553 | 388 field |
389 u : Ordinal | |
390 u<x : u o< x | |
554 | 391 zuy : odef (ZChain.chain (prev u u<x {y} ay )) z |
392 Uz : HOD | |
393 Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A ; <odmax = {!!} } | |
394 u-total : IsTotalOrderSet Uz | |
553 | 395 u-total {x} {y} ux uy = {!!} |
396 | |
551 | 397 zorn00 : Maximal A |
398 zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM | |
399 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where | |
400 -- yes we have the maximal | |
401 zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) | |
402 zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
403 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) | |
404 zorn01 = proj1 zorn03 | |
405 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) | |
406 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) | |
407 ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where | |
408 -- if we have no maximal, make ZChain, which contradict SUP condition | |
409 nmx : ¬ Maximal A | |
410 nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where | |
411 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) | |
412 zc5 = ⟪ Maximal.A∋maximal 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) ) ⟫ | |
413 zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO (& A) | |
414 zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain A ya f mf supO z } (ind f mf) (& A) | |
415 zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO (& A) | |
416 zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa ) | |
417 | |
516 | 418 -- usage (see filter.agda ) |
419 -- | |
497 | 420 -- _⊆'_ : ( A B : HOD ) → Set n |
421 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x | |
482 | 422 |
497 | 423 -- MaximumSubset : {L P : HOD} |
424 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
425 -- → IsPartialOrderSet P _⊆'_ | |
426 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
427 -- → Maximal P (_⊆'_) | |
428 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP |