|
478
|
1 {-# OPTIONS --allow-unsolved-metas #-}
|
|
431
|
2 open import Level
|
|
|
3 open import Ordinals
|
|
477
|
4 module zorn {n : Level } (O : Ordinals {n}) where
|
|
431
|
5
|
|
|
6 open import zf
|
|
477
|
7 open import logic
|
|
|
8 -- open import partfunc {n} O
|
|
|
9 import OD
|
|
|
10
|
|
|
11 open import Relation.Nullary
|
|
|
12 open import Relation.Binary
|
|
|
13 open import Data.Empty
|
|
431
|
14 open import Relation.Binary
|
|
|
15 open import Relation.Binary.Core
|
|
477
|
16 open import Relation.Binary.PropositionalEquality
|
|
|
17 import BAlgbra
|
|
431
|
18
|
|
|
19
|
|
|
20 open inOrdinal O
|
|
|
21 open OD O
|
|
|
22 open OD.OD
|
|
|
23 open ODAxiom odAxiom
|
|
477
|
24 import OrdUtil
|
|
|
25 import ODUtil
|
|
431
|
26 open Ordinals.Ordinals O
|
|
|
27 open Ordinals.IsOrdinals isOrdinal
|
|
|
28 open Ordinals.IsNext isNext
|
|
|
29 open OrdUtil O
|
|
477
|
30 open ODUtil O
|
|
|
31
|
|
|
32
|
|
|
33 import ODC
|
|
|
34
|
|
|
35
|
|
|
36 open _∧_
|
|
|
37 open _∨_
|
|
|
38 open Bool
|
|
431
|
39
|
|
|
40
|
|
|
41 open HOD
|
|
|
42
|
|
469
|
43 record Element (A : HOD) : Set (suc n) where
|
|
|
44 field
|
|
|
45 elm : HOD
|
|
|
46 is-elm : A ∋ elm
|
|
|
47
|
|
|
48 open Element
|
|
|
49
|
|
493
|
50 IsPartialOrderSet : ( A : HOD ) → (_<_ : (x y : HOD) → Set n ) → Set (suc n)
|
|
495
|
51 IsPartialOrderSet A _<_ = IsStrictPartialOrder _≡A_ _<A_ where
|
|
|
52 _<A_ : (x y : Element A ) → Set n
|
|
|
53 x <A y = elm x < elm y
|
|
490
|
54 _≡A_ : (x y : Element A ) → Set (suc n)
|
|
|
55 x ≡A y = elm x ≡ elm y
|
|
|
56
|
|
492
|
57 open _==_
|
|
|
58 open _⊆_
|
|
|
59
|
|
495
|
60 isA : { A B : HOD } → B ⊆ A → (x : Element B) → Element A
|
|
|
61 isA B⊆A x = record { elm = elm x ; is-elm = incl B⊆A (is-elm x) }
|
|
494
|
62
|
|
493
|
63 ⊆-IsPartialOrderSet : { A B : HOD } → B ⊆ A → {_<_ : (x y : HOD) → Set n } → IsPartialOrderSet A _<_ → IsPartialOrderSet B _<_
|
|
|
64 ⊆-IsPartialOrderSet {A} {B} B⊆A {_<_} PA = record {
|
|
495
|
65 isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; reflexive = λ eq → case1 eq ; trans = λ {x} {y} {z} → trans1 {x} {y} {z}
|
|
|
66 ; irrefl = λ {x} {y} → irrefl1 {x} {y} ; trans = trans1 ; <-resp-≈ = resp0
|
|
493
|
67 } where
|
|
495
|
68 _<B_ : (x y : Element B ) → Set n
|
|
|
69 x <B y = elm x < elm y
|
|
|
70 trans1 : {x y z : Element B} → x <B y → y <B z → x <B z
|
|
|
71 trans1 {x} {y} {z} x<y y<z = IsStrictPartialOrder.trans PA {isA B⊆A x} {isA B⊆A y} {isA B⊆A z} x<y y<z
|
|
|
72 irrefl1 : {x y : Element B} → elm x ≡ elm y → ¬ ( x <B y )
|
|
|
73 irrefl1 {x} {y} x=y x<y = IsStrictPartialOrder.irrefl PA {isA B⊆A x} {isA B⊆A y} x=y x<y
|
|
|
74 open import Data.Product
|
|
|
75 resp0 : ({x y z : Element B} → elm y ≡ elm z → x <B y → x <B z) × ({x y z : Element B} → elm y ≡ elm z → y <B x → z <B x)
|
|
|
76 resp0 = Data.Product._,_ (λ {x} {y} {z} → proj₁ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z })
|
|
|
77 (λ {x} {y} {z} → proj₂ (IsStrictPartialOrder.<-resp-≈ PA) {isA B⊆A x } {isA B⊆A y }{isA B⊆A z })
|
|
492
|
78
|
|
493
|
79 IsTotalOrderSet : ( A : HOD ) → (_<_ : (x y : HOD) → Set n ) → Set (suc n)
|
|
494
|
80 IsTotalOrderSet A _<_ = IsTotalOrder _≡A_ _≤A_ where
|
|
|
81 _≤A_ : (x y : Element A ) → Set (suc n)
|
|
|
82 x ≤A y = (elm x ≡ elm y) ∨ (elm x < elm y)
|
|
490
|
83 _≡A_ : (x y : Element A ) → Set (suc n)
|
|
|
84 x ≡A y = elm x ≡ elm y
|
|
|
85
|
|
469
|
86 me : { A a : HOD } → A ∋ a → Element A
|
|
|
87 me {A} {a} lt = record { elm = a ; is-elm = lt }
|
|
|
88
|
|
493
|
89 record SUP ( A B : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
|
|
464
|
90 field
|
|
|
91 sup : HOD
|
|
|
92 A∋maximal : A ∋ sup
|
|
494
|
93 x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive
|
|
464
|
94
|
|
493
|
95 record Maximal ( A : HOD ) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
|
|
464
|
96 field
|
|
|
97 maximal : HOD
|
|
472
|
98 A∋maximal : A ∋ maximal
|
|
493
|
99 ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative
|
|
464
|
100
|
|
477
|
101
|
|
493
|
102 record ZChain ( A : HOD ) (y : Ordinal) (_<_ : (x y : HOD) → Set n ) : Set (suc n) where
|
|
464
|
103 field
|
|
491
|
104 fb : (x : Ordinal ) → HOD
|
|
493
|
105 A∋fb : (ox : Ordinal ) → ox o< y → A ∋ fb ox
|
|
|
106 total : {ox oz : Ordinal } → (x<y : ox o< y ) → (z<y : oz o< y ) → ( ox ≡ oz ) ∨ ( fb ox < fb oz ) ∨ ( fb oz < fb ox )
|
|
|
107 monotonic : {ox oz : Ordinal } → (x<y : ox o< y ) → (z<y : oz o< y ) → ox o< oz → fb ox < fb oz
|
|
464
|
108
|
|
493
|
109 Zorn-lemma : { A : HOD } → { _<_ : (x y : HOD) → Set n }
|
|
464
|
110 → o∅ o< & A
|
|
493
|
111 → IsPartialOrderSet A _<_
|
|
|
112 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B _<_ → SUP A B _<_ ) -- SUP condition
|
|
|
113 → Maximal A _<_
|
|
|
114 Zorn-lemma {A} {_<_} 0<A PO supP = zorn00 where
|
|
472
|
115 someA : HOD
|
|
477
|
116 someA = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
|
|
473
|
117 isSomeA : A ∋ someA
|
|
477
|
118 isSomeA = ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
|
|
467
|
119 HasMaximal : HOD
|
|
493
|
120 HasMaximal = record { od = record { def = λ y → odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))} ; odmax = & A ; <odmax = z08 } where
|
|
|
121 z08 : {y : Ordinal} → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → y o< & A
|
|
482
|
122 z08 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P)))
|
|
493
|
123 no-maximal : HasMaximal =h= od∅ → (y : Ordinal) → (odef A y ∧ ((m : Ordinal) → odef A m → ¬ (* y < * m))) → ⊥
|
|
482
|
124 no-maximal nomx y P = ¬x<0 (eq→ nomx {y} ⟪ proj1 P , (λ m am → (proj2 P) m am ) ⟫ )
|
|
472
|
125 Gtx : { x : HOD} → A ∋ x → HOD
|
|
493
|
126 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z09 } where
|
|
|
127 z09 : {y : Ordinal} → (odef A y ∧ (x < (* y))) → y o< & A
|
|
482
|
128 z09 {y} P = subst (λ k → k o< & A) &iso (c<→o< {* y} (subst (λ k → odef A k) (sym &iso) (proj1 P)))
|
|
493
|
129 z01 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥
|
|
|
130 z01 {a} {b} A∋a A∋b (case1 a=b) b<a = {!!} -- proj1 (proj2 (PO (me A∋b) (me A∋a)) (sym a=b)) b<a
|
|
|
131 z01 {a} {b} A∋a A∋b (case2 a<b) b<a = {!!} -- proj1 (proj1 (PO (me A∋b) (me A∋a)) b<a) a<b
|
|
478
|
132 -- ZChain is not compatible with the SUP condition
|
|
491
|
133 record BX (x y : Ordinal) (fb : ( x : Ordinal ) → HOD ) : Set n where
|
|
484
|
134 field
|
|
|
135 bx : Ordinal
|
|
493
|
136 bx<y : bx o< y
|
|
491
|
137 is-fb : x ≡ & (fb bx )
|
|
493
|
138 bx<A : (z : ZChain A (& A) _<_ ) → {x : Ordinal } → (bx : BX x (& A) ( ZChain.fb z )) → BX.bx bx o< & A
|
|
|
139 bx<A z {x} bx = BX.bx<y bx
|
|
|
140 B : (z : ZChain A (& A) _<_ ) → HOD
|
|
|
141 B z = record { od = record { def = λ x → BX x (& A) ( ZChain.fb z ) } ; odmax = & A ; <odmax = {!!} }
|
|
|
142 z11 : (z : ZChain A (& A) _<_ ) → (x : Element (B z)) → elm x ≡ ZChain.fb z (BX.bx (is-elm x))
|
|
485
|
143 z11 z x = subst₂ (λ j k → j ≡ k ) *iso *iso ( cong (*) (BX.is-fb (is-elm x)) )
|
|
493
|
144 obx : (z : ZChain A (& A) _<_ ) → {x : HOD} → B z ∋ x → Ordinal
|
|
485
|
145 obx z {x} bx = BX.bx bx
|
|
493
|
146 obx=fb : (z : ZChain A (& A) _<_ ) → {x : HOD} → (bx : B z ∋ x ) → x ≡ ZChain.fb z (BX.bx bx)
|
|
485
|
147 obx=fb z {x} bx = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (BX.is-fb bx))
|
|
493
|
148 B⊆A : (z : ZChain A (& A) _<_ ) → B z ⊆ A
|
|
|
149 B⊆A z = record { incl = λ {x} bx → subst (λ k → odef A k ) (sym (BX.is-fb bx)) (ZChain.A∋fb z (BX.bx bx) (BX.bx<y bx) ) }
|
|
|
150 PO-B : (z : ZChain A (& A) _<_ ) → IsPartialOrderSet (B z) _<_
|
|
495
|
151 PO-B z = subst₂ (λ j k → IsStrictPartialOrder j k ) {!!} {!!} {!!} where
|
|
493
|
152 _<B_ = {!!}
|
|
492
|
153 _≡B_ = {!!}
|
|
|
154 -- a b = {!!} -- PO record { elm = elm a ; is-elm = incl ( B⊆A z) (is-elm a) } record { elm = elm b ; is-elm = incl ( B⊆A z) (is-elm b) }
|
|
493
|
155 bx-monotonic : (z : ZChain A (& A) _<_ ) → {x y : Element (B z)} → obx z (is-elm x) o< obx z (is-elm y) → elm x < elm y
|
|
|
156 bx-monotonic z {x} {y} a = subst₂ (λ j k → j < k ) (sym (z11 z x)) (sym (z11 z y)) (ZChain.monotonic z (bx<A z (is-elm x)) (bx<A z (is-elm y)) a )
|
|
488
|
157 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
|
|
493
|
158 z12 : (z : ZChain A (& A) _<_ ) → {a b : HOD } → (x : BX (& a) (& A) (ZChain.fb z)) (y : BX (& b) (& A) (ZChain.fb z))
|
|
|
159 → obx z x ≡ obx z y → bx<A z x ≅ bx<A z y
|
|
488
|
160 z12 z {a} {b} x y eq = {!!}
|
|
493
|
161 bx-inject : (z : ZChain A (& A) _<_ ) → {x y : Element (B z)} → BX.bx (is-elm x) ≡ BX.bx (is-elm y) → elm x ≡ elm y
|
|
488
|
162 bx-inject z {x} {y} eq = begin
|
|
489
|
163 elm x ≡⟨ {!!} ⟩
|
|
490
|
164 {!!} ≡⟨ cong (λ k → {!!} ) {!!} ⟩
|
|
489
|
165 {!!} ≡⟨ {!!} ⟩
|
|
488
|
166 elm y ∎ where open ≡-Reasoning
|
|
493
|
167 B-is-total : (z : ZChain A (& A) _<_ ) → IsTotalOrderSet (B z) _<_
|
|
492
|
168 B-is-total = {!!}
|
|
493
|
169 B-Tri : (z : ZChain A (& A) _<_ ) → Trichotomous (λ (x : Element A) y → elm x ≡ elm y ) (λ x y → elm x < elm y )
|
|
492
|
170 B-Tri z x y with trio< (obx z {!!}) (obx z {!!})
|
|
493
|
171 ... | tri< a ¬b ¬c = {!!} where -- tri< z10 (λ eq → proj1 (proj2 (PO-B z x y) eq ) z10) (λ ¬c → proj1 (proj1 (PO-B z y x) ¬c ) z10) where
|
|
|
172 z10 : elm x < elm y
|
|
492
|
173 z10 = {!!} -- bx-monotonic z {x} {y} a
|
|
|
174 ... | tri≈ ¬a b ¬c = {!!} -- tri≈ {!!} (bx-inject z {x} {y} b) {!!}
|
|
|
175 ... | tri> ¬a ¬b c = {!!} -- tri> (λ ¬a → proj1 (proj1 (PO-B z x y) ¬a ) (bx-monotonic z {y} {x} c) ) (λ eq → proj2 (proj2 (PO-B z x y) eq ) (bx-monotonic z {y} {x} c)) (bx-monotonic z {y} {x} c)
|
|
493
|
176 ZChain→¬SUP : (z : ZChain A (& A) _<_ ) → ¬ (SUP A (B z) _<_ )
|
|
485
|
177 ZChain→¬SUP z sp = ⊥-elim {!!} where
|
|
472
|
178 z03 : & (SUP.sup sp) o< osuc (& A)
|
|
|
179 z03 = ordtrans (c<→o< (SUP.A∋maximal sp)) <-osuc
|
|
493
|
180 z02 : (x : HOD) → B z ∋ x → SUP.sup sp < x → ⊥
|
|
494
|
181 z02 x xe s<x = z01 (incl (B⊆A z) xe) (SUP.A∋maximal sp) (SUP.x<sup sp xe) s<x
|
|
471
|
182 ind : HasMaximal =h= od∅
|
|
493
|
183 → (x : Ordinal) → ((y : Ordinal) → y o< x → ZChain A y _<_ )
|
|
|
184 → ZChain A x _<_
|
|
471
|
185 ind nomx x prev with Oprev-p x
|
|
477
|
186 ... | yes op with ODC.∋-p O A (* x)
|
|
483
|
187 ... | no ¬Ax = {!!} where
|
|
476
|
188 -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
|
|
471
|
189 px = Oprev.oprev op
|
|
493
|
190 zc1 : ZChain A px _<_
|
|
471
|
191 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
|
|
493
|
192 z04 : {!!} -- (sup : HOD) (as : A ∋ sup) → & sup o< osuc x → sup < ZChain.fb zc1 as
|
|
|
193 z04 sup as s<x with trio< (& sup) x
|
|
471
|
194 ... | tri≈ ¬a b ¬c = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans b (sym &iso)) as) )
|
|
494
|
195 ... | tri< a ¬b ¬c = {!!} -- ZChain.¬x<sup zc1 _ as ( subst (λ k → & sup o< k ) (sym (Oprev.oprev=x op)) a )
|
|
493
|
196 ... | tri> ¬a ¬b c with osuc-≡< s<x
|
|
472
|
197 ... | case1 eq = ⊥-elim (¬Ax (subst (λ k → odef A k) (trans eq (sym &iso)) as) )
|
|
|
198 ... | case2 lt = ⊥-elim (¬a lt )
|
|
476
|
199 ... | yes ax = z06 where -- we have previous ordinal and A ∋ x
|
|
472
|
200 px = Oprev.oprev op
|
|
493
|
201 zc1 : ZChain A px _<_
|
|
472
|
202 zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc)
|
|
493
|
203 z06 : ZChain A x _<_
|
|
473
|
204 z06 with is-o∅ (& (Gtx ax))
|
|
482
|
205 ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal
|
|
493
|
206 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m)
|
|
|
207 x-is-maximal m am = ¬x<m where
|
|
|
208 ¬x<m : ¬ (* x < * m)
|
|
|
209 ¬x<m x<m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt)
|
|
483
|
210 ... | no not = {!!} where
|
|
476
|
211 ind nomx x prev | no ¬ox with trio< (& A) x --- limit ordinal case
|
|
483
|
212 ... | tri< a ¬b ¬c = {!!} where
|
|
493
|
213 zc1 : ZChain A (& A) _<_
|
|
475
|
214 zc1 = prev (& A) a
|
|
483
|
215 ... | tri≈ ¬a b ¬c = {!!} where
|
|
478
|
216 ... | tri> ¬a ¬b c with ODC.∋-p O A (* x)
|
|
|
217 ... | no ¬Ax = {!!} where
|
|
|
218 ... | yes ax with is-o∅ (& (Gtx ax))
|
|
482
|
219 ... | yes nogt = ⊥-elim (no-maximal nomx x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal ⟫ ) where -- no larger element, so it is maximal
|
|
493
|
220 x-is-maximal : (m : Ordinal) → odef A m → ¬ (* x < * m)
|
|
|
221 x-is-maximal m am = ¬x<m where
|
|
|
222 ¬x<m : ¬ (* x < * m)
|
|
|
223 ¬x<m x<m = ∅< {Gtx ax} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt)
|
|
478
|
224 ... | no not = {!!} where
|
|
493
|
225 zorn00 : Maximal A _<_
|
|
467
|
226 zorn00 with is-o∅ ( & HasMaximal )
|
|
493
|
227 ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; A∋maximal = zorn01 ; ¬maximal<x = zorn02 } where
|
|
478
|
228 -- yes we have the maximal
|
|
483
|
229 hasm : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) )
|
|
|
230 hasm = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
|
|
477
|
231 zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
|
|
483
|
232 zorn01 = proj1 hasm
|
|
493
|
233 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
|
|
|
234 zorn02 {x} ax m<x = ((proj2 hasm) (& x) ax) (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x )
|
|
483
|
235 ... | yes ¬Maximal = ⊥-elim {!!} where
|
|
478
|
236 -- if we have no maximal, make ZChain, which contradict SUP condition
|
|
493
|
237 z : (x : Ordinal) → HasMaximal =h= od∅ → ZChain A x _<_
|
|
471
|
238 z x nomx = TransFinite (ind nomx) x
|
|
464
|
239
|
|
482
|
240 _⊆'_ : ( A B : HOD ) → Set n
|
|
|
241 _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x
|
|
|
242
|
|
|
243 MaximumSubset : {L P : HOD}
|
|
|
244 → o∅ o< & L → o∅ o< & P → P ⊆ L
|
|
491
|
245 → IsPartialOrderSet P _⊆'_
|
|
|
246 → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ )
|
|
482
|
247 → Maximal P (_⊆'_)
|
|
|
248 MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP
|
