Mercurial > hg > Members > kono > Proof > ZF-in-agda
comparison ODC.agda @ 276:6f10c47e4e7a
separate choice
fix sup-o
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Sat, 09 May 2020 09:02:52 +0900 |
parents | OD.agda@29a85a427ed2 |
children | d9d3654baee1 |
comparison
equal
deleted
inserted
replaced
275:455792eaa611 | 276:6f10c47e4e7a |
---|---|
1 open import Level | |
2 open import Ordinals | |
3 module ODC {n : Level } (O : Ordinals {n} ) where | |
4 | |
5 open import zf | |
6 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) | |
7 open import Relation.Binary.PropositionalEquality | |
8 open import Data.Nat.Properties | |
9 open import Data.Empty | |
10 open import Relation.Nullary | |
11 open import Relation.Binary | |
12 open import Relation.Binary.Core | |
13 | |
14 open import logic | |
15 open import nat | |
16 import OD | |
17 | |
18 open inOrdinal O | |
19 open OD O | |
20 open OD.OD | |
21 open OD._==_ | |
22 | |
23 postulate | |
24 -- mimimul and x∋minimal is an Axiom of choice | |
25 minimal : (x : OD ) → ¬ (x == od∅ )→ OD | |
26 -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox ( minimum of x ) | |
27 x∋minimal : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) ) | |
28 -- minimality (may proved by ε-induction ) | |
29 minimal-1 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) | |
30 | |
31 | |
32 -- | |
33 -- Axiom of choice in intutionistic logic implies the exclude middle | |
34 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog | |
35 -- | |
36 | |
37 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p | |
38 ppp {p} {a} d = d | |
39 | |
40 p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice | |
41 p∨¬p p with is-o∅ ( od→ord ( record { def = λ x → p } )) | |
42 p∨¬p p | yes eq = case2 (¬p eq) where | |
43 ps = record { def = λ x → p } | |
44 lemma : ps == od∅ → p → ⊥ | |
45 lemma eq p0 = ¬x<0 {od→ord ps} (eq→ eq p0 ) | |
46 ¬p : (od→ord ps ≡ o∅) → p → ⊥ | |
47 ¬p eq = lemma ( subst₂ (λ j k → j == k ) oiso o∅≡od∅ ( o≡→== eq )) | |
48 p∨¬p p | no ¬p = case1 (ppp {p} {minimal ps (λ eq → ¬p (eqo∅ eq))} lemma) where | |
49 ps = record { def = λ x → p } | |
50 eqo∅ : ps == od∅ → od→ord ps ≡ o∅ | |
51 eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq)) | |
52 lemma : ps ∋ minimal ps (λ eq → ¬p (eqo∅ eq)) | |
53 lemma = x∋minimal ps (λ eq → ¬p (eqo∅ eq)) | |
54 | |
55 decp : ( p : Set n ) → Dec p -- assuming axiom of choice | |
56 decp p with p∨¬p p | |
57 decp p | case1 x = yes x | |
58 decp p | case2 x = no x | |
59 | |
60 double-neg-eilm : {A : Set n} → ¬ ¬ A → A -- we don't have this in intutionistic logic | |
61 double-neg-eilm {A} notnot with decp A -- assuming axiom of choice | |
62 ... | yes p = p | |
63 ... | no ¬p = ⊥-elim ( notnot ¬p ) | |
64 | |
65 OrdP : ( x : Ordinal ) ( y : OD ) → Dec ( Ord x ∋ y ) | |
66 OrdP x y with trio< x (od→ord y) | |
67 OrdP x y | tri< a ¬b ¬c = no ¬c | |
68 OrdP x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl ) | |
69 OrdP x y | tri> ¬a ¬b c = yes c | |
70 | |
71 open import zfc | |
72 | |
73 OD→ZFC : ZFC | |
74 OD→ZFC = record { | |
75 ZFSet = OD | |
76 ; _∋_ = _∋_ | |
77 ; _≈_ = _==_ | |
78 ; ∅ = od∅ | |
79 ; Select = Select | |
80 ; isZFC = isZFC | |
81 } where | |
82 -- infixr 200 _∈_ | |
83 -- infixr 230 _∩_ _∪_ | |
84 isZFC : IsZFC (OD ) _∋_ _==_ od∅ Select | |
85 isZFC = record { | |
86 choice-func = choice-func ; | |
87 choice = choice | |
88 } where | |
89 -- Axiom of choice ( is equivalent to the existence of minimal in our case ) | |
90 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] | |
91 choice-func : (X : OD ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD | |
92 choice-func X {x} not X∋x = minimal x not | |
93 choice : (X : OD ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A | |
94 choice X {A} X∋A not = x∋minimal A not | |
95 | |
96 --- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice | |
97 --- | |
98 record choiced ( X : OD) : Set (suc n) where | |
99 field | |
100 a-choice : OD | |
101 is-in : X ∋ a-choice | |
102 | |
103 choice-func' : (X : OD ) → (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X | |
104 choice-func' X p∨¬p not = have_to_find where | |
105 ψ : ( ox : Ordinal ) → Set (suc n) | |
106 ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ def X x )) ∨ choiced X | |
107 lemma-ord : ( ox : Ordinal ) → ψ ox | |
108 lemma-ord ox = TransFinite {ψ} induction ox where | |
109 ∋-p : (A x : OD ) → Dec ( A ∋ x ) | |
110 ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x )) -- LEM | |
111 ∋-p A x | case1 (lift t) = yes t | |
112 ∋-p A x | case2 t = no (λ x → t (lift x )) | |
113 ∀-imply-or : {A : Ordinal → Set n } {B : Set (suc n) } | |
114 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B | |
115 ∀-imply-or {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x)) -- LEM | |
116 ∀-imply-or {A} {B} ∀AB | case1 (lift t) = case1 t | |
117 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x (lift not ))) where | |
118 lemma : ¬ ((x : Ordinal ) → A x) → B | |
119 lemma not with p∨¬p B | |
120 lemma not | case1 b = b | |
121 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) | |
122 induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x | |
123 induction x prev with ∋-p X ( ord→od x) | |
124 ... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } ) | |
125 ... | no ¬p = lemma where | |
126 lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X | |
127 lemma1 y with ∋-p X (ord→od y) | |
128 lemma1 y | yes y<X = case2 ( record { a-choice = ord→od y ; is-in = y<X } ) | |
129 lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → def X k ) (sym diso) y<X ) ) | |
130 lemma : ((y : Ordinals.ord O) → (O Ordinals.o< y) x → def X y → ⊥) ∨ choiced X | |
131 lemma = ∀-imply-or lemma1 | |
132 have_to_find : choiced X | |
133 have_to_find = dont-or (lemma-ord (od→ord X )) ¬¬X∋x where | |
134 ¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → def X x → ⊥) | |
135 ¬¬X∋x nn = not record { | |
136 eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt) | |
137 ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt ) | |
138 } | |
139 |