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annotate LEMC.agda @ 277:d9d3654baee1

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seperate choice from LEM
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 09 May 2020 09:38:21 +0900
parents ODC.agda@6f10c47e4e7a
children 197e0b3d39dc
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
16
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 15
diff changeset
1 open import Level
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
2 open import Ordinals
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
3 open import logic
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
4 open import Relation.Nullary
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
5 module LEMC {n : Level } (O : Ordinals {n} ) (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
7 open import zf
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 22
diff changeset
8 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
14
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
9 open import Relation.Binary.PropositionalEquality
e11e95d5ddee separete constructible set
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 11
diff changeset
10 open import Data.Nat.Properties
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
11 open import Data.Empty
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
12 open import Relation.Binary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
13 open import Relation.Binary.Core
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
14
213
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
15 open import nat
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
16 import OD
213
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 210
diff changeset
17
223
2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
18 open inOrdinal O
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
19 open OD O
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
20 open OD.OD
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
21 open OD._==_
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
22 open ODAxiom odAxiom
119
6e264c78e420 infinite
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 118
diff changeset
23
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
24 open import zfc
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
25
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
26 --- With assuption of OD is ordered, p ∨ ( ¬ p ) <=> axiom of choice
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
27 ---
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
28 record choiced ( X : OD) : Set (suc n) where
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
29 field
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
30 a-choice : OD
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
31 is-in : X ∋ a-choice
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
32
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
33 open choiced
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
34
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
35 OD→ZFC : ZFC
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
36 OD→ZFC = record {
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2e1f19c949dc sepration of ordinal from OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 219
diff changeset
37 ZFSet = OD
43
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
38 ; _∋_ = _∋_
0d9b9db14361 equalitu and internal parametorisity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
39 ; _≈_ = _==_
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
40 ; ∅ = od∅
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
41 ; Select = Select
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
42 ; isZFC = isZFC
28
f36e40d5d2c3 OD continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 27
diff changeset
43 } where
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6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
44 -- infixr 200 _∈_
96
f239ffc27fd0 Power Set and L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 95
diff changeset
45 -- infixr 230 _∩_ _∪_
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
46 isZFC : IsZFC (OD ) _∋_ _==_ od∅ Select
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 274
diff changeset
47 isZFC = record {
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
48 choice-func = λ A {X} not A∋X → a-choice (choice-func X not );
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
49 choice = λ A {X} A∋X not → is-in (choice-func X not)
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 28
diff changeset
50 } where
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
51 choice-func : (X : OD ) → ¬ ( X == od∅ ) → choiced X
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
52 choice-func X not = have_to_find where
234
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
53 ψ : ( ox : Ordinal ) → Set (suc n)
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
54 ψ ox = (( x : Ordinal ) → x o< ox → ( ¬ def X x )) ∨ choiced X
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
55 lemma-ord : ( ox : Ordinal ) → ψ ox
235
846e0926bb89 fix cardinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 234
diff changeset
56 lemma-ord ox = TransFinite {ψ} induction ox where
234
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
57 ∋-p : (A x : OD ) → Dec ( A ∋ x )
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
58 ∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x )) -- LEM
234
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
59 ∋-p A x | case1 (lift t) = yes t
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
60 ∋-p A x | case2 t = no (λ x → t (lift x ))
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
61 ∀-imply-or : {A : Ordinal → Set n } {B : Set (suc n) }
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
62 → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
63 ∀-imply-or {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x)) -- LEM
234
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
64 ∀-imply-or {A} {B} ∀AB | case1 (lift t) = case1 t
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
65 ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x (lift not ))) where
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
66 lemma : ¬ ((x : Ordinal ) → A x) → B
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
67 lemma not with p∨¬p B
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
68 lemma not | case1 b = b
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
69 lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b ))
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
70 induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
71 induction x prev with ∋-p X ( ord→od x)
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
72 ... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } )
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
73 ... | no ¬p = lemma where
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
74 lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
75 lemma1 y with ∋-p X (ord→od y)
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
76 lemma1 y | yes y<X = case2 ( record { a-choice = ord→od y ; is-in = y<X } )
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
77 lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → def X k ) (sym diso) y<X ) )
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
78 lemma : ((y : Ordinals.ord O) → (O Ordinals.o< y) x → def X y → ⊥) ∨ choiced X
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
79 lemma = ∀-imply-or lemma1
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
80 have_to_find : choiced X
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 262
diff changeset
81 have_to_find = dont-or (lemma-ord (od→ord X )) ¬¬X∋x where
234
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
82 ¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → def X x → ⊥)
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
83 ¬¬X∋x nn = not record {
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
84 eq→ = λ {x} lt → ⊥-elim (nn x (def→o< lt) lt)
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
85 ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt )
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
86 }
e06b76e5b682 ac from LEM in abstract ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 228
diff changeset
87