### view LEMC.agda @ 281:81d639ee9bfd

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author Shinji KONO Sat, 09 May 2020 22:03:17 +0900 a2991ce14ced 6630dab08784
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open import Level
open import Ordinals
open import logic
open import Relation.Nullary
module LEMC {n : Level } (O : Ordinals {n} ) (p∨¬p : ( p : Set (suc n)) → p ∨ ( ¬ p )) where

open import zf
open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ )
open import  Relation.Binary.PropositionalEquality
open import Data.Nat.Properties
open import Data.Empty
open import Relation.Binary
open import Relation.Binary.Core

open import nat
import OD

open inOrdinal O
open OD O
open OD.OD
open OD._==_
open ODAxiom odAxiom

open import zfc

--- With assuption of OD is ordered,  p ∨ ( ¬ p ) <=> axiom of choice
---
record choiced  ( X : OD) : Set (suc n) where
field
a-choice : OD
is-in : X ∋ a-choice

open choiced

OD→ZFC : ZFC
OD→ZFC   = record {
ZFSet = OD
; _∋_ = _∋_
; _≈_ = _==_
; ∅  = od∅
; Select = Select
; isZFC = isZFC
} where
-- infixr  200 _∈_
-- infixr  230 _∩_ _∪_
isZFC : IsZFC (OD )  _∋_  _==_ od∅ Select
isZFC = record {
choice-func = λ A {X} not A∋X → a-choice (choice-func X not );
choice = λ A {X} A∋X not → is-in (choice-func X not)
} where
choice-func :  (X : OD ) → ¬ ( X == od∅ ) → choiced X
choice-func  X not = have_to_find where
ψ : ( ox : Ordinal ) → Set (suc n)
ψ ox = (( x : Ordinal ) → x o< ox  → ( ¬ def X x )) ∨ choiced X
lemma-ord : ( ox : Ordinal  ) → ψ ox
lemma-ord  ox = TransFinite {ψ} induction ox where
∋-p : (A x : OD ) → Dec ( A ∋ x )
∋-p A x with p∨¬p (Lift (suc n) ( A ∋ x )) -- LEM
∋-p A x | case1 (lift t)  = yes t
∋-p A x | case2 t  = no (λ x → t (lift x ))
∀-imply-or :  {A : Ordinal  → Set n } {B : Set (suc n) }
→ ((x : Ordinal ) → A x ∨ B) →  ((x : Ordinal ) → A x) ∨ B
∀-imply-or  {A} {B} ∀AB with p∨¬p (Lift ( suc n ) ((x : Ordinal ) → A x)) -- LEM
∀-imply-or  {A} {B} ∀AB | case1 (lift t) = case1 t
∀-imply-or  {A} {B} ∀AB | case2 x  = case2 (lemma (λ not → x (lift not ))) where
lemma : ¬ ((x : Ordinal ) → A x) →  B
lemma not with p∨¬p B
lemma not | case1 b = b
lemma not | case2 ¬b = ⊥-elim  (not (λ x → dont-orb (∀AB x) ¬b ))
induction : (x : Ordinal) → ((y : Ordinal) → y o< x → ψ y) → ψ x
induction x prev with ∋-p X ( ord→od x)
... | yes p = case2 ( record { a-choice = ord→od x ; is-in = p } )
... | no ¬p = lemma where
lemma1 : (y : Ordinal) → (y o< x → def X y → ⊥) ∨ choiced X
lemma1 y with ∋-p X (ord→od y)
lemma1 y | yes y<X = case2 ( record { a-choice = ord→od y ; is-in = y<X } )
lemma1 y | no ¬y<X = case1 ( λ lt y<X → ¬y<X (subst (λ k → def X k ) (sym diso) y<X ) )
lemma :  ((y : Ordinals.ord O) → (O Ordinals.o< y) x → def X y → ⊥) ∨ choiced X
lemma = ∀-imply-or lemma1
have_to_find : choiced X
have_to_find = dont-or  (lemma-ord (od→ord X )) ¬¬X∋x where
¬¬X∋x : ¬ ((x : Ordinal) → x o< (od→ord X) → def X x → ⊥)
¬¬X∋x nn = not record {
eq→ = λ {x} lt → ⊥-elim  (nn x (def→o< lt) lt)
; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt )
}
record Minimal (x : OD) (ne : ¬ (x == od∅ ) )  : Set (suc n) where
field
min : OD
x∋min :   x ∋ min
min-empty :  (y : OD ) → ¬ ( min ∋ y) ∧ (x ∋ y)
open Minimal
open _∧_
induction : {x : OD} → ({y : OD} → x ∋ y → (ne : ¬ (y == od∅ )) → Minimal y ne ) →  (ne : ¬ (x == od∅ )) → Minimal x ne
induction {x} prev ne = c2
where
ch : choiced x
ch = choice-func x ne
c1 : OD
c1 = a-choice ch
c2 : Minimal x ne
c2 with p∨¬p ( (y : OD ) → ¬ ( def c1 (od→ord y) ∧ (def x (od→ord  y)))  )
c2 | case1 Yes = record {
min = c1
;  x∋min = is-in ch
;  min-empty = Yes
}
c2 | case2 No = c3 where
y1 : OD
y1 =  record { def = λ y → ( def c1 y ∧ def x y) }
noty1 : ¬ (y1 == od∅ )
noty1 not = ⊥-elim ( No (λ z n → ∅< n not ) )
ch1 : choiced y1
ch1 = choice-func y1 noty1
c3 : Minimal x ne
c3 with is-o∅ (od→ord (a-choice ch1))
... | yes eq =  record {
min = od∅
;  x∋min = def-subst {x} {od→ord (a-choice ch1)} {x} (proj2 (is-in ch1)) refl (trans eq (sym ord-od∅) )
;  min-empty = λ z p → ⊥-elim ( ¬x<0 (proj1 p) )
}
... | no n =  record {
min = min min3
;  x∋min = ?
;  min-empty = {!!}
}  where
lemma : (a-choice ch1 == od∅ ) → od→ord (a-choice ch1) ≡ o∅
lemma eq = begin
od→ord (a-choice ch1)
≡⟨ cong (λ k → od→ord k ) (==→o≡ eq ) ⟩
od→ord od∅
≡⟨ ord-od∅ ⟩
o∅
∎ where open ≡-Reasoning
ch1not : ¬ (a-choice ch1 == od∅)
ch1not ch1=0 = n (lemma ch1=0)
min3 : Minimal (a-choice ch1) ch1not
min3 = ( prev (proj2 (is-in ch1)) (λ ch1=0 → n (lemma ch1=0)))

Min1 : (x : OD) → (ne : ¬ (x == od∅ )) → Minimal x ne
Min1 x ne = (ε-induction {λ y → (ne : ¬ (y == od∅ ) ) → Minimal y ne } induction x ne )
minimal : (x : OD  ) → ¬ (x == od∅ ) → OD
minimal x not = min (Min1 x not )
x∋minimal : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimal x ne ) )
x∋minimal x ne = x∋min (Min1 x ne)
minimal-1 : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord  y) )
minimal-1 x ne y = min-empty (Min1 x ne ) y

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