### view LEMC.agda @ 331:12071f79f3cf

HOD done
author Shinji KONO Sun, 05 Jul 2020 16:56:21 +0900 0faa7120e4b5 2a8a51375e49
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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 HOD is ordered,  p ∨ ( ¬ p ) <=> axiom of choice
---
record choiced  ( X : HOD) : Set (suc n) where
field
a-choice : HOD
is-in : X ∋ a-choice

open HOD
_=h=_ : (x y : HOD) → Set n
x =h= y  = od x == od y

open choiced

OD→ZFC : ZFC
OD→ZFC   = record {
ZFSet = HOD
; _∋_ = _∋_
; _≈_ = _=h=_
; ∅  = od∅
; Select = Select
; isZFC = isZFC
} where
-- infixr  200 _∈_
-- infixr  230 _∩_ _∪_
isZFC : IsZFC (HOD )  _∋_  _=h=_ 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 : HOD ) → ¬ ( X =h= od∅ ) → choiced X
choice-func  X not = have_to_find where
ψ : ( ox : Ordinal ) → Set (suc n)
ψ ox = (( x : Ordinal ) → x o< ox  → ( ¬ odef X x )) ∨ choiced X
lemma-ord : ( ox : Ordinal  ) → ψ ox
lemma-ord  ox = TransFinite1 {ψ} induction ox where
∋-p : (A x : HOD ) → 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 → odef 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 → odef X k ) (sym diso) y<X ) )
lemma :  ((y : Ordinals.ord O) → (O Ordinals.o< y) x → odef 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) → odef X x → ⊥)
¬¬X∋x nn = not record {
eq→ = λ {x} lt → ⊥-elim  (nn x (odef→o< lt) lt)
; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt )
}
record Minimal (x : HOD)  : Set (suc n) where
field
min : HOD
x∋min :   x ∋ min
min-empty :  (y : HOD ) → ¬ ( min ∋ y) ∧ (x ∋ y)
open Minimal
open _∧_
--
--  from https://math.stackexchange.com/questions/2973777/is-it-possible-to-prove-regularity-with-transfinite-induction-only
--
induction : {x : HOD} → ({y : HOD} → x ∋ y → (u : HOD ) → (u∋x : u ∋ y) → Minimal u )
→  (u : HOD ) → (u∋x : u ∋ x) → Minimal u
induction {x} prev u u∋x with p∨¬p ((y : HOD) → ¬ (x ∋ y) ∧ (u ∋ y))
... | case1 P =
record { min = x
;     x∋min = u∋x
;     min-empty = P
}
... | case2 NP =  min2 where
p : HOD
p  = record { od = record { def = λ y1 → odef x  y1 ∧ odef u y1 } ; odmax = omin (odmax x) (odmax u) ; <odmax = lemma } where
lemma : {y : Ordinal} → OD.def (od x) y ∧ OD.def (od u) y → y o< omin (odmax x) (odmax u)
lemma {y} lt = min1 (<odmax x (proj1 lt)) (<odmax u (proj2 lt))
np : ¬ (p =h= od∅)
np p∅ =  NP (λ y p∋y → ∅< {p} {_} p∋y p∅ )
y1choice : choiced p
y1choice = choice-func p np
y1 : HOD
y1 = a-choice y1choice
y1prop : (x ∋ y1) ∧ (u ∋ y1)
y1prop = is-in y1choice
min2 : Minimal u
min2 = prev (proj1 y1prop) u (proj2 y1prop)
Min2 : (x : HOD) → (u : HOD ) → (u∋x : u ∋ x) → Minimal u
Min2 x u u∋x = (ε-induction1 {λ y →  (u : HOD ) → (u∋x : u ∋ y) → Minimal u  } induction x u u∋x )
cx : {x : HOD} →  ¬ (x =h= od∅ ) → choiced x
cx {x} nx = choice-func x nx
minimal : (x : HOD  ) → ¬ (x =h= od∅ ) → HOD
minimal x ne = min (Min2 (a-choice (cx {x} ne) ) x (is-in (cx ne)))
x∋minimal : (x : HOD  ) → ( ne : ¬ (x =h= od∅ ) ) → odef x ( od→ord ( minimal x ne ) )
x∋minimal x ne = x∋min (Min2 (a-choice (cx {x} ne) ) x (is-in (cx ne)))
minimal-1 : (x : HOD  ) → ( ne : ¬ (x =h= od∅ ) ) → (y : HOD ) → ¬ ( odef (minimal x ne) (od→ord y)) ∧ (odef x (od→ord  y) )
minimal-1 x ne y = min-empty (Min2 (a-choice (cx ne) ) x (is-in (cx ne))) y

```