{-# OPTIONS --allow-unsolved-metas #-}
open import Level
open import Ordinals
module ODUtil {n : Level } (O : Ordinals {n} ) where
open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
open import Relation.Binary.PropositionalEquality hiding ( [_] )
open import Data.Nat.Properties
open import Data.Empty
open import Relation.Nullary
open import Relation.Binary hiding ( _⇔_ )
open import logic
open import nat
open Ordinals.Ordinals O
open Ordinals.IsOrdinals isOrdinal
import OrdUtil
open OrdUtil O
import OD
open OD O
open OD.OD
open ODAxiom odAxiom
open HOD
open _∧_
open _==_
_⊂_ : ( A B : HOD) → Set n
_⊂_ A B = ( & A o< & B) ∧ ( A ⊆ B )
⊆∩-dist : {a b c : HOD} → a ⊆ b → a ⊆ c → a ⊆ ( b ∩ c )
⊆∩-dist {a} {b} {c} a<b a<c {z} az = ⟪ a<b az , a<c az ⟫
⊆∩-incl-1 : {a b c : HOD} → a ⊆ c → ( a ∩ b ) ⊆ c
⊆∩-incl-1 {a} {b} {c} a<c {z} ab = a<c (proj1 ab)
⊆∩-incl-2 : {a b c : HOD} → a ⊆ c → ( b ∩ a ) ⊆ c
⊆∩-incl-2 {a} {b} {c} a<c {z} ab = a<c (proj2 ab)
cseq : HOD → HOD
cseq x = record { od = record { def = λ y → odef x (osuc y) } ; odmax = osuc (odmax x) ; <odmax = lemma } where
lemma : {y : Ordinal} → def (od x) (osuc y) → y o< osuc (odmax x)
lemma {y} lt = ordtrans <-osuc (ordtrans (<odmax x lt) <-osuc )
∩-comm : { A B : HOD } → (A ∩ B) ≡ (B ∩ A)
∩-comm {A} {B} = ==→o≡ record { eq← = λ {x} lt → ⟪ proj2 lt , proj1 lt ⟫ ; eq→ = λ {x} lt → ⟪ proj2 lt , proj1 lt ⟫ }
_∪_ : ( A B : HOD ) → HOD
A ∪ B = record { od = record { def = λ x → odef A x ∨ odef B x } ;
odmax = omax (odmax A) (odmax B) ; <odmax = lemma } where
lemma : {y : Ordinal} → odef A y ∨ odef B y → y o< omax (odmax A) (odmax B)
lemma {y} (case1 a) = ordtrans (<odmax A a) (omax-x _ _)
lemma {y} (case2 b) = ordtrans (<odmax B b) (omax-y _ _)
x∪x≡x : { A : HOD } → (A ∪ A) ≡ A
x∪x≡x {A} = ==→o≡ record { eq← = λ {x} lt → case1 lt ; eq→ = lem00 } where
lem00 : {x : Ordinal} → odef A x ∨ odef A x → odef A x
lem00 {x} (case1 ax) = ax
lem00 {x} (case2 ax) = ax
_＼_ : ( A B : HOD ) → HOD
A ＼ B = record { od = record { def = λ x → odef A x ∧ ( ¬ ( odef B x ) ) }; odmax = odmax A ; <odmax = λ y → <odmax A (proj1 y) }
¬∅∋ : {x : HOD} → ¬ ( od∅ ∋ x )
¬∅∋ {x} = ¬x<0
pair-xx<xy : {x y : HOD} → & (x , x) o< osuc (& (x , y) )
pair-xx<xy {x} {y} = ⊆→o≤ lemma where
lemma : {z : Ordinal} → def (od (x , x)) z → def (od (x , y)) z
lemma {z} (case1 refl) = case1 refl
lemma {z} (case2 refl) = case1 refl
trans-⊆ : { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C
trans-⊆ A⊆B B⊆C ab = B⊆C (A⊆B ab)
trans-⊂ : { A B C : HOD} → A ⊂ B → B ⊂ C → A ⊂ C
trans-⊂ ⟪ A<B , A⊆B ⟫ ⟪ B<C , B⊆C ⟫ = ⟪ ordtrans A<B B<C , (λ ab → B⊆C (A⊆B ab)) ⟫
refl-⊆ : {A : HOD} → A ⊆ A
refl-⊆ x = x
od⊆→o≤ : {x y : HOD } → x ⊆ y → & x o< osuc (& y)
od⊆→o≤ {x} {y} lt = ⊆→o≤ {x} {y} (λ {z} x>z → subst (λ k → def (od y) k ) &iso (lt (d→∋ x x>z)))
⊆→= : {F U : HOD} → F ⊆ U → U ⊆ F → F =h= U
⊆→= {F} {U} FU UF = record { eq→ = λ {x} lt → subst (λ k → odef U k) &iso (FU (subst (λ k → odef F k) (sym &iso) lt) )
; eq← = λ {x} lt → subst (λ k → odef F k) &iso (UF (subst (λ k → odef U k) (sym &iso) lt) ) }
¬A∋x→A≡od∅ : (A : HOD) → {x : HOD} → A ∋ x → ¬ ( & A ≡ o∅ )
¬A∋x→A≡od∅ A {x} ax a=0 = ¬x<0 ( subst (λ k → & x o< k) a=0 (c<→o< ax ))
subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → (A ∩ x ) ∋ y ) ⇔ ( x ⊆ A )
subset-lemma {A} {x} = record {
proj1 = λ lt x∋z → subst (λ k → odef A k ) &iso ( proj1 (lt (subst (λ k → odef x k) (sym &iso) x∋z ) ))
; proj2 = λ x⊆A lt → ⟪ x⊆A lt , lt ⟫
}
nat→ω : Nat → HOD
nat→ω Zero = od∅
nat→ω (Suc y) = Union (nat→ω y , (nat→ω y , nat→ω y))
ω→nato : {y : Ordinal} → Omega-d y → Nat
ω→nato iφ = Zero
ω→nato (isuc lt) = Suc (ω→nato lt)
ω→nat : (n : HOD) → Omega ∋ n → Nat
ω→nat n = ω→nato
ω∋nat→ω : {n : Nat} → def (od Omega) (& (nat→ω n))
ω∋nat→ω {Zero} = subst (λ k → def (od Omega) k) (sym ord-od∅) iφ
ω∋nat→ω {Suc n} = subst (λ k → def (od Omega) k) lemma (isuc ( ω∋nat→ω {n})) where
lemma : & (Union (* (& (nat→ω n)) , (* (& (nat→ω n)) , * (& (nat→ω n))))) ≡ & (nat→ω (Suc n))
lemma = subst (λ k → & (Union (k , ( k , k ))) ≡ & (nat→ω (Suc n))) (sym *iso) refl
pair1 : { x y : HOD } → (x , y ) ∋ x
pair1 = case1 refl
pair2 : { x y : HOD } → (x , y ) ∋ y
pair2 = case2 refl
single : {x y : HOD } → (x , x ) ∋ y → x ≡ y
single (case1 eq) = ==→o≡ ( ord→== (sym eq) )
single (case2 eq) = ==→o≡ ( ord→== (sym eq) )
single& : {x y : Ordinal } → odef (* x , * x ) y → x ≡ y
single& (case1 eq) = sym (trans eq &iso)
single& (case2 eq) = sym (trans eq &iso)
open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
pair=∨ : {a b c : Ordinal } → odef (* a , * b) c → ( a ≡ c ) ∨ ( b ≡ c )
pair=∨ {a} {b} {c} (case1 c=a) = case1 ( sym (trans c=a &iso))
pair=∨ {a} {b} {c} (case2 c=b) = case2 ( sym (trans c=b &iso))
ω-prev-eq1 : {x y : Ordinal} → & (Union (* y , (* y , * y))) ≡ & (Union (* x , (* x , * x))) → ¬ (x o< y)
ω-prev-eq1 {x} {y} eq x<y with eq→ (ord→== eq) record { owner = & (* y , * y) ; ao = case2 refl
; ox = subst (λ k → odef k (& (* y))) (sym *iso) (case1 refl) }
... | record { owner = u ; ao = xxx∋u ; ox = uy } with xxx∋u
... | case1 u=x = ⊥-elim ( o<> x<y (osucprev (begin
osuc y ≡⟨ sym (cong osuc &iso) ⟩
osuc (& (* y)) ≤⟨ osucc (c<→o< {* y} {* u} uy) ⟩
& (* u) ≡⟨ &iso ⟩
u ≡⟨ u=x ⟩
& (* x) ≡⟨ &iso ⟩
x ∎ ))) where open o≤-Reasoning O
... | case2 u=xx = ⊥-elim (o<¬≡ ( begin
x ≡⟨ single& (subst₂ (λ j k → odef j k ) (begin
* u ≡⟨ cong (*) u=xx ⟩
* (& (* x , * x)) ≡⟨ *iso ⟩
(* x , * x ) ∎ ) &iso uy ) ⟩
y ∎ ) x<y) where open ≡-Reasoning
ω-prev-eq : {x y : Ordinal} → & (Union (* y , (* y , * y))) ≡ & (Union (* x , (* x , * x))) → y ≡ x
ω-prev-eq {x} {y} eq with trio< x y
ω-prev-eq {x} {y} eq | tri< a ¬b ¬c = ⊥-elim (ω-prev-eq1 eq a)
ω-prev-eq {x} {y} eq | tri≈ ¬a b ¬c = (sym b)
ω-prev-eq {x} {y} eq | tri> ¬a ¬b c = ⊥-elim (ω-prev-eq1 (sym eq) c)
ω-inject : {x y : HOD} → Union ( y , ( y , y)) ≡ Union ( x , ( x , x)) → y ≡ x
ω-inject {x} {y} eq = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) ( ω-prev-eq (cong (&) (subst₂ (λ j k → Union (j , (j , j)) ≡ Union (k , (k , k))) (sym *iso) (sym *iso) eq ))))
ω-∈s : (x : HOD) → Union ( x , (x , x)) ∋ x
ω-∈s x = record { owner = & ( x , x ) ; ao = case2 refl ; ox = subst₂ (λ j k → odef j k ) (sym *iso) refl (case2 refl) }
ωs≠0 : (x : HOD) → ¬ ( Union ( x , (x , x)) ≡ od∅ )
ωs≠0 y eq = ⊥-elim ( ¬x<0 (subst (λ k → & y o< k ) ord-od∅ (c<→o< (subst (λ k → odef k (& y )) eq (ω-∈s y) ))) )
ωs0 : o∅ ≡ & (nat→ω 0)
ωs0 = trans (sym ord-od∅) (cong (&) refl )
nat→ω-iso : {i : HOD} → (lt : Omega ∋ i ) → nat→ω ( ω→nat i lt ) ≡ i
nat→ω-iso {i} = ε-induction {λ i → (lt : Omega ∋ i ) → nat→ω ( ω→nat i lt ) ≡ i } ind i where
ind : {x : HOD} → ({y : HOD} → x ∋ y → (lt : Omega ∋ y) → nat→ω (ω→nat y lt) ≡ y) →
(lt : Omega ∋ x) → nat→ω (ω→nat x lt) ≡ x
ind {x} prev lt = ind1 lt *iso where
ind1 : {ox : Ordinal } → (ltd : Omega-d ox ) → * ox ≡ x → nat→ω (ω→nato ltd) ≡ x
ind1 {o∅} iφ refl = sym o∅≡od∅
ind1 (isuc {x₁} ltd) ox=x = begin
nat→ω (ω→nato (isuc ltd) )
≡⟨⟩
Union (nat→ω (ω→nato ltd) , (nat→ω (ω→nato ltd) , nat→ω (ω→nato ltd)))
≡⟨ cong (λ k → Union (k , (k , k ))) lemma ⟩
Union (* x₁ , (* x₁ , * x₁))
≡⟨ trans ( sym *iso) ox=x ⟩
x
∎ where
open ≡-Reasoning
lemma0 : x ∋ * x₁
lemma0 = subst (λ k → odef k (& (* x₁))) (trans (sym *iso) ox=x)
record { owner = & ( * x₁ , * x₁ ) ; ao = case2 refl ; ox = subst (λ k → odef k (& (* x₁))) (sym *iso) (case1 refl) }
lemma1 : Omega ∋ * x₁
lemma1 = subst (λ k → odef Omega k) (sym &iso) ltd
lemma3 : {x y : Ordinal} → (ltd : Omega-d x ) (ltd1 : Omega-d y ) → y ≡ x → ltd ≅ ltd1
lemma3 iφ iφ refl = HE.refl
lemma3 iφ (isuc {y} ltd1) eq = ⊥-elim ( ¬x<0 (subst₂ (λ j k → j o< k ) &iso eq (c<→o< (ω-∈s (* y)) )))
lemma3 (isuc {y} ltd) iφ eq = ⊥-elim ( ¬x<0 (subst₂ (λ j k → j o< k ) &iso (sym eq) (c<→o< (ω-∈s (* y)) )))
lemma3 (isuc {x} ltd) (isuc {y} ltd1) eq with lemma3 ltd ltd1 (ω-prev-eq (eq))
... | t = HE.cong₂ (λ j k → isuc {j} k ) (HE.≡-to-≅ (ω-prev-eq (sym eq))) t
lemma2 : {x y : Ordinal} → (ltd : Omega-d x ) (ltd1 : Omega-d y ) → y ≡ x → ω→nato ltd ≡ ω→nato ltd1
lemma2 {x} {y} ltd ltd1 eq = lemma6 eq (lemma3 {x} {y} ltd ltd1 eq) where
lemma6 : {x y : Ordinal} → {ltd : Omega-d x } {ltd1 : Omega-d y } → y ≡ x → ltd ≅ ltd1 → ω→nato ltd ≡ ω→nato ltd1
lemma6 refl HE.refl = refl
lemma : nat→ω (ω→nato ltd) ≡ * x₁
lemma = trans (cong (λ k → nat→ω k) (lemma2 {x₁} {_} ltd (subst (λ k → Omega-d k ) (sym &iso) ltd) &iso ) ) ( prev {* x₁} lemma0 lemma1 )
ω→nat-iso0 : (x : Nat) → {ox : Ordinal } → (ltd : Omega-d ox) → * ox ≡ nat→ω x → ω→nato ltd ≡ x
ω→nat-iso0 Zero iφ eq = refl
ω→nat-iso0 (Suc x) iφ eq = ⊥-elim ( ωs≠0 _ (trans (sym eq) o∅≡od∅ ))
ω→nat-iso0 Zero (isuc ltd) eq = ⊥-elim ( ωs≠0 _ (subst (λ k → k ≡ od∅ ) *iso eq ))
ω→nat-iso0 (Suc i) (isuc {x} ltd) eq = cong Suc ( ω→nat-iso0 i ltd (lemma1 eq) ) where
lemma1 : * (& (Union (* x , (* x , * x)))) ≡ Union (nat→ω i , (nat→ω i , nat→ω i)) → * x ≡ nat→ω i
lemma1 eq = subst (λ k → * x ≡ k ) *iso (cong (λ k → * k)
(sym ( ω-prev-eq (subst (λ k → _ ≡ k ) &iso (cong (λ k → & k ) (sym
(subst (λ k → _ ≡ Union ( k , ( k , k ))) (sym *iso ) eq )))))))
ω→nat-iso : {i : Nat} → ω→nat ( nat→ω i ) (ω∋nat→ω {i}) ≡ i
ω→nat-iso {i} = ω→nat-iso0 i (ω∋nat→ω {i}) *iso
nat→ω-inject : {i j : Nat} → nat→ω i ≡ nat→ω j → i ≡ j
nat→ω-inject {Zero} {Zero} eq = refl
nat→ω-inject {Zero} {Suc j} eq = ⊥-elim ( ¬0=ux (trans (trans (sym ord-od∅) (cong (&) eq)) refl ))
nat→ω-inject {Suc i} {Zero} eq = ⊥-elim ( ¬0=ux (trans (trans (sym ord-od∅) (cong (&) (sym eq))) refl ))
nat→ω-inject {Suc i} {Suc j} eq = cong Suc (nat→ω-inject {i} {j} ( ω-inject (eq) ))
Repl⊆ : {A B : HOD} (A⊆B : A ⊆ B) → { ψa : ( x : HOD) → A ∋ x → HOD } { ψb : ( x : HOD) → B ∋ x → HOD }
→ {Ca : HOD} → {rca : RXCod A Ca ψa }
→ {Cb : HOD} → {rcb : RXCod B Cb ψb }
→ ( {z : Ordinal } → (az : odef A z ) → (ψa (* z) (subst (odef A) (sym &iso) az) ≡ ψb (* z) (subst (odef B) (sym &iso) (A⊆B az))))
→ Replace' A ψa rca ⊆ Replace' B ψb rcb
Repl⊆ {A} {B} A⊆B {ψa} {ψb} eq record { z = z ; az = az ; x=ψz = x=ψz } = record { z = z ; az = A⊆B az
; x=ψz = trans x=ψz (cong (&) (eq az) ) }
PPP : {P : HOD} → Power P ∋ P
PPP {P} z pz = subst (λ k → odef k z ) *iso pz
UPower⊆Q : {P Q : HOD} → P ⊆ Q → Union (Power P) ⊆ Q
UPower⊆Q {P} {Q} P⊆Q {z} record { owner = y ; ao = ppy ; ox = yz } = P⊆Q (ppy _ yz)
UPower∩ : {P : HOD} → ({ p q : HOD } → P ∋ p → P ∋ q → P ∋ (p ∩ q))
→ { p q : HOD } → Union (Power P) ∋ p → Union (Power P) ∋ q → Union (Power P) ∋ (p ∩ q)
UPower∩ {P} each {p} {q} record { owner = x ; ao = ppx ; ox = xz } record { owner = y ; ao = ppy ; ox = yz }
= record { owner = & P ; ao = PPP ; ox = lem03 } where
lem03 : odef (* (& P)) (& (p ∩ q))
lem03 = subst (λ k → odef k (& (p ∩ q))) (sym *iso) ( each (ppx _ xz) (ppy _ yz) )