{-# OPTIONS --cubical-compatible --safe #-}
open import Level renaming (zero to Zero ; suc to Suc)
open import Ordinals
open import logic
open import Relation.Nullary
import HODBase
import OD
open import Relation.Binary
open import Relation.Binary.PropositionalEquality hiding ( [_] )
module zorn {n : Level } (O : Ordinals {n} ) (HODAxiom : HODBase.ODAxiom O) (ho< : OD.ODAxiom-ho< O HODAxiom )
(AC : OD.AxiomOfChoice O HODAxiom )
(_<_ : (x y : OD.HOD O HODAxiom) → Set n )
(<-cong : {x y w z : OD.HOD O HODAxiom}
→ HODBase._==_ O (HODBase.HOD.od x) (HODBase.HOD.od w) → HODBase._==_ O (HODBase.HOD.od y) (HODBase.HOD.od z) → x < y → w < z )
(PO : IsStrictPartialOrder _≡_ _<_ ) where
open import Data.Empty
open import Data.Nat hiding ( _≤_ ; _<_ )
import OrdUtil
open inOrdinal O
open Ordinals.Ordinals O
open Ordinals.IsOrdinals isOrdinal
import ODUtil
open import logic
open import nat
open OrdUtil O
open ODUtil O HODAxiom ho<
open _∧_
open _∨_
open Bool
open HODBase._==_
open HODBase.ODAxiom HODAxiom
open OD O HODAxiom
open HODBase.HOD
open AxiomOfChoice AC
open import ODC O HODAxiom AC as ODC
open import filter O HODAxiom ho< AC
open import ZProduct O HODAxiom ho<
open Filter
_<<_ : (x y : Ordinal ) → Set n
x << y = * x < * y
_≤_ : (x y : Ordinal ) → Set n
x ≤ y = (x ≡ y ) ∨ ( * x < * y )
POO : IsStrictPartialOrder _≡_ _<<_
POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
; trans = IsStrictPartialOrder.trans PO
; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y
; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } }
≤-ftrans : {x y z : Ordinal } → x ≤ y → y ≤ z → x ≤ z
≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl
≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z
≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y
≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z )
ftrans≤-< : {x y z : Ordinal } → x ≤ y → y << z → x << z
ftrans≤-< {x} {y} {z} (case1 eq) y<z = subst (λ k → k < * z) (sym (cong (*) eq)) y<z
ftrans≤-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z
ftrans<-≤ : {x y z : Ordinal } → x << y → y ≤ z → x << z
ftrans<-≤ {x} {y} {z} x<y (case1 eq) = subst (λ k → * x < k ) ((cong (*) eq)) x<y
ftrans<-≤ {x} {y} {z} x<y (case2 lt) = IsStrictPartialOrder.trans PO x<y lt
<-irr : {a b : Ordinal} → (a ≡ b ) ∨ (* a < * b ) → * b < * a → ⊥
<-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym (cong (*) a=b) ) b<a
<-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl
(IsStrictPartialOrder.trans PO b<a a<b)
<<-irr : {a b : Ordinal } → a ≤ b → b << a → ⊥
<<-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (cong (*) (sym a=b)) b<a
<<-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl (IsStrictPartialOrder.trans PO b<a a<b)
ptrans = IsStrictPartialOrder.trans PO
≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set n
≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( x ≤ (f x) ) ∧ odef A (f x )
<-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set n
<-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x < * (f x) ) ∧ odef A (f x )
data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where
init : {s1 : Ordinal } → odef A s → s ≡ s1 → FClosure A f s s1
fsuc : (x : Ordinal) ( p : FClosure A f s x ) → FClosure A f s (f x)
A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y
A∋fc {A} s f mf (init as refl ) = as
A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s f mf fcy ) )
A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s
A∋fcs {A} s f mf (init as refl) = as
A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy
s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → s ≤ y
s≤fc {A} s {.s} f mf (init x refl ) = case1 refl
s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) )
... | case1 x=fx = subst₂ (λ j k → j ≤ k ) refl x=fx (s≤fc s f mf fcy)
... | case2 x<fx with s≤fc {A} s f mf fcy
... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym (cong (*) s≡x )) refl x<fx )
... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx )
fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ
fcn s mf (init as refl) = zero
fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p))
... | case1 eq = fcn s mf p
... | case2 y<fy = suc (fcn s mf p )
fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f)
→ (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → x ≡ y
fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where
fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq )
fc06 {x} {y} refl {j} not = fc08 not where
fc08 : {j : ℕ} → ¬ suc j ≡ 0
fc08 ()
fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → s ≡ x
fc07 {x} (init as refl) eq = refl
fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) )
... | case1 x=fx = trans (fc07 cx eq ) x=fx
fc00 : (i j : ℕ ) → i ≡ j → {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx → j ≡ fcn s mf cy → x ≡ y
fc00 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ )
fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ )
fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl
fc00 zero zero refl (init as refl) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) )
... | case1 y=fy = trans (fc07 cy i=y) y=fy
fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) )
... | case1 x=fx = sym (trans (fc07 cx i=x) x=fx )
fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) )
... | case1 x=fx | case1 y=fy = trans (sym x=fx) (trans ( fc00 zero zero refl cx cy i=x i=y ) y=fy)
fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) )
... | case1 x=fx | case1 y=fy = trans (sym x=fx) (trans ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) y=fy )
... | case1 x=fx | case2 y<fy = trans (sym x=fx) (fc02 x cx i=x) where
fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) → suc i ≡ fcn s mf cx1 → x1 ≡ f y
fc02 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x)
fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) )
... | case1 eq = trans (sym eq ) ( fc02 x1 cx1 i=x1 )
... | case2 lt = cong f fc04 where
fc04 : x1 ≡ y
fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y)
... | case2 x<fx | case1 y=fy = trans (fc03 y cy j=y) y=fy where
fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) → suc j ≡ fcn s mf cy1 → f x ≡ y1
fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x)
fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) )
... | case1 eq = trans ( fc03 y1 cy1 j=y1 ) eq
... | case2 lt = cong f fc05 where
fc05 : x ≡ y1
fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1)
... | case2 x₁ | case2 x₂ = cong f (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))
fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f)
→ (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y
fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where
fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq )
fc06 {x} {y} refl {j} not = fc08 not where
fc08 : {j : ℕ} → ¬ suc j ≡ 0
fc08 ()
fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y
fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x)
fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) )
... | case1 y=fy = subst (λ k → * x < k ) (cong (*) y=fy) ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) )
... | case2 y<fy with <-cmp (fcn s mf cx ) i
... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c )
... | tri≈ ¬a b ¬c = subst (λ k → * k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy
... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where
fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy
fc03 eq = cong pred eq
fc02 : * x < * y1
fc02 = fc01 i cx cy (fc03 i=y ) a
fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f)
→ (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (x ≡ y) (* y < * x )
fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy )
... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where
fc11 : * x < * y
fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a
... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where
fc10 : x ≡ y
fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b
... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where
fc12 : * y < * x
fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c
IsTotalOrderSet : ( A : HOD ) → Set n
IsTotalOrderSet A = {a b : Ordinal } → odef A a → odef A b → Tri (* a < * b) (a ≡ b) (* b < * a )
⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B
⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (B⊆A ax) (B⊆A ay)
record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where
field
ax : odef A x
y : Ordinal
ay : odef B y
x=fy : x ≡ f y
record IsSUP (A B : HOD) (x : Ordinal ) : Set n where
field
ax : odef A x
x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x )
record SUP ( A B : HOD ) : Set (Level.suc n) where
field
sup : HOD
isSUP : IsSUP A B (& sup)
ax = IsSUP.ax isSUP
x≤sup = IsSUP.x≤sup isSUP
fc-stop : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) { a b : Ordinal }
→ (aa : odef A a ) →( {y : Ordinal} → FClosure A f a y → (y ≡ b ) ∨ (y << b )) → a ≡ b → f a ≡ a
fc-stop A f mf {a} {b} aa x≤sup a=b with x≤sup (fsuc a (init aa refl ))
... | case1 eq = trans eq (sym a=b)
... | case2 lt = ⊥-elim (<-irr (case1 (cong f (sym a=b))) (ftrans<-≤ lt fc00 ) ) where
fc00 : b ≤ (f b)
fc00 = proj1 (mf _ (subst (λ k → odef A k) a=b aa ))
∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A
∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
data UChain { A : HOD } { f : Ordinal → Ordinal } {supf : Ordinal → Ordinal} {y : Ordinal } (ay : odef A y )
(x : Ordinal) : (z : Ordinal) → Set n where
ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain ay x z
ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) (supu=u : supf u ≡ u) ( fc : FClosure A f (supf u) z ) → UChain ay x z
UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD
UnionCF A f ay supf x
= record { od = record { def = λ z → odef A z ∧ UChain {A} {f} {supf} ay x z } ;
odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy }
data IChain {A : HOD} { f : Ordinal → Ordinal } {y : Ordinal } (ay : odef A y )
{x : Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) : (z : Ordinal ) → Set n where
ic-init : {z : Ordinal } (fc : FClosure A f y z) → IChain ay supfz z
ic-isup : {z : Ordinal} (i : Ordinal) (i<x : i o< x) (s<x : supfz i<x o≤ i ) (fc : FClosure A f (supfz i<x) z) → IChain ay supfz z
UnionIC : ( A : HOD ) ( f : Ordinal → Ordinal ) { x : Ordinal } {y : Ordinal } (ay : odef A y ) (supfz : {z : Ordinal } → z o< x → Ordinal) → HOD
UnionIC A f ay supfz
= record { od = record { def = λ z → odef A z ∧ IChain {A} {f} ay supfz z } ;
odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy }
supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y )
→ supf x o< supf y → x o< y
supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y
... | tri< a ¬b ¬c = a
... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy )
... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) )
... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy )
... | case2 lt = ⊥-elim ( o<> sx<sy lt )
record IsMinSUP ( A B : HOD ) (sup : Ordinal) : Set n where
field
as : odef A sup
x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup )
minsup : { sup1 : Ordinal } → odef A sup1
→ ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1
record MinSUP ( A B : HOD ) : Set n where
field
sup : Ordinal
isMinSUP : IsMinSUP A B sup
as = IsMinSUP.as isMinSUP
x≤sup = IsMinSUP.x≤sup isMinSUP
minsup = IsMinSUP.minsup isMinSUP
chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal )
(supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b
→ odef (UnionCF A f ay supf a) c → odef (UnionCF A f ay supf b) c
chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = ⟪ ua , ch-init fc ⟫
chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-is-sup u u<x supu=u fc ⟫ = ⟪ ua , ch-is-sup u (ordtrans<-≤ u<x a≤b) supu=u fc ⟫
record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f)
{y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where
field
supf : Ordinal → Ordinal
asupf : {x : Ordinal } → odef A (supf x)
is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → IsMinSUP A (UnionCF A f ay supf x) (supf x)
supf-mono : {a b : Ordinal } → a o≤ b → supf a o≤ supf b
cfcs : {a b w : Ordinal } → a o< b → b o≤ z → supf a o< b → FClosure A f (supf a) w → odef (UnionCF A f ay supf b) w
zo≤sz : {x : Ordinal } → x o≤ z → x o≤ supf x
chain : HOD
chain = UnionCF A f ay supf z
chain⊆A : chain ⊆ A
chain⊆A = λ lt → proj1 lt
chain∋init : {x : Ordinal } → odef (UnionCF A f ay supf x) y
chain∋init {x} = ⟪ ay , ch-init (init ay refl) ⟫
mf : ≤-monotonic-f A f
mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
mf00 : * x < * (f x)
mf00 = proj1 ( mf< x ax )
f-next : {a z : Ordinal} → odef (UnionCF A f ay supf z) a → odef (UnionCF A f ay supf z) (f a)
f-next {a} ⟪ ua , ch-init fc ⟫ = ⟪ proj2 ( mf _ ua) , ch-init (fsuc _ fc) ⟫
f-next {a} ⟪ ua , ch-is-sup u su<x su=u fc ⟫ = ⟪ proj2 ( mf _ ua) , ch-is-sup u su<x su=u (fsuc _ fc) ⟫
supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y
supf-inject {x} {y} sx<sy with trio< x y
... | tri< a ¬b ¬c = a
... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy )
... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) )
... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy )
... | case2 lt = ⊥-elim ( o<> sx<sy lt )
csupf : {b : Ordinal } → supf b o< supf z → supf b o< z → odef (UnionCF A f ay supf z) (supf b)
csupf {b} sb<sz sb<z = cfcs (supf-inject sb<sz) o≤-refl sb<z (init asupf refl)
minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f ay supf x)
minsup {x} x≤z = record { sup = supf x ; isMinSUP = is-minsup x≤z }
supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup (minsup x≤z)
supf-is-minsup _ = refl
fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u )
fcy<sup {u} {w} u≤z fc with MinSUP.x≤sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc)
, ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫
... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso (trans eq (sym (supf-is-minsup u≤z ) ) ))
... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt )
initial : {x : Ordinal } → x o≤ z → odef (UnionCF A f ay supf x) x → y ≤ x
initial {x} x≤z ⟪ aa , ch-init fc ⟫ = s≤fc y f mf fc
initial {x} x≤z ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ≤-ftrans (fcy<sup (ordtrans u<x x≤z) (init ay refl)) (s≤fc _ f mf fc)
sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z
→ IsSUP A (UnionCF A f ay supf b) b → supf b ≡ b
sup=u {b} ab b≤z is-sup = z50 where
z48 : supf b o≤ b
z48 = IsMinSUP.minsup (is-minsup b≤z) ab (λ ux → IsSUP.x≤sup is-sup ux )
z50 : supf b ≡ b
z50 with trio< (supf b) b
... | tri< sb<b ¬b ¬c = ⊥-elim ( o≤> z47 sb<b ) where
z47 : b o≤ supf b
z47 = zo≤sz b≤z
... | tri≈ ¬a b ¬c = b
... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb )
supfeq : {a b : Ordinal } → a o≤ z → b o≤ z → UnionCF A f ay supf a =h= UnionCF A f ay supf b → supf a ≡ supf b
supfeq {a} {b} a≤z b≤z ua=ub with trio< (supf a) (supf b)
... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> (
IsMinSUP.minsup (is-minsup b≤z) asupf (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (eq← ua=ub uzb) )) sa<sb )
... | tri≈ ¬a b ¬c = b
... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> (
IsMinSUP.minsup (is-minsup a≤z) asupf (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (eq→ ua=ub uza) )) sb<sa )
union-max : {a b : Ordinal } → b o≤ supf a → b o≤ z → supf a o< supf b → UnionCF A f ay supf a =h= UnionCF A f ay supf b
union-max {a} {b} b≤sa b≤z sa<sb = record { eq→ = z47 ; eq← = z48 } where
z47 : {w : Ordinal } → odef (UnionCF A f ay supf a ) w → odef ( UnionCF A f ay supf b ) w
z47 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫
z47 {w} ⟪ aw , ch-is-sup u u<a su=u fc ⟫ = ⟪ aw , ch-is-sup u u<b su=u fc ⟫ where
u<b : u o< b
u<b = ordtrans u<a (supf-inject sa<sb )
z48 : {w : Ordinal } → odef (UnionCF A f ay supf b ) w → odef ( UnionCF A f ay supf a ) w
z48 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫
z48 {w} ⟪ aw , ch-is-sup u u<b su=u fc ⟫ = ⟪ aw , ch-is-sup u u<a su=u fc ⟫ where
u<a : u o< a
u<a = supf-inject ( osucprev (begin
osuc (supf u) ≡⟨ cong osuc su=u ⟩
osuc u ≤⟨ osucc u<b ⟩
b ≤⟨ b≤sa ⟩
supf a ∎ )) where open o≤-Reasoning
x≤supfx→¬sa<sa : {a b : Ordinal } → b o≤ z → b o≤ supf a → ¬ (supf a o< supf b )
x≤supfx→¬sa<sa {a} {b} b≤z b≤sa sa<sb = ⊥-elim ( o<¬≡ z27 sa<sb ) where
z27 : supf a ≡ supf b
z27 = supfeq (ordtrans (supf-inject sa<sb) b≤z) b≤z ( union-max b≤sa b≤z sa<sb)
order : {a b w : Ordinal } → b o≤ z → supf a o< supf b → FClosure A f (supf a) w → w ≤ supf b
order {a} {b} {w} b≤z sa<sb fc = IsMinSUP.x≤sup (is-minsup b≤z) (cfcs (supf-inject sa<sb) b≤z sa<b fc) where
sa<b : supf a o< b
sa<b with x<y∨y≤x (supf a) b
... | case1 lt = lt
... | case2 b≤sa = ⊥-elim (x≤supfx→¬sa<sa b≤z b≤sa sa<sb)
supf-idem : {b : Ordinal } → b o≤ z → supf b o≤ z → supf (supf b) ≡ supf b
supf-idem {b} b≤z sfb≤x = z52 where
z54 : {w : Ordinal} → odef (UnionCF A f ay supf (supf b)) w → (w ≡ supf b) ∨ (w << supf b)
z54 {w} ⟪ aw , ch-init fc ⟫ = fcy<sup b≤z fc
z54 {w} ⟪ aw , ch-is-sup u u<x su=u fc ⟫ = order b≤z (subst (λ k → k o< supf b) (sym su=u) u<x) fc where
u<b : u o< b
u<b = supf-inject (subst (λ k → k o< supf b ) (sym (su=u)) u<x )
z52 : supf (supf b) ≡ supf b
z52 = sup=u asupf sfb≤x record { ax = asupf ; x≤sup = z54 }
supf-mono< : {a b : Ordinal } → b o≤ z → supf a o< supf b → supf a << supf b
supf-mono< {a} {b} b≤z sa<sb with order {a} {b} b≤z sa<sb (init asupf refl)
... | case2 lt = lt
... | case1 eq = ⊥-elim ( o<¬≡ eq sa<sb )
f-total : IsTotalOrderSet chain
f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = fc-total where
fc-total : Tri (* a < * b) (a ≡ b) (* b < * a )
fc-total with trio< ua ub
... | tri< a₁ ¬b ¬c with ≤-ftrans (order (o<→≤ sub<x) (subst₂ (λ j k → j o< k) (sym sua=ua) (sym sub=ub) a₁) fca ) (s≤fc (supf ub) f mf fcb )
... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym eq1)) lt)) eq1 (λ lt → ⊥-elim (<-irr (case1 eq1) lt))
... | case2 a<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt)
fc-total | tri≈ _ refl _ = fcn-cmp _ f mf fca fcb
fc-total | tri> ¬a ¬b c with ≤-ftrans (order (o<→≤ sua<x) (subst₂ (λ j k → j o< k) (sym sub=ub) (sym sua=ua) c) fcb ) (s≤fc (supf ua) f mf fca )
... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 eq1) lt)) (sym eq1) (λ lt → ⊥-elim (<-irr (case1 (sym eq1)) lt))
... | case2 b<a = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a
f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = ft00 where
ft01 : a ≤ b → Tri ( * a < * b) ( a ≡ b) ( * b < * a )
ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym eq)) lt)) eq (λ lt → ⊥-elim (<-irr (case1 eq) lt))
ft01 (case2 a<b) = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt)
ft00 : Tri ( * a < * b) ( a ≡ b) ( * b < * a )
ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sub<x) fca) (s≤fc {A} _ f mf fcb))
f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-init fcb ⟫ = ft00 where
ft01 : b ≤ a → Tri ( * a < * b) ( a ≡ b) ( * b < * a )
ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 eq) lt)) (sym eq) (λ lt → ⊥-elim (<-irr (case1 (sym eq)) lt))
ft01 (case2 b<a) = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a
ft00 : Tri ( * a < * b) ( a ≡ b) ( * b < * a )
ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sua<x) fcb) (s≤fc {A} _ f mf fca))
f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-init fcb ⟫ = fcn-cmp y f mf fca fcb
record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f)
{y : Ordinal} (ay : odef A y) (zc : ZChain A f mf< ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where
supf = ZChain.supf zc
field
is-max : {a b : Ordinal } → (ca : odef (UnionCF A f ay supf z) a ) → b o< z → (ab : odef A b)
→ HasPrev A (UnionCF A f ay supf z) f b ∨ IsSUP A (UnionCF A f ay supf b) b
→ * a < * b → odef ((UnionCF A f ay supf z)) b
record Maximal ( A : HOD ) : Set (Level.suc n) where
field
maximal : HOD
as : A ∋ maximal
¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x
supf-unique : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf< : <-monotonic-f A f)
{y xa xb : Ordinal} → (ay : odef A y) → (xa o≤ xb ) → (za : ZChain A f mf< ay xa ) (zb : ZChain A f mf< ay xb )
→ {z : Ordinal } → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z
supf-unique A f mf< {y} {xa} {xb} ay xa≤xb za zb {z} z≤xa =
TransFinite0 {λ z → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z } ind z z≤xa where
supfa = ZChain.supf za
supfb = ZChain.supf zb
ind : (x : Ordinal) → ((w : Ordinal) → w o< x → w o≤ xa → supfa w ≡ supfb w) → x o≤ xa → supfa x ≡ supfb x
ind x prev x≤xa = sxa=sxb where
ma = ZChain.minsup za x≤xa
mb = ZChain.minsup zb (OrdTrans x≤xa xa≤xb )
spa = MinSUP.sup ma
spb = MinSUP.sup mb
sax=spa : supfa x ≡ spa
sax=spa = ZChain.supf-is-minsup za x≤xa
sbx=spb : supfb x ≡ spb
sbx=spb = ZChain.supf-is-minsup zb (OrdTrans x≤xa xa≤xb )
sxa=sxb : supfa x ≡ supfb x
sxa=sxb with trio< (supfa x) (supfb x)
... | tri≈ ¬a b ¬c = b
... | tri< a ¬b ¬c = ⊥-elim ( o≤> (
begin
supfb x ≡⟨ sbx=spb ⟩
spb ≤⟨ MinSUP.minsup mb (MinSUP.as ma) (λ {z} uzb → MinSUP.x≤sup ma (z53 uzb)) ⟩
spa ≡⟨ sym sax=spa ⟩
supfa x ∎ ) a ) where
open o≤-Reasoning
z53 : {z : Ordinal } → odef (UnionCF A f ay (ZChain.supf zb) x) z → odef (UnionCF A f ay (ZChain.supf za) x) z
z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫
z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ = ⟪ as , ch-is-sup u u<x (trans ua=ub su=u) z55 ⟫ where
ua=ub : supfa u ≡ supfb u
ua=ub = prev u u<x (ordtrans u<x x≤xa )
z55 : FClosure A f (ZChain.supf za u) z
z55 = subst (λ k → FClosure A f k z ) (sym ua=ub) fc
... | tri> ¬a ¬b c = ⊥-elim ( o≤> (
begin
supfa x ≡⟨ sax=spa ⟩
spa ≤⟨ MinSUP.minsup ma (MinSUP.as mb) (λ uza → MinSUP.x≤sup mb (z53 uza)) ⟩
spb ≡⟨ sym sbx=spb ⟩
supfb x ∎ ) c ) where
open o≤-Reasoning
z53 : {z : Ordinal } → odef (UnionCF A f ay (ZChain.supf za) x) z → odef (UnionCF A f ay (ZChain.supf zb) x) z
z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫
z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ = ⟪ as , ch-is-sup u u<x (trans ub=ua su=u) z55 ⟫ where
ub=ua : supfb u ≡ supfa u
ub=ua = sym ( prev u u<x (ordtrans u<x x≤xa ))
z55 : FClosure A f (ZChain.supf zb u) z
z55 = subst (λ k → FClosure A f k z ) (sym ub=ua) fc
Zorn-lemma : { A : HOD }
→ o∅ o< & A
→ ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B )
→ Maximal A
Zorn-lemma {A} 0<A supP = zorn00 where
z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A
z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
s : HOD
s = minimal A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
as : A ∋ s
as = x∋minimal A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
s<A : & s o< & A
s<A = c<→o< as
HasMaximal : HOD
HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 }
no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥
no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ )
Gtx : { x : HOD} → A ∋ x → HOD
Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 }
z08 : ¬ Maximal A → HasMaximal =h= od∅
z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt)
; ¬maximal<x = λ {y} ay x<y → proj2 lt (& y) ay (<-cong ==-refl (==-sym *iso) x<y) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )}
x-is-maximal : ¬ Maximal A → {x : Ordinal} → (ax : odef A x) → & (Gtx (subst (λ k → odef A k ) (sym &iso) ax)) ≡ o∅ → (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m))
x-is-maximal nmx {x} ax nogt m am = ⟪ subst (λ k → odef A k) &iso (subst (λ k → odef A k ) (sym &iso) ax) , ¬x<m ⟫ where
¬x<m : ¬ (* x < * m)
¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt)
minsupP : ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → MinSUP A B
minsupP B B⊆A total = m02 where
xsup : (sup : Ordinal ) → Set n
xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup )
∀-imply-or : {A : Ordinal → Set n } {B : Set n }
→ ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B
∀-imply-or {A} {B} ∀AB with p∨¬p ((x : Ordinal ) → A x)
∀-imply-or {A} {B} ∀AB | case1 t = case1 t
∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x 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 ))
m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B
m00 x = TransFinite0 ind x where
ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B)
→ ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B
ind x prev = ∀-imply-or m01 where
m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B
m01 z with trio< z x
... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) )
... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) )
... | tri< a ¬b ¬c with prev z a
... | case2 mins = case2 mins
... | case1 not with p∨¬p (odef A z ∧ xsup z)
... | case1 mins = case2 record { sup = z ; isMinSUP = record { as = proj1 mins ; x≤sup = proj2 mins ; minsup = m04 } } where
m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1
m04 {s} as lt with trio< z s
... | tri< a ¬b ¬c = o<→≤ a
... | tri≈ ¬a b ¬c = o≤-refl0 b
... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ )
... | case2 notz = case1 (λ _ → notz )
m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z)
m03 not = ⊥-elim ( not s1 (odef< (SUP.ax S)) ⟪ SUP.ax S , m05 ⟫ ) where
S : SUP A B
S = supP B B⊆A total
s1 = & (SUP.sup S)
m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 )
m05 {w} bw with SUP.x≤sup S bw
... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (trans &iso eq))
... | case2 lt = case2 lt
m02 : MinSUP A B
m02 = dont-or (m00 (& A)) m03
cf : ¬ Maximal A → Ordinal → Ordinal
cf nmx x with ∋-p A (* x)
... | no _ = o∅
... | yes ax with is-o∅ (& ( Gtx ax ))
... | yes nogt =
⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ )
... | no not = & (minimal (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)))
is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) )
is-cf nmx {x} ax with ∋-p A (* x)
... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax ))
... | yes ax with is-o∅ (& ( Gtx ax ))
... | yes nogt = ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ )
... | no not = x∋minimal (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))
cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) → odef A x → ( * x < * (cf nmx x) ) ∧ odef A (cf nmx x )
cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫
cf-is-≤-monotonic : (nmx : ¬ Maximal A ) → ≤-monotonic-f A ( cf nmx )
cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax )) , proj2 ( cf-is-<-monotonic nmx x ax ) ⟫
SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) (mf< : <-monotonic-f A f)
{init : Ordinal} (ay : odef A init) (zc : ZChain A f mf< ay (& A)) (x : Ordinal) → x o≤ & A → ZChain1 A f mf< ay zc x
SZ1 f mf mf< {y} ay zc x x≤A = zc1 x x≤A where
chain-mono1 : {a b c : Ordinal} → a o≤ b
→ odef (UnionCF A f ay (ZChain.supf zc) a) c → odef (UnionCF A f ay (ZChain.supf zc) b) c
chain-mono1 {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b
is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) x) a → (ab : odef A b)
→ HasPrev A (UnionCF A f ay (ZChain.supf zc) x) f b
→ * a < * b → odef (UnionCF A f ay (ZChain.supf zc) x) b
is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev
... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫
... | ⟪ ab0 , ch-is-sup u u<x su=u fc ⟫ = ⟪ ab , subst (λ k → UChain ay x k )
(sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x su=u (fsuc _ fc)) ⟫
supf = ZChain.supf zc
zc1 : (x : Ordinal ) → x o≤ & A → ZChain1 A f mf< ay zc x
zc1 x x≤A with Oprev-p x
... | yes op = record { is-max = is-max } where
px = Oprev.oprev op
is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a →
b o< x → (ab : odef A b) →
HasPrev A (UnionCF A f ay supf x) f b ∨ IsSUP A (UnionCF A f ay supf b) b →
* a < * b → odef (UnionCF A f ay supf x) b
is-max {a} {b} ua b<x ab P a<b with or-exclude P
is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b
is-max {a} {b} ua b<x ab P a<b | case2 is-sup with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x))
... | case2 sb<sx = m10 where
b<A : b o< & A
b<A = odef< ab
m05 : ZChain.supf zc b ≡ b
m05 = ZChain.sup=u zc ab (o<→≤ (odef< ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz }
m10 : odef (UnionCF A f ay supf x) b
m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05)
... | case1 sb=sx = ⊥-elim (<-irr (case1 m10) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where
m17 : MinSUP A (UnionCF A f ay supf x)
m17 = ZChain.minsup zc x≤A
m18 : supf x ≡ MinSUP.sup m17
m18 = ZChain.supf-is-minsup zc x≤A
m10 : f (supf b) ≡ supf b
m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where
m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x)
m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where
m04 : ¬ HasPrev A (UnionCF A f ay supf b) f b
m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay =
chain-mono1 (o<→≤ b<x) (HasPrev.ay nhp) ; x=fy = HasPrev.x=fy nhp } )
m05 : ZChain.supf zc b ≡ b
m05 = ZChain.sup=u zc ab (o<→≤ (odef< ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz }
m14 : ZChain.supf zc b o< x
m14 = subst (λ k → k o< x ) (sym m05) b<x
m13 : odef (UnionCF A f ay supf x) z
m13 = ZChain.cfcs zc b<x x≤A m14 fc
... | no lim = record { is-max = is-max } where
is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a →
b o< x → (ab : odef A b) →
HasPrev A (UnionCF A f ay supf x) f b ∨ IsSUP A (UnionCF A f ay supf b) b →
* a < * b → odef (UnionCF A f ay supf x) b
is-max {a} {b} ua b<x ab P a<b with or-exclude P
is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b
is-max {a} {b} ua b<x ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (ZChain.chain∋init zc )
... | case1 b=y = ⟪ subst (λ k → odef A k ) b=y ay , ch-init (subst (λ k → FClosure A f y k ) b=y (init ay refl )) ⟫
... | case2 y<b with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x))
... | case2 sb<sx = m10 where
m09 : b o< & A
m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab))
m05 : ZChain.supf zc b ≡ b
m05 = ZChain.sup=u zc ab (o<→≤ m09) record { ax = ab ; x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt }
m10 : odef (UnionCF A f ay supf x) b
m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05)
... | case1 sb=sx = ⊥-elim (<-irr (case1 m10) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where
m17 : MinSUP A (UnionCF A f ay supf x)
m17 = ZChain.minsup zc x≤A
m18 : supf x ≡ MinSUP.sup m17
m18 = ZChain.supf-is-minsup zc x≤A
m10 : f (supf b) ≡ supf b
m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where
m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x)
m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where
m05 = ZChain.sup=u zc ab (o<→≤ (odef< ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz }
m14 : ZChain.supf zc b o< x
m14 = subst (λ k → k o< x ) (sym m05) b<x
m13 : odef (UnionCF A f ay supf x) z
m13 = ZChain.cfcs zc b<x x≤A m14 fc
uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD
uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax =
λ {z} cz → subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (A∋fc y f mf cz ))) }
utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y)
→ IsTotalOrderSet (uchain f mf ay)
utotal f mf {y} ay {a} {b} ca cb = fcn-cmp y f mf ca cb
ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y)
→ MinSUP A (uchain f mf ay)
ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt) (utotal f mf ay)
ind : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal)
→ ((z : Ordinal) → z o< x → ZChain A f mf< ay z) → ZChain A f mf< ay x
ind f mf< {y} ay x prev with Oprev-p x
... | yes op = zc41 sup1 where
px = Oprev.oprev op
zc : ZChain A f mf< ay (Oprev.oprev op)
zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc )
px<x : px o< x
px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc
opx=x : osuc px ≡ x
opx=x = Oprev.oprev=x op
zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px
zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt
supf0 = ZChain.supf zc
pchain : HOD
pchain = UnionCF A f ay supf0 px
supf-mono = ZChain.supf-mono zc
zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x )
zc04 {b} b≤x with trio< b px
... | tri< a ¬b ¬c = case1 (o<→≤ a)
... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b)
... | tri> ¬a ¬b px<b with osuc-≡< b≤x
... | case1 eq = case2 eq
... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ )
mf : ≤-monotonic-f A f
mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
mf00 : * x < * (f x)
mf00 = proj1 ( mf< x ax )
pchainpx : HOD
pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain ay px z )
∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x)) } ; odmax = & A ; <odmax = zc00 } where
zc00 : {z : Ordinal } → (odef A z ∧ UChain ay px z ) ∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x) )→ z o< & A
zc00 {z} (case1 lt) = z07 lt
zc00 {z} (case2 fc) = odef< ( A∋fc (supf0 px) f mf (proj1 fc) )
zc02 : { a b : Ordinal } → odef A a ∧ UChain ay px a → FClosure A f (supf0 px) b ∧ ( supf0 px o< x) → a ≤ b
zc02 {a} {b} ca fb = zc05 (proj1 fb) where
zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a ≤ b
zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb ))
... | case1 eq = subst (λ k → a ≤ k ) eq (zc05 fb)
... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt)
zc05 (init b1 refl) = MinSUP.x≤sup (ZChain.minsup zc o≤-refl) ca
ptotal : IsTotalOrderSet pchainpx
ptotal (case1 a) (case1 b) = ZChain.f-total zc a b
ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b
... | case1 eq = tri≈ (<-irr (case1 (sym eq))) eq (<-irr (case1 eq))
... | case2 lt = tri< lt (λ eq → <-irr (case1 (sym eq)) lt) (<-irr (case2 lt))
ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b
... | case1 eq = tri≈ (<-irr (case1 eq)) (sym eq) (<-irr (case1 (sym eq)))
... | case2 lt = tri> (<-irr (case2 lt)) (λ eq → <-irr (case1 eq) lt) lt
ptotal (case2 a) (case2 b) = fcn-cmp (supf0 px) f mf (proj1 a) (proj1 b)
pcha : pchainpx ⊆ A
pcha (case1 lt) = proj1 lt
pcha (case2 fc) = A∋fc _ f mf (proj1 fc)
sup1 : MinSUP A pchainpx
sup1 = minsupP pchainpx pcha ptotal
zc41 : MinSUP A pchainpx → ZChain A f mf< ay x
zc41 sup1 = record { supf = supf1 ; asupf = asupf1 ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = cfcs ; supf-mono = supf1-mono } where
sp1 = MinSUP.sup sup1
supf1 : Ordinal → Ordinal
supf1 z with trio< z px
... | tri< a ¬b ¬c = supf0 z
... | tri≈ ¬a b ¬c = supf0 z
... | tri> ¬a ¬b c = sp1
sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z
sf1=sf0 {z} z≤px with trio< z px
... | tri< a ¬b ¬c = refl
... | tri≈ ¬a b ¬c = refl
... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c )
sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1
sf1=sp1 {z} px<z with trio< z px
... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a )
... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z )
... | tri> ¬a ¬b c = refl
sf=eq : { z : Ordinal } → z o< x → supf0 z ≡ supf1 z
sf=eq {z} z<x = sym (sf1=sf0 (subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ))
asupf1 : {z : Ordinal } → odef A (supf1 z)
asupf1 {z} with trio< z px
... | tri< a ¬b ¬c = ZChain.asupf zc
... | tri≈ ¬a b ¬c = ZChain.asupf zc
... | tri> ¬a ¬b c = MinSUP.as sup1
supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b
supf1-mono {a} {b} a≤b with trio< b px
... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b )
... | tri≈ ¬a b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (subst (λ k → a o≤ k) b a≤b))) refl ( supf-mono a≤b )
supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px
... | tri< a<px ¬b ¬c = zc19 where
zc21 : MinSUP A (UnionCF A f ay supf0 a)
zc21 = ZChain.minsup zc (o<→≤ a<px)
zc24 : {x₁ : Ordinal} → odef (UnionCF A f ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1)
zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) )
zc19 : supf0 a o≤ sp1
zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.as sup1) zc24 )
... | tri≈ ¬a b ¬c = zc18 where
zc21 : MinSUP A (UnionCF A f ay supf0 a)
zc21 = ZChain.minsup zc (o≤-refl0 b)
zc20 : MinSUP.sup zc21 ≡ supf0 a
zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b))
zc24 : {x₁ : Ordinal} → odef (UnionCF A f ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1)
zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) )
zc18 : supf0 a o≤ sp1
zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.as sup1) zc24 )
... | tri> ¬a ¬b c = o≤-refl
fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z
fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc
fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px → FClosure A f (supf1 u) z
fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc
cfcs : {a b w : Ordinal }
→ a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w
cfcs {a} {b} {w} a<b b≤x sa<b fc with x<y∨y≤x (supf0 a) px
... | case2 px≤sa = z50 where
a<x : a o< x
a<x = ordtrans<-≤ a<b b≤x
a≤px : a o≤ px
a≤px = subst (λ k → a o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ a<b b≤x)
z50 : odef (UnionCF A f ay supf1 b) w
z50 with osuc-≡< px≤sa
... | case1 px=sa = ⟪ A∋fc {A} _ f mf fc , cp ⟫ where
sa≤px : supf0 a o≤ px
sa≤px = subst₂ (λ j k → j o< k) px=sa (sym (Oprev.oprev=x op)) px<x
spx=sa : supf0 px ≡ supf0 a
spx=sa = begin
supf0 px ≡⟨ cong supf0 px=sa ⟩
supf0 (supf0 a) ≡⟨ ZChain.supf-idem zc a≤px sa≤px ⟩
supf0 a ∎ where open ≡-Reasoning
z51 : supf0 px o< b
z51 = subst (λ k → k o< b ) (sym ( begin supf0 px ≡⟨ spx=sa ⟩
supf0 a ≡⟨ sym (sf1=sf0 a≤px) ⟩
supf1 a ∎ )) sa<b where open ≡-Reasoning
z52 : supf1 a ≡ supf1 (supf0 px)
z52 = begin supf1 a ≡⟨ sf1=sf0 a≤px ⟩
supf0 a ≡⟨ sym (ZChain.supf-idem zc a≤px sa≤px ) ⟩
supf0 (supf0 a) ≡⟨ sym (sf1=sf0 sa≤px) ⟩
supf1 (supf0 a) ≡⟨ cong supf1 (sym spx=sa) ⟩
supf1 (supf0 px) ∎ where open ≡-Reasoning
z53 : supf1 (supf0 px) ≡ supf0 px
z53 = begin
supf1 (supf0 px) ≡⟨ cong supf1 spx=sa ⟩
supf1 (supf0 a) ≡⟨ sf1=sf0 sa≤px ⟩
supf0 (supf0 a) ≡⟨ sym ( cong supf0 px=sa ) ⟩
supf0 px ∎ where open ≡-Reasoning
cp : UChain ay b w
cp = ch-is-sup (supf0 px) z51 z53 (subst (λ k → FClosure A f k w) z52 fc)
... | case2 px<sa = ⊥-elim ( ¬p<x<op ⟪ px<sa , subst₂ (λ j k → j o< k ) (sf1=sf0 a≤px) (sym (Oprev.oprev=x op)) z53 ⟫ ) where
z53 : supf1 a o< x
z53 = ordtrans<-≤ sa<b b≤x
... | case1 sa<px with trio< a px
... | tri< a<px ¬b ¬c = z50 where
z50 : odef (UnionCF A f ay supf1 b) w
z50 with osuc-≡< b≤x
... | case2 lt with ZChain.cfcs zc a<b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) lt) sa<b fc
... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
... | ⟪ az , ch-is-sup u u<b su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc u≤px ) ⟫ where
u≤px : u o≤ px
u≤px = subst (λ k → u o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ u<b b≤x )
u<x : u o< x
u<x = ordtrans<-≤ u<b b≤x
z50 | case1 eq with ZChain.cfcs zc a<px o≤-refl sa<px fc
... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
... | ⟪ az , ch-is-sup u u<px su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc (o<→≤ u<px)) ⟫ where
u<b : u o< b
u<b = subst (λ k → u o< k ) (trans (Oprev.oprev=x op) (sym eq) ) (ordtrans u<px <-osuc )
u<x : u o< x
u<x = subst (λ k → u o< k ) (Oprev.oprev=x op) ( ordtrans u<px <-osuc )
... | tri≈ ¬a a=px ¬c = csupf1 where
px<b : px o< b
px<b = subst₂ (λ j k → j o< k) a=px refl a<b
b=x : b ≡ x
b=x with trio< b x
... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) a ⟫ )
... | tri≈ ¬a b ¬c = b
... | tri> ¬a ¬b c = ⊥-elim ( o≤> b≤x c )
z51 : FClosure A f (supf1 px) w
z51 = subst (λ k → FClosure A f k w) (sym (trans (cong supf1 (sym a=px)) (sf1=sf0 (o≤-refl0 a=px) ))) fc
z53 : odef A w
z53 = A∋fc {A} _ f mf fc
csupf1 : odef (UnionCF A f ay supf1 b) w
csupf1 with x<y∨y≤x px (supf0 px)
... | case2 spx≤px = ⟪ z53 , ch-is-sup (supf0 px) z54 z52 fc1 ⟫ where
z54 : supf0 px o< b
z54 = subst (λ k → supf0 px o< k ) (trans (Oprev.oprev=x op) (sym b=x) ) spx≤px
z52 : supf1 (supf0 px) ≡ supf0 px
z52 = trans (sf1=sf0 spx≤px ) ( ZChain.supf-idem zc o≤-refl spx≤px )
fc1 : FClosure A f (supf1 (supf0 px)) w
fc1 = subst (λ k → FClosure A f k w ) (trans (cong supf0 a=px) (sym z52) ) fc
... | case1 px<spx = ⊥-elim (¬p<x<op ⟪ px<spx , z54 ⟫ ) where
z54 : supf0 px o≤ px
z54 = subst₂ (λ j k → supf0 j o< k ) a=px (trans b=x (sym (Oprev.oprev=x op))) sa<b
... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → a o< k ) (sym (Oprev.oprev=x op)) ( ordtrans<-≤ a<b b≤x) ⟫ )
zc11 : {z : Ordinal} → odef (UnionCF A f ay supf1 x) z → odef pchainpx z
zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫
zc11 {z} ⟪ az , ch-is-sup u u<x su=u fc ⟫ = zc21 fc where
zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1
zc21 {z1} (fsuc z2 fc ) with zc21 fc
... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-init (fsuc _ fc₁) ⟫
... | case1 ⟪ ua1 , ch-is-sup u u<x su=u fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x su=u (fsuc _ fc₁) ⟫
... | case2 fc = case2 ⟪ fsuc _ (proj1 fc) , proj2 fc ⟫
zc21 (init asp refl ) with trio< (supf0 u) (supf0 px)
... | tri< a ¬b ¬c = case1 ⟪ asp , ch-is-sup u u<px (trans (sym (sf1=sf0 (o<→≤ u<px))) su=u )(init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where
u<px : u o< px
u<px = ZChain.supf-inject zc a
asp0 : odef A (supf0 u)
asp0 = ZChain.asupf zc
... | tri≈ ¬a b ¬c = case2 ⟪ (init (subst (λ k → odef A k) b (ZChain.asupf zc) )
(sym (trans (sf1=sf0 (zc-b<x _ u<x)) b ))) , spx<x ⟫ where
spx<x : supf0 px o< x
spx<x = osucprev ( begin
osuc (supf0 px) ≡⟨ cong osuc (sym b) ⟩
osuc (supf0 u) ≡⟨ cong osuc (sym (sf1=sf0 (zc-b<x _ u<x) )) ⟩
osuc (supf1 u) ≡⟨ cong osuc su=u ⟩
osuc u ≤⟨ osucc u<x ⟩
x ∎ ) where open o≤-Reasoning
... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ ZChain.supf-inject zc c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x ⟫ )
is-minsup : {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z)
is-minsup {z} z≤x with osuc-≡< z≤x
... | case1 z=x = record { as = zc22 ; x≤sup = z23 ; minsup = z24 } where
px<z : px o< z
px<z = subst (λ k → px o< k) (sym z=x) px<x
zc22 : odef A (supf1 z)
zc22 = subst (λ k → odef A k ) (sym (sf1=sp1 px<z )) ( MinSUP.as sup1 )
z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z
z23 {w} uz = subst (λ k → w ≤ k ) (sym (sf1=sp1 px<z)) ( MinSUP.x≤sup sup1 (
zc11 (subst (λ k → odef (UnionCF A f ay supf1 k) w) z=x uz )))
z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s )
→ supf1 z o≤ s
z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sp1 px<z)) ( MinSUP.minsup sup1 as z25 ) where
z25 : {w : Ordinal } → odef pchainpx w → w ≤ s
z25 {w} (case2 fc) = sup ⟪ A∋fc _ f mf (proj1 fc) , ch-is-sup (supf0 px) z28 z27 fc1 ⟫ where
z28 : supf0 px o< z
z28 = subst (λ k → supf0 px o< k) (sym z=x) (proj2 fc)
z29 : supf0 px o≤ px
z29 = zc-b<x _ (proj2 fc)
z27 : supf1 (supf0 px) ≡ supf0 px
z27 = trans (sf1=sf0 z29) ( ZChain.supf-idem zc o≤-refl z29 )
fc1 : FClosure A f (supf1 (supf0 px)) w
fc1 = subst (λ k → FClosure A f k w) (sym z27) (proj1 fc)
z25 {w} (case1 ⟪ ua , ch-init fc ⟫) = sup ⟪ ua , ch-init fc ⟫
z25 {w} (case1 ⟪ ua , ch-is-sup u u<x su=u fc ⟫) = sup ⟪ ua , ch-is-sup u u<z
(trans (sf1=sf0 u≤px) su=u) (fcpu fc u≤px) ⟫ where
u≤px : u o< osuc px
u≤px = ordtrans u<x <-osuc
u<z : u o< z
u<z = ordtrans u<x (subst (λ k → px o< k ) (sym z=x) px<x )
... | case2 z<x = record { as = zc22 ; x≤sup = z23 ; minsup = z24 } where
z≤px = zc-b<x _ z<x
m = ZChain.is-minsup zc z≤px
zc22 : odef A (supf1 z)
zc22 = subst (λ k → odef A k ) (sym (sf1=sf0 z≤px)) ( IsMinSUP.as m )
z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z
z23 {w} ⟪ ua , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px)) ( ZChain.fcy<sup zc z≤px fc )
z23 {w} ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px))
(IsMinSUP.x≤sup m ⟪ ua , ch-is-sup u u<x (trans (sym (sf1=sf0 u≤px )) su=u) (fcup fc u≤px ) ⟫ ) where
u≤px : u o≤ px
u≤px = ordtrans u<x z≤px
z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s )
→ supf1 z o≤ s
z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf0 z≤px)) ( IsMinSUP.minsup m as z25 ) where
z25 : {w : Ordinal } → odef ( UnionCF A f ay supf0 z ) w → w ≤ s
z25 {w} ⟪ ua , ch-init fc ⟫ = sup ⟪ ua , ch-init fc ⟫
z25 {w} ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = sup ⟪ ua , ch-is-sup u u<x
(trans (sf1=sf0 u≤px) su=u) (fcpu fc u≤px) ⟫ where
u≤px : u o≤ px
u≤px = ordtrans u<x z≤px
zo≤sz : {z : Ordinal} → z o≤ x → z o≤ supf1 z
zo≤sz {z} z≤x with osuc-≡< z≤x
... | case2 z<x = subst (λ k → z o≤ k) (sym (sf1=sf0 (zc-b<x _ z<x ))) (ZChain.zo≤sz zc (zc-b<x _ z<x ))
... | case1 refl with osuc-≡< (supf1-mono (o<→≤ (px<x)))
... | case2 lt = begin
x ≡⟨ sym (Oprev.oprev=x op) ⟩
osuc px ≤⟨ osucc (ZChain.zo≤sz zc o≤-refl) ⟩
osuc (supf0 px) ≡⟨ sym (cong osuc (sf1=sf0 o≤-refl )) ⟩
osuc (supf1 px) ≤⟨ osucc lt ⟩
supf1 x ∎ where open o≤-Reasoning
... | case1 spx=sx with osuc-≡< ( ZChain.zo≤sz zc o≤-refl )
... | case2 lt = begin
x ≡⟨ sym (Oprev.oprev=x op) ⟩
osuc px ≤⟨ osucc lt ⟩
supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩
supf1 px ≤⟨ supf1-mono (o<→≤ px<x) ⟩
supf1 x ∎ where open o≤-Reasoning
... | case1 px=spx = ⊥-elim ( <<-irr zc40 (proj1 ( mf< (supf0 px) (ZChain.asupf zc))) ) where
zc37 : supf0 px ≡ px
zc37 = sym px=spx
zc39 : supf0 px ≡ sp1
zc39 = begin
supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩
supf1 px ≡⟨ spx=sx ⟩
supf1 x ≡⟨ sf1=sp1 px<x ⟩
sp1 ∎ where open ≡-Reasoning
zc40 : f (supf0 px) ≤ supf0 px
zc40 = subst (λ k → f (supf0 px) ≤ k ) (sym zc39)
( MinSUP.x≤sup sup1 (case2 ⟪ fsuc _ (init (ZChain.asupf zc) refl) , subst (λ k → k o< x) (sym zc37) px<x ⟫ ))
... | no lim with trio< x o∅
... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a )
... | tri≈ ¬a x=0 ¬c = record { supf = λ _ → MinSUP.sup (ysup f mf ay) ; asupf = MinSUP.as (ysup f mf ay) ; supf-mono = λ _ → o≤-refl
; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = λ a<b b≤0 → ⊥-elim ( ¬x<0 (subst (λ k → _ o< k ) x=0 (ordtrans<-≤ a<b b≤0))) } where
mf : ≤-monotonic-f A f
mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
mf00 : * x < * (f x)
mf00 = proj1 ( mf< x ax )
ym = MinSUP.sup (ysup f mf ay)
zo≤sz : {z : Ordinal} → z o≤ x → z o≤ MinSUP.sup (ysup f mf ay)
zo≤sz {z} z≤x with osuc-≡< z≤x
... | case1 refl = subst (λ k → k o≤ _) (sym x=0) o∅≤z
... | case2 lt = ⊥-elim ( ¬x<0 (subst (λ k → z o< k ) x=0 lt ) )
is-minsup : {z : Ordinal} → z o≤ x →
IsMinSUP A (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) z) (MinSUP.sup (ysup f mf ay))
is-minsup {z} z≤x with osuc-≡< z≤x
... | case1 refl = record { as = MinSUP.as (ysup f mf ay) ; x≤sup = λ {w} uw → is00 uw ; minsup = λ {s} as sup → is01 as sup } where
is00 : {w : Ordinal } → odef (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) x ) w → w ≤ MinSUP.sup (ysup f mf ay)
is00 {w} ⟪ aw , ch-init fc ⟫ = MinSUP.x≤sup (ysup f mf ay) fc
is00 {w} ⟪ aw , ch-is-sup u u<z su=u fc ⟫ = ⊥-elim (¬x<0 (subst (λ k → u o< k ) x=0 u<z ))
is01 : { s : Ordinal } → odef A s → ( {w : Ordinal } → odef (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) x ) w → w ≤ s )
→ ym o≤ s
is01 {s} as sup = MinSUP.minsup (ysup f mf ay) as is02 where
is02 : {w : Ordinal } → odef (uchain f mf ay) w → (w ≡ s) ∨ (w << s)
is02 fc = sup ⟪ A∋fc _ f mf fc , ch-init fc ⟫
... | case2 lt = ⊥-elim ( ¬x<0 (subst (λ k → z o< k ) x=0 lt ) )
... | tri> ¬a ¬b 0<x = zc400 usup ssup where
mf : ≤-monotonic-f A f
mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
mf00 : * x < * (f x)
mf00 = proj1 ( mf< x ax )
pzc : {z : Ordinal} → z o< x → ZChain A f mf< ay z
pzc {z} z<x = prev z z<x
ysp = MinSUP.sup (ysup f mf ay)
supfz : {z : Ordinal } → z o< x → Ordinal
supfz {z} z<x = ZChain.supf (pzc (ob<x lim z<x)) z
pchainU : HOD
pchainU = UnionIC A f ay supfz
zeq : {xa xb z : Ordinal }
→ (xa<x : xa o< x) → (xb<x : xb o< x) → xa o≤ xb → z o≤ xa
→ ZChain.supf (pzc xa<x) z ≡ ZChain.supf (pzc xb<x) z
zeq {xa} {xb} {z} xa<x xb<x xa≤xb z≤xa = supf-unique A f mf< ay xa≤xb
(pzc xa<x) (pzc xb<x) z≤xa
IChain-i : {z : Ordinal } → IChain ay supfz z → Ordinal
IChain-i (ic-init fc) = o∅
IChain-i (ic-isup ia ia<x sa<x fca) = ia
pic<x : {z : Ordinal } → (ic : IChain ay supfz z ) → osuc (IChain-i ic) o< x
pic<x {z} (ic-init fc) = ob<x lim 0<x
pic<x {z} (ic-isup ia ia<x sa<x fca) = ob<x lim ia<x
pchainU⊆chain : {z : Ordinal } → (pz : odef pchainU z) → odef (ZChain.chain (pzc (pic<x (proj2 pz)))) z
pchainU⊆chain {z} ⟪ aw , ic-init fc ⟫ = ⟪ aw , ch-init fc ⟫
pchainU⊆chain {z} ⟪ aw , (ic-isup ia ia<x sa<x fca) ⟫ = ZChain.cfcs (pzc (ob<x lim ia<x) ) <-osuc o≤-refl uz03 fca where
uz02 : FClosure A f (ZChain.supf (pzc (ob<x lim ia<x)) ia ) z
uz02 = fca
uz03 : ZChain.supf (pzc (ob<x lim ia<x)) ia o≤ ia
uz03 = sa<x
chain⊆pchainU : {z w : Ordinal } → (z<x : z o< x) → odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim z<x))) z) w → odef pchainU w
chain⊆pchainU {z} {w} z<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫
chain⊆pchainU {z} {w} z<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫
= ⟪ aw , ic-isup u u<x (o≤-refl0 su≡u) (subst (λ k → FClosure A f k w ) su=su fc) ⟫ where
u<x : u o< x
u<x = ordtrans u<oz z<x
su=su : ZChain.supf (pzc (ob<x lim z<x)) u ≡ supfz u<x
su=su = sym ( zeq _ _ (o<→≤ (osucc u<oz)) (o<→≤ <-osuc) )
su≡u : supfz u<x ≡ u
su≡u = begin
ZChain.supf (pzc (ob<x lim u<x )) u ≡⟨ sym su=su ⟩
ZChain.supf (pzc (ob<x lim z<x)) u ≡⟨ su=u ⟩
u ∎ where open ≡-Reasoning
IC⊆ : {a b : Ordinal } (ia : IChain ay supfz a ) (ib : IChain ay supfz b )
→ IChain-i ia o< IChain-i ib → odef (ZChain.chain (pzc (pic<x ib))) a
IC⊆ {a} {b} (ic-init fc ) ib ia<ib = ⟪ A∋fc _ f mf fc , ch-init fc ⟫
IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-init fcb ) ia<ib = ⊥-elim ( ¬x<0 ia<ib )
IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-isup j j<x sb<x fcb ) ia<ib
= ZChain.cfcs (pzc (ob<x lim j<x) ) (o<→≤ ia<ib) o≤-refl (OrdTrans (ZChain.supf-mono (pzc (ob<x lim j<x)) (o<→≤ ia<ib)) sb<x)
(subst (λ k → FClosure A f k a) (zeq _ _ (osucc (o<→≤ ia<ib)) (o<→≤ <-osuc)) fc )
ptotalU : IsTotalOrderSet pchainU
ptotalU {a} {b} ia ib with trio< (IChain-i (proj2 ia)) (IChain-i (proj2 ib))
... | tri< ia<ib ¬b ¬c = ZChain.f-total (pzc (pic<x (proj2 ib))) (IC⊆ (proj2 ia) (proj2 ib) ia<ib) (pchainU⊆chain ib)
... | tri≈ ¬a ia=ib ¬c = pcmp (proj2 ia) (proj2 ib) ia=ib where
pcmp : (ia : IChain ay supfz a) → (ib : IChain ay supfz b) → IChain-i ia ≡ IChain-i ib
→ Tri (* a < * b) (a ≡ b) (* b < * a )
pcmp (ic-init fca) (ic-init fcb) eq = fcn-cmp _ f mf fca fcb
pcmp (ic-init fca) (ic-isup i i<x s<x fcb) eq with ZChain.fcy<sup (pzc i<x) o≤-refl fca
... | case1 eq1 = ct22 where
ct22 : Tri (* a < * b) (a ≡ b) (* b < * a )
ct22 with subst (λ k → k ≤ b) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fcb )
... | case1 eq2 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where
ct00 : a ≡ b
ct00 = trans eq1 eq2
... | case2 lt = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where
fc11 : * a < * b
fc11 = subst (λ k → k < * b ) (cong (*) (sym eq1)) lt
... | case2 lt = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where
fc11 : * a < * b
fc11 = ftrans<-≤ lt (subst (λ k → k ≤ b) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fcb ) )
pcmp (ic-isup i i<x s<x fca) (ic-init fcb) eq with ZChain.fcy<sup (pzc i<x) o≤-refl fcb
... | case1 eq1 = ct22 where
ct22 : Tri (* a < * b) (a ≡ b) (* b < * a )
ct22 with subst (λ k → k ≤ a) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fca )
... | case1 eq2 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where
ct00 : a ≡ b
ct00 = sym (trans eq1 eq2)
... | case2 lt = tri> (λ lt → <-irr (case2 fc11) lt) (λ eq → <-irr (case1 eq) fc11) fc11 where
fc11 : * b < * a
fc11 = subst (λ k → k < * a ) (cong (*) (sym eq1)) lt
... | case2 lt = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where
fc12 : * b < * a
fc12 = ftrans<-≤ lt (subst (λ k → k ≤ a) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fca ) )
pcmp (ic-isup i i<x s<x fca) (ic-isup i i<y s<y fcb) refl = fcn-cmp _ f mf fca (subst (λ k → FClosure A f k b) pc01 fcb ) where
pc01 : supfz i<y ≡ supfz i<x
pc01 = cong supfz o<-irr
... | tri> ¬a ¬b ib<ia = ZChain.f-total (pzc (pic<x (proj2 ia))) (pchainU⊆chain ia) (IC⊆ (proj2 ib) (proj2 ia) ib<ia)
usup : MinSUP A pchainU
usup = minsupP pchainU (λ ic → proj1 ic ) ptotalU
spu0 = MinSUP.sup usup
pchainS : HOD
pchainS = record { od = record { def = λ z → (odef A z ∧ IChain ay supfz z )
∨ (FClosure A f spu0 z ∧ (spu0 o< x)) } ; odmax = & A ; <odmax = zc00 } where
zc00 : {z : Ordinal } → (odef A z ∧ IChain ay supfz z ) ∨ (FClosure A f spu0 z ∧ (spu0 o< x) )→ z o< & A
zc00 {z} (case1 lt) = z07 lt
zc00 {z} (case2 fc) = odef< ( A∋fc spu0 f mf (proj1 fc) )
zc02 : { a b : Ordinal } → odef A a ∧ IChain ay supfz a → FClosure A f spu0 b ∧ ( spu0 o< x) → a ≤ b
zc02 {a} {b} ca fb = zc05 (proj1 fb) where
zc05 : {b : Ordinal } → FClosure A f spu0 b → a ≤ b
zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc spu0 f mf fb ))
... | case1 eq = subst (λ k → a ≤ k ) eq (zc05 fb)
... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt)
zc05 (init b1 refl) = MinSUP.x≤sup usup ca
ptotalS : IsTotalOrderSet pchainS
ptotalS (case1 a) (case1 b) = ptotalU a b
ptotalS {a0} {b0} (case1 a) (case2 b) with zc02 a b
... | case1 eq = tri≈ (<-irr (case1 (sym eq))) eq (<-irr (case1 eq))
... | case2 lt = tri< lt (λ eq → <-irr (case1 (sym eq)) lt) (<-irr (case2 lt))
ptotalS {b0} {a0} (case2 b) (case1 a) with zc02 a b
... | case1 eq = tri≈ (<-irr (case1 eq)) (sym eq) (<-irr (case1 (sym eq)))
... | case2 lt = tri> (<-irr (case2 lt)) (λ eq → <-irr (case1 eq) lt) lt
ptotalS (case2 a) (case2 b) = fcn-cmp spu0 f mf (proj1 a) (proj1 b)
S⊆A : pchainS ⊆ A
S⊆A (case1 lt) = proj1 lt
S⊆A (case2 fc) = A∋fc _ f mf (proj1 fc)
ssup : MinSUP A pchainS
ssup = minsupP pchainS S⊆A ptotalS
zc400 : MinSUP A pchainU → MinSUP A pchainS → ZChain A f mf< ay x
zc400 usup ssup = record { supf = supf1 ; asupf = asupf ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = cfcs ; supf-mono = supf-mono } where
spu = MinSUP.sup usup
sps = MinSUP.sup ssup
supf1 : Ordinal → Ordinal
supf1 z with trio< z x
... | tri< a ¬b ¬c = ZChain.supf (pzc (ob<x lim a)) z
... | tri≈ ¬a b ¬c = spu
... | tri> ¬a ¬b c = sps
pchain : HOD
pchain = UnionCF A f ay supf1 x
sf1=sf : {z : Ordinal } → (a : z o< x ) → supf1 z ≡ ZChain.supf (pzc (ob<x lim a)) z
sf1=sf {z} z<x with trio< z x
... | tri< a ¬b ¬c = cong ( λ k → ZChain.supf (pzc (ob<x lim k)) z) o<-irr
... | tri≈ ¬a b ¬c = ⊥-elim (¬a z<x)
... | tri> ¬a ¬b c = ⊥-elim (¬a z<x)
sf1=spu : {z : Ordinal } → x ≡ z → supf1 z ≡ spu
sf1=spu {z} eq with trio< z x
... | tri< a ¬b ¬c = ⊥-elim (¬b (sym eq))
... | tri≈ ¬a b ¬c = refl
... | tri> ¬a ¬b c = ⊥-elim (¬b (sym eq))
sf1=sps : {z : Ordinal } → (a : x o< z ) → supf1 z ≡ sps
sf1=sps {z} x<z with trio< z x
... | tri< a ¬b ¬c = ⊥-elim (o<> x<z a)
... | tri≈ ¬a b ¬c = ⊥-elim (¬c x<z )
... | tri> ¬a ¬b c = refl
asupf : {z : Ordinal } → odef A (supf1 z)
asupf {z} with trio< z x
... | tri< a ¬b ¬c = ZChain.asupf (pzc (ob<x lim a))
... | tri≈ ¬a b ¬c = MinSUP.as usup
... | tri> ¬a ¬b c = MinSUP.as ssup
supf-mono : {z y : Ordinal } → z o≤ y → supf1 z o≤ supf1 y
supf-mono {z} {y} z≤y with trio< y x
... | tri< y<x ¬b ¬c = zc01 where
open o≤-Reasoning
zc01 : supf1 z o≤ ZChain.supf (pzc (ob<x lim y<x)) y
zc01 = begin
supf1 z ≡⟨ sf1=sf (ordtrans≤-< z≤y y<x) ⟩
ZChain.supf (pzc (ob<x lim (ordtrans≤-< z≤y y<x))) z ≡⟨ zeq _ _ (osucc z≤y) (o<→≤ <-osuc) ⟩
ZChain.supf (pzc (ob<x lim y<x)) z ≤⟨ ZChain.supf-mono (pzc (ob<x lim y<x)) z≤y ⟩
ZChain.supf (pzc (ob<x lim y<x)) y ∎
... | tri≈ ¬a b ¬c = zc01 where
open o≤-Reasoning
zc01 : supf1 z o≤ spu
zc01 with osuc-≡< (subst (λ k → z o≤ k) b z≤y)
... | case1 z=x = o≤-refl0 (sf1=spu (sym z=x))
... | case2 z<x = subst (λ k → k o≤ spu ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) )
(MinSUP.as usup) (λ uw → MinSUP.x≤sup usup (chain⊆pchainU z<x uw)) )
... | tri> ¬a ¬b c = zc01 where
zc01 : supf1 z o≤ sps
zc01 with trio< z x
... | tri< z<x ¬b ¬c = IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) )
(MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 (chain⊆pchainU z<x uw)) )
... | tri≈ ¬a z=x ¬c = MinSUP.minsup usup (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 uw) )
... | tri> ¬a ¬b c = o≤-refl
is-minsup : {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z)
is-minsup {z} z≤x with osuc-≡< z≤x
... | case1 z=x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where
zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z
zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x))) ( MinSUP.x≤sup usup ⟪ az , ic-init fc ⟫ )
zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x)))
( MinSUP.x≤sup usup ⟪ az , ic-isup u u<x (o≤-refl0 zm05) (subst (λ k → FClosure A f k w) (sym zm06) fc) ⟫ ) where
u<x : u o< x
u<x = subst (λ k → u o< k) z=x u<b
zm06 : supfz (subst (λ k → u o< k) z=x u<b) ≡ supf1 u
zm06 = trans (zeq _ _ o≤-refl (o<→≤ <-osuc) ) (sym (sf1=sf u<x ))
zm05 : supfz (subst (λ k → u o< k) z=x u<b) ≡ u
zm05 = trans zm06 su=u
zm01 : { s : Ordinal } → odef A s → ( {x : Ordinal } → odef (UnionCF A f ay supf1 z) x → x ≤ s ) → supf1 z o≤ s
zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=spu (sym z=x))) ( MinSUP.minsup usup as zm02 ) where
zm02 : {w : Ordinal } → odef pchainU w → w ≤ s
zm02 {w} uw with pchainU⊆chain uw
... | ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫
... | ⟪ az , ch-is-sup u1 u<b su=u fc ⟫ = sup ⟪ az , ch-is-sup u1 (ordtrans u<b zm05) (trans zm03 su=u) zm04 ⟫ where
zm05 : osuc (IChain-i (proj2 uw)) o< z
zm05 = subst (λ k → osuc (IChain-i (proj2 uw)) o< k) (sym z=x) ( pic<x (proj2 uw) )
u<x : u1 o< x
u<x = subst (λ k → u1 o< k) z=x ( ordtrans u<b zm05 )
zm03 : supf1 u1 ≡ ZChain.supf (prev (osuc (IChain-i (proj2 uw))) (pic<x (proj2 uw))) u1
zm03 = trans (sf1=sf u<x) (zeq _ _ (osucc u<b) (o<→≤ <-osuc) )
zm04 : FClosure A f (supf1 u1) w
zm04 = subst (λ k → FClosure A f k w) (sym zm03) fc
... | case2 z<x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where
supf0 = ZChain.supf (pzc (ob<x lim z<x))
msup : IsMinSUP A (UnionCF A f ay supf0 z) (supf0 z)
msup = ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc)
s1=0 : {u : Ordinal } → u o< z → supf1 u ≡ supf0 u
s1=0 {u} u<z = trans (sf1=sf (ordtrans u<z z<x)) (zeq _ _ (o<→≤ (osucc u<z)) (o<→≤ <-osuc) )
zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z
zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x)) ( IsMinSUP.x≤sup msup ⟪ az , ch-init fc ⟫ )
zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x))
( IsMinSUP.x≤sup msup ⟪ az , ch-is-sup u u<b (trans (sym (s1=0 u<b)) su=u) (subst (λ k → FClosure A f k w) (s1=0 u<b) fc) ⟫ )
zm01 : { s : Ordinal } → odef A s → ( {x : Ordinal } → odef (UnionCF A f ay supf1 z) x → x ≤ s ) → supf1 z o≤ s
zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup msup as zm02 ) where
zm02 : {w : Ordinal } → odef (UnionCF A f ay supf0 z) w → w ≤ s
zm02 {w} ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫
zm02 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = sup
⟪ az , ch-is-sup u u<b (trans (s1=0 u<b) su=u) (subst (λ k → FClosure A f k w) (sym (s1=0 u<b)) fc) ⟫
cfcs : {a b w : Ordinal } → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w
cfcs {a} {b} {w} a<b b≤x sa<b fc with osuc-≡< b≤x
... | case1 b=x with trio< a x
... | tri< a<x ¬b ¬c = zc40 where
sa = ZChain.supf (pzc (ob<x lim a<x)) a
m = omax a sa
m<x : m o< x
m<x with trio< a sa
... | tri< a<sa ¬b ¬c = ob<x lim (ordtrans<-≤ sa<b b≤x )
... | tri≈ ¬a a=sa ¬c = zc41 where
zc41 : osuc a o< x
zc41 = osucprev ( begin
osuc ( osuc a ) ≤⟨ o<→≤ (ob<x lim (ob<x lim a<x)) ⟩
x ∎ ) where open o≤-Reasoning
... | tri> ¬a ¬b c = ob<x lim a<x
sam = ZChain.supf (pzc (ob<x lim m<x)) a
zc42 : osuc a o≤ osuc m
zc42 = osucc (o<→≤ ( omax-x _ _ ) )
sam<m : sam o< m
sam<m = subst (λ k → k o< m ) (supf-unique A f mf< ay zc42 (pzc (ob<x lim a<x)) (pzc (ob<x lim m<x)) (o<→≤ <-osuc)) ( omax-y _ _ )
fcm : FClosure A f (ZChain.supf (pzc (ob<x lim m<x)) a) w
fcm = subst (λ k → FClosure A f k w ) (zeq (ob<x lim a<x) (ob<x lim m<x) zc42 (o<→≤ <-osuc) ) fc
zcm : odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim m<x))) (osuc (omax a sa))) w
zcm = ZChain.cfcs (pzc (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm
zc40 : odef (UnionCF A f ay supf1 b) w
zc40 with ZChain.cfcs (pzc (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm
... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans zc45 su=u) zc44 ⟫ where
u<b : u o< b
u<b = osucprev ( begin
osuc u ≤⟨ osucc u<x ⟩
osuc m ≤⟨ osucc m<x ⟩
x ≡⟨ sym b=x ⟩
b ∎ ) where open o≤-Reasoning
zc45 : supf1 u ≡ ZChain.supf (pzc (ob<x lim m<x)) u
zc45 = begin
supf1 u ≡⟨ sf1=sf (subst (λ k → u o< k) b=x u<b ) ⟩
ZChain.supf (pzc (ob<x lim (subst (λ k → u o< k) b=x u<b ))) u ≡⟨ zeq _ _ (osucc u<x) (o<→≤ <-osuc) ⟩
ZChain.supf (pzc (ob<x lim m<x)) u ∎ where open ≡-Reasoning
zc44 : FClosure A f (supf1 u) w
zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc
... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x))
... | tri> ¬a ¬b c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x))
cfcs {a} {b} {w} a<b b≤x sa<b fc | case2 b<x = zc40 where
supfb = ZChain.supf (pzc (ob<x lim b<x))
sb=sa : {a : Ordinal } → a o< b → supf1 a ≡ ZChain.supf (pzc (ob<x lim b<x)) a
sb=sa {a} a<b = trans (sf1=sf (ordtrans<-≤ a<b b≤x)) (zeq _ _ (o<→≤ (osucc a<b)) (o<→≤ <-osuc) )
fcb : FClosure A f (supfb a) w
fcb = subst (λ k → FClosure A f k w) (sb=sa a<b) fc
zcb : odef (UnionCF A f ay supfb b) w
zcb = ZChain.cfcs (pzc (ob<x lim b<x)) a<b (o<→≤ <-osuc) (subst (λ k → k o< b) (sb=sa a<b) sa<b) fcb
zc40 : odef (UnionCF A f ay supf1 b) w
zc40 with zcb
... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<x (trans zc45 su=u) zc44 ⟫ where
zc45 : supf1 u ≡ ZChain.supf (pzc (ob<x lim b<x)) u
zc45 = begin
supf1 u ≡⟨ sf1=sf (ordtrans u<x b<x) ⟩
ZChain.supf (pzc (ob<x lim (ordtrans u<x b<x) )) u ≡⟨ zeq _ _ (o<→≤ (osucc u<x)) (o<→≤ <-osuc) ⟩
ZChain.supf (pzc (ob<x lim b<x )) u ∎ where open ≡-Reasoning
zc44 : FClosure A f (supf1 u) w
zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc
zo≤sz : {z : Ordinal} → z o≤ x → z o≤ supf1 z
zo≤sz {z} z≤x with osuc-≡< z≤x
... | case2 z<x = subst (λ k → z o≤ k) (sym (trans (sf1=sf z<x) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl)))) ( ZChain.zo≤sz (pzc z<x) o≤-refl )
... | case1 refl with x<y∨y≤x (supf1 spu) x
... | case2 x≤ssp = z40 where
z40 : z o≤ supf1 z
z40 with x<y∨y≤x z spu
... | case1 z<spu = o<→≤ ( subst (λ k → z o< k ) (sym (sf1=spu refl)) z<spu )
... | case2 spu≤z = begin
x ≤⟨ x≤ssp ⟩
supf1 spu ≤⟨ supf-mono spu≤z ⟩
supf1 x ∎ where open o≤-Reasoning
... | case1 ssp<x = subst (λ k → x o≤ k) (sym (sf1=spu refl)) z47 where
z47 : x o≤ spu
z47 with x<y∨y≤x spu x
... | case2 lt = lt
... | case1 spu<x = ⊥-elim ( <<-irr (MinSUP.x≤sup usup z48) (proj1 ( mf< spu (MinSUP.as usup)))) where
z70 : odef (UnionCF A f ay supf1 z) (supf1 spu)
z70 = cfcs spu<x o≤-refl ssp<x (init asupf refl )
z73 : IsSUP A (UnionCF A f ay (ZChain.supf (pzc (ob<x lim spu<x))) spu) spu
z73 = record { ax = MinSUP.as usup ; x≤sup = λ uw → MinSUP.x≤sup usup (chain⊆pchainU spu<x uw ) }
z49 : supfz spu<x ≡ spu
z49 = begin
supfz spu<x ≡⟨ ZChain.sup=u (pzc (ob<x lim spu<x)) (MinSUP.as usup) (o<→≤ <-osuc) z73 ⟩
spu ∎ where open ≡-Reasoning
z50 : supfz spu<x o≤ spu
z50 = o≤-refl0 z49
z48 : odef pchainU (f spu)
z48 = ⟪ proj2 (mf _ (MinSUP.as usup) ) , ic-isup _ (subst (λ k → k o< x) refl spu<x) z50
(fsuc _ (init (ZChain.asupf (pzc (ob<x lim spu<x))) z49)) ⟫
SZ : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf< ay x
SZ f mf< {y} ay x = TransFinite {λ z → ZChain A f mf< ay z } (λ x → ind f mf< ay x ) x
msp0 : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {x y : Ordinal} (ay : odef A y)
→ (zc : ZChain A f mf< ay x )
→ MinSUP A (UnionCF A f ay (ZChain.supf zc) x)
msp0 f mf< {x} ay zc = minsupP (UnionCF A f ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc)
fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (mf< : <-monotonic-f A f ) (zc : ZChain A f mf< as (& A) )
→ (sp1 : MinSUP A (ZChain.chain zc))
→ f (MinSUP.sup sp1) ≡ MinSUP.sup sp1
fixpoint f mf mf< zc sp1 = z14 where
chain = ZChain.chain zc
supf = ZChain.supf zc
sp : Ordinal
sp = MinSUP.sup sp1
asp : odef A sp
asp = MinSUP.as sp1
ay = as
z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< (& A) → (ab : odef A b )
→ HasPrev A chain f b ∨ IsSUP A (UnionCF A f ay (ZChain.supf zc) b) b
→ * a < * b → odef chain b
z10 = ZChain1.is-max (SZ1 f mf mf< as zc (& A) o≤-refl )
z22 : sp o< & A
z22 = odef< asp
z12 : odef chain sp
z12 with o≡? (& s) sp
... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc )
... | no ne = ZChain1.is-max (SZ1 f mf mf< as zc (& A) o≤-refl) {& s} {sp} ( ZChain.chain∋init zc ) (odef< asp) asp (case2 z19 ) z13 where
z13 : * (& s) < * sp
z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc )
... | case1 eq = ⊥-elim ( ne eq )
... | case2 lt = lt
z19 : IsSUP A (UnionCF A f ay (ZChain.supf zc) sp) sp
z19 = record { ax = asp ; x≤sup = z20 } where
z20 : {y : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp)
z20 {y} zy with MinSUP.x≤sup sp1
(subst (λ k → odef chain k ) (sym &iso) (chain-mono f mf as supf (ZChain.supf-mono zc) (o<→≤ z22) zy ))
... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p )
... | case2 y<p = case2 (subst (λ k → * k < _ ) &iso y<p )
z14 : f sp ≡ sp
z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) (subst (λ k → odef chain k) (sym &iso) z12 )
... | tri< a ¬b ¬c = ⊥-elim z16 where
z16 : ⊥
z16 with proj1 (mf (( MinSUP.sup sp1)) ( MinSUP.as sp1 ))
... | case1 eq = ⊥-elim (¬b (sym (trans &iso (trans eq (sym &iso) ))))
... | case2 lt = ⊥-elim (¬c (<-cong (==-sym *iso) (==-sym *iso) lt) )
... | tri≈ ¬a b ¬c = trans (sym &iso) (trans b &iso )
... | tri> ¬a ¬b c = ⊥-elim z17 where
z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) )
z15 = MinSUP.x≤sup sp1 (ZChain.f-next zc z12 )
z17 : ⊥
z17 with z15
... | case1 eq = ¬b (trans &iso (trans eq (sym &iso)))
... | case2 lt = ¬a (<-cong (==-sym *iso) (==-sym *iso) lt )
¬Maximal→¬cf-mono : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-<-monotonic nmx) as (& A)) → ⊥
¬Maximal→¬cf-mono nmx zc = <-irr {cf nmx c} {c}
(case1 ( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) (cf-is-<-monotonic nmx ) zc msp1 ))
(proj1 (cf-is-<-monotonic nmx c (MinSUP.as msp1))) where
supf = ZChain.supf zc
msp1 : MinSUP A (ZChain.chain zc)
msp1 = msp0 (cf nmx) (cf-is-<-monotonic nmx) as zc
c : Ordinal
c = MinSUP.sup msp1
zorn00 : Maximal A
zorn00 with is-o∅ ( & HasMaximal )
... | no not = record { maximal = minimal HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where
zorn03 : odef HasMaximal ( & ( minimal HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) )
zorn03 = x∋minimal HasMaximal (λ eq → not (=od∅→≡o∅ eq))
zorn01 : A ∋ minimal HasMaximal (λ eq → not (=od∅→≡o∅ eq))
zorn01 = proj1 zorn03
zorn02 : {x : HOD} → A ∋ x → ¬ (minimal HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (<-cong (==-sym *iso) (==-sym *iso) m<x )
... | yes ¬Maximal = ⊥-elim ( ¬Maximal→¬cf-mono nmx (SZ (cf nmx) (cf-is-<-monotonic nmx) as (& A) )) where
nmx : ¬ Maximal A
nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where
zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y))
zc5 = ⟪ Maximal.as mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (<-cong *iso ==-refl mx<y) ) ⟫