Mercurial > hg > Members > kono > Proof > ZF-in-agda
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sat, 26 Nov 2022 16:34:38 +0900 |
| parents | 8b3d7c461a84 |
| children | d1eecfc6cdfa |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level hiding ( suc ; zero ) open import Ordinals open import Relation.Binary open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality import OD module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where -- -- Zorn-lemma : { A : HOD } -- → o∅ o< & A -- → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B ) -- SUP condition -- → Maximal A -- open import zf open import logic -- open import partfunc {n} O open import Relation.Nullary open import Data.Empty import BAlgbra open import Data.Nat hiding ( _<_ ; _≤_ ) open import Data.Nat.Properties open import nat open inOrdinal O open OD O open OD.OD open ODAxiom odAxiom import OrdUtil import ODUtil open Ordinals.Ordinals O open Ordinals.IsOrdinals isOrdinal open Ordinals.IsNext isNext open OrdUtil O open ODUtil O import ODC open _∧_ open _∨_ open Bool open HOD -- -- Partial Order on HOD ( possibly limited in A ) -- _<<_ : (x y : Ordinal ) → Set n x << y = * x < * y _<=_ : (x y : Ordinal ) → Set n -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain 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 } } _≤_ : (x y : HOD) → Set (Level.suc n) x ≤ y = ( x ≡ y ) ∨ ( x < y ) ≤-ftrans : {x y z : HOD} → 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 ) <=-trans : {x y z : Ordinal } → x <= y → y <= z → x <= z <=-trans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl <=-trans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z <=-trans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y <=-trans {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 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y <=to≤ (case1 eq) = case1 (cong (*) eq) <=to≤ (case2 lt) = case2 lt ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) ≤to<= (case2 lt) = case2 lt <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (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 open _==_ open _⊆_ -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x -- <-TransFinite = ? -- -- Closure of ≤-monotonic function f has total order -- ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc 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 = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} 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 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 = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) -- ... | case2 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 = subst (λ k → * s ≡ k ) y=fy (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y ) 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 = subst (λ k → k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) i=x i=y ) 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 = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy i=x i=y ) 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 = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy i=x j=y ) ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) 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 ) -- derefence while f x ≡ x ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) 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 = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) 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 = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) 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 = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (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 ) 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 -- open import Relation.Binary.Properties.Poset as Poset IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) IsTotalOrderSet A = {a b : HOD} → 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 (incl B⊆A ax) (incl B⊆A ay) _⊆'_ : ( A B : HOD ) → Set n _⊆'_ A B = {x : Ordinal } → odef A x → odef B x -- -- inductive maxmum tree from x -- tree structure -- 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 } (xa : odef A x) : Set n where field x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) record IsMinSUP (A B : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (xa : odef A x) : Set n where field x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) minsup : { sup1 : Ordinal } → odef A sup1 → ( {z : Ordinal } → odef B z → (z ≡ sup1 ) ∨ (z << sup1 )) → x o≤ sup1 not-hp : ¬ ( HasPrev A B f x ) record SUP ( A B : HOD ) : Set (Level.suc n) where field sup : HOD as : A ∋ sup x≤sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive -- -- sup and its fclosure is in a chain HOD -- chain HOD is sorted by sup as Ordinal and <-ordered -- whole chain is a union of separated Chain -- minimum index is sup of y not ϕ -- record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where field fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) supu=u : supf u ≡ u data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : u o< x) ( is-sup : ChainP A f mf ay supf u ) ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z -- -- f (f ( ... (supf y))) f (f ( ... (supf z1))) -- / | / | -- / | / | -- supf y < supf z1 < supf z2 -- o< o< -- -- if sup z1 ≡ sup z2, the chain is stopped at sup z1, then f (sup z1) ≡ sup z1 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 (λ k → * (f k) ) (sym a=b))) (ftrans<-<= lt (≤to<= fc00 )) ) where fc00 : * b ≤ * (f b) fc00 = proj1 (mf _ (subst (λ k → odef A k) a=b aa )) -- -- data UChain is total chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) {s s1 a b : Ordinal } ( ca : UChain A f mf ay supf s a ) ( cb : UChain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) with ChainP.fcy<sup supb fca ... | case1 eq with s≤fc (supf ub) f mf fcb ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * a ≡ * b ct00 = trans (cong (*) eq) eq1 ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where ct01 : * a < * b ct01 = subst (λ k → * k < * b ) (sym eq) lt ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where ct00 : * a < * (supf ub) ct00 = lt ct01 : * a < * b ct01 with s≤fc (supf ub) f mf fcb ... | case1 eq = subst (λ k → * a < k ) eq ct00 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb ... | case1 eq with s≤fc (supf ua) f mf fca ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * a ≡ * b ct00 = sym (trans (cong (*) eq) eq1 ) ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where ct01 : * b < * a ct01 = subst (λ k → * k < * a ) (sym eq) lt ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where ct00 : * b < * (supf ua) ct00 = lt ct01 : * b < * a ct01 with s≤fc (supf ua) f mf fca ... | case1 eq = subst (λ k → * b < k ) eq ct00 ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁) fca ... | case1 eq with s≤fc (supf ub) f mf fcb ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * a ≡ * b ct00 = trans (cong (*) eq) eq1 ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where ct02 : * a < * b ct02 = subst (λ k → * k < * b ) (sym eq) lt ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where ct03 : * a < * (supf ub) ct03 = lt ct02 : * a < * b ct02 with s≤fc (supf ub) f mf fcb ... | case1 eq = subst (λ k → * a < k ) eq ct03 ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri≈ ¬a eq ¬c = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) fcb ... | case1 eq with s≤fc (supf ua) f mf fca ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * a ≡ * b ct00 = sym (trans (cong (*) eq) eq1) ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where ct02 : * b < * a ct02 = subst (λ k → * k < * a ) (sym eq) lt ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where ct05 : * b < * (supf ua) ct05 = lt ct04 : * b < * a ct04 with s≤fc (supf ua) f mf fca ... | case1 eq = subst (λ k → * b < k ) eq ct05 ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt ∈∧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 ))) -- Union of supf z which o< x -- UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD UnionCF A f mf ay supf x = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x 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 MinSUP ( A B : HOD ) : Set n where field sup : Ordinal asm : 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 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } → (supf : Ordinal → Ordinal ) → MinSUP A (UnionCF A f mf ay supf x) → SUP A (UnionCF A f mf ay supf x) M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x≤sup = ms00 } where msup = MinSUP.sup ms ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup) ms00 {z} uz with MinSUP.x≤sup ms uz ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq)) ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso refl lt ) 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 mf ay supf a) c → odef (UnionCF A f mf 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 ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x a≤b) is-sup 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 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z → IsSUP A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) f b ) → supf b ≡ b cfcs : (mf< : <-monotonic-f A f) {a b w : Ordinal } → a o< b → b o≤ z → supf a o< b → FClosure A f (supf a) w → odef (UnionCF A f mf ay supf b) w asupf : {x : Ordinal } → odef A (supf x) supf-<= : {x y w : Ordinal } → x o< y → FClosure A f (supf x) w → w <= supf y supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z ) chain : HOD chain = UnionCF A f mf ay supf z chain⊆A : chain ⊆' A chain⊆A = λ lt → proj1 lt sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) sup {x} x≤z = M→S supf (minsup x≤z) s=ms : {x : Ordinal } → (x≤z : x o≤ z ) → & (SUP.sup (sup x≤z)) ≡ MinSUP.sup (minsup x≤z) s=ms {x} x≤z = &iso chain∋init : odef chain y chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a) f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ f-next {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u<x is-sup (fsuc _ fc ) ⟫ initial : {z : Ordinal } → odef chain z → * y ≤ * z initial {a} ⟪ aa , ua ⟫ with ua ... | ch-init fc = s≤fc y f mf fc ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where zc7 : y <= supf u zc7 = ChainP.fcy<sup is-sup (init ay refl) f-total : IsTotalOrderSet chain f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) 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 ) supf<A : {x : Ordinal } → supf x o< & A supf<A = z09 asupf csupf : (mf< : <-monotonic-f A f) {b : Ordinal } → supf b o< supf z → supf b o< z → odef (UnionCF A f mf ay supf z) (supf b) -- supf z is not an element of this chain csupf mf< {b} sb<sz sb<z = cfcs mf< (supf-inject sb<sz) o≤-refl sb<z (init asupf refl) fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf 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 ) -- ordering is not proved here but in ZChain1 IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp → ({y : Ordinal} → odef (UnionCF A f mf ay supf x) y → (y ≡ sp ) ∨ (y << sp )) → ( {a : Ordinal } → odef A a → a << f a ) → ¬ ( HasPrev A (UnionCF A f mf ay supf x) f sp ) IsMinSUP→NotHasPrev {x} {sp} asp is-sup <-mono-f hp = ⊥-elim (<-irr ( <=to≤ fsp≤sp) sp<fsp ) where sp<fsp : sp << f sp sp<fsp = <-mono-f asp pr = HasPrev.y hp im00 : f (f pr) <= sp im00 = is-sup ( f-next (f-next (HasPrev.ay hp))) fsp≤sp : f sp <= sp fsp≤sp = subst (λ k → f k <= sp ) (sym (HasPrev.x=fy hp)) im00 supf-¬hp : {x : Ordinal } → x o≤ z → ( {a : Ordinal } → odef A a → a << f a ) → ¬ ( HasPrev A (UnionCF A f mf ay supf x) f (supf x) ) supf-¬hp {x} x≤z <-mono hp = IsMinSUP→NotHasPrev asupf (λ {w} uw → (subst (λ k → w <= k) (sym (supf-is-minsup x≤z)) ( MinSUP.x≤sup (minsup x≤z) uw) )) <-mono hp supf-idem : (mf< : <-monotonic-f A f) {b : Ordinal } → b o≤ z → supf b o≤ z → supf (supf b) ≡ supf b supf-idem mf< {b} b≤z sfb≤x = z52 where z54 : {w : Ordinal} → odef (UnionCF A f mf 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 is-sup fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k )) (sym (supf-is-minsup b≤z)) (MinSUP.x≤sup (minsup b≤z) (cfcs mf< u<b b≤z (subst (λ k → k o< b) (sym (ChainP.supu=u is-sup)) u<b) fc )) where u<b : u o< b u<b = supf-inject ( subst (λ k → k o< supf b) (sym (ChainP.supu=u is-sup)) u<x ) z52 : supf (supf b) ≡ supf b z52 = sup=u asupf sfb≤x ⟪ record { x≤sup = z54 } , IsMinSUP→NotHasPrev asupf z54 ( λ ax → proj1 (mf< _ ax)) ⟫ -- cp : (mf< : <-monotonic-f A f) {b : Ordinal } → b o≤ z → supf b o≤ z → ChainP A f mf ay supf (supf b) -- the condition of cfcs is satisfied, this is obvious 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 mf ay supf z) a ) → b o< z → (ab : odef A b) → HasPrev A (UnionCF A f mf ay supf z) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → * a < * b → odef ((UnionCF A f mf 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 -- A is Partial, use negative init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ record IChain (A : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) (z : Ordinal ) : Set n where field i : Ordinal i<x : i o< x fc : FClosure A f (supfz i<x) z -- -- supf in TransFinite indution may differ each other, but it is the same because of the minimul sup -- 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.asm ma) (λ {z} uzb → MinSUP.x≤sup ma (z53 uzb)) ⟩ spa ≡⟨ sym sax=spa ⟩ supfa x ∎ ) a ) where open o≤-Reasoning O z53 : {z : Ordinal } → odef (UnionCF A f mf ay (ZChain.supf zb) x) z → odef (UnionCF A f mf ay (ZChain.supf za) x) z z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ z53 {z} ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u u<x z54 z55 ⟫ where ua=ub : supfa u ≡ supfb u ua=ub = prev u u<x (ordtrans u<x x≤xa ) order : {s z1 : Ordinal} → ZChain.supf za s o< ZChain.supf za u → FClosure A f (ZChain.supf za s) z1 → (z1 ≡ ZChain.supf za u) ∨ (z1 << ZChain.supf za u) order {s} {z1} lt fc = subst (λ k → z1 <= k) (sym ua=ub) (ChainP.order is-sup (subst₂ ( λ j k → j o< k ) z56 ua=ub lt ) (subst (λ k → FClosure A f k z1 ) z56 fc )) where s<x : s o< x s<x = ordtrans (ZChain.supf-inject za lt) u<x z56 : supfa s ≡ supfb s z56 = prev s s<x (ordtrans s<x x≤xa) z54 : ChainP A f mf ay (ZChain.supf za) u z54 = record { fcy<sup = λ {w} fc → subst (λ k → w <= k ) (sym ua=ub) (ChainP.fcy<sup is-sup fc ) ; order = order ; supu=u = trans ua=ub (ChainP.supu=u is-sup) } 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.asm mb) (λ uza → MinSUP.x≤sup mb (z53 uza)) ⟩ spb ≡⟨ sym sbx=spb ⟩ supfb x ∎ ) c ) where open o≤-Reasoning O z53 : {z : Ordinal } → odef (UnionCF A f mf ay (ZChain.supf za) x) z → odef (UnionCF A f mf ay (ZChain.supf zb) x) z z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫ z53 {z} ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u u<x z54 z55 ⟫ where ub=ua : supfb u ≡ supfa u ub=ua = sym ( prev u u<x (ordtrans u<x x≤xa )) order : {s z1 : Ordinal} → ZChain.supf zb s o< ZChain.supf zb u → FClosure A f (ZChain.supf zb s) z1 → (z1 ≡ ZChain.supf zb u) ∨ (z1 << ZChain.supf zb u) order {s} {z1} lt fc = subst (λ k → z1 <= k) (sym ub=ua) (ChainP.order is-sup (subst₂ ( λ j k → j o< k ) z56 ub=ua lt ) (subst (λ k → FClosure A f k z1 ) z56 fc )) where s<x : s o< x s<x = ordtrans (ZChain.supf-inject zb lt) u<x z56 : supfb s ≡ supfa s z56 = sym (prev s s<x (ordtrans s<x x≤xa)) z54 : ChainP A f mf ay (ZChain.supf zb) u z54 = record { fcy<sup = λ {w} fc → subst (λ k → w <= k ) (sym ub=ua) (ChainP.fcy<sup is-sup fc ) ; order = order ; supu=u = trans ub=ua (ChainP.supu=u is-sup) } 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 ) -- SUP condition → Maximal A Zorn-lemma {A} 0<A supP = zorn00 where <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ <-irr0 {a} {b} A∋a A∋b = <-irr 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 = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) as : A ∋ * ( & s ) as = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) as0 : odef A (& s ) as0 = subst (λ k → odef A k ) &iso as s<A : & s o< & A s<A = c<→o< (subst (λ k → odef A (& k) ) *iso 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 → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; 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) -- -- we have minsup using LEM, this is similar to the proof of the axiom of choice -- 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 ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM ∀-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 ODC.p∨¬p O 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 ODC.p∨¬p O (odef A z ∧ xsup z) ... | case1 mins = case2 record { sup = z ; asm = 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 (z09 (SUP.as S)) ⟪ SUP.as 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 {* w} (subst (λ k → odef B k) (sym &iso) bw ) ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) m02 : MinSUP A B m02 = dont-or (m00 (& A)) m03 -- Uncountable ascending chain by axiom of choice cf : ¬ Maximal A → Ordinal → Ordinal cf nmx x with ODC.∋-p O A (* x) ... | no _ = o∅ ... | yes ax with is-o∅ (& ( Gtx ax )) ... | yes nogt = -- no larger element, so it is maximal ⊥-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 = & (ODC.minimal O (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 ODC.∋-p O 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 = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)) --- --- infintie ascention sequence of f --- 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 ) ⟫ -- -- maximality of chain -- -- supf is fixed for z ≡ & A , we can prove order and is-max -- we have supf-unique now, it is provable in the first Tranfinte induction 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 mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf 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 mf ay (ZChain.supf zc) x) a → (ab : odef A b) → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) f b → * a < * b → odef (UnionCF A f mf 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 is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay (ZChain.supf zc) x k ) (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (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 mf ay supf x) a → b o< x → (ab : odef A b) → HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → * a < * b → odef (UnionCF A f mf ay supf x) b is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O 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 = z09 ab m04 : ¬ HasPrev A (UnionCF A f mf 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<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ m10 : odef (UnionCF A f mf ay supf x) b m10 = ZChain.cfcs zc mf< 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 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where m17 : MinSUP A (UnionCF A f mf ay supf x) -- supf z o< supf ( 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 mf 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<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ m14 : ZChain.supf zc b o< x m14 = subst (λ k → k o< x ) (sym m05) b<x m13 : odef (UnionCF A f mf ay supf x) z m13 = ZChain.cfcs zc mf< b<x x≤A m14 fc ... | no lim = record { is-max = is-max } where is-max : {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a → b o< x → (ab : odef A b) → HasPrev A (UnionCF A f mf ay supf x) f b ∨ IsSUP A (UnionCF A f mf ay supf b) ab → * a < * b → odef (UnionCF A f mf ay supf x) b is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O 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) (init-uchain A f mf ay ) ... | 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)) m04 : ¬ HasPrev A (UnionCF A f mf 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<→≤ m09) ⟪ record { x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } , m04 ⟫ -- ZChain on x m10 : odef (UnionCF A f mf ay supf x) b m10 = ZChain.cfcs zc mf< 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 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where m17 : MinSUP A (UnionCF A f mf ay supf x) -- supf z o< supf ( 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 mf 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<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz } , m04 ⟫ m14 : ZChain.supf zc b o< x m14 = subst (λ k → k o< x ) (sym m05) b<x m13 : odef (UnionCF A f mf ay supf x) z m13 = ZChain.cfcs zc mf< 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 = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) uz01 = 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) SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x≤sup = λ lt → SUP.x≤sup sup (B⊆C lt) } record xSUP (B : HOD) (f : Ordinal → Ordinal ) (x : Ordinal) : Set n where field ax : odef A x is-sup : IsMinSUP A B f ax zc43 : (x sp1 : Ordinal ) → ( x o< sp1 ) ∨ ( sp1 o≤ x ) zc43 x sp1 with trio< x sp1 ... | tri< a ¬b ¬c = case1 a ... | tri≈ ¬a b ¬c = case2 (o≤-refl0 (sym b)) ... | tri> ¬a ¬b c = case2 (o<→≤ c) -- -- create all ZChains under o< x -- 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 where -- -- we have previous ordinal to use induction -- 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 mf ay supf0 px supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b 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 ⟫ ) -- -- find the next value of supf -- pchainpx : HOD pchainpx = record { od = record { def = λ z → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z } ; odmax = & A ; <odmax = zc00 } where zc00 : {z : Ordinal } → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z → z o< & A zc00 {z} (case1 lt) = z07 lt zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf fc ) zc02 : { a b : Ordinal } → odef A a ∧ UChain A f mf ay supf0 px a → FClosure A f (supf0 px) b → a <= b zc02 {a} {b} ca fb = zc05 fb where zc06 : MinSUP.sup (ZChain.minsup zc o≤-refl) ≡ supf0 px zc06 = trans (sym ( ZChain.supf-is-minsup zc o≤-refl )) refl 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 ) (subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) eq)) (zc05 fb) ... | case2 lt = <=-trans (zc05 fb) (case2 lt) zc05 (init b1 refl) with MinSUP.x≤sup (ZChain.minsup zc o≤-refl) (subst (λ k → odef A k ∧ UChain A f mf ay supf0 px k) (sym &iso) ca ) ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso zc06 eq ) ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso (cong (*) zc06) lt ) ptotal : IsTotalOrderSet pchainpx ptotal (case1 a) (case1 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (chain-total A f mf ay supf0 (proj2 a) (proj2 b)) ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where eq1 : a0 ≡ b0 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where lt1 : a0 < b0 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b ... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where eq1 : a0 ≡ b0 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq ) ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1 where lt1 : a0 < b0 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt ptotal (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp (supf0 px) f mf a b) pcha : pchainpx ⊆' A pcha (case1 lt) = proj1 lt pcha (case2 fc) = A∋fc _ f mf fc sup1 : MinSUP A pchainpx sup1 = minsupP pchainpx pcha ptotal sp1 = MinSUP.sup sup1 sfpx<=sp1 : supf0 px <= sp1 sfpx<=sp1 = MinSUP.x≤sup sup1 (case2 (init (ZChain.asupf zc {px}) refl )) sfpx≤sp1 : supf0 px o≤ sp1 sfpx≤sp1 = subst ( λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc o≤-refl )) ( MinSUP.minsup (ZChain.minsup zc o≤-refl) (MinSUP.asm sup1) (λ {x} ux → MinSUP.x≤sup sup1 (case1 ux)) ) -- -- supf0 px o≤ sp1 -- zc41 : ZChain A f mf ay x zc41 with zc43 x sp1 zc41 | (case2 sp≤x ) = record { supf = supf1 ; sup=u = ? ; asupf = asupf1 ; supf-mono = supf1-mono ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; cfcs = cfcs } where 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.asm 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 mf ay supf0 a) zc21 = ZChain.minsup zc (o<→≤ a<px) zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf 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.asm sup1) zc24 ) ... | tri≈ ¬a b ¬c = zc18 where zc21 : MinSUP A (UnionCF A f mf 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 mf 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.asm sup1) zc24 ) ... | tri> ¬a ¬b c = o≤-refl sf≤ : { z : Ordinal } → x o≤ z → supf0 x o≤ supf1 z sf≤ {z} x≤z with trio< z px ... | tri< a ¬b ¬c = ⊥-elim ( o<> (osucc a) (subst (λ k → k o≤ z) (sym (Oprev.oprev=x op)) x≤z ) ) ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) px<x )) ... | tri> ¬a ¬b c = subst₂ (λ j k → j o≤ k ) (trans (sf1=sf0 o≤-refl ) (sym (ZChain.supfmax zc px<x))) (sf1=sp1 c) (supf1-mono (o<→≤ c )) -- px o<z → supf x ≡ supf0 px ≡ supf1 px o≤ supf1 z 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 -- this is a kind of maximality, so we cannot prove this without <-monotonicity -- cfcs : (mf< : <-monotonic-f A f) {a b w : Ordinal } → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f mf ay supf1 b) w cfcs mf< {a} {b} {w} a<b b≤x sa<b fc with zc43 (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) -- supf0 a ≡ px we cannot use previous cfcs, it is in the chain because -- supf0 a ≡ supf0 (supf0 a) ≡ supf0 px o< x z50 : odef (UnionCF A f mf ay supf1 b) w z50 with osuc-≡< px≤sa ... | case1 px=sa = ⟪ A∋fc {A} _ f mf fc , ch-is-sup (supf0 px) z51 cp (subst (λ k → FClosure A f k w) z52 fc) ⟫ 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 mf< 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 mf< 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 m : MinSUP A (UnionCF A f mf ay supf0 px) m = ZChain.minsup zc o≤-refl m=spx : MinSUP.sup m ≡ supf1 (supf0 px) m=spx = begin MinSUP.sup m ≡⟨ sym ( ZChain.supf-is-minsup zc o≤-refl) ⟩ supf0 px ≡⟨ cong supf0 px=sa ⟩ 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 : ChainP A f mf ay supf1 (supf0 px) cp = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) m=spx (MinSUP.x≤sup m ⟪ A∋fc _ f mf fc , ch-init fc ⟫ ) ; order = order ; supu=u = z53 } where uz : {s z1 : Ordinal } → supf1 s o< supf1 (supf0 px) → FClosure A f (supf1 s) z1 → odef (UnionCF A f mf ay supf0 px) z1 uz {s} {z1} ss<sp fc = ZChain.cfcs zc mf< s<px o≤-refl ss<px (subst (λ k → FClosure A f k z1) (sf1=sf0 (o<→≤ s<px)) fc ) where s<spx : s o< supf0 px s<spx = supf-inject0 supf1-mono ss<sp s<px : s o< px s<px = osucprev ( begin osuc s ≤⟨ osucc s<spx ⟩ supf0 px ≡⟨ spx=sa ⟩ supf0 a ≡⟨ sym px=sa ⟩ px ∎ ) where open o≤-Reasoning O ss<px : supf0 s o< px ss<px = osucprev ( begin osuc (supf0 s) ≡⟨ cong osuc (sym (sf1=sf0 (o<→≤ s<px))) ⟩ osuc (supf1 s) ≤⟨ osucc ss<sp ⟩ supf1 (supf0 px) ≡⟨ sym z52 ⟩ supf1 a ≡⟨ sf1=sf0 a≤px ⟩ supf0 a ≡⟨ sym px=sa ⟩ px ∎ ) where open o≤-Reasoning O order : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 (supf0 px) → FClosure A f (supf1 s) z1 → (z1 ≡ supf1 (supf0 px)) ∨ (z1 << supf1 (supf0 px)) order {s} {z} s<u fc = subst (λ k → (z ≡ k) ∨ ( z << k ) ) m=spx (MinSUP.x≤sup m (uz s<u 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 mf ay supf1 b) w z50 with osuc-≡< b≤x ... | case2 lt with ZChain.cfcs zc mf< 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 is-sup fc ⟫ = ⟪ az , ch-is-sup u u<b cp1 (fcpu fc u≤px ) ⟫ where -- u o< px → u o< b ? 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 cp1 : ChainP A f mf ay supf1 u cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sf=eq u<x) (ChainP.fcy<sup is-sup fc ) ; order = λ {s} {z} s<u fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sf=eq u<x) (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (sf=eq (ordtrans (supf-inject0 supf1-mono s<u) u<x) )) (sym (sf=eq u<x)) s<u) (subst (λ k → FClosure A f k z ) (sym (sf=eq (ordtrans (supf-inject0 supf1-mono s<u) u<x) )) fc )) ; supu=u = trans (sym (sf=eq u<x)) (ChainP.supu=u is-sup) } z50 | case1 eq with ZChain.cfcs zc mf< a<px o≤-refl sa<px fc ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ ... | ⟪ az , ch-is-sup u u<px is-sup fc ⟫ = ⟪ az , ch-is-sup u u<b cp1 (fcpu fc (o<→≤ u<px)) ⟫ where -- u o< px → u o< b ? 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 ) cp1 : ChainP A f mf ay supf1 u cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sf=eq u<x) (ChainP.fcy<sup is-sup fc ) ; order = λ {s} {z} s<u fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sf=eq u<x) (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (sf=eq (ordtrans (supf-inject0 supf1-mono s<u) u<x) )) (sym (sf=eq u<x)) s<u) (subst (λ k → FClosure A f k z ) (sym (sf=eq (ordtrans (supf-inject0 supf1-mono s<u) u<x) )) fc )) ; supu=u = trans (sym (sf=eq u<x)) (ChainP.supu=u is-sup) } ... | tri≈ ¬a a=px ¬c = csupf1 where -- a ≡ px , b ≡ x, sp o≤ x 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 ⟫ ) -- px o< b o< x ... | tri≈ ¬a b ¬c = b ... | tri> ¬a ¬b c = ⊥-elim ( o≤> b≤x c ) -- x o< b 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 mf ay supf1 b) w csupf1 with trio< (supf0 px) x ... | tri< sfpx<x ¬b ¬c = ⟪ z53 , ch-is-sup spx (subst (λ k → spx o< k) (sym b=x) sfpx<x) cp1 fc1 ⟫ where spx = supf0 px spx≤px : supf0 px o≤ px spx≤px = zc-b<x _ sfpx<x z52 : supf1 (supf0 px) ≡ supf0 px z52 = trans (sf1=sf0 (zc-b<x _ sfpx<x)) ( ZChain.supf-idem zc mf< o≤-refl (zc-b<x _ sfpx<x ) ) fc1 : FClosure A f (supf1 spx) w fc1 = subst (λ k → FClosure A f k w ) (trans (cong supf0 a=px) (sym z52) ) fc order : {s z1 : Ordinal} → supf1 s o< supf1 spx → FClosure A f (supf1 s) z1 → (z1 ≡ supf1 spx) ∨ (z1 << supf1 spx) order {s} {z1} ss<spx fcs = subst (λ k → (z1 ≡ k) ∨ (z1 << k )) (trans (sym (ZChain.supf-is-minsup zc spx≤px )) (sym (sf1=sf0 spx≤px) ) ) (MinSUP.x≤sup (ZChain.minsup zc spx≤px) (ZChain.cfcs zc mf< (supf-inject0 supf1-mono ss<spx) spx≤px ss0<spx (fcup fcs (ordtrans (supf-inject0 supf1-mono ss<spx) spx≤px ) ))) where ss0<spx : supf0 s o< spx ss0<spx = osucprev ( begin osuc (supf0 s) ≡⟨ cong osuc (sym (sf1=sf0 ( begin s <⟨ supf-inject0 supf1-mono ss<spx ⟩ supf0 px ≤⟨ spx≤px ⟩ px ∎ ) )) ⟩ osuc (supf1 s) ≤⟨ osucc ss<spx ⟩ supf1 spx ≡⟨ sf1=sf0 spx≤px ⟩ supf0 spx ≤⟨ ZChain.supf-mono zc spx≤px ⟩ supf0 px ∎ ) where open o≤-Reasoning O cp1 : ChainP A f mf ay supf1 spx cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ (z << k )) (sym (sf1=sf0 spx≤px )) ( ZChain.fcy<sup zc spx≤px fc ) ; order = order ; supu=u = z52 } ... | tri≈ ¬a spx=x ¬c = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf0 px) (ZChain.asupf zc)))) where -- supf px ≡ x then the chain is stopped, which cannot happen when <-monotonic case m12 : supf0 px ≡ sp1 m12 with osuc-≡< sfpx≤sp1 ... | case1 eq = eq ... | case2 lt = ⊥-elim ( o≤> sp≤x (subst (λ k → k o< sp1) spx=x lt )) -- supf0 px o< sp1 , x o< sp1 m10 : f (supf0 px) ≡ supf0 px m10 = fc-stop A f mf (ZChain.asupf zc) m11 m12 where m11 : {z : Ordinal} → FClosure A f (supf0 px) z → (z ≡ sp1) ∨ (z << sp1) m11 {z} fc = MinSUP.x≤sup sup1 (case2 fc) ... | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (ordtrans<-≤ c (OrdTrans sfpx≤sp1 sp≤x))) -- x o< supf0 px o≤ sp1 ≤ x ... | 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) ⟫ ) -- px o< a o< b o≤ x zc11 : {z : Ordinal} → odef (UnionCF A f mf 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 is-sup fc ⟫ = zc21 fc where u≤px : u o≤ px u≤px = zc-b<x _ u<x 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 u1-is-sup fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ ... | case2 fc = case2 (fsuc _ fc) zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) | inspect supf1 u ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u u<px record {fcy<sup = zc13 ; order = zc17 ; supu=u = trans (sym (sf1=sf0 (o<→≤ u<px))) (ChainP.supu=u is-sup) } (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 zc17 : {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 u → FClosure A f (supf0 s) z1 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u) zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) ((sf1=sf0 u≤px)) ( ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 zc18)) (sym (sf1=sf0 u≤px)) ss<spx) (fcpu fc zc18) ) where zc18 : s o≤ px zc18 = ordtrans (ZChain.supf-inject zc ss<spx) u≤px zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf0 u) ∨ ( z << supf0 u ) zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sf1=sf0 (o<→≤ u<px)) ( ChainP.fcy<sup is-sup fc ) ... | tri≈ ¬a b ¬c | _ = case2 (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) (sym (trans (sf1=sf0 u≤px) b ))) ... | 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 ⟫ ) record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where field tsup : MinSUP A (UnionCF A f mf ay supf1 z) tsup=sup : supf1 z ≡ MinSUP.sup tsup sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x sup {z} z≤x with trio< z px ... | tri< a ¬b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o<→≤ a) ) (ZChain.supf-is-minsup zc (o<→≤ a)) } where m = ZChain.minsup zc (o<→≤ a) ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) ms00 {x} ux = MinSUP.x≤sup m ? ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 ms01 {sup2} us P = MinSUP.minsup m us ? ... | tri≈ ¬a b ¬c = record { tsup = record { sup = MinSUP.sup m ; asm = MinSUP.asm m ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o≤-refl0 b) ) (ZChain.supf-is-minsup zc (o≤-refl0 b)) } where m = ZChain.minsup zc (o≤-refl0 b) ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) ms00 {x} ux = MinSUP.x≤sup m ? ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 ms01 {sup2} us P = MinSUP.minsup m us ? ... | tri> ¬a ¬b px<z = record { tsup = record { sup = sp1 ; asm = MinSUP.asm sup1 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = sf1=sp1 px<z } where m = sup1 ms00 : {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m) ms00 {x} ux = MinSUP.x≤sup m ? ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2 ms01 {sup2} us P = MinSUP.minsup m us ? zc41 | (case1 x<sp ) = record { supf = supf0 ; sup=u = ? ; asupf = ZChain.asupf zc ; supf-mono = ZChain.supf-mono zc ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; cfcs = cfcs } where -- supf0 px not is included by the chain -- supf1 x ≡ supf0 px because of supfmax cfcs : (mf< : <-monotonic-f A f) {a b w : Ordinal } → a o< b → b o≤ x → supf0 a o< b → FClosure A f (supf0 a) w → odef (UnionCF A f mf ay supf0 b) w cfcs mf< {a} {b} {w} a<b b≤x sa<b fc with trio< b px ... | tri< a ¬b ¬c = ZChain.cfcs zc mf< a<b (o<→≤ a) sa<b fc ... | tri≈ ¬a refl ¬c = ZChain.cfcs zc mf< a<b o≤-refl sa<b fc ... | tri> ¬a ¬b px<b = cfcs1 where x=b : x ≡ b x=b with trio< x b ... | tri< a ¬b ¬c = ⊥-elim ( o≤> b≤x a ) ... | tri≈ ¬a b ¬c = b ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ px<b , zc-b<x _ c ⟫ ) -- px o< b o< x -- a o< x, supf a o< x -- a o< px , supf a o< px → odef U w -- a ≡ px -- supf0 px o< x → odef U w -- supf a ≡ px -- a o< px → odef U w -- a ≡ px → supf px ≡ px → odef U w cfcs0 : a ≡ px → odef (UnionCF A f mf ay supf0 b) w cfcs0 a=px = ⟪ A∋fc {A} _ f mf fc , ch-is-sup (supf0 px) spx<b cp fc1 ⟫ where spx<b : supf0 px o< b spx<b = subst (λ k → supf0 k o< b) a=px sa<b cs01 : supf0 a ≡ supf0 (supf0 px) cs01 = trans (cong supf0 a=px) ( sym ( ZChain.supf-idem zc mf< o≤-refl (subst (λ k → supf0 px o< k ) (sym (Oprev.oprev=x op)) (ordtrans<-≤ spx<b b≤x)))) fc1 : FClosure A f (supf0 (supf0 px)) w fc1 = subst (λ k → FClosure A f k w) cs01 fc m : MinSUP A (UnionCF A f mf ay supf0 (supf0 px)) m = ZChain.minsup zc (zc-b<x _ (ordtrans<-≤ spx<b b≤x)) m=sa : MinSUP.sup m ≡ supf0 (supf0 px) m=sa = begin MinSUP.sup m ≡⟨ sym ( ZChain.supf-is-minsup zc (zc-b<x _ (ordtrans<-≤ spx<b b≤x) )) ⟩ supf0 (supf0 px) ∎ where open ≡-Reasoning cp : ChainP A f mf ay supf0 (supf0 px) cp = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) m=sa (MinSUP.x≤sup m ⟪ A∋fc _ f mf fc , ch-init fc ⟫ ) ; order = order ; supu=u = sym (trans (cong supf0 (sym a=px)) cs01) } where uz : {s z1 : Ordinal } → supf0 s o< supf0 (supf0 px) → FClosure A f (supf0 s) z1 → odef (UnionCF A f mf ay supf0 (supf0 px)) z1 uz {s} {z1} ss<sp fc = ZChain.cfcs zc mf< s<spx spx≤px (subst (λ k → supf0 s o< k) (sym (trans (cong supf0 (sym a=px)) cs01) ) ss<sp) fc where s<spx : s o< supf0 px s<spx = ZChain.supf-inject zc ss<sp spx≤px : supf0 px o≤ px spx≤px = zc-b<x _ (ordtrans<-≤ spx<b b≤x) order : {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 (supf0 px) → FClosure A f (supf0 s) z1 → (z1 ≡ supf0 (supf0 px)) ∨ (z1 << supf0 (supf0 px)) order {s} {z} s<u fc = subst (λ k → (z ≡ k) ∨ ( z << k ) ) m=sa (MinSUP.x≤sup m (uz s<u fc) ) cfcs1 : odef (UnionCF A f mf ay supf0 b) w cfcs1 with trio< a px ... | tri< a<px ¬b ¬c = cfcs2 where sa<x : supf0 a o< x sa<x = ordtrans<-≤ sa<b b≤x cfcs2 : odef (UnionCF A f mf ay supf0 b) w cfcs2 with trio< (supf0 a) px ... | tri< sa<x ¬b ¬c = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) (o<→≤ px<b) ( ZChain.cfcs zc mf< a<px o≤-refl sa<x fc ) ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , (zc-b<x _ sa<x) ⟫ ) ... | tri≈ ¬a sa=px ¬c with trio< a px ... | tri< a<px ¬b ¬c = ⟪ A∋fc {A} _ f mf fc , ch-is-sup (supf0 a) sa<b cp fc1 ⟫ where cs01 : supf0 a ≡ supf0 (supf0 a) cs01 = sym ( ZChain.supf-idem zc mf< (zc-b<x _ (ordtrans<-≤ a<b b≤x)) (zc-b<x _ (ordtrans<-≤ sa<b b≤x))) fc1 : FClosure A f (supf0 (supf0 a)) w fc1 = subst (λ k → FClosure A f k w) cs01 fc m : MinSUP A (UnionCF A f mf ay supf0 (supf0 a)) m = ZChain.minsup zc (o≤-refl0 sa=px) m=sa : MinSUP.sup m ≡ supf0 (supf0 a) m=sa = begin MinSUP.sup m ≡⟨ sym ( ZChain.supf-is-minsup zc (o≤-refl0 sa=px) ) ⟩ supf0 (supf0 a) ∎ where open ≡-Reasoning cp : ChainP A f mf ay supf0 (supf0 a) cp = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) m=sa (MinSUP.x≤sup m ⟪ A∋fc _ f mf fc , ch-init fc ⟫ ) ; order = order ; supu=u = sym cs01 } where uz : {s z1 : Ordinal } → supf0 s o< supf0 (supf0 a) → FClosure A f (supf0 s) z1 → odef (UnionCF A f mf ay supf0 (supf0 a)) z1 uz {s} {z1} ss<sp fc = ZChain.cfcs zc mf< (ZChain.supf-inject zc ss<sp) (zc-b<x _ (ordtrans<-≤ sa<b b≤x)) ss<sa fc where ss<sa : supf0 s o< supf0 a ss<sa = subst (λ k → supf0 s o< k ) (sym cs01) ss<sp order : {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 (supf0 a) → FClosure A f (supf0 s) z1 → (z1 ≡ supf0 (supf0 a)) ∨ (z1 << supf0 (supf0 a)) order {s} {z} s<u fc = subst (λ k → (z ≡ k) ∨ ( z << k ) ) m=sa (MinSUP.x≤sup m (uz s<u fc) ) ... | tri≈ ¬a a=px ¬c = cfcs0 a=px ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , (zc-b<x _ (ordtrans<-≤ a<b b≤x) ) ⟫ ) ... | tri≈ ¬a a=px ¬c = cfcs0 a=px ... | tri> ¬a ¬b c = ⊥-elim ( o≤> (zc-b<x _ (ordtrans<-≤ a<b b≤x)) c ) zc17 : {z : Ordinal } → supf0 z o≤ supf0 px zc17 {z} with trio< z px ... | tri< a ¬b ¬c = ZChain.supf-mono zc (o<→≤ a) ... | tri≈ ¬a b ¬c = o≤-refl0 (cong supf0 b) ... | tri> ¬a ¬b px<z = o≤-refl0 zc177 where zc177 : supf0 z ≡ supf0 px zc177 = ZChain.supfmax zc px<z -- px o< z, px o< supf0 px zc11 : {z : Ordinal} → odef (UnionCF A f mf ay supf0 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 is-sup fc ⟫ = zc21 fc where u≤px : u o≤ px u≤px = zc-b<x _ u<x zc21 : {z1 : Ordinal } → FClosure A f (supf0 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 u1-is-sup fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1) , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫ ... | case2 fc = case2 (fsuc _ fc) zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) | inspect supf0 u ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u u<px is-sup (init asp refl ) ⟫ where u<px : u o< px u<px = ZChain.supf-inject zc a ... | tri≈ ¬a b ¬c | _ = case2 (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) (sym b )) ... | 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 ⟫ ) record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where field tsup : MinSUP A (UnionCF A f mf ay supf0 z) tsup=sup : supf0 z ≡ MinSUP.sup tsup sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x sup {z} z≤x with trio< z px ... | tri< a ¬b ¬c = ? -- jrecord { tsup = ZChain.minsup zc (o<→≤ a) ; tsup=sup = ZChain.supf-is-minsup zc (o<→≤ a) } ... | tri≈ ¬a b ¬c = ? -- record { tsup = ZChain.minsup zc (o≤-refl0 b) ; tsup=sup = ZChain.supf-is-minsup zc (o≤-refl0 b) } ... | tri> ¬a ¬b px<z = zc35 where zc30 : z ≡ x zc30 with osuc-≡< z≤x ... | case1 eq = eq ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ ) zc32 = ZChain.sup zc o≤-refl zc34 : ¬ (supf0 px ≡ px) → {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup zc32) ∨ (w < SUP.sup zc32) zc34 ne {w} lt = ? zc33 : supf0 z ≡ & (SUP.sup zc32) zc33 = ? -- trans (sym (supfx (o≤-refl0 (sym zc30)))) ( ZChain.supf-is-minsup zc o≤-refl ) zc36 : ¬ (supf0 px ≡ px) → STMP z≤x zc36 ne = ? -- record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x≤sup = zc34 ne } ; tsup=sup = zc33 } zc35 : STMP z≤x zc35 with trio< (supf0 px) px ... | tri< a ¬b ¬c = zc36 ¬b ... | tri> ¬a ¬b c = zc36 ¬b ... | tri≈ ¬a b ¬c = record { tsup = ? ; tsup=sup = ? } where zc37 : MinSUP A (UnionCF A f mf ay supf0 z) zc37 = record { sup = ? ; asm = ? ; x≤sup = ? } sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsMinSUP A (UnionCF A f mf ay supf0 b) supf0 ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf0 b) f b ) → supf0 b ≡ b sup=u {b} ab b≤x is-sup with trio< b px ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (o≤-refl0 b) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫ ... | tri> ¬a ¬b px<b = zc31 ? where zc30 : x ≡ b zc30 with osuc-≡< b≤x ... | case1 eq = sym (eq) ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) -- x o< sp supf0 b ≡ supf0 x o≤ supf0 sp -- supf0 sp ≡ sp (?) zcsup : xSUP (UnionCF A f mf ay supf0 px) supf0 x zcsup with zc30 ... | refl = record { ax = ab ; is-sup = record { x≤sup = λ {w} lt → IsMinSUP.x≤sup (proj1 is-sup) ? ; minsup = ? } } zc31 : ( (¬ xSUP (UnionCF A f mf ay supf0 px) supf0 x ) ∨ HasPrev A (UnionCF A f mf ay supf0 px) f x ) → supf0 b ≡ b zc31 (case1 ¬sp=x) with zc30 ... | refl = ⊥-elim (¬sp=x zcsup ) zc31 (case2 hasPrev ) with zc30 ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev ; ay = ? ; x=fy = HasPrev.x=fy hasPrev } ) ... | no lim = record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf-mono ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; cfcs = cfcs } where 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 pchainx : HOD pchainx = record { od = record { def = λ z → IChain A f supfz z } ; odmax = & A ; <odmax = zc00 } where zc00 : {z : Ordinal } → IChain A f supfz z → z o< & A zc00 {z} ic = z09 ( A∋fc (supfz (IChain.i<x ic)) f mf (IChain.fc ic) ) aic : {z : Ordinal } → IChain A f supfz z → odef A z aic {z} ic = A∋fc _ f mf (IChain.fc ic ) 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 iceq : {ix iy : Ordinal } → ix ≡ iy → {i<x : ix o< x } {i<y : iy o< x } → supfz i<x ≡ supfz i<y iceq refl = cong supfz o<-irr ifc≤ : {za zb : Ordinal } ( ia : IChain A f supfz za ) ( ib : IChain A f supfz zb ) → (ia≤ib : IChain.i ia o≤ IChain.i ib ) → FClosure A f (ZChain.supf (pzc (ob<x lim (IChain.i<x ib))) (IChain.i ia)) za ifc≤ {za} {zb} ia ib ia≤ib = subst (λ k → FClosure A f k _ ) (zeq _ _ (osucc ia≤ib) (o<→≤ <-osuc) ) (IChain.fc ia) ptotalx : IsTotalOrderSet pchainx ptotalx {a} {b} ia ib = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) uz01 with trio< (IChain.i ia) (IChain.i ib) ... | tri< ia<ib ¬b ¬c with <=-trans (ZChain.supf-<= (pzc (ob<x lim (IChain.i<x ib))) ia<ib (ifc≤ ia ib (o<→≤ ia<ib))) (≤to<= (s≤fc _ f mf (IChain.fc ib))) ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * (& a) ≡ * (& b) ct00 = cong (*) eq1 ... | case2 lt = tri< lt (λ eq → <-irr (case1 (sym eq)) lt) (λ lt1 → <-irr (case2 lt) lt1) uz01 | tri≈ ¬a ia=ib ¬c = fcn-cmp _ f mf (IChain.fc ia) (subst (λ k → FClosure A f k (& b)) (sym (iceq ia=ib)) (IChain.fc ib)) uz01 | tri> ¬a ¬b ib<ia with <=-trans (ZChain.supf-<= (pzc (ob<x lim (IChain.i<x ia))) ib<ia (ifc≤ ib ia (o<→≤ ib<ia))) (≤to<= (s≤fc _ f mf (IChain.fc ia))) ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * (& a) ≡ * (& b) ct00 = sym (cong (*) eq1) ... | case2 lt = tri> (λ lt1 → <-irr (case2 lt) lt1) (λ eq → <-irr (case1 eq) lt) lt usup : MinSUP A pchainx usup = minsupP pchainx (λ ic → A∋fc _ f mf (IChain.fc ic ) ) ptotalx spu = MinSUP.sup usup 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 = spu pchain : HOD pchain = UnionCF A f mf ay supf1 x -- pchain ⊆ pchainx ptotal : IsTotalOrderSet pchain ptotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) uz01 = chain-total A f mf ay supf1 ( (proj2 ca)) ( (proj2 cb)) 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 } → (a : x o≤ z ) → supf1 z ≡ spu sf1=spu {z} x≤z with trio< z x ... | tri< a ¬b ¬c = ⊥-elim (o≤> x≤z a) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = refl zc11 : {z : Ordinal } → odef pchain z → odef pchainx z zc11 {z} lt = ? sfpx<=spu : {z : Ordinal } → supf1 z <= spu sfpx<=spu {z} with trio< z x ... | tri< a ¬b ¬c = MinSUP.x≤sup usup ? -- (init (ZChain.asupf (pzc (ob<x lim a)) ) refl ) ... | tri≈ ¬a b ¬c = case1 refl ... | tri> ¬a ¬b c = case1 refl sfpx≤spu : {z : Ordinal } → supf1 z o≤ spu sfpx≤spu {z} with trio< z x ... | tri< a ¬b ¬c = subst ( λ k → k o≤ spu) ? ( MinSUP.minsup (ZChain.minsup ? o≤-refl) ? (λ {x} ux → MinSUP.x≤sup ? ?) ) ... | tri≈ ¬a b ¬c = ? ... | tri> ¬a ¬b c = ? supf-mono : {x y : Ordinal } → x o≤ y → supf1 x o≤ supf1 y supf-mono {x} {y} x≤y with trio< y x ... | tri< a ¬b ¬c = ? ... | tri≈ ¬a b ¬c = ? ... | tri> ¬a ¬b c = ? cfcs : (mf< : <-monotonic-f A f) {a b w : Ordinal } → a o< b → b o≤ x → supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f mf ay supf1 b) w cfcs mf< {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 -- x is limit ordinal, so we have sa o< m o< x m<x : m o< x m<x with trio< a sa | inspect (omax a) sa ... | tri< a<sa ¬b ¬c | record { eq = eq } = ob<x lim (ordtrans<-≤ sa<b b≤x ) ... | tri≈ ¬a a=sa ¬c | record { eq = eq } = subst (λ k → k o< x) eq zc41 where zc41 : omax a sa o< x zc41 = osucprev ( begin osuc ( omax a sa ) ≡⟨ cong (λ k → osuc (omax a k)) (sym a=sa) ⟩ osuc ( omax a a ) ≡⟨ cong osuc (omxx _) ⟩ osuc ( osuc a ) ≤⟨ o<→≤ (ob<x lim (ob<x lim a<x)) ⟩ x ∎ ) where open o≤-Reasoning O ... | tri> ¬a ¬b c | record { eq = eq } = 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 mf ay (ZChain.supf (pzc (ob<x lim m<x))) (osuc (omax a sa))) w zcm = ZChain.cfcs (pzc (ob<x lim m<x)) mf< (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm zc40 : odef (UnionCF A f mf ay supf1 b) w zc40 with ZChain.cfcs (pzc (ob<x lim m<x)) mf< (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ ... | ⟪ az , ch-is-sup u u<x is-sup fc1 ⟫ = ⟪ az , ch-is-sup u u<b cp fc2 ⟫ where zc55 : u o< osuc m zc55 = u<x u<b : u o< b u<b = subst (λ k → u o< k ) (sym b=x) ( ordtrans u<x (ob<x lim m<x)) fc1m : FClosure A f (ZChain.supf (pzc (ob<x lim m<x)) u) w fc1m = fc1 fc1a : FClosure A f (ZChain.supf (pzc (ob<x lim a<x)) a) w fc1a = fc fc2 : FClosure A f (supf1 u) w fc2 = subst (λ k → FClosure A f k w) (trans (sym (zeq _ _ zc57 (o<→≤ <-osuc))) (sym (sf1=sf (ordtrans≤-< u<x m<x))) ) fc1 where zc57 : osuc u o≤ osuc m zc57 = osucc u<x sb=sa : {a : Ordinal } → a o≤ m → supf1 a ≡ ZChain.supf (pzc (ob<x lim m<x)) a sb=sa {a} a≤m = trans (sf1=sf (ordtrans≤-< a≤m m<x)) (zeq _ _ (osucc a≤m) (o<→≤ <-osuc)) cp : ChainP A f mf ay supf1 u cp = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sym (sb=sa u<x)) (ChainP.fcy<sup is-sup fc ) ; order = order ; supu=u = trans (sb=sa u<x ) (ChainP.supu=u is-sup) } where order : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 u → FClosure A f (supf1 s) z1 → (z1 ≡ supf1 u) ∨ (z1 << supf1 u) order {s} {z} s<u fc = subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sym (sb=sa u<x)) (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sb=sa s≤m) (sb=sa u<x) s<u) (subst (λ k → FClosure A f k z) (sb=sa s≤m ) fc )) where s≤m : s o≤ m s≤m = ordtrans (supf-inject0 supf-mono s<u ) u<x ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) ... | tri> ¬a ¬b c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x)) cfcs mf< {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 -- supfb a o< b assures it is in Union b zcb : odef (UnionCF A f mf ay supfb b) w zcb = ZChain.cfcs (pzc (ob<x lim b<x)) mf< a<b (o<→≤ <-osuc) (subst (λ k → k o< b) (sb=sa a<b) sa<b) fcb zc40 : odef (UnionCF A f mf ay supf1 b) w zc40 with zcb ... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ ... | ⟪ az , ch-is-sup u u<x is-sup fc1 ⟫ = ⟪ az , ch-is-sup u u<x cp (subst (λ k → FClosure A f k w) (sym (sb=sa u<x)) fc1 ) ⟫ where cp : ChainP A f mf ay supf1 u cp = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sym (sb=sa u<x)) (ChainP.fcy<sup is-sup fc ) ; order = order ; supu=u = trans (sb=sa u<x) (ChainP.supu=u is-sup) } where order : {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 u → FClosure A f (supf1 s) z1 → (z1 ≡ supf1 u) ∨ (z1 << supf1 u) order {s} {z} s<u fc = subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sym (sb=sa u<x)) (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sb=sa s<b) (sb=sa u<x) s<u) (subst (λ k → FClosure A f k z) (sb=sa s<b ) fc )) where s<b : s o< b s<b = ordtrans (supf-inject0 supf-mono s<u ) u<x --- --- the maximum chain has fix point of any ≤-monotonic function --- 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 mf ay (ZChain.supf zc) x) msp0 f mf {x} ay zc = minsupP (UnionCF A f mf 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 as0 (& 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.asm sp1 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 mf as0 (ZChain.supf zc) b) ab → * a < * b → odef chain b z10 = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl ) z22 : sp o< & A z22 = z09 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< as0 zc (& A) o≤-refl) {& s} {sp} ( ZChain.chain∋init zc ) (z09 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 mf as0 (ZChain.supf zc) sp) asp z19 = record { x≤sup = z20 } where z20 : {y : Ordinal} → odef (UnionCF A f mf as0 (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 as0 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.asm sp1 )) ... | case1 eq = ⊥-elim (¬b (sym eq) ) ... | case2 lt = ⊥-elim (¬c lt ) ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) b ) ... | 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 (cong (*) eq) ... | case2 lt = ¬a lt tri : {n : Level} (u w : Ordinal ) { R : Set n } → ( u o< w → R ) → ( u ≡ w → R ) → ( w o< u → R ) → R tri {_} u w p q r with trio< u w ... | tri< a ¬b ¬c = p a ... | tri≈ ¬a b ¬c = q b ... | tri> ¬a ¬b c = r c or : {n m r : Level } {P : Set n } {Q : Set m} {R : Set r} → P ∨ Q → ( P → R ) → (Q → R ) → R or (case1 p) p→r q→r = p→r p or (case2 q) p→r q→r = q→r q -- ZChain contradicts ¬ Maximal -- -- ZChain forces fix point on any ≤-monotonic function (fixpoint) -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain -- z04 : (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥ z04 nmx zc = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.asm msp1 )))) (subst (λ k → odef A k) (sym &iso) (MinSUP.asm msp1) ) (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) (cf-is-<-monotonic nmx ) zc msp1 ))) -- x ≡ f x ̄ (proj1 (cf-is-<-monotonic nmx c (MinSUP.asm msp1 ))) where -- x < f x supf = ZChain.supf zc msp1 : MinSUP A (ZChain.chain zc) msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc c : Ordinal c = MinSUP.sup msp1 zorn00 : Maximal A zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where -- yes we have the maximal zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) zorn01 = proj1 zorn03 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where -- if we have no maximal, make ZChain, which contradict SUP condition 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) (subst (λ k → k < * y) *iso mx<y) ) ⟫ -- usage (see filter.agda ) -- -- _⊆'_ : ( A B : HOD ) → Set n -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x -- MaximumSubset : {L P : HOD} -- → o∅ o< & L → o∅ o< & P → P ⊆ L -- → IsPartialOrderSet P _⊆'_ -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) -- → Maximal P (_⊆'_) -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP