Mercurial > hg > Members > kono > Proof > ZF-in-agda
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 17 Jun 2022 21:45:33 +0900 |
| parents | 099ca2fea51c |
| children | eb9cf48f530a d0938f220648 |
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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 -- Set n order 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 ) <-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 _⊆_ -- -- 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 ) data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where init : odef A s → FClosure A f s s 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) = as A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {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) = 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) = 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 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 zero zero refl (init _) (init x₁) i=x i=y = refl fc00 zero zero refl (init as) (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 ( fc00 zero zero refl (init as) cy i=x i=y ) fc00 zero zero refl (fsuc x cx) (init as) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as) 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 .(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 .(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 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) {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 fcn-imm : {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 ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = fc21 where fc20 : fcn s mf cy Data.Nat.< suc (fcn s mf cx) → (fcn s mf cy ≡ fcn s mf cx) ∨ ( fcn s mf cy Data.Nat.< fcn s mf cx ) fc20 y<sx with <-cmp ( fcn s mf cy ) (fcn s mf cx ) ... | tri< a ¬b ¬c = case2 a ... | tri≈ ¬a b ¬c = case1 b ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> y<sx (s≤s c)) fc17 : {x y : Ordinal } → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → suc (fcn s mf cx) ≡ fcn s mf cy → * (f x ) ≡ * y fc17 {x} {y} cx cy sx=y = fc18 (fcn s mf cy) cx cy refl sx=y where fc18 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → suc (fcn s mf cx) ≡ i → * (f x) ≡ * y fc18 (suc i) {y} cx (fsuc y1 cy) i=y sx=i with proj1 (mf y1 (A∋fc s f mf cy ) ) ... | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy ( fc18 (suc i) {y1} cx cy i=y sx=i) -- dereference ... | case2 y<fy = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k) ) ) fc19) where fc19 : * x ≡ * y1 fc19 = fcn-inject s mf cx cy (cong pred ( trans sx=i i=y )) fc21 : ⊥ fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy ) ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy cxx : FClosure A f s (f x) cxx = fsuc x cx fc16 : (x : Ordinal ) → (cx : FClosure A f s x) → (fcn s mf cx ≡ fcn s mf (fsuc x cx)) ∨ ( suc (fcn s mf cx ) ≡ fcn s mf (fsuc x cx)) fc16 x (init as) with proj1 (mf s as ) ... | case1 _ = case1 refl ... | case2 _ = case2 refl fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) ) ... | case1 _ = case1 refl ... | case2 _ = case2 refl fc22 : (suc ( fcn s mf cx )) Data.Nat.< (fcn s mf cy ) → * (f x) < * y fc22 a with fc16 x cx ... | case1 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq (<-trans a<sa a)) ... | case2 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq a ) ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx y=x )) x<y ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) -- 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) {x : Ordinal } (xa : odef A x) ( f : Ordinal → Ordinal ) : Set n where field 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 SupF (A : HOD) : Set n where field chain : Ordinal -- sup : Ordinal -- asup : odef A sup -- is-sup : IsSup A (* chain) asup record ZChain ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) (supf : Ordinal → SupF A) ( z : Ordinal ) : Set (Level.suc n) where chain : HOD chain = * (SupF.chain (supf z )) field chain⊆A : chain ⊆' A chain∋x : odef chain x initial : {y : Ordinal } → odef chain y → * x ≤ * y f-total : IsTotalOrderSet chain f-next : {a : Ordinal } → odef chain a → odef chain (f a) f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) is-max : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc z → (ab : odef A b) → HasPrev A chain ab f ∨ IsSup A chain ab → * a < * b → odef chain b record ZChain1 ( A : HOD ) (x : Ordinal) ( f : Ordinal → Ordinal ) ( z : Ordinal ) : Set (Level.suc n) where field supf : Ordinal → SupF A zc : ZChain A x f supf z chain-mono : {x y : Ordinal} → x o≤ y → y o< osuc z → * (SupF.chain (supf x )) ⊆' * (SupF.chain (supf y )) record Maximal ( A : HOD ) : Set (Level.suc n) where field maximal : HOD A∋maximal : A ∋ maximal ¬maximal<x : {x : HOD} → A ∋ x → ¬ maximal < x -- A is Partial, use negative record SUP ( A B : HOD ) : Set (Level.suc n) where field sup : HOD A∋maximal : A ∋ sup x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n) SupCond A B _ _ = SUP A B 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 supO : (C : HOD ) → C ⊆' A → IsTotalOrderSet C → Ordinal supO C C⊆A TC = & ( SUP.sup ( supP C C⊆A TC )) <-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} → {P : Set n} → odef A y ∧ P → y o< & A z07 {y} 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 ; A∋maximal = 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) -- 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 ) ⟫ zsup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (zc1 : ZChain1 A (& s) f (& A) ) → SUP A (ZChain.chain (ZChain1.zc zc1)) zsup f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc ) where zc = ZChain1.zc zc1 A∋zsup : (nmx : ¬ Maximal A ) (zc1 : ZChain1 A (& s) (cf nmx) (& A) ) → A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc1 ))) A∋zsup nmx zc1 = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc1 ) ) sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc1 : ZChain1 A (& s) f (& A) ) → SUP A (ZChain.chain (ZChain1.zc zc1)) sp0 f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain.f-total zc) where zc = ZChain1.zc zc1 zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y) --- --- the maximum chain has fix point of any ≤-monotonic function --- fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc1 : ZChain1 A (& s) f (& A) ) → f (& (SUP.sup (sp0 f mf zc1 ))) ≡ & (SUP.sup (sp0 f mf zc1 )) fixpoint f mf zc1 = z14 where zc = ZChain1.zc zc1 chain = ZChain.chain zc sp1 = sp0 f mf zc1 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< osuc (& A) → (ab : odef A b ) → HasPrev A chain ab f ∨ IsSup A chain {b} ab -- (supO chain (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b ) → * a < * b → odef chain b z10 = ZChain.is-max zc z11 : & (SUP.sup sp1) o< & A z11 = c<→o< ( SUP.A∋maximal sp1) z12 : odef chain (& (SUP.sup sp1)) z12 with o≡? (& s) (& (SUP.sup sp1)) ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc ) ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) (ordtrans z11 <-osuc ) (SUP.A∋maximal sp1) (case2 z19 ) z13 where z13 : * (& s) < * (& (SUP.sup sp1)) z13 with SUP.x<sup sp1 ( ZChain.chain∋x zc ) ... | case1 eq = ⊥-elim ( ne (cong (&) eq) ) ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1) z19 = record { x<sup = z20 } where z20 : {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1)) z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy) ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p )) ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ? (SUP.x<sup sp1 ? ) } z14 : f (& (SUP.sup (sp0 f mf zc1))) ≡ & (SUP.sup (sp0 f mf zc1)) z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 )) z12 ... | tri< a ¬b ¬c = ⊥-elim z16 where z16 : ⊥ z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 )) ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) )) ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt )) ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b ) ... | tri> ¬a ¬b c = ⊥-elim z17 where z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) < SUP.sup sp1) z15 = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 )) z17 : ⊥ z17 with z15 ... | case1 eq = ¬b eq ... | case2 lt = ¬a lt -- 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 : ZChain1 A (& s) (cf nmx) (& A)) → ⊥ z04 nmx zc1 = <-irr0 {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal sp1)))) (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) ) (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc1 ))) -- x ≡ f x ̄ (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where -- x < f x zc = ZChain1.zc zc1 sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc1 c = & (SUP.sup sp1) -- -- create all ZChains under o< x -- ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → ((y : Ordinal) → y o< x → { y₁ : Ordinal} (ay : odef A y₁) → ZChain1 A y₁ f y) → {y : Ordinal} (ay : odef A y) → ZChain1 A y f x ind f mf x prev {y} ay with Oprev-p x ... | yes op = zc4 where -- -- we have previous ordinal to use induction -- open ZChain px = Oprev.oprev op supf : Ordinal → SupF A supf = ZChain1.supf (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay) zc1 : ZChain1 A y f (Oprev.oprev op) zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay zc0 : ZChain A y f (ZChain1.supf (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay)) (Oprev.oprev op) zc0 = ZChain1.zc (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay) zc0-b<x : (b : Ordinal ) → b o< x → b o< osuc px zc0-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt px<x : px o< x px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc -- if previous chain satisfies maximality, we caan reuse it -- no-extenion : ( {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → * a < * b → odef (ZChain.chain zc0) b ) → ZChain1 A y f x no-extenion is-max = record { supf = supf0 ; zc = record { chain⊆A = subst (λ k → k ⊆' A ) seq (ZChain.chain⊆A zc0) ; initial = subst (λ k → {y₁ : Ordinal} → odef k y₁ → * y ≤ * y₁ ) seq (ZChain.initial zc0) ; f-total = subst (λ k → IsTotalOrderSet k ) seq (ZChain.f-total zc0) ; f-next = subst (λ k → {a : Ordinal} → odef k a → odef k (f a) ) seq (ZChain.f-next zc0) ; f-immediate = subst (λ k → {x₁ : Ordinal} {y₁ : Ordinal} → odef k x₁ → odef k y₁ → ¬ (* x₁ < * y₁) ∧ (* y₁ < * (f x₁)) ) seq (ZChain.f-immediate zc0) ; chain∋x = subst (λ k → odef k y ) seq (ZChain.chain∋x zc0) ; is-max = subst (λ k → {a b : Ordinal} → odef k a → b o< osuc x → (ab : odef A b) → HasPrev A k ab f ∨ IsSup A k ab → * a < * b → odef k b ) seq is-max } ; chain-mono = mono } where supf0 : Ordinal → SupF A supf0 z with trio< z x ... | tri< a ¬b ¬c = supf z ... | tri≈ ¬a b ¬c = record { chain = & (ZChain.chain zc0) } ... | tri> ¬a ¬b c = record { chain = & (ZChain.chain zc0) } seq : ZChain.chain zc0 ≡ * (SupF.chain (supf0 x)) seq with trio< x x ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) ... | tri≈ ¬a b ¬c = sym *iso ... | tri> ¬a ¬b c = sym *iso seq<x : {b : Ordinal } → b o< x → * (SupF.chain (supf b)) ≡ * (SupF.chain (supf0 b)) seq<x {b} b<x with trio< b x ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) mono : {a b : Ordinal} → a o≤ b → b o< osuc x → * (SupF.chain (supf0 a )) ⊆' * (SupF.chain (supf0 b )) mono {a} {b} a≤b b<ox with osuc-≡< a≤b ... | case1 refl = λ x → x ... | case2 a<b with osuc-≡< b<ox ... | case1 b=x = subst₂ (λ j k → j ⊆' k ) (seq<x a<x) nc00 ( ZChain1.chain-mono zc1 a≤px <-osuc ) where a<x : a o< x a<x with osuc-≡< b<ox ... | case1 b=x = subst (λ k → a o< k ) b=x a<b ... | case2 b<x = ordtrans a<b b<x a≤px : a o≤ px a≤px = subst (λ k → a o< k ) (sym (Oprev.oprev=x op)) a<x nc00 : * (SupF.chain (supf px)) ≡ * (SupF.chain (supf0 b)) nc00 with trio< b x ... | tri< a ¬b ¬c = ⊥-elim ( ¬b b=x ) ... | tri≈ ¬a b ¬c = sym *iso ... | tri> ¬a ¬b c = ⊥-elim ( ¬b b=x ) ... | case2 b<x = subst₂ (λ j k → j ⊆' k ) (seq<x a<x ) (seq<x b<x ) ( ZChain1.chain-mono zc1 a≤b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x ) ) where a<x : a o< x a<x with osuc-≡< b<ox ... | case1 b=x = subst (λ k → a o< k ) b=x a<b ... | case2 b<x = ordtrans a<b b<x zc4 : ZChain1 A y f x zc4 with ODC.∋-p O A (* x) ... | no noax = no-extenion zc11 where -- ¬ A ∋ p, just skip zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → * a < * b → odef (ZChain.chain zc0) b zc11 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox ... | case1 eq = ⊥-elim ( noax (subst (λ k → odef A k) (trans eq (sym &iso)) ab ) ) ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc0) ax f ) -- we have to check adding x preserve is-max ZChain A y f mf supO x ... | case1 pr = no-extenion zc17 where -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next chain0 = ZChain.chain zc0 zc17 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → * a < * b → odef (ZChain.chain zc0) b zc17 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b ... | case1 b=x = subst (λ k → odef chain0 k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr)) ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax ) ... | case1 is-sup = -- x is a sup of zc0 record { zc = record { chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} ; chain∋sup = {!!} ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} ; fc∨sup = {!!} } ; supf = supf0 ; chain-mono = mono } where sup0 : SUP A (ZChain.chain zc0) sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where x21 : {y : HOD} → ZChain.chain zc0 ∋ y → (y ≡ * x) ∨ (y < * x) x21 {y} zy with IsSup.x<sup is-sup zy ... | case1 y=x = case1 ( subst₂ (λ j k → j ≡ * k ) *iso &iso ( cong (*) y=x) ) ... | case2 y<x = case2 (subst₂ (λ j k → j < * k ) *iso &iso y<x ) sp : HOD sp = SUP.sup sup0 x=sup : x ≡ & sp x=sup = sym &iso chain0 = ZChain.chain zc0 sc<A : {y : Ordinal} → odef chain0 y ∨ FClosure A f (& sp) y → y o< & A sc<A {y} (case1 zx) = subst (λ k → k o< (& A)) &iso ( c<→o< (ZChain.chain⊆A zc0 (subst (λ k → odef chain0 k) (sym &iso) zx ))) sc<A {y} (case2 fx) = subst (λ k → k o< (& A)) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso) (A∋fc (& sp) f mf fx )) ) schain : HOD schain = record { od = record { def = λ x → odef chain0 x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = λ {y} sy → sc<A {y} sy } A∋schain : {x : HOD } → schain ∋ x → A ∋ x A∋schain (case1 zx ) = ZChain.chain⊆A zc0 zx A∋schain {y} (case2 fx ) = A∋fc (& sp) f mf fx s⊆A : schain ⊆' A s⊆A {x} (case1 zx) = ZChain.chain⊆A zc0 zx s⊆A {x} (case2 fx) = A∋fc (& sp) f mf fx cmp : {a b : HOD} (za : odef chain0 (& a)) (fb : FClosure A f (& sp) (& b)) → Tri (a < b) (a ≡ b) (b < a ) cmp {a} {b} za fb with SUP.x<sup sup0 za | s≤fc (& sp) f mf fb ... | case1 sp=a | case1 sp=b = tri≈ (λ lt → <-irr (case1 (sym eq)) lt ) eq (λ lt → <-irr (case1 eq) lt ) where eq : a ≡ b eq = trans sp=a (subst₂ (λ j k → j ≡ k ) *iso *iso sp=b ) ... | case1 sp=a | case2 sp<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b) lt ) where a<b : a < b a<b = subst (λ k → k < b ) (sym sp=a) (subst₂ (λ j k → j < k ) *iso *iso sp<b ) ... | case2 a<sp | case1 sp=b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b) lt ) where a<b : a < b a<b = subst (λ k → a < k ) (trans sp=b *iso ) (subst (λ k → a < k ) (sym *iso) a<sp ) ... | case2 a<sp | case2 sp<b = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b) lt ) where a<b : a < b a<b = ptrans (subst (λ k → a < k ) (sym *iso) a<sp ) ( subst₂ (λ j k → j < k ) refl *iso sp<b ) scmp : {a b : HOD} → odef schain (& a) → odef schain (& b) → Tri (a < b) (a ≡ b) (b < a ) scmp (case1 za) (case1 zb) = ZChain.f-total zc0 za zb scmp {a} {b} (case1 za) (case2 fb) = cmp za fb scmp (case2 fa) (case1 zb) with cmp zb fa ... | tri< a ¬b ¬c = tri> ¬c (λ eq → ¬b (sym eq)) a ... | tri≈ ¬a b ¬c = tri≈ ¬c (sym b) ¬a ... | tri> ¬a ¬b c = tri< c (λ eq → ¬b (sym eq)) ¬a scmp (case2 fa) (case2 fb) = subst₂ (λ a b → Tri (a < b) (a ≡ b) (b < a ) ) *iso *iso (fcn-cmp (& sp) f mf fa fb) scnext : {a : Ordinal} → odef schain a → odef schain (f a) scnext {x} (case1 zx) = case1 (ZChain.f-next zc0 zx) scnext {x} (case2 sx) = case2 ( fsuc x sx ) scinit : {x : Ordinal} → odef schain x → * y ≤ * x scinit {x} (case1 zx) = ZChain.initial zc0 zx scinit {x} (case2 sx) with (s≤fc (& sp) f mf sx ) | SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) ( ZChain.chain∋x zc0 ) ) ... | case1 sp=x | case1 y=sp = case1 (trans y=sp (subst (λ k → k ≡ * x ) *iso sp=x ) ) ... | case1 sp=x | case2 y<sp = case2 (subst (λ k → * y < k ) (trans (sym *iso) sp=x) y<sp ) ... | case2 sp<x | case1 y=sp = case2 (subst (λ k → k < * x ) (trans *iso (sym y=sp )) sp<x ) ... | case2 sp<x | case2 y<sp = case2 (ptrans y<sp (subst (λ k → k < * x ) *iso sp<x) ) A∋za : {a : Ordinal } → odef chain0 a → odef A a A∋za za = ZChain.chain⊆A zc0 za za<sup : {a : Ordinal } → odef chain0 a → ( * a ≡ sp ) ∨ ( * a < sp ) za<sup za = SUP.x<sup sup0 (subst (λ k → odef chain0 k ) (sym &iso) za ) simm : {a b : Ordinal} → odef schain a → odef schain b → ¬ (* a < * b) ∧ (* b < * (f a)) simm {a} {b} (case1 za) (case1 zb) = ZChain.f-immediate zc0 za zb simm {a} {b} (case1 za) (case2 sb) p with proj1 (mf a (A∋za za) ) ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) ... | case2 a<fa with za<sup ( ZChain.f-next zc0 za ) | s≤fc (& sp) f mf sb ... | case1 fa=sp | case1 sp=b = <-irr (case1 (trans fa=sp (trans (sym *iso) sp=b )) ) ( proj2 p ) ... | case2 fa<sp | case1 sp=b = <-irr (case2 fa<sp) (subst (λ k → k < * (f a) ) (trans (sym sp=b) *iso) (proj2 p ) ) ... | case1 fa=sp | case2 sp<b = <-irr (case2 (proj2 p )) (subst (λ k → k < * b) (sym fa=sp) (subst (λ k → k < * b ) *iso sp<b ) ) ... | case2 fa<sp | case2 sp<b = <-irr (case2 (proj2 p )) (ptrans fa<sp (subst (λ k → k < * b ) *iso sp<b ) ) simm {a} {b} (case2 sa) (case1 zb) p with proj1 (mf a ( subst (λ k → odef A k) &iso ( A∋schain (case2 (subst (λ k → FClosure A f (& sp) k ) (sym &iso) sa) )) ) ) ... | case1 eq = <-irr (case2 (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) ... | case2 b<fb with s≤fc (& sp) f mf sa | za<sup zb ... | case1 sp=a | case1 b=sp = <-irr (case1 (trans b=sp (trans (sym *iso) sp=a )) ) (proj1 p ) ... | case1 sp=a | case2 b<sp = <-irr (case2 (subst (λ k → * b < k ) (trans (sym *iso) sp=a) b<sp ) ) (proj1 p ) ... | case2 sp<a | case1 b=sp = <-irr (case2 (subst ( λ k → k < * a ) (trans *iso (sym b=sp)) sp<a )) (proj1 p ) ... | case2 sp<a | case2 b<sp = <-irr (case2 (ptrans b<sp (subst (λ k → k < * a) *iso sp<a ))) (proj1 p ) simm {a} {b} (case2 sa) (case2 sb) p = fcn-imm {A} (& sp) {a} {b} f mf sa sb p s-ismax : {a b : Ordinal} → odef schain a → b o< osuc x → (ab : odef A b) → HasPrev A schain ab f ∨ IsSup A schain ab → * a < * b → odef schain b s-ismax {a} {b} sa b<ox ab p a<b with osuc-≡< b<ox -- b is x? ... | case1 b=x = case2 (subst (λ k → FClosure A f (& sp) k ) (sym (trans b=x x=sup )) (init (SUP.A∋maximal sup0) )) s-ismax {a} {b} (case1 za) b<ox ab (case1 p) a<b | case2 b<x = z21 p where -- has previous z21 : HasPrev A schain ab f → odef schain b z21 record { y = y ; ay = (case1 zy) ; x=fy = x=fy } = case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case1 record { y = y ; ay = zy ; x=fy = x=fy }) a<b ) z21 record { y = y ; ay = (case2 sy) ; x=fy = x=fy } = subst (λ k → odef schain k) (sym x=fy) (case2 (fsuc y sy) ) s-ismax {a} {b} (case1 za) b<ox ab (case2 p) a<b | case2 b<x = case1 (ZChain.is-max zc0 za (zc0-b<x b b<x) ab (case2 z22) a<b ) where -- previous sup z22 : IsSup A (ZChain.chain zc0) ab z22 = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } s-ismax {a} {b} (case2 sa) b<ox ab (case1 p) a<b | case2 b<x with HasPrev.ay p ... | case1 zy = case1 (subst (λ k → odef chain0 k ) (sym (HasPrev.x=fy p)) (ZChain.f-next zc0 zy )) -- in previous closure of f ... | case2 sy = case2 (subst (λ k → FClosure A f (& (* x)) k ) (sym (HasPrev.x=fy p)) (fsuc (HasPrev.y p) sy )) -- in current closure of f s-ismax {a} {b} (case2 sa) b<ox ab (case2 p) a<b | case2 b<x = case1 z23 where -- sup o< x is already in zc0 z24 : IsSup A schain ab → IsSup A (ZChain.chain zc0) ab z24 p = record { x<sup = λ {y} zy → IsSup.x<sup p (case1 zy ) } z23 : odef chain0 b z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋x zc0 ) ... | case1 y=b = subst (λ k → odef chain0 k ) y=b ( ZChain.chain∋x zc0 ) ... | case2 y<b = ZChain.is-max zc0 (ZChain.chain∋x zc0 ) (zc0-b<x b b<x) ab (case2 (z24 p)) y<b supf0 : Ordinal → SupF A supf0 z with trio< z x ... | tri< a ¬b ¬c = supf z ... | tri≈ ¬a b ¬c = record { chain = & schain } ... | tri> ¬a ¬b c = record { chain = & schain } seq : schain ≡ * (SupF.chain (supf0 x)) seq with trio< x x ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) ... | tri≈ ¬a b ¬c = sym *iso ... | tri> ¬a ¬b c = sym *iso seq<x : {b : Ordinal } → b o< x → * (SupF.chain (supf b)) ≡ * (SupF.chain (supf0 b)) seq<x {b} b<x with trio< b x ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) mono : {a b : Ordinal} → a o≤ b → b o< osuc x → * (SupF.chain (supf0 a )) ⊆' * (SupF.chain (supf0 b )) mono {a} {b} a≤b b<ox = {!!} ... | case2 ¬x=sup = no-extenion z18 where -- x is not f y' nor sup of former ZChain from y -- no extention z18 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) → HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0) ab → * a < * b → odef (ZChain.chain zc0) b z18 {a} {b} za b<x ab p a<b with osuc-≡< b<x ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab p a<b ... | case1 b=x with p ... | case1 pr = ⊥-elim ( ¬fy<x record {y = HasPrev.y pr ; ay = HasPrev.ay pr ; x=fy = trans (trans &iso (sym b=x) ) (HasPrev.x=fy pr ) } ) ... | case2 b=sup = ⊥-elim ( ¬x=sup record { x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy) } ) ... | no ¬ox = record { supf = supf0 ; chain-mono = {!!} ; zc = record { chain⊆A = {!!} ; f-total = {!!} ; f-next = {!!} ; initial = {!!} ; f-immediate = {!!} ; chain∋x = {!!} ; is-max = {!!} } } where --- limit ordinal case record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x field u : Ordinal u<x : u o< x chain∋z : odef (ZChain.chain (ZChain1.zc (prev u u<x {y} ay ))) z Uz⊆A : {z : Ordinal} → UZFChain z → odef A z Uz⊆A {z} u = ZChain.chain⊆A (ZChain1.zc ( prev (UZFChain.u u) (UZFChain.u<x u) {y} ay )) (UZFChain.chain∋z u) uzc1 : {z : Ordinal} → (u : UZFChain z) → ZChain1 A y f (UZFChain.u u) uzc1 {z} u = prev (UZFChain.u u) (UZFChain.u<x u) {y} ay uzc : {z : Ordinal} → (u : UZFChain z) → ZChain A y f (ZChain1.supf (uzc1 u)) (UZFChain.u u) uzc {z} u = ZChain1.zc (prev (UZFChain.u u) (UZFChain.u<x u) {y} ay) Uz : HOD Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A ; <odmax = λ lt → subst (λ k → k o< & A ) &iso (c<→o< (subst (λ k → odef A k ) (sym &iso) (Uz⊆A lt))) } u-next : {z : Ordinal} → odef Uz z → odef Uz (f z) u-next {z} u = record { u = UZFChain.u u ; u<x = UZFChain.u<x u ; chain∋z = ZChain.f-next ( uzc u ) (UZFChain.chain∋z u) } u-initial : {z : Ordinal} → odef Uz z → * y ≤ * z u-initial {z} u = ZChain.initial ( uzc u ) (UZFChain.chain∋z u) u-chain∋x : odef Uz y u-chain∋x = record { u = y ; u<x = {!!} ; chain∋z = ZChain.chain∋x (ZChain1.zc (prev y {!!} ay )) } supf0 : Ordinal → SupF A supf0 z with trio< z x ... | tri< a ¬b ¬c = ZChain1.supf (prev z a {y} ay) z ... | tri≈ ¬a b ¬c = record { chain = & Uz } ... | tri> ¬a ¬b c = record { chain = & Uz } seq : Uz ≡ * (SupF.chain (supf0 x)) seq with trio< x x ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl ) ... | tri≈ ¬a b ¬c = sym *iso ... | tri> ¬a ¬b c = sym *iso seq<x : {b : Ordinal } → (b<x : b o< x ) → * (SupF.chain (ZChain1.supf (prev b b<x {y} ay) b)) ≡ * (SupF.chain (supf0 b)) seq<x {b} b<x with trio< b x ... | tri< a ¬b ¬c = cong (λ k → * (SupF.chain (ZChain1.supf (prev b k {y} ay) b)) ) o<-irr -- b<x ≡ a ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a b<x ) ... | tri> ¬a ¬b c = ⊥-elim (¬a b<x ) u-mono : ( a b : Ordinal ) → b o< x → a o< osuc b → (za : ZChain1 A y f a) (zb : ZChain1 A y f b) → ZChain.chain (ZChain1.zc za) ⊆' ZChain.chain (ZChain1.zc zb) u-mono a b b<x a≤b za zb {i} zai = ZChain1.chain-mono zb a≤b <-osuc (uz01 zai) where uz01 : odef (* (SupF.chain (ZChain1.supf za a))) i → odef (* (SupF.chain (ZChain1.supf zb a))) i uz01 = {!!} u-total : IsTotalOrderSet Uz u-total {x} {y} ux uy with trio< (UZFChain.u ux) (UZFChain.u uy) ... | tri< a ¬b ¬c = ZChain.f-total (uzc uy) (u-mono (UZFChain.u ux) (UZFChain.u uy) (UZFChain.u<x uy) (ordtrans a <-osuc ) (uzc1 ux) (uzc1 uy) (UZFChain.chain∋z ux)) (UZFChain.chain∋z uy) ... | tri≈ ¬a b ¬c = ZChain.f-total (uzc ux) (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) (UZFChain.u<x ux) (subst (λ k → k o< osuc (UZFChain.u ux)) b <-osuc) (uzc1 uy) (uzc1 ux) (UZFChain.chain∋z uy)) ... | tri> ¬a ¬b c = ZChain.f-total (uzc ux) (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux) (UZFChain.u<x ux) (ordtrans c <-osuc) (uzc1 uy) (uzc1 ux) (UZFChain.chain∋z uy)) SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ya : odef A y) → ZChain1 A y f (& A) SZ f mf = TransFinite {λ z → {y : Ordinal } → (ay : odef A y ) → ZChain1 A y f z } (ind f mf) (& A) 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)) ; A∋maximal = 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 zorn04) 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.A∋maximal 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) ) ⟫ zorn04 : ZChain1 A (& s) (cf nmx) (& A) zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) -- 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