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 Aug 2023 08:37:08 +0900 |
| parents | 79cf38cc667b |
| children |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level hiding (suc ; zero ) open import Ordinals module cardinal {n : Level } (O : Ordinals {n}) where open import logic -- import OD import OD import ODC open import Data.Nat open import Relation.Binary.PropositionalEquality open import Data.Nat.Properties open import Data.Empty open import Relation.Nullary open import Relation.Binary open import Relation.Binary.Core open inOrdinal O open OD O open OD.OD open ODAxiom odAxiom open import ZProduct O import OrdUtil import ODUtil open Ordinals.Ordinals O open Ordinals.IsOrdinals isOrdinal -- open Ordinals.IsNext isNext open OrdUtil O open ODUtil O open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) open _∧_ open _∨_ open Bool open _==_ open HOD -- record HODBijection (A B : HOD ) : Set n where -- field -- fun→ : (x : Ordinal ) → odef B x → Ordinal -- fun← : (x : Ordinal ) → odef A x → Ordinal -- funB : (x : Ordinal ) → ( lt : odef A x ) → odef B ( fun← x lt ) -- funA : (x : Ordinal ) → ( lt : odef B x ) → odef A ( fun→ x lt ) -- fiso→ : (x : Ordinal ) → ( lt : odef A x ) → fun→ ( fun← x lt ) ( funB x lt ) ≡ x -- fiso← : (x : Ordinal ) → ( lt : odef B x ) → fun← ( fun→ x lt ) ( funA x lt ) ≡ x open Injection open HODBijection record IsImage0 (a b : HOD) (f : (x : Ordinal) → odef a x → Ordinal) (x : Ordinal ) : Set n where field y : Ordinal ay : odef a y x=fy : x ≡ f y ay IsImage : (a b : Ordinal) (iab : Injection a b ) (x : Ordinal ) → Set n IsImage a b iab x = IsImage0 (* a) (* b) (i→ iab) x Image : (a : Ordinal) { b : Ordinal } → Injection a b → HOD Image a {b} iab = record { od = record { def = λ x → IsImage a b iab x } ; odmax = b ; <odmax = im00 } where im00 : {x : Ordinal } → IsImage a b iab x → x o< b im00 {x} record { y = y ; ay = ay ; x=fy = x=fy } = subst₂ ( λ j k → j o< k) (trans &iso (sym x=fy)) &iso (c<→o< (subst ( λ k → odef (* b) k) (sym &iso) (iB iab y ay)) ) Image⊆b : { a b : Ordinal } → (iab : Injection a b) → Image a iab ⊆ (* b ) Image⊆b {a} {b} iab {x} record { y = y ; ay = ay ; x=fy = x=fy } = subst (λ k → odef (* b) k) (sym x=fy) (iB iab y ay) _=c=_ : ( A B : HOD ) → Set n A =c= B = HODBijection A B ≡→c= : {A B : HOD} → A ≡ B → A =c= B ≡→c= eq = hodbij-refl eq open import BAlgebra O _-_ : (a b : Ordinal ) → Ordinal a - b = & ( (* a) ＼ (* b) ) -→< : (a b : Ordinal ) → (a - b) o≤ a -→< a b = subst₂ (λ j k → j o≤ k) &iso &iso ( ⊆→o≤ ( λ {x} a-b → proj1 (subst ( λ k → odef k x) *iso a-b) ) ) b-a⊆b : {a b x : Ordinal } → odef (* (b - a)) x → odef (* b) x b-a⊆b {a} {b} {x} lt with subst (λ k → odef k x) *iso lt ... | ⟪ bx , ¬ax ⟫ = bx Injection-⊆ : {a b c : Ordinal } → * c ⊆ * a → Injection a b → Injection c b Injection-⊆ {a} {b} {c} le f = record { i→ = λ x cx → i→ f x (le cx) ; iB = λ x cx → iB f x (le cx) ; inject = λ x y ix iy eq → inject f x y (le ix) (le iy) eq } Injection-∙ : {a b c : Ordinal } → Injection a b → Injection b c → Injection a c Injection-∙ {a} {b} {c} f g = record { i→ = λ x ax → i→ g (i→ f x ax) (iB f x ax) ; iB = λ x ax → iB g (i→ f x ax) (iB f x ax) ; inject = λ x y ix iy eq → inject f x y ix iy (inject g (i→ f x ix) (i→ f y iy) (iB f x ix) (iB f y iy) eq) } -- -- Two injection can be joined -- A and C may overrap -- bi-∪ : {A B C D : HOD } → (ab : HODBijection A B) → (cd : HODBijection C D ) → HODBijection (A ∪ C) (B ∪ D) bi-∪ {A} {B} {C} {D} ab cd = record { fun→ = fa ; fun← = fb ; funB = fa-isb ; funA = fb-isa ; fiso→ = faiso ; fiso← = fbiso } where fa : (x : Ordinal) → odef (A ∪ C) x → Ordinal fa x (case1 a) = fun→ ab x a fa x (case2 c) = fun→ cd x c fb : (x : Ordinal) → odef (B ∪ D) x → Ordinal fb x (case1 b) = fun← ab x b fb x (case2 d) = fun← cd x d fa-isb : (x : Ordinal) (lt : odef (A ∪ C) x) → odef (B ∪ D) (fa x lt) fa-isb x (case1 a) = case1 (funB ab x a) fa-isb x (case2 c) = case2 (funB cd x c) fb-isa : (x : Ordinal) (lt : odef (B ∪ D) x) → odef (A ∪ C) (fb x lt) fb-isa x (case1 b) = case1 (funA ab x b) fb-isa x (case2 d) = case2 (funA cd x d) faiso : (x : Ordinal) (lt : odef (B ∪ D) x) → fa (fb x lt) (fb-isa x lt) ≡ x faiso x (case1 b) = fiso→ ab x b faiso x (case2 d) = fiso→ cd x d fbiso : (x : Ordinal) (lt : odef (A ∪ C) x) → fb (fa x lt) (fa-isb x lt) ≡ x fbiso x (case1 b) = fiso← ab x b fbiso x (case2 d) = fiso← cd x d -- -- f : A → B OrdBijection A (Image A f) -- g : B → A OrdBijection (Image B g) B -- UC (closure of g ∙ f from ¬ Image B g ) -- A = UC ∪ (A \ Image B gi ) -- B = (Image B g) UC -- Bernstein : {a b : Ordinal } → Injection a b → Injection b a → HODBijection (* a) (* b) Bernstein {a} {b} ( fi @ record { i→ = f ; iB = b∋f ; inject = f-inject }) ( gi @ record { i→ = g ; iB = a∋g ; inject = g-inject }) = subst₂ (λ j k → HODBijection j k ) (sym a=UC∨a-UC) (sym b=fUC∨b-fUC) (bi-∪ bi-UC bi-fUC) where gf : Injection a a gf = record { i→ = λ x ax → g (f x ax) (b∋f x ax) ; iB = λ x ax → a∋g _ (b∋f x ax) ; inject = λ x y ax ay eq → f-inject _ _ ax ay ( g-inject _ _ (b∋f _ ax) (b∋f _ ay) eq) } -- HOD UC is closure of gi ∙ fi starting from a - Image g -- and a-UC is the remaining part of a. Both sets have inverse functions. -- -- We cannot directly create h : * a → * b from these functions, because -- the choise of UC ∨ a-UC is non constructive and -- LEM cannot be used in this non positive context. -- -- We use the following trick: -- -- bi-UC : HODBijection UC fUC -- bi-fUC : HODBijection a-UC b-fUC -- -- The HODBijection (* a) (* b) is a merge of these bijections. -- The merging bi-UC and bi-fUC uses also LEM but use it positively. -- -- bijection on each side is easy, because these are images of fi and g. -- somehow we don't use injection of f. -- -- data gfImage : (x : Ordinal) → Set n where a-g : {x : Ordinal} → (ax : odef (* a) x ) → (¬ib : ¬ ( IsImage b a gi x )) → gfImage x next-gf : {x : Ordinal} → (ix : IsImage a a gf x) → (gfiy : gfImage (IsImage0.y ix) ) → gfImage x a∋gfImage : {x : Ordinal } → gfImage x → odef (* a) x a∋gfImage {x} (a-g ax ¬ib) = ax a∋gfImage {x} (next-gf record { y = y ; ay = ay ; x=fy = x=fy } lt ) = subst (λ k → odef (* a) k ) (sym x=fy) (a∋g _ (b∋f y ay) ) UC : HOD UC = record { od = record { def = λ x → gfImage x } ; odmax = & (* a) ; <odmax = λ lt → odef< (a∋gfImage lt) } a-UC : HOD a-UC = record { od = record { def = λ x → odef (* a) x ∧ (¬ gfImage x) } ; odmax = & (* a) ; <odmax = λ lt → odef< (proj1 lt) } a=UC∨a-UC : * a ≡ UC ∪ a-UC a=UC∨a-UC = ==→o≡ record { eq→ = be00 ; eq← = be01 } where be00 : {x : Ordinal} → odef (* a) x → odef (UC ∪ a-UC) x be00 {x} ax with ODC.p∨¬p O ( gfImage x) ... | case1 gfx = case1 gfx ... | case2 ngfx = case2 ⟪ ax , ngfx ⟫ be01 : {x : Ordinal} → odef (UC ∪ a-UC) x → odef (* a) x be01 {x} (case1 gfx) = a∋gfImage gfx be01 {x} (case2 ngfx) = proj1 ngfx FA : (x : Ordinal) → (ax : gfImage x) → Ordinal FA x ax = f x (a∋gfImage ax) b∋f⁻¹ : (x : Ordinal) → IsImage0 UC (* b) FA x → odef (* b) x b∋f⁻¹ x record { y = y ; ay = ay ; x=fy = x=fy } = subst (λ k → odef (* b) k ) (sym x=fy) (b∋f y (a∋gfImage ay)) fUC : HOD fUC = record { od = record { def = λ x → IsImage0 UC (* b) FA x } ; odmax = & (* b) ; <odmax = λ {x} lt → odef< (b∋f⁻¹ x lt)} b-fUC : HOD b-fUC = record { od = record { def = λ x → odef (* b) x ∧ (¬ IsImage0 UC (* b) FA x) } ; odmax = & (* b) ; <odmax = λ lt → odef∧< lt } b=fUC∨b-fUC : * b ≡ fUC ∪ b-fUC b=fUC∨b-fUC = ==→o≡ record { eq→ = be00 ; eq← = be01 } where be00 : {x : Ordinal} → odef (* b) x → odef (fUC ∪ b-fUC) x be00 {x} bx with ODC.p∨¬p O (IsImage0 UC (* b) FA x) ... | case1 gfx = case1 gfx ... | case2 ngfx = case2 ⟪ bx , ngfx ⟫ be01 : {x : Ordinal} → odef (fUC ∪ b-fUC) x → odef (* b) x be01 {x} (case1 record { y = y ; ay = ay ; x=fy = x=fy }) = subst (λ k → odef (* b) k) (sym x=fy) ( b∋f y (a∋gfImage ay)) be01 {x} (case2 ngfx) = proj1 ngfx nimg : {x : Ordinal } → (ax : odef (* a) x ) → ¬ (odef UC x) → IsImage b a gi x nimg {x} ax ncn with ODC.p∨¬p O (IsImage b a gi x) ... | case1 img = img ... | case2 nimg = ⊥-elim ( ncn (a-g ax nimg ) ) f-cong : {x y : Ordinal } → {ax : odef (* a) x} {ax1 : odef (* a) y} → x ≡ y → f x ax ≡ f y ax1 f-cong {x} {x} {ax} {ax1} refl = cong (λ k → f x k) ( HE.≅-to-≡ ( ∋-irr {* a} ax ax1 )) g-cong : {x y : Ordinal } → {bx : odef (* b) x} {bx1 : odef (* b) y} → x ≡ y → g x bx ≡ g y bx1 g-cong {x} {x} {bx} {bx1} refl = cong (λ k → g x k) ( HE.≅-to-≡ ( ∋-irr {* b} bx bx1 )) g⁻¹ : {x : Ordinal } → (ax : odef (* a) x) → (nc0 : IsImage b a gi x ) → Ordinal g⁻¹ {x} ax record { y = y ; ay = ay ; x=fy = x=fy } = y b∋g⁻¹ : {x : Ordinal } → (ax : odef (* a) x) (nc0 : IsImage b a gi x ) → odef (* b) (g⁻¹ ax nc0) b∋g⁻¹ {x} ax record { y = y ; ay = ay ; x=fy = x=fy } = ay g⁻¹-iso : {x : Ordinal } → (ax : odef (* a) x) → (nc0 : IsImage b a gi x ) → g (g⁻¹ ax nc0) (b∋g⁻¹ ax nc0) ≡ x g⁻¹-iso {x} ax record { y = y ; ay = ay ; x=fy = x=fy } = sym x=fy g⁻¹-iso1 : {x : Ordinal } → (bx : odef (* b) x) → (nc0 : IsImage b a gi (g x bx) ) → g⁻¹ (a∋g x bx) nc0 ≡ x g⁻¹-iso1 {x} bx nc0 = inject gi _ _ (b∋g⁻¹ (a∋g x bx) nc0) bx (g⁻¹-iso (a∋g x bx) nc0) g⁻¹-eq : {x : Ordinal } → (ax ax' : odef (* a) x) → {nc0 nc0' : IsImage b a gi x } → g⁻¹ ax nc0 ≡ g⁻¹ ax' nc0' g⁻¹-eq {x} ax ax' {record { y = y₁ ; ay = ay₁ ; x=fy = x=fy₁ }} {record { y = y ; ay = ay ; x=fy = x=fy }} = inject gi _ _ ay₁ ay (trans (sym x=fy₁) x=fy ) cc11-case2 : {x : Ordinal} (ax : odef (* a) x) → (ncn : ¬ gfImage x) → ¬ IsImage0 UC (* b) (λ x ax → f x (a∋gfImage ax)) (g⁻¹ ax (nimg ax ncn)) cc11-case2 {x} ax ncn record { y = y ; ay = ay ; x=fy = x=fy } with nimg ax ncn ... | record { y = y1 ; ay = ay1 ; x=fy = x=fy1 } = ncn ( subst (λ k → gfImage k) be113 (next-gf record { y = y ; ay = (a∋gfImage ay) ; x=fy = refl } ay ) ) where be113 : g (f y (a∋gfImage ay)) (b∋f y (a∋gfImage ay)) ≡ x be113 = begin g (f y (a∋gfImage ay)) (b∋f y (a∋gfImage ay)) ≡⟨ g-cong (sym x=fy) ⟩ g y1 ay1 ≡⟨ sym (x=fy1) ⟩ x ∎ where open ≡-Reasoning cc10-case2 : {x : Ordinal } → (bx : odef (* b) x ) → ¬ IsImage0 UC (* b) (λ x ax → f x (a∋gfImage ax)) x → ¬ gfImage (g x bx) cc10-case2 {x} bx nix (a-g ax ¬ib) = ¬ib record { y = _ ; ay = bx ; x=fy = refl } cc10-case2 {x} bx nix (next-gf record { y = y ; ay = ay ; x=fy = x=fy } gfy) = nix record { y = _ ; ay = gfy ; x=fy = inject gi _ _ bx (b∋f y (a∋gfImage gfy)) (trans x=fy (g-cong (f-cong refl))) } fU1 : (x : Ordinal) → odef UC x → Ordinal fU1 x gfx = f x (a∋gfImage gfx) f⁻¹ : (x : Ordinal) → IsImage0 UC (* b) FA x → Ordinal f⁻¹ x record { y = y ; ay = ay ; x=fy = x=fy } = y UC∋f⁻¹ : {x : Ordinal } → (lt : odef fUC x) → odef UC (f⁻¹ x lt ) UC∋f⁻¹ {x} record { y = y ; ay = ay ; x=fy = x=fy } = ay fU-iso1 : {x : Ordinal } → (lt : odef fUC x) → fU1 (f⁻¹ x lt) (UC∋f⁻¹ lt) ≡ x fU-iso1 {x} record { y = y ; ay = (a-g ax ¬ib) ; x=fy = x=fy } = trans (f-cong refl) (sym x=fy) fU-iso1 {x} record { y = y ; ay = (next-gf record { y = y₁ ; ay = ay₁ ; x=fy = x=fy₁ } ay) ; x=fy = x=fy } = trans (f-cong refl) (sym x=fy) fU-iso0 : {x : Ordinal } → (lt : odef UC x) → f⁻¹ (fU1 x lt) record { y = _ ; ay = lt ; x=fy = refl } ≡ x fU-iso0 {x} (a-g ax ¬ib) = refl fU-iso0 {x} (next-gf record { y = y ; ay = ay ; x=fy = x=fy } lt) = refl bi-UC : HODBijection UC fUC bi-UC = record { fun→ = λ x lt → f x (a∋gfImage lt) ; fun← = λ x lt → f⁻¹ x lt ; funB = λ x lt → record { y = _ ; ay = lt ; x=fy = refl } ; funA = λ x lt → UC∋f⁻¹ lt ; fiso→ = λ x lt → fU-iso1 lt ; fiso← = λ x lt → fU-iso0 lt } b-FUC∋g⁻¹ : {x : Ordinal } → (lt : odef a-UC x )→ odef b-fUC (g⁻¹ (proj1 lt) (nimg (proj1 lt) (proj2 lt))) b-FUC∋g⁻¹ {x} ⟪ ax , ngfix ⟫ = ⟪ b∋g⁻¹ ax (nimg ax ngfix) , cc11-case2 ax ngfix ⟫ a-UC∋g : {x : Ordinal } → (lt : odef b-fUC x) → odef a-UC (g x (proj1 lt )) a-UC∋g {x} ⟪ bx , ¬img ⟫ = ⟪ a∋g x bx , cc10-case2 bx ¬img ⟫ fUC-iso1 : {x : Ordinal } → (lt : odef b-fUC x ) → g⁻¹ (proj1 (a-UC∋g lt)) (nimg (proj1 (a-UC∋g lt)) (proj2 (a-UC∋g lt))) ≡ x fUC-iso1 {x} lt with nimg (proj1 (a-UC∋g lt)) (proj2 (a-UC∋g lt)) ... | record { y = y ; ay = ay ; x=fy = x=fy } = sym ( inject gi _ _ (proj1 lt) ay x=fy ) fUC-iso0 : {x : Ordinal} → (lt : odef a-UC x) → g (g⁻¹ (proj1 lt) (nimg (proj1 lt) (proj2 lt))) (proj1 (b-FUC∋g⁻¹ lt)) ≡ x fUC-iso0 {x} lt with nimg (proj1 lt) (proj2 lt) ... | record { y = y ; ay = ay ; x=fy = x=fy } = sym x=fy bi-fUC : HODBijection a-UC b-fUC bi-fUC = record { fun→ = λ x lt → g⁻¹ (proj1 lt) (nimg (proj1 lt) (proj2 lt)) ; fun← = λ x lt → g x (proj1 lt) ; funB = λ x lt → b-FUC∋g⁻¹ lt ; funA = λ x lt → a-UC∋g lt ; fiso→ = λ x lt → fUC-iso1 lt ; fiso← = λ x lt → fUC-iso0 lt } _c<_ : ( A B : HOD ) → Set n A c< B = ¬ ( Injection (& B) (& A) ) record Cardinal (a : Ordinal ) : Set (Level.suc n) where field card : Ordinal ciso : HODBijection (* a) (* card) cmax : (x : Ordinal) → card o< x → ¬ HODBijection (* a) (* x) -- Cardinal∈ : { s : HOD } → { t : Ordinal } → Ord t ∋ s → s c< Ord t -- Cardinal∈ = {!!} -- Cardinal⊆ : { s t : HOD } → s ⊆ t → ( s c< t ) ∨ ( s =c= t ) -- this is not valid -- Cardinal⊆ = {!!} -- we may have infinite sets with the same cardinality PtoF : {u : HOD} {x s : Ordinal } → odef (Power u) s → odef u x → Bool PtoF {u} {x} {s} su ux with ODC.p∨¬p O (odef (* s) x ) ... | case1 a = true ... | case2 b = false fun←cong : {P S : HOD} (b : HODBijection P S ) {x y : Ordinal } → {ax : odef S x} {ax1 : odef S y} → x ≡ y → fun← b x ax ≡ fun← b y ax1 fun←cong {P} {S} b {x} {x} {ax} {ax1} refl = cong (λ k → fun← b x k) ( HE.≅-to-≡ ( ∋-irr {S} ax ax1 )) fun→cong : {P S : HOD} (b : HODBijection P S ) {x y : Ordinal } → {ax : odef P x} {ax1 : odef P y} → x ≡ y → fun→ b x ax ≡ fun→ b y ax1 fun→cong {P} {S} b {x} {x} {ax} {ax1} refl = cong (λ k → fun→ b x k) ( HE.≅-to-≡ ( ∋-irr {P} ax ax1 )) -- S -- t ⊆ S ( Power S ∋ t ) -- S s₀ s₁ ... sn -- t true false ... false --- Cantor1 : ( S : HOD ) → S c< Power S Cantor1 S f = diag4 where f1 : Injection (& S) (& (Power S)) f1 = record { i→ = λ x sx → & (* x , * x) ; iB = c00 ; inject = c02 }where c02 : (x y : Ordinal) (ltx : odef (* (& S)) x) (lty : odef (* (& S)) y) → & (* x , * x) ≡ & (* y , * y) → x ≡ y c02 x y ltx lty eq = subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (xx=zy→x=y c03 )) where c03 : (* x , * x) =h= (* y , * y) c03 = ord→== eq c00 : (x : Ordinal) (lt : odef (* (& S)) x) → odef (* (& (Power S))) (& (* x , * x)) c00 x lt = subst₂ (λ j k → odef j (& k) ) (sym *iso) refl (λ y z → c01 y (subst (λ k → odef k y ) *iso z )) where c01 : (y : Ordinal ) → odef (* x , * x ) y → odef S y c01 y (case1 eq) = subst₂ (λ j k → odef j k ) *iso (trans (sym &iso) (sym eq) ) lt c01 y (case2 eq) = subst₂ (λ j k → odef j k ) *iso (trans (sym &iso) (sym eq) ) lt f2 : Injection (& (Power S)) (& S) f2 = f postulate -- yes we have a proof but it is very slow b : HODBijection (Power S) S -- b = subst₂ (λ j k → HODBijection j k) *iso *iso ( Bernstein f2 f1) -- this makes check very slow -- but postulate makes check weak eg. irrerevance of ∋ -- we have at least one element since Power S contains od∅ -- p0 : odef (Power S) o∅ p0 z xz = ⊥-elim (¬x<0 (subst (λ k → odef k z) o∅≡od∅ xz) ) s : Ordinal s = fun→ b o∅ p0 ss : odef S s ss = funB b o∅ p0 is-S : (S : HOD) → (x : Ordinal ) → Bool is-S S x with ODC.p∨¬p O (odef S x ) ... | case1 _ = true ... | case2 _ = false diag : {x : Ordinal} → (sx : odef S x) → Bool diag {x} sx = is-S (* (fun← b x sx)) x Diag : HOD Diag = record { od = record { def = λ x → odef S x ∧ ((sx : odef S x) → diag sx ≡ false) } ; odmax = & S ; <odmax = odef∧< } diag3 : odef (Power S) (& Diag) diag3 z xz with subst (λ k → odef k z) *iso xz ... | ⟪ sx , eq ⟫ = sx not-isD : (x : Ordinal) → (sn : odef S x) → not ( is-S (* (fun← b x sn )) x ) ≡ is-S Diag x not-isD x sn with ODC.p∨¬p O (odef (* (fun← b x sn )) x) | ODC.p∨¬p O (odef Diag x) | inspect (is-S (* (fun← b x sn ))) x ... | case1 lt | case1 ⟪ sx , eq ⟫ | record { eq = eq1 } = ⊥-elim (¬t=f false (trans (sym eq1) (eq sn )) ) ... | case1 lt | case2 lt1 | _ = refl ... | case2 lt | case1 lt1 | _ = refl ... | case2 lt | case2 neg | record { eq = eq1 } = ⊥-elim ( neg ⟪ sn , (λ sx → trans (cong diag ( HE.≅-to-≡ ( ∋-irr {S} sx sn ))) eq1 ) ⟫ ) diagn1 : (n : Ordinal ) → odef S n → ¬ (fun→ b (& Diag) diag3 ≡ n) diagn1 n sn dn = ¬t=f (is-S Diag n) (begin not (is-S Diag n) ≡⟨ cong (λ k → not (is-S k n)) (sym *iso) ⟩ not (is-S (* (& Diag)) n) ≡⟨ cong (λ k → not (is-S (* k) n)) (sym (fiso← b (& Diag) diag3 )) ⟩ not ( is-S (* (fun← b (fun→ b (& Diag) diag3) (funB b (& Diag) diag3 ))) n ) ≡⟨ cong (λ k → not (is-S (* k) n)) ( fun←cong b {_} {_} {funB b _ diag3} {sn} dn ) ⟩ not ( is-S (* (fun← b n sn )) n ) ≡⟨ not-isD _ sn ⟩ is-S Diag n ∎ ) where open ≡-Reasoning diag4 : ⊥ diag4 = diagn1 (fun→ b (& Diag) diag3) (funB b (& Diag) diag3) refl c<¬= : { u s : HOD } → u c< s → ¬ ( u =c= s ) c<¬= {u} {s} c<u ceq = c<u record { i→ = λ x lt → fun← ceq x (subst (λ k → odef k x) *iso lt) ; iB = λ x lt → subst₂ (λ j k → odef j k) (sym *iso) refl (funA ceq x (subst (λ k → odef k x) *iso lt)) ; inject = c04 } where c04 : (x y : Ordinal) (ltx : odef (* (& (s))) x) (lty : odef (* (& (s))) y) → fun← ceq x (subst (λ k → odef k x) *iso ltx) ≡ fun← ceq y (subst (λ k → odef k y) *iso lty) → x ≡ y c04 x y ltx lty eq = begin x ≡⟨ sym ( fiso→ ceq x c05 ) ⟩ fun→ ceq ( fun← ceq x c05 ) (funA ceq x c05) ≡⟨ fun←cong (hodbij-sym ceq) eq ⟩ fun→ ceq ( fun← ceq y c06 ) (funA ceq y c06) ≡⟨ fiso→ ceq y c06 ⟩ y ∎ where open ≡-Reasoning c05 : odef (s) x c05 = subst (λ k → odef k x) *iso ltx c06 : odef (s) y c06 = subst (λ k → odef k y) *iso lty Cantor2 : (u : HOD) → ¬ ( u =c= Power u ) Cantor2 u = c<¬= (Cantor1 u )