Mercurial > hg > Members > kono > Proof > ZF-in-agda
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Recursive record CN needs to be declared as either inductive or coinductive
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Fri, 30 Jun 2023 12:22:26 +0900 |
parents | cc76e2b1f3b5 |
children | 4b72bc3e2fab |
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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 hiding ( _⊆_ ) 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 _⊆_ : ( A B : HOD ) → Set n _⊆_ A B = {x : Ordinal } → odef A x → odef B x open _∧_ open _∨_ open Bool open _==_ open HOD record OrdBijection (A B : Ordinal ) : Set n where field fun← : (x : Ordinal ) → odef (* A) x → Ordinal fun→ : (x : Ordinal ) → odef (* B) 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 (* B) x ) → fun← ( fun→ x lt ) ( funA x lt ) ≡ x fiso→ : (x : Ordinal ) → ( lt : odef (* A) x ) → fun→ ( fun← x lt ) ( funB x lt ) ≡ x ordbij-refl : { a b : Ordinal } → a ≡ b → OrdBijection a b ordbij-refl {a} refl = record { fun← = λ x _ → x ; fun→ = λ x _ → x ; funB = λ x lt → lt ; funA = λ x lt → lt ; fiso← = λ x lt → refl ; fiso→ = λ x lt → refl } open Injection open OrdBijection record IsImage (a b : Ordinal) (iab : Injection a b ) (x : Ordinal ) : Set n where field y : Ordinal ay : odef (* a) y x=fy : x ≡ i→ iab _ ay 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)) ) record IsInverseImage (a b : Ordinal) (iab : Injection a b ) (x y : Ordinal ) : Set n where field ax : odef (* a) x x=fy : y ≡ i→ iab x ax InverseImage : {a : Ordinal} ( b : Ordinal ) → Injection a b → (y : Ordinal ) → HOD InverseImage {a} b iab y = record { od = record { def = λ x → IsInverseImage a b iab x y } ; odmax = & (* a) ; <odmax = im00 } where im00 : {x : Ordinal } → IsInverseImage a b iab x y → x o< & (* a) im00 {x} record { ax = ax ; x=fy = x=fy } = odef< ax 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 = OrdBijection ( & A ) ( & B ) c=→≡ : {A B : HOD} → A =c= B → (A ≡ ?) ∧ (B ≡ ?) c=→≡ = ? ≡→c= : {A B : HOD} → A ≡ B → A =c= B ≡→c= = ? 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 open Data.Nat 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) } Bernstein : {a b : Ordinal } → Injection a b → Injection b a → OrdBijection a b Bernstein {a} {b} (f @ record { i→ = fab ; iB = b∋fab ; inject = fab-inject }) ( g @ record { i→ = fba ; iB = a∋fba ; inject = fba-inject }) = record { fun← = λ x lt → h lt ; fun→ = λ x lt → h⁻¹ lt ; funB = be70 ; funA = be71 ; fiso← = ? ; fiso→ = ? } where gf : Injection a a gf = record { i→ = λ x ax → fba (fab x ax) (b∋fab x ax) ; iB = λ x ax → a∋fba _ (b∋fab x ax) ; inject = λ x y ax ay eq → fab-inject _ _ ax ay ( fba-inject _ _ (b∋fab _ ax) (b∋fab _ ay) eq) } data gfImage : (i : ℕ) (x : Ordinal) → Set n where a-g : {x : Ordinal} → (ax : odef (* a) x ) → (¬ib : ¬ ( IsImage b a g x )) → gfImage 0 x next-gf : {x : Ordinal} → {i : ℕ} → (ix : IsImage a a gf x) → (gfiy : gfImage i (IsImage.y ix) ) → gfImage (suc i) x a∋gfImage : (i : ℕ) → {x : Ordinal } → gfImage i x → odef (* a) x a∋gfImage 0 {x} (a-g ax ¬ib) = ax a∋gfImage (suc i) {x} (next-gf record { y = y ; ay = ay ; x=fy = x=fy } lt ) = subst (λ k → odef (* a) k ) (sym x=fy) (a∋fba _ (b∋fab y ay) ) C : ℕ → HOD -- Image {& (C i)} {a} (gf i) does not work C i = record { od = record { def = gfImage i } ; odmax = & (* a) ; <odmax = λ lt → odef< (a∋gfImage i lt) } record CN (x : Ordinal) : Set n where field i : ℕ gfix : gfImage i x UC : HOD UC = record { od = record { def = λ x → CN x } ; odmax = & (* a) ; <odmax = λ lt → odef< (a∋gfImage (CN.i lt) (CN.gfix lt)) } a-UC : HOD a-UC = record { od = record { def = λ x → odef (* a) x ∧ (¬ CN x) } ; odmax = & (* a) ; <odmax = λ lt → odef< (proj1 lt) } -- UC ⊆ * a -- f : UC → Image f UC is injection -- g : Image f UC → UC is injection UC⊆a : * (& UC) ⊆ * a UC⊆a {x} lt = a∋gfImage (CN.i be02) (CN.gfix be02) where be02 : CN x be02 = subst (λ k → odef k x) *iso lt a-UC⊆a : * (& a-UC) ⊆ * a a-UC⊆a {x} lt = proj1 ( subst (λ k → odef k x) *iso lt ) open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) fab-eq : {x y : Ordinal } → {ax : odef (* a) x} {ax1 : odef (* a) y} → x ≡ y → fab x ax ≡ fab y ax1 fab-eq {x} {x} {ax} {ax1} refl = cong (λ k → fab x k) ( HE.≅-to-≡ ( ∋-irr {* a} ax ax1 )) fba-eq : {x y : Ordinal } → {bx : odef (* b) x} {bx1 : odef (* b) y} → x ≡ y → fba x bx ≡ fba y bx1 fba-eq {x} {x} {bx} {bx1} refl = cong (λ k → fba x k) ( HE.≅-to-≡ ( ∋-irr {* b} bx bx1 )) UC∋gf : {y : Ordinal } → (uy : odef (* (& UC)) y ) → CN ( fba (fab y (UC⊆a uy) ) (b∋fab y (UC⊆a uy))) UC∋gf {y} uy = record { i = suc (CN.i uc00) ; gfix = next-gf record { y = _ ; ay = UC⊆a uy ; x=fy = fba-eq (fab-eq refl) } (CN.gfix uc00) } where uc00 : CN y uc00 = subst (λ k → odef k y) *iso uy g⁻¹ : {x : Ordinal } → (ax : odef (* a) x) → ¬ odef (C 0) x → Ordinal g⁻¹ {x} ax nc0 with ODC.p∨¬p O ( IsImage b a g x ) ... | case1 record { y = y ; ay = ay ; x=fy = x=fy } = y ... | case2 ¬ism = ⊥-elim ( nc0 ( a-g ax ¬ism ) ) b∋g⁻¹ : {x : Ordinal } → (ax : odef (* a) x) → (nc0 : ¬ odef (C 0) x) → odef (* b) (g⁻¹ ax nc0) b∋g⁻¹ {x} ax nc0 with ODC.p∨¬p O ( IsImage b a g x ) ... | case1 record { y = y ; ay = ay ; x=fy = x=fy } = ay ... | case2 ¬ism = ⊥-elim ( nc0 ( a-g ax ¬ism ) ) g⁻¹-iso : {x : Ordinal } → (ax : odef (* a) x) → (nc0 : ¬ odef (C 0) x ) → fba (g⁻¹ ax nc0) (b∋g⁻¹ ax nc0) ≡ x g⁻¹-iso {x} ax nc0 with ODC.p∨¬p O ( IsImage b a g x ) ... | case1 record { y = y ; ay = ay ; x=fy = x=fy } = sym x=fy ... | case2 ¬ism = ⊥-elim ( nc0 ( a-g ax ¬ism ) ) g⁻¹-iso1 : {x : Ordinal } → (bx : odef (* b) x) → (nc0 : ¬ odef (C 0) (fba x bx) ) → g⁻¹ (a∋fba x bx) nc0 ≡ x g⁻¹-iso1 {x} bx nc0 = inject g _ _ (b∋g⁻¹ (a∋fba x bx) nc0) bx (g⁻¹-iso (a∋fba x bx) nc0) be10 : Injection (& a-UC) (b - (& (Image (& UC) (Injection-⊆ UC⊆a f) )) ) -- g⁻¹ x be10 = record { i→ = λ x lt → g⁻¹ (be15 lt) (be16 lt) ; iB = be17 ; inject = be18 } where be15 : {x : Ordinal } → odef (* (& a-UC)) x → odef (* a) x be15 {x} lt with subst (λ k → odef k x) *iso lt ... | ⟪ ax , ncn ⟫ = ax be16 : {x : Ordinal } → odef (* (& a-UC)) x → ¬ odef (C 0) x be16 {x} lt nc0 with subst (λ k → odef k x) *iso lt ... | ⟪ ax , ncn ⟫ = ⊥-elim ( ncn record { i = 0 ; gfix = nc0 } ) be17 : (x : Ordinal) (lt : odef (* (& a-UC)) x) → odef (* (b - & (Image (& UC) (Injection-⊆ UC⊆a f)))) (g⁻¹ (be15 lt) (be16 lt)) be17 x lt = subst ( λ k → odef k (g⁻¹ (be15 lt) (be16 lt))) (sym *iso) ⟪ be19 , (λ img → be18 be14 (subst (λ k → odef k (g⁻¹ (be15 lt) (be16 lt))) *iso img) ) ⟫ where be14 : odef a-UC x be14 = subst (λ k → odef k x) *iso lt be19 : odef (* b) (g⁻¹ (be15 lt) (be16 lt)) be19 = b∋g⁻¹ (be15 lt) (be16 lt) be18 : odef a-UC x → ¬ odef (Image (& UC) (Injection-⊆ UC⊆a f)) (g⁻¹ (be15 lt) (be16 lt)) be18 ⟪ ax , ncn ⟫ record { y = y ; ay = ay ; x=fy = x=fy } = ncn ( subst (λ k → CN k) be13 (UC∋gf ay) ) where be13 : fba (fab y (UC⊆a ay)) (b∋fab y (UC⊆a ay)) ≡ x be13 = begin fba (fab y (UC⊆a ay)) (b∋fab y (UC⊆a ay)) ≡⟨ fba-eq (sym x=fy) ⟩ fba (g⁻¹ (be15 lt) (be16 lt)) be19 ≡⟨ g⁻¹-iso (be15 lt) (be16 lt) ⟩ x ∎ where open ≡-Reasoning be18 : (x y : Ordinal) (ltx : odef (* (& a-UC)) x) (lty : odef (* (& a-UC)) y) → g⁻¹ (be15 ltx) (be16 ltx) ≡ g⁻¹ (be15 lty) (be16 lty) → x ≡ y be18 = ? be11 : Injection (b - (& (Image (& UC) (Injection-⊆ UC⊆a f) ))) (& a-UC) -- g x be11 = record { i→ = be13 ; iB = be14 ; inject = ? } where be13 : (x : Ordinal) → odef (* (b - & (Image (& UC) (Injection-⊆ UC⊆a f)))) x → Ordinal be13 x lt = fba x ( proj1 ( subst (λ k → odef k x) (*iso) lt )) be14 : (x : Ordinal) (lt : odef (* (b - & (Image (& UC) (Injection-⊆ UC⊆a f)))) x) → odef (* (& a-UC)) (be13 x lt) be14 x lt = subst (λ k → odef k (be13 x lt)) (sym *iso) ⟪ a∋fba x ( proj1 ( subst (λ k → odef k x) (*iso) lt )) , be15 ⟫ where be16 : ¬ (odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x) be16 = proj2 ( subst (λ k → odef k x) (*iso) lt ) be15 : ¬ CN (be13 x lt) be15 cn with CN.i cn | CN.gfix cn ... | 0 | a-g ax ¬ib = ⊥-elim (¬ib record { y = _ ; ay = proj1 ( subst (λ k → odef k x) (*iso) lt ) ; x=fy = refl } ) ... | suc i | next-gf record { y = y ; ay = ay ; x=fy = x=fy } t = ⊥-elim (be16 (subst (λ k → odef k x) (sym *iso) record { y = y ; ay = subst (λ k → odef k y) (sym *iso) record { i = i ; gfix = t } ; x=fy = be17 })) where be17 : x ≡ fab y (UC⊆a (subst (λ k → odef k y) (sym *iso) (record { i = i ; gfix = t }))) be17 = trans (inject g _ _ (proj1 ( subst (λ k → odef k x) (*iso) lt )) (b∋fab y ay) x=fy) (fab-eq refl) a-UC-iso0 : (x : Ordinal ) → (cx : odef (* (& a-UC)) x ) → i→ be11 ( i→ be10 x cx ) (iB be10 x cx) ≡ x a-UC-iso0 x cx = trans (fba-eq refl) ( g⁻¹-iso (proj1 ( subst (λ k → odef k x) (*iso) cx )) (λ cn → ⊥-elim (proj2 ( subst (λ k → odef k x) (*iso) cx ) record { i = 0 ; gfix = cn} ) )) a-UC-iso1 : (x : Ordinal ) → (cx : odef (* (b - (& (Image (& UC) (Injection-⊆ UC⊆a f) )))) x ) → i→ be10 ( i→ be11 x cx ) (iB be11 x cx) ≡ x a-UC-iso1 x cx with ODC.p∨¬p O ( IsImage b a g (fba x (proj1 ( subst (λ k → odef k x) (*iso) cx ))) ) ... | case1 record { y = y ; ay = ay ; x=fy = x=fy } = sym ( inject g _ _ (proj1 ( subst (λ k → odef k x) (*iso) cx )) ay x=fy ) ... | case2 ¬ism = ⊥-elim (¬ism record { y = x ; ay = proj1 ( subst (λ k → odef k x) (*iso) cx ) ; x=fy = refl }) -- C n → f (C n) fU : (x : Ordinal) → ( odef (* (& UC)) x) → Ordinal fU x lt = be03 where be02 : CN x be02 = subst (λ k → odef k x) *iso lt be03 : Ordinal be03 with CN.i be02 | CN.gfix be02 ... | zero | a-g {x} ax ¬ib = fab x ax ... | suc i | next-gf record { y = y ; ay = ay ; x=fy = x=fy } gfiy = fab x (subst (λ k → odef (* a) k) (sym x=fy) (a∋fba _ (b∋fab y ay) )) -- f (C n) → g (f (C n) ) ≡ C (suc i) Uf : (x : Ordinal) → odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x → Ordinal Uf x lt with subst (λ k → odef k x ) *iso lt ... | record { y = y ; ay = ay ; x=fy = x=fy } = y be04 : {x : Ordinal } → (cx : odef (* (& UC)) x) → odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) (fU x cx) be04 {x} cx = subst (λ k → odef k (fU x cx) ) (sym *iso) be06 where be02 : CN x be02 = subst (λ k → odef k x) *iso cx be06 : odef (Image (& UC) (Injection-⊆ UC⊆a f)) (fU x cx) be06 with CN.i be02 | CN.gfix be02 ... | zero | a-g ax ¬ib = record { y = x ; ay = subst (λ k → odef k x) (sym *iso) be02 ; x=fy = fab-eq refl } ... | suc i | next-gf record { y = y ; ay = ay ; x=fy = x=fy } gfiy = record { y = x ; ay = subst (λ k → odef k x) (sym *iso) be02 ; x=fy = fab-eq refl } UC-iso0 : (x : Ordinal ) → (cx : odef (* (& UC)) x ) → Uf ( fU x cx ) (be04 cx) ≡ x UC-iso0 x cx = be03 where be02 : CN x be02 = subst (λ k → odef k x) *iso cx be03 : Uf ( fU x cx ) (be04 cx) ≡ x be03 with CN.i be02 | CN.gfix be02 | be04 cx ... | zero | a-g ax ¬ib | cb with subst (λ k → odef k _ ) *iso cb | inspect (fU x) cx ... | record { y = y ; ay = ay ; x=fy = x=fy } | record { eq = refl } = sym ( inject f _ _ ax (UC⊆a ay) x=fy ) be03 | suc i | next-gf record { y = x1 ; ay = ax1 ; x=fy = x=fx1 } s | cb with subst (λ k → odef k (fab x (subst (odef (* a)) (sym x=fx1) (a∋fba (fab x1 ax1) (b∋fab x1 ax1)))) ) *iso cb ... | record { y = y ; ay = ay ; x=fy = x=fy } = sym ( inject f _ _ ax (UC⊆a ay) x=fy ) where ax : odef (* a) x ax = subst (λ k → odef (* a) k ) (sym x=fx1) ( a∋fba _ (b∋fab x1 ax1) ) be08 : {x : Ordinal } → (cx : odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x) → odef (* (& UC)) (Uf x cx) be08 {x} cx with subst (λ k → odef k x) *iso cx ... | record { y = y ; ay = ay ; x=fy = x=fy } = ay UC-iso1 : (x : Ordinal ) → (cx : odef (* (& (Image (& UC) {b} (Injection-⊆ UC⊆a f)))) x ) → fU ( Uf x cx ) (be08 cx) ≡ x UC-iso1 x cx = be14 where be14 : fU ( Uf x cx ) (be08 cx) ≡ x be14 with subst (λ k → odef k x) *iso cx ... | record { y = y ; ay = ay ; x=fy = x=fy } with CN.i (subst (λ k → OD.def (od k) y) *iso ay) | CN.gfix (subst (λ k → OD.def (od k) y) *iso ay) ... | 0 | a-g ax ¬ib = sym x=fy ... | (suc i) | next-gf t ix = sym x=fy h : {x : Ordinal } → (ax : odef (* a) x) → Ordinal h {x} ax with ODC.∋-p O UC (* x) ... | yes cn = fU x (subst (λ k → odef k x ) (sym *iso) (subst (λ k → CN k) &iso cn) ) ... | no ncn = i→ be10 x (subst (λ k → odef k x ) (sym *iso) ⟪ ax , subst (λ k → ¬ (CN k)) &iso ncn ⟫ ) h⁻¹ : {x : Ordinal } → (bx : odef (* b) x) → Ordinal h⁻¹ {x} bx with ODC.p∨¬p O (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) ... | case1 cn = Uf x (subst (λ k → odef k x) (sym *iso) cn) -- x ≡ f y ... | case2 ncn = i→ be11 x (subst (λ k → odef k x ) (sym *iso) be60 ) where -- ¬ x ≡ f y be60 : odef (* b \ * (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x be60 = ⟪ bx , subst (λ k → ¬ odef k x ) (sym *iso) ncn ⟫ be70 : (x : Ordinal) (lt : odef (* a) x) → odef (* b) (h lt) be70 x ax = ? -- with ODC.p∨¬p O ( CN x ) --... | case1 cn = be03 (subst (λ k → odef k x) (sym *iso) cn) where -- make the same condition for Uf -- be03 : (cn : odef (* (& UC)) x) → odef (* b) (fU x cn ) -- be03 cn with CN.i (subst (λ k → odef k x) *iso cn) | CN.gfix (subst (λ k → odef k x) *iso cn ) -- ... | zero | a-g ax ¬ib = b∋fab x ax -- ... | suc i | next-gf record { y = y ; ay = ay ; x=fy = x=fy } gfiy = b∋fab x -- (subst (odef (* a)) (sym x=fy) (a∋fba (fab y ay) (b∋fab y ay))) --... | case2 ncn = proj1 (subst₂ (λ j k → odef j k ) *iso refl (iB be10 x (subst (λ k → odef k x ) (sym *iso) ⟪ ax , ncn ⟫ ))) be71 : (x : Ordinal) (bx : odef (* b) x) → odef (* a) (h⁻¹ bx) be71 x bx with ODC.p∨¬p O (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) ... | case1 cn = be03 (subst (λ k → odef k x) (sym *iso) cn) where be03 : (cn : odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x) → odef (* a) (Uf x cn ) be03 cn with subst (λ k → odef k x ) *iso cn ... | record { y = y ; ay = ay ; x=fy = x=fy } = UC⊆a ay ... | case2 ncn = proj1 (subst₂ (λ j k → odef j k ) *iso refl (iB be11 x (subst (λ k → odef k x) (sym *iso) be60) )) where be60 : odef (* b \ * (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x be60 = ⟪ bx , subst (λ k → ¬ odef k x ) (sym *iso) ncn ⟫ be72 : (x : Ordinal) (bx : odef (* b) x) → h (be71 x bx) ≡ x be72 x bx = ? where be76 : (cn : odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) → h⁻¹ bx ≡ Uf x (subst (λ k → odef k x) (sym *iso) cn) be76 cn with ODC.p∨¬p O (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) ... | case1 img = cong (λ k → Uf x k ) ( HE.≅-to-≡ ( ∋-irr {(* (& (Image (& UC) (Injection-⊆ UC⊆a f))))} b04 b05 )) where b04 : odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x b04 = subst (λ k → odef k x) (sym *iso) img b05 : odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x b05 = subst (λ k → odef k x) (sym *iso) cn ... | case2 nimg = ⊥-elim ( nimg cn) be73 : (cn : odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) → odef (* a) (Uf x (subst (λ k → odef k x) (sym *iso) cn)) be73 cn with ODC.p∨¬p O (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) ... | case1 img = be03 (subst (λ k → odef k x) (sym *iso) cn) where be03 : (cn : odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x) → odef (* a) (Uf x cn ) be03 cn with subst (λ k → odef k x ) *iso cn ... | record { y = y ; ay = ay ; x=fy = x=fy } = UC⊆a ay ... | case2 nimg = ⊥-elim ( nimg cn) be60 : (ncn : ¬ (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x)) → odef (* b \ * (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x be60 ncn = ⟪ bx , subst (λ k → ¬ odef k x ) (sym *iso) ncn ⟫ be74 : (ncn : ¬ (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x)) → odef (* a) (i→ be11 x (subst (λ k → odef k x ) (sym *iso) (be60 ncn) )) be74 ncn with ODC.p∨¬p O (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) ... | case1 img = ⊥-elim ( ncn img ) ... | case2 nimg = proj1 (subst₂ (λ j k → odef j k ) *iso refl (iB be11 x (subst (λ k → odef k x) (sym *iso) (be60 ncn)) )) be75 : h (be71 x bx) ≡ x be75 with ODC.p∨¬p O (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) ... | case1 cn = trans ? (be78 (be73 cn)) where be78 : (auf : odef (* a) (Uf x (subst (λ k → odef k x) (sym *iso) cn))) → h auf ≡ x be78 = ? ... | case2 ncn = ? _c<_ : ( A B : HOD ) → Set n A c< B = ¬ ( Injection (& A) (& B) ) Card : OD Card = record { def = λ x → (a : Ordinal) → a o< x → ¬ OrdBijection a x } record Cardinal (a : Ordinal ) : Set (Level.suc n) where field card : Ordinal ciso : OrdBijection a card cmax : (x : Ordinal) → card o< x → ¬ OrdBijection 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 ) Cardinal⊆ = {!!} Cantor1 : { u : HOD } → u c< Power u Cantor1 = {!!} Cantor2 : { u : HOD } → ¬ ( u =c= Power u ) Cantor2 = {!!}