Mercurial > hg > Members > kono > Proof > ZF-in-agda
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 01 Jul 2023 10:19:03 +0900 |
| parents | cdfe297f9a79 |
| children | 77a3de21ee50 |
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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 (cc0 x) ; fun→ = λ x lt → h⁻¹ lt (cc1 x) ; funB = be74 ; funA = be75 ; fiso← = λ x lt → be72 x lt (cc11 (a∋fba x lt) (cc20 lt)) ; fiso→ = λ x lt → be73 x lt (cc10 (b∋fab x lt) (cc21 lt)) } 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 : (x : Ordinal) → Set n where a-g : {x : Ordinal} → (ax : odef (* a) x ) → (¬ib : ¬ ( IsImage b a g x )) → gfImage x next-gf : {x : Ordinal} → (ix : IsImage a a gf x) → (gfiy : gfImage (IsImage.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∋fba _ (b∋fab 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) } nimg : {x : Ordinal } → (ax : odef (* a) x ) → ¬ (odef UC x) → IsImage b a g x nimg {x} ax ncn with ODC.p∨¬p O (IsImage b a g x) ... | case1 img = img ... | case2 nimg = ⊥-elim ( ncn (a-g ax nimg ) ) aucimg : {x : Ordinal } → (cx : odef (* (& a-UC)) x ) → IsImage b a g x aucimg {x} cx with subst (λ k → odef k x ) *iso cx ... | ⟪ ax , ncn ⟫ = nimg ax ncn -- 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 be02 where be02 : gfImage 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 ) → gfImage ( fba (fab y (UC⊆a uy) ) (b∋fab y (UC⊆a uy))) UC∋gf {y} uy = next-gf record { y = _ ; ay = UC⊆a uy ; x=fy = fba-eq (fab-eq refl) } uc00 where uc00 : gfImage y uc00 = subst (λ k → odef k y) *iso uy g⁻¹ : {x : Ordinal } → (ax : odef (* a) x) → (nc0 : IsImage b a g 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 g 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 g x ) → fba (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 g (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 → IsImage b a g x be16 {x} lt with subst (λ k → odef k x) *iso lt ... | ⟪ ax , ncn ⟫ = nimg ax ncn 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 → gfImage 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 : ¬ gfImage (be13 x lt) be15 cn with cn ... | a-g ax ¬ib = ⊥-elim (¬ib record { y = _ ; ay = proj1 ( subst (λ k → odef k x) (*iso) lt ) ; x=fy = refl } ) ... | 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) t ; x=fy = be17 })) where be17 : x ≡ fab y (UC⊆a (subst (λ k → odef k y) (sym *iso) 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 )) (aucimg cx)) 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 : gfImage x be02 = subst (λ k → odef k x) *iso lt be03 : Ordinal be03 with be02 ... | a-g {x} ax ¬ib = fab x ax ... | 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 : gfImage x be02 = subst (λ k → odef k x) *iso cx be06 : odef (Image (& UC) (Injection-⊆ UC⊆a f)) (fU x cx) be06 with be02 ... | a-g ax ¬ib = record { y = x ; ay = subst (λ k → odef k x) (sym *iso) be02 ; x=fy = fab-eq refl } ... | 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 : gfImage x be02 = subst (λ k → odef k x) *iso cx be03 : Uf ( fU x cx ) (be04 cx) ≡ x be03 with be02 | be04 cx ... | 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 | 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 subst (λ k → OD.def (od k) y) *iso ay ... | a-g ax ¬ib = sym x=fy ... | next-gf t ix = sym x=fy UC-iso11 : (x : Ordinal ) → (cx : odef (* (& (Image (& UC) {b} (Injection-⊆ UC⊆a f)))) x ) → (ux : odef (* (& UC)) (Uf x cx)) → fU ( Uf x cx ) ux ≡ x UC-iso11 x cx ux = subst (λ k → fU (Uf x cx) k ≡ x) ( HE.≅-to-≡ ( ∋-irr {* (& UC)} {_} (be08 cx) ux)) (UC-iso1 x cx) CC0 : (x : Ordinal) → Set n CC0 x = gfImage x ∨ (¬ gfImage x) CC1 : (x : Ordinal) → Set n CC1 x = odef (Image (& UC) (Injection-⊆ UC⊆a f)) x ∨ (¬ odef (Image (& UC) (Injection-⊆ UC⊆a f)) x) cc0 : (x : Ordinal) → CC0 x cc0 x = ODC.p∨¬p O (gfImage x) cc1 : (x : Ordinal) → CC1 x cc1 x = ODC.p∨¬p O (odef (Image (& UC) (Injection-⊆ UC⊆a f)) x) cc20 : {x : Ordinal } → (lt : odef (* b) x) → CC0 (fba x lt ) cc20 = ? cc21 : {x : Ordinal } → (lt : odef (* a) x) → CC1 (fab x lt ) cc21 = ? -- -- h : * a → * b -- h⁻¹ : * b → * a -- -- it have to accept any alement in a or b -- it is handle by different function fU or gU with exclusive conditon CC0 (p ∨ ¬ p) -- in OrdBijection record, LEM rules are used -- but we cannot use LEM in second function call on iso parts. -- so we have to do some devices. -- -- otherwise not strict positive on gfImage error will happen h : {x : Ordinal } → (ax : odef (* a) x) → gfImage x ∨ (¬ gfImage x) → Ordinal h {x} ax (case1 cn) = fU x (subst (λ k → odef k x ) (sym *iso) cn ) h {x} ax (case2 ncn) = i→ be10 x (subst (λ k → odef k x ) (sym *iso) ⟪ ax , ncn ⟫ ) h⁻¹ : {x : Ordinal } → (bx : odef (* b) x) → (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) ∨ ( ¬ (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x)) → Ordinal h⁻¹ {x} bx ( case1 cn) = Uf x (subst (λ k → odef k x) (sym *iso) cn) -- x ≡ f y h⁻¹ {x} bx ( 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) → (or : gfImage x ∨ (¬ gfImage x )) → odef (* b) (h lt or ) be70 x ax ( 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 subst (λ k → odef k x) *iso cn ... | a-g ax ¬ib = b∋fab x ax ... | 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))) be70 x ax ( 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) → (or : (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) ∨ ( ¬ (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x) )) → odef (* a) (h⁻¹ bx or ) be71 x bx ( 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 be71 x bx ( 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 ⟫ be71-1 : (x : Ordinal) (bx : odef (* b) x) → (or : (odef ( Image (& UC) (Injection-⊆ UC⊆a f)) x)) → gfImage (h⁻¹ bx (case1 or) ) be71-1 x bx cn = subst₂ (λ j k → odef j k ) *iso refl (be08 (subst (λ k → odef k x) (sym *iso) cn)) be70-1 : (x : Ordinal) (lt : odef (* a) x) → (or : gfImage x ∨ (¬ gfImage x )) → odef ( Image (& UC) (Injection-⊆ UC⊆a f)) (h lt or ) be70-1 = ? cc10 : {x : Ordinal} → (bx : odef (* b) x) → (cc1 : CC1 x) → CC0 (h⁻¹ bx cc1 ) cc10 {x} bx (case1 x₁) = case1 ? cc10 {x} bx (case2 x₁) = ? cc11 : {x : Ordinal} → (ax : odef (* a) x) → (cc0 : CC0 x) → CC1 (h ax cc0 ) cc11 {x} bx (case1 x₁) = case1 (subst₂ (λ j k → odef j k ) *iso refl (be04 (subst (λ k → odef k x) (sym *iso) x₁))) cc11 {x} bx (case2 x₁) = ? be74 : (x : Ordinal) (ax : odef (* a) x) → odef (* b) (h ax (cc0 x)) be74 x ax with cc0 x ... | case1 lt1 = ? ... | case2 lt1 with iB be10 ? ? ... | t = ? be75 : (x : Ordinal) (bx : odef (* b) x) → odef (* a) (h⁻¹ bx (cc1 x)) be75 x lt with cc1 x ... | case1 lt1 = ? ... | case2 lt1 = ? fba-image : {x : Ordinal } → (bx : odef (* b) x) → odef (Image b g) (fba x bx) fba-image {x} bx = record { y = _ ; ay = bx ; x=fy = refl } ImageUnique : {a b x : Ordinal } → {F : Injection a b} → (i j : odef (Image a F) x ) → IsImage.y i ≡ IsImage.y j ImageUnique {a} {b} {x} {F} i j = inject F _ _ (IsImage.ay i) (IsImage.ay j) (trans (sym (IsImage.x=fy i)) (IsImage.x=fy j)) be72 : (x : Ordinal) (bx : odef (* b) x) → (cc1 : CC1 x) → h (be71 x bx cc1 ) (cc10 bx cc1) ≡ x be72 x bx (case1 (x₁ @ record { y = y ; ay = ay ; x=fy = x=fy })) = UC-iso11 x be76 (subst (λ k → odef k (IsImage.y (subst (λ k → odef k x) *iso be76))) (sym *iso) be77 ) where be76 : odef (* (& (Image (& UC) (Injection-⊆ UC⊆a f)))) x be76 = subst (λ k → odef k x) (sym *iso) x₁ be78 : y ≡ IsImage.y (subst (λ k → odef k x) *iso be76) be78 = ImageUnique x₁ (subst (λ k → odef k x) *iso be76) be77 : odef UC (IsImage.y (subst (λ k → odef k x) *iso be76)) be77 = subst₂ (λ j k → odef j k ) *iso be78 ay be72 x bx (case2 x₁) with cc10 bx (case2 x₁) ... | case1 (a-g ax ¬ib) = ⊥-elim (¬ib record { y = x ; ay = bx ; x=fy = (fba-eq refl) } ) ... | case1 (next-gf record { y = y ; ay = ay ; x=fy = x=fy } c1) = ⊥-elim (x₁ record { y = y ; ay = subst (λ k → odef k y) (sym *iso) c1 ; x=fy = inject g _ _ _ (b∋fab y _) (trans x=fy (fba-eq (fab-eq refl))) }) ... | case2 c2 = be80 where be79 : odef (* (b - (& (Image (& UC) (Injection-⊆ UC⊆a f) )) )) x be79 = subst (λ k → odef k x) (sym *iso) ⟪ bx , subst (λ k → odef k x → ⊥) (sym *iso) x₁ ⟫ bx1 : odef (* b) x bx1 = proj1 (subst (λ k → odef k x) *iso be79) be77 : odef (Image b g) (fba x (proj1 (subst (λ k → odef k x) *iso be79)) ) be77 = nimg (a∋fba x (proj1 (subst (λ k → odef k x) *iso be79))) c2 be80 : g⁻¹ ? ? ≡ x be80 = ? -- with ODC.p∨¬p O ( IsImage b a g (fba x bx )) -- ... | case1 (img @ record { y = y ; ay = ay ; x=fy = x=fy }) = trans (ImageUnique ? ? ) (sym ( inject g _ _ bx ay x=fy )) where -- be79 : odef (* b) x -- be79 = proj1 (subst (λ k → OD.def (od k) x) *iso (subst (λ k → OD.def (od k) x) (sym *iso) -- ⟪ bx , subst (λ k → OD.def (od k) x → ⊥) (sym *iso) x₁ ⟫)) -- be78 : odef (Image b g) ? -- be78 = nimg (a∋fba ? ?) c2 -- be77 : odef (Image b g) (fba x (proj1 (subst (λ k → OD.def (od k) x) *iso (subst (λ k → OD.def (od k) x) (sym *iso) -- ⟪ bx , subst (λ k → OD.def (od k) x → ⊥) (sym *iso) x₁ ⟫)))) -- be77 = nimg (a∋fba ? ?) c2 -- ... | case2 ¬img = ? be73 : (x : Ordinal) (ax : odef (* a) x) → (cc0 : CC0 x ) → h⁻¹ (be70 x ax cc0) (cc11 ax cc0) ≡ x be73 x ax (case1 x₁) with cc11 ax (case1 x₁) ... | case1 c1 = ? where be76 : IsImage.y (subst (λ k → odef k (fU x (subst (λ k₁ → odef k₁ x) (sym *iso) x₁))) *iso (be04 (subst (λ k → odef k x) (sym *iso) x₁))) ≡ x be76 = UC-iso0 x (subst (λ k → odef k x) (sym *iso) x₁ ) ... | case2 c2 = ? be73 x ax (case2 x₁) with cc11 ax (case2 x₁) ... | case1 c1 = ⊥-elim ( x₁ ? ) ... | case2 c2 = trans ? (a-UC-iso0 x ? ) _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 = {!!}