Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 297:be6670af87fa
maxod try
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 22 Jun 2020 16:43:31 +0900 |
| parents | 42f89e5efb00 |
| children | 3795ffb127d0 |
| files | OD.agda |
| diffstat | 1 files changed, 69 insertions(+), 61 deletions(-) [+] |
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--- a/OD.agda Mon Jun 15 18:15:48 2020 +0900 +++ b/OD.agda Mon Jun 22 16:43:31 2020 +0900 @@ -70,15 +70,18 @@ record ODAxiom : Set (suc n) where -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) field + maxod : Ordinal od→ord : OD → Ordinal - ord→od : Ordinal → OD + ord→od : (x : Ordinal ) → x o< maxod → OD + o<max : {x : OD } → od→ord x o< maxod c<→o< : {x y : OD } → def y ( od→ord x ) → od→ord x o< od→ord y - oiso : {x : OD } → ord→od ( od→ord x ) ≡ x - diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x + oiso : {x : OD } → ord→od ( od→ord x ) o<max ≡ x + diso : {x : Ordinal } → (lt : x o< maxod) → od→ord ( ord→od x lt ) ≡ x ==→o≡ : { x y : OD } → (x == y) → x ≡ y -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum ) sup-o : ( OD → Ordinal ) → Ordinal sup-o< : { ψ : OD → Ordinal } → ∀ {x : OD } → ψ x o< sup-o ψ + sup-<od : { ψ : OD → OD } → ∀ {x : OD } → sup-o (λ x → od→ord (ψ x)) o< maxod -- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use -- sup-x : {n : Level } → ( OD → Ordinal ) → Ordinal -- sup-lb : {n : Level } → { ψ : OD → Ordinal } → {z : Ordinal } → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) @@ -93,8 +96,8 @@ Ords : OD Ords = record { def = λ x → One } -maxod : {x : OD} → od→ord x o< od→ord Ords -maxod {x} = c<→o< OneObj +-- maxod : {x : OD} → od→ord x o< od→ord Ords +-- maxod {x} = c<→o< OneObj -- Ordinal in OD ( and ZFSet ) Transitive Set Ord : ( a : Ordinal ) → OD @@ -104,12 +107,12 @@ od∅ = Ord o∅ -o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y) x ) → {x : OD } → x ≡ Ord (od→ord x) -o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where - lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y - lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → def x k ) (sym diso) lt)) - lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y - lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt ) +-- o<→c<→OD=Ord : ( {x y : Ordinal } → x o< y → def (ord→od y {!!} ) x ) → {x : OD } → x ≡ Ord (od→ord x) +-- o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where +-- lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y +-- lemma1 {y} lt = subst ( λ k → k o< od→ord x ) (diso {!!}) (c<→o< {ord→od y {!!} } {x} (subst (λ k → def x k ) (sym (diso {!!})) lt)) +-- lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y +-- lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt ) _∋_ : ( a x : OD ) → Set n _∋_ a x = def a ( od→ord x ) @@ -129,50 +132,55 @@ sup-c< : ( ψ : OD → OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x )) sup-c< ψ {x} = def-subst {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ x)) )} {od→ord ( ψ x )} lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where - lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ x)) - lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst (sup-o< ) refl (sym diso) ) + lemma : od→ord (ψ (ord→od (od→ord x) o<max )) o< sup-o (λ x → od→ord (ψ x)) + lemma = subst₂ (λ j k → j o< k ) refl (diso (sup-<od {ψ} {x}) ) (o<-subst (sup-o< ) refl (sym (diso sup-<od))) otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y otrans x<a y<x = ordtrans y<x x<a -def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X -def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym diso) )) diso diso +-- def→o< : {X : OD } → {x : Ordinal } → def X x → x o< od→ord X +-- def→o< {X} {x} lt = o<-subst {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {X} {x} lt (sym oiso) (sym (diso lemma)))) (diso lemma) (diso o<max) where +-- lemma : x o< maxod +-- lemma = subst (λ k → k o< maxod ) (diso {!!} ) (otrans o<max ( c<→o< lt )) -- avoiding lv != Zero error orefl : { x : OD } → { y : Ordinal } → od→ord x ≡ y → od→ord x ≡ y orefl refl = refl -==-iso : { x y : OD } → ord→od (od→ord x) == ord→od (od→ord y) → x == y +==-iso : { x y : OD } → ord→od (od→ord x) o<max == ord→od (od→ord y) o<max → x == y ==-iso {x} {y} eq = record { eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } where - lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z + lemma : {x : OD } {z : Ordinal } → def (ord→od (od→ord x) o<max ) z → def x z lemma {x} {z} d = def-subst d oiso refl -=-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) == y) +=-iso : {x y : OD } → (x == y) ≡ (ord→od (od→ord x) o<max == y) =-iso {_} {y} = cong ( λ k → k == y ) (sym oiso) +<-irr : {x y z : Ordinal } → x ≡ y → (x o< z) ≡ (y o< z) +<-irr refl = refl + ord→== : { x y : OD } → od→ord x ≡ od→ord y → x == y -ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where - lemma : ( ox oy : Ordinal ) → ox ≡ oy → (ord→od ox) == (ord→od oy) - lemma ox ox refl = ==-refl +ord→== {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq) o<max o<max ) where + lemma : ( ox oy : Ordinal ) → ox ≡ oy → (x<m : ox o< maxod) (y<m : oy o< maxod) → (ord→od ox x<m ) == (ord→od oy y<m ) + lemma ox ox refl x<m y<m = subst (λ k → ord→od ox x<m == ord→od ox k) {!!} ==-refl -o≡→== : { x y : Ordinal } → x ≡ y → ord→od x == ord→od y +o≡→== : { x y : Ordinal } → x ≡ y → ord→od x {!!} == ord→od y {!!} o≡→== {x} {.x} refl = ==-refl -o∅≡od∅ : ord→od (o∅ ) ≡ od∅ +o∅≡od∅ : ord→od (o∅ ) {!!} ≡ od∅ o∅≡od∅ = ==→o≡ lemma where - lemma0 : {x : Ordinal} → def (ord→od o∅) x → def od∅ x - lemma0 {x} lt = o<-subst (c<→o< {ord→od x} {ord→od o∅} (def-subst {ord→od o∅} {x} lt refl (sym diso)) ) diso diso - lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅) x + lemma0 : {x : Ordinal} → def (ord→od o∅ {!!} ) x → def od∅ x + lemma0 {x} lt = o<-subst (c<→o< {ord→od x {!!} } {ord→od o∅ {!!} } (def-subst {ord→od o∅ {!!} } {x} lt refl (sym (diso {!!} ))) ) (diso {!!}) (diso {!!}) + lemma1 : {x : Ordinal} → def od∅ x → def (ord→od o∅ {!!} ) x lemma1 {x} lt = ⊥-elim (¬x<0 lt) - lemma : ord→od o∅ == od∅ + lemma : ord→od o∅ {!!} == od∅ lemma = record { eq→ = lemma0 ; eq← = lemma1 } ord-od∅ : od→ord (od∅ ) ≡ o∅ -ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) ) +ord-od∅ = sym ( subst (λ k → k ≡ od→ord (od∅ ) ) (diso {!!}) (cong ( λ k → od→ord k ) o∅≡od∅ ) ) ∅0 : record { def = λ x → Lift n ⊥ } == od∅ eq→ ∅0 {w} (lift ()) @@ -201,7 +209,7 @@ -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n) in-codomain : (X : OD ) → ( ψ : OD → OD ) → OD -in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } +in-codomain X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧ ( x ≡ od→ord (ψ (ord→od y {!!} ))))) } -- Power Set of X ( or constructible by λ y → def X (od→ord y ) @@ -234,10 +242,10 @@ → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) → (x : OD ) → ψ x ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc ) where - induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox) - induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso ))) - ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy) - ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy + induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy {!!} )) → ψ (ord→od ox {!!} ) + induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl (diso {!!}) ))) + ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy {!!} ) + ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy {!!} )} induction oy -- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} ) @@ -258,13 +266,13 @@ } where ZFSet = OD -- is less than Ords because of maxod Select : (X : OD ) → ((x : OD ) → Set n ) → OD - Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } + Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x {!!} )) } Replace : OD → (OD → OD ) → OD Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x } _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = record { def = λ x → def A x ∧ def B x } Union : OD → OD - Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x))) } + Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u {!!} ) x))) } _∈_ : ( A B : ZFSet ) → Set n A ∈ B = B ∋ A Power : OD → OD @@ -275,7 +283,7 @@ data infinite-d : ( x : Ordinal ) → Set n where iφ : infinite-d o∅ isuc : {x : Ordinal } → infinite-d x → - infinite-d (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) )) + infinite-d (od→ord ( Union (ord→od x {!!} , (ord→od x {!!} , ord→od x {!!} ) ) )) infinite : OD infinite = record { def = λ x → infinite-d x } @@ -321,16 +329,16 @@ ⊆→o< {x} {y} lt with trio< x y ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc - ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym diso) refl ) - ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) + ⊆→o< {x} {y} lt | tri> ¬a ¬b c with (incl lt) (o<-subst c (sym (diso {!!})) refl ) + ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt (diso {!!}) refl )) union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } )) union← : (X z : OD) (X∋z : Union X ∋ z) → ¬ ( (u : OD ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) union← X z UX∋z = FExists _ lemma UX∋z where - lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) - lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } + lemma : {y : Ordinal} → def X y ∧ def (ord→od y {!!} ) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z)) + lemma {y} xx not = not (ord→od y {!!} ) record { proj1 = subst ( λ k → def X k ) (sym (diso {!!})) (proj1 xx ) ; proj2 = proj2 xx } ψiso : {ψ : OD → Set n} {x y : OD } → ψ x → x ≡ y → ψ y ψiso {ψ} t refl = t @@ -345,13 +353,13 @@ lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} )) replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y)) replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where - lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) - → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y))) + lemma2 : ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y {!!} )))) + → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) {!!} == ψ (ord→od y {!!} ))) lemma2 not not2 = not ( λ y d → not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where - lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y))) → (ord→od (od→ord x) == ψ (ord→od y)) - lemma3 {y} eq = subst (λ k → ord→od (od→ord x) == k ) oiso (o≡→== eq ) - lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) ) - lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso ( proj2 not2 )) + lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od y {!!} ))) → (ord→od (od→ord x) {!!} == ψ (ord→od y {!!} )) + lemma3 {y} eq = subst (λ k → ord→od (od→ord x) {!!} == k ) oiso (o≡→== eq ) + lemma : ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) {!!} == ψ (ord→od y {!!} )) ) + lemma not y not2 = not (ord→od y {!!} ) (subst (λ k → k == ψ (ord→od y {!!} )) oiso ( proj2 not2 )) --- --- Power Set @@ -364,7 +372,7 @@ ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) ∩-≡ {a} {b} inc = record { eq→ = λ {x} x<a → record { proj2 = x<a ; - proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym diso) )) refl diso } ; + proj1 = def-subst {_} {_} {b} {x} (inc (def-subst {_} {_} {a} {_} x<a refl (sym (diso {!!})) )) refl (diso {!!}) } ; eq← = λ {x} x<a∩b → proj2 x<a∩b } -- -- Transitive Set case @@ -379,11 +387,11 @@ eq→ lemma-eq {z} w = proj2 w eq← lemma-eq {z} w = record { proj2 = w ; proj1 = def-subst {_} {_} {(Ord a)} {z} - ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso } + ( t→A (def-subst {_} {_} {t} {od→ord (ord→od z {!!} )} w refl (sym (diso {!!})) )) refl (diso {!!}) } lemma1 : {a : Ordinal } { t : OD } - → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t + → (eq : ZFSubset (Ord a) t == t) → od→ord (ZFSubset (Ord a) (ord→od (od→ord t) o<max )) ≡ od→ord t lemma1 {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq )) - lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x)) + lemma : od→ord (ZFSubset (Ord a) (ord→od (od→ord t) o<max ) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) x)) lemma = sup-o< -- @@ -401,10 +409,10 @@ lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x) lemma3 y eq not = not (proj1 (eq→ eq t∋x)) - lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y))) + lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ (ord→od y {!!} )))) lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 )) - lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) → ¬ ¬ (def A (od→ord x)) - lemma5 {y} eq not = (lemma3 (ord→od y) eq) not + lemma5 : {y : Ordinal} → t == (A ∩ (ord→od y {!!})) → ¬ ¬ (def A (od→ord x)) + lemma5 {y} eq not = (lemma3 (ord→od y {!!} ) eq) not power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where @@ -413,9 +421,9 @@ lemma0 {x} t∋x = c<→o< (t→A t∋x) lemma3 : Def (Ord a) ∋ t lemma3 = ord-power← a t lemma0 - lemma4 : (A ∩ ord→od (od→ord t)) ≡ t + lemma4 : (A ∩ ord→od (od→ord t) {!!} ) ≡ t lemma4 = let open ≡-Reasoning in begin - A ∩ ord→od (od→ord t) + A ∩ ord→od (od→ord t) {!!} ≡⟨ cong (λ k → A ∩ k) oiso ⟩ A ∩ t ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩ @@ -426,7 +434,7 @@ lemma4 (sup-o< {λ x → od→ord (A ∩ x)} ) lemma2 : def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t) lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where - lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) + lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t) {!!} ) lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A ))) ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a)) @@ -440,8 +448,8 @@ continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a) extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B - eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym diso) (proj1 (eq (ord→od x))) d - eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym diso) (proj2 (eq (ord→od x))) d + eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {A} {B} (sym (diso {!!})) (proj1 (eq (ord→od x {!!} ))) d + eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {B} {A} (sym (diso {!!})) (proj2 (eq (ord→od x {!!} ))) d extensionality : {A B w : OD } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B) proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d @@ -452,14 +460,14 @@ lemma : o∅ ≡ od→ord od∅ lemma = let open ≡-Reasoning in begin o∅ - ≡⟨ sym diso ⟩ - od→ord ( ord→od o∅ ) + ≡⟨ sym (diso {!!}) ⟩ + od→ord ( ord→od o∅ {!!} ) ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩ od→ord od∅ ∎ infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) infinity x lt = def-subst {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where - lemma : od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x)))) + lemma : od→ord (Union (ord→od (od→ord x) {!!} , (ord→od (od→ord x) {!!} , ord→od (od→ord x) {!!} ))) ≡ od→ord (Union (x , (x , x))) lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso