# HG changeset patch # User Shinji KONO # Date 1593420966 -32400 # Node ID 7963b76df6e13a1703759ef7b56d546370c77275 # Parent 304c271b3d47d67290241766e7c6f6b4e2c9350d ¬odmax based HOD diff -r 304c271b3d47 -r 7963b76df6e1 OD.agda --- a/OD.agda Sun Jun 28 18:09:04 2020 +0900 +++ b/OD.agda Mon Jun 29 17:56:06 2020 +0900 @@ -67,10 +67,17 @@ -- In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic, -- we need explict assumption on sup. -record HOD (odmax : Ordinal) : Set (suc n) where +data One : Set n where + OneObj : One + +-- Ordinals in OD , the maximum +Ords : OD +Ords = record { def = λ x → One } + +record HOD : Set (suc n) where field - hmax : {x : Ordinal } → x o< odmax - hdef : Ordinal → Set n + od : OD + ¬odmax : ¬ (od ≡ Ords) record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where field @@ -88,34 +95,15 @@ record ODAxiom : Set (suc n) where -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) field - od→ord : OD → Ordinal - ord→od : Ordinal → OD - 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 + od→ord : HOD → Ordinal + ord→od : Ordinal → HOD + c<→o< : {x y : HOD } → def (od y) ( od→ord x ) → od→ord x o< od→ord y + oiso : {x : HOD } → ord→od ( od→ord x ) ≡ x diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x - ==→o≡ : { x y : OD } → (x == y) → x ≡ y + ==→o≡ : { x y : HOD } → (od x == od 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 - -- 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)) - -record HODAxiom : Set (suc n) where - -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) - field - mod : Ordinal - mod-limit : ¬ ((y : Ordinal) → mod ≡ osuc y) - os : OrdinalSubset mod - od→ord : HOD mod → Ordinal - ord→od : Ordinal → HOD mod - c<→o< : {x y : HOD mod } → hdef y (od→ord x) → od→ord x o< od→ord y - oiso : {x : HOD mod } → ord→od ( od→ord x ) ≡ x - diso : {x : Ordinal } → od→ord ( ord→od x ) ≡ x - ==→o≡ : { x y : OD } → (x == y) → x ≡ y - -- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum ) - sup-o : ( HOD mod → Ordinal ) → Ordinal - sup-o< : { ψ : HOD mod → Ordinal } → ∀ {x : HOD mod } → ψ x o< sup-o ψ + sup-od : ( HOD → HOD ) → HOD + sup-c< : ( ψ : HOD → HOD ) → ∀ {x : HOD } → def (od ( sup-od ψ )) (od→ord ( ψ x )) -- 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)) @@ -123,112 +111,103 @@ postulate odAxiom : ODAxiom open ODAxiom odAxiom -data One : Set n where - OneObj : One - --- Ordinals in OD , the maximum -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 -- we have to avoid this contradiction -bad-bad : ⊥ -bad-bad = osuc-< <-osuc (c<→o< { record { def = λ x → One }} OneObj) +-- bad-bad : ⊥ +-- bad-bad = osuc-< <-osuc (c<→o< { record { def = λ x → One }} OneObj) -- Ordinal in OD ( and ZFSet ) Transitive Set -Ord : ( a : Ordinal ) → OD -Ord a = record { def = λ y → y o< a } +Ord : ( a : Ordinal ) → HOD +Ord a = record { od = record { def = λ y → y o< a } ; ¬odmax = ? } -od∅ : OD +od∅ : HOD od∅ = Ord o∅ +sup-o : ( HOD → Ordinal ) → Ordinal +sup-o = ? +sup-o< : { ψ : HOD → Ordinal } → ∀ {x : HOD } → ψ x o< sup-o ψ +sup-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 ) +odef : HOD → Ordinal → Set n +odef A x = def ( od A ) x -_∋_ : ( a x : OD ) → Set n -_∋_ a x = def a ( od→ord x ) +o<→c<→HOD=Ord : ( {x y : Ordinal } → x o< y → odef (ord→od y) x ) → {x : HOD } → x ≡ Ord (od→ord x) +o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where + lemma1 : {y : Ordinal} → odef x y → odef (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 → odef x k ) (sym diso) lt)) + lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y + lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt ) -_c<_ : ( x a : OD ) → Set n +_∋_ : ( a x : HOD ) → Set n +_∋_ a x = odef a ( od→ord x ) + +_c<_ : ( x a : HOD ) → Set n x c< a = a ∋ x -cseq : {n : Level} → OD → OD -cseq x = record { def = λ y → def x (osuc y) } where - -def-subst : {Z : OD } {X : Ordinal }{z : OD } {x : Ordinal }→ def Z X → Z ≡ z → X ≡ x → def z x -def-subst df refl refl = df +cseq : {n : Level} → HOD → HOD +cseq x = record { od = record { def = λ y → odef x (osuc y) } ; ¬odmax = ? } where -sup-od : ( OD → OD ) → OD -sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) ) +odef-subst : {Z : HOD } {X : Ordinal }{z : HOD } {x : Ordinal }→ odef Z X → Z ≡ z → X ≡ x → odef z x +odef-subst df refl refl = df -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) ) - -otrans : {n : Level} {a x y : Ordinal } → def (Ord a) x → def (Ord x) y → def (Ord a) y +otrans : {n : Level} {a x y : Ordinal } → odef (Ord a) x → odef (Ord x) y → odef (Ord a) y otrans x ¬a ¬b c = no ¬b -_,_ : OD → OD → OD -x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } -- Ord (omax (od→ord x) (od→ord y)) +_,_ : HOD → HOD → HOD +x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; ¬odmax = ? } -- Ord (omax (od→ord x) (od→ord y)) -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- 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 : HOD ) → ( ψ : HOD → HOD ) → HOD +in-codomain X ψ = record { od = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧ ( x ≡ od→ord (ψ (ord→od y ))))) } ; ¬odmax = ? } --- Power Set of X ( or constructible by λ y → def X (od→ord y ) +-- Power Set of X ( or constructible by λ y → odef X (od→ord y ) -ZFSubset : (A x : OD ) → OD -ZFSubset A x = record { def = λ y → def A y ∧ def x y } -- roughly x = A → Set +ZFSubset : (A x : HOD ) → HOD +ZFSubset A x = record { od = record { def = λ y → odef A y ∧ odef x y } ; ¬odmax = ? } -- roughly x = A → Set -OPwr : (A : OD ) → OD +OPwr : (A : HOD ) → HOD OPwr A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A x) ) ) --- _⊆_ : ( A B : OD ) → ∀{ x : OD } → Set n +-- _⊆_ : ( A B : HOD ) → ∀{ x : HOD } → Set n -- _⊆_ A B {x} = A ∋ x → B ∋ x -record _⊆_ ( A B : OD ) : Set (suc n) where +record _⊆_ ( A B : HOD ) : Set (suc n) where field - incl : { x : OD } → A ∋ x → B ∋ x + incl : { x : HOD } → A ∋ x → B ∋ x open _⊆_ infixr 220 _⊆_ -subset-lemma : {A x : OD } → ( {y : OD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) +subset-lemma : {A x : HOD } → ( {y : HOD } → x ∋ y → ZFSubset A x ∋ y ) ⇔ ( x ⊆ A ) subset-lemma {A} {x} = record { proj1 = λ lt → record { incl = λ x∋z → proj1 (lt x∋z) } ; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt } @@ -272,23 +251,23 @@ open import Data.Unit -ε-induction : { ψ : OD → Set (suc n)} - → ( {x : OD } → ({ y : OD } → x ∋ y → ψ y ) → ψ x ) - → (x : OD ) → ψ x +ε-induction : { ψ : HOD → Set (suc n)} + → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x ) + → (x : HOD ) → 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 --- minimal-2 : (x : OD ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord y) ) +-- minimal-2 : (x : HOD ) → ( ne : ¬ (x == od∅ ) ) → (y : HOD ) → ¬ ( odef (minimal x ne) (od→ord y)) ∧ (odef x (od→ord y) ) -- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} ) -OD→ZF : ZF -OD→ZF = record { - ZFSet = OD +HOD→ZF : ZF +HOD→ZF = record { + ZFSet = HOD ; _∋_ = _∋_ - ; _≈_ = _==_ + ; _≈_ = _=h=_ ; ∅ = od∅ ; _,_ = _,_ ; Union = Union @@ -298,18 +277,18 @@ ; infinite = infinite ; isZF = isZF } 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 )) } - 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 } + ZFSet = HOD -- is less than Ords because of maxod + Select : (X : HOD ) → ((x : HOD ) → Set n ) → HOD + Select X ψ = record { od = record { def = λ x → ( odef X x ∧ ψ ( ord→od x )) } ; ¬odmax = ? } + Replace : HOD → (HOD → HOD ) → HOD + Replace X ψ = record { od = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ x))) ∧ odef (in-codomain X ψ) x } ; ¬odmax = ? } _∩_ : ( 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))) } + A ∩ B = record { od = record { def = λ x → odef A x ∧ odef B x } ; ¬odmax = ? } + Union : HOD → HOD + Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x))) } ; ¬odmax = ? } _∈_ : ( A B : ZFSet ) → Set n A ∈ B = B ∋ A - Power : OD → OD + Power : HOD → HOD Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x ) -- ｛_｝ : ZFSet → ZFSet -- ｛ x ｝ = ( x , x ) -- it works but we don't use @@ -319,12 +298,15 @@ isuc : {x : Ordinal } → infinite-d 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 } + infinite : HOD + infinite = record { od = record { def = λ x → infinite-d x } ; ¬odmax = ? } + + _=h=_ : (x y : HOD) → Set n + x =h= y = od x == od y infixr 200 _∈_ -- infixr 230 _∩_ _∪_ - isZF : IsZF (OD ) _∋_ _==_ od∅ _,_ Union Power Select Replace infinite + isZF : IsZF (HOD ) _∋_ _=h=_ od∅ _,_ Union Power Select Replace infinite isZF = record { isEquivalence = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans } ; pair→ = pair→ @@ -345,15 +327,15 @@ -- ; choice = choice } where - pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t == x ) ∨ ( t == y ) - pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x )) - pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y )) + pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t =h= x ) ∨ ( t =h= y ) + pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x )) + pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡y )) - pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t - pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x)) - pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y)) + pair← : ( x y t : ZFSet ) → ( t =h= x ) ∨ ( t =h= y ) → (x , y) ∋ t + pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x)) + pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y)) - empty : (x : OD ) → ¬ (od∅ ∋ x) + empty : (x : HOD ) → ¬ (od∅ ∋ x) empty x = ¬x<0 o<→c< : {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y) @@ -366,92 +348,92 @@ ⊆→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 : HOD) → (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 ))) + ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } )) + union← : (X z : HOD) (X∋z : Union X ∋ z) → ¬ ( (u : HOD ) → ¬ ((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} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z)) + lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef 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 : HOD → Set n} {x y : HOD } → ψ x → x ≡ y → ψ y ψiso {ψ} t refl = t - selection : {ψ : OD → Set n} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) + selection : {ψ : HOD → Set n} {X y : HOD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) selection {ψ} {X} {y} = record { proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } } - replacement← : {ψ : OD → OD} (X x : OD) → X ∋ x → Replace X ψ ∋ ψ x - replacement← {ψ} X x lt = record { proj1 = sup-c< ψ {x} ; proj2 = lemma } where - lemma : def (in-codomain X ψ) (od→ord (ψ x)) + replacement← : {ψ : HOD → HOD} (X x : HOD) → X ∋ x → Replace X ψ ∋ ψ x + replacement← {ψ} X x lt = record { proj1 = ? ; proj2 = lemma } where -- sup-c< ψ {x} + lemma : odef (in-codomain X ψ) (od→ord (ψ x)) 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 : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= 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) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y)))) + → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= 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) =h= ψ (ord→od y)) + lemma3 {y} eq = subst (λ k → ord→od (od→ord x) =h= k ) oiso (o≡→== eq ) + lemma : ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) ) + lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso ( proj2 not2 )) --- --- Power Set --- - --- First consider ordinals in OD + --- First consider ordinals in HOD --- - --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A + --- ZFSubset A x = record { def = λ y → odef A y ∧ odef x y } subset of A -- -- - ∩-≡ : { a b : OD } → ({x : OD } → (a ∋ x → b ∋ x)) → a == ( b ∩ a ) + ∩-≡ : { a b : HOD } → ({x : HOD } → (a ∋ x → b ∋ x)) → a =h= ( b ∩ a ) ∩-≡ {a} {b} inc = record { eq→ = λ {x} x