Mercurial > hg > Members > kono > Proof > automaton
view agda/finiteSet.agda @ 118:37c919cec9ac
fin-∨' almost finished
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 20 Nov 2019 13:34:34 +0900 |
parents | f00c990a24da |
children | eddc4ad8e99a |
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{-# OPTIONS --allow-unsolved-metas #-} module finiteSet where open import Data.Nat hiding ( _≟_ ) open import Data.Fin renaming ( _<_ to _<<_ ) hiding (_≤_) open import Data.Fin.Properties open import Data.Empty open import Relation.Nullary open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality open import logic open import nat open import Data.Nat.Properties hiding ( _≟_ ) open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) record Found ( Q : Set ) (p : Q → Bool ) : Set where field found-q : Q found-p : p found-q ≡ true lt0 : (n : ℕ) → n Data.Nat.≤ n lt0 zero = z≤n lt0 (suc n) = s≤s (lt0 n) lt2 : {m n : ℕ} → m < n → m Data.Nat.≤ n lt2 {zero} lt = z≤n lt2 {suc m} {zero} () lt2 {suc m} {suc n} (s≤s lt) = s≤s (lt2 lt) record FiniteSet ( Q : Set ) { n : ℕ } : Set where field Q←F : Fin n → Q F←Q : Q → Fin n finiso→ : (q : Q) → Q←F ( F←Q q ) ≡ q finiso← : (f : Fin n ) → F←Q ( Q←F f ) ≡ f finℕ : ℕ finℕ = n exists1 : (m : ℕ ) → m Data.Nat.≤ n → (Q → Bool) → Bool exists1 zero _ _ = false exists1 ( suc m ) m<n p = p (Q←F (fromℕ≤ {m} {n} m<n)) \/ exists1 m (lt2 m<n) p exists : ( Q → Bool ) → Bool exists p = exists1 n (lt0 n) p open import Data.List list1 : (m : ℕ ) → m Data.Nat.≤ n → (Q → Bool) → List Q list1 zero _ _ = [] list1 ( suc m ) m<n p with bool-≡-? (p (Q←F (fromℕ≤ {m} {n} m<n))) true ... | yes _ = Q←F (fromℕ≤ {m} {n} m<n) ∷ list1 m (lt2 m<n) p ... | no _ = list1 m (lt2 m<n) p to-list : ( Q → Bool ) → List Q to-list p = list1 n (lt0 n) p equal? : Q → Q → Bool equal? q0 q1 with F←Q q0 ≟ F←Q q1 ... | yes p = true ... | no ¬p = false equal→refl : { x y : Q } → equal? x y ≡ true → x ≡ y equal→refl {q0} {q1} eq with F←Q q0 ≟ F←Q q1 equal→refl {q0} {q1} refl | yes eq = begin q0 ≡⟨ sym ( finiso→ q0) ⟩ Q←F (F←Q q0) ≡⟨ cong (λ k → Q←F k ) eq ⟩ Q←F (F←Q q1) ≡⟨ finiso→ q1 ⟩ q1 ∎ where open ≡-Reasoning equal→refl {q0} {q1} () | no ne equal?-refl : {q : Q} → equal? q q ≡ true equal?-refl {q} with F←Q q ≟ F←Q q ... | yes p = refl ... | no ne = ⊥-elim (ne refl) fin<n : {n : ℕ} {f : Fin n} → toℕ f < n fin<n {_} {zero} = s≤s z≤n fin<n {suc n} {suc f} = s≤s (fin<n {n} {f}) i=j : {n : ℕ} (i j : Fin n) → toℕ i ≡ toℕ j → i ≡ j i=j {suc n} zero zero refl = refl i=j {suc n} (suc i) (suc j) eq = cong ( λ k → suc k ) ( i=j i j (cong ( λ k → Data.Nat.pred k ) eq) ) -- ¬∀⟶∃¬ : ∀ n {p} (P : Pred (Fin n) p) → Decidable P → ¬ (∀ i → P i) → (∃ λ i → ¬ P i) End : (m : ℕ ) → (p : Q → Bool ) → Set End m p = (i : Fin n) → m ≤ toℕ i → p (Q←F i ) ≡ false next-end : {m : ℕ } → ( p : Q → Bool ) → End (suc m) p → (m<n : m < n ) → p (Q←F (fromℕ≤ m<n )) ≡ false → End m p next-end {m} p prev m<n np i m<i with Data.Nat.Properties.<-cmp m (toℕ i) next-end p prev m<n np i m<i | tri< a ¬b ¬c = prev i a next-end p prev m<n np i m<i | tri> ¬a ¬b c = ⊥-elim ( nat-≤> m<i c ) next-end {m} p prev m<n np i m<i | tri≈ ¬a b ¬c = subst ( λ k → p (Q←F k) ≡ false) (m<n=i i b m<n ) np where m<n=i : {n : ℕ } (i : Fin n) {m : ℕ } → m ≡ (toℕ i) → (m<n : m < n ) → fromℕ≤ m<n ≡ i m<n=i i eq m<n = i=j (fromℕ≤ m<n) i (subst (λ k → k ≡ toℕ i) (sym (toℕ-fromℕ≤ m<n)) eq ) first-end : ( p : Q → Bool ) → End n p first-end p i i>n = ⊥-elim (nat-≤> i>n (fin<n {n} {i}) ) found : { p : Q → Bool } → (q : Q ) → p q ≡ true → exists p ≡ true found {p} q pt = found1 n (lt0 n) ( first-end p ) where found1 : (m : ℕ ) (m<n : m Data.Nat.≤ n ) → ((i : Fin n) → m ≤ toℕ i → p (Q←F i ) ≡ false ) → exists1 m m<n p ≡ true found1 0 m<n end = ⊥-elim ( ¬-bool (subst (λ k → k ≡ false ) (cong (λ k → p k) (finiso→ q) ) (end (F←Q q) z≤n )) pt ) found1 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ≤ m<n))) true found1 (suc m) m<n end | yes eq = subst (λ k → k \/ exists1 m (lt2 m<n) p ≡ true ) (sym eq) (bool-or-4 {exists1 m (lt2 m<n) p} ) found1 (suc m) m<n end | no np = begin p (Q←F (fromℕ≤ m<n)) \/ exists1 m (lt2 m<n) p ≡⟨ bool-or-1 (¬-bool-t np ) ⟩ exists1 m (lt2 m<n) p ≡⟨ found1 m (lt2 m<n) (next-end p end m<n (¬-bool-t np )) ⟩ true ∎ where open ≡-Reasoning not-found : { p : Q → Bool } → ( (q : Q ) → p q ≡ false ) → exists p ≡ false not-found {p} pn = not-found2 n (lt0 n) where not-found2 : (m : ℕ ) → (m<n : m Data.Nat.≤ n ) → exists1 m m<n p ≡ false not-found2 zero _ = refl not-found2 ( suc m ) m<n with pn (Q←F (fromℕ≤ {m} {n} m<n)) not-found2 (suc m) m<n | eq = begin p (Q←F (fromℕ≤ m<n)) \/ exists1 m (lt2 m<n) p ≡⟨ bool-or-1 eq ⟩ exists1 m (lt2 m<n) p ≡⟨ not-found2 m (lt2 m<n) ⟩ false ∎ where open ≡-Reasoning open import Level postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n n -- (Level.suc n) found← : { p : Q → Bool } → exists p ≡ true → Found Q p found← {p} exst = found2 n (lt0 n) (first-end p ) where found2 : (m : ℕ ) (m<n : m Data.Nat.≤ n ) → End m p → Found Q p found2 0 m<n end = ⊥-elim ( ¬-bool (not-found (λ q → end (F←Q q) z≤n ) ) (subst (λ k → exists k ≡ true) (sym lemma) exst ) ) where lemma : (λ z → p (Q←F (F←Q z))) ≡ p lemma = f-extensionality ( λ q → subst (λ k → p k ≡ p q ) (sym (finiso→ q)) refl ) found2 (suc m) m<n end with bool-≡-? (p (Q←F (fromℕ≤ m<n))) true found2 (suc m) m<n end | yes eq = record { found-q = Q←F (fromℕ≤ m<n) ; found-p = eq } found2 (suc m) m<n end | no np = found2 m (lt2 m<n) (next-end p end m<n (¬-bool-t np )) not-found← : { p : Q → Bool } → exists p ≡ false → (q : Q ) → p q ≡ false not-found← {p} np q = ¬-bool-t ( contra-position {_} {_} {_} {exists p ≡ true} (found q) (λ ep → ¬-bool np ep ) ) fin-∨' : {A B : Set} → { a b : ℕ } → FiniteSet A {a} → FiniteSet B {b} → FiniteSet (A ∨ B) {a Data.Nat.+ b} fin-∨' {A} {B} {a} {b} fa fb = record { Q←F = Q←F ; F←Q = F←Q ; finiso→ = finiso→ ; finiso← = finiso← } where n : ℕ n = a Data.Nat.+ b Q : Set Q = A ∨ B f-a : ∀{i b} → (f : Fin i ) → (a : ℕ ) → toℕ f > a → toℕ f < a Data.Nat.+ b → Fin b f-a {i} {b} f zero lt lt2 = fromℕ≤ lt2 f-a {suc i} {_} (suc f) (suc a) (s≤s lt) (s≤s lt2) = f-a f a lt lt2 f-a zero (suc x) () _ a<a+b : {f : Fin n} → toℕ f ≡ a → a < a Data.Nat.+ b a<a+b {f} eq = subst (λ k → k < a Data.Nat.+ b) eq ( toℕ<n f ) 0<b : (a : ℕ ) → a < a Data.Nat.+ b → 0 < b 0<b zero a<a+b = a<a+b 0<b (suc a) (s≤s a<a+b) = 0<b a a<a+b lemma3 : {a b : ℕ } → (lt : a < b ) → fromℕ≤ (s≤s lt) ≡ suc (fromℕ≤ lt) lemma3 (s≤s lt) = refl lemma4 : {i : ℕ } → { f : Fin i} → fromℕ≤ (toℕ<n f) ≡ f lemma4 {suc _} {zero} = refl lemma4 {suc i} {suc f} = begin fromℕ≤ (toℕ<n (suc f)) ≡⟨ lemma3 _ ⟩ suc (fromℕ≤ (toℕ<n f)) ≡⟨ cong (λ k → suc k ) (lemma4 {i} {f}) ⟩ suc f ∎ where open ≡-Reasoning lemma6 : {a b : ℕ } → {f : Fin a} → toℕ (inject+ b f) ≡ toℕ f lemma6 {suc a} {b} {zero} = refl lemma6 {suc a} {b} {suc f} = cong (λ k → suc k ) (lemma6 {a} {b} {f}) lemmaa : {a b c : ℕ } → {b<a : b < a } → {c<a : c < a} → b ≡ c → fromℕ≤ b<a ≡ fromℕ≤ c<a lemmaa {suc a} {zero} {zero} {s≤s z≤n} {s≤s z≤n} refl = refl lemmaa {suc a} {suc b} {suc b} {s≤s b<a} {s≤s c<a} refl = subst₂ ( λ j k → j ≡ k ) (sym (lemma3 _ )) (sym (lemma3 _ )) (cong (λ k → suc k ) ( lemmaa {a} {b} {b} {b<a} {c<a} refl )) Q←F : Fin n → Q Q←F f with Data.Nat.Properties.<-cmp (toℕ f) a Q←F f | tri< lt ¬b ¬c = case1 (FiniteSet.Q←F fa (fromℕ≤ lt )) Q←F f | tri≈ ¬a eq ¬c = case2 (FiniteSet.Q←F fb (fromℕ≤ (0<b a (a<a+b eq ) ))) where Q←F f | tri> ¬a ¬b c = case2 (FiniteSet.Q←F fb (f-a f a c (toℕ<n f) )) F←Q : Q → Fin n F←Q (case1 qa) = inject+ b (FiniteSet.F←Q fa qa) F←Q (case2 qb) = raise a (FiniteSet.F←Q fb qb) finiso→ : (q : Q) → Q←F ( F←Q q ) ≡ q finiso→ (case1 qa) = lemma7 where lemma5 : toℕ (inject+ b (FiniteSet.F←Q fa qa)) < a lemma5 = subst (λ k → k < a ) (sym lemma6) (toℕ<n (FiniteSet.F←Q fa qa)) lemma7 : Q←F (F←Q (case1 qa)) ≡ case1 qa lemma7 with Data.Nat.Properties.<-cmp (toℕ (inject+ b (FiniteSet.F←Q fa qa))) a lemma7 | tri< lt ¬b ¬c = begin case1 (FiniteSet.Q←F fa (fromℕ≤ lt )) ≡⟨ cong (λ k → case1 (FiniteSet.Q←F fa k )) (lemmaa lemma6 ) ⟩ case1 (FiniteSet.Q←F fa (fromℕ≤ (toℕ<n (FiniteSet.F←Q fa qa)) )) ≡⟨ cong (λ k → case1 (FiniteSet.Q←F fa k )) lemma4 ⟩ case1 (FiniteSet.Q←F fa (FiniteSet.F←Q fa qa) ) ≡⟨ cong (λ k → case1 k ) (FiniteSet.finiso→ fa _ ) ⟩ case1 qa ∎ where open ≡-Reasoning lemma7 | tri≈ ¬a b ¬c = ⊥-elim ( ¬a lemma5 ) lemma7 | tri> ¬a ¬b c = ⊥-elim ( ¬a lemma5 ) finiso→ (case2 qb) = lemma9 where lemmab : toℕ (raise a (FiniteSet.F←Q fb qb)) > a lemmab = {!!} lemmac : (f : Fin b) (f>a : toℕ (raise a f) > a ) → f-a (raise a f) a f>a (toℕ<n (raise a f)) ≡ f lemmac = {!!} lemmad : {qb : B } → 0 ≡ toℕ (FiniteSet.F←Q fb qb) lemmad = {!!} lemma9 : Q←F (F←Q (case2 qb)) ≡ case2 qb lemma9 with Data.Nat.Properties.<-cmp (toℕ (raise a (FiniteSet.F←Q fb qb))) a lemma9 | tri< a ¬b ¬c = ⊥-elim ( ¬c lemmab ) lemma9 | tri≈ ¬a eq ¬c = begin case2 (FiniteSet.Q←F fb (fromℕ≤ (0<b a (a<a+b eq ) ))) ≡⟨ cong (λ k → case2 (FiniteSet.Q←F fb k )) (lemmaa lemmad ) ⟩ case2 (FiniteSet.Q←F fb (fromℕ≤ (toℕ<n (FiniteSet.F←Q fb qb)))) ≡⟨ cong (λ k → case2 (FiniteSet.Q←F fb k )) lemma4 ⟩ case2 (FiniteSet.Q←F fb (FiniteSet.F←Q fb qb)) ≡⟨ cong (λ k → case2 k ) (FiniteSet.finiso→ fb _ ) ⟩ case2 qb ∎ where open ≡-Reasoning lemma9 | tri> ¬a ¬b c = begin case2 (FiniteSet.Q←F fb (f-a (raise a (FiniteSet.F←Q fb qb)) a c (toℕ<n (raise a (FiniteSet.F←Q fb qb)) ) )) ≡⟨ cong (λ k → case2 (FiniteSet.Q←F fb k) ) (lemmac (FiniteSet.F←Q fb qb) c ) ⟩ case2 (FiniteSet.Q←F fb (FiniteSet.F←Q fb qb)) ≡⟨ cong (λ k → case2 k ) (FiniteSet.finiso→ fb _ ) ⟩ case2 qb ∎ where open ≡-Reasoning finiso← : (f : Fin n ) → F←Q ( Q←F f ) ≡ f finiso← f with Data.Nat.Properties.<-cmp (toℕ f) a finiso← f | tri< lt ¬b ¬c = lemma11 where lemma14 : { a b : ℕ } { f : Fin ( a Data.Nat.+ b) } { lt : (toℕ f) < a } → inject+ b (fromℕ≤ lt ) ≡ f lemma14 {suc a} {b} {zero} {s≤s z≤n} = refl lemma14 {suc a} {b} {suc f} {s≤s lt} = begin inject+ b (fromℕ≤ (s≤s lt)) ≡⟨ cong (λ k → inject+ b k ) (lemma3 lt ) ⟩ inject+ b (suc (fromℕ≤ lt)) ≡⟨⟩ suc (inject+ b (fromℕ≤ lt)) ≡⟨ cong (λ k → suc k) (lemma14 {a} {b} {f} {lt} ) ⟩ suc f ∎ where open ≡-Reasoning lemma11 : inject+ b (FiniteSet.F←Q fa (FiniteSet.Q←F fa (fromℕ≤ lt ) )) ≡ f lemma11 = subst ( λ k → inject+ b k ≡ f ) (sym (FiniteSet.finiso← fa _) ) lemma14 finiso← f | tri≈ ¬a eq ¬c = lemma12 where lemma15 : {a b : ℕ } ( f : Fin ( a Data.Nat.+ b) ) ( eq : (toℕ f) ≡ a ) → (0<b : zero < b ) → raise a (fromℕ≤ 0<b) ≡ f lemma15 {zero} {suc b} zero refl (s≤s z≤n) = refl lemma15 {suc a} {suc b} (suc f) eq (s≤s z≤n) = cong (λ k → suc k ) ( lemma15 {a} {suc b} f (cong (λ k → Data.Nat.pred k) eq) (s≤s z≤n)) lemma12 : raise a (FiniteSet.F←Q fb (FiniteSet.Q←F fb (fromℕ≤ (0<b a (a<a+b eq ))))) ≡ f lemma12 = subst ( λ k → raise a k ≡ f ) (sym (FiniteSet.finiso← fb _) ) (lemma15 f eq (0<b a (a<a+b eq ))) finiso← f | tri> ¬a ¬b c = lemma13 where lemma16 : {a b : ℕ } (f : Fin (a Data.Nat.+ b)) → (lt : toℕ f > a ) → raise a (f-a f a lt (toℕ<n f)) ≡ f lemma16 {zero} {b} (suc f) (s≤s z≤n) = lemma17 where lemma17 : fromℕ≤ (s≤s (toℕ<n f)) ≡ suc f lemma17 = begin fromℕ≤ (s≤s (toℕ<n f)) ≡⟨ lemma3 _ ⟩ suc ( fromℕ≤ (toℕ<n f) ) ≡⟨ cong (λ k → suc k) lemma4 ⟩ suc f ∎ where open ≡-Reasoning lemma16 {suc a} {b} (suc f) (s≤s lt) = cong ( λ k → suc k ) (lemma16 {a} {b} f lt) lemma13 : raise a (FiniteSet.F←Q fb ((FiniteSet.Q←F fb (f-a f a c (toℕ<n f))))) ≡ f lemma13 = subst ( λ k → raise a k ≡ f ) (sym (FiniteSet.finiso← fb _) ) (lemma16 f c ) import Data.Nat.DivMod import Data.Nat.Properties open _∧_ open import Data.Vec import Data.Product exp2 : (n : ℕ ) → exp 2 (suc n) ≡ exp 2 n Data.Nat.+ exp 2 n exp2 n = begin exp 2 (suc n) ≡⟨⟩ 2 * ( exp 2 n ) ≡⟨ *-comm 2 (exp 2 n) ⟩ ( exp 2 n ) * 2 ≡⟨ +-*-suc ( exp 2 n ) 1 ⟩ (exp 2 n ) Data.Nat.+ ( exp 2 n ) * 1 ≡⟨ cong ( λ k → (exp 2 n ) Data.Nat.+ k ) (proj₂ *-identity (exp 2 n) ) ⟩ exp 2 n Data.Nat.+ exp 2 n ∎ where open ≡-Reasoning open Data.Product cast-iso : {n m : ℕ } → (eq : n ≡ m ) → (f : Fin m ) → cast eq ( cast (sym eq ) f) ≡ f cast-iso refl zero = refl cast-iso refl (suc f) = cong ( λ k → suc k ) ( cast-iso refl f ) fin2List : {n : ℕ } → FiniteSet (Vec Bool n) {exp 2 n } fin2List {zero} = record { Q←F = λ _ → Vec.[] ; F←Q = λ _ → # 0 ; finiso→ = finiso→ ; finiso← = finiso← } where Q = Vec Bool zero finiso→ : (q : Q) → [] ≡ q finiso→ [] = refl finiso← : (f : Fin (exp 2 zero)) → # 0 ≡ f finiso← zero = refl fin2List {suc n} = record { Q←F = Q←F ; F←Q = F←Q ; finiso→ = finiso→ ; finiso← = finiso← } where Q : Set Q = Vec Bool (suc n) QtoR : Vec Bool (suc n) → Vec Bool n ∨ Vec Bool n QtoR ( true ∷ x ) = case1 x QtoR ( false ∷ x ) = case2 x RtoQ : Vec Bool n ∨ Vec Bool n → Vec Bool (suc n) RtoQ ( case1 x ) = true ∷ x RtoQ ( case2 x ) = false ∷ x isoRQ : (x : Vec Bool (suc n) ) → RtoQ ( QtoR x ) ≡ x isoRQ (true ∷ _ ) = refl isoRQ (false ∷ _ ) = refl isoQR : (x : Vec Bool n ∨ Vec Bool n ) → QtoR ( RtoQ x ) ≡ x isoQR (case1 x) = refl isoQR (case2 x) = refl fin∨ = fin-∨' (fin2List {n}) (fin2List {n}) Q←F : Fin (exp 2 (suc n)) → Q Q←F f = RtoQ ( FiniteSet.Q←F fin∨ (cast (exp2 n) f )) F←Q : Q → Fin (exp 2 (suc n)) F←Q q = cast (sym (exp2 n)) ( FiniteSet.F←Q fin∨ ( QtoR q ) ) finiso→ : (q : Q) → Q←F ( F←Q q ) ≡ q finiso→ q = begin RtoQ ( FiniteSet.Q←F fin∨ (cast (exp2 n) (cast (sym (exp2 n)) ( FiniteSet.F←Q fin∨ (QtoR q) )) )) ≡⟨ cong (λ k → RtoQ ( FiniteSet.Q←F fin∨ k)) (cast-iso (exp2 n) _ ) ⟩ RtoQ ( FiniteSet.Q←F fin∨ ( FiniteSet.F←Q fin∨ (QtoR q) )) ≡⟨ cong ( λ k → RtoQ k ) ( FiniteSet.finiso→ fin∨ _ ) ⟩ RtoQ (QtoR _) ≡⟨ isoRQ q ⟩ q ∎ where open ≡-Reasoning finiso← : (f : Fin (exp 2 (suc n) )) → F←Q ( Q←F f ) ≡ f finiso← f = begin cast _ (FiniteSet.F←Q fin∨ (QtoR (RtoQ (FiniteSet.Q←F fin∨ (cast _ f )) ) )) ≡⟨ cong (λ k → cast (sym (exp2 n)) (FiniteSet.F←Q fin∨ k )) (isoQR (FiniteSet.Q←F fin∨ (cast _ f))) ⟩ cast (sym (exp2 n)) (FiniteSet.F←Q fin∨ (FiniteSet.Q←F fin∨ (cast (exp2 n) f ))) ≡⟨ cong (λ k → cast (sym (exp2 n)) k ) ( FiniteSet.finiso← fin∨ _ ) ⟩ cast _ (cast (exp2 n) f ) ≡⟨ cast-iso (sym (exp2 n)) _ ⟩ f ∎ where open ≡-Reasoning Func2List : { Q : Set } → {n m : ℕ } → n < suc m → FiniteSet Q {m} → ( Q → Bool ) → Vec Bool n Func2List {Q} {zero} _ fin Q→B = [] Func2List {Q} {suc n} {m} (s≤s n<m) fin Q→B = Q→B (FiniteSet.Q←F fin (fromℕ≤ n<m)) ∷ Func2List {Q} {n} {m} (Data.Nat.Properties.<-trans n<m a<sa ) fin Q→B List2Func : { Q : Set } → {n m : ℕ } → n < suc m → FiniteSet Q {m} → Vec Bool n → Q → Bool List2Func {Q} {zero} _ fin [] q = false List2Func {Q} {suc n} {m} (s≤s n<m) fin (h ∷ t) q with FiniteSet.F←Q fin q ≟ fromℕ≤ n<m ... | yes _ = h ... | no _ = List2Func {Q} {n} {m} (Data.Nat.Properties.<-trans n<m a<sa ) fin t q F2L-iso : { Q : Set } → {n : ℕ } → (fin : FiniteSet Q {n}) → (x : Vec Bool n ) → Func2List a<sa fin (List2Func a<sa fin x ) ≡ x F2L-iso = {!!} L2F-iso : { Q : Set } → {n : ℕ } → (fin : FiniteSet Q {n}) → (f : Q → Bool ) → (q : Q ) → (List2Func a<sa fin (Func2List a<sa fin f )) q ≡ f q L2F-iso = {!!}