--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/automaton-in-agda/src/finiteSetUtil.agda	Sun Jun 13 20:45:17 2021 +0900
@@ -0,0 +1,461 @@
+{-# OPTIONS --allow-unsolved-metas #-} 
+
+module finiteSetUtil  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.Definitions
+open import Relation.Binary.PropositionalEquality
+open import logic
+open import nat
+open import finiteSet
+open import fin
+open import Data.Nat.Properties as NatP  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
+
+module _ {Q : Set } (F : FiniteSet Q) where
+     open FiniteSet F
+     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
+     End : (m : ℕ ) → (p : Q → Bool ) → Set
+     End m p = (i : Fin finite) → m ≤ toℕ i → p (Q←F i )  ≡ false 
+     first-end :  ( p : Q → Bool ) → End finite p
+     first-end p i i>n = ⊥-elim (nat-≤> i>n (fin<n {finite} {i}) )
+     next-end : {m : ℕ } → ( p : Q → Bool ) → End (suc m) p
+              → (m<n : m < finite ) → p (Q←F (fromℕ< m<n )) ≡ false
+              → End m p
+     next-end {m} p prev m<n np i m<i with NatP.<-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 = {!!} -- toℕ-inject (fromℕ≤ ?) i (subst (λ k → k ≡ toℕ i) (sym (toℕ-fromℕ≤ m<n)) eq )
+     found : { p : Q → Bool } → (q : Q ) → p q ≡ true → exists p ≡ true
+     found {p} q pt = found1 finite  (NatP.≤-refl ) ( first-end p ) where
+         found1 : (m : ℕ ) (m<n : m Data.Nat.≤ finite ) → ((i : Fin finite) → 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 (≤to<  m<n) p ≡ true ) (sym eq) (bool-or-4 {exists1 m (≤to<  m<n) p} ) 
+         found1 (suc m)  m<n end | no np = begin
+                 p (Q←F (fromℕ< m<n)) \/ exists1 m (≤to<  m<n) p
+              ≡⟨ bool-or-1 (¬-bool-t np ) ⟩
+                 exists1 m (≤to<  m<n) p
+              ≡⟨ found1 m (≤to<  m<n) (next-end p end m<n (¬-bool-t np )) ⟩
+                 true
+              ∎  where open ≡-Reasoning
+
+
+
+record ISO (A B : Set) : Set where
+   field
+     A←B : B → A
+     B←A : A → B
+     iso← : (q : A) → A←B ( B←A q ) ≡ q
+     iso→ : (f : B) → B←A ( A←B f ) ≡ f
+
+iso-fin : {A B : Set} → FiniteSet A  → ISO A B → FiniteSet B 
+iso-fin {A} {B}  fin iso = record {
+   Q←F = λ f → ISO.B←A iso ( FiniteSet.Q←F fin f )
+ ; F←Q = λ b → FiniteSet.F←Q fin ( ISO.A←B iso b )
+ ; finiso→ = finiso→ 
+ ; finiso← = finiso← 
+   } where
+   finiso→ : (q : B) → ISO.B←A iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (ISO.A←B iso q))) ≡ q
+   finiso→ q = begin
+              ISO.B←A iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (ISO.A←B iso q))) 
+           ≡⟨ cong (λ k → ISO.B←A iso k ) (FiniteSet.finiso→ fin _ ) ⟩
+              ISO.B←A iso (ISO.A←B iso q)
+           ≡⟨ ISO.iso→ iso _ ⟩
+              q
+           ∎  where
+           open ≡-Reasoning
+   finiso← : (f : Fin (FiniteSet.finite fin ))→ FiniteSet.F←Q fin (ISO.A←B iso (ISO.B←A iso (FiniteSet.Q←F fin f))) ≡ f
+   finiso← f = begin
+              FiniteSet.F←Q fin (ISO.A←B iso (ISO.B←A iso (FiniteSet.Q←F fin f))) 
+           ≡⟨ cong (λ k → FiniteSet.F←Q fin k ) (ISO.iso← iso _) ⟩
+              FiniteSet.F←Q fin (FiniteSet.Q←F fin f) 
+           ≡⟨ FiniteSet.finiso← fin _  ⟩
+              f
+           ∎  where
+           open ≡-Reasoning
+
+data One : Set where
+   one : One
+
+fin-∨1 : {B : Set} → (fb : FiniteSet B ) → FiniteSet (One ∨ B) 
+fin-∨1 {B} fb =  record {
+   Q←F = Q←F
+ ; F←Q =  F←Q
+ ; finiso→ = finiso→
+ ; finiso← = finiso←
+   }  where
+   b = FiniteSet.finite fb
+   Q←F : Fin (suc b) → One ∨ B
+   Q←F zero = case1 one
+   Q←F (suc f) = case2 (FiniteSet.Q←F fb f)
+   F←Q : One ∨ B → Fin (suc b)
+   F←Q (case1 one) = zero
+   F←Q (case2 f ) = suc (FiniteSet.F←Q fb f) 
+   finiso→ : (q : One ∨ B) → Q←F (F←Q q) ≡ q
+   finiso→ (case1 one) = refl
+   finiso→ (case2 b) = cong (λ k → case2 k ) (FiniteSet.finiso→ fb b)
+   finiso← : (q : Fin (suc b)) → F←Q (Q←F q) ≡ q
+   finiso← zero = refl
+   finiso← (suc f) = cong ( λ k → suc k ) (FiniteSet.finiso← fb f)
+
+
+fin-∨2 : {B : Set} → ( a : ℕ ) → FiniteSet B  → FiniteSet (Fin a ∨ B) 
+fin-∨2 {B} zero  fb = iso-fin fb iso where
+   iso : ISO B (Fin zero ∨ B)
+   iso =  record {
+        A←B = A←B
+      ; B←A = λ b → case2 b
+      ; iso→ = iso→
+      ; iso← = λ _ → refl
+    } where
+     A←B : Fin zero ∨ B → B
+     A←B (case2 x) = x 
+     iso→ : (f : Fin zero ∨ B ) → case2 (A←B f) ≡ f
+     iso→ (case2 x)  = refl
+fin-∨2 {B} (suc a) fb =  iso-fin (fin-∨1 (fin-∨2 a fb) ) iso
+    where
+  iso : ISO (One ∨ (Fin a ∨ B) ) (Fin (suc a) ∨ B)
+  ISO.A←B iso (case1 zero) = case1 one
+  ISO.A←B iso (case1 (suc f)) = case2 (case1 f)
+  ISO.A←B iso (case2 b) = case2 (case2 b)
+  ISO.B←A iso (case1 one) = case1 zero
+  ISO.B←A iso (case2 (case1 f)) = case1 (suc f)
+  ISO.B←A iso (case2 (case2 b)) = case2 b
+  ISO.iso← iso (case1 one) = refl
+  ISO.iso← iso (case2 (case1 x)) = refl
+  ISO.iso← iso (case2 (case2 x)) = refl
+  ISO.iso→ iso (case1 zero) = refl
+  ISO.iso→ iso (case1 (suc x)) = refl
+  ISO.iso→ iso (case2 x) = refl
+
+
+FiniteSet→Fin : {A : Set} → (fin : FiniteSet A  ) → ISO (Fin (FiniteSet.finite fin)) A
+ISO.A←B (FiniteSet→Fin fin) f = FiniteSet.F←Q fin f
+ISO.B←A (FiniteSet→Fin fin) f = FiniteSet.Q←F fin f
+ISO.iso← (FiniteSet→Fin fin) = FiniteSet.finiso← fin
+ISO.iso→ (FiniteSet→Fin fin) =  FiniteSet.finiso→ fin
+   
+
+fin-∨ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∨ B) 
+fin-∨ {A} {B}  fa fb = iso-fin (fin-∨2 a  fb ) iso2 where
+    a = FiniteSet.finite fa
+    ia = FiniteSet→Fin fa
+    iso2 : ISO (Fin a ∨ B ) (A ∨ B)
+    ISO.A←B iso2 (case1 x) = case1 ( ISO.A←B ia x )
+    ISO.A←B iso2 (case2 x) = case2 x
+    ISO.B←A iso2 (case1 x) = case1 ( ISO.B←A ia x )
+    ISO.B←A iso2 (case2 x) = case2 x
+    ISO.iso← iso2 (case1 x) = cong ( λ k → case1 k ) (ISO.iso← ia x)
+    ISO.iso← iso2 (case2 x) = refl
+    ISO.iso→ iso2 (case1 x) = cong ( λ k → case1 k ) (ISO.iso→ ia x)
+    ISO.iso→ iso2 (case2 x) = refl
+
+open import Data.Product
+
+fin-× : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A × B) 
+fin-× {A} {B}  fa fb with FiniteSet→Fin fa
+... | a=f = iso-fin (fin-×-f a ) iso-1 where
+   a = FiniteSet.finite fa
+   b = FiniteSet.finite fb
+   iso-1 : ISO (Fin a × B) ( A × B )
+   ISO.A←B iso-1 x = ( FiniteSet.F←Q fa (proj₁ x)  , proj₂ x) 
+   ISO.B←A iso-1 x = ( FiniteSet.Q←F fa (proj₁ x)  , proj₂ x) 
+   ISO.iso← iso-1 x  =  lemma  where
+     lemma : (FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj₁ x)) , proj₂ x) ≡ ( proj₁ x , proj₂ x )
+     lemma = cong ( λ k → ( k ,  proj₂ x ) )  (FiniteSet.finiso← fa _ )
+   ISO.iso→ iso-1 x = cong ( λ k → ( k ,  proj₂ x ) )  (FiniteSet.finiso→ fa _ )
+
+   iso-2 : {a : ℕ } → ISO (B ∨ (Fin a × B)) (Fin (suc a) × B)
+   ISO.A←B iso-2 (zero , b ) = case1 b
+   ISO.A←B iso-2 (suc fst , b ) = case2 ( fst , b )
+   ISO.B←A iso-2 (case1 b) = ( zero , b )
+   ISO.B←A iso-2 (case2 (a , b )) = ( suc a , b )
+   ISO.iso← iso-2 (case1 x) = refl
+   ISO.iso← iso-2 (case2 x) = refl
+   ISO.iso→ iso-2 (zero , b ) = refl
+   ISO.iso→ iso-2 (suc a , b ) = refl
+
+   fin-×-f : ( a  : ℕ ) → FiniteSet ((Fin a) × B) 
+   fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 }
+   fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2
+
+open _∧_
+
+fin-∧ : {A B : Set} → FiniteSet A → FiniteSet B → FiniteSet (A ∧ B) 
+fin-∧ {A} {B} fa fb with FiniteSet→Fin fa    -- same thing for our tool
+... | a=f = iso-fin (fin-×-f a ) iso-1 where
+   a = FiniteSet.finite fa
+   b = FiniteSet.finite fb
+   iso-1 : ISO (Fin a ∧ B) ( A ∧ B )
+   ISO.A←B iso-1 x = record { proj1 = FiniteSet.F←Q fa (proj1 x)  ; proj2 =  proj2 x} 
+   ISO.B←A iso-1 x = record { proj1 = FiniteSet.Q←F fa (proj1 x)  ; proj2 =  proj2 x}
+   ISO.iso← iso-1 x  =  lemma  where
+     lemma : record { proj1 = FiniteSet.F←Q fa (FiniteSet.Q←F fa (proj1 x)) ; proj2 =  proj2 x} ≡ record {proj1 =  proj1 x ; proj2 =  proj2 x }
+     lemma = cong ( λ k → record {proj1 = k ;  proj2 = proj2 x } )  (FiniteSet.finiso← fa _ )
+   ISO.iso→ iso-1 x = cong ( λ k → record {proj1 =  k ; proj2 =  proj2 x } )  (FiniteSet.finiso→ fa _ )
+
+   iso-2 : {a : ℕ } → ISO (B ∨ (Fin a ∧ B)) (Fin (suc a) ∧ B)
+   ISO.A←B iso-2 (record { proj1 = zero ; proj2 =  b }) = case1 b
+   ISO.A←B iso-2 (record { proj1 = suc fst ; proj2 =  b }) = case2 ( record { proj1 = fst ; proj2 =  b } )
+   ISO.B←A iso-2 (case1 b) = record {proj1 =  zero ; proj2 =  b }
+   ISO.B←A iso-2 (case2 (record { proj1 = a ; proj2 =  b })) = record { proj1 =  suc a ; proj2 =  b }
+   ISO.iso← iso-2 (case1 x) = refl
+   ISO.iso← iso-2 (case2 x) = refl
+   ISO.iso→ iso-2 (record { proj1 = zero ; proj2 =  b }) = refl
+   ISO.iso→ iso-2 (record { proj1 = suc a ; proj2 =  b }) = refl
+
+   fin-×-f : ( a  : ℕ ) → FiniteSet ((Fin a) ∧ B) 
+   fin-×-f zero = record { Q←F = λ () ; F←Q = λ () ; finiso→ = λ () ; finiso← = λ () ; finite = 0 }
+   fin-×-f (suc a) = iso-fin ( fin-∨ fb ( fin-×-f a ) ) iso-2
+
+-- import Data.Nat.DivMod
+
+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) 
+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} = subst (λ k → FiniteSet (Vec Bool (suc n)) ) (sym (exp2 n)) ( iso-fin (fin-∨ (fin2List ) (fin2List )) iso )
+    where
+   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
+   iso : ISO (Vec Bool n ∨ Vec Bool n) (Vec Bool (suc n))
+   iso = record { A←B = QtoR ; B←A = RtoQ ; iso← = isoQR ; iso→ = isoRQ  }
+
+F2L : {Q : Set } {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin) → ( (q : Q) → toℕ (FiniteSet.F←Q fin q ) < n  → Bool ) → Vec Bool n
+F2L  {Q} {zero} fin _ Q→B = []
+F2L  {Q} {suc n} fin (s≤s n<m) Q→B = Q→B (FiniteSet.Q←F fin (fromℕ< n<m)) lemma6 ∷ F2L {Q} fin (NatP.<-trans n<m a<sa ) qb1 where
+   lemma6 : toℕ (FiniteSet.F←Q fin (FiniteSet.Q←F fin (fromℕ< n<m))) < suc n
+   lemma6 = subst (λ k → toℕ k < suc n ) (sym (FiniteSet.finiso← fin _ )) (subst (λ k → k < suc n) (sym (toℕ-fromℕ< n<m ))  a<sa )
+   qb1 : (q : Q) → toℕ (FiniteSet.F←Q fin q) < n → Bool
+   qb1 q q<n = Q→B q (NatP.<-trans q<n a<sa)
+
+List2Func : { Q : Set } → {n : ℕ } → (fin : FiniteSet Q ) → n < suc (FiniteSet.finite fin)  → Vec Bool n →  Q → Bool 
+List2Func {Q} {zero} fin (s≤s z≤n) [] q = false
+List2Func {Q} {suc n} fin (s≤s n<m) (h ∷ t) q with  FiniteSet.F←Q fin q ≟ fromℕ< n<m
+... | yes _ = h
+... | no _ = List2Func {Q} fin (NatP.<-trans n<m a<sa ) t q
+
+open import Level renaming ( suc to Suc ; zero to Zero) 
+open import Axiom.Extensionality.Propositional
+postulate f-extensionality : { n : Level}  →  Axiom.Extensionality.Propositional.Extensionality n n 
+
+F2L-iso : { Q : Set } → (fin : FiniteSet Q ) → (x : Vec Bool (FiniteSet.finite fin) ) → F2L fin a<sa (λ q _ → List2Func fin a<sa x q ) ≡ x
+F2L-iso {Q} fin x = f2l m a<sa x where
+  m = FiniteSet.finite fin
+  f2l : (n : ℕ ) → (n<m : n < suc m )→ (x : Vec Bool n ) → F2L fin n<m (λ q q<n → List2Func fin n<m x q )  ≡ x 
+  f2l zero (s≤s z≤n) [] = refl
+  f2l (suc n) (s≤s n<m) (h ∷ t ) = lemma1 lemma2 lemma3f where
+    lemma1 : {n : ℕ } → {h h1 : Bool } → {t t1 : Vec Bool n } → h ≡ h1 → t ≡ t1 →  h ∷ t ≡ h1 ∷ t1
+    lemma1 refl refl = refl
+    lemma2 : List2Func fin (s≤s n<m) (h ∷ t) (FiniteSet.Q←F fin (fromℕ< n<m)) ≡ h
+    lemma2 with FiniteSet.F←Q fin (FiniteSet.Q←F fin (fromℕ< n<m))  ≟ fromℕ< n<m
+    lemma2 | yes p = refl
+    lemma2 | no ¬p = ⊥-elim ( ¬p (FiniteSet.finiso← fin _) )
+    lemma4 : (q : Q ) → toℕ (FiniteSet.F←Q fin q ) < n → List2Func fin (s≤s n<m) (h ∷ t) q ≡ List2Func fin (NatP.<-trans n<m a<sa) t q
+    lemma4 q _ with FiniteSet.F←Q fin q ≟ fromℕ< n<m 
+    lemma4 q lt | yes p = ⊥-elim ( nat-≡< (toℕ-fromℕ< n<m) (lemma5 n lt (cong (λ k → toℕ k) p))) where 
+        lemma5 : {j k : ℕ } → ( n : ℕ) → suc j ≤ n → j ≡ k → k < n
+        lemma5 {zero} (suc n) (s≤s z≤n) refl = s≤s  z≤n
+        lemma5 {suc j} (suc n) (s≤s lt) refl = s≤s (lemma5 {j} n lt refl)
+    lemma4 q _ | no ¬p = refl
+    lemma3f :  F2L fin (NatP.<-trans n<m a<sa) (λ q q<n → List2Func fin (s≤s n<m) (h ∷ t) q  ) ≡ t
+    lemma3f = begin 
+         F2L fin (NatP.<-trans n<m a<sa) (λ q q<n → List2Func fin (s≤s n<m) (h ∷ t) q )
+       ≡⟨ cong (λ k → F2L fin (NatP.<-trans n<m a<sa) ( λ q q<n → k q q<n ))
+              (f-extensionality ( λ q →  
+              (f-extensionality ( λ q<n →  lemma4 q q<n )))) ⟩
+         F2L fin (NatP.<-trans n<m a<sa) (λ q q<n → List2Func fin (NatP.<-trans n<m a<sa) t q )
+       ≡⟨ f2l n (NatP.<-trans n<m a<sa ) t ⟩
+         t
+       ∎  where
+         open ≡-Reasoning
+
+
+L2F : {Q : Set } {n : ℕ } → (fin : FiniteSet Q )  → n < suc (FiniteSet.finite fin) → Vec Bool n → (q :  Q ) → toℕ (FiniteSet.F←Q fin q ) < n  → Bool
+L2F fin n<m x q q<n = List2Func fin n<m x q 
+
+L2F-iso : { Q : Set } → (fin : FiniteSet Q ) → (f : Q → Bool ) → (q : Q ) → (L2F fin a<sa (F2L fin a<sa (λ q _ → f q) )) q (toℕ<n _) ≡ f q
+L2F-iso {Q} fin f q = l2f m a<sa (toℕ<n _) where
+  m = FiniteSet.finite fin
+  lemma11f : {n : ℕ } → (n<m : n < m )  → ¬ ( FiniteSet.F←Q fin q ≡ fromℕ< n<m ) → toℕ (FiniteSet.F←Q fin q) ≤ n → toℕ (FiniteSet.F←Q fin q) < n
+  lemma11f  n<m ¬q=n q≤n = lemma13 n<m (contra-position (lemma12 n<m _) ¬q=n ) q≤n where
+     lemma13 : {n nq : ℕ } → (n<m : n < m )  → ¬ ( nq ≡ n ) → nq  ≤ n → nq < n
+     lemma13 {0} {0} (s≤s z≤n) nt z≤n = ⊥-elim ( nt refl )
+     lemma13 {suc _} {0} (s≤s (s≤s n<m)) nt z≤n = s≤s z≤n
+     lemma13 {suc n} {suc nq} n<m nt (s≤s nq≤n) = s≤s (lemma13 {n} {nq} (NatP.<-trans a<sa n<m ) (λ eq → nt ( cong ( λ k → suc k ) eq )) nq≤n)
+     lemma3f : {a b : ℕ } → (lt : a < b ) → fromℕ< (s≤s lt) ≡ suc (fromℕ< lt)
+     lemma3f (s≤s lt) = refl
+     lemma12f : {n m : ℕ } → (n<m : n < m ) → (f : Fin m )  → toℕ f ≡ n → f ≡ fromℕ< n<m 
+     lemma12f {zero} {suc m} (s≤s z≤n) zero refl = refl
+     lemma12f {suc n} {suc m} (s≤s n<m) (suc f) refl = subst ( λ k → suc f ≡ k ) (sym (lemma3f n<m) ) ( cong ( λ k → suc k ) ( lemma12f {n} {m} n<m f refl  ) )
+  l2f :  (n : ℕ ) → (n<m : n < suc m ) → (q<n : toℕ (FiniteSet.F←Q fin q ) < n )  →  (L2F fin n<m (F2L fin n<m  (λ q _ → f q))) q q<n ≡ f q
+  l2f zero (s≤s z≤n) ()
+  l2f (suc n) (s≤s n<m) (s≤s n<q) with FiniteSet.F←Q fin q ≟ fromℕ< n<m 
+  l2f (suc n) (s≤s n<m) (s≤s n<q) | yes p = begin 
+        f (FiniteSet.Q←F fin (fromℕ< n<m)) 
+     ≡⟨ cong ( λ k → f (FiniteSet.Q←F fin k )) (sym p)  ⟩
+        f (FiniteSet.Q←F fin ( FiniteSet.F←Q fin q ))
+     ≡⟨ cong ( λ k → f k ) (FiniteSet.finiso→ fin _ ) ⟩
+        f q 
+     ∎  where
+       open ≡-Reasoning
+  l2f (suc n) (s≤s n<m) (s≤s n<q) | no ¬p = l2f n (NatP.<-trans n<m a<sa) (lemma11f n<m ¬p n<q)
+
+fin→ : {A : Set} → FiniteSet A → FiniteSet (A → Bool ) 
+fin→ {A}  fin = iso-fin fin2List iso where
+    a = FiniteSet.finite fin
+    iso : ISO (Vec Bool a ) (A → Bool)
+    ISO.A←B iso x = F2L fin a<sa ( λ q _ → x q )
+    ISO.B←A iso x = List2Func fin a<sa x 
+    ISO.iso← iso x  =  F2L-iso fin x 
+    ISO.iso→ iso x = lemma where
+      lemma : List2Func fin a<sa (F2L fin a<sa (λ q _ → x q)) ≡ x
+      lemma = f-extensionality ( λ q → L2F-iso fin x q )
+    
+
+Fin2Finite : ( n : ℕ ) → FiniteSet (Fin n) 
+Fin2Finite n = record { F←Q = λ x → x ; Q←F = λ x → x ; finiso← = λ q → refl ; finiso→ = λ q → refl }
+
+data fin-less { n : ℕ } { A : Set }  (fa : FiniteSet A ) (n<m : n < FiniteSet.finite fa ) : Set where
+  elm1 : (elm : A ) → toℕ (FiniteSet.F←Q fa elm ) < n → fin-less fa n<m 
+
+get-elm : { n : ℕ }  { A : Set } {fa : FiniteSet A } {n<m : n < FiniteSet.finite fa } → fin-less fa n<m → A
+get-elm (elm1 a _ ) = a
+
+get-< : { n : ℕ }  { A : Set } {fa : FiniteSet A } {n<m : n < FiniteSet.finite fa }→ (f : fin-less fa n<m ) → toℕ (FiniteSet.F←Q fa (get-elm f )) < n
+get-< (elm1 _ b ) = b
+
+fin-less-cong : { n : ℕ }  { A : Set } (fa : FiniteSet A ) (n<m : n < FiniteSet.finite fa )
+   → (x y : fin-less fa n<m ) → get-elm {n} {A} {fa} x ≡ get-elm {n} {A} {fa} y → get-< x ≅  get-< y → x ≡ y
+fin-less-cong fa n<m (elm1 elm x) (elm1 elm x) refl HE.refl = refl
+
+fin-< : {A : Set} → { n : ℕ } → (fa : FiniteSet A ) → (n<m : n < FiniteSet.finite fa ) → FiniteSet (fin-less fa n<m ) 
+fin-< {A}  {n} fa n<m = iso-fin (Fin2Finite n) iso where
+    m = FiniteSet.finite fa
+    iso : ISO (Fin n) (fin-less fa n<m )
+    lemma8f : {i j n : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → i<n ≅ j<n  
+    lemma8f {zero} {zero} {suc n} refl {s≤s z≤n} {s≤s z≤n} = HE.refl
+    lemma8f {suc i} {suc i} {suc n} refl {s≤s i<n} {s≤s j<n} = HE.cong (λ k → s≤s k ) ( lemma8f {i} {i}  refl  )
+    lemma10f : {n i j  : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n }  → fromℕ< i<n ≡ fromℕ< j<n
+    lemma10f  refl  = HE.≅-to-≡ (HE.cong (λ k → fromℕ< k ) (lemma8f refl  ))
+    lemma3f : {a b c : ℕ } → { a<b : a < b } { b<c : b < c } { a<c : a < c } → NatP.<-trans a<b b<c ≡ a<c
+    lemma3f {a} {b} {c} {a<b} {b<c} {a<c} = HE.≅-to-≡ (lemma8f refl) 
+    lemma11f : {n : ℕ } {x : Fin n } → (n<m : n < m ) → toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m)) ≡ toℕ x
+    lemma11f {n} {x} n<m  = begin
+         toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m))
+      ≡⟨ toℕ-fromℕ< _ ⟩
+         toℕ x
+      ∎  where
+          open ≡-Reasoning
+    ISO.A←B iso (elm1 elm x) = fromℕ< x
+    ISO.B←A iso x = elm1 (FiniteSet.Q←F fa (fromℕ< (NatP.<-trans x<n n<m ))) to<n where
+      x<n : toℕ x < n
+      x<n = toℕ<n x
+      to<n : toℕ (FiniteSet.F←Q fa (FiniteSet.Q←F fa (fromℕ< (NatP.<-trans x<n n<m)))) < n
+      to<n = subst (λ k → toℕ k < n ) (sym (FiniteSet.finiso← fa _ )) (subst (λ k → k < n ) (sym ( toℕ-fromℕ< (NatP.<-trans x<n n<m) )) x<n )
+    ISO.iso← iso x  = lemma2 where
+      lemma2 : fromℕ< (subst (λ k → toℕ k < n) (sym
+       (FiniteSet.finiso← fa (fromℕ< (NatP.<-trans (toℕ<n x) n<m)))) (subst (λ k → k < n)
+       (sym (toℕ-fromℕ< (NatP.<-trans (toℕ<n x) n<m))) (toℕ<n x))) ≡ x
+      lemma2 = begin
+        fromℕ< (subst (λ k → toℕ k < n) (sym
+          (FiniteSet.finiso← fa (fromℕ< (NatP.<-trans (toℕ<n x) n<m)))) (subst (λ k → k < n)
+               (sym (toℕ-fromℕ< (NatP.<-trans (toℕ<n x) n<m))) (toℕ<n x))) 
+        ≡⟨⟩
+           fromℕ< ( subst (λ k → toℕ ( k ) < n ) (sym (FiniteSet.finiso← fa _ )) lemma6 )
+        ≡⟨ lemma10 (cong (λ k → toℕ k) (FiniteSet.finiso← fa _ ) ) ⟩
+           fromℕ< lemma6
+        ≡⟨ lemma10 (lemma11 n<m ) ⟩
+           fromℕ< ( toℕ<n x )
+        ≡⟨ fromℕ<-toℕ _ _ ⟩
+           x 
+        ∎  where
+          open ≡-Reasoning
+          lemma6 : toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m)) < n
+          lemma6 = subst ( λ k → k < n ) (sym (toℕ-fromℕ< (NatP.<-trans (toℕ<n x) n<m))) (toℕ<n x )
+    ISO.iso→ iso (elm1 elm x) = fin-less-cong fa n<m _ _ lemma (lemma8 (cong (λ k → toℕ (FiniteSet.F←Q fa k) ) lemma ) ) where
+      lemma13 : toℕ (fromℕ< x) ≡ toℕ (FiniteSet.F←Q fa elm)
+      lemma13 = begin
+            toℕ (fromℕ< x)
+         ≡⟨ toℕ-fromℕ< _ ⟩
+            toℕ (FiniteSet.F←Q fa elm)
+         ∎  where open ≡-Reasoning
+      lemma : FiniteSet.Q←F fa (fromℕ< (NatP.<-trans (toℕ<n (ISO.A←B iso (elm1 elm x))) n<m)) ≡ elm 
+      lemma = begin
+           FiniteSet.Q←F fa (fromℕ< (NatP.<-trans (toℕ<n (ISO.A←B iso (elm1 elm x))) n<m))
+         ≡⟨⟩
+           FiniteSet.Q←F fa (fromℕ< ( NatP.<-trans (toℕ<n ( fromℕ< x ) ) n<m))
+         ≡⟨ cong (λ k → FiniteSet.Q←F fa k) (lemma10 lemma13 ) ⟩
+            FiniteSet.Q←F fa (fromℕ< ( NatP.<-trans x n<m))
+         ≡⟨ cong (λ k → FiniteSet.Q←F fa (fromℕ< k ))  {!!} ⟩
+           FiniteSet.Q←F fa (fromℕ< ( toℕ<n (FiniteSet.F←Q fa elm)))
+         ≡⟨ cong (λ k → FiniteSet.Q←F fa k ) ( fromℕ<-toℕ _ _ ) ⟩
+           FiniteSet.Q←F fa (FiniteSet.F←Q fa elm )
+         ≡⟨ FiniteSet.finiso→ fa _ ⟩
+            elm 
+         ∎  where open ≡-Reasoning
+
+