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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 29 Dec 2020 15:32:57 +0900
parents b3f05cd08d24
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module FSetUtil  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 Data.Nat.Properties as NatP  hiding ( _≟_ )
open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 

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 lemma3 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
    lemma3 :  F2L fin (NatP.<-trans n<m a<sa) (λ q q<n → List2Func fin (s≤s n<m) (h ∷ t) q  ) ≡ t
    lemma3 = 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
  lemma11 : {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
  lemma11  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)
     lemma3 : {a b : ℕ } → (lt : a < b ) → fromℕ< (s≤s lt) ≡ suc (fromℕ< lt)
     lemma3 (s≤s lt) = refl
     lemma12 : {n m : ℕ } → (n<m : n < m ) → (f : Fin m )  → toℕ f ≡ n → f ≡ fromℕ< n<m 
     lemma12 {zero} {suc m} (s≤s z≤n) zero refl = refl
     lemma12 {suc n} {suc m} (s≤s n<m) (suc f) refl = subst ( λ k → suc f ≡ k ) (sym (lemma3 n<m) ) ( cong ( λ k → suc k ) ( lemma12 {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) (lemma11 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 )
    lemma8 : {i j n : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n } → i<n ≅ j<n  
    lemma8 {zero} {zero} {suc n} refl {s≤s z≤n} {s≤s z≤n} = HE.refl
    lemma8 {suc i} {suc i} {suc n} refl {s≤s i<n} {s≤s j<n} = HE.cong (λ k → s≤s k ) ( lemma8 {i} {i}  refl  )
    lemma10 : {n i j  : ℕ } → ( i ≡ j ) → {i<n : i < n } → {j<n : j < n }  → fromℕ< i<n ≡ fromℕ< j<n
    lemma10  refl  = HE.≅-to-≡ (HE.cong (λ k → fromℕ< k ) (lemma8 refl  ))
    lemma3 : {a b c : ℕ } → { a<b : a < b } { b<c : b < c } { a<c : a < c } → NatP.<-trans a<b b<c ≡ a<c
    lemma3 {a} {b} {c} {a<b} {b<c} {a<c} = HE.≅-to-≡ (lemma8 refl) 
    lemma11 : {n : ℕ } {x : Fin n } → (n<m : n < m ) → toℕ (fromℕ< (NatP.<-trans (toℕ<n x) n<m)) ≡ toℕ x
    lemma11 {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 ))  lemma3 ⟩
           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