Mercurial > hg > Members > kono > Proof > automaton
view automaton-in-agda/src/finiteSetUtil.agda @ 405:af8f630b7e60
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 24 Sep 2023 18:02:04 +0900 |
| parents | dfaf230f7b9a |
| children | a60132983557 |
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{-# OPTIONS --cubical-compatible --safe #-} module finiteSetUtil where open import Data.Nat hiding ( _≟_ ) open import Data.Fin renaming ( _<_ to _<<_ ; _>_ to _f>_ ; _≟_ to _f≟_ ) hiding (_≤_ ; pred ) open import Data.Fin.Properties hiding ( <-trans ; ≤-trans ; ≤-refl ; <-irrelevant ) renaming ( <-cmp to <-fcmp ) 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 NP hiding ( _≟_ ) record Found ( Q : Set ) (p : Q → Bool ) : Set where field found-q : Q found-p : p found-q ≡ true open Bijection open import Axiom.Extensionality.Propositional open import Level hiding (suc ; zero) module _ {Q : Set } (F : FiniteSet Q) where open FiniteSet F equal?-refl : { x : Q } → equal? x x ≡ true equal?-refl {x} with F←Q x ≟ F←Q x ... | yes eq = refl ... | no ne = ⊥-elim (ne refl) 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 eqP : (x y : Q) → Dec ( x ≡ y ) eqP x y with F←Q x ≟ F←Q y ... | yes eq = yes (subst₂ (λ j k → j ≡ k ) (finiso→ x) (finiso→ y) (cong Q←F eq) ) ... | no n = no (λ eq → n (cong F←Q eq)) 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 NP.<-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 refl m<n = fromℕ<-toℕ i m<n found : { p : Q → Bool } → (q : Q ) → p q ≡ true → exists p ≡ true found {p} q pt = found1 finite (NP.≤-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 not-found : { p : Q → Bool } → ( (q : Q ) → p q ≡ false ) → exists p ≡ false not-found {p} pn = not-found2 finite NP.≤-refl where not-found2 : (m : ℕ ) → (m<n : m Data.Nat.≤ finite ) → exists1 m m<n p ≡ false not-found2 zero _ = refl not-found2 ( suc m ) m<n with pn (Q←F (fromℕ< {m} {finite} m<n)) not-found2 (suc m) m<n | eq = begin p (Q←F (fromℕ< m<n)) \/ exists1 m (<to≤ m<n) p ≡⟨ bool-or-1 eq ⟩ exists1 m (<to≤ m<n) p ≡⟨ not-found2 m (<to≤ m<n) ⟩ false ∎ where open ≡-Reasoning found← : { p : Q → Bool } → exists p ≡ true → Found Q p found← {p} exst = found2 finite NP.≤-refl (first-end p ) where found2 : (m : ℕ ) (m<n : m Data.Nat.≤ finite ) → End m p → Found Q p found2 0 m<n end = ⊥-elim ( ¬-bool f01 exst ) where f01 : exists p ≡ false f01 = not-found (λ q → subst ( λ k → p k ≡ false ) (finiso→ _) (end (F←Q q) z≤n )) 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 (<to≤ 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 ) ) Q←F-inject : {x y : Fin finite} → Q←F x ≡ Q←F y → x ≡ y Q←F-inject eq = subst₂ (λ j k → j ≡ k ) (finiso← _) (finiso← _) (cong F←Q eq) F←Q-inject : {x y : Q } → F←Q x ≡ F←Q y → x ≡ y F←Q-inject eq = subst₂ (λ j k → j ≡ k ) (finiso→ _) (finiso→ _) (cong Q←F eq) iso-fin : {A B : Set} → FiniteSet A → Bijection A B → FiniteSet B iso-fin {A} {B} fin iso = record { Q←F = λ f → fun→ iso ( FiniteSet.Q←F fin f ) ; F←Q = λ b → FiniteSet.F←Q fin (fun← iso b ) ; finiso→ = finiso→ ; finiso← = finiso← } where finiso→ : (q : B) → fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) ≡ q finiso→ q = begin fun→ iso (FiniteSet.Q←F fin (FiniteSet.F←Q fin (Bijection.fun← iso q))) ≡⟨ cong (λ k → fun→ iso k ) (FiniteSet.finiso→ fin _ ) ⟩ fun→ iso (Bijection.fun← iso q) ≡⟨ fiso→ iso _ ⟩ q ∎ where open ≡-Reasoning finiso← : (f : Fin (FiniteSet.finite fin ))→ FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) ≡ f finiso← f = begin FiniteSet.F←Q fin (Bijection.fun← iso (Bijection.fun→ iso (FiniteSet.Q←F fin f))) ≡⟨ cong (λ k → FiniteSet.F←Q fin k ) (Bijection.fiso← iso _) ⟩ FiniteSet.F←Q fin (FiniteSet.Q←F fin f) ≡⟨ FiniteSet.finiso← fin _ ⟩ f ∎ where open ≡-Reasoning data One : Set where one : One finOne : FiniteSet One finOne = record { finite = 1 ; Q←F = λ _ → one ; F←Q = λ _ → # 0 ; finiso→ = fin00 ; finiso← = fin1≡0 } where fin00 : (q : One) → one ≡ q fin00 one = refl 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 : Bijection B (Fin zero ∨ B) iso = record { fun← = fun←1 ; fun→ = λ b → case2 b ; fiso→ = fiso→1 ; fiso← = λ _ → refl } where fun←1 : Fin zero ∨ B → B fun←1 (case2 x) = x fiso→1 : (f : Fin zero ∨ B ) → case2 (fun←1 f) ≡ f fiso→1 (case2 x) = refl fin-∨2 {B} (suc a) fb = iso-fin (fin-∨1 (fin-∨2 a fb) ) iso where iso : Bijection (One ∨ (Fin a ∨ B) ) (Fin (suc a) ∨ B) fun← iso (case1 zero) = case1 one fun← iso (case1 (suc f)) = case2 (case1 f) fun← iso (case2 b) = case2 (case2 b) fun→ iso (case1 one) = case1 zero fun→ iso (case2 (case1 f)) = case1 (suc f) fun→ iso (case2 (case2 b)) = case2 b fiso← iso (case1 one) = refl fiso← iso (case2 (case1 x)) = refl fiso← iso (case2 (case2 x)) = refl fiso→ iso (case1 zero) = refl fiso→ iso (case1 (suc x)) = refl fiso→ iso (case2 x) = refl FiniteSet→Fin : {A : Set} → (fin : FiniteSet A ) → Bijection (Fin (FiniteSet.finite fin)) A fun← (FiniteSet→Fin fin) f = FiniteSet.F←Q fin f fun→ (FiniteSet→Fin fin) f = FiniteSet.Q←F fin f fiso← (FiniteSet→Fin fin) = FiniteSet.finiso← fin fiso→ (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 : Bijection (Fin a ∨ B ) (A ∨ B) fun← iso2 (case1 x) = case1 (fun← ia x ) fun← iso2 (case2 x) = case2 x fun→ iso2 (case1 x) = case1 (fun→ ia x ) fun→ iso2 (case2 x) = case2 x fiso← iso2 (case1 x) = cong ( λ k → case1 k ) (Bijection.fiso← ia x) fiso← iso2 (case2 x) = refl fiso→ iso2 (case1 x) = cong ( λ k → case1 k ) (Bijection.fiso→ ia x) fiso→ iso2 (case2 x) = refl open import Data.Product hiding ( map ) 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 : Bijection (Fin a × B) ( A × B ) fun← iso-1 x = ( FiniteSet.F←Q fa (proj₁ x) , proj₂ x) fun→ iso-1 x = ( FiniteSet.Q←F fa (proj₁ x) , proj₂ x) fiso← 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 _ ) fiso→ iso-1 x = cong ( λ k → ( k , proj₂ x ) ) (FiniteSet.finiso→ fa _ ) iso-2 : {a : ℕ } → Bijection (B ∨ (Fin a × B)) (Fin (suc a) × B) fun← iso-2 (zero , b ) = case1 b fun← iso-2 (suc fst , b ) = case2 ( fst , b ) fun→ iso-2 (case1 b) = ( zero , b ) fun→ iso-2 (case2 (a , b )) = ( suc a , b ) fiso← iso-2 (case1 x) = refl fiso← iso-2 (case2 x) = refl fiso→ iso-2 (zero , b ) = refl fiso→ 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 : Bijection (Fin a ∧ B) ( A ∧ B ) fun← iso-1 x = record { proj1 = FiniteSet.F←Q fa (proj1 x) ; proj2 = proj2 x} fun→ iso-1 x = record { proj1 = FiniteSet.Q←F fa (proj1 x) ; proj2 = proj2 x} fiso← 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 _ ) fiso→ iso-1 x = cong ( λ k → record {proj1 = k ; proj2 = proj2 x } ) (FiniteSet.finiso→ fa _ ) iso-2 : {a : ℕ } → Bijection (B ∨ (Fin a ∧ B)) (Fin (suc a) ∧ B) fun← iso-2 (record { proj1 = zero ; proj2 = b }) = case1 b fun← iso-2 (record { proj1 = suc fst ; proj2 = b }) = case2 ( record { proj1 = fst ; proj2 = b } ) fun→ iso-2 (case1 b) = record {proj1 = zero ; proj2 = b } fun→ iso-2 (case2 (record { proj1 = a ; proj2 = b })) = record { proj1 = suc a ; proj2 = b } fiso← iso-2 (case1 x) = refl fiso← iso-2 (case2 x) = refl fiso→ iso-2 (record { proj1 = zero ; proj2 = b }) = refl fiso→ 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 hiding ( map ; length ) 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 ) → Data.Fin.cast eq ( Data.Fin.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 : Bijection (Vec Bool n ∨ Vec Bool n) (Vec Bool (suc n)) iso = record { fun← = QtoR ; fun→ = RtoQ ; fiso← = isoQR ; fiso→ = 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 (NP.<-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 (NP.<-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 (NP.<-trans n<m a<sa ) t q open import Level renaming ( suc to Suc ; zero to Zero) 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} (NP.<-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 (NP.<-trans n<m a<sa) (lemma11f n<m ¬p n<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 } open import Data.List open FiniteSet memberQ : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool memberQ {Q} finq q [] = false memberQ {Q} finq q (q0 ∷ qs) with equal? finq q q0 ... | true = true ... | false = memberQ finq q qs -- -- there is a duplicate element in finite list -- -- -- How about this? -- get list of Q -- remove one element for each Q from list -- there must be remaining list > 1 -- theses are duplicates -- actualy it is duplicate phase2 : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool phase2 finq q [] = false phase2 finq q (x ∷ qs) with equal? finq q x ... | true = true ... | false = phase2 finq q qs phase1 : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool phase1 finq q [] = false phase1 finq q (x ∷ qs) with equal? finq q x ... | true = phase2 finq q qs ... | false = phase1 finq q qs dup-in-list : { Q : Set } (finq : FiniteSet Q) (q : Q) (qs : List Q ) → Bool dup-in-list {Q} finq q qs = phase1 finq q qs -- -- if length of the list is longer than kinds of a finite set, there is a duplicate -- prove this based on the theorem on Data.Fin -- dup-in-list+fin : { Q : Set } (finq : FiniteSet Q) → (q : Q) (qs : List Q ) → fin-dup-in-list (F←Q finq q) (map (F←Q finq) qs) ≡ true → dup-in-list finq q qs ≡ true dup-in-list+fin {Q} finq q qs p = i-phase1 qs p where i-phase2 : (qs : List Q) → fin-phase2 (F←Q finq q) (map (F←Q finq) qs) ≡ true → phase2 finq q qs ≡ true i-phase2 (x ∷ qs) p with equal? finq q x in eq | <-fcmp (F←Q finq q) (F←Q finq x) ... | true | t = refl ... | false | tri< a ¬b ¬c = i-phase2 qs p ... | false | tri≈ ¬a b ¬c = ⊥-elim (¬-bool eq (subst₂ (λ j k → equal? finq j k ≡ true) (finiso→ finq q) (subst (λ k → Q←F finq k ≡ x) (sym b) (finiso→ finq x)) ( equal?-refl finq ))) ... | false | tri> ¬a ¬b c = i-phase2 qs p i-phase1 : (qs : List Q) → fin-phase1 (F←Q finq q) (map (F←Q finq) qs) ≡ true → phase1 finq q qs ≡ true i-phase1 (x ∷ qs) p with equal? finq q x in eq | <-fcmp (F←Q finq q) (F←Q finq x) ... | true | tri< a ¬b ¬c = ⊥-elim ( nat-≡< (cong (λ x → toℕ (F←Q finq x)) ( equal→refl finq eq )) a ) ... | true | tri≈ ¬a b ¬c = i-phase2 qs p ... | true | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (cong (λ x → toℕ (F←Q finq x)) (sym ( equal→refl finq eq ))) c ) ... | false | tri< a ¬b ¬c = i-phase1 qs p ... | false | tri≈ ¬a b ¬c = ⊥-elim (¬-bool eq (subst₂ (λ j k → equal? finq j k ≡ true) (finiso→ finq q) (subst (λ k → Q←F finq k ≡ x) (sym b) (finiso→ finq x)) ( equal?-refl finq ))) ... | false | tri> ¬a ¬b c = i-phase1 qs p record Dup-in-list {Q : Set } (finq : FiniteSet Q) (qs : List Q) : Set where field dup : Q is-dup : dup-in-list finq dup qs ≡ true dup-in-list>n : {Q : Set } → (finq : FiniteSet Q) → (qs : List Q) → (len> : length qs > finite finq ) → Dup-in-list finq qs dup-in-list>n {Q} finq qs lt = record { dup = Q←F finq (FDup-in-list.dup dl) ; is-dup = dup-in-list+fin finq (Q←F finq (FDup-in-list.dup dl)) qs dl01 } where maplen : (qs : List Q) → length (map (F←Q finq) qs) ≡ length qs maplen [] = refl maplen (x ∷ qs) = cong suc (maplen qs) dl : FDup-in-list (finite finq ) (map (F←Q finq) qs) dl = fin-dup-in-list>n (map (F←Q finq) qs) (subst (λ k → k > finite finq ) (sym (maplen qs)) lt) dl01 : fin-dup-in-list (F←Q finq (Q←F finq (FDup-in-list.dup dl))) (map (F←Q finq) qs) ≡ true dl01 = subst (λ k → fin-dup-in-list k (map (F←Q finq) qs) ≡ true ) (sym (finiso← finq _)) ( FDup-in-list.is-dup dl ) open import bijection using ( InjectiveF ; Is ) -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) inject-fin : {A B : Set} (fa : FiniteSet A ) → (fi : InjectiveF B A) → (is-B : (a : A ) → Dec (Is B A (InjectiveF.f fi) a) ) → FiniteSet B inject-fin {A} {B} fa fi is-B with finite fa in eq1 ... | zero = record { finite = 0 ; Q←F = λ () ; F←Q = λ b → ⊥-elim ( lem00 b) ; finiso→ = λ b → ⊥-elim ( lem00 b) ; finiso← = λ () } where lem00 : ( b : B) → ⊥ lem00 b with subst (λ k → Fin k ) eq1 (F←Q fa (InjectiveF.f fi b)) ... | () ... | suc pfa = record { finite = maxb ; Q←F = λ fb → CountB.b (cb00 _ (fin<n {_} fb)) ; F←Q = λ b → fromℕ< (cb<mb b) ; finiso→ = iso1 ; finiso← = iso0 } where f = InjectiveF.f fi pfa<fa : pfa < finite fa pfa<fa = subst (λ k → pfa < k ) (sym eq1) a<sa 0<fa : 0 < finite fa 0<fa = <-transˡ (s≤s z≤n) pfa<fa count-B : ℕ → ℕ count-B zero with is-B (Q←F fa ( fromℕ< {0} 0<fa )) ... | yes isb = 1 ... | no nisb = 0 count-B (suc n) with <-cmp (finite fa) (suc n) ... | tri< a ¬b ¬c = count-B n ... | tri≈ ¬a b ¬c = count-B n ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) ... | yes isb = suc (count-B n) ... | no nisb = count-B n record CountB (n : ℕ) : Set where field b : B cb : ℕ b=cn : cb ≡ toℕ (F←Q fa (f b)) cb=n : count-B cb ≡ suc n cb-inject : (cb1 : ℕ) → (c1<a : cb1 < finite fa) → Is B A f (Q←F fa (fromℕ< c1<a)) → count-B cb ≡ count-B cb1 → cb ≡ cb1 maxb : ℕ maxb = count-B (finite fa) count-B-mono : {i j : ℕ} → i ≤ j → count-B i ≤ count-B j count-B-mono {i} {j} i≤j with ≤-∨ i≤j ... | case1 refl = ≤-refl ... | case2 i<j = lem00 _ _ i<j where lem00 : (i j : ℕ) → i < j → count-B i ≤ count-B j lem00 i (suc j) (s≤s i<j) = ≤-trans (count-B-mono i<j) (lem01 j) where lem01 : (j : ℕ) → count-B j ≤ count-B (suc j) lem01 zero with <-cmp (finite fa) 1 lem01 zero | tri< a ¬b ¬c = ≤-refl lem01 zero | tri≈ ¬a b ¬c = ≤-refl lem01 zero | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa ( fromℕ< {0} 0<fa )) ... | yes isb1 | yes isb0 = s≤s z≤n ... | yes isb1 | no nisb0 = z≤n ... | no nisb1 | yes isb0 = refl-≤≡ (sym lem14 ) where lem14 : count-B 0 ≡ 1 -- in-equality does not work we have to repeat the proof lem14 with is-B (Q←F fa ( fromℕ< {0} 0<fa )) ... | yes isb = refl ... | no ne = ⊥-elim (ne isb0) ... | no nisb1 | no nisb0 = z≤n lem01 (suc i) with <-cmp (finite fa) (suc i) | <-cmp (finite fa) (suc (suc i)) ... | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = refl-≤≡ (sym lem14) where lem14 : count-B (suc i) ≡ count-B i lem14 with <-cmp (finite fa) (suc i) ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a a ) ... | tri> ¬a ¬b c = ⊥-elim ( ¬a a ) ... | tri< a ¬b ¬c | tri≈ ¬a b ¬c₁ = ⊥-elim (nat-≡< b (<-trans a a<sa)) ... | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = ⊥-elim (nat-<> a (<-trans a<sa c) ) ... | tri≈ ¬a b ¬c | tri< a ¬b ¬c₁ = refl-≤≡ (sym lem14 ) where lem14 : count-B (suc i) ≡ count-B i lem14 with <-cmp (finite fa) (suc i) ... | tri< a ¬b ¬c = refl ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = ⊥-elim ( ¬c c ) ... | tri≈ ¬a b ¬c | tri≈ ¬a₁ b₁ ¬c₁ = ⊥-elim (nat-≡< (sym b) (subst (λ k → _ < k ) (sym b₁) a<sa) ) ... | tri≈ ¬a b ¬c | tri> ¬a₁ ¬b c = ⊥-elim (nat-≡< (sym b) (<-trans a<sa c)) ... | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = ⊥-elim (nat-≤> a (<-transʳ c a<sa ) ) ... | tri> ¬a ¬b c | tri≈ ¬a₁ b ¬c with is-B (Q←F fa (fromℕ< c)) ... | yes isb = refl-≤≡ (sym lem14) where lem14 : count-B (suc i) ≡ suc (count-B i) lem14 with <-cmp (finite fa) (suc i) ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) ... | yes isb = refl ... | no ne = ⊥-elim (ne record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) ... | no nisb = refl-≤≡ (sym lem14) where lem14 : count-B (suc i) ≡ count-B i lem14 with <-cmp (finite fa) (suc i) ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) ... | yes isb = ⊥-elim (nisb record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) ... | no ne = refl lem01 (suc i) | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ with is-B (Q←F fa (fromℕ< c)) | is-B (Q←F fa (fromℕ< c₁)) ... | yes isb0 | yes isb1 = ≤-trans (refl-≤≡ (sym lem14)) a≤sa where lem14 : count-B (suc i) ≡ suc (count-B i) lem14 with <-cmp (finite fa) (suc i) ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) ... | no ne = ⊥-elim (ne record {a = Is.a isb0 ; fa=c = trans (Is.fa=c isb0) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) ... | yes isb = refl ... | yes isb0 | no nisb1 = refl-≤≡ (sym lem14) where lem14 : count-B (suc i) ≡ suc (count-B i) lem14 with <-cmp (finite fa) (suc i) ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) ... | no ne = ⊥-elim (ne record {a = Is.a isb0 ; fa=c = trans (Is.fa=c isb0) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) ... | yes isb = refl ... | no nisb0 | yes isb1 = ≤-trans (refl-≤≡ (sym lem14)) a≤sa where lem14 : count-B (suc i) ≡ count-B i lem14 with <-cmp (finite fa) (suc i) ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) ... | no ne = refl ... | yes isb = ⊥-elim (nisb0 record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) ... | no nisb0 | no nisb1 = refl-≤≡ (sym lem14) where lem14 : count-B (suc i) ≡ count-B i lem14 with <-cmp (finite fa) (suc i) ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) ... | no ne = refl ... | yes isb = ⊥-elim (nisb0 record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) lem31 : (b : B) → 0 < count-B (toℕ (F←Q fa (f b))) lem31 b = lem32 (toℕ (F←Q fa (f b))) refl where lem32 : (i : ℕ) → toℕ (F←Q fa (f b)) ≡ i → 0 < count-B i lem32 zero eq with is-B (Q←F fa ( fromℕ< {0} 0<fa )) ... | yes isb = s≤s z≤n ... | no nisb = ⊥-elim ( nisb record { a = b ; fa=c = lem33 } ) where lem33 : f b ≡ Q←F fa ( fromℕ< {0} 0<fa ) lem33 = begin f b ≡⟨ sym (finiso→ fa _) ⟩ Q←F fa ( F←Q fa (f b)) ≡⟨ sym (cong (λ k → Q←F fa k) ( fromℕ<-toℕ _ (fin<n _))) ⟩ Q←F fa ( fromℕ< (fin<n _) ) ≡⟨ cong (λ k → Q←F fa k) (fromℕ<-cong _ _ eq (fin<n _) 0<fa) ⟩ Q←F fa ( fromℕ< {0} 0<fa ) ∎ where open ≡-Reasoning lem32 (suc i) eq with <-cmp (finite fa) (suc i) ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< eq (<-trans (fin<n _) a) ) ... | tri≈ ¬a eq1 ¬c = ⊥-elim ( nat-≡< eq (subst (λ k → toℕ (F←Q fa (f b)) < k ) eq1 (fin<n _))) ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) ... | yes isb = s≤s z≤n ... | no nisb = ⊥-elim ( nisb record { a = b ; fa=c = lem33 } ) where lem33 : f b ≡ Q←F fa ( fromℕ< c) lem33 = begin f b ≡⟨ sym (finiso→ fa _) ⟩ Q←F fa ( F←Q fa (f b)) ≡⟨ sym (cong (λ k → Q←F fa k) ( fromℕ<-toℕ _ (fin<n _))) ⟩ Q←F fa ( fromℕ< (fin<n _) ) ≡⟨ cong (λ k → Q←F fa k) (fromℕ<-cong _ _ eq (fin<n _) c ) ⟩ Q←F fa ( fromℕ< c ) ∎ where open ≡-Reasoning cb<mb : (b : B) → pred (count-B (toℕ (F←Q fa (f b)))) < maxb cb<mb b = sx≤y→x<y ( begin suc ( pred (count-B (toℕ (F←Q fa (f b))))) ≡⟨ sucprd (lem31 b) ⟩ count-B (toℕ (F←Q fa (f b))) ≤⟨ lem02 ⟩ count-B (finite fa) ∎ ) where open ≤-Reasoning lem02 : count-B (toℕ (F←Q fa (f b))) ≤ count-B (finite fa) lem02 = count-B-mono (<to≤ (fin<n {_} (F←Q fa (f b)))) cb00 : (n : ℕ) → n < count-B (finite fa) → CountB n cb00 n n<cb = lem09 (finite fa) (count-B (finite fa)) (<-transˡ a<sa n<cb) refl where lem06 : (i j : ℕ) → (i<fa : i < finite fa) (j<fa : j < finite fa) → Is B A f (Q←F fa (fromℕ< i<fa)) → Is B A f (Q←F fa (fromℕ< j<fa)) → count-B i ≡ count-B j → i ≡ j lem06 i j i<fa j<fa bi bj eq = lem08 where lem20 : (i j : ℕ) → i < j → (i<fa : i < finite fa) (j<fa : j < finite fa) → Is B A f (Q←F fa (fromℕ< i<fa)) → Is B A f (Q←F fa (fromℕ< j<fa)) → count-B i < count-B j lem20 zero (suc j) i<j i<fa j<fa bi bj with <-cmp (finite fa) (suc j) ... | tri< a ¬b ¬c = ⊥-elim (¬c j<fa) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c j<fa) ... | tri> ¬a ¬b c with is-B (Q←F fa ( fromℕ< 0<fa )) | is-B (Q←F fa (fromℕ< c)) ... | no nisc | _ = ⊥-elim (nisc record { a = Is.a bi ; fa=c = lem26 } ) where lem26 : f (Is.a bi) ≡ Q←F fa (fromℕ< 0<fa) lem26 = trans (Is.fa=c bi) (cong (Q←F fa) (fromℕ<-cong _ _ refl i<fa 0<fa) ) ... | yes _ | no nisc = ⊥-elim (nisc record { a = Is.a bj ; fa=c = lem26 } ) where lem26 : f (Is.a bj) ≡ Q←F fa (fromℕ< c) lem26 = trans (Is.fa=c bj) (cong (Q←F fa) (fromℕ<-cong _ _ refl j<fa c) ) ... | yes isb1 | yes _ = lem25 where lem14 : count-B 0 ≡ 1 lem14 with is-B (Q←F fa ( fromℕ< 0<fa )) ... | no ne = ⊥-elim (ne record {a = Is.a isb1 ; fa=c = trans (Is.fa=c isb1) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) ... | yes isb = refl lem25 : 2 ≤ suc (count-B j) lem25 = begin 2 ≡⟨ cong suc (sym lem14) ⟩ suc (count-B 0) ≤⟨ s≤s (count-B-mono {0} {j} z≤n) ⟩ suc (count-B j) ∎ where open ≤-Reasoning lem20 (suc i) zero () i<fa j<fa bi bj lem20 (suc i) (suc j) (s≤s i<j) i<fa j<fa bi bj = lem21 where -- -- i < suc i ≤ j -- cb i < suc (cb i) < cb (suc i) ≤ cb j -- lem23 : suc (count-B j) ≡ count-B (suc j) lem23 with <-cmp (finite fa) (suc j) ... | tri< a ¬b ¬c = ⊥-elim (¬c j<fa) ... | tri≈ ¬a b ¬c = ⊥-elim (¬c j<fa) ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) ... | yes _ = refl ... | no nisa = ⊥-elim ( nisa record { a = Is.a bj ; fa=c = lem26 } ) where lem26 : f (Is.a bj) ≡ Q←F fa (fromℕ< c) lem26 = trans (Is.fa=c bj) (cong (Q←F fa) (fromℕ<-cong _ _ refl j<fa c) ) lem21 : count-B (suc i) < count-B (suc j) lem21 = sx≤py→x≤y ( begin suc (suc (count-B (suc i))) ≤⟨ s≤s ( s≤s ( count-B-mono i<j )) ⟩ suc (suc (count-B j)) ≡⟨ cong suc lem23 ⟩ suc (count-B (suc j)) ∎ ) where open ≤-Reasoning lem08 : i ≡ j lem08 with <-cmp i j ... | tri< a ¬b ¬c = ⊥-elim (nat-≡< eq ( lem20 i j a i<fa j<fa bi bj )) ... | tri≈ ¬a b ¬c = b ... | tri> ¬a ¬b c₁ = ⊥-elim (nat-≡< (sym eq) ( lem20 j i c₁ j<fa i<fa bj bi )) lem09 : (i j : ℕ) → suc n ≤ j → j ≡ count-B i → CountB n lem09 0 (suc j) (s≤s le) eq with is-B (Q←F fa (fromℕ< {0} 0<fa )) ... | no nisb = ⊥-elim ( nat-≡< (sym eq) (s≤s z≤n) ) ... | yes isb with ≤-∨ (s≤s le) ... | case1 eq2 = record { b = Is.a isb ; cb = 0 ; b=cn = lem10 ; cb=n = trans lem14 (sym (trans eq2 eq)) ; cb-inject = λ cb1 c1<fa b1 eq → lem06 0 cb1 0<fa c1<fa isb b1 eq } where lem14 : count-B 0 ≡ 1 lem14 with is-B (Q←F fa ( fromℕ< 0<fa )) ... | no ne = ⊥-elim (ne record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) ... | yes isb = refl lem10 : 0 ≡ toℕ (F←Q fa (f (Is.a isb))) lem10 = begin 0 ≡⟨ sym ( toℕ-fromℕ< 0<fa ) ⟩ toℕ (fromℕ< {0} 0<fa ) ≡⟨ cong toℕ (sym (finiso← fa _)) ⟩ toℕ (F←Q fa (Q←F fa (fromℕ< {0} 0<fa ))) ≡⟨ cong (λ k → toℕ ((F←Q fa k))) (sym (Is.fa=c isb)) ⟩ toℕ (F←Q fa (f (Is.a isb))) ∎ where open ≡-Reasoning ... | case2 (s≤s lt) = ⊥-elim ( nat-≡< (sym eq) (s≤s (<-transʳ z≤n lt) )) lem09 (suc i) (suc j) (s≤s le) eq with <-cmp (finite fa) (suc i) ... | tri< a ¬b ¬c = lem09 i (suc j) (s≤s le) eq ... | tri≈ ¬a b ¬c = lem09 i (suc j) (s≤s le) eq ... | tri> ¬a ¬b c with is-B (Q←F fa (fromℕ< c)) ... | no nisb = lem09 i (suc j) (s≤s le) eq ... | yes isb with ≤-∨ (s≤s le) ... | case1 eq2 = record { b = Is.a isb ; cb = suc i ; b=cn = lem11 ; cb=n = trans lem14 (sym (trans eq2 eq )) ; cb-inject = λ cb1 c1<fa b1 eq → lem06 (suc i) cb1 c c1<fa isb b1 eq } where lem14 : count-B (suc i) ≡ suc (count-B i) lem14 with <-cmp (finite fa) (suc i) ... | tri< a₂ ¬b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri≈ ¬a₂ b₂ ¬c₂ = ⊥-elim (¬c₂ c) ... | tri> ¬a₂ ¬b₂ c₂ with is-B (Q←F fa ( fromℕ< c₂ )) ... | yes isb = refl ... | no ne = ⊥-elim (ne record {a = Is.a isb ; fa=c = trans (Is.fa=c isb) (cong (λ k → Q←F fa k) (lemma10 refl )) } ) lem11 : suc i ≡ toℕ (F←Q fa (f (Is.a isb))) lem11 = begin suc i ≡⟨ sym ( toℕ-fromℕ< c) ⟩ toℕ (fromℕ< c) ≡⟨ cong toℕ (sym (finiso← fa _)) ⟩ toℕ (F←Q fa (Q←F fa (fromℕ< c ))) ≡⟨ cong (λ k → toℕ ((F←Q fa k))) (sym (Is.fa=c isb)) ⟩ toℕ (F←Q fa (f (Is.a isb))) ∎ where open ≡-Reasoning ... | case2 (s≤s lt) = lem09 i j lt (cong pred eq) iso0 : (x : Fin maxb) → fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡ x iso0 x = begin fromℕ< (cb<mb (CountB.b (cb00 (toℕ x) (fin<n _)))) ≡⟨ fromℕ<-cong _ _ ( begin pred (count-B (toℕ (F←Q fa (f (CountB.b (cb00 (toℕ x) (fin<n _))))))) ≡⟨ sym (cong (λ k → pred (count-B k)) (CountB.b=cn CB)) ⟩ pred (count-B (CountB.cb CB)) ≡⟨ cong pred (CountB.cb=n CB) ⟩ pred (suc (toℕ x)) ≡⟨ refl ⟩ toℕ x ∎ ) (cb<mb (CountB.b CB)) (fin<n _) ⟩ fromℕ< (fin<n {_} x) ≡⟨ fromℕ<-toℕ _ (fin<n {_} x) ⟩ x ∎ where open ≡-Reasoning CB = cb00 (toℕ x) (fin<n _) iso1 : (b : B) → CountB.b (cb00 (toℕ (fromℕ< (cb<mb b))) (fin<n _)) ≡ b iso1 b = begin CountB.b CB ≡⟨ InjectiveF.inject fi (F←Q-inject fa (toℕ-injective (begin toℕ (F←Q fa (f (CountB.b CB))) ≡⟨ sym (CountB.b=cn CB) ⟩ CountB.cb CB ≡⟨ CountB.cb-inject CB _ (fin<n _) isb lem30 ⟩ toℕ (F←Q fa (f b)) ∎ ) )) ⟩ b ∎ where open ≡-Reasoning CB = cb00 (toℕ (fromℕ< (cb<mb b))) (fin<n _) isb : Is B A f (Q←F fa (fromℕ< (fin<n {_} (F←Q fa (f b)) ))) isb = record { a = b ; fa=c = lem33 } where lem33 : f b ≡ Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) lem33 = begin f b ≡⟨ sym (finiso→ fa _) ⟩ Q←F fa (F←Q fa (f b)) ≡⟨ cong (Q←F fa) (sym (fromℕ<-toℕ _ (fin<n (F←Q fa (f b))))) ⟩ Q←F fa (fromℕ< (fin<n (F←Q fa (f b)))) ∎ lem30 : count-B (CountB.cb CB) ≡ count-B (toℕ (F←Q fa (InjectiveF.f fi b))) lem30 = begin count-B (CountB.cb CB) ≡⟨ CountB.cb=n CB ⟩ suc (toℕ (fromℕ< (cb<mb b))) ≡⟨ cong suc (toℕ-fromℕ< (cb<mb b)) ⟩ suc (pred (count-B (toℕ (F←Q fa (f b))))) ≡⟨ sucprd (lem31 b) ⟩ count-B (toℕ (F←Q fa (f b))) ∎ -- end