Mercurial > hg > Members > kono > Proof > automaton
view automaton-in-agda/src/bijection.agda @ 256:5aff0067b194
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 05 Jul 2021 11:10:15 +0900 |
| parents | 3fa72793620b |
| children | 246da8456ad1 |
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module bijection where open import Level renaming ( zero to Zero ; suc to Suc ) open import Data.Nat open import Data.Maybe open import Data.List hiding ([_] ; sum ) open import Data.Nat.Properties open import Relation.Nullary open import Data.Empty open import Data.Unit hiding ( _≤_ ) open import Relation.Binary.Core hiding (_⇔_) open import Relation.Binary.Definitions open import Relation.Binary.PropositionalEquality open import logic open import nat record Bijection {n m : Level} (R : Set n) (S : Set m) : Set (n Level.⊔ m) where field fun← : S → R fun→ : R → S fiso← : (x : R) → fun← ( fun→ x ) ≡ x fiso→ : (x : S ) → fun→ ( fun← x ) ≡ x injection : {n m : Level} (R : Set n) (S : Set m) (f : R → S ) → Set (n Level.⊔ m) injection R S f = (x y : R) → f x ≡ f y → x ≡ y open Bijection b→injection0 : {n m : Level} (R : Set n) (S : Set m) → (b : Bijection R S) → injection R S (fun→ b) b→injection0 R S b x y eq = begin x ≡⟨ sym ( fiso← b x ) ⟩ fun← b ( fun→ b x ) ≡⟨ cong (λ k → fun← b k ) eq ⟩ fun← b ( fun→ b y ) ≡⟨ fiso← b y ⟩ y ∎ where open ≡-Reasoning b→injection1 : {n m : Level} (R : Set n) (S : Set m) → (b : Bijection R S) → injection S R (fun← b) b→injection1 R S b x y eq = trans ( sym ( fiso→ b x ) ) (trans ( cong (λ k → fun→ b k ) eq ) ( fiso→ b y )) -- ¬ A = A → ⊥ diag : {S : Set } (b : Bijection ( S → Bool ) S) → S → Bool diag b n = not (fun← b n n) diagonal : { S : Set } → ¬ Bijection ( S → Bool ) S diagonal {S} b = diagn1 (fun→ b (diag b) ) refl where diagn1 : (n : S ) → ¬ (fun→ b (diag b) ≡ n ) diagn1 n dn = ¬t=f (diag b n ) ( begin not (diag b n) ≡⟨⟩ not (not fun← b n n) ≡⟨ cong (λ k → not (k n) ) (sym (fiso← b _)) ⟩ not (fun← b (fun→ b (diag b)) n) ≡⟨ cong (λ k → not (fun← b k n) ) dn ⟩ not (fun← b n n) ≡⟨⟩ diag b n ∎ ) where open ≡-Reasoning open _∧_ record NN ( i : ℕ) (nxn→n : ℕ → ℕ → ℕ) (n→nxn : ℕ → ℕ ∧ ℕ) : Set where field j k sum stage : ℕ nn : j + k ≡ sum ni : i ≡ j + stage k1 : nxn→n j k ≡ i k0 : n→nxn i ≡ ⟪ j , k ⟫ nn-unique : {j0 k0 : ℕ } → nxn→n j0 k0 ≡ i → ⟪ j , k ⟫ ≡ ⟪ j0 , k0 ⟫ i≤0→i≡0 : {i : ℕ } → i ≤ 0 → i ≡ 0 i≤0→i≡0 {0} z≤n = refl nxn : Bijection ℕ (ℕ ∧ ℕ) nxn = record { fun← = λ p → nxn→n (proj1 p) (proj2 p) ; fun→ = n→nxn ; fiso← = n-id ; fiso→ = λ x → nn-id (proj1 x) (proj2 x) } where nxn→n : ℕ → ℕ → ℕ nxn→n zero zero = zero nxn→n zero (suc j) = j + suc (nxn→n zero j) nxn→n (suc i) zero = suc i + suc (nxn→n i zero) nxn→n (suc i) (suc j) = suc i + suc j + suc (nxn→n i (suc j)) n→nxn : ℕ → ℕ ∧ ℕ n→nxn zero = ⟪ 0 , 0 ⟫ n→nxn (suc i) with n→nxn i ... | ⟪ x , zero ⟫ = ⟪ zero , suc x ⟫ ... | ⟪ x , suc y ⟫ = ⟪ suc x , y ⟫ nxn→n0 : { j k : ℕ } → nxn→n j k ≡ 0 → ( j ≡ 0 ) ∧ ( k ≡ 0 ) nxn→n0 {zero} {zero} eq = ⟪ refl , refl ⟫ nxn→n0 {zero} {(suc k)} eq = ⊥-elim ( nat-≡< (sym eq) (subst (λ k → 0 < k) (+-comm _ k) (s≤s z≤n))) nxn→n0 {(suc j)} {zero} eq = ⊥-elim ( nat-≡< (sym eq) (s≤s z≤n) ) nxn→n0 {(suc j)} {(suc k)} eq = ⊥-elim ( nat-≡< (sym eq) (s≤s z≤n) ) nid20 : (i : ℕ) → i + (nxn→n 0 i) ≡ nxn→n i 0 nid20 zero = refl -- suc (i + (i + suc (nxn→n 0 i))) ≡ suc (i + suc (nxn→n i 0)) nid20 (suc i) = begin suc (i + (i + suc (nxn→n 0 i))) ≡⟨ cong (λ k → suc (i + k)) (sym (+-assoc i 1 (nxn→n 0 i))) ⟩ suc (i + ((i + 1) + nxn→n 0 i)) ≡⟨ cong (λ k → suc (i + (k + nxn→n 0 i))) (+-comm i 1) ⟩ suc (i + suc (i + nxn→n 0 i)) ≡⟨ cong ( λ k → suc (i + suc k)) (nid20 i) ⟩ suc (i + suc (nxn→n i 0)) ∎ where open ≡-Reasoning nid4 : {i j : ℕ} → i + 1 + j ≡ i + suc j nid4 {zero} {j} = refl nid4 {suc i} {j} = cong suc (nid4 {i} {j} ) nid5 : {i j k : ℕ} → i + suc (suc j) + suc k ≡ i + suc j + suc (suc k ) nid5 {zero} {j} {k} = begin suc (suc j) + suc k ≡⟨ +-assoc 1 (suc j) _ ⟩ 1 + (suc j + suc k) ≡⟨ +-comm 1 _ ⟩ (suc j + suc k) + 1 ≡⟨ +-assoc (suc j) (suc k) _ ⟩ suc j + (suc k + 1) ≡⟨ cong (λ k → suc j + k ) (+-comm (suc k) 1) ⟩ suc j + suc (suc k) ∎ where open ≡-Reasoning nid5 {suc i} {j} {k} = cong suc (nid5 {i} {j} {k} ) nid2 : (i j : ℕ) → suc (nxn→n i (suc j)) ≡ nxn→n (suc i) j nid2 zero zero = refl nid2 zero (suc j) = refl nid2 (suc i) zero = begin suc (nxn→n (suc i) 1) ≡⟨ refl ⟩ suc (suc (i + 1 + suc (nxn→n i 1))) ≡⟨ cong (λ k → suc (suc k)) nid4 ⟩ suc (suc (i + suc (suc (nxn→n i 1)))) ≡⟨ cong (λ k → suc (suc (i + suc (suc k)))) (nid3 i) ⟩ suc (suc (i + suc (suc (i + suc (nxn→n i 0))))) ≡⟨ refl ⟩ nxn→n (suc (suc i)) zero ∎ where open ≡-Reasoning nid3 : (i : ℕ) → nxn→n i 1 ≡ i + suc (nxn→n i 0) nid3 zero = refl nid3 (suc i) = begin suc (i + 1 + suc (nxn→n i 1)) ≡⟨ cong suc nid4 ⟩ suc (i + suc (suc (nxn→n i 1))) ≡⟨ cong (λ k → suc (i + suc (suc k))) (nid3 i) ⟩ suc (i + suc (suc (i + suc (nxn→n i 0)))) ∎ nid2 (suc i) (suc j) = begin suc (nxn→n (suc i) (suc (suc j))) ≡⟨ refl ⟩ suc (suc (i + suc (suc j) + suc (nxn→n i (suc (suc j))))) ≡⟨ cong (λ k → suc (suc (i + suc (suc j) + k))) (nid2 i (suc j)) ⟩ suc (suc (i + suc (suc j) + suc (i + suc j + suc (nxn→n i (suc j))))) ≡⟨ cong ( λ k → suc (suc k)) nid5 ⟩ suc (suc (i + suc j + suc (suc (i + suc j + suc (nxn→n i (suc j)))))) ≡⟨ refl ⟩ nxn→n (suc (suc i)) (suc j) ∎ where open ≡-Reasoning nid00 : (i : ℕ) → suc (nxn→n i 0) ≡ nxn→n 0 (suc i) nid00 zero = refl nid00 (suc i) = begin suc (suc (i + suc (nxn→n i 0))) ≡⟨ cong (λ k → suc (suc (i + k ))) (nid00 i) ⟩ suc (suc (i + (nxn→n 0 (suc i)))) ≡⟨ refl ⟩ suc (suc (i + (i + suc (nxn→n 0 i)))) ≡⟨ cong suc (sym ( +-assoc 1 i (i + suc (nxn→n 0 i)))) ⟩ suc ((1 + i) + (i + suc (nxn→n 0 i))) ≡⟨ cong (λ k → suc (k + (i + suc (nxn→n 0 i)))) (+-comm 1 i) ⟩ suc ((i + 1) + (i + suc (nxn→n 0 i))) ≡⟨ cong suc (+-assoc i 1 (i + suc (nxn→n 0 i))) ⟩ suc (i + suc (i + suc (nxn→n 0 i))) ∎ where open ≡-Reasoning nn : ( i : ℕ) → NN i nxn→n n→nxn nn zero = record { j = 0 ; k = 0 ; sum = 0 ; stage = 0 ; nn = refl ; ni = refl ; k1 = refl ; k0 = refl ; nn-unique = λ {j0} {k0} eq → cong₂ (λ x y → ⟪ x , y ⟫) (sym (proj1 (nxn→n0 eq))) (sym (proj2 (nxn→n0 {j0} {k0} eq))) } nn (suc i) with NN.k (nn i) | inspect NN.k (nn i) ... | zero | record { eq = eq } = record { k = suc (NN.sum (nn i)) ; j = 0 ; sum = suc (NN.sum (nn i)) ; stage = suc (NN.sum (nn i)) + (NN.stage (nn i)) ; nn = refl ; ni = nn01 ; k1 = nn02 ; k0 = nn03 ; nn-unique = nn04 } where sum = NN.sum (nn i) stage = NN.stage (nn i) j = NN.j (nn i) nn01 : suc i ≡ 0 + (suc sum + stage ) nn01 = begin suc i ≡⟨ cong suc (NN.ni (nn i)) ⟩ suc ((NN.j (nn i) ) + stage ) ≡⟨ cong (λ k → suc (k + stage )) (+-comm 0 _ ) ⟩ suc ((NN.j (nn i) + 0 ) + stage ) ≡⟨ cong (λ k → suc ((NN.j (nn i) + k) + stage )) (sym eq) ⟩ suc ((NN.j (nn i) + NN.k (nn i)) + stage ) ≡⟨ cong (λ k → suc ( k + stage )) (NN.nn (nn i)) ⟩ 0 + (suc sum + stage ) ∎ where open ≡-Reasoning nn02 : nxn→n 0 (suc sum) ≡ suc i nn02 = begin sum + suc (nxn→n 0 sum) ≡⟨ sym (+-assoc sum 1 (nxn→n 0 sum)) ⟩ (sum + 1) + nxn→n 0 sum ≡⟨ cong (λ k → k + nxn→n 0 sum ) (+-comm sum 1 )⟩ suc (sum + nxn→n 0 sum) ≡⟨ cong suc (nid20 sum ) ⟩ suc (nxn→n sum 0) ≡⟨ cong (λ k → suc (nxn→n k 0 )) (sym (NN.nn (nn i))) ⟩ suc (nxn→n (NN.j (nn i) + (NN.k (nn i)) ) 0) ≡⟨ cong₂ (λ j k → suc (nxn→n (NN.j (nn i) + j) k )) eq (sym eq) ⟩ suc (nxn→n (NN.j (nn i) + 0 ) (NN.k (nn i))) ≡⟨ cong (λ k → suc ( nxn→n k (NN.k (nn i)))) (+-comm (NN.j (nn i)) 0) ⟩ suc (nxn→n (NN.j (nn i)) (NN.k (nn i))) ≡⟨ cong suc (NN.k1 (nn i) ) ⟩ suc i ∎ where open ≡-Reasoning nn03 : n→nxn (suc i) ≡ ⟪ 0 , suc (NN.sum (nn i)) ⟫ -- k0 : n→nxn i ≡ ⟪ NN.j (nn i) = sum , NN.k (nn i) = 0 ⟫ nn03 with n→nxn i | inspect n→nxn i ... | ⟪ x , zero ⟫ | record { eq = eq1 } = begin ⟪ zero , suc x ⟫ ≡⟨ cong (λ k → ⟪ zero , suc k ⟫) (sym (cong proj1 eq1)) ⟩ ⟪ zero , suc (proj1 (n→nxn i)) ⟫ ≡⟨ cong (λ k → ⟪ zero , suc k ⟫) (cong proj1 (NN.k0 (nn i))) ⟩ ⟪ zero , suc (NN.j (nn i)) ⟫ ≡⟨ cong (λ k → ⟪ zero , suc k ⟫) (+-comm 0 _ ) ⟩ ⟪ zero , suc (NN.j (nn i) + 0) ⟫ ≡⟨ cong (λ k → ⟪ zero , suc (NN.j (nn i) + k) ⟫ ) (sym eq) ⟩ ⟪ zero , suc (NN.j (nn i) + NN.k (nn i)) ⟫ ≡⟨ cong (λ k → ⟪ zero , suc k ⟫ ) (NN.nn (nn i)) ⟩ ⟪ 0 , suc sum ⟫ ∎ where open ≡-Reasoning ... | ⟪ x , suc y ⟫ | record { eq = eq1 } = ⊥-elim ( nat-≡< (sym (cong proj2 (NN.k0 (nn i)))) (begin suc (NN.k (nn i)) ≡⟨ cong suc eq ⟩ suc 0 ≤⟨ s≤s z≤n ⟩ suc y ≡⟨ sym (cong proj2 eq1) ⟩ proj2 (n→nxn i) ∎ )) where open ≤-Reasoning -- nid2 : (i j : ℕ) → suc (nxn→n i (suc j)) ≡ nxn→n (suc i) j nn04 : {j0 k0 : ℕ} → nxn→n j0 k0 ≡ suc i → ⟪ 0 , suc (NN.sum (nn i)) ⟫ ≡ ⟪ j0 , k0 ⟫ nn04 {zero} {suc k0} eq1 = cong (λ k → ⟪ 0 , k ⟫ ) (cong suc (sym nn08)) where -- eq : nxn→n zero (suc k0) ≡ suc i -- nn07 : nxn→n k0 0 ≡ i nn07 = cong pred ( begin suc ( nxn→n k0 0 ) ≡⟨ nid00 k0 ⟩ nxn→n 0 (suc k0 ) ≡⟨ eq1 ⟩ suc i ∎ ) where open ≡-Reasoning nn08 : k0 ≡ sum nn08 = begin k0 ≡⟨ cong proj1 (sym (NN.nn-unique (nn i) nn07)) ⟩ NN.j (nn i) ≡⟨ +-comm 0 _ ⟩ NN.j (nn i) + 0 ≡⟨ cong (λ k → NN.j (nn i) + k) (sym eq) ⟩ NN.j (nn i) + NN.k (nn i) ≡⟨ NN.nn (nn i) ⟩ sum ∎ where open ≡-Reasoning nn04 {suc j0} {k0} eq1 = ⊥-elim ( nat-≡< (cong proj2 (nn06 nn05)) (subst (λ k → k < suc k0) (sym eq) (s≤s z≤n))) where nn05 : nxn→n j0 (suc k0) ≡ i nn05 = begin nxn→n j0 (suc k0) ≡⟨ cong pred ( begin suc (nxn→n j0 (suc k0)) ≡⟨ nid2 j0 k0 ⟩ nxn→n (suc j0) k0 ≡⟨ eq1 ⟩ suc i ∎ ) ⟩ i ∎ where open ≡-Reasoning nn06 : nxn→n j0 (suc k0) ≡ i → ⟪ NN.j (nn i) , NN.k (nn i) ⟫ ≡ ⟪ j0 , suc k0 ⟫ nn06 = NN.nn-unique (nn i) ... | suc k | record {eq = eq} = record { k = k ; j = suc (NN.j (nn i)) ; sum = NN.sum (nn i) ; stage = NN.stage (nn i) ; nn = nn10 ; ni = cong suc (NN.ni (nn i)) ; k1 = nn11 ; k0 = nn12 ; nn-unique = nn13 } where nn10 : suc (NN.j (nn i)) + k ≡ NN.sum (nn i) nn10 = begin suc (NN.j (nn i)) + k ≡⟨ cong (λ x → x + k) (+-comm 1 _) ⟩ (NN.j (nn i) + 1) + k ≡⟨ +-assoc (NN.j (nn i)) 1 k ⟩ NN.j (nn i) + suc k ≡⟨ cong (λ k → NN.j (nn i) + k) (sym eq) ⟩ NN.j (nn i) + NN.k (nn i) ≡⟨ NN.nn (nn i) ⟩ NN.sum (nn i) ∎ where open ≡-Reasoning nn11 : nxn→n (suc (NN.j (nn i))) k ≡ suc i -- nxn→n ( NN.j (nn i)) (NN.k (nn i) ≡ i nn11 = begin nxn→n (suc (NN.j (nn i))) k ≡⟨ sym (nid2 (NN.j (nn i)) k) ⟩ suc (nxn→n (NN.j (nn i)) (suc k)) ≡⟨ cong (λ k → suc (nxn→n (NN.j (nn i)) k)) (sym eq) ⟩ suc (nxn→n ( NN.j (nn i)) (NN.k (nn i))) ≡⟨ cong suc (NN.k1 (nn i)) ⟩ suc i ∎ where open ≡-Reasoning nn18 : zero < NN.k (nn i) nn18 = subst (λ k → 0 < k ) ( begin suc k ≡⟨ sym eq ⟩ NN.k (nn i) ∎ ) (s≤s z≤n ) where open ≡-Reasoning nn12 : n→nxn (suc i) ≡ ⟪ suc (NN.j (nn i)) , k ⟫ -- n→nxn i ≡ ⟪ NN.j (nn i) , NN.k (nn i) ⟫ nn12 with n→nxn i | inspect n→nxn i ... | ⟪ x , zero ⟫ | record { eq = eq1 } = ⊥-elim ( nat-≡< (sym (cong proj2 eq1)) (subst (λ k → 0 < k ) ( begin suc k ≡⟨ sym eq ⟩ NN.k (nn i) ≡⟨ cong proj2 (sym (NN.k0 (nn i)) ) ⟩ proj2 (n→nxn i) ∎ ) (s≤s z≤n )) ) where open ≡-Reasoning -- eq1 n→nxn i ≡ ⟪ x , zero ⟫ ... | ⟪ x , suc y ⟫ | record { eq = eq1 } = begin -- n→nxn i ≡ ⟪ x , suc y ⟫ ⟪ suc x , y ⟫ ≡⟨ refl ⟩ ⟪ suc x , pred (suc y) ⟫ ≡⟨ cong (λ k → ⟪ suc (proj1 k) , pred (proj2 k) ⟫) ( begin ⟪ x , suc y ⟫ ≡⟨ sym eq1 ⟩ n→nxn i ≡⟨ NN.k0 (nn i) ⟩ ⟪ NN.j (nn i) , NN.k (nn i) ⟫ ∎ ) ⟩ ⟪ suc (NN.j (nn i)) , pred (NN.k (nn i)) ⟫ ≡⟨ cong (λ k → ⟪ suc (NN.j (nn i)) , pred k ⟫) eq ⟩ ⟪ suc (NN.j (nn i)) , k ⟫ ∎ where open ≡-Reasoning nn13 : {j0 k0 : ℕ} → nxn→n j0 k0 ≡ suc i → ⟪ suc (NN.j (nn i)) , k ⟫ ≡ ⟪ j0 , k0 ⟫ nn13 {zero} {suc k0} eq1 = ⊥-elim ( nat-≡< (sym (cong proj2 nn17)) nn18 ) where -- (nxn→n zero (suc k0)) ≡ suc i nn16 : nxn→n k0 zero ≡ i nn16 = cong pred ( subst (λ k → k ≡ suc i) (sym ( nid00 k0 )) eq1 ) nn17 : ⟪ NN.j (nn i) , NN.k (nn i) ⟫ ≡ ⟪ k0 , zero ⟫ nn17 = NN.nn-unique (nn i) nn16 nn13 {suc j0} {k0} eq1 = begin ⟪ suc (NN.j (nn i)) , pred (suc k) ⟫ ≡⟨ cong (λ k → ⟪ suc (NN.j (nn i)) , pred k ⟫ ) (sym eq) ⟩ ⟪ suc (NN.j (nn i)) , pred (NN.k (nn i)) ⟫ ≡⟨ cong (λ k → ⟪ suc (proj1 k) , pred (proj2 k) ⟫) ( begin ⟪ NN.j (nn i) , NN.k (nn i) ⟫ ≡⟨ nn15 ⟩ ⟪ j0 , suc k0 ⟫ ∎ ) ⟩ ⟪ suc j0 , k0 ⟫ ∎ where -- nxn→n (suc j0) k0 ≡ suc i open ≡-Reasoning nn14 : nxn→n j0 (suc k0) ≡ i nn14 = cong pred ( subst (λ k → k ≡ suc i) (sym ( nid2 j0 k0 )) eq1 ) nn15 : ⟪ NN.j (nn i) , NN.k (nn i) ⟫ ≡ ⟪ j0 , suc k0 ⟫ nn15 = NN.nn-unique (nn i) nn14 n-id : (i : ℕ) → nxn→n (proj1 (n→nxn i)) (proj2 (n→nxn i)) ≡ i n-id i = subst (λ k → nxn→n (proj1 k) (proj2 k) ≡ i ) (sym (NN.k0 (nn i))) (NN.k1 (nn i)) nn-id : (j k : ℕ) → n→nxn (nxn→n j k) ≡ ⟪ j , k ⟫ nn-id j k = begin n→nxn (nxn→n j k) ≡⟨ NN.k0 (nn (nxn→n j k)) ⟩ ⟪ NN.j (nn (nxn→n j k)) , NN.k (nn (nxn→n j k)) ⟫ ≡⟨ NN.nn-unique (nn (nxn→n j k)) refl ⟩ ⟪ j , k ⟫ ∎ where open ≡-Reasoning b1 : (b : Bijection ( ℕ → Bool ) ℕ) → ℕ b1 b = fun→ b (diag b) b-iso : (b : Bijection ( ℕ → Bool ) ℕ) → fun← b (b1 b) ≡ (diag b) b-iso b = fiso← b _ to1 : {n : Level} {R : Set n} → Bijection ℕ R → Bijection ℕ (⊤ ∨ R ) to1 {n} {R} b = record { fun← = to11 ; fun→ = to12 ; fiso← = to13 ; fiso→ = to14 } where to11 : ⊤ ∨ R → ℕ to11 (case1 tt) = 0 to11 (case2 x) = suc ( fun← b x ) to12 : ℕ → ⊤ ∨ R to12 zero = case1 tt to12 (suc n) = case2 ( fun→ b n) to13 : (x : ℕ) → to11 (to12 x) ≡ x to13 zero = refl to13 (suc x) = cong suc (fiso← b x) to14 : (x : ⊤ ∨ R) → to12 (to11 x) ≡ x to14 (case1 x) = refl to14 (case2 x) = cong case2 (fiso→ b x) open _∧_ open import nat open ≡-Reasoning -- [] 0 -- 0 → 1 -- 1 → 2 -- 01 → 3 -- 11 → 4 -- ... -- {-# TERMINATING #-} LBℕ : Bijection ℕ ( List Bool ) LBℕ = record { fun← = λ x → lton x ; fun→ = λ n → ntol n ; fiso← = lbiso0 ; fiso→ = lbisor } where lton1 : List Bool → ℕ lton1 [] = 0 lton1 (true ∷ t) = suc (lton1 t + lton1 t) lton1 (false ∷ t) = lton1 t + lton1 t lton : List Bool → ℕ lton [] = 0 lton x = suc (lton1 x) ntol1 : ℕ → List Bool ntol1 0 = [] ntol1 (suc x) with div2 (suc x) ... | ⟪ x1 , true ⟫ = true ∷ ntol1 x1 -- non terminating ... | ⟪ x1 , false ⟫ = false ∷ ntol1 x1 ntol : ℕ → List Bool ntol 0 = [] ntol 1 = false ∷ [] ntol (suc n) = ntol1 n xx : (x : ℕ ) → List Bool ∧ ℕ xx x = ⟪ (ntol x) , lton ((ntol x)) ⟫ add11 : (x1 : ℕ ) → suc x1 + suc x1 ≡ suc (suc (x1 + x1)) add11 zero = refl add11 (suc x) = cong (λ k → suc (suc k)) (trans (+-comm x _) (cong suc (+-comm _ x))) add12 : (x1 x : ℕ ) → suc x1 + x ≡ x1 + suc x add12 zero x = refl add12 (suc x1) x = cong suc (add12 x1 x) ---- div2-eq : (x : ℕ ) → div2-rev ( div2 x ) ≡ x div20 : (x x1 : ℕ ) → div2 (suc x) ≡ ⟪ x1 , false ⟫ → x1 + x1 ≡ suc x div20 x x1 eq = begin x1 + x1 ≡⟨ cong (λ k → div2-rev k ) (sym eq) ⟩ div2-rev (div2 (suc x)) ≡⟨ div2-eq _ ⟩ suc x ∎ div21 : (x x1 : ℕ ) → div2 (suc x) ≡ ⟪ x1 , true ⟫ → suc (x1 + x1) ≡ suc x div21 x x1 eq = begin suc (x1 + x1) ≡⟨ cong (λ k → div2-rev k ) (sym eq) ⟩ div2-rev (div2 (suc x)) ≡⟨ div2-eq _ ⟩ suc x ∎ lbiso1 : (x : ℕ) → suc (lton1 (ntol1 x)) ≡ suc x lbiso1 zero = refl lbiso1 (suc x) with div2 (suc x) | inspect div2 (suc x) ... | ⟪ x1 , true ⟫ | record { eq = eq1 } = begin suc (suc (lton1 (ntol1 x1) + lton1 (ntol1 x1))) ≡⟨ sym (add11 _) ⟩ suc (lton1 (ntol1 x1)) + suc (lton1 (ntol1 x1)) ≡⟨ cong ( λ k → k + k ) (lbiso1 x1) ⟩ suc x1 + suc x1 ≡⟨ add11 x1 ⟩ suc (suc (x1 + x1)) ≡⟨ cong suc (div21 x x1 eq1) ⟩ suc (suc x) ∎ ... | ⟪ x1 , false ⟫ | record { eq = eq1 } = begin suc (lton1 (ntol1 x1) + lton1 (ntol1 x1)) ≡⟨ cong ( λ k → k + lton1 (ntol1 x1) ) (lbiso1 x1) ⟩ suc x1 + lton1 (ntol1 x1) ≡⟨ add12 _ _ ⟩ x1 + suc (lton1 (ntol1 x1)) ≡⟨ cong ( λ k → x1 + k ) (lbiso1 x1) ⟩ x1 + suc x1 ≡⟨ +-comm x1 _ ⟩ suc (x1 + x1) ≡⟨ cong suc (div20 x x1 eq1) ⟩ suc (suc x) ∎ lbiso0 : (x : ℕ) → lton (ntol x) ≡ x lbiso0 zero = refl lbiso0 (suc zero) = refl lbiso0 (suc (suc x)) = subst (λ k → k ≡ suc (suc x)) (hh x) ( lbiso1 (suc x)) where hh : (x : ℕ ) → suc (lton1 (ntol1 (suc x))) ≡ lton (ntol (suc (suc x))) hh x with div2 (suc x) ... | ⟪ _ , true ⟫ = refl ... | ⟪ _ , false ⟫ = refl lbisor0 : (x : List Bool) → ntol1 (lton1 (true ∷ x)) ≡ true ∷ x lbisor0 = {!!} lbisor1 : (x : List Bool) → ntol1 (lton1 (false ∷ x)) ≡ false ∷ x lbisor1 = {!!} lbisor : (x : List Bool) → ntol (lton x) ≡ x lbisor [] = refl lbisor (false ∷ []) = refl lbisor (true ∷ []) = refl lbisor (false ∷ t) = trans {!!} ( lbisor1 t ) lbisor (true ∷ t) = trans {!!} ( lbisor0 t )