Mercurial > hg > Members > kono > Proof > automaton
view automaton-in-agda/src/bijection.agda @ 265:c47634c830f3
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 08 Jul 2021 08:41:05 +0900 |
| parents | d1e8e5eadc38 |
| children | e5cf49902db3 |
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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 bi-trans : {n m l : Level} (R : Set n) (S : Set m) (T : Set l) → Bijection R S → Bijection S T → Bijection R T bi-trans R S T rs st = record { fun← = λ x → fun← rs ( fun← st x ) ; fun→ = λ x → fun→ st ( fun→ rs x ) ; fiso← = λ x → trans (cong (λ k → fun← rs k) (fiso← st (fun→ rs x))) ( fiso← rs x) ; fiso→ = λ x → trans (cong (λ k → fun→ st k) (fiso→ rs (fun← st x))) ( fiso→ st x) } bid : {n : Level} (R : Set n) → Bijection R R bid {n} R = record { fun← = λ x → x ; fun→ = λ x → x ; fiso← = λ _ → refl ; fiso→ = λ _ → refl } bi-sym : {n m : Level} (R : Set n) (S : Set m) → Bijection R S → Bijection S R bi-sym R S eq = record { fun← = fun→ eq ; fun→ = fun← eq ; fiso← = fiso→ eq ; fiso→ = fiso← eq } open import Relation.Binary.Structures bijIsEquivalence : {n : Level } → IsEquivalence (Bijection {n} {n}) bijIsEquivalence = record { refl = λ {R} → bid R ; sym = λ {R} {S} → bi-sym R S ; trans = λ {R} {S} {T} → bi-trans R S T } -- ¬ A = A → ⊥ -- -- famous diagnostic function -- 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 (λ n → diag b n) ) refl where diagn1 : (n : S ) → ¬ (fun→ b (λ n → diag b n) ≡ 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 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 _ -- -- ℕ <=> ℕ + 1 -- 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 _∧_ 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 -- not used 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} ) -- increment in ths same stage 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 -- increment ths stage 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 ----- -- -- create the invariant NN for all n -- 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 --- --- increment the stage --- 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 --- --- increment in a stage --- 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 -- [] 0 -- 0 → 1 -- 1 → 2 -- 01 → 3 -- 11 → 4 -- ... -- -- binary invariant -- record LB (n : ℕ) (lton : List Bool → ℕ ) : Set where field nlist : List Bool isBin : lton nlist ≡ n isUnique : (x : List Bool) → lton x ≡ n → nlist ≡ x -- we don't need this lb+1 : List Bool → List Bool lb+1 [] = false ∷ [] lb+1 (false ∷ t) = true ∷ t lb+1 (true ∷ t) = false ∷ lb+1 t lb-1 : List Bool → List Bool lb-1 [] = [] lb-1 (true ∷ t) = false ∷ t lb-1 (false ∷ t) with lb-1 t ... | [] = true ∷ [] ... | x ∷ t1 = true ∷ x ∷ t1 LBℕ : Bijection ℕ ( List Bool ) LBℕ = record { fun← = λ x → lton x ; fun→ = λ n → LB.nlist (lb n) ; fiso← = λ n → LB.isBin (lb n) ; fiso→ = λ x → LB.isUnique (lb (lton x)) x refl } where lton1 : List Bool → ℕ lton1 [] = 1 lton1 (true ∷ t) = suc (lton1 t + lton1 t) lton1 (false ∷ t) = lton1 t + lton1 t lton : List Bool → ℕ lton x = pred (lton1 x) lton1>0 : (x : List Bool ) → 0 < lton1 x lton1>0 [] = a<sa lton1>0 (true ∷ x₁) = 0<s lton1>0 (false ∷ t) = ≤-trans (lton1>0 t) x≤x+y 2lton1>0 : (t : List Bool ) → 0 < lton1 t + lton1 t 2lton1>0 t = ≤-trans (lton1>0 t) x≤x+y lb=3 : {x y : ℕ} → 0 < x → 0 < y → 1 ≤ pred (x + y) lb=3 {suc x} {suc y} (s≤s 0<x) (s≤s 0<y) = subst (λ k → 1 ≤ k ) (+-comm (suc y) _ ) (s≤s z≤n) lton-cons>0 : {x : Bool} {y : List Bool } → 0 < lton (x ∷ y) lton-cons>0 {true} {[]} = refl-≤s lton-cons>0 {true} {x ∷ y} = ≤-trans ( lb=3 (lton1>0 (x ∷ y)) (lton1>0 (x ∷ y))) px≤x lton-cons>0 {false} {[]} = refl-≤ lton-cons>0 {false} {x ∷ y} = lb=3 (lton1>0 (x ∷ y)) (lton1>0 (x ∷ y)) lb=0 : {x y : ℕ } → pred x < pred y → suc (x + x ∸ 1) < suc (y + y ∸ 1) lb=0 {0} {suc y} lt = s≤s (subst (λ k → 0 < k ) (+-comm (suc y) y ) 0<s) lb=0 {suc x} {suc y} lt = begin suc (suc ((suc x + suc x) ∸ 1)) ≡⟨ refl ⟩ suc (suc x) + suc x ≡⟨ refl ⟩ suc (suc x + suc x) ≤⟨ <-plus (s≤s lt) ⟩ suc y + suc x <⟨ <-plus-0 {suc x} {suc y} {suc y} (s≤s lt) ⟩ suc y + suc y ≡⟨ refl ⟩ suc ((suc y + suc y) ∸ 1) ∎ where open ≤-Reasoning lb=2 : {x y : ℕ } → pred x < pred y → suc (x + x ) < suc (y + y ) lb=2 {zero} {suc y} lt = s≤s 0<s lb=2 {suc x} {suc y} lt = s≤s (lb=0 {suc x} {suc y} lt) lb=1 : {x y : List Bool} {z : Bool} → lton (z ∷ x) ≡ lton (z ∷ y) → lton x ≡ lton y lb=1 {x} {y} {true} eq with <-cmp (lton x) (lton y) ... | tri< a ¬b ¬c = ⊥-elim (nat-≡< (cong suc eq) (lb=2 {lton1 x} {lton1 y} a)) ... | tri≈ ¬a b ¬c = b ... | tri> ¬a ¬b c = ⊥-elim (nat-≡< (cong suc (sym eq)) (lb=2 {lton1 y} {lton1 x} c)) lb=1 {x} {y} {false} eq with <-cmp (lton x) (lton y) ... | tri< a ¬b ¬c = ⊥-elim (nat-≡< (cong suc eq) (lb=0 {lton1 x} {lton1 y} a)) ... | tri≈ ¬a b ¬c = b ... | tri> ¬a ¬b c = ⊥-elim (nat-≡< (cong suc (sym eq)) (lb=0 {lton1 y} {lton1 x} c)) --- --- lton is unique in a head --- lb-tf : {x y : List Bool } → ¬ (lton (true ∷ x) ≡ lton (false ∷ y)) lb-tf {x} {y} eq with <-cmp (lton1 x) (lton1 y) ... | tri< a ¬b ¬c = ⊥-elim ( nat-≡< eq (lb=01 a)) where lb=01 : {x y : ℕ } → x < y → x + x < (y + y ∸ 1) lb=01 {x} {y} x<y = begin suc (x + x) ≡⟨ refl ⟩ suc x + x ≤⟨ ≤-plus x<y ⟩ y + x ≡⟨ refl ⟩ pred (suc y + x) ≡⟨ cong (λ k → pred ( k + x)) (+-comm 1 y) ⟩ pred ((y + 1) + x ) ≡⟨ cong pred (+-assoc y 1 x) ⟩ pred (y + suc x) ≤⟨ px≤py (≤-plus-0 {suc x} {y} {y} x<y) ⟩ (y + y) ∸ 1 ∎ where open ≤-Reasoning ... | tri> ¬a ¬b c = ⊥-elim ( nat-≡< (sym eq) (lb=02 c) ) where lb=02 : {x y : ℕ } → x < y → x + x ∸ 1 < y + y lb=02 {0} {y} lt = ≤-trans lt x≤x+y lb=02 {suc x} {y} lt = begin suc ( suc x + suc x ∸ 1 ) ≡⟨ refl ⟩ suc x + suc x ≤⟨ ≤-plus {suc x} (<to≤ lt) ⟩ y + suc x ≤⟨ ≤-plus-0 (<to≤ lt) ⟩ y + y ∎ where open ≤-Reasoning ... | tri≈ ¬a b ¬c = ⊥-elim ( nat-≡< (sym eq) ( begin suc (lton1 y + lton1 y ∸ 1) ≡⟨ sucprd ( 2lton1>0 y) ⟩ lton1 y + lton1 y ≡⟨ cong (λ k → k + k ) (sym b) ⟩ lton1 x + lton1 x ∎ )) where open ≤-Reasoning --- --- lton uniqueness --- lb=b : (x y : List Bool) → lton x ≡ lton y → x ≡ y lb=b [] [] eq = refl lb=b [] (x ∷ y) eq = ⊥-elim ( nat-≡< eq (lton-cons>0 {x} {y} )) lb=b (x ∷ y) [] eq = ⊥-elim ( nat-≡< (sym eq) (lton-cons>0 {x} {y} )) lb=b (true ∷ x) (false ∷ y) eq = ⊥-elim ( lb-tf {x} {y} eq ) lb=b (false ∷ x) (true ∷ y) eq = ⊥-elim ( lb-tf {y} {x} (sym eq) ) lb=b (true ∷ x) (true ∷ y) eq = cong (λ k → true ∷ k ) (lb=b x y (lb=1 {x} {y} {true} eq)) lb=b (false ∷ x) (false ∷ y) eq = cong (λ k → false ∷ k ) (lb=b x y (lb=1 {x} {y} {false} eq)) lb : (n : ℕ) → LB n lton lb zero = record { nlist = [] ; isBin = refl ; isUnique = lb05 } where lb05 : (x : List Bool) → lton x ≡ zero → [] ≡ x lb05 x eq = lb=b [] x (sym eq) lb (suc n) with LB.nlist (lb n) | inspect LB.nlist (lb n) ... | [] | record { eq = eq } = record { nlist = false ∷ [] ; isUnique = lb06 ; isBin = lb10 } where open ≡-Reasoning lb10 : lton1 (false ∷ []) ∸ 1 ≡ suc n lb10 = begin lton (false ∷ []) ≡⟨ refl ⟩ suc 0 ≡⟨ refl ⟩ suc (lton []) ≡⟨ cong (λ k → suc (lton k)) (sym eq) ⟩ suc (lton (LB.nlist (lb n))) ≡⟨ cong suc (LB.isBin (lb n) ) ⟩ suc n ∎ lb06 : (x : List Bool) → pred (lton1 x ) ≡ suc n → false ∷ [] ≡ x lb06 x eq1 = lb=b (false ∷ []) x (trans lb10 (sym eq1)) -- lton (false ∷ []) ≡ lton x ... | false ∷ t | record { eq = eq } = record { nlist = true ∷ t ; isBin = lb01 ; isUnique = lb09 } where lb01 : lton (true ∷ t) ≡ suc n lb01 = begin lton (true ∷ t) ≡⟨ refl ⟩ lton1 t + lton1 t ≡⟨ sym ( sucprd (2lton1>0 t) ) ⟩ suc (pred (lton1 t + lton1 t )) ≡⟨ refl ⟩ suc (lton (false ∷ t)) ≡⟨ cong (λ k → suc (lton k )) (sym eq) ⟩ suc (lton (LB.nlist (lb n))) ≡⟨ cong suc (LB.isBin (lb n)) ⟩ suc n ∎ where open ≡-Reasoning lb09 : (x : List Bool) → lton1 x ∸ 1 ≡ suc n → true ∷ t ≡ x lb09 x eq1 = lb=b (true ∷ t) x (trans lb01 (sym eq1) ) -- lton (true ∷ t) ≡ lton x ... | true ∷ t | record { eq = eq } = record { nlist = lb+1 (true ∷ t) ; isBin = lb02 (true ∷ t) lb03 ; isUnique = lb07 } where lb03 : lton (true ∷ t) ≡ n lb03 = begin lton (true ∷ t) ≡⟨ cong (λ k → lton k ) (sym eq ) ⟩ lton (LB.nlist (lb n)) ≡⟨ LB.isBin (lb n) ⟩ n ∎ where open ≡-Reasoning 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))) lb04 : (t : List Bool) → suc (lton1 t) ≡ lton1 (lb+1 t) lb04 [] = refl lb04 (false ∷ t) = refl lb04 (true ∷ []) = refl lb04 (true ∷ t0 ) = begin suc (suc (lton1 t0 + lton1 t0)) ≡⟨ sym (add11 (lton1 t0)) ⟩ suc (lton1 t0) + suc (lton1 t0) ≡⟨ cong (λ k → k + k ) (lb04 t0 ) ⟩ lton1 (lb+1 t0) + lton1 (lb+1 t0) ∎ where open ≡-Reasoning lb02 : (t : List Bool) → lton t ≡ n → lton (lb+1 t) ≡ suc n lb02 [] refl = refl lb02 t eq1 = begin lton (lb+1 t) ≡⟨ refl ⟩ pred (lton1 (lb+1 t)) ≡⟨ cong pred (sym (lb04 t)) ⟩ pred (suc (lton1 t)) ≡⟨ sym (sucprd (lton1>0 t)) ⟩ suc (pred (lton1 t)) ≡⟨ refl ⟩ suc (lton t) ≡⟨ cong suc eq1 ⟩ suc n ∎ where open ≡-Reasoning lb07 : (x : List Bool) → pred (lton1 x ) ≡ suc n → lb+1 (true ∷ t) ≡ x lb07 x eq1 = lb=b (lb+1 (true ∷ t)) x (trans ( lb02 (true ∷ t) lb03 ) (sym eq1))