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Problem 1 of Project Euler (#85)
This commit is contained in:
Patrick Stevens
GitHub
@@ -12,343 +12,343 @@ open import Orders
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module Numbers.BinaryNaturals.Definition where
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data Bit : Set where
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zero : Bit
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one : Bit
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data Bit : Set where
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zero : Bit
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one : Bit
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BinNat : Set
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BinNat = List Bit
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BinNat : Set
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BinNat = List Bit
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::Inj : {xs ys : BinNat} {i : Bit} → i :: xs ≡ i :: ys → xs ≡ ys
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::Inj {i = zero} refl = refl
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::Inj {i = one} refl = refl
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::Inj : {xs ys : BinNat} {i : Bit} → i :: xs ≡ i :: ys → xs ≡ ys
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::Inj {i = zero} refl = refl
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::Inj {i = one} refl = refl
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nonEmptyNotEmpty : {a : _} {A : Set a} {l1 : List A} {i : A} → i :: l1 ≡ [] → False
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nonEmptyNotEmpty {l1 = l1} {i} ()
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nonEmptyNotEmpty : {a : _} {A : Set a} {l1 : List A} {i : A} → i :: l1 ≡ [] → False
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nonEmptyNotEmpty {l1 = l1} {i} ()
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-- TODO - maybe we should do the floating-point style of assuming there's a leading bit and not storing it.
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-- That way, everything is already canonical.
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canonical : BinNat → BinNat
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canonical [] = []
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canonical (zero :: n) with canonical n
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canonical (zero :: n) | [] = []
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canonical (zero :: n) | x :: bl = zero :: x :: bl
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canonical (one :: n) = one :: canonical n
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canonical : BinNat → BinNat
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canonical [] = []
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canonical (zero :: n) with canonical n
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canonical (zero :: n) | [] = []
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canonical (zero :: n) | x :: bl = zero :: x :: bl
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canonical (one :: n) = one :: canonical n
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Canonicalised : Set
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Canonicalised = Sg BinNat (λ i → canonical i ≡ i)
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Canonicalised : Set
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Canonicalised = Sg BinNat (λ i → canonical i ≡ i)
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binNatToN : BinNat → ℕ
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binNatToN [] = 0
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binNatToN (zero :: b) = 2 *N binNatToN b
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binNatToN (one :: b) = 1 +N (2 *N binNatToN b)
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binNatToN : BinNat → ℕ
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binNatToN [] = 0
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binNatToN (zero :: b) = 2 *N binNatToN b
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binNatToN (one :: b) = 1 +N (2 *N binNatToN b)
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incr : BinNat → BinNat
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incr [] = one :: []
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incr (zero :: n) = one :: n
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incr (one :: n) = zero :: (incr n)
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incr : BinNat → BinNat
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incr [] = one :: []
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incr (zero :: n) = one :: n
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incr (one :: n) = zero :: (incr n)
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incrNonzero : (x : BinNat) → canonical (incr x) ≡ [] → False
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incrNonzero : (x : BinNat) → canonical (incr x) ≡ [] → False
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incrPreservesCanonical : (x : BinNat) → (canonical x ≡ x) → canonical (incr x) ≡ incr x
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incrPreservesCanonical [] pr = refl
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incrPreservesCanonical (zero :: xs) pr with canonical xs
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incrPreservesCanonical (zero :: xs) pr | x :: t = applyEquality (one ::_) (::Inj pr)
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incrPreservesCanonical (one :: xs) pr with inspect (canonical (incr xs))
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incrPreservesCanonical (one :: xs) pr | [] with≡ x = exFalso (incrNonzero xs x)
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incrPreservesCanonical (one :: xs) pr | (x₁ :: y) with≡ x rewrite x = applyEquality (zero ::_) (transitivity (equalityCommutative x) (incrPreservesCanonical xs (::Inj pr)))
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incrPreservesCanonical : (x : BinNat) → (canonical x ≡ x) → canonical (incr x) ≡ incr x
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incrPreservesCanonical [] pr = refl
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incrPreservesCanonical (zero :: xs) pr with canonical xs
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incrPreservesCanonical (zero :: xs) pr | x :: t = applyEquality (one ::_) (::Inj pr)
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incrPreservesCanonical (one :: xs) pr with inspect (canonical (incr xs))
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incrPreservesCanonical (one :: xs) pr | [] with≡ x = exFalso (incrNonzero xs x)
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incrPreservesCanonical (one :: xs) pr | (x₁ :: y) with≡ x rewrite x = applyEquality (zero ::_) (transitivity (equalityCommutative x) (incrPreservesCanonical xs (::Inj pr)))
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incrPreservesCanonical' : (x : BinNat) → canonical (incr x) ≡ incr (canonical x)
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incrPreservesCanonical' : (x : BinNat) → canonical (incr x) ≡ incr (canonical x)
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incrC : Canonicalised → Canonicalised
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incrC (a , b) = incr a , incrPreservesCanonical a b
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incrC : Canonicalised → Canonicalised
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incrC (a , b) = incr a , incrPreservesCanonical a b
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NToBinNat : ℕ → BinNat
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NToBinNat zero = []
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NToBinNat (succ n) with NToBinNat n
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NToBinNat (succ n) | t = incr t
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NToBinNat : ℕ → BinNat
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NToBinNat zero = []
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NToBinNat (succ n) with NToBinNat n
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NToBinNat (succ n) | t = incr t
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NToBinNatC : ℕ → Canonicalised
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NToBinNatC zero = [] , refl
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NToBinNatC (succ n) = incrC (NToBinNatC n)
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NToBinNatC : ℕ → Canonicalised
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NToBinNatC zero = [] , refl
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NToBinNatC (succ n) = incrC (NToBinNatC n)
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incrInj : {x y : BinNat} → incr x ≡ incr y → canonical x ≡ canonical y
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incrInj : {x y : BinNat} → incr x ≡ incr y → canonical x ≡ canonical y
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incrNonzero' : (x : BinNat) → (incr x) ≡ [] → False
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incrNonzero' (zero :: xs) ()
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incrNonzero' (one :: xs) ()
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incrNonzero' : (x : BinNat) → (incr x) ≡ [] → False
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incrNonzero' (zero :: xs) ()
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incrNonzero' (one :: xs) ()
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canonicalRespectsIncr' : {x y : BinNat} → canonical (incr x) ≡ canonical (incr y) → canonical x ≡ canonical y
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canonicalRespectsIncr' : {x y : BinNat} → canonical (incr x) ≡ canonical (incr y) → canonical x ≡ canonical y
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binNatToNSucc : (n : BinNat) → binNatToN (incr n) ≡ succ (binNatToN n)
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NToBinNatSucc : (n : ℕ) → incr (NToBinNat n) ≡ NToBinNat (succ n)
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binNatToNSucc : (n : BinNat) → binNatToN (incr n) ≡ succ (binNatToN n)
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NToBinNatSucc : (n : ℕ) → incr (NToBinNat n) ≡ NToBinNat (succ n)
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binNatToNZero : (x : BinNat) → binNatToN x ≡ 0 → canonical x ≡ []
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binNatToNZero' : (x : BinNat) → canonical x ≡ [] → binNatToN x ≡ 0
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binNatToNZero : (x : BinNat) → binNatToN x ≡ 0 → canonical x ≡ []
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binNatToNZero' : (x : BinNat) → canonical x ≡ [] → binNatToN x ≡ 0
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NToBinNatZero : (n : ℕ) → NToBinNat n ≡ [] → n ≡ 0
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NToBinNatZero zero pr = refl
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NToBinNatZero (succ n) pr with NToBinNat n
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NToBinNatZero (succ n) pr | zero :: bl = exFalso (nonEmptyNotEmpty pr)
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NToBinNatZero (succ n) pr | one :: bl = exFalso (nonEmptyNotEmpty pr)
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NToBinNatZero : (n : ℕ) → NToBinNat n ≡ [] → n ≡ 0
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NToBinNatZero zero pr = refl
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NToBinNatZero (succ n) pr with NToBinNat n
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NToBinNatZero (succ n) pr | zero :: bl = exFalso (nonEmptyNotEmpty pr)
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NToBinNatZero (succ n) pr | one :: bl = exFalso (nonEmptyNotEmpty pr)
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canonicalAscends : {i : Bit} → (a : BinNat) → 0 <N binNatToN a → i :: canonical a ≡ canonical (i :: a)
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canonicalAscends : {i : Bit} → (a : BinNat) → 0 <N binNatToN a → i :: canonical a ≡ canonical (i :: a)
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canonicalAscends' : {i : Bit} → (a : BinNat) → (canonical a ≡ [] → False) → i :: canonical a ≡ canonical (i :: a)
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canonicalAscends' {i} a pr = canonicalAscends {i} a (t a pr)
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where
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t : (a : BinNat) → (canonical a ≡ [] → False) → 0 <N binNatToN a
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t a pr with TotalOrder.totality ℕTotalOrder 0 (binNatToN a)
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t a pr | inl (inl x) = x
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t a pr | inr x = exFalso (pr (binNatToNZero a (equalityCommutative x)))
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canonicalAscends' : {i : Bit} → (a : BinNat) → (canonical a ≡ [] → False) → i :: canonical a ≡ canonical (i :: a)
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canonicalAscends' {i} a pr = canonicalAscends {i} a (t a pr)
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where
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t : (a : BinNat) → (canonical a ≡ [] → False) → 0 <N binNatToN a
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t a pr with TotalOrder.totality ℕTotalOrder 0 (binNatToN a)
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t a pr | inl (inl x) = x
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t a pr | inr x = exFalso (pr (binNatToNZero a (equalityCommutative x)))
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canonicalIdempotent : (a : BinNat) → canonical a ≡ canonical (canonical a)
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canonicalIdempotent [] = refl
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canonicalIdempotent (zero :: a) with inspect (canonical a)
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canonicalIdempotent (zero :: a) | [] with≡ y rewrite y = refl
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canonicalIdempotent (zero :: a) | (x :: bl) with≡ y = transitivity (equalityCommutative (canonicalAscends' {zero} a λ p → contr p y)) (transitivity (applyEquality (zero ::_) (canonicalIdempotent a)) (equalityCommutative v))
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where
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contr : {a : _} {A : Set a} {l1 l2 : List A} → {x : A} → l1 ≡ [] → l1 ≡ x :: l2 → False
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contr {l1 = []} p1 ()
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contr {l1 = x :: l1} () p2
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u : canonical (canonical (zero :: a)) ≡ canonical (zero :: canonical a)
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u = applyEquality canonical (equalityCommutative (canonicalAscends' {zero} a λ p → contr p y))
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v : canonical (canonical (zero :: a)) ≡ zero :: canonical (canonical a)
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v = transitivity u (equalityCommutative (canonicalAscends' {zero} (canonical a) λ p → contr (transitivity (canonicalIdempotent a) p) y))
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canonicalIdempotent (one :: a) rewrite equalityCommutative (canonicalIdempotent a) = refl
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canonicalIdempotent : (a : BinNat) → canonical a ≡ canonical (canonical a)
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canonicalIdempotent [] = refl
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canonicalIdempotent (zero :: a) with inspect (canonical a)
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canonicalIdempotent (zero :: a) | [] with≡ y rewrite y = refl
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canonicalIdempotent (zero :: a) | (x :: bl) with≡ y = transitivity (equalityCommutative (canonicalAscends' {zero} a λ p → contr p y)) (transitivity (applyEquality (zero ::_) (canonicalIdempotent a)) (equalityCommutative v))
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where
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contr : {a : _} {A : Set a} {l1 l2 : List A} → {x : A} → l1 ≡ [] → l1 ≡ x :: l2 → False
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contr {l1 = []} p1 ()
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contr {l1 = x :: l1} () p2
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u : canonical (canonical (zero :: a)) ≡ canonical (zero :: canonical a)
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u = applyEquality canonical (equalityCommutative (canonicalAscends' {zero} a λ p → contr p y))
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v : canonical (canonical (zero :: a)) ≡ zero :: canonical (canonical a)
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v = transitivity u (equalityCommutative (canonicalAscends' {zero} (canonical a) λ p → contr (transitivity (canonicalIdempotent a) p) y))
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canonicalIdempotent (one :: a) rewrite equalityCommutative (canonicalIdempotent a) = refl
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canonicalAscends'' : {i : Bit} → (a : BinNat) → canonical (i :: canonical a) ≡ canonical (i :: a)
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canonicalAscends'' : {i : Bit} → (a : BinNat) → canonical (i :: canonical a) ≡ canonical (i :: a)
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binNatToNInj : (x y : BinNat) → binNatToN x ≡ binNatToN y → canonical x ≡ canonical y
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binNatToNInj : (x y : BinNat) → binNatToN x ≡ binNatToN y → canonical x ≡ canonical y
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NToBinNatInj : (x y : ℕ) → canonical (NToBinNat x) ≡ canonical (NToBinNat y) → x ≡ y
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NToBinNatInj : (x y : ℕ) → canonical (NToBinNat x) ≡ canonical (NToBinNat y) → x ≡ y
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NToBinNatIsCanonical : (x : ℕ) → NToBinNat x ≡ canonical (NToBinNat x)
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NToBinNatIsCanonical zero = refl
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NToBinNatIsCanonical (succ x) with NToBinNatC x
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NToBinNatIsCanonical (succ x) | a , b = equalityCommutative (incrPreservesCanonical (NToBinNat x) (equalityCommutative (NToBinNatIsCanonical x)))
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NToBinNatIsCanonical : (x : ℕ) → NToBinNat x ≡ canonical (NToBinNat x)
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NToBinNatIsCanonical zero = refl
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NToBinNatIsCanonical (succ x) with NToBinNatC x
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NToBinNatIsCanonical (succ x) | a , b = equalityCommutative (incrPreservesCanonical (NToBinNat x) (equalityCommutative (NToBinNatIsCanonical x)))
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contr' : {a : _} {A : Set a} {l1 l2 : List A} → {x : A} → l1 ≡ [] → l1 ≡ x :: l2 → False
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contr' {l1 = []} p1 ()
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contr' {l1 = x :: l1} () p2
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contr' : {a : _} {A : Set a} {l1 l2 : List A} → {x : A} → l1 ≡ [] → l1 ≡ x :: l2 → False
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contr' {l1 = []} p1 ()
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contr' {l1 = x :: l1} () p2
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binNatToNIsCanonical : (x : BinNat) → binNatToN (canonical x) ≡ binNatToN x
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binNatToNIsCanonical [] = refl
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binNatToNIsCanonical (zero :: x) with inspect (canonical x)
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binNatToNIsCanonical (zero :: x) | [] with≡ t rewrite t | binNatToNZero' x t = refl
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binNatToNIsCanonical (zero :: x) | (x₁ :: bl) with≡ t rewrite (equalityCommutative (canonicalAscends' {zero} x λ p → contr' p t)) | binNatToNIsCanonical x = refl
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binNatToNIsCanonical (one :: x) rewrite binNatToNIsCanonical x = refl
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binNatToNIsCanonical : (x : BinNat) → binNatToN (canonical x) ≡ binNatToN x
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binNatToNIsCanonical [] = refl
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binNatToNIsCanonical (zero :: x) with inspect (canonical x)
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binNatToNIsCanonical (zero :: x) | [] with≡ t rewrite t | binNatToNZero' x t = refl
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binNatToNIsCanonical (zero :: x) | (x₁ :: bl) with≡ t rewrite (equalityCommutative (canonicalAscends' {zero} x λ p → contr' p t)) | binNatToNIsCanonical x = refl
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binNatToNIsCanonical (one :: x) rewrite binNatToNIsCanonical x = refl
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-- The following two theorems demonstrate that Canonicalised is isomorphic to ℕ
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nToN : (x : ℕ) → binNatToN (NToBinNat x) ≡ x
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nToN : (x : ℕ) → binNatToN (NToBinNat x) ≡ x
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binToBin : (x : BinNat) → NToBinNat (binNatToN x) ≡ canonical x
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binToBin x = transitivity (NToBinNatIsCanonical (binNatToN x)) (binNatToNInj (NToBinNat (binNatToN x)) x (nToN (binNatToN x)))
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binToBin : (x : BinNat) → NToBinNat (binNatToN x) ≡ canonical x
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binToBin x = transitivity (NToBinNatIsCanonical (binNatToN x)) (binNatToNInj (NToBinNat (binNatToN x)) x (nToN (binNatToN x)))
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doubleIsBitShift' : (a : ℕ) → NToBinNat (2 *N succ a) ≡ zero :: NToBinNat (succ a)
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doubleIsBitShift' zero = refl
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doubleIsBitShift' (succ a) with doubleIsBitShift' a
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... | bl rewrite Semiring.commutative ℕSemiring a (succ (succ (a +N 0))) | Semiring.commutative ℕSemiring (succ (a +N 0)) a | Semiring.commutative ℕSemiring a (succ (a +N 0)) | Semiring.commutative ℕSemiring (a +N 0) a | bl = refl
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doubleIsBitShift' : (a : ℕ) → NToBinNat (2 *N succ a) ≡ zero :: NToBinNat (succ a)
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doubleIsBitShift' zero = refl
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doubleIsBitShift' (succ a) with doubleIsBitShift' a
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... | bl rewrite Semiring.commutative ℕSemiring a (succ (succ (a +N 0))) | Semiring.commutative ℕSemiring (succ (a +N 0)) a | Semiring.commutative ℕSemiring a (succ (a +N 0)) | Semiring.commutative ℕSemiring (a +N 0) a | bl = refl
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doubleIsBitShift : (a : ℕ) → (0 <N a) → NToBinNat (2 *N a) ≡ zero :: NToBinNat a
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doubleIsBitShift zero ()
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doubleIsBitShift (succ a) _ = doubleIsBitShift' a
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doubleIsBitShift : (a : ℕ) → (0 <N a) → NToBinNat (2 *N a) ≡ zero :: NToBinNat a
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doubleIsBitShift zero ()
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doubleIsBitShift (succ a) _ = doubleIsBitShift' a
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canonicalDescends : {a : Bit} (as : BinNat) → (prA : a :: as ≡ canonical (a :: as)) → as ≡ canonical as
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canonicalDescends {zero} as pr with canonical as
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canonicalDescends {zero} as pr | x :: bl = ::Inj pr
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canonicalDescends {one} as pr = ::Inj pr
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canonicalDescends : {a : Bit} (as : BinNat) → (prA : a :: as ≡ canonical (a :: as)) → as ≡ canonical as
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canonicalDescends {zero} as pr with canonical as
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canonicalDescends {zero} as pr | x :: bl = ::Inj pr
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canonicalDescends {one} as pr = ::Inj pr
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--- Proofs
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parity : (a b : ℕ) → succ (2 *N a) ≡ 2 *N b → False
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doubleInj : (a b : ℕ) → (2 *N a) ≡ (2 *N b) → a ≡ b
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parity : (a b : ℕ) → succ (2 *N a) ≡ 2 *N b → False
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doubleInj : (a b : ℕ) → (2 *N a) ≡ (2 *N b) → a ≡ b
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incrNonzeroTwice : (x : BinNat) → canonical (incr (incr x)) ≡ one :: [] → False
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incrNonzeroTwice (zero :: xs) pr with canonical (incr xs)
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incrNonzeroTwice (zero :: xs) () | []
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incrNonzeroTwice (zero :: xs) () | x :: bl
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incrNonzeroTwice (one :: xs) pr = exFalso (incrNonzero xs (::Inj pr))
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incrNonzeroTwice : (x : BinNat) → canonical (incr (incr x)) ≡ one :: [] → False
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incrNonzeroTwice (zero :: xs) pr with canonical (incr xs)
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incrNonzeroTwice (zero :: xs) () | []
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incrNonzeroTwice (zero :: xs) () | x :: bl
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incrNonzeroTwice (one :: xs) pr = exFalso (incrNonzero xs (::Inj pr))
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canonicalRespectsIncr' {[]} {[]} pr = refl
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canonicalRespectsIncr' {[]} {zero :: y} pr with canonical y
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canonicalRespectsIncr' {[]} {zero :: y} pr | [] = refl
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canonicalRespectsIncr' {[]} {one :: y} pr with canonical (incr y)
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canonicalRespectsIncr' {[]} {one :: y} () | []
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canonicalRespectsIncr' {[]} {one :: y} () | x :: bl
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canonicalRespectsIncr' {zero :: xs} {y} pr with canonical xs
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canonicalRespectsIncr' {zero :: xs} {[]} pr | [] = refl
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canonicalRespectsIncr' {zero :: xs} {zero :: y} pr | [] with canonical y
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canonicalRespectsIncr' {zero :: xs} {zero :: y} pr | [] | [] = refl
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canonicalRespectsIncr' {zero :: xs} {one :: y} pr | [] with canonical (incr y)
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canonicalRespectsIncr' {zero :: xs} {one :: y} () | [] | []
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canonicalRespectsIncr' {zero :: xs} {one :: y} () | [] | x :: bl
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canonicalRespectsIncr' {zero :: xs} {zero :: ys} pr | x :: bl with canonical ys
|
||||
canonicalRespectsIncr' {zero :: xs} {zero :: ys} pr | x :: bl | x₁ :: th = applyEquality (zero ::_) (::Inj pr)
|
||||
canonicalRespectsIncr' {zero :: xs} {one :: ys} pr | x :: bl with canonical (incr ys)
|
||||
canonicalRespectsIncr' {zero :: xs} {one :: ys} () | x :: bl | []
|
||||
canonicalRespectsIncr' {zero :: xs} {one :: ys} () | x :: bl | x₁ :: th
|
||||
canonicalRespectsIncr' {one :: xs} {[]} pr with canonical (incr xs)
|
||||
canonicalRespectsIncr' {one :: xs} {[]} () | []
|
||||
canonicalRespectsIncr' {one :: xs} {[]} () | x :: bl
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} pr with canonical ys
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} pr | [] with canonical (incr xs)
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} () | [] | []
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} () | [] | x :: t
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} pr | x :: bl with canonical (incr xs)
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} () | x :: bl | []
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} () | x :: bl | x₁ :: t
|
||||
canonicalRespectsIncr' {one :: xs} {one :: ys} pr with inspect (canonical (incr xs))
|
||||
canonicalRespectsIncr' {one :: xs} {one :: ys} pr | [] with≡ x = exFalso (incrNonzero xs x)
|
||||
canonicalRespectsIncr' {one :: xs} {one :: ys} pr | (x+1 :: x+1s) with≡ bad rewrite bad = applyEquality (one ::_) ans
|
||||
where
|
||||
ans : canonical xs ≡ canonical ys
|
||||
ans with inspect (canonical (incr ys))
|
||||
ans | [] with≡ x = exFalso (incrNonzero ys x)
|
||||
ans | (x₁ :: y) with≡ x rewrite x = canonicalRespectsIncr' {xs} {ys} (transitivity bad (transitivity (::Inj pr) (equalityCommutative x)))
|
||||
canonicalRespectsIncr' {[]} {[]} pr = refl
|
||||
canonicalRespectsIncr' {[]} {zero :: y} pr with canonical y
|
||||
canonicalRespectsIncr' {[]} {zero :: y} pr | [] = refl
|
||||
canonicalRespectsIncr' {[]} {one :: y} pr with canonical (incr y)
|
||||
canonicalRespectsIncr' {[]} {one :: y} () | []
|
||||
canonicalRespectsIncr' {[]} {one :: y} () | x :: bl
|
||||
canonicalRespectsIncr' {zero :: xs} {y} pr with canonical xs
|
||||
canonicalRespectsIncr' {zero :: xs} {[]} pr | [] = refl
|
||||
canonicalRespectsIncr' {zero :: xs} {zero :: y} pr | [] with canonical y
|
||||
canonicalRespectsIncr' {zero :: xs} {zero :: y} pr | [] | [] = refl
|
||||
canonicalRespectsIncr' {zero :: xs} {one :: y} pr | [] with canonical (incr y)
|
||||
canonicalRespectsIncr' {zero :: xs} {one :: y} () | [] | []
|
||||
canonicalRespectsIncr' {zero :: xs} {one :: y} () | [] | x :: bl
|
||||
canonicalRespectsIncr' {zero :: xs} {zero :: ys} pr | x :: bl with canonical ys
|
||||
canonicalRespectsIncr' {zero :: xs} {zero :: ys} pr | x :: bl | x₁ :: th = applyEquality (zero ::_) (::Inj pr)
|
||||
canonicalRespectsIncr' {zero :: xs} {one :: ys} pr | x :: bl with canonical (incr ys)
|
||||
canonicalRespectsIncr' {zero :: xs} {one :: ys} () | x :: bl | []
|
||||
canonicalRespectsIncr' {zero :: xs} {one :: ys} () | x :: bl | x₁ :: th
|
||||
canonicalRespectsIncr' {one :: xs} {[]} pr with canonical (incr xs)
|
||||
canonicalRespectsIncr' {one :: xs} {[]} () | []
|
||||
canonicalRespectsIncr' {one :: xs} {[]} () | x :: bl
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} pr with canonical ys
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} pr | [] with canonical (incr xs)
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} () | [] | []
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} () | [] | x :: t
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} pr | x :: bl with canonical (incr xs)
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} () | x :: bl | []
|
||||
canonicalRespectsIncr' {one :: xs} {zero :: ys} () | x :: bl | x₁ :: t
|
||||
canonicalRespectsIncr' {one :: xs} {one :: ys} pr with inspect (canonical (incr xs))
|
||||
canonicalRespectsIncr' {one :: xs} {one :: ys} pr | [] with≡ x = exFalso (incrNonzero xs x)
|
||||
canonicalRespectsIncr' {one :: xs} {one :: ys} pr | (x+1 :: x+1s) with≡ bad rewrite bad = applyEquality (one ::_) ans
|
||||
where
|
||||
ans : canonical xs ≡ canonical ys
|
||||
ans with inspect (canonical (incr ys))
|
||||
ans | [] with≡ x = exFalso (incrNonzero ys x)
|
||||
ans | (x₁ :: y) with≡ x rewrite x = canonicalRespectsIncr' {xs} {ys} (transitivity bad (transitivity (::Inj pr) (equalityCommutative x)))
|
||||
|
||||
binNatToNInj[] : (y : BinNat) → 0 ≡ binNatToN y → [] ≡ canonical y
|
||||
binNatToNInj[] [] pr = refl
|
||||
binNatToNInj[] (zero :: y) pr with productZeroImpliesOperandZero {2} {binNatToN y} (equalityCommutative pr)
|
||||
binNatToNInj[] (zero :: y) pr | inr x with inspect (canonical y)
|
||||
binNatToNInj[] (zero :: y) pr | inr x | [] with≡ canYPr rewrite canYPr = refl
|
||||
binNatToNInj[] (zero :: ys) pr | inr x | (y :: canY) with≡ canYPr with binNatToNZero ys x
|
||||
... | r with canonical ys
|
||||
binNatToNInj[] (zero :: ys) pr | inr x | (y :: canY) with≡ canYPr | r | [] = refl
|
||||
binNatToNInj[] (zero :: ys) pr | inr x | (y :: canY) with≡ canYPr | () | x₁ :: t
|
||||
binNatToNInj[] : (y : BinNat) → 0 ≡ binNatToN y → [] ≡ canonical y
|
||||
binNatToNInj[] [] pr = refl
|
||||
binNatToNInj[] (zero :: y) pr with productZeroImpliesOperandZero {2} {binNatToN y} (equalityCommutative pr)
|
||||
binNatToNInj[] (zero :: y) pr | inr x with inspect (canonical y)
|
||||
binNatToNInj[] (zero :: y) pr | inr x | [] with≡ canYPr rewrite canYPr = refl
|
||||
binNatToNInj[] (zero :: ys) pr | inr x | (y :: canY) with≡ canYPr with binNatToNZero ys x
|
||||
... | r with canonical ys
|
||||
binNatToNInj[] (zero :: ys) pr | inr x | (y :: canY) with≡ canYPr | r | [] = refl
|
||||
binNatToNInj[] (zero :: ys) pr | inr x | (y :: canY) with≡ canYPr | () | x₁ :: t
|
||||
|
||||
binNatToNInj [] y pr = binNatToNInj[] y pr
|
||||
binNatToNInj (zero :: xs) [] pr = equalityCommutative (binNatToNInj[] (zero :: xs) (equalityCommutative pr))
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr with doubleInj (binNatToN xs) (binNatToN ys) pr
|
||||
... | x=y with binNatToNInj xs ys x=y
|
||||
... | t with canonical xs
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr | x=y | t | [] with canonical ys
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr | x=y | t | [] | [] = refl
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr | x=y | t | x :: cxs with canonical ys
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr | x=y | t | x :: cxs | x₁ :: cys = applyEquality (zero ::_) t
|
||||
binNatToNInj (zero :: xs) (one :: ys) pr = exFalso (parity (binNatToN ys) (binNatToN xs) (equalityCommutative pr))
|
||||
binNatToNInj (one :: xs) (zero :: ys) pr = exFalso (parity (binNatToN xs) (binNatToN ys) pr)
|
||||
binNatToNInj (one :: xs) (one :: ys) pr = applyEquality (one ::_) (binNatToNInj xs ys (doubleInj (binNatToN xs) (binNatToN ys) (succInjective pr)))
|
||||
binNatToNInj [] y pr = binNatToNInj[] y pr
|
||||
binNatToNInj (zero :: xs) [] pr = equalityCommutative (binNatToNInj[] (zero :: xs) (equalityCommutative pr))
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr with doubleInj (binNatToN xs) (binNatToN ys) pr
|
||||
... | x=y with binNatToNInj xs ys x=y
|
||||
... | t with canonical xs
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr | x=y | t | [] with canonical ys
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr | x=y | t | [] | [] = refl
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr | x=y | t | x :: cxs with canonical ys
|
||||
binNatToNInj (zero :: xs) (zero :: ys) pr | x=y | t | x :: cxs | x₁ :: cys = applyEquality (zero ::_) t
|
||||
binNatToNInj (zero :: xs) (one :: ys) pr = exFalso (parity (binNatToN ys) (binNatToN xs) (equalityCommutative pr))
|
||||
binNatToNInj (one :: xs) (zero :: ys) pr = exFalso (parity (binNatToN xs) (binNatToN ys) pr)
|
||||
binNatToNInj (one :: xs) (one :: ys) pr = applyEquality (one ::_) (binNatToNInj xs ys (doubleInj (binNatToN xs) (binNatToN ys) (succInjective pr)))
|
||||
|
||||
NToBinNatInj zero zero pr = refl
|
||||
NToBinNatInj zero (succ y) pr with NToBinNat y
|
||||
NToBinNatInj zero (succ y) pr | bl = exFalso (incrNonzero bl (equalityCommutative pr))
|
||||
NToBinNatInj (succ x) zero pr with NToBinNat x
|
||||
... | bl = exFalso (incrNonzero bl pr)
|
||||
NToBinNatInj (succ x) (succ y) pr with inspect (NToBinNat x)
|
||||
NToBinNatInj (succ zero) (succ zero) pr | [] with≡ nToBinXPr = refl
|
||||
NToBinNatInj (succ zero) (succ (succ y)) pr | [] with≡ nToBinXPr with NToBinNat y
|
||||
NToBinNatInj (succ zero) (succ (succ y)) pr | [] with≡ nToBinXPr | y' = exFalso (incrNonzeroTwice y' (equalityCommutative pr))
|
||||
NToBinNatInj (succ (succ x)) (succ y) pr | [] with≡ nToBinXPr with NToBinNat x
|
||||
NToBinNatInj (succ (succ x)) (succ y) pr | [] with≡ nToBinXPr | t = exFalso (incrNonzero' t nToBinXPr)
|
||||
NToBinNatInj (succ x) (succ y) pr | (x₁ :: nToBinX) with≡ nToBinXPr with NToBinNatInj x y (canonicalRespectsIncr' {NToBinNat x} {NToBinNat y} pr)
|
||||
NToBinNatInj (succ x) (succ .x) pr | (x₁ :: nToBinX) with≡ nToBinXPr | refl = refl
|
||||
NToBinNatInj zero zero pr = refl
|
||||
NToBinNatInj zero (succ y) pr with NToBinNat y
|
||||
NToBinNatInj zero (succ y) pr | bl = exFalso (incrNonzero bl (equalityCommutative pr))
|
||||
NToBinNatInj (succ x) zero pr with NToBinNat x
|
||||
... | bl = exFalso (incrNonzero bl pr)
|
||||
NToBinNatInj (succ x) (succ y) pr with inspect (NToBinNat x)
|
||||
NToBinNatInj (succ zero) (succ zero) pr | [] with≡ nToBinXPr = refl
|
||||
NToBinNatInj (succ zero) (succ (succ y)) pr | [] with≡ nToBinXPr with NToBinNat y
|
||||
NToBinNatInj (succ zero) (succ (succ y)) pr | [] with≡ nToBinXPr | y' = exFalso (incrNonzeroTwice y' (equalityCommutative pr))
|
||||
NToBinNatInj (succ (succ x)) (succ y) pr | [] with≡ nToBinXPr with NToBinNat x
|
||||
NToBinNatInj (succ (succ x)) (succ y) pr | [] with≡ nToBinXPr | t = exFalso (incrNonzero' t nToBinXPr)
|
||||
NToBinNatInj (succ x) (succ y) pr | (x₁ :: nToBinX) with≡ nToBinXPr with NToBinNatInj x y (canonicalRespectsIncr' {NToBinNat x} {NToBinNat y} pr)
|
||||
NToBinNatInj (succ x) (succ .x) pr | (x₁ :: nToBinX) with≡ nToBinXPr | refl = refl
|
||||
|
||||
incrInj {[]} {[]} pr = refl
|
||||
incrInj {[]} {zero :: ys} pr rewrite equalityCommutative (::Inj pr) = refl
|
||||
incrInj {zero :: xs} {[]} pr rewrite ::Inj pr = refl
|
||||
incrInj {zero :: xs} {zero :: .xs} refl = refl
|
||||
incrInj {one :: xs} {one :: ys} pr = applyEquality (one ::_) (incrInj {xs} {ys} (l (incr xs) (incr ys) pr))
|
||||
where
|
||||
l : (a : BinNat) → (b : BinNat) → zero :: a ≡ zero :: b → a ≡ b
|
||||
l a .a refl = refl
|
||||
incrInj {[]} {[]} pr = refl
|
||||
incrInj {[]} {zero :: ys} pr rewrite equalityCommutative (::Inj pr) = refl
|
||||
incrInj {zero :: xs} {[]} pr rewrite ::Inj pr = refl
|
||||
incrInj {zero :: xs} {zero :: .xs} refl = refl
|
||||
incrInj {one :: xs} {one :: ys} pr = applyEquality (one ::_) (incrInj {xs} {ys} (l (incr xs) (incr ys) pr))
|
||||
where
|
||||
l : (a : BinNat) → (b : BinNat) → zero :: a ≡ zero :: b → a ≡ b
|
||||
l a .a refl = refl
|
||||
|
||||
doubleInj zero zero pr = refl
|
||||
doubleInj (succ a) (succ b) pr = applyEquality succ (doubleInj a b u)
|
||||
where
|
||||
t : a +N a ≡ b +N b
|
||||
t rewrite Semiring.commutative ℕSemiring (succ a) 0 | Semiring.commutative ℕSemiring (succ b) 0 | Semiring.commutative ℕSemiring a (succ a) | Semiring.commutative ℕSemiring b (succ b) = succInjective (succInjective pr)
|
||||
u : a +N (a +N zero) ≡ b +N (b +N zero)
|
||||
u rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 = t
|
||||
doubleInj zero zero pr = refl
|
||||
doubleInj (succ a) (succ b) pr = applyEquality succ (doubleInj a b u)
|
||||
where
|
||||
t : a +N a ≡ b +N b
|
||||
t rewrite Semiring.commutative ℕSemiring (succ a) 0 | Semiring.commutative ℕSemiring (succ b) 0 | Semiring.commutative ℕSemiring a (succ a) | Semiring.commutative ℕSemiring b (succ b) = succInjective (succInjective pr)
|
||||
u : a +N (a +N zero) ≡ b +N (b +N zero)
|
||||
u rewrite Semiring.commutative ℕSemiring a 0 | Semiring.commutative ℕSemiring b 0 = t
|
||||
|
||||
binNatToNZero [] pr = refl
|
||||
binNatToNZero (zero :: xs) pr with inspect (canonical xs)
|
||||
binNatToNZero (zero :: xs) pr | [] with≡ x rewrite x = refl
|
||||
binNatToNZero (zero :: xs) pr | (x₁ :: y) with≡ x with productZeroImpliesOperandZero {2} {binNatToN xs} pr
|
||||
binNatToNZero (zero :: xs) pr | (x₁ :: y) with≡ x | inr pr' with binNatToNZero xs pr'
|
||||
... | bl with canonical xs
|
||||
binNatToNZero (zero :: xs) pr | (x₁ :: y) with≡ x | inr pr' | bl | [] = refl
|
||||
binNatToNZero (zero :: xs) pr | (x₁ :: y) with≡ x | inr pr' | () | x₂ :: t
|
||||
binNatToNZero [] pr = refl
|
||||
binNatToNZero (zero :: xs) pr with inspect (canonical xs)
|
||||
binNatToNZero (zero :: xs) pr | [] with≡ x rewrite x = refl
|
||||
binNatToNZero (zero :: xs) pr | (x₁ :: y) with≡ x with productZeroImpliesOperandZero {2} {binNatToN xs} pr
|
||||
binNatToNZero (zero :: xs) pr | (x₁ :: y) with≡ x | inr pr' with binNatToNZero xs pr'
|
||||
... | bl with canonical xs
|
||||
binNatToNZero (zero :: xs) pr | (x₁ :: y) with≡ x | inr pr' | bl | [] = refl
|
||||
binNatToNZero (zero :: xs) pr | (x₁ :: y) with≡ x | inr pr' | () | x₂ :: t
|
||||
|
||||
binNatToNSucc [] = refl
|
||||
binNatToNSucc (zero :: n) = refl
|
||||
binNatToNSucc (one :: n) rewrite Semiring.commutative ℕSemiring (binNatToN n) zero | Semiring.commutative ℕSemiring (binNatToN (incr n)) 0 | binNatToNSucc n = applyEquality succ (Semiring.commutative ℕSemiring (binNatToN n) (succ (binNatToN n)))
|
||||
binNatToNSucc [] = refl
|
||||
binNatToNSucc (zero :: n) = refl
|
||||
binNatToNSucc (one :: n) rewrite Semiring.commutative ℕSemiring (binNatToN n) zero | Semiring.commutative ℕSemiring (binNatToN (incr n)) 0 | binNatToNSucc n = applyEquality succ (Semiring.commutative ℕSemiring (binNatToN n) (succ (binNatToN n)))
|
||||
|
||||
incrNonzero (one :: xs) pr with inspect (canonical (incr xs))
|
||||
incrNonzero (one :: xs) pr | [] with≡ x = incrNonzero xs x
|
||||
incrNonzero (one :: xs) pr | (y :: ys) with≡ x with canonical (incr xs)
|
||||
incrNonzero (one :: xs) pr | (y :: ys) with≡ () | []
|
||||
incrNonzero (one :: xs) () | (.x₁ :: .th) with≡ refl | x₁ :: th
|
||||
incrNonzero (one :: xs) pr with inspect (canonical (incr xs))
|
||||
incrNonzero (one :: xs) pr | [] with≡ x = incrNonzero xs x
|
||||
incrNonzero (one :: xs) pr | (y :: ys) with≡ x with canonical (incr xs)
|
||||
incrNonzero (one :: xs) pr | (y :: ys) with≡ () | []
|
||||
incrNonzero (one :: xs) () | (.x₁ :: .th) with≡ refl | x₁ :: th
|
||||
|
||||
nToN zero = refl
|
||||
nToN (succ x) with inspect (NToBinNat x)
|
||||
nToN (succ x) | [] with≡ pr = ans
|
||||
where
|
||||
t : x ≡ zero
|
||||
t = NToBinNatInj x 0 (applyEquality canonical pr)
|
||||
ans : binNatToN (incr (NToBinNat x)) ≡ succ x
|
||||
ans rewrite t | pr = refl
|
||||
nToN (succ x) | (bit :: xs) with≡ pr = transitivity (binNatToNSucc (NToBinNat x)) (applyEquality succ (nToN x))
|
||||
nToN zero = refl
|
||||
nToN (succ x) with inspect (NToBinNat x)
|
||||
nToN (succ x) | [] with≡ pr = ans
|
||||
where
|
||||
t : x ≡ zero
|
||||
t = NToBinNatInj x 0 (applyEquality canonical pr)
|
||||
ans : binNatToN (incr (NToBinNat x)) ≡ succ x
|
||||
ans rewrite t | pr = refl
|
||||
nToN (succ x) | (bit :: xs) with≡ pr = transitivity (binNatToNSucc (NToBinNat x)) (applyEquality succ (nToN x))
|
||||
|
||||
parity zero (succ b) pr rewrite Semiring.commutative ℕSemiring b (succ (b +N 0)) = bad pr
|
||||
where
|
||||
bad : (1 ≡ succ (succ ((b +N 0) +N b))) → False
|
||||
bad ()
|
||||
parity (succ a) (succ b) pr rewrite Semiring.commutative ℕSemiring b (succ (b +N 0)) | Semiring.commutative ℕSemiring a (succ (a +N 0)) | Semiring.commutative ℕSemiring (a +N 0) a | Semiring.commutative ℕSemiring (b +N 0) b = parity a b (succInjective (succInjective pr))
|
||||
parity zero (succ b) pr rewrite Semiring.commutative ℕSemiring b (succ (b +N 0)) = bad pr
|
||||
where
|
||||
bad : (1 ≡ succ (succ ((b +N 0) +N b))) → False
|
||||
bad ()
|
||||
parity (succ a) (succ b) pr rewrite Semiring.commutative ℕSemiring b (succ (b +N 0)) | Semiring.commutative ℕSemiring a (succ (a +N 0)) | Semiring.commutative ℕSemiring (a +N 0) a | Semiring.commutative ℕSemiring (b +N 0) b = parity a b (succInjective (succInjective pr))
|
||||
|
||||
binNatToNZero' [] pr = refl
|
||||
binNatToNZero' (zero :: xs) pr with inspect (canonical xs)
|
||||
binNatToNZero' (zero :: xs) pr | [] with≡ p2 = ans
|
||||
where
|
||||
t : binNatToN xs ≡ 0
|
||||
t = binNatToNZero' xs p2
|
||||
ans : 2 *N binNatToN xs ≡ 0
|
||||
ans rewrite t = refl
|
||||
binNatToNZero' (zero :: xs) pr | (y :: ys) with≡ p rewrite p = exFalso (bad pr)
|
||||
where
|
||||
bad : zero :: y :: ys ≡ [] → False
|
||||
bad ()
|
||||
binNatToNZero' (one :: xs) ()
|
||||
binNatToNZero' [] pr = refl
|
||||
binNatToNZero' (zero :: xs) pr with inspect (canonical xs)
|
||||
binNatToNZero' (zero :: xs) pr | [] with≡ p2 = ans
|
||||
where
|
||||
t : binNatToN xs ≡ 0
|
||||
t = binNatToNZero' xs p2
|
||||
ans : 2 *N binNatToN xs ≡ 0
|
||||
ans rewrite t = refl
|
||||
binNatToNZero' (zero :: xs) pr | (y :: ys) with≡ p rewrite p = exFalso (bad pr)
|
||||
where
|
||||
bad : zero :: y :: ys ≡ [] → False
|
||||
bad ()
|
||||
binNatToNZero' (one :: xs) ()
|
||||
|
||||
canonicalAscends {zero} a 0<a with inspect (canonical a)
|
||||
canonicalAscends {zero} (zero :: a) 0<a | [] with≡ x = exFalso (contr'' (binNatToN a) t v)
|
||||
where
|
||||
u : binNatToN (zero :: a) ≡ 0
|
||||
u = binNatToNZero' (zero :: a) x
|
||||
v : binNatToN a ≡ 0
|
||||
v with inspect (binNatToN a)
|
||||
v | zero with≡ x = x
|
||||
v | succ a' with≡ x with inspect (binNatToN (zero :: a))
|
||||
v | succ a' with≡ x | zero with≡ pr2 rewrite pr2 = exFalso (TotalOrder.irreflexive ℕTotalOrder 0<a)
|
||||
v | succ a' with≡ x | succ y with≡ pr2 rewrite u = exFalso (TotalOrder.irreflexive ℕTotalOrder 0<a)
|
||||
t : 0 <N binNatToN a
|
||||
t with binNatToN a
|
||||
t | succ bl rewrite Semiring.commutative ℕSemiring (succ bl) 0 = succIsPositive bl
|
||||
contr'' : (x : ℕ) → (0 <N x) → (x ≡ 0) → False
|
||||
contr'' x 0<x x=0 rewrite x=0 = TotalOrder.irreflexive ℕTotalOrder 0<x
|
||||
canonicalAscends {zero} a 0<a | (x₁ :: y) with≡ x rewrite x = refl
|
||||
canonicalAscends {one} a 0<a = refl
|
||||
canonicalAscends {zero} a 0<a with inspect (canonical a)
|
||||
canonicalAscends {zero} (zero :: a) 0<a | [] with≡ x = exFalso (contr'' (binNatToN a) t v)
|
||||
where
|
||||
u : binNatToN (zero :: a) ≡ 0
|
||||
u = binNatToNZero' (zero :: a) x
|
||||
v : binNatToN a ≡ 0
|
||||
v with inspect (binNatToN a)
|
||||
v | zero with≡ x = x
|
||||
v | succ a' with≡ x with inspect (binNatToN (zero :: a))
|
||||
v | succ a' with≡ x | zero with≡ pr2 rewrite pr2 = exFalso (TotalOrder.irreflexive ℕTotalOrder 0<a)
|
||||
v | succ a' with≡ x | succ y with≡ pr2 rewrite u = exFalso (TotalOrder.irreflexive ℕTotalOrder 0<a)
|
||||
t : 0 <N binNatToN a
|
||||
t with binNatToN a
|
||||
t | succ bl rewrite Semiring.commutative ℕSemiring (succ bl) 0 = succIsPositive bl
|
||||
contr'' : (x : ℕ) → (0 <N x) → (x ≡ 0) → False
|
||||
contr'' x 0<x x=0 rewrite x=0 = TotalOrder.irreflexive ℕTotalOrder 0<x
|
||||
canonicalAscends {zero} a 0<a | (x₁ :: y) with≡ x rewrite x = refl
|
||||
canonicalAscends {one} a 0<a = refl
|
||||
|
||||
canonicalAscends'' {i} a with inspect (canonical a)
|
||||
canonicalAscends'' {zero} a | [] with≡ x rewrite x = refl
|
||||
canonicalAscends'' {one} a | [] with≡ x rewrite x = refl
|
||||
canonicalAscends'' {i} a | (x₁ :: y) with≡ x = transitivity (applyEquality canonical (canonicalAscends' {i} a λ p → contr p x)) (equalityCommutative (canonicalIdempotent (i :: a)))
|
||||
where
|
||||
contr : {a : _} {A : Set a} {l1 l2 : List A} → {x : A} → l1 ≡ [] → l1 ≡ x :: l2 → False
|
||||
contr {l1 = []} p1 ()
|
||||
contr {l1 = x :: l1} () p2
|
||||
canonicalAscends'' {i} a with inspect (canonical a)
|
||||
canonicalAscends'' {zero} a | [] with≡ x rewrite x = refl
|
||||
canonicalAscends'' {one} a | [] with≡ x rewrite x = refl
|
||||
canonicalAscends'' {i} a | (x₁ :: y) with≡ x = transitivity (applyEquality canonical (canonicalAscends' {i} a λ p → contr p x)) (equalityCommutative (canonicalIdempotent (i :: a)))
|
||||
where
|
||||
contr : {a : _} {A : Set a} {l1 l2 : List A} → {x : A} → l1 ≡ [] → l1 ≡ x :: l2 → False
|
||||
contr {l1 = []} p1 ()
|
||||
contr {l1 = x :: l1} () p2
|
||||
|
||||
incrPreservesCanonical' [] = refl
|
||||
incrPreservesCanonical' (zero :: xs) with inspect (canonical xs)
|
||||
incrPreservesCanonical' (zero :: xs) | [] with≡ x rewrite x = refl
|
||||
incrPreservesCanonical' (zero :: xs) | (x₁ :: y) with≡ x rewrite x = refl
|
||||
incrPreservesCanonical' (one :: xs) with inspect (canonical (incr xs))
|
||||
... | [] with≡ pr = exFalso (incrNonzero xs pr)
|
||||
... | (_ :: _) with≡ pr rewrite pr = applyEquality (zero ::_) (transitivity (equalityCommutative pr) (incrPreservesCanonical' xs))
|
||||
incrPreservesCanonical' [] = refl
|
||||
incrPreservesCanonical' (zero :: xs) with inspect (canonical xs)
|
||||
incrPreservesCanonical' (zero :: xs) | [] with≡ x rewrite x = refl
|
||||
incrPreservesCanonical' (zero :: xs) | (x₁ :: y) with≡ x rewrite x = refl
|
||||
incrPreservesCanonical' (one :: xs) with inspect (canonical (incr xs))
|
||||
... | [] with≡ pr = exFalso (incrNonzero xs pr)
|
||||
... | (_ :: _) with≡ pr rewrite pr = applyEquality (zero ::_) (transitivity (equalityCommutative pr) (incrPreservesCanonical' xs))
|
||||
|
||||
NToBinNatSucc zero = refl
|
||||
NToBinNatSucc (succ n) with NToBinNat n
|
||||
... | [] = refl
|
||||
... | a :: as = refl
|
||||
NToBinNatSucc zero = refl
|
||||
NToBinNatSucc (succ n) with NToBinNat n
|
||||
... | [] = refl
|
||||
... | a :: as = refl
|
||||
|
||||
@@ -12,172 +12,172 @@ open import Semirings.Definition
|
||||
|
||||
module Numbers.BinaryNaturals.Multiplication where
|
||||
|
||||
_*Binherit_ : BinNat → BinNat → BinNat
|
||||
a *Binherit b = NToBinNat (binNatToN a *N binNatToN b)
|
||||
_*Binherit_ : BinNat → BinNat → BinNat
|
||||
a *Binherit b = NToBinNat (binNatToN a *N binNatToN b)
|
||||
|
||||
_*B_ : BinNat → BinNat → BinNat
|
||||
[] *B b = []
|
||||
(zero :: a) *B b = zero :: (a *B b)
|
||||
(one :: a) *B b = (zero :: (a *B b)) +B b
|
||||
_*B_ : BinNat → BinNat → BinNat
|
||||
[] *B b = []
|
||||
(zero :: a) *B b = zero :: (a *B b)
|
||||
(one :: a) *B b = (zero :: (a *B b)) +B b
|
||||
|
||||
contr : {a : _} {A : Set a} {l1 l2 : List A} → {x : A} → l1 ≡ [] → l1 ≡ x :: l2 → False
|
||||
contr {l1 = []} p1 ()
|
||||
contr {l1 = x :: l1} () p2
|
||||
contr : {a : _} {A : Set a} {l1 l2 : List A} → {x : A} → l1 ≡ [] → l1 ≡ x :: l2 → False
|
||||
contr {l1 = []} p1 ()
|
||||
contr {l1 = x :: l1} () p2
|
||||
|
||||
*BEmpty : (a : BinNat) → canonical (a *B []) ≡ []
|
||||
*BEmpty [] = refl
|
||||
*BEmpty (zero :: a) rewrite *BEmpty a = refl
|
||||
*BEmpty (one :: a) rewrite *BEmpty a = refl
|
||||
*BEmpty : (a : BinNat) → canonical (a *B []) ≡ []
|
||||
*BEmpty [] = refl
|
||||
*BEmpty (zero :: a) rewrite *BEmpty a = refl
|
||||
*BEmpty (one :: a) rewrite *BEmpty a = refl
|
||||
|
||||
canonicalDistributesPlus : (a b : BinNat) → canonical (a +B b) ≡ canonical a +B canonical b
|
||||
canonicalDistributesPlus a b = transitivity ans (+BIsInherited (canonical a) (canonical b) (canonicalIdempotent a) (canonicalIdempotent b))
|
||||
where
|
||||
ans : canonical (a +B b) ≡ NToBinNat (binNatToN (canonical a) +N binNatToN (canonical b))
|
||||
ans rewrite binNatToNIsCanonical a | binNatToNIsCanonical b = equalityCommutative (+BIsInherited' a b)
|
||||
canonicalDistributesPlus : (a b : BinNat) → canonical (a +B b) ≡ canonical a +B canonical b
|
||||
canonicalDistributesPlus a b = transitivity ans (+BIsInherited (canonical a) (canonical b) (canonicalIdempotent a) (canonicalIdempotent b))
|
||||
where
|
||||
ans : canonical (a +B b) ≡ NToBinNat (binNatToN (canonical a) +N binNatToN (canonical b))
|
||||
ans rewrite binNatToNIsCanonical a | binNatToNIsCanonical b = equalityCommutative (+BIsInherited' a b)
|
||||
|
||||
incrPullsOut : (a b : BinNat) → incr (a +B b) ≡ (incr a) +B b
|
||||
incrPullsOut [] [] = refl
|
||||
incrPullsOut [] (zero :: b) = refl
|
||||
incrPullsOut [] (one :: b) = refl
|
||||
incrPullsOut (zero :: a) [] = refl
|
||||
incrPullsOut (zero :: a) (zero :: b) = refl
|
||||
incrPullsOut (zero :: a) (one :: b) = refl
|
||||
incrPullsOut (one :: a) [] = refl
|
||||
incrPullsOut (one :: a) (zero :: b) = applyEquality (zero ::_) (incrPullsOut a b)
|
||||
incrPullsOut (one :: a) (one :: b) = applyEquality (one ::_) (incrPullsOut a b)
|
||||
incrPullsOut : (a b : BinNat) → incr (a +B b) ≡ (incr a) +B b
|
||||
incrPullsOut [] [] = refl
|
||||
incrPullsOut [] (zero :: b) = refl
|
||||
incrPullsOut [] (one :: b) = refl
|
||||
incrPullsOut (zero :: a) [] = refl
|
||||
incrPullsOut (zero :: a) (zero :: b) = refl
|
||||
incrPullsOut (zero :: a) (one :: b) = refl
|
||||
incrPullsOut (one :: a) [] = refl
|
||||
incrPullsOut (one :: a) (zero :: b) = applyEquality (zero ::_) (incrPullsOut a b)
|
||||
incrPullsOut (one :: a) (one :: b) = applyEquality (one ::_) (incrPullsOut a b)
|
||||
|
||||
timesZero : (a b : BinNat) → canonical a ≡ [] → canonical (a *B b) ≡ []
|
||||
timesZero [] b pr = refl
|
||||
timesZero (zero :: a) b pr with inspect (canonical a)
|
||||
timesZero (zero :: a) b pr | [] with≡ prA rewrite prA | timesZero a b prA = refl
|
||||
timesZero (zero :: a) b pr | (a1 :: as) with≡ prA rewrite prA = exFalso (nonEmptyNotEmpty pr)
|
||||
timesZero : (a b : BinNat) → canonical a ≡ [] → canonical (a *B b) ≡ []
|
||||
timesZero [] b pr = refl
|
||||
timesZero (zero :: a) b pr with inspect (canonical a)
|
||||
timesZero (zero :: a) b pr | [] with≡ prA rewrite prA | timesZero a b prA = refl
|
||||
timesZero (zero :: a) b pr | (a1 :: as) with≡ prA rewrite prA = exFalso (nonEmptyNotEmpty pr)
|
||||
|
||||
timesZero'' : (a b : BinNat) → canonical (a *B b) ≡ [] → (canonical a ≡ []) || (canonical b ≡ [])
|
||||
timesZero'' [] b pr = inl refl
|
||||
timesZero'' (x :: a) [] pr = inr refl
|
||||
timesZero'' (zero :: as) (zero :: bs) pr with inspect (canonical as)
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | y with≡ x with inspect (canonical bs)
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | [] with≡ prAs | y₁ with≡ prBs rewrite prAs = inl refl
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | [] with≡ prBs rewrite prBs = inr refl
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (x :: y) with≡ prBs with inspect (canonical (as *B (zero :: bs)))
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (b1 :: b's) with≡ prBs | [] with≡ pr' with timesZero'' as (zero :: bs) pr'
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (b1 :: b's) with≡ prBs | [] with≡ pr' | inl x rewrite prAs | prBs | pr' | x = exFalso (nonEmptyNotEmpty x)
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (b1 :: b's) with≡ prBs | [] with≡ pr' | inr x rewrite prBs | pr' | x = exFalso (nonEmptyNotEmpty x)
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (b1 :: b's) with≡ prBs | (x :: y) with≡ pr' rewrite prAs | prBs | pr' = exFalso (nonEmptyNotEmpty pr)
|
||||
timesZero'' (zero :: as) (one :: bs) pr with inspect (canonical as)
|
||||
timesZero'' (zero :: as) (one :: bs) pr | [] with≡ x rewrite x = inl refl
|
||||
timesZero'' (zero :: as) (one :: bs) pr | (a1 :: a's) with≡ x with inspect (canonical (as *B (one :: bs)))
|
||||
timesZero'' (zero :: as) (one :: bs) pr | (a1 :: a's) with≡ x | [] with≡ pr' with timesZero'' as (one :: bs) pr'
|
||||
timesZero'' (zero :: as) (one :: bs) pr | (a1 :: a's) with≡ x | [] with≡ pr' | inl pr'' rewrite x | pr' | pr'' = exFalso (nonEmptyNotEmpty pr'')
|
||||
timesZero'' (zero :: as) (one :: bs) pr | (a1 :: a's) with≡ x | (x₁ :: y) with≡ pr' rewrite x | pr' = exFalso (nonEmptyNotEmpty pr)
|
||||
timesZero'' (one :: as) (zero :: bs) pr with inspect (canonical bs)
|
||||
timesZero'' (one :: as) (zero :: bs) pr | [] with≡ x rewrite x = inr refl
|
||||
timesZero'' (one :: as) (zero :: bs) pr | (b1 :: b's) with≡ prB with inspect (canonical ((as *B (zero :: bs)) +B bs))
|
||||
timesZero'' (one :: as) (zero :: bs) pr | (b1 :: b's) with≡ prB | [] with≡ x rewrite equalityCommutative (+BIsInherited' (as *B (zero :: bs)) bs) = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prB) v))
|
||||
where
|
||||
t : binNatToN (as *B (zero :: bs)) +N binNatToN bs ≡ 0
|
||||
t = transitivity (equalityCommutative (nToN _)) (applyEquality binNatToN x)
|
||||
u : (binNatToN (as *B (zero :: bs)) ≡ 0) && (binNatToN bs ≡ 0)
|
||||
u = sumZeroImpliesSummandsZero t
|
||||
v : canonical bs ≡ []
|
||||
v with u
|
||||
... | fst ,, snd = binNatToNZero bs snd
|
||||
timesZero'' (one :: as) (zero :: bs) pr | (b1 :: b's) with≡ prB | (x₁ :: y) with≡ x rewrite prB | x = exFalso (nonEmptyNotEmpty pr)
|
||||
timesZero'' : (a b : BinNat) → canonical (a *B b) ≡ [] → (canonical a ≡ []) || (canonical b ≡ [])
|
||||
timesZero'' [] b pr = inl refl
|
||||
timesZero'' (x :: a) [] pr = inr refl
|
||||
timesZero'' (zero :: as) (zero :: bs) pr with inspect (canonical as)
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | y with≡ x with inspect (canonical bs)
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | [] with≡ prAs | y₁ with≡ prBs rewrite prAs = inl refl
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | [] with≡ prBs rewrite prBs = inr refl
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (x :: y) with≡ prBs with inspect (canonical (as *B (zero :: bs)))
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (b1 :: b's) with≡ prBs | [] with≡ pr' with timesZero'' as (zero :: bs) pr'
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (b1 :: b's) with≡ prBs | [] with≡ pr' | inl x rewrite prAs | prBs | pr' | x = exFalso (nonEmptyNotEmpty x)
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (b1 :: b's) with≡ prBs | [] with≡ pr' | inr x rewrite prBs | pr' | x = exFalso (nonEmptyNotEmpty x)
|
||||
timesZero'' (zero :: as) (zero :: bs) pr | (a1 :: a's) with≡ prAs | (b1 :: b's) with≡ prBs | (x :: y) with≡ pr' rewrite prAs | prBs | pr' = exFalso (nonEmptyNotEmpty pr)
|
||||
timesZero'' (zero :: as) (one :: bs) pr with inspect (canonical as)
|
||||
timesZero'' (zero :: as) (one :: bs) pr | [] with≡ x rewrite x = inl refl
|
||||
timesZero'' (zero :: as) (one :: bs) pr | (a1 :: a's) with≡ x with inspect (canonical (as *B (one :: bs)))
|
||||
timesZero'' (zero :: as) (one :: bs) pr | (a1 :: a's) with≡ x | [] with≡ pr' with timesZero'' as (one :: bs) pr'
|
||||
timesZero'' (zero :: as) (one :: bs) pr | (a1 :: a's) with≡ x | [] with≡ pr' | inl pr'' rewrite x | pr' | pr'' = exFalso (nonEmptyNotEmpty pr'')
|
||||
timesZero'' (zero :: as) (one :: bs) pr | (a1 :: a's) with≡ x | (x₁ :: y) with≡ pr' rewrite x | pr' = exFalso (nonEmptyNotEmpty pr)
|
||||
timesZero'' (one :: as) (zero :: bs) pr with inspect (canonical bs)
|
||||
timesZero'' (one :: as) (zero :: bs) pr | [] with≡ x rewrite x = inr refl
|
||||
timesZero'' (one :: as) (zero :: bs) pr | (b1 :: b's) with≡ prB with inspect (canonical ((as *B (zero :: bs)) +B bs))
|
||||
timesZero'' (one :: as) (zero :: bs) pr | (b1 :: b's) with≡ prB | [] with≡ x rewrite equalityCommutative (+BIsInherited' (as *B (zero :: bs)) bs) = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prB) v))
|
||||
where
|
||||
t : binNatToN (as *B (zero :: bs)) +N binNatToN bs ≡ 0
|
||||
t = transitivity (equalityCommutative (nToN _)) (applyEquality binNatToN x)
|
||||
u : (binNatToN (as *B (zero :: bs)) ≡ 0) && (binNatToN bs ≡ 0)
|
||||
u = sumZeroImpliesSummandsZero t
|
||||
v : canonical bs ≡ []
|
||||
v with u
|
||||
... | fst ,, snd = binNatToNZero bs snd
|
||||
timesZero'' (one :: as) (zero :: bs) pr | (b1 :: b's) with≡ prB | (x₁ :: y) with≡ x rewrite prB | x = exFalso (nonEmptyNotEmpty pr)
|
||||
|
||||
lemma : {i : Bit} → (a b : BinNat) → canonical a ≡ canonical b → canonical (i :: a) ≡ canonical (i :: b)
|
||||
lemma {zero} a b pr with inspect (canonical a)
|
||||
lemma {zero} a b pr | [] with≡ x rewrite x | equalityCommutative pr = refl
|
||||
lemma {zero} a b pr | (a1 :: as) with≡ x rewrite x | equalityCommutative pr = refl
|
||||
lemma {one} a b pr = applyEquality (one ::_) pr
|
||||
lemma : {i : Bit} → (a b : BinNat) → canonical a ≡ canonical b → canonical (i :: a) ≡ canonical (i :: b)
|
||||
lemma {zero} a b pr with inspect (canonical a)
|
||||
lemma {zero} a b pr | [] with≡ x rewrite x | equalityCommutative pr = refl
|
||||
lemma {zero} a b pr | (a1 :: as) with≡ x rewrite x | equalityCommutative pr = refl
|
||||
lemma {one} a b pr = applyEquality (one ::_) pr
|
||||
|
||||
binNatToNDistributesPlus : (a b : BinNat) → binNatToN (a +B b) ≡ binNatToN a +N binNatToN b
|
||||
binNatToNDistributesPlus a b = transitivity (equalityCommutative (binNatToNIsCanonical (a +B b))) (transitivity (applyEquality binNatToN (equalityCommutative (+BIsInherited' a b))) (nToN (binNatToN a +N binNatToN b)))
|
||||
binNatToNDistributesPlus : (a b : BinNat) → binNatToN (a +B b) ≡ binNatToN a +N binNatToN b
|
||||
binNatToNDistributesPlus a b = transitivity (equalityCommutative (binNatToNIsCanonical (a +B b))) (transitivity (applyEquality binNatToN (equalityCommutative (+BIsInherited' a b))) (nToN (binNatToN a +N binNatToN b)))
|
||||
|
||||
+BAssociative : (a b c : BinNat) → canonical (a +B (b +B c)) ≡ canonical ((a +B b) +B c)
|
||||
+BAssociative a b c rewrite equalityCommutative (+BIsInherited' a (b +B c)) | equalityCommutative (+BIsInherited' (a +B b) c) | binNatToNDistributesPlus a b | binNatToNDistributesPlus b c | equalityCommutative (Semiring.+Associative ℕSemiring (binNatToN a) (binNatToN b) (binNatToN c)) = refl
|
||||
+BAssociative : (a b c : BinNat) → canonical (a +B (b +B c)) ≡ canonical ((a +B b) +B c)
|
||||
+BAssociative a b c rewrite equalityCommutative (+BIsInherited' a (b +B c)) | equalityCommutative (+BIsInherited' (a +B b) c) | binNatToNDistributesPlus a b | binNatToNDistributesPlus b c | equalityCommutative (Semiring.+Associative ℕSemiring (binNatToN a) (binNatToN b) (binNatToN c)) = refl
|
||||
|
||||
timesOne : (a b : BinNat) → (canonical b) ≡ (one :: []) → canonical (a *B b) ≡ canonical a
|
||||
timesOne [] b pr = refl
|
||||
timesOne (zero :: a) b pr with inspect (canonical (a *B b))
|
||||
timesOne (zero :: a) b pr | [] with≡ prAB with timesZero'' a b prAB
|
||||
timesOne (zero :: a) b pr | [] with≡ prAB | inl a=0 rewrite a=0 | prAB = refl
|
||||
timesOne (zero :: a) b pr | [] with≡ prAB | inr b=0 = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative pr) b=0))
|
||||
timesOne (zero :: a) b pr | (ab1 :: abs) with≡ prAB with inspect (canonical a)
|
||||
timesOne (zero :: a) b pr | (ab1 :: abs) with≡ prAB | [] with≡ x = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prAB) (timesZero a b x)))
|
||||
timesOne (zero :: a) b pr | (ab1 :: abs) with≡ prAB | (x₁ :: y) with≡ x rewrite prAB | x | timesOne a b pr = applyEquality (zero ::_) (transitivity (equalityCommutative prAB) x)
|
||||
timesOne (one :: a) (zero :: b) pr with canonical b
|
||||
timesOne (one :: a) (zero :: b) () | []
|
||||
timesOne (one :: a) (zero :: b) () | x :: bl
|
||||
timesOne (one :: a) (one :: b) pr rewrite canonicalDistributesPlus (a *B (one :: b)) b | ::Inj pr | +BCommutative (canonical (a *B (one :: b))) [] | timesOne a (one :: b) pr = refl
|
||||
timesOne : (a b : BinNat) → (canonical b) ≡ (one :: []) → canonical (a *B b) ≡ canonical a
|
||||
timesOne [] b pr = refl
|
||||
timesOne (zero :: a) b pr with inspect (canonical (a *B b))
|
||||
timesOne (zero :: a) b pr | [] with≡ prAB with timesZero'' a b prAB
|
||||
timesOne (zero :: a) b pr | [] with≡ prAB | inl a=0 rewrite a=0 | prAB = refl
|
||||
timesOne (zero :: a) b pr | [] with≡ prAB | inr b=0 = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative pr) b=0))
|
||||
timesOne (zero :: a) b pr | (ab1 :: abs) with≡ prAB with inspect (canonical a)
|
||||
timesOne (zero :: a) b pr | (ab1 :: abs) with≡ prAB | [] with≡ x = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prAB) (timesZero a b x)))
|
||||
timesOne (zero :: a) b pr | (ab1 :: abs) with≡ prAB | (x₁ :: y) with≡ x rewrite prAB | x | timesOne a b pr = applyEquality (zero ::_) (transitivity (equalityCommutative prAB) x)
|
||||
timesOne (one :: a) (zero :: b) pr with canonical b
|
||||
timesOne (one :: a) (zero :: b) () | []
|
||||
timesOne (one :: a) (zero :: b) () | x :: bl
|
||||
timesOne (one :: a) (one :: b) pr rewrite canonicalDistributesPlus (a *B (one :: b)) b | ::Inj pr | +BCommutative (canonical (a *B (one :: b))) [] | timesOne a (one :: b) pr = refl
|
||||
|
||||
timesZero' : (a b : BinNat) → canonical b ≡ [] → canonical (a *B b) ≡ []
|
||||
timesZero' [] b pr = refl
|
||||
timesZero' (zero :: a) b pr with inspect (canonical (a *B b))
|
||||
timesZero' (zero :: a) b pr | [] with≡ x rewrite x = refl
|
||||
timesZero' (zero :: a) b pr | (ab1 :: abs) with≡ prAB rewrite prAB = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prAB) (timesZero' a b pr)))
|
||||
timesZero' (one :: a) b pr rewrite canonicalDistributesPlus (zero :: (a *B b)) b | pr | +BCommutative (canonical (zero :: (a *B b))) [] = ans
|
||||
where
|
||||
ans : canonical (zero :: (a *B b)) ≡ []
|
||||
ans with inspect (canonical (a *B b))
|
||||
ans | [] with≡ x rewrite x = refl
|
||||
ans | (x₁ :: y) with≡ x rewrite x = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative x) (timesZero' a b pr)))
|
||||
timesZero' : (a b : BinNat) → canonical b ≡ [] → canonical (a *B b) ≡ []
|
||||
timesZero' [] b pr = refl
|
||||
timesZero' (zero :: a) b pr with inspect (canonical (a *B b))
|
||||
timesZero' (zero :: a) b pr | [] with≡ x rewrite x = refl
|
||||
timesZero' (zero :: a) b pr | (ab1 :: abs) with≡ prAB rewrite prAB = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prAB) (timesZero' a b pr)))
|
||||
timesZero' (one :: a) b pr rewrite canonicalDistributesPlus (zero :: (a *B b)) b | pr | +BCommutative (canonical (zero :: (a *B b))) [] = ans
|
||||
where
|
||||
ans : canonical (zero :: (a *B b)) ≡ []
|
||||
ans with inspect (canonical (a *B b))
|
||||
ans | [] with≡ x rewrite x = refl
|
||||
ans | (x₁ :: y) with≡ x rewrite x = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative x) (timesZero' a b pr)))
|
||||
|
||||
canonicalDistributesTimes : (a b : BinNat) → canonical (a *B b) ≡ canonical ((canonical a) *B (canonical b))
|
||||
canonicalDistributesTimes [] b = refl
|
||||
canonicalDistributesTimes (zero :: a) b with inspect (canonical a)
|
||||
canonicalDistributesTimes (zero :: a) b | [] with≡ x rewrite timesZero a b x | x = refl
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA with inspect (canonical b)
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | [] with≡ prB rewrite prA | prB = ans
|
||||
where
|
||||
ans : canonical (zero :: (a *B b)) ≡ canonical (zero :: ((a1 :: as) *B []))
|
||||
ans with inspect (canonical ((a1 :: as) *B []))
|
||||
ans | [] with≡ x rewrite x | timesZero' a b prB = refl
|
||||
ans | (x₁ :: y) with≡ x = exFalso (nonEmptyNotEmpty (equalityCommutative (transitivity (equalityCommutative (timesZero' (a1 :: as) [] refl)) x)))
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB with inspect (canonical (a *B b))
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB | [] with≡ x with timesZero'' a b x
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB | [] with≡ x | inl a=0 = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prA) a=0))
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB | [] with≡ x | inr b=0 = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prB) b=0))
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB | (ab1 :: abs) with≡ x rewrite prA | prB | x = ans
|
||||
where
|
||||
ans : zero :: ab1 :: abs ≡ canonical (zero :: ((a1 :: as) *B (b1 :: bs)))
|
||||
ans rewrite equalityCommutative prA | equalityCommutative prB | equalityCommutative (canonicalDistributesTimes a b) | x = refl
|
||||
canonicalDistributesTimes (one :: a) b = transitivity (canonicalDistributesPlus (zero :: (a *B b)) b) (transitivity (transitivity (applyEquality (_+B canonical b) (transitivity (equalityCommutative (canonicalAscends'' (a *B b))) (transitivity (applyEquality (λ i → canonical (zero :: i)) (canonicalDistributesTimes a b)) (canonicalAscends'' (canonical a *B canonical b))))) (applyEquality (λ i → canonical (zero :: (canonical a *B canonical b)) +B i) (canonicalIdempotent b))) (equalityCommutative (canonicalDistributesPlus (zero :: (canonical a *B canonical b)) (canonical b))))
|
||||
canonicalDistributesTimes : (a b : BinNat) → canonical (a *B b) ≡ canonical ((canonical a) *B (canonical b))
|
||||
canonicalDistributesTimes [] b = refl
|
||||
canonicalDistributesTimes (zero :: a) b with inspect (canonical a)
|
||||
canonicalDistributesTimes (zero :: a) b | [] with≡ x rewrite timesZero a b x | x = refl
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA with inspect (canonical b)
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | [] with≡ prB rewrite prA | prB = ans
|
||||
where
|
||||
ans : canonical (zero :: (a *B b)) ≡ canonical (zero :: ((a1 :: as) *B []))
|
||||
ans with inspect (canonical ((a1 :: as) *B []))
|
||||
ans | [] with≡ x rewrite x | timesZero' a b prB = refl
|
||||
ans | (x₁ :: y) with≡ x = exFalso (nonEmptyNotEmpty (equalityCommutative (transitivity (equalityCommutative (timesZero' (a1 :: as) [] refl)) x)))
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB with inspect (canonical (a *B b))
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB | [] with≡ x with timesZero'' a b x
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB | [] with≡ x | inl a=0 = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prA) a=0))
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB | [] with≡ x | inr b=0 = exFalso (nonEmptyNotEmpty (transitivity (equalityCommutative prB) b=0))
|
||||
canonicalDistributesTimes (zero :: a) b | (a1 :: as) with≡ prA | (b1 :: bs) with≡ prB | (ab1 :: abs) with≡ x rewrite prA | prB | x = ans
|
||||
where
|
||||
ans : zero :: ab1 :: abs ≡ canonical (zero :: ((a1 :: as) *B (b1 :: bs)))
|
||||
ans rewrite equalityCommutative prA | equalityCommutative prB | equalityCommutative (canonicalDistributesTimes a b) | x = refl
|
||||
canonicalDistributesTimes (one :: a) b = transitivity (canonicalDistributesPlus (zero :: (a *B b)) b) (transitivity (transitivity (applyEquality (_+B canonical b) (transitivity (equalityCommutative (canonicalAscends'' (a *B b))) (transitivity (applyEquality (λ i → canonical (zero :: i)) (canonicalDistributesTimes a b)) (canonicalAscends'' (canonical a *B canonical b))))) (applyEquality (λ i → canonical (zero :: (canonical a *B canonical b)) +B i) (canonicalIdempotent b))) (equalityCommutative (canonicalDistributesPlus (zero :: (canonical a *B canonical b)) (canonical b))))
|
||||
|
||||
NToBinNatDistributesPlus : (a b : ℕ) → NToBinNat (a +N b) ≡ NToBinNat a +B NToBinNat b
|
||||
NToBinNatDistributesPlus zero b = refl
|
||||
NToBinNatDistributesPlus (succ a) b with inspect (NToBinNat a)
|
||||
... | bl with≡ prA with inspect (NToBinNat (a +N b))
|
||||
... | q with≡ prAB = transitivity (applyEquality incr (NToBinNatDistributesPlus a b)) (incrPullsOut (NToBinNat a) (NToBinNat b))
|
||||
NToBinNatDistributesPlus : (a b : ℕ) → NToBinNat (a +N b) ≡ NToBinNat a +B NToBinNat b
|
||||
NToBinNatDistributesPlus zero b = refl
|
||||
NToBinNatDistributesPlus (succ a) b with inspect (NToBinNat a)
|
||||
... | bl with≡ prA with inspect (NToBinNat (a +N b))
|
||||
... | q with≡ prAB = transitivity (applyEquality incr (NToBinNatDistributesPlus a b)) (incrPullsOut (NToBinNat a) (NToBinNat b))
|
||||
|
||||
timesCommutative : (a b : BinNat) → canonical (a *B b) ≡ canonical (b *B a)
|
||||
timesCommLemma : (a b : BinNat) → canonical (zero :: (b *B a)) ≡ canonical (b *B (zero :: a))
|
||||
timesCommLemma a b rewrite timesCommutative b (zero :: a) | equalityCommutative (canonicalAscends'' {zero} (b *B a)) | equalityCommutative (canonicalAscends'' {zero} (a *B b)) | timesCommutative b a = refl
|
||||
timesCommutative : (a b : BinNat) → canonical (a *B b) ≡ canonical (b *B a)
|
||||
timesCommLemma : (a b : BinNat) → canonical (zero :: (b *B a)) ≡ canonical (b *B (zero :: a))
|
||||
timesCommLemma a b rewrite timesCommutative b (zero :: a) | equalityCommutative (canonicalAscends'' {zero} (b *B a)) | equalityCommutative (canonicalAscends'' {zero} (a *B b)) | timesCommutative b a = refl
|
||||
|
||||
timesCommutative [] b rewrite timesZero' b [] refl = refl
|
||||
timesCommutative (x :: a) [] rewrite timesZero' (x :: a) [] refl = refl
|
||||
timesCommutative (zero :: as) (zero :: b) rewrite equalityCommutative (canonicalAscends'' {zero} (as *B (zero :: b))) | timesCommutative as (zero :: b) | canonicalAscends'' {zero} (zero :: b *B as) | equalityCommutative (canonicalAscends'' {zero} (b *B (zero :: as))) | timesCommutative b (zero :: as) | canonicalAscends'' {zero} (zero :: (as *B b)) = ans
|
||||
where
|
||||
ans : canonical (zero :: zero :: (b *B as)) ≡ canonical (zero :: zero :: (as *B b))
|
||||
ans rewrite equalityCommutative (canonicalAscends'' {zero} (zero :: (b *B as))) | equalityCommutative (canonicalAscends'' {zero} (b *B as)) | timesCommutative b as | canonicalAscends'' {zero} (as *B b) | canonicalAscends'' {zero} (zero :: (as *B b)) = refl
|
||||
timesCommutative (zero :: as) (one :: b) = transitivity (equalityCommutative (canonicalAscends'' (as *B (one :: b)))) (transitivity (applyEquality (λ i → canonical (zero :: i)) (timesCommutative as (one :: b))) (transitivity (applyEquality (λ i → canonical (zero :: i)) ans) (canonicalAscends'' ((b *B (zero :: as)) +B as))))
|
||||
where
|
||||
ans : canonical ((zero :: (b *B as)) +B as) ≡ canonical ((b *B (zero :: as)) +B as)
|
||||
ans rewrite canonicalDistributesPlus (zero :: (b *B as)) as | canonicalDistributesPlus (b *B (zero :: as)) as = applyEquality (_+B canonical as) (timesCommLemma as b)
|
||||
timesCommutative (one :: as) (zero :: bs) = transitivity (equalityCommutative (canonicalAscends'' ((as *B (zero :: bs)) +B bs))) (transitivity (applyEquality (λ i → canonical (zero :: i)) ans) (canonicalAscends'' (bs *B (one :: as))))
|
||||
where
|
||||
ans : canonical ((as *B (zero :: bs)) +B bs) ≡ canonical (bs *B (one :: as))
|
||||
ans rewrite timesCommutative bs (one :: as) | canonicalDistributesPlus (as *B (zero :: bs)) bs | canonicalDistributesPlus (zero :: (as *B bs)) bs = applyEquality (_+B canonical bs) (equalityCommutative (timesCommLemma bs as))
|
||||
timesCommutative (one :: as) (one :: bs) = applyEquality (one ::_) (transitivity (canonicalDistributesPlus (as *B (one :: bs)) bs) (transitivity (transitivity (applyEquality (_+B canonical bs) (timesCommutative as (one :: bs))) (transitivity ans (equalityCommutative (applyEquality (_+B canonical as) (timesCommutative bs (one :: as)))))) (equalityCommutative (canonicalDistributesPlus (bs *B (one :: as)) as))))
|
||||
where
|
||||
ans : canonical ((zero :: (bs *B as)) +B as) +B canonical bs ≡ canonical ((zero :: (as *B bs)) +B bs) +B canonical as
|
||||
ans rewrite equalityCommutative (canonicalDistributesPlus ((zero :: (bs *B as)) +B as) bs) | equalityCommutative (canonicalDistributesPlus ((zero :: (as *B bs)) +B bs) as) | equalityCommutative (+BAssociative (zero :: (bs *B as)) as bs) | equalityCommutative (+BAssociative (zero :: (as *B bs)) bs as) | canonicalDistributesPlus (zero :: (bs *B as)) (as +B bs) | canonicalDistributesPlus (zero :: (as *B bs)) (bs +B as) | equalityCommutative (canonicalAscends'' {zero} (bs *B as)) | timesCommutative bs as | canonicalAscends'' {zero} (as *B bs) | +BCommutative as bs = refl
|
||||
timesCommutative [] b rewrite timesZero' b [] refl = refl
|
||||
timesCommutative (x :: a) [] rewrite timesZero' (x :: a) [] refl = refl
|
||||
timesCommutative (zero :: as) (zero :: b) rewrite equalityCommutative (canonicalAscends'' {zero} (as *B (zero :: b))) | timesCommutative as (zero :: b) | canonicalAscends'' {zero} (zero :: b *B as) | equalityCommutative (canonicalAscends'' {zero} (b *B (zero :: as))) | timesCommutative b (zero :: as) | canonicalAscends'' {zero} (zero :: (as *B b)) = ans
|
||||
where
|
||||
ans : canonical (zero :: zero :: (b *B as)) ≡ canonical (zero :: zero :: (as *B b))
|
||||
ans rewrite equalityCommutative (canonicalAscends'' {zero} (zero :: (b *B as))) | equalityCommutative (canonicalAscends'' {zero} (b *B as)) | timesCommutative b as | canonicalAscends'' {zero} (as *B b) | canonicalAscends'' {zero} (zero :: (as *B b)) = refl
|
||||
timesCommutative (zero :: as) (one :: b) = transitivity (equalityCommutative (canonicalAscends'' (as *B (one :: b)))) (transitivity (applyEquality (λ i → canonical (zero :: i)) (timesCommutative as (one :: b))) (transitivity (applyEquality (λ i → canonical (zero :: i)) ans) (canonicalAscends'' ((b *B (zero :: as)) +B as))))
|
||||
where
|
||||
ans : canonical ((zero :: (b *B as)) +B as) ≡ canonical ((b *B (zero :: as)) +B as)
|
||||
ans rewrite canonicalDistributesPlus (zero :: (b *B as)) as | canonicalDistributesPlus (b *B (zero :: as)) as = applyEquality (_+B canonical as) (timesCommLemma as b)
|
||||
timesCommutative (one :: as) (zero :: bs) = transitivity (equalityCommutative (canonicalAscends'' ((as *B (zero :: bs)) +B bs))) (transitivity (applyEquality (λ i → canonical (zero :: i)) ans) (canonicalAscends'' (bs *B (one :: as))))
|
||||
where
|
||||
ans : canonical ((as *B (zero :: bs)) +B bs) ≡ canonical (bs *B (one :: as))
|
||||
ans rewrite timesCommutative bs (one :: as) | canonicalDistributesPlus (as *B (zero :: bs)) bs | canonicalDistributesPlus (zero :: (as *B bs)) bs = applyEquality (_+B canonical bs) (equalityCommutative (timesCommLemma bs as))
|
||||
timesCommutative (one :: as) (one :: bs) = applyEquality (one ::_) (transitivity (canonicalDistributesPlus (as *B (one :: bs)) bs) (transitivity (transitivity (applyEquality (_+B canonical bs) (timesCommutative as (one :: bs))) (transitivity ans (equalityCommutative (applyEquality (_+B canonical as) (timesCommutative bs (one :: as)))))) (equalityCommutative (canonicalDistributesPlus (bs *B (one :: as)) as))))
|
||||
where
|
||||
ans : canonical ((zero :: (bs *B as)) +B as) +B canonical bs ≡ canonical ((zero :: (as *B bs)) +B bs) +B canonical as
|
||||
ans rewrite equalityCommutative (canonicalDistributesPlus ((zero :: (bs *B as)) +B as) bs) | equalityCommutative (canonicalDistributesPlus ((zero :: (as *B bs)) +B bs) as) | equalityCommutative (+BAssociative (zero :: (bs *B as)) as bs) | equalityCommutative (+BAssociative (zero :: (as *B bs)) bs as) | canonicalDistributesPlus (zero :: (bs *B as)) (as +B bs) | canonicalDistributesPlus (zero :: (as *B bs)) (bs +B as) | equalityCommutative (canonicalAscends'' {zero} (bs *B as)) | timesCommutative bs as | canonicalAscends'' {zero} (as *B bs) | +BCommutative as bs = refl
|
||||
|
||||
*BIsInherited : (a b : BinNat) → a *Binherit b ≡ canonical (a *B b)
|
||||
*BIsInherited a b = transitivity (applyEquality NToBinNat t) (transitivity (binToBin (canonical (a *B b))) (equalityCommutative (canonicalIdempotent (a *B b))))
|
||||
where
|
||||
ans : (a b : BinNat) → binNatToN a *N binNatToN b ≡ binNatToN (a *B b)
|
||||
ans [] b = refl
|
||||
ans (zero :: a) b rewrite equalityCommutative (ans a b) = equalityCommutative (Semiring.*Associative ℕSemiring 2 (binNatToN a) (binNatToN b))
|
||||
ans (one :: a) b rewrite binNatToNDistributesPlus (zero :: (a *B b)) b | Semiring.commutative ℕSemiring (binNatToN b) ((2 *N (binNatToN a)) *N (binNatToN b)) | equalityCommutative (ans a b) = applyEquality (_+N binNatToN b) (equalityCommutative (Semiring.*Associative ℕSemiring 2 (binNatToN a) (binNatToN b)))
|
||||
t : (binNatToN a *N binNatToN b) ≡ binNatToN (canonical (a *B b))
|
||||
t = transitivity (ans a b) (equalityCommutative (binNatToNIsCanonical (a *B b)))
|
||||
*BIsInherited : (a b : BinNat) → a *Binherit b ≡ canonical (a *B b)
|
||||
*BIsInherited a b = transitivity (applyEquality NToBinNat t) (transitivity (binToBin (canonical (a *B b))) (equalityCommutative (canonicalIdempotent (a *B b))))
|
||||
where
|
||||
ans : (a b : BinNat) → binNatToN a *N binNatToN b ≡ binNatToN (a *B b)
|
||||
ans [] b = refl
|
||||
ans (zero :: a) b rewrite equalityCommutative (ans a b) = equalityCommutative (Semiring.*Associative ℕSemiring 2 (binNatToN a) (binNatToN b))
|
||||
ans (one :: a) b rewrite binNatToNDistributesPlus (zero :: (a *B b)) b | Semiring.commutative ℕSemiring (binNatToN b) ((2 *N (binNatToN a)) *N (binNatToN b)) | equalityCommutative (ans a b) = applyEquality (_+N binNatToN b) (equalityCommutative (Semiring.*Associative ℕSemiring 2 (binNatToN a) (binNatToN b)))
|
||||
t : (binNatToN a *N binNatToN b) ≡ binNatToN (canonical (a *B b))
|
||||
t = transitivity (ans a b) (equalityCommutative (binNatToNIsCanonical (a *B b)))
|
||||
|
||||
Reference in New Issue
Block a user