Please help。
解决方案这是@gallais说的，你可以明确地使用有根据的递归，但我不喜欢这种方法，因为它是完全不可读的。数据类型
record是{α} {A：Setα}（x：A）：Setαwhere $ b $b¡ = x 打开是 ！ ：∀{α} {A：Setα} - > （x：A）→>是x ！ _ = _允许将值提升到类型级别，例如，您可以定义类型安全 pred 函数：
pred + +：∀{n} - >是（成功） - > ℕ pred + = pred∘$然后
test-1：pred +（！1）≡0 test-1 = refl typechecks，而 失败：pred +（！0） ≡0 fail = refl没有。可以用相同的方式定义减数与正减数（以确保充分发现）：
_- +：：∀ {m} - > ℕ - >是（成功） - > ℕn-+ im = n∸im im然后使用我描述的这里，你可以重复减去另一个数字，直到差值小于第二个数字：
lem：∀{nm} {im：Is（suc m）} - > m < n - > n + 1 im n-+ im）< -well-founded lem（_≤？_（im））n 例如 test-1 ：iter-sub 10（！3）≡10∷7∷4∷[] test-1 = refl test-2：iter-sub 16（！4）≡16∷16∷ 12∷8∷4∷[] test-2 = refldiv + 然后就是
_div +_：∀{m} - > ℕ - >是（成功） - > ℕn div + im = length（iter-sub n im）和版本类似到 Data.Nat.DivMod 模块中的模块（仅包含 Mod 部分）：
_div_：ℕ - > （m：ℕ）{_：False（m≟0）} - > ℕn div 0 =λ{（）} n div（suc m）= n div +（！（suc m））一些测试：
test-3：map（λn - > n b $ b（0∷1∷2∷3∷4∷5∷6∷7∷8∷9∷[]）≡（0∷0∷0∷1∷1∷1∷ 2∷2∷2∷3∷[]） test-3 = refl注但是，标准库中的版本还包含健全性证明：
属性：dividend≡toℕremaining + quotient * divisor整个代码。
I want to divide two natural number. I have made function like this
_/_ : N -> N -> frac m / one = m / one (suc m) / n = ?? I dont know what to write here.Please help.
解决方案As @gallais says you can use well-founded recursion explicitly, but I don't like this approach, because it's totally unreadable.
This datatype
record Is {α} {A : Set α} (x : A) : Set α where ¡ = x open Is ! : ∀ {α} {A : Set α} -> (x : A) -> Is x ! _ = _allows to lift values to the type level, for example you can define a type-safe pred function:
pred⁺ : ∀ {n} -> Is (suc n) -> ℕ pred⁺ = pred ∘ ¡Then
test-1 : pred⁺ (! 1) ≡ 0 test-1 = refltypechecks, while
fail : pred⁺ (! 0) ≡ 0 fail = refldoesn't. It's possible to define subtraction with positive subtrahend (to ensure well-foundness) in the same way:
_-⁺_ : ∀ {m} -> ℕ -> Is (suc m) -> ℕ n -⁺ im = n ∸ ¡ imThen using stuff that I described here, you can repeatedly subtract one number from another until the difference is smaller than the second number:
lem : ∀ {n m} {im : Is (suc m)} -> m < n -> n -⁺ im <′ n lem {suc n} {m} (s≤s _) = s≤′s (≤⇒≤′ (n∸m≤n m n)) iter-sub : ∀ {m} -> ℕ -> Is (suc m) -> List ℕ iter-sub n im = calls (λ n -> n -⁺ im) <-well-founded lem (_≤?_ (¡ im)) nFor example
test-1 : iter-sub 10 (! 3) ≡ 10 ∷ 7 ∷ 4 ∷ [] test-1 = refl test-2 : iter-sub 16 (! 4) ≡ 16 ∷ 12 ∷ 8 ∷ 4 ∷ [] test-2 = refldiv⁺ then is simply
_div⁺_ : ∀ {m} -> ℕ -> Is (suc m) -> ℕ n div⁺ im = length (iter-sub n im)And a version similar to the one in the Data.Nat.DivMod module (only without the Mod part):
_div_ : ℕ -> (m : ℕ) {_ : False (m ≟ 0)} -> ℕ n div 0 = λ{()} n div (suc m) = n div⁺ (! (suc m))Some tests:
test-3 : map (λ n -> n div 3) (0 ∷ 1 ∷ 2 ∷ 3 ∷ 4 ∷ 5 ∷ 6 ∷ 7 ∷ 8 ∷ 9 ∷ []) ≡ (0 ∷ 0 ∷ 0 ∷ 1 ∷ 1 ∷ 1 ∷ 2 ∷ 2 ∷ 2 ∷ 3 ∷ []) test-3 = reflNote however, that the version in the standard library also contains the soundness proof:
property : dividend ≡ toℕ remainder + quotient * divisorThe whole code.
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