Basic

每次我们使用 Inductive 来声明数据类型时,Coq 会自动为这个类型生成 归纳原理
Every time we declare a new Inductive datatype, Coq automatically generates an induction principle for this type.

自然数的归纳原理:

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Check nat_ind. :

∀ P : nat → Prop,
P 0
(∀ n : nat, P n -> P (S n)) →
∀ n : nat, P n

written as inference rule:

                    P 0
  ∀ n : nat, P n -> P (S n)
  -------------------------
  ∀ n : nat,        P n

induction tactic is wrapper of apply t_ind

Coq 为每一个 Inductive 定义的数据类型生成了归纳原理,包括那些非递归的
Coq generates induction principles for every datatype defined with Inductive, including those that aren’t recursive.

尽管我们不需要使用归纳来证明非递归数据类型的性质
Although of course we don’t need induction to prove properties of non-recursive datatypes. (destruct would be sufficient)

归纳原理的概念仍然适用于它们: 它是一种证明一个对于这个类型所有值都成立的性质的方法。
the idea of an induction principle still makes sense for them: it gives a way to prove that a property holds for all values of the type.

Non-recursive

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Inductive yesno : Type :=
| yes
| no.

Check yesno_ind. :
yesno_ind : ∀ P : yesno → Prop,
P yes →
P no →
∀ y : yesno, P y
             P yes 
             P no
------------------
∀ y : yesno, P y 

Structural-Recursive

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Inductive natlist : Type :=
| nnil
| ncons (n : nat) (l : natlist).

Check natlist_ind. :
natlist_ind : ∀ P : natlist → Prop,
P nnil →
(∀ (n : nat) (l : natlist), P l -> P (ncons n l)) →
∀ l : natlist, P l
                                  P nnil 
∀ (n : nat) (l : natlist), P l -> P (ncons n l)
-----------------------------------------------
∀ l : natlist,                    P l 

P only need to fullfill l : the_type but not n:nat since we are proving property of the_type.

The Pattern

These generated principles follow a similar pattern.

  • induction on each cases
  • proof by exhaustiveness?
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Inductive t : Type := 
| c1 (x1 : a1) ... (xn : an)
...
| cn ...

t_ind : ∀P : t → Prop,
... case for c1 ... →
... case for c2 ... → ...
... case for cn ... →
∀n : t, P n

对于 t 的归纳原理是又所有对于 c 的归纳原理所组成的: (即所有 case 成立)

对于 c 的归纳原理则是

对于所有的类型为 a1...an 的值 x1...xn,如果 P 对每个 归纳的参数(每个具有类型 txi)都成立,那么 P 对于 c x1 ... xn 成立”

每个具有类型 t 的参数的地方即发生了「递归」与「子结构」,归纳假设 = 「对子结构成立」.

Polymorphism

接下来考虑多态列表:

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(* in ADT syntax *)
Inductive list (X:Type) : Type :=
| nil
| cons (x : X) (l': list X)

(* in GADT syntax *)
Inductive list (X:Type) : Type :=
| nil : list X
| cons : X → list X → list X.

here, the whole def is parameterized on a set X: that is, we are defining a family of inductive types list X, one for each X.

这里,整个定义都是被集合 X 参数化的:
也即,我们定义了一个族 list : X -> Type, 对于每个 X,我们都有一个对应的: list X, which is a Type, 可写作 list X : Type.

list_ind can be thought of as a polymorphic function that,
when applied to a type X, gives us back an induction principle specialized to the type list X.

因此,其归纳定理 list_ind 是一个被 X 参数化多态的函数。
当应用 X : Type 时,返回一个特化在 list X : Type 上的归纳原理

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list_ind : ∀(X : Type) (P : list X → Prop),
P [] →
(∀(x : X) (l : list X), P l → P (x :: l)) →
∀l : list X, P l
∀(X : Type), {

                           P []                   -- base structure holds
    ∀(x : X) (l : list X), P l → P (x :: l)       -- sub-structure holds -> structure holds
    ---------------------------------------
    ∀l : list X,           P l                    -- all structure holds

}

Induction Hypotheses 归纳假设

The induction hypothesis is the premise of this latter implication
— the assumption that P holds of n', which we are allowed to use in proving that P holds for S n'.

归纳假设就是 P n' -> P (S n') 这个蕴含式中的前提部分
使用 nat_ind 时需要显式得用 intros n IHn 引入,于是就变成了 proof context 中的假设.

More on the induction Tactic

“Re-generalize” 重新泛化

Noticed that in proofs using nat_ind, we need to keep n generailzed.
if we intros particular n first then apply nat_ind, it won’t works…

But we could intros n. induction n., that’s induction tactic internally “re-generalize” the n we perform induction on.

Automatic intros i.e. specialize variables before the variable we induction on

A canonical case is induction n vs induction m on theorem plus_comm'' : ∀n m : nat, n + m = m + n..
to keep a var generial…we can either change variable order under , or using generalize dependent.

Induction Principles in Prop

理解依赖类型的归纳假设 与 Coq 排除证据参数的原因

除了集合 Set,命题 Prop 也可以是归纳定义与 induction on 得.
难点在于:Inductive Prop 通常是 dependent type 的,这里会带来复杂度。

考虑命题 even:

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Inductive even : nat → Prop :=
| ev_0 : even 0
| ev_SS : ∀n : nat, even n → even (S (S n)).

我们可以猜测一个最 general 的归纳假设:

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ev_ind_max : ∀ P : (∀n : nat, even n → Prop),
P O ev_0 →
(∀(m : nat) (E : even m), P m E → P (S (S m)) (ev_SS m E)) →
∀(n : nat) (E : even n), P n E

即:

                                   P 0 ev_0                    -- base 
  ∀(m : nat) (E : even m), P m E → P (S (S m)) (ev_SS m E)     -- sub structure -> structure
  --------------------------------------------------------
  ∀(n : nat) (E : even n),         P n E                       -- all structure

注意这里:

  1. even is indexed by nat n (对比 list is parametrized by X)

    • 从族的角度: even : nat -> Prop, a family of Prop indexed by nat
    • 从实体角度: 每个 E : even n 对象都是一个 evidence that particular nat is even.
  2. 要证的性质 P is parametrized by E : even n 也因此连带着 by n. 也就是 P : (∀n : nat, even n → Prop) (对比 P : list X → Prop)

    • 所以其实关于 even n 的性质是同时关于数字 n 和证据 even n 这两件事的.

因此 sub structure -> structure 说得是:

whenever n is an even number and E is an evidence of its evenness, if P holds of n and E, then it also holds of S (S n) and ev_SS n E.
对于任意数字 n 与证据 E,如果 PnE 成立,那么它也对 S (S n)ev_SS n E 成立。

然而,当我们 induction (H : even n) 时,我们通常想证的性质并不包括「证据」,而是「满足该性质的这 Type 东西」的性质,
比如:

  1. nat 上的一元关系 (性质) 证明 nat 的性质 : ev_even : even n → ∃k, n = double k
  2. nat 上的二元关系 证明 nat 上的二元关系 : le_trans : ∀m n o, m ≤ n → n ≤ o → m ≤ o
  3. 二元关系 reg_exp × list T 证明 二元关系 reg_exp × T: in_re_match : ∀T (s : list T) (x : T) (re : reg_exp), s =~ re → In x s → In x (re_chars re).
    都是如此,

因此我们也不希望生成的归纳假设是包括证据的…
原来的归纳假设:

  ∀P : (∀n : nat, even n → Prop),
  ... →
  ∀(n : nat) (E : even n), P n E

可以被简化为只对 nat 参数化的归纳假设:

  ∀P : nat → Prop,
  ... →
  ∀(n : nat) (E: even n), P n

因此 coq 生成的归纳原理也是不包括证据的。注意 P 丢弃了参数 E:

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even_ind : ∀ P : nat -> Prop,
P 0
(∀ n : nat, even n -> P n -> P (S (S n))) →
∀ n : nat, even n -> P n *)

用人话说就是:

  1. P 对 0 成立,
  2. 对任意 n,如果 n 是偶数且 P 对 n 成立,那么 P 对 S (S n) 成立。
    => P 对所有偶数成立

“General Parameter”

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Inductive le : nat → nat → Prop :=
| le_n : ∀ n, le n n
| le_S : ∀ n m, (le n m) → (le n (S m)).
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Inductive le (n:nat) : nat → Prop :=
| le_n : le n n
| le_S m (H : le n m) : le n (S m).

两者虽然等价,但是共同的 ∀ n 可以被提升为 typecon 的参数, i.e. “General Parameter” to the whole definition.

其生成的归纳假设也会不同: (after renaming)

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le_ind : ∀ P : nat -> nat -> Prop,
(∀ n : nat, P n n) ->
(∀ n m : nat, le n m -> P n m -> P n (S m)) ->
∀ n m : nat, le n m -> P n m
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le_ind : ∀ (n : nat) (P : nat -> Prop),
P n ->
(∀ m : nat, n <= m -> P m -> P (S m)) ->
∀ m : nat, n <= m -> P m

The 1st one looks more symmetric but 2nd one is easier (for proving things).