筛子类型

{I: Int | I >= 0}
{S: StrWithLen N | N >= 1}
{T: (Ratio, Ratio) | T.0 >= 0; T.1 >= 0}

Erg 通过将 Enum 和 Interval 类型转换为筛选类型来实现类型确定

转换为筛型

在 [Refinement types] 一节中，我们说过区间类型和枚举类型是 refinement 类型的语法糖。每个转换如下

{0} -> {I: Int | I == 0}
{0, 1} -> {I: Int | I == 0 or I == 1}
1.._ -> {I: Int | I >= 1}
1<.._ -> {I: Int | I > 1} -> {I: Int | I >= 2}
{0} or 1.._ -> {I: Int | I == 0 or I >= 1}
{0} or {-3, -2} or 1.._ -> {I: Int | I == 0 or (I == -2 or I == -3) or I >= 1}
{0} and {-3, 0} -> {I: Int | I == 0 and (I == -3 or I == 0)}
{0} not {-3, 0} or 1.._ -> {I: Int | I == 0 and not (I == -3 or I == 0) or I >= 1}

筛型检测

描述了一种用于确定筛类型 A 是否是另一筛类型 B 的子类型的算法。正式地，(所有)子类型定义如下:

A <: B <=> ∀a∈A; a∈B

具体而言，应用以下推理规则。假定布尔表达式是简化的

间隔规则(从类型定义自动完成)
- Nat => {I: Int | I >= 0}
围捕规则
- {I: Int | I < n} => {I: Int | I <= n-1}
- {I: Int | I > n} => {I: Int | I >= n+1}
- {R: Ratio | R < n} => {R: Ratio | R <= n-ε}
- {R: Ratio | R > n} => {R: Ratio | R >= n+ε}
反转规则
- {A not B} => {A and (not B)}
德摩根规则
- {not (A or B)} => {not A and not B}
- {not (A and B)} => {not A or not B}
分配规则
- {A and (B or C)} <: D => {(A and B) or (A and C)} <: D => ({A and B} <: D) and ( {A and C} <: D)
- {(A or B) and C} <: D => {(C and A) or (C and B)} <: D => ({C and A} <: D) and ( {C and B} <: D)
- D <: {A or (B and C)} => D <: {(A or B) and (A or C)} => (D <: {A or B}) and ( D <: {A or C})
- D <: {(A and B) or C} => D <: {(C or A) and (C or B)} => (D <: {C or A}) and ( D <: {C or B})
- {A or B} <: C => ({A} <: C) and ({B} <: C)
- A <: {B and C} => (A <: {B}) and (A <: {C})
终止规则
- {I: T | ...} <: T = True
- {} <: _ = True
- _ <: {...} = True
- {...} <: _ = False
- _ <: {} == False
- {I >= a and I <= b} (a < b) <: {I >= c} = (a >= c)
- {I >= a and I <= b} (a < b) <: {I <= d} = (b <= d)
- {I >= a} <: {I >= c or I <= d} (c >= d) = (a >= c)
- {I <= b} <: {I >= c or I <= d} (c >= d) = (b <= d)
- {I >= a and I <= b} (a <= b) <: {I >= c or I <= d} (c > d) = ((a >= c) or (b <= d ))
- 基本公式
  - {I >= l} <: {I >= r} = (l >= r)
  - {I <= l} <: {I <= r} = (l <= r)
  - {I >= l} <: {I <= r} = False
  - {I <= l} <: {I >= r} = False

布尔表达式的简化规则如下。min, max 不能被删除。此外，多个 or, and 被转换为嵌套的 min, max

组合规则
- I == a => I >= a 和 I <= a
- i != a => I >= a+1 或 I <= a-1
一致性规则
- I >= a 或 I <= b (a < b) == {...}
恒常规则
- I >= a 和 I <= b (a > b) == {}
替换规则
- 以 I >= n 和 I <= n 的顺序替换顺序表达式
扩展规则
- I == n 或 I >= n+1 => I >= n
- I == n 或 I <= n-1 => I <= n
最大规则
- I <= m 或 I <= n => I <= max(m, n)
- I >= m 和 I >= n => I >= max(m, n)
最低规则
- I >= m 或 I >= n => I >= min(m, n)
- I <= m 和 I <= n => I <= min(m, n)
淘汰规则
- 当 I >= a (n >= a) 或 I <= b (n <= b) 或 I == n 在右侧时，左侧的 I == n 被删除能够
- 如果无法消除所有左手方程，则为 False

例如

1.._<: Nat
=> {I: Int | I >= 1} <: {I: Int | I >= 0}
=> {I >= 1} <: {I >= 0}
=> (I >= 0 => I >= 1)
=> 1 >= 0
=> True
# {I >= l} <: {I >= r} == (l >= r)
# {I <= l} <: {I <= r} == (l <= r)

{I: Int | I >= 0} <: {I: Int | I >= 1 or I <= -3}
=> {I >= 0} <: {I >= 1 or I <= -3}
=> {I >= 0} <: {I >= 1} or {I >= 0} <: {I <= -3}
=> False or False
=> False

{I: Int | I >= 0} <: {I: Int | I >= -3 and I <= 1}
=> {I >= 0} <: {I >= -3 and I <= 1}
=> {I >= 0} <: {I >= -3} and {I >= 0} <: {I <= 1}
=> True and False
=> False

{I: Int | I >= 2 or I == -2 or I <= -4} <: {I: Int | I >= 1 or I <= -1}
=> {I >= 2 or I <= -4 or I == -2} <: {I >= 1 or I <= -1}
=> {I >= 2 or I <= -4} <: {I >= 1 or I <= -1}
    and {I == -2} <: {I >= 1 or I <= -1}
=> {I >= 2} <: {I >= 1 or I <= -1}
        and {I <= -4} <: {I >= 1 or I <= -1}
    and
        {I == -2} <: {I >= 1}
        or {I == -2} <: {I <= -1}
=> {I >= 2} <: {I >= 1}
        or {I >= 2} <: {I <= -1}
    and
        {I <= -4} <: {I >= 1}
        or {I <= -4} <: {I <= -1}
    and
        False or True
=> True or False
    and
        False or True
    and
        True
=> True and True
=> True

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筛子类型

转换为筛型

筛型检测

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