erg/doc/EN/compiler/refinement_subtyping.md

Refinement type

The refinement type is the following type.

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

Erg enables type determination by converting Enum and Interval types into refinement types.

Convert to refinement type

In the section [Refinement types], we said that interval types and enum types are syntactic sugar for refinement types. Each is converted as follows.

• {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}

Refinement type detection

An algorithm for determining whether a refinement type A is a subtype of another refinement type B is described. Formally, (all) subtyping is defined as follows:

A <: B <=> ∀a∈A; a ∈ B

Specifically, the following inference rules are applied. Boolean expressions are assumed to be simplified.

• intervalization rules (done automatically from type definition)
  • Nat => {I: Int | I >= 0}
• Round-up rule
  • {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}
• reversal rule
  • {A not B} => {A and (not B)}
• De Morgan's Law
  • {not (A or B)} => {not A and not B}
  • {not (A and B)} => {not A or not B}
• Distribution rule
  • {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})
• termination rule
  • {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 ))
  • basic formula
    • {I >= l} <: {I >= r} = (l >= r)
    • {I <= l} <: {I <= r} = (l <= r)
    • {I >= l} <: {I <= r} = False
    • {I <= l} <: {I >= r} = False

The simplification rules for Boolean expressions are as follows. min, max may not be removed. Also, multiple or, and are converted to nested min, max.

• ordering rules
  • I == a => I >= a and I <= a
  • i != a => I >= a+1 or I <= a-1
• Consistency rule
  • I >= a or I <= b (a < b) == {...}
• Constancy rule
  • I >= a and I <= b (a > b) == {}
• replacement rule
  • Replace order expressions in the order I >= n and I <= n.
• Extension rule
  • I == n or I >= n+1 => I >= n
  • I == n or I <= n-1 => I <= n
• maximum rule
  • I <= m or I <= n => I <= max(m, n)
  • I >= m and I >= n => I >= max(m, n)
• minimum rule
  • I >= m or I >= n => I >= min(m, n)
  • I <= m and I <= n => I <= min(m, n)
• elimination rule
  • I == n on the left side is removed when I >= a (n >= a) or I <= b (n <= b) or I == n on the right side can.
  • False if all left-hand equations cannot be eliminated

e.g.

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