Lean 4 fork for HoTT-compatible kernel extensions (Path types, transport, HITs). Maintained against upstream leanprover/lean4.
e758c0e35c
This PR redefines the `String.isNat` function to use less state and
perform short-circuiting. It then verifies the `String.isNat` and
`String.toNat?` functions.
Recall that `isNat` and `toNat?` allow `_` as a digit separator. This is
why we get the complicated statement
```lean
public theorem isNat_iff {s : String} :
s.isNat = true ↔
s ≠ "" ∧
(∀ c ∈ s.toList, c.isDigit ∨ c = '_') ∧
¬ ['_', '_'] <:+: s.toList ∧
s.toList.head? ≠ some '_' ∧
s.toList.getLast? ≠ some '_'
```
For `toNat?`, we prove the fully general
```lean
public theorem toNat?_eq_some_ofDigitChars {s : String} (h : s.isNat = true) :
s.toNat? = some (Nat.ofDigitChars 10 (s.toList.filter (· != '_')) 0)
```
as well as the useful `(Nat.repr n).toNat? = some n` (and the corollary
that `Nat.repr` is injective).
For people implementing formatting routines that involve digit
separators, we have
```lean
public theorem isNat_of_isDigit {s : String} (hne : s ≠ "")
(hdigit : ∀ c ∈ s.toList, c.isDigit) : s.isNat = true
public theorem isNat_append_underscore_append {s t : String}
(hs : s.isNat = true) (ht : t.isNat = true) :
(s ++ "_" ++ t).isNat = true
public theorem toNat?_append_underscore_append_eq_some {s t : String} {n m : Nat}
(hs : s.toNat? = some n) (ht : t.toNat? = some m) :
(s ++ "_" ++ t).toNat? =
some (10 ^ (t.toList.filter (· != '_')).length * n + m)
```
The missing bit here is `(s.leftpad k '0').toNat? = s.toNat?`, which is
missing because we don't have `String.leftpad` (yet). For any reasonable
definition of `leftpad`, this will follow from
`toNat?_eq_some_ofDigitChars` since we prove the necessary ingredients
about `ofDigitChars`.
There are some rough edges around `ofDigitChars`, and in the future it
will be nice to connect this all to mathlib's `Nat.digits` and
`Nat.ofDigits`, which are similar but different.
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