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chore: generalize theorems about Nat.ofDigitChars #13098

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TwoFX merged 1 commit into from
Mar 24, 2026
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chore: generalize theorems about Nat.ofDigitChars #13098

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18 changes: 10 additions & 8 deletions src/Init/Data/Nat/ToString.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -298,7 +298,7 @@ theorem ofDigitChars_cons {c : Char} {cs : List Char} {init : Nat} :
simp [ofDigitChars]

theorem ofDigitChars_cons_digitChar_of_lt_ten {n : Nat} (hn : n < 10) {cs : List Char} {init : Nat} :
ofDigitChars 10 (n.digitChar :: cs) init = ofDigitChars 10 cs (10 * init + n) := by
ofDigitChars b (n.digitChar :: cs) init = ofDigitChars b cs (b * init + n) := by
simp [ofDigitChars_cons, Nat.toNat_digitChar_sub_48_of_lt_ten hn]

theorem ofDigitChars_eq_ofDigitChars_zero {l : List Char} {init : Nat} :
Expand All @@ -320,15 +320,17 @@ theorem ofDigitChars_replicate_zero {n : Nat} : ofDigitChars b (List.replicate n
| zero => simp
| succ n ih => simp [List.replicate_succ, ofDigitChars_cons, ih, Nat.pow_succ, Nat.mul_assoc]

@[simp]
theorem ofDigitChars_toDigits {n : Nat} : ofDigitChars 10 (toDigits 10 n) 0 = n := by
have : 1 < 10 := by decide
induction n using base_induction 10 this with
theorem ofDigitChars_toDigits {b n : Nat} (hb' : 1 < b) (hb : b ≤ 10) : ofDigitChars b (toDigits b n) 0 = n := by
induction n using base_induction b hb' with
| single m hm =>
simp [Nat.toDigits_of_lt_base hm, ofDigitChars_cons_digitChar_of_lt_ten hm]
simp [Nat.toDigits_of_lt_base hm, ofDigitChars_cons_digitChar_of_lt_ten (by omega : m < 10)]
| digit m k hk hm ih =>
rw [← Nat.toDigits_append_toDigits this hm hk,
rw [← Nat.toDigits_append_toDigits hb' hm hk,
ofDigitChars_append, ih, Nat.toDigits_of_lt_base hk,
ofDigitChars_cons_digitChar_of_lt_ten hk, ofDigitChars_nil]
ofDigitChars_cons_digitChar_of_lt_ten (Nat.lt_of_lt_of_le hk hb), ofDigitChars_nil]

@[simp]
theorem ofDigitChars_ten_toDigits {n : Nat} : ofDigitChars 10 (toDigits 10 n) 0 = n :=
ofDigitChars_toDigits (by decide) (by decide)

end Nat
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