[Merged by Bors] - feat: add Nat.digits_append_digits#6999
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[Merged by Bors] - feat: add Nat.digits_append_digits#6999
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This is a generalization of a lemma that I needed for a USAMO problem: https://github.com/dwrensha/math-puzzles-in-lean4/blob/918bb01e426d1c60ae2f9b841dfeb95f1f9f7841/MathPuzzles/Usa2003Q1.lean#L21-L30 cc @b-mehta, who minimized the proof of |
mo271
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Sep 7, 2023
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This also works theorem digits_append_digits {b m n : ℕ} (hb : 1 < b) :
digits b n ++ digits b m = digits b (n + b ^ (digits b n).length * m) := by
rw [← ofDigits_digits_append_digits]
refine' (digits_ofDigits b hb _ (fun l hl => _) (fun h_append => _)).symm
· rcases (List.mem_append.mp hl) with (h | h) <;> exact digits_lt_base hb h
· by_cases digits b m = []
· simp only [h, List.append_nil] at h_append ⊢
exact getLast_digit_ne_zero b <| digits_ne_nil_iff_ne_zero.mp h_append
· exact (List.getLast_append' _ _ h) ▸
(getLast_digit_ne_zero _ <| digits_ne_nil_iff_ne_zero.mp h) |
mo271
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@mo271 Thanks! Your proof looks nicer, so I switched to that. I also weakened |
mo271
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Sep 7, 2023
bors bot
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Sep 8, 2023
Similar to `Nat.ofDigits_digits_append_digits`, but with a `digits` on the RHS instead of an `ofDigits` on the LHS.
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* fix-lint: (463 commits) chore: lint-style.py was calling str.replace incorrectly chore: module doc for #find_home, #minimize_imports, import early (#7095) chore: reduce imports to Data.Rat.Cast.CharZero (#7091) chore: cleanup imports of Data/Rat/Cast/Defs (#7092) chore: linarith only needs ring1 (#7090) refactor(Data/Int/Bitwise): use `<<<` and `>>>` notation (#6789) chore: delete redundant commented-out positivity test (#7085) chore: fix docstrings, names and aligns about paracompacity of emetric spaces (#7064) feat(Data/Finsupp): add notation (#6367) refactor: re-home some meta code (#6921) fix: don't use `False` as a bool, use `false` (#7059) chore: fix reference to `compactCovering` in docstring (#7065) fix: fix link in docstring of IsWellFounded (#7063) chore: tidy various files (#7041) feat: roots in an algebra (#6740) chore: revert ProofWidgets bump in #7044 and #7056 (#7069) feat: super factorial (#6768) feat(LinearAlgebra/Matrix/Trace): add Matrix.trace_diagonal (#7061) chore: complete ProofWidgets bump (#7056) feat: More `Finset.sup'` lemmas (#7021) feat: self_lt_pow (#7058) chore: move some files to `MeasureTheory/MeasurableSpace/` (#7045) chore(RingTheory/Nilpotent): untwine two universes (#7057) feat: von Neumann Mean Ergodic Theorem (#6053) feat: (sSup, Iic) and (Ici, sInf) are Galois connections (#6951) feat: derivative along a vector (#7038) feat: check for some common git problems in CI (#7043) chore: bump ProofWidgets (#7044) feat(Topology/Algebra): add `DiscreteTopology Mˣ` (#7028) chore: simplify `by rfl` (#7039) chore: tidy various files (#7035) refactor(LinearAlgebra/CliffordAlgebra/Conjugation): expose implementation details of 'reverse' (#6783) feat: add `DiscreteTopology.of_continuous_injective` (#7029) feat: restore the module docstring code snippet (#7030) feat: flesh out the API for `realPart` and `imaginaryPart` (#7023) chore: missing simps lemmas for `Multiplicative` defs (#7016) feat: characterize isLittleO on principal filters (#7008) doc: cleanup documentation on Basis.constr (#7007) feat: cleanup API around differentiable functions (#7004) feat: add Nat.digits_append_digits (#6999) feat: definition of the homology of a short complex (#6994) chore: implement porting notes about Polish spaces (#6991) feat(Algebra/Category/ModuleCat): composition of restriction of scalar functors (#6915) feat: compute the integral of sqrt (1 - x ^ 2) (#6905) chore(*/TensorProduct): missing functorial lemmas (#6781) feat: a functor from a small category to a filtered category factors through a small filtered category (#6212) feat: expand API on locally integrable functions (#7006) chore: split Tactic.NormNum.Basic (#7002) feat: a few lemmas on continuous functions (#7005) feat: ZMod.castHom_self (#7013) ...
kodyvajjha
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Sep 22, 2023
Similar to `Nat.ofDigits_digits_append_digits`, but with a `digits` on the RHS instead of an `ofDigits` on the LHS.
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Nat.ofDigits_digits_append_digits, but with adigitson the RHS instead of anofDigitson the LHS.