Plain-English statement
- A continuous objective with an attained minimum has an actual regularized minimum in a proper normed group; its radius and approximate values transfer.
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Lean statement
theorem exists_minimizer_radius_and_accuracy
{E : Type*} [NormedAddCommGroup E] [ProperSpace E]
{f : E → ℝ} (hf : Continuous f) {z x₀ : E} (hz : IsMinOn f univ z)
{R ε : ℝ} (hR : 0 < R) (hε : 0 < ε) (hx : ‖z-x₀‖ ≤ R) :
let δ := ε/R^2
let W := fun x => f x + δ/2*‖x-x₀‖^2
∃ w, IsMinOn W univ w ∧ ‖w-x₀‖ ≤ ‖z-x₀‖ ∧
∀ x, W x - W w ≤ ε/2 → f x - f z ≤ ε := by
let δ := ε/R^2
let W := fun x => f x + δ/2*‖x-x₀‖^2
have hδ : 0 < δ := div_pos hε (sq_pos_of_pos hR)
have hn : Continuous (fun x : E => ‖x-x₀‖^2) :=
((continuous_id.sub continuous_const).norm).pow 2
have hW : Continuous W := hf.add (continuous_const.mul hn)
have hzball : z ∈ closedBall x₀ R := by simpa [mem_closedBall, dist_eq_norm] using hx
obtain ⟨w, _, hw⟩ := (isCompact_closedBall x₀ R).exists_isMinOn ⟨z,hzball⟩ hW.continuousOn
have hwz : W w ≤ W z := hw hzball
have hglobal : IsMinOn W univ w := by
intro x _
by_cases hb : x ∈ closedBall x₀ R
· exact hw hb
· have hr : R < ‖x-x₀‖ := by simpa [mem_closedBall, dist_eq_norm] using hb
have hs : ‖z-x₀‖^2 ≤ ‖x-x₀‖^2 := by
nlinarith [norm_nonneg (z-x₀), norm_nonneg (x-x₀)]
have hzfx : f z ≤ f x := hz (mem_univ x)
have hquad := mul_le_mul_of_nonneg_left hs (le_of_lt (div_pos hδ (by norm_num : (0:ℝ)<2)))
have hzW : W z ≤ W x := by dsimp [W]; linarith
exact hwz.trans hzW
have hradius : ‖w-x₀‖ ≤ ‖z-x₀‖ := by
have hzw : f z ≤ f w := hz (mem_univ w)
have hs : ‖w-x₀‖^2 ≤ ‖z-x₀‖^2 := by
dsimp [W] at hwz
nlinarith
nlinarith [norm_nonneg (w-x₀), norm_nonneg (z-x₀)]
refine ⟨w,hglobal,hradius,?_⟩
intro x he
have hbase : f x ≤ W x := by
dsimp [W]
exact le_add_of_nonneg_right (mul_nonneg (div_nonneg hδ.le (by norm_num)) (sq_nonneg _))
have hs : ‖z-x₀‖^2 ≤ R^2 := by nlinarith [norm_nonneg (z-x₀)]
have hquad := mul_le_mul_of_nonneg_left hs (le_of_lt (div_pos hδ (by norm_num : (0:ℝ)<2)))
have hid : δ/2*R^2 = ε/2 := by dsimp [δ]; field_simp
rw [hid] at hquad
dsimp [W] at hwz
change W x - W w ≤ ε/2 at he
have : W w ≤ f z + ε/2 := by dsimp [W]; linarith
linarith
end AutoSamplingTheory.TechnicalLemmas.Analysis.QuadraticRegularizationTransferOpen AutoSamplingTheory/TechnicalLemmas/Analysis/QuadraticRegularizationTransfer.lean:22published source at 0e31a3cda412
Proof architecture
Construct actual regularized minimum and transfer radius and objective approximation.
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `intro` introduces quantified hypotheses into the local proof context.
- `have` creates a named intermediate mathematical fact.
- `rw` rewrites by an established identity.
- `refine` instantiates a reusable theorem while leaving explicit subgoals.
- `exact` closes the current goal with an already typed term.
- `simpa` closes the goal after a controlled simplification of a typed result.
Why the statement has this shape
The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.
Hidden assumptions and non-claims
- No additional hidden-contract keyword was inferred; the exact Lean hypotheses remain controlling.