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theorem proved term proof

energy_bounded_below

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formal statement (Lean)

 105theorem energy_bounded_below (w_val f_val : ℝ) (hw : 1 ≤ w_val) (hf : 0 ≤ f_val ^ 2) :
 106    0 ≤ w_val * f_val ^ 2 := by

proof body

Term-mode proof.

 107  exact mul_nonneg (by linarith) hf
 108
 109/-! ## No-Retuning Theorem -/
 110
 111/-- The No-Retuning Theorem: if the ILG potential is the unique energy
 112    minimizer for a GLOBAL kernel (same w for all galaxies), then changing
 113    w per galaxy would produce a DIFFERENT potential that is NOT the minimizer.
 114
 115    Formally: if w is the unique minimizer kernel, any w' != w gives
 116    a strictly higher energy. -/

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