openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Spectrum /SpectralBranchCone.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Spectrum.SpectralShellNorm | |
| namespace OAI | |
| /-! A small weighted error around a graph branch controls both the value | |
| and its Robin residual. -/ | |
| namespace DefocusingNLS | |
| theorem spectralBranch_value_residual (k C A delta : ℝ) (hk : 0 < k) (hC : 0 ≤ C) | |
| (z L : ℂ) (D q : ℂ × ℂ) (hD : D.2 = L*D.1) | |
| (hA : k*‖D.1‖ = A) (hL : ‖L‖ ≤ C*k^2) | |
| (hclose : spectralShellNorm k (q-z • D) ≤ delta) : | |
| ‖z‖*A-delta ≤ k*‖q.1‖ ∧ | |
| ‖q.2-L*q.1‖ ≤ (1+C)*k*delta := by | |
| let e := q-z • D | |
| have hd : 0 ≤ delta := (spectralShellNorm_nonneg k hk.le _).trans hclose | |
| have hval : ‖e.1‖ ≤ delta/k := | |
| (spectralShellNorm_value k hk e).trans (div_le_div_of_nonneg_right hclose hk.le) | |
| have hder : ‖e.2‖ ≤ k*delta := | |
| (spectralShellNorm_slope k hk e).trans (mul_le_mul_of_nonneg_left hclose hk.le) | |
| have hfirst : ‖z*D.1‖ ≤ ‖q.1‖+‖e.1‖ := by | |
| have he : z*D.1 = q.1-e.1 := by | |
| dsimp only [e,Prod.fst_sub,Prod.smul_fst,smul_eq_mul] | |
| ring | |
| rw [he] | |
| exact norm_sub_le _ _ | |
| have hmult := mul_le_mul_of_nonneg_left hfirst hk.le | |
| have hscaled : k*‖z*D.1‖ = ‖z‖*A := by rw [norm_mul,← hA]; ring | |
| have hvalscaled : k*‖e.1‖ ≤ delta := by | |
| simpa only [mul_comm] using (le_div_iff₀ hk).mp hval | |
| refine ⟨by rw [hscaled] at hmult; nlinarith,?_⟩ | |
| have he : q.2-L*q.1 = e.2-L*e.1 := by | |
| dsimp only [e,Prod.fst_sub,Prod.snd_sub,Prod.smul_fst,Prod.smul_snd,smul_eq_mul] | |
| rw [hD] | |
| ring | |
| rw [he] | |
| calc | |
| _ ≤ ‖e.2‖+‖L‖*‖e.1‖ := by simpa only [norm_mul] using norm_sub_le e.2 (L*e.1) | |
| _ ≤ k*delta+(C*k^2)*(delta/k) := by gcongr | |
| _ = (1+C)*k*delta := by field_simp | |
| end DefocusingNLS | |
| end OAI | |