| import OAI.Geometry.Kahler.BaseLogFactor |
|
|
| namespace OAI |
|
|
| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open Set Filter Topology |
| open scoped ContDiff ENNReal Pointwise |
| open scoped ContDiff |
| open Set Filter Topology MeasureTheory |
| noncomputable section |
|
|
| open Set Filter Topology MeasureTheory |
| namespace PinchedHartogs.BaseConstruction |
|
|
| lemma logFactor_complex_identity (ε : ℝ) (v : ℂ) : |
| ‖1-v‖^2+ε^2 = logFactor ‖v‖ ε*‖1-v/(logFactor ‖v‖ ε:ℂ)‖^2 := by |
| have hl := (logFactor_pos ‖v‖ ε).ne' |
| have he := logFactor_equation ‖v‖ ε |
| have hn := Complex.sq_norm v |
| simp only [Complex.normSq_apply] at hn |
| have hdiv : v/(logFactor ‖v‖ ε:ℂ)=( (logFactor ‖v‖ ε)⁻¹:ℝ) * v := by |
| push_cast; ring |
| rw [hdiv] |
| simp only [Complex.sq_norm,Complex.normSq_apply,Complex.sub_re,Complex.sub_im,Complex.one_re, |
| Complex.one_im,Complex.mul_re,Complex.mul_im,Complex.ofReal_re,Complex.ofReal_im, |
| zero_mul,sub_zero,zero_sub,add_zero] |
| field_simp at he ⊢ |
| nlinarith |
| |
| lemma logFactor_ratio_bound {ε : ℝ} (he : ε ≠ 0) {A : ℝ} (hA : 0 ≤ A) : |
| ∃ θ : ℝ, 0 < θ ∧ θ < 1 ∧ ∀ q ∈ Icc 0 A, q/logFactor q ε ≤ θ := by |
| have hc : Continuous (fun q : ℝ => q/logFactor q ε) := |
| continuous_id.div (logFactor_continuous.comp (continuous_id.prodMk continuous_const)) |
| (fun q => (logFactor_pos q ε).ne') |
| obtain ⟨q,hq,hmax⟩ := isCompact_Icc.exists_isMaxOn (nonempty_Icc.mpr hA) hc.continuousOn |
| let θ := (1+q/logFactor q ε)/2 |
| have hr : q/logFactor q ε < 1 := (div_lt_one (logFactor_pos q ε)).mpr (logFactor_gt he) |
| have hr0 : 0 ≤ q/logFactor q ε := div_nonneg hq.1 (logFactor_pos q ε).le |
| refine ⟨θ,by dsimp [θ]; linarith,by dsimp [θ]; linarith,?_⟩ |
| intro r hr' |
| exact (hmax hr').trans (by dsimp [θ]; linarith) |
|
|
| def logMean (ε : ℝ) (v : ℂ) : ℝ := Real.log (logFactor ‖v‖ ε)-Real.log (1+ε^2) |
| def logNormalized (ε : ℝ) (v : ℂ) : ℂ := v/(logFactor ‖v‖ ε:ℂ) |
| def logTerm (ε : ℝ) (n : ℕ) (v : ℂ) : ℝ := -2*((logNormalized ε v)^(n+1)).re/(n+1) |
| def logTruncation (ε : ℝ) (M : ℕ) (v : ℂ) : ℝ := logMean ε v+∑ n ∈ Finset.range M, logTerm ε n v |
|
|
| lemma logNormalized_norm_lt {ε : ℝ} (he : ε ≠ 0) (v : ℂ) : ‖logNormalized ε v‖ < 1 := by |
| rw [logNormalized,norm_div,Complex.norm_real,Real.norm_eq_abs,abs_of_pos (logFactor_pos _ _)] |
| exact (div_lt_one (logFactor_pos _ _)).mpr (logFactor_gt he) |
|
|
| lemma regularizedLog_hasSum {ε : ℝ} (he : ε ≠ 0) (v : ℂ) : |
| HasSum (fun n => logTerm ε n v) (regularizedLog 1 ε v-logMean ε v) := by |
| have hs := (Complex.hasSum_re (Complex.hasSum_taylorSeries_neg_log' (logNormalized_norm_lt he v))).mul_left (-2) |
| have heq : Real.log (‖1-v‖^2+ε^2)=Real.log (logFactor ‖v‖ ε)+Real.log (‖1-logNormalized ε v‖^2) := by |
| rw [logFactor_complex_identity] |
| apply Real.log_mul (logFactor_pos _ _).ne' |
| apply pow_ne_zero |
| apply norm_ne_zero_iff.mpr |
| intro hz |
| have hh := congrArg norm (sub_eq_zero.mp hz) |
| rw [norm_one] at hh |
| exact (ne_of_lt (logNormalized_norm_lt he v)) hh.symm |
| have hval : -2 * (-Complex.log (1-logNormalized ε v)).re = regularizedLog 1 ε v-logMean ε v := by |
| rw [Complex.neg_re,Complex.log_re,regularizedLog_eq 1 ε v he] |
| simp only [Complex.ofReal_one,one_mul,logMean] |
| rw [heq,Real.log_pow] |
| ring |
| rw [hval] at hs |
| apply hs.congr_fun |
| intro n |
| simp only [logTerm,Complex.div_re,Complex.add_re,Complex.natCast_re,Complex.one_re, |
| Complex.add_im,Complex.natCast_im,Complex.one_im,add_zero,mul_zero,add_zero, |
| Complex.normSq_apply] |
| field_simp |
| ring |
|
|
| lemma logTerm_bound {ε θ : ℝ} (hθ : 0 ≤ θ) {v : ℂ} (hv : ‖logNormalized ε v‖ ≤ θ) (n : ℕ) : |
| |logTerm ε n v| ≤ 2*θ^(n+1) := by |
| have hn : (1:ℝ) ≤ n+1 := by have := Nat.cast_nonneg (α:=ℝ) n; linarith |
| calc |
| _ = 2*|((logNormalized ε v)^(n+1)).re|/(n+1) := by |
| simp only [logTerm,abs_div,abs_mul,abs_of_pos (by positivity : (0:ℝ)<n+1)] |
| norm_num |
| _ ≤ 2*‖(logNormalized ε v)^(n+1)‖/(n+1) := by gcongr; exact Complex.abs_re_le_norm _ |
| _ ≤ 2*θ^(n+1)/(n+1) := by rw [norm_pow]; gcongr |
| _ ≤ 2*θ^(n+1) := div_le_self (by positivity) hn |
|
|
| lemma logTruncation_error {ε θ : ℝ} (he : ε ≠ 0) (hθ : 0 ≤ θ) (hθ1 : θ < 1) |
| {v : ℂ} (hv : ‖logNormalized ε v‖ ≤ θ) (M : ℕ) : |
| |regularizedLog 1 ε v-logTruncation ε M v| ≤ 2*θ^(M+1)/(1-θ) := by |
| have hsum := regularizedLog_hasSum he v |
| have htail : regularizedLog 1 ε v-logTruncation ε M v = ∑' n : ℕ, logTerm ε (n+M) v := by |
| have h := hsum.summable.sum_add_tsum_nat_add M |
| rw [hsum.tsum_eq] at h |
| dsimp [logTruncation] |
| linarith |
| rw [htail,← Real.norm_eq_abs] |
| have hbound : ∀ n : ℕ, ‖logTerm ε (n+M) v‖ ≤ (2*θ^(M+1))*θ^n := by |
| intro n |
| rw [Real.norm_eq_abs] |
| calc |
| _ ≤ 2*θ^(n+M+1) := logTerm_bound hθ hv (n+M) |
| _ = _ := by rw [show n+M+1=(M+1)+n by omega,pow_add]; ring |
| have hg : Summable (fun n : ℕ => (2*θ^(M+1))*θ^n) := (summable_geometric_of_lt_one hθ hθ1).mul_left _ |
| have hn : Summable (fun n : ℕ => ‖logTerm ε (n+M) v‖) := |
| hg.of_nonneg_of_le (fun _ => norm_nonneg _) hbound |
| calc |
| _ ≤ ∑' n : ℕ, ‖logTerm ε (n+M) v‖ := norm_tsum_le_tsum_norm hn |
| _ ≤ ∑' n : ℕ, (2*θ^(M+1))*θ^n := hn.tsum_le_tsum hbound hg |
| _ = _ := by rw [tsum_mul_left,tsum_geometric_of_lt_one hθ hθ1]; rfl |
| |
| lemma logNormalized_continuous (ε : ℝ) : Continuous (logNormalized ε) := by |
| unfold logNormalized |
| exact continuous_id.div |
| (Complex.continuous_ofReal.comp (logFactor_continuous.comp (continuous_norm.prodMk continuous_const))) |
| (fun v => Complex.ofReal_ne_zero.mpr (logFactor_pos _ _).ne') |
|
|
| lemma logMean_continuous (ε : ℝ) : Continuous (logMean ε) := by |
| unfold logMean |
| exact ((logFactor_continuous.comp (continuous_norm.prodMk continuous_const)).log |
| (fun v => (logFactor_pos _ _).ne')).sub continuous_const |
| |
| lemma logTerm_continuous (ε : ℝ) (n : ℕ) : Continuous (logTerm ε n) := by |
| unfold logTerm |
| exact (continuous_const.mul (Complex.continuous_re.comp ((logNormalized_continuous ε).pow _))).div_const _ |
| |
| lemma logTruncation_continuous (ε : ℝ) (M : ℕ) : Continuous (logTruncation ε M) := by |
| unfold logTruncation |
| exact (logMean_continuous ε).add (continuous_finsetSum _ (fun n _ => logTerm_continuous ε n)) |
| |
| end PinchedHartogs.BaseConstruction |
| |
| end |
| |
| end OAI |
| |
