openAI-math-dataset / lean /OAI /MathematicalPhysics /DefocusingNLS /Profile /ProfileCertificateComparisons.lean
| import OAI.MathematicalPhysics.DefocusingNLS.Profile.ProfileCertificateValues | |
| namespace OAI | |
| /-! Exact rational comparisons for Lemma `free:profile-degree`. | |
| The state is obtained by nine checked blocks of the K=34 recurrence. -/ | |
| namespace DefocusingNLS.ProfileCertificate | |
| open RationalComplex | |
| /-- All entrywise comparisons in `free:profile-enclosures`, before taking roots. -/ | |
| def CentralBounds (s : State) : Prop := | |
| let m := s.value.b | |
| let u := quotient s.value.a m | |
| let x := quotient s.derivB.d m | |
| let y := quotient s.derivZ.d m | |
| 81 < normSq m ∧ normSq m < 100 ∧ | |
| 2835/1000 < u.re ∧ u.re < 2837/1000 ∧ | |
| 1426/1000 < u.im ∧ u.im < 1428/1000 ∧ | |
| normSq s.value.d < (1/1000000000000 : ℚ)^2 * normSq m ∧ | |
| -811/10000 < x.re ∧ x.re < -810/10000 ∧ | |
| 1361/10000 < x.im ∧ x.im < 1363/10000 ∧ | |
| -1/10000 < y.re ∧ y.re < 1/10000 ∧ | |
| 1237/10000 < y.im ∧ y.im < 1240/10000 ∧ | |
| (∀ v ∈ [s.derivB.a, s.derivB.b, s.derivB.c, s.derivB.d, | |
| s.derivZ.a, s.derivZ.b, s.derivZ.c, s.derivZ.d], normSq v < 400*normSq m) ∧ | |
| (∀ e ∈ [s.error.a, s.error.b, s.error.c, s.error.d], | |
| 0 ≤ e ∧ e^2 < 50000^2*normSq m) | |
| theorem central_bounds : CentralBounds result := by | |
| rw [result_eq_computed] | |
| simp only [CentralBounds, List.forall_mem_cons] | |
| decide +kernel | |
| /-- Sylvester's strict inequalities and the denominator/linear-size comparisons. -/ | |
| def LinearBounds (s : State) : Prop := | |
| let m := s.value.b | |
| let u := quotient s.value.a m | |
| let x := quotient s.derivB.d m | |
| let y := quotient s.derivZ.d m | |
| let κ : ℚ := (48/1000)^2 | |
| κ < normSq x ∧ | |
| (normSq x-κ)*(normSq y-κ) > (x.re*y.re+x.im*y.im)^2 ∧ | |
| normSq (1 - (⟨0,centerZ/5⟩ : RationalComplex)*u) > (234/100)^2 ∧ | |
| normSq u < (318/100)^2 ∧ | |
| normSq x < (159/1000)^2 ∧ normSq y < (124/1000)^2 ∧ | |
| x.re*y.im-x.im*y.re < 0 | |
| theorem linear_bounds : LinearBounds result := by | |
| rw [result_eq_computed] | |
| unfold LinearBounds | |
| decide +kernel | |
| /-- Factorwise rational upper bound for the determinant on the parameter disk. -/ | |
| def determinantSqBound : ℚ := | |
| ((List.range 34).map fun n => | |
| ((n : ℚ)^2+(centerB+radius)^2)*(((n : ℚ)-5)^2+(centerB+radius)^2)).prod / | |
| (Nat.factorial 34 : ℚ)^4 | |
| theorem determinant_bound : determinantSqBound < | |
| (62/10000000000000 : ℚ)^2 * (normSq result.value.b)^2 := by | |
| rw [result_eq_computed] | |
| decide +kernel | |
| theorem propagation_comparisons : | |
| 6/10 < denominatorGap ∧ changeBound < 5/10000000 ∧ | |
| jetError + tailError < 25/1000000000000 ∧ | |
| (283/1000)*radius+25/1000000000000 < 3/1000000000 ∧ | |
| (48/1000)*radius > 25/1000000000000 ∧ | |
| (48/1000)*radius-25/1000000000000 = 91/200000000000 := by | |
| decide +kernel | |
| end DefocusingNLS.ProfileCertificate | |
| end OAI | |