kim-em

import ChallengeDeps

open LeanEval.KnotTheory

theorem exists_chiral_knot : ∃ K : Knot, K.Chiral := by
  sorry

kim-em

import Mathlib

open Filter Topology

theorem cubic_decay_asymptotic (y : ℝ → ℝ) (hy_diff : ∀ t : ℝ, 0 < t → HasDerivAt y (-(y t) ^ 3) t)
    (hy_cont : ContinuousWithinAt y (Set.Ici 0) 0)
    (hy0 : y 0 = 1) :
    Tendsto (fun t : ℝ => y t * Real.sqrt t) atTop (𝓝 (1 / Real.sqrt 2)) := by
  sorry

kim-em

import Mathlib

theorem brauer_character_in_cyclotomic (G : Type) [Group G] [Fintype G] :
    ∃ φ : CyclotomicField (Monoid.exponent G) ℚ →+* ℂ,
      ∀ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V) (_ : FiniteDimensional ℂ V)
        (ρ : Representation ℂ G V) (g : G),
        LinearMap.trace ℂ V (ρ g) ∈ φ.range := by
  sorry

kim-em

import Mathlib

theorem bvp_comparison (J : Set ℝ) (hJ_open : IsOpen J) (hJ_sub : Set.Icc (0 : ℝ) 1 ⊆ J)
    (u v : ℝ → ℝ)
    (hu : ∀ x ∈ J, HasDerivAt u (deriv u x) x)
    (hu' : ∀ x ∈ J, HasDerivAt (deriv u) (deriv (deriv u) x) x)
    (hv : ∀ x ∈ J, HasDerivAt v (deriv v x) x)
    (hv' : ∀ x ∈ J, HasDerivAt (deriv v) (deriv (deriv v) x) x)
    (hineq : ∀ x ∈ Set.Ioo (0 : ℝ) 1, -deriv (deriv u) x ≤ -deriv (deriv v) x)
    (hu0 : u 0 ≤ v 0) (hu1 : u 1 ≤ v 1) :

kim-em

import ChallengeDeps

open LeanEval.Analysis

theorem gleason_theorem_separable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
      [CompleteSpace H] [TopologicalSpace.SeparableSpace H]
    (hdim : 3 ≤ Module.rank ℂ H)
    (f : SphereFrameFunction H) :
    ∃ ρ : H →L[ℂ] H,
      ContinuousLinearMap.IsPositive ρ ∧

kim-em

import ChallengeDeps

open LeanEval.Analysis

theorem gleason_theorem_finite {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
      [CompleteSpace H] [FiniteDimensional ℂ H]
    (hdim : 3 ≤ Module.finrank ℂ H)
    (f : FrameFunction H) :
    ∃! ρ : H →L[ℂ] H,
      ContinuousLinearMap.IsPositive ρ ∧

kim-em

import ChallengeDeps

open LeanEval.RepresentationTheory
open scoped TensorProduct

theorem e8_irrep_tensor_square_decomp :
    ∃ (V : Type) (_ : AddCommGroup V) (_ : Module ℂ V)
      (_ : LieRingModule (LieAlgebra.e₈ ℂ) V) (_ : LieModule ℂ (LieAlgebra.e₈ ℂ) V),
      Module.finrank ℂ V = 779247 ∧
      LieModule.IsIrreducible ℂ (LieAlgebra.e₈ ℂ) V ∧

kim-em

import ChallengeDeps

open LeanEval.RepresentationTheory
open scoped TensorProduct

theorem glAction_range_eq_centralizer_symAction {R : Type*} [Field R]
    {M : Type*} [AddCommGroup M] [Module R M] [FiniteDimensional R M]
    {k : ℕ} [Invertible (k.factorial : R)] :
    Algebra.adjoin R (Set.range (glAction R M k)) =
      Subalgebra.centralizer R (Set.range (symAction R M k)) := by

kim-em

import Mathlib

open scoped MatrixOrder Matrix

theorem oppenheim_inequality {n : Type*} [Fintype n] [DecidableEq n]
    {A B : Matrix n n ℝ} (hA : A.PosSemidef) (hB : B.PosSemidef) :
    A.det * ∏ i, B i i ≤ (A ⊙ B).det := by
  sorry

kim-em

import Mathlib

open scoped MatrixOrder Matrix

theorem posSemidef_map_exp {n : Type*} [Fintype n] [DecidableEq n]
    {A : Matrix n n ℝ} (hA : A.PosSemidef) :
    (A.map Real.exp).PosSemidef := by
  sorry