import Mathlib
import OAI.Probability.SKGap.Model

namespace OAI

section
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Real Set
open scoped NNReal ENNReal
open Filter
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius NNReal
noncomputable section
open MeasureTheory ProbabilityTheory Real Set Filter
open MeasureTheory.Measure
open scoped ENNReal NNReal MeasureTheory Topology
open MeasureTheory
noncomputable section
noncomputable section
open MeasureTheory Set NormedSpace
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius
noncomputable section
open Set Real
open scoped Topology
noncomputable section
open Matrix Set Filter
open scoped Topology Matrix.Norms.Frobenius
noncomputable section
open Matrix NormedSpace ContinuousLinearMap
open scoped Matrix.Norms.Frobenius
noncomputable section
open Matrix
namespace SKGap.ResolventIdentity
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
variable {𝕜 : Type*} [RCLike 𝕜]

lemma inverse_one_sub_square_mul (D V : Matrix ι ι 𝕜)
    (hS : IsUnit (1-D*V*D)) :
    (1-D*D*V)⁻¹ = 1+D*(1-D*V*D)⁻¹*D*V := by
  apply Matrix.inv_eq_right_inv
  have hSS : (1-D*V*D)*(1-D*V*D)⁻¹ = 1 :=
    Matrix.mul_nonsing_inv _ (((1-D*V*D).isUnit_iff_isUnit_det).mp hS)
  calc
    _ = (1-D*D*V)+D*((1-D*V*D)*(1-D*V*D)⁻¹)*D*V := by noncomm_ring
    _ = 1 := by rw [hSS]; noncomm_ring

lemma inverse_one_sub_square_mul_unit (D V : Matrix ι ι 𝕜)
    (hS : IsUnit (1-D*V*D)) : IsUnit (1-D*D*V) := by
  have hSS : (1-D*V*D)*(1-D*V*D)⁻¹ = 1 :=
    Matrix.mul_nonsing_inv _ (((1-D*V*D).isUnit_iff_isUnit_det).mp hS)
  apply ((1-D*D*V).isUnit_iff_isUnit_det).mpr
  have hh : (1-D*D*V)*(1+D*(1-D*V*D)⁻¹*D*V) = 1 := by
    calc
      _ = (1-D*D*V)+D*((1-D*V*D)*(1-D*V*D)⁻¹)*D*V := by noncomm_ring
      _ = 1 := by rw [hSS]; noncomm_ring
  exact Matrix.isUnit_det_of_right_inverse hh

lemma inverse_one_sub_square_mul_square (D V : Matrix ι ι 𝕜)
    (hS : IsUnit (1-D*V*D)) :
    (1-D*D*V)⁻¹*(D*D) = D*(1-D*V*D)⁻¹*D := by
  rw [inverse_one_sub_square_mul D V hS]
  have hSS : (1-D*V*D)⁻¹*(1-D*V*D) = 1 :=
    Matrix.nonsing_inv_mul _ (((1-D*V*D).isUnit_iff_isUnit_det).mp hS)
  calc
    _ = D*(1-D*V*D)⁻¹*D+D*(1-(1-D*V*D)⁻¹*(1-D*V*D))*D := by noncomm_ring
    _ = _ := by rw [hSS]; simp

end SKGap.ResolventIdentity

end
end
end
end
end
end
end
end
end
end
end
end

end OAI