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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import OAI.Analysis.GroupDeterminants.Model
namespace OAI
universe uR ud ul um ur
noncomputable section
open scoped BigOperators
namespace GroupRingDeterminant
def CompanionInput : Prop :=
∃ (K : Type) (_ : Field K) (_ : Finite K) (_ : CharP K 2)
(H : Type) (_ : Group H) (_ : Group.FG H)
(a b : MonoidAlgebra K H), a * b = 1 ∧ b * a ≠ 1
theorem no_finite_group_input (K H : Type) [Field K] [Finite K] [Group H] [Finite H]
(a b : MonoidAlgebra K H) (h : a * b = 1) : b * a = 1 := by
have : Finite (H →₀ K) :=
Finite.of_injective (fun f : H →₀ K => (f : H → K)) DFunLike.coe_injective
have : Finite (MonoidAlgebra K H) :=
Finite.of_equiv (H →₀ K) MonoidAlgebra.coeffEquiv.symm
exact mul_eq_one_symm h
theorem regularOperator_unique {G : Type} [Group G] {n : ℕ}
{A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G)}
{T S : Hilbert G n →L[ℂ] Hilbert G n}
(hT : IsRegularOperator A T) (hS : IsRegularOperator A S) : T = S := by
ext ξ ⟨i, h⟩
exact (hT ξ i h).trans (hS ξ i h).symm
theorem fkDet_pos {G : Type} [Group G] {n : ℕ}
(T : Hilbert G n →L[ℂ] Hilbert G n) : 0 < fkDet T :=
Real.exp_pos _
section Defect
variable {R : Type uR} [Ring R]
theorem defect_idempotent {x y : R} (h : x * y = 1) :
(1 - y * x) * (1 - y * x) = 1 - y * x := by
have hy : y * x * (y * x) = y * x := by
calc
y * x * (y * x) = y * (x * y) * x := by simp only [mul_assoc]
_ = y * x := by rw [h, mul_one]
calc
(1 - y * x) * (1 - y * x) = 1 - y * x - y * x + y * x * (y * x) := by
noncomm_ring
_ = 1 - y * x := by rw [hy]; noncomm_ring
theorem mul_defect {x y : R} (h : x * y = 1) : x * (1 - y * x) = 0 := by
simp only [mul_sub, mul_one, ← mul_assoc, h, one_mul, sub_self]
theorem defect_mul {x y : R} (h : x * y = 1) : (1 - y * x) * y = 0 := by
simp only [sub_mul, one_mul, mul_assoc, h, mul_one, sub_self]
theorem defect_ne_zero {x y : R} (h : y * x ≠ 1) : 1 - y * x ≠ 0 := by
exact sub_ne_zero.mpr h.symm
end Defect
section MatrixDefect
variable {R : Type uR} {d : Type ud} {l : Type ul} [Ring R] [Fintype d] [DecidableEq d]
theorem defect_factorization (X Y : Matrix d d R) (hXY : X * Y = 1)
(f : Matrix l d R) (w : Matrix d l R) :
Matrix.fromRows X (f * (1 - Y * X)) * Matrix.fromCols Y ((1 - Y * X) * w) =
Matrix.fromBlocks 1 0 0 (f * (1 - Y * X) * w) := by
rw [Matrix.fromRows_mul_fromCols, hXY]
congr 1
· rw [← Matrix.mul_assoc, mul_defect hXY, Matrix.zero_mul]
· rw [Matrix.mul_assoc, defect_mul hXY, Matrix.mul_zero]
· rw [← Matrix.mul_assoc, Matrix.mul_assoc f, defect_idempotent hXY]
end MatrixDefect
section CornerLifting
variable {R : Type uR} [Ring R]
def cornerSeries (e z : R) (k : ℕ) : R :=
e * ∑ i ∈ Finset.range k, (e - z) ^ i
theorem cornerSeries_spec (e z : R) (he : e * e = e)
(hez : e * z = z) (hze : z * e = z) (k : ℕ) (hk : (e - z) ^ k = 0) :
cornerSeries e z k * z = e ∧ z * cornerSeries e z k = e ∧
e * cornerSeries e z k = cornerSeries e z k ∧
cornerSeries e z k * e = cornerSeries e z k := by
let w := e - z
let s := ∑ i ∈ Finset.range k, w ^ i
have hew : e * w = w := by simp only [w, mul_sub, he, hez]
have hwe : w * e = w := by simp only [w, sub_mul, he, hze]
have hew' : Commute e w := hew.trans hwe.symm
have hes : Commute e s :=
Commute.sum_right _ _ _ (fun i _ => hew'.pow_right i)
have hz : z = e * (1 - w) := by simp only [mul_sub, mul_one, hew, w]; noncomm_ring
have hinv : s * (1 - w) = 1 := by
simpa only [w, hk, sub_zero] using geom_sum_mul_neg w k
have hinv' : (1 - w) * s = 1 := by
simpa only [w, hk, sub_zero] using mul_neg_geom_sum w k
change (e * s) * z = e ∧ z * (e * s) = e ∧
e * (e * s) = e * s ∧ (e * s) * e = e * s
have hq : (e * s) * e = e * s := by
rw [mul_assoc, ← hes.eq, ← mul_assoc, he]
refine ⟨?_, ?_, ?_, hq⟩
· rw [hz, ← mul_assoc, hq, mul_assoc, hinv, mul_one]
· calc
z * (e * s) = z * s := by rw [← mul_assoc, hze]
_ = e := by rw [hz, mul_assoc, hinv', mul_one]
· rw [← mul_assoc, he]
theorem lift_rectangular_factorization {m : Type um} {r : Type ur} [Fintype m] [DecidableEq m]
[Fintype r] (e : Matrix m m R) (he : e * e = e)
(u₀ : Matrix m r R) (v₀ : Matrix r m R) (k : ℕ)
(hk : (e - e * u₀ * v₀ * e) ^ k = 0) :
∃ u : Matrix m r R, ∃ v : Matrix r m R, u * v = e := by
let z := e * u₀ * v₀ * e
have hez : e * z = z := by
dsimp [z]
rw [← Matrix.mul_assoc, ← Matrix.mul_assoc, ← Matrix.mul_assoc, he]
have hze : z * e = z := by
dsimp [z]
rw [Matrix.mul_assoc (e * u₀ * v₀), he]
obtain ⟨hqz, _, _, _⟩ := cornerSeries_spec e z he hez hze k hk
refine ⟨cornerSeries e z k * e * u₀, v₀ * e, ?_⟩
simpa only [z, Matrix.mul_assoc] using hqz
end CornerLifting
end GroupRingDeterminant
end
end OAI
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