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[Merged by Bors] - chore(QuadraticForm): Fintype -> Finite, DecidableEq #10462

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[Merged by Bors] - chore(QuadraticForm): Fintype -> Finite, DecidableEq #10462

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chore(QuadraticForm): `Fintype -> Finite`, `DecidableEq`
urkud Feb 12, 2024
cc71c0a
Fix
urkud Feb 12, 2024
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9 changes: 6 additions & 3 deletions Mathlib/LinearAlgebra/QuadraticForm/Basic.lean
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Expand Up @@ -1270,20 +1270,23 @@ open Finset BilinForm

variable [CommSemiring R] [AddCommMonoid M] [Module R M]

variable {ι : Type*} [Fintype ι] {v : Basis ι R M}
variable {ι : Type*}

/-- Given a quadratic form `Q` and a basis, `basisRepr` is the basis representation of `Q`. -/
noncomputable def basisRepr (Q : QuadraticForm R M) (v : Basis ι R M) : QuadraticForm R (ι → R) :=
noncomputable def basisRepr [Finite ι] (Q : QuadraticForm R M) (v : Basis ι R M) :
QuadraticForm R (ι → R) :=
Q.comp v.equivFun.symm
#align quadratic_form.basis_repr QuadraticForm.basisRepr

@[simp]
theorem basisRepr_apply (Q : QuadraticForm R M) (w : ι → R) :
theorem basisRepr_apply [Fintype ι] {v : Basis ι R M} (Q : QuadraticForm R M) (w : ι → R) :
Q.basisRepr v w = Q (∑ i : ι, w i • v i) := by
rw [← v.equivFun_symm_apply]
rfl
#align quadratic_form.basis_repr_apply QuadraticForm.basisRepr_apply

variable [Fintype ι] {v : Basis ι R M}

section

variable (R)
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5 changes: 3 additions & 2 deletions Mathlib/LinearAlgebra/QuadraticForm/Complex.lean
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Expand Up @@ -27,7 +27,7 @@ variable {ι : Type*} [Fintype ι]

/-- The isometry between a weighted sum of squares on the complex numbers and the
sum of squares, i.e. `weightedSumSquares` with weights 1 or 0. -/
noncomputable def isometryEquivSumSquares [DecidableEq ι] (w' : ι → ℂ) :
noncomputable def isometryEquivSumSquares (w' : ι → ℂ) :
IsometryEquiv (weightedSumSquares ℂ w')
(weightedSumSquares ℂ (fun i => if w' i = 0 then 0 else 1 : ι → ℂ)) := by
let w i := if h : w' i = 0 then (1 : Units ℂ) else Units.mk0 (w' i) h
Expand All @@ -41,6 +41,7 @@ noncomputable def isometryEquivSumSquares [DecidableEq ι] (w' : ι → ℂ) :
refine' sum_congr rfl fun j hj => _
have hsum : (∑ i : ι, v i • ((isUnit_iff_ne_zero.2 <| hw' i).unit : ℂ) • (Pi.basisFun ℂ ι) i) j =
v j • w j ^ (-(1 / 2 : ℂ)) := by
classical
rw [Finset.sum_apply, sum_eq_single j, Pi.basisFun_apply, IsUnit.unit_spec,
LinearMap.stdBasis_apply, Pi.smul_apply, Pi.smul_apply, Function.update_same, smul_eq_mul,
smul_eq_mul, smul_eq_mul, mul_one]
Expand All @@ -64,7 +65,7 @@ noncomputable def isometryEquivSumSquares [DecidableEq ι] (w' : ι → ℂ) :

/-- The isometry between a weighted sum of squares on the complex numbers and the
sum of squares, i.e. `weightedSumSquares` with weight `fun (i : ι) => 1`. -/
noncomputable def isometryEquivSumSquaresUnits [DecidableEq ι] (w : ι → Units ℂ) :
noncomputable def isometryEquivSumSquaresUnits (w : ι → Units ℂ) :
IsometryEquiv (weightedSumSquares ℂ w) (weightedSumSquares ℂ (1 : ι → ℂ)) := by
simpa using isometryEquivSumSquares ((↑) ∘ w)
#align quadratic_form.isometry_sum_squares_units QuadraticForm.isometryEquivSumSquaresUnits
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12 changes: 5 additions & 7 deletions Mathlib/LinearAlgebra/QuadraticForm/Real.lean
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Expand Up @@ -32,7 +32,7 @@ variable {ι : Type*} [Fintype ι]

/-- The isometry between a weighted sum of squares with weights `u` on the
(non-zero) real numbers and the weighted sum of squares with weights `sign ∘ u`. -/
noncomputable def isometryEquivSignWeightedSumSquares [DecidableEq ι] (w : ι → ℝ) :
noncomputable def isometryEquivSignWeightedSumSquares (w : ι → ℝ) :
IsometryEquiv (weightedSumSquares ℝ w) (weightedSumSquares ℝ (Real.sign ∘ w)) := by
let u i := if h : w i = 0 then (1 : ℝˣ) else Units.mk0 (w i) h
have hu' : ∀ i : ι, (Real.sign (u i) * u i) ^ (-(1 / 2 : ℝ)) ≠ 0 := by
Expand All @@ -47,6 +47,7 @@ noncomputable def isometryEquivSignWeightedSumSquares [DecidableEq ι] (w : ι
have hsum :
(∑ i : ι, v i • ((isUnit_iff_ne_zero.2 <| hu' i).unit : ℝ) • (Pi.basisFun ℝ ι) i) j =
v j • (Real.sign (u j) * u j) ^ (-(1 / 2 : ℝ)) := by
classical
rw [Finset.sum_apply, sum_eq_single j, Pi.basisFun_apply, IsUnit.unit_spec,
LinearMap.stdBasis_apply, Pi.smul_apply, Pi.smul_apply, Function.update_same, smul_eq_mul,
smul_eq_mul, smul_eq_mul, mul_one]
Expand All @@ -63,13 +64,10 @@ noncomputable def isometryEquivSignWeightedSumSquares [DecidableEq ι] (w : ι
have hwu : w j = u j := by simp only [dif_neg h, Units.val_mk0]
simp only [Units.val_mk0]
rw [hwu]
suffices
(u j : ℝ).sign * v j * v j =
suffices (u j : ℝ).sign * v j * v j =
(Real.sign (u j) * u j) ^ (-(1 / 2 : ℝ)) * (Real.sign (u j) * u j) ^ (-(1 / 2 : ℝ)) *
u j *
v j *
v j
by erw [← mul_assoc, this]; ring
u j * v j * v j by
erw [← mul_assoc, this]; ring
rw [← Real.rpow_add (sign_mul_pos_of_ne_zero _ <| Units.ne_zero _),
show -(1 / 2 : ℝ) + -(1 / 2) = -1 by ring, Real.rpow_neg_one, mul_inv, inv_sign,
mul_assoc (Real.sign (u j)) (u j)⁻¹, inv_mul_cancel (Units.ne_zero _), mul_one]
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