refactor: Quad Lin equations
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jstoobysmith
parent
772e78ca77
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b5dd319eed
14 changed files with 245 additions and 299 deletions
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@ -20,11 +20,9 @@ In particular a HomogeneousQuadratic should be a map `V →ₗ[ℚ] V →ₗ[ℚ
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-/
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/-- The structure defining a homogeneous quadratic equation. -/
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structure HomogeneousQuadratic (V : Type) [AddCommMonoid V] [Module ℚ V] where
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/-- The quadratic equation. -/
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toFun : V → ℚ
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/-- The equation is homogeneous. -/
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map_smul' : ∀ a S, toFun (a • S) = a ^ 2 * toFun S
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@[simp]
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def HomogeneousQuadratic (V : Type) [AddCommMonoid V] [Module ℚ V] : Type :=
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V →ₑ[((fun a => a ^ 2) : ℚ → ℚ)] ℚ
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namespace HomogeneousQuadratic
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@ -37,71 +35,15 @@ instance instFun : FunLike (HomogeneousQuadratic V) V ℚ where
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cases g
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simp_all
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/-- Given an equivariant function from `V` to `ℚ` we get a `HomogeneousQuadratic V`. -/
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def fromEquivariant (f : V →ₑ[((fun a => a ^ 2) : ℚ → ℚ)] ℚ) : HomogeneousQuadratic V where
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toFun := f
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map_smul' a S := by
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rw [map_smulₛₗ]
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rfl
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lemma map_smul (f : HomogeneousQuadratic V) (a : ℚ) (S : V) : f (a • S) = a ^ 2 * f S :=
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f.map_smul' a S
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end HomogeneousQuadratic
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/-- The structure of a bilinear map. -/
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structure BiLinear (V : Type) [AddCommMonoid V] [Module ℚ V] where
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/-- The underling function. -/
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toFun : V × V → ℚ
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map_smul₁' : ∀ a S T, toFun (a • S, T) = a * toFun (S, T)
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map_smul₂' : ∀ a S T , toFun (S, a • T) = a * toFun (S, T)
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map_add₁' : ∀ S1 S2 T, toFun (S1 + S2, T) = toFun (S1, T) + toFun (S2, T)
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map_add₂' : ∀ S T1 T2, toFun (S, T1 + T2) = toFun (S, T1) + toFun (S, T2)
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namespace BiLinear
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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instance instFun : FunLike (BiLinear V) (V × V) ℚ where
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coe f := f.toFun
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coe_injective' f g h := by
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cases f
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cases g
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simp_all
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/-- A bilinear map from a linear function ` V →ₗ[ℚ] V →ₗ[ℚ] ℚ`. -/
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def fromLinearToHom (f : V →ₗ[ℚ] V →ₗ[ℚ] ℚ) : BiLinear V where
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toFun S := f S.1 S.2
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map_smul₁' a S T := by
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simp only [LinearMapClass.map_smul]
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rfl
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map_smul₂' a S T := (f S).map_smul a T
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map_add₁' S1 S2 T := by
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simp only [f.map_add]
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rfl
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map_add₂' S T1 T2 := (f S).map_add T1 T2
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lemma map_smul₁ (f : BiLinear V) (a : ℚ) (S T : V) : f (a • S, T) = a * f (S, T) :=
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f.map_smul₁' a S T
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lemma map_smul₂ (f : BiLinear V) (S : V) (a : ℚ) (T : V) : f (S, a • T) = a * f (S, T) :=
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f.map_smul₂' a S T
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lemma map_add₁ (f : BiLinear V) (S1 S2 T : V) : f (S1 + S2, T) = f (S1, T) + f (S2, T) :=
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f.map_add₁' S1 S2 T
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lemma map_add₂ (f : BiLinear V) (S : V) (T1 T2 : V) : f (S, T1 + T2) = f (S, T1) + f (S, T2) :=
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f.map_add₂' S T1 T2
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end BiLinear
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/-- The structure of a symmetric bilinear function. -/
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structure BiLinearSymm (V : Type) [AddCommMonoid V] [Module ℚ V] where
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/-- The underlying function. -/
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toFun : V × V → ℚ
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map_smul₁' : ∀ a S T, toFun (a • S, T) = a * toFun (S, T)
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map_add₁' : ∀ S1 S2 T, toFun (S1 + S2, T) = toFun (S1, T) + toFun (S2, T)
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swap' : ∀ S T, toFun (S, T) = toFun (T, S)
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structure BiLinearSymm (V : Type) [AddCommMonoid V] [Module ℚ V] extends V →ₗ[ℚ] V →ₗ[ℚ] ℚ where
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swap' : ∀ S T, toFun S T = toFun T S
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/-- A symmetric bilinear function. -/
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class IsSymmetric {V : Type} [AddCommMonoid V] [Module ℚ V] (f : V →ₗ[ℚ] V →ₗ[ℚ] ℚ) : Prop where
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@ -113,61 +55,79 @@ open BigOperators
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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instance instFun (V : Type) [AddCommMonoid V] [Module ℚ V] :
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FunLike (BiLinearSymm V) (V × V) ℚ where
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FunLike (BiLinearSymm V) (V) (V →ₗ[ℚ] ℚ ) where
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coe f := f.toFun
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coe_injective' f g h := by
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cases f
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cases g
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simp_all
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/-- A bilinear symmetric function from a symemtric `V →ₗ[ℚ] V →ₗ[ℚ] ℚ`. -/
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def fromSymmToHom (f : V →ₗ[ℚ] V →ₗ[ℚ] ℚ) [IsSymmetric f] : BiLinearSymm V where
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toFun S := f S.1 S.2
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map_smul₁' a S T := by
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simp only [LinearMapClass.map_smul]
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rfl
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map_add₁' S1 S2 T := by
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simp only [f.map_add]
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rfl
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swap' S T := IsSymmetric.swap S T
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@[simps!]
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def mk₂ (f : V × V → ℚ) (map_smul : ∀ a S T, f (a • S, T) = a * f (S, T))
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(map_add : ∀ S1 S2 T, f (S1 + S2, T) = f (S1, T) + f (S2, T))
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(swap : ∀ S T, f (S, T) = f (T, S)) : BiLinearSymm V where
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toFun := fun S => {
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toFun := fun T => f (S, T)
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map_add' := by
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intro T1 T2
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simp
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rw [swap, map_add]
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rw [swap T1 S, swap T2 S]
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map_smul' :=by
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intro a T
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simp
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rw [swap, map_smul]
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rw [swap T S]
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}
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map_smul' := by
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intro a S
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apply LinearMap.ext
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intro T
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exact map_smul a S T
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map_add' := by
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intro S1 S2
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apply LinearMap.ext
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intro T
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exact map_add S1 S2 T
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swap' := swap
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lemma toFun_eq_coe (f : BiLinearSymm V) : f.toFun = f := rfl
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lemma map_smul₁ (f : BiLinearSymm V) (a : ℚ) (S T : V) : f (a • S) T = a * f S T := by
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erw [f.map_smul a S]
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rfl
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lemma map_smul₁ (f : BiLinearSymm V) (a : ℚ) (S T : V) : f (a • S, T) = a * f (S, T) :=
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f.map_smul₁' a S T
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lemma swap (f : BiLinearSymm V) (S T : V) : f (S, T) = f (T, S) :=
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lemma swap (f : BiLinearSymm V) (S T : V) : f S T = f T S :=
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f.swap' S T
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lemma map_smul₂ (f : BiLinearSymm V) (a : ℚ) (S : V) (T : V) : f (S, a • T) = a * f (S, T) := by
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lemma map_smul₂ (f : BiLinearSymm V) (a : ℚ) (S : V) (T : V) : f S (a • T) = a * f S T := by
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rw [f.swap, f.map_smul₁, f.swap]
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lemma map_add₁ (f : BiLinearSymm V) (S1 S2 T : V) : f (S1 + S2, T) = f (S1, T) + f (S2, T) :=
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f.map_add₁' S1 S2 T
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lemma map_add₁ (f : BiLinearSymm V) (S1 S2 T : V) : f (S1 + S2) T = f S1 T + f S2 T := by
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erw [f.map_add]
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rfl
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lemma map_add₂ (f : BiLinearSymm V) (S : V) (T1 T2 : V) :
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f (S, T1 + T2) = f (S, T1) + f (S, T2) := by
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f S (T1 + T2) = f S T1 + f S T2 := by
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rw [f.swap, f.map_add₁, f.swap T1 S, f.swap T2 S]
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/-- Fixing the second input vectors, the resulting linear map. -/
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def toLinear₁ (f : BiLinearSymm V) (T : V) : V →ₗ[ℚ] ℚ where
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toFun S := f (S, T)
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toFun S := f S T
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map_add' S1 S2 := by
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simp only [f.map_add₁]
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map_smul' a S := by
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simp only [f.map_smul₁]
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rfl
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lemma toLinear₁_apply (f : BiLinearSymm V) (S T : V) : f (S, T) = f.toLinear₁ T S := rfl
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lemma toLinear₁_apply (f : BiLinearSymm V) (S T : V) : f S T = f.toLinear₁ T S := rfl
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lemma map_sum₁ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
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f (∑ i, S i, T) = ∑ i, f (S i, T) := by
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f (∑ i, S i) T = ∑ i, f (S i) T := by
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rw [f.toLinear₁_apply]
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rw [map_sum]
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rfl
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lemma map_sum₂ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
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f ( T, ∑ i, S i) = ∑ i, f (T, S i) := by
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f T (∑ i, S i) = ∑ i, f T (S i) := by
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rw [swap, map_sum₁]
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apply Fintype.sum_congr
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intro i
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@ -178,28 +138,28 @@ lemma map_sum₂ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
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@[simps!]
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def toHomogeneousQuad {V : Type} [AddCommMonoid V] [Module ℚ V]
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(τ : BiLinearSymm V) : HomogeneousQuadratic V where
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toFun S := τ (S, S)
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toFun S := τ S S
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map_smul' a S := by
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simp only
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rw [τ.map_smul₁, τ.map_smul₂]
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ring
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ring_nf
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rfl
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lemma toHomogeneousQuad_add {V : Type} [AddCommMonoid V] [Module ℚ V]
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(τ : BiLinearSymm V) (S T : V) :
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τ.toHomogeneousQuad (S + T) = τ.toHomogeneousQuad S +
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τ.toHomogeneousQuad T + 2 * τ (S, T) := by
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simp only [toHomogeneousQuad_toFun]
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rw [τ.map_add₁, τ.map_add₂, τ.map_add₂, τ.swap T S]
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τ.toHomogeneousQuad T + 2 * τ S T := by
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simp [toHomogeneousQuad_apply]
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rw [τ.map_add₁, τ.map_add₁, τ.swap T S]
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ring
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end BiLinearSymm
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/-- The structure of a homogeneous cubic equation. -/
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structure HomogeneousCubic (V : Type) [AddCommMonoid V] [Module ℚ V] where
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/-- The underlying function. -/
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toFun : V → ℚ
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map_smul' : ∀ a S, toFun (a • S) = a ^ 3 * toFun S
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@[simp]
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def HomogeneousCubic (V : Type) [AddCommMonoid V] [Module ℚ V] : Type :=
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V →ₑ[((fun a => a ^ 3) : ℚ → ℚ)] ℚ
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namespace HomogeneousCubic
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@ -217,29 +177,6 @@ lemma map_smul (f : HomogeneousCubic V) (a : ℚ) (S : V) : f (a • S) = a ^ 3
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end HomogeneousCubic
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/-- The structure of a trilinear function. -/
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structure TriLinear (V : Type) [AddCommMonoid V] [Module ℚ V] where
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/-- The underlying function. -/
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toFun : V × V × V → ℚ
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map_smul₁' : ∀ a S T L, toFun (a • S, T, L) = a * toFun (S, T, L)
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map_smul₂' : ∀ a S T L, toFun (S, a • T, L) = a * toFun (S, T, L)
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map_smul₃' : ∀ a S T L, toFun (S, T, a • L) = a * toFun (S, T, L)
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map_add₁' : ∀ S1 S2 T L, toFun (S1 + S2, T, L) = toFun (S1, T, L) + toFun (S2, T, L)
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map_add₂' : ∀ S T1 T2 L, toFun (S, T1 + T2, L) = toFun (S, T1, L) + toFun (S, T2, L)
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map_add₃' : ∀ S T L1 L2, toFun (S, T, L1 + L2) = toFun (S, T, L1) + toFun (S, T, L2)
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namespace TriLinear
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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instance instFun : FunLike (TriLinear V) (V × V × V) ℚ where
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coe f := f.toFun
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coe_injective' f g h := by
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cases f
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cases g
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simp_all
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end TriLinear
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/-- The structure of a symmetric trilinear function. -/
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structure TriLinearSymm' (V : Type) [AddCommMonoid V] [Module ℚ V] extends
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V →ₗ[ℚ] V →ₗ[ℚ] V →ₗ[ℚ] ℚ where
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@ -350,7 +287,7 @@ def toCubic {charges : Type} [AddCommMonoid charges] [Module ℚ charges]
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(τ : TriLinearSymm charges) : HomogeneousCubic charges where
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toFun S := τ (S, S, S)
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map_smul' a S := by
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simp only
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simp
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rw [τ.map_smul₁, τ.map_smul₂, τ.map_smul₃]
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ring
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@ -358,7 +295,7 @@ lemma toCubic_add {charges : Type} [AddCommMonoid charges] [Module ℚ charges]
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(τ : TriLinearSymm charges) (S T : charges) :
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τ.toCubic (S + T) = τ.toCubic S +
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τ.toCubic T + 3 * τ (S, S, T) + 3 * τ (T, T, S) := by
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simp only [toCubic_toFun]
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simp only [HomogeneousCubic, toCubic_apply]
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rw [τ.map_add₁, τ.map_add₂, τ.map_add₂, τ.map_add₃, τ.map_add₃, τ.map_add₃, τ.map_add₃]
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rw [τ.swap₂ S T S, τ.swap₁ T S S, τ.swap₂ S T S, τ.swap₂ T S T, τ.swap₁ S T T, τ.swap₂ T S T]
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ring
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@ -270,13 +270,13 @@ lemma accYY_ext {S T : MSSMCharges.charges}
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rw [hd, hu]
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rfl
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/-- The symmetric bilinear form used to define the quadratic ACC. -/
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@[simps!]
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def quadBiLin : BiLinearSymm MSSMCharges.charges where
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toFun S := ∑ i, (Q S.1 i * Q S.2 i + (- 2) * (U S.1 i * U S.2 i) +
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def quadBiLin : BiLinearSymm MSSMCharges.charges := BiLinearSymm.mk₂ (
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fun S => ∑ i, (Q S.1 i * Q S.2 i + (- 2) * (U S.1 i * U S.2 i) +
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D S.1 i * D S.2 i + (- 1) * (L S.1 i * L S.2 i) + E S.1 i * E S.2 i) +
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(- Hd S.1 * Hd S.2 + Hu S.1 * Hu S.2)
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map_smul₁' a S T := by
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(- Hd S.1 * Hd S.2 + Hu S.1 * Hu S.2))
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(by
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intro a S T
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simp only
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rw [mul_add]
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congr 1
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@ -287,8 +287,9 @@ def quadBiLin : BiLinearSymm MSSMCharges.charges where
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simp only [HSMul.hSMul, SMul.smul, toSMSpecies_apply, Fin.isValue, neg_mul, one_mul]
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ring
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simp only [map_smul, Hd_apply, Fin.reduceFinMk, Fin.isValue, smul_eq_mul, neg_mul, Hu_apply]
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ring
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map_add₁' S T R := by
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ring)
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(by
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intro S T R
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simp only
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rw [add_assoc, ← add_assoc (-Hd S * Hd R + Hu S * Hu R) _ _]
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rw [add_comm (-Hd S * Hd R + Hu S * Hu R) _]
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@ -303,15 +304,17 @@ def quadBiLin : BiLinearSymm MSSMCharges.charges where
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one_mul]
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ring
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rw [Hd.map_add, Hu.map_add]
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ring
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swap' S L := by
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ring)
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(by
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intro S L
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simp only [MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue, neg_mul, one_mul,
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Hd_apply, Fin.reduceFinMk, Hu_apply]
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congr 1
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rw [Fin.sum_univ_three, Fin.sum_univ_three]
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simp only [Fin.isValue]
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ring
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ring
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ring)
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/-- The quadratic ACC. -/
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@[simp]
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@ -323,10 +326,9 @@ lemma accQuad_ext {S T : (MSSMCharges).charges}
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∑ i, ((fun a => a^2) ∘ toSMSpecies j T) i)
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(hd : Hd S = Hd T) (hu : Hu S = Hu T) :
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accQuad S = accQuad T := by
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simp only [accQuad, BiLinearSymm.toHomogeneousQuad_toFun]
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simp only [HomogeneousQuadratic, accQuad, BiLinearSymm.toHomogeneousQuad_apply]
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erw [← quadBiLin.toFun_eq_coe]
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rw [quadBiLin]
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simp only
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simp only [quadBiLin, BiLinearSymm.mk₂, AddHom.toFun_eq_coe, AddHom.coe_mk, LinearMap.coe_mk]
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repeat erw [Finset.sum_add_distrib]
|
||||
repeat erw [← Finset.mul_sum]
|
||||
ring_nf
|
||||
|
|
@ -337,6 +339,7 @@ lemma accQuad_ext {S T : (MSSMCharges).charges}
|
|||
repeat rw [h1]
|
||||
rw [hd, hu]
|
||||
|
||||
|
||||
/-- The function underlying the symmetric trilinear form used to define the cubic ACC. -/
|
||||
@[simp]
|
||||
def cubeTriLinToFun
|
||||
|
|
@ -461,7 +464,7 @@ open MSSMCharges
|
|||
|
||||
lemma quadSol (S : MSSMACC.QuadSols) : accQuad S.val = 0 := by
|
||||
have hS := S.quadSol
|
||||
simp [MSSMACCs.accQuad, HomogeneousQuadratic.toFun] at hS
|
||||
simp only [MSSMACC_numberQuadratic, HomogeneousQuadratic, MSSMACC_quadraticACCs] at hS
|
||||
exact hS 0
|
||||
|
||||
/-- A solution from a charge satisfying the ACCs. -/
|
||||
|
|
@ -523,30 +526,33 @@ lemma AnomalyFreeMk''_val (S : MSSMACC.QuadSols)
|
|||
|
||||
/-- The dot product on the vector space of charges. -/
|
||||
@[simps!]
|
||||
def dot : BiLinearSymm MSSMCharges.charges where
|
||||
toFun S := ∑ i, (Q S.1 i * Q S.2 i + U S.1 i * U S.2 i +
|
||||
def dot : BiLinearSymm MSSMCharges.charges := BiLinearSymm.mk₂
|
||||
(fun S => ∑ i, (Q S.1 i * Q S.2 i + U S.1 i * U S.2 i +
|
||||
D S.1 i * D S.2 i + L S.1 i * L S.2 i + E S.1 i * E S.2 i
|
||||
+ N S.1 i * N S.2 i) + Hd S.1 * Hd S.2 + Hu S.1 * Hu S.2
|
||||
map_smul₁' a S T := by
|
||||
+ N S.1 i * N S.2 i) + Hd S.1 * Hd S.2 + Hu S.1 * Hu S.2)
|
||||
(by
|
||||
intro a S T
|
||||
simp only [MSSMSpecies_numberCharges]
|
||||
repeat rw [(toSMSpecies _).map_smul]
|
||||
rw [Hd.map_smul, Hu.map_smul]
|
||||
rw [Fin.sum_univ_three, Fin.sum_univ_three]
|
||||
simp only [HSMul.hSMul, SMul.smul, Fin.isValue, toSMSpecies_apply, Hd_apply, Fin.reduceFinMk,
|
||||
Hu_apply]
|
||||
ring
|
||||
map_add₁' S T R := by
|
||||
ring)
|
||||
(by
|
||||
intro S1 S2 T
|
||||
simp only [MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue,
|
||||
ACCSystemCharges.chargesAddCommMonoid_add, map_add, Hd_apply, Fin.reduceFinMk, Hu_apply]
|
||||
repeat erw [AddHom.map_add]
|
||||
rw [Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
|
||||
simp only [Fin.isValue]
|
||||
ring
|
||||
swap' S L := by
|
||||
ring)
|
||||
(by
|
||||
intro S T
|
||||
simp only [MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue, Hd_apply, Fin.reduceFinMk,
|
||||
Hu_apply]
|
||||
rw [Fin.sum_univ_three, Fin.sum_univ_three]
|
||||
simp only [Fin.isValue]
|
||||
ring
|
||||
ring)
|
||||
|
||||
end MSSMACC
|
||||
|
|
|
|||
|
|
@ -40,7 +40,7 @@ lemma lineY₃B₃Charges_quad (a b : ℚ) : accQuad (lineY₃B₃Charges a b).v
|
|||
rw [quadBiLin.toHomogeneousQuad.map_smul]
|
||||
rw [quadBiLin.map_smul₁, quadBiLin.map_smul₂]
|
||||
erw [quadSol Y₃.1, quadSol B₃.1]
|
||||
simp only [mul_zero, add_zero, quadBiLin_toFun, Fin.isValue, Fin.reduceFinMk, zero_add,
|
||||
simp only [mul_zero, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add,
|
||||
mul_eq_zero, OfNat.ofNat_ne_zero, false_or]
|
||||
apply Or.inr ∘ Or.inr
|
||||
rfl
|
||||
|
|
|
|||
|
|
@ -29,42 +29,42 @@ open BigOperators
|
|||
|
||||
/-- The type of linear solutions orthogonal to $Y_3$ and $B_3$. -/
|
||||
structure AnomalyFreePerp extends MSSMACC.LinSols where
|
||||
perpY₃ : dot (Y₃.val, val) = 0
|
||||
perpB₃ : dot (B₃.val, val) = 0
|
||||
perpY₃ : dot Y₃.val val = 0
|
||||
perpB₃ : dot B₃.val val = 0
|
||||
|
||||
/-- The projection of an object in `MSSMACC.AnomalyFreeLinear` onto the subspace
|
||||
orthgonal to `Y₃` and`B₃`. -/
|
||||
def proj (T : MSSMACC.LinSols) : MSSMACC.AnomalyFreePerp :=
|
||||
⟨(dot (B₃.val, T.val) - dot (Y₃.val, T.val)) • Y₃.1.1
|
||||
+ (dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val)) • B₃.1.1
|
||||
+ dot (Y₃.val, B₃.val) • T,
|
||||
⟨(dot B₃.val T.val - dot Y₃.val T.val) • Y₃.1.1
|
||||
+ (dot Y₃.val T.val - 2 * dot B₃.val T.val) • B₃.1.1
|
||||
+ dot Y₃.val B₃.val • T,
|
||||
by
|
||||
change dot (_, _ • Y₃.val + _ • B₃.val + _ • T.val) = 0
|
||||
change dot _ (_ • Y₃.val + _ • B₃.val + _ • T.val) = 0
|
||||
rw [dot.map_add₂, dot.map_add₂]
|
||||
rw [dot.map_smul₂, dot.map_smul₂, dot.map_smul₂]
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl]
|
||||
rw [show dot (Y₃.val, Y₃.val) = 216 by rfl]
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl]
|
||||
rw [show dot Y₃.val Y₃.val = 216 by rfl]
|
||||
ring,
|
||||
by
|
||||
change dot (_, _ • Y₃.val + _ • B₃.val + _ • T.val) = 0
|
||||
change dot _ (_ • Y₃.val + _ • B₃.val + _ • T.val) = 0
|
||||
rw [dot.map_add₂, dot.map_add₂]
|
||||
rw [dot.map_smul₂, dot.map_smul₂, dot.map_smul₂]
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl]
|
||||
rw [show dot (B₃.val, Y₃.val) = 108 by rfl]
|
||||
rw [show dot (B₃.val, B₃.val) = 108 by rfl]
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl]
|
||||
rw [show dot B₃.val Y₃.val = 108 by rfl]
|
||||
rw [show dot B₃.val B₃.val = 108 by rfl]
|
||||
ring⟩
|
||||
|
||||
lemma proj_val (T : MSSMACC.LinSols) :
|
||||
(proj T).val = (dot (B₃.val, T.val) - dot (Y₃.val, T.val)) • Y₃.val +
|
||||
( (dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val))) • B₃.val +
|
||||
dot (Y₃.val, B₃.val) • T.val := by
|
||||
(proj T).val = (dot B₃.val T.val - dot Y₃.val T.val) • Y₃.val +
|
||||
( (dot Y₃.val T.val - 2 * dot B₃.val T.val)) • B₃.val +
|
||||
dot Y₃.val B₃.val • T.val := by
|
||||
rfl
|
||||
|
||||
lemma Y₃_plus_B₃_plus_proj (T : MSSMACC.LinSols) (a b c : ℚ) :
|
||||
a • Y₃.val + b • B₃.val + c • (proj T).val =
|
||||
(a + c * (dot (B₃.val, T.val) - dot (Y₃.val, T.val))) • Y₃.val
|
||||
+ (b + c * (dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val))) • B₃.val
|
||||
+ (dot (Y₃.val, B₃.val) * c) • T.val:= by
|
||||
(a + c * (dot B₃.val T.val - dot Y₃.val T.val)) • Y₃.val
|
||||
+ (b + c * (dot Y₃.val T.val - 2 * dot B₃.val T.val)) • B₃.val
|
||||
+ (dot Y₃.val B₃.val * c) • T.val:= by
|
||||
rw [proj_val]
|
||||
rw [DistribMulAction.smul_add, DistribMulAction.smul_add]
|
||||
rw [add_assoc (_ • _ • Y₃.val), ← add_assoc (_ • Y₃.val + _ • B₃.val), add_assoc (_ • Y₃.val)]
|
||||
|
|
@ -81,27 +81,27 @@ lemma Y₃_plus_B₃_plus_proj (T : MSSMACC.LinSols) (a b c : ℚ) :
|
|||
|
||||
|
||||
lemma quad_Y₃_proj (T : MSSMACC.LinSols) :
|
||||
quadBiLin (Y₃.val, (proj T).val) = dot (Y₃.val, B₃.val) * quadBiLin (Y₃.val, T.val) := by
|
||||
quadBiLin Y₃.val (proj T).val = dot Y₃.val B₃.val * quadBiLin Y₃.val T.val := by
|
||||
rw [proj_val]
|
||||
rw [quadBiLin.map_add₂, quadBiLin.map_add₂]
|
||||
rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂]
|
||||
rw [show quadBiLin (Y₃.val, B₃.val) = 0 by rfl]
|
||||
rw [show quadBiLin (Y₃.val, Y₃.val) = 0 by rfl]
|
||||
rw [show quadBiLin Y₃.val B₃.val = 0 by rfl]
|
||||
rw [show quadBiLin Y₃.val Y₃.val = 0 by rfl]
|
||||
ring
|
||||
|
||||
lemma quad_B₃_proj (T : MSSMACC.LinSols) :
|
||||
quadBiLin (B₃.val, (proj T).val) = dot (Y₃.val, B₃.val) * quadBiLin (B₃.val, T.val) := by
|
||||
quadBiLin B₃.val (proj T).val = dot Y₃.val B₃.val * quadBiLin B₃.val T.val := by
|
||||
rw [proj_val]
|
||||
rw [quadBiLin.map_add₂, quadBiLin.map_add₂]
|
||||
rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂]
|
||||
rw [show quadBiLin (B₃.val, Y₃.val) = 0 by rfl]
|
||||
rw [show quadBiLin (B₃.val, B₃.val) = 0 by rfl]
|
||||
rw [show quadBiLin B₃.val Y₃.val = 0 by rfl]
|
||||
rw [show quadBiLin B₃.val B₃.val = 0 by rfl]
|
||||
ring
|
||||
|
||||
lemma quad_self_proj (T : MSSMACC.Sols) :
|
||||
quadBiLin (T.val, (proj T.1.1).val) =
|
||||
(dot (B₃.val, T.val) - dot (Y₃.val, T.val)) * quadBiLin (Y₃.val, T.val) +
|
||||
(dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val)) * quadBiLin (B₃.val, T.val) := by
|
||||
quadBiLin T.val (proj T.1.1).val =
|
||||
(dot B₃.val T.val - dot Y₃.val T.val) * quadBiLin Y₃.val T.val +
|
||||
(dot Y₃.val T.val - 2 * dot B₃.val T.val) * quadBiLin B₃.val T.val := by
|
||||
rw [proj_val]
|
||||
rw [quadBiLin.map_add₂, quadBiLin.map_add₂]
|
||||
rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂]
|
||||
|
|
@ -110,9 +110,9 @@ lemma quad_self_proj (T : MSSMACC.Sols) :
|
|||
ring
|
||||
|
||||
lemma quad_proj (T : MSSMACC.Sols) :
|
||||
quadBiLin ((proj T.1.1).val, (proj T.1.1).val) = 2 * (dot (Y₃.val, B₃.val)) *
|
||||
((dot (B₃.val, T.val) - dot (Y₃.val, T.val)) * quadBiLin (Y₃.val, T.val) +
|
||||
(dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val)) * quadBiLin (B₃.val, T.val) ) := by
|
||||
quadBiLin (proj T.1.1).val (proj T.1.1).val = 2 * dot Y₃.val B₃.val *
|
||||
((dot B₃.val T.val - dot Y₃.val T.val) * quadBiLin Y₃.val T.val +
|
||||
(dot Y₃.val T.val - 2 * dot B₃.val T.val) * quadBiLin B₃.val T.val ) := by
|
||||
nth_rewrite 1 [proj_val]
|
||||
repeat rw [quadBiLin.map_add₁]
|
||||
repeat rw [quadBiLin.map_smul₁]
|
||||
|
|
@ -122,7 +122,7 @@ lemma quad_proj (T : MSSMACC.Sols) :
|
|||
|
||||
lemma cube_proj_proj_Y₃ (T : MSSMACC.LinSols) :
|
||||
cubeTriLin ((proj T).val, (proj T).val, Y₃.val) =
|
||||
(dot (Y₃.val, B₃.val))^2 * cubeTriLin (T.val, T.val, Y₃.val):= by
|
||||
(dot Y₃.val B₃.val)^2 * cubeTriLin (T.val, T.val, Y₃.val) := by
|
||||
rw [proj_val]
|
||||
rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂]
|
||||
erw [lineY₃B₃_doublePoint]
|
||||
|
|
@ -144,7 +144,7 @@ lemma cube_proj_proj_Y₃ (T : MSSMACC.LinSols) :
|
|||
|
||||
lemma cube_proj_proj_B₃ (T : MSSMACC.LinSols) :
|
||||
cubeTriLin ((proj T).val, (proj T).val, B₃.val) =
|
||||
(dot (Y₃.val, B₃.val))^2 * cubeTriLin (T.val, T.val, B₃.val):= by
|
||||
(dot Y₃.val B₃.val)^2 * cubeTriLin (T.val, T.val, B₃.val) := by
|
||||
rw [proj_val]
|
||||
rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂]
|
||||
erw [lineY₃B₃_doublePoint]
|
||||
|
|
@ -160,9 +160,9 @@ lemma cube_proj_proj_B₃ (T : MSSMACC.LinSols) :
|
|||
|
||||
lemma cube_proj_proj_self (T : MSSMACC.Sols) :
|
||||
cubeTriLin ((proj T.1.1).val, (proj T.1.1).val, T.val) =
|
||||
2 * dot (Y₃.val, B₃.val) *
|
||||
((dot (B₃.val, T.val) - dot (Y₃.val, T.val)) * cubeTriLin (T.val, T.val, Y₃.val) +
|
||||
( dot (Y₃.val, T.val)- 2 * dot (B₃.val, T.val)) * cubeTriLin (T.val, T.val, B₃.val)) := by
|
||||
2 * dot Y₃.val B₃.val *
|
||||
((dot B₃.val T.val - dot Y₃.val T.val) * cubeTriLin (T.val, T.val, Y₃.val) +
|
||||
( dot Y₃.val T.val- 2 * dot B₃.val T.val) * cubeTriLin (T.val, T.val, B₃.val)) := by
|
||||
rw [proj_val]
|
||||
rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂]
|
||||
erw [lineY₃B₃_doublePoint]
|
||||
|
|
@ -177,9 +177,9 @@ lemma cube_proj_proj_self (T : MSSMACC.Sols) :
|
|||
|
||||
lemma cube_proj (T : MSSMACC.Sols) :
|
||||
cubeTriLin ((proj T.1.1).val, (proj T.1.1).val, (proj T.1.1).val) =
|
||||
3 * dot (Y₃.val, B₃.val) ^ 2 *
|
||||
((dot (B₃.val, T.val) - dot (Y₃.val, T.val)) * cubeTriLin (T.val, T.val, Y₃.val) +
|
||||
(dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val)) * cubeTriLin (T.val, T.val, B₃.val)) := by
|
||||
3 * dot Y₃.val B₃.val ^ 2 *
|
||||
((dot B₃.val T.val - dot Y₃.val T.val) * cubeTriLin (T.val, T.val, Y₃.val) +
|
||||
(dot Y₃.val T.val - 2 * dot B₃.val T.val) * cubeTriLin (T.val, T.val, B₃.val)) := by
|
||||
nth_rewrite 3 [proj_val]
|
||||
repeat rw [cubeTriLin.map_add₃]
|
||||
repeat rw [cubeTriLin.map_smul₃]
|
||||
|
|
|
|||
|
|
@ -53,8 +53,8 @@ lemma planeY₃B₃_val_eq' (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (hR' : R
|
|||
(h : (planeY₃B₃ R a b c).val = (planeY₃B₃ R a' b' c').val) :
|
||||
a = a' ∧ b = b' ∧ c = c' := by
|
||||
rw [planeY₃B₃_val, planeY₃B₃_val] at h
|
||||
have h1 := congrArg (fun S => dot (Y₃.val, S)) h
|
||||
have h2 := congrArg (fun S => dot (B₃.val, S)) h
|
||||
have h1 := congrArg (fun S => dot Y₃.val S) h
|
||||
have h2 := congrArg (fun S => dot B₃.val S) h
|
||||
simp only [ Fin.isValue, ACCSystemCharges.chargesAddCommMonoid_add, Fin.reduceFinMk] at h1 h2
|
||||
erw [dot.map_add₂, dot.map_add₂] at h1 h2
|
||||
erw [dot.map_add₂ Y₃.val (a' • Y₃.val + b' • B₃.val) (c' • R.val)] at h1
|
||||
|
|
@ -64,10 +64,10 @@ lemma planeY₃B₃_val_eq' (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (hR' : R
|
|||
rw [dot.map_smul₂, dot.map_smul₂, dot.map_smul₂] at h1 h2
|
||||
rw [R.perpY₃] at h1
|
||||
rw [R.perpB₃] at h2
|
||||
rw [show dot (Y₃.val, Y₃.val) = 216 by rfl] at h1
|
||||
rw [show dot (B₃.val, B₃.val) = 108 by rfl] at h2
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h1
|
||||
rw [show dot (B₃.val, Y₃.val) = 108 by rfl] at h2
|
||||
rw [show dot Y₃.val Y₃.val = 216 by rfl] at h1
|
||||
rw [show dot B₃.val B₃.val = 108 by rfl] at h2
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl] at h1
|
||||
rw [show dot B₃.val Y₃.val = 108 by rfl] at h2
|
||||
simp_all
|
||||
have ha : a = a' := by
|
||||
linear_combination h1 / 108 + -1 * h2 / 108
|
||||
|
|
@ -102,15 +102,15 @@ lemma planeY₃B₃_val_eq' (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (hR' : R
|
|||
|
||||
|
||||
lemma planeY₃B₃_quad (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
|
||||
accQuad (planeY₃B₃ R a b c).val = c * (2 * a * quadBiLin (Y₃.val, R.val)
|
||||
+ 2 * b * quadBiLin (B₃.val, R.val) + c * quadBiLin (R.val, R.val)) := by
|
||||
accQuad (planeY₃B₃ R a b c).val = c * (2 * a * quadBiLin Y₃.val R.val
|
||||
+ 2 * b * quadBiLin B₃.val R.val + c * quadBiLin R.val R.val) := by
|
||||
rw [planeY₃B₃_val]
|
||||
erw [BiLinearSymm.toHomogeneousQuad_add]
|
||||
erw [lineY₃B₃Charges_quad]
|
||||
rw [quadBiLin.toHomogeneousQuad.map_smul]
|
||||
rw [quadBiLin.map_add₁, quadBiLin.map_smul₁, quadBiLin.map_smul₁]
|
||||
rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂]
|
||||
rw [show (BiLinearSymm.toHomogeneousQuad quadBiLin) R.val = quadBiLin (R.val, R.val) by rfl]
|
||||
rw [show (BiLinearSymm.toHomogeneousQuad quadBiLin) R.val = quadBiLin R.val R.val by rfl]
|
||||
ring
|
||||
|
||||
lemma planeY₃B₃_cubic (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
|
||||
|
|
@ -130,9 +130,9 @@ lemma planeY₃B₃_cubic (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
|
|||
/-- The line in the plane spanned by $Y_3$, $B_3$ and $R$ which is in the quadratic,
|
||||
as `LinSols`. -/
|
||||
def lineQuadAFL (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) : MSSMACC.LinSols :=
|
||||
planeY₃B₃ R (c2 * quadBiLin (R.val, R.val) - 2 * c3 * quadBiLin (B₃.val, R.val))
|
||||
(2 * c3 * quadBiLin (Y₃.val, R.val) - c1 * quadBiLin (R.val, R.val))
|
||||
(2 * c1 * quadBiLin (B₃.val, R.val) - 2 * c2 * quadBiLin (Y₃.val, R.val))
|
||||
planeY₃B₃ R (c2 * quadBiLin R.val R.val - 2 * c3 * quadBiLin B₃.val R.val)
|
||||
(2 * c3 * quadBiLin Y₃.val R.val - c1 * quadBiLin R.val R.val)
|
||||
(2 * c1 * quadBiLin B₃.val R.val - 2 * c2 * quadBiLin Y₃.val R.val)
|
||||
|
||||
lemma lineQuadAFL_quad (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) :
|
||||
accQuad (lineQuadAFL R c1 c2 c3).val = 0 := by
|
||||
|
|
@ -146,10 +146,10 @@ def lineQuad (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) : MSSMACC.QuadSols :
|
|||
AnomalyFreeQuadMk' (lineQuadAFL R c1 c2 c3) (lineQuadAFL_quad R c1 c2 c3)
|
||||
|
||||
lemma lineQuad_val (R : MSSMACC.AnomalyFreePerp) (c1 c2 c3 : ℚ) :
|
||||
(lineQuad R c1 c2 c3).val = (planeY₃B₃ R
|
||||
(c2 * quadBiLin (R.val, R.val) - 2 * c3 * quadBiLin (B₃.val, R.val))
|
||||
(2 * c3 * quadBiLin (Y₃.val, R.val) - c1 * quadBiLin (R.val, R.val))
|
||||
(2 * c1 * quadBiLin (B₃.val, R.val) - 2 * c2 * quadBiLin (Y₃.val, R.val))).val := by
|
||||
(lineQuad R c1 c2 c3).val = (planeY₃B₃ R
|
||||
(c2 * quadBiLin R.val R.val - 2 * c3 * quadBiLin B₃.val R.val)
|
||||
(2 * c3 * quadBiLin Y₃.val R.val - c1 * quadBiLin R.val R.val)
|
||||
(2 * c1 * quadBiLin B₃.val R.val - 2 * c2 * quadBiLin Y₃.val R.val)).val := by
|
||||
rfl
|
||||
|
||||
lemma lineQuad_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
|
||||
|
|
@ -163,26 +163,26 @@ lemma lineQuad_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
|
|||
|
||||
/-- A helper function to simplify following expressions. -/
|
||||
def α₁ (T : MSSMACC.AnomalyFreePerp) : ℚ :=
|
||||
(3 * cubeTriLin (T.val, T.val, B₃.val) * quadBiLin (T.val, T.val) -
|
||||
2 * cubeTriLin (T.val, T.val, T.val) * quadBiLin (B₃.val, T.val))
|
||||
(3 * cubeTriLin (T.val, T.val, B₃.val) * quadBiLin T.val T.val -
|
||||
2 * cubeTriLin (T.val, T.val, T.val) * quadBiLin B₃.val T.val)
|
||||
|
||||
/-- A helper function to simplify following expressions. -/
|
||||
def α₂ (T : MSSMACC.AnomalyFreePerp) : ℚ :=
|
||||
(2 * cubeTriLin (T.val, T.val, T.val) * quadBiLin (Y₃.val, T.val) -
|
||||
3 * cubeTriLin (T.val, T.val, Y₃.val) * quadBiLin (T.val, T.val))
|
||||
/-- A helper function to simplify following expressions. -/
|
||||
def α₂ (T : MSSMACC.AnomalyFreePerp) : ℚ :=
|
||||
(2 * cubeTriLin (T.val, T.val, T.val) * quadBiLin Y₃.val T.val -
|
||||
3 * cubeTriLin (T.val, T.val, Y₃.val) * quadBiLin T.val T.val)
|
||||
|
||||
/-- A helper function to simplify following expressions. -/
|
||||
def α₃ (T : MSSMACC.AnomalyFreePerp) : ℚ :=
|
||||
6 * ((cubeTriLin (T.val, T.val, Y₃.val)) * quadBiLin (B₃.val, T.val) -
|
||||
(cubeTriLin (T.val, T.val, B₃.val)) * quadBiLin (Y₃.val, T.val))
|
||||
/-- A helper function to simplify following expressions. -/
|
||||
def α₃ (T : MSSMACC.AnomalyFreePerp) : ℚ :=
|
||||
6 * ((cubeTriLin (T.val, T.val, Y₃.val)) * quadBiLin B₃.val T.val -
|
||||
(cubeTriLin (T.val, T.val, B₃.val)) * quadBiLin Y₃.val T.val)
|
||||
|
||||
lemma lineQuad_cube (R : MSSMACC.AnomalyFreePerp) (c₁ c₂ c₃ : ℚ) :
|
||||
accCube (lineQuad R c₁ c₂ c₃).val =
|
||||
- 4 * ( c₁ * quadBiLin (B₃.val, R.val) - c₂ * quadBiLin (Y₃.val, R.val)) ^ 2 *
|
||||
( α₁ R * c₁ + α₂ R * c₂ + α₃ R * c₃ ) := by
|
||||
rw [lineQuad_val]
|
||||
rw [planeY₃B₃_cubic, α₁, α₂, α₃]
|
||||
ring
|
||||
lemma lineQuad_cube (R : MSSMACC.AnomalyFreePerp) (c₁ c₂ c₃ : ℚ) :
|
||||
accCube (lineQuad R c₁ c₂ c₃).val =
|
||||
- 4 * ( c₁ * quadBiLin B₃.val R.val - c₂ * quadBiLin Y₃.val R.val) ^ 2 *
|
||||
( α₁ R * c₁ + α₂ R * c₂ + α₃ R * c₃ ) := by
|
||||
rw [lineQuad_val]
|
||||
rw [planeY₃B₃_cubic, α₁, α₂, α₃]
|
||||
ring
|
||||
|
||||
/-- The line in the plane spanned by $Y_3$, $B_3$ and $R$ which is in the cubic. -/
|
||||
def lineCube (R : MSSMACC.AnomalyFreePerp) (a₁ a₂ a₃ : ℚ) :
|
||||
|
|
@ -220,21 +220,21 @@ lemma lineCube_quad (R : MSSMACC.AnomalyFreePerp) (a₁ a₂ a₃ : ℚ) :
|
|||
section proj
|
||||
|
||||
lemma α₃_proj (T : MSSMACC.Sols) : α₃ (proj T.1.1) =
|
||||
6 * dot (Y₃.val, B₃.val) ^ 3 * (
|
||||
cubeTriLin (T.val, T.val, Y₃.val) * quadBiLin (B₃.val, T.val) -
|
||||
cubeTriLin (T.val, T.val, B₃.val) * quadBiLin (Y₃.val, T.val)) := by
|
||||
6 * dot Y₃.val B₃.val ^ 3 * (
|
||||
cubeTriLin (T.val, T.val, Y₃.val) * quadBiLin B₃.val T.val -
|
||||
cubeTriLin (T.val, T.val, B₃.val) * quadBiLin Y₃.val T.val) := by
|
||||
rw [α₃]
|
||||
rw [cube_proj_proj_Y₃, cube_proj_proj_B₃, quad_B₃_proj, quad_Y₃_proj]
|
||||
ring
|
||||
|
||||
lemma α₂_proj (T : MSSMACC.Sols) : α₂ (proj T.1.1) =
|
||||
- α₃ (proj T.1.1) * (dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val)) := by
|
||||
- α₃ (proj T.1.1) * (dot Y₃.val T.val - 2 * dot B₃.val T.val) := by
|
||||
rw [α₃_proj, α₂]
|
||||
rw [cube_proj_proj_Y₃, quad_Y₃_proj, quad_proj, cube_proj]
|
||||
ring
|
||||
|
||||
lemma α₁_proj (T : MSSMACC.Sols) : α₁ (proj T.1.1) =
|
||||
- α₃ (proj T.1.1) * (dot (B₃.val, T.val) - dot (Y₃.val, T.val)) := by
|
||||
- α₃ (proj T.1.1) * (dot B₃.val T.val - dot Y₃.val T.val) := by
|
||||
rw [α₃_proj, α₁]
|
||||
rw [cube_proj_proj_B₃, quad_B₃_proj, quad_proj, cube_proj]
|
||||
ring
|
||||
|
|
|
|||
|
|
@ -46,12 +46,12 @@ instance (R : MSSMACC.AnomalyFreePerp) : Decidable (lineEqProp R) := by
|
|||
/-- A condition on `Sols` which we will show in `linEqPropSol_iff_proj_linEqProp` that is equivalent
|
||||
to the condition that the `proj` of the solution satisfies `lineEqProp`. -/
|
||||
def lineEqPropSol (R : MSSMACC.Sols) : Prop :=
|
||||
cubeTriLin (R.val, R.val, Y₃.val) * quadBiLin (B₃.val, R.val) -
|
||||
cubeTriLin (R.val, R.val, B₃.val) * quadBiLin (Y₃.val, R.val) = 0
|
||||
cubeTriLin (R.val, R.val, Y₃.val) * quadBiLin B₃.val R.val -
|
||||
cubeTriLin (R.val, R.val, B₃.val) * quadBiLin Y₃.val R.val = 0
|
||||
|
||||
/-- A rational which appears in `toSolNS` acting on sols, and which been zero is
|
||||
equivalent to satisfying `lineEqPropSol`. -/
|
||||
def lineEqCoeff (T : MSSMACC.Sols) : ℚ := dot (Y₃.val, B₃.val) * α₃ (proj T.1.1)
|
||||
def lineEqCoeff (T : MSSMACC.Sols) : ℚ := dot Y₃.val B₃.val * α₃ (proj T.1.1)
|
||||
|
||||
lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
|
||||
lineEqPropSol T ↔ lineEqCoeff T = 0 := by
|
||||
|
|
@ -59,7 +59,7 @@ lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
|
|||
simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero,
|
||||
false_or]
|
||||
rw [cube_proj_proj_B₃, cube_proj_proj_Y₃, quad_B₃_proj, quad_Y₃_proj]
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl]
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl]
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, OfNat.ofNat_ne_zero, false_or]
|
||||
ring_nf
|
||||
rw [mul_comm _ 1259712, mul_comm _ 1259712, ← mul_sub]
|
||||
|
|
@ -70,10 +70,10 @@ lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
|
|||
rw [lineEqPropSol_iff_lineEqCoeff_zero, lineEqCoeff, lineEqProp]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl] at h
|
||||
simp at h
|
||||
rw [α₁_proj, α₂_proj, h]
|
||||
simp only [neg_zero, dot_toFun, Fin.isValue, Fin.reduceFinMk, zero_mul, and_self]
|
||||
simp only [neg_zero, Fin.isValue, Fin.reduceFinMk, zero_mul, and_self]
|
||||
intro h
|
||||
rw [h.2.2]
|
||||
simp
|
||||
|
|
@ -82,7 +82,7 @@ lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
|
|||
/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies
|
||||
entirely in the quadratic surface. -/
|
||||
def inQuadProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
|
||||
quadBiLin (R.val, R.val) = 0 ∧ quadBiLin (Y₃.val, R.val) = 0 ∧ quadBiLin (B₃.val, R.val) = 0
|
||||
quadBiLin R.val R.val = 0 ∧ quadBiLin Y₃.val R.val = 0 ∧ quadBiLin B₃.val R.val = 0
|
||||
|
||||
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inQuadProp R) := by
|
||||
apply And.decidable
|
||||
|
|
@ -90,23 +90,23 @@ instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inQuadProp R) := by
|
|||
/-- A condition which is satisfied if the plane spanned by the solutions `R`, `Y₃` and `B₃`
|
||||
lies entirely in the quadratic surface. -/
|
||||
def inQuadSolProp (R : MSSMACC.Sols) : Prop :=
|
||||
quadBiLin (Y₃.val, R.val) = 0 ∧ quadBiLin (B₃.val, R.val) = 0
|
||||
quadBiLin Y₃.val R.val = 0 ∧ quadBiLin B₃.val R.val = 0
|
||||
|
||||
/-- A rational which has two properties. It is zero for a solution `T` if and only if
|
||||
that solution satisfies `inQuadSolProp`. It appears in the definition of `inQuadProj`. -/
|
||||
def quadCoeff (T : MSSMACC.Sols) : ℚ :=
|
||||
2 * dot (Y₃.val, B₃.val) ^ 2 *
|
||||
(quadBiLin (Y₃.val, T.val) ^ 2 + quadBiLin (B₃.val, T.val) ^ 2)
|
||||
2 * dot Y₃.val B₃.val ^ 2 *
|
||||
(quadBiLin Y₃.val T.val ^ 2 + quadBiLin B₃.val T.val ^ 2)
|
||||
|
||||
lemma inQuadSolProp_iff_quadCoeff_zero (T : MSSMACC.Sols) : inQuadSolProp T ↔ quadCoeff T = 0 := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [quadCoeff, h.1, h.2]
|
||||
simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
|
||||
zero_pow, add_zero, mul_zero]
|
||||
intro h
|
||||
rw [quadCoeff] at h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl] at h
|
||||
simp only [ Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
|
||||
not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
|
||||
apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
|
||||
|
|
@ -123,10 +123,10 @@ lemma inQuadSolProp_iff_proj_inQuadProp (R : MSSMACC.Sols) :
|
|||
apply Iff.intro
|
||||
intro h
|
||||
rw [h.1, h.2]
|
||||
simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
|
||||
intro h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk , mul_eq_zero,
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl] at h
|
||||
simp only [Fin.isValue, Fin.reduceFinMk , mul_eq_zero,
|
||||
OfNat.ofNat_ne_zero, or_self, false_or] at h
|
||||
rw [h.2.1, h.2.2]
|
||||
simp
|
||||
|
|
@ -149,7 +149,7 @@ def inCubeSolProp (R : MSSMACC.Sols) : Prop :=
|
|||
/-- A rational which has two properties. It is zero for a solution `T` if and only if
|
||||
that solution satisfies `inCubeSolProp`. It appears in the definition of `inLineEqProj`. -/
|
||||
def cubicCoeff (T : MSSMACC.Sols) : ℚ :=
|
||||
3 * dot (Y₃.val, B₃.val) ^ 3 * (cubeTriLin (T.val, T.val, Y₃.val) ^ 2 +
|
||||
3 * (dot Y₃.val B₃.val) ^ 3 * (cubeTriLin (T.val, T.val, Y₃.val) ^ 2 +
|
||||
cubeTriLin (T.val, T.val, B₃.val) ^ 2 )
|
||||
|
||||
lemma inCubeSolProp_iff_cubicCoeff_zero (T : MSSMACC.Sols) :
|
||||
|
|
@ -157,11 +157,11 @@ lemma inCubeSolProp_iff_cubicCoeff_zero (T : MSSMACC.Sols) :
|
|||
apply Iff.intro
|
||||
intro h
|
||||
rw [cubicCoeff, h.1, h.2]
|
||||
simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
|
||||
zero_pow, add_zero, mul_zero]
|
||||
intro h
|
||||
rw [cubicCoeff] at h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl] at h
|
||||
simp only [ Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
|
||||
not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
|
||||
apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
|
||||
|
|
@ -176,10 +176,10 @@ lemma inCubeSolProp_iff_proj_inCubeProp (R : MSSMACC.Sols) :
|
|||
apply Iff.intro
|
||||
intro h
|
||||
rw [h.1, h.2]
|
||||
simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
|
||||
intro h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl] at h
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
|
||||
not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
|
||||
rw [h.2.1, h.2.2]
|
||||
simp
|
||||
|
|
@ -254,7 +254,7 @@ lemma toSolNS_proj (T : notInLineEqSol) : toSolNS (toSolNSProj T.val) = T.val :=
|
|||
rw [α₁_proj, α₂_proj]
|
||||
ring_nf
|
||||
simp only [zero_smul, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add]
|
||||
have h1 : α₃ (proj T.val.toLinSols) * dot (Y₃.val, B₃.val) = lineEqCoeff T.val := by
|
||||
have h1 : α₃ (proj T.val.toLinSols) * dot Y₃.val B₃.val = lineEqCoeff T.val := by
|
||||
rw [lineEqCoeff]
|
||||
ring
|
||||
rw [h1]
|
||||
|
|
@ -273,12 +273,11 @@ def inLineEqToSol : inLineEq × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, c
|
|||
/-- On elements of `inLineEqSol` a right-inverse to `inLineEqSol`. -/
|
||||
def inLineEqProj (T : inLineEqSol) : inLineEq × ℚ × ℚ × ℚ :=
|
||||
(⟨proj T.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1 ⟩,
|
||||
(quadCoeff T.val)⁻¹ * quadBiLin (B₃.val, T.val.val),
|
||||
(quadCoeff T.val)⁻¹ * (- quadBiLin (Y₃.val, T.val.val)),
|
||||
(quadCoeff T.val)⁻¹ * quadBiLin B₃.val T.val.val,
|
||||
(quadCoeff T.val)⁻¹ * (- quadBiLin Y₃.val T.val.val),
|
||||
(quadCoeff T.val)⁻¹ * (
|
||||
quadBiLin (B₃.val, T.val.val) * (dot (B₃.val, T.val.val) - dot (Y₃.val, T.val.val))
|
||||
- quadBiLin (Y₃.val, T.val.val) * (dot (Y₃.val, T.val.val) - 2 * dot (B₃.val, T.val.val))))
|
||||
|
||||
quadBiLin B₃.val T.val.val * (dot B₃.val T.val.val - dot Y₃.val T.val.val)
|
||||
- quadBiLin Y₃.val T.val.val * (dot Y₃.val T.val.val - 2 * dot B₃.val T.val.val)))
|
||||
|
||||
lemma inLineEqTo_smul (R : inLineEq) (c₁ c₂ c₃ d : ℚ) :
|
||||
inLineEqToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inLineEqToSol (R, c₁, c₂, c₃) := by
|
||||
|
|
@ -297,8 +296,8 @@ lemma inLineEqToSol_proj (T : inLineEqSol) : inLineEqToSol (inLineEqProj T) = T.
|
|||
rw [quad_proj, quad_Y₃_proj, quad_B₃_proj]
|
||||
ring_nf
|
||||
simp only [zero_smul, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add]
|
||||
have h1 : (quadBiLin (Y₃.val, (T).val.val) ^ 2 * dot (Y₃.val, B₃.val) ^ 2 * 2 +
|
||||
dot (Y₃.val, B₃.val) ^ 2 * quadBiLin (B₃.val, (T).val.val) ^ 2 * 2) = quadCoeff T.val := by
|
||||
have h1 : (quadBiLin Y₃.val T.val.val ^ 2 * dot Y₃.val B₃.val ^ 2 * 2 +
|
||||
dot Y₃.val B₃.val ^ 2 * quadBiLin B₃.val T.val.val ^ 2 * 2) = quadCoeff T.val := by
|
||||
rw [quadCoeff]
|
||||
ring
|
||||
rw [h1]
|
||||
|
|
@ -329,9 +328,9 @@ def inQuadProj (T : inQuadSol) : inQuad × ℚ × ℚ × ℚ :=
|
|||
(cubicCoeff T.val)⁻¹ * (cubeTriLin (T.val.val, T.val.val, B₃.val)),
|
||||
(cubicCoeff T.val)⁻¹ * (- cubeTriLin (T.val.val, T.val.val, Y₃.val)),
|
||||
(cubicCoeff T.val)⁻¹ *
|
||||
(cubeTriLin (T.val.val, T.val.val, B₃.val) * (dot (B₃.val, T.val.val) - dot (Y₃.val, T.val.val))
|
||||
(cubeTriLin (T.val.val, T.val.val, B₃.val) * (dot B₃.val T.val.val - dot Y₃.val T.val.val)
|
||||
- cubeTriLin (T.val.val, T.val.val, Y₃.val)
|
||||
* (dot (Y₃.val, T.val.val) - 2 * dot (B₃.val, T.val.val))))
|
||||
* (dot Y₃.val T.val.val - 2 * dot B₃.val T.val.val)))
|
||||
|
||||
|
||||
lemma inQuadToSol_proj (T : inQuadSol) : inQuadToSol (inQuadProj T) = T.val := by
|
||||
|
|
@ -343,8 +342,8 @@ lemma inQuadToSol_proj (T : inQuadSol) : inQuadToSol (inQuadProj T) = T.val := b
|
|||
rw [cube_proj, cube_proj_proj_B₃, cube_proj_proj_Y₃]
|
||||
ring_nf
|
||||
simp only [zero_smul, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add]
|
||||
have h1 : (cubeTriLin (T.val.val, T.val.val, Y₃.val) ^ 2 * dot (Y₃.val, B₃.val) ^ 3 * 3 +
|
||||
dot (Y₃.val, B₃.val) ^ 3 * cubeTriLin (T.val.val, T.val.val, B₃.val) ^ 2
|
||||
have h1 : (cubeTriLin (T.val.val, T.val.val, Y₃.val) ^ 2 * dot Y₃.val B₃.val ^ 3 * 3 +
|
||||
dot Y₃.val B₃.val ^ 3 * cubeTriLin (T.val.val, T.val.val, B₃.val) ^ 2
|
||||
* 3) = cubicCoeff T.val := by
|
||||
rw [cubicCoeff]
|
||||
ring
|
||||
|
|
@ -378,9 +377,9 @@ def inQuadCubeProj (T : inQuadCubeSol) : inQuadCube × ℚ × ℚ × ℚ :=
|
|||
(⟨⟨⟨proj T.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1 ⟩,
|
||||
(inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1⟩,
|
||||
(inCubeSolProp_iff_proj_inCubeProp T.val).mp T.prop.2.2⟩,
|
||||
(dot (Y₃.val, B₃.val))⁻¹ * (dot (Y₃.val, T.val.val) - dot (B₃.val, T.val.val)),
|
||||
(dot (Y₃.val, B₃.val))⁻¹ * (2 * dot (B₃.val, T.val.val) - dot (Y₃.val, T.val.val)),
|
||||
(dot (Y₃.val, B₃.val))⁻¹ * 1)
|
||||
(dot Y₃.val B₃.val)⁻¹ * (dot Y₃.val T.val.val - dot B₃.val T.val.val),
|
||||
(dot Y₃.val B₃.val)⁻¹ * (2 * dot B₃.val T.val.val - dot Y₃.val T.val.val),
|
||||
(dot Y₃.val B₃.val)⁻¹ * 1)
|
||||
|
||||
lemma inQuadCubeToSol_proj (T : inQuadCubeSol) :
|
||||
inQuadCubeToSol (inQuadCubeProj T) = T.val := by
|
||||
|
|
@ -393,7 +392,7 @@ lemma inQuadCubeToSol_proj (T : inQuadCubeSol) :
|
|||
simp only [Fin.isValue, Fin.reduceFinMk, zero_smul, add_zero, zero_add]
|
||||
rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel]
|
||||
simp only [one_smul]
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl]
|
||||
rw [show dot Y₃.val B₃.val = 108 by rfl]
|
||||
simp
|
||||
|
||||
/-- Given an element of `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` a solution. We will
|
||||
|
|
|
|||
|
|
@ -208,21 +208,23 @@ lemma accYY_ext {S T : (SMCharges n).charges}
|
|||
|
||||
/-- The quadratic bilinear map. -/
|
||||
@[simps!]
|
||||
def quadBiLin : BiLinearSymm (SMCharges n).charges where
|
||||
toFun S := ∑ i, (Q S.1 i * Q S.2 i +
|
||||
def quadBiLin : BiLinearSymm (SMCharges n).charges := BiLinearSymm.mk₂
|
||||
(fun S => ∑ i, (Q S.1 i * Q S.2 i +
|
||||
- 2 * (U S.1 i * U S.2 i) +
|
||||
D S.1 i * D S.2 i +
|
||||
(- 1) * (L S.1 i * L S.2 i) +
|
||||
E S.1 i * E S.2 i)
|
||||
map_smul₁' a S T := by
|
||||
E S.1 i * E S.2 i))
|
||||
(by
|
||||
intro a S T
|
||||
simp only
|
||||
rw [Finset.mul_sum]
|
||||
apply Fintype.sum_congr
|
||||
intro i
|
||||
repeat erw [map_smul]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
ring
|
||||
map_add₁' S T R := by
|
||||
ring)
|
||||
(by
|
||||
intro S1 S2 T
|
||||
simp only
|
||||
rw [← Finset.sum_add_distrib]
|
||||
apply Fintype.sum_congr
|
||||
|
|
@ -230,12 +232,13 @@ def quadBiLin : BiLinearSymm (SMCharges n).charges where
|
|||
repeat erw [map_add]
|
||||
simp only [ACCSystemCharges.chargesAddCommMonoid_add, toSpecies_apply, Fin.isValue, neg_mul,
|
||||
one_mul]
|
||||
ring
|
||||
swap' S T := by
|
||||
ring)
|
||||
(by
|
||||
intro S T
|
||||
simp only [SMSpecies_numberCharges, toSpecies_apply, Fin.isValue, neg_mul, one_mul]
|
||||
apply Fintype.sum_congr
|
||||
intro i
|
||||
ring
|
||||
ring)
|
||||
|
||||
/-- The quadratic anomaly cancellation condition. -/
|
||||
@[simp]
|
||||
|
|
@ -247,10 +250,10 @@ lemma accQuad_ext {S T : (SMCharges n).charges}
|
|||
(h : ∀ j, ∑ i, ((fun a => a^2) ∘ toSpecies j S) i =
|
||||
∑ i, ((fun a => a^2) ∘ toSpecies j T) i) :
|
||||
accQuad S = accQuad T := by
|
||||
simp only [accQuad, BiLinearSymm.toHomogeneousQuad_toFun, Fin.isValue]
|
||||
simp only [HomogeneousQuadratic, accQuad, BiLinearSymm.toHomogeneousQuad_apply]
|
||||
erw [← quadBiLin.toFun_eq_coe]
|
||||
rw [quadBiLin]
|
||||
simp only
|
||||
simp only [quadBiLin, BiLinearSymm.mk₂, AddHom.toFun_eq_coe, AddHom.coe_mk, LinearMap.coe_mk]
|
||||
repeat erw [Finset.sum_add_distrib]
|
||||
repeat erw [← Finset.mul_sum]
|
||||
ring_nf
|
||||
|
|
@ -301,7 +304,7 @@ lemma accCube_ext {S T : (SMCharges n).charges}
|
|||
(h : ∀ j, ∑ i, ((fun a => a^3) ∘ toSpecies j S) i =
|
||||
∑ i, ((fun a => a^3) ∘ toSpecies j T) i) :
|
||||
accCube S = accCube T := by
|
||||
simp only [accCube, TriLinearSymm.toCubic_toFun, Fin.isValue]
|
||||
simp only [HomogeneousCubic, accCube, TriLinearSymm.toCubic_apply]
|
||||
erw [← cubeTriLin.toFun_eq_coe]
|
||||
rw [cubeTriLin]
|
||||
simp only
|
||||
|
|
|
|||
|
|
@ -84,9 +84,7 @@ lemma asLinear_val (S : linearParameters) : S.asLinear.val = S.asCharges := by
|
|||
|
||||
lemma cubic (S : linearParameters) :
|
||||
accCube (S.asCharges) = - 54 * S.Q'^3 - 18 * S.Q' * S.Y ^ 2 + S.E'^3 := by
|
||||
simp only [accCube, TriLinearSymm.toCubic_toFun, Finset.univ_unique,
|
||||
Fin.default_eq_zero, Fin.isValue, asCharges, SMNoGrav_numberCharges, neg_add_rev, neg_mul,
|
||||
Finset.sum_singleton]
|
||||
simp only [HomogeneousCubic, accCube, TriLinearSymm.toCubic_apply, neg_mul]
|
||||
erw [← TriLinearSymm.toFun_eq_coe]
|
||||
rw [cubeTriLin]
|
||||
simp only [SMSpecies_numberCharges, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
|
||||
|
|
|
|||
|
|
@ -230,21 +230,23 @@ lemma accYY_ext {S T : (SMνCharges n).charges}
|
|||
|
||||
/-- The quadratic bilinear map. -/
|
||||
@[simps!]
|
||||
def quadBiLin : BiLinearSymm (SMνCharges n).charges where
|
||||
toFun S := ∑ i, (Q S.1 i * Q S.2 i +
|
||||
def quadBiLin : BiLinearSymm (SMνCharges n).charges := BiLinearSymm.mk₂
|
||||
(fun S => ∑ i, (Q S.1 i * Q S.2 i +
|
||||
- 2 * (U S.1 i * U S.2 i) +
|
||||
D S.1 i * D S.2 i +
|
||||
(- 1) * (L S.1 i * L S.2 i) +
|
||||
E S.1 i * E S.2 i)
|
||||
map_smul₁' a S T := by
|
||||
E S.1 i * E S.2 i))
|
||||
(by
|
||||
intro a S T
|
||||
simp only
|
||||
rw [Finset.mul_sum]
|
||||
apply Fintype.sum_congr
|
||||
intro i
|
||||
repeat erw [map_smul]
|
||||
simp [HSMul.hSMul, SMul.smul]
|
||||
ring
|
||||
map_add₁' S T R := by
|
||||
ring)
|
||||
(by
|
||||
intro S T R
|
||||
simp only
|
||||
rw [← Finset.sum_add_distrib]
|
||||
apply Fintype.sum_congr
|
||||
|
|
@ -252,19 +254,20 @@ def quadBiLin : BiLinearSymm (SMνCharges n).charges where
|
|||
repeat erw [map_add]
|
||||
simp only [ACCSystemCharges.chargesAddCommMonoid_add, toSpecies_apply, Fin.isValue, neg_mul,
|
||||
one_mul]
|
||||
ring
|
||||
swap' S T := by
|
||||
ring)
|
||||
(by
|
||||
intro S T
|
||||
simp only [SMνSpecies_numberCharges, toSpecies_apply, Fin.isValue, neg_mul, one_mul]
|
||||
apply Fintype.sum_congr
|
||||
intro i
|
||||
ring
|
||||
ring)
|
||||
|
||||
lemma quadBiLin_decomp (S T : (SMνCharges n).charges) :
|
||||
quadBiLin (S, T) = ∑ i, Q S i * Q T i - 2 * ∑ i, U S i * U T i +
|
||||
quadBiLin S T = ∑ i, Q S i * Q T i - 2 * ∑ i, U S i * U T i +
|
||||
∑ i, D S i * D T i - ∑ i, L S i * L T i + ∑ i, E S i * E T i := by
|
||||
erw [← quadBiLin.toFun_eq_coe]
|
||||
rw [quadBiLin]
|
||||
simp only
|
||||
simp only [quadBiLin, BiLinearSymm.mk₂, AddHom.toFun_eq_coe, AddHom.coe_mk, LinearMap.coe_mk]
|
||||
repeat erw [Finset.sum_add_distrib]
|
||||
repeat erw [← Finset.mul_sum]
|
||||
simp only [SMνSpecies_numberCharges, toSpecies_apply, Fin.isValue, neg_mul, one_mul, add_left_inj]
|
||||
|
|
|
|||
|
|
@ -190,7 +190,7 @@ lemma familyUniversal_accYY (S : (SMνCharges 1).charges) :
|
|||
ring
|
||||
|
||||
lemma familyUniversal_quadBiLin (S : (SMνCharges 1).charges) (T : (SMνCharges n).charges) :
|
||||
quadBiLin (familyUniversal n S, T) =
|
||||
quadBiLin (familyUniversal n S) T =
|
||||
S (0 : Fin 6) * ∑ i, Q T i - 2 * S (1 : Fin 6) * ∑ i, U T i + S (2 : Fin 6) *∑ i, D T i -
|
||||
S (3 : Fin 6) * ∑ i, L T i + S (4 : Fin 6) * ∑ i, E T i := by
|
||||
rw [quadBiLin_decomp]
|
||||
|
|
|
|||
|
|
@ -58,14 +58,14 @@ namespace BL
|
|||
variable {n : ℕ}
|
||||
|
||||
lemma on_quadBiLin (S : (PlusU1 n).charges) :
|
||||
quadBiLin ((BL n).val, S) = 1/2 * accYY S + 3/2 * accSU2 S - 2 * accSU3 S := by
|
||||
quadBiLin (BL n).val S = 1/2 * accYY S + 3/2 * accSU2 S - 2 * accSU3 S := by
|
||||
erw [familyUniversal_quadBiLin]
|
||||
rw [accYY_decomp, accSU2_decomp, accSU3_decomp]
|
||||
simp only [Fin.isValue, BL₁_val, SMνSpecies_numberCharges, toSpecies_apply, one_mul, mul_neg,
|
||||
mul_one, neg_mul, sub_neg_eq_add, one_div]
|
||||
ring
|
||||
|
||||
lemma on_quadBiLin_AFL (S : (PlusU1 n).LinSols) : quadBiLin ((BL n).val, S.val) = 0 := by
|
||||
lemma on_quadBiLin_AFL (S : (PlusU1 n).LinSols) : quadBiLin (BL n).val S.val = 0 := by
|
||||
rw [on_quadBiLin]
|
||||
rw [YYsol S, SU2Sol S, SU3Sol S]
|
||||
simp
|
||||
|
|
@ -110,8 +110,8 @@ lemma add_AFL_cube (S : (PlusU1 n).LinSols) (a b : ℚ) :
|
|||
erw [TriLinearSymm.toCubic_add, cubeSol (b • (BL n)), accCube.map_smul]
|
||||
repeat rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
|
||||
rw [on_cubeTriLin_AFL]
|
||||
simp only [accCube, TriLinearSymm.toCubic_toFun, cubeTriLin_toFun, Fin.isValue, add_zero, BL_val,
|
||||
mul_zero]
|
||||
simp only [HomogeneousCubic, accCube, TriLinearSymm.toCubic_apply, cubeTriLin_toFun, Fin.isValue,
|
||||
add_zero, BL_val, mul_zero]
|
||||
ring
|
||||
|
||||
|
||||
|
|
|
|||
|
|
@ -56,7 +56,7 @@ namespace Y
|
|||
variable {n : ℕ}
|
||||
|
||||
lemma on_quadBiLin (S : (PlusU1 n).charges) :
|
||||
quadBiLin ((Y n).val, S) = accYY S := by
|
||||
quadBiLin (Y n).val S = accYY S := by
|
||||
erw [familyUniversal_quadBiLin]
|
||||
rw [accYY_decomp]
|
||||
simp only [Fin.isValue, Y₁_val, SMνSpecies_numberCharges, toSpecies_apply, one_mul, mul_neg,
|
||||
|
|
@ -64,7 +64,7 @@ lemma on_quadBiLin (S : (PlusU1 n).charges) :
|
|||
ring_nf
|
||||
simp
|
||||
|
||||
lemma on_quadBiLin_AFL (S : (PlusU1 n).LinSols) : quadBiLin ((Y n).val, S.val) = 0 := by
|
||||
lemma on_quadBiLin_AFL (S : (PlusU1 n).LinSols) : quadBiLin (Y n).val S.val = 0 := by
|
||||
rw [on_quadBiLin]
|
||||
rw [YYsol S]
|
||||
|
||||
|
|
@ -123,8 +123,8 @@ lemma add_AFL_cube (S : (PlusU1 n).LinSols) (a b : ℚ) :
|
|||
erw [TriLinearSymm.toCubic_add, cubeSol (b • (Y n)), accCube.map_smul]
|
||||
repeat rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
|
||||
rw [on_cubeTriLin_AFL]
|
||||
simp only [accCube, TriLinearSymm.toCubic_toFun, cubeTriLin_toFun, Fin.isValue, add_zero, Y_val,
|
||||
mul_zero]
|
||||
simp only [HomogeneousCubic, accCube, TriLinearSymm.toCubic_apply, cubeTriLin_toFun, Fin.isValue,
|
||||
add_zero, Y_val, mul_zero]
|
||||
ring
|
||||
|
||||
lemma add_AFQ_cube (S : (PlusU1 n).QuadSols) (a b : ℚ) :
|
||||
|
|
|
|||
|
|
@ -74,13 +74,13 @@ def B : Fin 11 → (PlusU1 3).charges := fun i =>
|
|||
| 9 => B₉
|
||||
| 10 => B₁₀
|
||||
|
||||
lemma Bi_Bj_quad {i j : Fin 11} (hi : i ≠ j) : quadBiLin (B i, B j) = 0 := by
|
||||
lemma Bi_Bj_quad {i j : Fin 11} (hi : i ≠ j) : quadBiLin (B i) (B j) = 0 := by
|
||||
fin_cases i <;> fin_cases j
|
||||
any_goals rfl
|
||||
all_goals simp at hi
|
||||
|
||||
lemma Bi_sum_quad (i : Fin 11) (f : Fin 11 → ℚ) :
|
||||
quadBiLin (B i, ∑ k, f k • B k) = f i * quadBiLin (B i, B i) := by
|
||||
quadBiLin (B i) (∑ k, f k • B k) = f i * quadBiLin (B i) (B i) := by
|
||||
rw [quadBiLin.map_sum₂]
|
||||
rw [Fintype.sum_eq_single i]
|
||||
rw [quadBiLin.map_smul₂]
|
||||
|
|
@ -92,13 +92,13 @@ lemma Bi_sum_quad (i : Fin 11) (f : Fin 11 → ℚ) :
|
|||
@[simp]
|
||||
def quadCoeff : Fin 11 → ℚ := ![1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0]
|
||||
|
||||
lemma quadCoeff_eq_bilinear (i : Fin 11) : quadCoeff i = quadBiLin (B i, B i) := by
|
||||
lemma quadCoeff_eq_bilinear (i : Fin 11) : quadCoeff i = quadBiLin (B i) (B i) := by
|
||||
fin_cases i
|
||||
all_goals rfl
|
||||
|
||||
lemma on_accQuad (f : Fin 11 → ℚ) :
|
||||
accQuad (∑ i, f i • B i) = ∑ i, quadCoeff i * (f i)^2 := by
|
||||
change quadBiLin _ = _
|
||||
change quadBiLin _ _ = _
|
||||
rw [quadBiLin.map_sum₁]
|
||||
apply Fintype.sum_congr
|
||||
intro i
|
||||
|
|
|
|||
|
|
@ -33,14 +33,14 @@ variable (C : (PlusU1 n).QuadSols)
|
|||
|
||||
lemma add_AFL_quad (S : (PlusU1 n).LinSols) (a b : ℚ) :
|
||||
accQuad (a • S.val + b • C.val) =
|
||||
a * (a * accQuad S.val + 2 * b * quadBiLin (S.val, C.val)) := by
|
||||
a * (a * accQuad S.val + 2 * b * quadBiLin S.val C.val) := by
|
||||
erw [BiLinearSymm.toHomogeneousQuad_add, quadSol (b • C)]
|
||||
rw [quadBiLin.map_smul₁, quadBiLin.map_smul₂]
|
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erw [accQuad.map_smul]
|
||||
ring
|
||||
|
||||
/-- A helper function for what comes later. -/
|
||||
def α₁ (S : (PlusU1 n).LinSols) : ℚ := - 2 * quadBiLin (S.val, C.val)
|
||||
def α₁ (S : (PlusU1 n).LinSols) : ℚ := - 2 * quadBiLin S.val C.val
|
||||
|
||||
/-- A helper function for what comes later. -/
|
||||
def α₂ (S : (PlusU1 n).LinSols) : ℚ := accQuad S.val
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue