262 lines
8.6 KiB
Text
262 lines
8.6 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license.
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Authors: Joseph Tooby-Smith
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-/
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import Mathlib.Tactic.Polyrith
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import Mathlib.Algebra.Module.LinearMap.Basic
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/-!
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# Linear maps
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Some definitions and properites of linear, bilinear, and trilinear maps, along with homogeneous
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quadratic and cubic equations.
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## TODO
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Use definitions in `Mathlib4` for definitions where possible.
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In particular a HomogeneousQuadratic should be a map `V →ₗ[ℚ] V →ₗ[ℚ] ℚ` etc.
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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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namespace HomogeneousQuadratic
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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instance instFun : FunLike (HomogeneousQuadratic 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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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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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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namespace BiLinearSymm
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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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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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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) :=
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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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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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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_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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rw [f.swap, f.map_add₁, f.swap T1 S, f.swap T2 S]
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/-- The homogenous quadratic equation obtainable from a bilinear function. -/
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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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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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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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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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namespace HomogeneousCubic
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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instance instFun : FunLike (HomogeneousCubic 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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lemma map_smul (f : HomogeneousCubic V) (a : ℚ) (S : V) : f (a • S) = a ^ 3 * f S :=
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f.map_smul' a S
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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] 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_add₁' : ∀ S1 S2 T L, toFun (S1 + S2, T, L) = toFun (S1, T, L) + toFun (S2, T, L)
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swap₁' : ∀ S T L, toFun (S, T, L) = toFun (T, S, L)
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swap₂' : ∀ S T L, toFun (S, T, L) = toFun (S, L, T)
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namespace TriLinearSymm
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variable {V : Type} [AddCommMonoid V] [Module ℚ V]
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instance instFun : FunLike (TriLinearSymm 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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lemma toFun_eq_coe (f : TriLinearSymm V) : f.toFun = f := rfl
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lemma swap₁ (f : TriLinearSymm V) (S T L : V) : f (S, T, L) = f (T, S, L) :=
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f.swap₁' S T L
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lemma swap₂ (f : TriLinearSymm V) (S T L : V) : f (S, T, L) = f (S, L, T) :=
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f.swap₂' S T L
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lemma swap₃ (f : TriLinearSymm V) (S T L : V) : f (S, T, L) = f (L, T, S) := by
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rw [f.swap₁, f.swap₂, f.swap₁]
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lemma map_smul₁ (f : TriLinearSymm V) (a : ℚ) (S T L : V) :
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f (a • S, T, L) = a * f (S, T, L) :=
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f.map_smul₁' a S T L
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lemma map_smul₂ (f : TriLinearSymm V) (S : V) (a : ℚ) (T L : V) :
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f (S, a • T, L) = a * f (S, T, L) := by
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rw [f.swap₁, f.map_smul₁, f.swap₁]
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lemma map_smul₃ (f : TriLinearSymm V) (S T : V) (a : ℚ) (L : V) :
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f (S, T, a • L) = a * f (S, T, L) := by
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rw [f.swap₃, f.map_smul₁, f.swap₃]
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lemma map_add₁ (f : TriLinearSymm V) (S1 S2 T L : V) :
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f (S1 + S2, T, L) = f (S1, T, L) + f (S2, T, L) :=
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f.map_add₁' S1 S2 T L
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lemma map_add₂ (f : TriLinearSymm V) (S T1 T2 L : V) :
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f (S, T1 + T2, L) = f (S, T1, L) + f (S, T2, L) := by
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rw [f.swap₁, f.map_add₁, f.swap₁ S T1, f.swap₁ S T2]
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lemma map_add₃ (f : TriLinearSymm V) (S T L1 L2 : V) :
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f (S, T, L1 + L2) = f (S, T, L1) + f (S, T, L2) := by
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rw [f.swap₃, f.map_add₁, f.swap₃, f.swap₃ L2 T S]
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/-- The homogenous cubic equation obtainable from a symmetric trilinear function. -/
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@[simps!]
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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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rw [τ.map_smul₁, τ.map_smul₂, τ.map_smul₃]
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ring
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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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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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end TriLinearSymm
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