refactor: def of symmetric trilin function
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jstoobysmith
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748bcb61ae
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e36c61b331
24 changed files with 279 additions and 246 deletions
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@ -471,7 +471,7 @@ lemma P!_accCube (f : Fin n → ℚ) : accCube (2 * n +1) (P! f) = 0 := by
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ring
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lemma P_P_P!_accCube (g : Fin n → ℚ) (j : Fin n) :
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accCubeTriLinSymm.toFun (P g, P g, basis!AsCharges j)
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accCubeTriLinSymm (P g) (P g) (basis!AsCharges j)
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= (P g (δ!₁ j))^2 - (g j)^2 := by
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simp [accCubeTriLinSymm]
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rw [sum_δ!, basis!_on_δ!₃]
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@ -42,8 +42,8 @@ def lineInCubic (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
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lemma lineInCubic_expand {S : (PureU1 (2 * n + 1)).LinSols} (h : lineInCubic S) :
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∀ (g : Fin n → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f) (a b : ℚ) ,
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3 * a * b * (a * accCubeTriLinSymm (P g, P g, P! f)
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+ b * accCubeTriLinSymm (P! f, P! f, P g)) = 0 := by
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3 * a * b * (a * accCubeTriLinSymm (P g) (P g) (P! f)
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+ b * accCubeTriLinSymm (P! f) (P! f) (P g)) = 0 := by
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intro g f hS a b
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have h1 := h g f hS a b
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change accCubeTriLinSymm.toCubic (a • P g + b • P! f) = 0 at h1
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@ -57,7 +57,7 @@ lemma lineInCubic_expand {S : (PureU1 (2 * n + 1)).LinSols} (h : lineInCubic S)
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lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n + 1)).LinSols} (h : lineInCubic S) :
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∀ (g : Fin n → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f),
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accCubeTriLinSymm (P g, P g, P! f) = 0 := by
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accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
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intro g f hS
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linear_combination 2 / 3 * (lineInCubic_expand h g f hS 1 1 ) -
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(lineInCubic_expand h g f hS 1 2 ) / 6
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@ -91,7 +91,7 @@ lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ + 1)).LinSols}
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(LIC : lineInCubicPerm S) :
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∀ (j : Fin n.succ) (g f : Fin n.succ → ℚ) (_ : S.val = Pa g f) ,
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(S.val (δ!₂ j) - S.val (δ!₁ j))
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* accCubeTriLinSymm.toFun (P g, P g, basis!AsCharges j) = 0 := by
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* accCubeTriLinSymm (P g) (P g) (basis!AsCharges j) = 0 := by
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intro j g f h
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let S' := (FamilyPermutations (2 * n.succ + 1)).linSolRep
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(pairSwap (δ!₁ j) (δ!₂ j)) S
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@ -107,7 +107,7 @@ lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ + 1)).LinSols}
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lemma P_P_P!_accCube' {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
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(f g : Fin n.succ.succ → ℚ) (hS : S.val = Pa f g) :
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accCubeTriLinSymm.toFun (P f, P f, basis!AsCharges 0) =
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accCubeTriLinSymm (P f) (P f) (basis!AsCharges 0) =
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(S.val (δ!₁ 0) + S.val (δ!₂ 0)) * (2 * S.val δ!₃ + S.val (δ!₁ 0) + S.val (δ!₂ 0)) := by
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rw [P_P_P!_accCube f 0]
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rw [← Pa_δa₁ f g]
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@ -33,13 +33,13 @@ open VectorLikeOddPlane
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show that this can be extended to a complete solution. -/
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def parameterizationAsLinear (g f : Fin n → ℚ) (a : ℚ) :
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(PureU1 (2 * n + 1)).LinSols :=
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a • ((accCubeTriLinSymm (P! f, P! f, P g)) • P' g +
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(- accCubeTriLinSymm (P g, P g, P! f)) • P!' f)
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a • ((accCubeTriLinSymm (P! f) (P! f) (P g)) • P' g +
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(- accCubeTriLinSymm (P g) (P g) (P! f)) • P!' f)
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lemma parameterizationAsLinear_val (g f : Fin n → ℚ) (a : ℚ) :
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(parameterizationAsLinear g f a).val =
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a • ((accCubeTriLinSymm (P! f, P! f, P g)) • P g +
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(- accCubeTriLinSymm (P g, P g, P! f)) • P! f) := by
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a • ((accCubeTriLinSymm (P! f) (P! f) (P g)) • P g +
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(- accCubeTriLinSymm (P g) (P g) (P! f)) • P! f) := by
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rw [parameterizationAsLinear]
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change a • (_ • (P' g).val + _ • (P!' f).val) = _
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rw [P'_val, P!'_val]
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@ -65,8 +65,8 @@ def parameterization (g f : Fin n → ℚ) (a : ℚ) :
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lemma anomalyFree_param {S : (PureU1 (2 * n + 1)).Sols}
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(g f : Fin n → ℚ) (hS : S.val = P g + P! f) :
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accCubeTriLinSymm (P g, P g, P! f) =
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- accCubeTriLinSymm (P! f, P! f, P g) := by
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accCubeTriLinSymm (P g) (P g) (P! f) =
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- accCubeTriLinSymm (P! f) (P! f) (P g) := by
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have hC := S.cubicSol
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rw [hS] at hC
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change (accCube (2 * n + 1)) (P g + P! f) = 0 at hC
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@ -79,11 +79,11 @@ lemma anomalyFree_param {S : (PureU1 (2 * n + 1)).Sols}
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In this case our parameterization above will be able to recover this point. -/
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def genericCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
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∀ (g f : Fin n.succ → ℚ) (_ : S.val = P g + P! f) ,
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accCubeTriLinSymm (P g, P g, P! f) ≠ 0
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accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0
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lemma genericCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
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(hs : ∃ (g f : Fin n.succ → ℚ), S.val = P g + P! f ∧
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accCubeTriLinSymm (P g, P g, P! f) ≠ 0) : genericCase S := by
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accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : genericCase S := by
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intro g f hS hC
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obtain ⟨g', f', hS', hC'⟩ := hs
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rw [hS] at hS'
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@ -95,11 +95,11 @@ lemma genericCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
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In this case we will show that S is zero if it is true for all permutations. -/
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def specialCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
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∀ (g f : Fin n.succ → ℚ) (_ : S.val = P g + P! f) ,
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accCubeTriLinSymm (P g, P g, P! f) = 0
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accCubeTriLinSymm (P g) (P g) (P! f) = 0
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lemma specialCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
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(hs : ∃ (g f : Fin n.succ → ℚ), S.val = P g + P! f ∧
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accCubeTriLinSymm (P g, P g, P! f) = 0) : specialCase S := by
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accCubeTriLinSymm (P g) (P g) (P! f) = 0) : specialCase S := by
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intro g f hS
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obtain ⟨g', f', hS', hC'⟩ := hs
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rw [hS] at hS'
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@ -110,8 +110,8 @@ lemma specialCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
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lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
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genericCase S ∨ specialCase S := by
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obtain ⟨g, f, h⟩ := span_basis S.1.1
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have h1 : accCubeTriLinSymm (P g, P g, P! f) ≠ 0 ∨
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accCubeTriLinSymm (P g, P g, P! f) = 0 := by
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have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
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accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
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exact ne_or_eq _ _
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cases h1 <;> rename_i h1
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exact Or.inl (genericCase_exists S ⟨g, f, h, h1⟩)
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@ -120,7 +120,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
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theorem generic_case {S : (PureU1 (2 * n.succ + 1)).Sols} (h : genericCase S) :
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∃ g f a, S = parameterization g f a := by
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obtain ⟨g, f, hS⟩ := span_basis S.1.1
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use g, f, (accCubeTriLinSymm (P! f, P! f, P g))⁻¹
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use g, f, (accCubeTriLinSymm (P! f) (P! f) (P g))⁻¹
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rw [parameterization]
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apply ACCSystem.Sols.ext
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rw [parameterizationAsLinear_val]
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@ -149,7 +149,7 @@ lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ + 1)).Sols}
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rw [h]
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rw [anomalyFree_param _ _ hS] at h
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simp at h
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change accCubeTriLinSymm (P! f, P! f, P g) = 0 at h
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change accCubeTriLinSymm (P! f) (P! f) (P g) = 0 at h
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erw [h]
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simp
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