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refactor: def of symmetric trilin function

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jstoobysmith 2024-04-22 09:48:44 -04:00
parent 748bcb61ae
commit e36c61b331
24 changed files with 279 additions and 246 deletions
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28
HepLean/AnomalyCancellation/PureU1/Odd/Parameterization.lean
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@ -33,13 +33,13 @@ open VectorLikeOddPlane
show that this can be extended to a complete solution. -/
def parameterizationAsLinear (g f : Fin n → ℚ) (a : ℚ) :
(PureU1 (2 * n + 1)).LinSols :=
a • ((accCubeTriLinSymm (P! f, P! f, P g)) • P' g +
(- accCubeTriLinSymm (P g, P g, P! f)) • P!' f)
a • ((accCubeTriLinSymm (P! f) (P! f) (P g)) • P' g +
(- accCubeTriLinSymm (P g) (P g) (P! f)) • P!' f)
lemma parameterizationAsLinear_val (g f : Fin n → ℚ) (a : ℚ) :
(parameterizationAsLinear g f a).val =
a • ((accCubeTriLinSymm (P! f, P! f, P g)) • P g +
(- accCubeTriLinSymm (P g, P g, P! f)) • P! f) := by
a • ((accCubeTriLinSymm (P! f) (P! f) (P g)) • P g +
(- accCubeTriLinSymm (P g) (P g) (P! f)) • P! f) := by
rw [parameterizationAsLinear]
change a • (_ • (P' g).val + _ • (P!' f).val) = _
rw [P'_val, P!'_val]
@ -65,8 +65,8 @@ def parameterization (g f : Fin n → ℚ) (a : ℚ) :
lemma anomalyFree_param {S : (PureU1 (2 * n + 1)).Sols}
(g f : Fin n → ℚ) (hS : S.val = P g + P! f) :
accCubeTriLinSymm (P g, P g, P! f) =
- accCubeTriLinSymm (P! f, P! f, P g) := by
accCubeTriLinSymm (P g) (P g) (P! f) =
- accCubeTriLinSymm (P! f) (P! f) (P g) := by
have hC := S.cubicSol
rw [hS] at hC
change (accCube (2 * n + 1)) (P g + P! f) = 0 at hC
@ -79,11 +79,11 @@ lemma anomalyFree_param {S : (PureU1 (2 * n + 1)).Sols}
In this case our parameterization above will be able to recover this point. -/
def genericCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
∀ (g f : Fin n.succ → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g, P g, P! f) ≠ 0
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0
lemma genericCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
(hs : ∃ (g f : Fin n.succ → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g, P g, P! f) ≠ 0) : genericCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : genericCase S := by
intro g f hS hC
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -95,11 +95,11 @@ lemma genericCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
In this case we will show that S is zero if it is true for all permutations. -/
def specialCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
∀ (g f : Fin n.succ → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g, P g, P! f) = 0
accCubeTriLinSymm (P g) (P g) (P! f) = 0
lemma specialCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
(hs : ∃ (g f : Fin n.succ → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g, P g, P! f) = 0) : specialCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) = 0) : specialCase S := by
intro g f hS
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -110,8 +110,8 @@ lemma specialCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
genericCase S ∨ specialCase S := by
obtain ⟨g, f, h⟩ := span_basis S.1.1
have h1 : accCubeTriLinSymm (P g, P g, P! f) ≠ 0 ∨
accCubeTriLinSymm (P g, P g, P! f) = 0 := by
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
exact ne_or_eq _ _
cases h1 <;> rename_i h1
exact Or.inl (genericCase_exists S ⟨g, f, h, h1⟩)
@ -120,7 +120,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
theorem generic_case {S : (PureU1 (2 * n.succ + 1)).Sols} (h : genericCase S) :
∃ g f a, S = parameterization g f a := by
obtain ⟨g, f, hS⟩ := span_basis S.1.1
use g, f, (accCubeTriLinSymm (P! f, P! f, P g))⁻¹
use g, f, (accCubeTriLinSymm (P! f) (P! f) (P g))⁻¹
rw [parameterization]
apply ACCSystem.Sols.ext
rw [parameterizationAsLinear_val]
@ -149,7 +149,7 @@ lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ + 1)).Sols}
rw [h]
rw [anomalyFree_param _ _ hS] at h
simp at h
change accCubeTriLinSymm (P! f, P! f, P g) = 0 at h
change accCubeTriLinSymm (P! f) (P! f) (P g) = 0 at h
erw [h]
simp