reactor: Removal of double spaces
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jstoobysmith
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64 changed files with 550 additions and 546 deletions
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@ -14,7 +14,7 @@ import Mathlib.RepresentationTheory.Basic
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/-!
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# Parameterization in odd case
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Given maps `g : Fin n → ℚ`, `f : Fin n → ℚ` and `a : ℚ` we form a solution to the anomaly
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Given maps `g : Fin n → ℚ`, `f : Fin n → ℚ` and `a : ℚ` we form a solution to the anomaly
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equations. We show that every solution can be got in this way, up to permutation, unless it is zero.
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The main reference is:
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@ -31,7 +31,7 @@ open VectorLikeOddPlane
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/-- Given a `g f : Fin n → ℚ` and a `a : ℚ` we form a linear solution. We will later
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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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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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@ -44,7 +44,7 @@ lemma parameterizationAsLinear_val (g f : Fin n → ℚ) (a : ℚ) :
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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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lemma parameterizationCharge_cube (g f : Fin n → ℚ) (a : ℚ):
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lemma parameterizationCharge_cube (g f : Fin n → ℚ) (a : ℚ):
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(accCube (2 * n + 1)) (parameterizationAsLinear g f a).val = 0 := by
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change accCubeTriLinSymm.toCubic _ = 0
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rw [parameterizationAsLinear_val]
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@ -64,7 +64,7 @@ def parameterization (g f : Fin n → ℚ) (a : ℚ) :
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parameterizationCharge_cube g f a⟩
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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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(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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have hC := S.cubicSol
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@ -75,15 +75,15 @@ lemma anomalyFree_param {S : (PureU1 (2 * n + 1)).Sols}
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erw [P!_accCube] at hC
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linear_combination hC / 3
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/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
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/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
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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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∀ (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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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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@ -91,15 +91,15 @@ lemma genericCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
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rw [hS'.1, hS'.2] at hC
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exact hC' hC
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/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
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/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
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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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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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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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@ -110,7 +110,7 @@ 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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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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@ -118,7 +118,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
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exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
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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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∃ 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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rw [parameterization]
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@ -164,7 +164,7 @@ theorem special_case {S : (PureU1 (2 * n.succ.succ + 1)).Sols}
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(h : ∀ (M : (FamilyPermutations (2 * n.succ.succ + 1)).group),
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SpecialCase ((FamilyPermutations (2 * n.succ.succ + 1)).solAction.toFun S M)) :
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S.1.1 = 0 := by
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have ht := special_case_lineInCubic_perm h
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have ht := special_case_lineInCubic_perm h
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exact lineInCubicPerm_zero ht
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end Odd
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end PureU1
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