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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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32
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/ToSols.lean
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@ -46,8 +46,8 @@ 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`. -/
@ -134,8 +134,8 @@ lemma inQuadSolProp_iff_proj_inQuadProp (R : MSSMACC.Sols) :
/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies
entirely in the cubic surface. -/
def inCubeProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
cubeTriLin (R.val, R.val, R.val) = 0 ∧ cubeTriLin (R.val, R.val, B₃.val) = 0 ∧
cubeTriLin (R.val, R.val, Y₃.val) = 0
cubeTriLin R.val R.val R.val = 0 ∧ cubeTriLin R.val R.val B₃.val = 0 ∧
cubeTriLin R.val R.val Y₃.val = 0
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inCubeProp R) := by
@ -144,13 +144,13 @@ instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inCubeProp R) := by
/-- A condition which is satisfied if the plane spanned by the solutions `R`, `Y₃` and `B₃`
lies entirely in the cubic surface. -/
def inCubeSolProp (R : MSSMACC.Sols) : Prop :=
cubeTriLin (R.val, R.val, B₃.val) = 0 ∧ cubeTriLin (R.val, R.val, Y₃.val) = 0
cubeTriLin R.val R.val B₃.val = 0 ∧ cubeTriLin R.val R.val Y₃.val = 0
/-- 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 +
cubeTriLin (T.val, T.val, B₃.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) :
inCubeSolProp T ↔ cubicCoeff T = 0 := by
@ -214,9 +214,9 @@ def inQuadCubeSol : Type :=
/-- Given a `R` perpendicular to `Y₃` and `B₃` a quadratic solution. -/
def toSolNSQuad (R : MSSMACC.AnomalyFreePerp) : MSSMACC.QuadSols :=
lineQuad R
(3 * cubeTriLin (R.val, R.val, Y₃.val))
(3 * cubeTriLin (R.val, R.val, B₃.val))
(cubeTriLin (R.val, R.val, R.val))
(3 * cubeTriLin R.val R.val Y₃.val)
(3 * cubeTriLin R.val R.val B₃.val)
(cubeTriLin R.val R.val R.val)
lemma toSolNSQuad_cube (R : MSSMACC.AnomalyFreePerp) :
accCube (toSolNSQuad R).val = 0 := by
@ -325,11 +325,11 @@ lemma inQuadToSol_smul (R : inQuad) (c₁ c₂ c₃ d : ℚ) :
def inQuadProj (T : inQuadSol) : inQuad × 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⟩,
(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),
(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, Y₃.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)))
@ -342,8 +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