refactor: Parameterization in MSSM ACC
This commit is contained in:
jstoobysmith
parent
1f573395cc
commit
6d4078f979
1 changed files with 294 additions and 500 deletions
|
|
@ -34,606 +34,400 @@ open MSSMCharges
|
|||
open MSSMACCs
|
||||
open BigOperators
|
||||
|
||||
namespace AnomalyFreePerp
|
||||
|
||||
def lineEqProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
|
||||
α₁ R = 0 ∧ α₂ R = 0 ∧ α₃ R = 0
|
||||
|
||||
|
||||
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (lineEqProp R) := by
|
||||
apply And.decidable
|
||||
|
||||
|
||||
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
|
||||
|
||||
def lineEqCoeff (T : MSSMACC.Sols) : ℚ := dot (Y₃.val, B₃.val) * α₃ (proj T.1.1)
|
||||
|
||||
lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
|
||||
lineEqPropSol T ↔ lineEqCoeff T = 0 := by
|
||||
rw [lineEqPropSol, lineEqCoeff, α₃]
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero,
|
||||
false_or]
|
||||
rw [cube_proj_proj_B₃, cube_proj_proj_Y₃, quad_B₃_proj, quad_Y₃_proj]
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl]
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, OfNat.ofNat_ne_zero, false_or]
|
||||
have h1 : 108 ^ 2 * cubeTriLin (T.val, T.val, Y₃.val) * (108 * quadBiLin (B₃.val, T.val)) -
|
||||
108 ^ 2 * cubeTriLin (T.val, T.val, B₃.val) * (108 * quadBiLin (Y₃.val, T.val)) =
|
||||
108 ^ 3 * (cubeTriLin (T.val, T.val, Y₃.val) * quadBiLin (B₃.val, T.val) -
|
||||
cubeTriLin (T.val, T.val, B₃.val) * quadBiLin (Y₃.val, T.val) ) := by
|
||||
ring
|
||||
rw [h1]
|
||||
simp
|
||||
|
||||
lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
|
||||
lineEqPropSol R ↔ lineEqProp (proj R.1.1) := by
|
||||
rw [lineEqPropSol_iff_lineEqCoeff_zero, lineEqCoeff, lineEqProp]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
simp at h
|
||||
rw [α₁_proj, α₂_proj, h]
|
||||
simp
|
||||
intro h
|
||||
rw [h.2.2]
|
||||
simp
|
||||
|
||||
|
||||
/-- Case₂ is defined when the plane lies entirely within the quadratic. -/
|
||||
def inQuadProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
|
||||
quadBiLin (R.val, R.val) = 0 ∧ quadBiLin (Y₃.val, R.val) = 0 ∧ quadBiLin (B₃.val, R.val) = 0
|
||||
|
||||
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inQuadProp R) := by
|
||||
apply And.decidable
|
||||
|
||||
def inQuadSolProp (R : MSSMACC.Sols) : Prop :=
|
||||
quadBiLin (Y₃.val, R.val) = 0 ∧ quadBiLin (B₃.val, R.val) = 0
|
||||
|
||||
/-- The coefficent which multiplies a solution on passing through `case₁`. -/
|
||||
def quadCoeff (T : MSSMACC.Sols) : ℚ :=
|
||||
2 * dot (Y₃.val, B₃.val) ^ 2 *
|
||||
(quadBiLin (Y₃.val, T.val) ^ 2 + quadBiLin (B₃.val, T.val) ^ 2)
|
||||
|
||||
lemma inQuadSolProp_iff_quadCoeff_zero (T : MSSMACC.Sols) : inQuadSolProp T ↔ quadCoeff T = 0 := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [quadCoeff, h.1, h.2]
|
||||
simp
|
||||
intro h
|
||||
rw [quadCoeff] at h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
simp only [ Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
|
||||
not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
|
||||
apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero,
|
||||
not_false_eq_true, pow_eq_zero_iff] at h
|
||||
exact h
|
||||
|
||||
lemma inQuadSolProp_iff_proj_inQuadProp (R : MSSMACC.Sols) :
|
||||
inQuadSolProp R ↔ inQuadProp (proj R.1.1) := by
|
||||
rw [inQuadSolProp, inQuadProp]
|
||||
rw [quad_proj, quad_Y₃_proj, quad_B₃_proj]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [h.1, h.2]
|
||||
simp
|
||||
intro h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk , mul_eq_zero,
|
||||
OfNat.ofNat_ne_zero, or_self, false_or] at h
|
||||
rw [h.2.1, h.2.2]
|
||||
simp
|
||||
|
||||
/-- Case₃ is defined when the plane lies entirely within the quadratic and cubic. -/
|
||||
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
|
||||
|
||||
|
||||
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inCubeProp R) := by
|
||||
apply And.decidable
|
||||
|
||||
def inCubeSolProp (R : MSSMACC.Sols) : Prop :=
|
||||
cubeTriLin (R.val, R.val, B₃.val) = 0 ∧ cubeTriLin (R.val, R.val, Y₃.val) = 0
|
||||
|
||||
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 )
|
||||
|
||||
lemma inCubeSolProp_iff_cubicCoeff_zero (T : MSSMACC.Sols) :
|
||||
inCubeSolProp T ↔ cubicCoeff T = 0 := by
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [cubicCoeff, h.1, h.2]
|
||||
simp
|
||||
intro h
|
||||
rw [cubicCoeff] at h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
simp only [ Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
|
||||
not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
|
||||
apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero,
|
||||
not_false_eq_true, pow_eq_zero_iff] at h
|
||||
exact h.symm
|
||||
|
||||
lemma inCubeSolProp_iff_proj_inCubeProp (R : MSSMACC.Sols) :
|
||||
inCubeSolProp R ↔ inCubeProp (proj R.1.1) := by
|
||||
rw [inCubeSolProp, inCubeProp]
|
||||
rw [cube_proj, cube_proj_proj_Y₃, cube_proj_proj_B₃]
|
||||
apply Iff.intro
|
||||
intro h
|
||||
rw [h.1, h.2]
|
||||
simp
|
||||
intro h
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
|
||||
simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
|
||||
not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
|
||||
rw [h.2.1, h.2.2]
|
||||
simp
|
||||
|
||||
|
||||
/-- Given a `R ∈ LinSols` perpendicular to $Y_3$, and $B_3$, a solution to the quadratic. -/
|
||||
def genericQuad (R : MSSMACC.AnomalyFreePerp) : MSSMACC.QuadSols :=
|
||||
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))
|
||||
|
||||
lemma genericQuad_cube (R : MSSMACC.AnomalyFreePerp) :
|
||||
accCube (genericQuad R).val = 0 := by
|
||||
rw [genericQuad]
|
||||
lemma toSolNSQuad_cube (R : MSSMACC.AnomalyFreePerp) :
|
||||
accCube (toSolNSQuad R).val = 0 := by
|
||||
rw [toSolNSQuad]
|
||||
rw [lineQuad_val]
|
||||
rw [planeY₃B₃_cubic]
|
||||
ring
|
||||
|
||||
/-- Given a `R ∈ LinSols` perpendicular to $Y_3$, and $B_3$, a element of `Sols`. -/
|
||||
def generic (R : MSSMACC.AnomalyFreePerp) : MSSMACC.Sols :=
|
||||
AnomalyFreeMk'' (genericQuad R) (genericQuad_cube R)
|
||||
|
||||
lemma generic_eq_planeY₃B₃_on_α (R : MSSMACC.AnomalyFreePerp) :
|
||||
(generic R).1.1 = planeY₃B₃ R (α₁ R) (α₂ R) (α₃ R) := by
|
||||
lemma toSolNSQuad_eq_planeY₃B₃_on_α (R : MSSMACC.AnomalyFreePerp) :
|
||||
(toSolNSQuad R).1 = planeY₃B₃ R (α₁ R) (α₂ R) (α₃ R) := by
|
||||
change (planeY₃B₃ _ _ _ _) = _
|
||||
apply planeY₃B₃_eq
|
||||
rw [α₁, α₂, α₃]
|
||||
ring_nf
|
||||
simp
|
||||
|
||||
/-- Case₁ is when the quadratic and cubic lines in the plane agree, which occurs when
|
||||
`α₁ R = 0`, `α₂ R = 0` and `α₃ R = 0` (if they exist uniquely). -/
|
||||
def case₁prop (R : MSSMACC.AnomalyFreePerp) : Prop :=
|
||||
α₁ R = 0 ∧ α₂ R = 0 ∧ α₃ R = 0
|
||||
/-- Given a `R ∈ LinSols` perpendicular to $Y_3$, and $B_3$, a element of `Sols`. -/
|
||||
def toSolNS : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a, _ , _) =>
|
||||
a • AnomalyFreeMk'' (toSolNSQuad R) (toSolNSQuad_cube R)
|
||||
|
||||
/-- Case₂ is defined when the plane lies entirely within the quadratic. -/
|
||||
def case₂prop (R : MSSMACC.AnomalyFreePerp) : Prop :=
|
||||
quadBiLin (R.val, R.val) = 0 ∧ quadBiLin (Y₃.val, R.val) = 0 ∧ quadBiLin (B₃.val, R.val) = 0
|
||||
def toSolNSProj (T : MSSMACC.Sols) : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ :=
|
||||
(proj T.1.1, (lineEqCoeff T)⁻¹, 0, 0)
|
||||
|
||||
/-- Case₃ is defined when the plane lies entirely within the quadratic and cubic. -/
|
||||
def case₃prop (R : MSSMACC.AnomalyFreePerp) : Prop :=
|
||||
quadBiLin (R.val, R.val) = 0 ∧ quadBiLin (Y₃.val, R.val) = 0 ∧ quadBiLin (B₃.val, R.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 (case₁prop R) := by
|
||||
apply And.decidable
|
||||
|
||||
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (case₂prop R) := by
|
||||
apply And.decidable
|
||||
|
||||
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (case₃prop R) := by
|
||||
apply And.decidable
|
||||
|
||||
section proj
|
||||
|
||||
/-- On elements of `Sols`, `generic (proj _)` is equivalent to multiplying
|
||||
by `genericProjCoeff`. -/
|
||||
def genericProjCoeff (T : MSSMACC.Sols) : ℚ :=
|
||||
dot (Y₃.val, B₃.val) * α₃ (proj T.1.1)
|
||||
|
||||
lemma generic_proj (T : MSSMACC.Sols) :
|
||||
generic (proj T.1.1) = (genericProjCoeff T) • T := by
|
||||
lemma toSolNS_proj (T : MSSMACC.Sols) (h : lineEqCoeff T ≠ 0) :
|
||||
toSolNS (toSolNSProj T) = T := by
|
||||
apply ACCSystem.Sols.ext
|
||||
erw [generic_eq_planeY₃B₃_on_α]
|
||||
rw [toSolNS, toSolNSProj]
|
||||
change (lineEqCoeff T)⁻¹ • (toSolNSQuad _).1.1 = _
|
||||
rw [toSolNSQuad_eq_planeY₃B₃_on_α]
|
||||
rw [planeY₃B₃_val]
|
||||
rw [Y₃_plus_B₃_plus_proj]
|
||||
rw [α₁_proj, α₂_proj]
|
||||
simp
|
||||
rfl
|
||||
|
||||
lemma genericProjCoeff_ne_zero (T : MSSMACC.Sols) (hT : genericProjCoeff T ≠ 0 ) :
|
||||
(genericProjCoeff T)⁻¹ • generic (proj T.1.1) = T := by
|
||||
rw [generic_proj, ← MulAction.mul_smul, mul_comm, mul_inv_cancel hT]
|
||||
ring_nf
|
||||
simp only [zero_smul, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add]
|
||||
have h1 : α₃ (proj T.toLinSols) * dot (Y₃.val, B₃.val) = lineEqCoeff T := by
|
||||
rw [lineEqCoeff]
|
||||
ring
|
||||
rw [h1]
|
||||
rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h]
|
||||
simp
|
||||
|
||||
lemma genericProjCoeff_zero_α₃ (T : MSSMACC.Sols) (hT : genericProjCoeff T = 0) :
|
||||
α₃ (proj T.1.1) = 0 := by
|
||||
rw [genericProjCoeff, mul_eq_zero] at hT
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at hT
|
||||
simp at hT
|
||||
exact hT
|
||||
def inLineEq : Type := {R : MSSMACC.AnomalyFreePerp // lineEqProp R}
|
||||
|
||||
lemma genericProjCoeff_zero_α₂ (T : MSSMACC.Sols) (hT : genericProjCoeff T = 0) :
|
||||
α₂ (proj T.1.1) = 0 := by
|
||||
rw [α₂_proj, genericProjCoeff_zero_α₃ T hT]
|
||||
simp
|
||||
def inLineEqSol : Type := {R : MSSMACC.Sols // lineEqPropSol R}
|
||||
|
||||
lemma genericProjCoeff_zero_α₁ (T : MSSMACC.Sols) (hT : genericProjCoeff T = 0) :
|
||||
α₁ (proj T.1.1) = 0 := by
|
||||
rw [α₁_proj, genericProjCoeff_zero_α₃ T hT]
|
||||
simp
|
||||
def inLineEqProj (T : inLineEqSol) : inLineEq × ℚ × ℚ × ℚ :=
|
||||
(⟨proj T.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop ⟩,
|
||||
(quadCoeff T.val)⁻¹ * quadBiLin (B₃.val, T.val.val),
|
||||
(quadCoeff T.val)⁻¹ * (- quadBiLin (Y₃.val, T.val.val)),
|
||||
(quadCoeff T.val)⁻¹ * (- quadBiLin (B₃.val, T.val.val) * ( dot (Y₃.val, T.val.val)
|
||||
- dot (B₃.val, T.val.val))
|
||||
- quadBiLin (Y₃.val, T.val.val) * ( dot (Y₃.val, T.val.val) - 2 * dot (B₃.val, T.val.val))))
|
||||
|
||||
lemma genericProjCoeff_zero (T : MSSMACC.Sols) :
|
||||
genericProjCoeff T = 0 ↔ case₁prop (proj T.1.1) := by
|
||||
apply Iff.intro
|
||||
intro hT
|
||||
rw [case₁prop]
|
||||
rw [genericProjCoeff_zero_α₁ T hT, genericProjCoeff_zero_α₂ T hT, genericProjCoeff_zero_α₃ T hT]
|
||||
simp only [and_self]
|
||||
intro h
|
||||
rw [case₁prop] at h
|
||||
rw [genericProjCoeff]
|
||||
rw [h.2.2]
|
||||
simp
|
||||
|
||||
lemma genericProjCoeff_neq_zero_case₁ (T : MSSMACC.Sols) (hT : genericProjCoeff T ≠ 0) :
|
||||
¬ case₁prop (proj T.1.1) :=
|
||||
(genericProjCoeff_zero T).mpr.mt hT
|
||||
|
||||
lemma genericProjCoeff_neq_zero_case₂ (T : MSSMACC.Sols) (hT : genericProjCoeff T ≠ 0) :
|
||||
¬ case₂prop (proj T.1.1) := by
|
||||
by_contra hn
|
||||
rw [case₂prop] at hn
|
||||
rw [genericProjCoeff, α₃] at hT
|
||||
simp_all
|
||||
|
||||
lemma genericProjCoeff_neq_zero_case₃ (T : MSSMACC.Sols) (hT : genericProjCoeff T ≠ 0) :
|
||||
¬ case₃prop (proj T.1.1) := by
|
||||
by_contra hn
|
||||
rw [case₃prop] at hn
|
||||
rw [genericProjCoeff, α₃] at hT
|
||||
simp_all
|
||||
|
||||
end proj
|
||||
|
||||
/-- The case when the quadratic and cubic lines agree (if they exist uniquely). -/
|
||||
def case₁ (R : MSSMACC.AnomalyFreePerp) (c₁ c₂ c₃ : ℚ)
|
||||
(h : case₁prop R) : MSSMACC.Sols :=
|
||||
AnomalyFreeMk'' (lineQuad R c₁ c₂ c₃)
|
||||
def inLineEqToSol : inLineEq × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, c₁, c₂, c₃) =>
|
||||
AnomalyFreeMk'' (lineQuad R.val c₁ c₂ c₃)
|
||||
(by
|
||||
rw [lineQuad_cube]
|
||||
rw [h.1, h.2.1, h.2.2]
|
||||
rw [R.prop.1, R.prop.2.1, R.prop.2.2]
|
||||
simp)
|
||||
|
||||
lemma case₁_smul (R : MSSMACC.AnomalyFreePerp) (c₁ c₂ c₃ d : ℚ)
|
||||
(h : case₁prop R) : case₁ R (d * c₁) (d * c₂) (d * c₃) h = d • case₁ R c₁ c₂ c₃ h := by
|
||||
lemma inLineEqTo_smul (R : inLineEq) (c₁ c₂ c₃ d : ℚ) :
|
||||
inLineEqToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inLineEqToSol (R, c₁, c₂, c₃) := by
|
||||
apply ACCSystem.Sols.ext
|
||||
change (lineQuad _ _ _ _).val = _
|
||||
rw [lineQuad_smul]
|
||||
rfl
|
||||
|
||||
section proj
|
||||
|
||||
/-- The coefficent which multiplies a solution on passing through `case₁`. -/
|
||||
def case₁ProjCoeff (T : MSSMACC.Sols) : ℚ :=
|
||||
2 * (quadBiLin (Y₃.val, (proj T.1.1).val) ^ 2 +
|
||||
quadBiLin (B₃.val, (proj T.1.1).val) ^ 2)
|
||||
|
||||
/-- The first parameter in case₁ needed to form an inverse on Proj. -/
|
||||
def case₁ProjC₁ (T : MSSMACC.Sols) : ℚ := (quadBiLin (B₃.val, T.val))
|
||||
|
||||
/-- The second parameter in case₁ needed to form an inverse on Proj. -/
|
||||
def case₁ProjC₂ (T : MSSMACC.Sols) : ℚ := (- quadBiLin (Y₃.val, T.val))
|
||||
|
||||
/-- The third parameter in case₁ needed to form an inverse on Proj. -/
|
||||
def case₁ProjC₃ (T : MSSMACC.Sols) : ℚ :=
|
||||
- quadBiLin (B₃.val, T.val) * ( dot (Y₃.val, T.val)- dot (B₃.val, T.val) )
|
||||
- quadBiLin (Y₃.val, T.val) * ( dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val) )
|
||||
|
||||
lemma case₁_proj (T : MSSMACC.Sols) (h1 : genericProjCoeff T = 0) :
|
||||
case₁ (proj T.1.1)
|
||||
(case₁ProjC₁ T)
|
||||
(case₁ProjC₂ T)
|
||||
(case₁ProjC₃ T)
|
||||
((genericProjCoeff_zero T).mp h1)
|
||||
= (case₁ProjCoeff T) • T := by
|
||||
lemma inLineEqToSol_proj (T : inLineEqSol) (h : quadCoeff T.val ≠ 0) :
|
||||
inLineEqToSol (inLineEqProj T) = T.val := by
|
||||
rw [inLineEqProj, inLineEqTo_smul]
|
||||
apply ACCSystem.Sols.ext
|
||||
change (lineQuad _ _ _ _).val = _
|
||||
change _ • (lineQuad _ _ _ _).val = _
|
||||
rw [lineQuad_val]
|
||||
rw [planeY₃B₃_val]
|
||||
rw [Y₃_plus_B₃_plus_proj]
|
||||
rw [case₁ProjCoeff, case₁ProjC₁, case₁ProjC₂, case₁ProjC₃, quad_proj, quad_Y₃_proj, quad_B₃_proj]
|
||||
rw [quad_proj, quad_Y₃_proj, quad_B₃_proj]
|
||||
ring_nf
|
||||
simp
|
||||
rfl
|
||||
|
||||
lemma case₁ProjCoeff_ne_zero (T : MSSMACC.Sols) (h1 : genericProjCoeff T = 0)
|
||||
(hT : case₁ProjCoeff T ≠ 0 ) :
|
||||
(case₁ProjCoeff T)⁻¹ • case₁ (proj T.1.1)
|
||||
(case₁ProjC₁ T)
|
||||
(case₁ProjC₂ T)
|
||||
(case₁ProjC₃ T)
|
||||
((genericProjCoeff_zero T).mp h1) = T := by
|
||||
rw [case₁_proj T h1, ← MulAction.mul_smul, mul_comm, mul_inv_cancel hT]
|
||||
simp only [zero_smul, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add]
|
||||
have h1 : (quadBiLin (Y₃.val, (T).val.val) ^ 2 * dot (Y₃.val, B₃.val) ^ 2 * 2 +
|
||||
dot (Y₃.val, B₃.val) ^ 2 * quadBiLin (B₃.val, (T).val.val) ^ 2 * 2) = quadCoeff T.val := by
|
||||
rw [quadCoeff]
|
||||
ring
|
||||
rw [h1]
|
||||
rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h]
|
||||
simp
|
||||
|
||||
lemma case₁ProjCoeff_zero_Y₃_B₃ (T : MSSMACC.Sols) (h1 : case₁ProjCoeff T = 0) :
|
||||
quadBiLin (Y₃.val, (proj T.1.1).val) = 0 ∧
|
||||
quadBiLin (B₃.val, (proj T.1.1).val) = 0 := by
|
||||
rw [case₁ProjCoeff, mul_eq_zero] at h1
|
||||
simp only [OfNat.ofNat_ne_zero, Fin.isValue, Fin.reduceFinMk, false_or] at h1
|
||||
have h : quadBiLin (Y₃.val, (proj T.1.1).val) ^ 2 = 0 ∧
|
||||
quadBiLin (B₃.val, (proj T.1.1).val) ^ 2 = 0 :=
|
||||
(add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp h1
|
||||
simp only [ Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero,
|
||||
not_false_eq_true, pow_eq_zero_iff] at h
|
||||
exact h
|
||||
|
||||
lemma case₁ProjCoeff_zero_Y₃ (T : MSSMACC.Sols) (h1 : case₁ProjCoeff T = 0) :
|
||||
quadBiLin (Y₃.val, (proj T.1.1).val) = 0 :=
|
||||
(case₁ProjCoeff_zero_Y₃_B₃ T h1).left
|
||||
def inQuad : Type := {R : inLineEq // inQuadProp R.val}
|
||||
|
||||
lemma case₁ProjCoeff_zero_B₃ (T : MSSMACC.Sols) (h1 : case₁ProjCoeff T = 0) :
|
||||
quadBiLin (B₃.val, (proj T.1.1).val) = 0 :=
|
||||
(case₁ProjCoeff_zero_Y₃_B₃ T h1).right
|
||||
def inQuadSol : Type := {R : inLineEqSol // inQuadSolProp R.val}
|
||||
|
||||
lemma case₁ProjCoeff_zero_T (T : MSSMACC.Sols) (h1 : case₁ProjCoeff T = 0) :
|
||||
quadBiLin (T.val, (proj T.1.1).val) = 0 := by
|
||||
have hY3 : quadBiLin (T.val, Y₃.val) = 0 := by
|
||||
have h11 := case₁ProjCoeff_zero_Y₃ T h1
|
||||
rw [quad_Y₃_proj] at h11
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h11
|
||||
simp only [ Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero,
|
||||
false_or] at h11
|
||||
erw [quadBiLin.swap] at h11
|
||||
exact h11
|
||||
have hB3 : quadBiLin (T.val, B₃.val) = 0 := by
|
||||
have h11 := case₁ProjCoeff_zero_B₃ T h1
|
||||
rw [quad_B₃_proj] at h11
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h11
|
||||
simp only [ Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero,
|
||||
false_or] at h11
|
||||
erw [quadBiLin.swap] at h11
|
||||
exact h11
|
||||
rw [proj_val]
|
||||
rw [quadBiLin.map_add₂, quadBiLin.map_add₂]
|
||||
rw [quadBiLin.map_smul₂, quadBiLin.map_smul₂, quadBiLin.map_smul₂]
|
||||
rw [hY3, hB3]
|
||||
erw [quadSol T.1]
|
||||
simp
|
||||
|
||||
lemma case₁ProjCoeff_zero_self (T : MSSMACC.Sols) (h1 : case₁ProjCoeff T = 0) :
|
||||
quadBiLin ((proj T.1.1).val, (proj T.1.1).val) = 0 := by
|
||||
nth_rewrite 1 [proj_val]
|
||||
rw [quadBiLin.map_add₁, quadBiLin.map_add₁]
|
||||
rw [quadBiLin.map_smul₁, quadBiLin.map_smul₁, quadBiLin.map_smul₁]
|
||||
rw [case₁ProjCoeff_zero_Y₃ T h1, case₁ProjCoeff_zero_B₃ T h1, case₁ProjCoeff_zero_T T h1]
|
||||
simp
|
||||
|
||||
lemma case₁ProjCoeff_zero (T : MSSMACC.Sols) :
|
||||
case₁ProjCoeff T = 0 ↔ case₂prop (proj T.1.1) := by
|
||||
apply Iff.intro
|
||||
intro h1
|
||||
rw [case₂prop]
|
||||
rw [case₁ProjCoeff_zero_self T h1, case₁ProjCoeff_zero_Y₃ T h1, case₁ProjCoeff_zero_B₃ T h1]
|
||||
simp only [and_self]
|
||||
intro h
|
||||
rw [case₂prop] at h
|
||||
rw [case₁ProjCoeff]
|
||||
rw [h.2.1, h.2.2]
|
||||
simp
|
||||
|
||||
lemma case₁ProjCoeff_ne_zero_case₂ (T : MSSMACC.Sols) (h1 : case₁ProjCoeff T ≠ 0) :
|
||||
¬ case₂prop (proj T.1.1) :=
|
||||
(case₁ProjCoeff_zero T).mpr.mt h1
|
||||
|
||||
lemma case₁ProjCoeff_ne_zero_case₃ (T : MSSMACC.Sols) (h1 : case₁ProjCoeff T ≠ 0) :
|
||||
¬ case₃prop (proj T.1.1) := by
|
||||
by_contra hn
|
||||
rw [case₃prop] at hn
|
||||
rw [case₁ProjCoeff] at h1
|
||||
simp_all
|
||||
|
||||
|
||||
end proj
|
||||
|
||||
/-- The case where the plane lies entirely within the quadratic. -/
|
||||
def case₂ (R : MSSMACC.AnomalyFreePerp) (a₁ a₂ a₃ : ℚ)
|
||||
(h : case₂prop R) : MSSMACC.Sols :=
|
||||
AnomalyFreeMk' (lineCube R a₁ a₂ a₃)
|
||||
def inQuadToSol : inQuad × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a₁, a₂, a₃) =>
|
||||
AnomalyFreeMk' (lineCube R.val.val a₁ a₂ a₃)
|
||||
(by
|
||||
erw [planeY₃B₃_quad]
|
||||
rw [h.1, h.2.1, h.2.2]
|
||||
rw [R.prop.1, R.prop.2.1, R.prop.2.2]
|
||||
simp)
|
||||
(lineCube_cube R a₁ a₂ a₃)
|
||||
(lineCube_cube R.val.val a₁ a₂ a₃)
|
||||
|
||||
lemma case₂_smul (R : MSSMACC.AnomalyFreePerp) (c₁ c₂ c₃ d : ℚ)
|
||||
(h : case₂prop R) : case₂ R (d * c₁) (d * c₂) (d * c₃) h = d • case₂ R c₁ c₂ c₃ h := by
|
||||
lemma inQuadToSol_smul (R : inQuad) (c₁ c₂ c₃ d : ℚ) :
|
||||
inQuadToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inQuadToSol (R, c₁, c₂, c₃) := by
|
||||
apply ACCSystem.Sols.ext
|
||||
change (lineCube _ _ _ _).val = _
|
||||
rw [lineCube_smul]
|
||||
rfl
|
||||
|
||||
section proj
|
||||
def inQuadProj (T : inQuadSol) : inQuad × ℚ × ℚ × ℚ :=
|
||||
(⟨⟨proj T.val.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val.val).mp T.val.prop ⟩,
|
||||
(inQuadSolProp_iff_proj_inQuadProp T.val.val).mp T.prop⟩,
|
||||
(cubicCoeff T.val.val)⁻¹ * (cubeTriLin (T.val.val.val, T.val.val.val, B₃.val)),
|
||||
(cubicCoeff T.val.val)⁻¹ * (- cubeTriLin (T.val.val.val, T.val.val.val, Y₃.val)),
|
||||
(cubicCoeff T.val.val)⁻¹ * (- cubeTriLin (T.val.val.val, T.val.val.val, B₃.val)
|
||||
* (dot (Y₃.val, T.val.val.val) - dot (B₃.val, T.val.val.val))
|
||||
- cubeTriLin (T.val.val.val, T.val.val.val, Y₃.val)
|
||||
* (dot (Y₃.val, T.val.val.val) - 2 * dot (B₃.val, T.val.val.val))))
|
||||
|
||||
/-- The coefficent which multiplies a solution on passing through `case₂`. -/
|
||||
def case₂ProjCoeff (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 )
|
||||
|
||||
/-- The first parameter in `case₂` needed to form an inverse on `Proj`. -/
|
||||
def case₂ProjC₁ (T : MSSMACC.Sols) : ℚ := cubeTriLin (T.val, T.val, B₃.val)
|
||||
|
||||
/-- The second parameter in `case₂` needed to form an inverse on `Proj`. -/
|
||||
def case₂ProjC₂ (T : MSSMACC.Sols) : ℚ := - cubeTriLin (T.val, T.val, Y₃.val)
|
||||
|
||||
/-- The third parameter in `case₂` needed to form an inverse on `Proj`. -/
|
||||
def case₂ProjC₃ (T : MSSMACC.Sols) : ℚ :=
|
||||
(- cubeTriLin (T.val, T.val, B₃.val) * (dot (Y₃.val, T.val) - dot (B₃.val, T.val))
|
||||
- cubeTriLin (T.val, T.val, Y₃.val) * (dot (Y₃.val, T.val) - 2 * dot (B₃.val, T.val)))
|
||||
|
||||
lemma case₂_proj (T : MSSMACC.Sols) (h1 : case₁ProjCoeff T = 0) :
|
||||
case₂ (proj T.1.1)
|
||||
(case₂ProjC₁ T)
|
||||
(case₂ProjC₂ T)
|
||||
(case₂ProjC₃ T)
|
||||
((case₁ProjCoeff_zero T).mp h1) = (case₂ProjCoeff T) • T := by
|
||||
lemma inQuadToSol_proj (T : inQuadSol) (h : cubicCoeff T.val.val ≠ 0) :
|
||||
inQuadToSol (inQuadProj T) = T.val.val := by
|
||||
rw [inQuadProj, inQuadToSol_smul]
|
||||
apply ACCSystem.Sols.ext
|
||||
change (planeY₃B₃ _ _ _ _).val = _
|
||||
change _ • (planeY₃B₃ _ _ _ _).val = _
|
||||
rw [planeY₃B₃_val]
|
||||
rw [Y₃_plus_B₃_plus_proj]
|
||||
rw [case₂ProjCoeff, case₂ProjC₁, case₂ProjC₂, case₂ProjC₃, cube_proj, cube_proj_proj_B₃,
|
||||
cube_proj_proj_Y₃]
|
||||
rw [cube_proj, cube_proj_proj_B₃, cube_proj_proj_Y₃]
|
||||
ring_nf
|
||||
simp
|
||||
rfl
|
||||
|
||||
lemma case₂ProjCoeff_ne_zero (T : MSSMACC.Sols) (h1 : case₁ProjCoeff T = 0)
|
||||
(hT : case₂ProjCoeff T ≠ 0 ) :
|
||||
(case₂ProjCoeff T)⁻¹ • case₂ (proj T.1.1)
|
||||
(case₂ProjC₁ T)
|
||||
(case₂ProjC₂ T)
|
||||
(case₂ProjC₃ T)
|
||||
((case₁ProjCoeff_zero T).mp h1) = T := by
|
||||
rw [case₂_proj T h1, ← MulAction.mul_smul, mul_comm, mul_inv_cancel hT]
|
||||
simp only [zero_smul, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add]
|
||||
have h1 : (cubeTriLin (T.val.val.val, T.val.val.val, Y₃.val) ^ 2 * dot (Y₃.val, B₃.val) ^ 3 * 3 +
|
||||
dot (Y₃.val, B₃.val) ^ 3 * cubeTriLin (T.val.val.val, T.val.val.val, B₃.val) ^ 2
|
||||
* 3) = cubicCoeff T.val.val := by
|
||||
rw [cubicCoeff]
|
||||
ring
|
||||
rw [h1]
|
||||
rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h]
|
||||
simp
|
||||
|
||||
lemma case₂ProjCoeff_zero_Y₃_B₃ (T : MSSMACC.Sols) (h1 : case₂ProjCoeff T = 0) :
|
||||
cubeTriLin ((proj T.1.1).val, (proj T.1.1).val, Y₃.val) = 0 ∧
|
||||
cubeTriLin ((proj T.1.1).val, (proj T.1.1).val, B₃.val) = 0 := by
|
||||
rw [case₂ProjCoeff, mul_eq_zero] at h1
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h1
|
||||
simp at h1
|
||||
have h : cubeTriLin (T.val, T.val, Y₃.val) ^ 2 = 0 ∧
|
||||
cubeTriLin (T.val, T.val, B₃.val) ^ 2 = 0 :=
|
||||
(add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp h1
|
||||
simp at h
|
||||
have h1 := cube_proj_proj_B₃ T.1.1
|
||||
erw [h.2] at h1
|
||||
have h2 := cube_proj_proj_Y₃ T.1.1
|
||||
erw [h.1] at h2
|
||||
simp_all
|
||||
def inQuadCube : Type := {R : inQuad // inCubeProp R.val.val}
|
||||
|
||||
|
||||
lemma case₂ProjCoeff_zero_Y₃ (T : MSSMACC.Sols) (h1 : case₂ProjCoeff T = 0) :
|
||||
cubeTriLin ((proj T.1.1).val, (proj T.1.1).val, Y₃.val) = 0 :=
|
||||
(case₂ProjCoeff_zero_Y₃_B₃ T h1).left
|
||||
|
||||
lemma case₂ProjCoeff_zero_B₃ (T : MSSMACC.Sols) (h1 : case₂ProjCoeff T = 0) :
|
||||
cubeTriLin ((proj T.1.1).val, (proj T.1.1).val, B₃.val) = 0 :=
|
||||
(case₂ProjCoeff_zero_Y₃_B₃ T h1).right
|
||||
|
||||
|
||||
lemma case₂ProjCoeff_zero_T (T : MSSMACC.Sols) (h1 : case₂ProjCoeff T = 0) :
|
||||
cubeTriLin ((proj T.1.1).val, (proj T.1.1).val, T.val) = 0 := by
|
||||
rw [cube_proj_proj_self]
|
||||
have hr : cubeTriLin (T.val, T.val, Y₃.val) = 0 := by
|
||||
have h11 := case₂ProjCoeff_zero_Y₃ T h1
|
||||
rw [cube_proj_proj_Y₃] at h11
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h11
|
||||
simp at h11
|
||||
exact h11
|
||||
have h2 : cubeTriLin (T.val, T.val, B₃.val) = 0 := by
|
||||
have h11 := case₂ProjCoeff_zero_B₃ T h1
|
||||
rw [cube_proj_proj_B₃] at h11
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h11
|
||||
simp at h11
|
||||
exact h11
|
||||
rw [hr, h2]
|
||||
simp
|
||||
|
||||
lemma case₂ProjCoeff_zero_self (T : MSSMACC.Sols) (h1 : case₂ProjCoeff T = 0) :
|
||||
cubeTriLin ((proj T.1.1).val, (proj T.1.1).val, (proj T.1.1).val) = 0 := by
|
||||
nth_rewrite 3 [proj_val]
|
||||
rw [cubeTriLin.map_add₃, cubeTriLin.map_add₃]
|
||||
rw [cubeTriLin.map_smul₃, cubeTriLin.map_smul₃, cubeTriLin.map_smul₃]
|
||||
rw [case₂ProjCoeff_zero_Y₃ T h1, case₂ProjCoeff_zero_B₃ T h1, case₂ProjCoeff_zero_T T h1]
|
||||
simp
|
||||
|
||||
|
||||
lemma case₂ProjCoeff_zero (T : MSSMACC.Sols) :
|
||||
(case₁ProjCoeff T = 0 ∧ case₂ProjCoeff T = 0) ↔ case₃prop (proj T.1.1) := by
|
||||
apply Iff.intro
|
||||
intro h1
|
||||
rw [case₃prop]
|
||||
rw [case₂ProjCoeff_zero_self T h1.2, case₂ProjCoeff_zero_Y₃ T h1.2, case₂ProjCoeff_zero_B₃ T h1.2]
|
||||
rw [case₁ProjCoeff_zero_self T h1.1, case₁ProjCoeff_zero_Y₃ T h1.1, case₁ProjCoeff_zero_B₃ T h1.1]
|
||||
simp only [and_self]
|
||||
intro h
|
||||
rw [case₃prop] at h
|
||||
rw [case₁ProjCoeff, case₂ProjCoeff]
|
||||
simp_all only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, add_zero,
|
||||
mul_zero, mul_eq_zero, pow_eq_zero_iff, false_or, true_and]
|
||||
erw [show dot (Y₃.val, B₃.val) = 108 by rfl]
|
||||
simp only [OfNat.ofNat_ne_zero, false_or]
|
||||
have h1' := cube_proj_proj_B₃ T.1.1
|
||||
have h2' := cube_proj_proj_Y₃ T.1.1
|
||||
erw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h1' h2'
|
||||
simp_all
|
||||
|
||||
lemma case₂ProjCoeff_ne_zero_case₃ (T : MSSMACC.Sols) (h1 : case₂ProjCoeff T ≠ 0) :
|
||||
¬ case₃prop (proj T.1.1) := by
|
||||
have h1 : ¬ (case₁ProjCoeff T = 0 ∧ case₂ProjCoeff T = 0) := by
|
||||
simp_all
|
||||
exact (case₂ProjCoeff_zero T).mpr.mt h1
|
||||
|
||||
end proj
|
||||
def inQuadCubeSol : Type := {R : inQuadSol // inCubeSolProp R.val.val}
|
||||
|
||||
/-- The case where the plane lies entirely within the quadratic and cubic. -/
|
||||
def case₃ (R : MSSMACC.AnomalyFreePerp) (b₁ b₂ b₃ : ℚ)
|
||||
(h₃ : case₃prop R) :
|
||||
MSSMACC.Sols :=
|
||||
AnomalyFreeMk' (planeY₃B₃ R b₁ b₂ b₃)
|
||||
def inQuadCubeToSol : inQuadCube × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, b₁, b₂, b₃) =>
|
||||
AnomalyFreeMk' (planeY₃B₃ R.val.val.val b₁ b₂ b₃)
|
||||
(by
|
||||
rw [planeY₃B₃_quad]
|
||||
rw [h₃.1, h₃.2.1, h₃.2.2.1]
|
||||
rw [R.val.prop.1, R.val.prop.2.1, R.val.prop.2.2]
|
||||
simp)
|
||||
(by
|
||||
rw [planeY₃B₃_cubic]
|
||||
rw [h₃.2.2.2.1, h₃.2.2.2.2.1, h₃.2.2.2.2.2]
|
||||
rw [R.prop.1, R.prop.2.1, R.prop.2.2]
|
||||
simp)
|
||||
|
||||
lemma case₃_smul (R : MSSMACC.AnomalyFreePerp) (c₁ c₂ c₃ d : ℚ)
|
||||
(h : case₃prop R) : case₃ R (d * c₁) (d * c₂) (d * c₃) h = d • case₃ R c₁ c₂ c₃ h := by
|
||||
lemma inQuadCubeToSol_smul (R : inQuadCube) (c₁ c₂ c₃ d : ℚ) :
|
||||
inQuadCubeToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inQuadCubeToSol (R, c₁, c₂, c₃):= by
|
||||
apply ACCSystem.Sols.ext
|
||||
change (planeY₃B₃ _ _ _ _).val = _
|
||||
rw [planeY₃B₃_smul]
|
||||
rfl
|
||||
|
||||
|
||||
section proj
|
||||
def inQuadCubeProj (T : inQuadCubeSol) : inQuadCube × ℚ × ℚ × ℚ :=
|
||||
(⟨⟨⟨proj T.val.val.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val.val.val).mp T.val.val.prop ⟩,
|
||||
(inQuadSolProp_iff_proj_inQuadProp T.val.val.val).mp T.val.prop⟩,
|
||||
(inCubeSolProp_iff_proj_inCubeProp T.val.val.val).mp T.prop⟩,
|
||||
(dot (Y₃.val, B₃.val)) ⁻¹ * (dot (Y₃.val, T.val.val.val.val) - dot (B₃.val, T.val.val.val.val)),
|
||||
(dot (Y₃.val, B₃.val)) ⁻¹ * (2 * dot (B₃.val, T.val.val.val.val) -
|
||||
dot (Y₃.val, T.val.val.val.val)), (dot (Y₃.val, B₃.val)) ⁻¹ * 1)
|
||||
|
||||
/-- The coefficent which multiplies a solution on passing through `case₃`. -/
|
||||
def case₃ProjCoeff : ℚ := dot (Y₃.val, B₃.val)
|
||||
|
||||
/-- The first parameter in `case₃` needed to form an inverse on `Proj`. -/
|
||||
def case₃ProjC₁ (T : MSSMACC.Sols) : ℚ := (dot (Y₃.val, T.val) - dot (B₃.val, T.val))
|
||||
|
||||
/-- The second parameter in `case₃` needed to form an inverse on `Proj`. -/
|
||||
def case₃ProjC₂ (T : MSSMACC.Sols) : ℚ := (2 * dot (B₃.val, T.val) - dot (Y₃.val, T.val))
|
||||
|
||||
lemma case₃_proj (T : MSSMACC.Sols) (h0 : case₁ProjCoeff T = 0) (h1 : case₂ProjCoeff T = 0) :
|
||||
case₃ (proj T.1.1)
|
||||
(case₃ProjC₁ T)
|
||||
(case₃ProjC₂ T)
|
||||
1
|
||||
((case₂ProjCoeff_zero T).mp (And.intro h0 h1)) = case₃ProjCoeff • T := by
|
||||
lemma inQuadCubeToSol_proj (T : inQuadCubeSol) :
|
||||
inQuadCubeToSol (inQuadCubeProj T) = T.val.val.val := by
|
||||
rw [inQuadCubeProj, inQuadCubeToSol_smul]
|
||||
apply ACCSystem.Sols.ext
|
||||
change (planeY₃B₃ _ _ _ _).val = _
|
||||
change _ • (planeY₃B₃ _ _ _ _).val = _
|
||||
rw [planeY₃B₃_val]
|
||||
rw [Y₃_plus_B₃_plus_proj]
|
||||
rw [case₃ProjC₁, case₃ProjC₂]
|
||||
ring_nf
|
||||
simp
|
||||
rfl
|
||||
|
||||
lemma case₃_smul_coeff (T : MSSMACC.Sols) (h0 : case₁ProjCoeff T = 0) (h1 : case₂ProjCoeff T = 0) :
|
||||
case₃ProjCoeff⁻¹ • case₃ (proj T.1.1)
|
||||
(case₃ProjC₁ T)
|
||||
(case₃ProjC₂ T)
|
||||
1
|
||||
((case₂ProjCoeff_zero T).mp (And.intro h0 h1)) = T := by
|
||||
rw [case₃_proj T h0 h1]
|
||||
simp only [Fin.isValue, Fin.reduceFinMk, zero_smul, add_zero, zero_add]
|
||||
rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel]
|
||||
simp only [one_smul]
|
||||
rw [case₃ProjCoeff]
|
||||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl]
|
||||
simp only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true]
|
||||
simp
|
||||
|
||||
end proj
|
||||
|
||||
/-- A map from `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ ` to `MSSMACC.Sols`.
|
||||
This allows generation of solutions given elements of `MSSMACC.AnomalyFreePerp` and
|
||||
three rational numbers. -/
|
||||
def parameterization :
|
||||
MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → MSSMACC.Sols := fun A =>
|
||||
if h₃ : case₃prop A.1 then
|
||||
case₃ A.1 A.2.1 A.2.2.1 A.2.2.2 h₃
|
||||
def toSol :
|
||||
MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a, b, c) =>
|
||||
if h₃ : lineEqProp R ∧ inQuadProp R ∧ inCubeProp R then
|
||||
inQuadCubeToSol (⟨⟨⟨R, h₃.1⟩, h₃.2.1⟩, h₃.2.2⟩, a, b, c)
|
||||
else
|
||||
if h₂ : case₂prop A.1 then
|
||||
case₂ A.1 A.2.1 A.2.2.1 A.2.2.2 h₂
|
||||
if h₂ : lineEqProp R ∧ inQuadProp R then
|
||||
inQuadToSol (⟨⟨R, h₂.1⟩, h₂.2⟩, a, b, c)
|
||||
else
|
||||
if h₁ : case₁prop A.1 then
|
||||
case₁ A.1 A.2.1 A.2.2.1 A.2.2.2 h₁
|
||||
if h₁ : lineEqProp R then
|
||||
inLineEqToSol (⟨R, h₁⟩, a, b, c)
|
||||
else
|
||||
A.2.1 • generic A.1
|
||||
toSolNS ⟨R, a, b, c⟩
|
||||
|
||||
lemma parameterization_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
|
||||
parameterization (R, d * a, d * b, d * c) = d • parameterization (R, a, b, c) := by
|
||||
rw [parameterization, parameterization]
|
||||
by_cases h₃ : case₃prop R
|
||||
rw [dif_pos h₃, dif_pos h₃]
|
||||
rw [case₃_smul]
|
||||
rw [dif_neg h₃, dif_neg h₃]
|
||||
by_cases h₂ : case₂prop R
|
||||
rw [dif_pos h₂, dif_pos h₂]
|
||||
rw [case₂_smul]
|
||||
rw [dif_neg h₂, dif_neg h₂]
|
||||
by_cases h₁ : case₁prop R
|
||||
rw [dif_pos h₁, dif_pos h₁]
|
||||
rw [case₁_smul]
|
||||
rw [dif_neg h₁, dif_neg h₁]
|
||||
rw [mul_smul]
|
||||
|
||||
lemma parameterization_not₁₂₃ (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ)
|
||||
(h1 : ¬ case₁prop R) (h2 : ¬ case₂prop R) (h3 : ¬ case₃prop R) :
|
||||
parameterization (R, a, b, c) = a • generic R := by
|
||||
rw [parameterization]
|
||||
rw [dif_neg h3]
|
||||
rw [dif_neg h2]
|
||||
rw [dif_neg h1]
|
||||
|
||||
lemma parameterization_is₁_not₂₃ (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ)
|
||||
(h1 : case₁prop R) (h2 : ¬ case₂prop R) (h3 : ¬ case₃prop R) :
|
||||
parameterization (R, a, b, c) = case₁ R a b c h1:= by
|
||||
rw [parameterization]
|
||||
rw [dif_neg h3]
|
||||
rw [dif_neg h2]
|
||||
rw [dif_pos h1]
|
||||
|
||||
lemma parameterization_is₁₂_not₃ (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (h2 : case₂prop R)
|
||||
(h3 : ¬ case₃prop R) : parameterization (R, a, b, c) = case₂ R a b c h2 := by
|
||||
rw [parameterization]
|
||||
rw [dif_neg h3]
|
||||
rw [dif_pos h2]
|
||||
|
||||
lemma parameterization_is₃ (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (h3 : case₃prop R) :
|
||||
parameterization (R, a, b, c) = case₃ R a b c h3 := by
|
||||
rw [parameterization]
|
||||
rw [dif_pos h3]
|
||||
|
||||
/-- A right inverse of `parameterizaiton`. -/
|
||||
def inverse (R : MSSMACC.Sols) : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ :=
|
||||
if genericProjCoeff R ≠ 0 then
|
||||
(proj R.1.1, (genericProjCoeff R)⁻¹, 0, 0)
|
||||
else
|
||||
if case₁ProjCoeff R ≠ 0 then
|
||||
(proj R.1.1, (case₁ProjCoeff R)⁻¹ * case₁ProjC₁ R, (case₁ProjCoeff R)⁻¹ * case₁ProjC₂ R,
|
||||
(case₁ProjCoeff R)⁻¹ * case₁ProjC₃ R)
|
||||
else
|
||||
if case₂ProjCoeff R ≠ 0 then
|
||||
(proj R.1.1, (case₂ProjCoeff R)⁻¹ * case₂ProjC₁ R, (case₂ProjCoeff R)⁻¹ * case₂ProjC₂ R,
|
||||
(case₂ProjCoeff R)⁻¹ * case₂ProjC₃ R)
|
||||
else
|
||||
(proj R.1.1, (case₃ProjCoeff)⁻¹ * case₃ProjC₁ R, (case₃ProjCoeff)⁻¹ * case₃ProjC₂ R,
|
||||
(case₃ProjCoeff)⁻¹ * 1)
|
||||
|
||||
lemma inverse_generic (R : MSSMACC.Sols) (h : genericProjCoeff R ≠ 0) :
|
||||
inverse R = (proj R.1.1, (genericProjCoeff R)⁻¹, 0, 0) := by
|
||||
rw [inverse, if_pos h]
|
||||
|
||||
lemma inverse_case₁ (R : MSSMACC.Sols) (h0 : genericProjCoeff R = 0)
|
||||
(h1 : case₁ProjCoeff R ≠ 0) : inverse R = (proj R.1.1, (case₁ProjCoeff R)⁻¹ * case₁ProjC₁ R,
|
||||
(case₁ProjCoeff R)⁻¹ * case₁ProjC₂ R, (case₁ProjCoeff R)⁻¹ * case₁ProjC₃ R) := by
|
||||
rw [inverse]
|
||||
lemma toSol_toSolNSProj (T : MSSMACC.Sols) (h₁ : ¬ lineEqPropSol T) :
|
||||
toSol (toSolNSProj T) = T := by
|
||||
have h1 : ¬ lineEqProp (toSolNSProj T).1 := (linEqPropSol_iff_proj_linEqProp T).mpr.mt h₁
|
||||
rw [toSol]
|
||||
simp_all
|
||||
exact toSolNS_proj T (mt (lineEqPropSol_iff_lineEqCoeff_zero T).mpr h₁)
|
||||
|
||||
lemma inverse_case₂ (R : MSSMACC.Sols) (h0 : genericProjCoeff R = 0)
|
||||
(h1 : case₁ProjCoeff R = 0) (h2 : case₂ProjCoeff R ≠ 0) : inverse R = (proj R.1.1,
|
||||
(case₂ProjCoeff R)⁻¹ * case₂ProjC₁ R,
|
||||
(case₂ProjCoeff R)⁻¹ * case₂ProjC₂ R, (case₂ProjCoeff R)⁻¹ * case₂ProjC₃ R) := by
|
||||
rw [inverse]
|
||||
lemma toSol_inLineEq (T : MSSMACC.Sols) (h₁ : lineEqPropSol T) (h₂ : ¬ inQuadSolProp T) :
|
||||
∃ X, toSol X = T := by
|
||||
let X := inLineEqProj ⟨T, h₁⟩
|
||||
use ⟨X.1.val, X.2.1, X.2.2⟩
|
||||
have ha₁ : ¬ inQuadProp X.1.val := (inQuadSolProp_iff_proj_inQuadProp T).mpr.mt h₂
|
||||
have ha₂ : lineEqProp X.1.val := (linEqPropSol_iff_proj_linEqProp T).mp h₁
|
||||
rw [toSol]
|
||||
simp_all
|
||||
exact inLineEqToSol_proj ⟨T, h₁⟩ (mt (inQuadSolProp_iff_quadCoeff_zero T).mpr h₂)
|
||||
|
||||
lemma inverse_case₃ (R : MSSMACC.Sols) (h0 : genericProjCoeff R = 0)
|
||||
(h1 : case₁ProjCoeff R = 0) (h2 : case₂ProjCoeff R = 0) :
|
||||
inverse R = (proj R.1.1, (case₃ProjCoeff)⁻¹ * case₃ProjC₁ R,
|
||||
(case₃ProjCoeff)⁻¹ * case₃ProjC₂ R,
|
||||
(case₃ProjCoeff)⁻¹ * 1) := by
|
||||
rw [inverse]
|
||||
lemma toSol_inQuad (T : MSSMACC.Sols) (h₁ : lineEqPropSol T) (h₂ : inQuadSolProp T)
|
||||
(h₃ : ¬ inCubeSolProp T) : ∃ X, toSol X = T := by
|
||||
let X := inQuadProj ⟨⟨T, h₁⟩, h₂⟩
|
||||
use ⟨X.1.val.val, X.2.1, X.2.2⟩
|
||||
have ha₁ : ¬ inCubeProp X.1.val.val := (inCubeSolProp_iff_proj_inCubeProp T).mpr.mt h₃
|
||||
have ha₂ : inQuadProp X.1.val.val := (inQuadSolProp_iff_proj_inQuadProp T).mp h₂
|
||||
have ha₃ : lineEqProp X.1.val.val := (linEqPropSol_iff_proj_linEqProp T).mp h₁
|
||||
rw [toSol]
|
||||
simp_all
|
||||
exact inQuadToSol_proj ⟨⟨T, h₁⟩, h₂⟩ (mt (inCubeSolProp_iff_cubicCoeff_zero T).mpr h₃)
|
||||
|
||||
lemma inverse_parameterization (R : MSSMACC.Sols) :
|
||||
parameterization (inverse R) = R := by
|
||||
by_cases h0 : genericProjCoeff R ≠ 0
|
||||
rw [inverse_generic R h0]
|
||||
rw [parameterization_not₁₂₃ _ _ _ _ (genericProjCoeff_neq_zero_case₁ R h0)
|
||||
(genericProjCoeff_neq_zero_case₂ R h0) (genericProjCoeff_neq_zero_case₃ R h0)]
|
||||
rw [genericProjCoeff_ne_zero R h0]
|
||||
by_cases h1 : case₁ProjCoeff R ≠ 0
|
||||
simp at h0
|
||||
rw [inverse_case₁ R h0 h1]
|
||||
rw [parameterization_smul]
|
||||
rw [parameterization_is₁_not₂₃ _ _ _ _ ((genericProjCoeff_zero R).mp h0)
|
||||
(case₁ProjCoeff_ne_zero_case₂ R h1) (case₁ProjCoeff_ne_zero_case₃ R h1)]
|
||||
rw [case₁ProjCoeff_ne_zero R h0 h1]
|
||||
simp at h0 h1
|
||||
by_cases h2 : case₂ProjCoeff R ≠ 0
|
||||
rw [inverse_case₂ R h0 h1 h2]
|
||||
rw [parameterization_smul]
|
||||
rw [parameterization_is₁₂_not₃ _ _ _ _ ((case₁ProjCoeff_zero R).mp h1)
|
||||
(case₂ProjCoeff_ne_zero_case₃ R h2)]
|
||||
rw [case₂ProjCoeff_ne_zero R h1 h2]
|
||||
simp at h2
|
||||
rw [inverse_case₃ R h0 h1 h2]
|
||||
rw [parameterization_smul]
|
||||
rw [parameterization_is₃ _ _ _ _ ((case₂ProjCoeff_zero R).mp (And.intro h1 h2))]
|
||||
rw [case₃_smul_coeff R h1 h2]
|
||||
lemma toSol_inQuadCube (T : MSSMACC.Sols) (h₁ : lineEqPropSol T) (h₂ : inQuadSolProp T)
|
||||
(h₃ : inCubeSolProp T) : ∃ X, toSol X = T := by
|
||||
let X := inQuadCubeProj ⟨⟨⟨T, h₁⟩, h₂⟩, h₃⟩
|
||||
use ⟨X.1.val.val.val, X.2.1, X.2.2⟩
|
||||
have ha₁ : inCubeProp X.1.val.val.val := (inCubeSolProp_iff_proj_inCubeProp T).mp h₃
|
||||
have ha₂ : inQuadProp X.1.val.val.val := (inQuadSolProp_iff_proj_inQuadProp T).mp h₂
|
||||
have ha₃ : lineEqProp X.1.val.val.val := (linEqPropSol_iff_proj_linEqProp T).mp h₁
|
||||
rw [toSol]
|
||||
simp_all
|
||||
exact inQuadCubeToSol_proj ⟨⟨⟨T, h₁⟩, h₂⟩, h₃⟩
|
||||
|
||||
theorem parameterization_surjective : Function.Surjective parameterization := by
|
||||
theorem toSol_surjective : Function.Surjective toSol := by
|
||||
intro T
|
||||
use inverse T
|
||||
exact inverse_parameterization T
|
||||
by_cases h₁ : ¬ lineEqPropSol T
|
||||
use toSolNSProj T
|
||||
exact toSol_toSolNSProj T h₁
|
||||
simp at h₁
|
||||
by_cases h₂ : ¬ inQuadSolProp T
|
||||
exact toSol_inLineEq T h₁ h₂
|
||||
simp at h₂
|
||||
by_cases h₃ : ¬ inCubeSolProp T
|
||||
exact toSol_inQuad T h₁ h₂ h₃
|
||||
simp at h₃
|
||||
exact toSol_inQuadCube T h₁ h₂ h₃
|
||||
|
||||
|
||||
end AnomalyFreePerp
|
||||
|
||||
end MSSMACC
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue