refactor: Quad Lin equations
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jstoobysmith
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772e78ca77
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b5dd319eed
14 changed files with 245 additions and 299 deletions
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@ -46,12 +46,12 @@ instance (R : MSSMACC.AnomalyFreePerp) : Decidable (lineEqProp R) := by
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/-- A condition on `Sols` which we will show in `linEqPropSol_iff_proj_linEqProp` that is equivalent
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to the condition that the `proj` of the solution satisfies `lineEqProp`. -/
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def lineEqPropSol (R : MSSMACC.Sols) : Prop :=
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cubeTriLin (R.val, R.val, Y₃.val) * quadBiLin (B₃.val, R.val) -
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cubeTriLin (R.val, R.val, B₃.val) * quadBiLin (Y₃.val, R.val) = 0
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cubeTriLin (R.val, R.val, Y₃.val) * quadBiLin B₃.val R.val -
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cubeTriLin (R.val, R.val, B₃.val) * quadBiLin Y₃.val R.val = 0
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/-- A rational which appears in `toSolNS` acting on sols, and which been zero is
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equivalent to satisfying `lineEqPropSol`. -/
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def lineEqCoeff (T : MSSMACC.Sols) : ℚ := dot (Y₃.val, B₃.val) * α₃ (proj T.1.1)
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def lineEqCoeff (T : MSSMACC.Sols) : ℚ := dot Y₃.val B₃.val * α₃ (proj T.1.1)
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lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
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lineEqPropSol T ↔ lineEqCoeff T = 0 := by
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@ -59,7 +59,7 @@ lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero,
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false_or]
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rw [cube_proj_proj_B₃, cube_proj_proj_Y₃, quad_B₃_proj, quad_Y₃_proj]
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rw [show dot (Y₃.val, B₃.val) = 108 by rfl]
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rw [show dot Y₃.val B₃.val = 108 by rfl]
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simp only [Fin.isValue, Fin.reduceFinMk, OfNat.ofNat_ne_zero, false_or]
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ring_nf
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rw [mul_comm _ 1259712, mul_comm _ 1259712, ← mul_sub]
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@ -70,10 +70,10 @@ lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
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rw [lineEqPropSol_iff_lineEqCoeff_zero, lineEqCoeff, lineEqProp]
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apply Iff.intro
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intro h
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rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp at h
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rw [α₁_proj, α₂_proj, h]
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simp only [neg_zero, dot_toFun, Fin.isValue, Fin.reduceFinMk, zero_mul, and_self]
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simp only [neg_zero, Fin.isValue, Fin.reduceFinMk, zero_mul, and_self]
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intro h
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rw [h.2.2]
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simp
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@ -82,7 +82,7 @@ lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
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/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies
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entirely in the quadratic surface. -/
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def inQuadProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
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quadBiLin (R.val, R.val) = 0 ∧ quadBiLin (Y₃.val, R.val) = 0 ∧ quadBiLin (B₃.val, R.val) = 0
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quadBiLin R.val R.val = 0 ∧ quadBiLin Y₃.val R.val = 0 ∧ quadBiLin B₃.val R.val = 0
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instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inQuadProp R) := by
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apply And.decidable
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@ -90,23 +90,23 @@ instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inQuadProp R) := by
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/-- A condition which is satisfied if the plane spanned by the solutions `R`, `Y₃` and `B₃`
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lies entirely in the quadratic surface. -/
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def inQuadSolProp (R : MSSMACC.Sols) : Prop :=
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quadBiLin (Y₃.val, R.val) = 0 ∧ quadBiLin (B₃.val, R.val) = 0
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quadBiLin Y₃.val R.val = 0 ∧ quadBiLin B₃.val R.val = 0
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/-- A rational which has two properties. It is zero for a solution `T` if and only if
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that solution satisfies `inQuadSolProp`. It appears in the definition of `inQuadProj`. -/
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def quadCoeff (T : MSSMACC.Sols) : ℚ :=
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2 * dot (Y₃.val, B₃.val) ^ 2 *
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(quadBiLin (Y₃.val, T.val) ^ 2 + quadBiLin (B₃.val, T.val) ^ 2)
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2 * dot Y₃.val B₃.val ^ 2 *
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(quadBiLin Y₃.val T.val ^ 2 + quadBiLin B₃.val T.val ^ 2)
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lemma inQuadSolProp_iff_quadCoeff_zero (T : MSSMACC.Sols) : inQuadSolProp T ↔ quadCoeff T = 0 := by
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apply Iff.intro
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intro h
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rw [quadCoeff, h.1, h.2]
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simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
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zero_pow, add_zero, mul_zero]
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intro h
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rw [quadCoeff] at h
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rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [ Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
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apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
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@ -123,10 +123,10 @@ lemma inQuadSolProp_iff_proj_inQuadProp (R : MSSMACC.Sols) :
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apply Iff.intro
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intro h
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rw [h.1, h.2]
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simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
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simp only [Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
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intro h
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rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
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simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk , mul_eq_zero,
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk , mul_eq_zero,
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OfNat.ofNat_ne_zero, or_self, false_or] at h
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rw [h.2.1, h.2.2]
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simp
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@ -149,7 +149,7 @@ def inCubeSolProp (R : MSSMACC.Sols) : Prop :=
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/-- A rational which has two properties. It is zero for a solution `T` if and only if
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that solution satisfies `inCubeSolProp`. It appears in the definition of `inLineEqProj`. -/
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def cubicCoeff (T : MSSMACC.Sols) : ℚ :=
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3 * dot (Y₃.val, B₃.val) ^ 3 * (cubeTriLin (T.val, T.val, Y₃.val) ^ 2 +
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3 * (dot Y₃.val B₃.val) ^ 3 * (cubeTriLin (T.val, T.val, Y₃.val) ^ 2 +
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cubeTriLin (T.val, T.val, B₃.val) ^ 2 )
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lemma inCubeSolProp_iff_cubicCoeff_zero (T : MSSMACC.Sols) :
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@ -157,11 +157,11 @@ lemma inCubeSolProp_iff_cubicCoeff_zero (T : MSSMACC.Sols) :
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apply Iff.intro
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intro h
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rw [cubicCoeff, h.1, h.2]
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simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
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simp only [Fin.isValue, Fin.reduceFinMk, ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true,
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zero_pow, add_zero, mul_zero]
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intro h
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rw [cubicCoeff] at h
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rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [ Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
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apply (add_eq_zero_iff' (sq_nonneg _) (sq_nonneg _)).mp at h
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@ -176,10 +176,10 @@ lemma inCubeSolProp_iff_proj_inCubeProp (R : MSSMACC.Sols) :
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apply Iff.intro
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intro h
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rw [h.1, h.2]
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simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
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simp only [Fin.isValue, Fin.reduceFinMk, mul_zero, add_zero, and_self]
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intro h
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rw [show dot (Y₃.val, B₃.val) = 108 by rfl] at h
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simp only [dot_toFun, Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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rw [show dot Y₃.val B₃.val = 108 by rfl] at h
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simp only [Fin.isValue, Fin.reduceFinMk, mul_eq_zero, OfNat.ofNat_ne_zero, ne_eq,
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not_false_eq_true, pow_eq_zero_iff, or_self, false_or] at h
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rw [h.2.1, h.2.2]
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simp
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@ -254,7 +254,7 @@ lemma toSolNS_proj (T : notInLineEqSol) : toSolNS (toSolNSProj T.val) = T.val :=
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rw [α₁_proj, α₂_proj]
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ring_nf
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simp only [zero_smul, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add]
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have h1 : α₃ (proj T.val.toLinSols) * dot (Y₃.val, B₃.val) = lineEqCoeff T.val := by
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have h1 : α₃ (proj T.val.toLinSols) * dot Y₃.val B₃.val = lineEqCoeff T.val := by
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rw [lineEqCoeff]
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ring
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rw [h1]
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@ -273,12 +273,11 @@ def inLineEqToSol : inLineEq × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, c
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/-- On elements of `inLineEqSol` a right-inverse to `inLineEqSol`. -/
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def inLineEqProj (T : inLineEqSol) : inLineEq × ℚ × ℚ × ℚ :=
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(⟨proj T.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1 ⟩,
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(quadCoeff T.val)⁻¹ * quadBiLin (B₃.val, T.val.val),
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(quadCoeff T.val)⁻¹ * (- quadBiLin (Y₃.val, T.val.val)),
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(quadCoeff T.val)⁻¹ * quadBiLin B₃.val T.val.val,
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(quadCoeff T.val)⁻¹ * (- quadBiLin Y₃.val T.val.val),
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(quadCoeff T.val)⁻¹ * (
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quadBiLin (B₃.val, T.val.val) * (dot (B₃.val, T.val.val) - dot (Y₃.val, T.val.val))
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- quadBiLin (Y₃.val, T.val.val) * (dot (Y₃.val, T.val.val) - 2 * dot (B₃.val, T.val.val))))
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quadBiLin B₃.val T.val.val * (dot B₃.val T.val.val - dot Y₃.val T.val.val)
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- quadBiLin Y₃.val T.val.val * (dot Y₃.val T.val.val - 2 * dot B₃.val T.val.val)))
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lemma inLineEqTo_smul (R : inLineEq) (c₁ c₂ c₃ d : ℚ) :
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inLineEqToSol (R, (d * c₁), (d * c₂), (d * c₃)) = d • inLineEqToSol (R, c₁, c₂, c₃) := by
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@ -297,8 +296,8 @@ lemma inLineEqToSol_proj (T : inLineEqSol) : inLineEqToSol (inLineEqProj T) = T.
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rw [quad_proj, quad_Y₃_proj, quad_B₃_proj]
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ring_nf
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simp only [zero_smul, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add]
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have h1 : (quadBiLin (Y₃.val, (T).val.val) ^ 2 * dot (Y₃.val, B₃.val) ^ 2 * 2 +
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dot (Y₃.val, B₃.val) ^ 2 * quadBiLin (B₃.val, (T).val.val) ^ 2 * 2) = quadCoeff T.val := by
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have h1 : (quadBiLin Y₃.val T.val.val ^ 2 * dot Y₃.val B₃.val ^ 2 * 2 +
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dot Y₃.val B₃.val ^ 2 * quadBiLin B₃.val T.val.val ^ 2 * 2) = quadCoeff T.val := by
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rw [quadCoeff]
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ring
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rw [h1]
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@ -329,9 +328,9 @@ def inQuadProj (T : inQuadSol) : inQuad × ℚ × ℚ × ℚ :=
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(cubicCoeff T.val)⁻¹ * (cubeTriLin (T.val.val, T.val.val, B₃.val)),
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(cubicCoeff T.val)⁻¹ * (- cubeTriLin (T.val.val, T.val.val, Y₃.val)),
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(cubicCoeff T.val)⁻¹ *
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(cubeTriLin (T.val.val, T.val.val, B₃.val) * (dot (B₃.val, T.val.val) - dot (Y₃.val, T.val.val))
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(cubeTriLin (T.val.val, T.val.val, B₃.val) * (dot B₃.val T.val.val - dot Y₃.val T.val.val)
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- cubeTriLin (T.val.val, T.val.val, Y₃.val)
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* (dot (Y₃.val, T.val.val) - 2 * dot (B₃.val, T.val.val))))
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* (dot Y₃.val T.val.val - 2 * dot B₃.val T.val.val)))
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lemma inQuadToSol_proj (T : inQuadSol) : inQuadToSol (inQuadProj T) = T.val := by
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@ -343,8 +342,8 @@ lemma inQuadToSol_proj (T : inQuadSol) : inQuadToSol (inQuadProj T) = T.val := b
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rw [cube_proj, cube_proj_proj_B₃, cube_proj_proj_Y₃]
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ring_nf
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simp only [zero_smul, add_zero, Fin.isValue, Fin.reduceFinMk, zero_add]
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have h1 : (cubeTriLin (T.val.val, T.val.val, Y₃.val) ^ 2 * dot (Y₃.val, B₃.val) ^ 3 * 3 +
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dot (Y₃.val, B₃.val) ^ 3 * cubeTriLin (T.val.val, T.val.val, B₃.val) ^ 2
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have h1 : (cubeTriLin (T.val.val, T.val.val, Y₃.val) ^ 2 * dot Y₃.val B₃.val ^ 3 * 3 +
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dot Y₃.val B₃.val ^ 3 * cubeTriLin (T.val.val, T.val.val, B₃.val) ^ 2
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* 3) = cubicCoeff T.val := by
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rw [cubicCoeff]
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ring
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@ -378,9 +377,9 @@ def inQuadCubeProj (T : inQuadCubeSol) : inQuadCube × ℚ × ℚ × ℚ :=
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(⟨⟨⟨proj T.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1 ⟩,
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(inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1⟩,
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(inCubeSolProp_iff_proj_inCubeProp T.val).mp T.prop.2.2⟩,
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(dot (Y₃.val, B₃.val))⁻¹ * (dot (Y₃.val, T.val.val) - dot (B₃.val, T.val.val)),
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(dot (Y₃.val, B₃.val))⁻¹ * (2 * dot (B₃.val, T.val.val) - dot (Y₃.val, T.val.val)),
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(dot (Y₃.val, B₃.val))⁻¹ * 1)
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(dot Y₃.val B₃.val)⁻¹ * (dot Y₃.val T.val.val - dot B₃.val T.val.val),
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(dot Y₃.val B₃.val)⁻¹ * (2 * dot B₃.val T.val.val - dot Y₃.val T.val.val),
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(dot Y₃.val B₃.val)⁻¹ * 1)
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lemma inQuadCubeToSol_proj (T : inQuadCubeSol) :
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inQuadCubeToSol (inQuadCubeProj T) = T.val := by
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@ -393,7 +392,7 @@ lemma inQuadCubeToSol_proj (T : inQuadCubeSol) :
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simp only [Fin.isValue, Fin.reduceFinMk, zero_smul, add_zero, zero_add]
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rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel]
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simp only [one_smul]
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rw [show dot (Y₃.val, B₃.val) = 108 by rfl]
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rw [show dot Y₃.val B₃.val = 108 by rfl]
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simp
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/-- Given an element of `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` a solution. We will
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