feat: Change file structure, added comments
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3 changed files with 140 additions and 108 deletions
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@ -5,7 +5,7 @@ Authors: Joseph Tooby-Smith
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-/
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import HepLean.AnomalyCancellation.MSSMNu.Basic
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import HepLean.AnomalyCancellation.MSSMNu.LineY3B3
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import HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3
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import HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.Basic
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import Mathlib.Tactic.Polyrith
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/-!
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# Plane Y₃ B₃ and an orthogonal third point
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@ -5,19 +5,19 @@ Authors: Joseph Tooby-Smith
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-/
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import HepLean.AnomalyCancellation.MSSMNu.Basic
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import HepLean.AnomalyCancellation.MSSMNu.LineY3B3
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import HepLean.AnomalyCancellation.MSSMNu.PlaneY3B3Orthog
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import HepLean.AnomalyCancellation.MSSMNu.OrthogY3B3.PlaneWithY3B3
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import Mathlib.Tactic.Polyrith
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/-!
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# Parameterization of solutions to the MSSM anomaly cancellation equations
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# From charges perpendicular to `Y₃` and `B₃` to solutions
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This file uses planes through $Y_3$ and $B_3$ to form solutions to the anomaly cancellation
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conditions.
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The main aim of this file is to take charge assignments perpendicular to `Y₃` and `B₃` and
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produce solutions to the anomaly cancellation conditions. In this regard we will define
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a surjective map `toSol` from `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` to `MSSMACC.Sols`.
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To define `toSols` we define a series of other maps from various subtypes of
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`MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` to `MSSMACC.Sols`. And show that these maps form a
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surjection on certain subtypes of `MSSMACC.Sols`.
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Split into four cases:
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- The generic case.
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- `case₁`: The case when the quadratic and cubic lines agree (if they exist uniquely).
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- `case₂`: The case where the plane lies entirely within the quadratic.
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- `case₃`: The case where the plane lies entirely within the cubic and quadratic.
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# References
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@ -36,18 +36,21 @@ open BigOperators
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namespace AnomalyFreePerp
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def lineEqProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
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α₁ R = 0 ∧ α₂ R = 0 ∧ α₃ R = 0
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/-- A condition for the quad line in the plane spanned by R, Y₃ and B₃ to sit in the cubic,
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and for the cube line to sit in the quad. -/
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def lineEqProp (R : MSSMACC.AnomalyFreePerp) : Prop := α₁ R = 0 ∧ α₂ R = 0 ∧ α₃ R = 0
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instance (R : MSSMACC.AnomalyFreePerp) : Decidable (lineEqProp R) := by
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apply And.decidable
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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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/-- 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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lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
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@ -58,12 +61,8 @@ lemma lineEqPropSol_iff_lineEqCoeff_zero (T : MSSMACC.Sols) :
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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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simp only [Fin.isValue, Fin.reduceFinMk, OfNat.ofNat_ne_zero, false_or]
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have h1 : 108 ^ 2 * cubeTriLin (T.val, T.val, Y₃.val) * (108 * quadBiLin (B₃.val, T.val)) -
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108 ^ 2 * cubeTriLin (T.val, T.val, B₃.val) * (108 * quadBiLin (Y₃.val, T.val)) =
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108 ^ 3 * (cubeTriLin (T.val, T.val, Y₃.val) * quadBiLin (B₃.val, T.val) -
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cubeTriLin (T.val, T.val, B₃.val) * quadBiLin (Y₃.val, T.val) ) := by
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ring
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rw [h1]
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ring_nf
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rw [mul_comm _ 1259712, mul_comm _ 1259712, ← mul_sub]
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simp
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lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
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@ -80,17 +79,21 @@ lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
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simp
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/-- Case₂ is defined when the plane lies entirely within the quadratic. -/
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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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instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inQuadProp R) := by
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apply And.decidable
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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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/-- The coefficent which multiplies a solution on passing through `case₁`. -/
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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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@ -110,6 +113,8 @@ lemma inQuadSolProp_iff_quadCoeff_zero (T : MSSMACC.Sols) : inQuadSolProp T ↔
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not_false_eq_true, pow_eq_zero_iff] at h
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exact h
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/-- The conditions `inQuadSolProp R` and `inQuadProp (proj R.1.1)` are equivalent. This is to be
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expected since both `R` and `proj R.1.1` define the same plane with `Y₃` and `B₃`. -/
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lemma inQuadSolProp_iff_proj_inQuadProp (R : MSSMACC.Sols) :
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inQuadSolProp R ↔ inQuadProp (proj R.1.1) := by
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rw [inQuadSolProp, inQuadProp]
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@ -125,7 +130,8 @@ lemma inQuadSolProp_iff_proj_inQuadProp (R : MSSMACC.Sols) :
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rw [h.2.1, h.2.2]
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simp
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/-- Case₃ is defined when the plane lies entirely within the quadratic and cubic. -/
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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 cubic surface. -/
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def inCubeProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
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cubeTriLin (R.val, R.val, R.val) = 0 ∧ cubeTriLin (R.val, R.val, B₃.val) = 0 ∧
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cubeTriLin (R.val, R.val, Y₃.val) = 0
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@ -134,9 +140,13 @@ def inCubeProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
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instance (R : MSSMACC.AnomalyFreePerp) : Decidable (inCubeProp R) := by
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apply And.decidable
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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 cubic surface. -/
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def inCubeSolProp (R : MSSMACC.Sols) : Prop :=
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cubeTriLin (R.val, R.val, B₃.val) = 0 ∧ cubeTriLin (R.val, R.val, Y₃.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 `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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cubeTriLin (T.val, T.val, B₃.val) ^ 2 )
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@ -172,8 +182,34 @@ lemma inCubeSolProp_iff_proj_inCubeProp (R : MSSMACC.Sols) :
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rw [h.2.1, h.2.2]
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simp
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/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the condition
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`lineEqProp`. -/
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def inLineEq : Type := {R : MSSMACC.AnomalyFreePerp // lineEqProp R}
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/-- Given a `R ∈ LinSols` perpendicular to $Y_3$, and $B_3$, a solution to the quadratic. -/
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/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the condition
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`lineEqProp` and `inQuadProp`. -/
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def inQuad : Type := {R : inLineEq // inQuadProp R.val}
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/-- Those charge assignments perpendicular to `Y₃` and `B₃` which satisfy the condition
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`lineEqProp`, `inQuadProp` and `inCubeProp`. -/
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def inQuadCube : Type := {R : inQuad // inCubeProp R.val.val}
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/-- Those solutions which do not satisfy the condition `lineEqPropSol`. -/
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def notInLineEqSol : Type := {R : MSSMACC.Sols // ¬ lineEqPropSol R}
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/-- Those solutions which satisfy the condition `lineEqPropSol` by not `inQuadSolProp`. -/
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def inLineEqSol : Type := {R : MSSMACC.Sols // lineEqPropSol R ∧ ¬ inQuadSolProp R}
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/-- Those solutions which satisfy the condition `lineEqPropSol` and `inQuadSolProp` but
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not `inCubeSolProp`. -/
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def inQuadSol : Type := {R : MSSMACC.Sols // lineEqPropSol R ∧ inQuadSolProp R ∧ ¬ inCubeSolProp R}
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/-- Those solutions which satisfy the condition all the conditions `lineEqPropSol`, `inQuadSolProp`
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and `inCubeSolProp`. -/
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def inQuadCubeSol : Type :=
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{R : MSSMACC.Sols // lineEqPropSol R ∧ inQuadSolProp R ∧ inCubeSolProp R}
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/-- Given a `R` perpendicular to `Y₃` and `B₃` a quadratic solution. -/
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def toSolNSQuad (R : MSSMACC.AnomalyFreePerp) : MSSMACC.QuadSols :=
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lineQuad R
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(3 * cubeTriLin (R.val, R.val, Y₃.val))
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@ -195,43 +231,36 @@ lemma toSolNSQuad_eq_planeY₃B₃_on_α (R : MSSMACC.AnomalyFreePerp) :
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ring_nf
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simp
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/-- Given a `R ∈ LinSols` perpendicular to $Y_3$, and $B_3$, a element of `Sols`. -/
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def toSolNS : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a, _ , _) =>
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/-- Given a `R ` perpendicular to $Y_3$, and $B_3$, a element of `Sols`. This map is
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not surjective. -/
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def toSolNS : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a, _ , _) =>
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a • AnomalyFreeMk'' (toSolNSQuad R) (toSolNSQuad_cube R)
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/-- A map from `Sols` to `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` which on elements of
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`notInLineEqSol` will produce a right inverse to `toSolNS`. -/
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def toSolNSProj (T : MSSMACC.Sols) : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ :=
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(proj T.1.1, (lineEqCoeff T)⁻¹, 0, 0)
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lemma toSolNS_proj (T : MSSMACC.Sols) (h : lineEqCoeff T ≠ 0) :
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toSolNS (toSolNSProj T) = T := by
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lemma toSolNS_proj (T : notInLineEqSol) : toSolNS (toSolNSProj T.val) = T.val := by
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apply ACCSystem.Sols.ext
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rw [toSolNS, toSolNSProj]
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change (lineEqCoeff T)⁻¹ • (toSolNSQuad _).1.1 = _
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change (lineEqCoeff T.val)⁻¹ • (toSolNSQuad _).1.1 = _
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rw [toSolNSQuad_eq_planeY₃B₃_on_α]
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rw [planeY₃B₃_val]
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rw [Y₃_plus_B₃_plus_proj]
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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.toLinSols) * dot (Y₃.val, B₃.val) = lineEqCoeff T := 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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rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h]
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have h1 := (lineEqPropSol_iff_lineEqCoeff_zero T.val).mpr.mt T.prop
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rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h1]
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simp
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def inLineEq : Type := {R : MSSMACC.AnomalyFreePerp // lineEqProp R}
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def inLineEqSol : Type := {R : MSSMACC.Sols // lineEqPropSol R}
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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 ⟩,
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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) * ( dot (Y₃.val, T.val.val)
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- dot (B₃.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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/-- Given a element of `inLineEq × ℚ × ℚ × ℚ`, a solution to the ACCs. -/
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def inLineEqToSol : inLineEq × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, c₁, c₂, c₃) =>
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AnomalyFreeMk'' (lineQuad R.val c₁ c₂ c₃)
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(by
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@ -239,6 +268,16 @@ def inLineEqToSol : inLineEq × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, c
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rw [R.prop.1, R.prop.2.1, R.prop.2.2]
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simp)
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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 * (
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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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apply ACCSystem.Sols.ext
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@ -246,8 +285,7 @@ lemma inLineEqTo_smul (R : inLineEq) (c₁ c₂ c₃ d : ℚ) :
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rw [lineQuad_smul]
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rfl
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lemma inLineEqToSol_proj (T : inLineEqSol) (h : quadCoeff T.val ≠ 0) :
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inLineEqToSol (inLineEqProj T) = T.val := by
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lemma inLineEqToSol_proj (T : inLineEqSol) : inLineEqToSol (inLineEqProj T) = T.val := by
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rw [inLineEqProj, inLineEqTo_smul]
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apply ACCSystem.Sols.ext
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change _ • (lineQuad _ _ _ _).val = _
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@ -262,14 +300,11 @@ lemma inLineEqToSol_proj (T : inLineEqSol) (h : quadCoeff T.val ≠ 0) :
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rw [quadCoeff]
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ring
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rw [h1]
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rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h]
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have h2 := (inQuadSolProp_iff_quadCoeff_zero T.val).mpr.mt T.prop.2
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rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h2]
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simp
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def inQuad : Type := {R : inLineEq // inQuadProp R.val}
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def inQuadSol : Type := {R : inLineEqSol // inQuadSolProp R.val}
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/-- Given a element of `inQuad × ℚ × ℚ × ℚ`, a solution to the ACCs. -/
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def inQuadToSol : inQuad × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a₁, a₂, a₃) =>
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AnomalyFreeMk' (lineCube R.val.val a₁ a₂ a₃)
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(by
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@ -285,19 +320,19 @@ lemma inQuadToSol_smul (R : inQuad) (c₁ c₂ c₃ d : ℚ) :
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rw [lineCube_smul]
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rfl
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/-- On elements of `inQuadSol` a right-inverse to `inQuadToSol`. -/
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def inQuadProj (T : inQuadSol) : inQuad × ℚ × ℚ × ℚ :=
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(⟨⟨proj T.val.val.1.1, (linEqPropSol_iff_proj_linEqProp T.val.val).mp T.val.prop ⟩,
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(inQuadSolProp_iff_proj_inQuadProp T.val.val).mp T.prop⟩,
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(cubicCoeff T.val.val)⁻¹ * (cubeTriLin (T.val.val.val, T.val.val.val, B₃.val)),
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(cubicCoeff T.val.val)⁻¹ * (- cubeTriLin (T.val.val.val, T.val.val.val, Y₃.val)),
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(cubicCoeff T.val.val)⁻¹ * (- cubeTriLin (T.val.val.val, T.val.val.val, B₃.val)
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* (dot (Y₃.val, T.val.val.val) - dot (B₃.val, T.val.val.val))
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- cubeTriLin (T.val.val.val, T.val.val.val, Y₃.val)
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* (dot (Y₃.val, T.val.val.val) - 2 * dot (B₃.val, T.val.val.val))))
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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⟩,
|
||||
(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)
|
||||
* (dot (Y₃.val, T.val.val) - 2 * dot (B₃.val, T.val.val))))
|
||||
|
||||
|
||||
lemma inQuadToSol_proj (T : inQuadSol) (h : cubicCoeff T.val.val ≠ 0) :
|
||||
inQuadToSol (inQuadProj T) = T.val.val := by
|
||||
lemma inQuadToSol_proj (T : inQuadSol) : inQuadToSol (inQuadProj T) = T.val := by
|
||||
rw [inQuadProj, inQuadToSol_smul]
|
||||
apply ACCSystem.Sols.ext
|
||||
change _ • (planeY₃B₃ _ _ _ _).val = _
|
||||
|
|
@ -306,20 +341,17 @@ lemma inQuadToSol_proj (T : inQuadSol) (h : cubicCoeff T.val.val ≠ 0) :
|
|||
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.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
|
||||
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
|
||||
rw [h1]
|
||||
rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h]
|
||||
have h2 := (inCubeSolProp_iff_cubicCoeff_zero T.val).mpr.mt T.prop.2.2
|
||||
rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h2]
|
||||
simp
|
||||
|
||||
def inQuadCube : Type := {R : inQuad // inCubeProp R.val.val}
|
||||
|
||||
def inQuadCubeSol : Type := {R : inQuadSol // inCubeSolProp R.val.val}
|
||||
|
||||
/-- The case where the plane lies entirely within the quadratic and cubic. -/
|
||||
/-- Given a element of `inQuadCube × ℚ × ℚ × ℚ`, a solution to the ACCs. -/
|
||||
def inQuadCubeToSol : inQuadCube × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, b₁, b₂, b₃) =>
|
||||
AnomalyFreeMk' (planeY₃B₃ R.val.val.val b₁ b₂ b₃)
|
||||
(by
|
||||
|
|
@ -339,16 +371,17 @@ lemma inQuadCubeToSol_smul (R : inQuadCube) (c₁ c₂ c₃ d : ℚ) :
|
|||
rfl
|
||||
|
||||
|
||||
/-- On elements of `inQuadCubeSol` a right-inverse to `inQuadCubeToSol`. -/
|
||||
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)
|
||||
(⟨⟨⟨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⟩,
|
||||
(inCubeSolProp_iff_proj_inCubeProp T.val).mp T.prop.2.2⟩,
|
||||
(dot (Y₃.val, B₃.val))⁻¹ * (dot (Y₃.val, T.val.val) - dot (B₃.val, T.val.val)),
|
||||
(dot (Y₃.val, B₃.val))⁻¹ * (2 * dot (B₃.val, T.val.val) - dot (Y₃.val, T.val.val)),
|
||||
(dot (Y₃.val, B₃.val))⁻¹ * 1)
|
||||
|
||||
lemma inQuadCubeToSol_proj (T : inQuadCubeSol) :
|
||||
inQuadCubeToSol (inQuadCubeProj T) = T.val.val.val := by
|
||||
inQuadCubeToSol (inQuadCubeProj T) = T.val := by
|
||||
rw [inQuadCubeProj, inQuadCubeToSol_smul]
|
||||
apply ACCSystem.Sols.ext
|
||||
change _ • (planeY₃B₃ _ _ _ _).val = _
|
||||
|
|
@ -361,8 +394,9 @@ lemma inQuadCubeToSol_proj (T : inQuadCubeSol) :
|
|||
rw [show dot (Y₃.val, B₃.val) = 108 by rfl]
|
||||
simp
|
||||
|
||||
def toSol :
|
||||
MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a, b, c) =>
|
||||
/-- Given an element of `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` a solution. We will
|
||||
show that this map is a surjection. -/
|
||||
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
|
||||
|
|
@ -374,58 +408,56 @@ def toSol :
|
|||
else
|
||||
toSolNS ⟨R, a, b, c⟩
|
||||
|
||||
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
|
||||
lemma toSol_toSolNSProj (T : notInLineEqSol) :
|
||||
∃ X, toSol X = T.val := by
|
||||
use toSolNSProj T.val
|
||||
have h1 : ¬ lineEqProp (toSolNSProj T.val).1 :=
|
||||
(linEqPropSol_iff_proj_linEqProp T.val).mpr.mt T.prop
|
||||
rw [toSol]
|
||||
simp_all
|
||||
exact toSolNS_proj T (mt (lineEqPropSol_iff_lineEqCoeff_zero T).mpr h₁)
|
||||
exact toSolNS_proj T
|
||||
|
||||
lemma toSol_inLineEq (T : MSSMACC.Sols) (h₁ : lineEqPropSol T) (h₂ : ¬ inQuadSolProp T) :
|
||||
∃ X, toSol X = T := by
|
||||
let X := inLineEqProj ⟨T, h₁⟩
|
||||
lemma toSol_inLineEq (T : inLineEqSol) : ∃ X, toSol X = T.val := by
|
||||
let X := inLineEqProj T
|
||||
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
|
||||
have : ¬ inQuadProp X.1.val := (inQuadSolProp_iff_proj_inQuadProp T.val).mpr.mt T.prop.2
|
||||
have : lineEqProp X.1.val := (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1
|
||||
rw [toSol]
|
||||
simp_all
|
||||
exact inLineEqToSol_proj ⟨T, h₁⟩ (mt (inQuadSolProp_iff_quadCoeff_zero T).mpr h₂)
|
||||
exact inLineEqToSol_proj T
|
||||
|
||||
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⟩
|
||||
lemma toSol_inQuad (T : inQuadSol) : ∃ X, toSol X = T.val := by
|
||||
let X := inQuadProj T
|
||||
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
|
||||
have : ¬ inCubeProp X.1.val.val := (inCubeSolProp_iff_proj_inCubeProp T.val).mpr.mt T.prop.2.2
|
||||
have : inQuadProp X.1.val.val := (inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1
|
||||
have : lineEqProp X.1.val.val := (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1
|
||||
rw [toSol]
|
||||
simp_all
|
||||
exact inQuadToSol_proj ⟨⟨T, h₁⟩, h₂⟩ (mt (inCubeSolProp_iff_cubicCoeff_zero T).mpr h₃)
|
||||
exact inQuadToSol_proj T
|
||||
|
||||
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⟩
|
||||
lemma toSol_inQuadCube (T : inQuadCubeSol) : ∃ X, toSol X = T.val := by
|
||||
let X := inQuadCubeProj T
|
||||
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
|
||||
have : inCubeProp X.1.val.val.val := (inCubeSolProp_iff_proj_inCubeProp T.val).mp T.prop.2.2
|
||||
have : inQuadProp X.1.val.val.val := (inQuadSolProp_iff_proj_inQuadProp T.val).mp T.prop.2.1
|
||||
have : lineEqProp X.1.val.val.val := (linEqPropSol_iff_proj_linEqProp T.val).mp T.prop.1
|
||||
rw [toSol]
|
||||
simp_all
|
||||
exact inQuadCubeToSol_proj ⟨⟨⟨T, h₁⟩, h₂⟩, h₃⟩
|
||||
exact inQuadCubeToSol_proj T
|
||||
|
||||
theorem toSol_surjective : Function.Surjective toSol := by
|
||||
intro T
|
||||
by_cases h₁ : ¬ lineEqPropSol T
|
||||
use toSolNSProj T
|
||||
exact toSol_toSolNSProj T h₁
|
||||
exact toSol_toSolNSProj ⟨T, h₁⟩
|
||||
simp at h₁
|
||||
by_cases h₂ : ¬ inQuadSolProp T
|
||||
exact toSol_inLineEq T h₁ h₂
|
||||
exact toSol_inLineEq ⟨T, And.intro h₁ h₂⟩
|
||||
simp at h₂
|
||||
by_cases h₃ : ¬ inCubeSolProp T
|
||||
exact toSol_inQuad T h₁ h₂ h₃
|
||||
exact toSol_inQuad ⟨T, And.intro h₁ (And.intro h₂ h₃)⟩
|
||||
simp at h₃
|
||||
exact toSol_inQuadCube T h₁ h₂ h₃
|
||||
exact toSol_inQuadCube ⟨T, And.intro h₁ (And.intro h₂ h₃)⟩
|
||||
|
||||
|
||||
end AnomalyFreePerp
|
||||
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Add a link
Reference in a new issue