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reactor: Removal of double spaces

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jstoobysmith 2024-07-12 11:23:02 -04:00
parent ce92e1d649
commit 13f62a50eb
64 changed files with 550 additions and 546 deletions
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8
HepLean/AnomalyCancellation/Basic.lean
View file

@ -197,7 +197,7 @@ instance quadSolsMulAction (χ : ACCSystemQuad) : MulAction ℚ χ.QuadSols wher
exact one_smul _ _
/-- The inclusion of quadratic solutions into linear solutions. -/
def quadSolsInclLinSols (χ : ACCSystemQuad) : χ.QuadSols →[ℚ] χ.LinSols where
def quadSolsInclLinSols (χ : ACCSystemQuad) : χ.QuadSols →[ℚ] χ.LinSols where
toFun := QuadSols.toLinSols
map_smul' _ _ := rfl
@ -229,7 +229,7 @@ structure Sols (χ : ACCSystem) extends χ.QuadSols where
/-- Two solutions are equal if the underlying charges are equal. -/
lemma Sols.ext {χ : ACCSystem} {S T : χ.Sols} (h : S.val = T.val) :
S = T := by
have h := ACCSystemQuad.QuadSols.ext h
have h := ACCSystemQuad.QuadSols.ext h
cases' S
simp_all only
@ -251,8 +251,8 @@ instance solsMulAction (χ : ACCSystem) : MulAction ℚ χ.Sols where
exact one_smul _ _
/-- The inclusion of `Sols` into `QuadSols`. -/
def solsInclQuadSols (χ : ACCSystem) : χ.Sols →[ℚ] χ.QuadSols where
toFun := Sols.toQuadSols
def solsInclQuadSols (χ : ACCSystem) : χ.Sols →[ℚ] χ.QuadSols where
toFun := Sols.toQuadSols
map_smul' _ _ := rfl
/-- The inclusion of `Sols` into `LinSols`. -/

4
HepLean/AnomalyCancellation/GroupActions.lean
View file

@ -104,7 +104,7 @@ lemma linSolRep_quadSolAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction
G.linSolRep g (χ.quadSolsInclLinSols S) := rfl
lemma rep_quadSolAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
(S : χ.QuadSols) : χ.quadSolsIncl (G.quadSolAction.toFun S g) =
(S : χ.QuadSols) : χ.quadSolsIncl (G.quadSolAction.toFun S g) =
G.rep g (χ.quadSolsIncl S) := rfl
/-- The group action acting on solutions to the anomaly cancellation conditions. -/
@ -126,7 +126,7 @@ instance solAction {χ : ACCSystem} (G : ACCSystemGroupAction χ) : MulAction G.
rfl
lemma quadSolAction_solAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
(S : χ.Sols) : χ.solsInclQuadSols (G.solAction.toFun S g) =
(S : χ.Sols) : χ.solsInclQuadSols (G.solAction.toFun S g) =
G.quadSolAction.toFun (χ.solsInclQuadSols S) g := rfl
lemma linSolRep_solAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)

2
HepLean/AnomalyCancellation/MSSMNu/B3.lean
View file

@ -34,7 +34,7 @@ def B₃AsCharge : MSSMACC.Charges := toSpecies.symm
| 0, 2 => - 1
| 1, 0 => - 1
| 1, 1 => - 1
| 1, 2 => 1
| 1, 2 => 1
| 2, 0 => - 1
| 2, 1 => - 1
| 2, 2 => 1

32
HepLean/AnomalyCancellation/MSSMNu/Basic.lean
View file

@ -43,7 +43,7 @@ def toSMPlusH : MSSMCharges.Charges ≃ (Fin 18 ⊕ Fin 2 → ℚ) :=
@[simps!]
def splitSMPlusH : (Fin 18 ⊕ Fin 2 → ℚ) ≃ (Fin 18 → ℚ) × (Fin 2 → ℚ) where
toFun f := (f ∘ Sum.inl , f ∘ Sum.inr)
invFun f := Sum.elim f.1 f.2
invFun f := Sum.elim f.1 f.2
left_inv f := Sum.elim_comp_inl_inr f
right_inv _ := rfl
@ -107,7 +107,7 @@ def Hu : MSSMCharges.Charges →ₗ[ℚ] ℚ where
map_smul' _ _ := by rfl
lemma charges_eq_toSpecies_eq (S T : MSSMCharges.Charges) :
S = T ↔ (∀ i, toSMSpecies i S = toSMSpecies i T) ∧ Hd S = Hd T ∧ Hu S = Hu T := by
S = T ↔ (∀ i, toSMSpecies i S = toSMSpecies i T) ∧ Hd S = Hd T ∧ Hu S = Hu T := by
apply Iff.intro
intro h
rw [h]
@ -159,7 +159,7 @@ def accGrav : MSSMCharges.Charges →ₗ[ℚ] ℚ where
/-- Extensionality lemma for `accGrav`. -/
lemma accGrav_ext {S T : MSSMCharges.Charges}
(hj : ∀ (j : Fin 6), ∑ i, (toSMSpecies j) S i = ∑ i, (toSMSpecies j) T i)
(hj : ∀ (j : Fin 6), ∑ i, (toSMSpecies j) S i = ∑ i, (toSMSpecies j) T i)
(hd : Hd S = Hd T) (hu : Hu S = Hu T) :
accGrav S = accGrav T := by
simp only [accGrav, MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue,
@ -192,7 +192,7 @@ def accSU2 : MSSMCharges.Charges →ₗ[ℚ] ℚ where
/-- Extensionality lemma for `accSU2`. -/
lemma accSU2_ext {S T : MSSMCharges.Charges}
(hj : ∀ (j : Fin 6), ∑ i, (toSMSpecies j) S i = ∑ i, (toSMSpecies j) T i)
(hj : ∀ (j : Fin 6), ∑ i, (toSMSpecies j) S i = ∑ i, (toSMSpecies j) T i)
(hd : Hd S = Hd T) (hu : Hu S = Hu T) :
accSU2 S = accSU2 T := by
simp only [accSU2, MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue,
@ -224,7 +224,7 @@ def accSU3 : MSSMCharges.Charges →ₗ[ℚ] ℚ where
/-- Extensionality lemma for `accSU3`. -/
lemma accSU3_ext {S T : MSSMCharges.Charges}
(hj : ∀ (j : Fin 6), ∑ i, (toSMSpecies j) S i = ∑ i, (toSMSpecies j) T i) :
(hj : ∀ (j : Fin 6), ∑ i, (toSMSpecies j) S i = ∑ i, (toSMSpecies j) T i) :
accSU3 S = accSU3 T := by
simp only [accSU3, MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue, LinearMap.coe_mk,
AddHom.coe_mk]
@ -256,7 +256,7 @@ def accYY : MSSMCharges.Charges →ₗ[ℚ] ℚ where
/-- Extensionality lemma for `accGrav`. -/
lemma accYY_ext {S T : MSSMCharges.Charges}
(hj : ∀ (j : Fin 6), ∑ i, (toSMSpecies j) S i = ∑ i, (toSMSpecies j) T i)
(hj : ∀ (j : Fin 6), ∑ i, (toSMSpecies j) S i = ∑ i, (toSMSpecies j) T i)
(hd : Hd S = Hd T) (hu : Hu S = Hu T) :
accYY S = accYY T := by
simp only [accYY, MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue,
@ -269,9 +269,9 @@ lemma accYY_ext {S T : MSSMCharges.Charges}
/-- The symmetric bilinear function used to define the quadratic ACC. -/
@[simps!]
def quadBiLin : BiLinearSymm MSSMCharges.Charges := BiLinearSymm.mk₂ (
fun (S, T) => ∑ i, (Q S i * Q T i + (- 2) * (U S i * U T i) +
D S i * D T i + (- 1) * (L S i * L T i) + E S i * E T i) +
def quadBiLin : BiLinearSymm MSSMCharges.Charges := BiLinearSymm.mk₂ (
fun (S, T) => ∑ i, (Q S i * Q T i + (- 2) * (U S i * U T i) +
D S i * D T i + (- 1) * (L S i * L T i) + E S i * E T i) +
(- Hd S * Hd T + Hu S * Hu T))
(by
intro a S T
@ -346,7 +346,7 @@ def cubeTriLinToFun
+ (2 * Hd S.1 * Hd S.2.1 * Hd S.2.2
+ 2 * Hu S.1 * Hu S.2.1 * Hu S.2.2)
lemma cubeTriLinToFun_map_smul₁ (a : ℚ) (S T R : MSSMCharges.Charges) :
lemma cubeTriLinToFun_map_smul₁ (a : ℚ) (S T R : MSSMCharges.Charges) :
cubeTriLinToFun (a • S, T, R) = a * cubeTriLinToFun (S, T, R) := by
simp only [cubeTriLinToFun]
rw [mul_add]
@ -366,7 +366,7 @@ lemma cubeTriLinToFun_map_add₁ (S T R L : MSSMCharges.Charges) :
rw [add_assoc, ← add_assoc (2 * Hd S * Hd R * Hd L + 2 * Hu S * Hu R * Hu L) _ _]
rw [add_comm (2 * Hd S * Hd R * Hd L + 2 * Hu S * Hu R * Hu L) _]
rw [add_assoc]
rw [← add_assoc _ _ (2 * Hd S * Hd R * Hd L + 2 * Hu S * Hu R * Hu L +
rw [← add_assoc _ _ (2 * Hd S * Hd R * Hd L + 2 * Hu S * Hu R * Hu L +
(2 * Hd T * Hd R * Hd L + 2 * Hu T * Hu R * Hu L))]
congr 1
rw [← Finset.sum_add_distrib]
@ -415,7 +415,7 @@ def accCube : HomogeneousCubic MSSMCharges.Charges := cubeTriLin.toCubic
lemma accCube_ext {S T : MSSMCharges.Charges}
(h : ∀ j, ∑ i, ((fun a => a^3) ∘ toSMSpecies j S) i =
∑ i, ((fun a => a^3) ∘ toSMSpecies j T) i)
(hd : Hd S = Hd T) (hu : Hu S = Hu T) :
(hd : Hd S = Hd T) (hu : Hu S = Hu T) :
accCube S = accCube T := by
simp only [HomogeneousCubic, accCube, cubeTriLin, TriLinearSymm.toCubic_apply,
TriLinearSymm.mk₃_toFun_apply_apply, cubeTriLinToFun]
@ -452,7 +452,7 @@ def MSSMACC : ACCSystem where
namespace MSSMACC
open MSSMCharges
lemma quadSol (S : MSSMACC.QuadSols) : accQuad S.val = 0 := by
lemma quadSol (S : MSSMACC.QuadSols) : accQuad S.val = 0 := by
have hS := S.quadSol
simp only [MSSMACC_numberQuadratic, HomogeneousQuadratic, MSSMACC_quadraticACCs] at hS
exact hS 0
@ -516,9 +516,9 @@ lemma AnomalyFreeMk''_val (S : MSSMACC.QuadSols)
/-- The dot product on the vector space of charges. -/
@[simps!]
def dot : BiLinearSymm MSSMCharges.Charges := BiLinearSymm.mk₂
(fun S => ∑ i, (Q S.1 i * Q S.2 i + U S.1 i * U S.2 i +
D S.1 i * D S.2 i + L S.1 i * L S.2 i + E S.1 i * E S.2 i
def dot : BiLinearSymm MSSMCharges.Charges := BiLinearSymm.mk₂
(fun S => ∑ i, (Q S.1 i * Q S.2 i + U S.1 i * U S.2 i +
D S.1 i * D S.2 i + L S.1 i * L S.2 i + E S.1 i * E S.2 i
+ N S.1 i * N S.2 i) + Hd S.1 * Hd S.2 + Hu S.1 * Hu S.2)
(by
intro a S T

4
HepLean/AnomalyCancellation/MSSMNu/LineY3B3.lean
View file

@ -11,7 +11,7 @@ import Mathlib.Tactic.Polyrith
# The line through B₃ and Y₃
We give properties of lines through `B₃` and `Y₃`. We show that every point on this line
is a solution to the quadratic `lineY₃B₃Charges_quad` and a double point of the cubic
is a solution to the quadratic `lineY₃B₃Charges_quad` and a double point of the cubic
`lineY₃B₃_doublePoint`.
# References
@ -52,7 +52,7 @@ lemma lineY₃B₃Charges_cubic (a b : ℚ) : accCube (lineY₃B₃Charges a b).
rw [cubeTriLin.toCubic.map_smul]
rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
rw [cubeTriLin.map_smul₁, cubeTriLin.map_smul₂, cubeTriLin.map_smul₃]
erw [Y₃.cubicSol, B₃.cubicSol]
erw [Y₃.cubicSol, B₃.cubicSol]
rw [show cubeTriLin Y₃.val Y₃.val B₃.val = 0 by rfl]
rw [show cubeTriLin B₃.val B₃.val Y₃.val = 0 by rfl]
simp

8
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/Basic.lean
View file

@ -158,7 +158,7 @@ lemma cube_proj_proj_B₃ (T : MSSMACC.LinSols) :
lemma cube_proj_proj_self (T : MSSMACC.Sols) :
cubeTriLin (proj T.1.1).val (proj T.1.1).val T.val =
2 * dot Y₃.val B₃.val *
((dot B₃.val T.val - dot Y₃.val T.val) * cubeTriLin T.val T.val Y₃.val +
((dot B₃.val T.val - dot Y₃.val T.val) * cubeTriLin T.val T.val Y₃.val +
( dot Y₃.val T.val- 2 * dot B₃.val T.val) * cubeTriLin T.val T.val B₃.val) := by
rw [proj_val]
rw [cubeTriLin.map_add₁, cubeTriLin.map_add₂]
@ -168,13 +168,13 @@ lemma cube_proj_proj_self (T : MSSMACC.Sols) :
repeat rw [cubeTriLin.map_add₂]
repeat rw [cubeTriLin.map_smul₂]
erw [T.cubicSol]
rw [cubeTriLin.swap₁ Y₃.val T.val T.val, cubeTriLin.swap₂ T.val Y₃.val T.val]
rw [cubeTriLin.swap₁ B₃.val T.val T.val, cubeTriLin.swap₂ T.val B₃.val T.val]
rw [cubeTriLin.swap₁ Y₃.val T.val T.val, cubeTriLin.swap₂ T.val Y₃.val T.val]
rw [cubeTriLin.swap₁ B₃.val T.val T.val, cubeTriLin.swap₂ T.val B₃.val T.val]
ring
lemma cube_proj (T : MSSMACC.Sols) :
cubeTriLin (proj T.1.1).val (proj T.1.1).val (proj T.1.1).val =
3 * dot Y₃.val B₃.val ^ 2 *
3 * dot Y₃.val B₃.val ^ 2 *
((dot B₃.val T.val - dot Y₃.val T.val) * cubeTriLin T.val T.val Y₃.val +
(dot Y₃.val T.val - 2 * dot B₃.val T.val) * cubeTriLin T.val T.val B₃.val) := by
nth_rewrite 3 [proj_val]

16
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/PlaneWithY3B3.lean
View file

@ -42,7 +42,7 @@ lemma planeY₃B₃_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
rw [smul_add, smul_add]
rw [smul_smul, smul_smul, smul_smul]
lemma planeY₃B₃_eq (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (h : a = a' ∧ b = b' ∧ c = c') :
lemma planeY₃B₃_eq (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (h : a = a' ∧ b = b' ∧ c = c') :
(planeY₃B₃ R a b c) = (planeY₃B₃ R a' b' c') := by
rw [h.1, h.2.1, h.2.2]
@ -82,7 +82,7 @@ lemma planeY₃B₃_val_eq' (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (hR' : R
have h2 := congrArg (fun S => S i) h1i
change _ = 0 at h2
simp [HSMul.hSMul] at h2
have hc : c + -c' = 0 := by
have hc : c + -c' = 0 := by
cases h2 <;> rename_i h2
exact h2
exact (hi h2).elim
@ -105,8 +105,8 @@ lemma planeY₃B₃_quad (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
lemma planeY₃B₃_cubic (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
accCube (planeY₃B₃ R a b c).val = c ^ 2 *
(3 * a * cubeTriLin R.val R.val Y₃.val
+ 3 * b * cubeTriLin R.val R.val B₃.val + c * cubeTriLin R.val R.val R.val) := by
(3 * a * cubeTriLin R.val R.val Y₃.val
+ 3 * b * cubeTriLin R.val R.val B₃.val + c * cubeTriLin R.val R.val R.val) := by
rw [planeY₃B₃_val]
erw [TriLinearSymm.toCubic_add]
erw [lineY₃B₃Charges_cubic]
@ -178,7 +178,7 @@ def lineCube (R : MSSMACC.AnomalyFreePerp) (a₁ a₂ a₃ : ℚ) :
MSSMACC.LinSols :=
planeY₃B₃ R
(a₂ * cubeTriLin R.val R.val R.val - 3 * a₃ * cubeTriLin R.val R.val B₃.val)
(3 * a₃ * cubeTriLin R.val R.val Y₃.val - a₁ * cubeTriLin R.val R.val R.val)
(3 * a₃ * cubeTriLin R.val R.val Y₃.val - a₁ * cubeTriLin R.val R.val R.val)
(3 * (a₁ * cubeTriLin R.val R.val B₃.val - a₂ * cubeTriLin R.val R.val Y₃.val))
lemma lineCube_smul (R : MSSMACC.AnomalyFreePerp) (a b c d : ℚ) :
@ -205,7 +205,7 @@ lemma lineCube_quad (R : MSSMACC.AnomalyFreePerp) (a₁ a₂ a₃ : ℚ) :
section proj
lemma α₃_proj (T : MSSMACC.Sols) : α₃ (proj T.1.1) =
lemma α₃_proj (T : MSSMACC.Sols) : α₃ (proj T.1.1) =
6 * dot Y₃.val B₃.val ^ 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
@ -214,13 +214,13 @@ lemma α₃_proj (T : MSSMACC.Sols) : α₃ (proj T.1.1) =
ring
lemma α₂_proj (T : MSSMACC.Sols) : α₂ (proj T.1.1) =
- α₃ (proj T.1.1) * (dot Y₃.val T.val - 2 * dot B₃.val T.val) := by
- α₃ (proj T.1.1) * (dot Y₃.val T.val - 2 * dot B₃.val T.val) := by
rw [α₃_proj, α₂]
rw [cube_proj_proj_Y₃, quad_Y₃_proj, quad_proj, cube_proj]
ring
lemma α₁_proj (T : MSSMACC.Sols) : α₁ (proj T.1.1) =
- α₃ (proj T.1.1) * (dot B₃.val T.val - dot Y₃.val T.val) := by
- α₃ (proj T.1.1) * (dot B₃.val T.val - dot Y₃.val T.val) := by
rw [α₃_proj, α₁]
rw [cube_proj_proj_B₃, quad_B₃_proj, quad_proj, cube_proj]
ring

12
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/ToSols.lean
View file

@ -146,7 +146,7 @@ def InCubeSolProp (R : MSSMACC.Sols) : Prop :=
/-- A rational which has two properties. It is zero for a solution `T` if and only if
that solution satisfies `inCubeSolProp`. It appears in the definition of `inLineEqProj`. -/
def cubicCoeff (T : MSSMACC.Sols) : ℚ :=
3 * (dot Y₃.val B₃.val) ^ 3 * (cubeTriLin T.val T.val Y₃.val ^ 2 +
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) :
@ -243,7 +243,7 @@ def toSolNSProj (T : MSSMACC.Sols) : MSSMACC.AnomalyFreePerp × ℚ × ℚ ×
lemma toSolNS_proj (T : NotInLineEqSol) : toSolNS (toSolNSProj T.val) = T.val := by
apply ACCSystem.Sols.ext
rw [toSolNS, toSolNSProj]
change (lineEqCoeff T.val)⁻¹ • (toSolNSQuad _).1.1 = _
change (lineEqCoeff T.val)⁻¹ • (toSolNSQuad _).1.1 = _
rw [toSolNSQuad_eq_planeY₃B₃_on_α]
rw [planeY₃B₃_val]
rw [Y₃_plus_B₃_plus_proj]
@ -254,7 +254,7 @@ lemma toSolNS_proj (T : NotInLineEqSol) : toSolNS (toSolNSProj T.val) = T.val :=
rw [lineEqCoeff]
ring
rw [h1]
have h1 := (lineEqPropSol_iff_lineEqCoeff_zero T.val).mpr.mt T.prop
have h1 := (lineEqPropSol_iff_lineEqCoeff_zero T.val).mpr.mt T.prop
rw [← MulAction.mul_smul, mul_comm, mul_inv_cancel h1]
simp
@ -302,7 +302,7 @@ lemma inLineEqToSol_proj (T : InLineEqSol) : inLineEqToSol (inLineEqProj T) = T.
simp
/-- Given a element of `inQuad × ℚ × ℚ × ℚ`, a solution to the ACCs. -/
def inQuadToSol : InQuad × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (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]
@ -391,8 +391,8 @@ lemma inQuadCubeToSol_proj (T : InQuadCubeSol) :
/-- 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
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 : LineEqProp R ∧ InQuadProp R then

24
HepLean/AnomalyCancellation/MSSMNu/Permutations.lean
View file

@ -17,7 +17,7 @@ and its action on the MSSM.
universe v u
open Nat
open Finset
open Finset
namespace MSSM
@ -30,7 +30,7 @@ open BigOperators
def PermGroup := Fin 6 → Equiv.Perm (Fin 3)
@[simp]
instance : Group PermGroup := Pi.group
instance : Group PermGroup := Pi.group
/-- The image of an element of `permGroup` under the representation on charges. -/
@[simps!]
@ -65,7 +65,7 @@ def repCharges : Representation ℚ PermGroup (MSSMCharges).Charges where
apply LinearMap.ext
intro S
rw [charges_eq_toSpecies_eq]
refine And.intro ?_ $ Prod.mk.inj_iff.mp rfl
refine And.intro ?_ $ Prod.mk.inj_iff.mp rfl
intro i
simp only [ Pi.mul_apply, Pi.inv_apply, Equiv.Perm.coe_mul, LinearMap.mul_apply]
rw [chargeMap_toSpecies, chargeMap_toSpecies]
@ -76,7 +76,7 @@ def repCharges : Representation ℚ PermGroup (MSSMCharges).Charges where
apply LinearMap.ext
intro S
rw [charges_eq_toSpecies_eq]
refine And.intro ?_ $ Prod.mk.inj_iff.mp rfl
refine And.intro ?_ $ Prod.mk.inj_iff.mp rfl
intro i
erw [toSMSpecies_toSpecies_inv]
rfl
@ -91,46 +91,46 @@ lemma toSpecies_sum_invariant (m : ℕ) (f : PermGroup) (S : MSSMCharges.Charges
rw [repCharges_toSMSpecies]
exact Equiv.sum_comp (f⁻¹ j) ((fun a => a ^ m) ∘ (toSMSpecies j) S)
lemma Hd_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
lemma Hd_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
Hd (repCharges f S) = Hd S := rfl
lemma Hu_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
lemma Hu_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
Hu (repCharges f S) = Hu S := rfl
lemma accGrav_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
lemma accGrav_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
accGrav (repCharges f S) = accGrav S :=
accGrav_ext
(by simpa using toSpecies_sum_invariant 1 f S)
(Hd_invariant f S)
(Hu_invariant f S)
lemma accSU2_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
lemma accSU2_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
accSU2 (repCharges f S) = accSU2 S :=
accSU2_ext
(by simpa using toSpecies_sum_invariant 1 f S)
(Hd_invariant f S)
(Hu_invariant f S)
lemma accSU3_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
lemma accSU3_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
accSU3 (repCharges f S) = accSU3 S :=
accSU3_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accYY_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
lemma accYY_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
accYY (repCharges f S) = accYY S :=
accYY_ext
(by simpa using toSpecies_sum_invariant 1 f S)
(Hd_invariant f S)
(Hu_invariant f S)
lemma accQuad_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
lemma accQuad_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
accQuad (repCharges f S) = accQuad S :=
accQuad_ext
(toSpecies_sum_invariant 2 f S)
(Hd_invariant f S)
(Hu_invariant f S)
lemma accCube_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
lemma accCube_invariant (f : PermGroup) (S : MSSMCharges.Charges) :
accCube (repCharges f S) = accCube S :=
accCube_ext
(fun j => toSpecies_sum_invariant 3 f S j)

10
HepLean/AnomalyCancellation/PureU1/Basic.lean
View file

@ -16,7 +16,7 @@ We define the anomaly cancellation conditions for a pure U(1) gauge theory with
universe v u
open Nat
open Finset
open Finset
namespace PureU1
open BigOperators
@ -37,7 +37,7 @@ def accGrav (n : ℕ) : ((PureU1Charges n).Charges →ₗ[ℚ] ℚ) where
/-- The symmetric trilinear form used to define the cubic anomaly. -/
@[simps!]
def accCubeTriLinSymm {n : ℕ} : TriLinearSymm (PureU1Charges n).Charges := TriLinearSymm.mk₃
(fun S => ∑ i, S.1 i * S.2.1 i * S.2.2 i)
(fun S => ∑ i, S.1 i * S.2.1 i * S.2.2 i)
(by
intro a S L T
simp [HSMul.hSMul]
@ -70,13 +70,13 @@ def accCubeTriLinSymm {n : ℕ} : TriLinearSymm (PureU1Charges n).Charges := Tri
/-- The cubic anomaly equation. -/
@[simp]
def accCube (n : ℕ) : HomogeneousCubic ((PureU1Charges n).Charges) :=
def accCube (n : ℕ) : HomogeneousCubic ((PureU1Charges n).Charges) :=
(accCubeTriLinSymm).toCubic
lemma accCube_explicit (n : ℕ) (S : (PureU1Charges n).Charges) :
accCube n S = ∑ i : Fin n, S i ^ 3:= by
rw [accCube, TriLinearSymm.toCubic]
change accCubeTriLinSymm S S S = _
change accCubeTriLinSymm S S S = _
rw [accCubeTriLinSymm]
simp only [PureU1Charges_numberCharges, TriLinearSymm.mk₃_toFun_apply_apply]
apply Finset.sum_congr
@ -99,7 +99,7 @@ def PureU1 (n : ℕ) : ACCSystem where
/-- An equivalence of vector spaces of charges when the number of fermions is equal. -/
def pureU1EqCharges {n m : ℕ} (h : n = m):
(PureU1 n).Charges ≃ₗ[ℚ] (PureU1 m).Charges where
(PureU1 n).Charges ≃ₗ[ℚ] (PureU1 m).Charges where
toFun f := f ∘ Fin.cast h.symm
invFun f := f ∘ Fin.cast h
map_add' _ _ := rfl

6
HepLean/AnomalyCancellation/PureU1/BasisLinear.lean
View file

@ -84,7 +84,7 @@ noncomputable
def coordinateMap : ((PureU1 n.succ).LinSols) ≃ₗ[ℚ] Fin n →₀ ℚ where
toFun S := Finsupp.equivFunOnFinite.invFun (S.1 ∘ Fin.castSucc)
map_add' S T := Finsupp.ext (congrFun rfl)
map_smul' a S := Finsupp.ext (congrFun rfl)
map_smul' a S := Finsupp.ext (congrFun rfl)
invFun f := ∑ i : Fin n, f i • asLinSols i
left_inv S := by
simp only [PureU1_numberCharges, Equiv.invFun_as_coe, Finsupp.equivFunOnFinite_symm_apply_toFun,
@ -92,7 +92,7 @@ def coordinateMap : ((PureU1 n.succ).LinSols) ≃ₗ[ℚ] Fin n →₀ ℚ where
apply pureU1_anomalyFree_ext
intro j
rw [sum_of_vectors]
simp only [HSMul.hSMul, SMul.smul, PureU1_numberCharges,
simp only [HSMul.hSMul, SMul.smul, PureU1_numberCharges,
asLinSols_val, Equiv.toFun_as_coe,
Fin.coe_eq_castSucc, mul_ite, mul_one, mul_neg, mul_zero, Equiv.invFun_as_coe]
rw [Finset.sum_eq_single j]
@ -127,7 +127,7 @@ instance : Module.Finite ℚ ((PureU1 n.succ).LinSols) :=
lemma finrank_AnomalyFreeLinear :
FiniteDimensional.finrank ℚ (((PureU1 n.succ).LinSols)) = n := by
have h := Module.mk_finrank_eq_card_basis (@asBasis n)
have h := Module.mk_finrank_eq_card_basis (@asBasis n)
simp only [Nat.succ_eq_add_one, finrank_eq_rank, Cardinal.mk_fintype, Fintype.card_fin] at h
exact FiniteDimensional.finrank_eq_of_rank_eq h

26
HepLean/AnomalyCancellation/PureU1/ConstAbs.lean
View file

@ -14,7 +14,7 @@ We look at charge assignments in which all charges have the same absolute value.
universe v u
open Nat
open Finset
open Finset
open BigOperators
namespace PureU1
@ -53,7 +53,7 @@ section charges
variable {S : (PureU1 n.succ).Charges} {A : (PureU1 n.succ).LinSols}
variable (hS : ConstAbsSorted S) (hA : ConstAbsSorted A.val)
lemma lt_eq {k i : Fin n.succ} (hk : S k ≤ 0) (hik : i ≤ k) : S i = S k := by
lemma lt_eq {k i : Fin n.succ} (hk : S k ≤ 0) (hik : i ≤ k) : S i = S k := by
have hSS := hS.2 i k hik
have ht := hS.1 i k
rw [sq_eq_sq_iff_eq_or_eq_neg] at ht
@ -61,7 +61,7 @@ lemma lt_eq {k i : Fin n.succ} (hk : S k ≤ 0) (hik : i ≤ k) : S i = S k :=
exact h
linarith
lemma val_le_zero {i : Fin n.succ} (hi : S i ≤ 0) : S i = S (0 : Fin n.succ) := by
lemma val_le_zero {i : Fin n.succ} (hi : S i ≤ 0) : S i = S (0 : Fin n.succ) := by
symm
apply lt_eq hS hi
simp
@ -108,7 +108,7 @@ lemma boundary_succ {k : Fin n} (hk : Boundary S k) : S k.succ = - S (0 : Fin n.
rw [opposite_signs_eq_neg hS (le_of_lt hk.left) (le_of_lt hk.right)] at hn
linear_combination -(1 * hn)
lemma boundary_split (k : Fin n) : k.succ.val + (n.succ - k.succ.val) = n.succ := by
lemma boundary_split (k : Fin n) : k.succ.val + (n.succ - k.succ.val) = n.succ := by
omega
lemma boundary_accGrav' (k : Fin n) : accGrav n.succ S =
@ -128,7 +128,7 @@ lemma boundary_accGrav'' (k : Fin n) (hk : Boundary S k) :
S (Fin.cast (boundary_split k) (Fin.castAdd (n.succ - k.succ.val) i)) = S k.castSucc := by
apply lt_eq hS (le_of_lt hk.left) (by rw [Fin.le_def]; simp; omega)
have hsnd (i : Fin (n.succ - k.succ.val)) :
S (Fin.cast (boundary_split k) (Fin.natAdd (k.succ.val) i)) = S k.succ := by
S (Fin.cast (boundary_split k) (Fin.natAdd (k.succ.val) i)) = S k.succ := by
apply gt_eq hS (le_of_lt hk.right) (by rw [Fin.le_def]; simp)
simp only [hfst, hsnd]
simp only [Fin.val_succ, sum_const, card_fin, nsmul_eq_mul, cast_add, cast_one,
@ -160,7 +160,7 @@ lemma not_hasBoundry_zero (hnot : ¬ (HasBoundary S)) (i : Fin n.succ) :
simp at hi
exact zero_gt hS hi i
lemma not_hasBoundary_grav (hnot : ¬ (HasBoundary S)) :
lemma not_hasBoundary_grav (hnot : ¬ (HasBoundary S)) :
accGrav n.succ S = n.succ * S (0 : Fin n.succ) := by
simp [accGrav, ← not_hasBoundry_zero hS hnot]
@ -196,7 +196,7 @@ lemma AFL_odd_zero {A : (PureU1 (2 * n + 1)).LinSols} (h : ConstAbsSorted A.val)
theorem AFL_odd (A : (PureU1 (2 * n + 1)).LinSols) (h : ConstAbsSorted A.val) :
A = 0 := by
apply ACCSystemLinear.LinSols.ext
exact is_zero h (AFL_odd_zero h)
exact is_zero h (AFL_odd_zero h)
lemma AFL_even_Boundary {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.val)
(hA : A.val (0 : Fin (2 * n.succ)) ≠ 0) {k : Fin (2 * n + 1)} (hk : Boundary A.val k) :
@ -209,8 +209,8 @@ lemma AFL_even_Boundary {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted
linear_combination h0 / 2
lemma AFL_even_below' {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.val)
(hA : A.val (0 : Fin (2 * n.succ)) ≠ 0) (i : Fin n.succ) :
A.val (Fin.cast (split_equal n.succ) (Fin.castAdd n.succ i)) = A.val (0 : Fin (2*n.succ)) := by
(hA : A.val (0 : Fin (2 * n.succ)) ≠ 0) (i : Fin n.succ) :
A.val (Fin.cast (split_equal n.succ) (Fin.castAdd n.succ i)) = A.val (0 : Fin (2*n.succ)) := by
obtain ⟨k, hk⟩ := AFL_hasBoundary h hA
rw [← boundary_castSucc h hk]
apply lt_eq h (le_of_lt hk.left)
@ -221,7 +221,7 @@ lemma AFL_even_below' {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.
lemma AFL_even_below (A : (PureU1 (2 * n.succ)).LinSols) (h : ConstAbsSorted A.val)
(i : Fin n.succ) :
A.val (Fin.cast (split_equal n.succ) (Fin.castAdd n.succ i))
A.val (Fin.cast (split_equal n.succ) (Fin.castAdd n.succ i))
= A.val (0 : Fin (2*n.succ)) := by
by_cases hA : A.val (0 : Fin (2*n.succ)) = 0
rw [is_zero h hA]
@ -230,8 +230,8 @@ lemma AFL_even_below (A : (PureU1 (2 * n.succ)).LinSols) (h : ConstAbsSorted A.v
exact AFL_even_below' h hA i
lemma AFL_even_above' {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.val)
(hA : A.val (0 : Fin (2*n.succ)) ≠ 0) (i : Fin n.succ) :
A.val (Fin.cast (split_equal n.succ) (Fin.natAdd n.succ i)) =
(hA : A.val (0 : Fin (2*n.succ)) ≠ 0) (i : Fin n.succ) :
A.val (Fin.cast (split_equal n.succ) (Fin.natAdd n.succ i)) =
- A.val (0 : Fin (2*n.succ)) := by
obtain ⟨k, hk⟩ := AFL_hasBoundary h hA
rw [← boundary_succ h hk]
@ -243,7 +243,7 @@ lemma AFL_even_above' {A : (PureU1 (2 * n.succ)).LinSols} (h : ConstAbsSorted A.
lemma AFL_even_above (A : (PureU1 (2 * n.succ)).LinSols) (h : ConstAbsSorted A.val)
(i : Fin n.succ) :
A.val (Fin.cast (split_equal n.succ) (Fin.natAdd n.succ i)) =
A.val (Fin.cast (split_equal n.succ) (Fin.natAdd n.succ i)) =
- A.val (0 : Fin (2*n.succ)) := by
by_cases hA : A.val (0 : Fin (2*n.succ)) = 0
rw [is_zero h hA]

42
HepLean/AnomalyCancellation/PureU1/Even/BasisLinear.lean
View file

@ -16,7 +16,7 @@ that splits into two planes on which every point is a solution to the ACCs.
universe v u
open Nat
open Finset
open Finset
open BigOperators
namespace PureU1
@ -51,7 +51,7 @@ def δ!₃ : Fin (2 * n.succ) := (Fin.cast (n_cond₂ n) (Fin.castAdd ((n + n) +
def δ!₄ : Fin (2 * n.succ) := (Fin.cast (n_cond₂ n) (Fin.natAdd 1 (Fin.natAdd (n + n) 0)))
lemma ext_δ (S T : Fin (2 * n.succ) → ℚ) (h1 : ∀ i, S (δ₁ i) = T (δ₁ i))
(h2 : ∀ i, S (δ₂ i) = T (δ₂ i)) : S = T := by
(h2 : ∀ i, S (δ₂ i) = T (δ₂ i)) : S = T := by
funext i
by_cases hi : i.val < n.succ
let j : Fin n.succ := ⟨i, hi⟩
@ -68,7 +68,7 @@ lemma ext_δ (S T : Fin (2 * n.succ) → ℚ) (h1 : ∀ i, S (δ₁ i) = T (δ
rw [h3] at h2
exact h2
lemma sum_δ₁_δ₂ (S : Fin (2 * n.succ) → ℚ) :
lemma sum_δ₁_δ₂ (S : Fin (2 * n.succ) → ℚ) :
∑ i, S i = ∑ i : Fin n.succ, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
have h1 : ∑ i, S i = ∑ i : Fin (n.succ + n.succ), S (Fin.cast (split_equal n.succ) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (split_equal n.succ)).symm.toEquiv]
@ -80,7 +80,7 @@ lemma sum_δ₁_δ₂ (S : Fin (2 * n.succ) → ℚ) :
rw [Fin.sum_univ_add, Finset.sum_add_distrib]
rfl
lemma sum_δ₁_δ₂' (S : Fin (2 * n.succ) → ℚ) :
lemma sum_δ₁_δ₂' (S : Fin (2 * n.succ) → ℚ) :
∑ i, S i = ∑ i : Fin n.succ, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
have h1 : ∑ i, S i = ∑ i : Fin (n.succ + n.succ), S (Fin.cast (split_equal n.succ) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (split_equal n.succ)).symm.toEquiv]
@ -92,8 +92,8 @@ lemma sum_δ₁_δ₂' (S : Fin (2 * n.succ) → ℚ) :
rw [Fin.sum_univ_add, Finset.sum_add_distrib]
rfl
lemma sum_δ!₁_δ!₂ (S : Fin (2 * n.succ) → ℚ) :
∑ i, S i = S δ!₃ + S δ!₄ + ∑ i : Fin n, ((S ∘ δ!₁) i + (S ∘ δ!₂) i) := by
lemma sum_δ!₁_δ!₂ (S : Fin (2 * n.succ) → ℚ) :
∑ i, S i = S δ!₃ + S δ!₄ + ∑ i : Fin n, ((S ∘ δ!₁) i + (S ∘ δ!₂) i) := by
have h1 : ∑ i, S i = ∑ i : Fin (1 + ((n + n) + 1)), S (Fin.cast (n_cond₂ n) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (n_cond₂ n)).symm.toEquiv]
intro i
@ -180,12 +180,12 @@ lemma basis_on_δ₁_other {k j : Fin n.succ} (h : k ≠ j) :
omega
rfl
lemma basis_on_other {k : Fin n.succ} {j : Fin (2 * n.succ)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
lemma basis_on_other {k : Fin n.succ} {j : Fin (2 * n.succ)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
basisAsCharges k j = 0 := by
simp [basisAsCharges]
simp_all only [ne_eq, ↓reduceIte]
lemma basis!_on_other {k : Fin n} {j : Fin (2 * n.succ)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
lemma basis!_on_other {k : Fin n} {j : Fin (2 * n.succ)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
basis!AsCharges k j = 0 := by
simp [basis!AsCharges]
simp_all only [ne_eq, ↓reduceIte]
@ -338,11 +338,11 @@ def basisa : (Fin n.succ) ⊕ (Fin n) → (PureU1 (2 * n.succ)).LinSols := fun i
/-- Swapping the elements δ!₁ j and δ!₂ j is equivalent to adding a vector basis!AsCharges j. -/
lemma swap!_as_add {S S' : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
(hS : ((FamilyPermutations (2 * n.succ)).linSolRep
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
S'.val = S.val + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j := by
funext i
rw [← hS, FamilyPermutations_anomalyFreeLinear_apply]
by_cases hi : i = δ!₁ j
by_cases hi : i = δ!₁ j
subst hi
simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
by_cases hi2 : i = δ!₂ j
@ -350,7 +350,7 @@ lemma swap!_as_add {S S' : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
simp [HSMul.hSMul, basis!_on_δ!₂_self, pairSwap_inv_snd]
simp [HSMul.hSMul]
rw [basis!_on_other hi hi2]
change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
simp
@ -374,7 +374,7 @@ lemma P_δ₁ (f : Fin n.succ → ℚ) (j : Fin n.succ) : P f (δ₁ j) = f j :=
simp only [mul_zero]
simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
rw [P!, sum_of_charges]
simp [HSMul.hSMul, SMul.smul]
rw [Finset.sum_eq_single j]
@ -424,7 +424,7 @@ lemma P!_δ!₃ (f : Fin n → ℚ) : P! f (δ!₃) = 0 := by
rw [P!, sum_of_charges]
simp [HSMul.hSMul, SMul.smul, basis!_on_δ!₃]
lemma Pa_δ!₃ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₃) = f 0 := by
lemma Pa_δ!₃ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₃) = f 0 := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
rw [P!_δ!₃, δ!₃_δ₁0, P_δ₁]
@ -434,13 +434,13 @@ lemma P!_δ!₄ (f : Fin n → ℚ) : P! f (δ!₄) = 0 := by
rw [P!, sum_of_charges]
simp [HSMul.hSMul, SMul.smul, basis!_on_δ!₄]
lemma Pa_δ!₄ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₄) = - f (Fin.last n) := by
lemma Pa_δ!₄ (f : Fin n.succ → ℚ) (g : Fin n → ℚ) : Pa f g (δ!₄) = - f (Fin.last n) := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
rw [P!_δ!₄, δ!₄_δ₂Last, P_δ₂]
simp
lemma P_δ₁_δ₂ (f : Fin n.succ → ℚ) : P f ∘ δ₂ = - P f ∘ δ₁ := by
lemma P_δ₁_δ₂ (f : Fin n.succ → ℚ) : P f ∘ δ₂ = - P f ∘ δ₁ := by
funext j
simp only [PureU1_numberCharges, Function.comp_apply, Pi.neg_apply]
rw [P_δ₁, P_δ₂]
@ -484,7 +484,7 @@ lemma P_P_P!_accCube (g : Fin n.succ → ℚ) (j : Fin n) :
lemma P_P!_P!_accCube (g : Fin n → ℚ) (j : Fin n.succ) :
accCubeTriLinSymm (P! g) (P! g) (basisAsCharges j)
= (P! g (δ₁ j))^2 - (P! g (δ₂ j))^2 := by
= (P! g (δ₁ j))^2 - (P! g (δ₂ j))^2 := by
simp [accCubeTriLinSymm]
rw [sum_δ₁_δ₂]
simp only [Function.comp_apply]
@ -604,7 +604,7 @@ theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n) := by
simp_all
simp_all
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
apply Iff.intro
intro h
funext i
@ -625,7 +625,7 @@ lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n) → ℚ) : Pa' f = Pa' f' ↔ f =
/-! TODO: Replace the definition of `join` with a Mathlib definition, most likely `Sum.elim`. -/
/-- A helper function for what follows. -/
def join (g : Fin n.succ → ℚ) (f : Fin n → ℚ) : (Fin n.succ) ⊕ (Fin n) → ℚ := fun i =>
def join (g : Fin n.succ → ℚ) (f : Fin n → ℚ) : (Fin n.succ) ⊕ (Fin n) → ℚ := fun i =>
match i with
| .inl i => g i
| .inr i => f i
@ -661,7 +661,7 @@ lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n → ℚ) :
rw [← join_ext]
exact Pa'_eq _ _
lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n)) =
lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n)) =
FiniteDimensional.finrank ℚ (PureU1 (2 * n.succ)).LinSols := by
erw [BasisLinear.finrank_AnomalyFreeLinear]
simp only [Fintype.card_sum, Fintype.card_fin, mul_eq]
@ -673,7 +673,7 @@ noncomputable def basisaAsBasis :
basisOfLinearIndependentOfCardEqFinrank (@basisa_linear_independent n) basisa_card
lemma span_basis (S : (PureU1 (2 * n.succ)).LinSols) :
∃ (g : Fin n.succ → ℚ) (f : Fin n → ℚ), S.val = P g + P! f := by
∃ (g : Fin n.succ → ℚ) (f : Fin n → ℚ), S.val = P g + P! f := by
have h := (mem_span_range_iff_exists_fun ℚ).mp (Basis.mem_span basisaAsBasis S)
obtain ⟨f, hf⟩ := h
simp [basisaAsBasis] at hf
@ -706,7 +706,7 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ)).LinSols} (j : Fin n)
S'.val = P g' + P! f' ∧ P! f' = P! f +
(S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j ∧ g' = g := by
let X := P! f + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j
have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
apply Submodule.add_mem
exact (P!_in_span f)
exact (smul_basis!AsCharges_in_span S j)

32
HepLean/AnomalyCancellation/PureU1/Even/LineInCubic.lean
View file

@ -34,7 +34,7 @@ open BigOperators
variable {n : ℕ}
open VectorLikeEvenPlane
/-- A property on `LinSols`, satisfied if every point on the line between the two planes
/-- A property on `LinSols`, satisfied if every point on the line between the two planes
in the basis through that point is in the cubic. -/
def LineInCubic (S : (PureU1 (2 * n.succ)).LinSols) : Prop :=
∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ) ,
@ -61,18 +61,18 @@ lemma lineInCubic_expand {S : (PureU1 (2 * n.succ)).LinSols} (h : LineInCubic S)
then `accCubeTriLinSymm.toFun (P g, P g, P! f) = 0`.
-/
lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n.succ)).LinSols} (h : LineInCubic S) :
∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f),
∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f),
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
intro g f hS
linear_combination 2 / 3 * (lineInCubic_expand h g f hS 1 1) -
(lineInCubic_expand h g f hS 1 2) / 6
/-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
/-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
def LineInCubicPerm (S : (PureU1 (2 * n.succ)).LinSols) : Prop :=
∀ (M : (FamilyPermutations (2 * n.succ)).group ),
LineInCubic ((FamilyPermutations (2 * n.succ)).linSolRep M S)
/-- If `lineInCubicPerm S` then `lineInCubic S`. -/
/-- If `lineInCubicPerm S` then `lineInCubic S`. -/
lemma lineInCubicPerm_self {S : (PureU1 (2 * n.succ)).LinSols}
(hS : LineInCubicPerm S) : LineInCubic S := hS 1
@ -94,7 +94,7 @@ lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ)).LinSols}
(S.val (δ!₂ j) - S.val (δ!₁ j))
* accCubeTriLinSymm (P g) (P g) (basis!AsCharges j) = 0 := by
intro j g f h
let S' := (FamilyPermutations (2 * n.succ)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j)) S
let S' := (FamilyPermutations (2 * n.succ)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j)) S
have hSS' : ((FamilyPermutations (2 * n.succ)).linSolRep (pairSwap (δ!₁ j) (δ!₂ j))) S = S' := rfl
obtain ⟨g', f', hall⟩ := span_basis_swap! j hSS' g f h
have h1 := line_in_cubic_P_P_P! (lineInCubicPerm_self LIC) g f h
@ -106,21 +106,21 @@ lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ)).LinSols}
lemma P_P_P!_accCube' {S : (PureU1 (2 * n.succ.succ )).LinSols}
(f : Fin n.succ.succ → ℚ) (g : Fin n.succ → ℚ) (hS : S.val = Pa f g) :
accCubeTriLinSymm (P f) (P f) (basis!AsCharges (Fin.last n)) =
accCubeTriLinSymm (P f) (P f) (basis!AsCharges (Fin.last n)) =
- (S.val (δ!₂ (Fin.last n)) + S.val (δ!₁ (Fin.last n))) * (2 * S.val δ!₄ +
S.val (δ!₂ (Fin.last n)) + S.val (δ!₁ (Fin.last n))) := by
rw [P_P_P!_accCube f (Fin.last n)]
S.val (δ!₂ (Fin.last n)) + S.val (δ!₁ (Fin.last n))) := by
rw [P_P_P!_accCube f (Fin.last n)]
have h1 := Pa_δ!₄ f g
have h2 := Pa_δ!₁ f g (Fin.last n)
have h3 := Pa_δ!₂ f g (Fin.last n)
simp at h1 h2 h3
have hl : f (Fin.succ (Fin.last (n ))) = - Pa f g δ!₄ := by
have hl : f (Fin.succ (Fin.last (n ))) = - Pa f g δ!₄ := by
simp_all only [Fin.succ_last, neg_neg]
erw [hl] at h2
have hg : g (Fin.last n) = Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄ := by
have hg : g (Fin.last n) = Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄ := by
linear_combination -(1 * h2)
have hll : f (Fin.castSucc (Fin.last (n ))) =
- (Pa f g (δ!₂ (Fin.last n)) + Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄) := by
have hll : f (Fin.castSucc (Fin.last (n ))) =
- (Pa f g (δ!₂ (Fin.last n)) + Pa f g (δ!₁ (Fin.last n)) + Pa f g δ!₄) := by
linear_combination h3 - 1 * hg
rw [← hS] at hl hll
rw [hl, hll]
@ -145,7 +145,7 @@ lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ)).LinSols}
apply Or.inr
exact h1
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ)).LinSols}
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ)).LinSols}
(LIC : LineInCubicPerm S) : LineInPlaneCond S := by
refine @Prop_three (2 * n.succ.succ) LineInPlaneProp S (δ!₂ (Fin.last n)) (δ!₁ (Fin.last n))
δ!₄ ?_ ?_ ?_ ?_
@ -156,15 +156,15 @@ lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ)).LinSols}
intro M
exact lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ)).Sols}
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ)).Sols}
(LIC : LineInCubicPerm S.1.1) : ConstAbs S.val :=
linesInPlane_constAbs_AF S (lineInCubicPerm_last_perm LIC)
theorem lineInCubicPerm_vectorLike {S : (PureU1 (2 * n.succ.succ)).Sols}
theorem lineInCubicPerm_vectorLike {S : (PureU1 (2 * n.succ.succ)).Sols}
(LIC : LineInCubicPerm S.1.1) : VectorLikeEven S.val :=
ConstAbs.boundary_value_even S.1.1 (lineInCubicPerm_constAbs LIC)
theorem lineInCubicPerm_in_plane (S : (PureU1 (2 * n.succ.succ)).Sols)
theorem lineInCubicPerm_in_plane (S : (PureU1 (2 * n.succ.succ)).Sols)
(LIC : LineInCubicPerm S.1.1) : ∃ (M : (FamilyPermutations (2 * n.succ.succ)).group),
(FamilyPermutations (2 * n.succ.succ)).linSolRep M S.1.1
∈ Submodule.span ℚ (Set.range basis) :=

18
HepLean/AnomalyCancellation/PureU1/Even/Parameterization.lean
View file

@ -14,7 +14,7 @@ import Mathlib.Tactic.Polyrith
/-!
# Parameterization in even case
Given maps `g : Fin n.succ → ℚ`, `f : Fin n → ℚ` and `a : ℚ` we form a solution to the anomaly
Given maps `g : Fin n.succ → ℚ`, `f : Fin n → ℚ` and `a : ℚ` we form a solution to the anomaly
equations. We show that every solution can be got in this way, up to permutation, unless it, up to
permutation, lives in the plane spanned by the first part of the basis vector.
@ -60,7 +60,7 @@ lemma parameterizationCharge_cube (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (
accCubeTriLinSymm.map_smul₃]
ring
/-- The construction of a `Sol` from a `Fin n.succ → ℚ`, a `Fin n → ℚ` and a `ℚ`. -/
/-- The construction of a `Sol` from a `Fin n.succ → ℚ`, a `Fin n → ℚ` and a `ℚ`. -/
def parameterization (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (a : ℚ) :
(PureU1 (2 * n.succ)).Sols :=
⟨⟨parameterizationAsLinear g f a, by intro i; simp at i; exact Fin.elim0 i⟩,
@ -77,15 +77,15 @@ lemma anomalyFree_param {S : (PureU1 (2 * n.succ)).Sols}
erw [P!_accCube] at hC
linear_combination hC / 3
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
In this case our parameterization above will be able to recover this point. -/
def GenericCase (S : (PureU1 (2 * n.succ)).Sols) : Prop :=
∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0
lemma genericCase_exists (S : (PureU1 (2 * n.succ)).Sols)
(hs : ∃ (g : Fin n.succ → ℚ) (f : Fin n → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : GenericCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : GenericCase S := by
intro g f hS hC
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -94,13 +94,13 @@ lemma genericCase_exists (S : (PureU1 (2 * n.succ)).Sols)
exact hC' hC
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) = 0`.-/
def SpecialCase (S : (PureU1 (2 * n.succ)).Sols) : Prop :=
def SpecialCase (S : (PureU1 (2 * n.succ)).Sols) : Prop :=
∀ (g : Fin n.succ → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g) (P g) (P! f) = 0
lemma specialCase_exists (S : (PureU1 (2 * n.succ)).Sols)
(hs : ∃ (g : Fin n.succ → ℚ) (f : Fin n → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g) (P g) (P! f) = 0) : SpecialCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) = 0) : SpecialCase S := by
intro g f hS
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -111,7 +111,7 @@ lemma specialCase_exists (S : (PureU1 (2 * n.succ)).Sols)
lemma generic_or_special (S : (PureU1 (2 * n.succ)).Sols) :
GenericCase S ∨ SpecialCase S := by
obtain ⟨g, f, h⟩ := span_basis S.1.1
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
exact ne_or_eq _ _
cases h1 <;> rename_i h1
@ -119,7 +119,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ)).Sols) :
exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
theorem generic_case {S : (PureU1 (2 * n.succ)).Sols} (h : GenericCase S) :
∃ g f a, S = parameterization g f a := by
∃ g f a, S = parameterization g f a := by
obtain ⟨g, f, hS⟩ := span_basis S.1.1
use g, f, (accCubeTriLinSymm (P! f) (P! f) (P g))⁻¹
rw [parameterization]

16
HepLean/AnomalyCancellation/PureU1/LineInPlaneCond.lean
View file

@ -14,7 +14,7 @@ import Mathlib.RepresentationTheory.Basic
We say a `LinSol` satisfies the `line in plane` condition if for all distinct `i1`, `i2`, `i3` in
`Fin n`, we have
`S i1 = S i2` or `S i1 = - S i2` or `2 S i3 + S i1 + S i2 = 0`.
`S i1 = S i2` or `S i1 = - S i2` or `2 S i3 + S i1 + S i2 = 0`.
We look at various consequences of this.
The main reference for this material is
@ -60,7 +60,7 @@ lemma lineInPlaneCond_eq_last' {S : (PureU1 (n.succ.succ)).LinSols} (hS : LineIn
simp only [LineInPlaneProp] at hS
simp [not_or] at h
have h1 (i : Fin n) : S.val i.castSucc.castSucc =
- (S.val ((Fin.last n).castSucc) + (S.val ((Fin.last n).succ))) / 2 := by
- (S.val ((Fin.last n).castSucc) + (S.val ((Fin.last n).succ))) / 2 := by
have h1S := hS (Fin.last n).castSucc ((Fin.last n).succ) i.castSucc.castSucc
(by simp; rw [Fin.ext_iff]; simp)
(by simp; rw [Fin.ext_iff]; simp; omega)
@ -82,14 +82,14 @@ lemma lineInPlaneCond_eq_last {S : (PureU1 (n.succ.succ.succ.succ.succ)).LinSols
have h := lineInPlaneCond_eq_last' hS hn
rw [sq_eq_sq_iff_eq_or_eq_neg] at hn
simp [or_not] at hn
have hx : ((2 : ℚ) - ↑(n + 3)) ≠ 0 := by
have hx : ((2 : ℚ) - ↑(n + 3)) ≠ 0 := by
rw [Nat.cast_add]
simp only [Nat.cast_ofNat, ne_eq]
apply Not.intro
intro a
linarith
have ht : S.val ((Fin.last n.succ.succ.succ).succ) =
- S.val ((Fin.last n.succ.succ.succ).castSucc) := by
have ht : S.val ((Fin.last n.succ.succ.succ).succ) =
- S.val ((Fin.last n.succ.succ.succ).castSucc) := by
rw [← mul_right_inj' hx]
simp [Nat.succ_eq_add_one]
simp at h
@ -115,7 +115,7 @@ theorem linesInPlane_constAbs {S : (PureU1 (n.succ.succ.succ.succ.succ)).LinSols
rw [hij]
lemma linesInPlane_four (S : (PureU1 4).Sols) (hS : LineInPlaneCond S.1.1) :
ConstAbsProp (S.val (0 : Fin 4), S.val (1 : Fin 4)) := by
ConstAbsProp (S.val (0 : Fin 4), S.val (1 : Fin 4)) := by
simp [ConstAbsProp]
by_contra hn
have hLin := pureU1_linear S.1.1
@ -128,7 +128,7 @@ lemma linesInPlane_four (S : (PureU1 4).Sols) (hS : LineInPlaneCond S.1.1) :
have l013 := hS 0 1 3 (by simp) (by simp) (by simp)
have l023 := hS 0 2 3 (by simp) (by simp) (by simp)
simp_all [LineInPlaneProp]
have h1 : S.val (2 : Fin 4) = S.val (3 : Fin 4) := by
have h1 : S.val (2 : Fin 4) = S.val (3 : Fin 4) := by
linear_combination l012 / 2 + -1 * l013 / 2
by_cases h2 : S.val (0 : Fin 4) = S.val (2 : Fin 4)
simp_all
@ -161,7 +161,7 @@ lemma linesInPlane_eq_sq_four {S : (PureU1 4).Sols}
refine Prop_two ConstAbsProp (by simp : (0 : Fin 4) ≠ 1) ?_
intro M
let S' := (FamilyPermutations 4).solAction.toFun S M
have hS' : LineInPlaneCond S'.1.1 :=
have hS' : LineInPlaneCond S'.1.1 :=
(lineInPlaneCond_perm hS M)
exact linesInPlane_four S' hS'

2
HepLean/AnomalyCancellation/PureU1/LowDim/One.lean
View file

@ -13,7 +13,7 @@ We show that in this case the charge must be zero.
universe v u
open Nat
open Finset
open Finset
namespace PureU1

2
HepLean/AnomalyCancellation/PureU1/LowDim/Three.lean
View file

@ -14,7 +14,7 @@ We define a surjective map from `LinSols` with a charge equal to zero to `Sols`.
universe v u
open Nat
open Finset
open Finset
namespace PureU1

2
HepLean/AnomalyCancellation/PureU1/LowDim/Two.lean
View file

@ -13,7 +13,7 @@ We define an equivalence between `LinSols` and `Sols`.
universe v u
open Nat
open Finset
open Finset
namespace PureU1

34
HepLean/AnomalyCancellation/PureU1/Odd/BasisLinear.lean
View file

@ -16,7 +16,7 @@ that splits into two planes on which every point is a solution to the ACCs.
universe v u
open Nat
open Finset
open Finset
open BigOperators
namespace PureU1
@ -111,7 +111,7 @@ lemma δ₂_δ!₂ (j : Fin n) : δ₂ j = δ!₂ j := by
simp [δ₂, δ!₂]
omega
lemma sum_δ (S : Fin (2 * n + 1) → ℚ) :
lemma sum_δ (S : Fin (2 * n + 1) → ℚ) :
∑ i, S i = S δ₃ + ∑ i : Fin n, ((S ∘ δ₁) i + (S ∘ δ₂) i) := by
have h1 : ∑ i, S i = ∑ i : Fin (n + 1 + n), S (Fin.cast (split_odd n) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (split_odd n)).symm.toEquiv]
@ -127,7 +127,7 @@ lemma sum_δ (S : Fin (2 * n + 1) → ℚ) :
rw [Finset.sum_add_distrib]
rfl
lemma sum_δ! (S : Fin (2 * n + 1) → ℚ) :
lemma sum_δ! (S : Fin (2 * n + 1) → ℚ) :
∑ i, S i = S δ!₃ + ∑ i : Fin n, ((S ∘ δ!₁) i + (S ∘ δ!₂) i) := by
have h1 : ∑ i, S i = ∑ i : Fin ((1+n)+n), S (Fin.cast (split_odd! n) i) := by
rw [Finset.sum_equiv (Fin.castOrderIso (split_odd! n)).symm.toEquiv]
@ -210,12 +210,12 @@ lemma basis!_on_δ!₁_other {k j : Fin n} (h : k ≠ j) :
omega
rfl
lemma basis_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
lemma basis_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ₁ k) (h2 : j ≠ δ₂ k) :
basisAsCharges k j = 0 := by
simp [basisAsCharges]
simp_all only [ne_eq, ↓reduceIte]
lemma basis!_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
lemma basis!_on_other {k : Fin n} {j : Fin (2 * n + 1)} (h1 : j ≠ δ!₁ k) (h2 : j ≠ δ!₂ k) :
basis!AsCharges k j = 0 := by
simp [basis!AsCharges]
simp_all only [ne_eq, ↓reduceIte]
@ -333,11 +333,11 @@ end theBasisVectors
/-- Swapping the elements δ!₁ j and δ!₂ j is equivalent to adding a vector basis!AsCharges j. -/
lemma swap!_as_add {S S' : (PureU1 (2 * n + 1)).LinSols} (j : Fin n)
(hS : ((FamilyPermutations (2 * n + 1)).linSolRep
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
(pairSwap (δ!₁ j) (δ!₂ j))) S = S') :
S'.val = S.val + (S.val (δ!₂ j) - S.val (δ!₁ j)) • basis!AsCharges j := by
funext i
rw [← hS, FamilyPermutations_anomalyFreeLinear_apply]
by_cases hi : i = δ!₁ j
by_cases hi : i = δ!₁ j
subst hi
simp [HSMul.hSMul, basis!_on_δ!₁_self, pairSwap_inv_fst]
by_cases hi2 : i = δ!₂ j
@ -345,7 +345,7 @@ lemma swap!_as_add {S S' : (PureU1 (2 * n + 1)).LinSols} (j : Fin n)
simp [HSMul.hSMul,basis!_on_δ!₂_self, pairSwap_inv_snd]
simp [HSMul.hSMul]
rw [basis!_on_other hi hi2]
change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
change S.val ((pairSwap (δ!₁ j) (δ!₂ j)).invFun i) =_
erw [pairSwap_inv_other (Ne.symm hi) (Ne.symm hi2)]
simp
@ -369,7 +369,7 @@ lemma P_δ₁ (f : Fin n → ℚ) (j : Fin n) : P f (δ₁ j) = f j := by
simp only [mul_zero]
simp only [mem_univ, not_true_eq_false, _root_.mul_eq_zero, IsEmpty.forall_iff]
lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
lemma P!_δ!₁ (f : Fin n → ℚ) (j : Fin n) : P! f (δ!₁ j) = f j := by
rw [P!, sum_of_charges]
simp [HSMul.hSMul, SMul.smul]
rw [Finset.sum_eq_single j]
@ -418,14 +418,14 @@ lemma Pa_δa₁ (f g : Fin n.succ → ℚ) : Pa f g δa₁ = f 0 := by
rw [P_δ₁, P!_δ!₃]
simp
lemma Pa_δa₂ (f g : Fin n.succ → ℚ) (j : Fin n) : Pa f g (δa₂ j) = f j.succ + g j.castSucc := by
lemma Pa_δa₂ (f g : Fin n.succ → ℚ) (j : Fin n) : Pa f g (δa₂ j) = f j.succ + g j.castSucc := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
nth_rewrite 1 [δa₂_δ₁]
rw [δa₂_δ!₁]
rw [P_δ₁, P!_δ!₁]
lemma Pa_δa₃ (f g : Fin n.succ → ℚ) : Pa f g (δa₃) = g (Fin.last n) := by
lemma Pa_δa₃ (f g : Fin n.succ → ℚ) : Pa f g (δa₃) = g (Fin.last n) := by
rw [Pa]
simp only [ACCSystemCharges.chargesAddCommMonoid_add]
nth_rewrite 1 [δa₃_δ₃]
@ -495,7 +495,7 @@ lemma P!_zero (f : Fin n → ℚ) (h : P! f = 0) : ∀ i, f i = 0 := by
rw [h]
rfl
lemma Pa_zero (f g : Fin n.succ → ℚ) (h : Pa f g = 0) :
lemma Pa_zero (f g : Fin n.succ → ℚ) (h : Pa f g = 0) :
∀ i, f i = 0 := by
have h₃ := Pa_δa₁ f g
rw [h] at h₃
@ -589,7 +589,7 @@ theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n.succ) := by
simp_all
simp_all
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n.succ) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n.succ) → ℚ) : Pa' f = Pa' f' ↔ f = f' := by
apply Iff.intro
intro h
funext i
@ -610,7 +610,7 @@ lemma Pa'_eq (f f' : (Fin n.succ) ⊕ (Fin n.succ) → ℚ) : Pa' f = Pa' f'
/-! TODO: Replace the definition of `join` with a Mathlib definition, most likely `Sum.elim`. -/
/-- A helper function for what follows. -/
def join (g f : Fin n → ℚ) : Fin n ⊕ Fin n → ℚ := fun i =>
def join (g f : Fin n → ℚ) : Fin n ⊕ Fin n → ℚ := fun i =>
match i with
| .inl i => g i
| .inr i => f i
@ -646,7 +646,7 @@ lemma Pa_eq (g g' : Fin n.succ → ℚ) (f f' : Fin n.succ → ℚ) :
rw [← join_ext]
exact Pa'_eq _ _
lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n.succ)) =
lemma basisa_card : Fintype.card ((Fin n.succ) ⊕ (Fin n.succ)) =
FiniteDimensional.finrank ℚ (PureU1 (2 * n.succ + 1)).LinSols := by
erw [BasisLinear.finrank_AnomalyFreeLinear]
simp only [Fintype.card_sum, Fintype.card_fin]
@ -658,7 +658,7 @@ noncomputable def basisaAsBasis :
basisOfLinearIndependentOfCardEqFinrank (@basisa_linear_independent n) basisa_card
lemma span_basis (S : (PureU1 (2 * n.succ + 1)).LinSols) :
∃ (g f : Fin n.succ → ℚ) , S.val = P g + P! f := by
∃ (g f : Fin n.succ → ℚ) , S.val = P g + P! f := by
have h := (mem_span_range_iff_exists_fun ℚ).mp (Basis.mem_span basisaAsBasis S)
obtain ⟨f, hf⟩ := h
simp [basisaAsBasis] at hf
@ -686,7 +686,7 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ + 1)).LinSols} (j : Fin n.succ)
apply SetLike.mem_of_subset
apply Submodule.subset_span
simp_all only [Set.mem_range, exists_apply_eq_apply]
have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
have hX : X ∈ Submodule.span ℚ (Set.range (basis!AsCharges)) := by
apply Submodule.add_mem
exact hf
exact hP

22
HepLean/AnomalyCancellation/PureU1/Odd/LineInCubic.lean
View file

@ -34,10 +34,10 @@ open BigOperators
variable {n : ℕ}
open VectorLikeOddPlane
/-- A property on `LinSols`, satisfied if every point on the line between the two planes
/-- A property on `LinSols`, satisfied if every point on the line between the two planes
in the basis through that point is in the cubic. -/
def LineInCubic (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
∀ (g f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ) ,
∀ (g f : Fin n → ℚ) (_ : S.val = Pa g f) (a b : ℚ) ,
accCube (2 * n + 1) (a • P g + b • P! f) = 0
lemma lineInCubic_expand {S : (PureU1 (2 * n + 1)).LinSols} (h : LineInCubic S) :
@ -55,18 +55,18 @@ lemma lineInCubic_expand {S : (PureU1 (2 * n + 1)).LinSols} (h : LineInCubic S)
ring
lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n + 1)).LinSols} (h : LineInCubic S) :
∀ (g : Fin n → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f),
∀ (g : Fin n → ℚ) (f : Fin n → ℚ) (_ : S.val = P g + P! f),
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
intro g f hS
linear_combination 2 / 3 * (lineInCubic_expand h g f hS 1 1 ) -
(lineInCubic_expand h g f hS 1 2 ) / 6
/-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
/-- We say a `LinSol` satisfies `lineInCubicPerm` if all its permutations satisfy `lineInCubic`. -/
def LineInCubicPerm (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
∀ (M : (FamilyPermutations (2 * n + 1)).group ),
LineInCubic ((FamilyPermutations (2 * n + 1)).linSolRep M S)
/-- If `lineInCubicPerm S` then `lineInCubic S`. -/
/-- If `lineInCubicPerm S` then `lineInCubic S`. -/
lemma lineInCubicPerm_self {S : (PureU1 (2 * n + 1)).LinSols} (hS : LineInCubicPerm S) :
LineInCubic S := hS 1
@ -89,7 +89,7 @@ lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ + 1)).LinSols}
(S.val (δ!₂ j) - S.val (δ!₁ j))
* accCubeTriLinSymm (P g) (P g) (basis!AsCharges j) = 0 := by
intro j g f h
let S' := (FamilyPermutations (2 * n.succ + 1)).linSolRep
let S' := (FamilyPermutations (2 * n.succ + 1)).linSolRep
(pairSwap (δ!₁ j) (δ!₂ j)) S
have hSS' : ((FamilyPermutations (2 * n.succ + 1)).linSolRep
(pairSwap (δ!₁ j) (δ!₂ j))) S = S' := rfl
@ -119,7 +119,7 @@ lemma P_P_P!_accCube' {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
have h2 := Pa_δa₂ f g 0
rw [← hS] at h1 h2 h4
simp at h2
have h5 : f 1 = S.val (δa₂ 0) + S.val δa₁ + S.val (δa₄ 0):= by
have h5 : f 1 = S.val (δa₂ 0) + S.val δa₁ + S.val (δa₄ 0):= by
linear_combination -(1 * h1) - 1 * h4 - 1 * h2
rw [h5]
rw [δa₄_δ!₂]
@ -133,7 +133,7 @@ lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ+1)).LinSols}
(LIC : LineInCubicPerm S) :
LineInPlaneProp ((S.val (δ!₂ 0)), ((S.val (δ!₁ 0)), (S.val δ!₃))) := by
obtain ⟨g, f, hfg⟩ := span_basis S
have h1 := lineInCubicPerm_swap LIC 0 g f hfg
have h1 := lineInCubicPerm_swap LIC 0 g f hfg
rw [P_P_P!_accCube' g f hfg] at h1
simp at h1
cases h1 <;> rename_i h1
@ -147,7 +147,7 @@ lemma lineInCubicPerm_last_cond {S : (PureU1 (2 * n.succ.succ+1)).LinSols}
apply Or.inr
linear_combination h1
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
(LIC : LineInCubicPerm S) : LineInPlaneCond S := by
refine @Prop_three (2 * n.succ.succ + 1) LineInPlaneProp S (δ!₂ 0) (δ!₁ 0)
δ!₃ ?_ ?_ ?_ ?_
@ -157,11 +157,11 @@ lemma lineInCubicPerm_last_perm {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
intro M
exact lineInCubicPerm_last_cond (lineInCubicPerm_permute LIC M)
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
lemma lineInCubicPerm_constAbs {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
(LIC : LineInCubicPerm S) : ConstAbs S.val :=
linesInPlane_constAbs (lineInCubicPerm_last_perm LIC)
theorem lineInCubicPerm_zero {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
theorem lineInCubicPerm_zero {S : (PureU1 (2 * n.succ.succ + 1)).LinSols}
(LIC : LineInCubicPerm S) : S = 0 :=
ConstAbs.boundary_value_odd S (lineInCubicPerm_constAbs LIC)

28
HepLean/AnomalyCancellation/PureU1/Odd/Parameterization.lean
View file

@ -14,7 +14,7 @@ import Mathlib.RepresentationTheory.Basic
/-!
# Parameterization in odd case
Given maps `g : Fin n → ℚ`, `f : Fin n → ℚ` and `a : ℚ` we form a solution to the anomaly
Given maps `g : Fin n → ℚ`, `f : Fin n → ℚ` and `a : ℚ` we form a solution to the anomaly
equations. We show that every solution can be got in this way, up to permutation, unless it is zero.
The main reference is:
@ -31,7 +31,7 @@ open VectorLikeOddPlane
/-- Given a `g f : Fin n → ℚ` and a `a : ℚ` we form a linear solution. We will later
show that this can be extended to a complete solution. -/
def parameterizationAsLinear (g f : Fin n → ℚ) (a : ℚ) :
def parameterizationAsLinear (g f : Fin n → ℚ) (a : ℚ) :
(PureU1 (2 * n + 1)).LinSols :=
a • ((accCubeTriLinSymm (P! f) (P! f) (P g)) • P' g +
(- accCubeTriLinSymm (P g) (P g) (P! f)) • P!' f)
@ -44,7 +44,7 @@ lemma parameterizationAsLinear_val (g f : Fin n → ℚ) (a : ℚ) :
change a • (_ • (P' g).val + _ • (P!' f).val) = _
rw [P'_val, P!'_val]
lemma parameterizationCharge_cube (g f : Fin n → ℚ) (a : ℚ):
lemma parameterizationCharge_cube (g f : Fin n → ℚ) (a : ℚ):
(accCube (2 * n + 1)) (parameterizationAsLinear g f a).val = 0 := by
change accCubeTriLinSymm.toCubic _ = 0
rw [parameterizationAsLinear_val]
@ -64,7 +64,7 @@ def parameterization (g f : Fin n → ℚ) (a : ℚ) :
parameterizationCharge_cube g f a⟩
lemma anomalyFree_param {S : (PureU1 (2 * n + 1)).Sols}
(g f : Fin n → ℚ) (hS : S.val = P g + P! f) :
(g f : Fin n → ℚ) (hS : S.val = P g + P! f) :
accCubeTriLinSymm (P g) (P g) (P! f) =
- accCubeTriLinSymm (P! f) (P! f) (P g) := by
have hC := S.cubicSol
@ -75,15 +75,15 @@ lemma anomalyFree_param {S : (PureU1 (2 * n + 1)).Sols}
erw [P!_accCube] at hC
linear_combination hC / 3
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
In this case our parameterization above will be able to recover this point. -/
def GenericCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
∀ (g f : Fin n.succ → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0
∀ (g f : Fin n.succ → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0
lemma genericCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
(hs : ∃ (g f : Fin n.succ → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : GenericCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0) : GenericCase S := by
intro g f hS hC
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -91,15 +91,15 @@ lemma genericCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
rw [hS'.1, hS'.2] at hC
exact hC' hC
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
/-- A proposition on a solution which is true if `accCubeTriLinSymm (P g, P g, P! f) ≠ 0`.
In this case we will show that S is zero if it is true for all permutations. -/
def SpecialCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
def SpecialCase (S : (PureU1 (2 * n.succ + 1)).Sols) : Prop :=
∀ (g f : Fin n.succ → ℚ) (_ : S.val = P g + P! f) ,
accCubeTriLinSymm (P g) (P g) (P! f) = 0
lemma specialCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
(hs : ∃ (g f : Fin n.succ → ℚ), S.val = P g + P! f ∧
accCubeTriLinSymm (P g) (P g) (P! f) = 0) : SpecialCase S := by
accCubeTriLinSymm (P g) (P g) (P! f) = 0) : SpecialCase S := by
intro g f hS
obtain ⟨g', f', hS', hC'⟩ := hs
rw [hS] at hS'
@ -110,7 +110,7 @@ lemma specialCase_exists (S : (PureU1 (2 * n.succ + 1)).Sols)
lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
GenericCase S ∨ SpecialCase S := by
obtain ⟨g, f, h⟩ := span_basis S.1.1
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
have h1 : accCubeTriLinSymm (P g) (P g) (P! f) ≠ 0 ∨
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
exact ne_or_eq _ _
cases h1 <;> rename_i h1
@ -118,7 +118,7 @@ lemma generic_or_special (S : (PureU1 (2 * n.succ + 1)).Sols) :
exact Or.inr (specialCase_exists S ⟨g, f, h, h1⟩)
theorem generic_case {S : (PureU1 (2 * n.succ + 1)).Sols} (h : GenericCase S) :
∃ g f a, S = parameterization g f a := by
∃ g f a, S = parameterization g f a := by
obtain ⟨g, f, hS⟩ := span_basis S.1.1
use g, f, (accCubeTriLinSymm (P! f) (P! f) (P g))⁻¹
rw [parameterization]
@ -164,7 +164,7 @@ theorem special_case {S : (PureU1 (2 * n.succ.succ + 1)).Sols}
(h : ∀ (M : (FamilyPermutations (2 * n.succ.succ + 1)).group),
SpecialCase ((FamilyPermutations (2 * n.succ.succ + 1)).solAction.toFun S M)) :
S.1.1 = 0 := by
have ht := special_case_lineInCubic_perm h
have ht := special_case_lineInCubic_perm h
exact lineInCubicPerm_zero ht
end Odd
end PureU1

44
HepLean/AnomalyCancellation/PureU1/Permutations.lean
View file

@ -18,13 +18,13 @@ We further define the action on the ACC System.
universe v u
open Nat
open Finset
open Finset
namespace PureU1
/-- The permutation group of the n-fermions. -/
@[simp]
def PermGroup (n : ℕ) := Equiv.Perm (Fin n)
def PermGroup (n : ℕ) := Equiv.Perm (Fin n)
instance {n : ℕ} : Group (PermGroup n) := by
simp [PermGroup]
@ -35,7 +35,7 @@ section Charges
/-- The image of an element of `permGroup` under the representation on charges. -/
@[simps!]
def chargeMap {n : ℕ} (f : PermGroup n) :
(PureU1 n).Charges →ₗ[ℚ] (PureU1 n).Charges where
(PureU1 n).Charges →ₗ[ℚ] (PureU1 n).Charges where
toFun S := S ∘ f.toFun
map_add' S T := by
funext i
@ -73,7 +73,7 @@ open BigOperators
lemma accCube_invariant {n : ℕ} (f : (PermGroup n)) (S : (PureU1 n).Charges) :
accCube n (permCharges f S) = accCube n S := by
rw [accCube_explicit, accCube_explicit]
change ∑ i : Fin n, ((((fun a => a^3) ∘ S) (f.symm i))) = _
change ∑ i : Fin n, ((((fun a => a^3) ∘ S) (f.symm i))) = _
apply (Equiv.Perm.sum_comp _ _ _ ?_)
simp
@ -130,22 +130,22 @@ def pairSwap {n : ℕ} (i j : Fin n) : (FamilyPermutations n).group where
right_inv s := by
aesop
lemma pairSwap_self_inv {n : ℕ} (i j : Fin n) : (pairSwap i j)⁻¹ = (pairSwap i j) := by
lemma pairSwap_self_inv {n : ℕ} (i j : Fin n) : (pairSwap i j)⁻¹ = (pairSwap i j) := by
rfl
lemma pairSwap_fst {n : ℕ} (i j : Fin n) : (pairSwap i j).toFun i = j := by
lemma pairSwap_fst {n : ℕ} (i j : Fin n) : (pairSwap i j).toFun i = j := by
simp [pairSwap]
lemma pairSwap_snd {n : ℕ} (i j : Fin n) : (pairSwap i j).toFun j = i := by
lemma pairSwap_snd {n : ℕ} (i j : Fin n) : (pairSwap i j).toFun j = i := by
simp [pairSwap]
lemma pairSwap_inv_fst {n : ℕ} (i j : Fin n) : (pairSwap i j).invFun i = j := by
lemma pairSwap_inv_fst {n : ℕ} (i j : Fin n) : (pairSwap i j).invFun i = j := by
simp [pairSwap]
lemma pairSwap_inv_snd {n : ℕ} (i j : Fin n) : (pairSwap i j).invFun j = i := by
lemma pairSwap_inv_snd {n : ℕ} (i j : Fin n) : (pairSwap i j).invFun j = i := by
simp [pairSwap]
lemma pairSwap_other {n : ℕ} (i j k : Fin n) (hik : i ≠ k) (hjk : j ≠ k) :
lemma pairSwap_other {n : ℕ} (i j k : Fin n) (hik : i ≠ k) (hjk : j ≠ k) :
(pairSwap i j).toFun k = k := by
simp [pairSwap]
split
@ -154,7 +154,7 @@ lemma pairSwap_other {n : ℕ} (i j k : Fin n) (hik : i ≠ k) (hjk : j ≠ k)
simp_all
rfl
lemma pairSwap_inv_other {n : ℕ} {i j k : Fin n} (hik : i ≠ k) (hjk : j ≠ k) :
lemma pairSwap_inv_other {n : ℕ} {i j k : Fin n} (hik : i ≠ k) (hjk : j ≠ k) :
(pairSwap i j).invFun k = k := by
simp [pairSwap]
split
@ -164,12 +164,12 @@ lemma pairSwap_inv_other {n : ℕ} {i j k : Fin n} (hik : i ≠ k) (hjk : j
rfl
/-- A permutation of fermions which takes one ordered subset into another. -/
noncomputable def permOfInjection (f g : Fin m ↪ Fin n) : (FamilyPermutations n).group :=
noncomputable def permOfInjection (f g : Fin m ↪ Fin n) : (FamilyPermutations n).group :=
Equiv.extendSubtype (g.toEquivRange.symm.trans f.toEquivRange)
section permTwo
variable {i j i' j' : Fin n} (hij : i ≠ j) (hij' : i' ≠ j')
variable {i j i' j' : Fin n} (hij : i ≠ j) (hij' : i' ≠ j')
/-- Given two distinct elements, an embedding of `Fin 2` into `Fin n`. -/
def permTwoInj : Fin 2 ↪ Fin n where
@ -185,7 +185,7 @@ lemma permTwoInj_fst : i ∈ Set.range ⇑(permTwoInj hij) := by
rfl
lemma permTwoInj_fst_apply :
(Function.Embedding.toEquivRange (permTwoInj hij)).symm ⟨i, permTwoInj_fst hij⟩ = 0 := by
(Function.Embedding.toEquivRange (permTwoInj hij)).symm ⟨i, permTwoInj_fst hij⟩ = 0 := by
exact (Equiv.symm_apply_eq (Function.Embedding.toEquivRange (permTwoInj hij))).mpr rfl
lemma permTwoInj_snd : j ∈ Set.range ⇑(permTwoInj hij) := by
@ -195,12 +195,12 @@ lemma permTwoInj_snd : j ∈ Set.range ⇑(permTwoInj hij) := by
lemma permTwoInj_snd_apply :
(Function.Embedding.toEquivRange (permTwoInj hij)).symm
⟨j, permTwoInj_snd hij⟩ = 1 := by
⟨j, permTwoInj_snd hij⟩ = 1 := by
exact (Equiv.symm_apply_eq (Function.Embedding.toEquivRange (permTwoInj hij))).mpr rfl
/-- A permutation which swaps `i` with `i'` and `j` with `j'`. -/
noncomputable def permTwo : (FamilyPermutations n).group :=
permOfInjection (permTwoInj hij) (permTwoInj hij')
permOfInjection (permTwoInj hij) (permTwoInj hij')
lemma permTwo_fst : (permTwo hij hij').toFun i' = i := by
rw [permTwo, permOfInjection]
@ -243,7 +243,7 @@ lemma permThreeInj_fst : i ∈ Set.range ⇑(permThreeInj hij hjk hik) := by
lemma permThreeInj_fst_apply :
(Function.Embedding.toEquivRange (permThreeInj hij hjk hik)).symm
⟨i, permThreeInj_fst hij hjk hik⟩ = 0 := by
⟨i, permThreeInj_fst hij hjk hik⟩ = 0 := by
exact (Equiv.symm_apply_eq (Function.Embedding.toEquivRange (permThreeInj hij hjk hik))).mpr rfl
lemma permThreeInj_snd : j ∈ Set.range ⇑(permThreeInj hij hjk hik) := by
@ -253,7 +253,7 @@ lemma permThreeInj_snd : j ∈ Set.range ⇑(permThreeInj hij hjk hik) := by
lemma permThreeInj_snd_apply :
(Function.Embedding.toEquivRange (permThreeInj hij hjk hik)).symm
⟨j, permThreeInj_snd hij hjk hik⟩ = 1 := by
⟨j, permThreeInj_snd hij hjk hik⟩ = 1 := by
exact (Equiv.symm_apply_eq (Function.Embedding.toEquivRange (permThreeInj hij hjk hik))).mpr rfl
lemma permThreeInj_thd : k ∈ Set.range ⇑(permThreeInj hij hjk hik) := by
@ -268,7 +268,7 @@ lemma permThreeInj_thd_apply :
/-- A permutation which swaps three distinct elements with another three. -/
noncomputable def permThree : (FamilyPermutations n).group :=
permOfInjection (permThreeInj hij hjk hik) (permThreeInj hij' hjk' hik')
permOfInjection (permThreeInj hij hjk hik) (permThreeInj hij' hjk' hik')
lemma permThree_fst : (permThree hij hjk hik hij' hjk' hik').toFun i' = i := by
rw [permThree, permOfInjection]
@ -310,7 +310,7 @@ lemma Prop_two (P : ℚ × ℚ → Prop) {S : (PureU1 n).LinSols}
have h1 := h (permTwo hij hab ).symm
rw [FamilyPermutations_anomalyFreeLinear_apply, FamilyPermutations_anomalyFreeLinear_apply] at h1
simp at h1
change P
change P
(S.val ((permTwo hij hab ).toFun a),
S.val ((permTwo hij hab ).toFun b)) at h1
erw [permTwo_fst,permTwo_snd] at h1
@ -322,14 +322,14 @@ lemma Prop_three (P : ℚ × ℚ × ℚ → Prop) {S : (PureU1 n).LinSols}
P ((((FamilyPermutations n).linSolRep f S).val a),(
(((FamilyPermutations n).linSolRep f S).val b),
(((FamilyPermutations n).linSolRep f S).val c)
))) : ∀ (i j k : Fin n) (_ : i ≠ j) (_ : j ≠ k) (_ : i ≠ k),
))) : ∀ (i j k : Fin n) (_ : i ≠ j) (_ : j ≠ k) (_ : i ≠ k),
P (S.val i, (S.val j, S.val k)) := by
intro i j k hij hjk hik
have h1 := h (permThree hij hjk hik hab hbc hac).symm
rw [FamilyPermutations_anomalyFreeLinear_apply, FamilyPermutations_anomalyFreeLinear_apply,
FamilyPermutations_anomalyFreeLinear_apply] at h1
simp at h1
change P
change P
(S.val ((permThree hij hjk hik hab hbc hac).toFun a),
S.val ((permThree hij hjk hik hab hbc hac).toFun b),
S.val ((permThree hij hjk hik hab hbc hac).toFun c)) at h1

16
HepLean/AnomalyCancellation/PureU1/Sort.lean
View file

@ -15,7 +15,7 @@ We define the sort function for Pure U(1) charges, and prove some basic properti
universe v u
open Nat
open Finset
open Finset
namespace PureU1
@ -23,15 +23,15 @@ variable {n : ℕ}
/-- We say a charge is shorted if for all `i ≤ j`, then `S i ≤ S j`. -/
@[simp]
def Sorted {n : ℕ} (S : (PureU1 n).Charges) : Prop :=
∀ i j (_ : i ≤ j), S i ≤ S j
def Sorted {n : ℕ} (S : (PureU1 n).Charges) : Prop :=
∀ i j (_ : i ≤ j), S i ≤ S j
/-- Given a charge assignment `S`, the corresponding sorted charge assignment. -/
@[simp]
def sort {n : ℕ} (S : (PureU1 n).Charges) : (PureU1 n).Charges :=
((FamilyPermutations n).rep (Tuple.sort S).symm S)
lemma sort_sorted {n : ℕ} (S : (PureU1 n).Charges) : Sorted (sort S) := by
lemma sort_sorted {n : ℕ} (S : (PureU1 n).Charges) : Sorted (sort S) := by
simp only [Sorted, PureU1_numberCharges, sort, FamilyPermutations, PermGroup, permCharges,
MonoidHom.coe_mk, OneHom.coe_mk, chargeMap_apply]
intro i j hij
@ -45,7 +45,7 @@ lemma sort_apply {n : ℕ} (S : (PureU1 n).Charges) (j : Fin n) :
sort S j = S ((Tuple.sort S) j) := by
rfl
lemma sort_zero {n : ℕ} (S : (PureU1 n).Charges) (hS : sort S = 0) : S = 0 := by
lemma sort_zero {n : ℕ} (S : (PureU1 n).Charges) (hS : sort S = 0) : S = 0 := by
funext i
have hj : ∀ j, sort S j = 0 := by
rw [hS]
@ -61,13 +61,13 @@ lemma sort_projection {n : ℕ} (S : (PureU1 n).Charges) : sort (sort S) = sort
sort_perm S (Tuple.sort S).symm
/-- The sort function acting on `LinSols`. -/
def sortAFL {n : ℕ} (S : (PureU1 n).LinSols) : (PureU1 n).LinSols :=
def sortAFL {n : ℕ} (S : (PureU1 n).LinSols) : (PureU1 n).LinSols :=
((FamilyPermutations n).linSolRep (Tuple.sort S.val).symm S)
lemma sortAFL_val {n : ℕ} (S : (PureU1 n).LinSols) : (sortAFL S).val = sort S.val := by
lemma sortAFL_val {n : ℕ} (S : (PureU1 n).LinSols) : (sortAFL S).val = sort S.val := by
rfl
lemma sortAFL_zero {n : ℕ} (S : (PureU1 n).LinSols) (hS : sortAFL S = 0) : S = 0 := by
lemma sortAFL_zero {n : ℕ} (S : (PureU1 n).LinSols) (hS : sortAFL S = 0) : S = 0 := by
apply ACCSystemLinear.LinSols.ext
have h1 : sort S.val = 0 := by
rw [← sortAFL_val]

10
HepLean/AnomalyCancellation/PureU1/VectorLike.lean
View file

@ -15,7 +15,7 @@ For the `n`-even case we define the property of a charge assignment being vector
universe v u
open Nat
open Finset
open Finset
open BigOperators
namespace PureU1
@ -30,10 +30,10 @@ lemma split_equal (n : ℕ) : n + n = 2 * n := (Nat.two_mul n).symm
lemma split_odd (n : ℕ) : n + 1 + n = 2 * n + 1 := by omega
/-- A charge configuration for n even is vector like if when sorted the `i`th element
is equal to the negative of the `n + i`th element. -/
is equal to the negative of the `n + i`th element. -/
@[simp]
def VectorLikeEven (S : (PureU1 (2 * n)).Charges) : Prop :=
∀ (i : Fin n), (sort S) (Fin.cast (split_equal n) (Fin.castAdd n i))
= - (sort S) (Fin.cast (split_equal n) (Fin.natAdd n i))
def VectorLikeEven (S : (PureU1 (2 * n)).Charges) : Prop :=
∀ (i : Fin n), (sort S) (Fin.cast (split_equal n) (Fin.castAdd n i))
= - (sort S) (Fin.cast (split_equal n) (Fin.natAdd n i))
end PureU1

20
HepLean/AnomalyCancellation/SM/Basic.lean
View file

@ -31,7 +31,7 @@ def SMSpecies (n : ℕ) : ACCSystemCharges := ACCSystemChargesMk n
namespace SMCharges
variable {n : ℕ}
variable {n : ℕ}
/-- An equivalence between the set `(SMCharges n).charges` and the set
`(Fin 5 → Fin n → ℚ)`. -/
@ -54,7 +54,7 @@ lemma charges_eq_toSpecies_eq (S T : (SMCharges n).Charges) :
apply toSpeciesEquiv.injective
exact (Set.eqOn_univ (toSpeciesEquiv S) (toSpeciesEquiv T)).mp fun ⦃x⦄ _ => h x
lemma toSMSpecies_toSpecies_inv (i : Fin 5) (f : (Fin 5 → Fin n → ℚ) ) :
lemma toSMSpecies_toSpecies_inv (i : Fin 5) (f : (Fin 5 → Fin n → ℚ) ) :
(toSpecies i) (toSpeciesEquiv.symm f) = f i := by
change (toSpeciesEquiv ∘ toSpeciesEquiv.symm ) _ i= f i
simp
@ -80,7 +80,7 @@ namespace SMACCs
open SMCharges
variable {n : ℕ}
variable {n : ℕ}
/-- The gravitational anomaly equation. -/
@[simp]
@ -89,7 +89,7 @@ def accGrav : (SMCharges n).Charges →ₗ[ℚ] ℚ where
map_add' S T := by
simp only
repeat rw [map_add]
simp [Pi.add_apply, mul_add]
simp [Pi.add_apply, mul_add]
repeat erw [Finset.sum_add_distrib]
ring
map_smul' a S := by
@ -103,7 +103,7 @@ def accGrav : (SMCharges n).Charges →ₗ[ℚ] ℚ where
/-- Extensionality lemma for `accGrav`. -/
lemma accGrav_ext {S T : (SMCharges n).Charges}
(hj : ∀ (j : Fin 5), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
(hj : ∀ (j : Fin 5), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
accGrav S = accGrav T := by
simp only [accGrav, SMSpecies_numberCharges, toSpecies_apply, Fin.isValue, LinearMap.coe_mk,
AddHom.coe_mk]
@ -133,7 +133,7 @@ def accSU2 : (SMCharges n).Charges →ₗ[ℚ] ℚ where
/-- Extensionality lemma for `accSU2`. -/
lemma accSU2_ext {S T : (SMCharges n).Charges}
(hj : ∀ (j : Fin 5), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
(hj : ∀ (j : Fin 5), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
accSU2 S = accSU2 T := by
simp only [accSU2, SMSpecies_numberCharges, toSpecies_apply, Fin.isValue, LinearMap.coe_mk,
AddHom.coe_mk]
@ -162,7 +162,7 @@ def accSU3 : (SMCharges n).Charges →ₗ[ℚ] ℚ where
/-- Extensionality lemma for `accSU3`. -/
lemma accSU3_ext {S T : (SMCharges n).Charges}
(hj : ∀ (j : Fin 5), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
(hj : ∀ (j : Fin 5), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
accSU3 S = accSU3 T := by
simp only [accSU3, SMSpecies_numberCharges, toSpecies_apply, Fin.isValue, LinearMap.coe_mk,
AddHom.coe_mk]
@ -193,7 +193,7 @@ def accYY : (SMCharges n).Charges →ₗ[ℚ] ℚ where
/-- Extensionality lemma for `accYY`. -/
lemma accYY_ext {S T : (SMCharges n).Charges}
(hj : ∀ (j : Fin 5), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
(hj : ∀ (j : Fin 5), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
accYY S = accYY T := by
simp only [accYY, SMSpecies_numberCharges, toSpecies_apply, Fin.isValue, LinearMap.coe_mk,
AddHom.coe_mk]
@ -238,7 +238,7 @@ def quadBiLin : BiLinearSymm (SMCharges n).Charges := BiLinearSymm.mk₂
/-- The quadratic anomaly cancellation condition. -/
@[simp]
def accQuad : HomogeneousQuadratic (SMCharges n).Charges :=
def accQuad : HomogeneousQuadratic (SMCharges n).Charges :=
(@quadBiLin n).toHomogeneousQuad
/-- Extensionality lemma for `accQuad`. -/
@ -263,7 +263,7 @@ def cubeTriLin : TriLinearSymm (SMCharges n).Charges := TriLinearSymm.mk₃
+ 3 * ((U S.1 i) * (U S.2.1 i) * (U S.2.2 i))
+ 3 * ((D S.1 i) * (D S.2.1 i) * (D S.2.2 i))
+ 2 * ((L S.1 i) * (L S.2.1 i) * (L S.2.2 i))
+ ((E S.1 i) * (E S.2.1 i) * (E S.2.2 i))))
+ ((E S.1 i) * (E S.2.1 i) * (E S.2.2 i))))
(by
intro a S T R
simp only

8
HepLean/AnomalyCancellation/SM/FamilyMaps.lean
View file

@ -37,7 +37,7 @@ def chargesMapOfSpeciesMap {n m : ℕ} (f : (SMSpecies n).Charges →ₗ[ℚ] (S
rw [map_smul]
rfl
/-- The projection of the `m`-family charges onto the first `n`-family charges for species. -/
/-- The projection of the `m`-family charges onto the first `n`-family charges for species. -/
@[simps!]
def speciesFamilyProj {m n : ℕ} (h : n ≤ m) :
(SMSpecies m).Charges →ₗ[ℚ] (SMSpecies n).Charges where
@ -50,12 +50,12 @@ def speciesFamilyProj {m n : ℕ} (h : n ≤ m) :
simp [HSMul.hSMul]
--rw [show Rat.cast a = a from rfl]
/-- The projection of the `m`-family charges onto the first `n`-family charges. -/
/-- The projection of the `m`-family charges onto the first `n`-family charges. -/
def familyProjection {m n : ℕ} (h : n ≤ m) : (SMCharges m).Charges →ₗ[ℚ] (SMCharges n).Charges :=
chargesMapOfSpeciesMap (speciesFamilyProj h)
/-- For species, the embedding of the `m`-family charges onto the `n`-family charges, with all
other charges zero. -/
other charges zero. -/
@[simps!]
def speciesEmbed (m n : ℕ) :
(SMSpecies m).Charges →ₗ[ℚ] (SMSpecies n).Charges where
@ -80,7 +80,7 @@ def speciesEmbed (m n : ℕ) :
simp
/-- The embedding of the `m`-family charges onto the `n`-family charges, with all
other charges zero. -/
other charges zero. -/
def familyEmbedding (m n : ℕ) : (SMCharges m).Charges →ₗ[ℚ] (SMCharges n).Charges :=
chargesMapOfSpeciesMap (speciesEmbed m n)

8
HepLean/AnomalyCancellation/SM/NoGrav/Basic.lean
View file

@ -36,17 +36,17 @@ namespace SMNoGrav
variable {n : ℕ}
lemma SU2Sol (S : (SMNoGrav n).LinSols) : accSU2 S.val = 0 := by
lemma SU2Sol (S : (SMNoGrav n).LinSols) : accSU2 S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 0
lemma SU3Sol (S : (SMNoGrav n).LinSols) : accSU3 S.val = 0 := by
lemma SU3Sol (S : (SMNoGrav n).LinSols) : accSU3 S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 1
lemma cubeSol (S : (SMNoGrav n).Sols) : accCube S.val = 0 := by
lemma cubeSol (S : (SMNoGrav n).Sols) : accCube S.val = 0 := by
exact S.cubicSol
/-- An element of `charges` which satisfies the linear ACCs
@ -84,7 +84,7 @@ def chargeToAF (S : (SMNoGrav n).Charges) (hSU2 : accSU2 S = 0) (hSU3 : accSU3 S
(hc : accCube S = 0) : (SMNoGrav n).Sols :=
quadToAF (chargeToQuad S hSU2 hSU3) hc
/-- An element of `AnomalyFreeLinear` which satisfies the quadratic and cubic ACCs
/-- An element of `AnomalyFreeLinear` which satisfies the quadratic and cubic ACCs
gives us a element of `AnomalyFree`. -/
def linearToAF (S : (SMNoGrav n).LinSols)
(hc : accCube S.val = 0) : (SMNoGrav n).Sols :=

10
HepLean/AnomalyCancellation/SM/NoGrav/One/Lemmas.lean
View file

@ -25,11 +25,11 @@ open SMACCs
open BigOperators
lemma E_zero_iff_Q_zero {S : (SMNoGrav 1).Sols} : Q S.val (0 : Fin 1) = 0 ↔
E S.val (0 : Fin 1) = 0 := by
E S.val (0 : Fin 1) = 0 := by
let S' := linearParameters.bijection.symm S.1.1
have hC := cubeSol S
have hS' := congrArg (fun S => S.val) (linearParameters.bijection.right_inv S.1.1)
change S'.asCharges = S.val at hS'
change S'.asCharges = S.val at hS'
rw [← hS'] at hC
apply Iff.intro
intro hQ
@ -37,7 +37,7 @@ lemma E_zero_iff_Q_zero {S : (SMNoGrav 1).Sols} : Q S.val (0 : Fin 1) = 0 ↔
intro hE
exact S'.cubic_zero_E'_zero hC hE
lemma accGrav_Q_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) = 0) :
lemma accGrav_Q_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) = 0) :
accGrav S.val = 0 := by
rw [accGrav]
simp only [SMSpecies_numberCharges, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,
@ -59,12 +59,12 @@ lemma accGrav_Q_neq_zero {S : (SMNoGrav 1).Sols} (hQ : Q S.val (0 : Fin 1) ≠ 0
have hC := cubeSol S
have hS' := congrArg (fun S => S.val.val)
(linearParametersQENeqZero.bijection.right_inv ⟨S.1.1, And.intro hQ hE⟩)
change _ = S.val at hS'
change _ = S.val at hS'
rw [← hS'] at hC
rw [← hS']
exact S'.grav_of_cubic hC FLTThree
/-- Any solution to the ACCs without gravity satisfies the gravitational ACC. -/
/-- Any solution to the ACCs without gravity satisfies the gravitational ACC. -/
theorem accGravSatisfied {S : (SMNoGrav 1).Sols} (FLTThree : FermatLastTheoremWith ℚ 3) :
accGrav S.val = 0 := by
by_cases hQ : Q S.val (0 : Fin 1)= 0

18
HepLean/AnomalyCancellation/SM/NoGrav/One/LinearParameterization.lean
View file

@ -198,15 +198,15 @@ namespace linearParametersQENeqZero
@[ext]
lemma ext {S T : linearParametersQENeqZero} (hx : S.x = T.x) (hv : S.v = T.v)
(hw : S.w = T.w) : S = T := by
(hw : S.w = T.w) : S = T := by
cases' S
simp_all only
/-- A map from `linearParametersQENeqZero` to `linearParameters`. -/
@[simps!]
def toLinearParameters (S : linearParametersQENeqZero) :
{S : linearParameters // S.Q' ≠ 0 ∧ S.E' ≠ 0} :=
⟨⟨S.x, 3 * S.x * (S.v - S.w) / (S.v + S.w), - 6 * S.x / (S.v + S.w)⟩,
{S : linearParameters // S.Q' ≠ 0 ∧ S.E' ≠ 0} :=
⟨⟨S.x, 3 * S.x * (S.v - S.w) / (S.v + S.w), - 6 * S.x / (S.v + S.w)⟩,
by
apply And.intro S.hx
simp only [neg_mul, ne_eq, div_eq_zero_iff, neg_eq_zero, mul_eq_zero, OfNat.ofNat_ne_zero,
@ -217,9 +217,9 @@ def toLinearParameters (S : linearParametersQENeqZero) :
/-- A map from `linearParameters` to `linearParametersQENeqZero` in the special case when
`Q'` and `E'` of the linear parameters are non-zero. -/
@[simps!]
def tolinearParametersQNeqZero (S : {S : linearParameters // S.Q' ≠ 0 ∧ S.E' ≠ 0}) :
def tolinearParametersQNeqZero (S : {S : linearParameters // S.Q' ≠ 0 ∧ S.E' ≠ 0}) :
linearParametersQENeqZero :=
⟨S.1.Q', - (3 * S.1.Q' + S.1.Y) / S.1.E', - (3 * S.1.Q' - S.1.Y)/ S.1.E', S.2.1,
⟨S.1.Q', - (3 * S.1.Q' + S.1.Y) / S.1.E', - (3 * S.1.Q' - S.1.Y)/ S.1.E', S.2.1,
by
simp only [ne_eq, neg_add_rev, neg_sub]
field_simp
@ -231,7 +231,7 @@ def tolinearParametersQNeqZero (S : {S : linearParameters // S.Q' ≠ 0 ∧ S.
with `Q'` and `E'` non-zero. -/
@[simps!]
def bijectionLinearParameters :
linearParametersQENeqZero ≃ {S : linearParameters // S.Q' ≠ 0 ∧ S.E' ≠ 0} where
linearParametersQENeqZero ≃ {S : linearParameters // S.Q' ≠ 0 ∧ S.E' ≠ 0} where
toFun := toLinearParameters
invFun := tolinearParametersQNeqZero
left_inv S := by
@ -260,7 +260,7 @@ def bijectionLinearParameters :
/-- The bijection between `linearParametersQENeqZero` and `LinSols` with `Q` and `E` non-zero. -/
def bijection : linearParametersQENeqZero ≃
{S : (SMNoGrav 1).LinSols // Q S.val (0 : Fin 1) ≠ 0 ∧ E S.val (0 : Fin 1) ≠ 0} :=
{S : (SMNoGrav 1).LinSols // Q S.val (0 : Fin 1) ≠ 0 ∧ E S.val (0 : Fin 1) ≠ 0} :=
bijectionLinearParameters.trans (linearParameters.bijectionQEZero)
lemma cubic (S : linearParametersQENeqZero) :
@ -297,7 +297,7 @@ lemma cubic_v_zero (S : linearParametersQENeqZero) (h : accCube (bijection S).1.
have h' : (S.w + 1) * (1 * S.w * S.w + (-1) * S.w + 1) = 0 := by
ring_nf
exact add_eq_zero_iff_neg_eq.mpr (id (Eq.symm h))
have h'' : (1 * S.w * S.w + (-1) * S.w + 1) ≠ 0 := by
have h'' : (1 * S.w * S.w + (-1) * S.w + 1) ≠ 0 := by
refine quadratic_ne_zero_of_discrim_ne_sq ?_ S.w
intro s
by_contra hn
@ -315,7 +315,7 @@ lemma cube_w_zero (S : linearParametersQENeqZero) (h : accCube (bijection S).1.v
have h' : (S.v + 1) * (1 * S.v * S.v + (-1) * S.v + 1) = 0 := by
ring_nf
exact add_eq_zero_iff_neg_eq.mpr (id (Eq.symm h))
have h'' : (1 * S.v * S.v + (-1) * S.v + 1) ≠ 0 := by
have h'' : (1 * S.v * S.v + (-1) * S.v + 1) ≠ 0 := by
refine quadratic_ne_zero_of_discrim_ne_sq ?_ S.v
intro s
by_contra hn

16
HepLean/AnomalyCancellation/SM/Permutations.lean
View file

@ -16,7 +16,7 @@ We define the group of permutations for the SM charges with no RHN.
universe v u
open Nat
open Finset
open Finset
namespace SM
@ -26,7 +26,7 @@ open BigOperators
/-- The group of `Sₙ` permutations for each species. -/
@[simp]
def PermGroup (n : ℕ) := ∀ (_ : Fin 5), Equiv.Perm (Fin n)
def PermGroup (n : ℕ) := ∀ (_ : Fin 5), Equiv.Perm (Fin n)
variable {n : ℕ}
@ -72,32 +72,32 @@ lemma toSpecies_sum_invariant (m : ℕ) (f : PermGroup n) (S : (SMCharges n).Cha
exact Fintype.sum_equiv (f⁻¹ j) (fun x => ((fun a => a ^ m) ∘ (toSpecies j) S ∘ ⇑(f⁻¹ j)) x)
(fun x => ((fun a => a ^ m) ∘ (toSpecies j) S) x) (congrFun rfl)
lemma accGrav_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
lemma accGrav_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
accGrav (repCharges f S) = accGrav S :=
accGrav_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accSU2_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
lemma accSU2_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
accSU2 (repCharges f S) = accSU2 S :=
accSU2_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accSU3_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
lemma accSU3_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
accSU3 (repCharges f S) = accSU3 S :=
accSU3_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accYY_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
lemma accYY_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
accYY (repCharges f S) = accYY S :=
accYY_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accQuad_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
lemma accQuad_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
accQuad (repCharges f S) = accQuad S :=
accQuad_ext
(toSpecies_sum_invariant 2 f S)
lemma accCube_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
lemma accCube_invariant (f : PermGroup n) (S : (SMCharges n).Charges) :
accCube (repCharges f S) = accCube S :=
accCube_ext (fun j => toSpecies_sum_invariant 3 f S j)

38
HepLean/AnomalyCancellation/SMNu/Basic.lean
View file

@ -28,7 +28,7 @@ def SMνSpecies (n : ℕ) : ACCSystemCharges := ACCSystemChargesMk n
namespace SMνCharges
variable {n : ℕ}
variable {n : ℕ}
/-- An equivalence between `(SMνCharges n).charges` and `(Fin 6 → Fin n → ℚ)`
splitting the charges into species.-/
@ -54,12 +54,12 @@ lemma charges_eq_toSpecies_eq (S T : (SMνCharges n).Charges) :
funext i
exact h i
lemma toSMSpecies_toSpecies_inv (i : Fin 6) (f : (Fin 6 → Fin n → ℚ) ) :
lemma toSMSpecies_toSpecies_inv (i : Fin 6) (f : (Fin 6 → Fin n → ℚ) ) :
(toSpecies i) (toSpeciesEquiv.symm f) = f i := by
change (toSpeciesEquiv ∘ toSpeciesEquiv.symm ) _ i = f i
simp
lemma toSpecies_one (S : (SMνCharges 1).Charges) (j : Fin 6) :
lemma toSpecies_one (S : (SMνCharges 1).Charges) (j : Fin 6) :
toSpecies j S ⟨0, by simp⟩ = S j := by
match j with
| 0 => rfl
@ -88,7 +88,7 @@ namespace SMνACCs
open SMνCharges
variable {n : ℕ}
variable {n : ℕ}
/-- The gravitational anomaly equation. -/
@[simp]
@ -97,7 +97,7 @@ def accGrav : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
map_add' S T := by
simp only
repeat rw [map_add]
simp [Pi.add_apply, mul_add]
simp [Pi.add_apply, mul_add]
repeat erw [Finset.sum_add_distrib]
ring
map_smul' a S := by
@ -119,7 +119,7 @@ lemma accGrav_decomp (S : (SMνCharges n).Charges) :
/-- Extensionality lemma for `accGrav`. -/
lemma accGrav_ext {S T : (SMνCharges n).Charges}
(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
accGrav S = accGrav T := by
rw [accGrav_decomp, accGrav_decomp]
repeat erw [hj]
@ -131,7 +131,7 @@ def accSU2 : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
map_add' S T := by
simp only
repeat rw [map_add]
simp [Pi.add_apply, mul_add]
simp [Pi.add_apply, mul_add]
repeat erw [Finset.sum_add_distrib]
ring
map_smul' a S := by
@ -152,7 +152,7 @@ lemma accSU2_decomp (S : (SMνCharges n).Charges) :
/-- Extensionality lemma for `accSU2`. -/
lemma accSU2_ext {S T : (SMνCharges n).Charges}
(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
accSU2 S = accSU2 T := by
rw [accSU2_decomp, accSU2_decomp]
repeat erw [hj]
@ -164,7 +164,7 @@ def accSU3 : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
map_add' S T := by
simp only
repeat rw [map_add]
simp [ Pi.add_apply, mul_add]
simp [ Pi.add_apply, mul_add]
repeat erw [Finset.sum_add_distrib]
ring
map_smul' a S := by
@ -185,7 +185,7 @@ lemma accSU3_decomp (S : (SMνCharges n).Charges) :
/-- Extensionality lemma for `accSU3`. -/
lemma accSU3_ext {S T : (SMνCharges n).Charges}
(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
accSU3 S = accSU3 T := by
rw [accSU3_decomp, accSU3_decomp]
repeat rw [hj]
@ -198,7 +198,7 @@ def accYY : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
map_add' S T := by
simp only
repeat rw [map_add]
simp [Pi.add_apply, mul_add]
simp [Pi.add_apply, mul_add]
repeat erw [Finset.sum_add_distrib]
ring
map_smul' a S := by
@ -219,7 +219,7 @@ lemma accYY_decomp (S : (SMνCharges n).Charges) :
/-- Extensionality lemma for `accYY`. -/
lemma accYY_ext {S T : (SMνCharges n).Charges}
(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
(hj : ∀ (j : Fin 6), ∑ i, (toSpecies j) S i = ∑ i, (toSpecies j) T i) :
accYY S = accYY T := by
rw [accYY_decomp, accYY_decomp]
repeat rw [hj]
@ -259,8 +259,8 @@ def quadBiLin : BiLinearSymm (SMνCharges n).Charges := BiLinearSymm.mk₂
ring)
lemma quadBiLin_decomp (S T : (SMνCharges n).Charges) :
quadBiLin S T = ∑ i, Q S i * Q T i - 2 * ∑ i, U S i * U T i +
∑ i, D S i * D T i - ∑ i, L S i * L T i + ∑ i, E S i * E T i := by
quadBiLin S T = ∑ i, Q S i * Q T i - 2 * ∑ i, U S i * U T i +
∑ i, D S i * D T i - ∑ i, L S i * L T i + ∑ i, E S i * E T i := by
erw [← quadBiLin.toFun_eq_coe]
rw [quadBiLin]
simp only [quadBiLin, BiLinearSymm.mk₂, AddHom.toFun_eq_coe, AddHom.coe_mk, LinearMap.coe_mk]
@ -271,7 +271,7 @@ lemma quadBiLin_decomp (S T : (SMνCharges n).Charges) :
/-- The quadratic anomaly cancellation condition. -/
@[simp]
def accQuad : HomogeneousQuadratic (SMνCharges n).Charges :=
def accQuad : HomogeneousQuadratic (SMνCharges n).Charges :=
(@quadBiLin n).toHomogeneousQuad
lemma accQuad_decomp (S : (SMνCharges n).Charges) :
@ -296,8 +296,8 @@ def cubeTriLin : TriLinearSymm (SMνCharges n).Charges := TriLinearSymm.mk₃
+ 3 * ((U S.1 i) * (U S.2.1 i) * (U S.2.2 i))
+ 3 * ((D S.1 i) * (D S.2.1 i) * (D S.2.2 i))
+ 2 * ((L S.1 i) * (L S.2.1 i) * (L S.2.2 i))
+ ((E S.1 i) * (E S.2.1 i) * (E S.2.2 i))
+ ((N S.1 i) * (N S.2.1 i) * (N S.2.2 i))))
+ ((E S.1 i) * (E S.2.1 i) * (E S.2.2 i))
+ ((N S.1 i) * (N S.2.1 i) * (N S.2.2 i))))
(by
intro a S T R
simp only
@ -330,8 +330,8 @@ def cubeTriLin : TriLinearSymm (SMνCharges n).Charges := TriLinearSymm.mk₃
ring)
lemma cubeTriLin_decomp (S T R : (SMνCharges n).Charges) :
cubeTriLin S T R = 6 * ∑ i, (Q S i * Q T i * Q R i) + 3 * ∑ i, (U S i * U T i * U R i) +
3 * ∑ i, (D S i * D T i * D R i) + 2 * ∑ i, (L S i * L T i * L R i) +
cubeTriLin S T R = 6 * ∑ i, (Q S i * Q T i * Q R i) + 3 * ∑ i, (U S i * U T i * U R i) +
3 * ∑ i, (D S i * D T i * D R i) + 2 * ∑ i, (L S i * L T i * L R i) +
∑ i, (E S i * E T i * E R i) + ∑ i, (N S i * N T i * N R i) := by
erw [← cubeTriLin.toFun_eq_coe]
rw [cubeTriLin]

30
HepLean/AnomalyCancellation/SMNu/FamilyMaps.lean
View file

@ -5,7 +5,7 @@ Authors: Joseph Tooby-Smith
-/
import HepLean.AnomalyCancellation.SMNu.Basic
/-!
# Family maps for the Standard Model for RHN ACCs
# Family maps for the Standard Model for RHN ACCs
We define the a series of maps between the charges for different numbers of families.
@ -39,10 +39,10 @@ def chargesMapOfSpeciesMap {n m : ℕ} (f : (SMνSpecies n).Charges →ₗ[ℚ]
lemma chargesMapOfSpeciesMap_toSpecies {n m : ℕ}
(f : (SMνSpecies n).Charges →ₗ[ℚ] (SMνSpecies m).Charges)
(S : (SMνCharges n).Charges) (j : Fin 6) :
toSpecies j (chargesMapOfSpeciesMap f S) = (LinearMap.comp f (toSpecies j)) S := by
toSpecies j (chargesMapOfSpeciesMap f S) = (LinearMap.comp f (toSpecies j)) S := by
erw [toSMSpecies_toSpecies_inv]
/-- The projection of the `m`-family charges onto the first `n`-family charges for species. -/
/-- The projection of the `m`-family charges onto the first `n`-family charges for species. -/
@[simps!]
def speciesFamilyProj {m n : ℕ} (h : n ≤ m) :
(SMνSpecies m).Charges →ₗ[ℚ] (SMνSpecies n).Charges where
@ -50,12 +50,12 @@ def speciesFamilyProj {m n : ℕ} (h : n ≤ m) :
map_add' _ _ := rfl
map_smul' _ _ := rfl
/-- The projection of the `m`-family charges onto the first `n`-family charges. -/
/-- The projection of the `m`-family charges onto the first `n`-family charges. -/
def familyProjection {m n : ℕ} (h : n ≤ m) : (SMνCharges m).Charges →ₗ[ℚ] (SMνCharges n).Charges :=
chargesMapOfSpeciesMap (speciesFamilyProj h)
/-- For species, the embedding of the `m`-family charges onto the `n`-family charges, with all
other charges zero. -/
other charges zero. -/
@[simps!]
def speciesEmbed (m n : ℕ) :
(SMνSpecies m).Charges →ₗ[ℚ] (SMνSpecies n).Charges where
@ -80,7 +80,7 @@ def speciesEmbed (m n : ℕ) :
exact Eq.symm (Rat.mul_zero a)
/-- The embedding of the `m`-family charges onto the `n`-family charges, with all
other charges zero. -/
other charges zero. -/
def familyEmbedding (m n : ℕ) : (SMνCharges m).Charges →ₗ[ℚ] (SMνCharges n).Charges :=
chargesMapOfSpeciesMap (speciesEmbed m n)
@ -118,7 +118,7 @@ lemma sum_familyUniversal {n : ℕ} (m : ℕ) (S : (SMνCharges 1).Charges) (j :
intro i _
erw [toSpecies_familyUniversal]
lemma sum_familyUniversal_one {n : ℕ} (S : (SMνCharges 1).Charges) (j : Fin 6) :
lemma sum_familyUniversal_one {n : ℕ} (S : (SMνCharges 1).Charges) (j : Fin 6) :
∑ i, toSpecies j (familyUniversal n S) i = n * (toSpecies j S ⟨0, by simp⟩) := by
simpa using @sum_familyUniversal n 1 S j
@ -134,7 +134,7 @@ lemma sum_familyUniversal_two {n : ℕ} (S : (SMνCharges 1).Charges)
erw [toSpecies_familyUniversal]
rfl
lemma sum_familyUniversal_three {n : ℕ} (S : (SMνCharges 1).Charges)
lemma sum_familyUniversal_three {n : ℕ} (S : (SMνCharges 1).Charges)
(T L : (SMνCharges n).Charges) (j : Fin 6) :
∑ i, (toSpecies j (familyUniversal n S) i * toSpecies j T i * toSpecies j L i) =
(toSpecies j S ⟨0, by simp⟩) * ∑ i, toSpecies j T i * toSpecies j L i := by
@ -148,7 +148,7 @@ lemma sum_familyUniversal_three {n : ℕ} (S : (SMνCharges 1).Charges)
ring
lemma familyUniversal_accGrav (S : (SMνCharges 1).Charges) :
accGrav (familyUniversal n S) = n * (accGrav S) := by
accGrav (familyUniversal n S) = n * (accGrav S) := by
rw [accGrav_decomp, accGrav_decomp]
repeat rw [sum_familyUniversal_one]
simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
@ -156,7 +156,7 @@ lemma familyUniversal_accGrav (S : (SMνCharges 1).Charges) :
ring
lemma familyUniversal_accSU2 (S : (SMνCharges 1).Charges) :
accSU2 (familyUniversal n S) = n * (accSU2 S) := by
accSU2 (familyUniversal n S) = n * (accSU2 S) := by
rw [accSU2_decomp, accSU2_decomp]
repeat rw [sum_familyUniversal_one]
simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
@ -164,7 +164,7 @@ lemma familyUniversal_accSU2 (S : (SMνCharges 1).Charges) :
ring
lemma familyUniversal_accSU3 (S : (SMνCharges 1).Charges) :
accSU3 (familyUniversal n S) = n * (accSU3 S) := by
accSU3 (familyUniversal n S) = n * (accSU3 S) := by
rw [accSU3_decomp, accSU3_decomp]
repeat rw [sum_familyUniversal_one]
simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
@ -172,7 +172,7 @@ lemma familyUniversal_accSU3 (S : (SMνCharges 1).Charges) :
ring
lemma familyUniversal_accYY (S : (SMνCharges 1).Charges) :
accYY (familyUniversal n S) = n * (accYY S) := by
accYY (familyUniversal n S) = n * (accYY S) := by
rw [accYY_decomp, accYY_decomp]
repeat rw [sum_familyUniversal_one]
simp only [Fin.isValue, SMνSpecies_numberCharges, Fin.zero_eta, toSpecies_apply,
@ -182,7 +182,7 @@ lemma familyUniversal_accYY (S : (SMνCharges 1).Charges) :
lemma familyUniversal_quadBiLin (S : (SMνCharges 1).Charges) (T : (SMνCharges n).Charges) :
quadBiLin (familyUniversal n S) T =
S (0 : Fin 6) * ∑ i, Q T i - 2 * S (1 : Fin 6) * ∑ i, U T i + S (2 : Fin 6) *∑ i, D T i -
S (3 : Fin 6) * ∑ i, L T i + S (4 : Fin 6) * ∑ i, E T i := by
S (3 : Fin 6) * ∑ i, L T i + S (4 : Fin 6) * ∑ i, E T i := by
rw [quadBiLin_decomp]
repeat rw [sum_familyUniversal_two]
repeat rw [toSpecies_one]
@ -191,7 +191,7 @@ lemma familyUniversal_quadBiLin (S : (SMνCharges 1).Charges) (T : (SMνCharges
ring
lemma familyUniversal_accQuad (S : (SMνCharges 1).Charges) :
accQuad (familyUniversal n S) = n * (accQuad S) := by
accQuad (familyUniversal n S) = n * (accQuad S) := by
rw [accQuad_decomp]
repeat erw [sum_familyUniversal]
rw [accQuad_decomp]
@ -201,7 +201,7 @@ lemma familyUniversal_accQuad (S : (SMνCharges 1).Charges) :
lemma familyUniversal_cubeTriLin (S : (SMνCharges 1).Charges) (T R : (SMνCharges n).Charges) :
cubeTriLin (familyUniversal n S) T R = 6 * S (0 : Fin 6) * ∑ i, (Q T i * Q R i) +
3 * S (1 : Fin 6) * ∑ i, (U T i * U R i) + 3 * S (2 : Fin 6) * ∑ i, (D T i * D R i)
3 * S (1 : Fin 6) * ∑ i, (U T i * U R i) + 3 * S (2 : Fin 6) * ∑ i, (D T i * D R i)
+ 2 * S (3 : Fin 6) * ∑ i, (L T i * L R i) +
S (4 : Fin 6) * ∑ i, (E T i * E R i) + S (5 : Fin 6) * ∑ i, (N T i * N R i) := by
rw [cubeTriLin_decomp]

8
HepLean/AnomalyCancellation/SMNu/NoGrav/Basic.lean
View file

@ -38,17 +38,17 @@ namespace SMNoGrav
variable {n : ℕ}
lemma SU2Sol (S : (SMNoGrav n).LinSols) : accSU2 S.val = 0 := by
lemma SU2Sol (S : (SMNoGrav n).LinSols) : accSU2 S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 0
lemma SU3Sol (S : (SMNoGrav n).LinSols) : accSU3 S.val = 0 := by
lemma SU3Sol (S : (SMNoGrav n).LinSols) : accSU3 S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 1
lemma cubeSol (S : (SMNoGrav n).Sols) : accCube S.val = 0 := by
lemma cubeSol (S : (SMNoGrav n).Sols) : accCube S.val = 0 := by
exact S.cubicSol
/-- An element of `charges` which satisfies the linear ACCs
@ -86,7 +86,7 @@ def chargeToAF (S : (SMNoGrav n).Charges) (hSU2 : accSU2 S = 0) (hSU3 : accSU3 S
(hc : accCube S = 0) : (SMNoGrav n).Sols :=
quadToAF (chargeToQuad S hSU2 hSU3) hc
/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs
/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs
gives us a element of `Sols`. -/
def linearToAF (S : (SMNoGrav n).LinSols)
(hc : accCube S.val = 0) : (SMNoGrav n).Sols :=

12
HepLean/AnomalyCancellation/SMNu/Ordinary/Basic.lean
View file

@ -39,22 +39,22 @@ namespace SM
variable {n : ℕ}
lemma gravSol (S : (SM n).LinSols) : accGrav S.val = 0 := by
lemma gravSol (S : (SM n).LinSols) : accGrav S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 0
lemma SU2Sol (S : (SM n).LinSols) : accSU2 S.val = 0 := by
lemma SU2Sol (S : (SM n).LinSols) : accSU2 S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 1
lemma SU3Sol (S : (SM n).LinSols) : accSU3 S.val = 0 := by
lemma SU3Sol (S : (SM n).LinSols) : accSU3 S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 2
lemma cubeSol (S : (SM n).Sols) : accCube S.val = 0 := by
lemma cubeSol (S : (SM n).Sols) : accCube S.val = 0 := by
exact S.cubicSol
/-- An element of `charges` which satisfies the linear ACCs
@ -94,13 +94,13 @@ def chargeToAF (S : (SM n).Charges) (hGrav : accGrav S = 0) (hSU2 : accSU2 S = 0
(hSU3 : accSU3 S = 0) (hc : accCube S = 0) : (SM n).Sols :=
quadToAF (chargeToQuad S hGrav hSU2 hSU3) hc
/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs
/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs
gives us a element of `Sols`. -/
def linearToAF (S : (SM n).LinSols)
(hc : accCube S.val = 0) : (SM n).Sols :=
quadToAF (linearToQuad S) hc
/-- The permutations acting on the ACC system corresponding to the SM with RHN. -/
/-- The permutations acting on the ACC system corresponding to the SM with RHN. -/
def perm (n : ℕ) : ACCSystemGroupAction (SM n) where
group := PermGroup n
groupInst := inferInstance

16
HepLean/AnomalyCancellation/SMNu/Ordinary/DimSevenPlane.lean
View file

@ -32,7 +32,7 @@ def B₀ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
)
lemma B₀_cubic (S T : (SM 3).Charges) : cubeTriLin B₀ S T =
6 * (S (0 : Fin 18) * T (0 : Fin 18) - S (1 : Fin 18) * T (1 : Fin 18)) := by
6 * (S (0 : Fin 18) * T (0 : Fin 18) - S (1 : Fin 18) * T (1 : Fin 18)) := by
simp [Fin.sum_univ_three, B₀, Fin.divNat, Fin.modNat, finProdFinEquiv]
ring
@ -45,7 +45,7 @@ def B₁ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
)
lemma B₁_cubic (S T : (SM 3).Charges) : cubeTriLin B₁ S T =
3 * (S (3 : Fin 18) * T (3 : Fin 18) - S (4 : Fin 18) * T (4 : Fin 18)) := by
3 * (S (3 : Fin 18) * T (3 : Fin 18) - S (4 : Fin 18) * T (4 : Fin 18)) := by
simp [Fin.sum_univ_three, B₁, Fin.divNat, Fin.modNat, finProdFinEquiv]
ring
@ -58,7 +58,7 @@ def B₂ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
)
lemma B₂_cubic (S T : (SM 3).Charges) : cubeTriLin B₂ S T =
3 * (S (6 : Fin 18) * T (6 : Fin 18) - S (7 : Fin 18) * T (7 : Fin 18)) := by
3 * (S (6 : Fin 18) * T (6 : Fin 18) - S (7 : Fin 18) * T (7 : Fin 18)) := by
simp [Fin.sum_univ_three, B₂, Fin.divNat, Fin.modNat, finProdFinEquiv]
ring
@ -71,7 +71,7 @@ def B₃ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
)
lemma B₃_cubic (S T : (SM 3).Charges) : cubeTriLin B₃ S T =
2 * (S (9 : Fin 18) * T (9 : Fin 18) - S (10 : Fin 18) * T (10 : Fin 18)) := by
2 * (S (9 : Fin 18) * T (9 : Fin 18) - S (10 : Fin 18) * T (10 : Fin 18)) := by
simp [Fin.sum_univ_three, B₃, Fin.divNat, Fin.modNat, finProdFinEquiv]
ring_nf
rfl
@ -85,7 +85,7 @@ def B₄ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
)
lemma B₄_cubic (S T : (SM 3).Charges) : cubeTriLin B₄ S T =
(S (12 : Fin 18) * T (12 : Fin 18) - S (13 : Fin 18) * T (13 : Fin 18)) := by
(S (12 : Fin 18) * T (12 : Fin 18) - S (13 : Fin 18) * T (13 : Fin 18)) := by
simp [Fin.sum_univ_three, B₄, Fin.divNat, Fin.modNat, finProdFinEquiv]
ring_nf
rfl
@ -99,7 +99,7 @@ def B₅ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
)
lemma B₅_cubic (S T : (SM 3).Charges) : cubeTriLin B₅ S T =
(S (15 : Fin 18) * T (15 : Fin 18) - S (16 : Fin 18) * T (16 : Fin 18)) := by
(S (15 : Fin 18) * T (15 : Fin 18) - S (16 : Fin 18) * T (16 : Fin 18)) := by
simp [Fin.sum_univ_three, B₅, Fin.divNat, Fin.modNat, finProdFinEquiv]
ring_nf
rfl
@ -113,7 +113,7 @@ def B₆ : (SM 3).Charges := toSpeciesEquiv.invFun ( fun s => fun i =>
)
lemma B₆_cubic (S T : (SM 3).Charges) : cubeTriLin B₆ S T =
3* (S (5 : Fin 18) * T (5 : Fin 18) - S (8 : Fin 18) * T (8 : Fin 18)) := by
3 * (S (5 : Fin 18) * T (5 : Fin 18) - S (8 : Fin 18) * T (8 : Fin 18)) := by
simp [Fin.sum_univ_three, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
ring_nf
@ -336,7 +336,7 @@ theorem basis_linear_independent : LinearIndependent ℚ B := by
end PlaneSeven
theorem seven_dim_plane_exists : ∃ (B : Fin 7 → (SM 3).Charges),
LinearIndependent ℚ B ∧ ∀ (f : Fin 7 → ℚ), (SM 3).IsSolution (∑ i, f i • B i) := by
LinearIndependent ℚ B ∧ ∀ (f : Fin 7 → ℚ), (SM 3).IsSolution (∑ i, f i • B i) := by
use PlaneSeven.B
apply And.intro
exact PlaneSeven.basis_linear_independent

16
HepLean/AnomalyCancellation/SMNu/Permutations.lean
View file

@ -16,7 +16,7 @@ We define the group of permutations for the SM charges with RHN.
universe v u
open Nat
open Finset
open Finset
namespace SMRHN
@ -69,36 +69,36 @@ lemma toSpecies_sum_invariant (m : ℕ) (f : PermGroup n) (S : (SMνCharges n).C
∑ i, ((fun a => a ^ m) ∘ toSpecies j (repCharges f S)) i =
∑ i, ((fun a => a ^ m) ∘ toSpecies j S) i := by
erw [repCharges_toSpecies]
change ∑ i : Fin n, ((fun a => a ^ m) ∘ _) (⇑(f⁻¹ _) i) = ∑ i : Fin n, ((fun a => a ^ m) ∘ _) i
change ∑ i : Fin n, ((fun a => a ^ m) ∘ _) (⇑(f⁻¹ _) i) = ∑ i : Fin n, ((fun a => a ^ m) ∘ _) i
refine Equiv.Perm.sum_comp _ _ _ ?_
simp only [PermGroup, Fin.isValue, Pi.inv_apply, ne_eq, coe_univ, Set.subset_univ]
lemma accGrav_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
lemma accGrav_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
accGrav (repCharges f S) = accGrav S :=
accGrav_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accSU2_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
lemma accSU2_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
accSU2 (repCharges f S) = accSU2 S :=
accSU2_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accSU3_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
lemma accSU3_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
accSU3 (repCharges f S) = accSU3 S :=
accSU3_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accYY_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
lemma accYY_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
accYY (repCharges f S) = accYY S :=
accYY_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accQuad_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
lemma accQuad_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
accQuad (repCharges f S) = accQuad S :=
accQuad_ext
(toSpecies_sum_invariant 2 f S)
lemma accCube_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
lemma accCube_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
accCube (repCharges f S) = accCube S :=
accCube_ext
(by simpa using toSpecies_sum_invariant 3 f S)

16
HepLean/AnomalyCancellation/SMNu/PlusU1/Basic.lean
View file

@ -39,32 +39,32 @@ namespace PlusU1
variable {n : ℕ}
lemma gravSol (S : (PlusU1 n).LinSols) : accGrav S.val = 0 := by
lemma gravSol (S : (PlusU1 n).LinSols) : accGrav S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 0
lemma SU2Sol (S : (PlusU1 n).LinSols) : accSU2 S.val = 0 := by
lemma SU2Sol (S : (PlusU1 n).LinSols) : accSU2 S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 1
lemma SU3Sol (S : (PlusU1 n).LinSols) : accSU3 S.val = 0 := by
lemma SU3Sol (S : (PlusU1 n).LinSols) : accSU3 S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 2
lemma YYsol (S : (PlusU1 n).LinSols) : accYY S.val = 0 := by
lemma YYsol (S : (PlusU1 n).LinSols) : accYY S.val = 0 := by
have hS := S.linearSol
simp at hS
exact hS 3
lemma quadSol (S : (PlusU1 n).QuadSols) : accQuad S.val = 0 := by
lemma quadSol (S : (PlusU1 n).QuadSols) : accQuad S.val = 0 := by
have hS := S.quadSol
simp at hS
exact hS 0
lemma cubeSol (S : (PlusU1 n).Sols) : accCube S.val = 0 := by
lemma cubeSol (S : (PlusU1 n).Sols) : accCube S.val = 0 := by
exact S.cubicSol
/-- An element of `charges` which satisfies the linear ACCs
@ -110,13 +110,13 @@ def chargeToAF (S : (PlusU1 n).Charges) (hGrav : accGrav S = 0) (hSU2 : accSU2 S
(PlusU1 n).Sols :=
quadToAF (chargeToQuad S hGrav hSU2 hSU3 hYY hQ) hc
/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs
/-- An element of `LinSols` which satisfies the quadratic and cubic ACCs
gives us a element of `Sols`. -/
def linearToAF (S : (PlusU1 n).LinSols) (hQ : accQuad S.val = 0)
(hc : accCube S.val = 0) : (PlusU1 n).Sols :=
quadToAF (linearToQuad S hQ) hc
/-- The permutations acting on the ACC system corresponding to the SM with RHN. -/
/-- The permutations acting on the ACC system corresponding to the SM with RHN. -/
def perm (n : ℕ) : ACCSystemGroupAction (PlusU1 n) where
group := PermGroup n
groupInst := inferInstance

16
HepLean/AnomalyCancellation/SMNu/PlusU1/HyperCharge.lean
View file

@ -26,11 +26,11 @@ def Y₁ : (PlusU1 1).Sols where
val := fun i =>
match i with
| (0 : Fin 6) => 1
| (1 : Fin 6) => -4
| (2 : Fin 6) => 2
| (3 : Fin 6) => -3
| (4 : Fin 6) => 6
| (5 : Fin 6) => 0
| (1 : Fin 6) => -4
| (2 : Fin 6) => 2
| (3 : Fin 6) => -3
| (4 : Fin 6) => 6
| (5 : Fin 6) => 0
linearSol := by
intro i
simp at i
@ -89,7 +89,7 @@ lemma addQuad_zero (S : (PlusU1 n).QuadSols) (a : ℚ): addQuad S a 0 = a • S
rfl
lemma on_cubeTriLin (S : (PlusU1 n).Charges) :
cubeTriLin (Y n).val (Y n).val S = 6 * accYY S := by
cubeTriLin (Y n).val (Y n).val S = 6 * accYY S := by
erw [familyUniversal_cubeTriLin']
rw [accYY_decomp]
simp only [Fin.isValue, Y₁_val, mul_one, SMνSpecies_numberCharges, toSpecies_apply, mul_neg,
@ -103,7 +103,7 @@ lemma on_cubeTriLin_AFL (S : (PlusU1 n).LinSols) :
simp
lemma on_cubeTriLin' (S : (PlusU1 n).Charges) :
cubeTriLin (Y n).val S S = 6 * accQuad S := by
cubeTriLin (Y n).val S S = 6 * accQuad S := by
erw [familyUniversal_cubeTriLin]
rw [accQuad_decomp]
simp only [Fin.isValue, Y₁_val, mul_one, SMνSpecies_numberCharges, toSpecies_apply, mul_neg,
@ -127,7 +127,7 @@ lemma add_AFL_cube (S : (PlusU1 n).LinSols) (a b : ℚ) :
ring
lemma add_AFQ_cube (S : (PlusU1 n).QuadSols) (a b : ℚ) :
accCube (a • S.val + b • (Y n).val) = a ^ 3 * accCube S.val := by
accCube (a • S.val + b • (Y n).val) = a ^ 3 * accCube S.val := by
rw [add_AFL_cube]
rw [cubeTriLin.swap₃]
rw [on_cubeTriLin'_ALQ]

10
HepLean/AnomalyCancellation/SMNu/PlusU1/PlaneNonSols.lean
View file

@ -50,7 +50,7 @@ def B₆ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0,
def B₇ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0]
/-- A charge assignment forming one of the basis elements of the plane. -/
def B₈ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
def B₈ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0]
/-- A charge assignment forming one of the basis elements of the plane. -/
def B₉ : (PlusU1 3).Charges := ![0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0]
@ -79,7 +79,7 @@ lemma Bi_Bj_quad {i j : Fin 11} (hi : i ≠ j) : quadBiLin (B i) (B j) = 0 := by
all_goals simp at hi
lemma Bi_sum_quad (i : Fin 11) (f : Fin 11 → ℚ) :
quadBiLin (B i) (∑ k, f k • B k) = f i * quadBiLin (B i) (B i) := by
quadBiLin (B i) (∑ k, f k • B k) = f i * quadBiLin (B i) (B i) := by
rw [quadBiLin.map_sum₂]
rw [Fintype.sum_eq_single i]
rw [quadBiLin.map_smul₂]
@ -96,7 +96,7 @@ lemma quadCoeff_eq_bilinear (i : Fin 11) : quadCoeff i = quadBiLin (B i) (B i) :
all_goals rfl
lemma on_accQuad (f : Fin 11 → ℚ) :
accQuad (∑ i, f i • B i) = ∑ i, quadCoeff i * (f i)^2 := by
accQuad (∑ i, f i • B i) = ∑ i, quadCoeff i * (f i)^2 := by
change quadBiLin _ _ = _
rw [quadBiLin.map_sum₁]
apply Fintype.sum_congr
@ -105,7 +105,7 @@ lemma on_accQuad (f : Fin 11 → ℚ) :
ring
lemma isSolution_quadCoeff_f_sq_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i))
(k : Fin 11) : quadCoeff k * (f k)^2 = 0 := by
(k : Fin 11) : quadCoeff k * (f k)^2 = 0 := by
obtain ⟨S, hS⟩ := hS
have hQ := quadSol S.1
rw [hS, on_accQuad] at hQ
@ -169,7 +169,7 @@ lemma isSolution_sum_part (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑
simp
rfl
lemma isSolution_grav (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
lemma isSolution_grav (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
f 10 = - 3 * f 9 := by
have hx := isSolution_sum_part f hS
obtain ⟨S, hS'⟩ := hS

8
HepLean/AnomalyCancellation/SMNu/PlusU1/QuadSol.lean
View file

@ -61,7 +61,7 @@ lemma accQuad_α₁_α₂_zero (S : (PlusU1 n).LinSols) (h1 : α₁ C S = 0)
/-- The construction of a `QuadSol` from a `LinSols` in the generic case. -/
def genericToQuad (S : (PlusU1 n).LinSols) :
(PlusU1 n).QuadSols :=
linearToQuad ((α₁ C S) • S + α₂ S • C.1) (accQuad_α₁_α₂ C S)
linearToQuad ((α₁ C S) • S + α₂ S • C.1) (accQuad_α₁_α₂ C S)
lemma genericToQuad_on_quad (S : (PlusU1 n).QuadSols) :
genericToQuad C S.1 = (α₁ C S.1) • S := by
@ -80,7 +80,7 @@ def specialToQuad (S : (PlusU1 n).LinSols) (a b : ℚ) (h1 : α₁ C S = 0)
(h2 : α₂ S = 0) : (PlusU1 n).QuadSols :=
linearToQuad (a • S + b • C.1) (accQuad_α₁_α₂_zero C S h1 h2 a b)
lemma special_on_quad (S : (PlusU1 n).QuadSols) (h1 : α₁ C S.1 = 0) :
lemma special_on_quad (S : (PlusU1 n).QuadSols) (h1 : α₁ C S.1 = 0) :
specialToQuad C S.1 1 0 h1 (α₂_AFQ S) = S := by
apply ACCSystemQuad.QuadSols.ext
change (1 • S.val + 0 • C.val) = S.val
@ -123,9 +123,9 @@ lemma toQuadInv_special (S : (PlusU1 n).QuadSols) (h : α₁ C S.1 = 0) :
rw [special_on_quad]
lemma toQuadInv_generic (S : (PlusU1 n).QuadSols) (h : α₁ C S.1 ≠ 0) :
(toQuadInv C S).2.1 • genericToQuad C (toQuadInv C S).1 = S := by
(toQuadInv C S).2.1 • genericToQuad C (toQuadInv C S).1 = S := by
simp only [toQuadInv_fst]
rw [show (toQuadInv C S).2.1 = (α₁ C S.1)⁻¹ by rw [toQuadInv, if_neg h]]
rw [show (toQuadInv C S).2.1 = (α₁ C S.1)⁻¹ by rw [toQuadInv, if_neg h]]
rw [genericToQuad_neq_zero C S h]
lemma toQuad_rightInverse : Function.RightInverse (@toQuadInv n C) (toQuad C) := by

20
HepLean/AnomalyCancellation/SMNu/PlusU1/QuadSolToSol.lean
View file

@ -71,7 +71,7 @@ def special (S : (PlusU1 n).QuadSols) (a b : ℚ) (h1 : α₁ S = 0) (h2 : α₂
(PlusU1 n).Sols :=
quadToAF (BL.addQuad S a b) (cube_α₁_α₂_zero S a b h1 h2)
lemma special_on_AF (S : (PlusU1 n).Sols) (h1 : α₁ S.1 = 0) :
lemma special_on_AF (S : (PlusU1 n).Sols) (h1 : α₁ S.1 = 0) :
special S.1 1 0 h1 (α₂_AF S) = S := by
apply ACCSystem.Sols.ext
change (BL.addQuad S.1 1 0).val = _
@ -84,7 +84,7 @@ open QuadSolToSol
/-- A map from `QuadSols × ℚ × ℚ` to `Sols` taking account of the special and generic cases.
We will show that this map is a surjection. -/
def quadSolToSol {n : ℕ} : (PlusU1 n).QuadSols × ℚ × ℚ → (PlusU1 n).Sols := fun S =>
if h1 : α₁ S.1 = 0 ∧ α₂ S.1 = 0 then
if h1 : α₁ S.1 = 0 ∧ α₂ S.1 = 0 then
special S.1 S.2.1 S.2.2 h1.1 h1.2
else
S.2.1 • generic S.1
@ -98,7 +98,7 @@ def quadSolToSolInv {n : ℕ} : (PlusU1 n).Sols → (PlusU1 n).QuadSols × ℚ
else
(S.1, (α₁ S.1)⁻¹, 0)
lemma quadSolToSolInv_1 (S : (PlusU1 n).Sols) :
lemma quadSolToSolInv_1 (S : (PlusU1 n).Sols) :
(quadSolToSolInv S).1 = S.1 := by
simp [quadSolToSolInv]
split
@ -110,24 +110,24 @@ lemma quadSolToSolInv_α₁_α₂_zero (S : (PlusU1 n).Sols) (h : α₁ S.1 = 0)
rw [quadSolToSolInv_1, α₂_AF S, h]
simp
lemma quadSolToSolInv_α₁_α₂_neq_zero (S : (PlusU1 n).Sols) (h : α₁ S.1 ≠ 0) :
¬ (α₁ (quadSolToSolInv S).1 = 0 ∧ α₂ (quadSolToSolInv S).1 = 0) := by
lemma quadSolToSolInv_α₁_α₂_neq_zero (S : (PlusU1 n).Sols) (h : α₁ S.1 ≠ 0) :
¬ (α₁ (quadSolToSolInv S).1 = 0 ∧ α₂ (quadSolToSolInv S).1 = 0) := by
rw [not_and, quadSolToSolInv_1, α₂_AF S]
intro hn
simp_all
lemma quadSolToSolInv_special (S : (PlusU1 n).Sols) (h : α₁ S.1 = 0) :
special (quadSolToSolInv S).1 (quadSolToSolInv S).2.1 (quadSolToSolInv S).2.2
(quadSolToSolInv_α₁_α₂_zero S h).1 (quadSolToSolInv_α₁_α₂_zero S h).2 = S := by
(quadSolToSolInv_α₁_α₂_zero S h).1 (quadSolToSolInv_α₁_α₂_zero S h).2 = S := by
simp [quadSolToSolInv_1]
rw [show (quadSolToSolInv S).2.1 = 1 by rw [quadSolToSolInv, if_pos h]]
rw [show (quadSolToSolInv S).2.2 = 0 by rw [quadSolToSolInv, if_pos h]]
rw [show (quadSolToSolInv S).2.1 = 1 by rw [quadSolToSolInv, if_pos h]]
rw [show (quadSolToSolInv S).2.2 = 0 by rw [quadSolToSolInv, if_pos h]]
rw [special_on_AF]
lemma quadSolToSolInv_generic (S : (PlusU1 n).Sols) (h : α₁ S.1 ≠ 0) :
(quadSolToSolInv S).2.1 • generic (quadSolToSolInv S).1 = S := by
(quadSolToSolInv S).2.1 • generic (quadSolToSolInv S).1 = S := by
simp [quadSolToSolInv_1]
rw [show (quadSolToSolInv S).2.1 = (α₁ S.1)⁻¹ by rw [quadSolToSolInv, if_neg h]]
rw [show (quadSolToSolInv S).2.1 = (α₁ S.1)⁻¹ by rw [quadSolToSolInv, if_neg h]]
rw [generic_on_AF_α₁_ne_zero S h]
lemma quadSolToSolInv_rightInverse : Function.RightInverse (@quadSolToSolInv n) quadSolToSol := by