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Merge pull request #84 from HEPLean/Update-versions

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refactor: Remove non-initial double spaces
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Joseph Tooby-Smith 2024-07-12 11:41:54 -04:00 • committed by GitHub
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74 changed files with 709 additions and 690 deletions
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20
HepLean/AnomalyCancellation/Basic.lean
View file

@ -108,7 +108,7 @@ instance linSolsAddCommMonoid (χ : ACCSystemLinear) :
add_zero S := by
apply LinSols.ext
exact χ.chargesAddCommMonoid.add_zero _
nsmul n S := ⟨n • S.val, by
nsmul n S := ⟨n • S.val, by
intro i
rw [nsmul_eq_smul_cast ℚ]
erw [(χ.linearACCs i).map_smul, S.linearSol i]
@ -122,7 +122,7 @@ instance linSolsAddCommMonoid (χ : ACCSystemLinear) :
/-- An instance providing the operations and properties for `LinSols` to form an
module over `ℚ`. -/
@[simps!]
instance linSolsModule (χ : ACCSystemLinear) : Module ℚ χ.LinSols where
instance linSolsModule (χ : ACCSystemLinear) : Module ℚ χ.LinSols where
smul a S := ⟨a • S.val, by
intro i
rw [(χ.linearACCs i).map_smul, S.linearSol i]
@ -177,13 +177,13 @@ structure QuadSols (χ : ACCSystemQuad) extends χ.LinSols where
@[ext]
lemma QuadSols.ext {χ : ACCSystemQuad} {S T : χ.QuadSols} (h : S.val = T.val) :
S = T := by
have h := ACCSystemLinear.LinSols.ext h
have h := ACCSystemLinear.LinSols.ext h
cases' S
simp_all only
/-- An instance giving the properties and structures to define an action of `ℚ` on `QuadSols`. -/
instance quadSolsMulAction (χ : ACCSystemQuad) : MulAction ℚ χ.QuadSols where
smul a S := ⟨a • S.toLinSols , by
smul a S := ⟨a • S.toLinSols , by
intro i
erw [(χ.quadraticACCs i).map_smul]
rw [S.quadSol i]
@ -197,8 +197,8 @@ 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
toFun := QuadSols.toLinSols
def quadSolsInclLinSols (χ : ACCSystemQuad) : χ.QuadSols →[ℚ] χ.LinSols where
toFun := QuadSols.toLinSols
map_smul' _ _ := rfl
/-- If there are no quadratic equations (i.e. no U(1)'s in the underlying gauge group. The inclusion
@ -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
@ -239,7 +239,7 @@ def IsSolution (χ : ACCSystem) (S : χ.Charges) : Prop :=
/-- An instance giving the properties and structures to define an action of `ℚ` on `Sols`. -/
instance solsMulAction (χ : ACCSystem) : MulAction ℚ χ.Sols where
smul a S := ⟨a • S.toQuadSols , by
smul a S := ⟨a • S.toQuadSols, by
erw [(χ.cubicACC).map_smul]
rw [S.cubicSol]
simp⟩
@ -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`. -/

8
HepLean/AnomalyCancellation/GroupActions.lean
View file

@ -22,7 +22,7 @@ From this we define
the vector spaces of charges under which the anomaly equations are invariant.
-/
structure ACCSystemGroupAction (χ : ACCSystem) where
/-- The underlying type of the group-/
/-- The underlying type of the group. -/
group : Type
/-- An instance given group the structure of a group. -/
groupInst : Group group
@ -100,11 +100,11 @@ instance quadSolAction {χ : ACCSystem} (G : ACCSystemGroupAction χ) :
rfl
lemma linSolRep_quadSolAction_commute {χ : ACCSystem} (G : ACCSystemGroupAction χ) (g : G.group)
(S : χ.QuadSols) : χ.quadSolsInclLinSols (G.quadSolAction.toFun S g) =
(S : χ.QuadSols) : χ.quadSolsInclLinSols (G.quadSolAction.toFun S g) =
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)

4
HepLean/AnomalyCancellation/MSSMNu/B3.lean
View file

@ -34,10 +34,10 @@ 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
| 2, 2 => 1
| 3, 0 => - 3
| 3, 1 => - 3
| 3, 2 => 3

36
HepLean/AnomalyCancellation/MSSMNu/Basic.lean
View file

@ -43,14 +43,14 @@ 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
/-- An equivalence between `MSSMCharges.charges` and `(Fin 18 → ℚ) × (Fin 2 → ℚ)`. This
splits the charges up into the SM and the additional ones for the MSSM. -/
@[simps!]
def toSplitSMPlusH : MSSMCharges.Charges ≃ (Fin 18 → ℚ) × (Fin 2 → ℚ) :=
def toSplitSMPlusH : MSSMCharges.Charges ≃ (Fin 18 → ℚ) × (Fin 2 → ℚ) :=
toSMPlusH.trans splitSMPlusH
/-- An equivalence between `(Fin 18 → ℚ)` and `(Fin 6 → Fin 3 → ℚ)`. -/
@ -74,7 +74,7 @@ def toSMSpecies (i : Fin 6) : MSSMCharges.Charges →ₗ[ℚ] MSSMSpecies.Charge
map_add' _ _ := by rfl
map_smul' _ _ := by rfl
lemma toSMSpecies_toSpecies_inv (i : Fin 6) (f : (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)) :
lemma toSMSpecies_toSpecies_inv (i : Fin 6) (f : (Fin 6 → Fin 3 → ℚ) × (Fin 2 → ℚ)) :
(toSMSpecies i) (toSpecies.symm f) = f.1 i := by
change (Prod.fst ∘ toSpecies ∘ toSpecies.symm ) _ i= f.1 i
simp
@ -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

8
HepLean/BeyondTheStandardModel/TwoHDM/Basic.lean
View file

@ -25,10 +25,10 @@ noncomputable section
/-- The potential of the two Higgs doublet model. -/
def potential (m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ : ℝ)
(m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ : ℂ) (Φ1 Φ2 : HiggsField) (x : SpaceTime) : ℝ :=
m₁₁2 * ‖Φ1‖_H ^ 2 x + m₂₂2 * ‖Φ2‖_H ^ 2 x - (m₁₂2 * ⟪Φ1, Φ2⟫_H x + conj m₁₂2 * ⟪Φ2, Φ1⟫_H x).re
+ 1/2 * 𝓵₁ * ‖Φ1‖_H ^ 2 x * ‖Φ1‖_H ^ 2 x + 1/2 * 𝓵₂ * ‖Φ2‖_H ^ 2 x * ‖Φ2‖_H ^ 2 x
m₁₁2 * ‖Φ1‖_H ^ 2 x + m₂₂2 * ‖Φ2‖_H ^ 2 x - (m₁₂2 * ⟪Φ1, Φ2⟫_H x + conj m₁₂2 * ⟪Φ2, Φ1⟫_H x).re
+ 1/2 * 𝓵₁ * ‖Φ1‖_H ^ 2 x * ‖Φ1‖_H ^ 2 x + 1/2 * 𝓵₂ * ‖Φ2‖_H ^ 2 x * ‖Φ2‖_H ^ 2 x
+ 𝓵₃ * ‖Φ1‖_H ^ 2 x * ‖Φ2‖_H ^ 2 x
+ 𝓵₄ * ‖⟪Φ1, Φ2⟫_H x‖ ^ 2 + (1/2 * 𝓵₅ * ⟪Φ1, Φ2⟫_H x ^ 2 + 1/2 * conj 𝓵₅ * ⟪Φ2, Φ1⟫_H x ^ 2).re
+ 𝓵₄ * ‖⟪Φ1, Φ2⟫_H x‖ ^ 2 + (1/2 * 𝓵₅ * ⟪Φ1, Φ2⟫_H x ^ 2 + 1/2 * conj 𝓵₅ * ⟪Φ2, Φ1⟫_H x ^ 2).re
+ (𝓵₆ * ‖Φ1‖_H ^ 2 x * ⟪Φ1, Φ2⟫_H x + conj 𝓵₆ * ‖Φ1‖_H ^ 2 x * ⟪Φ2, Φ1⟫_H x).re
+ (𝓵₇ * ‖Φ2‖_H ^ 2 x * ⟪Φ1, Φ2⟫_H x + conj 𝓵₇ * ‖Φ2‖_H ^ 2 x * ⟪Φ2, Φ1⟫_H x).re
@ -61,7 +61,7 @@ lemma swap_fields :
/-- If `Φ₂` is zero the potential reduces to the Higgs potential on `Φ₁`. -/
lemma right_zero : potential m₁₁2 m₂₂2 𝓵₁ 𝓵₂ 𝓵₃ 𝓵₄ m₁₂2 𝓵₅ 𝓵₆ 𝓵₇ Φ1 0 =
StandardModel.HiggsField.potential (- m₁₁2) (𝓵₁/2) Φ1 := by
StandardModel.HiggsField.potential (- m₁₁2) (𝓵₁/2) Φ1 := by
funext x
simp only [potential, normSq, ContMDiffSection.coe_zero, Pi.zero_apply, norm_zero, ne_eq,
OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, mul_zero, add_zero, innerProd_right_zero,

82
HepLean/FeynmanDiagrams/Basic.lean
View file

@ -72,14 +72,14 @@ def toEdge {𝓔 𝓥 : Type} : Over (P.HalfEdgeLabel × 𝓔 × 𝓥) ⥤ Over
/-- The functor from `Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)`
to `Over (P.HalfEdgeLabel)` induced by projections on products. -/
@[simps!]
def toHalfEdge {𝓔 𝓥 : Type} : Over (P.HalfEdgeLabel × 𝓔 × 𝓥) ⥤ Over P.HalfEdgeLabel :=
def toHalfEdge {𝓔 𝓥 : Type} : Over (P.HalfEdgeLabel × 𝓔 × 𝓥) ⥤ Over P.HalfEdgeLabel :=
Over.map Prod.fst
/-- The functor from `Over P.VertexLabel` to `Type` induced by pull-back along insertion of
`v : P.VertexLabel`. -/
@[simps!]
def preimageType' {𝓥 : Type} (v : 𝓥) : Over 𝓥 ⥤ Type where
obj := fun f => f.hom ⁻¹' {v}
obj := fun f => f.hom ⁻¹' {v}
map {f g} F := fun x =>
⟨F.left x.1, by
have h := congrFun F.w x
@ -88,19 +88,19 @@ def preimageType' {𝓥 : Type} (v : 𝓥) : Over 𝓥 ⥤ Type where
simpa [h] using x.2⟩
/-- The functor from `Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)` to
`Over P.HalfEdgeLabel` induced by pull-back along insertion of `v : P.VertexLabel`. -/
def preimageVertex {𝓔 𝓥 : Type} (v : 𝓥) :
`Over P.HalfEdgeLabel` induced by pull-back along insertion of `v : P.VertexLabel`. -/
def preimageVertex {𝓔 𝓥 : Type} (v : 𝓥) :
Over (P.HalfEdgeLabel × 𝓔 × 𝓥) ⥤ Over P.HalfEdgeLabel where
obj f := Over.mk (fun x => Prod.fst (f.hom x.1) :
obj f := Over.mk (fun x => Prod.fst (f.hom x.1) :
(P.toVertex ⋙ preimageType' v).obj f ⟶ P.HalfEdgeLabel)
map {f g} F := Over.homMk ((P.toVertex ⋙ preimageType' v).map F)
(funext <| fun x => congrArg Prod.fst <| congrFun F.w x.1)
/-- The functor from `Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)` to
`Over P.HalfEdgeLabel` induced by pull-back along insertion of `v : P.EdgeLabel`. -/
`Over P.HalfEdgeLabel` induced by pull-back along insertion of `v : P.EdgeLabel`. -/
def preimageEdge {𝓔 𝓥 : Type} (v : 𝓔) :
Over (P.HalfEdgeLabel × 𝓔 × 𝓥) ⥤ Over P.HalfEdgeLabel where
obj f := Over.mk (fun x => Prod.fst (f.hom x.1) :
Over (P.HalfEdgeLabel × 𝓔 × 𝓥) ⥤ Over P.HalfEdgeLabel where
obj f := Over.mk (fun x => Prod.fst (f.hom x.1) :
(P.toEdge ⋙ preimageType' v).obj f ⟶ P.HalfEdgeLabel)
map {f g} F := Over.homMk ((P.toEdge ⋙ preimageType' v).map F)
(funext <| fun x => congrArg Prod.fst <| congrFun F.w x.1)
@ -117,7 +117,7 @@ properties.
-/
/-- A set of conditions on `PreFeynmanRule` for it to be considered finite. -/
/-- A set of conditions on `PreFeynmanRule` for it to be considered finite. -/
class IsFinitePreFeynmanRule (P : PreFeynmanRule) where
/-- The type of edge labels is decidable. -/
edgeLabelDecidable : DecidableEq P.EdgeLabel
@ -173,31 +173,31 @@ instance preimageEdgeDecidablePred {𝓔 𝓥 : Type} [DecidableEq 𝓔] (v :
| isFalse h => isFalse h
instance preimageVertexDecidable {𝓔 𝓥 : Type} (v : 𝓥)
(F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [DecidableEq F.left] :
(F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [DecidableEq F.left] :
DecidableEq ((P.preimageVertex v).obj F).left := Subtype.instDecidableEq
instance preimageEdgeDecidable {𝓔 𝓥 : Type} (v : 𝓔)
(F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [DecidableEq F.left] :
(F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [DecidableEq F.left] :
DecidableEq ((P.preimageEdge v).obj F).left := Subtype.instDecidableEq
instance preimageVertexFintype {𝓔 𝓥 : Type} [DecidableEq 𝓥]
(v : 𝓥) (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [h : Fintype F.left] :
Fintype ((P.preimageVertex v).obj F).left := @Subtype.fintype _ _ _ h
instance preimageVertexFintype {𝓔 𝓥 : Type} [DecidableEq 𝓥]
(v : 𝓥) (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [h : Fintype F.left] :
Fintype ((P.preimageVertex v).obj F).left := @Subtype.fintype _ _ _ h
instance preimageEdgeFintype {𝓔 𝓥 : Type} [DecidableEq 𝓔]
(v : 𝓔) (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [h : Fintype F.left] :
Fintype ((P.preimageEdge v).obj F).left := @Subtype.fintype _ _ _ h
(v : 𝓔) (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [h : Fintype F.left] :
Fintype ((P.preimageEdge v).obj F).left := @Subtype.fintype _ _ _ h
instance preimageVertexMapFintype [IsFinitePreFeynmanRule P] {𝓔 𝓥 : Type}
[DecidableEq 𝓥] (v : 𝓥) (f : 𝓥 ⟶ P.VertexLabel) (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥))
instance preimageVertexMapFintype [IsFinitePreFeynmanRule P] {𝓔 𝓥 : Type}
[DecidableEq 𝓥] (v : 𝓥) (f : 𝓥 ⟶ P.VertexLabel) (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥))
[Fintype F.left] :
Fintype ((P.vertexLabelMap (f v)).left → ((P.preimageVertex v).obj F).left) :=
Fintype ((P.vertexLabelMap (f v)).left → ((P.preimageVertex v).obj F).left) :=
Pi.fintype
instance preimageEdgeMapFintype [IsFinitePreFeynmanRule P] {𝓔 𝓥 : Type}
[DecidableEq 𝓔] (v : 𝓔) (f : 𝓔 ⟶ P.EdgeLabel) (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥))
instance preimageEdgeMapFintype [IsFinitePreFeynmanRule P] {𝓔 𝓥 : Type}
[DecidableEq 𝓔] (v : 𝓔) (f : 𝓔 ⟶ P.EdgeLabel) (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥))
[Fintype F.left] :
Fintype ((P.edgeLabelMap (f v)).left → ((P.preimageEdge v).obj F).left) :=
Fintype ((P.edgeLabelMap (f v)).left → ((P.preimageEdge v).obj F).left) :=
Pi.fintype
/-!
@ -214,11 +214,11 @@ We will show that for `IsFinitePreFeynmanRule` the condition of been external is
def External {P : PreFeynmanRule} (v : P.VertexLabel) : Prop :=
IsIsomorphic (P.vertexLabelMap v).left (Fin 1)
lemma external_iff_exists_bijection {P : PreFeynmanRule} (v : P.VertexLabel) :
lemma external_iff_exists_bijection {P : PreFeynmanRule} (v : P.VertexLabel) :
External v ↔ ∃ (κ : (P.vertexLabelMap v).left → Fin 1), Function.Bijective κ := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
obtain ⟨κ, κm1, h1, h2⟩ := h
let f : (P.vertexLabelMap v).left ≅ (Fin 1) := ⟨κ, κm1, h1, h2⟩
let f : (P.vertexLabelMap v).left ≅ (Fin 1) := ⟨κ, κm1, h1, h2⟩
exact ⟨f.hom, (isIso_iff_bijective f.hom).mp $ Iso.isIso_hom f⟩
obtain ⟨κ, h1⟩ := h
let f : (P.vertexLabelMap v).left ⟶ (Fin 1) := κ
@ -248,7 +248,7 @@ def DiagramVertexProp {𝓔 𝓥 : Type} (F : Over (P.HalfEdgeLabel × 𝓔 ×
`F : Over (P.HalfEdgeLabel × P.EdgeLabel × P.VertexLabel)` for it to be a Feynman diagram.
This condition corresponds to the edges of `F` having the correct half-edges associated
with them. -/
def DiagramEdgeProp {𝓔 𝓥 : Type} (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥))
def DiagramEdgeProp {𝓔 𝓥 : Type} (F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥))
(f : 𝓔 ⟶ P.EdgeLabel) :=
∀ v, IsIsomorphic (P.edgeLabelMap (f v)) ((P.preimageEdge v).obj F)
@ -274,7 +274,7 @@ lemma diagramEdgeProp_iff {𝓔 𝓥 : Type} (F : Over (P.HalfEdgeLabel × 𝓔
∧ ((P.preimageEdge v).obj F).hom ∘ κ = (P.edgeLabelMap (f v)).hom := by
refine Iff.intro (fun h v => ?_) (fun h v => ?_)
obtain ⟨κ, κm1, h1, h2⟩ := h v
let f := (Over.forget P.HalfEdgeLabel).mapIso ⟨κ, κm1, h1, h2⟩
let f := (Over.forget P.HalfEdgeLabel).mapIso ⟨κ, κm1, h1, h2⟩
refine ⟨f.hom, (isIso_iff_bijective f.hom).mp $ Iso.isIso_hom f, κ.w⟩
obtain ⟨κ, h1, h2⟩ := h v
let f : (P.edgeLabelMap (f v)) ⟶ ((P.preimageEdge v).obj F) := Over.homMk κ h2
@ -289,17 +289,17 @@ instance diagramVertexPropDecidable
@decidable_of_decidable_of_iff _ _
(@Fintype.decidableForallFintype _ _ (fun _ => @Fintype.decidableExistsFintype _ _
(fun _ => @And.decidable _ _ _ (@Fintype.decidablePiFintype _ _
(fun _ => P.preFeynmanRuleDecEq𝓱𝓔) _ _ _)) _ ) _)
(fun _ => P.preFeynmanRuleDecEq𝓱𝓔) _ _ _)) _ ) _)
(P.diagramVertexProp_iff F f).symm
instance diagramEdgePropDecidable
[IsFinitePreFeynmanRule P] {𝓔 𝓥 : Type} [Fintype 𝓔] [DecidableEq 𝓔]
[IsFinitePreFeynmanRule P] {𝓔 𝓥 : Type} [Fintype 𝓔] [DecidableEq 𝓔]
(F : Over (P.HalfEdgeLabel × 𝓔 × 𝓥)) [DecidableEq F.left] [Fintype F.left]
(f : 𝓔 ⟶ P.EdgeLabel) : Decidable (P.DiagramEdgeProp F f) :=
@decidable_of_decidable_of_iff _ _
(@Fintype.decidableForallFintype _ _ (fun _ => @Fintype.decidableExistsFintype _ _
(fun _ => @And.decidable _ _ _ (@Fintype.decidablePiFintype _ _
(fun _ => P.preFeynmanRuleDecEq𝓱𝓔) _ _ _)) _ ) _)
(fun _ => P.preFeynmanRuleDecEq𝓱𝓔) _ _ _)) _ ) _)
(P.diagramEdgeProp_iff F f).symm
end PreFeynmanRule
@ -358,8 +358,8 @@ def 𝓱𝓔To𝓥 : Over F.𝓥 := P.toVertex.obj F.𝓱𝓔To𝓔𝓥
/-- The condition which must be satisfied by maps to form a Feynman diagram. -/
def Cond {𝓔 𝓥 𝓱𝓔 : Type} (𝓔𝓞 : 𝓔 → P.EdgeLabel) (𝓥𝓞 : 𝓥 → P.VertexLabel)
(𝓱𝓔To𝓔𝓥 : 𝓱𝓔 → P.HalfEdgeLabel × 𝓔 × 𝓥) : Prop :=
P.DiagramEdgeProp (Over.mk 𝓱𝓔To𝓔𝓥) 𝓔𝓞 ∧
(𝓱𝓔To𝓔𝓥 : 𝓱𝓔 → P.HalfEdgeLabel × 𝓔 × 𝓥) : Prop :=
P.DiagramEdgeProp (Over.mk 𝓱𝓔To𝓔𝓥) 𝓔𝓞 ∧
P.DiagramVertexProp (Over.mk 𝓱𝓔To𝓔𝓥) 𝓥𝓞
lemma cond_self : Cond F.𝓔𝓞.hom F.𝓥𝓞.hom F.𝓱𝓔To𝓔𝓥.hom :=
@ -441,7 +441,7 @@ instance {F : FeynmanDiagram P} [IsFiniteDiagram F] : DecidableEq F.𝓱𝓔 :=
IsFiniteDiagram.𝓱𝓔DecidableEq
instance {F : FeynmanDiagram P} [IsFiniteDiagram F] (i : F.𝓱𝓔) (j : F.𝓔) :
Decidable (F.𝓱𝓔To𝓔.hom i = j) := IsFiniteDiagram.𝓔DecidableEq (F.𝓱𝓔To𝓔.hom i) j
Decidable (F.𝓱𝓔To𝓔.hom i = j) := IsFiniteDiagram.𝓔DecidableEq (F.𝓱𝓔To𝓔.hom i) j
instance fintypeProdHalfEdgeLabel𝓔𝓥 {F : FeynmanDiagram P} [IsFinitePreFeynmanRule P]
[IsFiniteDiagram F] : DecidableEq (P.HalfEdgeLabel × F.𝓔 × F.𝓥) :=
@ -457,15 +457,15 @@ This will be used to define the category of Feynman diagrams.
-/
/-- Given two maps `F.𝓔 ⟶ G.𝓔` and `F.𝓥 ⟶ G.𝓥` the corresponding map
`P.HalfEdgeLabel × F.𝓔 × F.𝓥 → P.HalfEdgeLabel × G.𝓔 × G.𝓥`. -/
`P.HalfEdgeLabel × F.𝓔 × F.𝓥 → P.HalfEdgeLabel × G.𝓔 × G.𝓥`. -/
@[simps!]
def edgeVertexMap {F G : FeynmanDiagram P} (𝓔 : F.𝓔 ⟶ G.𝓔) (𝓥 : F.𝓥 ⟶ G.𝓥) :
P.HalfEdgeLabel × F.𝓔 × F.𝓥 → P.HalfEdgeLabel × G.𝓔 × G.𝓥 :=
P.HalfEdgeLabel × F.𝓔 × F.𝓥 → P.HalfEdgeLabel × G.𝓔 × G.𝓥 :=
fun x => ⟨x.1, 𝓔 x.2.1, 𝓥 x.2.2⟩
/-- Given equivalences `F.𝓔 ≃ G.𝓔` and `F.𝓥 ≃ G.𝓥`, the induced equivalence between
`P.HalfEdgeLabel × F.𝓔 × F.𝓥` and `P.HalfEdgeLabel × G.𝓔 × G.𝓥 ` -/
def edgeVertexEquiv {F G : FeynmanDiagram P} (𝓔 : F.𝓔 ≃ G.𝓔) (𝓥 : F.𝓥 ≃ G.𝓥) :
def edgeVertexEquiv {F G : FeynmanDiagram P} (𝓔 : F.𝓔 ≃ G.𝓔) (𝓥 : F.𝓥 ≃ G.𝓥) :
P.HalfEdgeLabel × F.𝓔 × F.𝓥 ≃ P.HalfEdgeLabel × G.𝓔 × G.𝓥 where
toFun := edgeVertexMap 𝓔.toFun 𝓥.toFun
invFun := edgeVertexMap 𝓔.invFun 𝓥.invFun
@ -539,7 +539,7 @@ lemma ext {F G : FeynmanDiagram P} {f g : Hom F G} (h𝓔 : f.𝓔 = g.𝓔)
def Cond {F G : FeynmanDiagram P} (𝓔 : F.𝓔 → G.𝓔) (𝓥 : F.𝓥 → G.𝓥) (𝓱𝓔 : F.𝓱𝓔 → G.𝓱𝓔) : Prop :=
(∀ x, G.𝓔𝓞.hom (𝓔 x) = F.𝓔𝓞.hom x) ∧
(∀ x, G.𝓥𝓞.hom (𝓥 x) = F.𝓥𝓞.hom x) ∧
(∀ x, G.𝓱𝓔To𝓔𝓥.hom (𝓱𝓔 x) = edgeVertexMap 𝓔 𝓥 (F.𝓱𝓔To𝓔𝓥.hom x))
(∀ x, G.𝓱𝓔To𝓔𝓥.hom (𝓱𝓔 x) = edgeVertexMap 𝓔 𝓥 (F.𝓱𝓔To𝓔𝓥.hom x))
lemma cond_satisfied {F G : FeynmanDiagram P} (f : Hom F G) :
Cond f.𝓔 f.𝓥 f.𝓱𝓔 :=
@ -564,9 +564,9 @@ instance {F G : FeynmanDiagram P} [IsFiniteDiagram F] [IsFinitePreFeynmanRule P]
@Fintype.decidableForallFintype _ _ (fun _ => preFeynmanRuleDecEq𝓥 P _ _) _
instance {F G : FeynmanDiagram P} [IsFiniteDiagram F] [IsFiniteDiagram G] [IsFinitePreFeynmanRule P]
(𝓔 : F.𝓔 → G.𝓔) (𝓥 : F.𝓥 → G.𝓥) (𝓱𝓔 : F.𝓱𝓔 → G.𝓱𝓔) :
Decidable (∀ x, G.𝓱𝓔To𝓔𝓥.hom (𝓱𝓔 x) = edgeVertexMap 𝓔 𝓥 (F.𝓱𝓔To𝓔𝓥.hom x)) :=
@Fintype.decidableForallFintype _ _ (fun _ => fintypeProdHalfEdgeLabel𝓔𝓥 _ _) _
(𝓔 : F.𝓔 → G.𝓔) (𝓥 : F.𝓥 → G.𝓥) (𝓱𝓔 : F.𝓱𝓔 → G.𝓱𝓔) :
Decidable (∀ x, G.𝓱𝓔To𝓔𝓥.hom (𝓱𝓔 x) = edgeVertexMap 𝓔 𝓥 (F.𝓱𝓔To𝓔𝓥.hom x)) :=
@Fintype.decidableForallFintype _ _ (fun _ => fintypeProdHalfEdgeLabel𝓔𝓥 _ _) _
instance {F G : FeynmanDiagram P} [IsFiniteDiagram F] [IsFiniteDiagram G] [IsFinitePreFeynmanRule P]
(𝓔 : F.𝓔 → G.𝓔) (𝓥 : F.𝓥 → G.𝓥) (𝓱𝓔 : F.𝓱𝓔 → G.𝓱𝓔) : Decidable (Cond 𝓔 𝓥 𝓱𝓔) :=
@ -678,7 +678,7 @@ instance [IsFinitePreFeynmanRule P] [IsFiniteDiagram F] : Fintype F.SymmetryType
@[simp]
def cardSymmetryFactor : Cardinal := Cardinal.mk (F.SymmetryType)
/-- The symmetry factor of a Finite Feynman diagram, as a natural number. -/
/-- The symmetry factor of a Finite Feynman diagram, as a natural number. -/
@[simp]
def symmetryFactor [IsFinitePreFeynmanRule P] [IsFiniteDiagram F] : ℕ :=
(Fintype.card F.SymmetryType)
@ -714,7 +714,7 @@ def AdjRelation : F.𝓥 → F.𝓥 → Prop := fun x y =>
instance [IsFiniteDiagram F] : DecidableRel F.AdjRelation := fun _ _ =>
@And.decidable _ _ _ $
@Fintype.decidableExistsFintype _ _ (fun _ => @Fintype.decidableExistsFintype _ _ (
@Fintype.decidableExistsFintype _ _ (fun _ => @Fintype.decidableExistsFintype _ _ (
fun _ => @And.decidable _ _ (instDecidableEq𝓔OfIsFiniteDiagram _ _) $
@And.decidable _ _ (instDecidableEq𝓥OfIsFiniteDiagram _ _)
(instDecidableEq𝓥OfIsFiniteDiagram _ _)) _ ) _

10
HepLean/FeynmanDiagrams/Instances/ComplexScalar.lean
View file

@ -42,22 +42,22 @@ instance (a : ℕ) : OfNat complexScalarFeynmanRules.VertexLabel a where
ofNat := (a : Fin _)
instance : IsFinitePreFeynmanRule complexScalarFeynmanRules where
edgeLabelDecidable := instDecidableEqFin _
edgeLabelDecidable := instDecidableEqFin _
vertexLabelDecidable := instDecidableEqFin _
halfEdgeLabelDecidable := instDecidableEqFin _
vertexMapFintype := fun v =>
match v with
| 0 => Fin.fintype _
| 1 => Fin.fintype _
| 2 => Fin.fintype _
| 1 => Fin.fintype _
| 2 => Fin.fintype _
edgeMapFintype := fun v =>
match v with
| 0 => Fin.fintype _
vertexMapDecidable := fun v =>
match v with
| 0 => instDecidableEqFin _
| 1 => instDecidableEqFin _
| 2 => instDecidableEqFin _
| 1 => instDecidableEqFin _
| 2 => instDecidableEqFin _
edgeMapDecidable := fun v =>
match v with
| 0 => instDecidableEqFin _

6
HepLean/FeynmanDiagrams/Instances/Phi4.lean
View file

@ -44,20 +44,20 @@ instance (a : ℕ) : OfNat phi4PreFeynmanRules.VertexLabel a where
ofNat := (a : Fin _)
instance : IsFinitePreFeynmanRule phi4PreFeynmanRules where
edgeLabelDecidable := instDecidableEqFin _
edgeLabelDecidable := instDecidableEqFin _
vertexLabelDecidable := instDecidableEqFin _
halfEdgeLabelDecidable := instDecidableEqFin _
vertexMapFintype := fun v =>
match v with
| 0 => Fin.fintype _
| 1 => Fin.fintype _
| 1 => Fin.fintype _
edgeMapFintype := fun v =>
match v with
| 0 => Fin.fintype _
vertexMapDecidable := fun v =>
match v with
| 0 => instDecidableEqFin _
| 1 => instDecidableEqFin _
| 1 => instDecidableEqFin _
edgeMapDecidable := fun v =>
match v with
| 0 => instDecidableEqFin _

20
HepLean/FeynmanDiagrams/Momentum.lean
View file

@ -45,8 +45,8 @@ We define the direct sum of the edge and vertex momentum spaces.
-/
/-- The type which assocaites to each half-edge a `1`-dimensional vector space.
Corresponding to that spanned by its momentum. -/
def HalfEdgeMomenta : Type := F.𝓱𝓔 → ℝ
Corresponding to that spanned by its momentum. -/
def HalfEdgeMomenta : Type := F.𝓱𝓔 → ℝ
instance : AddCommGroup F.HalfEdgeMomenta := Pi.addCommGroup
@ -54,14 +54,14 @@ instance : Module ℝ F.HalfEdgeMomenta := Pi.module _ _ _
/-- An auxillary function used to define the Euclidean inner product on `F.HalfEdgeMomenta`. -/
def euclidInnerAux (x : F.HalfEdgeMomenta) : F.HalfEdgeMomenta →ₗ[ℝ] ℝ where
toFun y := ∑ i, (x i) * (y i)
toFun y := ∑ i, (x i) * (y i)
map_add' z y :=
show (∑ i, (x i) * (z i + y i)) = (∑ i, x i * z i) + ∑ i, x i * (y i) by
simp only [mul_add, Finset.sum_add_distrib]
map_smul' c y :=
show (∑ i, x i * (c * y i)) = c * ∑ i, x i * y i by
rw [Finset.mul_sum]
refine Finset.sum_congr rfl (fun _ _ => by ring)
refine Finset.sum_congr rfl (fun _ _ => by ring)
lemma euclidInnerAux_symm (x y : F.HalfEdgeMomenta) :
F.euclidInnerAux x y = F.euclidInnerAux y x := Finset.sum_congr rfl (fun _ _ => by ring)
@ -70,15 +70,15 @@ lemma euclidInnerAux_symm (x y : F.HalfEdgeMomenta) :
def euclidInner : F.HalfEdgeMomenta →ₗ[ℝ] F.HalfEdgeMomenta →ₗ[ℝ] ℝ where
toFun x := F.euclidInnerAux x
map_add' x y := by
refine LinearMap.ext (fun z => ?_)
refine LinearMap.ext (fun z => ?_)
simp only [euclidInnerAux_symm, map_add, LinearMap.add_apply]
map_smul' c x := by
refine LinearMap.ext (fun z => ?_)
refine LinearMap.ext (fun z => ?_)
simp only [euclidInnerAux_symm, LinearMapClass.map_smul, smul_eq_mul, RingHom.id_apply,
LinearMap.smul_apply]
/-- The type which assocaites to each ege a `1`-dimensional vector space.
Corresponding to that spanned by its total outflowing momentum. -/
Corresponding to that spanned by its total outflowing momentum. -/
def EdgeMomenta : Type := F.𝓔 → ℝ
instance : AddCommGroup F.EdgeMomenta := Pi.addCommGroup
@ -86,7 +86,7 @@ instance : AddCommGroup F.EdgeMomenta := Pi.addCommGroup
instance : Module ℝ F.EdgeMomenta := Pi.module _ _ _
/-- The type which assocaites to each ege a `1`-dimensional vector space.
Corresponding to that spanned by its total inflowing momentum. -/
Corresponding to that spanned by its total inflowing momentum. -/
def VertexMomenta : Type := F.𝓥 → ℝ
instance : AddCommGroup F.VertexMomenta := Pi.addCommGroup
@ -96,8 +96,8 @@ instance : Module ℝ F.VertexMomenta := Pi.module _ _ _
/-- The map from `Fin 2` to `Type` landing on `EdgeMomenta` and `VertexMomenta`. -/
def EdgeVertexMomentaMap : Fin 2 → Type := fun i =>
match i with
| 0 => F.EdgeMomenta
| 1 => F.VertexMomenta
| 0 => F.EdgeMomenta
| 1 => F.VertexMomenta
instance (i : Fin 2) : AddCommGroup (EdgeVertexMomentaMap F i) :=
match i with

16
HepLean/FlavorPhysics/CKMMatrix/Basic.lean
View file

@ -27,7 +27,7 @@ noncomputable section
leading diagonal. -/
@[simp]
def phaseShiftMatrix (a b c : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
![![cexp (I * a), 0, 0], ![0, cexp (I * b), 0], ![0, 0, cexp (I * c)]]
![![cexp (I * a), 0, 0], ![0, cexp (I * b), 0], ![0, 0, cexp (I * c)]]
lemma phaseShiftMatrix_one : phaseShiftMatrix 0 0 0 = 1 := by
ext i j
@ -49,7 +49,7 @@ lemma phaseShiftMatrix_mul (a b c d e f : ℝ) :
fin_cases i <;> fin_cases j <;>
simp [Matrix.mul_apply, Fin.sum_univ_three]
any_goals rw [mul_add, exp_add]
change cexp (I * ↑c) * 0 = 0
change cexp (I * ↑c) * 0 = 0
simp
/-- Given three real numbers `a b c` the unitary matrix with `exp (I * a)` etc on the
@ -161,7 +161,7 @@ lemma equiv (V : CKMMatrix) (a b c d e f : ℝ) :
rfl
lemma ud (V : CKMMatrix) (a b c d e f : ℝ) :
(phaseShiftApply V a b c d e f).1 0 0 = cexp (a * I + d * I) * V.1 0 0 := by
(phaseShiftApply V a b c d e f).1 0 0 = cexp (a * I + d * I) * V.1 0 0 := by
simp only [Fin.isValue, phaseShiftApply_coe]
rw [mul_apply, Fin.sum_univ_three]
rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
@ -173,7 +173,7 @@ lemma ud (V : CKMMatrix) (a b c d e f : ℝ) :
ring_nf
lemma us (V : CKMMatrix) (a b c d e f : ℝ) :
(phaseShiftApply V a b c d e f).1 0 1 = cexp (a * I + e * I) * V.1 0 1 := by
(phaseShiftApply V a b c d e f).1 0 1 = cexp (a * I + e * I) * V.1 0 1 := by
simp only [Fin.isValue, phaseShiftApply_coe]
rw [mul_apply, Fin.sum_univ_three]
rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
@ -184,7 +184,7 @@ lemma us (V : CKMMatrix) (a b c d e f : ℝ) :
ring_nf
lemma ub (V : CKMMatrix) (a b c d e f : ℝ) :
(phaseShiftApply V a b c d e f).1 0 2 = cexp (a * I + f * I) * V.1 0 2 := by
(phaseShiftApply V a b c d e f).1 0 2 = cexp (a * I + f * I) * V.1 0 2 := by
simp only [Fin.isValue, phaseShiftApply_coe]
rw [mul_apply, Fin.sum_univ_three]
rw [mul_apply, mul_apply, mul_apply, Fin.sum_univ_three, Fin.sum_univ_three, Fin.sum_univ_three]
@ -236,7 +236,7 @@ lemma td (V : CKMMatrix) (a b c d e f : ℝ) :
simp only [Fin.isValue, cons_val', cons_val_zero, empty_val', cons_val_fin_one, vecCons_const,
cons_val_two, tail_val', head_val', cons_val_one, head_cons, tail_cons, head_fin_const,
zero_mul, add_zero, mul_zero]
change (0 * _ + _ ) * _ + (0 * _ + _ ) * 0 = _
change (0 * _ + _ ) * _ + (0 * _ + _ ) * 0 = _
simp only [Fin.isValue, zero_mul, zero_add, mul_zero, add_zero]
rw [exp_add]
ring_nf
@ -272,7 +272,7 @@ end phaseShiftApply
def VAbs' (V : unitaryGroup (Fin 3) ℂ) (i j : Fin 3) : ℝ := Complex.abs (V i j)
lemma VAbs'_equiv (i j : Fin 3) (V U : CKMMatrix) (h : V ≈ U) :
VAbs' V i j = VAbs' U i j := by
VAbs' V i j = VAbs' U i j := by
simp only [VAbs']
obtain ⟨a, b, c, e, f, g, h⟩ := h
rw [h]
@ -288,7 +288,7 @@ lemma VAbs'_equiv (i j : Fin 3) (V U : CKMMatrix) (h : V ≈ U) :
all_goals change Complex.abs (0 * _ + _) = _
all_goals simp [Complex.abs_exp]
/-- The absolute value of the `(i,j)`th any representative of `⟦V⟧`. -/
/-- The absolute value of the `(i,j)`th any representative of `⟦V⟧`. -/
def VAbs (i j : Fin 3) : Quotient CKMMatrixSetoid → ℝ :=
Quotient.lift (fun V => VAbs' V i j) (VAbs'_equiv i j)

8
HepLean/FlavorPhysics/CKMMatrix/Invariants.lean
View file

@ -28,10 +28,10 @@ namespace Invariant
def jarlskogℂCKM (V : CKMMatrix) : ℂ :=
[V]us * [V]cb * conj [V]ub * conj [V]cs
lemma jarlskogℂCKM_equiv (V U : CKMMatrix) (h : V ≈ U) :
lemma jarlskogℂCKM_equiv (V U : CKMMatrix) (h : V ≈ U) :
jarlskogℂCKM V = jarlskogℂCKM U := by
obtain ⟨a, b, c, e, f, g, h⟩ := h
change V = phaseShiftApply U a b c e f g at h
change V = phaseShiftApply U a b c e f g at h
rw [h]
simp only [jarlskogℂCKM, Fin.isValue, phaseShiftApply.ub,
phaseShiftApply.us, phaseShiftApply.cb, phaseShiftApply.cs]
@ -51,8 +51,8 @@ def jarlskogℂ : Quotient CKMMatrixSetoid → ℂ :=
/-- An invariant for CKM mtrices corresponding to the square of the absolute values
of the `us`, `ub` and `cb` elements multipled together divied by `(VudAbs V ^ 2 + VusAbs V ^2)` .
-/
def VusVubVcdSq (V : Quotient CKMMatrixSetoid) : ℝ :=
VusAbs V ^ 2 * VubAbs V ^ 2 * VcbAbs V ^2 / (VudAbs V ^ 2 + VusAbs V ^2)
def VusVubVcdSq (V : Quotient CKMMatrixSetoid) : ℝ :=
VusAbs V ^ 2 * VubAbs V ^ 2 * VcbAbs V ^2 / (VudAbs V ^ 2 + VusAbs V ^2)
/-- An invariant for CKM matrices. The argument of this invariant is `δ₁₃` in the
standard parameterization. -/

40
HepLean/FlavorPhysics/CKMMatrix/PhaseFreedom.lean
View file

@ -126,7 +126,7 @@ lemma shift_cross_product_phase_zero {V : CKMMatrix} {τ : ℝ}
have hτ0 := congrFun hτ 0
simp [tRow] at hτ0
rw [← hτ0]
rw [← mul_assoc, ← exp_add, h1]
rw [← mul_assoc, ← exp_add, h1]
congr 2
simp only [ofReal_sub, ofReal_neg]
ring
@ -166,15 +166,15 @@ def FstRowThdColRealCond (U : CKMMatrix) : Prop := [U]ud = VudAbs ⟦U⟧ ∧ [U
the cross product of the conjugates of the `u`th and `c`th rows, and the `cd`th and `cs`th
elements are real and related in a set way.
-/
def ubOnePhaseCond (U : CKMMatrix) : Prop :=
def ubOnePhaseCond (U : CKMMatrix) : Prop :=
[U]ud = 0 ∧ [U]us = 0 ∧ [U]cb = 0 ∧ [U]ub = 1 ∧ [U]t = conj [U]u ×₃ conj [U]c
∧ [U]cd = - VcdAbs ⟦U⟧ ∧ [U]cs = √(1 - VcdAbs ⟦U⟧ ^ 2)
lemma fstRowThdColRealCond_shift_solution {V : CKMMatrix} (h1 : a + d = - arg [V]ud)
(h2 : a + e = - arg [V]us) (h3 : b + f = - arg [V]cb)
(h2 : a + e = - arg [V]us) (h3 : b + f = - arg [V]cb)
(h4 : c + f = - arg [V]tb) (h5 : τ = - a - b - c - d - e - f) :
b = - τ + arg [V]ud + arg [V]us + arg [V]tb + a ∧
c = - τ + arg [V]cb + arg [V]ud + arg [V]us + a ∧
b = - τ + arg [V]ud + arg [V]us + arg [V]tb + a ∧
c = - τ + arg [V]cb + arg [V]ud + arg [V]us + a ∧
d = - arg [V]ud - a ∧
e = - arg [V]us - a ∧
f = τ - arg [V]ud - arg [V]us - arg [V]cb - arg [V]tb - a := by
@ -202,11 +202,11 @@ lemma fstRowThdColRealCond_shift_solution {V : CKMMatrix} (h1 : a + d = - arg [V
lemma ubOnePhaseCond_shift_solution {V : CKMMatrix} (h1 : a + f = - arg [V]ub)
(h2 : 0 = - a - b - c - d - e - f)
(h3 : b + d = Real.pi - arg [V]cd) (h5 : b + e = - arg [V]cs) :
c = - Real.pi + arg [V]cd + arg [V]cs + arg [V]ub + b ∧
d = Real.pi - arg [V]cd - b ∧
e = - arg [V]cs - b ∧
f = - arg [V]ub - a := by
(h3 : b + d = Real.pi - arg [V]cd) (h5 : b + e = - arg [V]cs) :
c = - Real.pi + arg [V]cd + arg [V]cs + arg [V]ub + b ∧
d = Real.pi - arg [V]cd - b ∧
e = - arg [V]cs - b ∧
f = - arg [V]ub - a := by
have hf : f = - arg [V]ub - a := by
linear_combination h1
subst hf
@ -227,21 +227,21 @@ lemma fstRowThdColRealCond_holds_up_to_equiv (V : CKMMatrix) :
obtain ⟨τ, hτ⟩ := V.uRow_cross_cRow_eq_tRow
let U : CKMMatrix := phaseShiftApply V
0
(- τ + arg [V]ud + arg [V]us + arg [V]tb )
(- τ + arg [V]cb + arg [V]ud + arg [V]us )
(- τ + arg [V]ud + arg [V]us + arg [V]tb )
(- τ + arg [V]cb + arg [V]ud + arg [V]us )
(- arg [V]ud )
(- arg [V]us)
(τ - arg [V]ud - arg [V]us - arg [V]cb - arg [V]tb)
have hUV : Quotient.mk CKMMatrixSetoid U = ⟦V⟧ := by
simp only [Quotient.eq]
symm
exact phaseShiftApply.equiv _ _ _ _ _ _ _
exact phaseShiftApply.equiv _ _ _ _ _ _ _
use U
apply And.intro
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
rw [hUV]
apply shift_ud_phase_zero _ _ _ _ _ _ _
apply shift_ud_phase_zero _ _ _ _ _ _ _
ring
apply And.intro
rw [hUV]
@ -259,15 +259,15 @@ lemma fstRowThdColRealCond_holds_up_to_equiv (V : CKMMatrix) :
ring
lemma ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0))
(hV : FstRowThdColRealCond V) :
(hV : FstRowThdColRealCond V) :
∃ (U : CKMMatrix), V ≈ U ∧ ubOnePhaseCond U:= by
let U : CKMMatrix := phaseShiftApply V 0 0 (- Real.pi + arg [V]cd + arg [V]cs + arg [V]ub)
(Real.pi - arg [V]cd ) (- arg [V]cs) (- arg [V]ub )
(Real.pi - arg [V]cd ) (- arg [V]cs) (- arg [V]ub )
use U
have hUV : Quotient.mk CKMMatrixSetoid U= ⟦V⟧ := by
simp only [Quotient.eq]
symm
exact phaseShiftApply.equiv _ _ _ _ _ _ _
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
exact phaseShiftApply.equiv _ _ _ _ _ _ _
apply And.intro
@ -294,7 +294,7 @@ lemma ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬ ([V]ud
exact h1
apply And.intro
· have hU1 : [U]ub = VubAbs ⟦V⟧ := by
apply shift_ub_phase_zero _ _ _ _ _ _ _
apply shift_ub_phase_zero _ _ _ _ _ _ _
ring
rw [hU1]
have h1:= (ud_us_neq_zero_iff_ub_neq_one V).mpr.mt hb
@ -307,7 +307,7 @@ lemma ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬ ([V]ud
ring
apply And.intro
· rw [hUV]
apply shift_cd_phase_pi _ _ _ _ _ _ _
apply shift_cd_phase_pi _ _ _ _ _ _ _
ring
have hcs : [U]cs = VcsAbs ⟦U⟧ := by
rw [hUV]
@ -338,7 +338,7 @@ lemma cd_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠
lemma cs_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
(hV : FstRowThdColRealCond V) :
[V]cs = (VtbAbs ⟦V⟧ * VudAbs ⟦V⟧ / (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2))
+ (- VubAbs ⟦V⟧ * VusAbs ⟦V⟧ * VcbAbs ⟦V⟧/ (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2))
+ (- VubAbs ⟦V⟧ * VusAbs ⟦V⟧ * VcbAbs ⟦V⟧/ (VudAbs ⟦V⟧ ^2 + VusAbs ⟦V⟧ ^2))
* cexp (- arg [V]ub * I) := by
have hτ : [V]t = cexp ((0 : ℝ) * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
simp only [ofReal_zero, zero_mul, exp_zero, one_smul]

30
HepLean/FlavorPhysics/CKMMatrix/Relations.lean
View file

@ -64,7 +64,7 @@ lemma normSq_Vud_plus_normSq_Vus (V : CKMMatrix) :
normSq [V]ud + normSq [V]us = 1 - normSq [V]ub := by
linear_combination (fst_row_normalized_normSq V)
lemma VudAbs_sq_add_VusAbs_sq : VudAbs V ^ 2 + VusAbs V ^2 = 1 - VubAbs V ^2 := by
lemma VudAbs_sq_add_VusAbs_sq : VudAbs V ^ 2 + VusAbs V ^2 = 1 - VubAbs V ^2 := by
linear_combination (VAbs_sum_sq_row_eq_one V 0)
lemma ud_us_neq_zero_iff_ub_neq_one (V : CKMMatrix) :
@ -104,7 +104,7 @@ lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℝ {V : CKMMatrix} (hb : [V]ud ≠ 0
exact h2 h3
lemma VAbsub_neq_zero_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid}
(hV : VAbs 0 2 V ≠ 1) :(VudAbs V ^ 2 + VusAbs V ^ 2) ≠ 0 := by
(hV : VAbs 0 2 V ≠ 1) :(VudAbs V ^ 2 + VusAbs V ^ 2) ≠ 0 := by
obtain ⟨V⟩ := V
change VubAbs ⟦V⟧ ≠ 1 at hV
simp only [VubAbs, VAbs, VAbs', Fin.isValue, Quotient.lift_mk] at hV
@ -112,7 +112,7 @@ lemma VAbsub_neq_zero_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid}
simpa [← Complex.sq_abs] using (normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hV)
lemma VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid}
(hV : VAbs 0 2 V ≠ 1) : √(VudAbs V ^ 2 + VusAbs V ^ 2) ≠ 0 := by
(hV : VAbs 0 2 V ≠ 1) : √(VudAbs V ^ 2 + VusAbs V ^ 2) ≠ 0 := by
obtain ⟨V⟩ := V
rw [Real.sqrt_ne_zero (Left.add_nonneg (sq_nonneg _) (sq_nonneg _))]
change VubAbs ⟦V⟧ ≠ 1 at hV
@ -120,7 +120,7 @@ lemma VAbsub_neq_zero_sqrt_Vud_Vus_neq_zero {V : Quotient CKMMatrixSetoid}
rw [← ud_us_neq_zero_iff_ub_neq_one V] at hV
simpa [← Complex.sq_abs] using (normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hV)
lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℂ {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℂ {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
(normSq [V]ud : ℂ) + normSq [V]us ≠ 0 := by
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hb
simp at h1
@ -178,9 +178,9 @@ lemma VAbs_thd_eq_one_snd_eq_zero {V : Quotient CKMMatrixSetoid} {i : Fin 3} (hV
section crossProduct
lemma conj_Vtb_cross_product {V : CKMMatrix} {τ : ℝ}
lemma conj_Vtb_cross_product {V : CKMMatrix} {τ : ℝ}
(hτ : [V]t = cexp (τ * I) • (conj [V]u ×₃ conj [V]c)) :
conj [V]tb = cexp (- τ * I) * ([V]cs * [V]ud - [V]us * [V]cd) := by
conj [V]tb = cexp (- τ * I) * ([V]cs * [V]ud - [V]us * [V]cd) := by
have h1 := congrFun hτ 2
simp [crossProduct, tRow, uRow, cRow] at h1
apply congrArg conj at h1
@ -201,7 +201,7 @@ lemma conj_Vtb_mul_Vud {V : CKMMatrix} {τ : ℝ}
simp [exp_neg]
have h1 := exp_ne_zero (τ * I)
field_simp
have h2 : ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]ud = [V]cs
have h2 : ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]ud = [V]cs
* [V]ud * conj [V]ud - [V]us * ([V]cd * conj [V]ud) := by
ring
rw [h2, V.Vcd_mul_conj_Vud]
@ -217,7 +217,7 @@ lemma conj_Vtb_mul_Vus {V : CKMMatrix} {τ : ℝ}
simp [exp_neg]
have h1 := exp_ne_zero (τ * I)
field_simp
have h2 : ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]us = ([V]cs
have h2 : ([V]cs * [V]ud - [V]us * [V]cd) * conj [V]us = ([V]cs
* conj [V]us) * [V]ud - [V]us * [V]cd * conj [V]us := by
ring
rw [h2, V.Vcs_mul_conj_Vus]
@ -225,18 +225,18 @@ lemma conj_Vtb_mul_Vus {V : CKMMatrix} {τ : ℝ}
simp only [Fin.isValue, neg_mul]
ring
lemma cs_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
lemma cs_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cs = (- conj [V]ub * [V]us * [V]cb +
cexp (τ * I) * conj [V]tb * conj [V]ud) / (normSq [V]ud + normSq [V]us) := by
[V]cs = (- conj [V]ub * [V]us * [V]cb +
cexp (τ * I) * conj [V]tb * conj [V]ud) / (normSq [V]ud + normSq [V]us) := by
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℂ h
rw [conj_Vtb_mul_Vud hτ]
field_simp
ring
lemma cd_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
lemma cd_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cd = - (conj [V]ub * [V]ud * [V]cb + cexp (τ * I) * conj [V]tb * conj [V]us) /
[V]cd = - (conj [V]ub * [V]ud * [V]cb + cexp (τ * I) * conj [V]tb * conj [V]us) /
(normSq [V]ud + normSq [V]us) := by
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℂ h
rw [conj_Vtb_mul_Vus hτ]
@ -247,7 +247,7 @@ end rows
section individual
lemma VAbs_ge_zero (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : 0 ≤ VAbs i j V := by
lemma VAbs_ge_zero (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : 0 ≤ VAbs i j V := by
obtain ⟨V, hV⟩ := Quot.exists_rep V
rw [← hV]
exact Complex.abs.nonneg _
@ -291,7 +291,7 @@ lemma thd_col_normalized_normSq (V : CKMMatrix) :
repeat rw [Complex.sq_abs] at h1
exact h1
lemma cb_eq_zero_of_ud_us_zero {V : CKMMatrix} (h : [V]ud = 0 ∧ [V]us = 0) :
lemma cb_eq_zero_of_ud_us_zero {V : CKMMatrix} (h : [V]ud = 0 ∧ [V]us = 0) :
[V]cb = 0 := by
have h1 := fst_row_normalized_abs V
rw [← thd_col_normalized_abs V] at h1

54
HepLean/FlavorPhysics/CKMMatrix/Rows.lean
View file

@ -130,13 +130,13 @@ lemma tRow_cRow_orthog (V : CKMMatrix) : conj [V]t ⬝ᵥ [V]c = 0 := by
rw [mul_comm (V.1 _ 0) _, mul_comm (V.1 _ 1) _, mul_comm (V.1 _ 2) _] at ht
exact ht
lemma uRow_cross_cRow_conj (V : CKMMatrix) : conj (conj [V]u ×₃ conj [V]c) = [V]u ×₃ [V]c := by
lemma uRow_cross_cRow_conj (V : CKMMatrix) : conj (conj [V]u ×₃ conj [V]c) = [V]u ×₃ [V]c := by
simp only [crossProduct, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, LinearMap.mk₂_apply,
Pi.conj_apply]
funext i
fin_cases i <;> simp
lemma cRow_cross_tRow_conj (V : CKMMatrix) : conj (conj [V]c ×₃ conj [V]t) = [V]c ×₃ [V]t := by
lemma cRow_cross_tRow_conj (V : CKMMatrix) : conj (conj [V]c ×₃ conj [V]t) = [V]c ×₃ [V]t := by
simp only [crossProduct, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue, LinearMap.mk₂_apply,
Pi.conj_apply]
funext i
@ -169,9 +169,9 @@ lemma rows_linearly_independent (V : CKMMatrix) : LinearIndependent ℂ (rows V)
intro g hg
rw [Fin.sum_univ_three] at hg
simp only [Fin.isValue, rows] at hg
have h0 := congrArg (fun X => conj [V]u ⬝ᵥ X) hg
have h1 := congrArg (fun X => conj [V]c ⬝ᵥ X) hg
have h2 := congrArg (fun X => conj [V]t ⬝ᵥ X) hg
have h0 := congrArg (fun X => conj [V]u ⬝ᵥ X) hg
have h1 := congrArg (fun X => conj [V]c ⬝ᵥ X) hg
have h2 := congrArg (fun X => conj [V]t ⬝ᵥ X) hg
simp only [Fin.isValue, dotProduct_add, dotProduct_smul, Pi.conj_apply,
smul_eq_mul, dotProduct_zero] at h0 h1 h2
rw [uRow_normalized, uRow_cRow_orthog, uRow_tRow_orthog] at h0
@ -184,7 +184,7 @@ lemma rows_linearly_independent (V : CKMMatrix) : LinearIndependent ℂ (rows V)
· exact h1
· exact h2
lemma rows_card : Fintype.card (Fin 3) = FiniteDimensional.finrank ℂ (Fin 3 → ℂ) := by
lemma rows_card : Fintype.card (Fin 3) = FiniteDimensional.finrank ℂ (Fin 3 → ℂ) := by
simp
/-- The rows of a CKM matrix as a basis of `ℂ³`. -/
@ -198,24 +198,24 @@ lemma cRow_cross_tRow_eq_uRow (V : CKMMatrix) :
(conj [V]c ×₃ conj [V]t))
simp only [Fin.sum_univ_three, rowBasis, Fin.isValue,
coe_basisOfLinearIndependentOfCardEqFinrank, rows] at hg
have h0 := congrArg (fun X => conj [V]c ⬝ᵥ X) hg
have h1 := congrArg (fun X => conj [V]t ⬝ᵥ X) hg
have h0 := congrArg (fun X => conj [V]c ⬝ᵥ X) hg
have h1 := congrArg (fun X => conj [V]t ⬝ᵥ X) hg
simp only [Fin.isValue, dotProduct_add, dotProduct_smul, Pi.conj_apply,
smul_eq_mul, dotProduct_zero] at h0 h1
rw [cRow_normalized, cRow_uRow_orthog, cRow_tRow_orthog, dot_self_cross] at h0
rw [tRow_normalized, tRow_uRow_orthog, tRow_cRow_orthog, dot_cross_self] at h1
simp only [Fin.isValue, mul_zero, mul_one, zero_add, add_zero] at h0 h1 hg
simp only [h0, h1, zero_smul, add_zero] at hg
have h2 := congrArg (fun X => conj X ⬝ᵥ X) hg
have h2 := congrArg (fun X => conj X ⬝ᵥ X) hg
simp only [Fin.isValue, dotProduct_smul, Pi.conj_apply, Pi.smul_apply,
smul_eq_mul, _root_.map_mul, cRow_cross_tRow_normalized] at h2
have h3 : conj (g 0 • [V]u) = conj (g 0) • conj [V]u := by
have h3 : conj (g 0 • [V]u) = conj (g 0) • conj [V]u := by
funext i
fin_cases i <;> simp
simp only [h3, Fin.isValue, smul_dotProduct, Pi.conj_apply, smul_eq_mul,
uRow_normalized, Fin.isValue, mul_one, mul_conj, ← Complex.sq_abs] at h2
simp at h2
cases' h2 with h2 h2
cases' h2 with h2 h2
swap
have hx : Complex.abs (g 0) = -1 := by
simp [← ofReal_inj, Fin.isValue, ofReal_neg, ofReal_one, h2]
@ -223,7 +223,7 @@ lemma cRow_cross_tRow_eq_uRow (V : CKMMatrix) :
simp_all
have h4 : (0 : ℝ) < 1 := by norm_num
· exact False.elim (lt_iff_not_le.mp h4 h3)
have hx : [V]u = (g 0)⁻¹ • (conj ([V]c) ×₃ conj ([V]t)) := by
have hx : [V]u = (g 0)⁻¹ • (conj ([V]c) ×₃ conj ([V]t)) := by
rw [← hg, smul_smul, inv_mul_cancel, one_smul]
by_contra hn
simp [hn] at h2
@ -238,30 +238,30 @@ lemma cRow_cross_tRow_eq_uRow (V : CKMMatrix) :
rw [hx, hτ]
lemma uRow_cross_cRow_eq_tRow (V : CKMMatrix) :
∃ (τ : ℝ), [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
∃ (τ : ℝ), [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c)) := by
obtain ⟨g, hg⟩ := (mem_span_range_iff_exists_fun ℂ).mp (Basis.mem_span (rowBasis V)
(conj ([V]u) ×₃ conj ([V]c)) )
rw [Fin.sum_univ_three, rowBasis] at hg
simp at hg
have h0 := congrArg (fun X => conj [V]u ⬝ᵥ X) hg
have h1 := congrArg (fun X => conj [V]c ⬝ᵥ X) hg
have h0 := congrArg (fun X => conj [V]u ⬝ᵥ X) hg
have h1 := congrArg (fun X => conj [V]c ⬝ᵥ X) hg
simp only [Fin.isValue, dotProduct_add, dotProduct_smul, Pi.conj_apply,
smul_eq_mul, dotProduct_zero] at h0 h1
rw [uRow_normalized, uRow_cRow_orthog, uRow_tRow_orthog, dot_self_cross] at h0
rw [cRow_normalized, cRow_uRow_orthog, cRow_tRow_orthog, dot_cross_self] at h1
simp only [Fin.isValue, mul_one, mul_zero, add_zero, zero_add] at h0 h1
simp only [Fin.isValue, h0, zero_smul, h1, add_zero, zero_add] at hg
have h2 := congrArg (fun X => conj X ⬝ᵥ X) hg
have h2 := congrArg (fun X => conj X ⬝ᵥ X) hg
simp only [Fin.isValue, dotProduct_smul, Pi.conj_apply, Pi.smul_apply,
smul_eq_mul, _root_.map_mul] at h2
rw [uRow_cross_cRow_normalized] at h2
have h3 : conj (g 2 • [V]t) = conj (g 2) • conj [V]t := by
have h3 : conj (g 2 • [V]t) = conj (g 2) • conj [V]t := by
funext i
fin_cases i <;> simp
simp only [h3, Fin.isValue, smul_dotProduct, Pi.conj_apply, smul_eq_mul, tRow_normalized,
Fin.isValue, mul_one, mul_conj, ← Complex.sq_abs, ofReal_pow, sq_eq_one_iff,
ofReal_eq_one] at h2
cases' h2 with h2 h2
cases' h2 with h2 h2
swap
have hx : Complex.abs (g 2) = -1 := by
simp [h2, ← ofReal_inj, Fin.isValue, ofReal_neg, ofReal_one]
@ -269,7 +269,7 @@ lemma uRow_cross_cRow_eq_tRow (V : CKMMatrix) :
simp_all
have h4 : (0 : ℝ) < 1 := by norm_num
exact False.elim (lt_iff_not_le.mp h4 h3)
have hx : [V]t = (g 2)⁻¹ • (conj ([V]u) ×₃ conj ([V]c)) := by
have hx : [V]t = (g 2)⁻¹ • (conj ([V]u) ×₃ conj ([V]c)) := by
rw [← hg, @smul_smul, inv_mul_cancel, one_smul]
by_contra hn
simp [hn] at h2
@ -303,30 +303,30 @@ open CKMMatrix
variable (V : CKMMatrix) (a b c d e f : ℝ)
/-- The cross product of the conjugate of the `u` and `c` rows of a CKM matrix. -/
def ucCross : Fin 3 → ℂ :=
def ucCross : Fin 3 → ℂ :=
conj [phaseShiftApply V a b c d e f]u ×₃ conj [phaseShiftApply V a b c d e f]c
lemma ucCross_fst (V : CKMMatrix) : (ucCross V a b c d e f) 0 =
cexp ((- a * I) + (- b * I) + ( - e * I) + (- f * I)) * (conj [V]u ×₃ conj [V]c) 0 := by
cexp ((- a * I) + (- b * I) + ( - e * I) + (- f * I)) * (conj [V]u ×₃ conj [V]c) 0 := by
simp [ucCross, crossProduct, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue,
LinearMap.mk₂_apply, Pi.conj_apply, cons_val_zero, neg_mul, uRow, us, ub, cRow, cs, cb,
exp_add, exp_sub, ← exp_conj, conj_ofReal]
ring
lemma ucCross_snd (V : CKMMatrix) : (ucCross V a b c d e f) 1 =
cexp ((- a * I) + (- b * I) + ( - d * I) + (- f * I)) * (conj [V]u ×₃ conj [V]c) 1 := by
cexp ((- a * I) + (- b * I) + ( - d * I) + (- f * I)) * (conj [V]u ×₃ conj [V]c) 1 := by
simp [ucCross, crossProduct, uRow, us, ub, cRow, cs, cb, ud, cd, exp_add,
exp_sub, ← exp_conj, conj_ofReal]
ring
lemma ucCross_thd (V : CKMMatrix) : (ucCross V a b c d e f) 2 =
cexp ((- a * I) + (- b * I) + ( - d * I) + (- e * I)) * (conj [V]u ×₃ conj [V]c) 2 := by
cexp ((- a * I) + (- b * I) + ( - d * I) + (- e * I)) * (conj [V]u ×₃ conj [V]c) 2 := by
simp [ucCross, crossProduct, uRow, us, ub, cRow, cs, cb, ud, cd, exp_add, exp_sub,
← exp_conj, conj_ofReal]
ring
lemma uRow_mul (V : CKMMatrix) (a b c : ℝ) :
[phaseShiftApply V a b c 0 0 0]u = cexp (a * I) • [V]u := by
[phaseShiftApply V a b c 0 0 0]u = cexp (a * I) • [V]u := by
funext i
simp only [Pi.smul_apply, smul_eq_mul]
fin_cases i <;>
@ -336,7 +336,7 @@ lemma uRow_mul (V : CKMMatrix) (a b c : ℝ) :
simp [ub, uRow]
lemma cRow_mul (V : CKMMatrix) (a b c : ℝ) :
[phaseShiftApply V a b c 0 0 0]c = cexp (b * I) • [V]c := by
[phaseShiftApply V a b c 0 0 0]c = cexp (b * I) • [V]c := by
funext i
simp only [Pi.smul_apply, smul_eq_mul]
fin_cases i <;>
@ -346,7 +346,7 @@ lemma cRow_mul (V : CKMMatrix) (a b c : ℝ) :
simp [cb, cRow]
lemma tRow_mul (V : CKMMatrix) (a b c : ℝ) :
[phaseShiftApply V a b c 0 0 0]t = cexp (c * I) • [V]t := by
[phaseShiftApply V a b c 0 0 0]t = cexp (c * I) • [V]t := by
funext i
simp only [Pi.smul_apply, smul_eq_mul]
fin_cases i <;>
@ -355,6 +355,6 @@ lemma tRow_mul (V : CKMMatrix) (a b c : ℝ) :
simp [ts, tRow, ofReal_zero, zero_mul, add_zero, Fin.isValue, Fin.mk_one, cons_val_one, head_cons]
simp [tb, tRow]
end phaseShiftApply
end phaseShiftApply
end

14
HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/Basic.lean
View file

@ -25,7 +25,7 @@ noncomputable section
/-- Given four reals `θ₁₂ θ₁₃ θ₂₃ δ₁₃` the standard paramaterization of the CKM matrix
as a `3×3` complex matrix. -/
def standParamAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
def standParamAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin 3) (Fin 3) ℂ :=
![![Real.cos θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₂ * Real.cos θ₁₃, Real.sin θ₁₃ * exp (-I * δ₁₃)],
![(-Real.sin θ₁₂ * Real.cos θ₂₃) - (Real.cos θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃)),
Real.cos θ₁₂ * Real.cos θ₂₃ - Real.sin θ₁₂ * Real.sin θ₁₃ * Real.sin θ₂₃ * exp (I * δ₁₃),
@ -36,8 +36,8 @@ def standParamAsMatrix (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) : Matrix (Fin
open CKMMatrix
lemma standParamAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
((standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by
lemma standParamAsMatrix_unitary (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
((standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃)ᴴ * standParamAsMatrix θ₁₂ θ₁₃ θ₂₃ δ₁₃) = 1 := by
funext j i
simp only [standParamAsMatrix, neg_mul, Fin.isValue]
rw [mul_apply]
@ -139,7 +139,7 @@ lemma cross_product_t (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) :
ring
lemma eq_rows (U : CKMMatrix) {θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ} (hu : [U]u = [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]u)
(hc : [U]c = [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) (hU : [U]t = conj [U]u ×₃ conj [U]c) :
(hc : [U]c = [standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃]c) (hU : [U]t = conj [U]u ×₃ conj [U]c) :
U = standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
apply ext_Rows hu hc
rw [hU, cross_product_t, hu, hc]
@ -149,7 +149,7 @@ lemma eq_exp_of_phases (θ₁₂ θ₁₃ θ₂₃ δ₁₃ δ₁₃' : ℝ) (h
simp [standParam, standParamAsMatrix]
apply CKMMatrix_ext
simp only
rw [show exp (I * δ₁₃) = exp (I * δ₁₃') by rw [mul_comm, h, mul_comm]]
rw [show exp (I * δ₁₃) = exp (I * δ₁₃') by rw [mul_comm, h, mul_comm]]
rw [show cexp (-(I * ↑δ₁₃)) = cexp (-(I * ↑δ₁₃')) by rw [exp_neg, exp_neg, mul_comm, h, mul_comm]]
open Invariant in
@ -179,9 +179,9 @@ lemma VusVubVcdSq_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Rea
open Invariant in
lemma mulExpδ₁₃_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤ Real.sin θ₁₂)
(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
(h2 : 0 ≤ Real.cos θ₁₃) (h3 : 0 ≤ Real.sin θ₂₃) (h4 : 0 ≤ Real.cos θ₁₂) :
mulExpδ₁₃ ⟦standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃⟧ =
sin θ₁₂ * cos θ₁₃ ^ 2 * sin θ₂₃ * sin θ₁₃ * cos θ₁₂ * cos θ₂₃ * cexp (I * δ₁₃) := by
sin θ₁₂ * cos θ₁₃ ^ 2 * sin θ₂₃ * sin θ₁₃ * cos θ₁₂ * cos θ₂₃ * cexp (I * δ₁₃) := by
rw [mulExpδ₁₃, VusVubVcdSq_eq _ _ _ _ h1 h2 h3 h4 ]
simp only [jarlskogℂ, standParam, standParamAsMatrix, neg_mul,
Quotient.lift_mk, jarlskogℂCKM, Fin.isValue, cons_val', cons_val_one, head_cons,

58
HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/StandardParameters.lean
View file

@ -12,7 +12,7 @@ import Mathlib.Analysis.SpecialFunctions.Complex.Arg
/-!
# Standard parameters for the CKM Matrix
Given a CKM matrix `V` we can extract four real numbers `θ₁₂`, `θ₁₃`, `θ₂₃` and `δ₁₃`.
Given a CKM matrix `V` we can extract four real numbers `θ₁₂`, `θ₁₃`, `θ₂₃` and `δ₁₃`.
These, when used in the standard parameterization return `V` up to equivalence.
This leads to the theorem `standParam.exists_for_CKMatrix` which says that up to equivalence every
@ -26,45 +26,45 @@ open CKMMatrix
noncomputable section
/-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₂` in the
standard parameterization. --/
standard parameterization. --/
def S₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := VusAbs V / (√ (VudAbs V ^ 2 + VusAbs V ^ 2))
/-- Given a CKM matrix `V` the real number corresponding to `sin θ₁₃` in the
standard parameterization. --/
standard parameterization. --/
def S₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := VubAbs V
/-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the
standard parameterization. --/
standard parameterization. --/
def S₂₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
if VubAbs V = 1 then VcdAbs V
else VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2)
/-- Given a CKM matrix `V` the real number corresponding to `θ₁₂` in the
standard parameterization. --/
standard parameterization. --/
def θ₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₂ V)
/-- Given a CKM matrix `V` the real number corresponding to `θ₁₃` in the
standard parameterization. --/
standard parameterization. --/
def θ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₁₃ V)
/-- Given a CKM matrix `V` the real number corresponding to `θ₂₃` in the
standard parameterization. --/
standard parameterization. --/
def θ₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.arcsin (S₂₃ V)
/-- Given a CKM matrix `V` the real number corresponding to `cos θ₁₂` in the
standard parameterization. --/
standard parameterization. --/
def C₁₂ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₂ V)
/-- Given a CKM matrix `V` the real number corresponding to `cos θ₁₃` in the
standard parameterization. --/
standard parameterization. --/
def C₁₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₁₃ V)
/-- Given a CKM matrix `V` the real number corresponding to `sin θ₂₃` in the
standard parameterization. --/
standard parameterization. --/
def C₂₃ (V : Quotient CKMMatrixSetoid) : ℝ := Real.cos (θ₂₃ V)
/-- Given a CKM matrix `V` the real number corresponding to the phase `δ₁₃` in the
standard parameterization. --/
standard parameterization. --/
def δ₁₃ (V : Quotient CKMMatrixSetoid) : ℝ :=
arg (Invariant.mulExpδ₁₃ V)
@ -126,7 +126,7 @@ lemma S₂₃_leq_one (V : Quotient CKMMatrixSetoid) : S₂₃ V ≤ 1 := by
lemma S₁₂_eq_sin_θ₁₂ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₂ V) = S₁₂ V :=
Real.sin_arcsin (le_trans (by simp) (S₁₂_nonneg V)) (S₁₂_leq_one V)
lemma S₁₃_eq_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₃ V) = S₁₃ V :=
lemma S₁₃_eq_sin_θ₁₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₁₃ V) = S₁₃ V :=
Real.sin_arcsin (le_trans (by simp) (S₁₃_nonneg V)) (S₁₃_leq_one V)
lemma S₂₃_eq_sin_θ₂₃ (V : Quotient CKMMatrixSetoid) : Real.sin (θ₂₃ V) = S₂₃ V :=
@ -168,7 +168,7 @@ lemma S₂₃_of_Vub_eq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V = 1) :
rw [S₂₃, if_pos ha]
lemma S₂₃_of_Vub_neq_one {V : Quotient CKMMatrixSetoid} (ha : VubAbs V ≠ 1) :
S₂₃ V = VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := by
S₂₃ V = VcbAbs V / √ (VudAbs V ^ 2 + VusAbs V ^ 2) := by
rw [S₂₃, if_neg ha]
end sines
@ -336,9 +336,9 @@ namespace standParam
open Invariant
lemma mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) :
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ =
sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := by
* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) * cexp (I * δ₁₃) := by
refine mulExpδ₁₃_eq _ _ _ _ ?_ ?_ ?_ ?_
rw [S₁₂_eq_sin_θ₁₂]
exact S₁₂_nonneg _
@ -348,11 +348,11 @@ lemma mulExpδ₁₃_on_param_δ₁₃ (V : CKMMatrix) (δ₁₃ : ℝ) :
exact Real.cos_arcsin_nonneg _
lemma mulExpδ₁₃_on_param_eq_zero_iff (V : CKMMatrix) (δ₁₃ : ℝ) :
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = 0 ↔
VudAbs ⟦V⟧ = 0 ∨ VubAbs ⟦V⟧ = 0 ∨ VusAbs ⟦V⟧ = 0 ∨ VcbAbs ⟦V⟧ = 0 ∨ VtbAbs ⟦V⟧ = 0 := by
rw [VudAbs_eq_C₁₂_mul_C₁₃, VubAbs_eq_S₁₃, VusAbs_eq_S₁₂_mul_C₁₃, VcbAbs_eq_S₂₃_mul_C₁₃,
VtbAbs_eq_C₂₃_mul_C₁₃, ← ofReal_inj,
← ofReal_inj, ← ofReal_inj, ← ofReal_inj, ← ofReal_inj]
← ofReal_inj, ← ofReal_inj, ← ofReal_inj, ← ofReal_inj]
simp only [ofReal_mul]
rw [← S₁₃_eq_ℂsin_θ₁₃, ← S₁₂_eq_ℂsin_θ₁₂, ← S₂₃_eq_ℂsin_θ₂₃,
← C₁₃_eq_ℂcos_θ₁₃, ← C₂₃_eq_ℂcos_θ₂₃,← C₁₂_eq_ℂcos_θ₁₂]
@ -364,7 +364,7 @@ lemma mulExpδ₁₃_on_param_eq_zero_iff (V : CKMMatrix) (δ₁₃ : ℝ) :
aesop
lemma mulExpδ₁₃_on_param_abs (V : CKMMatrix) (δ₁₃ : ℝ) :
Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) =
Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) =
sin (θ₁₂ ⟦V⟧) * cos (θ₁₃ ⟦V⟧) ^ 2 * sin (θ₂₃ ⟦V⟧) * sin (θ₁₃ ⟦V⟧)
* cos (θ₁₂ ⟦V⟧) * cos (θ₂₃ ⟦V⟧) := by
rw [mulExpδ₁₃_on_param_δ₁₃]
@ -373,19 +373,19 @@ lemma mulExpδ₁₃_on_param_abs (V : CKMMatrix) (δ₁₃ : ℝ) :
complexAbs_sin_θ₂₃, complexAbs_cos_θ₂₃]
lemma mulExpδ₁₃_on_param_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
(h1 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
cexp (arg ( mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ) * I) =
(h1 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ≠ 0 ) :
cexp (arg ( mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ ) * I) =
cexp (δ₁₃ * I) := by
have h1a := mulExpδ₁₃_on_param_δ₁₃ V δ₁₃
have habs := mulExpδ₁₃_on_param_abs V δ₁₃
have h2 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = Complex.abs
(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
have h2 : mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ = Complex.abs
(mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) * exp (δ₁₃ * I) := by
rw [habs, h1a]
ring_nf
nth_rewrite 1 [← abs_mul_exp_arg_mul_I (mulExpδ₁₃
⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧ )] at h2
have habs_neq_zero :
(Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
(Complex.abs (mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃⟧) : ℂ) ≠ 0 := by
simp only [ne_eq, ofReal_eq_zero, map_eq_zero]
exact h1
rw [← mul_right_inj' habs_neq_zero]
@ -393,7 +393,7 @@ lemma mulExpδ₁₃_on_param_neq_zero_arg (V : CKMMatrix) (δ₁₃ : ℝ)
lemma on_param_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₃ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
have hS13 := congrArg ofReal (S₁₃_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
have hS13 := congrArg ofReal (S₁₃_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
simp [← S₁₃_eq_ℂsin_θ₁₃] at hS13
have hC12 := congrArg ofReal (C₁₂_of_Vub_one (VubAbs_of_cos_θ₁₃_zero h))
simp [← C₁₂_eq_ℂcos_θ₁₂] at hC12
@ -514,7 +514,7 @@ lemma eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0
funext i
fin_cases i
simp only [uRow, Fin.isValue, Fin.zero_eta, cons_val_zero, standParam, standParamAsMatrix,
ofReal_cos, ofReal_sin, ofReal_neg, mul_neg, neg_mul, neg_neg, cons_val', empty_val',
ofReal_cos, ofReal_sin, ofReal_neg, mul_neg, neg_mul, neg_neg, cons_val', empty_val',
cons_val_fin_one, cons_val_one, head_cons, cons_val_two, tail_cons]
rw [hV.1, VudAbs_eq_C₁₂_mul_C₁₃ ⟦V⟧]
simp [C₁₂, C₁₃]
@ -613,7 +613,7 @@ theorem exists_δ₁₃ (V : CKMMatrix) :
rw [hUV] at hna
simp [VAbs] at hna
simp_all
have hU' := eq_standParam_of_fstRowThdColRealCond haU hU.2
have hU' := eq_standParam_of_fstRowThdColRealCond haU hU.2
rw [hU'] at hU
use (- arg ([U]ub))
rw [← hUV]
@ -638,8 +638,8 @@ theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (δ₁₃ ⟦V⟧) := by
obtain ⟨δ₁₃', hδ₃⟩ := exists_δ₁₃ V
have hSV := (Quotient.eq.mpr (hδ₃))
by_cases h : Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
have h2 := eq_exp_of_phases (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
by_cases h : Invariant.mulExpδ₁₃ ⟦standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'⟧ ≠ 0
have h2 := eq_exp_of_phases (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃'
(δ₁₃ ⟦V⟧) (by rw [← mulExpδ₁₃_on_param_neq_zero_arg V δ₁₃' h, ← hSV, δ₁₃, Invariant.mulExpδ₁₃])
rw [h2] at hδ₃
exact hδ₃

16
HepLean/Mathematics/LinearMaps.lean
View file

@ -76,8 +76,8 @@ def mk₂ (f : V × V → ℚ) (map_smul : ∀ a S T, f (a • S, T) = a * f (S,
rw [swap, map_smul]
exact congrArg (HMul.hMul a) (swap T S)
}
map_smul' := fun a S => LinearMap.ext fun T => map_smul a S T
map_add' := fun S1 S2 => LinearMap.ext fun T => map_add S1 S2 T
map_smul' := fun a S => LinearMap.ext fun T => map_smul a S T
map_add' := fun S1 S2 => LinearMap.ext fun T => map_add S1 S2 T
swap' := swap
lemma map_smul₁ (f : BiLinearSymm V) (a : ℚ) (S T : V) : f (a • S) T = a * f S T := by
@ -87,7 +87,7 @@ lemma map_smul₁ (f : BiLinearSymm V) (a : ℚ) (S T : V) : f (a • S) T = a *
lemma swap (f : BiLinearSymm V) (S T : V) : f S T = f T S :=
f.swap' S T
lemma map_smul₂ (f : BiLinearSymm V) (a : ℚ) (S : V) (T : V) : f S (a • T) = a * f S T := by
lemma map_smul₂ (f : BiLinearSymm V) (a : ℚ) (S : V) (T : V) : f S (a • T) = a * f S T := by
rw [f.swap, f.map_smul₁, f.swap]
lemma map_add₁ (f : BiLinearSymm V) (S1 S2 T : V) : f (S1 + S2) T = f S1 T + f S2 T := by
@ -99,7 +99,7 @@ lemma map_add₂ (f : BiLinearSymm V) (S : V) (T1 T2 : V) :
rw [f.swap, f.map_add₁, f.swap T1 S, f.swap T2 S]
/-- Fixing the second input vectors, the resulting linear map. -/
def toLinear₁ (f : BiLinearSymm V) (T : V) : V →ₗ[ℚ] ℚ where
def toLinear₁ (f : BiLinearSymm V) (T : V) : V →ₗ[ℚ] ℚ where
toFun S := f S T
map_add' S1 S2 := map_add₁ f S1 S2 T
map_smul' a S := by
@ -108,7 +108,7 @@ def toLinear₁ (f : BiLinearSymm V) (T : V) : V →ₗ[ℚ] ℚ where
lemma toLinear₁_apply (f : BiLinearSymm V) (S T : V) : f S T = f.toLinear₁ T S := rfl
lemma map_sum₁ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
lemma map_sum₁ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
f (∑ i, S i) T = ∑ i, f (S i) T := by
rw [f.toLinear₁_apply]
rw [map_sum]
@ -166,14 +166,14 @@ end HomogeneousCubic
/-- The structure of a symmetric trilinear function. -/
structure TriLinearSymm (V : Type) [AddCommMonoid V] [Module ℚ V] extends
V →ₗ[ℚ] V →ₗ[ℚ] V →ₗ[ℚ] ℚ where
swap₁' : ∀ S T L, toFun S T L = toFun T S L
swap₁' : ∀ S T L, toFun S T L = toFun T S L
swap₂' : ∀ S T L, toFun S T L = toFun S L T
namespace TriLinearSymm
open BigOperators
variable {V : Type} [AddCommMonoid V] [Module ℚ V]
instance instFun : FunLike (TriLinearSymm V) V (V →ₗ[ℚ] V →ₗ[ℚ] ℚ ) where
instance instFun : FunLike (TriLinearSymm V) V (V →ₗ[ℚ] V →ₗ[ℚ] ℚ ) where
coe f := f.toFun
coe_injective' f g h := by
cases f
@ -251,7 +251,7 @@ lemma map_add₃ (f : TriLinearSymm V) (S T L1 L2 : V) :
rw [f.swap₃, f.map_add₁, f.swap₃, f.swap₃ L2 T S]
/-- Fixing the second and third input vectors, the resulting linear map. -/
def toLinear₁ (f : TriLinearSymm V) (T L : V) : V →ₗ[ℚ] ℚ where
def toLinear₁ (f : TriLinearSymm V) (T L : V) : V →ₗ[ℚ] ℚ where
toFun S := f S T L
map_add' S1 S2 := by
simp only [f.map_add₁]

20
HepLean/Mathematics/SO3/Basic.lean
View file

@ -59,7 +59,7 @@ def toGL : SO(3) →* GL (Fin 3) ℝ where
map_one' := by
simp
rfl
map_mul' x y := by
map_mul' x y := by
simp only [_root_.mul_inv_rev, coe_inv]
ext
rfl
@ -81,7 +81,7 @@ lemma toGL_injective : Function.Injective toGL := by
def toProd : SO(3) →* (Matrix (Fin 3) (Fin 3) ℝ) × (Matrix (Fin 3) (Fin 3) ℝ)ᵐᵒᵖ :=
MonoidHom.comp (Units.embedProduct _) toGL
lemma toProd_eq_transpose : toProd A = (A.1, ⟨A.1ᵀ⟩) := by
lemma toProd_eq_transpose : toProd A = (A.1, ⟨A.1ᵀ⟩) := by
simp only [toProd, Units.embedProduct, coe_units_inv, MulOpposite.op_inv, toGL, coe_inv,
MonoidHom.coe_comp, MonoidHom.coe_mk, OneHom.coe_mk, Function.comp_apply, Prod.mk.injEq,
true_and]
@ -143,16 +143,16 @@ instance : TopologicalGroup SO(3) :=
Inducing.topologicalGroup toGL toGL_embedding.toInducing
lemma det_minus_id (A : SO(3)) : det (A.1 - 1) = 0 := by
have h1 : det (A.1 - 1) = - det (A.1 - 1) :=
have h1 : det (A.1 - 1) = - det (A.1 - 1) :=
calc
det (A.1 - 1) = det (A.1 - A.1 * A.1ᵀ) := by simp [A.2.2]
_ = det A.1 * det (1 - A.1ᵀ) := by rw [← det_mul, mul_sub, mul_one]
det (A.1 - 1) = det (A.1 - A.1 * A.1ᵀ) := by simp [A.2.2]
_ = det A.1 * det (1 - A.1ᵀ) := by rw [← det_mul, mul_sub, mul_one]
_ = det (1 - A.1ᵀ):= by simp [A.2.1]
_ = det (1 - A.1ᵀ)ᵀ := by rw [det_transpose]
_ = det (1 - A.1) := by simp
_ = det (- (A.1 - 1)) := by simp
_ = (- 1) ^ 3 * det (A.1 - 1) := by simp only [det_neg, Fintype.card_fin, neg_mul, one_mul]
_ = - det (A.1 - 1) := by simp [pow_three]
_ = (- 1) ^ 3 * det (A.1 - 1) := by simp only [det_neg, Fintype.card_fin, neg_mul, one_mul]
_ = - det (A.1 - 1) := by simp [pow_three]
simpa using h1
@[simp]
@ -160,8 +160,8 @@ lemma det_id_minus (A : SO(3)) : det (1 - A.1) = 0 := by
have h1 : det (1 - A.1) = - det (A.1 - 1) := by
calc
det (1 - A.1) = det (- (A.1 - 1)) := by simp
_ = (- 1) ^ 3 * det (A.1 - 1) := by simp only [det_neg, Fintype.card_fin, neg_mul, one_mul]
_ = - det (A.1 - 1) := by simp [pow_three]
_ = (- 1) ^ 3 * det (A.1 - 1) := by simp only [det_neg, Fintype.card_fin, neg_mul, one_mul]
_ = - det (A.1 - 1) := by simp [pow_three]
rw [h1, det_minus_id]
simp only [neg_zero]
@ -216,7 +216,7 @@ lemma exists_basis_preserved (A : SO(3)) :
obtain ⟨v, hv⟩ := exists_stationary_vec A
have h3 : FiniteDimensional.finrank ℝ (EuclideanSpace ℝ (Fin 3)) = Fintype.card (Fin 3) := by
simp_all only [toEnd_apply, finrank_euclideanSpace, Fintype.card_fin]
obtain ⟨b, hb⟩ := Orthonormal.exists_orthonormalBasis_extension_of_card_eq h3 hv.1
obtain ⟨b, hb⟩ := Orthonormal.exists_orthonormalBasis_extension_of_card_eq h3 hv.1
simp at hb
use b
rw [hb, hv.2]

20
HepLean/SpaceTime/LorentzAlgebra/Basic.lean
View file

@ -13,7 +13,7 @@ We define
- Define `lorentzAlgebra` via `LieAlgebra.Orthogonal.so'` as a subalgebra of
`Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`.
- In `mem_iff` prove that a matrix is in the Lorentz algebra if and only if it satisfies the
condition `Aᵀ * η = - η * A`.
condition `Aᵀ * η = - η * A`.
-/
@ -21,29 +21,29 @@ namespace SpaceTime
open Matrix
open TensorProduct
/-- The Lorentz algebra as a subalgebra of `Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`. -/
/-- The Lorentz algebra as a subalgebra of `Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ`. -/
def lorentzAlgebra : LieSubalgebra ℝ (Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :=
(LieAlgebra.Orthogonal.so' (Fin 1) (Fin 3) ℝ)
namespace lorentzAlgebra
open minkowskiMatrix
lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
have h1 := A.2
erw [mem_skewAdjointMatricesLieSubalgebra] at h1
simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h1
lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ}
(h : Aᵀ * η = - η * A) : A ∈ lorentzAlgebra := by
(h : Aᵀ * η = - η * A) : A ∈ lorentzAlgebra := by
erw [mem_skewAdjointMatricesLieSubalgebra]
simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h
lemma mem_iff {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ} :
A ∈ lorentzAlgebra ↔ Aᵀ * η = - η * A :=
A ∈ lorentzAlgebra ↔ Aᵀ * η = - η * A :=
Iff.intro (fun h => transpose_eta ⟨A, h⟩) (fun h => mem_of_transpose_eta_eq_eta_mul_self h)
lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
A ∈ lorentzAlgebra ↔ A = - η * Aᵀ * η := by
lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
A ∈ lorentzAlgebra ↔ A = - η * Aᵀ * η := by
rw [mem_iff]
refine Iff.intro (fun h => ?_) (fun h => ?_)
· trans -η * (Aᵀ * η)
@ -52,7 +52,7 @@ lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
rw [minkowskiMatrix.sq]
all_goals noncomm_ring
· nth_rewrite 2 [h]
trans (η * η) * Aᵀ * η
trans (η * η) * Aᵀ * η
rw [minkowskiMatrix.sq]
all_goals noncomm_ring
@ -80,7 +80,7 @@ end lorentzAlgebra
@[simps!]
instance lorentzVectorAsLieRingModule : LieRingModule lorentzAlgebra (LorentzVector 3) where
bracket Λ x := Λ.1.mulVec x
bracket Λ x := Λ.1.mulVec x
add_lie Λ1 Λ2 x := by
simp [add_mulVec]
lie_add Λ x1 x2 := by
@ -91,7 +91,7 @@ instance lorentzVectorAsLieRingModule : LieRingModule lorentzAlgebra (LorentzVec
@[simps!]
instance spaceTimeAsLieModule : LieModule ℝ lorentzAlgebra (LorentzVector 3) where
smul_lie r Λ x := by
smul_lie r Λ x := by
simp [Bracket.bracket, smul_mulVec_assoc]
lie_smul r Λ x := by
simp [Bracket.bracket]

18
HepLean/SpaceTime/LorentzGroup/Basic.lean
View file

@ -30,7 +30,7 @@ We start studying the properties of matrices which preserve `ηLin`.
These matrices form the Lorentz group, which we will define in the next section at `lorentzGroup`.
-/
variable {d : ℕ}
variable {d : ℕ}
open minkowskiMetric in
/-- The Lorentz group is the subset of matrices which preserve the minkowski metric. -/
@ -44,7 +44,7 @@ scoped[LorentzGroup] notation (name := lorentzGroup_notation) "𝓛" => LorentzG
open minkowskiMetric
variable {Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ}
variable {Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ}
/-!
@ -52,7 +52,7 @@ variable {Λ Λ' : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ}
-/
lemma mem_iff_norm : Λ ∈ LorentzGroup d ↔
lemma mem_iff_norm : Λ ∈ LorentzGroup d ↔
∀ (x : LorentzVector d), ⟪Λ *ᵥ x, Λ *ᵥ x⟫ₘ = ⟪x, x⟫ₘ := by
refine Iff.intro (fun h x => h x x) (fun h x y => ?_)
have hp := h (x + y)
@ -74,11 +74,11 @@ lemma mem_iff_on_right : Λ ∈ LorentzGroup d ↔
rw [← dual_mulVec_right, mulVec_mulVec]
exact h x y
lemma mem_iff_dual_mul_self : Λ ∈ LorentzGroup d ↔ dual Λ * Λ = 1 := by
lemma mem_iff_dual_mul_self : Λ ∈ LorentzGroup d ↔ dual Λ * Λ = 1 := by
rw [mem_iff_on_right, matrix_eq_id_iff]
exact forall_comm
lemma mem_iff_self_mul_dual : Λ ∈ LorentzGroup d ↔ Λ * dual Λ = 1 := by
lemma mem_iff_self_mul_dual : Λ ∈ LorentzGroup d ↔ Λ * dual Λ = 1 := by
rw [mem_iff_dual_mul_self]
exact mul_eq_one_comm
@ -145,9 +145,9 @@ namespace LorentzGroup
open minkowskiMetric
variable {Λ Λ' : LorentzGroup d}
variable {Λ Λ' : LorentzGroup d}
lemma coe_inv : (Λ⁻¹).1 = Λ.1⁻¹:= by
lemma coe_inv : (Λ⁻¹).1 = Λ.1⁻¹:= by
refine (inv_eq_left_inv ?h).symm
exact mem_iff_dual_mul_self.mp Λ.2
@ -172,7 +172,7 @@ def toGL : LorentzGroup d →* GL (Fin 1 ⊕ Fin d) ℝ where
map_one' := by
simp
rfl
map_mul' x y := by
map_mul' x y := by
simp only [lorentzGroupIsGroup, _root_.mul_inv_rev, coe_inv]
ext
rfl
@ -190,7 +190,7 @@ def toProd : LorentzGroup d →* (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ
(Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ)ᵐᵒᵖ :=
MonoidHom.comp (Units.embedProduct _) toGL
lemma toProd_eq_transpose_η : toProd Λ = (Λ.1, MulOpposite.op $ minkowskiMetric.dual Λ.1) := rfl
lemma toProd_eq_transpose_η : toProd Λ = (Λ.1, MulOpposite.op $ minkowskiMetric.dual Λ.1) := rfl
lemma toProd_injective : Function.Injective (@toProd d) := by
intro A B h

4
HepLean/SpaceTime/LorentzGroup/Boosts.lean
View file

@ -99,8 +99,8 @@ lemma toMatrix_mulVec (u v : FuturePointing d) (x : LorentzVector d) :
open minkowskiMatrix LorentzVector in
@[simp]
lemma toMatrix_apply (u v : FuturePointing d) (μ ν : Fin 1 ⊕ Fin d) :
(toMatrix u v) μ ν = η μ μ * (⟪e μ, e ν⟫ₘ + 2 * ⟪e ν, u⟫ₘ * ⟪e μ, v⟫ₘ
lemma toMatrix_apply (u v : FuturePointing d) (μ ν : Fin 1 ⊕ Fin d) :
(toMatrix u v) μ ν = η μ μ * (⟪e μ, e ν⟫ₘ + 2 * ⟪e ν, u⟫ₘ * ⟪e μ, v⟫ₘ
- ⟪e μ, u + v⟫ₘ * ⟪e ν, u + v⟫ₘ / (1 + ⟪u, v.1.1⟫ₘ)) := by
rw [matrix_apply_stdBasis (toMatrix u v) μ ν, toMatrix_mulVec]
simp only [genBoost, genBoostAux₁, genBoostAux₂, map_add, smul_add, neg_smul, LinearMap.add_apply,

18
HepLean/SpaceTime/LorentzGroup/Orthochronous.lean
View file

@ -27,7 +27,7 @@ variable (Λ : LorentzGroup d)
open LorentzVector
open minkowskiMetric
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
/-- A Lorentz transformation is `orthochronous` if its `0 0` element is non-negative. -/
def IsOrthochronous : Prop := 0 ≤ timeComp Λ
lemma IsOrthochronous_iff_futurePointing :
@ -39,7 +39,7 @@ lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) := by rfl
lemma IsOrthochronous_iff_ge_one :
IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
IsOrthochronous Λ ↔ 1 ≤ timeComp Λ := by
rw [IsOrthochronous_iff_futurePointing, NormOneLorentzVector.FuturePointing.mem_iff,
NormOneLorentzVector.time_pos_iff]
simp only [time, toNormOneLorentzVector, timeVec, Fin.isValue]
@ -62,20 +62,20 @@ lemma not_orthochronous_iff_le_zero :
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ ,
def timeCompCont : C(LorentzGroup d, ℝ) := ⟨fun Λ => timeComp Λ,
Continuous.matrix_elem (continuous_iff_le_induced.mpr fun _ a => a) (Sum.inl 0) (Sum.inl 0)⟩
/-- An auxillary function used in the definition of `orthchroMapReal`. -/
def stepFunction : ℝ → ℝ := fun t =>
if t ≤ -1 then -1 else
if 1 ≤ t then 1 else t
if 1 ≤ t then 1 else t
lemma stepFunction_continuous : Continuous stepFunction := by
apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)
<;> intro a ha
rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha
rw [ha]
simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
simp [neg_lt_self_iff, zero_lt_one, ↓reduceIte]
have h1 : ¬ (1 : ℝ) ≤ 0 := by simp
exact Eq.symm (if_neg h1)
rw [Set.Ici_def, @frontier_Ici, @Set.mem_singleton_iff] at ha
@ -105,20 +105,20 @@ lemma orthchroMapReal_minus_one_or_one (Λ : LorentzGroup d) :
apply Or.inr $ orthchroMapReal_on_IsOrthochronous h
apply Or.inl $ orthchroMapReal_on_not_IsOrthochronous h
local notation "ℤ₂" => Multiplicative (ZMod 2)
local notation "ℤ₂" => Multiplicative (ZMod 2)
/-- A continuous map from `lorentzGroup` to `ℤ₂` whose kernal are the Orthochronous elements. -/
def orthchroMap : C(LorentzGroup d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨orthchroMapReal Λ, orthchroMapReal_minus_one_or_one Λ⟩,
continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
continuous_toFun := Continuous.subtype_mk (ContinuousMap.continuous orthchroMapReal) _}
lemma orthchroMap_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMap Λ = 1 := by
simp [orthchroMap, orthchroMapReal_on_IsOrthochronous h]
lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
orthchroMap Λ = Additive.toMul (1 : ZMod 2) := by
simp [orthchroMap, orthchroMapReal_on_not_IsOrthochronous h]
rfl
@ -136,7 +136,7 @@ lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h :
rw [IsOrthochronous, timeComp_mul]
exact NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_not_mem h h'
lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
(h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') := by
rw [not_orthochronous_iff_le_zero, timeComp_mul]
rw [IsOrthochronous_iff_transpose] at h

20
HepLean/SpaceTime/LorentzGroup/Proper.lean
View file

@ -31,7 +31,7 @@ lemma det_eq_one_or_neg_one (Λ : 𝓛 d) : Λ.1.det = 1 ∨ Λ.1.det = -1 := by
simp [det_mul, det_dual] at h1
exact mul_self_eq_one_iff.mp h1
local notation "ℤ₂" => Multiplicative (ZMod 2)
local notation "ℤ₂" => Multiplicative (ZMod 2)
instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
@ -42,7 +42,7 @@ instance : TopologicalGroup ℤ₂ := TopologicalGroup.mk
/-- A continuous function from `({-1, 1} : Set ℝ)` to `ℤ₂`. -/
@[simps!]
def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
toFun x := if x = ⟨1, Set.mem_insert_of_mem (-1) rfl⟩
then (Additive.toMul 0) else (Additive.toMul (1 : ZMod 2))
continuous_toFun := by
@ -50,8 +50,8 @@ def coeForℤ₂ : C(({-1, 1} : Set ℝ), ℤ₂) where
exact continuous_of_discreteTopology
/-- The continuous map taking a Lorentz matrix to its determinant. -/
def detContinuous : C(𝓛 d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
def detContinuous : C(𝓛 d, ℤ₂) :=
ContinuousMap.comp coeForℤ₂ {
toFun := fun Λ => ⟨Λ.1.det, Or.symm (LorentzGroup.det_eq_one_or_neg_one _)⟩,
continuous_toFun := by
refine Continuous.subtype_mk ?_ _
@ -64,8 +64,8 @@ lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
apply Iff.intro
intro h
simp [detContinuous] at h
cases' det_eq_one_or_neg_one Λ with h1 h1
<;> cases' det_eq_one_or_neg_one Λ' with h2 h2
cases' det_eq_one_or_neg_one Λ with h1 h1
<;> cases' det_eq_one_or_neg_one Λ' with h2 h2
<;> simp_all [h1, h2, h]
rw [← toMul_zero, @Equiv.apply_eq_iff_eq] at h
· change (0 : Fin 2) = (1 : Fin 2) at h
@ -92,16 +92,16 @@ def detRep : 𝓛 d →* ℤ₂ where
lemma detRep_continuous : Continuous (@detRep d) := detContinuous.2
lemma det_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
lemma det_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
Λ.1.det = Λ'.1.det := by
obtain ⟨s, hs, hΛ'⟩ := h
let f : ContinuousMap s ℤ₂ := ContinuousMap.restrict s detContinuous
haveI : PreconnectedSpace s := isPreconnected_iff_preconnectedSpace.mp hs.1
simpa [f, detContinuous_eq_iff_det_eq] using
(@IsPreconnected.subsingleton ℤ₂ _ _ _ (isPreconnected_range f.2))
(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
(Set.mem_range_self ⟨Λ, hs.2⟩) (Set.mem_range_self ⟨Λ', hΛ'⟩)
lemma detRep_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
lemma detRep_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
detRep Λ = detRep Λ' := by
simp [detRep_apply, detRep_apply, detContinuous]
rw [det_on_connected_component h]
@ -125,7 +125,7 @@ lemma IsProper_iff (Λ : LorentzGroup d) : IsProper Λ ↔ detRep Λ = 1 := by
lemma id_IsProper : (@IsProper d) 1 := by
simp [IsProper]
lemma isProper_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
lemma isProper_on_connected_component {Λ Λ' : LorentzGroup d} (h : Λ' ∈ connectedComponent Λ) :
IsProper Λ ↔ IsProper Λ' := by
simp [detRep_apply, detRep_apply, detContinuous]
rw [det_on_connected_component h]

20
HepLean/SpaceTime/LorentzTensor/Basic.lean
View file

@ -47,7 +47,7 @@ instance (d : ℕ) (μ : RealLorentzTensor.Colors) : Fintype (RealLorentzTensor.
| RealLorentzTensor.Colors.down => instFintypeSum (Fin 1) (Fin d)
/-- An `IndexValue` is a set of actual values an index can take. e.g. for a
3-tensor (0, 1, 2). -/
3-tensor (0, 1, 2). -/
@[simp]
def RealLorentzTensor.IndexValue {X : FintypeCat} (d : ℕ ) (c : X → RealLorentzTensor.Colors) :
Type 0 := (x : X) → RealLorentzTensor.ColorsIndex d (c x)
@ -107,9 +107,9 @@ def congrColorsDual {μ ν : Colors} (h : μ = τ ν) :
ColorsIndex d μ ≃ ColorsIndex d ν :=
(castColorsIndex h).trans dualColorsIndex.symm
lemma congrColorsDual_symm {μ ν : Colors} (h : μ = τ ν) :
lemma congrColorsDual_symm {μ ν : Colors} (h : μ = τ ν) :
(congrColorsDual h).symm =
@congrColorsDual d _ _ ((Function.Involutive.eq_iff τ_involutive).mp h.symm) := by
@congrColorsDual d _ _ ((Function.Involutive.eq_iff τ_involutive).mp h.symm) := by
match μ, ν with
| Colors.up, Colors.down => rfl
| Colors.down, Colors.up => rfl
@ -124,7 +124,7 @@ lemma color_eq_dual_symm {μ ν : Colors} (h : μ = τ ν) : ν = τ μ :=
-/
/-- An equivalence of Index values from an equality of color maps. -/
def castIndexValue {X : FintypeCat} {T S : X → Colors} (h : T = S) :
def castIndexValue {X : FintypeCat} {T S : X → Colors} (h : T = S) :
IndexValue d T ≃ IndexValue d S where
toFun i := (fun μ => castColorsIndex (congrFun h μ) (i μ))
invFun i := (fun μ => (castColorsIndex (congrFun h μ)).symm (i μ))
@ -133,7 +133,7 @@ def castIndexValue {X : FintypeCat} {T S : X → Colors} (h : T = S) :
right_inv i := by
simp
lemma indexValue_eq {T₁ T₂ : X → RealLorentzTensor.Colors} (d : ℕ) (h : T₁ = T₂) :
lemma indexValue_eq {T₁ T₂ : X → RealLorentzTensor.Colors} (d : ℕ) (h : T₁ = T₂) :
IndexValue d T₁ = IndexValue d T₂ :=
pi_congr fun a => congrArg (ColorsIndex d) (congrFun h a)
@ -176,7 +176,7 @@ lemma ext' {T₁ T₂ : RealLorentzTensor d X} (h : T₁.color = T₂.color)
/-- An equivalence between `X → Fin 1 ⊕ Fin d` and `Y → Fin 1 ⊕ Fin d` given an isomorphism
between `X` and `Y`. -/
def congrSetIndexValue (d : ℕ) (f : X ≃ Y) (i : X → Colors) :
IndexValue d i ≃ IndexValue d (i ∘ f.symm) :=
IndexValue d i ≃ IndexValue d (i ∘ f.symm) :=
Equiv.piCongrLeft' _ f
/-- Given an equivalence of indexing sets, a map on Lorentz tensors. -/
@ -185,8 +185,8 @@ def congrSetMap (f : X ≃ Y) (T : RealLorentzTensor d X) : RealLorentzTensor d
color := T.color ∘ f.symm
coord := T.coord ∘ (congrSetIndexValue d f T.color).symm
lemma congrSetMap_trans (f : X ≃ Y) (g : Y ≃ Z) (T : RealLorentzTensor d X) :
congrSetMap g (congrSetMap f T) = congrSetMap (f.trans g) T := by
lemma congrSetMap_trans (f : X ≃ Y) (g : Y ≃ Z) (T : RealLorentzTensor d X) :
congrSetMap g (congrSetMap f T) = congrSetMap (f.trans g) T := by
apply ext (by rfl)
have h1 : (congrSetIndexValue d (f.trans g) T.color) = (congrSetIndexValue d f T.color).trans
(congrSetIndexValue d g ((Equiv.piCongrLeft' (fun _ => Colors) f) T.color)) := by
@ -259,7 +259,7 @@ def inrIndexValue {Tc : X → Colors} {Sc : Y → Colors}
def sumCommIndexValue {X Y : FintypeCat} (Tc : X → Colors) (Sc : Y → Colors) :
IndexValue d (sumElimIndexColor Tc Sc) ≃ IndexValue d (sumElimIndexColor Sc Tc) :=
(congrSetIndexValue d (sumCommFintypeCat X Y) (sumElimIndexColor Tc Sc)).trans
(castIndexValue ((sumElimIndexColor_symm Sc Tc).symm))
(castIndexValue ((sumElimIndexColor_symm Sc Tc).symm))
lemma sumCommIndexValue_inlIndexValue {X Y : FintypeCat} {Tc : X → Colors} {Sc : Y → Colors}
(i : IndexValue d (sumElimIndexColor Tc Sc)) :
@ -300,7 +300,7 @@ def unmarkedColor (T : Marked d X n) : X → Colors :=
T.color ∘ Sum.inl
/-- The colors of marked indices. -/
def markedColor (T : Marked d X n) : FintypeCat.of (Σ _ : Fin n, PUnit) → Colors :=
def markedColor (T : Marked d X n) : FintypeCat.of (Σ _ : Fin n, PUnit) → Colors :=
T.color ∘ Sum.inr
/-- The index values restricted to unmarked indices. -/

18
HepLean/SpaceTime/LorentzVector/AsSelfAdjointMatrix.lean
View file

@ -81,7 +81,7 @@ noncomputable def toSelfAdjointMatrix :
rw [← h01, RCLike.conj_eq_re_sub_im]
rfl
exact conj_eq_iff_re.mp (congrArg (fun M => M 1 1) $ selfAdjoint.mem_iff.mp x.2 )
map_add' x y := by
map_add' x y := by
ext i j : 2
simp only [toSelfAdjointMatrix'_coe, add_apply, ofReal_add, of_apply, cons_val', empty_val',
cons_val_fin_one, AddSubmonoid.coe_add, AddSubgroup.coe_toAddSubmonoid, Matrix.add_apply]
@ -109,22 +109,22 @@ noncomputable def toSelfAdjointMatrix :
simp only [toSelfAdjointMatrix'_coe, Fin.isValue, of_apply, cons_val', empty_val',
cons_val_fin_one, RingHom.id_apply, selfAdjoint.val_smul, smul_apply, real_smul]
fin_cases i <;> fin_cases j
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero]
ring
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, Fin.zero_eta,
cons_val_zero]
ring
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
· rw [show (r • x) (Sum.inr 0) = r * x (Sum.inr 0) from rfl]
rw [show (r • x) (Sum.inr 1) = r * x (Sum.inr 1) from rfl]
simp only [Fin.isValue, ofReal_mul, Fin.zero_eta, cons_val_zero, Fin.mk_one, cons_val_one,
head_fin_const]
ring
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
· rw [show (r • x) (Sum.inl 0) = r * x (Sum.inl 0) from rfl]
rw [show (r • x) (Sum.inr 2) = r * x (Sum.inr 2) from rfl]
simp only [Fin.isValue, ofReal_mul, Fin.mk_one, cons_val_one, head_cons, head_fin_const]
ring

2
HepLean/SpaceTime/LorentzVector/Basic.lean
View file

@ -43,7 +43,7 @@ variable (v : LorentzVector d)
/-- The space components. -/
@[simp]
def space : EuclideanSpace ℝ (Fin d) := v ∘ Sum.inr
def space : EuclideanSpace ℝ (Fin d) := v ∘ Sum.inr
/-- The time component. -/
@[simp]

14
HepLean/SpaceTime/LorentzVector/NormOne.lean
View file

@ -36,7 +36,7 @@ def neg : NormOneLorentzVector d := ⟨- v, by
simp only [map_neg, LinearMap.neg_apply, neg_neg]
exact v.2⟩
lemma time_sq : v.1.time ^ 2 = 1 + ‖v.1.space‖ ^ 2 := by
lemma time_sq : v.1.time ^ 2 = 1 + ‖v.1.space‖ ^ 2 := by
rw [time_sq_eq_metric_add_space, v.2]
lemma abs_time_ge_one : 1 ≤ |v.1.time| := by
@ -48,7 +48,7 @@ lemma norm_space_le_abs_time : ‖v.1.space‖ < |v.1.time| := by
rw [(abs_norm _).symm, ← @sq_lt_sq, time_sq]
exact lt_one_add (‖(v.1).space‖ ^ 2)
lemma norm_space_leq_abs_time : ‖v.1.space‖ ≤ |v.1.time| :=
lemma norm_space_leq_abs_time : ‖v.1.space‖ ≤ |v.1.time| :=
le_of_lt (norm_space_le_abs_time v)
lemma time_le_minus_one_or_ge_one : v.1.time ≤ -1 ∨ 1 ≤ v.1.time :=
@ -77,7 +77,7 @@ lemma time_pos_iff : 0 < v.1.time ↔ 1 ≤ v.1.time := by
· exact (time_nonneg_iff v).mp (le_of_lt h)
· linarith
lemma time_abs_sub_space_norm :
lemma time_abs_sub_space_norm :
0 ≤ |v.1.time| * |w.1.time| - ‖v.1.space‖ * ‖w.1.space‖ := by
apply sub_nonneg.mpr
apply mul_le_mul (norm_space_leq_abs_time v) (norm_space_leq_abs_time w) ?_ ?_
@ -167,7 +167,7 @@ lemma metric_reflect_mem_mem (h : v ∈ FuturePointing d) (hw : w ∈ FuturePoin
exact abs_real_inner_le_norm (v.1).space (w.1).space
lemma metric_reflect_not_mem_not_mem (h : v ∉ FuturePointing d) (hw : w ∉ FuturePointing d) :
0 ≤ ⟪v.1, w.1.spaceReflection⟫ₘ := by
0 ≤ ⟪v.1, w.1.spaceReflection⟫ₘ := by
have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) ((not_mem_iff_neg w).mp hw)
apply le_of_le_of_eq h1 ?_
simp [neg]
@ -179,10 +179,10 @@ lemma metric_reflect_mem_not_mem (h : v ∈ FuturePointing d) (hw : w ∉ Future
apply le_of_le_of_eq h1 ?_
simp [neg]
lemma metric_reflect_not_mem_mem (h : v ∉ FuturePointing d) (hw : w ∈ FuturePointing d) :
lemma metric_reflect_not_mem_mem (h : v ∉ FuturePointing d) (hw : w ∈ FuturePointing d) :
⟪v.1, w.1.spaceReflection⟫ₘ ≤ 0 := by
rw [show (0 : ℝ) = - 0 from zero_eq_neg.mpr rfl, le_neg]
have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) hw
have h1 := metric_reflect_mem_mem ((not_mem_iff_neg v).mp h) hw
apply le_of_le_of_eq h1 ?_
simp [neg]
@ -264,7 +264,7 @@ noncomputable def pathFromTime (u : FuturePointing d) : Path timeVecNormOneFutur
lemma isPathConnected : IsPathConnected (@Set.univ (FuturePointing d)) := by
use timeVecNormOneFuture
apply And.intro trivial ?_
apply And.intro trivial ?_
intro y a
use pathFromTime y
exact fun _ => a

30
HepLean/SpaceTime/MinkowskiMetric.lean
View file

@ -35,7 +35,7 @@ variable {d : ℕ}
scoped[minkowskiMatrix] notation "η" => minkowskiMatrix
@[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
ext1 i j
rcases i with i | i <;> rcases j with j | j
@ -58,7 +58,7 @@ lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
lemma as_block : @minkowskiMatrix d = (
lemma as_block : @minkowskiMatrix d = (
Matrix.fromBlocks (1 : Matrix (Fin 1) (Fin 1) ℝ) 0 0 (-1 : Matrix (Fin d) (Fin d) ℝ)) := by
rw [minkowskiMatrix]
simp [LieAlgebra.Orthogonal.indefiniteDiagonal]
@ -77,7 +77,7 @@ end minkowskiMatrix
-/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w` -/
/-- Given a Lorentz vector `v` we define the the linear map `w ↦ v * η * w`. -/
@[simps!]
def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ[ℝ] ℝ where
toFun w := v ⬝ᵥ (minkowskiMatrix *ᵥ w)
@ -90,7 +90,7 @@ def minkowskiLinearForm {d : ℕ} (v : LorentzVector d) : LorentzVector d →ₗ
rfl
/-- The Minkowski metric as a bilinear map. -/
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
def minkowskiMetric {d : ℕ} : LorentzVector d →ₗ[ℝ] LorentzVector d →ₗ[ℝ] ℝ where
toFun v := minkowskiLinearForm v
map_add' y z := by
ext1 x
@ -134,7 +134,7 @@ lemma as_sum_self : ⟪v, v⟫ₘ = v.time ^ 2 - ‖v.space‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, PiLp.inner_apply, as_sum]
noncomm_ring
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ := by
rw [as_sum, @PiLp.inner_apply]
noncomm_ring
@ -153,7 +153,7 @@ lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^
/-!
# Minkowski metric and space reflections
# Minkowski metric and space reflections
-/
@ -195,7 +195,7 @@ lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
-/
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
lemma leq_time_sq : ⟪v, v⟫ₘ ≤ v.time ^ 2 := by
rw [time_sq_eq_metric_add_space]
exact (le_add_iff_nonneg_right _).mpr $ sq_nonneg ‖v.space‖
@ -227,7 +227,7 @@ lemma dual_id : @dual d 1 = 1 := by
@[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ := by
simp only [dual, transpose_mul]
trans η * Λ'ᵀ * (η * η) * Λᵀ * η
trans η * Λ'ᵀ * (η * η) * Λᵀ * η
noncomm_ring [minkowskiMatrix.sq]
noncomm_ring
@ -256,7 +256,7 @@ lemma det_dual : (dual Λ).det = Λ.det := by
simp
@[simp]
lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ := by
lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ := by
simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, dual, minkowskiLinearForm_apply,
mulVec_mulVec, ← mul_assoc, minkowskiMatrix.sq, one_mul, (vecMul_transpose Λ x).symm, ←
dotProduct_mulVec]
@ -265,7 +265,7 @@ lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ
lemma dual_mulVec_left : ⟪(dual Λ) *ᵥ x, y⟫ₘ = ⟪x, Λ *ᵥ y⟫ₘ := by
rw [symm, dual_mulVec_right, symm]
lemma matrix_apply_eq_iff_sub : ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ ↔ ⟪v, (Λ - Λ') *ᵥ w⟫ₘ = 0 := by
lemma matrix_apply_eq_iff_sub : ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ ↔ ⟪v, (Λ - Λ') *ᵥ w⟫ₘ = 0 := by
rw [← sub_eq_zero, ← LinearMap.map_sub, sub_mulVec]
lemma matrix_eq_iff_eq_forall' : (∀ v, Λ *ᵥ v= Λ' *ᵥ v) ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ := by
@ -278,7 +278,7 @@ lemma matrix_eq_iff_eq_forall' : (∀ v, Λ *ᵥ v= Λ' *ᵥ v) ↔ ∀ w v, ⟪
have h1 := h v
rw [nondegenerate] at h1
simp [sub_mulVec] at h1
exact h1
exact h1
lemma matrix_eq_iff_eq_forall : Λ = Λ' ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ := by
rw [← matrix_eq_iff_eq_forall']
@ -302,7 +302,7 @@ lemma matrix_eq_id_iff : Λ = 1 ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, w⟫
-/
@[simp]
lemma basis_left (μ : Fin 1 ⊕ Fin d) : ⟪e μ, v⟫ₘ = η μ μ * v μ := by
lemma basis_left (μ : Fin 1 ⊕ Fin d) : ⟪e μ, v⟫ₘ = η μ μ * v μ := by
rw [as_sum]
rcases μ with μ | μ
· fin_cases μ
@ -314,7 +314,7 @@ lemma on_timeVec : ⟪timeVec, @timeVec d⟫ₘ = 1 := by
LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_apply_eq, Sum.elim_inl, stdBasis_apply,
↓reduceIte, mul_one]
lemma on_basis_mulVec (μ ν : Fin 1 ⊕ Fin d) : ⟪e μ, Λ *ᵥ e ν⟫ₘ = η μ μ * Λ μ ν := by
lemma on_basis_mulVec (μ ν : Fin 1 ⊕ Fin d) : ⟪e μ, Λ *ᵥ e ν⟫ₘ = η μ μ * Λ μ ν := by
simp [basis_left, mulVec, dotProduct, stdBasis_apply]
lemma on_basis (μ ν : Fin 1 ⊕ Fin d) : ⟪e μ, e ν⟫ₘ = η μ ν := by
@ -324,8 +324,8 @@ lemma on_basis (μ ν : Fin 1 ⊕ Fin d) : ⟪e μ, e ν⟫ₘ = η μ ν := by
· simp [h, LieAlgebra.Orthogonal.indefiniteDiagonal, minkowskiMatrix]
exact fun a => False.elim (h (id (Eq.symm a)))
lemma matrix_apply_stdBasis (ν μ : Fin 1 ⊕ Fin d):
Λ ν μ = η ν ν * ⟪e ν, Λ *ᵥ e μ⟫ₘ := by
lemma matrix_apply_stdBasis (ν μ : Fin 1 ⊕ Fin d):
Λ ν μ = η ν ν * ⟪e ν, Λ *ᵥ e μ⟫ₘ := by
rw [on_basis_mulVec, ← mul_assoc]
have h1 : η ν ν * η ν ν = 1 := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]

12
HepLean/SpaceTime/SL2C/Basic.lean
View file

@ -33,8 +33,8 @@ we can define a representation a representation of `SL(2, ℂ)` on spacetime.
-/
/-- Given an element `M ∈ SL(2, ℂ)` the linear map from `selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)` to
itself defined by `A ↦ M * A * Mᴴ`. -/
/-- Given an element `M ∈ SL(2, ℂ)` the linear map from `selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)` to
itself defined by `A ↦ M * A * Mᴴ`. -/
@[simps!]
def toLinearMapSelfAdjointMatrix (M : SL(2, ℂ)) :
selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) →ₗ[ℝ] selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) where
@ -50,7 +50,7 @@ def toLinearMapSelfAdjointMatrix (M : SL(2, ℂ)) :
noncomm_ring [selfAdjoint.val_smul, Algebra.mul_smul_comm, Algebra.smul_mul_assoc,
RingHom.id_apply]
/-- The representation of `SL(2, ℂ)` on `selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)` given by
/-- The representation of `SL(2, ℂ)` on `selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)` given by
`M A ↦ M * A * Mᴴ`. -/
@[simps!]
def repSelfAdjointMatrix : Representation ℝ SL(2, ℂ) $ selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) where
@ -63,7 +63,7 @@ def repSelfAdjointMatrix : Representation ℝ SL(2, ℂ) $ selfAdjoint (Matrix (
noncomm_ring [toLinearMapSelfAdjointMatrix, SpecialLinearGroup.coe_mul, mul_assoc,
conjTranspose_mul, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply]
/-- The representation of `SL(2, ℂ)` on `spaceTime` obtained from `toSelfAdjointMatrix` and
/-- The representation of `SL(2, ℂ)` on `spaceTime` obtained from `toSelfAdjointMatrix` and
`repSelfAdjointMatrix`. -/
def repLorentzVector : Representation ℝ SL(2, ℂ) (LorentzVector 3) where
toFun M := toSelfAdjointMatrix.symm.comp ((repSelfAdjointMatrix M).comp
@ -85,7 +85,7 @@ In the next section we will restrict this homomorphism to the restricted Lorentz
-/
lemma iff_det_selfAdjoint (Λ : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ): Λ ∈ LorentzGroup 3 ↔
lemma iff_det_selfAdjoint (Λ : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) : Λ ∈ LorentzGroup 3 ↔
∀ (x : selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)),
det ((toSelfAdjointMatrix ∘
toLin LorentzVector.stdBasis LorentzVector.stdBasis Λ ∘ toSelfAdjointMatrix.symm) x).1
@ -107,7 +107,7 @@ def toLorentzGroupElem (M : SL(2, ℂ)) : LorentzGroup 3 :=
/-- The group homomorphism from ` SL(2, ℂ)` to the Lorentz group `𝓛`. -/
@[simps!]
def toLorentzGroup : SL(2, ℂ) →* LorentzGroup 3 where
def toLorentzGroup : SL(2, ℂ) →* LorentzGroup 3 where
toFun := toLorentzGroupElem
map_one' := by
simp only [toLorentzGroupElem, _root_.map_one, LinearMap.toMatrix_one]

6
HepLean/StandardModel/HiggsBoson/Basic.lean
View file

@ -72,7 +72,7 @@ We also define the Higgs bundle.
/-- The trivial vector bundle 𝓡² × ℂ². -/
abbrev HiggsBundle := Bundle.Trivial SpaceTime HiggsVec
instance : SmoothVectorBundle HiggsVec HiggsBundle SpaceTime.asSmoothManifold :=
instance : SmoothVectorBundle HiggsVec HiggsBundle SpaceTime.asSmoothManifold :=
Bundle.Trivial.smoothVectorBundle HiggsVec 𝓘(ℝ, SpaceTime)
/-- A Higgs field is a smooth section of the Higgs bundle. -/
@ -133,11 +133,11 @@ lemma apply_smooth (φ : HiggsField) :
(smooth_pi_space).mp (φ.toVec_smooth)
lemma apply_re_smooth (φ : HiggsField) (i : Fin 2) :
Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (reCLM ∘ (fun (x : SpaceTime) => (φ x i))) :=
Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (reCLM ∘ (fun (x : SpaceTime) => (φ x i))) :=
(ContinuousLinearMap.smooth reCLM).comp (φ.apply_smooth i)
lemma apply_im_smooth (φ : HiggsField) (i : Fin 2) :
Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (imCLM ∘ (fun (x : SpaceTime) => (φ x i))) :=
Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (imCLM ∘ (fun (x : SpaceTime) => (φ x i))) :=
(ContinuousLinearMap.smooth imCLM).comp (φ.apply_smooth i)
/-!

22
HepLean/StandardModel/HiggsBoson/GaugeAction.lean
View file

@ -34,9 +34,9 @@ open ComplexConjugate
@[simps!]
noncomputable def higgsRepUnitary : GaugeGroup →* unitaryGroup (Fin 2) ℂ where
toFun g := repU1 g.2.2 * fundamentalSU2 g.2.1
map_mul' := by
map_mul' := by
intro ⟨_, a2, a3⟩ ⟨_, b2, b3⟩
change repU1 (a3 * b3) * fundamentalSU2 (a2 * b2) = _
change repU1 (a3 * b3) * fundamentalSU2 (a2 * b2) = _
rw [repU1.map_mul, fundamentalSU2.map_mul, mul_assoc, mul_assoc,
← mul_assoc (repU1 b3) _ _, repU1_fundamentalSU2_commute]
repeat rw [mul_assoc]
@ -97,31 +97,31 @@ def rotateMatrix (φ : HiggsVec) : Matrix (Fin 2) (Fin 2) ℂ :=
lemma rotateMatrix_star (φ : HiggsVec) :
star φ.rotateMatrix =
![![conj (φ 1) /‖φ‖ , φ 0 /‖φ‖], ![- conj (φ 0) / ‖φ‖ , φ 1 / ‖φ‖] ] := by
![![conj (φ 1) /‖φ‖ , φ 0 /‖φ‖], ![- conj (φ 0) / ‖φ‖ , φ 1 / ‖φ‖] ] := by
simp_rw [star, rotateMatrix, conjTranspose]
ext i j
fin_cases i <;> fin_cases j <;> simp [conj_ofReal]
lemma rotateMatrix_det {φ : HiggsVec} (hφ : φ ≠ 0) : (rotateMatrix φ).det = 1 := by
have h1 : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
have h1 : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
field_simp [rotateMatrix, det_fin_two]
rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
lemma rotateMatrix_unitary {φ : HiggsVec} (hφ : φ ≠ 0) :
(rotateMatrix φ) ∈ unitaryGroup (Fin 2) ℂ := by
rw [mem_unitaryGroup_iff', rotateMatrix_star, rotateMatrix]
erw [mul_fin_two, one_fin_two]
have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
ext i j
fin_cases i <;> fin_cases j <;> field_simp
<;> rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm, add_comm]
· ring_nf
· ring_nf
· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
· simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
lemma rotateMatrix_specialUnitary {φ : HiggsVec} (hφ : φ ≠ 0) :
@ -147,10 +147,10 @@ lemma rotateGuageGroup_apply {φ : HiggsVec} (hφ : φ ≠ 0) :
· simp only [Fin.mk_one, Fin.isValue, cons_val_one, head_cons, mulVec, Fin.isValue,
cons_val', empty_val', cons_val_fin_one, vecHead, cons_dotProduct, vecTail, Nat.succ_eq_add_one,
Nat.reduceAdd, Function.comp_apply, Fin.succ_zero_eq_one, dotProduct_empty, add_zero]
have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
have : (‖φ‖ : ℂ) ≠ 0 := ofReal_inj.mp.mt (norm_ne_zero_iff.mpr hφ)
field_simp
rw [← ofReal_mul, ← sq, ← @real_inner_self_eq_norm_sq]
simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
simp [PiLp.inner_apply, Complex.inner, neg_mul, sub_neg_eq_add,
Fin.sum_univ_two, ofReal_add, ofReal_mul, mul_conj, mul_comm]
theorem rotate_fst_zero_snd_real (φ : HiggsVec) :
@ -166,7 +166,7 @@ theorem rotate_fst_zero_snd_real (φ : HiggsVec) :
end HiggsVec
/-! TODO: Define the global gauge action on HiggsField. -/
/-! TODO: Prove `⟪φ1, φ2⟫_H` invariant under the global gauge action. (norm_map_of_mem_unitary) -/
/-! TODO: Prove `⟪φ1, φ2⟫_H` invariant under the global gauge action. (norm_map_of_mem_unitary) -/
/-! TODO: Prove invariance of potential under global gauge action. -/
end StandardModel

14
HepLean/StandardModel/HiggsBoson/PointwiseInnerProd.lean
View file

@ -55,13 +55,13 @@ lemma innerProd_left_zero (φ : HiggsField) : ⟪0, φ⟫_H = 0 := by
lemma innerProd_right_zero (φ : HiggsField) : ⟪φ, 0⟫_H = 0 := by
funext x
simp [innerProd]
example (x : ℝ): RCLike.ofReal x = ofReal' x := by
example (x : ℝ): RCLike.ofReal x = ofReal' x := by
rfl
lemma innerProd_expand (φ1 φ2 : HiggsField) :
⟪φ1, φ2⟫_H = fun x => equivRealProdCLM.symm (((φ1 x 0).re * (φ2 x 0).re
⟪φ1, φ2⟫_H = fun x => equivRealProdCLM.symm (((φ1 x 0).re * (φ2 x 0).re
+ (φ1 x 1).re * (φ2 x 1).re+ (φ1 x 0).im * (φ2 x 0).im + (φ1 x 1).im * (φ2 x 1).im),
((φ1 x 0).re * (φ2 x 0).im + (φ1 x 1).re * (φ2 x 1).im
- (φ1 x 0).im * (φ2 x 0).re - (φ1 x 1).im * (φ2 x 1).re)) := by
- (φ1 x 0).im * (φ2 x 0).re - (φ1 x 1).im * (φ2 x 1).re)) := by
funext x
simp only [innerProd, PiLp.inner_apply, RCLike.inner_apply, Fin.sum_univ_two,
equivRealProdCLM_symm_apply, ofReal_add, ofReal_mul, ofReal_sub]
@ -69,7 +69,7 @@ lemma innerProd_expand (φ1 φ2 : HiggsField) :
nth_rewrite 1 [← RCLike.re_add_im (φ2 x 0)]
nth_rewrite 1 [← RCLike.re_add_im (φ2 x 1)]
ring_nf
repeat rw [show RCLike.ofReal _ = ofReal' _ by rfl]
repeat rw [show RCLike.ofReal _ = ofReal' _ by rfl]
simp only [Algebra.id.map_eq_id, RCLike.re_to_complex, RingHom.id_apply, RCLike.I_to_complex,
RCLike.im_to_complex, I_sq, mul_neg, mul_one, neg_mul, sub_neg_eq_add, one_mul]
ring
@ -124,10 +124,10 @@ lemma normSq_eq_innerProd_self (φ : HiggsField) (x : SpaceTime) :
-/
lemma toHiggsVec_norm (φ : HiggsField) (x : SpaceTime) :
‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl
‖φ x‖ = ‖φ.toHiggsVec x‖ := rfl
lemma normSq_expand (φ : HiggsField) :
φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1) ).re := by
lemma normSq_expand (φ : HiggsField) :
φ.normSq = fun x => (conj (φ x 0) * (φ x 0) + conj (φ x 1) * (φ x 1)).re := by
funext x
simp [normSq, add_re, mul_re, conj_re, conj_im, neg_mul, sub_neg_eq_add, @norm_sq_eq_inner ℂ]

46
HepLean/StandardModel/HiggsBoson/Potential.lean
View file

@ -34,7 +34,7 @@ open SpaceTime
/-- The Higgs potential of the form `- μ² * |φ|² + 𝓵 * |φ|⁴`. -/
@[simp]
def potential (μ2 𝓵 : ℝ ) (φ : HiggsField) (x : SpaceTime) : ℝ :=
def potential (μ2 𝓵 : ℝ ) (φ : HiggsField) (x : SpaceTime) : ℝ :=
- μ2 * ‖φ‖_H ^ 2 x + 𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x
/-!
@ -43,7 +43,7 @@ def potential (μ2 𝓵 : ℝ ) (φ : HiggsField) (x : SpaceTime) : ℝ :=
-/
lemma potential_smooth (μSq lambda : ℝ) (φ : HiggsField) :
lemma potential_smooth (μSq lambda : ℝ) (φ : HiggsField) :
Smooth 𝓘(ℝ, SpaceTime) 𝓘(ℝ, ℝ) (fun x => φ.potential μSq lambda x) := by
simp only [potential, normSq, neg_mul]
exact (smooth_const.smul φ.normSq_smooth).neg.add
@ -56,7 +56,7 @@ namespace potential
-/
lemma complete_square (μ2 𝓵 : ℝ) (h : 𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
lemma complete_square (μ2 𝓵 : ℝ) (h : 𝓵 ≠ 0) (φ : HiggsField) (x : SpaceTime) :
potential μ2 𝓵 φ x = 𝓵 * (‖φ‖_H ^ 2 x - μ2 / (2 * 𝓵)) ^ 2 - μ2 ^ 2 / (4 * 𝓵) := by
simp only [potential]
field_simp
@ -69,7 +69,7 @@ lemma complete_square (μ2 𝓵 : ℝ) (h : 𝓵 ≠ 0) (φ : HiggsField) (x : S
-/
/-- The proposition on the coefficents for a potential to be bounded. -/
def IsBounded (μ2 𝓵 : ℝ) : Prop :=
def IsBounded (μ2 𝓵 : ℝ) : Prop :=
∃ c, ∀ Φ x, c ≤ potential μ2 𝓵 Φ x
/-! TODO: Show when 𝓵 < 0, the potential is not bounded. -/
@ -87,19 +87,19 @@ variable (h𝓵 : 0 < 𝓵)
/-- The second term of the potential is non-negative. -/
lemma snd_term_nonneg (φ : HiggsField) (x : SpaceTime) :
0 ≤ 𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x := by
0 ≤ 𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x := by
rw [mul_nonneg_iff]
apply Or.inl
simp_all only [normSq, gt_iff_lt, mul_nonneg_iff_of_pos_left, ge_iff_le, norm_nonneg, pow_nonneg,
and_self]
lemma as_quad (μ2 𝓵 : ℝ) (φ : HiggsField) (x : SpaceTime) :
𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x + (- μ2 ) * ‖φ‖_H ^ 2 x + (- potential μ2 𝓵 φ x) = 0 := by
𝓵 * ‖φ‖_H ^ 2 x * ‖φ‖_H ^ 2 x + (- μ2 ) * ‖φ‖_H ^ 2 x + (- potential μ2 𝓵 φ x) = 0 := by
simp only [normSq, neg_mul, potential, neg_add_rev, neg_neg]
ring
/-- The discriminant of the quadratic formed by the potential is non-negative. -/
lemma discrim_nonneg (φ : HiggsField) (x : SpaceTime) :
lemma discrim_nonneg (φ : HiggsField) (x : SpaceTime) :
0 ≤ discrim 𝓵 (- μ2) (- potential μ2 𝓵 φ x) := by
have h1 := as_quad μ2 𝓵 φ x
rw [quadratic_eq_zero_iff_discrim_eq_sq] at h1
@ -111,7 +111,7 @@ lemma eq_zero_at (φ : HiggsField) (x : SpaceTime)
(hV : potential μ2 𝓵 φ x = 0) : φ x = 0 ∨ ‖φ‖_H ^ 2 x = μ2 / 𝓵 := by
have h1 := as_quad μ2 𝓵 φ x
rw [hV] at h1
have h2 : ‖φ‖_H ^ 2 x * (𝓵 * ‖φ‖_H ^ 2 x + - μ2) = 0 := by
have h2 : ‖φ‖_H ^ 2 x * (𝓵 * ‖φ‖_H ^ 2 x + - μ2) = 0 := by
linear_combination h1
simp at h2
cases' h2 with h2 h2
@ -121,13 +121,13 @@ lemma eq_zero_at (φ : HiggsField) (x : SpaceTime)
ring_nf
linear_combination h2
lemma eq_zero_at_of_μSq_nonpos {μ2 : ℝ} (hμ2 : μ2 ≤ 0)
(φ : HiggsField) (x : SpaceTime) (hV : potential μ2 𝓵 φ x = 0) : φ x = 0 := by
lemma eq_zero_at_of_μSq_nonpos {μ2 : ℝ} (hμ2 : μ2 ≤ 0)
(φ : HiggsField) (x : SpaceTime) (hV : potential μ2 𝓵 φ x = 0) : φ x = 0 := by
cases' (eq_zero_at μ2 h𝓵 φ x hV) with h1 h1
exact h1
by_cases hμSqZ : μ2 = 0
simpa [hμSqZ] using h1
refine ((?_ : ¬ 0 ≤ μ2 / 𝓵) (?_)).elim
refine ((?_ : ¬ 0 ≤ μ2 / 𝓵) (?_)).elim
· simp_all [div_nonneg_iff]
intro h
exact lt_imp_lt_of_le_imp_le (fun _ => h) (lt_of_le_of_ne hμ2 hμSqZ)
@ -135,17 +135,17 @@ lemma eq_zero_at_of_μSq_nonpos {μ2 : ℝ} (hμ2 : μ2 ≤ 0)
exact normSq_nonneg φ x
lemma bounded_below (φ : HiggsField) (x : SpaceTime) :
- μ2 ^ 2 / (4 * 𝓵) ≤ potential μ2 𝓵 φ x := by
- μ2 ^ 2 / (4 * 𝓵) ≤ potential μ2 𝓵 φ x := by
have h1 := discrim_nonneg μ2 h𝓵 φ x
simp only [discrim, even_two, Even.neg_pow, normSq, neg_mul, neg_add_rev, neg_neg] at h1
ring_nf at h1
rw [← neg_le_iff_add_nonneg'] at h1
rw [show 𝓵 * potential μ2 𝓵 φ x * 4 = (4 * 𝓵) * potential μ2 𝓵 φ x by ring] at h1
have h2 := (div_le_iff' (by simp [h𝓵] : 0 < 4 * 𝓵)).mpr h1
have h2 := (div_le_iff' (by simp [h𝓵] : 0 < 4 * 𝓵)).mpr h1
ring_nf at h2 ⊢
exact h2
lemma bounded_below_of_μSq_nonpos {μ2 : ℝ}
lemma bounded_below_of_μSq_nonpos {μ2 : ℝ}
(hμSq : μ2 ≤ 0) (φ : HiggsField) (x : SpaceTime) : 0 ≤ potential μ2 𝓵 φ x := by
refine add_nonneg ?_ (snd_term_nonneg h𝓵 φ x)
field_simp [mul_nonpos_iff]
@ -165,13 +165,13 @@ variable (h𝓵 : 0 < 𝓵)
-/
lemma discrim_eq_zero_of_eq_bound (φ : HiggsField) (x : SpaceTime)
(hV : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
(hV : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
discrim 𝓵 (- μ2) (- potential μ2 𝓵 φ x) = 0 := by
rw [discrim, hV]
field_simp
lemma normSq_of_eq_bound (φ : HiggsField) (x : SpaceTime)
(hV : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
(hV : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
‖φ‖_H ^ 2 x = μ2 / (2 * 𝓵) := by
have h1 := as_quad μ2 𝓵 φ x
rw [quadratic_eq_zero_iff_of_discrim_eq_zero _
@ -180,36 +180,36 @@ lemma normSq_of_eq_bound (φ : HiggsField) (x : SpaceTime)
exact ne_of_gt h𝓵
lemma eq_bound_iff (φ : HiggsField) (x : SpaceTime) :
potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵) ↔ ‖φ‖_H ^ 2 x = μ2 / (2 * 𝓵) :=
potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵) ↔ ‖φ‖_H ^ 2 x = μ2 / (2 * 𝓵) :=
Iff.intro (normSq_of_eq_bound μ2 h𝓵 φ x)
(fun h ↦ by
rw [potential, h]
field_simp
ring_nf)
lemma eq_bound_iff_of_μSq_nonpos {μ2 : ℝ}
lemma eq_bound_iff_of_μSq_nonpos {μ2 : ℝ}
(hμ2 : μ2 ≤ 0) (φ : HiggsField) (x : SpaceTime) :
potential μ2 𝓵 φ x = 0 ↔ φ x = 0 :=
Iff.intro (fun h ↦ eq_zero_at_of_μSq_nonpos h𝓵 hμ2 φ x h)
(fun h ↦ by simp [potential, h])
lemma eq_bound_IsMinOn (φ : HiggsField) (x : SpaceTime)
(hv : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
(hv : potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵)) :
IsMinOn (fun (φ, x) => potential μ2 𝓵 φ x) Set.univ (φ, x) := by
rw [isMinOn_univ_iff]
simp only [normSq, neg_mul, le_neg_add_iff_add_le, ge_iff_le, hv]
exact fun (φ', x') ↦ bounded_below μ2 h𝓵 φ' x'
lemma eq_bound_IsMinOn_of_μSq_nonpos {μ2 : ℝ}
lemma eq_bound_IsMinOn_of_μSq_nonpos {μ2 : ℝ}
(hμ2 : μ2 ≤ 0) (φ : HiggsField) (x : SpaceTime) (hv : potential μ2 𝓵 φ x = 0) :
IsMinOn (fun (φ, x) => potential μ2 𝓵 φ x) Set.univ (φ, x) := by
rw [isMinOn_univ_iff]
simp only [normSq, neg_mul, le_neg_add_iff_add_le, ge_iff_le, hv]
exact fun (φ', x') ↦ bounded_below_of_μSq_nonpos h𝓵 hμ2 φ' x'
lemma bound_reached_of_μSq_nonneg {μ2 : ℝ} (hμ2 : 0 ≤ μ2) :
lemma bound_reached_of_μSq_nonneg {μ2 : ℝ} (hμ2 : 0 ≤ μ2) :
∃ (φ : HiggsField) (x : SpaceTime),
potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵) := by
potential μ2 𝓵 φ x = - μ2 ^ 2 / (4 * 𝓵) := by
use HiggsVec.toField ![√(μ2/(2 * 𝓵)), 0], 0
refine (eq_bound_iff μ2 h𝓵 (HiggsVec.toField ![√(μ2/(2 * 𝓵)), 0]) 0).mpr ?_
simp only [normSq, HiggsVec.toField, ContMDiffSection.coeFn_mk, PiLp.norm_sq_eq_of_L2,
@ -218,7 +218,7 @@ lemma bound_reached_of_μSq_nonneg {μ2 : ℝ} (hμ2 : 0 ≤ μ2) :
not_false_eq_true, zero_pow, add_zero]
field_simp [mul_pow]
lemma IsMinOn_iff_of_μSq_nonneg {μ2 : ℝ} (hμ2 : 0 ≤ μ2) :
lemma IsMinOn_iff_of_μSq_nonneg {μ2 : ℝ} (hμ2 : 0 ≤ μ2) :
IsMinOn (fun (φ, x) => potential μ2 𝓵 φ x) Set.univ (φ, x) ↔
‖φ‖_H ^ 2 x = μ2 /(2 * 𝓵) := by
apply Iff.intro <;> rw [← eq_bound_iff μ2 h𝓵 φ]

2
lakefile.toml
View file

@ -39,7 +39,7 @@ name = "type_former_lint"
srcDir = "scripts"
[[lean_exe]]
name = "double_line_lint"
name = "hepLean_style_lint"
srcDir = "scripts"
[[lean_exe]]

8
scripts/README.md
View file

@ -7,14 +7,16 @@ Taken from Mathlib's linters. These perform text-based linting of the code.
These are to be slowly replaced with code written in Lean.
## double_line_lint
## hepLean_style_lint
Checks for double empty lines in HepLean.
Checks the following in HepLean
- There are no double empty lines.
- There are no non-initial double spaces.
Passing this linter is currently not required to pass CI on github.
Run using
```
lake exe double_line_lint
lake exe hepLean_style_lint
```
## check_file_imports.lean

31
scripts/double_line_lint.lean → scripts/hepLean_style_lint.lean
View file

@ -7,9 +7,11 @@ import Lean
import Mathlib.Tactic.Linter.TextBased
/-!
# Double line Lint
# HepLean style linter
This linter double empty lines in files.
A number of linters on HepLean to enforce a consistent style.
There are currently not enforced at the GitHub action level.
## Note
@ -19,7 +21,7 @@ Parts of this file are adapted from `Mathlib.Tactic.Linter.TextBased`,
open Lean System Meta
/-- Given a list of lines, outputs an error message and a line number. -/
def HepLeanTextLinter : Type := Array String → Array (String × ℕ)
def HepLeanTextLinter : Type := Array String → Array (String × ℕ × ℕ)
/-- Checks if there are two consecutive empty lines. -/
def doubleEmptyLineLinter : HepLeanTextLinter := fun lines ↦ Id.run do
@ -27,7 +29,20 @@ def doubleEmptyLineLinter : HepLeanTextLinter := fun lines ↦ Id.run do
let pairLines := List.zip enumLines (List.tail! enumLines)
let errors := pairLines.filterMap (fun ((lno1, l1), _, l2) ↦
if l1.length == 0 && l2.length == 0 then
some (s!" Double empty line. ", lno1)
some (s!" Double empty line. ", lno1, 1)
else none)
errors.toArray
/-- Checks if there is a double space in the line, which is not at the start. -/
def doubleSpaceLinter : HepLeanTextLinter := fun lines ↦ Id.run do
let enumLines := (lines.toList.enumFrom 1)
let errors := enumLines.filterMap (fun (lno, l) ↦
if String.containsSubstr l.trimLeft " " then
let k := (Substring.findAllSubstr l " ").toList.getLast?
let col := match k with
| none => 1
| some k => String.offsetOfPos l k.stopPos
some (s!" Non-initial double space in line.", lno, col)
else none)
errors.toArray
@ -36,18 +51,20 @@ structure HepLeanErrorContext where
error : String
/-- The line number -/
lineNumber : ℕ
/-- The column number -/
columnNumber : ℕ
/-- The file path -/
path : FilePath
def printErrors (errors : Array HepLeanErrorContext) : IO Unit := do
for e in errors do
IO.println (s!"error: {e.path}:{e.lineNumber}: {e.error}")
IO.println (s!"error: {e.path}:{e.lineNumber}:{e.columnNumber}: {e.error}")
def hepLeanLintFile (path : FilePath) : IO Bool := do
let lines ← IO.FS.lines path
let allOutput := (Array.map (fun lint ↦
(Array.map (fun (e, n) ↦ HepLeanErrorContext.mk e n path)) (lint lines)))
#[doubleEmptyLineLinter]
(Array.map (fun (e, n, c) ↦ HepLeanErrorContext.mk e n c path)) (lint lines)))
#[doubleEmptyLineLinter, doubleSpaceLinter]
let errors := allOutput.flatten
printErrors errors
return errors.size > 0