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Merge pull request #72 from HEPLean/Update-versions

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chore: Remove double empty lines
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Joseph Tooby-Smith 2024-07-03 08:30:58 -04:00 • committed by GitHub
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65 changed files with 75 additions and 249 deletions
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5
HepLean/AnomalyCancellation/MSSMNu/Basic.lean
View file

@ -22,7 +22,6 @@ universe v u
open Nat
open BigOperators
/-- The vector space of charges corresponding to the MSSM fermions. -/
@[simps!]
def MSSMCharges : ACCSystemCharges := ACCSystemChargesMk 20
@ -314,7 +313,6 @@ def quadBiLin : BiLinearSymm MSSMCharges.Charges := BiLinearSymm.mk₂ (
ring
ring)
/-- The quadratic ACC. -/
@[simp]
def accQuad : HomogeneousQuadratic MSSMCharges.Charges := quadBiLin.toHomogeneousQuad
@ -335,7 +333,6 @@ lemma accQuad_ext {S T : (MSSMCharges).Charges}
repeat rw [h1]
rw [hd, hu]
/-- The function underlying the symmetric trilinear form used to define the cubic ACC. -/
@[simp]
def cubeTriLinToFun
@ -363,7 +360,6 @@ lemma cubeTriLinToFun_map_smul₁ (a : ℚ) (S T R : MSSMCharges.Charges) :
simp only [map_smul, Hd_apply, Fin.reduceFinMk, Fin.isValue, smul_eq_mul, Hu_apply]
ring
lemma cubeTriLinToFun_map_add₁ (S T R L : MSSMCharges.Charges) :
cubeTriLinToFun (S + T, R, L) = cubeTriLinToFun (S, R, L) + cubeTriLinToFun (T, R, L) := by
simp only [cubeTriLinToFun]
@ -382,7 +378,6 @@ lemma cubeTriLinToFun_map_add₁ (S T R L : MSSMCharges.Charges) :
rw [Hd.map_add, Hu.map_add]
ring
lemma cubeTriLinToFun_swap1 (S T R : MSSMCharges.Charges) :
cubeTriLinToFun (S, T, R) = cubeTriLinToFun (T, S, R) := by
simp only [cubeTriLinToFun, MSSMSpecies_numberCharges, toSMSpecies_apply, Fin.isValue, Hd_apply,

1
HepLean/AnomalyCancellation/MSSMNu/HyperCharge.lean
View file

@ -50,5 +50,4 @@ def YAsCharge : MSSMACC.Charges := toSpecies.invFun
def Y : MSSMACC.Sols :=
MSSMACC.AnomalyFreeMk YAsCharge (by rfl) (by rfl) (by rfl) (by rfl) (by rfl) (by rfl)
end MSSMACC

5
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/Basic.lean
View file

@ -78,8 +78,6 @@ lemma Y₃_plus_B₃_plus_proj (T : MSSMACC.LinSols) (a b c : ℚ) :
funext i
linarith
lemma quad_Y₃_proj (T : MSSMACC.LinSols) :
quadBiLin Y₃.val (proj T).val = dot Y₃.val B₃.val * quadBiLin Y₃.val T.val := by
rw [proj_val]
@ -119,7 +117,6 @@ lemma quad_proj (T : MSSMACC.Sols) :
rw [quad_Y₃_proj, quad_B₃_proj, quad_self_proj]
ring
lemma cube_proj_proj_Y₃ (T : MSSMACC.LinSols) :
cubeTriLin (proj T).val (proj T).val Y₃.val =
(dot Y₃.val B₃.val)^2 * cubeTriLin T.val T.val Y₃.val := by
@ -186,6 +183,4 @@ lemma cube_proj (T : MSSMACC.Sols) :
rw [cube_proj_proj_Y₃, cube_proj_proj_B₃, cube_proj_proj_self]
ring
end MSSMACC

6
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/PlaneWithY3B3.lean
View file

@ -18,8 +18,6 @@ The plane spanned by Y₃, B₃ and third orthogonal point.
-/
universe v u
namespace MSSMACC
@ -44,7 +42,6 @@ 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') :
(planeY₃B₃ R a b c) = (planeY₃B₃ R a' b' c') := by
rw [h.1, h.2.1, h.2.2]
@ -94,7 +91,6 @@ lemma planeY₃B₃_val_eq' (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) (hR' : R
rw [ha, hb, hc]
simp
lemma planeY₃B₃_quad (R : MSSMACC.AnomalyFreePerp) (a b c : ℚ) :
accQuad (planeY₃B₃ R a b c).val = c * (2 * a * quadBiLin Y₃.val R.val
+ 2 * b * quadBiLin B₃.val R.val + c * quadBiLin R.val R.val) := by
@ -185,7 +181,6 @@ def lineCube (R : MSSMACC.AnomalyFreePerp) (a₁ a₂ a₃ : ℚ) :
(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) :
lineCube R (d * a) (d * b) (d * c) = d • lineCube R a b c := by
apply ACCSystemLinear.LinSols.ext
@ -242,5 +237,4 @@ lemma α₂_proj_zero (T : MSSMACC.Sols) (h1 : α₃ (proj T.1.1) = 0) :
end proj
end MSSMACC

5
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/ToSols.lean
View file

@ -18,7 +18,6 @@ To define `toSols` we define a series of other maps from various subtypes of
`MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` to `MSSMACC.Sols`. And show that these maps form a
surjection on certain subtypes of `MSSMACC.Sols`.
# References
The main reference for the material in this file is:
@ -78,7 +77,6 @@ lemma linEqPropSol_iff_proj_linEqProp (R : MSSMACC.Sols) :
rw [h.2.2]
simp
/-- A condition which is satisfied if the plane spanned by `R`, `Y₃` and `B₃` lies
entirely in the quadratic surface. -/
def InQuadProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
@ -137,7 +135,6 @@ def InCubeProp (R : MSSMACC.AnomalyFreePerp) : Prop :=
cubeTriLin R.val R.val R.val = 0 ∧ cubeTriLin R.val R.val B₃.val = 0 ∧
cubeTriLin R.val R.val Y₃.val = 0
instance (R : MSSMACC.AnomalyFreePerp) : Decidable (InCubeProp R) := by
apply And.decidable
@ -238,7 +235,6 @@ not surjective. -/
def toSolNS : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ → MSSMACC.Sols := fun (R, a, _ , _) =>
a • AnomalyFreeMk'' (toSolNSQuad R) (toSolNSQuad_cube R)
/-- A map from `Sols` to `MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ` which on elements of
`notInLineEqSol` will produce a right inverse to `toSolNS`. -/
def toSolNSProj (T : MSSMACC.Sols) : MSSMACC.AnomalyFreePerp × ℚ × ℚ × ℚ :=
@ -458,7 +454,6 @@ theorem toSol_surjective : Function.Surjective toSol := by
simp at h₃
exact toSol_inQuadCube ⟨T, And.intro h₁ (And.intro h₂ h₃)⟩
end AnomalyFreePerp
end MSSMACC

7
HepLean/AnomalyCancellation/PureU1/Basic.lean
View file

@ -16,7 +16,6 @@ We define the anomaly cancellation conditions for a pure U(1) gauge theory with
universe v u
open Nat
open Finset
namespace PureU1
@ -35,7 +34,6 @@ def accGrav (n : ℕ) : ((PureU1Charges n).Charges →ₗ[ℚ] ℚ) where
simp [HSMul.hSMul, SMul.smul]
rw [← Finset.mul_sum]
/-- The symmetric trilinear form used to define the cubic anomaly. -/
@[simps!]
def accCubeTriLinSymm {n : ℕ} : TriLinearSymm (PureU1Charges n).Charges := TriLinearSymm.mk₃
@ -70,14 +68,11 @@ def accCubeTriLinSymm {n : ℕ} : TriLinearSymm (PureU1Charges n).Charges := Tri
ring
)
/-- The cubic anomaly equation. -/
@[simp]
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]
@ -165,7 +160,6 @@ lemma sum_of_charges {n : ℕ} (f : Fin k → (PureU1 n).Charges) (j : Fin n) :
erw [← hlt]
simp
lemma sum_of_anomaly_free_linear {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j : Fin n) :
(∑ i : Fin k, (f i)).1 j = (∑ i : Fin k, (f i).1 j) := by
induction k
@ -177,5 +171,4 @@ lemma sum_of_anomaly_free_linear {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j
erw [← hlt]
simp
end PureU1

2
HepLean/AnomalyCancellation/PureU1/BasisLinear.lean
View file

@ -75,7 +75,6 @@ def asLinSols (j : Fin n) : (PureU1 n.succ).LinSols :=
intro hk
simp at hk⟩
lemma sum_of_vectors {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j : Fin n) :
(∑ i : Fin k, (f i)).1 j = (∑ i : Fin k, (f i).1 j) :=
sum_of_anomaly_free_linear (fun i => f i) j
@ -134,5 +133,4 @@ lemma finrank_AnomalyFreeLinear :
end BasisLinear
end PureU1

5
HepLean/AnomalyCancellation/PureU1/ConstAbs.lean
View file

@ -120,7 +120,6 @@ lemma boundary_accGrav' (k : Fin n) : accGrav n.succ S =
intro i
simp
lemma boundary_accGrav'' (k : Fin n) (hk : Boundary S k) :
accGrav n.succ S = (2 * ↑↑k + 1 - ↑n) * S (0 : Fin n.succ) := by
rw [boundary_accGrav' k]
@ -165,7 +164,6 @@ 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]
lemma AFL_hasBoundary (h : A.val (0 : Fin n.succ) ≠ 0) : HasBoundary A.val := by
by_contra hn
have h0 := not_hasBoundary_grav hA hn
@ -253,12 +251,10 @@ lemma AFL_even_above (A : (PureU1 (2 * n.succ)).LinSols) (h : ConstAbsSorted A.v
rfl
exact AFL_even_above' h hA i
end charges
end ConstAbsSorted
namespace ConstAbs
theorem boundary_value_odd (S : (PureU1 (2 * n + 1)).LinSols) (hs : ConstAbs S.val) :
@ -266,7 +262,6 @@ theorem boundary_value_odd (S : (PureU1 (2 * n + 1)).LinSols) (hs : ConstAbs S.v
have hS := And.intro (constAbs_sort hs) (sort_sorted S.val)
sortAFL_zero S (ConstAbsSorted.AFL_odd (sortAFL S) hS)
theorem boundary_value_even (S : (PureU1 (2 * n.succ)).LinSols) (hs : ConstAbs S.val) :
VectorLikeEven S.val := by
have hS := And.intro (constAbs_sort hs) (sort_sorted S.val)

5
HepLean/AnomalyCancellation/PureU1/Even/BasisLinear.lean
View file

@ -190,7 +190,6 @@ lemma basis!_on_other {k : Fin n} {j : Fin (2 * n.succ)} (h1 : j ≠ δ!₁ k)
simp [basis!AsCharges]
simp_all only [ne_eq, ↓reduceIte]
lemma basis!_on_δ!₁_other {k j : Fin n} (h : k ≠ j) :
basis!AsCharges k (δ!₁ j) = 0 := by
simp [basis!AsCharges]
@ -310,7 +309,6 @@ lemma basis!_accCube (j : Fin n) :
simp [basis!_δ!₂_eq_minus_δ!₁]
ring
/-- The first part of the basis as `LinSols`. -/
@[simps!]
def basis (j : Fin n.succ) : (PureU1 (2 * n.succ)).LinSols :=
@ -587,7 +585,6 @@ theorem basis!_linear_independent : LinearIndependent ℚ (@basis! n) := by
rw [P!'_val] at h1
exact P!_zero f h1
theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n) := by
apply Fintype.linearIndependent_iff.mpr
intro f h
@ -663,7 +660,6 @@ 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)) =
FiniteDimensional.finrank ℚ (PureU1 (2 * n.succ)).LinSols := by
erw [BasisLinear.finrank_AnomalyFreeLinear]
@ -675,7 +671,6 @@ noncomputable def basisaAsBasis :
Basis (Fin (succ n) ⊕ Fin n) ℚ (PureU1 (2 * succ n)).LinSols :=
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
have h := (mem_span_range_iff_exists_fun ℚ).mp (Basis.mem_span basisaAsBasis S)

3
HepLean/AnomalyCancellation/PureU1/Even/Parameterization.lean
View file

@ -134,7 +134,6 @@ theorem generic_case {S : (PureU1 (2 * n.succ)).Sols} (h : GenericCase S) :
simp at h
exact h
lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ)).Sols}
(h : SpecialCase S) : LineInCubic S.1.1 := by
intro g f hS a b
@ -152,7 +151,6 @@ lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ)).Sols}
erw [h]
simp
lemma special_case_lineInCubic_perm {S : (PureU1 (2 * n.succ)).Sols}
(h : ∀ (M : (FamilyPermutations (2 * n.succ)).group),
SpecialCase ((FamilyPermutations (2 * n.succ)).solAction.toFun S M)) :
@ -160,7 +158,6 @@ lemma special_case_lineInCubic_perm {S : (PureU1 (2 * n.succ)).Sols}
intro M
exact special_case_lineInCubic (h M)
theorem special_case {S : (PureU1 (2 * n.succ.succ)).Sols}
(h : ∀ (M : (FamilyPermutations (2 * n.succ.succ)).group),
SpecialCase ((FamilyPermutations (2 * n.succ.succ)).solAction.toFun S M)) :

5
HepLean/AnomalyCancellation/PureU1/LineInPlaneCond.lean
View file

@ -26,7 +26,6 @@ We will show that `n ≥ 4` the `line in plane` condition on solutions implies t
-/
namespace PureU1
open BigOperators
@ -52,7 +51,6 @@ lemma lineInPlaneCond_perm {S : (PureU1 (n)).LinSols} (hS : LineInPlaneCond S)
all_goals simp_all only [ne_eq, Equiv.invFun_as_coe, EmbeddingLike.apply_eq_iff_eq,
not_false_eq_true]
lemma lineInPlaneCond_eq_last' {S : (PureU1 (n.succ.succ)).LinSols} (hS : LineInPlaneCond S)
(h : ¬ (S.val ((Fin.last n).castSucc))^2 = (S.val ((Fin.last n).succ))^2 ) :
(2 - n) * S.val (Fin.last (n + 1)) =
@ -157,7 +155,6 @@ lemma linesInPlane_four (S : (PureU1 4).Sols) (hS : LineInPlaneCond S.1.1) :
simp at h6
simp_all
lemma linesInPlane_eq_sq_four {S : (PureU1 4).Sols}
(hS : LineInPlaneCond S.1.1) : ∀ (i j : Fin 4) (_ : i ≠ j),
ConstAbsProp (S.val i, S.val j) := by
@ -168,7 +165,6 @@ lemma linesInPlane_eq_sq_four {S : (PureU1 4).Sols}
(lineInPlaneCond_perm hS M)
exact linesInPlane_four S' hS'
lemma linesInPlane_constAbs_four (S : (PureU1 4).Sols)
(hS : LineInPlaneCond S.1.1) : ConstAbs S.val := by
intro i j
@ -183,5 +179,4 @@ theorem linesInPlane_constAbs_AF (S : (PureU1 (n.succ.succ.succ.succ)).Sols)
exact linesInPlane_constAbs_four S hS
exact linesInPlane_constAbs hS
end PureU1

1
HepLean/AnomalyCancellation/PureU1/LowDim/Three.lean
View file

@ -50,7 +50,6 @@ def solOfLinear (S : (PureU1 3).LinSols)
⟨⟨S, by intro i; simp at i; exact Fin.elim0 i⟩,
(cube_for_linSol S).mp hS⟩
theorem solOfLinear_surjects (S : (PureU1 3).Sols) :
∃ (T : (PureU1 3).LinSols) (hT : T.val (0 : Fin 3) = 0 ∨ T.val (1 : Fin 3) = 0
∨ T.val (2 : Fin 3) = 0), solOfLinear T hT = S := by

7
HepLean/AnomalyCancellation/PureU1/Odd/BasisLinear.lean
View file

@ -23,7 +23,6 @@ namespace PureU1
variable {n : ℕ}
namespace VectorLikeOddPlane
lemma split_odd! (n : ℕ) : (1 + n) + n = 2 * n +1 := by
@ -484,7 +483,6 @@ lemma P_P_P!_accCube (g : Fin n → ℚ) (j : Fin n) :
simp only [mul_zero, add_zero]
simp
lemma P_zero (f : Fin n → ℚ) (h : P f = 0) : ∀ i, f i = 0 := by
intro i
erw [← P_δ₁ f]
@ -572,7 +570,6 @@ theorem basis!_linear_independent : LinearIndependent ℚ (@basis! n) := by
rw [P!'_val] at h1
exact P!_zero f h1
theorem basisa_linear_independent : LinearIndependent ℚ (@basisa n.succ) := by
apply Fintype.linearIndependent_iff.mpr
intro f h
@ -648,8 +645,6 @@ 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)) =
FiniteDimensional.finrank ℚ (PureU1 (2 * n.succ + 1)).LinSols := by
erw [BasisLinear.finrank_AnomalyFreeLinear]
@ -706,8 +701,6 @@ lemma span_basis_swap! {S : (PureU1 (2 * n.succ + 1)).LinSols} (j : Fin n.succ)
apply swap!_as_add at hS
exact hS
end VectorLikeOddPlane
end PureU1

4
HepLean/AnomalyCancellation/PureU1/Odd/LineInCubic.lean
View file

@ -54,7 +54,6 @@ lemma lineInCubic_expand {S : (PureU1 (2 * n + 1)).LinSols} (h : LineInCubic S)
rw [← h1]
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),
accCubeTriLinSymm (P g) (P g) (P! f) = 0 := by
@ -62,8 +61,6 @@ lemma line_in_cubic_P_P_P! {S : (PureU1 (2 * n + 1)).LinSols} (h : LineInCubic S
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`. -/
def LineInCubicPerm (S : (PureU1 (2 * n + 1)).LinSols) : Prop :=
∀ (M : (FamilyPermutations (2 * n + 1)).group ),
@ -86,7 +83,6 @@ lemma lineInCubicPerm_permute {S : (PureU1 (2 * n + 1)).LinSols}
rw [ht]
exact hS (M * M')
lemma lineInCubicPerm_swap {S : (PureU1 (2 * n.succ + 1)).LinSols}
(LIC : LineInCubicPerm S) :
∀ (j : Fin n.succ) (g f : Fin n.succ → ℚ) (_ : S.val = Pa g f) ,

1
HepLean/AnomalyCancellation/PureU1/Odd/Parameterization.lean
View file

@ -133,7 +133,6 @@ theorem generic_case {S : (PureU1 (2 * n.succ + 1)).Sols} (h : GenericCase S) :
simp at h
exact h
lemma special_case_lineInCubic {S : (PureU1 (2 * n.succ + 1)).Sols}
(h : SpecialCase S) :
LineInCubic S.1.1 := by

4
HepLean/AnomalyCancellation/PureU1/Permutations.lean
View file

@ -96,7 +96,6 @@ def FamilyPermutations (n : ℕ) : ACCSystemGroupAction (PureU1 n) where
exact Fin.elim0 i
cubicInvariant := accCube_invariant
lemma FamilyPermutations_charges_apply (S : (PureU1 n).Charges)
(i : Fin n) (f : (FamilyPermutations n).group) :
((FamilyPermutations n).rep f S) i = S (f.invFun i) := by
@ -107,7 +106,6 @@ lemma FamilyPermutations_anomalyFreeLinear_apply (S : (PureU1 n).LinSols)
((FamilyPermutations n).linSolRep f S).val i = S.val (f.invFun i) := by
rfl
/-- The permutation which swaps i and j. TODO: Replace with: `Equiv.swap`. -/
def pairSwap {n : ℕ} (i j : Fin n) : (FamilyPermutations n).group where
toFun s :=
@ -247,7 +245,6 @@ lemma permThreeInj_fst_apply :
⟨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
simp only [Set.mem_range]
use 1
@ -338,5 +335,4 @@ lemma Prop_three (P : ℚ × ℚ × ℚ → Prop) {S : (PureU1 n).LinSols}
erw [permThree_fst,permThree_snd, permThree_thd] at h1
exact h1
end PureU1

1
HepLean/AnomalyCancellation/PureU1/Sort.lean
View file

@ -67,7 +67,6 @@ def sortAFL {n : ℕ} (S : (PureU1 n).LinSols) : (PureU1 n).LinSols :=
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
apply ACCSystemLinear.LinSols.ext
have h1 : sort S.val = 0 := by

2
HepLean/AnomalyCancellation/PureU1/VectorLike.lean
View file

@ -30,7 +30,6 @@ variable {n : ℕ}
-/
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
@ -40,5 +39,4 @@ 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

1
HepLean/AnomalyCancellation/SM/Basic.lean
View file

@ -319,5 +319,4 @@ lemma accCube_ext {S T : (SMCharges n).Charges}
rfl
repeat rw [h1]
end SMACCs

4
HepLean/AnomalyCancellation/SM/NoGrav/One/Lemmas.lean
View file

@ -24,7 +24,6 @@ open SMCharges
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
let S' := linearParameters.bijection.symm S.1.1
@ -38,8 +37,6 @@ 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) :
accGrav S.val = 0 := by
rw [accGrav]
@ -74,7 +71,6 @@ theorem accGravSatisfied {S : (SMNoGrav 1).Sols} (FLTThree : FermatLastTheoremWi
exact accGrav_Q_zero hQ
exact accGrav_Q_neq_zero hQ FLTThree
end One
end SMNoGrav
end SM

3
HepLean/AnomalyCancellation/SM/NoGrav/One/LinearParameterization.lean
View file

@ -183,7 +183,6 @@ lemma grav (S : linearParameters) :
end linearParameters
/-- The parameters for solutions to the linear ACCs with the condition that Q and E are non-zero.-/
structure linearParametersQENeqZero where
/-- The parameter `x`. -/
@ -280,7 +279,6 @@ lemma cubic (S : linearParametersQENeqZero) :
simp_all
exact add_eq_zero_iff_eq_neg
lemma cubic_v_or_w_zero (S : linearParametersQENeqZero) (h : accCube (bijection S).1.val = 0)
(FLTThree : FermatLastTheoremWith ℚ 3) :
S.v = 0 ∨ S.w = 0 := by
@ -364,7 +362,6 @@ lemma grav_of_cubic (S : linearParametersQENeqZero) (h : accCube (bijection S).1
end linearParametersQENeqZero
end One
end SMNoGrav
end SM

6
HepLean/AnomalyCancellation/SM/Permutations.lean
View file

@ -33,7 +33,6 @@ variable {n : ℕ}
@[simp]
instance : Group (PermGroup n) := Pi.group
/-- The image of an element of `permGroup n` under the representation on charges. -/
@[simps!]
def chargeMap (f : PermGroup n) : (SMCharges n).Charges →ₗ[ℚ] (SMCharges n).Charges where
@ -66,7 +65,6 @@ lemma repCharges_toSpecies (f : PermGroup n) (S : (SMCharges n).Charges) (j : Fi
toSpecies j (repCharges f S) = toSpecies j S ∘ f⁻¹ j := by
erw [toSMSpecies_toSpecies_inv]
lemma toSpecies_sum_invariant (m : ℕ) (f : PermGroup n) (S : (SMCharges n).Charges) (j : Fin 5) :
∑ i, ((fun a => a ^ m) ∘ toSpecies j (repCharges f S)) i =
∑ i, ((fun a => a ^ m) ∘ toSpecies j S) i := by
@ -74,20 +72,16 @@ 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) :
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) :
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) :
accSU3 (repCharges f S) = accSU3 S :=
accSU3_ext

5
HepLean/AnomalyCancellation/SMNu/Basic.lean
View file

@ -14,7 +14,6 @@ import Mathlib.Logic.Equiv.Fin
# Anomaly cancellation conditions for n family SM.
-/
universe v u
open Nat
open BigOperators
@ -83,7 +82,6 @@ abbrev E := @toSpecies n 4
/-- The `N` charges as a map `Fin n → ℚ`. -/
abbrev N := @toSpecies n 5
end SMνCharges
namespace SMνACCs
@ -111,7 +109,6 @@ def accGrav : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
-- rw [show Rat.cast a = a from rfl]
ring
lemma accGrav_decomp (S : (SMνCharges n).Charges) :
accGrav S = 6 * ∑ i, Q S i + 3 * ∑ i, U S i + 3 * ∑ i, D S i + 2 * ∑ i, L S i + ∑ i, E S i +
∑ i, N S i := by
@ -127,7 +124,6 @@ lemma accGrav_ext {S T : (SMνCharges n).Charges}
rw [accGrav_decomp, accGrav_decomp]
repeat erw [hj]
/-- The `SU(2)` anomaly equation. -/
@[simp]
def accSU2 : (SMνCharges n).Charges →ₗ[ℚ] ℚ where
@ -344,7 +340,6 @@ lemma cubeTriLin_decomp (S T R : (SMνCharges n).Charges) :
repeat erw [Finset.sum_add_distrib]
repeat erw [← Finset.mul_sum]
/-- The cubic ACC. -/
@[simp]
def accCube : HomogeneousCubic (SMνCharges n).Charges := cubeTriLin.toCubic

1
HepLean/AnomalyCancellation/SMNu/FamilyMaps.lean
View file

@ -79,7 +79,6 @@ def speciesEmbed (m n : ℕ) :
erw [dif_neg hi, dif_neg hi]
exact Eq.symm (Rat.mul_zero a)
/-- The embedding of the `m`-family charges onto the `n`-family charges, with all
other charges zero. -/
def familyEmbedding (m n : ℕ) : (SMνCharges m).Charges →ₗ[ℚ] (SMνCharges n).Charges :=

1
HepLean/AnomalyCancellation/SMNu/NoGrav/Basic.lean
View file

@ -110,7 +110,6 @@ def perm (n : ℕ) : ACCSystemGroupAction (SMNoGrav n) where
exact Fin.elim0 i
cubicInvariant := accCube_invariant
end SMNoGrav
end SMRHN

2
HepLean/AnomalyCancellation/SMNu/Ordinary/Basic.lean
View file

@ -37,7 +37,6 @@ def SM (n : ℕ) : ACCSystem where
namespace SM
variable {n : ℕ}
lemma gravSol (S : (SM n).LinSols) : accGrav S.val = 0 := by
@ -119,7 +118,6 @@ def perm (n : ℕ) : ACCSystemGroupAction (SM n) where
exact Fin.elim0 i
cubicInvariant := accCube_invariant
end SM
end SMRHN

3
HepLean/AnomalyCancellation/SMNu/Ordinary/DimSevenPlane.lean
View file

@ -137,7 +137,6 @@ lemma B₀_Bi_cubic {i : Fin 7} (hi : 0 ≠ i) (S : (SM 3).Charges) :
simp at hi <;>
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
lemma B₁_Bi_cubic {i : Fin 7} (hi : 1 ≠ i) (S : (SM 3).Charges) :
cubeTriLin (B 1) (B i) S = 0 := by
change cubeTriLin B₁ (B i) S = 0
@ -246,7 +245,6 @@ lemma B₆_B₆_Bi_cubic {i : Fin 7} :
fin_cases i <;>
simp [B₀, B₁, B₂, B₃, B₄, B₅, B₆, Fin.divNat, Fin.modNat, finProdFinEquiv]
lemma Bi_Bi_Bj_cubic (i j : Fin 7) :
cubeTriLin (B i) (B i) (B j) = 0 := by
fin_cases i
@ -344,6 +342,5 @@ theorem seven_dim_plane_exists : ∃ (B : Fin 7 → (SM 3).Charges),
exact PlaneSeven.basis_linear_independent
exact PlaneSeven.B_sum_is_sol
end SM
end SMRHN

7
HepLean/AnomalyCancellation/SMNu/Permutations.lean
View file

@ -65,7 +65,6 @@ lemma repCharges_toSpecies (f : PermGroup n) (S : (SMνCharges n).Charges) (j :
toSpecies j (repCharges f S) = toSpecies j S ∘ f⁻¹ j := by
erw [toSMSpecies_toSpecies_inv]
lemma toSpecies_sum_invariant (m : ℕ) (f : PermGroup n) (S : (SMνCharges n).Charges) (j : Fin 6) :
∑ i, ((fun a => a ^ m) ∘ toSpecies j (repCharges f S)) i =
∑ i, ((fun a => a ^ m) ∘ toSpecies j S) i := by
@ -74,19 +73,16 @@ lemma toSpecies_sum_invariant (m : ℕ) (f : PermGroup n) (S : (SMνCharges n).C
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) :
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) :
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) :
accSU3 (repCharges f S) = accSU3 S :=
accSU3_ext
@ -97,7 +93,6 @@ lemma accYY_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
accYY_ext
(by simpa using toSpecies_sum_invariant 1 f S)
lemma accQuad_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
accQuad (repCharges f S) = accQuad S :=
accQuad_ext
@ -108,6 +103,4 @@ lemma accCube_invariant (f : PermGroup n) (S : (SMνCharges n).Charges) :
accCube_ext
(by simpa using toSpecies_sum_invariant 3 f S)
end SMRHN

1
HepLean/AnomalyCancellation/SMNu/PlusU1/BMinusL.lean
View file

@ -114,7 +114,6 @@ lemma add_AFL_cube (S : (PlusU1 n).LinSols) (a b : ℚ) :
add_zero, BL_val, mul_zero]
ring
end BL
end PlusU1
end SMRHN

1
HepLean/AnomalyCancellation/SMNu/PlusU1/Basic.lean
View file

@ -37,7 +37,6 @@ def PlusU1 (n : ℕ) : ACCSystem where
namespace PlusU1
variable {n : ℕ}
lemma gravSol (S : (PlusU1 n).LinSols) : accGrav S.val = 0 := by

2
HepLean/AnomalyCancellation/SMNu/PlusU1/BoundPlaneDim.lean
View file

@ -58,7 +58,6 @@ lemma exists_plane_exists_basis {n : ℕ} (hE : ExistsPlane n) :
| Sum.inl i => exact h3 i
| Sum.inr i => exact h4 i
theorem plane_exists_dim_le_7 {n : ℕ} (hn : ExistsPlane n) : n ≤ 7 := by
obtain ⟨B, hB⟩ := exists_plane_exists_basis hn
have h1 := LinearIndependent.fintype_card_le_finrank hB
@ -67,6 +66,5 @@ theorem plane_exists_dim_le_7 {n : ℕ} (hn : ExistsPlane n) : n ≤ 7 := by
FiniteDimensional.finrank_fin_fun ℚ] at h1
exact Nat.le_of_add_le_add_left h1
end PlusU1
end SMRHN

2
HepLean/AnomalyCancellation/SMNu/PlusU1/HyperCharge.lean
View file

@ -68,7 +68,6 @@ lemma on_quadBiLin_AFL (S : (PlusU1 n).LinSols) : quadBiLin (Y n).val S.val = 0
rw [on_quadBiLin]
rw [YYsol S]
lemma add_AFL_quad (S : (PlusU1 n).LinSols) (a b : ℚ) :
accQuad (a • S.val + b • (Y n).val) = a ^ 2 * accQuad S.val := by
erw [BiLinearSymm.toHomogeneousQuad_add, quadSol (b • (Y n)).1]
@ -144,7 +143,6 @@ lemma add_AF_cube (S : (PlusU1 n).Sols) (a b : ℚ) :
def addCube (S : (PlusU1 n).Sols) (a b : ℚ) : (PlusU1 n).Sols :=
quadToAF (addQuad S.1 a b) (add_AF_cube S a b)
end Y
end PlusU1
end SMRHN

3
HepLean/AnomalyCancellation/SMNu/PlusU1/PlaneNonSols.lean
View file

@ -104,7 +104,6 @@ lemma on_accQuad (f : Fin 11 → ℚ) :
rw [quadBiLin.map_smul₁, Bi_sum_quad, quadCoeff_eq_bilinear]
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
obtain ⟨S, hS⟩ := hS
@ -231,7 +230,6 @@ lemma isSolution_f_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i,
exact isSolution_f9 f hS
exact isSolution_f10 f hS
lemma isSolution_only_if_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (∑ i, f i • B i)) :
∑ i, f i • B i = 0 := by
rw [isSolution_sum_part f hS]
@ -239,7 +237,6 @@ lemma isSolution_only_if_zero (f : Fin 11 → ℚ) (hS : (PlusU1 3).IsSolution (
rw [isSolution_f9 f hS]
simp
theorem basis_linear_independent : LinearIndependent ℚ B := by
apply Fintype.linearIndependent_iff.mpr
intro f h

1
HepLean/AnomalyCancellation/SMNu/PlusU1/QuadSol.lean
View file

@ -74,7 +74,6 @@ lemma genericToQuad_neq_zero (S : (PlusU1 n).QuadSols) (h : α₁ C S.1 ≠ 0) :
(α₁ C S.1)⁻¹ • genericToQuad C S.1 = S := by
rw [genericToQuad_on_quad, smul_smul, inv_mul_cancel h, one_smul]
/-- The construction of a `QuadSol` from a `LinSols` in the special case when `α₁ C S = 0` and
`α₂ S = 0`. -/
def specialToQuad (S : (PlusU1 n).LinSols) (a b : ℚ) (h1 : α₁ C S = 0)

2
HepLean/AnomalyCancellation/SMNu/PlusU1/QuadSolToSol.lean
View file

@ -78,10 +78,8 @@ lemma special_on_AF (S : (PlusU1 n).Sols) (h1 : α₁ S.1 = 0) :
rw [BL.addQuad_zero]
simp
end QuadSolToSol
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. -/

10
HepLean/FeynmanDiagrams/Basic.lean
View file

@ -22,7 +22,6 @@ import Mathlib.SetTheory.Cardinal.Basic
Feynman diagrams are a graphical representation of the terms in the perturbation expansion of
a quantum field theory.
-/
open CategoryTheory
@ -106,7 +105,6 @@ def preimageEdge {𝓔 𝓥 : Type} (v : 𝓔) :
map {f g} F := Over.homMk ((P.toEdge ⋙ preimageType' v).map F)
(funext <| fun x => congrArg Prod.fst <| congrFun F.w x.1)
/-!
## Finitness of pre-Feynman rules
@ -304,7 +302,6 @@ instance diagramEdgePropDecidable
(fun _ => P.preFeynmanRuleDecEq𝓱𝓔) _ _ _)) _ ) _)
(P.diagramEdgeProp_iff F f).symm
end PreFeynmanRule
/-!
@ -378,7 +375,6 @@ instance CondDecidable [IsFinitePreFeynmanRule P] {𝓔 𝓥 𝓱𝓔 : Type} (
(@diagramEdgePropDecidable P _ _ _ _ _ (Over.mk 𝓱𝓔To𝓔𝓥) _ h 𝓔𝓞)
(@diagramVertexPropDecidable P _ _ _ _ _ (Over.mk 𝓱𝓔To𝓔𝓥) _ h 𝓥𝓞)
/-- Making a Feynman diagram from maps of edges, vertices and half-edges. -/
def mk' {𝓔 𝓥 𝓱𝓔 : Type} (𝓔𝓞 : 𝓔 → P.EdgeLabel) (𝓥𝓞 : 𝓥 → P.VertexLabel)
(𝓱𝓔To𝓔𝓥 : 𝓱𝓔 → P.HalfEdgeLabel × 𝓔 × 𝓥)
@ -491,7 +487,6 @@ structure Hom (F G : FeynmanDiagram P) where
/-- The morphism of half-edge objects. -/
𝓱𝓔To𝓔𝓥 : (edgeVertexFunc 𝓔𝓞.left 𝓥𝓞.left).obj F.𝓱𝓔To𝓔𝓥 ⟶ G.𝓱𝓔To𝓔𝓥
namespace Hom
variable {F G : FeynmanDiagram P}
variable (f : Hom F G)
@ -508,12 +503,10 @@ def 𝓥 : F.𝓥 → G.𝓥 := f.𝓥𝓞.left
@[simp]
def 𝓱𝓔 : F.𝓱𝓔 → G.𝓱𝓔 := f.𝓱𝓔To𝓔𝓥.left
/-- The morphism `F.𝓱𝓔𝓞 ⟶ G.𝓱𝓔𝓞` induced by a homomorphism of Feynman diagrams. -/
@[simp]
def 𝓱𝓔𝓞 : F.𝓱𝓔𝓞 ⟶ G.𝓱𝓔𝓞 := P.toHalfEdge.map f.𝓱𝓔To𝓔𝓥
/-- The identity morphism for a Feynman diagram. -/
def id (F : FeynmanDiagram P) : Hom F F where
𝓔𝓞 := 𝟙 F.𝓔𝓞
@ -579,7 +572,6 @@ instance {F G : FeynmanDiagram P} [IsFiniteDiagram F] [IsFiniteDiagram G] [IsFin
(𝓔 : F.𝓔 → G.𝓔) (𝓥 : F.𝓥 → G.𝓥) (𝓱𝓔 : F.𝓱𝓔 → G.𝓱𝓔) : Decidable (Cond 𝓔 𝓥 𝓱𝓔) :=
And.decidable
/-- Making a Feynman diagram from maps of edges, vertices and half-edges. -/
@[simps! 𝓔𝓞_left 𝓥𝓞_left 𝓱𝓔To𝓔𝓥_left]
def mk' {F G : FeynmanDiagram P} (𝓔 : F.𝓔 → G.𝓔) (𝓥 : F.𝓥 → G.𝓥) (𝓱𝓔 : F.𝓱𝓔 → G.𝓱𝓔)
@ -667,7 +659,6 @@ We show that the symmetry factor for a finite Feynman diagram is finite.
/-- The type of isomorphisms of a Feynman diagram. -/
def SymmetryType : Type := F ≅ F
/-- An equivalence between `SymmetryType` and permutation of edges, vertices and half-edges
satisfying `Hom.Cond`. -/
def symmetryTypeEquiv :
@ -728,7 +719,6 @@ instance [IsFiniteDiagram F] : DecidableRel F.AdjRelation := fun _ _ =>
@And.decidable _ _ (instDecidableEq𝓥OfIsFiniteDiagram _ _)
(instDecidableEq𝓥OfIsFiniteDiagram _ _)) _ ) _
/-- From a Feynman diagram the simple graph showing those vertices which are connected. -/
def toSimpleGraph : SimpleGraph F.𝓥 where
Adj := AdjRelation F

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

@ -7,11 +7,8 @@ import HepLean.FeynmanDiagrams.Basic
/-!
# Feynman diagrams in a complex scalar field theory
-/
namespace PhiFour
open CategoryTheory
open FeynmanDiagram
@ -65,7 +62,4 @@ instance : IsFinitePreFeynmanRule complexScalarFeynmanRules where
match v with
| 0 => instDecidableEqFin _
end PhiFour

4
HepLean/FeynmanDiagrams/Instances/Phi4.lean
View file

@ -10,10 +10,8 @@ import HepLean.FeynmanDiagrams.Basic
The aim of this file is to start building up the theory of Feynman diagrams in the context of
Phi^4 theory.
-/
namespace PhiFour
open CategoryTheory
open FeynmanDiagram
@ -45,7 +43,6 @@ instance (a : ℕ) : OfNat phi4PreFeynmanRules.HalfEdgeLabel a where
instance (a : ℕ) : OfNat phi4PreFeynmanRules.VertexLabel a where
ofNat := (a : Fin _)
instance : IsFinitePreFeynmanRule phi4PreFeynmanRules where
edgeLabelDecidable := instDecidableEqFin _
vertexLabelDecidable := instDecidableEqFin _
@ -92,5 +89,4 @@ lemma figureEight_symmetryFactor : symmetryFactor figureEight = 8 := by decide
end Example
end PhiFour

6
HepLean/FeynmanDiagrams/Momentum.lean
View file

@ -26,7 +26,6 @@ vector space.
This section is non-computable as we depend on the norm on `F.HalfEdgeMomenta`.
-/
namespace FeynmanDiagram
open CategoryTheory
@ -78,7 +77,6 @@ def euclidInner : F.HalfEdgeMomenta →ₗ[ℝ] F.HalfEdgeMomenta →ₗ[ℝ]
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. -/
def EdgeMomenta : Type := F.𝓔 → ℝ
@ -87,7 +85,6 @@ 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. -/
def VertexMomenta : Type := F.𝓥 → ℝ
@ -119,7 +116,6 @@ instance : AddCommGroup F.EdgeVertexMomenta := DirectSum.instAddCommGroup _
instance : Module ℝ F.EdgeVertexMomenta := DirectSum.instModule
/-!
## Linear maps between the vector spaces.
@ -199,7 +195,6 @@ for specific Feynman diagrams.
- Complete this section.
-/
/-!
@ -210,7 +205,6 @@ for specific Feynman diagrams.
- Complete this section.
-/
end FeynmanDiagram

7
HepLean/FlavorPhysics/CKMMatrix/Basic.lean
View file

@ -17,12 +17,10 @@ related by phase shifts.
The notation `[V]ud` etc can be used for the elements of a CKM matrix, and
`[V]ud|us` etc for the ratios of elements.
-/
open Matrix Complex
noncomputable section
/-- Given three real numbers `a b c` the complex matrix with `exp (I * a)` etc on the
@ -107,7 +105,6 @@ lemma phaseShiftRelation_trans {U V W : unitaryGroup (Fin 3) ℂ} :
rw [phaseShiftMatrix_mul]
rw [add_comm k e, add_comm l f, add_comm m g]
lemma phaseShiftRelation_equiv : Equivalence PhaseShiftRelation where
refl := phaseShiftRelation_refl
symm := phaseShiftRelation_symm
@ -270,7 +267,6 @@ lemma tb (V : CKMMatrix) (a b c d e f : ℝ) :
end phaseShiftApply
/-- The aboslute value of the `(i,j)`th element of `V`. -/
@[simp]
def VAbs' (V : unitaryGroup (Fin 3) ℂ) (i j : Fin 3) : ℝ := Complex.abs (V i j)
@ -341,11 +337,9 @@ abbrev VtsAbs := VAbs 2 1
@[simp]
abbrev VtbAbs := VAbs 2 2
namespace CKMMatrix
open ComplexConjugate
section ratios
/-- The ratio of the `ub` and `ud` elements of a CKM matrix. -/
@ -394,7 +388,6 @@ lemma Rcscb_mul_cb {V : CKMMatrix} (h : [V]cb ≠ 0): [V]cs = Rcscb V * [V]cb :=
end ratios
end CKMMatrix
end

4
HepLean/FlavorPhysics/CKMMatrix/Invariants.lean
View file

@ -15,17 +15,14 @@ this equivalence.
Of note, this file defines the complex jarlskog invariant.
-/
open Matrix Complex
open ComplexConjugate
open CKMMatrix
noncomputable section
namespace Invariant
/-- The complex jarlskog invariant for a CKM matrix. -/
@[simps!]
def jarlskogℂCKM (V : CKMMatrix) : ℂ :=
@ -62,6 +59,5 @@ standard parameterization. -/
def mulExpδ₁₃ (V : Quotient CKMMatrixSetoid) : ℂ :=
jarlskogℂ V + VusVubVcdSq V
end Invariant
end

3
HepLean/FlavorPhysics/CKMMatrix/PhaseFreedom.lean
View file

@ -19,11 +19,9 @@ In this file we define two sets of conditions on the CKM matrices
and `ubOnePhaseCond` which we show can be satisfied by any CKM matrix up to equivalence as long as
the ub element as absolute value 1.
-/
open Matrix Complex
noncomputable section
namespace CKMMatrix
open ComplexConjugate
@ -260,7 +258,6 @@ lemma fstRowThdColRealCond_holds_up_to_equiv (V : CKMMatrix) :
apply shift_cross_product_phase_zero _ _ _ _ _ _ hτ.symm
ring
lemma ubOnePhaseCond_hold_up_to_equiv_of_ub_one {V : CKMMatrix} (hb : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0))
(hV : FstRowThdColRealCond V) :
∃ (U : CKMMatrix), V ≈ U ∧ ubOnePhaseCond U:= by

13
HepLean/FlavorPhysics/CKMMatrix/Relations.lean
View file

@ -16,7 +16,6 @@ matrix.
open Matrix Complex
noncomputable section
namespace CKMMatrix
@ -121,9 +120,6 @@ 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) :
(normSq [V]ud : ℂ) + normSq [V]us ≠ 0 := by
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℝ hb
@ -131,7 +127,6 @@ lemma normSq_Vud_plus_normSq_Vus_neq_zero_ℂ {V : CKMMatrix} (hb : [V]ud ≠ 0
rw [← ofReal_inj] at h1
simp_all
lemma Vabs_sq_add_neq_zero {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0) :
((VudAbs ⟦V⟧ : ℂ) * ↑(VudAbs ⟦V⟧) + ↑(VusAbs ⟦V⟧) * ↑(VusAbs ⟦V⟧)) ≠ 0 := by
have h1 := normSq_Vud_plus_normSq_Vus_neq_zero_ℂ hb
@ -230,7 +225,6 @@ 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)
{τ : ℝ} (hτ : [V]t = cexp (τ * I) • (conj ([V]u) ×₃ conj ([V]c))) :
[V]cs = (- conj [V]ub * [V]us * [V]cb +
@ -249,10 +243,8 @@ lemma cd_of_ud_us_ub_cb_tb {V : CKMMatrix} (h : [V]ud ≠ 0 ∨ [V]us ≠ 0)
field_simp
ring
end rows
section individual
lemma VAbs_ge_zero (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : 0 ≤ VAbs i j V := by
@ -270,12 +262,10 @@ lemma VAbs_leq_one (i j : Fin 3) (V : Quotient CKMMatrixSetoid) : VAbs i j V ≤
change VAbs i 2 V ≤ 1
nlinarith
end individual
section columns
lemma VAbs_sum_sq_col_eq_one (V : Quotient CKMMatrixSetoid) (i : Fin 3) :
(VAbs 0 i V) ^ 2 + (VAbs 1 i V) ^ 2 + (VAbs 2 i V) ^ 2 = 1 := by
obtain ⟨V, hV⟩ := Quot.exists_rep V
@ -312,7 +302,6 @@ lemma cb_eq_zero_of_ud_us_zero {V : CKMMatrix} (h : [V]ud = 0 ∧ [V]us = 0) :
simp at h1
exact h1.1
lemma cs_of_ud_us_zero {V : CKMMatrix} (ha : ¬ ([V]ud ≠ 0 ∨ [V]us ≠ 0)) :
VcsAbs ⟦V⟧ = √(1 - VcdAbs ⟦V⟧ ^ 2) := by
have h1 := snd_row_normalized_abs V
@ -336,8 +325,6 @@ lemma VcbAbs_sq_add_VtbAbs_sq (V : Quotient CKMMatrixSetoid) :
VcbAbs V ^ 2 + VtbAbs V ^ 2 = 1 - VubAbs V ^2 := by
linear_combination (VAbs_sum_sq_col_eq_one V 2)
lemma cb_tb_neq_zero_iff_ub_neq_one (V : CKMMatrix) :
[V]cb ≠ 0 ∨ [V]tb ≠ 0 ↔ abs [V]ub ≠ 1 := by
have h2 := V.thd_col_normalized_abs

2
HepLean/FlavorPhysics/CKMMatrix/Rows.lean
View file

@ -14,7 +14,6 @@ proves some properties between them.
The first row can be extracted as `[V]u` for a CKM matrix `V`.
-/
open Matrix Complex
@ -24,7 +23,6 @@ noncomputable section
namespace CKMMatrix
/-- The `u`th row of the CKM matrix. -/
def uRow (V : CKMMatrix) : Fin 3 → ℂ := ![[V]ud, [V]us, [V]ub]

3
HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/Basic.lean
View file

@ -16,7 +16,6 @@ four real numbers `θ₁₂`, `θ₁₃`, `θ₂₃` and `δ₁₃`.
We will show that every CKM matrix can be written within this standard parameterization
in the file `FlavorPhysics.CKMMatrix.StandardParameters`.
-/
open Matrix Complex
open ComplexConjugate
@ -194,7 +193,5 @@ lemma mulExpδ₁₃_eq (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ) (h1 : 0 ≤
have h1 : cexp (I * δ₁₃) ≠ 0 := exp_ne_zero _
field_simp
end standParam
end

16
HepLean/FlavorPhysics/CKMMatrix/StandardParameterization/StandardParameters.lean
View file

@ -18,7 +18,6 @@ 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
CKM matrix can be written using the standard parameterization.
-/
open Matrix Complex
open ComplexConjugate
@ -333,7 +332,6 @@ lemma Vs_zero_iff_cos_sin_zero (V : CKMMatrix) :
end VAbs
namespace standParam
open Invariant
@ -409,7 +407,6 @@ lemma on_param_cos_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.c
rfl
rfl
lemma on_param_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₁₂ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, δ₁₃, δ₁₃, -δ₁₃, 0, - δ₁₃
@ -434,7 +431,6 @@ lemma on_param_cos_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.c
ring_nf
field_simp
lemma on_param_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.cos (θ₂₃ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, δ₁₃, 0, 0, 0, - δ₁₃
@ -451,7 +447,6 @@ lemma on_param_cos_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.c
ring
field_simp
lemma on_param_sin_θ₁₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₁₃ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, 0, 0, 0, 0, 0
@ -488,8 +483,6 @@ lemma on_param_sin_θ₁₂_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.s
ring_nf
field_simp
lemma on_param_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.sin (θ₂₃ ⟦V⟧) = 0) :
standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₁₃ ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
use 0, 0, δ₁₃, 0, 0, - δ₁₃
@ -506,9 +499,6 @@ lemma on_param_sin_θ₂₃_eq_zero {V : CKMMatrix} (δ₁₃ : ℝ) (h : Real.s
ring
field_simp
lemma eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0 ∨ [V]us ≠ 0)
(hV : FstRowThdColRealCond V) : V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) (- arg [V]ub) := by
have hb' : VubAbs ⟦V⟧ ≠ 1 := by
@ -565,7 +555,6 @@ lemma eq_standParam_of_fstRowThdColRealCond {V : CKMMatrix} (hb : [V]ud ≠ 0
rw [VcbAbs_eq_S₂₃_mul_C₁₃ ⟦V⟧, S₂₃_eq_ℂsin_θ₂₃ ⟦V⟧, C₁₃]
simp
lemma eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : ubOnePhaseCond V) :
V = standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) 0 := by
have h1 : VubAbs ⟦V⟧ = 1 := by
@ -610,7 +599,6 @@ lemma eq_standParam_of_ubOnePhaseCond {V : CKMMatrix} (hV : ubOnePhaseCond V) :
rw [C₁₃_eq_ℂcos_θ₁₃ ⟦V⟧, C₁₃_of_Vub_eq_one h1, hV.2.2.1]
simp
theorem exists_δ₁₃ (V : CKMMatrix) :
∃ (δ₃ : ℝ), V ≈ standParam (θ₁₂ ⟦V⟧) (θ₁₃ ⟦V⟧) (θ₂₃ ⟦V⟧) δ₃ := by
obtain ⟨U, hU⟩ := fstRowThdColRealCond_holds_up_to_equiv V
@ -670,7 +658,6 @@ theorem eq_standardParameterization_δ₃ (V : CKMMatrix) :
exact on_param_sin_θ₁₃_eq_zero δ₁₃' h
exact on_param_sin_θ₂₃_eq_zero δ₁₃' h
theorem exists_for_CKMatrix (V : CKMMatrix) :
∃ (θ₁₂ θ₁₃ θ₂₃ δ₁₃ : ℝ), V ≈ standParam θ₁₂ θ₁₃ θ₂₃ δ₁₃ := by
use θ₁₂ ⟦V⟧, θ₁₃ ⟦V⟧, θ₂₃ ⟦V⟧, δ₁₃ ⟦V⟧
@ -678,9 +665,6 @@ theorem exists_for_CKMatrix (V : CKMMatrix) :
end standParam
open CKMMatrix
end

5
HepLean/Mathematics/LinearMaps.lean
View file

@ -40,7 +40,6 @@ lemma map_smul (f : HomogeneousQuadratic V) (a : ℚ) (S : V) : f (a • S) = a
end HomogeneousQuadratic
/-- The structure of a symmetric bilinear function. -/
structure BiLinearSymm (V : Type) [AddCommMonoid V] [Module ℚ V] extends V →ₗ[ℚ] V →ₗ[ℚ] ℚ where
swap' : ∀ S T, toFun S T = toFun T S
@ -126,7 +125,6 @@ lemma map_sum₂ {n : ℕ} (f : BiLinearSymm V) (S : Fin n → V) (T : V) :
intro i
rw [swap]
/-- The homogenous quadratic equation obtainable from a bilinear function. -/
@[simps!]
def toHomogeneousQuad {V : Type} [AddCommMonoid V] [Module ℚ V]
@ -146,7 +144,6 @@ lemma toHomogeneousQuad_add {V : Type} [AddCommMonoid V] [Module ℚ V]
rw [τ.map_add₁, τ.map_add₁, τ.swap T S]
ring
end BiLinearSymm
/-- The structure of a homogeneous cubic equation. -/
@ -222,7 +219,6 @@ def mk₃ (f : V × V × V→ ℚ) (map_smul : ∀ a S T L, f (a • S, T, L) =
swap₁' := swap₁
swap₂' := swap₂
lemma swap₁ (f : TriLinearSymm V) (S T L : V) : f S T L = f T S L :=
f.swap₁' S T L
@ -300,7 +296,6 @@ lemma map_sum₁₂₃ {n1 n2 n3 : ℕ} (f : TriLinearSymm V) (S : Fin n1 → V)
intro i
rw [map_sum₃]
/-- The homogenous cubic equation obtainable from a symmetric trilinear function. -/
@[simps!]
def toCubic {charges : Type} [AddCommMonoid charges] [Module ℚ charges]

5
HepLean/Mathematics/SO3/Basic.lean
View file

@ -11,7 +11,6 @@ import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# The group SO(3)
-/
namespace GroupTheory
@ -107,7 +106,6 @@ lemma toProd_continuous : Continuous toProd := by
refine Continuous.matrix_transpose ?_
exact continuous_iff_le_induced.mpr fun U a => a
/-- The embedding of `SO(3)` into the monoid of matrices times the opposite of
the monoid of matrices. -/
lemma toProd_embedding : Embedding toProd where
@ -223,10 +221,7 @@ lemma exists_basis_preserved (A : SO(3)) :
use b
rw [hb, hv.2]
end action
end SO3
end GroupTheory

1
HepLean/SpaceTime/Basic.lean
View file

@ -29,7 +29,6 @@ instance euclideanNormedAddCommGroup : NormedAddCommGroup SpaceTime := Pi.normed
instance euclideanNormedSpace : NormedSpace ℝ SpaceTime := Pi.normedSpace
namespace SpaceTime
open Manifold

8
HepLean/SpaceTime/LorentzAlgebra/Basic.lean
View file

@ -17,7 +17,6 @@ We define
-/
namespace SpaceTime
open Matrix
open TensorProduct
@ -34,7 +33,6 @@ lemma transpose_eta (A : lorentzAlgebra) : A.1ᵀ * η = - η * A.1 := by
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
erw [mem_skewAdjointMatricesLieSubalgebra]
@ -58,7 +56,6 @@ lemma mem_iff' (A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ) :
rw [minkowskiMatrix.sq]
all_goals noncomm_ring
lemma diag_comp (Λ : lorentzAlgebra) (μ : Fin 1 ⊕ Fin 3) : Λ.1 μ μ = 0 := by
have h := congrArg (fun M ↦ M μ μ) $ mem_iff.mp Λ.2
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
@ -67,22 +64,18 @@ lemma diag_comp (Λ : lorentzAlgebra) (μ : Fin 1 ⊕ Fin 3) : Λ.1 μ μ = 0 :=
simpa using h
simpa using h
lemma time_comps (Λ : lorentzAlgebra) (i : Fin 3) :
Λ.1 (Sum.inr i) (Sum.inl 0) = Λ.1 (Sum.inl 0) (Sum.inr i) := by
simpa only [Fin.isValue, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,
transpose_apply, Sum.elim_inr, mul_neg, mul_one, diagonal_neg, diagonal_mul, Sum.elim_inl,
neg_mul, one_mul, neg_inj] using congrArg (fun M ↦ M (Sum.inl 0) (Sum.inr i)) $ mem_iff.mp Λ.2
lemma space_comps (Λ : lorentzAlgebra) (i j : Fin 3) :
Λ.1 (Sum.inr i) (Sum.inr j) = - Λ.1 (Sum.inr j) (Sum.inr i) := by
simpa only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_neg, diagonal_mul,
Sum.elim_inr, neg_neg, one_mul, mul_diagonal, transpose_apply, mul_neg, mul_one] using
(congrArg (fun M ↦ M (Sum.inr i) (Sum.inr j)) $ mem_iff.mp Λ.2).symm
end lorentzAlgebra
@[simps!]
@ -104,5 +97,4 @@ instance spaceTimeAsLieModule : LieModule ℝ lorentzAlgebra (LorentzVector 3) w
simp [Bracket.bracket]
rw [mulVec_smul]
end SpaceTime

1
HepLean/SpaceTime/LorentzGroup/Basic.lean
View file

@ -44,7 +44,6 @@ def LorentzGroup (d : ℕ) : Set (Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ
{Λ : Matrix (Fin 1 ⊕ Fin d) (Fin 1 ⊕ Fin d) ℝ |
∀ (x y : LorentzVector d), ⟪Λ *ᵥ x, Λ *ᵥ y⟫ₘ = ⟪x, y⟫ₘ}
namespace LorentzGroup
/-- Notation for the Lorentz group. -/
scoped[LorentzGroup] notation (name := lorentzGroup_notation) "𝓛" => LorentzGroup

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

@ -140,7 +140,6 @@ lemma toMatrix_continuous (u : FuturePointing d) : Continuous (toMatrix u) := by
exact FuturePointing.metric_continuous _
exact fun x => FuturePointing.one_add_metric_non_zero u x
lemma toMatrix_in_lorentzGroup (u v : FuturePointing d) : (toMatrix u v) ∈ LorentzGroup d:= by
intro x y
rw [toMatrix_mulVec, toMatrix_mulVec, genBoost, genBoostAux₁, genBoostAux₂]
@ -161,7 +160,6 @@ lemma toLorentz_continuous (u : FuturePointing d) : Continuous (toLorentz u) :=
refine Continuous.subtype_mk ?_ (fun x => toMatrix_in_lorentzGroup u x)
exact toMatrix_continuous u
lemma toLorentz_joined_to_1 (u v : FuturePointing d) : Joined 1 (toLorentz u v) := by
obtain ⟨f, _⟩ := FuturePointing.isPathConnected.joinedIn u trivial v trivial
use ContinuousMap.comp ⟨toLorentz u, toLorentz_continuous u⟩ f
@ -180,8 +178,6 @@ lemma isProper (u v : FuturePointing d) : IsProper (toLorentz u v) :=
end genBoost
end LorentzGroup
end

3
HepLean/SpaceTime/LorentzGroup/Orthochronous.lean
View file

@ -17,10 +17,8 @@ matrices.
-/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
@ -66,7 +64,6 @@ lemma not_orthochronous_iff_le_zero :
rw [IsOrthochronous_iff_ge_one]
linarith
/-- The continuous map taking a Lorentz transformation to its `0 0` element. -/
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)⟩

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

@ -11,10 +11,8 @@ We define the give a series of lemmas related to the determinant of the lorentz
-/
noncomputable section
open Matrix
open Complex
open ComplexConjugate
@ -31,7 +29,6 @@ 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)
instance : TopologicalSpace ℤ₂ := instTopologicalSpaceFin
@ -76,7 +73,6 @@ lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
· intro h
simp [detContinuous, h]
/-- The representation taking a lorentz matrix to its determinant. -/
@[simps!]
def detRep : 𝓛 d →* ℤ₂ where

2
HepLean/SpaceTime/LorentzGroup/Rotations.lean
View file

@ -54,8 +54,6 @@ def SO3ToLorentz : SO(3) →* LorentzGroup 3 where
apply Subtype.eq
simp [Matrix.fromBlocks_multiply]
end LorentzGroup
end

1
HepLean/SpaceTime/LorentzVector/AsSelfAdjointMatrix.lean
View file

@ -52,7 +52,6 @@ noncomputable def fromSelfAdjointMatrix' (x : selfAdjoint (Matrix (Fin 2) (Fin 2
| Sum.inr 1 => (x.1 1 0).im
| Sum.inr 2 => 1/2 * (x.1 0 0 - x.1 1 1).re
/-- The linear equivalence between the vector-space `spaceTime` and self-adjoint
2×2-complex matrices. -/
noncomputable def toSelfAdjointMatrix :

1
HepLean/SpaceTime/LorentzVector/Basic.lean
View file

@ -77,7 +77,6 @@ lemma timeVec_space : (@timeVec d).space = 0 := by
lemma timeVec_time: (@timeVec d).time = 1 := by
simp only [time, Fin.isValue, stdBasis_apply, ↓reduceIte]
/-!
# Reflection of space

10
HepLean/SpaceTime/LorentzVector/NormOne.lean
View file

@ -63,7 +63,6 @@ lemma time_nonpos_iff : v.1.time ≤ 0 ↔ v.1.time ≤ - 1 := by
· intro h
linarith
lemma time_nonneg_iff : 0 ≤ v.1.time ↔ 1 ≤ v.1.time := by
apply Iff.intro
· intro h
@ -167,14 +166,12 @@ lemma metric_reflect_mem_mem (h : v ∈ FuturePointing d) (hw : w ∈ FuturePoin
apply le_trans (neg_le_abs _ : _ ≤ |⟪(v.1).space, (w.1).space⟫_ℝ|) ?_
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
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]
lemma metric_reflect_mem_not_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]
@ -209,7 +206,6 @@ noncomputable def timeVecNormOneFuture : FuturePointing d := ⟨⟨timeVec, by
rw [NormOneLorentzVector.mem_iff, on_timeVec]⟩, by
rw [mem_iff, timeVec_time]; exact Real.zero_lt_one⟩
/-- A continuous path from `timeVecNormOneFuture` to any other. -/
noncomputable def pathFromTime (u : FuturePointing d) : Path timeVecNormOneFuture u where
toFun t := ⟨
@ -266,9 +262,6 @@ noncomputable def pathFromTime (u : FuturePointing d) : Path timeVecNormOneFutur
simp only [Set.Icc.coe_one, one_pow, space, one_mul, Fin.isValue]
exact rfl
lemma isPathConnected : IsPathConnected (@Set.univ (FuturePointing d)) := by
use timeVecNormOneFuture
apply And.intro trivial ?_
@ -276,7 +269,6 @@ lemma isPathConnected : IsPathConnected (@Set.univ (FuturePointing d)) := by
use pathFromTime y
exact fun _ => a
lemma metric_continuous (u : LorentzVector d) :
Continuous (fun (a : FuturePointing d) => ⟪u, a.1.1⟫ₘ) := by
simp only [minkowskiMetric.eq_time_minus_inner_prod]
@ -289,8 +281,6 @@ lemma metric_continuous (u : LorentzVector d) :
(Continuous.comp' (Pi.continuous_precomp Sum.inr) (Continuous.comp'
continuous_subtype_val continuous_subtype_val))
end FuturePointing
end NormOneLorentzVector

9
HepLean/SpaceTime/MinkowskiMetric.lean
View file

@ -16,7 +16,6 @@ of Lorentz vectors in d dimensions.
-/
open Matrix
/-!
@ -55,7 +54,6 @@ lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 := by
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d := by
simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]
@[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d := by
simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]
@ -71,10 +69,8 @@ lemma as_block : @minkowskiMatrix d = (
rw [← diagonal_neg]
rfl
end minkowskiMatrix
/-!
# The definition of the Minkowski Metric
@ -133,7 +129,6 @@ lemma as_sum :
Function.comp_apply, minkowskiMatrix]
ring
/-- The Minkowski metric expressed as a sum for a single vector. -/
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]
@ -167,7 +162,6 @@ lemma right_spaceReflection : ⟪v, w.spaceReflection⟫ₘ = v.time * w.time +
simp only [time, Fin.isValue, space, inner_neg_right, PiLp.inner_apply, Function.comp_apply,
RCLike.inner_apply, conj_trivial, sub_neg_eq_add]
lemma self_spaceReflection_eq_zero_iff : ⟪v, v.spaceReflection⟫ₘ = 0 ↔ v = 0 := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
· rw [right_spaceReflection] at h
@ -195,7 +189,6 @@ lemma nondegenerate : (∀ w, ⟪w, v⟫ₘ = 0) ↔ v = 0 := by
· exact (self_spaceReflection_eq_zero_iff _ ).mp ((symm _ _).trans $ h v.spaceReflection)
· simp [h]
/-!
# Inequalitites involving the Minkowski metric
@ -215,7 +208,6 @@ lemma ge_sub_norm : v.time * w.time - ‖v.space‖ * ‖w.space‖ ≤ ⟪v, w
rw [sub_le_sub_iff_left]
exact norm_inner_le_norm v.space w.space
/-!
# The Minkowski metric and matrices
@ -303,7 +295,6 @@ lemma matrix_eq_id_iff : Λ = 1 ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, w⟫
rw [matrix_eq_iff_eq_forall]
simp
/-!
# The Minkowski metric and the standard basis

2
HepLean/SpaceTime/SL2C/Basic.lean
View file

@ -130,8 +130,6 @@ Complete this section.
-/
end
end SL2C

2
HepLean/StandardModel/HiggsBoson/Basic.lean
View file

@ -98,7 +98,6 @@ lemma apply_im_smooth (φ : HiggsField) (i : Fin 2):
/-- Given two `higgsField`, the map `spaceTime → ℂ` obtained by taking their inner product. -/
def innerProd (φ1 φ2 : HiggsField) : SpaceTime → ℂ := fun x => ⟪φ1 x, φ2 x⟫_ℂ
/-- Given a `higgsField`, the map `spaceTime → ℝ` obtained by taking the square norm of the
higgs vector. -/
@[simp]
@ -154,7 +153,6 @@ lemma potential_apply (φ : HiggsField) (μSq lambda : ℝ) (x : SpaceTime) :
simp [HiggsVec.potential, toHiggsVec_norm]
ring
/-- A higgs field is constant if it is equal for all `x` `y` in `spaceTime`. -/
def IsConst (Φ : HiggsField) : Prop := ∀ x y, Φ x = Φ y

2
HepLean/StandardModel/HiggsBoson/TargetSpace.lean
View file

@ -39,7 +39,6 @@ open ComplexConjugate
/-- The complex vector space in which the Higgs field takes values. -/
abbrev HiggsVec := EuclideanSpace ℂ (Fin 2)
/-- The continuous linear map from the vector space `higgsVec` to `(Fin 2 → ℂ)` achieved by
casting vectors. -/
def higgsVecToFin2ℂ : HiggsVec →L[ℝ] (Fin 2 → ℂ) where
@ -144,7 +143,6 @@ lemma potential_snd_term_nonneg (φ : HiggsVec) :
simp_all only [ge_iff_le, norm_nonneg, pow_nonneg, and_true]
exact le_of_lt hLam
lemma zero_le_potential_discrim (φ : HiggsVec) :
0 ≤ discrim (λ) (- μSq ) (- potential μSq (λ) φ) := by
have h1 := potential_as_quad μSq (λ) φ

1
HepLean/StandardModel/Representations.lean
View file

@ -53,5 +53,4 @@ lemma repU1_fundamentalSU2_commute (u1 : unitary ℂ) (g : specialUnitaryGroup (
apply Subtype.ext
simp
end StandardModel

4
lakefile.toml
View file

@ -37,3 +37,7 @@ srcDir = "scripts"
[[lean_exe]]
name = "type_former_lint"
srcDir = "scripts"
[[lean_exe]]
name = "double_line_lint"
srcDir = "scripts"

71
scripts/double_line_lint.lean Normal file
View file

@ -0,0 +1,71 @@
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license.
Authors: Joseph Tooby-Smith
-/
import Lean
import Mathlib.Tactic.Linter.TextBased
/-!
# Double line Lint
This linter double empty lines in files.
## Note
Parts of this file are adapted from `Mathlib.Tactic.Linter.TextBased`,
authored by Michael Rothgang.
-/
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)
/-- Checks if there are two consecutive empty lines. -/
def doubleEmptyLineLinter : HepLeanTextLinter := fun lines ↦ Id.run do
let enumLines := (lines.toList.enumFrom 1)
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)
else none)
errors.toArray
structure HepLeanErrorContext where
/-- The underlying `message`. -/
error : String
/-- The line number -/
lineNumber : 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}")
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]
let errors := allOutput.flatten
printErrors errors
return errors.size > 0
def main (_ : List String) : IO UInt32 := do
initSearchPath (← findSysroot)
let mods : Name := `HepLean
let imp : Import := {module := mods}
let mFile ← findOLean imp.module
unless (← mFile.pathExists) do
throw <| IO.userError s!"object file '{mFile}' of module {imp.module} does not exist"
let (hepLeanMod, _) ← readModuleData mFile
let mut warned := false
for imp in hepLeanMod.imports do
if imp.module == `Init then continue
let filePath := (mkFilePath (imp.module.toString.split (· == '.'))).addExtension "lean"
if ← hepLeanLintFile filePath then
warned := true
if warned then
throw <| IO.userError s!"Errors found."
return 0