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refactor: Remove double empty lines

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jstoobysmith 2024-07-03 07:56:30 -04:00
parent ae18a2196d
commit f03d063c86
60 changed files with 0 additions and 232 deletions
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5
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/PlaneWithY3B3.lean
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@ -18,8 +18,6 @@ The plane spanned by Y₃, B₃ and third orthogonal point.
-/
universe v u
namespace MSSMACC
@ -93,7 +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
@ -184,7 +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
@ -241,5 +237,4 @@ lemma α₂_proj_zero (T : MSSMACC.Sols) (h1 : α₃ (proj T.1.1) = 0) :
end proj
end MSSMACC

4
HepLean/AnomalyCancellation/MSSMNu/OrthogY3B3/ToSols.lean
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@ -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 :=
@ -237,7 +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 :=
@ -457,7 +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

3
HepLean/AnomalyCancellation/PureU1/Basic.lean
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@ -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
@ -162,7 +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

2
HepLean/AnomalyCancellation/PureU1/BasisLinear.lean
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@ -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
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@ -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
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@ -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
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@ -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
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@ -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
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@ -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
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@ -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. -/