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refactor: lint

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jstoobysmith 2025-02-03 11:42:56 +00:00
parent 6433259bc4
commit 70f617096b
16 changed files with 103 additions and 87 deletions
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2
HepLean/AnomalyCancellation/PureU1/BasisLinear.lean
View file

@ -76,7 +76,7 @@ lemma sum_of_vectors {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j : Fin n) :
sum_of_anomaly_free_linear (fun i => f i) j
TODO "The definition of `coordinateMap` here may be improved by replacing
`Finsupp.equivFunOnFinite` with `Finsupp.linearEquivFunOnFinite`. Investigate this."
`Finsupp.equivFunOnFinite` with `Finsupp.linearEquivFunOnFinite`. Investigate this."
/-- The coordinate map for the basis. -/
noncomputable
def coordinateMap : (PureU1 n.succ).LinSols ≃ₗ[ℚ] Fin n →₀ ℚ where

15
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/Basic.lean
View file

@ -203,7 +203,8 @@ lemma ι_eq_zero_iff_mem_ideal (x : FieldOpFreeAlgebra 𝓕) :
simp only
rfl
lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕)
(hx : x ∈ 𝓕.fieldOpIdealSet) :
x.bosonicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.bosonicProj = 0 := by
have hx' := hx
simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
@ -234,7 +235,8 @@ lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (hx
· right
rw [bosonicProj_of_mem_fermionic _ h]
lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕)
(hx : x ∈ 𝓕.fieldOpIdealSet) :
x.fermionicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProj = 0 := by
have hx' := hx
simp only [fieldOpIdealSet, exists_prop, Set.mem_setOf_eq] at hx
@ -265,10 +267,12 @@ lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : FieldOpFreeAlgebra 𝓕) (h
rw [fermionicProj_of_mem_fermionic _ h]
simpa using hx'
lemma bosonicProj_mem_ideal (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
lemma bosonicProj_mem_ideal (x : FieldOpFreeAlgebra 𝓕)
(hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
x.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at hx
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) : Prop :=
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕)
(h : a ∈ AddSubgroup.closure k) : Prop :=
a.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet
change p x hx
apply AddSubgroup.closure_induction
@ -401,7 +405,8 @@ lemma bosonicProj_mem_ideal (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ TwoSidedId
· intro x hx
simp [p]
lemma fermionicProj_mem_ideal (x : FieldOpFreeAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
lemma fermionicProj_mem_ideal (x : FieldOpFreeAlgebra 𝓕)
(hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
x.fermionicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
have hb := bosonicProj_mem_ideal x hx
rw [← ι_eq_zero_iff_mem_ideal] at hx hb ⊢

10
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/NormalOrder.lean
View file

@ -52,7 +52,8 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero
lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero
(φa φa' : 𝓕.CrAnFieldOp) (φs : List 𝓕.CrAnFieldOp)
(a : 𝓕.FieldOpFreeAlgebra) : ι 𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * a) = 0 := by
(a : 𝓕.FieldOpFreeAlgebra) :
ι 𝓝ᶠ(ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca * a) = 0 := by
have hf : ι.toLinearMap ∘ₗ normalOrderF ∘ₗ
mulLinearMap (ofCrAnListF φs * [ofCrAnOpF φa, ofCrAnOpF φa']ₛca) = 0 := by
apply ofCrAnListFBasis.ext
@ -126,7 +127,7 @@ lemma ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero_mul
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF (ofCrAnListF φs)) = 0 := by
apply ofCrAnListFBasis.ext
intro φs'
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, FieldOpFreeAlgebra.ofListBasis_eq_ofList,
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
LinearMap.zero_apply]
rw [ι_normalOrderF_superCommuteF_ofCrAnListF_ofCrAnListF_eq_zero_mul]
@ -142,7 +143,7 @@ lemma ι_normalOrderF_superCommuteF_eq_zero_mul
mulLinearMap.flip b ∘ₗ mulLinearMap a ∘ₗ superCommuteF.flip c) = 0 := by
apply ofCrAnListFBasis.ext
intro φs
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, FieldOpFreeAlgebra.ofListBasis_eq_ofList,
simp only [mulLinearMap, LinearMap.coe_mk, AddHom.coe_mk, ofListBasis_eq_ofList,
LinearMap.coe_comp, Function.comp_apply, LinearMap.flip_apply, AlgHom.toLinearMap_apply,
LinearMap.zero_apply]
rw [ι_normalOrderF_superCommuteF_ofCrAnListF_eq_zero_mul]
@ -182,7 +183,8 @@ lemma ι_normalOrderF_superCommuteF_eq_zero (c d : 𝓕.FieldOpFreeAlgebra) : ι
lemma ι_normalOrderF_zero_of_mem_ideal (a : 𝓕.FieldOpFreeAlgebra)
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓝ᶠ(a) = 0 := by
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓝ᶠ(a) = 0
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕)
(h : a ∈ AddSubgroup.closure k) := ι 𝓝ᶠ(a) = 0
change p a h
apply AddSubgroup.closure_induction
· intro x hx

6
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeOrder.lean
View file

@ -153,7 +153,8 @@ lemma ι_timeOrderF_superCommuteF_superCommuteF_ofCrAnListF {φ1 φ2 φ3 : 𝓕.
simp
@[simp]
lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnFieldOp} (a b : 𝓕.FieldOpFreeAlgebra) :
lemma ι_timeOrderF_superCommuteF_superCommuteF {φ1 φ2 φ3 : 𝓕.CrAnFieldOp}
(a b : 𝓕.FieldOpFreeAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * b) = 0 := by
let pb (b : 𝓕.FieldOpFreeAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnOpF φ1, [ofCrAnOpF φ2, ofCrAnOpF φ3]ₛca]ₛca * b) = 0
@ -303,7 +304,8 @@ lemma ι_timeOrderF_superCommuteF_neq_time {φ ψ : 𝓕.CrAnFieldOp}
lemma ι_timeOrderF_zero_of_mem_ideal (a : 𝓕.FieldOpFreeAlgebra)
(h : a ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) : ι 𝓣ᶠ(a) = 0 := by
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at h
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) := ι 𝓣ᶠ(a) = 0
let p {k : Set 𝓕.FieldOpFreeAlgebra} (a : FieldOpFreeAlgebra 𝓕)
(h : a ∈ AddSubgroup.closure k) := ι 𝓣ᶠ(a) = 0
change p a h
apply AddSubgroup.closure_induction
· intro x hx

3
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/Basic.lean
View file

@ -198,7 +198,8 @@ lemma ofListBasis_eq_ofList (φs : List 𝓕.CrAnFieldOp) :
-/
/-- The bi-linear map associated with multiplication in `FieldOpFreeAlgebra`. -/
noncomputable def mulLinearMap : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 where
noncomputable def mulLinearMap : FieldOpFreeAlgebra 𝓕 →ₗ[ℂ] FieldOpFreeAlgebra 𝓕 →ₗ[ℂ]
FieldOpFreeAlgebra 𝓕 where
toFun a := {
toFun := fun b => a * b,
map_add' := fun c d => by simp [mul_add]

12
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/Grading.lean
View file

@ -30,7 +30,8 @@ lemma ofCrAnListF_mem_statisticSubmodule_of (φs : List 𝓕.CrAnFieldOp) (f : F
refine Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩
lemma ofCrAnListF_bosonic_or_fermionic (φs : List 𝓕.CrAnFieldOp) :
ofCrAnListF φs ∈ statisticSubmodule bosonic ∨ ofCrAnListF φs ∈ statisticSubmodule fermionic := by
ofCrAnListF φs ∈ statisticSubmodule bosonic ∨
ofCrAnListF φs ∈ statisticSubmodule fermionic := by
by_cases h : (𝓕 |>ₛ φs) = bosonic
· left
exact ofCrAnListF_mem_statisticSubmodule_of φs bosonic h
@ -73,7 +74,8 @@ lemma bosonicProj_of_mem_bosonic (a : 𝓕.FieldOpFreeAlgebra) (h : a ∈ statis
· intro a x hx hy
simp_all [p]
lemma bosonicProj_of_mem_fermionic (a : 𝓕.FieldOpFreeAlgebra) (h : a ∈ statisticSubmodule fermionic) :
lemma bosonicProj_of_mem_fermionic (a : 𝓕.FieldOpFreeAlgebra)
(h : a ∈ statisticSubmodule fermionic) :
bosonicProj a = 0 := by
let p (a : 𝓕.FieldOpFreeAlgebra) (hx : a ∈ statisticSubmodule fermionic) : Prop :=
bosonicProj a = 0
@ -127,7 +129,8 @@ lemma fermionicProj_ofCrAnListF_if_bosonic (φs : List 𝓕.CrAnFieldOp) :
simp only [neq_fermionic_iff_eq_bosonic] at h1
simp [h1]
lemma fermionicProj_of_mem_fermionic (a : 𝓕.FieldOpFreeAlgebra) (h : a ∈ statisticSubmodule fermionic) :
lemma fermionicProj_of_mem_fermionic (a : 𝓕.FieldOpFreeAlgebra)
(h : a ∈ statisticSubmodule fermionic) :
fermionicProj a = ⟨a, h⟩ := by
let p (a : 𝓕.FieldOpFreeAlgebra) (hx : a ∈ statisticSubmodule fermionic) : Prop :=
fermionicProj a = ⟨a, hx⟩
@ -235,7 +238,8 @@ lemma directSum_eq_bosonic_plus_fermionic
abel
/-- The instance of a graded algebra on `FieldOpFreeAlgebra`. -/
instance fieldOpFreeAlgebraGrade : GradedAlgebra (A := 𝓕.FieldOpFreeAlgebra) statisticSubmodule where
instance fieldOpFreeAlgebraGrade :
GradedAlgebra (A := 𝓕.FieldOpFreeAlgebra) statisticSubmodule where
one_mem := by
simp only [statisticSubmodule]
refine Submodule.mem_span.mpr fun p a => a ?_

33
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/NormalOrder.lean
View file

@ -69,7 +69,8 @@ lemma normalOrderF_normalOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) :
obtain ⟨φs', rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pb]
let pa (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
Prop := 𝓝ᶠ(a * ofCrAnListF φs' * ofCrAnListF φs) = 𝓝ᶠ(a * 𝓝ᶠ(ofCrAnListF φs') * ofCrAnListF φs)
Prop := 𝓝ᶠ(a * ofCrAnListF φs' * ofCrAnListF φs) =
𝓝ᶠ(a * 𝓝ᶠ(ofCrAnListF φs') * ofCrAnListF φs)
change pa a (Basis.mem_span _ a)
apply Submodule.span_induction
· intro x hx
@ -101,13 +102,15 @@ lemma normalOrderF_normalOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) :
· intro x hx h hp
simp_all [pc]
lemma normalOrderF_normalOrderF_right (a b : 𝓕.FieldOpFreeAlgebra) : 𝓝ᶠ(a * b) = 𝓝ᶠ(a * 𝓝ᶠ(b)) := by
lemma normalOrderF_normalOrderF_right (a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * b) = 𝓝ᶠ(a * 𝓝ᶠ(b)) := by
trans 𝓝ᶠ(a * b * 1)
· simp
· rw [normalOrderF_normalOrderF_mid]
simp
lemma normalOrderF_normalOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) : 𝓝ᶠ(a * b) = 𝓝ᶠ(𝓝ᶠ(a) * b) := by
lemma normalOrderF_normalOrderF_left (a b : 𝓕.FieldOpFreeAlgebra) :
𝓝ᶠ(a * b) = 𝓝ᶠ(𝓝ᶠ(a) * b) := by
trans 𝓝ᶠ(1 * a * b)
· simp
· rw [normalOrderF_normalOrderF_mid]
@ -374,7 +377,8 @@ lemma normalOrderF_ofFieldOpF_mul_ofFieldOpF (φ φ' : 𝓕.FieldOp) :
(crPartF φ' * anPartF φ) +
crPartF φ * anPartF φ' +
anPartF φ * anPartF φ' := by
rw [ofFieldOpF_eq_crPartF_add_anPartF, ofFieldOpF_eq_crPartF_add_anPartF, mul_add, add_mul, add_mul]
rw [ofFieldOpF_eq_crPartF_add_anPartF, ofFieldOpF_eq_crPartF_add_anPartF, mul_add, add_mul,
add_mul]
simp only [map_add, normalOrderF_crPartF_mul_crPartF, normalOrderF_anPartF_mul_crPartF,
instCommGroup.eq_1, normalOrderF_crPartF_mul_anPartF, normalOrderF_anPartF_mul_anPartF]
abel
@ -397,8 +401,8 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_create_create_ofCrAnListF
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, mul_sub, sub_mul, map_sub]
conv_lhs =>
lhs; rhs
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append, ← ofCrAnListF_append,
← ofCrAnListF_append]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append,
← ofCrAnListF_append, ← ofCrAnListF_append]
conv_lhs =>
lhs
rw [normalOrderF_ofCrAnListF, normalOrderList_eq_createFilter_append_annihilateFilter]
@ -414,8 +418,8 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_create_create_ofCrAnListF
rhs; rhs
rw [smul_mul_assoc, Algebra.mul_smul_comm, smul_mul_assoc]
rhs
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append, ← ofCrAnListF_append,
← ofCrAnListF_append]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append,
← ofCrAnListF_append, ← ofCrAnListF_append]
conv_lhs =>
rhs
rw [map_smul]
@ -459,8 +463,8 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
rw [superCommuteF_ofCrAnOpF_ofCrAnOpF, mul_sub, sub_mul, map_sub]
conv_lhs =>
lhs; rhs
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append, ← ofCrAnListF_append,
← ofCrAnListF_append]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append,
← ofCrAnListF_append, ← ofCrAnListF_append]
conv_lhs =>
lhs
rw [normalOrderF_ofCrAnListF, normalOrderList_eq_createFilter_append_annihilateFilter]
@ -477,8 +481,8 @@ lemma normalOrderF_superCommuteF_ofCrAnListF_annihilate_annihilate_ofCrAnListF
rw [smul_mul_assoc]
rw [Algebra.mul_smul_comm, smul_mul_assoc]
rhs
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append, ← ofCrAnListF_append,
← ofCrAnListF_append]
rw [← ofCrAnListF_singleton, ← ofCrAnListF_singleton, ← ofCrAnListF_append,
← ofCrAnListF_append, ← ofCrAnListF_append]
conv_lhs =>
rhs
rw [map_smul]
@ -524,8 +528,9 @@ lemma ofCrAnListF_superCommuteF_normalOrderF_ofCrAnListF (φs φs' : List 𝓕.C
[ofCrAnListF φs, 𝓝ᶠ(ofCrAnListF φs')]ₛca =
ofCrAnListF φs * 𝓝ᶠ(ofCrAnListF φs') -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • 𝓝ᶠ(ofCrAnListF φs') * ofCrAnListF φs := by
simp [normalOrderF_ofCrAnListF, map_smul, superCommuteF_ofCrAnListF_ofCrAnListF, ofCrAnListF_append,
smul_sub, smul_smul, mul_comm]
simp only [normalOrderF_ofCrAnListF, map_smul, superCommuteF_ofCrAnListF_ofCrAnListF,
ofCrAnListF_append, instCommGroup.eq_1, normalOrderList_statistics, smul_sub, smul_smul,
Algebra.mul_smul_comm, mul_comm, Algebra.smul_mul_assoc]
lemma ofCrAnListF_superCommuteF_normalOrderF_ofFieldOpListF (φs : List 𝓕.CrAnFieldOp)
(φs' : List 𝓕.FieldOp) : [ofCrAnListF φs, 𝓝ᶠ(ofFieldOpListF φs')]ₛca =

36
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/SuperCommute.lean
View file

@ -26,7 +26,8 @@ open FieldStatistic
/-- The super commutor on the creation and annihlation algebra. For two bosonic operators
or a bosonic and fermionic operator this corresponds to the usual commutator
whilst for two fermionic operators this corresponds to the anti-commutator. -/
noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra :=
noncomputable def superCommuteF : 𝓕.FieldOpFreeAlgebra →ₗ[ℂ] 𝓕.FieldOpFreeAlgebra →ₗ[ℂ]
𝓕.FieldOpFreeAlgebra :=
Basis.constr ofCrAnListFBasis ℂ fun φs =>
Basis.constr ofCrAnListFBasis ℂ fun φs' =>
ofCrAnListF (φs ++ φs') - 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofCrAnListF (φs' ++ φs)
@ -86,13 +87,15 @@ lemma superCommuteF_ofFieldOpListF_ofFieldOpFsList (φ : List 𝓕.FieldOp) (φs
lemma superCommuteF_ofFieldOpF_ofFieldOpFsList (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) :
[ofFieldOpF φ, ofFieldOpListF φs]ₛca = ofFieldOpF φ * ofFieldOpListF φs -
𝓢(𝓕 |>ₛ φ, 𝓕 |>ₛ φs) • ofFieldOpListF φs * ofFieldOpF φ := by
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList,
ofFieldOpListF_singleton]
simp
lemma superCommuteF_ofFieldOpListF_ofFieldOpF (φs : List 𝓕.FieldOp) (φ : 𝓕.FieldOp) :
[ofFieldOpListF φs, ofFieldOpF φ]ₛca = ofFieldOpListF φs * ofFieldOpF φ -
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φ) • ofFieldOpF φ * ofFieldOpListF φs := by
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList, ofFieldOpListF_singleton]
rw [← ofFieldOpListF_singleton, superCommuteF_ofFieldOpListF_ofFieldOpFsList,
ofFieldOpListF_singleton]
simp
lemma superCommuteF_anPartF_crPartF (φ φ' : 𝓕.FieldOp) :
@ -269,7 +272,8 @@ lemma ofCrAnOpF_mul_ofCrAnListF_eq_superCommuteF (φ : 𝓕.CrAnFieldOp) (φs' :
simp
lemma ofFieldOpListF_mul_ofFieldOpListF_eq_superCommuteF (φs φs' : List 𝓕.FieldOp) :
ofFieldOpListF φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpListF φs
ofFieldOpListF φs * ofFieldOpListF φs' =
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofFieldOpListF φs
+ [ofFieldOpListF φs, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofFieldOpListF_ofFieldOpFsList]
simp
@ -314,8 +318,9 @@ lemma anPartF_mul_anPartF_eq_superCommuteF (φ φ' : 𝓕.FieldOp) :
rw [superCommuteF_anPartF_anPartF]
simp
lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnFieldOp) (φs' : List 𝓕.FieldOp) :
ofCrAnListF φs * ofFieldOpListF φs' = 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnListF φs
lemma ofCrAnListF_mul_ofFieldOpListF_eq_superCommuteF (φs : List 𝓕.CrAnFieldOp)
(φs' : List 𝓕.FieldOp) : ofCrAnListF φs * ofFieldOpListF φs' =
𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs') • ofFieldOpListF φs' * ofCrAnListF φs
+ [ofCrAnListF φs, ofFieldOpListF φs']ₛca := by
rw [superCommuteF_ofCrAnListF_ofFieldOpFsList]
simp
@ -365,7 +370,8 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_cons (φ : 𝓕.CrAnFieldOp) (φs φ
rw [superCommuteF_ofCrAnListF_ofCrAnListF]
conv_rhs =>
lhs
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF, sub_mul, ← ofCrAnListF_append]
rw [← ofCrAnListF_singleton, superCommuteF_ofCrAnListF_ofCrAnListF, sub_mul,
← ofCrAnListF_append]
rhs
rw [FieldStatistic.ofList_singleton, ofCrAnListF_append, ofCrAnListF_singleton, smul_mul_assoc,
mul_assoc, ← ofCrAnListF_append]
@ -410,14 +416,16 @@ lemma superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum (φs : List 𝓕.CrAnFieldOp)
| [] => by
simp [← ofCrAnListF_nil, superCommuteF_ofCrAnListF_ofCrAnListF]
| φ :: φs' => by
rw [superCommuteF_ofCrAnListF_ofCrAnListF_cons, superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum φs φs']
rw [superCommuteF_ofCrAnListF_ofCrAnListF_cons,
superCommuteF_ofCrAnListF_ofCrAnListF_eq_sum φs φs']
conv_rhs => erw [Fin.sum_univ_succ]
congr 1
· simp
· simp [Finset.mul_sum, smul_smul, ofCrAnListF_cons, mul_assoc,
FieldStatistic.ofList_cons_eq_mul, mul_comm]
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnFieldOp) : (φs' : List 𝓕.FieldOp) →
lemma superCommuteF_ofCrAnListF_ofFieldOpListF_eq_sum (φs : List 𝓕.CrAnFieldOp) :
(φs' : List 𝓕.FieldOp) →
[ofCrAnListF φs, ofFieldOpListF φs']ₛca =
∑ (n : Fin φs'.length), 𝓢(𝓕 |>ₛ φs, 𝓕 |>ₛ φs'.take n) •
ofFieldOpListF (φs'.take n) * [ofCrAnListF φs, ofFieldOpF (φs'.get n)]ₛca *
@ -643,12 +651,14 @@ lemma bosonic_superCommuteF {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisti
simp only [add_mul, mul_add]
abel
lemma superCommuteF_bonsonic_symm {a b : 𝓕.FieldOpFreeAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
lemma superCommuteF_bonsonic_symm {a b : 𝓕.FieldOpFreeAlgebra}
(hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = - [b, a]ₛca := by
rw [bosonic_superCommuteF hb, superCommuteF_bonsonic hb]
simp
lemma bonsonic_superCommuteF_symm {a b : 𝓕.FieldOpFreeAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
lemma bonsonic_superCommuteF_symm {a b : 𝓕.FieldOpFreeAlgebra}
(ha : a ∈ statisticSubmodule bosonic) :
[a, b]ₛca = - [b, a]ₛca := by
rw [bosonic_superCommuteF ha, superCommuteF_bonsonic ha]
simp
@ -808,8 +818,8 @@ lemma superCommuteF_bosonic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φ
simp_all [p, Finset.smul_sum]
· exact ha
lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra) (φs : List 𝓕.CrAnFieldOp)
(ha : a ∈ statisticSubmodule fermionic) :
lemma superCommuteF_fermionic_ofCrAnListF_eq_sum (a : 𝓕.FieldOpFreeAlgebra)
(φs : List 𝓕.CrAnFieldOp) (ha : a ∈ statisticSubmodule fermionic) :
[a, ofCrAnListF φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
ofCrAnListF (φs.take n) * [a, ofCrAnOpF (φs.get n)]ₛca *
ofCrAnListF (φs.drop (n + 1)) := by

9
HepLean/PerturbationTheory/Algebras/FieldOpFreeAlgebra/TimeOrder.lean
View file

@ -39,7 +39,8 @@ lemma timeOrderF_ofCrAnListF (φs : List 𝓕.CrAnFieldOp) :
rw [← ofListBasis_eq_ofList]
simp only [timeOrderF, Basis.constr_basis]
lemma timeOrderF_timeOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) : 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c) := by
lemma timeOrderF_timeOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) :
𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c) := by
let pc (c : 𝓕.FieldOpFreeAlgebra) (hc : c ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
Prop := 𝓣ᶠ(a * b * c) = 𝓣ᶠ(a * 𝓣ᶠ(b) * c)
change pc c (Basis.mem_span _ c)
@ -55,7 +56,8 @@ lemma timeOrderF_timeOrderF_mid (a b c : 𝓕.FieldOpFreeAlgebra) : 𝓣ᶠ(a *
obtain ⟨φs', rfl⟩ := hx
simp only [ofListBasis_eq_ofList, pb]
let pa (a : 𝓕.FieldOpFreeAlgebra) (ha : a ∈ Submodule.span ℂ (Set.range ofCrAnListFBasis)) :
Prop := 𝓣ᶠ(a * ofCrAnListF φs' * ofCrAnListF φs) = 𝓣ᶠ(a * 𝓣ᶠ(ofCrAnListF φs') * ofCrAnListF φs)
Prop := 𝓣ᶠ(a * ofCrAnListF φs' * ofCrAnListF φs) =
𝓣ᶠ(a * 𝓣ᶠ(ofCrAnListF φs') * ofCrAnListF φs)
change pa a (Basis.mem_span _ a)
apply Submodule.span_induction
· intro x hx
@ -116,7 +118,8 @@ lemma timeOrderF_ofFieldOpListF_nil : timeOrderF (𝓕 := 𝓕) (ofFieldOpListF
simp [timeOrderSign, Wick.koszulSign, timeOrderList]
@[simp]
lemma timeOrderF_ofFieldOpListF_singleton (φ : 𝓕.FieldOp) : 𝓣ᶠ(ofFieldOpListF [φ]) = ofFieldOpListF [φ] := by
lemma timeOrderF_ofFieldOpListF_singleton (φ : 𝓕.FieldOp) :
𝓣ᶠ(ofFieldOpListF [φ]) = ofFieldOpListF [φ] := by
simp [timeOrderF_ofFieldOpListF, timeOrderSign, timeOrderList]
lemma timeOrderF_ofFieldOpF_ofFieldOpF_ordered {φ ψ : 𝓕.FieldOp} (h : timeOrderRel φ ψ) :

12
HepLean/PerturbationTheory/WickContraction/Join.lean
View file

@ -532,9 +532,9 @@ lemma join_getDual?_apply_uncontractedListEmb_some {φs : List 𝓕.FieldOp}
simp
@[simp]
lemma join_getDual?_apply_uncontractedListEmb {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) (i : Fin [φsΛ]ᵘᶜ.length) :
((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)) =
lemma join_getDual?_apply_uncontractedListEmb {φs : List 𝓕.FieldOp}
(φsΛ : WickContraction φs.length) (φsucΛ : WickContraction [φsΛ]ᵘᶜ.length)
(i : Fin [φsΛ]ᵘᶜ.length) : ((join φsΛ φsucΛ).getDual? (uncontractedListEmd i)) =
Option.map uncontractedListEmd (φsucΛ.getDual? i) := by
by_cases h : (φsucΛ.getDual? i).isSome
· rw [join_getDual?_apply_uncontractedListEmb_some]
@ -608,9 +608,8 @@ lemma join_singleton_getDual?_right {φs : List 𝓕.FieldOp}
left
exact Finset.pair_comm j i
lemma exists_contraction_pair_of_card_ge_zero {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(h : 0 < φsΛ.1.card) :
lemma exists_contraction_pair_of_card_ge_zero {φs : List 𝓕.FieldOp}
(φsΛ : WickContraction φs.length) (h : 0 < φsΛ.1.card) :
∃ a, a ∈ φsΛ.1 := by
simpa using h
@ -656,5 +655,4 @@ lemma join_not_gradingCompliant_of_left_not_gradingCompliant {φs : List 𝓕.Fi
join_sndFieldOfContract_joinLift]
exact ha2
end WickContraction

4
HepLean/PerturbationTheory/WickContraction/Sign/InsertNone.lean
View file

@ -158,8 +158,8 @@ lemma signInsertNone_eq_prod_prod (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
erw [hG a]
rfl
lemma sign_insert_none_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) :
(φsΛ ↩Λ φ 0 none).sign = φsΛ.sign := by
lemma sign_insert_none_zero (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) : (φsΛ ↩Λ φ 0 none).sign = φsΛ.sign := by
rw [sign_insert_none]
simp [signInsertNone]

3
HepLean/PerturbationTheory/WickContraction/Sign/InsertSome.lean
View file

@ -19,14 +19,12 @@ variable {n : ℕ} (c : WickContraction n)
open HepLean.List
open FieldStatistic
/-!
## Sign insert some
-/
lemma stat_ofFinset_eq_one_of_gradingCompliant (φs : List 𝓕.FieldOp)
(a : Finset (Fin φs.length)) (φsΛ : WickContraction φs.length) (hg : GradingCompliant φs φsΛ)
(hnon : ∀ i, φsΛ.getDual? i = none → i ∉ a)
@ -63,7 +61,6 @@ lemma stat_ofFinset_eq_one_of_gradingCompliant (φs : List 𝓕.FieldOp)
exact False.elim (h1 hsom')
rfl
lemma signFinset_insertAndContract_some (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (i1 i2 : Fin φs.length)
(j : φsΛ.uncontracted) :

25
HepLean/PerturbationTheory/WickContraction/Sign/Join.lean
View file

@ -205,10 +205,8 @@ lemma join_singleton_sign_right {φs : List 𝓕.FieldOp}
rw [sign_right_eq_prod_mul_prod]
rfl
lemma joinSignRightExtra_eq_i_j_finset_eq_if {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j)
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
{i j : Fin φs.length} (h : i < j) (φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
joinSignRightExtra h φsucΛ = ∏ a,
𝓢((𝓕|>ₛ [singleton h]ᵘᶜ[φsucΛ.sndFieldOfContract a]),
𝓕 |>ₛ ⟨φs.get, (if uncontractedListEmd (φsucΛ.fstFieldOfContract a) < j ∧
@ -303,13 +301,10 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.FieldOp}
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
joinSignLeftExtra h φsucΛ = joinSignRightExtra h φsucΛ := by
/- Simplifying joinSignLeftExtra. -/
rw [joinSignLeftExtra]
rw [ofFinset_eq_prod]
rw [map_prod]
let e2 : Fin φs.length ≃ {x // (((singleton h).join φsucΛ).getDual? x).isSome} ⊕
{x // ¬ (((singleton h).join φsucΛ).getDual? x).isSome} := by
exact (Equiv.sumCompl fun a => (((singleton h).join φsucΛ).getDual? a).isSome = true).symm
rw [← e2.symm.prod_comp]
rw [joinSignLeftExtra, ofFinset_eq_prod, map_prod, ← e2.symm.prod_comp]
simp only [Fin.getElem_fin, Fintype.prod_sum_type, instCommGroup]
conv_lhs =>
enter [2, 2, x]
@ -326,8 +321,7 @@ lemma joinSignLeftExtra_eq_joinSignRightExtra {φs : List 𝓕.FieldOp}
enter [2, a]
rw [prod_finset_eq_mul_fst_snd]
simp [e2, sigmaContractedEquiv]
rw [prod_join]
rw [singleton_prod]
rw [prod_join, singleton_prod]
simp only [join_fstFieldOfContract_joinLiftLeft, singleton_fstFieldOfContract,
join_sndFieldOfContract_joinLift, singleton_sndFieldOfContract, lt_self_iff_false, and_false,
↓reduceIte, map_one, mul_one, join_fstFieldOfContract_joinLiftRight,
@ -384,17 +378,14 @@ lemma join_sign_singleton {φs : List 𝓕.FieldOp}
{i j : Fin φs.length} (h : i < j) (hs : (𝓕 |>ₛ φs[i]) = (𝓕 |>ₛ φs[j]))
(φsucΛ : WickContraction [singleton h]ᵘᶜ.length) :
(join (singleton h) φsucΛ).sign = (singleton h).sign * φsucΛ.sign := by
rw [join_singleton_sign_right]
rw [join_singleton_sign_left h φsucΛ]
rw [join_singleton_sign_right, join_singleton_sign_left h φsucΛ]
rw [joinSignLeftExtra_eq_joinSignRightExtra h hs φsucΛ]
rw [← mul_assoc]
rw [mul_assoc _ _ (joinSignRightExtra h φsucΛ)]
rw [← mul_assoc, mul_assoc _ _ (joinSignRightExtra h φsucΛ)]
have h1 : (joinSignRightExtra h φsucΛ * joinSignRightExtra h φsucΛ) = 1 := by
rw [← joinSignLeftExtra_eq_joinSignRightExtra h hs φsucΛ]
simp [joinSignLeftExtra]
simp only [instCommGroup, Fin.getElem_fin, h1, mul_one]
rw [sign]
rw [prod_join]
rw [sign, prod_join]
congr
· rw [singleton_prod]
simp
@ -414,9 +405,7 @@ lemma join_sign_induction {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs
| Nat.succ n, hn => by
obtain ⟨i, j, hij, φsucΛ', rfl, h1, h2, h3⟩ :=
exists_join_singleton_of_card_ge_zero φsΛ (by simp [hn]) hc
rw [join_assoc]
rw [join_sign_singleton hij h1]
rw [join_sign_singleton hij h1]
rw [join_assoc, join_sign_singleton hij h1, join_sign_singleton hij h1]
have hn : φsucΛ'.1.card = n := by
omega
rw [join_sign_induction φsucΛ' (congr (by simp [join_uncontractedListGet]) φsucΛ) h2

4
HepLean/PerturbationTheory/WickContraction/TimeCond.lean
View file

@ -199,8 +199,8 @@ lemma timeOrder_timeContract_mul_of_eqTimeOnly_left {φs : List 𝓕.FieldOp}
rw [timeOrder_timeContract_mul_of_eqTimeOnly_mid φsΛ hl]
simp
lemma exists_join_singleton_of_not_eqTimeOnly {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
(h1 : ¬ φsΛ.EqTimeOnly) :
lemma exists_join_singleton_of_not_eqTimeOnly {φs : List 𝓕.FieldOp}
(φsΛ : WickContraction φs.length) (h1 : ¬ φsΛ.EqTimeOnly) :
∃ (i j : Fin φs.length) (h : i < j) (φsucΛ : WickContraction [singleton h]ᵘᶜ.length),
φsΛ = join (singleton h) φsucΛ ∧ (¬ timeOrderRel φs[i] φs[j] ∨ ¬ timeOrderRel φs[j] φs[i]) := by
rw [eqTimeOnly_iff_forall_finset] at h1

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

@ -325,7 +325,7 @@ lemma isBounded_of_𝓵_pos (h : 0 < P.𝓵) : P.IsBounded := by
/-- When there is no quartic coupling, the potential is bounded iff the mass squared is
non-positive, i.e., for `P : Potential` then `P.IsBounded` iff `P.μ2 ≤ 0`. That is to say
`- P.μ2 * ‖φ‖_H^2 x` is bounded below ifff `P.μ2 ≤ 0`.-/
`- P.μ2 * ‖φ‖_H^2 x` is bounded below ifff `P.μ2 ≤ 0`. -/
informal_lemma isBounded_iff_of_𝓵_zero where
deps := [`StandardModel.HiggsField.Potential.IsBounded, `StandardModel.HiggsField.Potential]