DibyashanuPati/PhysLean
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

refactor: Lint

Browse source
This commit is contained in:
jstoobysmith 2025-01-29 16:06:28 +00:00
parent a329661c24
commit e5c85ac109
11 changed files with 179 additions and 182 deletions
Download patch file Download diff file Expand all files Collapse all files

15
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/Grading.lean
View file

@ -21,12 +21,12 @@ namespace CrAnAlgebra
noncomputable section
/-- The submodule of `CrAnAlgebra` spanned by lists of field statistic `f`. -/
def statisticSubmodule (f : FieldStatistic) : Submodule ℂ 𝓕.CrAnAlgebra :=
def statisticSubmodule (f : FieldStatistic) : Submodule ℂ 𝓕.CrAnAlgebra :=
Submodule.span ℂ {a | ∃ φs, a = ofCrAnList φs ∧ (𝓕 |>ₛ φs) = f}
lemma ofCrAnList_mem_statisticSubmodule_of (φs : List 𝓕.CrAnStates) (f : FieldStatistic)
(h : (𝓕 |>ₛ φs) = f) :
ofCrAnList φs ∈ statisticSubmodule f := by
ofCrAnList φs ∈ statisticSubmodule f := by
refine Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl, h⟩⟩
lemma ofCrAnList_bosonic_or_fermionic (φs : List 𝓕.CrAnStates) :
@ -119,7 +119,7 @@ lemma fermionicProj_ofCrAnList (φs : List 𝓕.CrAnStates) :
lemma fermionicProj_ofCrAnList_if_bosonic (φs : List 𝓕.CrAnStates) :
fermionicProj (ofCrAnList φs) = if h : (𝓕 |>ₛ φs) = bosonic then
0 else ⟨ofCrAnList φs, Submodule.mem_span.mpr fun _ a => a ⟨φs, ⟨rfl,
by simpa using h ⟩⟩⟩ := by
by simpa using h⟩⟩⟩ := by
rw [fermionicProj_ofCrAnList]
by_cases h1 : (𝓕 |>ₛ φs) = fermionic
· simp [h1]
@ -188,7 +188,6 @@ lemma bosonicProj_add_fermionicProj (a : 𝓕.CrAnAlgebra) :
· simp [h]
· simp [h]
lemma coeAddMonoidHom_apply_eq_bosonic_plus_fermionic
(a : DirectSum FieldStatistic (fun i => (statisticSubmodule (𝓕 := 𝓕) i))) :
DirectSum.coeAddMonoidHom statisticSubmodule a = a.1 bosonic + a.1 fermionic := by
@ -245,7 +244,7 @@ instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmo
rfl
mul_mem f1 f2 a1 a2 h1 h2 := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
a1 * a2 ∈ statisticSubmodule (f1 + f2)
a1 * a2 ∈ statisticSubmodule (f1 + f2)
change p a2 h2
apply Submodule.span_induction (p := p)
· intro x hx
@ -281,10 +280,10 @@ instance crAnAlgebraGrade : GradedAlgebra (A := 𝓕.CrAnAlgebra) statisticSubmo
simp only [Algebra.mul_smul_comm, p]
exact Submodule.smul_mem _ _ h1
· exact h2
decompose' a := DirectSum.of (fun i => (statisticSubmodule (𝓕 := 𝓕) i)) bosonic (bosonicProj a)
+ DirectSum.of (fun i => (statisticSubmodule (𝓕 := 𝓕) i)) fermionic (fermionicProj a)
decompose' a := DirectSum.of (fun i => (statisticSubmodule (𝓕 := 𝓕) i)) bosonic (bosonicProj a)
+ DirectSum.of (fun i => (statisticSubmodule (𝓕 := 𝓕) i)) fermionic (fermionicProj a)
left_inv a := by
trans a.bosonicProj + fermionicProj a
trans a.bosonicProj + fermionicProj a
· simp
· exact bosonicProj_add_fermionicProj a
right_inv a := by

106
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/SuperCommute.lean
View file

@ -441,38 +441,38 @@ lemma superCommute_ofCrAnList_ofStateList_eq_sum (φs : List 𝓕.CrAnStates) :
FieldStatistic.ofList_cons_eq_mul, mul_comm]
lemma summerCommute_jacobi_ofCrAnList (φs1 φs2 φs3 : List 𝓕.CrAnStates) :
[ofCrAnList φs1, [ofCrAnList φs2, ofCrAnList φs3]ₛca]ₛca =
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3 ) • [ofCrAnList φs3, [ofCrAnList φs1, ofCrAnList φs2]ₛca]ₛca -
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnList φs2, [ofCrAnList φs3, ofCrAnList φs1]ₛca]ₛca) := by
repeat rw [superCommute_ofCrAnList_ofCrAnList]
simp
repeat rw [superCommute_ofCrAnList_ofCrAnList]
simp only [instCommGroup.eq_1, ofList_append_eq_mul, List.append_assoc]
by_cases h1 : (𝓕 |>ₛ φs1) = bosonic <;>
by_cases h2 : (𝓕 |>ₛ φs2) = bosonic <;>
by_cases h3 : (𝓕 |>ₛ φs3) = bosonic
· simp [h1, h2, exchangeSign_bosonic, h3, mul_one, one_smul]
abel
· simp [h1, h2, exchangeSign_bosonic, bosonic_exchangeSign, mul_one, one_smul]
abel
· simp [h1, bosonic_exchangeSign, h3, exchangeSign_bosonic, mul_one, one_smul]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
[ofCrAnList φs1, [ofCrAnList φs2, ofCrAnList φs3]ₛca]ₛca =
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs3) •
(- 𝓢(𝓕 |>ₛ φs2, 𝓕 |>ₛ φs3) • [ofCrAnList φs3, [ofCrAnList φs1, ofCrAnList φs2]ₛca]ₛca -
𝓢(𝓕 |>ₛ φs1, 𝓕 |>ₛ φs2) • [ofCrAnList φs2, [ofCrAnList φs3, ofCrAnList φs1]ₛca]ₛca) := by
repeat rw [superCommute_ofCrAnList_ofCrAnList]
simp
repeat rw [superCommute_ofCrAnList_ofCrAnList]
simp only [instCommGroup.eq_1, ofList_append_eq_mul, List.append_assoc]
by_cases h1 : (𝓕 |>ₛ φs1) = bosonic <;>
by_cases h2 : (𝓕 |>ₛ φs2) = bosonic <;>
by_cases h3 : (𝓕 |>ₛ φs3) = bosonic
· simp [h1, h2, exchangeSign_bosonic, h3, mul_one, one_smul]
abel
· simp [h1, h2, exchangeSign_bosonic, bosonic_exchangeSign, mul_one, one_smul]
abel
· simp [h1, bosonic_exchangeSign, h3, exchangeSign_bosonic, mul_one, one_smul]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
· simp at h1 h2 h3
simp [h1, h2, h3]
abel
/-!
## Interaction with grading.
@ -483,14 +483,14 @@ lemma superCommute_grade {a b : 𝓕.CrAnAlgebra} {f1 f2 : FieldStatistic}
(ha : a ∈ statisticSubmodule f1) (hb : b ∈ statisticSubmodule f2) :
[a, b]ₛca ∈ statisticSubmodule (f1 + f2) := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f2) : Prop :=
[a, a2]ₛca ∈ statisticSubmodule (f1 + f2)
[a, a2]ₛca ∈ statisticSubmodule (f1 + f2)
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
simp [p]
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule f1) : Prop :=
[a2 , ofCrAnList φs]ₛca ∈ statisticSubmodule (f1 + f2)
[a2, ofCrAnList φs]ₛca ∈ statisticSubmodule (f1 + f2)
change p a ha
apply Submodule.span_induction (p := p)
· intro x hx
@ -525,13 +525,13 @@ lemma superCommute_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
[a, a2]ₛca = a * a2 - a2 * a
[a, a2]ₛca = a * a2 - a2 * a
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
change p a ha
apply Submodule.span_induction (p := p)
· intro x hx
@ -554,18 +554,17 @@ lemma superCommute_bosonic_bosonic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommute_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
(ha : a ∈ statisticSubmodule bosonic) (hb : b ∈ statisticSubmodule fermionic) :
[a, b]ₛca = a * b - b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
[a, a2]ₛca = a * a2 - a2 * a
[a, a2]ₛca = a * a2 - a2 * a
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
change p a ha
apply Submodule.span_induction (p := p)
· intro x hx
@ -588,18 +587,17 @@ lemma superCommute_bosonic_fermionic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommute_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule bosonic) : Prop :=
[a, a2]ₛca = a * a2 - a2 * a
[a, a2]ₛca = a * a2 - a2 * a
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs - ofCrAnList φs * a2
change p a ha
apply Submodule.span_induction (p := p)
· intro x hx
@ -622,7 +620,7 @@ lemma superCommute_fermionic_bonsonic {a b : 𝓕.CrAnAlgebra}
simp_all [p, smul_sub]
· exact hb
lemma superCommute_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
lemma superCommute_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
rw [← bosonicProj_add_fermionicProj a]
simp only [map_add, LinearMap.add_apply]
@ -630,7 +628,7 @@ lemma superCommute_bonsonic {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmo
simp only [add_mul, mul_add]
abel
lemma bosonic_superCommute {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
lemma bosonic_superCommute {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
[a, b]ₛca = a * b - b * a := by
rw [← bosonicProj_add_fermionicProj b]
simp only [map_add, LinearMap.add_apply]
@ -638,12 +636,12 @@ lemma bosonic_superCommute {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmod
simp only [add_mul, mul_add]
abel
lemma superCommute_bonsonic_symm {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
lemma superCommute_bonsonic_symm {a b : 𝓕.CrAnAlgebra} (hb : b ∈ statisticSubmodule bosonic) :
[a, b]ₛca = - [b, a]ₛca := by
rw [bosonic_superCommute hb, superCommute_bonsonic hb]
simp
lemma bonsonic_superCommute_symm {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
lemma bonsonic_superCommute_symm {a b : 𝓕.CrAnAlgebra} (ha : a ∈ statisticSubmodule bosonic) :
[a, b]ₛca = - [b, a]ₛca := by
rw [bosonic_superCommute ha, superCommute_bonsonic ha]
simp
@ -652,13 +650,13 @@ lemma superCommute_fermionic_fermionic {a b : 𝓕.CrAnAlgebra}
(ha : a ∈ statisticSubmodule fermionic) (hb : b ∈ statisticSubmodule fermionic) :
[a, b]ₛca = a * b + b * a := by
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
[a, a2]ₛca = a * a2 + a2 * a
[a, a2]ₛca = a * a2 + a2 * a
change p b hb
apply Submodule.span_induction (p := p)
· intro x hx
obtain ⟨φs, rfl, hφs⟩ := hx
let p (a2 : 𝓕.CrAnAlgebra) (hx : a2 ∈ statisticSubmodule fermionic) : Prop :=
[a2 , ofCrAnList φs]ₛca = a2 * ofCrAnList φs + ofCrAnList φs * a2
[a2, ofCrAnList φs]ₛca = a2 * ofCrAnList φs + ofCrAnList φs * a2
change p a ha
apply Submodule.span_induction (p := p)
· intro x hx
@ -701,7 +699,7 @@ lemma superCommute_expand_bosonicProj_fermionicProj (a b : 𝓕.CrAnAlgebra) :
abel
lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List 𝓕.CrAnStates) :
[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic := by
by_cases h1 : (𝓕 |>ₛ φs) = bosonic <;> by_cases h2 : (𝓕 |>ₛ φs') = bosonic
· left
@ -738,7 +736,7 @@ lemma superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic (φs φs' : List
apply ofCrAnList_mem_statisticSubmodule_of _ _ (by simpa using h2)
lemma superCommute_ofCrAnState_ofCrAnState_bosonic_or_fermionic (φ φ' : 𝓕.CrAnStates) :
[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule bosonic ∨
[ofCrAnState φ, ofCrAnState φ']ₛca ∈ statisticSubmodule fermionic := by
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
exact superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [φ']
@ -779,7 +777,7 @@ lemma superCommute_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
ofCrAnList (φs.drop (n + 1)) := by
let p (a : 𝓕.CrAnAlgebra) (ha : a ∈ statisticSubmodule bosonic) : Prop :=
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length),
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
ofCrAnList (φs.drop (n + 1))
change p a ha
@ -802,14 +800,13 @@ lemma superCommute_bosonic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List
simp_all [p, Finset.smul_sum]
· exact ha
lemma superCommute_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : List 𝓕.CrAnStates)
(ha : a ∈ statisticSubmodule fermionic) :
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
ofCrAnList (φs.drop (n + 1)) := by
let p (a : 𝓕.CrAnAlgebra) (ha : a ∈ statisticSubmodule fermionic) : Prop :=
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
[a, ofCrAnList φs]ₛca = ∑ (n : Fin φs.length), 𝓢(fermionic, 𝓕 |>ₛ φs.take n) •
ofCrAnList (φs.take n) * [a, ofCrAnState (φs.get n)]ₛca *
ofCrAnList (φs.drop (n + 1))
change p a ha
@ -835,7 +832,6 @@ lemma superCommute_fermionic_ofCrAnList_eq_sum (a : 𝓕.CrAnAlgebra) (φs : Lis
simp [smul_smul, mul_comm]
· exact ha
lemma statistic_neq_of_superCommute_fermionic {φs φs' : List 𝓕.CrAnStates}
(h : [ofCrAnList φs, ofCrAnList φs']ₛca ∈ statisticSubmodule fermionic) :
(𝓕 |>ₛ φs) ≠ (𝓕 |>ₛ φs') ∨ [ofCrAnList φs, ofCrAnList φs']ₛca = 0 := by

14
HepLean/PerturbationTheory/Algebras/CrAnAlgebra/TimeOrder.lean
View file

@ -66,7 +66,7 @@ lemma timeOrder_timeOrder_mid (a b c : 𝓕.CrAnAlgebra) : 𝓣ᶠ(a * b * c) =
Algebra.smul_mul_assoc, map_smul]
rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList, smul_smul]
congr 1
· simp only [crAnTimeOrderSign, crAnTimeOrderList]
· simp only [crAnTimeOrderSign, crAnTimeOrderList]
rw [Wick.koszulSign_of_append_eq_insertionSort, mul_comm]
· congr 1
simp only [crAnTimeOrderList]
@ -167,7 +167,6 @@ lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
simp
lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca * a) = 0 := by
@ -183,7 +182,7 @@ lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
simp
lemma timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel
{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.CrAnAlgebra):
{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.CrAnAlgebra) :
𝓣ᶠ([a, [ofCrAnState φ1, ofCrAnState φ2]ₛca]ₛca) = 0 := by
rw [← bosonicProj_add_fermionicProj a]
simp
@ -213,7 +212,7 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel
rw [summerCommute_jacobi_ofCrAnList]
simp [ofCrAnList_singleton]
right
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12 ]
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12]
rw [superCommute_ofCrAnState_ofCrAnState_symm φ3]
simp
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
@ -229,14 +228,14 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel'
right
rw [superCommute_ofCrAnState_ofCrAnState_symm φ1]
simp
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12 ]
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12]
simp
rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
simp
lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
(φ1 φ2 φ3 : 𝓕.CrAnStates) (h : ¬ (
crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
(φ1 φ2 φ3 : 𝓕.CrAnStates) (h : ¬
(crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2)) :
𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
@ -272,7 +271,6 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
refine False.elim (h ?_)
exact IsTrans.trans φ3 φ2 φ1 h32 h21
lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_eq_time
{φ ψ : 𝓕.CrAnStates} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = [ofCrAnState φ, ofCrAnState ψ]ₛca := by

11
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/Basic.lean
View file

@ -112,8 +112,8 @@ lemma ι_superCommute_zero_of_fermionic (φ ψ : 𝓕.CrAnStates)
· simp [h]
lemma ι_superCommute_ofCrAnState_ofCrAnState_bosonic_or_zero (φ ψ : 𝓕.CrAnStates) :
[ofCrAnState φ, ofCrAnState ψ]ₛca ∈ statisticSubmodule bosonic ∨
ι [ofCrAnState φ, ofCrAnState ψ]ₛca = 0 := by
[ofCrAnState φ, ofCrAnState ψ]ₛca ∈ statisticSubmodule bosonic ∨
ι [ofCrAnState φ, ofCrAnState ψ]ₛca = 0 := by
rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ] [ψ] with h | h
· simp_all [ofCrAnList_singleton]
· simp_all [ofCrAnList_singleton]
@ -236,7 +236,6 @@ lemma bosonicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈
· right
rw [bosonicProj_of_mem_fermionic _ h]
lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x ∈ 𝓕.fieldOpIdealSet) :
x.fermionicProj.1 ∈ 𝓕.fieldOpIdealSet ∨ x.fermionicProj = 0 := by
have hx' := hx
@ -269,7 +268,7 @@ lemma fermionicProj_mem_fieldOpIdealSet_or_zero (x : CrAnAlgebra 𝓕) (hx : x
simpa using hx'
lemma bosonicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
x.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
x.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
rw [TwoSidedIdeal.mem_span_iff_mem_addSubgroup_closure] at hx
let p {k : Set 𝓕.CrAnAlgebra} (a : CrAnAlgebra 𝓕) (h : a ∈ AddSubgroup.closure k) : Prop :=
a.bosonicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet
@ -405,7 +404,7 @@ lemma bosonicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.spa
simp [p]
lemma fermionicProj_mem_ideal (x : CrAnAlgebra 𝓕) (hx : x ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet) :
x.fermionicProj.1 ∈ TwoSidedIdeal.span 𝓕.fieldOpIdealSet := by
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 ⊢
rw [← bosonicProj_add_fermionicProj x] at hx
@ -425,7 +424,5 @@ lemma ι_eq_zero_iff_ι_bosonicProj_fermonicProj_zero (x : CrAnAlgebra 𝓕) :
rw [← bosonicProj_add_fermionicProj x]
simp_all
end FieldOpAlgebra
end FieldSpecification

2
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/SuperCommute.lean
View file

@ -80,7 +80,7 @@ lemma superCommuteRight_eq_of_equiv (a1 a2 : 𝓕.CrAnAlgebra) (h : a1 ≈ a2) :
apply ι_superCommute_eq_zero_of_ι_left_zero
exact (ι_eq_zero_iff_mem_ideal (a1 - a2)).mpr h
simp_all [superCommuteRight_apply_ι]
trans ι ((superCommute a2) b) + 0
trans ι ((superCommute a2) b) + 0
rw [← ha1b1]
simp
simp

24
HepLean/PerturbationTheory/Algebras/FieldOpAlgebra/TimeOrder.lean
View file

@ -20,7 +20,7 @@ namespace FieldOpAlgebra
variable {𝓕 : FieldSpecification}
lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
(φs1 φs2 : List 𝓕.CrAnStates) (h :
(φs1 φs2 : List 𝓕.CrAnStates) (h :
crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
crAnTimeOrderRel φ3 φ1 ∧ crAnTimeOrderRel φ3 φ2):
@ -115,12 +115,12 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
rw [← smul_sub, ← smul_sub, smul_smul, mul_comm, ← smul_smul, ← smul_sub]
simp
right
rw [← smul_mul_assoc, ← mul_smul_comm, mul_assoc]
rw [← smul_mul_assoc, ← mul_smul_comm, mul_assoc]
rw [← smul_mul_assoc, ← mul_smul_comm]
rw [← smul_mul_assoc, ← mul_smul_comm]
rw [smul_sub]
rw [← smul_mul_assoc, ← mul_smul_comm]
rw [← smul_mul_assoc, ← mul_smul_comm]
rw [← smul_mul_assoc, ← mul_smul_comm]
rw [← smul_mul_assoc, ← mul_smul_comm]
repeat rw [mul_assoc]
rw [← mul_sub, ← mul_sub, ← mul_sub]
rw [← sub_mul, ← sub_mul, ← sub_mul]
@ -181,7 +181,7 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b)) := by
let pb (b : 𝓕.CrAnAlgebra) (hc : b ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) =
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a * b))
change pb b (Basis.mem_span _ b)
apply Submodule.span_induction
@ -201,7 +201,7 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
simp [mul_sub, sub_mul, ← ofCrAnList_append]
rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList]
have h1 : crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) = crAnTimeOrderSign (φs' ++ ψ :: φ :: φs) := by
trans crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs)
trans crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs)
simp
rw [crAnTimeOrderSign]
have hp : List.Perm [φ,ψ] [ψ,φ] := by exact List.Perm.swap ψ φ []
@ -244,7 +244,7 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
rw [← map_mul, ← map_mul, ← map_mul, ← map_mul]
rw [← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append]
have h1 := insertionSort_of_takeWhile_filter 𝓕.crAnTimeOrderRel φ φs' φs
simp at h1 ⊢
simp at h1 ⊢
rw [← h1]
rw [← crAnTimeOrderList]
by_cases hq : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)
@ -269,7 +269,7 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
lemma ι_timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
(hφψ : ¬ (crAnTimeOrderRel φ ψ ∧ crAnTimeOrderRel ψ φ)) (a b : 𝓕.CrAnAlgebra) :
ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
rw [timeOrder_timeOrder_mid]
have hφψ : ¬ (crAnTimeOrderRel φ ψ) ∨ ¬ (crAnTimeOrderRel ψ φ) := by
@ -309,7 +309,7 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
simp
| Or.inr (Or.inl hc) =>
obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
· rw [ι_timeOrder_superCommute_eq_time]
simp
rw [ι_superCommute_of_create_create]
@ -321,7 +321,7 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
· rw [ι_timeOrder_superCommute_neq_time heqt]
| Or.inr (Or.inr (Or.inl hc)) =>
obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
· rw [ι_timeOrder_superCommute_eq_time]
simp
rw [ι_superCommute_of_annihilate_annihilate]
@ -333,7 +333,7 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
· rw [ι_timeOrder_superCommute_neq_time heqt]
| Or.inr (Or.inr (Or.inr hc)) =>
obtain ⟨φa, φb, hdiff, rfl⟩ := hc
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
· rw [ι_timeOrder_superCommute_eq_time]
simp
rw [ι_superCommute_of_diff_statistic]

2
HepLean/PerturbationTheory/FieldStatistics/ExchangeSign.lean
View file

@ -72,7 +72,7 @@ lemma bosonic_exchangeSign (a : FieldStatistic) : 𝓢(bosonic, a) = 1 := by
rw [exchangeSign_symm, exchangeSign_bosonic]
@[simp]
lemma fermionic_exchangeSign_fermionic : 𝓢(fermionic, fermionic) = - 1 := by
lemma fermionic_exchangeSign_fermionic : 𝓢(fermionic, fermionic) = - 1 := by
rfl
lemma exchangeSign_eq_if (a b : FieldStatistic) :

31
HepLean/PerturbationTheory/Koszul/KoszulSign.lean
View file

@ -261,7 +261,7 @@ lemma koszulSign_eraseIdx_insertionSortMinPos [IsTotal 𝓕 le] [IsTrans 𝓕 le
rfl
lemma koszulSign_swap_eq_rel_cons {ψ φ : 𝓕}
(h1 : le φ ψ) (h2 : le ψ φ) (φs' : List 𝓕):
(h1 : le φ ψ) (h2 : le ψ φ) (φs' : List 𝓕) :
koszulSign q le (φ :: ψ :: φs') = koszulSign q le (ψ :: φ :: φs') := by
simp only [Wick.koszulSign, ← mul_assoc, mul_eq_mul_right_iff]
left
@ -285,7 +285,7 @@ lemma koszulSign_eq_rel_eq_stat_append {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal
koszulSign q le (φ :: ψ :: φs) = koszulSign q le φs := by
intro φs
simp [koszulSign, ← mul_assoc]
trans 1 * koszulSign q le φs
trans 1 * koszulSign q le φs
swap
simp
congr
@ -305,11 +305,11 @@ lemma koszulSign_eq_rel_eq_stat {ψ φ : 𝓕} [IsTrans 𝓕 le] [IsTotal 𝓕 l
rw [koszulSign_eq_rel_eq_stat h1 h2 hq φs' φs]
simp
left
trans koszulSignInsert q le φ'' (φ :: ψ :: (φs' ++ φs) )
trans koszulSignInsert q le φ'' (φ :: ψ :: (φs' ++ φs))
apply koszulSignInsert_eq_perm
refine List.Perm.symm (List.perm_cons_append_cons φ ?_)
exact List.Perm.symm List.perm_middle
rw [koszulSignInsert_eq_remove_same_stat_append q le ]
rw [koszulSignInsert_eq_remove_same_stat_append q le]
simp_all
simp_all
simp_all
@ -331,15 +331,16 @@ lemma koszulSign_of_insertionSort [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φs : Lis
apply koszulSign_of_sorted
exact List.sorted_insertionSort le φs
lemma koszulSign_of_append_eq_insertionSort_left [IsTotal 𝓕 le] [IsTrans 𝓕 le] : (φs φs' : List 𝓕) →
koszulSign q le (φs ++ φs') =
koszulSign q le (List.insertionSort le φs ++ φs') * koszulSign q le φs
lemma koszulSign_of_append_eq_insertionSort_left [IsTotal 𝓕 le] [IsTrans 𝓕 le] :
(φs φs' : List 𝓕) → koszulSign q le (φs ++ φs') =
koszulSign q le (List.insertionSort le φs ++ φs') * koszulSign q le φs
| φs, [] => by
simp
| φs, φ :: φs' => by
have h1 : (φs ++ φ :: φs') = List.insertIdx φs.length φ (φs ++ φs') := by
rw [insertIdx_length_fst_append]
have h2 : (List.insertionSort le φs ++ φ :: φs') = List.insertIdx (List.insertionSort le φs).length φ (List.insertionSort le φs ++ φs') := by
have h2 : (List.insertionSort le φs ++ φ :: φs') =
List.insertIdx (List.insertionSort le φs).length φ (List.insertionSort le φs ++ φs') := by
rw [insertIdx_length_fst_append]
rw [h1, h2]
rw [koszulSign_insertIdx]
@ -353,7 +354,8 @@ lemma koszulSign_of_append_eq_insertionSort_left [IsTotal 𝓕 le] [IsTrans 𝓕
simp [mul_comm]
left
congr 3
· have h2 : (List.insertionSort le φs ++ φ :: φs') = List.insertIdx φs.length φ (List.insertionSort le φs ++ φs') := by
· have h2 : (List.insertionSort le φs ++ φ :: φs') =
List.insertIdx φs.length φ (List.insertionSort le φs ++ φs') := by
rw [← insertIdx_length_fst_append]
simp
rw [insertionSortEquiv_congr _ _ h2.symm]
@ -363,16 +365,16 @@ lemma koszulSign_of_append_eq_insertionSort_left [IsTotal 𝓕 le] [IsTrans 𝓕
rw [insertionSortEquiv_congr _ _ h1.symm]
simp
· rw [insertIdx_length_fst_append]
rw [show φs.length = (List.insertionSort le φs).length by simp]
rw [show φs.length = (List.insertionSort le φs).length by simp]
rw [insertIdx_length_fst_append]
symm
apply insertionSort_insertionSort_append
· simp
· simp
lemma koszulSign_of_append_eq_insertionSort [IsTotal 𝓕 le] [IsTrans 𝓕 le] : (φs'' φs φs' : List 𝓕) →
lemma koszulSign_of_append_eq_insertionSort [IsTotal 𝓕 le] [IsTrans 𝓕 le] : (φs'' φs φs' : List 𝓕) →
koszulSign q le (φs'' ++ φs ++ φs') =
koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') * koszulSign q le φs
koszulSign q le (φs'' ++ List.insertionSort le φs ++ φs') * koszulSign q le φs
| [], φs, φs'=> by
simp
exact koszulSign_of_append_eq_insertionSort_left q le φs φs'
@ -391,10 +393,10 @@ lemma koszulSign_of_append_eq_insertionSort [IsTotal 𝓕 le] [IsTrans 𝓕 le]
-/
lemma koszulSign_perm_eq_append [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) ( φs φs' φs2 : List 𝓕)
lemma koszulSign_perm_eq_append [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) (φs φs' φs2 : List 𝓕)
(hp : φs.Perm φs') : (h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) →
koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2) := by
let motive (φs φs' : List 𝓕) (hp : φs.Perm φs') : Prop :=
let motive (φs φs' : List 𝓕) (hp : φs.Perm φs') : Prop :=
(h : ∀ φ' ∈ φs, le φ φ' ∧ le φ' φ) →
koszulSign q le (φs ++ φs2) = koszulSign q le (φs' ++ φs2)
change motive φs φs' hp
@ -433,5 +435,4 @@ lemma koszulSign_perm_eq [IsTotal 𝓕 le] [IsTrans 𝓕 le] (φ : 𝓕) : (φs1
refine (List.perm_append_right_iff φs2).mpr ?_
exact List.Perm.append_left φs1 hp
end Wick

7
HepLean/PerturbationTheory/Koszul/KoszulSignInsert.lean
View file

@ -235,7 +235,7 @@ lemma koszulSignInsert_cons (r0 r1 : 𝓕) (r : List 𝓕) :
koszulSignInsert q le r0 r := by
simp [koszulSignInsert, koszulSignCons]
lemma koszulSignInsert_of_le_mem (φ0 : 𝓕) : (φs : List 𝓕) → (h : ∀ b ∈ φs, le φ0 b) →
lemma koszulSignInsert_of_le_mem (φ0 : 𝓕) : (φs : List 𝓕) → (h : ∀ b ∈ φs, le φ0 b) →
koszulSignInsert q le φ0 φs = 1
| [], _ => by
simp [koszulSignInsert]
@ -247,7 +247,6 @@ lemma koszulSignInsert_of_le_mem (φ0 : 𝓕) : (φs : List 𝓕) → (h : ∀
exact h b (List.mem_cons_of_mem _ hb)
· exact h φ1 (List.mem_cons_self _ _)
lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTotal 𝓕 le] [IsTrans 𝓕 le]
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
koszulSignInsert q le φ φs = koszulSignInsert q le ψ φs
@ -270,7 +269,7 @@ lemma koszulSignInsert_eq_rel_eq_stat {ψ φ : 𝓕} [IsTotal 𝓕 le] [IsTrans
rw [koszulSignInsert_eq_rel_eq_stat h1 h2 hq φs]
lemma koszulSignInsert_eq_remove_same_stat_append {ψ φ φ' : 𝓕} [IsTotal 𝓕 le] [IsTrans 𝓕 le]
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : ( φs : List 𝓕) →
(h1 : le φ ψ) (h2 : le ψ φ) (hq : q ψ = q φ) : (φs : List 𝓕) →
koszulSignInsert q le φ' (φ :: ψ :: φs) = koszulSignInsert q le φ' φs := by
intro φs
simp_all [koszulSignInsert]
@ -284,6 +283,4 @@ lemma koszulSignInsert_eq_remove_same_stat_append {ψ φ φ' : 𝓕} [IsTotal
apply IsTrans.trans φ' ψ φ hφ'ψ h2
simp_all [hφ'φ, hφ'ψ]
end Wick