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refactor: Text based Lint

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jstoobysmith 2024-10-29 11:23:08 +00:00
parent 319089ad54
commit 7010a1dae2
12 changed files with 54 additions and 52 deletions
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5
HepLean/Tensors/ComplexLorentz/Basic.lean
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@ -209,10 +209,10 @@ lemma basis_contr (c : complexLorentzTensor.C) (i : Fin (complexLorentzTensor.re
| Color.up => Lorentz.contrCoContraction_basis _ _
| Color.down => Lorentz.coContrContraction_basis _ _
instance {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
instance {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
DecidableEq (OverColor.mk c).left := instDecidableEqFin n
instance {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
instance {n : ℕ} {c : Fin n → complexLorentzTensor.C} :
Fintype (OverColor.mk c).left := Fin.fintype n
instance {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
@ -220,6 +220,5 @@ instance {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
Decidable (σ = σ') :=
decidable_of_iff _ (OverColor.Hom.ext_iff σ σ')
end complexLorentzTensor
end

34
HepLean/Tensors/ComplexLorentz/Bispinors/Basic.lean
View file

@ -76,7 +76,7 @@ lemma tensorNode_contrBispinorDown (p : complexContr) :
rw [contrBispinorDown, tensorNode_tensor]
/-- The definitional tensor node relation for `coBispinorUp`. -/
lemma tensorNode_coBispinorUp (p : complexCo) :
lemma tensorNode_coBispinorUp (p : complexCo) :
{coBispinorUp p | α β}ᵀ.tensor = {pauliContr | μ α β ⊗ p | μ}ᵀ.tensor := by
rw [coBispinorUp, tensorNode_tensor]
@ -94,23 +94,26 @@ lemma tensorNode_coBispinorDown (p : complexCo) :
-/
lemma contrBispinorDown_expand (p : complexContr) :
{contrBispinorDown p | α β}ᵀ.tensor = {Fermion.altLeftMetric | α α' ⊗ Fermion.altRightMetric | β β' ⊗
{contrBispinorDown p | α β}ᵀ.tensor =
{Fermion.altLeftMetric | α α' ⊗ Fermion.altRightMetric | β β' ⊗
(pauliCo | μ α β ⊗ p | μ)}ᵀ.tensor := by
rw [tensorNode_contrBispinorDown p]
rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_contrBispinorUp p]
lemma coBispinorDown_expand (p : complexCo) :
{coBispinorDown p | α β}ᵀ.tensor = {Fermion.altLeftMetric | α α' ⊗ Fermion.altRightMetric | β β' ⊗
{coBispinorDown p | α β}ᵀ.tensor =
{Fermion.altLeftMetric | α α' ⊗ Fermion.altRightMetric | β β' ⊗
(pauliContr | μ α β ⊗ p | μ)}ᵀ.tensor := by
rw [tensorNode_coBispinorDown p]
rw [contr_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_snd <| tensorNode_coBispinorUp p]
set_option maxRecDepth 5000 in
lemma contrBispinorDown_eq_pauliCoDown_contr (p : complexContr) :
{contrBispinorDown p | α β = pauliCoDown | μ α β ⊗ p | μ}ᵀ := by
{contrBispinorDown p | α β = pauliCoDown | μ α β ⊗ p | μ}ᵀ := by
conv =>
rhs
rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_fst <| pauliCoDown_eq_metric_mul_pauliCo]
rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_fst <|
pauliCoDown_eq_metric_mul_pauliCo]
rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_left _ _ _ _]
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
@ -118,12 +121,14 @@ lemma contrBispinorDown_eq_pauliCoDown_contr (p : complexContr) :
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
apply (perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _).trans
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_assoc' _ _ _ _ _ _]
erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <|
perm_eq_id _ rfl _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <|
prod_assoc' _ _ _ _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_contr_congr 0 4]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
rw [perm_tensor_eq <| perm_contr_congr 2 2]
@ -144,10 +149,11 @@ lemma contrBispinorDown_eq_pauliCoDown_contr (p : complexContr) :
set_option maxRecDepth 5000 in
lemma coBispinorDown_eq_pauliContrDown_contr (p : complexCo) :
{coBispinorDown p | α β = pauliContrDown | μ α β ⊗ p | μ}ᵀ := by
{coBispinorDown p | α β = pauliContrDown | μ α β ⊗ p | μ}ᵀs := by
conv =>
rhs
rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_fst <| pauliContrDown_eq_metric_mul_pauliContr]
rw [perm_tensor_eq <| contr_tensor_eq <| prod_tensor_eq_fst <|
pauliContrDown_eq_metric_mul_pauliContr]
rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_left _ _ _ _]
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
@ -155,12 +161,14 @@ lemma coBispinorDown_eq_pauliContrDown_contr (p : complexCo) :
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
apply (perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _).trans
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_prod _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
rw [perm_tensor_eq <| perm_contr_congr 2 2]
rw [perm_perm]
erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_eq_id _ rfl _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_assoc' _ _ _ _ _ _]
erw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <|
perm_eq_id _ rfl _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <|
prod_assoc' _ _ _ _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| perm_contr_congr 0 4]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr_congr 1 3]
rw [perm_tensor_eq <| perm_contr_congr 2 2]

8
HepLean/Tensors/ComplexLorentz/PauliMatrices/Basic.lean
View file

@ -63,7 +63,7 @@ lemma tensorNode_pauliContr : {pauliContr | μ α β}ᵀ.tensor =
rfl
/-- The definitional tensor node relation for `pauliCo`. -/
lemma tensorNode_pauliCo : {pauliCo | μ α β}ᵀ.tensor =
lemma tensorNode_pauliCo : {pauliCo | μ α β}ᵀ.tensor =
{Lorentz.coMetric | μ ν ⊗ PauliMatrix.asConsTensor | ν α β}ᵀ.tensor := by
rfl
@ -111,7 +111,7 @@ lemma pauliCoDown_eq_metric_mul_pauliCo :
rw [perm_perm]
rw [perm_tensor_eq <| contr_congr 1 2]
rw [perm_perm]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_comm _ _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_comm _ _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]
@ -138,7 +138,7 @@ lemma pauliCoDown_eq_metric_mul_pauliCo :
set_option maxRecDepth 5000 in
/-- A rearanging of `pauliContrDown` to place the pauli matrices on the right. -/
lemma pauliContrDown_eq_metric_mul_pauliContr :
{pauliContrDown | μ α' β' = Fermion.altLeftMetric | α α' ⊗
{pauliContrDown | μ α' β' = Fermion.altLeftMetric | α α' ⊗
Fermion.altRightMetric | β β' ⊗ pauliContr | μ α β}ᵀ := by
conv =>
lhs
@ -161,7 +161,7 @@ lemma pauliContrDown_eq_metric_mul_pauliContr :
rw [perm_perm]
rw [perm_tensor_eq <| contr_congr 1 2]
rw [perm_perm]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_comm _ _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| contr_tensor_eq <| prod_comm _ _ _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_contr _ _]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_tensor_eq <| contr_congr 5 0]
rw [perm_tensor_eq <| contr_tensor_eq <| perm_perm _ _ _]

4
HepLean/Tensors/ComplexLorentz/PauliMatrices/Basis.lean
View file

@ -26,7 +26,6 @@ noncomputable section
namespace complexLorentzTensor
open Fermion
/-!
## Expanding pauliContr in a basis.
@ -94,7 +93,6 @@ lemma pauliContr_basis_expand_tree : {pauliContr | μ α β}ᵀ.tensor =
smul_tensor, neg_smul, one_smul]
rfl
/-- The map to colors one gets when contracting with Pauli matrices on the right. -/
abbrev pauliMatrixContrMap {n : ℕ} (c : Fin n → complexLorentzTensor.C) :=
(Sum.elim c ![Color.up, Color.upL, Color.upR] ∘ ⇑finSumFinEquiv.symm)
@ -325,14 +323,12 @@ lemma basis_contr_pauliMatrix_basis_tree_expand_tensor {n : ℕ} {c : Fin n →
simp_all only [Function.comp_apply, Nat.succ_eq_add_one, Nat.reduceAdd, Fin.isValue]
rfl
/-!
## Expanding pauliCo in a basis.
-/
/-- The map to color one gets when lowering the indices of pauli matrices. -/
def pauliCoMap := ((Sum.elim ![Color.down, Color.down] ![Color.up, Color.upL, Color.upR] ∘
⇑finSumFinEquiv.symm) ∘ Fin.succAbove 1 ∘ Fin.succAbove 1)

10
HepLean/Tensors/Tree/NodeIdentities/Basic.lean
View file

@ -116,15 +116,16 @@ lemma perm_eq_of_eq_perm {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
simp only [Iso.map_hom_inv_id, Action.id_hom, ModuleCat.id_apply]
lemma perm_eq_iff_eq_perm {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
(σ : (OverColor.mk c) ⟶ (OverColor.mk c1))
(σ : (OverColor.mk c) ⟶ (OverColor.mk c1))
{t : TensorTree S c} {t2 : TensorTree S c1} :
(perm σ t).tensor = t2.tensor ↔ t.tensor =
(perm (equivToHomEq (Hom.toEquiv σ).symm (fun x => Hom.toEquiv_comp_apply σ x)) t2).tensor := by
refine Iff.intro (fun h => ?_) (fun h => ?_)
refine Iff.intro (fun h => ?_) (fun h => ?_)
· simp [perm_tensor, ← h]
change _ = (S.F.map _ ≫ S.F.map _).hom _
rw [← S.F.map_comp]
have h1 : (σ ≫ equivToHomEq (Hom.toEquiv σ).symm (fun x => Hom.toEquiv_comp_apply σ x)) = 𝟙 _ := by
have h1 : (σ ≫ equivToHomEq (Hom.toEquiv σ).symm
(fun x => Hom.toEquiv_comp_apply σ x)) = 𝟙 _ := by
apply Hom.ext
ext x
change (Hom.toEquiv σ).symm ((Hom.toEquiv σ) x) = x
@ -134,7 +135,8 @@ lemma perm_eq_iff_eq_perm {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
· rw [perm_tensor, h]
change (S.F.map _ ≫ S.F.map _).hom _ = _
rw [← S.F.map_comp]
have h1 : (equivToHomEq (Hom.toEquiv σ).symm (fun x => Hom.toEquiv_comp_apply σ x) ≫ σ) = 𝟙 _ := by
have h1 : (equivToHomEq (Hom.toEquiv σ).symm
(fun x => Hom.toEquiv_comp_apply σ x) ≫ σ) = 𝟙 _ := by
apply Hom.ext
ext x
change (Hom.toEquiv σ) ((Hom.toEquiv σ).symm x) = x

11
HepLean/Tensors/Tree/NodeIdentities/Congr.lean
View file

@ -30,21 +30,20 @@ variable {n m : ℕ}
lemma perm_congr {c1 : Fin n → S.C} {c2 : Fin m → S.C} {T T' : TensorTree S c1}
{σ σ' : OverColor.mk c1 ⟶ OverColor.mk c2}
(h : σ = σ') (hT : T.tensor = T'.tensor):
(h : σ = σ') (hT : T.tensor = T'.tensor) :
(perm σ T).tensor = (perm σ' T').tensor := by
rw [h]
simp only [perm_tensor, hT]
lemma perm_update {c1 : Fin n → S.C} {c2 : Fin m → S.C} {T : TensorTree S c1}
lemma perm_update {c1 : Fin n → S.C} {c2 : Fin m → S.C} {T : TensorTree S c1}
{σ : OverColor.mk c1 ⟶ OverColor.mk c2} (σ' : OverColor.mk c1 ⟶ OverColor.mk c2)
(h : σ = σ') :
(perm σ T).tensor = (perm σ' T).tensor := by rw [h]
lemma contr_congr {n : ℕ} {c : Fin n.succ.succ → S.C} {i : Fin n.succ.succ}
(i' : Fin n.succ.succ) {j : Fin n.succ} (j' : Fin n.succ)
{h : c (i.succAbove j) = S.τ (c i)} {t : TensorTree S c}
(hi : i = i' := by decide) (hj : j = j' := by decide)
:(contr i j h t).tensor =
(i' : Fin n.succ.succ) {j : Fin n.succ} (j' : Fin n.succ){h : c (i.succAbove j) = S.τ (c i)}
{t : TensorTree S c} (hi : i = i' := by decide) (hj : j = j' := by decide) :
(contr i j h t).tensor =
(perm (mkIso (by rw [hi, hj])).hom (contr i' j' (by rw [← hi, ← hj, h]) t)).tensor := by
subst hi
subst hj

14
HepLean/Tensors/Tree/NodeIdentities/ContrContr.lean
View file

@ -285,15 +285,13 @@ def contrContrPerm {n : ℕ} {c : Fin n.succ.succ.succ.succ → S.C} {i : Fin n.
{j : Fin n.succ.succ.succ} {k : Fin n.succ.succ} {l : Fin n.succ}
(hij : c (i.succAbove j) = S.τ (c i)) (hkl : (c ∘ i.succAbove ∘ j.succAbove) (k.succAbove l) =
S.τ ((c ∘ i.succAbove ∘ j.succAbove) k)) :
OverColor.mk
((c ∘
(ContrQuartet.mk i j k l hij hkl).swapI.succAbove ∘
(ContrQuartet.mk i j k l hij hkl).swapJ.succAbove) ∘
OverColor.mk ((c ∘ (ContrQuartet.mk i j k l hij hkl).swapI.succAbove ∘
(ContrQuartet.mk i j k l hij hkl).swapJ.succAbove) ∘
(ContrQuartet.mk i j k l hij hkl).swapK.succAbove ∘
(ContrQuartet.mk i j k l hij hkl).swapL.succAbove) ⟶
OverColor.mk
((c ∘ i.succAbove ∘ j.succAbove) ∘ k.succAbove ∘ l.succAbove)
:= (ContrQuartet.mk i j k l hij hkl).contrSwapHom
(ContrQuartet.mk i j k l hij hkl).swapL.succAbove) ⟶
OverColor.mk
((c ∘ i.succAbove ∘ j.succAbove) ∘ k.succAbove ∘ l.succAbove) :=
(ContrQuartet.mk i j k l hij hkl).contrSwapHom
/-- Contraction nodes commute on adjusting indices. -/
theorem contr_contr {n : ℕ} {c : Fin n.succ.succ.succ.succ → S.C} {i : Fin n.succ.succ.succ.succ}

4
HepLean/Tensors/Tree/NodeIdentities/PermContr.lean
View file

@ -264,8 +264,8 @@ lemma perm_contr_congr_mkIso_cond {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 :
{i' : Fin n.succ.succ} {j' : Fin n.succ}
(hi : i' = ((Hom.toEquiv σ).symm i))
(hj : j' = (((Hom.toEquiv (extractOne i σ))).symm j)) :
c ∘ i'.succAbove ∘ j'.succAbove =
c ∘ Fin.succAbove ((Hom.toEquiv σ).symm i) ∘ Fin.succAbove ((Hom.toEquiv (extractOne i σ)).symm j) := by
c ∘ i'.succAbove ∘ j'.succAbove = c ∘ Fin.succAbove ((Hom.toEquiv σ).symm i) ∘
Fin.succAbove ((Hom.toEquiv (extractOne i σ)).symm j) := by
rw [hi, hj]
lemma perm_contr_congr_contr_cond {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ → S.C}