feat: Lots of stuff about koszul contract
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jstoobysmith
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4 changed files with 1618 additions and 105 deletions
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@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.PerturbationTheory.Wick.Species
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import HepLean.PerturbationTheory.Wick.Koszul
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import HepLean.Lorentz.RealVector.Basic
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import HepLean.Mathematics.Fin
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import HepLean.SpaceTime.Basic
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@ -123,106 +124,6 @@ and $K(σ)$ is the Koszul sign factor. (see e.g. PSE:24157)
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There are two types $I$ we are intrested in.
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"
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/-- Gives a factor of `-1` when inserting `a` into a list `List I` in the ordered position
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for each fermion-fermion cross. -/
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def koszulSignInsert {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2) (a : I) :
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List I → ℂ
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| [] => 1
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| b :: l => if r a b then 1 else
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if q a = 1 ∧ q b = 1 then - koszulSignInsert r q a l else koszulSignInsert r q a l
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/-- When inserting a boson the `koszulSignInsert` is always `1`. -/
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lemma koszulSignInsert_boson {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2) (a : I)
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(ha : q a = 0) : (l : List I) → koszulSignInsert r q a l = 1
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| [] => by
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simp [koszulSignInsert]
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| b :: l => by
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simp only [koszulSignInsert, Fin.isValue, ite_eq_left_iff]
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intro _
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simp only [ha, Fin.isValue, zero_ne_one, false_and, ↓reduceIte]
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exact koszulSignInsert_boson r q a ha l
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/-- Gives a factor of `- 1` for every fermion-fermion (`q` is `1`) crossing that occurs when sorting
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a list of based on `r`. -/
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def koszulSign {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2) :
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List I → ℂ
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| [] => 1
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| a :: l => koszulSignInsert r q a l * koszulSign r q l
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@[simp]
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lemma koszulSign_freeMonoid_of {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2)
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(i : I) : koszulSign r q (FreeMonoid.of i) = 1 := by
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change koszulSign r q [i] = 1
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simp only [koszulSign, mul_one]
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rfl
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noncomputable section
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/-- Given a relation `r` on `I` sorts elements of `MonoidAlgebra ℂ (FreeMonoid I)` by `r` giving it
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a signed dependent on `q`. -/
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def koszulOrderMonoidAlgebra {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2) :
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MonoidAlgebra ℂ (FreeMonoid I) →ₗ[ℂ] MonoidAlgebra ℂ (FreeMonoid I) :=
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Finsupp.lift (MonoidAlgebra ℂ (FreeMonoid I)) ℂ (List I)
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(fun i => Finsupp.lsingle (R := ℂ) (List.insertionSort r i) (koszulSign r q i))
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lemma koszulOrderMonoidAlgebra_ofList {I : Type} (r : I → I → Prop) [DecidableRel r]
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(q : I → Fin 2) (i : List I) :
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koszulOrderMonoidAlgebra r q (MonoidAlgebra.of ℂ (FreeMonoid I) i) =
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(koszulSign r q i) • (MonoidAlgebra.of ℂ (FreeMonoid I) (List.insertionSort r i)) := by
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simp only [koszulOrderMonoidAlgebra, Finsupp.lsingle_apply, MonoidAlgebra.of_apply,
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MonoidAlgebra.smul_single', mul_one]
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rw [MonoidAlgebra.ext_iff]
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intro x
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erw [Finsupp.lift_apply]
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simp only [MonoidAlgebra.smul_single', zero_mul, Finsupp.single_zero, Finsupp.sum_single_index,
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one_mul]
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@[simp]
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lemma koszulOrderMonoidAlgebra_single {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2)
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(l : FreeMonoid I) (x : ℂ) :
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koszulOrderMonoidAlgebra r q (MonoidAlgebra.single l x)
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= (koszulSign r q l) • (MonoidAlgebra.single (List.insertionSort r l) x) := by
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simp only [koszulOrderMonoidAlgebra, Finsupp.lsingle_apply, MonoidAlgebra.smul_single']
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rw [MonoidAlgebra.ext_iff]
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intro x'
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erw [Finsupp.lift_apply]
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simp only [MonoidAlgebra.smul_single', zero_mul, Finsupp.single_zero, Finsupp.sum_single_index,
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one_mul, MonoidAlgebra.single]
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congr 2
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rw [NonUnitalNormedCommRing.mul_comm]
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/-- Given a relation `r` on `I` sorts elements of `FreeAlgebra ℂ I` by `r` giving it
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a signed dependent on `q`. -/
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def koszulOrder {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2) :
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FreeAlgebra ℂ I →ₗ[ℂ] FreeAlgebra ℂ I :=
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FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm.toAlgHom.toLinearMap
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∘ₗ koszulOrderMonoidAlgebra r q
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∘ₗ FreeAlgebra.equivMonoidAlgebraFreeMonoid.toAlgHom.toLinearMap
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lemma koszulOrder_ι {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2) (i : I) :
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koszulOrder r q (FreeAlgebra.ι ℂ i) = FreeAlgebra.ι ℂ i := by
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simp only [koszulOrder, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_toLinearMap,
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koszulOrderMonoidAlgebra, Finsupp.lsingle_apply, LinearMap.coe_comp, Function.comp_apply,
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AlgEquiv.toLinearMap_apply]
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rw [AlgEquiv.symm_apply_eq]
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simp only [FreeAlgebra.equivMonoidAlgebraFreeMonoid, MonoidAlgebra.of_apply,
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AlgEquiv.ofAlgHom_apply, FreeAlgebra.lift_ι_apply]
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rw [@MonoidAlgebra.ext_iff]
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intro x
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erw [Finsupp.lift_apply]
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simp only [MonoidAlgebra.smul_single', List.insertionSort, List.orderedInsert,
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koszulSign_freeMonoid_of, mul_one, Finsupp.single_zero, Finsupp.sum_single_index]
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rfl
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@[simp]
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lemma koszulOrder_single {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2)
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(l : FreeMonoid I) :
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koszulOrder r q (FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm (MonoidAlgebra.single l x))
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= FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm
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(MonoidAlgebra.single (List.insertionSort r l) (koszulSign r q l * x)) := by
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simp [koszulOrder]
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end
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/-- The indexing set of constructive and destructive position based operators. -/
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def ConstDestAlgebra.index : Type := Fin 2 × S.𝓯 × SpaceTime
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@ -259,7 +160,7 @@ lemma toWickAlgebra_ι_one (x : S.𝓯 × SpaceTime) :
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simp [toWickAlgebra]
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/-- The time ordering relation on constructive and destructive operators. -/
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def timeOrderRel : index S → index S → Prop := fun x y => x.2.2 0 ≤ y.2.2 0
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def timeOrderRel : index S → index S → Prop := fun x y => y.2.2 0 ≤ x.2.2 0
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/-- The normal ordering relation on constructive and destructive operators. -/
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def normalOrderRel : index S → index S → Prop := fun x y => x.1 ≤ y.1
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@ -271,7 +172,7 @@ noncomputable section
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/-- The time ordering relation of constructive and destructive operators is decidable. -/
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instance : DecidableRel (@timeOrderRel S) :=
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fun a b => Real.decidableLE (a.2.2 0) (b.2.2 0)
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fun a b => Real.decidableLE (b.2.2 0) (a.2.2 0)
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/-- The time ordering of constructive and destructive operators. -/
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def timeOrder (q : index S → Fin 2) : S.ConstDestAlgebra →ₗ[ℂ] S.ConstDestAlgebra :=
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@ -281,6 +182,14 @@ def timeOrder (q : index S → Fin 2) : S.ConstDestAlgebra →ₗ[ℂ] S.ConstDe
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def normalOrder (q : index S → Fin 2) : S.ConstDestAlgebra →ₗ[ℂ] S.ConstDestAlgebra :=
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koszulOrder normalOrderRel q
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lemma constructive_mul_normalOrder (q : index S → Fin 2) (i : FieldAlgebra.index S)
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(A : S.ConstDestAlgebra) :
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(FreeAlgebra.ι ℂ ((0, i) : index S) : S.ConstDestAlgebra) * normalOrder q A =
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normalOrder q ((FreeAlgebra.ι ℂ (0, i) : S.ConstDestAlgebra) * A) := by
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apply mul_koszulOrder_le
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intro j
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simp [normalOrderRel]
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/-- Contraction of constructive and destructive operators, defined as their time
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ordering minus their normal ordering. -/
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def contract (q : index S → Fin 2) : S.ConstDestAlgebra →ₗ[ℂ] S.ConstDestAlgebra :=
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@ -303,11 +212,11 @@ lemma toWickAlgebra_ι (i : index S) : toWickAlgebra 𝓞 (FreeAlgebra.ι ℂ i)
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simp [toWickAlgebra]
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/-- The time ordering relation in the field algebra. -/
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def timeOrderRel : index S → index S → Prop := fun x y => x.2 0 ≤ y.2 0
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def timeOrderRel : index S → index S → Prop := fun x y => y.2 0 ≤ x.2 0
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/-- The time ordering relation in the field algebra is decidable. -/
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noncomputable instance : DecidableRel (@timeOrderRel S) :=
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fun a b => Real.decidableLE (a.2 0) (b.2 0)
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fun a b => Real.decidableLE (b.2 0) (a.2 0)
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/-- The time ordering in the field algebra. -/
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noncomputable def timeOrder (q : index S → Fin 2) : S.FieldAlgebra →ₗ[ℂ] S.FieldAlgebra :=
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@ -645,6 +554,67 @@ lemma timeOrder_comm_toConstDestAlgebra (q : index S → Fin 2) :
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noncomputable def contract (q : index S → Fin 2) : FieldAlgebra S →ₗ[ℂ] ConstDestAlgebra S :=
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ConstDestAlgebra.contract (fun i => q i.2) ∘ₗ toConstDestAlgebra.toLinearMap
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lemma contract_timeOrderRel (q : index S → Fin 2) {i j : index S} (h : timeOrderRel i j) :
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contract q (FreeAlgebra.ι ℂ i * FreeAlgebra.ι ℂ j) =
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superCommute (fun i => q i.2) (FreeAlgebra.ι ℂ (1, i)) (FreeAlgebra.ι ℂ (0, j)) := by
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simp only [contract, LinearMap.coe_comp, Function.comp_apply, AlgHom.toLinearMap_apply, map_mul,
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toConstDestAlgebra_ι, Fin.isValue]
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trans (ConstDestAlgebra.contract fun i => q i.2)
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(FreeAlgebra.ι ℂ (0, i) * FreeAlgebra.ι ℂ (0, j)
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+ FreeAlgebra.ι ℂ (0, i) * FreeAlgebra.ι ℂ (1, j)
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+ FreeAlgebra.ι ℂ (1, i) * FreeAlgebra.ι ℂ (0, j)
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+ FreeAlgebra.ι ℂ (1, i) * FreeAlgebra.ι ℂ (1, j))
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· congr
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simp [mul_add, add_mul]
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abel
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simp only [ConstDestAlgebra.contract, ConstDestAlgebra.timeOrder, ConstDestAlgebra.normalOrder,
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Fin.isValue, map_add, LinearMap.sub_apply, koszulOrder_ι_pair, ConstDestAlgebra.normalOrderRel,
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le_refl, ↓reduceIte, Fin.zero_le, Fin.le_zero_iff, Nat.reduceAdd, one_ne_zero]
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have h1 : ConstDestAlgebra.timeOrderRel (0, i) (0, j) := h
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rw [if_pos h1]
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repeat rw [if_pos]
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abel_nf
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simp [koszulSign, koszulSignInsert , ConstDestAlgebra.normalOrderRel]
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· exact h1
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· exact h1
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· exact h1
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lemma contract_not_timeOrderRel (q : index S → Fin 2) {i j : index S} (h : ¬ timeOrderRel i j) :
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contract q (FreeAlgebra.ι ℂ i * FreeAlgebra.ι ℂ j) = -
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(superCommute (fun i => q i.2) (FreeAlgebra.ι ℂ (0, i)) (FreeAlgebra.ι ℂ (0, j)) +
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superCommute (fun i => q i.2) (FreeAlgebra.ι ℂ (1, i)) (FreeAlgebra.ι ℂ (1, j)) +
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superCommute (fun i => q i.2) (FreeAlgebra.ι ℂ (0, i)) (FreeAlgebra.ι ℂ (1, j))) := by
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simp [contract]
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trans (ConstDestAlgebra.contract fun i => q i.2)
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(FreeAlgebra.ι ℂ (0, i) * FreeAlgebra.ι ℂ (0, j)
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+ FreeAlgebra.ι ℂ (0, i) * FreeAlgebra.ι ℂ (1, j)
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+ FreeAlgebra.ι ℂ (1, i) * FreeAlgebra.ι ℂ (0, j)
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+ FreeAlgebra.ι ℂ (1, i) * FreeAlgebra.ι ℂ (1, j))
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· congr
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simp [mul_add, add_mul]
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abel
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simp [ConstDestAlgebra.contract, ConstDestAlgebra.timeOrder,
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ConstDestAlgebra.normalOrder, ConstDestAlgebra.normalOrderRel, ConstDestAlgebra.normalOrderRel]
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rw [if_neg, if_neg, if_neg, if_neg]
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simp [koszulSign, koszulSignInsert , ConstDestAlgebra.normalOrderRel]
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rw [if_neg]
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nth_rewrite 2 [if_neg]
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nth_rewrite 3 [if_neg]
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nth_rewrite 4 [if_neg]
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by_cases hq : q i = 1 ∧ q j = 1
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· simp [hq, superCommute]
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abel
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· simp [hq, superCommute]
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abel
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· exact h
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· exact h
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· exact h
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· exact h
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· exact h
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· exact h
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· exact h
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· exact h
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end FieldAlgebra
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end Species
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