253 lines
11 KiB
Text
253 lines
11 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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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 Mathlib.Algebra.FreeAlgebra
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import Mathlib.Algebra.Lie.OfAssociative
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import Mathlib.Analysis.Complex.Basic
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import HepLean.PerturbationTheory.Wick.Signs.StaticWickCoef
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/-!
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# Koszul signs and ordering for lists and algebras
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-/
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namespace Wick
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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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@[simp]
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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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@[simp]
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lemma koszulOrder_ι_pair {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2) (i j : I) :
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koszulOrder r q (FreeAlgebra.ι ℂ i * FreeAlgebra.ι ℂ j) =
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if r i j then FreeAlgebra.ι ℂ i * FreeAlgebra.ι ℂ j else
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(koszulSign r q [i, j]) • (FreeAlgebra.ι ℂ j * FreeAlgebra.ι ℂ i) := by
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have h1 : FreeAlgebra.ι ℂ i * FreeAlgebra.ι ℂ j =
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FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm (MonoidAlgebra.single [i, j] 1) := by
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simp only [FreeAlgebra.equivMonoidAlgebraFreeMonoid, MonoidAlgebra.of_apply,
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AlgEquiv.ofAlgHom_symm_apply, MonoidAlgebra.lift_single, one_smul]
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rfl
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conv_lhs => rw [h1]
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simp only [koszulOrder, AlgEquiv.toAlgHom_eq_coe, AlgEquiv.toAlgHom_toLinearMap,
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LinearMap.coe_comp, Function.comp_apply, AlgEquiv.toLinearMap_apply, AlgEquiv.apply_symm_apply,
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koszulOrderMonoidAlgebra_single, List.insertionSort, List.orderedInsert,
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MonoidAlgebra.smul_single', mul_one]
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by_cases hr : r i j
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· rw [if_pos hr, if_pos hr]
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simp only [FreeAlgebra.equivMonoidAlgebraFreeMonoid, MonoidAlgebra.of_apply,
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AlgEquiv.ofAlgHom_symm_apply, MonoidAlgebra.lift_single]
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have hKS : koszulSign r q [i, j] = 1 := by
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simp only [koszulSign, koszulSignInsert, Fin.isValue, mul_one, ite_eq_left_iff,
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ite_eq_right_iff, and_imp]
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exact fun a a_1 a_2 => False.elim (a hr)
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rw [hKS]
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simp only [one_smul]
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rfl
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· rw [if_neg hr, if_neg hr]
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simp only [FreeAlgebra.equivMonoidAlgebraFreeMonoid, MonoidAlgebra.of_apply,
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AlgEquiv.ofAlgHom_symm_apply, MonoidAlgebra.lift_single]
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rfl
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@[simp]
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lemma koszulOrder_one {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2) :
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koszulOrder r q 1 = 1 := by
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trans koszulOrder r q (FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm (MonoidAlgebra.single [] 1))
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congr
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· simp only [FreeAlgebra.equivMonoidAlgebraFreeMonoid, MonoidAlgebra.of_apply,
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AlgEquiv.ofAlgHom_symm_apply, MonoidAlgebra.lift_single, one_smul]
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rfl
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· simp only [koszulOrder_single, List.insertionSort, mul_one, EmbeddingLike.map_eq_one_iff]
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rfl
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lemma mul_koszulOrder_le {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2)
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(i : I) (A : FreeAlgebra ℂ I) (hi : ∀ j, r i j) :
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FreeAlgebra.ι ℂ i * koszulOrder r q A = koszulOrder r q (FreeAlgebra.ι ℂ i * A) := by
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let f : FreeAlgebra ℂ I →ₗ[ℂ] FreeAlgebra ℂ I := {
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toFun := fun x => FreeAlgebra.ι ℂ i * x,
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map_add' := fun x y => by
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simp [mul_add],
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map_smul' := by simp}
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change (f ∘ₗ koszulOrder r q) A = (koszulOrder r q ∘ₗ f) _
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have f_single (l : FreeMonoid I) (x : ℂ) :
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f ((FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm (MonoidAlgebra.single l x)))
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= (FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm (MonoidAlgebra.single (i :: l) x)) := by
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simp only [LinearMap.coe_mk, AddHom.coe_mk, f]
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have hf : FreeAlgebra.ι ℂ i = FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm
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(MonoidAlgebra.single [i] 1) := by
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simp only [FreeAlgebra.equivMonoidAlgebraFreeMonoid, MonoidAlgebra.of_apply,
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AlgEquiv.ofAlgHom_symm_apply, MonoidAlgebra.lift_single, one_smul]
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rfl
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rw [hf]
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rw [@AlgEquiv.eq_symm_apply]
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simp only [map_mul, AlgEquiv.apply_symm_apply, MonoidAlgebra.single_mul_single, one_mul]
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rfl
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have h1 : f ∘ₗ koszulOrder r q = koszulOrder r q ∘ₗ f := by
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let e : FreeAlgebra ℂ I ≃ₗ[ℂ] MonoidAlgebra ℂ (FreeMonoid I) :=
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FreeAlgebra.equivMonoidAlgebraFreeMonoid.toLinearEquiv
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apply (LinearEquiv.eq_comp_toLinearMap_iff (e₁₂ := e.symm) _ _).mp
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apply MonoidAlgebra.lhom_ext'
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intro l
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apply LinearMap.ext
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intro x
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simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
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MonoidAlgebra.lsingle_apply]
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erw [koszulOrder_single]
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rw [f_single]
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erw [f_single]
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rw [koszulOrder_single]
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congr 2
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· simp only [List.insertionSort]
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have hi (l : List I) : i :: l = List.orderedInsert r i l := by
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induction l with
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| nil => rfl
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| cons j l ih =>
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refine (List.orderedInsert_of_le r l (hi j)).symm
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exact hi _
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· congr 1
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rw [koszulSign]
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have h1 (l : List I) : koszulSignInsert r q i l = 1 := by
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exact koszulSignInsert_le_forall r q i l hi
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rw [h1]
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simp
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rw [h1]
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lemma koszulOrder_mul_ge {I : Type} (r : I → I → Prop) [DecidableRel r] (q : I → Fin 2)
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(i : I) (A : FreeAlgebra ℂ I) (hi : ∀ j, r j i) :
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koszulOrder r q A * FreeAlgebra.ι ℂ i = koszulOrder r q (A * FreeAlgebra.ι ℂ i) := by
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let f : FreeAlgebra ℂ I →ₗ[ℂ] FreeAlgebra ℂ I := {
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toFun := fun x => x * FreeAlgebra.ι ℂ i,
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map_add' := fun x y => by
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simp [add_mul],
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map_smul' := by simp}
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change (f ∘ₗ koszulOrder r q) A = (koszulOrder r q ∘ₗ f) A
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have f_single (l : FreeMonoid I) (x : ℂ) :
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f ((FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm (MonoidAlgebra.single l x)))
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= (FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm
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(MonoidAlgebra.single (l.toList ++ [i]) x)) := by
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simp only [LinearMap.coe_mk, AddHom.coe_mk, f]
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have hf : FreeAlgebra.ι ℂ i = FreeAlgebra.equivMonoidAlgebraFreeMonoid.symm
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(MonoidAlgebra.single [i] 1) := by
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simp only [FreeAlgebra.equivMonoidAlgebraFreeMonoid, MonoidAlgebra.of_apply,
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AlgEquiv.ofAlgHom_symm_apply, MonoidAlgebra.lift_single, one_smul]
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rfl
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rw [hf]
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rw [@AlgEquiv.eq_symm_apply]
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simp only [map_mul, AlgEquiv.apply_symm_apply, MonoidAlgebra.single_mul_single, mul_one]
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rfl
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have h1 : f ∘ₗ koszulOrder r q = koszulOrder r q ∘ₗ f := by
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let e : FreeAlgebra ℂ I ≃ₗ[ℂ] MonoidAlgebra ℂ (FreeMonoid I) :=
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FreeAlgebra.equivMonoidAlgebraFreeMonoid.toLinearEquiv
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apply (LinearEquiv.eq_comp_toLinearMap_iff (e₁₂ := e.symm) _ _).mp
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apply MonoidAlgebra.lhom_ext'
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intro l
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apply LinearMap.ext
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intro x
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simp only [LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply,
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MonoidAlgebra.lsingle_apply]
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erw [koszulOrder_single]
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rw [f_single]
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erw [f_single]
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rw [koszulOrder_single]
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congr 3
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· change (List.insertionSort r l) ++ [i] = List.insertionSort r (l.toList ++ [i])
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have hoi (l : List I) (j : I) : List.orderedInsert r j (l ++ [i]) =
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List.orderedInsert r j l ++ [i] := by
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induction l with
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| nil => simp [hi]
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| cons b l ih =>
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simp only [List.orderedInsert, List.append_eq]
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by_cases hr : r j b
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· rw [if_pos hr, if_pos hr]
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rfl
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· rw [if_neg hr, if_neg hr]
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rw [ih]
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rfl
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have hI (l : List I) : (List.insertionSort r l) ++ [i] = List.insertionSort r (l ++ [i]) := by
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induction l with
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| nil => rfl
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| cons j l ih =>
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simp only [List.insertionSort, List.append_eq]
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rw [← ih]
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rw [hoi]
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rw [hI]
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rfl
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· have hI (l : List I) : koszulSign r q l = koszulSign r q (l ++ [i]) := by
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induction l with
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| nil => simp [koszulSign, koszulSignInsert]
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| cons j l ih =>
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simp only [koszulSign, List.append_eq]
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rw [ih]
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simp only [mul_eq_mul_right_iff]
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apply Or.inl
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rw [koszulSignInsert_ge_forall_append r q l j i hi]
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rw [hI]
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rfl
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rw [h1]
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end
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end Wick
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