2025-01-20 15:17:48 +00:00
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/-
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Copyright (c) 2025 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 HepLean.PerturbationTheory.WickContraction.Insert
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/-!
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# Equivalence extracting element from contraction
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-/
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2025-01-21 06:11:47 +00:00
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open FieldSpecification
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variable {𝓕 : FieldSpecification}
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2025-01-20 15:17:48 +00:00
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namespace WickContraction
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variable {n : ℕ} (c : WickContraction n)
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open HepLean.List
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open HepLean.Fin
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lemma extractEquiv_equiv {c1 c2 : (c : WickContraction n) × Option c.uncontracted}
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(h : c1.1 = c2.1) (ho : c1.2 = uncontractedCongr (by rw [h]) c2.2) : c1 = c2 := by
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cases c1
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cases c2
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simp_all only [Sigma.mk.inj_iff]
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simp only at h
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subst h
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simp only [uncontractedCongr, Equiv.optionCongr_apply, heq_eq_eq, true_and]
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rename_i a
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match a with
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| none => simp
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| some a =>
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simp only [Option.map_some', Option.some.injEq]
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ext
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simp
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/-- The equivalence between `WickContraction n.succ` and the sigma type
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`(c : WickContraction n) × Option c.uncontracted` formed by inserting
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and erasing elements from a contraction. -/
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def extractEquiv (i : Fin n.succ) : WickContraction n.succ ≃
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(c : WickContraction n) × Option c.uncontracted where
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toFun := fun c => ⟨erase c i, getDualErase c i⟩
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invFun := fun ⟨c, j⟩ => insert c i j
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left_inv f := by
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simp
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right_inv f := by
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refine extractEquiv_equiv ?_ ?_
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simp only [insert_erase]
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simp only [Nat.succ_eq_add_one]
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have h1 := insert_getDualErase f.fst i f.snd
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exact insert_getDualErase _ i _
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lemma extractEquiv_symm_none_uncontracted (i : Fin n.succ) (c : WickContraction n) :
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((extractEquiv i).symm ⟨c, none⟩).uncontracted =
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(Insert.insert i (c.uncontracted.map i.succAboveEmb)) := by
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exact insert_none_uncontracted c i
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@[simp]
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lemma extractEquiv_apply_congr_symm_apply {n m : ℕ} (k : ℕ)
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(hnk : k < n.succ) (hkm : k < m.succ) (hnm : n = m) (c : WickContraction n)
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(i : c.uncontracted) : congr (by rw [hnm]) ((extractEquiv ⟨k, hkm⟩
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(congr (by rw [hnm]) ((extractEquiv ⟨k, hnk⟩).symm ⟨c, i⟩)))).1 = c := by
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subst hnm
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simp
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/-- The fintype instance of `WickContraction 0` defined through its single
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element `empty`. -/
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instance fintype_zero : Fintype (WickContraction 0) where
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elems := {empty}
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complete := by
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intro c
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simp only [Finset.mem_singleton]
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apply Subtype.eq
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simp only [empty]
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ext a
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apply Iff.intro
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· intro h
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have hc := c.2.1 a h
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rw [Finset.card_eq_two] at hc
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obtain ⟨x, y, hxy, ha⟩ := hc
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exact Fin.elim0 x
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· simp
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lemma sum_WickContraction_nil (f : WickContraction 0 → M) [AddCommMonoid M] :
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∑ c, f c = f empty := by
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rw [Finset.sum_eq_single_of_mem]
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simp only [Finset.mem_univ]
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intro b hb
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fin_cases b
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simp
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/-- The fintype instance of `WickContraction n`, for `n.succ` this is defined
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through the equivalence `extractEquiv`. -/
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instance fintype_succ : (n : ℕ) → Fintype (WickContraction n)
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| 0 => fintype_zero
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| Nat.succ n => by
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letI := fintype_succ n
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exact Fintype.ofEquiv _ (extractEquiv 0).symm
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lemma sum_extractEquiv_congr [AddCommMonoid M] {n m : ℕ} (i : Fin n) (f : WickContraction n → M)
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(h : n = m.succ) :
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∑ c, f c = ∑ (c : WickContraction m), ∑ (k : Option c.uncontracted),
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f (congr h.symm ((extractEquiv (finCongr h i)).symm ⟨c, k⟩)) := by
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subst h
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simp only [finCongr_refl, Equiv.refl_apply, congr_refl]
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rw [← (extractEquiv i).symm.sum_comp]
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rw [Finset.sum_sigma']
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rfl
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end WickContraction
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