feat: Join of Wick contractions
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137
HepLean/PerturbationTheory/WickContraction/SubContraction.lean
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HepLean/PerturbationTheory/WickContraction/SubContraction.lean
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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.TimeContract
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import HepLean.PerturbationTheory.WickContraction.StaticContract
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import HepLean.PerturbationTheory.Algebras.FieldOpAlgebra.TimeContraction
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/-!
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# Sub contractions
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-/
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open FieldSpecification
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variable {𝓕 : FieldSpecification}
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namespace WickContraction
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variable {n : ℕ} {φs : List 𝓕.States} {φsΛ : WickContraction φs.length}
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open HepLean.List
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open FieldOpAlgebra
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def subContraction (S : Finset (Finset (Fin φs.length))) (ha : S ⊆ φsΛ.1) : WickContraction φs.length :=
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⟨S, by
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intro i hi
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exact φsΛ.2.1 i (ha hi),
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by
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intro i hi j hj
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exact φsΛ.2.2 i (ha hi) j (ha hj)⟩
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lemma mem_of_mem_subContraction {S : Finset (Finset (Fin φs.length))} {hs : S ⊆ φsΛ.1}
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{a : Finset (Fin φs.length)} (ha : a ∈ (φsΛ.subContraction S hs).1) : a ∈ φsΛ.1 := by
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exact hs ha
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def quotContraction (S : Finset (Finset (Fin φs.length))) (ha : S ⊆ φsΛ.1) :
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WickContraction [φsΛ.subContraction S ha]ᵘᶜ.length :=
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⟨Finset.filter (fun a => Finset.map uncontractedListEmd a ∈ φsΛ.1) Finset.univ,
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by
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intro a ha'
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simp at ha'
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simpa using φsΛ.2.1 (Finset.map uncontractedListEmd a) ha'
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, by
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intro a ha b hb
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simp at ha hb
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by_cases hab : a = b
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· simp [hab]
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· simp [hab]
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have hx := φsΛ.2.2 (Finset.map uncontractedListEmd a) ha (Finset.map uncontractedListEmd b) hb
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simp_all⟩
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lemma mem_of_mem_quotContraction {S : Finset (Finset (Fin φs.length))} {hs : S ⊆ φsΛ.1}
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{a : Finset (Fin [φsΛ.subContraction S hs]ᵘᶜ.length)}
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(ha : a ∈ (quotContraction S hs).1) : Finset.map uncontractedListEmd a ∈ φsΛ.1 := by
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simp [quotContraction] at ha
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exact ha
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lemma mem_subContraction_or_quotContraction {S : Finset (Finset (Fin φs.length))} {hs : S ⊆ φsΛ.1}
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{a : Finset (Fin φs.length)} (ha : a ∈ φsΛ.1) :
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a ∈ (φsΛ.subContraction S hs).1 ∨
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∃ a', Finset.map uncontractedListEmd a' = a ∧ a' ∈ (quotContraction S hs).1 := by
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by_cases h1 : a ∈ (φsΛ.subContraction S hs).1
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· simp [h1]
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simp [h1]
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simp [subContraction] at h1
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have h2 := φsΛ.2.1 a ha
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rw [@Finset.card_eq_two] at h2
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obtain ⟨i, j, hij, rfl⟩ := h2
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have hi : i ∈ (φsΛ.subContraction S hs).uncontracted := by
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rw [mem_uncontracted_iff_not_contracted]
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intro p hp
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have hp' : p ∈ φsΛ.1 := hs hp
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have hp2 := φsΛ.2.2 p hp' {i, j} ha
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simp [subContraction] at hp
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rcases hp2 with hp2 | hp2
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· simp_all
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simp at hp2
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exact hp2.1
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have hj : j ∈ (φsΛ.subContraction S hs).uncontracted := by
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rw [mem_uncontracted_iff_not_contracted]
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intro p hp
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have hp' : p ∈ φsΛ.1 := hs hp
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have hp2 := φsΛ.2.2 p hp' {i, j} ha
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simp [subContraction] at hp
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rcases hp2 with hp2 | hp2
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· simp_all
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simp at hp2
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exact hp2.2
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obtain ⟨i, rfl⟩ := uncontractedListEmd_surjective_mem_uncontracted i hi
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obtain ⟨j, rfl⟩ := uncontractedListEmd_surjective_mem_uncontracted j hj
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use {i, j}
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simp [quotContraction]
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exact ha
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@[simp]
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lemma subContraction_uncontractedList_get {S : Finset (Finset (Fin φs.length))} {hs : S ⊆ φsΛ.1}
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{a : Fin [subContraction S hs]ᵘᶜ.length} :
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[subContraction S hs]ᵘᶜ[a] = φs[uncontractedListEmd a] := by
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erw [← getElem_uncontractedListEmd]
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rfl
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@[simp]
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lemma quotContraction_fstFieldOfContract_uncontractedListEmd {S : Finset (Finset (Fin φs.length))}
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{hs : S ⊆ φsΛ.1} (a : (quotContraction S hs).1) :
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uncontractedListEmd ((quotContraction S hs).fstFieldOfContract a) =
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(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a.1, mem_of_mem_quotContraction a.2⟩) := by
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symm
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apply eq_fstFieldOfContract_of_mem _ _ _ (uncontractedListEmd ((quotContraction S hs).sndFieldOfContract a) )
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· simp only [Finset.mem_map', fstFieldOfContract_mem]
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· simp
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· apply uncontractedListEmd_strictMono
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exact fstFieldOfContract_lt_sndFieldOfContract (quotContraction S hs) a
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@[simp]
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lemma quotContraction_sndFieldOfContract_uncontractedListEmd {S : Finset (Finset (Fin φs.length))}
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{hs : S ⊆ φsΛ.1} (a : (quotContraction S hs).1) :
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uncontractedListEmd ((quotContraction S hs).sndFieldOfContract a) =
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(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a.1, mem_of_mem_quotContraction a.2⟩) := by
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symm
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apply eq_sndFieldOfContract_of_mem _ _ (uncontractedListEmd ((quotContraction S hs).fstFieldOfContract a) )
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· simp only [Finset.mem_map', fstFieldOfContract_mem]
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· simp
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· apply uncontractedListEmd_strictMono
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exact fstFieldOfContract_lt_sndFieldOfContract (quotContraction S hs) a
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lemma quotContraction_gradingCompliant {S : Finset (Finset (Fin φs.length))} {hs : S ⊆ φsΛ.1}
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(hsΛ : φsΛ.GradingCompliant) :
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GradingCompliant [φsΛ.subContraction S hs]ᵘᶜ (quotContraction S hs) := by
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simp [GradingCompliant]
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intro a ha
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have h1' := mem_of_mem_quotContraction ha
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erw [subContraction_uncontractedList_get]
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erw [subContraction_uncontractedList_get]
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simp
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apply hsΛ
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end WickContraction
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