Physlib.QFT.PerturbationTheory.WickContraction.TimeCond

Imports

/-
Copyright (c) 2025 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
module

public import Physlib.QFT.PerturbationTheory.WickContraction.Join

Time contractions

@[expose] public section

The condition on a Wick contraction which is true iff and only if every contraction is between two fields of equal time.

def EqTimeOnly {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : Prop :=
  ∀ (i j), {i, j} ∈ φsΛ.1 → timeOrderRel φs[i] φs[j]

              instance {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) :
    Decidable (EqTimeOnly φsΛ) :=
    inferInstanceAs (Decidable (∀ (i j), {i, j} ∈ φsΛ.1 → timeOrderRel φs[i] φs[j]))lemma timeOrderRel_of_eqTimeOnly_pair {i j : Fin φs.length} (h : {i, j} ∈ φsΛ.1)
    (hc : EqTimeOnly φsΛ) :
    timeOrderRel φs[i] φs[j] := hc i j hlemma timeOrderRel_both_of_eqTimeOnly {i j : Fin φs.length} (h : {i, j} ∈ φsΛ.1)
    (hc : EqTimeOnly φsΛ) :
    timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i] :=
  ⟨timeOrderRel_of_eqTimeOnly_pair φsΛ h hc,
    timeOrderRel_of_eqTimeOnly_pair φsΛ (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthh:{i, j} ∈ ↑φsΛhc:φsΛ.EqTimeOnly⊢ {j, i} ∈ ↑φsΛ simpa [Finset.pair_comm] using hAll goals completed! 🐙) hc⟩

lemma eqTimeOnly_iff_forall_finset {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) :
    φsΛ.EqTimeOnly ↔ ∀ (a : φsΛ.1),
      timeOrderRel (φs[φsΛ.fstFieldOfContract a]) (φs[φsΛ.sndFieldOfContract a])
      ∧ timeOrderRel (φs[φsΛ.sndFieldOfContract a]) (φs[φsΛ.fstFieldOfContract a]) := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ φsΛ.EqTimeOnly ↔
  ∀ (a : ↥↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]
  apply Iff.intromp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ φsΛ.EqTimeOnly →
  ∀ (a : ↥↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]mpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ (∀ (a : ↥↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]) →
  φsΛ.EqTimeOnly
  ·mp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ φsΛ.EqTimeOnly →
  ∀ (a : ↥↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a] intro h amp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]
    apply timeOrderRel_both_of_eqTimeOnly φsΛ _ h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ {φsΛ.fstFieldOfContract a, φsΛ.sndFieldOfContract a} ∈ ↑φsΛ
    rw [← finset_eq_fstFieldOfContract_sndFieldOfContract𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ ↑a ∈ ↑φsΛ𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ ↑a ∈ ↑φsΛ]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ ↑a ∈ ↑φsΛ
    simpAll goals completed! 🐙
  ·mpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ (∀ (a : ↥↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]) →
  φsΛ.EqTimeOnly intro hmpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]⊢ φsΛ.EqTimeOnly
    simp only [EqTimeOnly, Fin.getElem_fin]mpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]⊢ ∀ (i j : Fin φs.length), {i, j} ∈ ↑φsΛ → timeOrderRel φs[↑i] φs[↑j]
    intro i j h1mpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛ⊢ timeOrderRel φs[↑i] φs[↑j]
    have h' := h ⟨{i, j}, h1⟩mpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]⊢ timeOrderRel φs[↑i] φs[↑j]
    by_cases hij: i < jpos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < j⊢ timeOrderRel φs[↑i] φs[↑j]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:¬i < j⊢ timeOrderRel φs[↑i] φs[↑j]
    ·pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < j⊢ timeOrderRel φs[↑i] φs[↑j] have hi : φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i :=
        eq_fstFieldOfContract_of_mem _ _ i j (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < j⊢ i ∈ ↑⟨{i, j}, h1⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < j⊢ j ∈ ↑⟨{i, j}, h1⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j]) hijpos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j]
      have hj : φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j :=
        eq_sndFieldOfContract_of_mem _ _ i j (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ i ∈ ↑⟨{i, j}, h1⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ihj:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ihj:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ j ∈ ↑⟨{i, j}, h1⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ihj:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ihj:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j]) hijpos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:i < jhi:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ihj:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j]
      simp_allAll goals completed! 🐙
    ·neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij:¬i < j⊢ timeOrderRel φs[↑i] φs[↑j] have hij : i ≠ j := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ φsΛ.EqTimeOnly ↔
  ∀ (a : ↥↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ j⊢ timeOrderRel φs[↑i] φs[↑j]
        by_contra hij𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i = j⊢ Falseneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ j⊢ timeOrderRel φs[↑i] φs[↑j]
        subst hij𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthh1:{i, i} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, i}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, i}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, i}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, i}, h1⟩]hij:¬i < i⊢ Falseneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ j⊢ timeOrderRel φs[↑i] φs[↑j]
        have h2 := φsΛ.2.1 {i, i} h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthh1:{i, i} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, i}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, i}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, i}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, i}, h1⟩]hij:¬i < ih2:{i, i}.card = 2⊢ Falseneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ j⊢ timeOrderRel φs[↑i] φs[↑j]
        simp at h2neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ j⊢ timeOrderRel φs[↑i] φs[↑j]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ j⊢ timeOrderRel φs[↑i] φs[↑j]
      have hj : φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j :=
        eq_fstFieldOfContract_of_mem _ _ j i (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ j⊢ j ∈ ↑⟨{i, j}, h1⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ j⊢ i ∈ ↑⟨{i, j}, h1⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ j⊢ j < ineg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j] omegaAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j])neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑i] φs[↑j]
      have hi : φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i :=
        eq_sndFieldOfContract_of_mem _ _ j i (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ j ∈ ↑⟨{i, j}, h1⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jhi:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jhi:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ i ∈ ↑⟨{i, j}, h1⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jhi:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jhi:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ j < ineg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jhi:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j] omegaAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jhi:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j])neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]i:Fin φs.lengthj:Fin φs.lengthh1:{i, j} ∈ ↑φsΛh':timeOrderRel φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] ∧
  timeOrderRel φs[φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩] φs[φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩]hij✝:¬i < jhij:i ≠ jhj:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jhi:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑i] φs[↑j]
      simp_allAll goals completed! 🐙

@[simp]
lemma empty_mem {φs : List 𝓕.FieldOp} : empty (n := φs.length).EqTimeOnly := by𝓕:FieldSpecificationφs:List 𝓕.FieldOp⊢ empty.EqTimeOnly
  rw [eqTimeOnly_iff_forall_finset𝓕:FieldSpecificationφs:List 𝓕.FieldOp⊢ ∀ (a : ↥↑empty),
  timeOrderRel φs[empty.fstFieldOfContract a] φs[empty.sndFieldOfContract a] ∧
    timeOrderRel φs[empty.sndFieldOfContract a] φs[empty.fstFieldOfContract a]𝓕:FieldSpecificationφs:List 𝓕.FieldOp⊢ ∀ (a : ↥↑empty),
  timeOrderRel φs[empty.fstFieldOfContract a] φs[empty.sndFieldOfContract a] ∧
    timeOrderRel φs[empty.sndFieldOfContract a] φs[empty.fstFieldOfContract a]]𝓕:FieldSpecificationφs:List 𝓕.FieldOp⊢ ∀ (a : ↥↑empty),
  timeOrderRel φs[empty.fstFieldOfContract a] φs[empty.sndFieldOfContract a] ∧
    timeOrderRel φs[empty.sndFieldOfContract a] φs[empty.fstFieldOfContract a]
  simp [empty]All goals completed! 🐙

Let φs be a list of 𝓕.FieldOp and φsΛ a WickContraction of φs within which every contraction involves two 𝓕.FieldOps that have the same time, then φsΛ.staticContract = φsΛ.timeContract.


lemma staticContract_eq_timeContract_of_eqTimeOnly (h : φsΛ.EqTimeOnly) :
    φsΛ.staticContract = φsΛ.timeContract := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnly⊢ φsΛ.staticContract = φsΛ.timeContract
  simp only [staticContract, timeContract]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnly⊢ ∏ a, ⟨(superCommute (anPart (φs.get (φsΛ.fstFieldOfContract a)))) (ofFieldOp (φs.get (φsΛ.sndFieldOfContract a))), ⋯⟩ =
  ∏ a, ⟨WickAlgebra.timeContract (φs.get (φsΛ.fstFieldOfContract a)) (φs.get (φsΛ.sndFieldOfContract a)), ⋯⟩
  apply congrArg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnly⊢ (fun a =>
    ⟨(superCommute (anPart (φs.get (φsΛ.fstFieldOfContract a)))) (ofFieldOp (φs.get (φsΛ.sndFieldOfContract a))), ⋯⟩) =
  fun a => ⟨WickAlgebra.timeContract (φs.get (φsΛ.fstFieldOfContract a)) (φs.get (φsΛ.sndFieldOfContract a)), ⋯⟩
  funext a𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ ⟨(superCommute (anPart (φs.get (φsΛ.fstFieldOfContract a)))) (ofFieldOp (φs.get (φsΛ.sndFieldOfContract a))), ⋯⟩ =
  ⟨WickAlgebra.timeContract (φs.get (φsΛ.fstFieldOfContract a)) (φs.get (φsΛ.sndFieldOfContract a)), ⋯⟩
  ext𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ ↑⟨(superCommute (anPart (φs.get (φsΛ.fstFieldOfContract a)))) (ofFieldOp (φs.get (φsΛ.sndFieldOfContract a))), ⋯⟩ =
  ↑⟨WickAlgebra.timeContract (φs.get (φsΛ.fstFieldOfContract a)) (φs.get (φsΛ.sndFieldOfContract a)), ⋯⟩
  simp only [List.get_eq_getElem]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ (superCommute (anPart φs[↑(φsΛ.fstFieldOfContract a)])) (ofFieldOp φs[↑(φsΛ.sndFieldOfContract a)]) =
  WickAlgebra.timeContract φs[↑(φsΛ.fstFieldOfContract a)] φs[↑(φsΛ.sndFieldOfContract a)]
  rw [timeContract_of_timeOrderRel𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ (superCommute (anPart φs[↑(φsΛ.fstFieldOfContract a)])) (ofFieldOp φs[↑(φsΛ.sndFieldOfContract a)]) =
  (superCommute (anPart φs[↑(φsΛ.fstFieldOfContract a)])) (ofFieldOp φs[↑(φsΛ.sndFieldOfContract a)])h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract a)] φs[↑(φsΛ.sndFieldOfContract a)]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract a)] φs[↑(φsΛ.sndFieldOfContract a)]]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract a)] φs[↑(φsΛ.sndFieldOfContract a)]
  refine timeOrderRel_of_eqTimeOnly_pair φsΛ ?_ hh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:↥↑φsΛ⊢ {φsΛ.fstFieldOfContract a, φsΛ.sndFieldOfContract a} ∈ ↑φsΛ
  simp [← finset_eq_fstFieldOfContract_sndFieldOfContract, a.2]All goals completed! 🐙

              lemma eqTimeOnly_congr {φs φs' : List 𝓕.FieldOp} (h : φs = φs') (φsΛ : WickContraction φs.length) :
    (congr (by𝓕:FieldSpecificationn:ℕc:WickContraction nφs✝:List 𝓕.FieldOpφsΛ✝:WickContraction φs✝.lengthφs:List 𝓕.FieldOpφs':List 𝓕.FieldOph:φs = φs'φsΛ:WickContraction φs.length⊢ φs.length = φs'.length simp [h]All goals completed! 🐙) φsΛ).EqTimeOnly (φs := φs') ↔ φsΛ.EqTimeOnly := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφs':List 𝓕.FieldOph:φs = φs'φsΛ:WickContraction φs.length⊢ ((congr ⋯) φsΛ).EqTimeOnly ↔ φsΛ.EqTimeOnly
  subst h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ ((congr ⋯) φsΛ).EqTimeOnly ↔ φsΛ.EqTimeOnly
  simpAll goals completed! 🐙

lemma quotContraction_eqTimeOnly {φs : List 𝓕.FieldOp} {φsΛ : WickContraction φs.length}
    (h : φsΛ.EqTimeOnly) (S : Finset (Finset (Fin φs.length))) (ha : S ⊆ φsΛ.1) :
    (φsΛ.quotContraction S ha).EqTimeOnly := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛ⊢ (quotContraction S ha).EqTimeOnly
  rw [eqTimeOnly_iff_forall_finset𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛ⊢ ∀ (a : ↥↑(quotContraction S ha)),
  timeOrderRel [subContraction S ha]ᵘᶜ[(quotContraction S ha).fstFieldOfContract a]
      [subContraction S ha]ᵘᶜ[(quotContraction S ha).sndFieldOfContract a] ∧
    timeOrderRel [subContraction S ha]ᵘᶜ[(quotContraction S ha).sndFieldOfContract a]
      [subContraction S ha]ᵘᶜ[(quotContraction S ha).fstFieldOfContract a]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛ⊢ ∀ (a : ↥↑(quotContraction S ha)),
  timeOrderRel [subContraction S ha]ᵘᶜ[(quotContraction S ha).fstFieldOfContract a]
      [subContraction S ha]ᵘᶜ[(quotContraction S ha).sndFieldOfContract a] ∧
    timeOrderRel [subContraction S ha]ᵘᶜ[(quotContraction S ha).sndFieldOfContract a]
      [subContraction S ha]ᵘᶜ[(quotContraction S ha).fstFieldOfContract a]]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛ⊢ ∀ (a : ↥↑(quotContraction S ha)),
  timeOrderRel [subContraction S ha]ᵘᶜ[(quotContraction S ha).fstFieldOfContract a]
      [subContraction S ha]ᵘᶜ[(quotContraction S ha).sndFieldOfContract a] ∧
    timeOrderRel [subContraction S ha]ᵘᶜ[(quotContraction S ha).sndFieldOfContract a]
      [subContraction S ha]ᵘᶜ[(quotContraction S ha).fstFieldOfContract a]
  intro a𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛa:↥↑(quotContraction S ha)⊢ timeOrderRel [subContraction S ha]ᵘᶜ[(quotContraction S ha).fstFieldOfContract a]
    [subContraction S ha]ᵘᶜ[(quotContraction S ha).sndFieldOfContract a] ∧
  timeOrderRel [subContraction S ha]ᵘᶜ[(quotContraction S ha).sndFieldOfContract a]
    [subContraction S ha]ᵘᶜ[(quotContraction S ha).fstFieldOfContract a]
  simp only [Fin.getElem_fin]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛa:↥↑(quotContraction S ha)⊢ timeOrderRel [subContraction S ha]ᵘᶜ[↑((quotContraction S ha).fstFieldOfContract a)]
    [subContraction S ha]ᵘᶜ[↑((quotContraction S ha).sndFieldOfContract a)] ∧
  timeOrderRel [subContraction S ha]ᵘᶜ[↑((quotContraction S ha).sndFieldOfContract a)]
    [subContraction S ha]ᵘᶜ[↑((quotContraction S ha).fstFieldOfContract a)]
  erw [subContraction_uncontractedList_get,𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛa:↥↑(quotContraction S ha)⊢ timeOrderRel φs[uncontractedListEmd ((quotContraction S ha).fstFieldOfContract a)]
    [subContraction S ha]ᵘᶜ[↑((quotContraction S ha).sndFieldOfContract a)] ∧
  timeOrderRel [subContraction S ha]ᵘᶜ[↑((quotContraction S ha).sndFieldOfContract a)]
    φs[uncontractedListEmd ((quotContraction S ha).fstFieldOfContract a)] subContraction_uncontractedList_get𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛa:↥↑(quotContraction S ha)⊢ timeOrderRel φs[uncontractedListEmd ((quotContraction S ha).fstFieldOfContract a)]
    φs[uncontractedListEmd ((quotContraction S ha).sndFieldOfContract a)] ∧
  timeOrderRel φs[uncontractedListEmd ((quotContraction S ha).sndFieldOfContract a)]
    φs[uncontractedListEmd ((quotContraction S ha).fstFieldOfContract a)]]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛa:↥↑(quotContraction S ha)⊢ timeOrderRel φs[uncontractedListEmd ((quotContraction S ha).fstFieldOfContract a)]
    φs[uncontractedListEmd ((quotContraction S ha).sndFieldOfContract a)] ∧
  timeOrderRel φs[uncontractedListEmd ((quotContraction S ha).sndFieldOfContract a)]
    φs[uncontractedListEmd ((quotContraction S ha).fstFieldOfContract a)]
  simp only [quotContraction_fstFieldOfContract_uncontractedListEmd, Fin.getElem_fin,
    quotContraction_sndFieldOfContract_uncontractedListEmd]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlyS:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛa:↥↑(quotContraction S ha)⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
    φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
  rw [eqTimeOnly_iff_forall_finset𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]S:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛa:↥↑(quotContraction S ha)⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
    φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]S:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛa:↥↑(quotContraction S ha)⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
    φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]] at h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]S:Finset (Finset (Fin φs.length))ha:S ⊆ ↑φsΛa:↥↑(quotContraction S ha)⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
    φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd ↑a, ⋯⟩)]
  apply hAll goals completed! 🐙

set_option backward.isDefEq.respectTransparency false in
lemma exists_join_singleton_of_card_ge_zero {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
    (h : 0 < φsΛ.1.card) (h1 : φsΛ.EqTimeOnly) :
    ∃ (i j : Fin φs.length) (h : i < j) (φsucΛ : WickContraction [singleton h]ᵘᶜ.length),
    φsΛ = join (singleton h) φsucΛ ∧ (timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i])
    ∧ φsucΛ.EqTimeOnly ∧ φsucΛ.1.card + 1 = φsΛ.1.card := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnly⊢ ∃ i j,
  ∃ (h : i < j),
    ∃ φsucΛ,
      φsΛ = (singleton h).join φsucΛ ∧
        (timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]) ∧ φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card
  obtain ⟨a, ha⟩ := exists_contraction_pair_of_card_ge_zero φsΛ h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛ⊢ ∃ i j,
  ∃ (h : i < j),
    ∃ φsucΛ,
      φsΛ = (singleton h).join φsucΛ ∧
        (timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]) ∧ φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card
  use φsΛ.fstFieldOfContract ⟨a, ha⟩h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛ⊢ ∃ j,
  ∃ (h : φsΛ.fstFieldOfContract ⟨a, ha⟩ < j),
    ∃ φsucΛ,
      φsΛ = (singleton h).join φsucΛ ∧
        (timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[j] ∧
            timeOrderRel φs[j] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) ∧
          φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card
  use φsΛ.sndFieldOfContract ⟨a, ha⟩h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛ⊢ ∃ (h : φsΛ.fstFieldOfContract ⟨a, ha⟩ < φsΛ.sndFieldOfContract ⟨a, ha⟩),
  ∃ φsucΛ,
    φsΛ = (singleton h).join φsucΛ ∧
      (timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∧
          timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) ∧
        φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card
  use φsΛ.fstFieldOfContract_lt_sndFieldOfContract ⟨a, ha⟩h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛ⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) ∧
      φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card
  let φsucΛ :
    WickContraction [singleton (φsΛ.fstFieldOfContract_lt_sndFieldOfContract ⟨a, ha⟩)]ᵘᶜ.length :=
    congr (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛ⊢ [subContraction {a} ?m.175]ᵘᶜ.length = [singleton ⋯]ᵘᶜ.lengthh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) ∧
      φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card simp [← subContraction_singleton_eq_singleton]All goals completed! 🐙h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) ∧
      φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card)
    (φsΛ.quotContraction {a} (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛ⊢ {a} ⊆ ↑φsΛh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) ∧
      φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card simpa using haAll goals completed! 🐙h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) ∧
      φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card))h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) ∧
      φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card
  use φsucΛh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsΛ = (singleton ⋯).join φsucΛ ∧
  (timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) ∧
    φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card
  simp only [Fin.getElem_fin]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsΛ = (singleton ⋯).join φsucΛ ∧
  (timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)]) ∧
    φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card
  apply And.introh.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsΛ = (singleton ⋯).join φsucΛh.right𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ (timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] ∧
    timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)]) ∧
  φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card
  ·h.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsΛ = (singleton ⋯).join φsucΛ have h1 := join_congr (subContraction_singleton_eq_singleton _ ⟨a, ha⟩).symm (φsucΛ := φsucΛ)h.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1✝:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:(singleton ⋯).join φsucΛ = (subContraction {↑⟨a, ha⟩} ⋯).join ((congr ⋯) φsucΛ)⊢ φsΛ = (singleton ⋯).join φsucΛ
    simp only [h1, congr_trans_apply, congr_refl, φsucΛ]h.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1✝:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:(singleton ⋯).join φsucΛ = (subContraction {↑⟨a, ha⟩} ⋯).join ((congr ⋯) φsucΛ)⊢ φsΛ = (subContraction {a} ⋯).join (quotContraction {a} ⋯)
    rw [join_sub_quoth.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1✝:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:(singleton ⋯).join φsucΛ = (subContraction {↑⟨a, ha⟩} ⋯).join ((congr ⋯) φsucΛ)⊢ φsΛ = φsΛAll goals completed! 🐙]All goals completed! 🐙
  ·h.right𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ (timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] ∧
    timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)]) ∧
  φsucΛ.EqTimeOnly ∧ (↑φsucΛ).card + 1 = (↑φsΛ).card refine ⟨?_, ?_, ?_⟩h.right.refine_1𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)]h.right.refine_2𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsucΛ.EqTimeOnlyh.right.refine_3𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ (↑φsucΛ).card + 1 = (↑φsΛ).card
    ·h.right.refine_1𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] apply timeOrderRel_both_of_eqTimeOnly φsΛ _ h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ {φsΛ.fstFieldOfContract ⟨a, ha⟩, φsΛ.sndFieldOfContract ⟨a, ha⟩} ∈ ↑φsΛ
      simp [← finset_eq_fstFieldOfContract_sndFieldOfContract, ha]All goals completed! 🐙
    ·h.right.refine_2𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsucΛ.EqTimeOnly simp only [φsucΛ]h.right.refine_2𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ((congr ⋯) (quotContraction {a} ⋯)).EqTimeOnly
      rw [eqTimeOnly_congr (φs := [(φsΛ.subContraction {a} (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ {a} ⊆ ↑φsΛh.right.refine_2𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ (quotContraction {a} ⋯).EqTimeOnlyh.right.refine_2.h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ [subContraction {a} ⋯]ᵘᶜ = [singleton ⋯]ᵘᶜ simpa using haAll goals completed! 🐙h.right.refine_2𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ (quotContraction {a} ⋯).EqTimeOnlyh.right.refine_2.h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ [subContraction {a} ⋯]ᵘᶜ = [singleton ⋯]ᵘᶜ))]ᵘᶜ)]h.right.refine_2𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ (quotContraction {a} ⋯).EqTimeOnlyh.right.refine_2.h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ [subContraction {a} ⋯]ᵘᶜ = [singleton ⋯]ᵘᶜ
      exact quotContraction_eqTimeOnly h1 _ _h.right.refine_2.h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ [subContraction {a} ⋯]ᵘᶜ = [singleton ⋯]ᵘᶜ
      rw [← subContraction_singleton_eq_singletonh.right.refine_2.h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ [subContraction {a} ⋯]ᵘᶜ = [subContraction {↑⟨a, ha⟩} ⋯]ᵘᶜAll goals completed! 🐙]All goals completed! 🐙
    ·h.right.refine_3𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ (↑φsucΛ).card + 1 = (↑φsΛ).card simp only [card_congr, φsucΛ]h.right.refine_3𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ (↑(quotContraction {a} ⋯)).card + 1 = (↑φsΛ).card
      have h1 := subContraction_card_plus_quotContraction_card_eq _ {a} (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ {a} ⊆ ↑?m.263h.right.refine_3𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1✝:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:(↑(subContraction {a} ⋯)).card + (↑(quotContraction {a} ⋯)).card = (↑φsΛ).card⊢ (↑(quotContraction {a} ⋯)).card + 1 = (↑φsΛ).card simpa using haAll goals completed! 🐙h.right.refine_3𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1✝:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:(↑(subContraction {a} ⋯)).card + (↑(quotContraction {a} ⋯)).card = (↑φsΛ).card⊢ (↑(quotContraction {a} ⋯)).card + 1 = (↑φsΛ).card)h.right.refine_3𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1✝:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:(↑(subContraction {a} ⋯)).card + (↑(quotContraction {a} ⋯)).card = (↑φsΛ).card⊢ (↑(quotContraction {a} ⋯)).card + 1 = (↑φsΛ).card
      simp only [subContraction, Finset.card_singleton] at h1h.right.refine_3𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:0 < (↑φsΛ).cardh1✝:φsΛ.EqTimeOnlya:Finset (Fin φs.length)ha:a ∈ ↑φsΛφsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:1 + (↑(quotContraction {a} ⋯)).card = (↑φsΛ).card⊢ (↑(quotContraction {a} ⋯)).card + 1 = (↑φsΛ).card
      omegaAll goals completed! 🐙

lemma timeOrder_timeContract_mul_of_eqTimeOnly_mid_induction {φs : List 𝓕.FieldOp}
    (φsΛ : WickContraction φs.length)
    (hl : φsΛ.EqTimeOnly) (a b: 𝓕.WickAlgebra) : (n : ℕ) → (hn : φsΛ.1.card = n) →
    𝓣(a * φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(a * b)
  | 0, hn =>𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlya:𝓕.WickAlgebrab:𝓕.WickAlgebrahn:(↑φsΛ).card = 0⊢ timeOrder (a * ↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder (a * b) by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlya:𝓕.WickAlgebrab:𝓕.WickAlgebrahn:(↑φsΛ).card = 0⊢ timeOrder (a * ↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder (a * b)
    rw [@card_zero_iff_empty𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlya:𝓕.WickAlgebrab:𝓕.WickAlgebrahn:φsΛ = empty⊢ timeOrder (a * ↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder (a * b)𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlya:𝓕.WickAlgebrab:𝓕.WickAlgebrahn:φsΛ = empty⊢ timeOrder (a * ↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder (a * b)] at hn𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlya:𝓕.WickAlgebrab:𝓕.WickAlgebrahn:φsΛ = empty⊢ timeOrder (a * ↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder (a * b)
    subst hn𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebrahl:empty.EqTimeOnly⊢ timeOrder (a * ↑empty.timeContract * b) = ↑empty.timeContract * timeOrder (a * b)
    simpAll goals completed! 🐙
  | Nat.succ n, hn =>𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlya:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕhn:(↑φsΛ).card = n.succ⊢ timeOrder (a * ↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder (a * b) by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlya:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕhn:(↑φsΛ).card = n.succ⊢ timeOrder (a * ↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder (a * b)
    obtain ⟨i, j, hij, φsucΛ, rfl, h2, h3, h4⟩ :=
      exists_join_singleton_of_card_ge_zero φsΛ (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlya:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕhn:(↑φsΛ).card = n.succ⊢ 0 < (↑φsΛ).card𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * ↑((singleton hij).join φsucΛ).timeContract * b) =
  ↑((singleton hij).join φsucΛ).timeContract * timeOrder (a * b) simp [hn]All goals completed! 🐙𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * ↑((singleton hij).join φsucΛ).timeContract * b) =
  ↑((singleton hij).join φsucΛ).timeContract * timeOrder (a * b)) hl𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * ↑((singleton hij).join φsucΛ).timeContract * b) =
  ↑((singleton hij).join φsucΛ).timeContract * timeOrder (a * b)
    rw [join_timeContract𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * ↑((singleton hij).timeContract * φsucΛ.timeContract) * b) =
  ↑((singleton hij).timeContract * φsucΛ.timeContract) * timeOrder (a * b)𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * ↑((singleton hij).timeContract * φsucΛ.timeContract) * b) =
  ↑((singleton hij).timeContract * φsucΛ.timeContract) * timeOrder (a * b)]𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * ↑((singleton hij).timeContract * φsucΛ.timeContract) * b) =
  ↑((singleton hij).timeContract * φsucΛ.timeContract) * timeOrder (a * b)
    rw [singleton_timeContract𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * ↑(⟨WickAlgebra.timeContract φs[i] φs[j], ⋯⟩ * φsucΛ.timeContract) * b) =
  ↑(⟨WickAlgebra.timeContract φs[i] φs[j], ⋯⟩ * φsucΛ.timeContract) * timeOrder (a * b)𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * ↑(⟨WickAlgebra.timeContract φs[i] φs[j], ⋯⟩ * φsucΛ.timeContract) * b) =
  ↑(⟨WickAlgebra.timeContract φs[i] φs[j], ⋯⟩ * φsucΛ.timeContract) * timeOrder (a * b)]𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * ↑(⟨WickAlgebra.timeContract φs[i] φs[j], ⋯⟩ * φsucΛ.timeContract) * b) =
  ↑(⟨WickAlgebra.timeContract φs[i] φs[j], ⋯⟩ * φsucΛ.timeContract) * timeOrder (a * b)
    simp only [Fin.getElem_fin, MulMemClass.coe_mul]𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * (WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract) * b) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)
    trans timeOrder (a * WickAlgebra.timeContract φs[↑i] φs[↑j] * (φsucΛ.timeContract.1 * b))𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * (WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract) * b) =
  timeOrder (a * WickAlgebra.timeContract φs[i] φs[j] * (↑φsucΛ.timeContract * b))𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * WickAlgebra.timeContract φs[i] φs[j] * (↑φsucΛ.timeContract * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)
    simp only [mul_assoc, Fin.getElem_fin]𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrder (a * WickAlgebra.timeContract φs[i] φs[j] * (↑φsucΛ.timeContract * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)
    rw [timeOrder_timeContract_eq_time_mid𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ WickAlgebra.timeContract φs[i] φs[j] * timeOrder (a * (↑φsucΛ.timeContract * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i]𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ WickAlgebra.timeContract φs[i] φs[j] * timeOrder (a * (↑φsucΛ.timeContract * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i]]𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ WickAlgebra.timeContract φs[i] φs[j] * timeOrder (a * (↑φsucΛ.timeContract * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i]
    have ih := timeOrder_timeContract_mul_of_eqTimeOnly_mid_induction φsucΛ h3 a b n (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ (↑φsucΛ).card = n𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).cardih:timeOrder (a * ↑φsucΛ.timeContract * b) = ↑φsucΛ.timeContract * timeOrder (a * b)⊢ WickAlgebra.timeContract φs[i] φs[j] * timeOrder (a * (↑φsucΛ.timeContract * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i] omegaAll goals completed! 🐙𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).cardih:timeOrder (a * ↑φsucΛ.timeContract * b) = ↑φsucΛ.timeContract * timeOrder (a * b)⊢ WickAlgebra.timeContract φs[i] φs[j] * timeOrder (a * (↑φsucΛ.timeContract * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i])𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).cardih:timeOrder (a * ↑φsucΛ.timeContract * b) = ↑φsucΛ.timeContract * timeOrder (a * b)⊢ WickAlgebra.timeContract φs[i] φs[j] * timeOrder (a * (↑φsucΛ.timeContract * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i]
    rw [← mul_assoc,𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).cardih:timeOrder (a * ↑φsucΛ.timeContract * b) = ↑φsucΛ.timeContract * timeOrder (a * b)⊢ WickAlgebra.timeContract φs[i] φs[j] * timeOrder (a * ↑φsucΛ.timeContract * b) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i]𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).cardih:timeOrder (a * ↑φsucΛ.timeContract * b) = ↑φsucΛ.timeContract * timeOrder (a * b)⊢ WickAlgebra.timeContract φs[i] φs[j] * (↑φsucΛ.timeContract * timeOrder (a * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i] ih𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).cardih:timeOrder (a * ↑φsucΛ.timeContract * b) = ↑φsucΛ.timeContract * timeOrder (a * b)⊢ WickAlgebra.timeContract φs[i] φs[j] * (↑φsucΛ.timeContract * timeOrder (a * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i]𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).cardih:timeOrder (a * ↑φsucΛ.timeContract * b) = ↑φsucΛ.timeContract * timeOrder (a * b)⊢ WickAlgebra.timeContract φs[i] φs[j] * (↑φsucΛ.timeContract * timeOrder (a * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i]]𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).cardih:timeOrder (a * ↑φsucΛ.timeContract * b) = ↑φsucΛ.timeContract * timeOrder (a * b)⊢ WickAlgebra.timeContract φs[i] φs[j] * (↑φsucΛ.timeContract * timeOrder (a * b)) =
  WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract * timeOrder (a * b)h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i]
    simp only [Fin.getElem_fin, mul_assoc]h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[i] φs[j]h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpa:𝓕.WickAlgebrab:𝓕.WickAlgebran:ℕi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhl:((singleton hij).join φsucΛ).EqTimeOnlyhn:(↑((singleton hij).join φsucΛ)).card = n.succh2:timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h3:φsucΛ.EqTimeOnlyh4:(↑φsucΛ).card + 1 = (↑((singleton hij).join φsucΛ)).card⊢ timeOrderRel φs[j] φs[i]
    all_goals simp_allAll goals completed! 🐙lemma timeOrder_timeContract_mul_of_eqTimeOnly_mid {φs : List 𝓕.FieldOp}
    (φsΛ : WickContraction φs.length)
    (hl : φsΛ.EqTimeOnly) (a b : 𝓕.WickAlgebra) :
    𝓣(a * φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(a * b) :=
  timeOrder_timeContract_mul_of_eqTimeOnly_mid_induction φsΛ hl a b φsΛ.1.card rfl

Let φs be a list of 𝓕.FieldOp, φsΛ a WickContraction of φs within which every contraction involves two 𝓕.FieldOps that have the same time and b a general element in 𝓕.WickAlgebra. Then 𝓣(φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(b).

This follows from properties of orderings and the ideal defining 𝓕.WickAlgebra.


lemma timeOrder_timeContract_mul_of_eqTimeOnly_left {φs : List 𝓕.FieldOp}
    (φsΛ : WickContraction φs.length)
    (hl : φsΛ.EqTimeOnly) (b : 𝓕.WickAlgebra) :
    𝓣(φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(b) := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlyb:𝓕.WickAlgebra⊢ timeOrder (↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder b
  trans 𝓣(1 * φsΛ.timeContract.1 * b)𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlyb:𝓕.WickAlgebra⊢ timeOrder (↑φsΛ.timeContract * b) = timeOrder (1 * ↑φsΛ.timeContract * b)𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlyb:𝓕.WickAlgebra⊢ timeOrder (1 * ↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder b
  simp only [one_mul]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlyb:𝓕.WickAlgebra⊢ timeOrder (1 * ↑φsΛ.timeContract * b) = ↑φsΛ.timeContract * timeOrder b
  rw [timeOrder_timeContract_mul_of_eqTimeOnly_mid φsΛ hl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlyb:𝓕.WickAlgebra⊢ ↑φsΛ.timeContract * timeOrder (1 * b) = ↑φsΛ.timeContract * timeOrder b𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlyb:𝓕.WickAlgebra⊢ ↑φsΛ.timeContract * timeOrder (1 * b) = ↑φsΛ.timeContract * timeOrder b]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:φsΛ.EqTimeOnlyb:𝓕.WickAlgebra⊢ ↑φsΛ.timeContract * timeOrder (1 * b) = ↑φsΛ.timeContract * timeOrder b
  simpAll goals completed! 🐙



set_option maxHeartbeats 400000 in
lemma exists_join_singleton_of_not_eqTimeOnly {φs : List 𝓕.FieldOp}
    (φsΛ : WickContraction φs.length) (h1 : ¬ φsΛ.EqTimeOnly) :
    ∃ (i j : Fin φs.length) (h : i < j) (φsucΛ : WickContraction [singleton h]ᵘᶜ.length),
    φsΛ = join (singleton h) φsucΛ ∧ (¬ timeOrderRel φs[i] φs[j] ∨ ¬ timeOrderRel φs[j] φs[i]) := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:¬φsΛ.EqTimeOnly⊢ ∃ i j, ∃ (h : i < j), ∃ φsucΛ, φsΛ = (singleton h).join φsucΛ ∧ (¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i])
  rw [eqTimeOnly_iff_forall_finset𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:¬∀ (a : ↥↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]⊢ ∃ i j, ∃ (h : i < j), ∃ φsucΛ, φsΛ = (singleton h).join φsucΛ ∧ (¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i])𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:¬∀ (a : ↥↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]⊢ ∃ i j, ∃ (h : i < j), ∃ φsucΛ, φsΛ = (singleton h).join φsucΛ ∧ (¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i])] at h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:¬∀ (a : ↥↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]⊢ ∃ i j, ∃ (h : i < j), ∃ φsucΛ, φsΛ = (singleton h).join φsucΛ ∧ (¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i])
  simp only [Fin.getElem_fin, Subtype.forall, not_forall, not_and] at h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:∃ x,
  ∃ (h : x ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨x, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨x, ⋯⟩)] →
      ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨x, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨x, ⋯⟩)]⊢ ∃ i j, ∃ (h : i < j), ∃ φsucΛ, φsΛ = (singleton h).join φsucΛ ∧ (¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i])
  obtain ⟨a, ha, hr⟩ := h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ∃ i j, ∃ (h : i < j), ∃ φsucΛ, φsΛ = (singleton h).join φsucΛ ∧ (¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i])
  use φsΛ.fstFieldOfContract ⟨a, ha⟩h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ∃ j,
  ∃ (h : φsΛ.fstFieldOfContract ⟨a, ha⟩ < j),
    ∃ φsucΛ,
      φsΛ = (singleton h).join φsucΛ ∧
        (¬timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[j] ∨
          ¬timeOrderRel φs[j] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩])
  use φsΛ.sndFieldOfContract ⟨a, ha⟩h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ∃ (h : φsΛ.fstFieldOfContract ⟨a, ha⟩ < φsΛ.sndFieldOfContract ⟨a, ha⟩),
  ∃ φsucΛ,
    φsΛ = (singleton h).join φsucΛ ∧
      (¬timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∨
        ¬timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩])
  use φsΛ.fstFieldOfContract_lt_sndFieldOfContract ⟨a, ha⟩h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (¬timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∨
      ¬timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩])
  let φsucΛ :
    WickContraction [singleton (φsΛ.fstFieldOfContract_lt_sndFieldOfContract ⟨a, ha⟩)]ᵘᶜ.length :=
    congr (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ [subContraction {a} ?m.167]ᵘᶜ.length = [singleton ⋯]ᵘᶜ.lengthh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (¬timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∨
      ¬timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) simp [← subContraction_singleton_eq_singleton]All goals completed! 🐙h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (¬timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∨
      ¬timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]))
      (φsΛ.quotContraction {a} (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ {a} ⊆ ↑φsΛh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (¬timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∨
      ¬timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩]) simpa using haAll goals completed! 🐙h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (¬timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∨
      ¬timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩])))h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ∃ φsucΛ,
  φsΛ = (singleton ⋯).join φsucΛ ∧
    (¬timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∨
      ¬timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩])
  use φsucΛh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsΛ = (singleton ⋯).join φsucΛ ∧
  (¬timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, ha⟩] φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] ∨
    ¬timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, ha⟩] φs[φsΛ.fstFieldOfContract ⟨a, ha⟩])
  simp only [Fin.getElem_fin]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsΛ = (singleton ⋯).join φsucΛ ∧
  (¬timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] ∨
    ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)])
  apply And.introh.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsΛ = (singleton ⋯).join φsucΛh.right𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ¬timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] ∨
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)]
  ·h.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ φsΛ = (singleton ⋯).join φsucΛ have h1 := join_congr (subContraction_singleton_eq_singleton _ ⟨a, ha⟩).symm (φsucΛ := φsucΛ)h.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:(singleton ⋯).join φsucΛ = (subContraction {↑⟨a, ha⟩} ⋯).join ((congr ⋯) φsucΛ)⊢ φsΛ = (singleton ⋯).join φsucΛ
    simp only [h1, congr_trans_apply, congr_refl, φsucΛ]h.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:(singleton ⋯).join φsucΛ = (subContraction {↑⟨a, ha⟩} ⋯).join ((congr ⋯) φsucΛ)⊢ φsΛ = (subContraction {a} ⋯).join (quotContraction {a} ⋯)
    rw [join_sub_quoth.left𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)h1:(singleton ⋯).join φsucΛ = (subContraction {↑⟨a, ha⟩} ⋯).join ((congr ⋯) φsucΛ)⊢ φsΛ = φsΛAll goals completed! 🐙]All goals completed! 🐙
  ·h.right𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛhr:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]φsucΛ:WickContraction [singleton ⋯]ᵘᶜ.length := (congr ⋯) (quotContraction {a} ⋯)⊢ ¬timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] ∨
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ha⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ha⟩)] tautoAll goals completed! 🐙



lemma timeOrder_timeContract_of_not_eqTimeOnly {φs : List 𝓕.FieldOp}
    (φsΛ : WickContraction φs.length)
    (hl : ¬ φsΛ.EqTimeOnly) : 𝓣(φsΛ.timeContract.1) = 0 := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:¬φsΛ.EqTimeOnly⊢ timeOrder ↑φsΛ.timeContract = 0
  obtain ⟨i, j, hij, φsucΛ, rfl, hr⟩ := exists_join_singleton_of_not_eqTimeOnly φsΛ hl𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑((singleton hij).join φsucΛ).timeContract = 0
  rw [join_timeContract𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑((singleton hij).timeContract * φsucΛ.timeContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑((singleton hij).timeContract * φsucΛ.timeContract) = 0]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑((singleton hij).timeContract * φsucΛ.timeContract) = 0
  rw [singleton_timeContract𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑(⟨WickAlgebra.timeContract φs[i] φs[j], ⋯⟩ * φsucΛ.timeContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑(⟨WickAlgebra.timeContract φs[i] φs[j], ⋯⟩ * φsucΛ.timeContract) = 0]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑(⟨WickAlgebra.timeContract φs[i] φs[j], ⋯⟩ * φsucΛ.timeContract) = 0
  simp only [Fin.getElem_fin, MulMemClass.coe_mul]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (WickAlgebra.timeContract φs[↑i] φs[↑j] * ↑φsucΛ.timeContract) = 0
  rw [timeOrder_timeOrder_left𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (timeOrder (WickAlgebra.timeContract φs[↑i] φs[↑j]) * ↑φsucΛ.timeContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (timeOrder (WickAlgebra.timeContract φs[↑i] φs[↑j]) * ↑φsucΛ.timeContract) = 0]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (timeOrder (WickAlgebra.timeContract φs[↑i] φs[↑j]) * ↑φsucΛ.timeContract) = 0
  rw [timeOrder_timeContract_ne_time𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (0 * ↑φsucΛ.timeContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ ¬(timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i])𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (0 * ↑φsucΛ.timeContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ ¬(timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i])]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (0 * ↑φsucΛ.timeContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ ¬(timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i])
  simp only [zero_mul, map_zero]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ ¬(timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i])
  tautoAll goals completed! 🐙

Let φs be a list of 𝓕.FieldOp and φsΛ a WickContraction with at least one contraction between 𝓕.FieldOp that do not have the same time. Then 𝓣(φsΛ.staticContract.1) = 0.


lemma timeOrder_staticContract_of_not_mem {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
    (hl : ¬ φsΛ.EqTimeOnly) : 𝓣(φsΛ.staticContract.1) = 0 := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthhl:¬φsΛ.EqTimeOnly⊢ timeOrder ↑φsΛ.staticContract = 0
  obtain ⟨i, j, hij, φsucΛ, rfl, hr⟩ := exists_join_singleton_of_not_eqTimeOnly φsΛ hl𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑((singleton hij).join φsucΛ).staticContract = 0
  rw [join_staticContract𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑((singleton hij).staticContract * φsucΛ.staticContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑((singleton hij).staticContract * φsucΛ.staticContract) = 0]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ↑((singleton hij).staticContract * φsucΛ.staticContract) = 0
  simp only [MulMemClass.coe_mul]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (↑(singleton hij).staticContract * ↑φsucΛ.staticContract) = 0
  rw [singleton_staticContract𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ((superCommute (anPart φs[i])) (ofFieldOp φs[j]) * ↑φsucΛ.staticContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ((superCommute (anPart φs[i])) (ofFieldOp φs[j]) * ↑φsucΛ.staticContract) = 0]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder ((superCommute (anPart φs[i])) (ofFieldOp φs[j]) * ↑φsucΛ.staticContract) = 0
  rw [timeOrder_timeOrder_left𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (timeOrder ((superCommute (anPart φs[i])) (ofFieldOp φs[j])) * ↑φsucΛ.staticContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (timeOrder ((superCommute (anPart φs[i])) (ofFieldOp φs[j])) * ↑φsucΛ.staticContract) = 0]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (timeOrder ((superCommute (anPart φs[i])) (ofFieldOp φs[j])) * ↑φsucΛ.staticContract) = 0
  rw [timeOrder_superCommute_anPart_ofFieldOp_ne_time𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (0 * ↑φsucΛ.staticContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ ¬(timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i])𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (0 * ↑φsucΛ.staticContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ ¬(timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i])]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ timeOrder (0 * ↑φsucΛ.staticContract) = 0𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ ¬(timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i])
  simp only [zero_mul, map_zero]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthhij:i < jφsucΛ:WickContraction [singleton hij]ᵘᶜ.lengthhr:¬timeOrderRel φs[i] φs[j] ∨ ¬timeOrderRel φs[j] φs[i]hl:¬((singleton hij).join φsucΛ).EqTimeOnly⊢ ¬(timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i])
  tautoAll goals completed! 🐙

The condition on a Wick contraction which is true if it has at least one contraction which is between two equal time fields.

def HaveEqTime {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) : Prop :=
  ∃ (i j : Fin φs.length) (h : {i, j} ∈ φsΛ.1),
  timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]



lemma haveEqTime_iff_finset {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) :
    HaveEqTime φsΛ ↔ ∃ (a : Finset (Fin φs.length)) (h : a ∈ φsΛ.1),
      timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, h⟩] φs[φsΛ.sndFieldOfContract ⟨a, h⟩]
    ∧ timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, h⟩] φs[φsΛ.fstFieldOfContract ⟨a, h⟩] := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ φsΛ.HaveEqTime ↔
  ∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, h⟩] φs[φsΛ.sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, h⟩] φs[φsΛ.fstFieldOfContract ⟨a, h⟩]
  simp only [HaveEqTime, Fin.getElem_fin, exists_and_left, exists_prop]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ (∃ i j, timeOrderRel φs[↑i] φs[↑j] ∧ {i, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑i]) ↔
  ∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]
  apply Iff.intromp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ (∃ i j, timeOrderRel φs[↑i] φs[↑j] ∧ {i, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑i]) →
  ∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]mpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ (∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]) →
  ∃ i j, timeOrderRel φs[↑i] φs[↑j] ∧ {i, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑i]
  ·mp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ (∃ i j, timeOrderRel φs[↑i] φs[↑j] ∧ {i, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑i]) →
  ∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] intro hmp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∃ i j, timeOrderRel φs[↑i] φs[↑j] ∧ {i, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑i]⊢ ∃ a,
  ∃ (h : a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]
    obtain ⟨i, j, hij, h1, h2⟩ := hmp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]⊢ ∃ a,
  ∃ (h : a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]
    use {i, j}, h1h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
    by_cases hij : i < jpos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
    ·pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] have h1n := eq_fstFieldOfContract_of_mem φsΛ ⟨{i,j}, h1⟩ i j (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < j⊢ i ∈ ↑⟨{i, j}, h1⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < j⊢ j ∈ ↑⟨{i, j}, h1⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]) hijpos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
      have h2n := eq_sndFieldOfContract_of_mem φsΛ ⟨{i,j}, h1⟩ i j (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ i ∈ ↑⟨{i, j}, h1⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ih2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ih2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = i⊢ j ∈ ↑⟨{i, j}, h1⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ih2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ih2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]) hijpos𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:i < jh1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = ih2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
      simp_all only [true_and]All goals completed! 🐙
    ·neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] have hineqj : i ≠ j := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ φsΛ.HaveEqTime ↔
  ∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, h⟩] φs[φsΛ.sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, h⟩] φs[φsΛ.fstFieldOfContract ⟨a, h⟩]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
        by_contra hineqj𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i = j⊢ Falseneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
        subst hineqj𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑i]h1:{i, i} ∈ ↑φsΛh2:timeOrderRel φs[↑i] φs[↑i]hij:¬i < i⊢ Falseneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
        have h2 := φsΛ.2.1 {i, i} h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑i]h1:{i, i} ∈ ↑φsΛh2✝:timeOrderRel φs[↑i] φs[↑i]hij:¬i < ih2:{i, i}.card = 2⊢ Falseneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
        simp_allneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
      have hji : j < i := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ φsΛ.HaveEqTime ↔
  ∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, h⟩] φs[φsΛ.sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, h⟩] φs[φsΛ.fstFieldOfContract ⟨a, h⟩]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] omeganeg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
      have h1n := eq_fstFieldOfContract_of_mem φsΛ ⟨{i,j}, h1⟩ j i (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < i⊢ j ∈ ↑⟨{i, j}, h1⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < i⊢ i ∈ ↑⟨{i, j}, h1⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]) hjineg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
      have h2n := eq_sndFieldOfContract_of_mem φsΛ ⟨{i,j}, h1⟩ j i (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ j ∈ ↑⟨{i, j}, h1⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jh2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jh2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = j⊢ i ∈ ↑⟨{i, j}, h1⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jh2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jh2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]) hjineg𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij✝:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]hij:¬i < jhineqj:i ≠ jhji:j < ih1n:φsΛ.fstFieldOfContract ⟨{i, j}, h1⟩ = jh2n:φsΛ.sndFieldOfContract ⟨{i, j}, h1⟩ = i⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, ⋯⟩)]
      simp_allAll goals completed! 🐙
  ·mpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ (∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]) →
  ∃ i j, timeOrderRel φs[↑i] φs[↑j] ∧ {i, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑i] intro hmpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∃ a,
  ∃ (h : a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ∃ i j, timeOrderRel φs[↑i] φs[↑j] ∧ {i, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑i]
    obtain ⟨a, h1, h2, h3⟩ := hmpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)h1:a ∈ ↑φsΛh2:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)]h3:timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ∃ i j, timeOrderRel φs[↑i] φs[↑j] ∧ {i, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑i]
    use φsΛ.fstFieldOfContract ⟨a, h1⟩h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)h1:a ∈ ↑φsΛh2:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)]h3:timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ∃ j,
  timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, h1⟩)] φs[↑j] ∧
    {φsΛ.fstFieldOfContract ⟨a, h1⟩, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑(φsΛ.fstFieldOfContract ⟨a, h1⟩)]
    use φsΛ.sndFieldOfContract ⟨a, h1⟩h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)h1:a ∈ ↑φsΛh2:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)]h3:timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, h1⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, h1⟩)] ∧
  {φsΛ.fstFieldOfContract ⟨a, h1⟩, φsΛ.sndFieldOfContract ⟨a, h1⟩} ∈ ↑φsΛ ∧
    timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, h1⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, h1⟩)]
    simp_all only [and_true, true_and]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)h1:a ∈ ↑φsΛh2:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)]h3:timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ {φsΛ.fstFieldOfContract ⟨a, h1⟩, φsΛ.sndFieldOfContract ⟨a, h1⟩} ∈ ↑φsΛ
    rw [← finset_eq_fstFieldOfContract_sndFieldOfContracth𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)h1:a ∈ ↑φsΛh2:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)]h3:timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ↑⟨a, h1⟩ ∈ ↑φsΛh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)h1:a ∈ ↑φsΛh2:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)]h3:timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ↑⟨a, h1⟩ ∈ ↑φsΛ]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)h1:a ∈ ↑φsΛh2:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)]h3:timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, ⋯⟩)]⊢ ↑⟨a, h1⟩ ∈ ↑φsΛ
    exact h1All goals completed! 🐙



@[simp]
lemma empty_not_haveEqTime {φs : List 𝓕.FieldOp} :
    ¬ HaveEqTime (empty : WickContraction φs.length) := by𝓕:FieldSpecificationφs:List 𝓕.FieldOp⊢ ¬empty.HaveEqTime
  rw [haveEqTime_iff_finset𝓕:FieldSpecificationφs:List 𝓕.FieldOp⊢ ¬∃ a,
    ∃ (h : a ∈ ↑empty),
      timeOrderRel φs[empty.fstFieldOfContract ⟨a, h⟩] φs[empty.sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel φs[empty.sndFieldOfContract ⟨a, h⟩] φs[empty.fstFieldOfContract ⟨a, h⟩]𝓕:FieldSpecificationφs:List 𝓕.FieldOp⊢ ¬∃ a,
    ∃ (h : a ∈ ↑empty),
      timeOrderRel φs[empty.fstFieldOfContract ⟨a, h⟩] φs[empty.sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel φs[empty.sndFieldOfContract ⟨a, h⟩] φs[empty.fstFieldOfContract ⟨a, h⟩]]𝓕:FieldSpecificationφs:List 𝓕.FieldOp⊢ ¬∃ a,
    ∃ (h : a ∈ ↑empty),
      timeOrderRel φs[empty.fstFieldOfContract ⟨a, h⟩] φs[empty.sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel φs[empty.sndFieldOfContract ⟨a, h⟩] φs[empty.fstFieldOfContract ⟨a, h⟩]
  simp [empty]All goals completed! 🐙

Given a Wick contraction the subset of contracted pairs between equal time fields.

def eqTimeContractSet {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) :
    Finset (Finset (Fin φs.length)) :=
  Finset.univ.filter (fun a =>
    a ∈ φsΛ.1 ∧ ∀ (h : a ∈ φsΛ.1),
    timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, h⟩] φs[φsΛ.sndFieldOfContract ⟨a, h⟩]
    ∧ timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, h⟩] φs[φsΛ.fstFieldOfContract ⟨a, h⟩])

              lemma eqTimeContractSet_subset {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) :
    eqTimeContractSet φsΛ ⊆ φsΛ.1 := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ φsΛ.eqTimeContractSet ⊆ ↑φsΛ
  intro a ha𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ φsΛ.eqTimeContractSet⊢ a ∈ ↑φsΛ
  simp only [eqTimeContractSet, Finset.mem_filter, Finset.mem_univ, true_and] at ha𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)ha:a ∈ ↑φsΛ ∧
  ∀ (h : a ∈ ↑φsΛ),
    timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, h⟩] φs[φsΛ.sndFieldOfContract ⟨a, h⟩] ∧
      timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, h⟩] φs[φsΛ.fstFieldOfContract ⟨a, h⟩]⊢ a ∈ ↑φsΛ
  exact ha.1All goals completed! 🐙lemma mem_of_mem_eqTimeContractSet{φs : List 𝓕.FieldOp} {φsΛ : WickContraction φs.length}
    {a : Finset (Fin φs.length)} (h : a ∈ eqTimeContractSet φsΛ) : a ∈ φsΛ.1 :=
  eqTimeContractSet_subset φsΛ h

lemma join_eqTimeContractSet {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
    (φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) :
    eqTimeContractSet (join φsΛ φsucΛ) = φsΛ.eqTimeContractSet ∪
    φsucΛ.eqTimeContractSet.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.length⊢ (φsΛ.join φsucΛ).eqTimeContractSet =
  φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet
  ext a𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSet ↔
  a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet
  apply Iff.intromp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSet →
  a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSetmpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet →
  a ∈ (φsΛ.join φsucΛ).eqTimeContractSet
  ·mp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSet →
  a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet intro hmp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ (φsΛ.join φsucΛ).eqTimeContractSet⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet
    have hmem := mem_of_mem_eqTimeContractSet hmp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:a ∈ ↑(φsΛ.join φsucΛ)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet
    have ht := joinLiftLeft_or_joinLiftRight_of_mem_join (φsucΛ := φsucΛ) _ hmemmp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:a ∈ ↑(φsΛ.join φsucΛ)ht:(∃ b, a = ↑(joinLiftLeft b)) ∨ ∃ b, a = ↑(joinLiftRight b)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet
    simp only [Finset.mem_union, Finset.mem_map, RelEmbedding.coe_toEmbedding]mp𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:a ∈ ↑(φsΛ.join φsucΛ)ht:(∃ b, a = ↑(joinLiftLeft b)) ∨ ∃ b, a = ↑(joinLiftRight b)⊢ a ∈ φsΛ.eqTimeContractSet ∨ ∃ a_1 ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a_1 = a
    rcases ht with ht | htmp.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:a ∈ ↑(φsΛ.join φsucΛ)ht:∃ b, a = ↑(joinLiftLeft b)⊢ a ∈ φsΛ.eqTimeContractSet ∨ ∃ a_1 ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a_1 = amp.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:a ∈ ↑(φsΛ.join φsucΛ)ht:∃ b, a = ↑(joinLiftRight b)⊢ a ∈ φsΛ.eqTimeContractSet ∨ ∃ a_1 ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a_1 = a
    ·mp.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:a ∈ ↑(φsΛ.join φsucΛ)ht:∃ b, a = ↑(joinLiftLeft b)⊢ a ∈ φsΛ.eqTimeContractSet ∨ ∃ a_1 ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a_1 = a obtain ⟨b, rfl⟩ := htmp.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsΛh:↑(joinLiftLeft b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftLeft b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑(joinLiftLeft b) ∈ φsΛ.eqTimeContractSet ∨
  ∃ a ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a = ↑(joinLiftLeft b)
      leftmp.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsΛh:↑(joinLiftLeft b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftLeft b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑(joinLiftLeft b) ∈ φsΛ.eqTimeContractSet
      simp only [eqTimeContractSet, Fin.getElem_fin, Finset.mem_filter, Finset.mem_univ, true_and]mp.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsΛh:↑(joinLiftLeft b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftLeft b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑(joinLiftLeft b) ∈ ↑φsΛ ∧
  ∀ (h : ↑(joinLiftLeft b) ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨↑(joinLiftLeft b), h⟩)]
        φs[↑(φsΛ.sndFieldOfContract ⟨↑(joinLiftLeft b), h⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨↑(joinLiftLeft b), h⟩)]
        φs[↑(φsΛ.fstFieldOfContract ⟨↑(joinLiftLeft b), h⟩)]
      apply And.intro (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsΛh:↑(joinLiftLeft b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftLeft b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑(joinLiftLeft b) ∈ ↑φsΛ simp [joinLiftLeft]All goals completed! 🐙)
      intro h'mp.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsΛh:↑(joinLiftLeft b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftLeft b) ∈ ↑(φsΛ.join φsucΛ)h':↑(joinLiftLeft b) ∈ ↑φsΛ⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨↑(joinLiftLeft b), h'⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨↑(joinLiftLeft b), h'⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨↑(joinLiftLeft b), h'⟩)]
    φs[↑(φsΛ.fstFieldOfContract ⟨↑(joinLiftLeft b), h'⟩)]
      simp only [eqTimeContractSet, Fin.getElem_fin, Finset.mem_filter, Finset.mem_univ,
        Finset.coe_mem, Subtype.coe_eta, join_fstFieldOfContract_joinLiftLeft,
        join_sndFieldOfContract_joinLift, forall_true_left, true_and] at hmp.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsΛhmem:↑(joinLiftLeft b) ∈ ↑(φsΛ.join φsucΛ)h':↑(joinLiftLeft b) ∈ ↑φsΛh:timeOrderRel φs[↑(φsΛ.fstFieldOfContract b)] φs[↑(φsΛ.sndFieldOfContract b)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract b)] φs[↑(φsΛ.fstFieldOfContract b)]⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨↑(joinLiftLeft b), h'⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨↑(joinLiftLeft b), h'⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨↑(joinLiftLeft b), h'⟩)]
    φs[↑(φsΛ.fstFieldOfContract ⟨↑(joinLiftLeft b), h'⟩)]
      exact hAll goals completed! 🐙
    ·mp.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:a ∈ ↑(φsΛ.join φsucΛ)ht:∃ b, a = ↑(joinLiftRight b)⊢ a ∈ φsΛ.eqTimeContractSet ∨ ∃ a_1 ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a_1 = a obtain ⟨b, rfl⟩ := htmp.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsucΛh:↑(joinLiftRight b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftRight b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑(joinLiftRight b) ∈ φsΛ.eqTimeContractSet ∨
  ∃ a ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a = ↑(joinLiftRight b)
      rightmp.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsucΛh:↑(joinLiftRight b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftRight b) ∈ ↑(φsΛ.join φsucΛ)⊢ ∃ a ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a = ↑(joinLiftRight b)
      use bh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsucΛh:↑(joinLiftRight b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftRight b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑b ∈ φsucΛ.eqTimeContractSet ∧ (Finset.mapEmbedding uncontractedListEmd) ↑b = ↑(joinLiftRight b)
      rw [Finset.mapEmbedding_applyh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsucΛh:↑(joinLiftRight b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftRight b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑b ∈ φsucΛ.eqTimeContractSet ∧ Finset.map uncontractedListEmd ↑b = ↑(joinLiftRight b)h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsucΛh:↑(joinLiftRight b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftRight b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑b ∈ φsucΛ.eqTimeContractSet ∧ Finset.map uncontractedListEmd ↑b = ↑(joinLiftRight b)]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsucΛh:↑(joinLiftRight b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftRight b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑b ∈ φsucΛ.eqTimeContractSet ∧ Finset.map uncontractedListEmd ↑b = ↑(joinLiftRight b)
      simp only [joinLiftRight, and_true]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:↥↑φsucΛh:↑(joinLiftRight b) ∈ (φsΛ.join φsucΛ).eqTimeContractSethmem:↑(joinLiftRight b) ∈ ↑(φsΛ.join φsucΛ)⊢ ↑b ∈ φsucΛ.eqTimeContractSet
      simpa [eqTimeContractSet] using hAll goals completed! 🐙
  ·mpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet →
  a ∈ (φsΛ.join φsucΛ).eqTimeContractSet intro hmpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSet
    simp only [Finset.mem_union, Finset.mem_map,
      RelEmbedding.coe_toEmbedding] at hmpr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ φsΛ.eqTimeContractSet ∨ ∃ a_1 ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a_1 = a⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSet
    rcases h with h | hmpr.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ φsΛ.eqTimeContractSet⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSetmpr.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:∃ a_1 ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a_1 = a⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSet
    ·mpr.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ φsΛ.eqTimeContractSet⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSet simp only [eqTimeContractSet, Fin.getElem_fin, Finset.mem_filter, Finset.mem_univ,
        true_and] at h ⊢mpr.inl𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ ↑φsΛ ∧
  ∀ (h : a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)]⊢ a ∈ ↑(φsΛ.join φsucΛ) ∧
  ∀ (h : a ∈ ↑(φsΛ.join φsucΛ)),
    timeOrderRel φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨a, h⟩)] φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨a, h⟩)] ∧
      timeOrderRel φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨a, h⟩)] φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨a, h⟩)]
      exact ⟨by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:a ∈ ↑φsΛ ∧
  ∀ (h : a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)]⊢ a ∈ ↑(φsΛ.join φsucΛ) simp [join, h.1]All goals completed! 🐙, fun _ => h.2 h.1⟩
    ·mpr.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:∃ a_1 ∈ φsucΛ.eqTimeContractSet, (Finset.mapEmbedding uncontractedListEmd) a_1 = a⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSet simp only [eqTimeContractSet, Fin.getElem_fin, Finset.mem_filter, Finset.mem_univ,
        true_and] at h ⊢mpr.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengtha:Finset (Fin φs.length)h:∃ a_1,
  (a_1 ∈ ↑φsucΛ ∧
      ∀ (h : a_1 ∈ ↑φsucΛ),
        timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨a_1, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨a_1, h⟩)] ∧
          timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨a_1, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨a_1, h⟩)]) ∧
    (Finset.mapEmbedding uncontractedListEmd) a_1 = a⊢ a ∈ ↑(φsΛ.join φsucΛ) ∧
  ∀ (h : a ∈ ↑(φsΛ.join φsucΛ)),
    timeOrderRel φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨a, h⟩)] φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨a, h⟩)] ∧
      timeOrderRel φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨a, h⟩)] φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨a, h⟩)]
      obtain ⟨b, h1, h2, rfl⟩ := hmpr.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:Finset (Fin [φsΛ]ᵘᶜ.length)h1:b ∈ ↑φsucΛ ∧
  ∀ (h : b ∈ ↑φsucΛ),
    timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] ∧
      timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)]⊢ (Finset.mapEmbedding uncontractedListEmd) b ∈ ↑(φsΛ.join φsucΛ) ∧
  ∀ (h : (Finset.mapEmbedding uncontractedListEmd) b ∈ ↑(φsΛ.join φsucΛ)),
    timeOrderRel φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h⟩)]
        φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h⟩)] ∧
      timeOrderRel φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h⟩)]
        φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h⟩)]
      refine ⟨by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:Finset (Fin [φsΛ]ᵘᶜ.length)h1:b ∈ ↑φsucΛ ∧
  ∀ (h : b ∈ ↑φsucΛ),
    timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] ∧
      timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)]⊢ (Finset.mapEmbedding uncontractedListEmd) b ∈ ↑(φsΛ.join φsucΛ) simp [join, h1]All goals completed! 🐙, fun h' => ?_⟩
      have h2 := h1.2 h1.1mpr.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:Finset (Fin [φsΛ]ᵘᶜ.length)h1:b ∈ ↑φsucΛ ∧
  ∀ (h : b ∈ ↑φsucΛ),
    timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] ∧
      timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)]h':(Finset.mapEmbedding uncontractedListEmd) b ∈ ↑(φsΛ.join φsucΛ)h2:timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] ∧
  timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)]⊢ timeOrderRel φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
    φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)] ∧
  timeOrderRel φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
    φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
      have hj : ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩
        = joinLiftRight ⟨b, h1.1⟩ := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.length⊢ (φsΛ.join φsucΛ).eqTimeContractSet =
  φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSetmpr.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:Finset (Fin [φsΛ]ᵘᶜ.length)h1:b ∈ ↑φsucΛ ∧
  ∀ (h : b ∈ ↑φsucΛ),
    timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] ∧
      timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)]h':(Finset.mapEmbedding uncontractedListEmd) b ∈ ↑(φsΛ.join φsucΛ)h2:timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] ∧
  timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)]hj:⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩ = joinLiftRight ⟨b, ⋯⟩⊢ timeOrderRel φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
    φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)] ∧
  timeOrderRel φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
    φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)] rflmpr.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:Finset (Fin [φsΛ]ᵘᶜ.length)h1:b ∈ ↑φsucΛ ∧
  ∀ (h : b ∈ ↑φsucΛ),
    timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] ∧
      timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)]h':(Finset.mapEmbedding uncontractedListEmd) b ∈ ↑(φsΛ.join φsucΛ)h2:timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] ∧
  timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)]hj:⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩ = joinLiftRight ⟨b, ⋯⟩⊢ timeOrderRel φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
    φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)] ∧
  timeOrderRel φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
    φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]mpr.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:Finset (Fin [φsΛ]ᵘᶜ.length)h1:b ∈ ↑φsucΛ ∧
  ∀ (h : b ∈ ↑φsucΛ),
    timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] ∧
      timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)]h':(Finset.mapEmbedding uncontractedListEmd) b ∈ ↑(φsΛ.join φsucΛ)h2:timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] ∧
  timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)]hj:⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩ = joinLiftRight ⟨b, ⋯⟩⊢ timeOrderRel φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
    φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)] ∧
  timeOrderRel φs[↑((φsΛ.join φsucΛ).sndFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
    φs[↑((φsΛ.join φsucΛ).fstFieldOfContract ⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩)]
      simp only [hj, join_fstFieldOfContract_joinLiftRight, getElem_uncontractedListEmd,
        join_sndFieldOfContract_joinLiftRight]mpr.inr𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthb:Finset (Fin [φsΛ]ᵘᶜ.length)h1:b ∈ ↑φsucΛ ∧
  ∀ (h : b ∈ ↑φsucΛ),
    timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] ∧
      timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, h⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, h⟩)]h':(Finset.mapEmbedding uncontractedListEmd) b ∈ ↑(φsΛ.join φsucΛ)h2:timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] ∧
  timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)]hj:⟨(Finset.mapEmbedding uncontractedListEmd) b, h'⟩ = joinLiftRight ⟨b, ⋯⟩⊢ timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] ∧
  timeOrderRel [φsΛ]ᵘᶜ[↑(φsucΛ.sndFieldOfContract ⟨b, ⋯⟩)] [φsΛ]ᵘᶜ[↑(φsucΛ.fstFieldOfContract ⟨b, ⋯⟩)]
      simpa using h2All goals completed! 🐙

lemma eqTimeContractSet_of_not_haveEqTime {φs : List 𝓕.FieldOp} {φsΛ : WickContraction φs.length}
    (h : ¬ HaveEqTime φsΛ) : eqTimeContractSet φsΛ = ∅ := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:¬φsΛ.HaveEqTime⊢ φsΛ.eqTimeContractSet = ∅
  ext a𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:¬φsΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ↔ a ∈ ∅
  simp only [Finset.notMem_empty, iff_false]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:¬φsΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∉ φsΛ.eqTimeContractSet
  by_contra hn𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:¬φsΛ.HaveEqTimea:Finset (Fin φs.length)hn:a ∈ φsΛ.eqTimeContractSet⊢ False
  rw [haveEqTime_iff_finset𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:¬∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, h⟩] φs[φsΛ.sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, h⟩] φs[φsΛ.fstFieldOfContract ⟨a, h⟩]a:Finset (Fin φs.length)hn:a ∈ φsΛ.eqTimeContractSet⊢ False𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:¬∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, h⟩] φs[φsΛ.sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, h⟩] φs[φsΛ.fstFieldOfContract ⟨a, h⟩]a:Finset (Fin φs.length)hn:a ∈ φsΛ.eqTimeContractSet⊢ False] at h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:¬∃ a,
    ∃ (h : a ∈ ↑φsΛ),
      timeOrderRel φs[φsΛ.fstFieldOfContract ⟨a, h⟩] φs[φsΛ.sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel φs[φsΛ.sndFieldOfContract ⟨a, h⟩] φs[φsΛ.fstFieldOfContract ⟨a, h⟩]a:Finset (Fin φs.length)hn:a ∈ φsΛ.eqTimeContractSet⊢ False
  simp only [Fin.getElem_fin, not_exists, not_and] at h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)hn:a ∈ φsΛ.eqTimeContractSeth:∀ (x : Finset (Fin φs.length)) (x_1 : x ∈ ↑φsΛ),
  timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨x, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨x, ⋯⟩)] →
    ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨x, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨x, ⋯⟩)]⊢ False
  simp only [eqTimeContractSet, Fin.getElem_fin, Finset.mem_filter, Finset.mem_univ, true_and] at hn𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin φs.length)h:∀ (x : Finset (Fin φs.length)) (x_1 : x ∈ ↑φsΛ),
  timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨x, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨x, ⋯⟩)] →
    ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨x, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨x, ⋯⟩)]hn:a ∈ ↑φsΛ ∧
  ∀ (h : a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)]⊢ False
  exact h a hn.1 (hn.2 hn.1).1 (hn.2 hn.1).2All goals completed! 🐙

lemma eqTimeContractSet_of_mem_eqTimeOnly {φs : List 𝓕.FieldOp} {φsΛ : WickContraction φs.length}
    (h : φsΛ.EqTimeOnly) : eqTimeContractSet φsΛ = φsΛ.1 := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnly⊢ φsΛ.eqTimeContractSet = ↑φsΛ
  ext a𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ↔ a ∈ ↑φsΛ
  simp only [eqTimeContractSet, Fin.getElem_fin, Finset.mem_filter, Finset.mem_univ, true_and,
    and_iff_left_iff_imp, imp_forall_iff_forall]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.EqTimeOnlya:Finset (Fin φs.length)⊢ ∀ (h : a ∈ ↑φsΛ),
  timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] ∧
    timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)]
  rw [EqTimeOnly.eqTimeOnly_iff_forall_finset𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]a:Finset (Fin φs.length)⊢ ∀ (h : a ∈ ↑φsΛ),
  timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] ∧
    timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]a:Finset (Fin φs.length)⊢ ∀ (h : a ∈ ↑φsΛ),
  timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] ∧
    timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)]] at h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∀ (a : ↥↑φsΛ),
  timeOrderRel φs[φsΛ.fstFieldOfContract a] φs[φsΛ.sndFieldOfContract a] ∧
    timeOrderRel φs[φsΛ.sndFieldOfContract a] φs[φsΛ.fstFieldOfContract a]a:Finset (Fin φs.length)⊢ ∀ (h : a ∈ ↑φsΛ),
  timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] ∧
    timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨a, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨a, h⟩)]
  exact fun h_1 => h ⟨a, h_1⟩All goals completed! 🐙

lemma subContraction_eqTimeContractSet_eqTimeOnly {φs : List 𝓕.FieldOp}
    (φsΛ : WickContraction φs.length) :
    (φsΛ.subContraction (eqTimeContractSet φsΛ) (eqTimeContractSet_subset φsΛ)).EqTimeOnly := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ (subContraction φsΛ.eqTimeContractSet ⋯).EqTimeOnly
  rw [EqTimeOnly.eqTimeOnly_iff_forall_finset𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ ∀ (a : ↥↑(subContraction φsΛ.eqTimeContractSet ⋯)),
  timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
      φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a] ∧
    timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a]
      φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ ∀ (a : ↥↑(subContraction φsΛ.eqTimeContractSet ⋯)),
  timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
      φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a] ∧
    timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a]
      φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ ∀ (a : ↥↑(subContraction φsΛ.eqTimeContractSet ⋯)),
  timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
      φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a] ∧
    timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a]
      φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
  intro a𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:↥↑(subContraction φsΛ.eqTimeContractSet ⋯)⊢ timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
    φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a] ∧
  timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a]
    φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
  have ha2 := a.2𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:↥↑(subContraction φsΛ.eqTimeContractSet ⋯)ha2:↑a ∈ ↑(subContraction φsΛ.eqTimeContractSet ⋯)⊢ timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
    φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a] ∧
  timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a]
    φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
  simp only [subContraction, eqTimeContractSet, Fin.getElem_fin, Finset.mem_filter, Finset.mem_univ,
    true_and] at ha2𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:↥↑(subContraction φsΛ.eqTimeContractSet ⋯)ha2:↑a ∈ ↑φsΛ ∧
  ∀ (h : ↑a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨↑a, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨↑a, h⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨↑a, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨↑a, h⟩)]⊢ timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
    φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a] ∧
  timeOrderRel φs[(subContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract a]
    φs[(subContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract a]
  simp only [subContraction_fstFieldOfContract, Fin.getElem_fin, subContraction_sndFieldOfContract]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:↥↑(subContraction φsΛ.eqTimeContractSet ⋯)ha2:↑a ∈ ↑φsΛ ∧
  ∀ (h : ↑a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨↑a, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨↑a, h⟩)] ∧
      timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨↑a, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨↑a, h⟩)]⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨↑a, ⋯⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨↑a, ⋯⟩)] ∧
  timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨↑a, ⋯⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨↑a, ⋯⟩)]
  exact ha2.2 ha2.1All goals completed! 🐙

lemma pair_mem_eqTimeContractSet_iff {φs : List 𝓕.FieldOp} {i j : Fin φs.length}
    (φsΛ : WickContraction φs.length) (h : {i, j} ∈ φsΛ.1) :
    {i, j} ∈ φsΛ.eqTimeContractSet ↔ timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i] := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛ⊢ {i, j} ∈ φsΛ.eqTimeContractSet ↔ timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]
  simp only [eqTimeContractSet, Fin.getElem_fin, Finset.mem_filter, Finset.mem_univ, true_and]𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛ⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
  by_cases hij : i < jpos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
  ·pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] have h1 := eq_fstFieldOfContract_of_mem φsΛ ⟨{i,j}, h⟩ i j (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < j⊢ i ∈ ↑⟨{i, j}, h⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < j⊢ j ∈ ↑⟨{i, j}, h⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]) hijpos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
    have h2 := eq_sndFieldOfContract_of_mem φsΛ ⟨{i,j}, h⟩ i j (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = i⊢ i ∈ ↑⟨{i, j}, h⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = ih2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = ih2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = i⊢ j ∈ ↑⟨{i, j}, h⟩pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = ih2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] simpAll goals completed! 🐙pos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = ih2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]) hijpos𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:i < jh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = ih2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
    simp_all only [forall_true_left, true_and]All goals completed! 🐙
  ·neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] have hineqj : i ≠ j := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛ⊢ {i, j} ∈ φsΛ.eqTimeContractSet ↔ timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
      by_contra hineqj𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i = j⊢ Falseneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
      subst hineqj𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, i} ∈ ↑φsΛhij:¬i < i⊢ Falseneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
      have h2 := φsΛ.2.1 {i, i} h𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, i} ∈ ↑φsΛhij:¬i < ih2:{i, i}.card = 2⊢ Falseneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
      simp_allneg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
    have hji : j < i := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛ⊢ {i, j} ∈ φsΛ.eqTimeContractSet ↔ timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] omeganeg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
    have h1 := eq_fstFieldOfContract_of_mem φsΛ ⟨{i,j}, h⟩ j i (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < i⊢ j ∈ ↑⟨{i, j}, h⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < i⊢ i ∈ ↑⟨{i, j}, h⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]) hjineg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = j⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
    have h2 := eq_sndFieldOfContract_of_mem φsΛ ⟨{i,j}, h⟩ j i (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = j⊢ j ∈ ↑⟨{i, j}, h⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = jh2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = jh2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]) (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = j⊢ i ∈ ↑⟨{i, j}, h⟩neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = jh2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i] simpAll goals completed! 🐙neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = jh2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]) hjineg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = jh2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧
    ∀ (h : {i, j} ∈ ↑φsΛ),
      timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] ∧
        timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨{i, j}, h⟩)] φs[↑(φsΛ.fstFieldOfContract ⟨{i, j}, h⟩)]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
    simp only [h1, h2]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛhij:¬i < jhineqj:i ≠ jhji:j < ih1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = jh2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = i⊢ ({i, j} ∈ ↑φsΛ ∧ ∀ (h : {i, j} ∈ ↑φsΛ), timeOrderRel φs[↑j] φs[↑i] ∧ timeOrderRel φs[↑i] φs[↑j]) ↔
  timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
    simp_all only [not_lt, ne_eq, forall_true_left, true_and]neg𝓕:FieldSpecificationφs:List 𝓕.FieldOpi:Fin φs.lengthj:Fin φs.lengthφsΛ:WickContraction φs.lengthh:{i, j} ∈ ↑φsΛh1:φsΛ.fstFieldOfContract ⟨{i, j}, h⟩ = jh2:φsΛ.sndFieldOfContract ⟨{i, j}, h⟩ = ihij:j ≤ ihineqj:¬i = jhji:j < i⊢ timeOrderRel φs[↑j] φs[↑i] ∧ timeOrderRel φs[↑i] φs[↑j] ↔ timeOrderRel φs[↑i] φs[↑j] ∧ timeOrderRel φs[↑j] φs[↑i]
    exact and_commAll goals completed! 🐙

lemma subContraction_eqTimeContractSet_not_empty_of_haveEqTime
    {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length) (h : HaveEqTime φsΛ) :
    φsΛ.subContraction (eqTimeContractSet φsΛ) (eqTimeContractSet_subset φsΛ) ≠ empty := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.HaveEqTime⊢ subContraction φsΛ.eqTimeContractSet ⋯ ≠ empty
  simp only [ne_eq]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.HaveEqTime⊢ ¬subContraction φsΛ.eqTimeContractSet ⋯ = empty
  erw [Subtype.ext_iff𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.HaveEqTime⊢ ¬↑(subContraction φsΛ.eqTimeContractSet ⋯) = ↑empty]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.HaveEqTime⊢ ¬↑(subContraction φsΛ.eqTimeContractSet ⋯) = ↑empty
  simp only [subContraction, empty]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.HaveEqTime⊢ ¬φsΛ.eqTimeContractSet = ∅
  rw [Finset.eq_empty_iff_forall_notMem𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.HaveEqTime⊢ ¬∀ (x : Finset (Fin φs.length)), x ∉ φsΛ.eqTimeContractSet𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.HaveEqTime⊢ ¬∀ (x : Finset (Fin φs.length)), x ∉ φsΛ.eqTimeContractSet]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:φsΛ.HaveEqTime⊢ ¬∀ (x : Finset (Fin φs.length)), x ∉ φsΛ.eqTimeContractSet
  simp only [HaveEqTime, Fin.getElem_fin, exists_and_left, exists_prop] at h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh:∃ i j, timeOrderRel φs[↑i] φs[↑j] ∧ {i, j} ∈ ↑φsΛ ∧ timeOrderRel φs[↑j] φs[↑i]⊢ ¬∀ (x : Finset (Fin φs.length)), x ∉ φsΛ.eqTimeContractSet
  obtain ⟨i, j, hij, h1, h2⟩ := h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]⊢ ¬∀ (x : Finset (Fin φs.length)), x ∉ φsΛ.eqTimeContractSet
  simp only [not_forall, Decidable.not_not]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]⊢ ∃ x, x ∈ φsΛ.eqTimeContractSet
  use {i, j}h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]⊢ {i, j} ∈ φsΛ.eqTimeContractSet
  rw [pair_mem_eqTimeContractSet_iff φsΛ h1h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]⊢ timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]⊢ timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthi:Fin φs.lengthj:Fin φs.lengthhij:timeOrderRel φs[↑i] φs[↑j]h1:{i, j} ∈ ↑φsΛh2:timeOrderRel φs[↑j] φs[↑i]⊢ timeOrderRel φs[i] φs[j] ∧ timeOrderRel φs[j] φs[i]
  simp_all only [Fin.getElem_fin, and_self]All goals completed! 🐙

set_option backward.isDefEq.respectTransparency false in
lemma quotContraction_eqTimeContractSet_not_haveEqTime {φs : List 𝓕.FieldOp}
    (φsΛ : WickContraction φs.length) :
    ¬ HaveEqTime (φsΛ.quotContraction (eqTimeContractSet φsΛ) (eqTimeContractSet_subset φsΛ)) := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ ¬(quotContraction φsΛ.eqTimeContractSet ⋯).HaveEqTime
  rw [haveEqTime_iff_finset𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ ¬∃ a,
    ∃ (h : a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)),
      timeOrderRel
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, h⟩]
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, h⟩]
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, h⟩]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ ¬∃ a,
    ∃ (h : a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)),
      timeOrderRel
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, h⟩]
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, h⟩]
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, h⟩]]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ ¬∃ a,
    ∃ (h : a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)),
      timeOrderRel
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, h⟩]
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, h⟩] ∧
        timeOrderRel
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, h⟩]
          [subContraction φsΛ.eqTimeContractSet
                ⋯]ᵘᶜ[(quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, h⟩]
  simp only [Fin.getElem_fin, not_exists, not_and]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.length⊢ ∀ (x : Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length))
  (x_1 : x ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)),
  timeOrderRel
      [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨x, ⋯⟩)]
      [subContraction φsΛ.eqTimeContractSet
            ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨x, ⋯⟩)] →
    ¬timeOrderRel
        [subContraction φsΛ.eqTimeContractSet
              ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨x, ⋯⟩)]
        [subContraction φsΛ.eqTimeContractSet
              ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨x, ⋯⟩)]
  intro a ha𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length)ha:a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)⊢ timeOrderRel
    [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, ⋯⟩)]
    [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel
      [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, ⋯⟩)]
      [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, ⋯⟩)]
  erw [subContraction_uncontractedList_get,𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length)ha:a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)⊢ timeOrderRel φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, ⋯⟩)]
    [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel
      [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ[↑((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, ⋯⟩)]
      φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, ⋯⟩)] subContraction_uncontractedList_get𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length)ha:a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)⊢ timeOrderRel φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, ⋯⟩)]
    φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, ⋯⟩)]
      φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, ⋯⟩)]]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length)ha:a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)⊢ timeOrderRel φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, ⋯⟩)]
    φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, ⋯⟩)] →
  ¬timeOrderRel φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).sndFieldOfContract ⟨a, ⋯⟩)]
      φs[uncontractedListEmd ((quotContraction φsΛ.eqTimeContractSet ⋯).fstFieldOfContract ⟨a, ⋯⟩)]
  simp only [quotContraction_fstFieldOfContract_uncontractedListEmd, Fin.getElem_fin,
    quotContraction_sndFieldOfContract_uncontractedListEmd]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length)ha:a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
      φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
  simp only [quotContraction, Finset.mem_filter, Finset.mem_univ, true_and] at ha𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length)ha✝:a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)ha:Finset.map uncontractedListEmd a ∈ ↑φsΛ⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
      φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
  have hn' : Finset.map uncontractedListEmd a ∉
      (φsΛ.subContraction (eqTimeContractSet φsΛ) (eqTimeContractSet_subset φsΛ)).1 :=
    uncontractedListEmd_finset_not_mem a𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length)ha✝:a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)ha:Finset.map uncontractedListEmd a ∈ ↑φsΛhn':Finset.map uncontractedListEmd a ∉ ↑(subContraction φsΛ.eqTimeContractSet ⋯)⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
      φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
  simp only [subContraction, eqTimeContractSet, Fin.getElem_fin, Finset.mem_filter, Finset.mem_univ,
    true_and, not_and, not_forall] at hn'𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length)ha✝:a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)ha:Finset.map uncontractedListEmd a ∈ ↑φsΛhn':Finset.map uncontractedListEmd a ∈ ↑φsΛ →
  ∃ (h : Finset.map uncontractedListEmd a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
        φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)] →
      ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
          φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
      φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
  obtain ⟨h, h1⟩ := hn' ha𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengtha:Finset (Fin [subContraction φsΛ.eqTimeContractSet ⋯]ᵘᶜ.length)ha✝:a ∈ ↑(quotContraction φsΛ.eqTimeContractSet ⋯)ha:Finset.map uncontractedListEmd a ∈ ↑φsΛhn':Finset.map uncontractedListEmd a ∈ ↑φsΛ →
  ∃ (h : Finset.map uncontractedListEmd a ∈ ↑φsΛ),
    timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
        φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)] →
      ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
          φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]h:Finset.map uncontractedListEmd a ∈ ↑φsΛh1:timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
      φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]⊢ timeOrderRel φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
    φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)] →
  ¬timeOrderRel φs[↑(φsΛ.sndFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
      φs[↑(φsΛ.fstFieldOfContract ⟨Finset.map uncontractedListEmd a, ⋯⟩)]
  simp_allAll goals completed! 🐙lemma join_haveEqTime_of_eqTimeOnly_nonEmpty {φs : List 𝓕.FieldOp} (φsΛ : WickContraction φs.length)
    (h1 : φsΛ.EqTimeOnly) (h2 : φsΛ ≠ empty)
    (φsucΛ : WickContraction [φsΛ]ᵘᶜ.length) :
    HaveEqTime (join φsΛ φsucΛ) := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnlyh2:φsΛ ≠ emptyφsucΛ:WickContraction [φsΛ]ᵘᶜ.length⊢ (φsΛ.join φsucΛ).HaveEqTime
  simp only [HaveEqTime, Fin.getElem_fin, join, Finset.mem_union,
    Finset.mem_map, RelEmbedding.coe_toEmbedding, exists_and_left, exists_prop]𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnlyh2:φsΛ ≠ emptyφsucΛ:WickContraction [φsΛ]ᵘᶜ.length⊢ ∃ i j,
  timeOrderRel φs[↑i] φs[↑j] ∧
    ({i, j} ∈ ↑φsΛ ∨ ∃ a ∈ ↑φsucΛ, (Finset.mapEmbedding uncontractedListEmd) a = {i, j}) ∧ timeOrderRel φs[↑j] φs[↑i]
  simp only [EqTimeOnly, Fin.getElem_fin] at h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:∀ (i j : Fin φs.length), {i, j} ∈ ↑φsΛ → timeOrderRel φs[↑i] φs[↑j]h2:φsΛ ≠ emptyφsucΛ:WickContraction [φsΛ]ᵘᶜ.length⊢ ∃ i j,
  timeOrderRel φs[↑i] φs[↑j] ∧
    ({i, j} ∈ ↑φsΛ ∨ ∃ a ∈ ↑φsucΛ, (Finset.mapEmbedding uncontractedListEmd) a = {i, j}) ∧ timeOrderRel φs[↑j] φs[↑i]
  obtain ⟨i, j, h⟩ := exists_pair_of_not_eq_empty _ h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:∀ (i j : Fin φs.length), {i, j} ∈ ↑φsΛ → timeOrderRel φs[↑i] φs[↑j]h2:φsΛ ≠ emptyφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthi:Fin φs.lengthj:Fin φs.lengthh:{i, j} ∈ ↑φsΛ⊢ ∃ i j,
  timeOrderRel φs[↑i] φs[↑j] ∧
    ({i, j} ∈ ↑φsΛ ∨ ∃ a ∈ ↑φsucΛ, (Finset.mapEmbedding uncontractedListEmd) a = {i, j}) ∧ timeOrderRel φs[↑j] φs[↑i]
  use i, jh𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:∀ (i j : Fin φs.length), {i, j} ∈ ↑φsΛ → timeOrderRel φs[↑i] φs[↑j]h2:φsΛ ≠ emptyφsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthi:Fin φs.lengthj:Fin φs.lengthh:{i, j} ∈ ↑φsΛ⊢ timeOrderRel φs[↑i] φs[↑j] ∧
  ({i, j} ∈ ↑φsΛ ∨ ∃ a ∈ ↑φsucΛ, (Finset.mapEmbedding uncontractedListEmd) a = {i, j}) ∧ timeOrderRel φs[↑j] φs[↑i]
  simp_all only [ne_eq, true_or, true_and]h𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:∀ (i j : Fin φs.length), {i, j} ∈ ↑φsΛ → timeOrderRel φs[↑i] φs[↑j]φsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthi:Fin φs.lengthj:Fin φs.lengthh2:¬φsΛ = emptyh:{i, j} ∈ ↑φsΛ⊢ timeOrderRel φs[↑j] φs[↑i]
  exact h1 j i (by𝓕:FieldSpecificationφs:List 𝓕.FieldOpφsΛ:WickContraction φs.lengthh1:∀ (i j : Fin φs.length), {i, j} ∈ ↑φsΛ → timeOrderRel φs[↑i] φs[↑j]φsucΛ:WickContraction [φsΛ]ᵘᶜ.lengthi:Fin φs.lengthj:Fin φs.lengthh2:¬φsΛ = emptyh:{i, j} ∈ ↑φsΛ⊢ {j, i} ∈ ↑φsΛ simpa [Finset.pair_comm] using hAll goals completed! 🐙)lemma hasEqTimeEquiv_ext_sigma {φs : List 𝓕.FieldOp} {x1 x2 :
    Σ (φsΛ : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty}),
    {φssucΛ : WickContraction [φsΛ.1]ᵘᶜ.length // ¬ HaveEqTime φssucΛ}}
    (h1 : x1.1.1 = x2.1.1) (h2 : x1.2.1 = congr (by𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpx1:(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }x2:(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }h1:↑x1.fst = ↑x2.fst⊢ [↑x2.fst]ᵘᶜ.length = [↑x1.fst]ᵘᶜ.length simp [h1]All goals completed! 🐙) x2.2.1) : x1 = x2 := by𝓕:FieldSpecificationφs:List 𝓕.FieldOpx1:(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }x2:(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }h1:↑x1.fst = ↑x2.fsth2:↑x1.snd = (congr ⋯) ↑x2.snd⊢ x1 = x2
  rcases x1 with ⟨⟨a1, b1⟩, ⟨c1, d1⟩⟩𝓕:FieldSpecificationφs:List 𝓕.FieldOpx2:(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }a1:WickContraction φs.lengthb1:a1.EqTimeOnly ∧ a1 ≠ emptyc1:WickContraction [↑⟨a1, b1⟩]ᵘᶜ.lengthd1:¬c1.HaveEqTimeh1:↑⟨⟨a1, b1⟩, ⟨c1, d1⟩⟩.fst = ↑x2.fsth2:↑⟨⟨a1, b1⟩, ⟨c1, d1⟩⟩.snd = (congr ⋯) ↑x2.snd⊢ ⟨⟨a1, b1⟩, ⟨c1, d1⟩⟩ = x2
  rcases x2 with ⟨b2, h2⟩𝓕:FieldSpecificationφs:List 𝓕.FieldOpa1:WickContraction φs.lengthb1:a1.EqTimeOnly ∧ a1 ≠ emptyc1:WickContraction [↑⟨a1, b1⟩]ᵘᶜ.lengthd1:¬c1.HaveEqTimeb2:{ φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }h2✝:{ φssucΛ // ¬φssucΛ.HaveEqTime }h1:↑⟨⟨a1, b1⟩, ⟨c1, d1⟩⟩.fst = ↑⟨b2, h2✝⟩.fsth2:↑⟨⟨a1, b1⟩, ⟨c1, d1⟩⟩.snd = (congr ⋯) ↑⟨b2, h2✝⟩.snd⊢ ⟨⟨a1, b1⟩, ⟨c1, d1⟩⟩ = ⟨b2, h2✝⟩
  simp only at h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpa1:WickContraction φs.lengthb1:a1.EqTimeOnly ∧ a1 ≠ emptyc1:WickContraction [↑⟨a1, b1⟩]ᵘᶜ.lengthd1:¬c1.HaveEqTimeb2:{ φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }h2✝:{ φssucΛ // ¬φssucΛ.HaveEqTime }h1:a1 = ↑b2h2:↑⟨⟨a1, b1⟩, ⟨c1, d1⟩⟩.snd = (congr ⋯) ↑⟨b2, h2✝⟩.snd⊢ ⟨⟨a1, b1⟩, ⟨c1, d1⟩⟩ = ⟨b2, h2✝⟩
  subst h1𝓕:FieldSpecificationφs:List 𝓕.FieldOpb2:{ φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }h2✝:{ φssucΛ // ¬φssucΛ.HaveEqTime }b1:(↑b2).EqTimeOnly ∧ ↑b2 ≠ emptyc1:WickContraction [↑⟨↑b2, b1⟩]ᵘᶜ.lengthd1:¬c1.HaveEqTimeh2:↑⟨⟨↑b2, b1⟩, ⟨c1, d1⟩⟩.snd = (congr ⋯) ↑⟨b2, h2✝⟩.snd⊢ ⟨⟨↑b2, b1⟩, ⟨c1, d1⟩⟩ = ⟨b2, h2✝⟩
  simp only [ne_eq, congr_refl] at h2𝓕:FieldSpecificationφs:List 𝓕.FieldOpb2:{ φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }h2✝:{ φssucΛ // ¬φssucΛ.HaveEqTime }b1:(↑b2).EqTimeOnly ∧ ↑b2 ≠ emptyc1:WickContraction [↑⟨↑b2, b1⟩]ᵘᶜ.lengthd1:¬c1.HaveEqTimeh2:c1 = ↑h2✝⊢ ⟨⟨↑b2, b1⟩, ⟨c1, d1⟩⟩ = ⟨b2, h2✝⟩
  simp [h2]All goals completed! 🐙

The equivalence which separates a Wick contraction which has an equal time contraction into a non-empty contraction only between equal-time fields and a Wick contraction which does not have equal time contractions.


def hasEqTimeEquiv (φs : List 𝓕.FieldOp) :
    {φsΛ : WickContraction φs.length // HaveEqTime φsΛ} ≃
    Σ (φsΛ : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty}),
    {φssucΛ : WickContraction [φsΛ.1]ᵘᶜ.length // ¬ HaveEqTime φssucΛ} where
  toFun φsΛ := ⟨⟨φsΛ.1.subContraction (eqTimeContractSet φsΛ.1) (eqTimeContractSet_subset φsΛ.1),
    subContraction_eqTimeContractSet_eqTimeOnly φsΛ.1,
    subContraction_eqTimeContractSet_not_empty_of_haveEqTime φsΛ.1 φsΛ.2⟩,
    ⟨φsΛ.1.quotContraction (eqTimeContractSet φsΛ.1) (eqTimeContractSet_subset φsΛ.1),
    quotContraction_eqTimeContractSet_not_haveEqTime φsΛ.1⟩⟩
  invFun φsΛ := ⟨join φsΛ.1.1 φsΛ.2.1,
    join_haveEqTime_of_eqTimeOnly_nonEmpty φsΛ.1.1 φsΛ.1.2.1 φsΛ.1.2.2 φsΛ.2⟩
  left_inv φsΛ := by𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ:{ φsΛ // φsΛ.HaveEqTime }⊢ (fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩)
    ((fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
      φsΛ) =
  φsΛ
    match φsΛ with
    | ⟨φsΛ, h1, h2⟩ =>𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ✝:{ φsΛ // φsΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:Fin φs.lengthh2:∃ j, ∃ (_ : {h1, j} ∈ ↑φsΛ), timeOrderRel φs[h1] φs[j] ∧ timeOrderRel φs[j] φs[h1]⊢ (fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩)
    ((fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
      ⟨φsΛ, ⋯⟩) =
  ⟨φsΛ, ⋯⟩
      simp only [ne_eq, Subtype.mk.injEq]𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ✝:{ φsΛ // φsΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:Fin φs.lengthh2:∃ j, ∃ (_ : {h1, j} ∈ ↑φsΛ), timeOrderRel φs[h1] φs[j] ∧ timeOrderRel φs[j] φs[h1]⊢ (subContraction φsΛ.eqTimeContractSet ⋯).join (quotContraction φsΛ.eqTimeContractSet ⋯) = φsΛ
      exact join_sub_quot φsΛ φsΛ.eqTimeContractSet (eqTimeContractSet_subset φsΛ)All goals completed! 🐙
  right_inv φsΛ' := by𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) φsΛ') =
  φsΛ'
    match φsΛ' with
    | ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩ =>𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTime⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩
      have hs : subContraction (φsΛ.join φsucΛ).eqTimeContractSet
        (eqTimeContractSet_subset (φsΛ.join φsucΛ)) = φsΛ := by𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) φsΛ') =
  φsΛ'𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩
        apply Subtype.ext𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTime⊢ ↑(subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯) = ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩
        ext a𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ ↑(subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯) ↔ a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩
        simp only [subContraction]𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ (φsΛ.join φsucΛ).eqTimeContractSet ↔ a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩
        rw [join_eqTimeContractSet𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet ↔
  a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet ↔
  a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩]𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding φsucΛ.eqTimeContractSet ↔
  a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩
        rw [eqTimeContractSet_of_not_haveEqTime h2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding ∅ ↔ a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding ∅ ↔ a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩]𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ∪ Finset.map (Finset.mapEmbedding uncontractedListEmd).toEmbedding ∅ ↔ a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩
        simp only [ne_eq, Finset.map_empty, Finset.union_empty]𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ φsΛ.eqTimeContractSet ↔ a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩
        rw [eqTimeContractSet_of_mem_eqTimeOnly h1.1𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimea:Finset (Fin φs.length)⊢ a ∈ ↑φsΛ ↔ a ∈ ↑φsΛ𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩]𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ (fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
    ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩) =
  ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩
      refine hasEqTimeEquiv_ext_sigma ?_ ?_refine_1𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ ↑((fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
        ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩)).fst =
  ↑⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩.fstrefine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ ↑((fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
        ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩)).snd =
  (congr ⋯) ↑⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩.snd
      ·refine_1𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ ↑((fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
        ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩)).fst =
  ↑⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩.fst simp only [ne_eq]refine_1𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ
        exact hsAll goals completed! 🐙
      ·refine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ ↑((fun φsΛ => ⟨⟨subContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩, ⟨quotContraction (↑φsΛ).eqTimeContractSet ⋯, ⋯⟩⟩)
        ((fun φsΛ => ⟨(↑φsΛ.fst).join ↑φsΛ.snd, ⋯⟩) ⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩)).snd =
  (congr ⋯) ↑⟨⟨φsΛ, h1⟩, ⟨φsucΛ, h2⟩⟩.snd simp only [ne_eq]refine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ quotContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = (congr ⋯) φsucΛ
        apply Subtype.extrefine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛ⊢ ↑(quotContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯) = ↑((congr ⋯) φsucΛ)
        ext arefine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ a ∈ ↑(quotContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯) ↔ a ∈ ↑((congr ⋯) φsucΛ)
        simp only [quotContraction, Finset.mem_filter, Finset.mem_univ, true_and]refine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map uncontractedListEmd a ∈ ↑(φsΛ.join φsucΛ) ↔ a ∈ ↑((congr ⋯) φsucΛ)
        rw [mem_congr_iffrefine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map uncontractedListEmd a ∈ ↑(φsΛ.join φsucΛ) ↔ Finset.map (finCongr ⋯).toEmbedding a ∈ ↑φsucΛrefine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map uncontractedListEmd a ∈ ↑(φsΛ.join φsucΛ) ↔ Finset.map (finCongr ⋯).toEmbedding a ∈ ↑φsucΛ]refine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map uncontractedListEmd a ∈ ↑(φsΛ.join φsucΛ) ↔ Finset.map (finCongr ⋯).toEmbedding a ∈ ↑φsucΛ
        rw [mem_join_right_iffrefine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map uncontractedListEmd a ∈ ↑(φsΛ.join φsucΛ) ↔
  Finset.map uncontractedListEmd (Finset.map (finCongr ⋯).toEmbedding a) ∈ ↑((↑⟨φsΛ, h1⟩).join φsucΛ)refine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map uncontractedListEmd a ∈ ↑(φsΛ.join φsucΛ) ↔
  Finset.map uncontractedListEmd (Finset.map (finCongr ⋯).toEmbedding a) ∈ ↑((↑⟨φsΛ, h1⟩).join φsucΛ)]refine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map uncontractedListEmd a ∈ ↑(φsΛ.join φsucΛ) ↔
  Finset.map uncontractedListEmd (Finset.map (finCongr ⋯).toEmbedding a) ∈ ↑((↑⟨φsΛ, h1⟩).join φsucΛ)
        simp only [ne_eq]refine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map uncontractedListEmd a ∈ ↑(φsΛ.join φsucΛ) ↔
  Finset.map uncontractedListEmd (Finset.map (finCongr ⋯).toEmbedding a) ∈ ↑(φsΛ.join φsucΛ)
        rw [uncontractedListEmd_congr hsrefine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map ((finCongr ⋯).toEmbedding.trans uncontractedListEmd) a ∈ ↑(φsΛ.join φsucΛ) ↔
  Finset.map uncontractedListEmd (Finset.map (finCongr ⋯).toEmbedding a) ∈ ↑(φsΛ.join φsucΛ)refine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map ((finCongr ⋯).toEmbedding.trans uncontractedListEmd) a ∈ ↑(φsΛ.join φsucΛ) ↔
  Finset.map uncontractedListEmd (Finset.map (finCongr ⋯).toEmbedding a) ∈ ↑(φsΛ.join φsucΛ)]refine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map ((finCongr ⋯).toEmbedding.trans uncontractedListEmd) a ∈ ↑(φsΛ.join φsucΛ) ↔
  Finset.map uncontractedListEmd (Finset.map (finCongr ⋯).toEmbedding a) ∈ ↑(φsΛ.join φsucΛ)
        rw [Finset.map_maprefine_2𝓕:FieldSpecificationn:ℕc:WickContraction nφs:List 𝓕.FieldOpφsΛ':(φsΛ : { φsΛ // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty }) × { φssucΛ // ¬φssucΛ.HaveEqTime }φsΛ:WickContraction φs.lengthh1:φsΛ.EqTimeOnly ∧ φsΛ ≠ emptyφsucΛ:WickContraction [↑⟨φsΛ, h1⟩]ᵘᶜ.lengthh2:¬φsucΛ.HaveEqTimehs:subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯ = φsΛa:Finset (Fin [subContraction (φsΛ.join φsucΛ).eqTimeContractSet ⋯]ᵘᶜ.length)⊢ Finset.map ((finCongr ⋯).toEmbedding.trans uncontractedListEmd) a ∈ ↑(φsΛ.join φsucΛ) ↔
  Finset.map ((finCongr ⋯).toEmbedding.trans uncontractedListEmd) a ∈ ↑(φsΛ.join φsucΛ)All goals completed! 🐙]All goals completed! 🐙



lemma sum_haveEqTime (φs : List 𝓕.FieldOp)
    (f : WickContraction φs.length → M) [AddCommMonoid M]:
  ∑ (φsΛ : {φsΛ : WickContraction φs.length // HaveEqTime φsΛ}), f φsΛ =
  ∑ (φsΛ : {φsΛ : WickContraction φs.length // φsΛ.EqTimeOnly ∧ φsΛ ≠ empty}),
  ∑ (φssucΛ : {φssucΛ : WickContraction [φsΛ.1]ᵘᶜ.length // ¬ HaveEqTime φssucΛ}),
    f (join φsΛ.1 φssucΛ.1) := by𝓕:FieldSpecificationM:Type u_1φs:List 𝓕.FieldOpf:WickContraction φs.length → Minst✝:AddCommMonoid M⊢ ∑ φsΛ, f ↑φsΛ = ∑ φsΛ, ∑ φssucΛ, f ((↑φsΛ).join ↑φssucΛ)
  rw [← (hasEqTimeEquiv φs).symm.sum_comp𝓕:FieldSpecificationM:Type u_1φs:List 𝓕.FieldOpf:WickContraction φs.length → Minst✝:AddCommMonoid M⊢ ∑ i, f ↑((hasEqTimeEquiv φs).symm i) = ∑ φsΛ, ∑ φssucΛ, f ((↑φsΛ).join ↑φssucΛ)𝓕:FieldSpecificationM:Type u_1φs:List 𝓕.FieldOpf:WickContraction φs.length → Minst✝:AddCommMonoid M⊢ ∑ i, f ↑((hasEqTimeEquiv φs).symm i) = ∑ φsΛ, ∑ φssucΛ, f ((↑φsΛ).join ↑φssucΛ)]𝓕:FieldSpecificationM:Type u_1φs:List 𝓕.FieldOpf:WickContraction φs.length → Minst✝:AddCommMonoid M⊢ ∑ i, f ↑((hasEqTimeEquiv φs).symm i) = ∑ φsΛ, ∑ φssucΛ, f ((↑φsΛ).join ↑φssucΛ)
  erw [Finset.sum_sigma𝓕:FieldSpecificationM:Type u_1φs:List 𝓕.FieldOpf:WickContraction φs.length → Minst✝:AddCommMonoid M⊢ ∑ a, ∑ s, f ↑((hasEqTimeEquiv φs).symm ⟨a, s⟩) = ∑ φsΛ, ∑ φssucΛ, f ((↑φsΛ).join ↑φssucΛ)]𝓕:FieldSpecificationM:Type u_1φs:List 𝓕.FieldOpf:WickContraction φs.length → Minst✝:AddCommMonoid M⊢ ∑ a, ∑ s, f ↑((hasEqTimeEquiv φs).symm ⟨a, s⟩) = ∑ φsΛ, ∑ φssucΛ, f ((↑φsΛ).join ↑φssucΛ)
  rflAll goals completed! 🐙