Physlib.Particles.BeyondTheStandardModel.TwoHDM.Potential

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.Particles.BeyondTheStandardModel.TwoHDM.GramMatrix

The potential of the Two Higgs doublet model

i. Overview

In this module we give the define the parameters of the 2HDM potential, and give stability properties of the potential.

ii. Key results

PotentialParameters : The parameters of the 2HDM potential.

massTerm : The mass term of the 2HDM potential.

quarticTerm : The quartic term of the 2HDM potential.

potential : The full potential of the 2HDM.

PotentialIsStable : The condition that the potential is stable.

iii. Table of contents

A. The parameters of the potential

A.1. The potential parameters corresponding to zero

A.2. Gram parameters

A.3. Specific cases

B. The mass term

C. The quartic term

D. The full potential

E. Stability of the potential

E.1. The stability condition

E.2. Instability of the stabilityCounterExample potential

E.3. The reduced mass term

E.4. The reduced quartic term

E.5. Stability in terms of the gram vectors

E.6. Strong stability implies stability

E.7. Showing step in hep-ph/0605184 is invalid

iv. References

For the parameterization of the potential we follow the convention of

https://arxiv.org/pdf/1605.03237

Stability arguments of the potential follow, in part, those from

https://arxiv.org/abs/hep-ph/0605184 Although we note that we explicitly prove that one of the steps in this paper is not valid.

@[expose] public section

A. The parameters of the potential

We define a type for the parameters of the Higgs potential in the 2HDM.

We follow the convention of 1605.03237, which is highlighted in the explicit construction of the potential itself.

We relate these parameters to the ξ and η parameters used in the gram vector formalism given in arXiv:hep-ph/0605184.

The parameters of the Two Higgs doublet model potential. Following the convention of https://arxiv.org/pdf/1605.03237.

The parameter corresponding to m₁₁² in the 2HDM potential.

The parameter corresponding to m₂₂² in the 2HDM potential.

The parameter corresponding to m₁₂² in the 2HDM potential.

The parameter corresponding to λ₁ in the 2HDM potential.

The parameter corresponding to λ₂ in the 2HDM potential.

The parameter corresponding to λ₃ in the 2HDM potential.

The parameter corresponding to λ₄ in the 2HDM potential.

The parameter corresponding to λ₅ in the 2HDM potential.

The parameter corresponding to λ₆ in the 2HDM potential.

The parameter corresponding to λ₇ in the 2HDM potential.

              structure PotentialParameters where  m₁₁2 : ℝ  m₂₂2 : ℝ  m₁₂2 : ℂ  𝓵₁ : ℝ  𝓵₂ : ℝ  𝓵₃ : ℝ  𝓵₄ : ℝ  𝓵₅ : ℂ  𝓵₆ : ℂ  𝓵₇ : ℂ

A.1. The potential parameters corresponding to zero

We define an instance of Zero for the potential parameters, corresponding to all parameters being zero, and therefore the potential itself being zero.

              instance : Zero PotentialParameters where
  zero :=
    { m₁₁2 := 0
      m₂₂2 := 0
      m₁₂2 := 0
      𝓵₁ := 0
      𝓵₂ := 0
      𝓵₃ := 0
      𝓵₄ := 0
      𝓵₅ := 0
      𝓵₆ := 0
      𝓵₇ := 0 }@[simp] lemma zero_m₁₁2 : (0 : PotentialParameters).m₁₁2 = 0 := rfl@[simp] lemma zero_m₂₂2 : (0 : PotentialParameters).m₂₂2 = 0 := rfl@[simp] lemma zero_m₁₂2 : (0 : PotentialParameters).m₁₂2 = 0 := rfl@[simp] lemma zero_𝓵₁ : (0 : PotentialParameters).𝓵₁ = 0 := rfl@[simp] lemma zero_𝓵₂ : (0 : PotentialParameters).𝓵₂ = 0 := rfl@[simp] lemma zero_𝓵₃ : (0 : PotentialParameters).𝓵₃ = 0 := rfl@[simp] lemma zero_𝓵₄ : (0 : PotentialParameters).𝓵₄ = 0 := rfl@[simp] lemma zero_𝓵₅ : (0 : PotentialParameters).𝓵₅ = 0 := rfl@[simp] lemma zero_𝓵₆ : (0 : PotentialParameters).𝓵₆ = 0 := rfl@[simp] lemma zero_𝓵₇ : (0 : PotentialParameters).𝓵₇ = 0 := rfl

A.2. Gram parameters

A reparameterization of the potential parameters corresponding to ξ and η in arXiv:hep-ph/0605184.

              @[simp]
lemma ξ_zero : (0 : PotentialParameters).ξ = 0 := by⊢ ξ 0 = 0
  ext μμ:Fin 1 ⊕ Fin 3⊢ ξ 0 μ = 0 μ
  fin_cases μ«0»⊢ ξ 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) = 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«1»⊢ ξ 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) = 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«2»⊢ ξ 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) = 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«3»⊢ ξ 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) = 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) <;>«0»⊢ ξ 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) = 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«1»⊢ ξ 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) = 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«2»⊢ ξ 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) = 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«3»⊢ ξ 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) = 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) simp [ξ]All goals completed! 🐙lemma η_symm (P : PotentialParameters) (μ ν : Fin 1 ⊕ Fin 3) :
    P.η μ ν = P.η ν μ := byP:PotentialParametersμ:Fin 1 ⊕ Fin 3ν:Fin 1 ⊕ Fin 3⊢ P.η μ ν = P.η ν μ
  fin_cases μ«0»P:PotentialParametersν:Fin 1 ⊕ Fin 3⊢ P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ν = P.η ν (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«1»P:PotentialParametersν:Fin 1 ⊕ Fin 3⊢ P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ν = P.η ν (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«2»P:PotentialParametersν:Fin 1 ⊕ Fin 3⊢ P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ν = P.η ν (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«3»P:PotentialParametersν:Fin 1 ⊕ Fin 3⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ν = P.η ν (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) <;>«0»P:PotentialParametersν:Fin 1 ⊕ Fin 3⊢ P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ν = P.η ν (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«1»P:PotentialParametersν:Fin 1 ⊕ Fin 3⊢ P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ν = P.η ν (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«2»P:PotentialParametersν:Fin 1 ⊕ Fin 3⊢ P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ν = P.η ν (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«3»P:PotentialParametersν:Fin 1 ⊕ Fin 3⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ν = P.η ν (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) fin_cases ν«3».«0»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩))«3».«1»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩))«3».«2»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩))«3».«3»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) <;>«0».«0»P:PotentialParameters⊢ P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«0».«1»P:PotentialParameters⊢ P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«0».«2»P:PotentialParameters⊢ P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«0».«3»P:PotentialParameters⊢ P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«1».«0»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«1».«1»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«1».«2»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«1».«3»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«2».«0»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«2».«1»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«2».«2»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«2».«3»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«3».«0»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩))«3».«1»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩))«3».«2»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩))«3».«3»P:PotentialParameters⊢ P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  P.η (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) simp [η]All goals completed! 🐙@[simp]
lemma η_zero : (0 : PotentialParameters).η = 0 := by⊢ η 0 = 0
  ext μ νμ:Fin 1 ⊕ Fin 3ν:Fin 1 ⊕ Fin 3⊢ η 0 μ ν = 0 μ ν
  fin_cases μ«0»ν:Fin 1 ⊕ Fin 3⊢ η 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ν = 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ν«1»ν:Fin 1 ⊕ Fin 3⊢ η 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ν = 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ν«2»ν:Fin 1 ⊕ Fin 3⊢ η 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ν = 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ν«3»ν:Fin 1 ⊕ Fin 3⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ν = 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ν <;>«0»ν:Fin 1 ⊕ Fin 3⊢ η 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ν = 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) ν«1»ν:Fin 1 ⊕ Fin 3⊢ η 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ν = 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) ν«2»ν:Fin 1 ⊕ Fin 3⊢ η 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ν = 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) ν«3»ν:Fin 1 ⊕ Fin 3⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ν = 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) ν fin_cases ν«3».«0»⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«3».«1»⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«3».«2»⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«3».«3»⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) <;>«0».«0»⊢ η 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«0».«1»⊢ η 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«0».«2»⊢ η 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«0».«3»⊢ η 0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  0 (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩))«1».«0»⊢ η 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«1».«1»⊢ η 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«1».«2»⊢ η 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«1».«3»⊢ η 0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩))«2».«0»⊢ η 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«2».«1»⊢ η 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«2».«2»⊢ η 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«2».«3»⊢ η 0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩))«3».«0»⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inl ((fun i => i) ⟨0, ⋯⟩))«3».«1»⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨0, ⋯⟩))«3».«2»⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨1, ⋯⟩))«3».«3»⊢ η 0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) =
  0 (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) (Sum.inr ((fun i => i) ⟨2, ⋯⟩)) simp [η]All goals completed! 🐙

A.3. Specific cases

An example of potential parameters that serve as a counterexample to the stability condition given in arXiv:hep-ph/0605184. This corresponds to the potential: 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + ‖H.Φ1 - H.Φ2‖ ^ 4 which has the property that the quartic term is non-negative and only zero if the mass term is also zero, but the potential is not stable. In the proof that stabilityCounterExample_not_potentialIsStable, we give explicit vectors H.Φ1 and H.Φ2 that show this potential is not stable.

This is the first occurrence of such a counterexample in the literature to the best of the author's knowledge.

def stabilityCounterExample : PotentialParameters := {(0 : PotentialParameters) with
    m₁₂2 := Complex.I
    𝓵₁ := 2
    𝓵₂ := 2
    𝓵₃ := 2
    𝓵₄ := 2
    𝓵₅ := 2
    𝓵₆ := -2
    𝓵₇ := -2}

lemma stabilityCounterExample_ξ :
    stabilityCounterExample.ξ = fun
      | .inl 0 => 0
      | .inr 0 => 0
      | .inr 1 => 1
      | .inr 2 => 0 := by⊢ stabilityCounterExample.ξ = fun x =>
  match x with
  | Sum.inl 0 => 0
  | Sum.inr 0 => 0
  | Sum.inr 1 => 1
  | Sum.inr 2 => 0
  funext μμ:Fin 1 ⊕ Fin 3⊢ stabilityCounterExample.ξ μ =
  match μ with
  | Sum.inl 0 => 0
  | Sum.inr 0 => 0
  | Sum.inr 1 => 1
  | Sum.inr 2 => 0
  simp [stabilityCounterExample, ξ]All goals completed! 🐙

lemma stabilityCounterExample_η :
    stabilityCounterExample.η = fun μ => fun ν =>
    match μ, ν with
    | .inl 0, .inl 0 => 1
    | .inl 0, .inr 0 => -1
    | .inl 0, .inr 1 => 0
    | .inl 0, .inr 2 => 0
    | .inr 0, .inl 0 => -1
    | .inr 1, .inl 0 => 0
    | .inr 2, .inl 0 => 0
    | .inr 0, .inr 0 => 1
    | .inr 1, .inr 1 => 0
    | .inr 2, .inr 2 => 0
    | .inr 0, .inr 1 => 0
    | .inr 2, .inr 0 => 0
    | .inr 2, .inr 1 => 0
    | .inr 1, .inr 0 => 0
    | .inr 0, .inr 2 => 0
    | .inr 1, .inr 2 => 0 := by⊢ stabilityCounterExample.η = fun μ ν =>
  match μ, ν with
  | Sum.inl 0, Sum.inl 0 => 1
  | Sum.inl 0, Sum.inr 0 => -1
  | Sum.inl 0, Sum.inr 1 => 0
  | Sum.inl 0, Sum.inr 2 => 0
  | Sum.inr 0, Sum.inl 0 => -1
  | Sum.inr 1, Sum.inl 0 => 0
  | Sum.inr 2, Sum.inl 0 => 0
  | Sum.inr 0, Sum.inr 0 => 1
  | Sum.inr 1, Sum.inr 1 => 0
  | Sum.inr 2, Sum.inr 2 => 0
  | Sum.inr 0, Sum.inr 1 => 0
  | Sum.inr 2, Sum.inr 0 => 0
  | Sum.inr 2, Sum.inr 1 => 0
  | Sum.inr 1, Sum.inr 0 => 0
  | Sum.inr 0, Sum.inr 2 => 0
  | Sum.inr 1, Sum.inr 2 => 0
  funext μ νμ:Fin 1 ⊕ Fin 3ν:Fin 1 ⊕ Fin 3⊢ stabilityCounterExample.η μ ν =
  match μ, ν with
  | Sum.inl 0, Sum.inl 0 => 1
  | Sum.inl 0, Sum.inr 0 => -1
  | Sum.inl 0, Sum.inr 1 => 0
  | Sum.inl 0, Sum.inr 2 => 0
  | Sum.inr 0, Sum.inl 0 => -1
  | Sum.inr 1, Sum.inl 0 => 0
  | Sum.inr 2, Sum.inl 0 => 0
  | Sum.inr 0, Sum.inr 0 => 1
  | Sum.inr 1, Sum.inr 1 => 0
  | Sum.inr 2, Sum.inr 2 => 0
  | Sum.inr 0, Sum.inr 1 => 0
  | Sum.inr 2, Sum.inr 0 => 0
  | Sum.inr 2, Sum.inr 1 => 0
  | Sum.inr 1, Sum.inr 0 => 0
  | Sum.inr 0, Sum.inr 2 => 0
  | Sum.inr 1, Sum.inr 2 => 0
  simp [stabilityCounterExample, η]μ:Fin 1 ⊕ Fin 3ν:Fin 1 ⊕ Fin 3⊢ (match μ, ν with
  | Sum.inl 0, Sum.inl 0 => (2 + 2 + 2 * 2) / 8
  | Sum.inl 0, Sum.inr 0 => (-2 + -2) / 4
  | Sum.inl 0, Sum.inr 1 => 0
  | Sum.inl 0, Sum.inr 2 => 0
  | Sum.inr 0, Sum.inl 0 => (-2 + -2) / 4
  | Sum.inr 1, Sum.inl 0 => 0
  | Sum.inr 2, Sum.inl 0 => 0
  | Sum.inr 0, Sum.inr 0 => (2 + 2) / 4
  | Sum.inr 1, Sum.inr 1 => 0
  | Sum.inr 2, Sum.inr 2 => (2 + 2 - 2 * 2) / 8
  | Sum.inr 0, Sum.inr 1 => 0
  | Sum.inr 2, Sum.inr 0 => 0
  | Sum.inr 2, Sum.inr 1 => 0
  | Sum.inr 1, Sum.inr 0 => 0
  | Sum.inr 0, Sum.inr 2 => 0
  | Sum.inr 1, Sum.inr 2 => 0) =
  match μ, ν with
  | Sum.inl 0, Sum.inl 0 => 1
  | Sum.inl 0, Sum.inr 0 => -1
  | Sum.inl 0, Sum.inr 1 => 0
  | Sum.inl 0, Sum.inr 2 => 0
  | Sum.inr 0, Sum.inl 0 => -1
  | Sum.inr 1, Sum.inl 0 => 0
  | Sum.inr 2, Sum.inl 0 => 0
  | Sum.inr 0, Sum.inr 0 => 1
  | Sum.inr 1, Sum.inr 1 => 0
  | Sum.inr 2, Sum.inr 2 => 0
  | Sum.inr 0, Sum.inr 1 => 0
  | Sum.inr 2, Sum.inr 0 => 0
  | Sum.inr 2, Sum.inr 1 => 0
  | Sum.inr 1, Sum.inr 0 => 0
  | Sum.inr 0, Sum.inr 2 => 0
  | Sum.inr 1, Sum.inr 2 => 0
  ring_nfAll goals completed! 🐙

B. The mass term

We define the mass term of the potential, write it in terms of the gram vector, and prove that it is gauge invariant.

              lemma massTerm_eq_gramVector (P : PotentialParameters) (H : TwoHiggsDoublet) :
    massTerm P H = ∑ μ, P.ξ μ * H.gramVector μ := byP:PotentialParametersH:TwoHiggsDoublet⊢ massTerm P H = ∑ μ, P.ξ μ * H.gramVector μ
  simp [massTerm, Fin.sum_univ_three, PotentialParameters.ξ, normSq_Φ1_eq_gramVector,
    normSq_Φ2_eq_gramVector, Φ1_inner_Φ2_eq_gramVector, Φ2_inner_Φ1_eq_gramVector]P:PotentialParametersH:TwoHiggsDoublet⊢ P.m₁₁2 * (2⁻¹ * (H.gramVector (Sum.inl 0) + H.gramVector (Sum.inr 2))) +
      P.m₂₂2 * (2⁻¹ * (H.gramVector (Sum.inl 0) - H.gramVector (Sum.inr 2))) -
    (P.m₁₂2.re * (2⁻¹ * H.gramVector (Sum.inr 0)) - P.m₁₂2.im * (2⁻¹ * H.gramVector (Sum.inr 1)) +
      (P.m₁₂2.re * (2⁻¹ * H.gramVector (Sum.inr 0)) - P.m₁₂2.im * (2⁻¹ * H.gramVector (Sum.inr 1)))) =
  (P.m₁₁2 + P.m₂₂2) / 2 * H.gramVector (Sum.inl 0) +
    (-(P.m₁₂2.re * H.gramVector (Sum.inr 0)) + P.m₁₂2.im * H.gramVector (Sum.inr 1) +
      (P.m₁₁2 - P.m₂₂2) / 2 * H.gramVector (Sum.inr 2))
  ringAll goals completed! 🐙@[simp]
lemma gaugeGroupI_smul_massTerm (g : StandardModel.GaugeGroupI) (P : PotentialParameters)
    (H : TwoHiggsDoublet) :
    massTerm P (g • H) = massTerm P H := byg:GaugeGroupIP:PotentialParametersH:TwoHiggsDoublet⊢ massTerm P (g • H) = massTerm P H
  simp [massTerm_eq_gramVector]All goals completed! 🐙@[simp]
lemma massTerm_zero : massTerm 0 = 0 := by⊢ massTerm 0 = 0
  ext HH:TwoHiggsDoublet⊢ massTerm 0 H = 0 H
  simp [massTerm]All goals completed! 🐙

lemma massTerm_stabilityCounterExample (H : TwoHiggsDoublet) :
    massTerm PotentialParameters.stabilityCounterExample H =
    2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im := byH:TwoHiggsDoublet⊢ massTerm PotentialParameters.stabilityCounterExample H = 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im
  simp [massTerm, PotentialParameters.stabilityCounterExample]H:TwoHiggsDoublet⊢ -(⟪H.Φ2, H.Φ1⟫_ℂ).im + (⟪H.Φ1, H.Φ2⟫_ℂ).im = 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im
  rw [← inner_conj_symm H.Φ2 H.Φ1,H:TwoHiggsDoublet⊢ -((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).im + (⟪H.Φ1, H.Φ2⟫_ℂ).im = 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).imH:TwoHiggsDoublet⊢ - -(⟪H.Φ1, H.Φ2⟫_ℂ).im + (⟪H.Φ1, H.Φ2⟫_ℂ).im = 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im Complex.conj_imH:TwoHiggsDoublet⊢ - -(⟪H.Φ1, H.Φ2⟫_ℂ).im + (⟪H.Φ1, H.Φ2⟫_ℂ).im = 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).imH:TwoHiggsDoublet⊢ - -(⟪H.Φ1, H.Φ2⟫_ℂ).im + (⟪H.Φ1, H.Φ2⟫_ℂ).im = 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im]H:TwoHiggsDoublet⊢ - -(⟪H.Φ1, H.Φ2⟫_ℂ).im + (⟪H.Φ1, H.Φ2⟫_ℂ).im = 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im
  ring_nfAll goals completed! 🐙

C. The quartic term

We define the quartic term of the potential, write it in terms of the gram vector, and prove that it is gauge invariant.

              

lemma quarticTerm_𝓵₄_expand (P : PotentialParameters) (H : TwoHiggsDoublet) :
    H.quarticTerm P =
    1/2 * P.𝓵₁ * ‖H.Φ1‖ ^ 2 * ‖H.Φ1‖ ^ 2 + 1/2 * P.𝓵₂ * ‖H.Φ2‖ ^ 2 * ‖H.Φ2‖ ^ 2
    + P.𝓵₃ * ‖H.Φ1‖ ^ 2 * ‖H.Φ2‖ ^ 2
    + P.𝓵₄ * (⟪H.Φ1, H.Φ2⟫_ℂ * ⟪H.Φ2, H.Φ1⟫_ℂ).re
    + (1/2 * P.𝓵₅ * ⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + 1/2 * conj P.𝓵₅ * ⟪H.Φ2, H.Φ1⟫_ℂ ^ 2).re
    + (P.𝓵₆ * ‖H.Φ1‖ ^ 2 * ⟪H.Φ1, H.Φ2⟫_ℂ + conj P.𝓵₆ * ‖H.Φ1‖ ^ 2 * ⟪H.Φ2, H.Φ1⟫_ℂ).re
    + (P.𝓵₇ * ‖H.Φ2‖ ^ 2 * ⟪H.Φ1, H.Φ2⟫_ℂ + conj P.𝓵₇ * ‖H.Φ2‖ ^ 2 * ⟪H.Φ2, H.Φ1⟫_ℂ).re := byP:PotentialParametersH:TwoHiggsDoublet⊢ quarticTerm P H =
  1 / 2 * P.𝓵₁ * ‖H.Φ1‖ ^ 2 * ‖H.Φ1‖ ^ 2 + 1 / 2 * P.𝓵₂ * ‖H.Φ2‖ ^ 2 * ‖H.Φ2‖ ^ 2 + P.𝓵₃ * ‖H.Φ1‖ ^ 2 * ‖H.Φ2‖ ^ 2 +
          P.𝓵₄ * (⟪H.Φ1, H.Φ2⟫_ℂ * ⟪H.Φ2, H.Φ1⟫_ℂ).re +
        (1 / 2 * P.𝓵₅ * ⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + 1 / 2 * (starRingEnd ℂ) P.𝓵₅ * ⟪H.Φ2, H.Φ1⟫_ℂ ^ 2).re +
      (P.𝓵₆ * ↑‖H.Φ1‖ ^ 2 * ⟪H.Φ1, H.Φ2⟫_ℂ + (starRingEnd ℂ) P.𝓵₆ * ↑‖H.Φ1‖ ^ 2 * ⟪H.Φ2, H.Φ1⟫_ℂ).re +
    (P.𝓵₇ * ↑‖H.Φ2‖ ^ 2 * ⟪H.Φ1, H.Φ2⟫_ℂ + (starRingEnd ℂ) P.𝓵₇ * ↑‖H.Φ2‖ ^ 2 * ⟪H.Φ2, H.Φ1⟫_ℂ).re
  simp [quarticTerm]P:PotentialParametersH:TwoHiggsDoublet⊢ ‖⟪H.Φ1, H.Φ2⟫_ℂ‖ ^ 2 = (⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ2, H.Φ1⟫_ℂ).re - (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ2, H.Φ1⟫_ℂ).im ∨ P.𝓵₄ = 0
  leftP:PotentialParametersH:TwoHiggsDoublet⊢ ‖⟪H.Φ1, H.Φ2⟫_ℂ‖ ^ 2 = (⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ2, H.Φ1⟫_ℂ).re - (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ2, H.Φ1⟫_ℂ).im
  rw [Complex.sq_norm,P:PotentialParametersH:TwoHiggsDoublet⊢ Complex.normSq ⟪H.Φ1, H.Φ2⟫_ℂ = (⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ2, H.Φ1⟫_ℂ).re - (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ2, H.Φ1⟫_ℂ).imAll goals completed! 🐙 ← inner_conj_symm H.Φ2 H.Φ1,P:PotentialParametersH:TwoHiggsDoublet⊢ Complex.normSq ⟪H.Φ1, H.Φ2⟫_ℂ =
  (⟪H.Φ1, H.Φ2⟫_ℂ).re * ((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).re - (⟪H.Φ1, H.Φ2⟫_ℂ).im * ((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).imAll goals completed! 🐙 ← Complex.mul_re,P:PotentialParametersH:TwoHiggsDoublet⊢ Complex.normSq ⟪H.Φ1, H.Φ2⟫_ℂ = (⟪H.Φ1, H.Φ2⟫_ℂ * (starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).reAll goals completed! 🐙 Complex.mul_conj,P:PotentialParametersH:TwoHiggsDoublet⊢ Complex.normSq ⟪H.Φ1, H.Φ2⟫_ℂ = (↑(Complex.normSq ⟪H.Φ1, H.Φ2⟫_ℂ)).reAll goals completed! 🐙
    Complex.ofReal_reP:PotentialParametersH:TwoHiggsDoublet⊢ Complex.normSq ⟪H.Φ1, H.Φ2⟫_ℂ = Complex.normSq ⟪H.Φ1, H.Φ2⟫_ℂAll goals completed! 🐙]All goals completed! 🐙lemma quarticTerm_eq_gramVector (P : PotentialParameters) (H : TwoHiggsDoublet) :
    quarticTerm P H = ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b := byP:PotentialParametersH:TwoHiggsDoublet⊢ quarticTerm P H = ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b
  simp [quarticTerm_𝓵₄_expand, Fin.sum_univ_three, PotentialParameters.η, normSq_Φ1_eq_gramVector,
    normSq_Φ2_eq_gramVector, Φ1_inner_Φ2_eq_gramVector, Φ2_inner_Φ1_eq_gramVector]P:PotentialParametersH:TwoHiggsDoublet⊢ 2⁻¹ * P.𝓵₁ * (2⁻¹ * (H.gramVector (Sum.inl 0) + H.gramVector (Sum.inr 2))) *
                (2⁻¹ * (H.gramVector (Sum.inl 0) + H.gramVector (Sum.inr 2))) +
              2⁻¹ * P.𝓵₂ * (2⁻¹ * (H.gramVector (Sum.inl 0) - H.gramVector (Sum.inr 2))) *
                (2⁻¹ * (H.gramVector (Sum.inl 0) - H.gramVector (Sum.inr 2))) +
            P.𝓵₃ * (2⁻¹ * (H.gramVector (Sum.inl 0) + H.gramVector (Sum.inr 2))) *
              (2⁻¹ * (H.gramVector (Sum.inl 0) - H.gramVector (Sum.inr 2))) +
          P.𝓵₄ *
            (2⁻¹ * H.gramVector (Sum.inr 0) * (2⁻¹ * H.gramVector (Sum.inr 0)) +
              2⁻¹ * H.gramVector (Sum.inr 1) * (2⁻¹ * H.gramVector (Sum.inr 1))) +
        (2⁻¹ * P.𝓵₅.re * ((2⁻¹ * (↑(H.gramVector (Sum.inr 0)) + Complex.I * ↑(H.gramVector (Sum.inr 1)))) ^ 2).re -
            2⁻¹ * P.𝓵₅.im * ((2⁻¹ * (↑(H.gramVector (Sum.inr 0)) + Complex.I * ↑(H.gramVector (Sum.inr 1)))) ^ 2).im +
          (2⁻¹ * P.𝓵₅.re * ((2⁻¹ * (↑(H.gramVector (Sum.inr 0)) - Complex.I * ↑(H.gramVector (Sum.inr 1)))) ^ 2).re +
            2⁻¹ * P.𝓵₅.im * ((2⁻¹ * (↑(H.gramVector (Sum.inr 0)) - Complex.I * ↑(H.gramVector (Sum.inr 1)))) ^ 2).im)) +
      ((P.𝓵₆.re * (↑‖H.Φ1‖ ^ 2).re - P.𝓵₆.im * (↑‖H.Φ1‖ ^ 2).im) * (2⁻¹ * H.gramVector (Sum.inr 0)) -
          (P.𝓵₆.re * (↑‖H.Φ1‖ ^ 2).im + P.𝓵₆.im * (↑‖H.Φ1‖ ^ 2).re) * (2⁻¹ * H.gramVector (Sum.inr 1)) +
        ((P.𝓵₆.re * (↑‖H.Φ1‖ ^ 2).re + P.𝓵₆.im * (↑‖H.Φ1‖ ^ 2).im) * (2⁻¹ * H.gramVector (Sum.inr 0)) +
          (P.𝓵₆.re * (↑‖H.Φ1‖ ^ 2).im + -(P.𝓵₆.im * (↑‖H.Φ1‖ ^ 2).re)) * (2⁻¹ * H.gramVector (Sum.inr 1)))) +
    ((P.𝓵₇.re * (↑‖H.Φ2‖ ^ 2).re - P.𝓵₇.im * (↑‖H.Φ2‖ ^ 2).im) * (2⁻¹ * H.gramVector (Sum.inr 0)) -
        (P.𝓵₇.re * (↑‖H.Φ2‖ ^ 2).im + P.𝓵₇.im * (↑‖H.Φ2‖ ^ 2).re) * (2⁻¹ * H.gramVector (Sum.inr 1)) +
      ((P.𝓵₇.re * (↑‖H.Φ2‖ ^ 2).re + P.𝓵₇.im * (↑‖H.Φ2‖ ^ 2).im) * (2⁻¹ * H.gramVector (Sum.inr 0)) +
        (P.𝓵₇.re * (↑‖H.Φ2‖ ^ 2).im + -(P.𝓵₇.im * (↑‖H.Φ2‖ ^ 2).re)) * (2⁻¹ * H.gramVector (Sum.inr 1)))) =
  H.gramVector (Sum.inl 0) * H.gramVector (Sum.inl 0) * ((P.𝓵₁ + P.𝓵₂ + 2 * P.𝓵₃) / 8) +
      (H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 0) * ((P.𝓵₆.re + P.𝓵₇.re) / 4) +
          H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 1) * ((-P.𝓵₇.im + -P.𝓵₆.im) / 4) +
        H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 2) * ((P.𝓵₁ - P.𝓵₂) / 8)) +
    (H.gramVector (Sum.inr 0) * H.gramVector (Sum.inl 0) * ((P.𝓵₆.re + P.𝓵₇.re) / 4) +
          (H.gramVector (Sum.inr 0) * H.gramVector (Sum.inr 0) * ((P.𝓵₅.re + P.𝓵₄) / 4) +
              H.gramVector (Sum.inr 0) * H.gramVector (Sum.inr 1) * (-P.𝓵₅.im / 4) +
            H.gramVector (Sum.inr 0) * H.gramVector (Sum.inr 2) * ((P.𝓵₆.re - P.𝓵₇.re) / 4)) +
        (H.gramVector (Sum.inr 1) * H.gramVector (Sum.inl 0) * ((-P.𝓵₇.im + -P.𝓵₆.im) / 4) +
          (H.gramVector (Sum.inr 1) * H.gramVector (Sum.inr 0) * (-P.𝓵₅.im / 4) +
              H.gramVector (Sum.inr 1) * H.gramVector (Sum.inr 1) * ((P.𝓵₄ - P.𝓵₅.re) / 4) +
            H.gramVector (Sum.inr 1) * H.gramVector (Sum.inr 2) * ((P.𝓵₇.im - P.𝓵₆.im) / 4))) +
      (H.gramVector (Sum.inr 2) * H.gramVector (Sum.inl 0) * ((P.𝓵₁ - P.𝓵₂) / 8) +
        (H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 0) * ((P.𝓵₆.re - P.𝓵₇.re) / 4) +
            H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 1) * ((P.𝓵₇.im - P.𝓵₆.im) / 4) +
          H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 2) * ((P.𝓵₁ + P.𝓵₂ - 2 * P.𝓵₃) / 8))))
  ring_nfP:PotentialParametersH:TwoHiggsDoublet⊢ P.𝓵₁ * H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 2) * (1 / 4) + P.𝓵₁ * H.gramVector (Sum.inl 0) ^ 2 * (1 / 8) +
                                  P.𝓵₁ * H.gramVector (Sum.inr 2) ^ 2 * (1 / 8) +
                                H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 2) * P.𝓵₂ * (-1 / 4) +
                              H.gramVector (Sum.inl 0) ^ 2 * P.𝓵₂ * (1 / 8) +
                            H.gramVector (Sum.inl 0) ^ 2 * P.𝓵₃ * (1 / 4) +
                          H.gramVector (Sum.inr 2) ^ 2 * P.𝓵₂ * (1 / 8) +
                        H.gramVector (Sum.inr 2) ^ 2 * P.𝓵₃ * (-1 / 4) +
                      P.𝓵₄ * H.gramVector (Sum.inr 0) ^ 2 * (1 / 4) +
                    P.𝓵₄ * H.gramVector (Sum.inr 1) ^ 2 * (1 / 4) +
                  H.gramVector (Sum.inr 0) * P.𝓵₆.re * (↑‖H.Φ1‖ ^ 2).re +
                H.gramVector (Sum.inr 0) * P.𝓵₇.re * (↑‖H.Φ2‖ ^ 2).re -
              H.gramVector (Sum.inr 1) * (↑‖H.Φ1‖ ^ 2).re * P.𝓵₆.im -
            H.gramVector (Sum.inr 1) * (↑‖H.Φ2‖ ^ 2).re * P.𝓵₇.im +
          P.𝓵₅.re *
              (↑(H.gramVector (Sum.inr 0)) * Complex.I * ↑(H.gramVector (Sum.inr 1)) * (1 / 2) +
                    ↑(H.gramVector (Sum.inr 0)) ^ 2 * (1 / 4) +
                  Complex.I ^ 2 * ↑(H.gramVector (Sum.inr 1)) ^ 2 * (1 / 4)).re *
            (1 / 2) +
        P.𝓵₅.re *
            (↑(H.gramVector (Sum.inr 0)) * Complex.I * ↑(H.gramVector (Sum.inr 1)) * (-1 / 2) +
                  ↑(H.gramVector (Sum.inr 0)) ^ 2 * (1 / 4) +
                Complex.I ^ 2 * ↑(H.gramVector (Sum.inr 1)) ^ 2 * (1 / 4)).re *
          (1 / 2) +
      P.𝓵₅.im *
          (↑(H.gramVector (Sum.inr 0)) * Complex.I * ↑(H.gramVector (Sum.inr 1)) * (1 / 2) +
                ↑(H.gramVector (Sum.inr 0)) ^ 2 * (1 / 4) +
              Complex.I ^ 2 * ↑(H.gramVector (Sum.inr 1)) ^ 2 * (1 / 4)).im *
        (-1 / 2) +
    P.𝓵₅.im *
        (↑(H.gramVector (Sum.inr 0)) * Complex.I * ↑(H.gramVector (Sum.inr 1)) * (-1 / 2) +
              ↑(H.gramVector (Sum.inr 0)) ^ 2 * (1 / 4) +
            Complex.I ^ 2 * ↑(H.gramVector (Sum.inr 1)) ^ 2 * (1 / 4)).im *
      (1 / 2) =
  P.𝓵₁ * H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 2) * (1 / 4) + P.𝓵₁ * H.gramVector (Sum.inl 0) ^ 2 * (1 / 8) +
                                        P.𝓵₁ * H.gramVector (Sum.inr 2) ^ 2 * (1 / 8) +
                                      H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 2) * P.𝓵₂ * (-1 / 4) +
                                    H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 0) * P.𝓵₆.re * (1 / 2) +
                                  H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 0) * P.𝓵₇.re * (1 / 2) +
                                H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 1) * P.𝓵₆.im * (-1 / 2) +
                              H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 1) * P.𝓵₇.im * (-1 / 2) +
                            H.gramVector (Sum.inl 0) ^ 2 * P.𝓵₂ * (1 / 8) +
                          H.gramVector (Sum.inl 0) ^ 2 * P.𝓵₃ * (1 / 4) +
                        H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 0) * P.𝓵₆.re * (1 / 2) +
                      H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 0) * P.𝓵₇.re * (-1 / 2) +
                    H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 1) * P.𝓵₆.im * (-1 / 2) +
                  H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 1) * P.𝓵₇.im * (1 / 2) +
                H.gramVector (Sum.inr 2) ^ 2 * P.𝓵₂ * (1 / 8) +
              H.gramVector (Sum.inr 2) ^ 2 * P.𝓵₃ * (-1 / 4) +
            P.𝓵₄ * H.gramVector (Sum.inr 0) ^ 2 * (1 / 4) +
          P.𝓵₄ * H.gramVector (Sum.inr 1) ^ 2 * (1 / 4) +
        H.gramVector (Sum.inr 0) * H.gramVector (Sum.inr 1) * P.𝓵₅.im * (-1 / 2) +
      H.gramVector (Sum.inr 0) ^ 2 * P.𝓵₅.re * (1 / 4) +
    H.gramVector (Sum.inr 1) ^ 2 * P.𝓵₅.re * (-1 / 4)
  simp [← Complex.ofReal_pow, Complex.ofReal_re, normSq_Φ1_eq_gramVector,
    normSq_Φ2_eq_gramVector]P:PotentialParametersH:TwoHiggsDoublet⊢ P.𝓵₁ * H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 2) * 4⁻¹ + P.𝓵₁ * H.gramVector (Sum.inl 0) ^ 2 * 8⁻¹ +
                                  P.𝓵₁ * H.gramVector (Sum.inr 2) ^ 2 * 8⁻¹ +
                                H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 2) * P.𝓵₂ * (-1 / 4) +
                              H.gramVector (Sum.inl 0) ^ 2 * P.𝓵₂ * 8⁻¹ +
                            H.gramVector (Sum.inl 0) ^ 2 * P.𝓵₃ * 4⁻¹ +
                          H.gramVector (Sum.inr 2) ^ 2 * P.𝓵₂ * 8⁻¹ +
                        H.gramVector (Sum.inr 2) ^ 2 * P.𝓵₃ * (-1 / 4) +
                      P.𝓵₄ * H.gramVector (Sum.inr 0) ^ 2 * 4⁻¹ +
                    P.𝓵₄ * H.gramVector (Sum.inr 1) ^ 2 * 4⁻¹ +
                  H.gramVector (Sum.inr 0) * P.𝓵₆.re * (2⁻¹ * (H.gramVector (Sum.inl 0) + H.gramVector (Sum.inr 2))) +
                H.gramVector (Sum.inr 0) * P.𝓵₇.re * (2⁻¹ * (H.gramVector (Sum.inl 0) - H.gramVector (Sum.inr 2))) -
              H.gramVector (Sum.inr 1) * (2⁻¹ * (H.gramVector (Sum.inl 0) + H.gramVector (Sum.inr 2))) * P.𝓵₆.im -
            H.gramVector (Sum.inr 1) * (2⁻¹ * (H.gramVector (Sum.inl 0) - H.gramVector (Sum.inr 2))) * P.𝓵₇.im +
          P.𝓵₅.re * (H.gramVector (Sum.inr 0) ^ 2 * 4⁻¹ + -(H.gramVector (Sum.inr 1) ^ 2 * 4⁻¹)) * 2⁻¹ +
        P.𝓵₅.re * (H.gramVector (Sum.inr 0) ^ 2 * 4⁻¹ + -(H.gramVector (Sum.inr 1) ^ 2 * 4⁻¹)) * 2⁻¹ +
      P.𝓵₅.im * (H.gramVector (Sum.inr 0) * H.gramVector (Sum.inr 1) * 2⁻¹) * (-1 / 2) +
    P.𝓵₅.im * (H.gramVector (Sum.inr 0) * H.gramVector (Sum.inr 1) * (-1 / 2)) * 2⁻¹ =
  P.𝓵₁ * H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 2) * 4⁻¹ + P.𝓵₁ * H.gramVector (Sum.inl 0) ^ 2 * 8⁻¹ +
                                        P.𝓵₁ * H.gramVector (Sum.inr 2) ^ 2 * 8⁻¹ +
                                      H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 2) * P.𝓵₂ * (-1 / 4) +
                                    H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 0) * P.𝓵₆.re * 2⁻¹ +
                                  H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 0) * P.𝓵₇.re * 2⁻¹ +
                                H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 1) * P.𝓵₆.im * (-1 / 2) +
                              H.gramVector (Sum.inl 0) * H.gramVector (Sum.inr 1) * P.𝓵₇.im * (-1 / 2) +
                            H.gramVector (Sum.inl 0) ^ 2 * P.𝓵₂ * 8⁻¹ +
                          H.gramVector (Sum.inl 0) ^ 2 * P.𝓵₃ * 4⁻¹ +
                        H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 0) * P.𝓵₆.re * 2⁻¹ +
                      H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 0) * P.𝓵₇.re * (-1 / 2) +
                    H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 1) * P.𝓵₆.im * (-1 / 2) +
                  H.gramVector (Sum.inr 2) * H.gramVector (Sum.inr 1) * P.𝓵₇.im * 2⁻¹ +
                H.gramVector (Sum.inr 2) ^ 2 * P.𝓵₂ * 8⁻¹ +
              H.gramVector (Sum.inr 2) ^ 2 * P.𝓵₃ * (-1 / 4) +
            P.𝓵₄ * H.gramVector (Sum.inr 0) ^ 2 * 4⁻¹ +
          P.𝓵₄ * H.gramVector (Sum.inr 1) ^ 2 * 4⁻¹ +
        H.gramVector (Sum.inr 0) * H.gramVector (Sum.inr 1) * P.𝓵₅.im * (-1 / 2) +
      H.gramVector (Sum.inr 0) ^ 2 * P.𝓵₅.re * 4⁻¹ +
    H.gramVector (Sum.inr 1) ^ 2 * P.𝓵₅.re * (-1 / 4)
  ringAll goals completed! 🐙@[simp]
lemma gaugeGroupI_smul_quarticTerm (g : StandardModel.GaugeGroupI) (P : PotentialParameters)
    (H : TwoHiggsDoublet) :
    quarticTerm P (g • H) = quarticTerm P H := byg:GaugeGroupIP:PotentialParametersH:TwoHiggsDoublet⊢ quarticTerm P (g • H) = quarticTerm P H
  simp [quarticTerm_eq_gramVector]All goals completed! 🐙@[simp]
lemma quarticTerm_zero : quarticTerm 0 = 0 := by⊢ quarticTerm 0 = 0
  ext HH:TwoHiggsDoublet⊢ quarticTerm 0 H = 0 H
  simp [quarticTerm]All goals completed! 🐙

lemma quarticTerm_stabilityCounterExample (H : TwoHiggsDoublet) :
    quarticTerm .stabilityCounterExample H =
    (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2:= byH:TwoHiggsDoublet⊢ quarticTerm PotentialParameters.stabilityCounterExample H = (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2
  calc _ = (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2
    + 2 * ‖⟪H.Φ1, H.Φ2⟫_ℂ‖ ^ 2
    + (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + ⟪H.Φ2, H.Φ1⟫_ℂ ^ 2).re
    - 2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ2, H.Φ1⟫_ℂ).re) := byH:TwoHiggsDoublet⊢ quarticTerm PotentialParameters.stabilityCounterExample H =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 + 2 * ‖⟪H.Φ1, H.Φ2⟫_ℂ‖ ^ 2 + (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + ⟪H.Φ2, H.Φ1⟫_ℂ ^ 2).re -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ2, H.Φ1⟫_ℂ).re)
        simp [quarticTerm, PotentialParameters.stabilityCounterExample, Complex.add_re,
          ← Complex.ofReal_pow]H:TwoHiggsDoublet⊢ ‖H.Φ1‖ ^ 2 * ‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 * ‖H.Φ2‖ ^ 2 + 2 * ‖H.Φ1‖ ^ 2 * ‖H.Φ2‖ ^ 2 + 2 * ‖⟪H.Φ1, H.Φ2⟫_ℂ‖ ^ 2 +
        ((⟪H.Φ1, H.Φ2⟫_ℂ ^ 2).re + (⟪H.Φ2, H.Φ1⟫_ℂ ^ 2).re) +
      (-(2 * ‖H.Φ1‖ ^ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) + -(2 * ‖H.Φ1‖ ^ 2 * (⟪H.Φ2, H.Φ1⟫_ℂ).re)) +
    (-(2 * ‖H.Φ2‖ ^ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) + -(2 * ‖H.Φ2‖ ^ 2 * (⟪H.Φ2, H.Φ1⟫_ℂ).re)) =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 + 2 * ‖⟪H.Φ1, H.Φ2⟫_ℂ‖ ^ 2 + ((⟪H.Φ1, H.Φ2⟫_ℂ ^ 2).re + (⟪H.Φ2, H.Φ1⟫_ℂ ^ 2).re) -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ2, H.Φ1⟫_ℂ).re)
        ringAll goals completed! 🐙
      _ = (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 := byH:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 + 2 * ‖⟪H.Φ1, H.Φ2⟫_ℂ‖ ^ 2 + (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + ⟪H.Φ2, H.Φ1⟫_ℂ ^ 2).re -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ2, H.Φ1⟫_ℂ).re) =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2
        rw [← inner_conj_symm H.Φ2 H.Φ1,H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 + 2 * ‖⟪H.Φ1, H.Φ2⟫_ℂ‖ ^ 2 +
      (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + (starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ ^ 2).re -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + ((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).re) =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 +
        2 * ((⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ1, H.Φ2⟫_ℂ).im) +
      (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + (starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ ^ 2).re -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + ((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).re) =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 ← Complex.normSq_eq_norm_sq,H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 + 2 * Complex.normSq ⟪H.Φ1, H.Φ2⟫_ℂ +
      (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + (starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ ^ 2).re -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + ((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).re) =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 +
        2 * ((⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ1, H.Φ2⟫_ℂ).im) +
      (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + (starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ ^ 2).re -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + ((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).re) =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 Complex.normSq_applyH:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 +
        2 * ((⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ1, H.Φ2⟫_ℂ).im) +
      (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + (starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ ^ 2).re -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + ((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).re) =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 +
        2 * ((⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ1, H.Φ2⟫_ℂ).im) +
      (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + (starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ ^ 2).re -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + ((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).re) =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2]H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) ^ 2 +
        2 * ((⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ1, H.Φ2⟫_ℂ).im) +
      (⟪H.Φ1, H.Φ2⟫_ℂ ^ 2 + (starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ ^ 2).re -
    2 * (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + ((starRingEnd ℂ) ⟪H.Φ1, H.Φ2⟫_ℂ).re) =
  (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2
        simp only [Complex.add_re, Complex.mul_re, Complex.conj_re, Complex.conj_im, pow_two]H:TwoHiggsDoublet⊢ (‖H.Φ1‖ * ‖H.Φ1‖ + ‖H.Φ2‖ * ‖H.Φ2‖) * (‖H.Φ1‖ * ‖H.Φ1‖ + ‖H.Φ2‖ * ‖H.Φ2‖) +
        2 * ((⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ1, H.Φ2⟫_ℂ).im) +
      ((⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ1, H.Φ2⟫_ℂ).re - (⟪H.Φ1, H.Φ2⟫_ℂ).im * (⟪H.Φ1, H.Φ2⟫_ℂ).im +
        ((⟪H.Φ1, H.Φ2⟫_ℂ).re * (⟪H.Φ1, H.Φ2⟫_ℂ).re - -(⟪H.Φ1, H.Φ2⟫_ℂ).im * -(⟪H.Φ1, H.Φ2⟫_ℂ).im)) -
    2 * (‖H.Φ1‖ * ‖H.Φ1‖ + ‖H.Φ2‖ * ‖H.Φ2‖) * ((⟪H.Φ1, H.Φ2⟫_ℂ).re + (⟪H.Φ1, H.Φ2⟫_ℂ).re) =
  (‖H.Φ1‖ * ‖H.Φ1‖ + ‖H.Φ2‖ * ‖H.Φ2‖ - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) *
    (‖H.Φ1‖ * ‖H.Φ1‖ + ‖H.Φ2‖ * ‖H.Φ2‖ - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re)
        ringAll goals completed! 🐙

lemma quarticTerm_stabilityCounterExample_eq_norm_pow_four (H : TwoHiggsDoublet) :
    quarticTerm .stabilityCounterExample H = ‖H.Φ1 - H.Φ2‖ ^ 4 := byH:TwoHiggsDoublet⊢ quarticTerm PotentialParameters.stabilityCounterExample H = ‖H.Φ1 - H.Φ2‖ ^ 4
  rw [quarticTerm_stabilityCounterExampleH:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = ‖H.Φ1 - H.Φ2‖ ^ 4H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = ‖H.Φ1 - H.Φ2‖ ^ 4]H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = ‖H.Φ1 - H.Φ2‖ ^ 4
  rw [show ‖H.Φ1 - H.Φ2‖ ^ 4 = (‖H.Φ1 - H.Φ2‖ ^ 2) ^ 2 from byH:TwoHiggsDoublet⊢ ‖H.Φ1 - H.Φ2‖ ^ 4 = (‖H.Φ1 - H.Φ2‖ ^ 2) ^ 2H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = (‖H.Φ1‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re + ‖H.Φ2‖ ^ 2) ^ 2 ringAll goals completed! 🐙H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = (‖H.Φ1‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re + ‖H.Φ2‖ ^ 2) ^ 2, norm_sub_sq (𝕜 := ℂ),H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = (‖H.Φ1‖ ^ 2 - 2 * RCLike.re ⟪H.Φ1, H.Φ2⟫_ℂ + ‖H.Φ2‖ ^ 2) ^ 2H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = (‖H.Φ1‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re + ‖H.Φ2‖ ^ 2) ^ 2
    RCLike.re_to_complexH:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = (‖H.Φ1‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re + ‖H.Φ2‖ ^ 2) ^ 2H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = (‖H.Φ1‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re + ‖H.Φ2‖ ^ 2) ^ 2]H:TwoHiggsDoublet⊢ (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = (‖H.Φ1‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re + ‖H.Φ2‖ ^ 2) ^ 2
  ringAll goals completed! 🐙

lemma quarticTerm_stabilityCounterExample_nonneg (H : TwoHiggsDoublet) :
    0 ≤ quarticTerm .stabilityCounterExample H := byH:TwoHiggsDoublet⊢ 0 ≤ quarticTerm PotentialParameters.stabilityCounterExample H
  rw [quarticTerm_stabilityCounterExample_eq_norm_pow_fourH:TwoHiggsDoublet⊢ 0 ≤ ‖H.Φ1 - H.Φ2‖ ^ 4H:TwoHiggsDoublet⊢ 0 ≤ ‖H.Φ1 - H.Φ2‖ ^ 4]H:TwoHiggsDoublet⊢ 0 ≤ ‖H.Φ1 - H.Φ2‖ ^ 4
  positivityAll goals completed! 🐙

lemma massTerm_zero_of_quarticTerm_zero_stabilityCounterExample (H : TwoHiggsDoublet)
    (h : quarticTerm .stabilityCounterExample H = 0) :
    massTerm .stabilityCounterExample H = 0 := byH:TwoHiggsDoubleth:quarticTerm PotentialParameters.stabilityCounterExample H = 0⊢ massTerm PotentialParameters.stabilityCounterExample H = 0
  rw [quarticTerm_stabilityCounterExample_eq_norm_pow_fourH:TwoHiggsDoubleth:‖H.Φ1 - H.Φ2‖ ^ 4 = 0⊢ massTerm PotentialParameters.stabilityCounterExample H = 0H:TwoHiggsDoubleth:‖H.Φ1 - H.Φ2‖ ^ 4 = 0⊢ massTerm PotentialParameters.stabilityCounterExample H = 0] at hH:TwoHiggsDoubleth:‖H.Φ1 - H.Φ2‖ ^ 4 = 0⊢ massTerm PotentialParameters.stabilityCounterExample H = 0
  rw [massTerm_stabilityCounterExampleH:TwoHiggsDoubleth:‖H.Φ1 - H.Φ2‖ ^ 4 = 0⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im = 0H:TwoHiggsDoubleth:‖H.Φ1 - H.Φ2‖ ^ 4 = 0⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im = 0]H:TwoHiggsDoubleth:‖H.Φ1 - H.Φ2‖ ^ 4 = 0⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im = 0
  have h1 : H.Φ1 = H.Φ2 := byH:TwoHiggsDoubleth:quarticTerm PotentialParameters.stabilityCounterExample H = 0⊢ massTerm PotentialParameters.stabilityCounterExample H = 0H:TwoHiggsDoubleth:‖H.Φ1 - H.Φ2‖ ^ 4 = 0h1:H.Φ1 = H.Φ2⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im = 0 simpa [sub_eq_zero] using hH:TwoHiggsDoubleth:‖H.Φ1 - H.Φ2‖ ^ 4 = 0h1:H.Φ1 = H.Φ2⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im = 0H:TwoHiggsDoubleth:‖H.Φ1 - H.Φ2‖ ^ 4 = 0h1:H.Φ1 = H.Φ2⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im = 0
  simp [← Complex.ofReal_pow, h1]All goals completed! 🐙

D. The full potential

We define the full potential as the sum of the mass and quartic terms, and prove that it is gauge invariant.

              @[simp]
lemma gaugeGroupI_smul_potential (g : StandardModel.GaugeGroupI)
    (P : PotentialParameters) (H : TwoHiggsDoublet) :
    potential P (g • H) = potential P H := byg:GaugeGroupIP:PotentialParametersH:TwoHiggsDoublet⊢ potential P (g • H) = potential P H
  simp [potential]All goals completed! 🐙@[simp]
lemma potential_zero : potential 0 = 0 := by⊢ potential 0 = 0
  ext HH:TwoHiggsDoublet⊢ potential 0 H = 0 H
  simp [potential]All goals completed! 🐙lemma potential_stabilityCounterExample (H : TwoHiggsDoublet) :
    potential .stabilityCounterExample H = 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + ‖H.Φ1 - H.Φ2‖ ^ 4 := byH:TwoHiggsDoublet⊢ potential PotentialParameters.stabilityCounterExample H = 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + ‖H.Φ1 - H.Φ2‖ ^ 4
  simp [potential, massTerm_stabilityCounterExample,
    quarticTerm_stabilityCounterExample_eq_norm_pow_four]All goals completed! 🐙

lemma potential_eq_gramVector (P : PotentialParameters) (H : TwoHiggsDoublet) :
    potential P H = ∑ μ, P.ξ μ * H.gramVector μ +
    ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b := byP:PotentialParametersH:TwoHiggsDoublet⊢ potential P H = ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b
  rw [potential,P:PotentialParametersH:TwoHiggsDoublet⊢ massTerm P H + quarticTerm P H = ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a bAll goals completed! 🐙 massTerm_eq_gramVector,P:PotentialParametersH:TwoHiggsDoublet⊢ ∑ μ, P.ξ μ * H.gramVector μ + quarticTerm P H =
  ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a bAll goals completed! 🐙 quarticTerm_eq_gramVectorP:PotentialParametersH:TwoHiggsDoublet⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b =
  ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a bAll goals completed! 🐙]All goals completed! 🐙TODO "Define a general effective potential for the two Higgs doublet model, mirroring
  `StandardModel.HiggsField.EffectivePotential`"

E. Stability of the potential

E.1. The stability condition

We define the condition that the potential is stable, that is, bounded from below.

The condition that the potential is stable.

def PotentialIsStable (P : PotentialParameters) : Prop :=
  ∃ c : ℝ, ∀ H : TwoHiggsDoublet, c ≤ potential P H

E.2. Instability of the stabilityCounterExample potential

The potential stabilityCounterExample is not stable.


lemma stabilityCounterExample_not_potentialIsStable :
    ¬ PotentialIsStable .stabilityCounterExample := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExample
  simp [PotentialIsStable]⊢ ∀ (x : ℝ), ∃ x_1, potential PotentialParameters.stabilityCounterExample x_1 < x
  intro cc:ℝ⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  /- The angle t and properties thereof. -/
  let t := Real.arctan (2 * Real.sqrt (|c| + 1))⁻¹c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have t_pos : 0 < t := arctan_pos.mpr (byc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹⊢ 0 < (2 * √(|c| + 1))⁻¹c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < t⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c positivityAll goals completed! 🐙c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < t⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < t⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have t_le_pi_div_2 : t ≤ Real.pi / 2 := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    simpa [t] using le_of_lt <| arctan_lt_pi_div_two ((√(|c| + 1))⁻¹ * 2⁻¹)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have t_ne_zero : t ≠ 0 := t_pos.ne'c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have sin_t_pos : 0 < sin t :=
    Real.sin_pos_of_pos_of_lt_pi t_pos (byc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0⊢ t < πc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin t⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c nlinarith [Real.pi_pos]All goals completed! 🐙c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin t⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin t⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have cos_t_pos : 0 < cos t := cos_arctan_pos (2 * Real.sqrt (|c| + 1))⁻¹c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos t⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have t_mul_sin_t_nonneg : 0 ≤ 2 * t * sin t - t ^ 2 := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    have hs := Real.mul_le_sin t_pos.le t_le_pi_div_2c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos ths:2 / π * t ≤ sin t⊢ 0 ≤ 2 * t * sin t - t ^ 2c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [div_mul_eq_mul_div,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos ths:2 * t / π ≤ sin t⊢ 0 ≤ 2 * t * sin t - t ^ 2c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos ths:2 * t ≤ sin t * π⊢ 0 ≤ 2 * t * sin t - t ^ 2c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c div_le_iff₀ Real.pi_posc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos ths:2 * t ≤ sin t * π⊢ 0 ≤ 2 * t * sin t - t ^ 2c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos ths:2 * t ≤ sin t * π⊢ 0 ≤ 2 * t * sin t - t ^ 2c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c] at hsc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos ths:2 * t ≤ sin t * π⊢ 0 ≤ 2 * t * sin t - t ^ 2c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    nlinarith [mul_le_mul_of_nonneg_left hs t_pos.le, Real.pi_le_four]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  /- The Two Higgs doublet violating stability.
    The two Higgs doublet is constructed so that for the gram vector
    `v` we have:
    - `v₀ = cos t/(2 * t * (sin t)^2)`
    - `v₁/v₀ = (1 - t * sin t)`
    - `v₂/v₀ = - t * cos t`
    - `v₃ = 0` -/
  let H : TwoHiggsDoublet := {
    Φ1 := !2[√(cos t/(4 * t * (sin t)^2)), 0]
    Φ2 := √(cos t/(4 * t * (sin t)^2)) • !2[1 - t * sin t - Complex.I * t * cos t,
      √(2 * t * sin t - t ^ 2)] }c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have Φ1_norm_sq : ‖H.Φ1‖ ^ 2 = cos t/(4 * t * (sin t)^2) := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    simp [H, PiLp.norm_sq_eq_of_L2]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }⊢ √(cos t / (4 * t * sin t ^ 2)) ^ 2 = cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [sq_sqrt (byc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }⊢ 0 ≤ cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c positivityAll goals completed! 🐙c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c)]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have Φ2_norm_sq : ‖H.Φ2‖ ^ 2 = cos t/(4 * t * (sin t)^2) := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    simp [H, norm_smul, mul_pow]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ √(cos t / (4 * t * sin t ^ 2)) ^ 2 *
    ‖!₂[1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t, ↑√(2 * t * sin t - t ^ 2)]‖ ^ 2 =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [sq_sqrt (byc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ 0 ≤ cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ cos t / (4 * t * sin t ^ 2) *
    ‖!₂[1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t, ↑√(2 * t * sin t - t ^ 2)]‖ ^ 2 =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c positivityAll goals completed! 🐙c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ cos t / (4 * t * sin t ^ 2) *
    ‖!₂[1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t, ↑√(2 * t * sin t - t ^ 2)]‖ ^ 2 =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c)]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ cos t / (4 * t * sin t ^ 2) *
    ‖!₂[1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t, ↑√(2 * t * sin t - t ^ 2)]‖ ^ 2 =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    simp [PiLp.norm_sq_eq_of_L2]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + √(2 * t * sin t - t ^ 2) ^ 2) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [sq_sqrt (byc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ 0 ≤ 2 * t * sin t - t ^ 2c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c positivityAll goals completed! 🐙c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c)]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    have h0 : ‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 =
        1 + t ^ 2 - 2 * t * sin t := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
      rw [← Complex.normSq_eq_norm_sq,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ Complex.normSq (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) = 1 + t ^ 2 - 2 * t * sin tc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).re *
      (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).re +
    (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).im *
      (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).im =
  1 + t ^ 2 - 2 * t * sin tc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Complex.normSq_applyc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).re *
      (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).re +
    (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).im *
      (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).im =
  1 + t ^ 2 - 2 * t * sin tc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).re *
      (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).re +
    (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).im *
      (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).im =
  1 + t ^ 2 - 2 * t * sin tc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).re *
      (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).re +
    (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).im *
      (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t).im =
  1 + t ^ 2 - 2 * t * sin tc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
      simp only [Complex.sub_re, Complex.sub_im, Complex.mul_re, Complex.mul_im, Complex.I_re,
        Complex.I_im, Complex.ofReal_re, Complex.ofReal_im, Complex.one_re, Complex.one_im,
        Complex.sin_ofReal_re, Complex.sin_ofReal_im, Complex.cos_ofReal_re, Complex.cos_ofReal_im]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ (1 - (t * sin t - 0 * 0) - ((0 * t - 1 * 0) * cos t - (0 * 0 + 1 * t) * 0)) *
      (1 - (t * sin t - 0 * 0) - ((0 * t - 1 * 0) * cos t - (0 * 0 + 1 * t) * 0)) +
    (0 - (t * 0 + 0 * sin t) - ((0 * t - 1 * 0) * 0 + (0 * 0 + 1 * t) * cos t)) *
      (0 - (t * 0 + 0 * sin t) - ((0 * t - 1 * 0) * 0 + (0 * 0 + 1 * t) * cos t)) =
  1 + t ^ 2 - 2 * t * sin tc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
      nlinarith [Real.sin_sq_add_cos_sq t]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) *
    (‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 + (2 * t * sin t - t ^ 2)) =
  cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [h0c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) * (1 + t ^ 2 - 2 * t * sin t + (2 * t * sin t - t ^ 2)) = cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) * (1 + t ^ 2 - 2 * t * sin t + (2 * t * sin t - t ^ 2)) = cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ cos t / (4 * t * sin t ^ 2) * (1 + t ^ 2 - 2 * t * sin t + (2 * t * sin t - t ^ 2)) = cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    field_simpc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)h0:‖1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t‖ ^ 2 = 1 + t ^ 2 - 2 * t * sin t⊢ 1 + t ^ 2 - t * sin t * 2 + t * (sin t * 2 - t) = 1c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    ringc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have Φ1_inner_Φ2 : ⟪H.Φ1, H.Φ2⟫_ℂ = Complex.ofReal (cos t/(4 * t * (sin t)^2) *
      (1 - t * sin t)) + Complex.I *
      Complex.ofReal (cos t/(4 * t * (sin t)^2) * (- t * cos t)) := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    simp [H, PiLp.inner_apply]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ↑√(cos t / (4 * t * sin t ^ 2)) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) *
    ↑√(cos t / (4 * t * sin t ^ 2)) =
  Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (1 - ↑t * Complex.sin ↑t) +
    -(Complex.I * (Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (↑t * Complex.cos ↑t)))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    trans Complex.ofReal ((√(cos t / (4 * t * sin t ^ 2))) ^ 2) *
        (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ↑√(cos t / (4 * t * sin t ^ 2)) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) *
    ↑√(cos t / (4 * t * sin t ^ 2)) =
  ↑(√(cos t / (4 * t * sin t ^ 2)) ^ 2) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ↑(√(cos t / (4 * t * sin t ^ 2)) ^ 2) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) =
  Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (1 - ↑t * Complex.sin ↑t) +
    -(Complex.I * (Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (↑t * Complex.cos ↑t)))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    ·c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ↑√(cos t / (4 * t * sin t ^ 2)) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) *
    ↑√(cos t / (4 * t * sin t ^ 2)) =
  ↑(√(cos t / (4 * t * sin t ^ 2)) ^ 2) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c simpc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ↑√(cos t / (4 * t * sin t ^ 2)) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) *
    ↑√(cos t / (4 * t * sin t ^ 2)) =
  ↑√(cos t / (4 * t * sin t ^ 2)) ^ 2 * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
      ringAll goals completed! 🐙c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [sq_sqrt (byc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ 0 ≤ cos t / (4 * t * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ↑(cos t / (4 * t * sin t ^ 2)) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) =
  Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (1 - ↑t * Complex.sin ↑t) +
    -(Complex.I * (Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (↑t * Complex.cos ↑t)))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c positivityAll goals completed! 🐙c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ↑(cos t / (4 * t * sin t ^ 2)) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) =
  Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (1 - ↑t * Complex.sin ↑t) +
    -(Complex.I * (Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (↑t * Complex.cos ↑t)))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c)]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ ↑(cos t / (4 * t * sin t ^ 2)) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) =
  Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (1 - ↑t * Complex.sin ↑t) +
    -(Complex.I * (Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (↑t * Complex.cos ↑t)))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    simp only [Complex.ofReal_div, Complex.ofReal_cos, Complex.ofReal_mul, Complex.ofReal_ofNat,
      Complex.ofReal_pow, Complex.ofReal_sin]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)⊢ Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (1 - ↑t * Complex.sin ↑t - Complex.I * ↑t * Complex.cos ↑t) =
  Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (1 - ↑t * Complex.sin ↑t) +
    -(Complex.I * (Complex.cos ↑t / (4 * ↑t * Complex.sin ↑t ^ 2) * (↑t * Complex.cos ↑t)))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    ringc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have Φ1_inner_Φ2_re : (⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t/(4 * t * (sin t)^2) * (1 - t * sin t) := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [Φ1_inner_Φ2,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ (↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))).re =
  cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ cos t / (4 * t * sin t ^ 2) * (1 - t * sin t) + Complex.I.re * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) =
  cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Complex.add_re,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ (↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))).re + (Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))).re =
  cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ cos t / (4 * t * sin t ^ 2) * (1 - t * sin t) + Complex.I.re * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) =
  cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Complex.ofReal_re,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ cos t / (4 * t * sin t ^ 2) * (1 - t * sin t) + (Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))).re =
  cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ cos t / (4 * t * sin t ^ 2) * (1 - t * sin t) + Complex.I.re * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) =
  cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Complex.re_mul_ofRealc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ cos t / (4 * t * sin t ^ 2) * (1 - t * sin t) + Complex.I.re * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) =
  cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ cos t / (4 * t * sin t ^ 2) * (1 - t * sin t) + Complex.I.re * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) =
  cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))⊢ cos t / (4 * t * sin t ^ 2) * (1 - t * sin t) + Complex.I.re * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) =
  cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    simpc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have Φ1_inner_Φ2_im : (⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t/(4 * t * (sin t)^2) * (- t * cos t) := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [Φ1_inner_Φ2,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ (↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))).im =
  cos t / (4 * t * sin t ^ 2) * (-t * cos t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ 0 + Complex.I.im * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) = cos t / (4 * t * sin t ^ 2) * (-t * cos t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Complex.add_im,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ (↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))).im + (Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))).im =
  cos t / (4 * t * sin t ^ 2) * (-t * cos t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ 0 + Complex.I.im * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) = cos t / (4 * t * sin t ^ 2) * (-t * cos t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Complex.im_mul_ofReal,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ (↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))).im + Complex.I.im * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) =
  cos t / (4 * t * sin t ^ 2) * (-t * cos t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ 0 + Complex.I.im * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) = cos t / (4 * t * sin t ^ 2) * (-t * cos t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Complex.ofReal_imc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ 0 + Complex.I.im * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) = cos t / (4 * t * sin t ^ 2) * (-t * cos t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ 0 + Complex.I.im * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) = cos t / (4 * t * sin t ^ 2) * (-t * cos t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)⊢ 0 + Complex.I.im * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) = cos t / (4 * t * sin t ^ 2) * (-t * cos t)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    simpc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have potential_H_cos_sin : potential .stabilityCounterExample H =
      - (cos t) ^ 2/ (4 * (sin t)^2) := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [potential,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ massTerm PotentialParameters.stabilityCounterExample H + quarticTerm PotentialParameters.stabilityCounterExample H =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c massTerm_stabilityCounterExample,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + quarticTerm PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c quarticTerm_stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + (‖H.Φ1‖ ^ 2 + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 = -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [Φ1_norm_sq,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + (cos t / (4 * t * sin t ^ 2) + ‖H.Φ2‖ ^ 2 - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) +
    (cos t / (4 * t * sin t ^ 2) + cos t / (4 * t * sin t ^ 2) - 2 * (cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))) ^
      2 =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Φ2_norm_sq,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im + (cos t / (4 * t * sin t ^ 2) + cos t / (4 * t * sin t ^ 2) - 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).re) ^ 2 =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) +
    (cos t / (4 * t * sin t ^ 2) + cos t / (4 * t * sin t ^ 2) - 2 * (cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))) ^
      2 =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Φ1_inner_Φ2_re,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (⟪H.Φ1, H.Φ2⟫_ℂ).im +
    (cos t / (4 * t * sin t ^ 2) + cos t / (4 * t * sin t ^ 2) - 2 * (cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))) ^
      2 =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) +
    (cos t / (4 * t * sin t ^ 2) + cos t / (4 * t * sin t ^ 2) - 2 * (cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))) ^
      2 =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c Φ1_inner_Φ2_imc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) +
    (cos t / (4 * t * sin t ^ 2) + cos t / (4 * t * sin t ^ 2) - 2 * (cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))) ^
      2 =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) +
    (cos t / (4 * t * sin t ^ 2) + cos t / (4 * t * sin t ^ 2) - 2 * (cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))) ^
      2 =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)⊢ 2 * (cos t / (4 * t * sin t ^ 2) * (-t * cos t)) +
    (cos t / (4 * t * sin t ^ 2) + cos t / (4 * t * sin t ^ 2) - 2 * (cos t / (4 * t * sin t ^ 2) * (1 - t * sin t))) ^
      2 =
  -cos t ^ 2 / (4 * sin t ^ 2)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    fieldc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  have potential_eq_c : potential .stabilityCounterExample H = - (|c| + 1) := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    have htan : sin t / cos t = (2 * √(|c| + 1))⁻¹ := by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
      rw [← tan_eq_sin_div_cosc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ tan t = (2 * √(|c| + 1))⁻¹c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ tan t = (2 * √(|c| + 1))⁻¹c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)⊢ tan t = (2 * √(|c| + 1))⁻¹c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
      exact tan_arctan _c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [potential_H_cos_sin,c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -cos t ^ 2 / (4 * sin t ^ 2) = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -1 / (4 * (2 * √(|c| + 1))⁻¹ ^ 2) = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c show -cos t ^ 2 / (4 * sin t ^ 2) = -1 / (4 * (sin t / cos t) ^ 2) by⊢ ¬PotentialIsStable PotentialParameters.stabilityCounterExamplec:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -1 / (4 * (2 * √(|c| + 1))⁻¹ ^ 2) = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
      fieldAll goals completed! 🐙c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -1 / (4 * (2 * √(|c| + 1))⁻¹ ^ 2) = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c, htanc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -1 / (4 * (2 * √(|c| + 1))⁻¹ ^ 2) = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -1 / (4 * (2 * √(|c| + 1))⁻¹ ^ 2) = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -1 / (4 * (2 * √(|c| + 1))⁻¹ ^ 2) = -(|c| + 1)c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    field_simpc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -(2 ^ 2 * √(|c| + 1) ^ 2) = -(4 * (|c| + 1))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    rw [sq_sqrt (byc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ 0 ≤ |c| + 1c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -(2 ^ 2 * (|c| + 1)) = -(4 * (|c| + 1))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c positivityAll goals completed! 🐙c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -(2 ^ 2 * (|c| + 1)) = -(4 * (|c| + 1))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c)]c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)htan:sin t / cos t = (2 * √(|c| + 1))⁻¹⊢ -(2 ^ 2 * (|c| + 1)) = -(4 * (|c| + 1))c:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
    ringc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < cc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ ∃ x, potential PotentialParameters.stabilityCounterExample x < c
  /- Proving potential is unbounded. -/
  use Hhc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ potential PotentialParameters.stabilityCounterExample H < c
  rw [potential_eq_chc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ -(|c| + 1) < chc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ -(|c| + 1) < c]hc:ℝt:ℝ := arctan (2 * √(|c| + 1))⁻¹t_pos:0 < tt_le_pi_div_2:t ≤ π / 2t_ne_zero:t ≠ 0sin_t_pos:0 < sin tcos_t_pos:0 < cos tt_mul_sin_t_nonneg:0 ≤ 2 * t * sin t - t ^ 2H:TwoHiggsDoublet := 
  { Φ1 := !₂[↑√(cos t / (4 * t * sin t ^ 2)), 0],
    Φ2 := √(cos t / (4 * t * sin t ^ 2)) • !₂[1 - ↑t * ↑(sin t) - Complex.I * ↑t * ↑(cos t), ↑√(2 * t * sin t - t ^ 2)] }Φ1_norm_sq:‖H.Φ1‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ2_norm_sq:‖H.Φ2‖ ^ 2 = cos t / (4 * t * sin t ^ 2)Φ1_inner_Φ2:⟪H.Φ1, H.Φ2⟫_ℂ =
  ↑(cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)) + Complex.I * ↑(cos t / (4 * t * sin t ^ 2) * (-t * cos t))Φ1_inner_Φ2_re:(⟪H.Φ1, H.Φ2⟫_ℂ).re = cos t / (4 * t * sin t ^ 2) * (1 - t * sin t)Φ1_inner_Φ2_im:(⟪H.Φ1, H.Φ2⟫_ℂ).im = cos t / (4 * t * sin t ^ 2) * (-t * cos t)potential_H_cos_sin:potential PotentialParameters.stabilityCounterExample H = -cos t ^ 2 / (4 * sin t ^ 2)potential_eq_c:potential PotentialParameters.stabilityCounterExample H = -(|c| + 1)⊢ -(|c| + 1) < c
  grindAll goals completed! 🐙

E.3. The reduced mass term

The reduced mass term is a function that helps express the stability condition. It is the function J2 in https://arxiv.org/abs/hep-ph/0605184.

              

lemma massTermReduced_lower_bound (P : PotentialParameters) (k : EuclideanSpace ℝ (Fin 3))
    (hk : ‖k‖ ^ 2 ≤ 1) : P.ξ (Sum.inl 0) - √(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ massTermReduced P k := byP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ P.ξ (Sum.inl 0) - √(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ massTermReduced P k
  simp only [Fin.isValue, massTermReduced]P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ P.ξ (Sum.inl 0) - √(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ P.ξ (Sum.inl 0) + ∑ x, P.ξ (Sum.inr x) * k.ofLp x
  have h1 (a b c : ℝ) (h : - b ≤ c) : a - b ≤ a + c:= byP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ P.ξ (Sum.inl 0) - √(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ massTermReduced P kP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + c⊢ P.ξ (Sum.inl 0) - √(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ P.ξ (Sum.inl 0) + ∑ x, P.ξ (Sum.inr x) * k.ofLp x grindP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + c⊢ P.ξ (Sum.inl 0) - √(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ P.ξ (Sum.inl 0) + ∑ x, P.ξ (Sum.inr x) * k.ofLp xP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + c⊢ P.ξ (Sum.inl 0) - √(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ P.ξ (Sum.inl 0) + ∑ x, P.ξ (Sum.inr x) * k.ofLp x
  apply h1P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + c⊢ -√(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ ∑ x, P.ξ (Sum.inr x) * k.ofLp x
  let ξEuclid : EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 (fun a => P.ξ (Sum.inr a))P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -√(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ ∑ x, P.ξ (Sum.inr x) * k.ofLp x
  trans - ‖ξEuclid‖P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -√(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ -‖ξEuclid‖P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -‖ξEuclid‖ ≤ ∑ x, P.ξ (Sum.inr x) * k.ofLp x
  ·P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -√(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ -‖ξEuclid‖ simp [PiLp.norm_eq_of_L2, ξEuclid]All goals completed! 🐙
  trans - (‖k‖ * ‖ξEuclid‖)P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -‖ξEuclid‖ ≤ -(‖k‖ * ‖ξEuclid‖)P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -(‖k‖ * ‖ξEuclid‖) ≤ ∑ x, P.ξ (Sum.inr x) * k.ofLp x
  ·P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -‖ξEuclid‖ ≤ -(‖k‖ * ‖ξEuclid‖) simpP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ ‖k‖ * ‖ξEuclid‖ ≤ ‖ξEuclid‖
    simp at hkP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)hk:‖k‖ ≤ 1⊢ ‖k‖ * ‖ξEuclid‖ ≤ ‖ξEuclid‖
    have ha (a b : ℝ) (h : a ≤ 1) (ha : 0 ≤ a) (hb : 0 ≤ b) : a * b ≤ b := byP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ P.ξ (Sum.inl 0) - √(∑ a, |P.ξ (Sum.inr a)| ^ 2) ≤ massTermReduced P kP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)hk:‖k‖ ≤ 1ha:∀ (a b : ℝ), a ≤ 1 → 0 ≤ a → 0 ≤ b → a * b ≤ b⊢ ‖k‖ * ‖ξEuclid‖ ≤ ‖ξEuclid‖ nlinarithP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)hk:‖k‖ ≤ 1ha:∀ (a b : ℝ), a ≤ 1 → 0 ≤ a → 0 ≤ b → a * b ≤ b⊢ ‖k‖ * ‖ξEuclid‖ ≤ ‖ξEuclid‖P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)hk:‖k‖ ≤ 1ha:∀ (a b : ℝ), a ≤ 1 → 0 ≤ a → 0 ≤ b → a * b ≤ b⊢ ‖k‖ * ‖ξEuclid‖ ≤ ‖ξEuclid‖
    exact ha _ _ hk (norm_nonneg k) (norm_nonneg ξEuclid)All goals completed! 🐙
  trans - ‖⟪k, ξEuclid⟫_ℝ‖P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -(‖k‖ * ‖ξEuclid‖) ≤ -‖⟪k, ξEuclid⟫_ℝ‖P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -‖⟪k, ξEuclid⟫_ℝ‖ ≤ ∑ x, P.ξ (Sum.inr x) * k.ofLp x
  ·P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -(‖k‖ * ‖ξEuclid‖) ≤ -‖⟪k, ξEuclid⟫_ℝ‖ simpP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ |⟪k, ξEuclid⟫_ℝ| ≤ ‖k‖ * ‖ξEuclid‖
    exact abs_real_inner_le_norm k ξEuclidAll goals completed! 🐙
  trans ⟪k, ξEuclid⟫_ℝP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -‖⟪k, ξEuclid⟫_ℝ‖ ≤ ⟪k, ξEuclid⟫_ℝP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ ⟪k, ξEuclid⟫_ℝ ≤ ∑ x, P.ξ (Sum.inr x) * k.ofLp x
  ·P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -‖⟪k, ξEuclid⟫_ℝ‖ ≤ ⟪k, ξEuclid⟫_ℝ simpP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1h1:∀ (a b c : ℝ), -b ≤ c → a - b ≤ a + cξEuclid:EuclideanSpace ℝ (Fin 3) := WithLp.toLp 2 fun a => P.ξ (Sum.inr a)⊢ -|⟪k, ξEuclid⟫_ℝ| ≤ ⟪k, ξEuclid⟫_ℝ
    grindAll goals completed! 🐙
  simp [PiLp.inner_apply, ξEuclid]All goals completed! 🐙@[simp]
lemma massTermReduced_zero : massTermReduced 0 = 0 := by⊢ massTermReduced 0 = 0
  ext kk:EuclideanSpace ℝ (Fin 3)⊢ massTermReduced 0 k = 0 k
  simp [massTermReduced]All goals completed! 🐙lemma massTermReduced_stabilityCounterExample (k : EuclideanSpace ℝ (Fin 3)) :
    massTermReduced .stabilityCounterExample k = k 1 := byk:EuclideanSpace ℝ (Fin 3)⊢ massTermReduced PotentialParameters.stabilityCounterExample k = k.ofLp 1
  simp [massTermReduced, PotentialParameters.ξ, Fin.isValue,
    PotentialParameters.stabilityCounterExample, Fin.sum_univ_three]All goals completed! 🐙

E.4. The reduced quartic term

The reduced quartic term is a function that helps express the stability condition. It is the function J4 in https://arxiv.org/abs/hep-ph/0605184.

              @[simp]
lemma quarticTermReduced_zero : quarticTermReduced 0 = 0 := by⊢ quarticTermReduced 0 = 0
  ext kk:EuclideanSpace ℝ (Fin 3)⊢ quarticTermReduced 0 k = 0 k
  simp [quarticTermReduced]All goals completed! 🐙lemma quarticTermReduced_stabilityCounterExample (k : EuclideanSpace ℝ (Fin 3)) :
    quarticTermReduced .stabilityCounterExample k = (1 - k 0) ^ 2 := byk:EuclideanSpace ℝ (Fin 3)⊢ quarticTermReduced PotentialParameters.stabilityCounterExample k = (1 - k.ofLp 0) ^ 2
  simp [quarticTermReduced, PotentialParameters.η, Fin.isValue,
    PotentialParameters.stabilityCounterExample, Fin.sum_univ_three]k:EuclideanSpace ℝ (Fin 3)⊢ (2 + 2 + 2 * 2) / 8 + 2 * (k.ofLp 0 * ((-2 + -2) / 4)) +
    (k.ofLp 0 * k.ofLp 0 * ((2 + 2) / 4) + k.ofLp 2 * k.ofLp 2 * ((2 + 2 - 2 * 2) / 8)) =
  (1 - k.ofLp 0) ^ 2
  ringAll goals completed! 🐙

lemma quarticTermReduced_stabilityCounterExample_nonneg (k : EuclideanSpace ℝ (Fin 3)) :
    0 ≤ quarticTermReduced .stabilityCounterExample k := byk:EuclideanSpace ℝ (Fin 3)⊢ 0 ≤ quarticTermReduced PotentialParameters.stabilityCounterExample k
  rw [quarticTermReduced_stabilityCounterExamplek:EuclideanSpace ℝ (Fin 3)⊢ 0 ≤ (1 - k.ofLp 0) ^ 2k:EuclideanSpace ℝ (Fin 3)⊢ 0 ≤ (1 - k.ofLp 0) ^ 2]k:EuclideanSpace ℝ (Fin 3)⊢ 0 ≤ (1 - k.ofLp 0) ^ 2
  apply sq_nonnegAll goals completed! 🐙

E.5. Stability in terms of the gram vectors

We give some necessary and sufficient conditions for the potential to be stable in terms of the gram vectors.

This follows the analysis in https://arxiv.org/abs/hep-ph/0605184.

We also give some necessary conditions.

              

lemma potentialIsStable_iff_forall_gramVector (P : PotentialParameters) :
    PotentialIsStable P ↔ ∃ c : ℝ, ∀ K : Fin 1 ⊕ Fin 3 → ℝ, 0 ≤ K (Sum.inl 0) →
      ∑ μ : Fin 3, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 →
      c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c,
    ∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
      0 ≤ K (Sum.inl 0) →
        ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b
  apply Iff.intrompP:PotentialParameters⊢ PotentialIsStable P →
  ∃ c,
    ∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
      0 ≤ K (Sum.inl 0) →
        ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a bmprP:PotentialParameters⊢ (∃ c,
    ∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
      0 ≤ K (Sum.inl 0) →
        ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b) →
  PotentialIsStable P
  ·mpP:PotentialParameters⊢ PotentialIsStable P →
  ∃ c,
    ∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
      0 ≤ K (Sum.inl 0) →
        ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b rintro ⟨c, hc⟩mpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P H⊢ ∃ c,
  ∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
    0 ≤ K (Sum.inl 0) →
      ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b
    refine ⟨c, fun v hv₀ hv_sum => ?_⟩mpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2⊢ c ≤ ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a b
    obtain ⟨H, hH⟩ := gramVector_surjective v hv₀ hv_summpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2H:TwoHiggsDoublethH:H.gramVector = v⊢ c ≤ ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a b
    refine (hc H).trans (le_of_eq ?_)mpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2H:TwoHiggsDoublethH:H.gramVector = v⊢ potential P H = ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a b
    rw [potential,mpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2H:TwoHiggsDoublethH:H.gramVector = v⊢ massTerm P H + quarticTerm P H = ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a bmpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2H:TwoHiggsDoublethH:H.gramVector = v⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b =
  ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a b massTerm_eq_gramVector,mpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2H:TwoHiggsDoublethH:H.gramVector = v⊢ ∑ μ, P.ξ μ * H.gramVector μ + quarticTerm P H = ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a bmpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2H:TwoHiggsDoublethH:H.gramVector = v⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b =
  ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a b quarticTerm_eq_gramVectormpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2H:TwoHiggsDoublethH:H.gramVector = v⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b =
  ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a bmpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2H:TwoHiggsDoublethH:H.gramVector = v⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b =
  ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a b]mpP:PotentialParametersc:ℝhc:∀ (H : TwoHiggsDoublet), c ≤ potential P Hv:Fin 1 ⊕ Fin 3 → ℝhv₀:0 ≤ v (Sum.inl 0)hv_sum:∑ μ, v (Sum.inr μ) ^ 2 ≤ v (Sum.inl 0) ^ 2H:TwoHiggsDoublethH:H.gramVector = v⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b =
  ∑ μ, P.ξ μ * v μ + ∑ a, ∑ b, v a * v b * P.η a b
    simp [hH]All goals completed! 🐙
  ·mprP:PotentialParameters⊢ (∃ c,
    ∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
      0 ≤ K (Sum.inl 0) →
        ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b) →
  PotentialIsStable P rintro ⟨c, hc⟩mprP:PotentialParametersc:ℝhc:∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
  0 ≤ K (Sum.inl 0) → ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b⊢ PotentialIsStable P
    refine ⟨c, fun H => (hc H.gramVector (gramVector_inl_nonneg H)
      (gramVector_inr_sum_sq_le_inl H)).trans (le_of_eq ?_)⟩mprP:PotentialParametersc:ℝhc:∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
  0 ≤ K (Sum.inl 0) → ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a bH:TwoHiggsDoublet⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b = potential P H
    rw [potential,mprP:PotentialParametersc:ℝhc:∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
  0 ≤ K (Sum.inl 0) → ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a bH:TwoHiggsDoublet⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b = massTerm P H + quarticTerm P HAll goals completed! 🐙 massTerm_eq_gramVector,mprP:PotentialParametersc:ℝhc:∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
  0 ≤ K (Sum.inl 0) → ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a bH:TwoHiggsDoublet⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b =
  ∑ μ, P.ξ μ * H.gramVector μ + quarticTerm P HAll goals completed! 🐙 quarticTerm_eq_gramVectormprP:PotentialParametersc:ℝhc:∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
  0 ≤ K (Sum.inl 0) → ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a bH:TwoHiggsDoublet⊢ ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a b =
  ∑ μ, P.ξ μ * H.gramVector μ + ∑ a, ∑ b, H.gramVector a * H.gramVector b * P.η a bAll goals completed! 🐙]All goals completed! 🐙

lemma potentialIsStable_iff_forall_euclid (P : PotentialParameters) :
    PotentialIsStable P ↔ ∃ c, ∀ K0 : ℝ, ∀ K : EuclideanSpace ℝ (Fin 3), 0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 → c ≤ P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K μ
      + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0)
      + 2 * K0 * ∑ b, K b * P.η (Sum.inl 0) (Sum.inr b) +
      ∑ a, ∑ b, K a * K b * P.η (Sum.inr a) (Sum.inr b) := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
  rw [potentialIsStable_iff_forall_gramVectorP:PotentialParameters⊢ (∃ c,
    ∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
      0 ≤ K (Sum.inl 0) →
        ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b) ↔
  ∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)P:PotentialParameters⊢ (∃ c,
    ∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
      0 ≤ K (Sum.inl 0) →
        ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b) ↔
  ∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)]P:PotentialParameters⊢ (∃ c,
    ∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
      0 ≤ K (Sum.inl 0) →
        ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b) ↔
  ∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
  refine exists_congr (fun c => ?_)P:PotentialParametersc:ℝ⊢ (∀ (K : Fin 1 ⊕ Fin 3 → ℝ),
    0 ≤ K (Sum.inl 0) →
      ∑ μ, K (Sum.inr μ) ^ 2 ≤ K (Sum.inl 0) ^ 2 → c ≤ ∑ μ, P.ξ μ * K μ + ∑ a, ∑ b, K a * K b * P.η a b) ↔
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
  rw [Equiv.forall_congr_left (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ)P:PotentialParametersc:ℝ⊢ (∀ (b : (Fin 1 → ℝ) × (Fin 3 → ℝ)),
    0 ≤ (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b (Sum.inl 0) →
      ∑ μ, (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b (Sum.inr μ) ^ 2 ≤
          (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b (Sum.inl 0) ^ 2 →
        c ≤
          ∑ μ, P.ξ μ * (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b μ +
            ∑ a,
              ∑ b_1,
                (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b a *
                    (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b b_1 *
                  P.η a b_1) ↔
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)P:PotentialParametersc:ℝ⊢ (∀ (b : (Fin 1 → ℝ) × (Fin 3 → ℝ)),
    0 ≤ (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b (Sum.inl 0) →
      ∑ μ, (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b (Sum.inr μ) ^ 2 ≤
          (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b (Sum.inl 0) ^ 2 →
        c ≤
          ∑ μ, P.ξ μ * (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b μ +
            ∑ a,
              ∑ b_1,
                (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b a *
                    (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b b_1 *
                  P.η a b_1) ↔
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)]P:PotentialParametersc:ℝ⊢ (∀ (b : (Fin 1 → ℝ) × (Fin 3 → ℝ)),
    0 ≤ (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b (Sum.inl 0) →
      ∑ μ, (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b (Sum.inr μ) ^ 2 ≤
          (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b (Sum.inl 0) ^ 2 →
        c ≤
          ∑ μ, P.ξ μ * (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b μ +
            ∑ a,
              ∑ b_1,
                (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b a *
                    (Equiv.sumArrowEquivProdArrow (Fin 1) (Fin 3) ℝ).symm b b_1 *
                  P.η a b_1) ↔
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
  simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,
    Finset.sum_singleton, Prod.forall, Equiv.sumArrowEquivProdArrow_symm_apply_inl,
    Equiv.sumArrowEquivProdArrow_symm_apply_inr]P:PotentialParametersc:ℝ⊢ (∀ (a : Fin 1 → ℝ) (b : Fin 3 → ℝ),
    0 ≤ a 0 →
      ∑ x, b x ^ 2 ≤ a 0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * a 0 + ∑ x, P.ξ (Sum.inr x) * b x +
            (a 0 * a 0 * P.η (Sum.inl 0) (Sum.inl 0) + ∑ x, a 0 * b x * P.η (Sum.inl 0) (Sum.inr x) +
              ∑ x, (b x * a 0 * P.η (Sum.inr x) (Sum.inl 0) + ∑ x_1, b x * b x_1 * P.η (Sum.inr x) (Sum.inr x_1)))) ↔
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  rw [Equiv.forall_congr_left <| Equiv.funUnique (Fin 1) ℝP:PotentialParametersc:ℝ⊢ (∀ (b : ℝ) (b_1 : Fin 3 → ℝ),
    0 ≤ (Equiv.funUnique (Fin 1) ℝ).symm b 0 →
      ∑ x, b_1 x ^ 2 ≤ (Equiv.funUnique (Fin 1) ℝ).symm b 0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * (Equiv.funUnique (Fin 1) ℝ).symm b 0 + ∑ x, P.ξ (Sum.inr x) * b_1 x +
            ((Equiv.funUnique (Fin 1) ℝ).symm b 0 * (Equiv.funUnique (Fin 1) ℝ).symm b 0 * P.η (Sum.inl 0) (Sum.inl 0) +
                ∑ x, (Equiv.funUnique (Fin 1) ℝ).symm b 0 * b_1 x * P.η (Sum.inl 0) (Sum.inr x) +
              ∑ x,
                (b_1 x * (Equiv.funUnique (Fin 1) ℝ).symm b 0 * P.η (Sum.inr x) (Sum.inl 0) +
                  ∑ x_1, b_1 x * b_1 x_1 * P.η (Sum.inr x) (Sum.inr x_1)))) ↔
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)P:PotentialParametersc:ℝ⊢ (∀ (b : ℝ) (b_1 : Fin 3 → ℝ),
    0 ≤ (Equiv.funUnique (Fin 1) ℝ).symm b 0 →
      ∑ x, b_1 x ^ 2 ≤ (Equiv.funUnique (Fin 1) ℝ).symm b 0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * (Equiv.funUnique (Fin 1) ℝ).symm b 0 + ∑ x, P.ξ (Sum.inr x) * b_1 x +
            ((Equiv.funUnique (Fin 1) ℝ).symm b 0 * (Equiv.funUnique (Fin 1) ℝ).symm b 0 * P.η (Sum.inl 0) (Sum.inl 0) +
                ∑ x, (Equiv.funUnique (Fin 1) ℝ).symm b 0 * b_1 x * P.η (Sum.inl 0) (Sum.inr x) +
              ∑ x,
                (b_1 x * (Equiv.funUnique (Fin 1) ℝ).symm b 0 * P.η (Sum.inr x) (Sum.inl 0) +
                  ∑ x_1, b_1 x * b_1 x_1 * P.η (Sum.inr x) (Sum.inr x_1)))) ↔
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)]P:PotentialParametersc:ℝ⊢ (∀ (b : ℝ) (b_1 : Fin 3 → ℝ),
    0 ≤ (Equiv.funUnique (Fin 1) ℝ).symm b 0 →
      ∑ x, b_1 x ^ 2 ≤ (Equiv.funUnique (Fin 1) ℝ).symm b 0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * (Equiv.funUnique (Fin 1) ℝ).symm b 0 + ∑ x, P.ξ (Sum.inr x) * b_1 x +
            ((Equiv.funUnique (Fin 1) ℝ).symm b 0 * (Equiv.funUnique (Fin 1) ℝ).symm b 0 * P.η (Sum.inl 0) (Sum.inl 0) +
                ∑ x, (Equiv.funUnique (Fin 1) ℝ).symm b 0 * b_1 x * P.η (Sum.inl 0) (Sum.inr x) +
              ∑ x,
                (b_1 x * (Equiv.funUnique (Fin 1) ℝ).symm b 0 * P.η (Sum.inr x) (Sum.inl 0) +
                  ∑ x_1, b_1 x * b_1 x_1 * P.η (Sum.inr x) (Sum.inr x_1)))) ↔
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  refine forall_congr' fun K0 => ?_P:PotentialParametersc:ℝK0:ℝ⊢ (∀ (b : Fin 3 → ℝ),
    0 ≤ (Equiv.funUnique (Fin 1) ℝ).symm K0 0 →
      ∑ x, b x ^ 2 ≤ (Equiv.funUnique (Fin 1) ℝ).symm K0 0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 + ∑ x, P.ξ (Sum.inr x) * b x +
            ((Equiv.funUnique (Fin 1) ℝ).symm K0 0 * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                  P.η (Sum.inl 0) (Sum.inl 0) +
                ∑ x, (Equiv.funUnique (Fin 1) ℝ).symm K0 0 * b x * P.η (Sum.inl 0) (Sum.inr x) +
              ∑ x,
                (b x * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 * P.η (Sum.inr x) (Sum.inl 0) +
                  ∑ x_1, b x * b x_1 * P.η (Sum.inr x) (Sum.inr x_1)))) ↔
  ∀ (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  rw [Equiv.forall_congr_left <| (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symmP:PotentialParametersc:ℝK0:ℝ⊢ (∀ (b : WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)),
    0 ≤ (Equiv.funUnique (Fin 1) ℝ).symm K0 0 →
      ∑ x, (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x ^ 2 ≤
          (Equiv.funUnique (Fin 1) ℝ).symm K0 0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 +
              ∑ x, P.ξ (Sum.inr x) * (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x +
            ((Equiv.funUnique (Fin 1) ℝ).symm K0 0 * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                  P.η (Sum.inl 0) (Sum.inl 0) +
                ∑ x,
                  (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                      (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x *
                    P.η (Sum.inl 0) (Sum.inr x) +
              ∑ x,
                ((WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                    P.η (Sum.inr x) (Sum.inl 0) +
                  ∑ x_1,
                    (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x *
                        (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x_1 *
                      P.η (Sum.inr x) (Sum.inr x_1)))) ↔
  ∀ (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)P:PotentialParametersc:ℝK0:ℝ⊢ (∀ (b : WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)),
    0 ≤ (Equiv.funUnique (Fin 1) ℝ).symm K0 0 →
      ∑ x, (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x ^ 2 ≤
          (Equiv.funUnique (Fin 1) ℝ).symm K0 0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 +
              ∑ x, P.ξ (Sum.inr x) * (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x +
            ((Equiv.funUnique (Fin 1) ℝ).symm K0 0 * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                  P.η (Sum.inl 0) (Sum.inl 0) +
                ∑ x,
                  (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                      (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x *
                    P.η (Sum.inl 0) (Sum.inr x) +
              ∑ x,
                ((WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                    P.η (Sum.inr x) (Sum.inl 0) +
                  ∑ x_1,
                    (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x *
                        (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x_1 *
                      P.η (Sum.inr x) (Sum.inr x_1)))) ↔
  ∀ (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)]P:PotentialParametersc:ℝK0:ℝ⊢ (∀ (b : WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)),
    0 ≤ (Equiv.funUnique (Fin 1) ℝ).symm K0 0 →
      ∑ x, (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x ^ 2 ≤
          (Equiv.funUnique (Fin 1) ℝ).symm K0 0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 +
              ∑ x, P.ξ (Sum.inr x) * (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x +
            ((Equiv.funUnique (Fin 1) ℝ).symm K0 0 * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                  P.η (Sum.inl 0) (Sum.inl 0) +
                ∑ x,
                  (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                      (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x *
                    P.η (Sum.inl 0) (Sum.inr x) +
              ∑ x,
                ((WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                    P.η (Sum.inr x) (Sum.inl 0) +
                  ∑ x_1,
                    (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x *
                        (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm b x_1 *
                      P.η (Sum.inr x) (Sum.inr x_1)))) ↔
  ∀ (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  refine forall_congr' fun K => ?_P:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)⊢ 0 ≤ (Equiv.funUnique (Fin 1) ℝ).symm K0 0 →
    ∑ x, (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm K x ^ 2 ≤ (Equiv.funUnique (Fin 1) ℝ).symm K0 0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 +
            ∑ x, P.ξ (Sum.inr x) * (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm K x +
          ((Equiv.funUnique (Fin 1) ℝ).symm K0 0 * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 * P.η (Sum.inl 0) (Sum.inl 0) +
              ∑ x,
                (Equiv.funUnique (Fin 1) ℝ).symm K0 0 * (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm K x *
                  P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x,
              ((WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm K x * (Equiv.funUnique (Fin 1) ℝ).symm K0 0 *
                  P.η (Sum.inr x) (Sum.inl 0) +
                ∑ x_1,
                  (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm K x *
                      (WithLp.equiv 2 ((i : Fin 3) → (fun x => ℝ) i)).symm.symm K x_1 *
                    P.η (Sum.inr x) (Sum.inr x_1))) ↔
  0 ≤ K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
          ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  simp only [Fin.isValue, Equiv.funUnique_symm_apply, uniqueElim_const, Equiv.symm_symm,
    WithLp.equiv_apply]P:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)⊢ 0 ≤ K0 →
    ∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x +
          (K0 * K0 * P.η (Sum.inl 0) (Sum.inl 0) + ∑ x, K0 * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
            ∑ x,
              (K.ofLp x * K0 * P.η (Sum.inr x) (Sum.inl 0) +
                ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1))) ↔
  0 ≤ K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
          ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  refine imp_congr_right fun _ => ?_P:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝:0 ≤ K0⊢ ∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2 →
    c ≤
      P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x +
        (K0 * K0 * P.η (Sum.inl 0) (Sum.inl 0) + ∑ x, K0 * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
          ∑ x,
            (K.ofLp x * K0 * P.η (Sum.inr x) (Sum.inl 0) +
              ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1))) ↔
  ‖K‖ ^ 2 ≤ K0 ^ 2 →
    c ≤
      P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
          2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
        ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  simp only [PiLp.norm_sq_eq_of_L2, Fin.isValue, Real.norm_eq_abs, sq_abs]P:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝:0 ≤ K0⊢ ∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2 →
    c ≤
      P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x +
        (K0 * K0 * P.η (Sum.inl 0) (Sum.inl 0) + ∑ x, K0 * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
          ∑ x,
            (K.ofLp x * K0 * P.η (Sum.inr x) (Sum.inl 0) +
              ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1))) ↔
  ∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2 →
    c ≤
      P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
          2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
        ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  refine imp_congr_right fun _ => ?_P:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝¹:0 ≤ K0x✝:∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2⊢ c ≤
    P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x +
      (K0 * K0 * P.η (Sum.inl 0) (Sum.inl 0) + ∑ x, K0 * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
        ∑ x,
          (K.ofLp x * K0 * P.η (Sum.inr x) (Sum.inl 0) +
            ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1))) ↔
  c ≤
    P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
        2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
      ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  apply le_iff_le_of_cmp_eq_cmpP:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝¹:0 ≤ K0x✝:∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2⊢ cmp c
    (P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x +
      (K0 * K0 * P.η (Sum.inl 0) (Sum.inl 0) + ∑ x, K0 * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
        ∑ x,
          (K.ofLp x * K0 * P.η (Sum.inr x) (Sum.inl 0) +
            ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)))) =
  cmp c
    (P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
        2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
      ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1))
  congr 1e_bP:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝¹:0 ≤ K0x✝:∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2⊢ P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x +
    (K0 * K0 * P.η (Sum.inl 0) (Sum.inl 0) + ∑ x, K0 * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
      ∑ x,
        (K.ofLp x * K0 * P.η (Sum.inr x) (Sum.inl 0) + ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1))) =
  P.ξ (Sum.inl 0) * K0 + ∑ x, P.ξ (Sum.inr x) * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
    ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  simp [add_assoc, sq, Finset.sum_add_distrib]e_bP:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝¹:0 ≤ K0x✝:∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2⊢ ∑ x, K0 * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
    (∑ x, K.ofLp x * K0 * P.η (Sum.inr x) (Sum.inl 0) +
      ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)) =
  2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
    ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  ring_nfe_bP:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝¹:0 ≤ K0x✝:∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2⊢ ∑ x, K0 * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) + ∑ x, K0 * K.ofLp x * P.η (Sum.inr x) (Sum.inl 0) +
    ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1) =
  (K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 +
    ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
  simp [mul_assoc, ← Finset.mul_sum]e_bP:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝¹:0 ≤ K0x✝:∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2⊢ K0 * ∑ i, K.ofLp i * P.η (Sum.inl 0) (Sum.inr i) + K0 * ∑ i, K.ofLp i * P.η (Sum.inr i) (Sum.inl 0) =
  K0 * ((∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2)
  conv_lhs =>
    enter [2, 2, 2, i]P:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝¹:0 ≤ K0x✝:∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2i:Fin 3| K.ofLp i * P.η (Sum.inr i) (Sum.inl 0)
    rw [PotentialParameters.η_symm]P:PotentialParametersc:ℝK0:ℝK:WithLp 2 ((i : Fin 3) → (fun x => ℝ) i)x✝¹:0 ≤ K0x✝:∑ x, K.ofLp x ^ 2 ≤ K0 ^ 2i:Fin 3| K.ofLp i * P.η (Sum.inl 0) (Sum.inr i)
  ringAll goals completed! 🐙

lemma potentialIsStable_iff_forall_euclid_lt (P : PotentialParameters) :
    PotentialIsStable P ↔ ∃ c ≤ 0, ∀ K0 : ℝ, ∀ K : EuclideanSpace ℝ (Fin 3), 0 < K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 → c ≤ P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K μ
      + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0)
      + 2 * K0 * ∑ b, K b * P.η (Sum.inl 0) (Sum.inr b) +
      ∑ a, ∑ b, K a * K b * P.η (Sum.inr a) (Sum.inr b) := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
  rw [potentialIsStable_iff_forall_euclidP:PotentialParameters⊢ (∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) ↔
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)P:PotentialParameters⊢ (∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) ↔
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)]P:PotentialParameters⊢ (∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) ↔
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
  apply Iff.intrompP:PotentialParameters⊢ (∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) →
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)mprP:PotentialParameters⊢ (∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) →
  ∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
  ·mpP:PotentialParameters⊢ (∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) →
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b) rintro ⟨c, hc⟩mpP:PotentialParametersc:ℝhc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 ≤ K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)⊢ ∃ c ≤ 0,
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 < K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
    refine ⟨c, ?_, fun K0 K hk0 hle => hc K0 K hk0.le hle⟩mpP:PotentialParametersc:ℝhc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 ≤ K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)⊢ c ≤ 0
    simpa using hc 0 0 (byP:PotentialParametersc:ℝhc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 ≤ K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)⊢ 0 ≤ 0 simpAll goals completed! 🐙) (byP:PotentialParametersc:ℝhc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 ≤ K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)⊢ ‖0‖ ^ 2 ≤ 0 ^ 2 simpAll goals completed! 🐙)
  ·mprP:PotentialParameters⊢ (∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) →
  ∃ c,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 ≤ K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b) rintro ⟨c, hc₀, hc⟩mprP:PotentialParametersc:ℝhc₀:c ≤ 0hc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)⊢ ∃ c,
  ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
    0 ≤ K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
    refine ⟨c, fun K0 K hK0 hle => ?_⟩mprP:PotentialParametersc:ℝhc₀:c ≤ 0hc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)K0:ℝK:EuclideanSpace ℝ (Fin 3)hK0:0 ≤ K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ c ≤
  P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
    by_cases hK0' : K0 = 0posP:PotentialParametersc:ℝhc₀:c ≤ 0hc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)K0:ℝK:EuclideanSpace ℝ (Fin 3)hK0:0 ≤ K0hle:‖K‖ ^ 2 ≤ K0 ^ 2hK0':K0 = 0⊢ c ≤
  P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)negP:PotentialParametersc:ℝhc₀:c ≤ 0hc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)K0:ℝK:EuclideanSpace ℝ (Fin 3)hK0:0 ≤ K0hle:‖K‖ ^ 2 ≤ K0 ^ 2hK0':¬K0 = 0⊢ c ≤
  P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
    ·posP:PotentialParametersc:ℝhc₀:c ≤ 0hc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)K0:ℝK:EuclideanSpace ℝ (Fin 3)hK0:0 ≤ K0hle:‖K‖ ^ 2 ≤ K0 ^ 2hK0':K0 = 0⊢ c ≤
  P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b) subst hK0'posP:PotentialParametersc:ℝhc₀:c ≤ 0hc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)K:EuclideanSpace ℝ (Fin 3)hK0:0 ≤ 0hle:‖K‖ ^ 2 ≤ 0 ^ 2⊢ c ≤
  P.ξ (Sum.inl 0) * 0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + 0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * 0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
      simp_allAll goals completed! 🐙
    ·negP:PotentialParametersc:ℝhc₀:c ≤ 0hc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)K0:ℝK:EuclideanSpace ℝ (Fin 3)hK0:0 ≤ K0hle:‖K‖ ^ 2 ≤ K0 ^ 2hK0':¬K0 = 0⊢ c ≤
  P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b) refine hc K0 K ?_ hlenegP:PotentialParametersc:ℝhc₀:c ≤ 0hc:∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)K0:ℝK:EuclideanSpace ℝ (Fin 3)hK0:0 ≤ K0hle:‖K‖ ^ 2 ≤ K0 ^ 2hK0':¬K0 = 0⊢ 0 < K0
      grindAll goals completed! 🐙

lemma potentialIsStable_iff_exists_forall_forall_reduced (P : PotentialParameters) :
    PotentialIsStable P ↔ ∃ c ≤ 0, ∀ K0 : ℝ, ∀ k : EuclideanSpace ℝ (Fin 3), 0 < K0 →
      ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
      0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k
  rw [potentialIsStable_iff_forall_euclid_ltP:PotentialParameters⊢ (∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) ↔
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
      0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kP:PotentialParameters⊢ (∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) ↔
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
      0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k]P:PotentialParameters⊢ (∃ c ≤ 0,
    ∀ (K0 : ℝ) (K : EuclideanSpace ℝ (Fin 3)),
      0 < K0 →
        ‖K‖ ^ 2 ≤ K0 ^ 2 →
          c ≤
            P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
                2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
              ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) ↔
  ∃ c ≤ 0,
    ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
      0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k
  refine exists_congr <| fun c => and_congr_right <| fun hc => forall_congr' <| fun K0 => ?_P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝ⊢ (∀ (K : EuclideanSpace ℝ (Fin 3)),
    0 < K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) ↔
  ∀ (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k
  apply Iff.intrompP:PotentialParametersc:ℝhc:c ≤ 0K0:ℝ⊢ (∀ (K : EuclideanSpace ℝ (Fin 3)),
    0 < K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) →
  ∀ (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kmprP:PotentialParametersc:ℝhc:c ≤ 0K0:ℝ⊢ (∀ (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k) →
  ∀ (K : EuclideanSpace ℝ (Fin 3)),
    0 < K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
  ·mpP:PotentialParametersc:ℝhc:c ≤ 0K0:ℝ⊢ (∀ (K : EuclideanSpace ℝ (Fin 3)),
    0 < K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)) →
  ∀ (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k refine fun h k hK0 k_le_one => (h (K0 • k) hK0 ?_).trans (le_of_eq ?_)mp.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ ‖K0 • k‖ ^ 2 ≤ K0 ^ 2mp.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * (K0 • k).ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, (K0 • k).ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, (K0 • k).ofLp a * (K0 • k).ofLp b * P.η (Sum.inr a) (Sum.inr b) =
  K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k
    ·mp.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ ‖K0 • k‖ ^ 2 ≤ K0 ^ 2 simp [norm_smul]mp.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ (|K0| * ‖k‖) ^ 2 ≤ K0 ^ 2
      rw [abs_of_nonneg (byP:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ K0mp.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ K0 ^ 2 * ‖k‖ ^ 2 ≤ K0 ^ 2 positivityAll goals completed! 🐙mp.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ K0 ^ 2 * ‖k‖ ^ 2 ≤ K0 ^ 2), mul_powmp.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ K0 ^ 2 * ‖k‖ ^ 2 ≤ K0 ^ 2mp.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ K0 ^ 2 * ‖k‖ ^ 2 ≤ K0 ^ 2]mp.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ K0 ^ 2 * ‖k‖ ^ 2 ≤ K0 ^ 2
      nlinarithAll goals completed! 🐙
    ·mp.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * (K0 • k).ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, (K0 • k).ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, (K0 • k).ofLp a * (K0 • k).ofLp b * P.η (Sum.inr a) (Sum.inr b) =
  K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k simp [add_assoc, massTermReduced, quarticTermReduced]mp.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ P.ξ (Sum.inl 0) * K0 +
    (∑ x, P.ξ (Sum.inr x) * (K0 * k.ofLp x) +
      (K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
        (2 * K0 * ∑ x, K0 * k.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
          ∑ x, ∑ x_1, K0 * k.ofLp x * (K0 * k.ofLp x_1) * P.η (Sum.inr x) (Sum.inr x_1)))) =
  K0 * (P.ξ (Sum.inl 0) + ∑ x, P.ξ (Sum.inr x) * k.ofLp x) +
    K0 ^ 2 *
      (P.η (Sum.inl 0) (Sum.inl 0) +
        (2 * ∑ x, k.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
          ∑ x, ∑ x_1, k.ofLp x * k.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)))
      ring_nfmp.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ P.ξ (Sum.inl 0) * K0 + (K0 * ∑ x, K0 * k.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 +
        K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      ∑ x, K0 * P.ξ (Sum.inr x) * k.ofLp x +
    ∑ x, ∑ x_1, K0 ^ 2 * k.ofLp x * k.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1) =
  P.ξ (Sum.inl 0) * K0 + K0 * ∑ x, P.ξ (Sum.inr x) * k.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      (K0 ^ 2 * ∑ x, k.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 +
    K0 ^ 2 * ∑ x, ∑ x_1, k.ofLp x * k.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
      simp [add_assoc, mul_assoc, ← Finset.mul_sum, sq]mp.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (K : EuclideanSpace ℝ (Fin 3)),
  0 < K0 →
    ‖K‖ ^ 2 ≤ K0 ^ 2 →
      c ≤
        P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
            2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
          ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)k:EuclideanSpace ℝ (Fin 3)hK0:0 < K0k_le_one:‖k‖ ^ 2 ≤ 1⊢ K0 * (K0 * ((∑ i, k.ofLp i * P.η (Sum.inl 0) (Sum.inr i)) * 2)) +
    (K0 * (K0 * P.η (Sum.inl 0) (Sum.inl 0)) +
      (K0 * ∑ i, P.ξ (Sum.inr i) * k.ofLp i +
        K0 * (K0 * ∑ i, k.ofLp i * ∑ i_1, k.ofLp i_1 * P.η (Sum.inr i) (Sum.inr i_1)))) =
  K0 * ∑ x, P.ξ (Sum.inr x) * k.ofLp x +
    (K0 * (K0 * P.η (Sum.inl 0) (Sum.inl 0)) +
      (K0 * (K0 * ((∑ x, k.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2)) +
        K0 * (K0 * ∑ i, k.ofLp i * ∑ i_1, k.ofLp i_1 * P.η (Sum.inr i) (Sum.inr i_1))))
      ringAll goals completed! 🐙
  ·mprP:PotentialParametersc:ℝhc:c ≤ 0K0:ℝ⊢ (∀ (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k) →
  ∀ (K : EuclideanSpace ℝ (Fin 3)),
    0 < K0 →
      ‖K‖ ^ 2 ≤ K0 ^ 2 →
        c ≤
          P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
              2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
            ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b) intro h K hK0 hlemprP:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ c ≤
  P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
    refine (h ((1 / K0) • K) hK0 ?_).trans (le_of_eq ?_)mpr.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ ‖(1 / K0) • K‖ ^ 2 ≤ 1mpr.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ K0 * massTermReduced P ((1 / K0) • K) + K0 ^ 2 * quarticTermReduced P ((1 / K0) • K) =
  P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b)
    ·mpr.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ ‖(1 / K0) • K‖ ^ 2 ≤ 1 simp [norm_smul]mpr.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ |K0|⁻¹ * ‖K‖ ≤ 1
      field_simpmpr.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ ‖K‖ ≤ |K0|
      rw [sq_le_sqmpr.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:|‖K‖| ≤ |K0|⊢ ‖K‖ ≤ |K0|mpr.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:|‖K‖| ≤ |K0|⊢ ‖K‖ ≤ |K0|] at hlempr.refine_1P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:|‖K‖| ≤ |K0|⊢ ‖K‖ ≤ |K0|
      simpa using hleAll goals completed! 🐙
    ·mpr.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ K0 * massTermReduced P ((1 / K0) • K) + K0 ^ 2 * quarticTermReduced P ((1 / K0) • K) =
  P.ξ (Sum.inl 0) * K0 + ∑ μ, P.ξ (Sum.inr μ) * K.ofLp μ + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      2 * K0 * ∑ b, K.ofLp b * P.η (Sum.inl 0) (Sum.inr b) +
    ∑ a, ∑ b, K.ofLp a * K.ofLp b * P.η (Sum.inr a) (Sum.inr b) simp [add_assoc, massTermReduced, quarticTermReduced]mpr.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ K0 * (P.ξ (Sum.inl 0) + ∑ x, P.ξ (Sum.inr x) * (K0⁻¹ * K.ofLp x)) +
    K0 ^ 2 *
      (P.η (Sum.inl 0) (Sum.inl 0) +
        (2 * ∑ x, K0⁻¹ * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
          ∑ x, ∑ x_1, K0⁻¹ * K.ofLp x * (K0⁻¹ * K.ofLp x_1) * P.η (Sum.inr x) (Sum.inr x_1))) =
  P.ξ (Sum.inl 0) * K0 +
    (∑ x, P.ξ (Sum.inr x) * K.ofLp x +
      (K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
        (2 * K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x) +
          ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1))))
      ring_nfmpr.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ K0 * P.ξ (Sum.inl 0) + K0 * ∑ x, P.ξ (Sum.inr x) * K0⁻¹ * K.ofLp x + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      (K0 ^ 2 * ∑ x, K0⁻¹ * K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 +
    K0 ^ 2 * ∑ x, ∑ x_1, K0⁻¹ ^ 2 * K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1) =
  K0 * P.ξ (Sum.inl 0) + (K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      ∑ x, P.ξ (Sum.inr x) * K.ofLp x +
    ∑ x, ∑ x_1, K.ofLp x * K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)
      simp [add_assoc, mul_assoc, ← Finset.mul_sum, sq]mpr.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ K0 * ∑ x, P.ξ (Sum.inr x) * (K0⁻¹ * K.ofLp x) +
    (K0 * (K0 * P.η (Sum.inl 0) (Sum.inl 0)) +
      (K0 * (K0 * (K0⁻¹ * ((∑ i, K.ofLp i * P.η (Sum.inl 0) (Sum.inr i)) * 2))) +
        K0 * (K0 * (K0⁻¹ * (K0⁻¹ * ∑ i, K.ofLp i * ∑ i_1, K.ofLp i_1 * P.η (Sum.inr i) (Sum.inr i_1)))))) =
  K0 * ((∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2) +
    (K0 * (K0 * P.η (Sum.inl 0) (Sum.inl 0)) +
      (∑ x, P.ξ (Sum.inr x) * K.ofLp x + ∑ i, K.ofLp i * ∑ i_1, K.ofLp i_1 * P.η (Sum.inr i) (Sum.inr i_1)))
      field_simpmpr.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ K0 * ∑ x, P.ξ (Sum.inr x) * K.ofLp x / K0 +
    (K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      ((K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 +
        ∑ x, K.ofLp x * ∑ x_1, K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1))) =
  (K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 +
    (K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      (∑ x, P.ξ (Sum.inr x) * K.ofLp x + ∑ x, K.ofLp x * ∑ x_1, K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1)))
      ring_nfmpr.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ K0 * ∑ x, P.ξ (Sum.inr x) * K.ofLp x * K0⁻¹ + (K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 +
      K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
    ∑ x, K.ofLp x * ∑ x_1, K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1) =
  (K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      ∑ x, K.ofLp x * ∑ x_1, K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1) +
    ∑ x, P.ξ (Sum.inr x) * K.ofLp x
      simp only [← Finset.sum_mul, Fin.isValue]mpr.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ K0 * ((∑ i, P.ξ (Sum.inr i) * K.ofLp i) * K0⁻¹) + (K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 +
      K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
    ∑ x, K.ofLp x * ∑ x_1, K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1) =
  (K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 + K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
      ∑ x, K.ofLp x * ∑ x_1, K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1) +
    ∑ i, P.ξ (Sum.inr i) * K.ofLp i
      field_simpmpr.refine_2P:PotentialParametersc:ℝhc:c ≤ 0K0:ℝh:∀ (k : EuclideanSpace ℝ (Fin 3)), 0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kK:EuclideanSpace ℝ (Fin 3)hK0:0 < K0hle:‖K‖ ^ 2 ≤ K0 ^ 2⊢ ∑ x, P.ξ (Sum.inr x) * K.ofLp x + (K0 * ∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 +
      K0 ^ 2 * P.η (Sum.inl 0) (Sum.inl 0) +
    ∑ x, K.ofLp x * ∑ x_1, K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1) =
  K0 * ((∑ x, K.ofLp x * P.η (Sum.inl 0) (Sum.inr x)) * 2 + K0 * P.η (Sum.inl 0) (Sum.inl 0)) +
      ∑ x, K.ofLp x * ∑ x_1, K.ofLp x_1 * P.η (Sum.inr x) (Sum.inr x_1) +
    ∑ x, P.ξ (Sum.inr x) * K.ofLp x
      ringAll goals completed! 🐙

lemma quarticTermReduced_nonneg_of_potentialIsStable (P : PotentialParameters)
    (hP : PotentialIsStable P) (k : EuclideanSpace ℝ (Fin 3))
    (hk : ‖k‖ ^ 2 ≤ 1) : 0 ≤ quarticTermReduced P k := byP:PotentialParametershP:PotentialIsStable Pk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P k
  rw [potentialIsStable_iff_exists_forall_forall_reducedP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P kP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P k] at hPP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P k
  suffices hp : ∀ (a b : ℝ), (∃ c ≤ 0, ∀ x, 0 < x → c ≤ a * x + b * x ^ 2) →
      0 ≤ b ∧ (b = 0 → 0 ≤ a) byP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hp:∀ (a b : ℝ), (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)⊢ 0 ≤ quarticTermReduced P kP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ ∀ (a b : ℝ), (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)
    obtain ⟨c, hc, h⟩ := hPP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hp:∀ (a b : ℝ), (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)c:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P kP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ ∀ (a b : ℝ), (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)
    refine (hp (massTermReduced P k) (quarticTermReduced P k) ⟨c, hc, ?_⟩).1P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hp:∀ (a b : ℝ), (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)c:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k⊢ ∀ (x : ℝ), 0 < x → c ≤ massTermReduced P k * x + quarticTermReduced P k * x ^ 2P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ ∀ (a b : ℝ), (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)
    grindP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ ∀ (a b : ℝ), (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ ∀ (a b : ℝ), (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)
  intro a bP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝ⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)
  by_cases hb : b = 0posP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:b = 0⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)
  /- The case of b = 0. -/
  ·posP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:b = 0⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a) subst hbposP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝ⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + 0 * x ^ 2) → 0 ≤ 0 ∧ (0 = 0 → 0 ≤ a)
    by_cases ha : a = 0posP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝha:a = 0⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + 0 * x ^ 2) → 0 ≤ 0 ∧ (0 = 0 → 0 ≤ a)negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝha:¬a = 0⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + 0 * x ^ 2) → 0 ≤ 0 ∧ (0 = 0 → 0 ≤ a)
    ·posP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝha:a = 0⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + 0 * x ^ 2) → 0 ≤ 0 ∧ (0 = 0 → 0 ≤ a) subst haposP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ 0 * x + 0 * x ^ 2) → 0 ≤ 0 ∧ (0 = 0 → 0 ≤ 0)
      simpAll goals completed! 🐙
    ·negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝha:¬a = 0⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + 0 * x ^ 2) → 0 ≤ 0 ∧ (0 = 0 → 0 ≤ a) simp only [zero_mul, add_zero, le_refl, forall_const, true_and]negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝha:¬a = 0⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x) → 0 ≤ a
      rintro ⟨c, hc, hx⟩negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝha:¬a = 0c:ℝhc:c ≤ 0hx:∀ (x : ℝ), 0 < x → c ≤ a * x⊢ 0 ≤ a
      by_contra h2negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝha:¬a = 0c:ℝhc:c ≤ 0hx:∀ (x : ℝ), 0 < x → c ≤ a * xh2:¬0 ≤ a⊢ False
      simp_allnegP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)a:ℝc:ℝhP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khk:‖k‖ ≤ 1ha:¬a = 0hc:c ≤ 0hx:∀ (x : ℝ), 0 < x → c ≤ a * xh2:a < 0⊢ False
      refine not_lt_of_ge (hx (c/a + 1) ?_) ?_neg.refine_1P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)a:ℝc:ℝhP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khk:‖k‖ ≤ 1ha:¬a = 0hc:c ≤ 0hx:∀ (x : ℝ), 0 < x → c ≤ a * xh2:a < 0⊢ 0 < c / a + 1neg.refine_2P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)a:ℝc:ℝhP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khk:‖k‖ ≤ 1ha:¬a = 0hc:c ≤ 0hx:∀ (x : ℝ), 0 < x → c ≤ a * xh2:a < 0⊢ a * (c / a + 1) < c
      ·neg.refine_1P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)a:ℝc:ℝhP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khk:‖k‖ ≤ 1ha:¬a = 0hc:c ≤ 0hx:∀ (x : ℝ), 0 < x → c ≤ a * xh2:a < 0⊢ 0 < c / a + 1 exact add_pos_of_nonneg_of_pos (div_nonneg_of_nonpos hc (Std.le_of_lt h2))
          Real.zero_lt_oneAll goals completed! 🐙
      ·neg.refine_2P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)a:ℝc:ℝhP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khk:‖k‖ ≤ 1ha:¬a = 0hc:c ≤ 0hx:∀ (x : ℝ), 0 < x → c ≤ a * xh2:a < 0⊢ a * (c / a + 1) < c field_simpneg.refine_2P:PotentialParametersk:EuclideanSpace ℝ (Fin 3)a:ℝc:ℝhP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khk:‖k‖ ≤ 1ha:¬a = 0hc:c ≤ 0hx:∀ (x : ℝ), 0 < x → c ≤ a * xh2:a < 0⊢ c + a < c
        grindAll goals completed! 🐙
  /- The case of b ≠ 0. -/
  have h1 (x : ℝ) : a * x + b * x ^ 2 = b * (x + a / (2 * b)) ^ 2 - a ^ 2 / (4 * b) := byP:PotentialParametershP:PotentialIsStable Pk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P knegP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0h1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + a / (2 * b)) ^ 2 - a ^ 2 / (4 * b)⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a) grindnegP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0h1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + a / (2 * b)) ^ 2 - a ^ 2 / (4 * b)⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0h1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + a / (2 * b)) ^ 2 - a ^ 2 / (4 * b)⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)
  generalize a ^ 2 / (4 * b) = c1 at h1negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + a / (2 * b)) ^ 2 - c1⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)
  generalize a / (2 * b) = d at h1negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b ∧ (b = 0 → 0 ≤ a)
  simp only [hb, IsEmpty.forall_iff, and_true]negP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b
  have hlt (c : ℝ) (x : ℝ) : (c ≤ a * x + b * x ^ 2) ↔ c + c1 ≤ b * (x + d) ^ 2 := byP:PotentialParametershP:PotentialIsStable Pk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P knegP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b grindnegP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ bnegP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c ≤ a * x + b * x ^ 2) → 0 ≤ b
  conv_lhs => enter [1, c, 2, x]P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝx:ℝ| 0 < x → c ≤ a * x + b * x ^ 2; rw [hlt c]P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝx:ℝ| 0 < x → c + c1 ≤ b * (x + d) ^ 2
  trans ∃ c, ∀ x, 0 < x → c ≤ b * (x + d) ^ 2P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c + c1 ≤ b * (x + d) ^ 2) → ∃ c, ∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2⊢ (∃ c, ∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2) → 0 ≤ b
  ·P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2⊢ (∃ c ≤ 0, ∀ (x : ℝ), 0 < x → c + c1 ≤ b * (x + d) ^ 2) → ∃ c, ∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2 rintro ⟨c, hc, hx⟩P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:c ≤ 0hx:∀ (x : ℝ), 0 < x → c + c1 ≤ b * (x + d) ^ 2⊢ ∃ c, ∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2
    use c + c1All goals completed! 🐙
  rintro ⟨c, hc⟩P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2⊢ 0 ≤ b
  by_contra hnP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ False
  suffices hs : ∀ x, x ^ 2 ≤ c/b from not_lt_of_ge (hs √(|c/b| + 1)) <| byP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), x ^ 2 ≤ c / b⊢ c / b < √(|c / b| + 1) ^ 2P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), x ^ 2 ≤ c / b
    rw [sq_sqrt (byP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), x ^ 2 ≤ c / b⊢ 0 ≤ |c / b| + 1P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), x ^ 2 ≤ c / b⊢ c / b < |c / b| + 1P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), x ^ 2 ≤ c / b positivityAll goals completed! 🐙P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), x ^ 2 ≤ c / b⊢ c / b < |c / b| + 1P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), x ^ 2 ≤ c / b)]P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), x ^ 2 ≤ c / b⊢ c / b < |c / b| + 1P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), x ^ 2 ≤ c / b
    grindP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), x ^ 2 ≤ c / bP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), x ^ 2 ≤ c / b
  suffices hs : ∀ x, 0 < x → (x + d) ^ 2 ≤ c/b from
    fun x => le_trans ((Real.sqrt_le_left (byP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / bx:ℝ⊢ 0 ≤ |x| + |d| + 1 + dP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / b grindAll goals completed! 🐙P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / b)).mp
      (byP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / bx:ℝ⊢ √(x ^ 2) ≤ |x| + |d| + 1 + dP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / b rw [sqrt_sq_eq_absP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / bx:ℝ⊢ |x| ≤ |x| + |d| + 1 + dP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / bx:ℝ⊢ |x| ≤ |x| + |d| + 1 + dP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / b]P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / bx:ℝ⊢ |x| ≤ |x| + |d| + 1 + dP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / b; grindAll goals completed! 🐙P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / b)) (hs (|x| + |d| + 1) (byP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bhs:∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / bx:ℝ⊢ 0 < |x| + |d| + 1P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / b positivityAll goals completed! 🐙P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / b))P:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ b⊢ ∀ (x : ℝ), 0 < x → (x + d) ^ 2 ≤ c / b
  exact fun x hx => (le_div_iff_of_neg (byP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bx:ℝhx:0 < x⊢ b < 0 grindAll goals completed! 🐙)).mpr (byP:PotentialParametershP:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1a:ℝb:ℝhb:¬b = 0c1:ℝd:ℝh1:∀ (x : ℝ), a * x + b * x ^ 2 = b * (x + d) ^ 2 - c1hlt:∀ (c x : ℝ), c ≤ a * x + b * x ^ 2 ↔ c + c1 ≤ b * (x + d) ^ 2c:ℝhc:∀ (x : ℝ), 0 < x → c ≤ b * (x + d) ^ 2hn:¬0 ≤ bx:ℝhx:0 < x⊢ c ≤ (x + d) ^ 2 * b grindAll goals completed! 🐙)

lemma potentialIsStable_iff_massTermReduced_sq_le_quarticTermReduced (P : PotentialParameters) :
    PotentialIsStable P ↔ ∃ c, 0 ≤ c ∧ ∀ k : EuclideanSpace ℝ (Fin 3), ‖k‖ ^ 2 ≤ 1 →
      0 ≤ quarticTermReduced P k ∧
      (massTermReduced P k < 0 →
      massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c) := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c,
    0 ≤ c ∧
      ∀ (k : EuclideanSpace ℝ (Fin 3)),
        ‖k‖ ^ 2 ≤ 1 →
          0 ≤ quarticTermReduced P k ∧
            (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
  rw [potentialIsStable_iff_exists_forall_forall_reducedP:PotentialParameters⊢ (∃ c ≤ 0,
    ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
      0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k) ↔
  ∃ c,
    0 ≤ c ∧
      ∀ (k : EuclideanSpace ℝ (Fin 3)),
        ‖k‖ ^ 2 ≤ 1 →
          0 ≤ quarticTermReduced P k ∧
            (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)P:PotentialParameters⊢ (∃ c ≤ 0,
    ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
      0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k) ↔
  ∃ c,
    0 ≤ c ∧
      ∀ (k : EuclideanSpace ℝ (Fin 3)),
        ‖k‖ ^ 2 ≤ 1 →
          0 ≤ quarticTermReduced P k ∧
            (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)]P:PotentialParameters⊢ (∃ c ≤ 0,
    ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
      0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k) ↔
  ∃ c,
    0 ≤ c ∧
      ∀ (k : EuclideanSpace ℝ (Fin 3)),
        ‖k‖ ^ 2 ≤ 1 →
          0 ≤ quarticTermReduced P k ∧
            (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
  refine Iff.intro (fun ⟨c, hc, h⟩ => ⟨-c, byP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k⊢ 0 ≤ -c grindAll goals completed! 🐙, fun k hk => ?_⟩)
    (fun ⟨c, hc, h⟩ => ⟨-c, byP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ ch:∀ (k : EuclideanSpace ℝ (Fin 3)),
  ‖k‖ ^ 2 ≤ 1 →
    0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)⊢ -c ≤ 0 grindAll goals completed! 🐙, fun K0 k hk0 hk => ?_⟩)
  ·refine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * -c) have hJ4_nonneg : 0 ≤ quarticTermReduced P k := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c,
    0 ≤ c ∧
      ∀ (k : EuclideanSpace ℝ (Fin 3)),
        ‖k‖ ^ 2 ≤ 1 →
          0 ≤ quarticTermReduced P k ∧
            (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)refine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hJ4_nonneg:0 ≤ quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * -c)
      refine quarticTermReduced_nonneg_of_potentialIsStable P ?_ k hkP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ PotentialIsStable Prefine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hJ4_nonneg:0 ≤ quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * -c)
      rw [potentialIsStable_iff_exists_forall_forall_reducedP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ ∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ ∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P krefine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hJ4_nonneg:0 ≤ quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * -c)]P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ ∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P krefine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hJ4_nonneg:0 ≤ quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * -c)
      exact ⟨c, hc, h⟩refine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hJ4_nonneg:0 ≤ quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * -c)refine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hJ4_nonneg:0 ≤ quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * -c)
    have h0 : ∀ K0, 0 < K0 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k :=
      fun K0 a => h K0 k a hkrefine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0h:∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
  0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hJ4_nonneg:0 ≤ quarticTermReduced P kh0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * -c)
    clear hrefine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hJ4_nonneg:0 ≤ quarticTermReduced P kh0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * -c)
    generalize massTermReduced P k = j2 at *refine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hJ4_nonneg:0 ≤ quarticTermReduced P kj2:ℝh0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * quarticTermReduced P k⊢ 0 ≤ quarticTermReduced P k ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * quarticTermReduced P k * -c)
    generalize quarticTermReduced P k = j4 at *refine_1P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c)
    by_cases j4_zero : j4 = 0posP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:j4 = 0⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c)negP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c)
    ·posP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:j4 = 0⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c) subst j4_zeroposP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝhJ4_nonneg:0 ≤ 0h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * 0⊢ 0 ≤ 0 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * 0 * -c)
      simp_allposP:PotentialParametersc:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khc:c ≤ 0hk:‖k‖ ≤ 1h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2⊢ j2 < 0 → j2 = 0
      intro hj2posP:PotentialParametersc:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khc:c ≤ 0hk:‖k‖ ≤ 1h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2hj2:j2 < 0⊢ j2 = 0
      by_contra hnposP:PotentialParametersc:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khc:c ≤ 0hk:‖k‖ ≤ 1h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2hj2:j2 < 0hn:¬j2 = 0⊢ False
      specialize h0 ((c - 1) / j2) <| byP:PotentialParametersc:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khc:c ≤ 0hk:‖k‖ ≤ 1h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2hj2:j2 < 0hn:¬j2 = 0⊢ 0 < (c - 1) / j2
        refine div_pos_iff.mpr (Or.inr ?_)P:PotentialParametersc:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khc:c ≤ 0hk:‖k‖ ≤ 1h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2hj2:j2 < 0hn:¬j2 = 0⊢ c - 1 < 0 ∧ j2 < 0
        grindAll goals completed! 🐙
      field_simp at h0posP:PotentialParametersc:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P khc:c ≤ 0hk:‖k‖ ≤ 1hj2:j2 < 0hn:¬j2 = 0h0:c ≤ c - 1⊢ False
      linarithAll goals completed! 🐙
    ·negP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c) have hsq (K0 : ℝ) : K0 * j2 + K0 ^ 2 * j4 =
            j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4) := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c,
    0 ≤ c ∧
      ∀ (k : EuclideanSpace ℝ (Fin 3)),
        ‖k‖ ^ 2 ≤ 1 →
          0 ≤ quarticTermReduced P k ∧
            (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)negP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c)
        grindnegP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c)negP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c)
      have hj_pos : 0 < j4 := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c,
    0 ≤ c ∧
      ∀ (k : EuclideanSpace ℝ (Fin 3)),
        ‖k‖ ^ 2 ≤ 1 →
          0 ≤ quarticTermReduced P k ∧
            (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)negP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c) grindnegP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c)negP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4⊢ 0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c)
      apply And.introneg.leftP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4⊢ 0 ≤ j4neg.rightP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4⊢ j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c
      ·neg.leftP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4⊢ 0 ≤ j4 grindAll goals completed! 🐙
      ·neg.rightP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4⊢ j2 < 0 → j2 ^ 2 ≤ 4 * j4 * -c intro j2_negneg.rightP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4j2_neg:j2 < 0⊢ j2 ^ 2 ≤ 4 * j4 * -c
        conv at h0 =>P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4j2_neg:j2 < 0| ∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4 enter [2]P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4j2_neg:j2 < 0K0✝:ℝ| 0 < K0✝ → c ≤ K0✝ * j2 + K0✝ ^ 2 * j4; rw [hsq]P:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (K0 : ℝ), 0 < K0 → c ≤ K0 * j2 + K0 ^ 2 * j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4j2_neg:j2 < 0K0✝:ℝ| 0 < K0✝ → c ≤ j4 * (K0✝ + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)
        specialize h0 (- j2 / (2 * j4)) <| byP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (a : ℝ), 0 < a → c ≤ j4 * (a + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4j2_neg:j2 < 0⊢ 0 < -j2 / (2 * j4)
          field_simpP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4h0:∀ (a : ℝ), 0 < a → c ≤ j4 * (a + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4j2_neg:j2 < 0⊢ 0 * 2 * j4 < -j2
          grindAll goals completed! 🐙
        ring_nf at h0neg.rightP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4j2_neg:j2 < 0h0:c ≤ j2 ^ 2 * j4⁻¹ * (-1 / 4)⊢ j2 ^ 2 ≤ 4 * j4 * -c
        field_simp at h0neg.rightP:PotentialParametersx✝:∃ c ≤ 0,
  ∀ (K0 : ℝ) (k : EuclideanSpace ℝ (Fin 3)),
    0 < K0 → ‖k‖ ^ 2 ≤ 1 → c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P kc:ℝhc:c ≤ 0k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝhJ4_nonneg:0 ≤ j4j4_zero:¬j4 = 0hsq:∀ (K0 : ℝ), K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hj_pos:0 < j4j2_neg:j2 < 0h0:c * j4 * 4 ≤ -j2 ^ 2⊢ j2 ^ 2 ≤ 4 * j4 * -c
        grindAll goals completed! 🐙
  ·refine_2P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ ch:∀ (k : EuclideanSpace ℝ (Fin 3)),
  ‖k‖ ^ 2 ≤ 1 →
    0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)K0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1⊢ -c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k specialize h k hkrefine_2P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)h:0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1⊢ -c ≤ K0 * massTermReduced P k + K0 ^ 2 * quarticTermReduced P k
    generalize massTermReduced P k = j2 at *refine_2P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝh:0 ≤ quarticTermReduced P k ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * quarticTermReduced P k * c)⊢ -c ≤ K0 * j2 + K0 ^ 2 * quarticTermReduced P k
    generalize quarticTermReduced P k = j4 at *refine_2P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4
    by_cases hJ4 : j4 = 0posP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:j4 = 0⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4
    ·posP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:j4 = 0⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4 subst j4posP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝh:0 ≤ 0 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * 0 * c)⊢ -c ≤ K0 * j2 + K0 ^ 2 * 0
      simp_allposP:PotentialParametersc:ℝK0:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hc:0 ≤ chk0:0 < K0hk:‖k‖ ≤ 1h:j2 < 0 → j2 = 0⊢ -c ≤ K0 * j2
      trans 0P:PotentialParametersc:ℝK0:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hc:0 ≤ chk0:0 < K0hk:‖k‖ ≤ 1h:j2 < 0 → j2 = 0⊢ -c ≤ 0P:PotentialParametersc:ℝK0:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hc:0 ≤ chk0:0 < K0hk:‖k‖ ≤ 1h:j2 < 0 → j2 = 0⊢ 0 ≤ K0 * j2
      ·P:PotentialParametersc:ℝK0:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hc:0 ≤ chk0:0 < K0hk:‖k‖ ≤ 1h:j2 < 0 → j2 = 0⊢ -c ≤ 0 grindAll goals completed! 🐙
      ·P:PotentialParametersc:ℝK0:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hc:0 ≤ chk0:0 < K0hk:‖k‖ ≤ 1h:j2 < 0 → j2 = 0⊢ 0 ≤ K0 * j2 by_cases hJ2 : j2 = 0posP:PotentialParametersc:ℝK0:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hc:0 ≤ chk0:0 < K0hk:‖k‖ ≤ 1h:j2 < 0 → j2 = 0hJ2:j2 = 0⊢ 0 ≤ K0 * j2negP:PotentialParametersc:ℝK0:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hc:0 ≤ chk0:0 < K0hk:‖k‖ ≤ 1h:j2 < 0 → j2 = 0hJ2:¬j2 = 0⊢ 0 ≤ K0 * j2
        ·posP:PotentialParametersc:ℝK0:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hc:0 ≤ chk0:0 < K0hk:‖k‖ ≤ 1h:j2 < 0 → j2 = 0hJ2:j2 = 0⊢ 0 ≤ K0 * j2 simp_allAll goals completed! 🐙
        ·negP:PotentialParametersc:ℝK0:ℝk:EuclideanSpace ℝ (Fin 3)j2:ℝx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)hc:0 ≤ chk0:0 < K0hk:‖k‖ ≤ 1h:j2 < 0 → j2 = 0hJ2:¬j2 = 0⊢ 0 ≤ K0 * j2 simp_allAll goals completed! 🐙
    ·negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4 have hJ4_pos : 0 < j4 := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c,
    0 ≤ c ∧
      ∀ (k : EuclideanSpace ℝ (Fin 3)),
        ‖k‖ ^ 2 ≤ 1 →
          0 ≤ quarticTermReduced P k ∧
            (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4 grindnegP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4
      have h0 : K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4) := byP:PotentialParameters⊢ PotentialIsStable P ↔
  ∃ c,
    0 ≤ c ∧
      ∀ (k : EuclideanSpace ℝ (Fin 3)),
        ‖k‖ ^ 2 ≤ 1 →
          0 ≤ quarticTermReduced P k ∧
            (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4
          grindnegP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)⊢ -c ≤ K0 * j2 + K0 ^ 2 * j4
      rw [h0negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)⊢ -c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)⊢ -c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)]negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)⊢ -c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)
      by_cases hJ2_neg : j2 < 0posP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:j2 < 0⊢ -c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:¬j2 < 0⊢ -c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)
      ·posP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:j2 < 0⊢ -c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4) trans j4 * (K0 + j2 / (2 * j4)) ^ 2 - cP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:j2 < 0⊢ -c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - cP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:j2 < 0⊢ j4 * (K0 + j2 / (2 * j4)) ^ 2 - c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)
        ·P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:j2 < 0⊢ -c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - c nlinarithAll goals completed! 🐙
        ·P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:j2 < 0⊢ j4 * (K0 + j2 / (2 * j4)) ^ 2 - c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4) field_simpP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:j2 < 0⊢ ((j4 * K0 * 2 + j2) ^ 2 - j4 * 2 ^ 2 * c) * 4 ≤ (j4 * K0 * 2 + j2) ^ 2 * 4 - j2 ^ 2 * 2 ^ 2
          grindAll goals completed! 🐙
      ·negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:¬j2 < 0⊢ -c ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4) refine neg_le_sub_iff_le_add.mpr ?_negP:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:¬j2 < 0⊢ j2 ^ 2 / (4 * j4) ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 + c
        trans j4 * (K0 + j2 / (2 * j4)) ^ 2P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:¬j2 < 0⊢ j2 ^ 2 / (4 * j4) ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:¬j2 < 0⊢ j4 * (K0 + j2 / (2 * j4)) ^ 2 ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 + c
        ·P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:¬j2 < 0⊢ j2 ^ 2 / (4 * j4) ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 nlinarithAll goals completed! 🐙
        ·P:PotentialParametersx✝:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)c:ℝhc:0 ≤ cK0:ℝk:EuclideanSpace ℝ (Fin 3)hk0:0 < K0hk:‖k‖ ^ 2 ≤ 1j2:ℝj4:ℝh:0 ≤ j4 ∧ (j2 < 0 → j2 ^ 2 ≤ 4 * j4 * c)hJ4:¬j4 = 0hJ4_pos:0 < j4h0:K0 * j2 + K0 ^ 2 * j4 = j4 * (K0 + j2 / (2 * j4)) ^ 2 - j2 ^ 2 / (4 * j4)hJ2_neg:¬j2 < 0⊢ j4 * (K0 + j2 / (2 * j4)) ^ 2 ≤ j4 * (K0 + j2 / (2 * j4)) ^ 2 + c grindAll goals completed! 🐙

lemma massTermReduced_pos_of_quarticTermReduced_zero_potentialIsStable (P : PotentialParameters)
    (hP : PotentialIsStable P) (k : EuclideanSpace ℝ (Fin 3))
    (hk : ‖k‖ ^ 2 ≤ 1) (hq : quarticTermReduced P k = 0) : 0 ≤ massTermReduced P k := byP:PotentialParametershP:PotentialIsStable Pk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0⊢ 0 ≤ massTermReduced P k
  rw [potentialIsStable_iff_massTermReduced_sq_le_quarticTermReducedP:PotentialParametershP:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0⊢ 0 ≤ massTermReduced P kP:PotentialParametershP:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0⊢ 0 ≤ massTermReduced P k] at hPP:PotentialParametershP:∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0⊢ 0 ≤ massTermReduced P k
  obtain ⟨c, hc₀, hc⟩ := hPP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0c:ℝhc₀:0 ≤ chc:∀ (k : EuclideanSpace ℝ (Fin 3)),
  ‖k‖ ^ 2 ≤ 1 →
    0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)⊢ 0 ≤ massTermReduced P k
  specialize hc k hkP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0c:ℝhc₀:0 ≤ chc:0 ≤ quarticTermReduced P k ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)⊢ 0 ≤ massTermReduced P k
  rw [hqP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0c:ℝhc₀:0 ≤ chc:0 ≤ 0 ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * 0 * c)⊢ 0 ≤ massTermReduced P kP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0c:ℝhc₀:0 ≤ chc:0 ≤ 0 ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * 0 * c)⊢ 0 ≤ massTermReduced P k] at hcP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0c:ℝhc₀:0 ≤ chc:0 ≤ 0 ∧ (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * 0 * c)⊢ 0 ≤ massTermReduced P k
  simp only [le_refl, mul_zero, zero_mul, sq_nonpos_iff, true_and] at hcP:PotentialParametersk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced P k = 0c:ℝhc₀:0 ≤ chc:massTermReduced P k < 0 → massTermReduced P k = 0⊢ 0 ≤ massTermReduced P k
  grindAll goals completed! 🐙

E.6. Strong stability implies stability

Stability in terms of the positivity of the quartic term, implies that the whole potential is stable.

The potential is stable if it is strongly stable, i.e. its quartic term is always positive. The proof of this result relies on the compactness of the closed unit ball in EuclideanSpace ℝ (Fin 3), and the extreme value theorem.


lemma potentialIsStable_of_strong (P : PotentialParameters)
    (h : ∀ k, ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P k) :
    PotentialIsStable P := byP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P k⊢ PotentialIsStable P
  rw [potentialIsStable_iff_massTermReduced_sq_le_quarticTermReducedP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P k⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P k⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)]P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P k⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
  let S := Metric.closedBall (0 : EuclideanSpace ℝ (Fin 3)) 1P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
  have S_nonEmpty : S.Nonempty := ⟨0, byP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1⊢ 0 ∈ SP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c) simp [S]All goals completed! 🐙P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)⟩P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
  obtain ⟨kmax, kmax_S, kmax_isMax⟩ := IsCompact.exists_isMaxOn
    (isCompact_closedBall 0 1) S_nonEmpty
    (f := fun k => (massTermReduced P k ^ 2) / (4 * quarticTermReduced P k)) <| byP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ContinuousOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1)P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
    apply ContinuousOn.div₀hfP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ContinuousOn (fun x => massTermReduced P x ^ 2) (Metric.closedBall 0 1)hgP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ContinuousOn (fun x => 4 * quarticTermReduced P x) (Metric.closedBall 0 1)h₀P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ∀ x ∈ Metric.closedBall 0 1, 4 * quarticTermReduced P x ≠ 0P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
    ·hfP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ContinuousOn (fun x => massTermReduced P x ^ 2) (Metric.closedBall 0 1)P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c) simp only [massTermReduced, Fin.isValue]hfP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ContinuousOn (fun x => (P.ξ (Sum.inl 0) + ∑ x_1, P.ξ (Sum.inr x_1) * x.ofLp x_1) ^ 2) (Metric.closedBall 0 1)P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
      fun_propAll goals completed! 🐙P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
    ·hgP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ContinuousOn (fun x => 4 * quarticTermReduced P x) (Metric.closedBall 0 1)P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c) simp only [quarticTermReduced, Fin.isValue]hgP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ContinuousOn
  (fun x =>
    4 *
      (P.η (Sum.inl 0) (Sum.inl 0) + 2 * ∑ x_1, x.ofLp x_1 * P.η (Sum.inl 0) (Sum.inr x_1) +
        ∑ x_1, ∑ x_2, x.ofLp x_1 * x.ofLp x_2 * P.η (Sum.inr x_1) (Sum.inr x_2)))
  (Metric.closedBall 0 1)P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
      fun_propAll goals completed! 🐙P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
    ·h₀P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonempty⊢ ∀ x ∈ Metric.closedBall 0 1, 4 * quarticTermReduced P x ≠ 0P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c) intro x hxh₀P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptyx:EuclideanSpace ℝ (Fin 3)hx:x ∈ Metric.closedBall 0 1⊢ 4 * quarticTermReduced P x ≠ 0P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
      specialize h x (byP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptyx:EuclideanSpace ℝ (Fin 3)hx:x ∈ Metric.closedBall 0 1⊢ ‖x‖ ^ 2 ≤ 1P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c) simpa using hxAll goals completed! 🐙P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c))
      linarithP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∃ c,
  0 ≤ c ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 →
        0 ≤ quarticTermReduced P k ∧
          (massTermReduced P k < 0 → massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * c)
  use (massTermReduced P kmax) ^ 2 / (4 * quarticTermReduced P kmax)hP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ 0 ≤ massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax) ∧
  ∀ (k : EuclideanSpace ℝ (Fin 3)),
    ‖k‖ ^ 2 ≤ 1 →
      0 ≤ quarticTermReduced P k ∧
        (massTermReduced P k < 0 →
          massTermReduced P k ^ 2 ≤
            4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax)))
  apply And.introh.leftP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ 0 ≤ massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax)h.rightP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∀ (k : EuclideanSpace ℝ (Fin 3)),
  ‖k‖ ^ 2 ≤ 1 →
    0 ≤ quarticTermReduced P k ∧
      (massTermReduced P k < 0 →
        massTermReduced P k ^ 2 ≤
          4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax)))
  ·h.leftP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ 0 ≤ massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax) refine (le_div_iff₀ ?_).mpr ?_h.left.refine_1P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ 0 < 4 * quarticTermReduced P kmaxh.left.refine_2P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ 0 * (4 * quarticTermReduced P kmax) ≤ massTermReduced P kmax ^ 2
    ·h.left.refine_1P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ 0 < 4 * quarticTermReduced P kmax specialize h kmax (byP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ‖kmax‖ ^ 2 ≤ 1 simpa using kmax_SAll goals completed! 🐙)
      linarithAll goals completed! 🐙
    ·h.left.refine_2P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ 0 * (4 * quarticTermReduced P kmax) ≤ massTermReduced P kmax ^ 2 simp only [zero_mul]h.left.refine_2P:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ 0 ≤ massTermReduced P kmax ^ 2
      exact sq_nonneg (massTermReduced P kmax)All goals completed! 🐙
  ·h.rightP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmax⊢ ∀ (k : EuclideanSpace ℝ (Fin 3)),
  ‖k‖ ^ 2 ≤ 1 →
    0 ≤ quarticTermReduced P k ∧
      (massTermReduced P k < 0 →
        massTermReduced P k ^ 2 ≤
          4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax))) intro k hkh.rightP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmaxk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P k ∧
  (massTermReduced P k < 0 →
    massTermReduced P k ^ 2 ≤
      4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax)))
    apply And.introh.right.leftP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmaxk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P kh.right.rightP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmaxk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ massTermReduced P k < 0 →
  massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax))
    ·h.right.leftP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmaxk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P k specialize h k hkh.right.leftP:PotentialParametersS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmaxk:EuclideanSpace ℝ (Fin 3)h:0 < quarticTermReduced P khk:‖k‖ ^ 2 ≤ 1⊢ 0 ≤ quarticTermReduced P k
      linarithAll goals completed! 🐙
    ·h.right.rightP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmaxk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1⊢ massTermReduced P k < 0 →
  massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax)) intro hqh.right.rightP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:IsMaxOn (fun k => massTermReduced P k ^ 2 / (4 * quarticTermReduced P k)) (Metric.closedBall 0 1) kmaxk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:massTermReduced P k < 0⊢ massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax))
      rw [isMaxOn_iffh.right.rightP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:∀ x ∈ Metric.closedBall 0 1,
  massTermReduced P x ^ 2 / (4 * quarticTermReduced P x) ≤ massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax)k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:massTermReduced P k < 0⊢ massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax))h.right.rightP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:∀ x ∈ Metric.closedBall 0 1,
  massTermReduced P x ^ 2 / (4 * quarticTermReduced P x) ≤ massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax)k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:massTermReduced P k < 0⊢ massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax))] at kmax_isMaxh.right.rightP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:∀ x ∈ Metric.closedBall 0 1,
  massTermReduced P x ^ 2 / (4 * quarticTermReduced P x) ≤ massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax)k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:massTermReduced P k < 0⊢ massTermReduced P k ^ 2 ≤ 4 * quarticTermReduced P k * (massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax))
      refine (div_le_iff₀' ?_).mp (kmax_isMax k (byP:PotentialParametersh:∀ (k : EuclideanSpace ℝ (Fin 3)), ‖k‖ ^ 2 ≤ 1 → 0 < quarticTermReduced P kS:Set (EuclideanSpace ℝ (Fin 3)) := Metric.closedBall 0 1S_nonEmpty:S.Nonemptykmax:EuclideanSpace ℝ (Fin 3)kmax_S:kmax ∈ Metric.closedBall 0 1kmax_isMax:∀ x ∈ Metric.closedBall 0 1,
  massTermReduced P x ^ 2 / (4 * quarticTermReduced P x) ≤ massTermReduced P kmax ^ 2 / (4 * quarticTermReduced P kmax)k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:massTermReduced P k < 0⊢ k ∈ Metric.closedBall 0 1 simpa using hkAll goals completed! 🐙))
      grindAll goals completed! 🐙

E.7. Showing step in hep-ph/0605184 is invalid

A lemma invalidating the step in https://arxiv.org/pdf/hep-ph/0605184 leading to equation (4.4).


lemma forall_reduced_exists_not_potentialIsStable :
    ∃ P, ¬ PotentialIsStable P ∧ (∀ k : EuclideanSpace ℝ (Fin 3), ‖k‖ ^ 2 ≤ 1 →
    0 ≤ quarticTermReduced P k ∧ (quarticTermReduced P k = 0 → 0 ≤ massTermReduced P k)) := by⊢ ∃ P,
  ¬PotentialIsStable P ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 → 0 ≤ quarticTermReduced P k ∧ (quarticTermReduced P k = 0 → 0 ≤ massTermReduced P k)
  /- Construction of the explicit counter example.
    The reason that this counter example works is that:
    - There is a zero of the quartic term `z` on the boundary.
    - The quartic term is equal to `((k - z) · z)²`, as `k - z` approaches orthogonal to `z`,
      this becomes small on two accounts: the abs of `k - z` has to become small as `z` is on
      the boundary, and the angle between `k - z` and `z` also becomes small.
    - The mass term is of the form `-(k - z) · w` for some `w` orthogonal to `z`, so as `k - z`
      approaches orthogonal to `z`, the mass term becomes small only on the account that the abs of
      `k - z` becomes small. -/
  refine ⟨.stabilityCounterExample, stabilityCounterExample_not_potentialIsStable,
    fun k hk => ⟨quarticTermReduced_stabilityCounterExample_nonneg k, fun hq => ?_⟩⟩k:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:quarticTermReduced PotentialParameters.stabilityCounterExample k = 0⊢ 0 ≤ massTermReduced PotentialParameters.stabilityCounterExample k
  simp [quarticTermReduced_stabilityCounterExample] at hqk:EuclideanSpace ℝ (Fin 3)hk:‖k‖ ^ 2 ≤ 1hq:1 - k.ofLp 0 = 0⊢ 0 ≤ massTermReduced PotentialParameters.stabilityCounterExample k
  simp only [PiLp.norm_sq_eq_of_L2, Real.norm_eq_abs, sq_abs, Fin.sum_univ_three,
    Fin.isValue] at hkk:EuclideanSpace ℝ (Fin 3)hq:1 - k.ofLp 0 = 0hk:k.ofLp 0 ^ 2 + k.ofLp 1 ^ 2 + k.ofLp 2 ^ 2 ≤ 1⊢ 0 ≤ massTermReduced PotentialParameters.stabilityCounterExample k
  have hk1 : k 1 = 0 := by⊢ ∃ P,
  ¬PotentialIsStable P ∧
    ∀ (k : EuclideanSpace ℝ (Fin 3)),
      ‖k‖ ^ 2 ≤ 1 → 0 ≤ quarticTermReduced P k ∧ (quarticTermReduced P k = 0 → 0 ≤ massTermReduced P k)k:EuclideanSpace ℝ (Fin 3)hq:1 - k.ofLp 0 = 0hk:k.ofLp 0 ^ 2 + k.ofLp 1 ^ 2 + k.ofLp 2 ^ 2 ≤ 1hk1:k.ofLp 1 = 0⊢ 0 ≤ massTermReduced PotentialParameters.stabilityCounterExample k nlinarithk:EuclideanSpace ℝ (Fin 3)hq:1 - k.ofLp 0 = 0hk:k.ofLp 0 ^ 2 + k.ofLp 1 ^ 2 + k.ofLp 2 ^ 2 ≤ 1hk1:k.ofLp 1 = 0⊢ 0 ≤ massTermReduced PotentialParameters.stabilityCounterExample kk:EuclideanSpace ℝ (Fin 3)hq:1 - k.ofLp 0 = 0hk:k.ofLp 0 ^ 2 + k.ofLp 1 ^ 2 + k.ofLp 2 ^ 2 ≤ 1hk1:k.ofLp 1 = 0⊢ 0 ≤ massTermReduced PotentialParameters.stabilityCounterExample k
  rw [massTermReduced_stabilityCounterExample,k:EuclideanSpace ℝ (Fin 3)hq:1 - k.ofLp 0 = 0hk:k.ofLp 0 ^ 2 + k.ofLp 1 ^ 2 + k.ofLp 2 ^ 2 ≤ 1hk1:k.ofLp 1 = 0⊢ 0 ≤ k.ofLp 1All goals completed! 🐙 hk1k:EuclideanSpace ℝ (Fin 3)hq:1 - k.ofLp 0 = 0hk:k.ofLp 0 ^ 2 + k.ofLp 1 ^ 2 + k.ofLp 2 ^ 2 ≤ 1hk1:k.ofLp 1 = 0⊢ 0 ≤ 0All goals completed! 🐙]All goals completed! 🐙