Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.Map

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.SuperSymmetry.SU5.ChargeSpectrum.Yukawa
public import Physlib.Particles.SuperSymmetry.SU5.ChargeSpectrum.Completions
public import Mathlib.Tactic.FinCases

Mapping charge spectra values

i. Overview

In this module we define a function map which takes an additive monoid homomorphism f : 𝓩 →+ 𝓩1 and a charge spectra x : ChargeSpectrum 𝓩, and returns the charge x.map f : ChargeSpectrum 𝓩1 obtained by mapping the elements of x by f.

There are various properties which are preserved under this mapping:

Anomaly cancellation.

The presence of a specific term in the potential.

Being complete.

There are some properties which are reflected under this mapping:

Not being pheno-constrained.

Not regenerating dangerous Yukawa terms at a given level.

We define the preimage of this mapping within a subset ofFinset S5 S10 of Charges 𝓩 in a computationally efficient way.

ii. Key results

map : The mapping of charge spectra under an additive monoid homomorphism.

map_allowsTerm : If a charge spectrum allows a potential term, then so does its mapping.

map_isPhenoConstrained : If a charge spectrum is pheno-constrained, then so is its mapping.

map_isComplete_iff : A charge spectrum is complete if and only if its mapping is complete.

map_yukawaGeneratesDangerousAtLevel : A charge spectrum regenerates dangerous Yukawa terms at a given level then so does its mapping.

preimageOfFinset : The preimage of a charge spectrum in ofFinset S5 S10 under a mapping.

preimageOfFinsetCard : The cardinality of the preimage of a charge spectrum in ofFinset S5 S10 under a mapping.

iii. Table of contents

A. The mapping of charge spectra

A.1. Mapping the empty charge spectrum gives the empty charge spectrum

A.2. Mapping of charge spectra obeys composing maps

A.3. Mapping of charge spectra obeys the identity

A.4. The charges of a field label commute with mapping of charge spectra

A.5. Mappings of charge spectra preserve the subset relation

A.6. Mappings of charge spectra and charges of potential terms

A.7. Mapping charge spectra of `allowsTermForm

A.8. Mapping preserves whether a charge spectrum allows a potential term

A.9. Mapping preserves if a charge spectrum is pheno-constrained

A.10. Mapping preserves completeness of charge spectra

A.11. Mapping commutes with charges of Yukawa terms

A.12. Mapping of charge spectra and regenerating dangerous Yukawa terms

B. Preimage of a charge spectrum under a mapping

B.1. preimageOfFinset gives the actual preimage

B.2. Efficient definition for the cardinality of the preimage

B.3. Definition for the cardinality equals cardinality of the preimage

iv. References

There are no known references for the material in this module.

@[expose] public section

A. The mapping of charge spectra

Given an additive monoid homomorphisms f : 𝓩 →+ 𝓩1, for a charge x : Charges 𝓩, x.map f is the charge of Charges 𝓩1 obtained by mapping the elements of x by f.

def map (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩) : ChargeSpectrum 𝓩1 where
  qHd := f <$> x.qHd
  qHu := f <$> x.qHu
  Q5 := x.Q5.image f
  Q10 := x.Q10.image f

/-

### A.1. Mapping the empty charge spectrum gives the empty charge spectrum

-/

@[simp]
lemma map_empty (f : 𝓩 →+ 𝓩1) : map f (∅ : ChargeSpectrum 𝓩) = ∅ := by𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1⊢ map f ∅ = ∅
  simp only [map, empty_qHd, Option.map_eq_map, Option.map_none, empty_qHu, empty_Q5,
    Finset.image_empty, empty_Q10]𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1⊢ { qHd := none, qHu := none, Q5 := ∅, Q10 := ∅ } = ∅
  rflAll goals completed! 🐙

A.2. Mapping of charge spectra obeys composing maps

lemma map_map (f : 𝓩 →+ 𝓩1) (g : 𝓩1 →+ 𝓩2) (x : ChargeSpectrum 𝓩) :
    map g (map f x) = map (g.comp f) x := by𝓩:Type𝓩1:Type𝓩2:Typeinst✝⁴:AddCommGroup 𝓩inst✝³:AddCommGroup 𝓩1inst✝²:DecidableEq 𝓩1inst✝¹:AddCommGroup 𝓩2inst✝:DecidableEq 𝓩2f:𝓩 →+ 𝓩1g:𝓩1 →+ 𝓩2x:ChargeSpectrum 𝓩⊢ map g (map f x) = map (g.comp f) x
  simp [map, Option.map_map, Finset.image_image]All goals completed! 🐙

A.3. Mapping of charge spectra obeys the identity

@[simp]
lemma map_id [DecidableEq 𝓩] (x : ChargeSpectrum 𝓩) : map (AddMonoidHom.id 𝓩) x = x := by𝓩:Typeinst✝¹:AddCommGroup 𝓩inst✝:DecidableEq 𝓩x:ChargeSpectrum 𝓩⊢ map (AddMonoidHom.id 𝓩) x = x
  simp [map, Finset.image_id]All goals completed! 🐙

A.4. The charges of a field label commute with mapping of charge spectra



lemma map_ofFieldLabel (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩) (F : FieldLabel) :
    ofFieldLabel (map f x) F = (ofFieldLabel x F).image f := by𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩F:FieldLabel⊢ (map f x).ofFieldLabel F = Finset.image (⇑f) (x.ofFieldLabel F)
  simp [ofFieldLabel, map]𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩F:FieldLabel⊢ (match F with
  | FieldLabel.fiveBarHd => (Option.map (⇑f) x.qHd).toFinset
  | FieldLabel.fiveBarHu => (Option.map (⇑f) x.qHu).toFinset
  | FieldLabel.fiveBarMatter => Finset.image (⇑f) x.Q5
  | FieldLabel.tenMatter => Finset.image (⇑f) x.Q10
  | FieldLabel.fiveHd => Finset.map { toFun := Neg.neg, inj' := ⋯ } (Option.map (⇑f) x.qHd).toFinset
  | FieldLabel.fiveHu => Finset.map { toFun := Neg.neg, inj' := ⋯ } (Option.map (⇑f) x.qHu).toFinset
  | FieldLabel.fiveMatter => Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image (⇑f) x.Q5)) =
  Finset.image (⇑f)
    (match F with
    | FieldLabel.fiveBarHd => x.qHd.toFinset
    | FieldLabel.fiveBarHu => x.qHu.toFinset
    | FieldLabel.fiveBarMatter => x.Q5
    | FieldLabel.tenMatter => x.Q10
    | FieldLabel.fiveHd => Finset.map { toFun := Neg.neg, inj' := ⋯ } x.qHd.toFinset
    | FieldLabel.fiveHu => Finset.map { toFun := Neg.neg, inj' := ⋯ } x.qHu.toFinset
    | FieldLabel.fiveMatter => Finset.map { toFun := Neg.neg, inj' := ⋯ } x.Q5)
  match x with
  | ⟨qHd, qHu, Q5, Q10⟩ =>𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩F:FieldLabelqHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (match F with
  | FieldLabel.fiveBarHd => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveBarHu => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveBarMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
  | FieldLabel.tenMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
  | FieldLabel.fiveHd =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveHu =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveMatter =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)) =
  Finset.image (⇑f)
    (match F with
    | FieldLabel.fiveBarHd => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveBarHu => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveBarMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
    | FieldLabel.tenMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
    | FieldLabel.fiveHd =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveHu =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveMatter =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)
  fin_cases F«0»𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (match FieldLabel.fiveBarHu with
  | FieldLabel.fiveBarHd => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveBarHu => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveBarMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
  | FieldLabel.tenMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
  | FieldLabel.fiveHd =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveHu =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveMatter =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)) =
  Finset.image (⇑f)
    (match FieldLabel.fiveBarHu with
    | FieldLabel.fiveBarHd => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveBarHu => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveBarMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
    | FieldLabel.tenMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
    | FieldLabel.fiveHd =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveHu =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveMatter =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)«1»𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (match FieldLabel.fiveHu with
  | FieldLabel.fiveBarHd => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveBarHu => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveBarMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
  | FieldLabel.tenMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
  | FieldLabel.fiveHd =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveHu =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveMatter =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)) =
  Finset.image (⇑f)
    (match FieldLabel.fiveHu with
    | FieldLabel.fiveBarHd => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveBarHu => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveBarMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
    | FieldLabel.tenMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
    | FieldLabel.fiveHd =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveHu =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveMatter =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)«2»𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (match FieldLabel.fiveBarHd with
  | FieldLabel.fiveBarHd => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveBarHu => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveBarMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
  | FieldLabel.tenMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
  | FieldLabel.fiveHd =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveHu =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveMatter =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)) =
  Finset.image (⇑f)
    (match FieldLabel.fiveBarHd with
    | FieldLabel.fiveBarHd => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveBarHu => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveBarMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
    | FieldLabel.tenMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
    | FieldLabel.fiveHd =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveHu =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveMatter =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)«3»𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (match FieldLabel.fiveHd with
  | FieldLabel.fiveBarHd => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveBarHu => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveBarMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
  | FieldLabel.tenMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
  | FieldLabel.fiveHd =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveHu =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveMatter =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)) =
  Finset.image (⇑f)
    (match FieldLabel.fiveHd with
    | FieldLabel.fiveBarHd => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveBarHu => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveBarMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
    | FieldLabel.tenMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
    | FieldLabel.fiveHd =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveHu =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveMatter =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)«4»𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (match FieldLabel.fiveBarMatter with
  | FieldLabel.fiveBarHd => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveBarHu => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveBarMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
  | FieldLabel.tenMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
  | FieldLabel.fiveHd =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveHu =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveMatter =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)) =
  Finset.image (⇑f)
    (match FieldLabel.fiveBarMatter with
    | FieldLabel.fiveBarHd => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveBarHu => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveBarMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
    | FieldLabel.tenMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
    | FieldLabel.fiveHd =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveHu =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveMatter =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)«5»𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (match FieldLabel.fiveMatter with
  | FieldLabel.fiveBarHd => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveBarHu => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveBarMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
  | FieldLabel.tenMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
  | FieldLabel.fiveHd =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveHu =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveMatter =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)) =
  Finset.image (⇑f)
    (match FieldLabel.fiveMatter with
    | FieldLabel.fiveBarHd => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveBarHu => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveBarMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
    | FieldLabel.tenMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
    | FieldLabel.fiveHd =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveHu =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveMatter =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)«6»𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (match FieldLabel.tenMatter with
  | FieldLabel.fiveBarHd => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveBarHu => (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveBarMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
  | FieldLabel.tenMatter => Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
  | FieldLabel.fiveHd =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd).toFinset
  | FieldLabel.fiveHu =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ }
      (Option.map ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu).toFinset
  | FieldLabel.fiveMatter =>
    Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image ⇑f { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)) =
  Finset.image (⇑f)
    (match FieldLabel.tenMatter with
    | FieldLabel.fiveBarHd => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveBarHu => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveBarMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5
    | FieldLabel.tenMatter => { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q10
    | FieldLabel.fiveHd =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHd.toFinset
    | FieldLabel.fiveHu =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.qHu.toFinset
    | FieldLabel.fiveMatter =>
      Finset.map { toFun := Neg.neg, inj' := ⋯ } { qHd := qHd, qHu := qHu, Q5 := Q5, Q10 := Q10 }.Q5)
  all_goals simpAll goals completed! 🐙
  case «0» | «1» =>𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Option.map (⇑f) qHu).toFinset =
  Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } qHu.toFinset)
    match qHu with
    | some a =>𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩a:𝓩⊢ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Option.map (⇑f) (some a)).toFinset =
  Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } (some a).toFinset)𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩a:𝓩⊢ (Option.map (⇑f) (some a)).toFinset = Finset.image (⇑f) (some a).toFinset simpAll goals completed! 🐙
    | none =>𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Option.map (⇑f) none).toFinset =
  Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } none.toFinset)𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (Option.map (⇑f) none).toFinset = Finset.image (⇑f) none.toFinset simpAll goals completed! 🐙
  case «2» | «3» =>𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Option.map (⇑f) qHd).toFinset =
  Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } qHd.toFinset)
    match qHd with
    | some a =>𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩a:𝓩⊢ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Option.map (⇑f) (some a)).toFinset =
  Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } (some a).toFinset)𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩a:𝓩⊢ (Option.map (⇑f) (some a)).toFinset = Finset.image (⇑f) (some a).toFinset simpAll goals completed! 🐙
    | none =>𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Option.map (⇑f) none).toFinset =
  Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } none.toFinset)𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ (Option.map (⇑f) none).toFinset = Finset.image (⇑f) none.toFinset simpAll goals completed! 🐙
  ·«5»𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image (⇑f) Q5) =
  Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5) trans (Finset.image (⇑f) Q5).image Neg.neg𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image (⇑f) Q5) = Finset.image Neg.neg (Finset.image (⇑f) Q5)𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.image Neg.neg (Finset.image (⇑f) Q5) = Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5)
    ·𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image (⇑f) Q5) = Finset.image Neg.neg (Finset.image (⇑f) Q5) ext a𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩a:𝓩1⊢ a ∈ Finset.map { toFun := Neg.neg, inj' := ⋯ } (Finset.image (⇑f) Q5) ↔ a ∈ Finset.image Neg.neg (Finset.image (⇑f) Q5)
      simpAll goals completed! 🐙
    ·𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.image Neg.neg (Finset.image (⇑f) Q5) = Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5) rw [Finset.image_image𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.image (Neg.neg ∘ ⇑f) Q5 = Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5)𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.image (Neg.neg ∘ ⇑f) Q5 = Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5)]𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.image (Neg.neg ∘ ⇑f) Q5 = Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5)
      symm𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5) = Finset.image (Neg.neg ∘ ⇑f) Q5
      trans Finset.image (⇑f ∘ Neg.neg) (Q5)𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5) = Finset.image (⇑f ∘ Neg.neg) Q5𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.image (⇑f ∘ Neg.neg) Q5 = Finset.image (Neg.neg ∘ ⇑f) Q5
      ·𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5) = Finset.image (⇑f ∘ Neg.neg) Q5 ext a𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩a:𝓩1⊢ a ∈ Finset.image (⇑f) (Finset.map { toFun := Neg.neg, inj' := ⋯ } Q5) ↔ a ∈ Finset.image (⇑f ∘ Neg.neg) Q5
        simpAll goals completed! 🐙
      congr 1e_f𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩⊢ ⇑f ∘ Neg.neg = Neg.neg ∘ ⇑f
      funext ae_f𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩qHd:Option 𝓩qHu:Option 𝓩Q5:Finset 𝓩Q10:Finset 𝓩a:𝓩⊢ (⇑f ∘ Neg.neg) a = (Neg.neg ∘ ⇑f) a
      simpAll goals completed! 🐙

A.5. Mappings of charge spectra preserve the subset relation

lemma map_subset {f : 𝓩 →+ 𝓩1} {x y : ChargeSpectrum 𝓩} (h : x ⊆ y) :
    map f x ⊆ map f y := by𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩h:x ⊆ y⊢ map f x ⊆ map f y
  simp [map, subset_def] at *𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩h:x.qHd.toFinset ⊆ y.qHd.toFinset ∧ x.qHu.toFinset ⊆ y.qHu.toFinset ∧ x.Q5 ⊆ y.Q5 ∧ x.Q10 ⊆ y.Q10⊢ (Option.map (⇑f) x.qHd).toFinset ⊆ (Option.map (⇑f) y.qHd).toFinset ∧
  (Option.map (⇑f) x.qHu).toFinset ⊆ (Option.map (⇑f) y.qHu).toFinset ∧
    Finset.image (⇑f) x.Q5 ⊆ Finset.image (⇑f) y.Q5 ∧ Finset.image (⇑f) x.Q10 ⊆ Finset.image (⇑f) y.Q10
  obtain ⟨hHd, hHu, hQ5, hQ10⟩ := h𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩hHd:x.qHd.toFinset ⊆ y.qHd.toFinsethHu:x.qHu.toFinset ⊆ y.qHu.toFinsethQ5:x.Q5 ⊆ y.Q5hQ10:x.Q10 ⊆ y.Q10⊢ (Option.map (⇑f) x.qHd).toFinset ⊆ (Option.map (⇑f) y.qHd).toFinset ∧
  (Option.map (⇑f) x.qHu).toFinset ⊆ (Option.map (⇑f) y.qHu).toFinset ∧
    Finset.image (⇑f) x.Q5 ⊆ Finset.image (⇑f) y.Q5 ∧ Finset.image (⇑f) x.Q10 ⊆ Finset.image (⇑f) y.Q10
  refine ⟨?_, ?_, ?_, ?_⟩refine_1𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩hHd:x.qHd.toFinset ⊆ y.qHd.toFinsethHu:x.qHu.toFinset ⊆ y.qHu.toFinsethQ5:x.Q5 ⊆ y.Q5hQ10:x.Q10 ⊆ y.Q10⊢ (Option.map (⇑f) x.qHd).toFinset ⊆ (Option.map (⇑f) y.qHd).toFinsetrefine_2𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩hHd:x.qHd.toFinset ⊆ y.qHd.toFinsethHu:x.qHu.toFinset ⊆ y.qHu.toFinsethQ5:x.Q5 ⊆ y.Q5hQ10:x.Q10 ⊆ y.Q10⊢ (Option.map (⇑f) x.qHu).toFinset ⊆ (Option.map (⇑f) y.qHu).toFinsetrefine_3𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩hHd:x.qHd.toFinset ⊆ y.qHd.toFinsethHu:x.qHu.toFinset ⊆ y.qHu.toFinsethQ5:x.Q5 ⊆ y.Q5hQ10:x.Q10 ⊆ y.Q10⊢ Finset.image (⇑f) x.Q5 ⊆ Finset.image (⇑f) y.Q5refine_4𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩hHd:x.qHd.toFinset ⊆ y.qHd.toFinsethHu:x.qHu.toFinset ⊆ y.qHu.toFinsethQ5:x.Q5 ⊆ y.Q5hQ10:x.Q10 ⊆ y.Q10⊢ Finset.image (⇑f) x.Q10 ⊆ Finset.image (⇑f) y.Q10
  ·refine_1𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩hHd:x.qHd.toFinset ⊆ y.qHd.toFinsethHu:x.qHu.toFinset ⊆ y.qHu.toFinsethQ5:x.Q5 ⊆ y.Q5hQ10:x.Q10 ⊆ y.Q10⊢ (Option.map (⇑f) x.qHd).toFinset ⊆ (Option.map (⇑f) y.qHd).toFinset match x, y with
    | ⟨a, _, _, _⟩, ⟨b, _, _, _⟩ =>𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩a:Option 𝓩qHu✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩b:Option 𝓩qHu✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩hHd:{ qHd := a, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := a, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := a, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆ { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := a, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆ { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := a, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd).toFinset ⊆
  (Option.map ⇑f { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd).toFinset
      cases anone𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHu✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩b:Option 𝓩qHu✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩hHd:{ qHd := none, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := none, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := none, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆ { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := none, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆ { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := none, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd).toFinset ⊆
  (Option.map ⇑f { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd).toFinsetsome𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHu✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩b:Option 𝓩qHu✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝:𝓩hHd:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆ { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆ { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd).toFinset ⊆
  (Option.map ⇑f { qHd := b, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd).toFinset
      all_goals cases bsome.none𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHu✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHu✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝:𝓩hHd:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := none, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := none, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := none, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := none, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd).toFinset ⊆
  (Option.map ⇑f { qHd := none, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd).toFinsetsome.some𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHu✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHu✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝¹:𝓩val✝:𝓩hHd:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd).toFinset ⊆
  (Option.map ⇑f { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd).toFinset
      all_goals simpsome.some𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHu✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHu✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝¹:𝓩val✝:𝓩hHd:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ f val✝¹ = f val✝
      all_goals simp at hHdsome.some𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHu✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHu✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝¹:𝓩val✝:𝓩hHu:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := some val✝¹, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10hHd:val✝¹ = val✝⊢ f val✝¹ = f val✝
      subst hHdsome.some𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHu✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHu✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝:𝓩hHu:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := some val✝, qHu := qHu✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := some val✝, qHu := qHu✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ f val✝ = f val✝
      rflAll goals completed! 🐙
  ·refine_2𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩hHd:x.qHd.toFinset ⊆ y.qHd.toFinsethHu:x.qHu.toFinset ⊆ y.qHu.toFinsethQ5:x.Q5 ⊆ y.Q5hQ10:x.Q10 ⊆ y.Q10⊢ (Option.map (⇑f) x.qHu).toFinset ⊆ (Option.map (⇑f) y.qHu).toFinset match x, y with
    | ⟨_, a, _, _⟩, ⟨_, b, _, _⟩ =>𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHd✝¹:Option 𝓩a:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHd✝:Option 𝓩b:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩hHd:{ qHd := qHd✝¹, qHu := a, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := qHd✝¹, qHu := a, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := qHd✝¹, qHu := a, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆ { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := qHd✝¹, qHu := a, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆ { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := qHd✝¹, qHu := a, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu).toFinset ⊆
  (Option.map ⇑f { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.qHu).toFinset
      cases anone𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHd✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHd✝:Option 𝓩b:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩hHd:{ qHd := qHd✝¹, qHu := none, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := qHd✝¹, qHu := none, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := qHd✝¹, qHu := none, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆ { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := qHd✝¹, qHu := none, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆ { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := qHd✝¹, qHu := none, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu).toFinset ⊆
  (Option.map ⇑f { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.qHu).toFinsetsome𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHd✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHd✝:Option 𝓩b:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝:𝓩hHd:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆ { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆ { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu).toFinset ⊆
  (Option.map ⇑f { qHd := qHd✝, qHu := b, Q5 := Q5✝, Q10 := Q10✝ }.qHu).toFinset
      all_goals cases bsome.none𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHd✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHd✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝:𝓩hHd:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := qHd✝, qHu := none, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := qHd✝, qHu := none, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := qHd✝, qHu := none, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := qHd✝, qHu := none, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu).toFinset ⊆
  (Option.map ⇑f { qHd := qHd✝, qHu := none, Q5 := Q5✝, Q10 := Q10✝ }.qHu).toFinsetsome.some𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHd✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHd✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝¹:𝓩val✝:𝓩hHd:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ (Option.map ⇑f { qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu).toFinset ⊆
  (Option.map ⇑f { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu).toFinset
      all_goals simpsome.some𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHd✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHd✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝¹:𝓩val✝:𝓩hHd:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethHu:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHu.toFinset ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.qHu.toFinsethQ5:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ f val✝¹ = f val✝
      all_goals simp at hHusome.some𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHd✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHd✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝¹:𝓩val✝:𝓩hHd:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethQ5:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := qHd✝¹, qHu := some val✝¹, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10hHu:val✝¹ = val✝⊢ f val✝¹ = f val✝
      subst hHusome.some𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩qHd✝¹:Option 𝓩Q5✝¹:Finset 𝓩Q10✝¹:Finset 𝓩qHd✝:Option 𝓩Q5✝:Finset 𝓩Q10✝:Finset 𝓩val✝:𝓩hHd:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.qHd.toFinset ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.qHd.toFinsethQ5:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q5 ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.Q5hQ10:{ qHd := qHd✝¹, qHu := some val✝, Q5 := Q5✝¹, Q10 := Q10✝¹ }.Q10 ⊆
  { qHd := qHd✝, qHu := some val✝, Q5 := Q5✝, Q10 := Q10✝ }.Q10⊢ f val✝ = f val✝
      rflAll goals completed! 🐙
  ·refine_3𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩hHd:x.qHd.toFinset ⊆ y.qHd.toFinsethHu:x.qHu.toFinset ⊆ y.qHu.toFinsethQ5:x.Q5 ⊆ y.Q5hQ10:x.Q10 ⊆ y.Q10⊢ Finset.image (⇑f) x.Q5 ⊆ Finset.image (⇑f) y.Q5 exact Finset.image_subset_image hQ5All goals completed! 🐙
  ·refine_4𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩y:ChargeSpectrum 𝓩hHd:x.qHd.toFinset ⊆ y.qHd.toFinsethHu:x.qHu.toFinset ⊆ y.qHu.toFinsethQ5:x.Q5 ⊆ y.Q5hQ10:x.Q10 ⊆ y.Q10⊢ Finset.image (⇑f) x.Q10 ⊆ Finset.image (⇑f) y.Q10 exact Finset.image_subset_image hQ10All goals completed! 🐙

A.6. Mappings of charge spectra and charges of potential terms

              

lemma map_ofPotentialTerm_toFinset [DecidableEq 𝓩]
    (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩) (T : PotentialTerm) :
    (ofPotentialTerm (map f x) T).toFinset = (ofPotentialTerm x T).toFinset.image f := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ ((map f x).ofPotentialTerm T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm T).toFinset
  have heq : ∀ {W : Type} [AddCommGroup W] [DecidableEq W] (y : ChargeSpectrum W)
      (S : PotentialTerm), (ofPotentialTerm y S).toFinset = (ofPotentialTerm' y S).toFinset := by
    intro W _ _ y S𝓩:Type𝓩1:Typeinst✝⁵:AddCommGroup 𝓩inst✝⁴:AddCommGroup 𝓩1inst✝³:DecidableEq 𝓩1inst✝²:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermW:Typeinst✝¹:AddCommGroup Winst✝:DecidableEq Wy:ChargeSpectrum WS:PotentialTerm⊢ (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm T).toFinset
    ext i𝓩:Type𝓩1:Typeinst✝⁵:AddCommGroup 𝓩inst✝⁴:AddCommGroup 𝓩1inst✝³:DecidableEq 𝓩1inst✝²:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermW:Typeinst✝¹:AddCommGroup Winst✝:DecidableEq Wy:ChargeSpectrum WS:PotentialTermi:W⊢ i ∈ (y.ofPotentialTerm S).toFinset ↔ i ∈ (y.ofPotentialTerm' S).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm T).toFinset
    simp only [Multiset.mem_toFinset]𝓩:Type𝓩1:Typeinst✝⁵:AddCommGroup 𝓩inst✝⁴:AddCommGroup 𝓩1inst✝³:DecidableEq 𝓩1inst✝²:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermW:Typeinst✝¹:AddCommGroup Winst✝:DecidableEq Wy:ChargeSpectrum WS:PotentialTermi:W⊢ i ∈ y.ofPotentialTerm S ↔ i ∈ y.ofPotentialTerm' S𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm T).toFinset
    rw [mem_ofPotentialTerm_iff_mem_ofPotentialTerm𝓩:Type𝓩1:Typeinst✝⁵:AddCommGroup 𝓩inst✝⁴:AddCommGroup 𝓩1inst✝³:DecidableEq 𝓩1inst✝²:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermW:Typeinst✝¹:AddCommGroup Winst✝:DecidableEq Wy:ChargeSpectrum WS:PotentialTermi:W⊢ i ∈ y.ofPotentialTerm' S ↔ i ∈ y.ofPotentialTerm' S𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm T).toFinset]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm T).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm T).toFinset
  rw [heq,𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm' T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm T).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm' T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm' T).toFinset heq𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm' T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm' T).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm' T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm' T).toFinset]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinset⊢ ((map f x).ofPotentialTerm' T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm' T).toFinset
  rcases x with ⟨_ | qHd, _ | qHu, Q5, Q10⟩none.none𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' T).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' T).toFinsetnone.some𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' T).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' T).toFinsetsome.none𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' T).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' T).toFinsetsome.some𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' T).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' T).toFinset <;>none.none𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' T).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' T).toFinsetnone.some𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' T).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' T).toFinsetsome.none𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' T).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' T).toFinsetsome.some𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1T:PotentialTermheq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' T).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' T).toFinset cases Tsome.some.μ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.μ).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.μ).toFinsetsome.some.β𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.β).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.β).toFinsetsome.some.Λ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.Λ).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.Λ).toFinsetsome.some.W1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W1).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W1).toFinsetsome.some.W2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W2).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W2).toFinsetsome.some.W3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W3).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W3).toFinsetsome.some.W4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W4).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W4).toFinsetsome.some.K1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K1).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K1).toFinsetsome.some.K2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K2).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K2).toFinsetsome.some.topYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.topYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.topYukawa).toFinsetsome.some.bottomYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm'
      PotentialTerm.bottomYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset <;>none.none.μ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.μ).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.μ).toFinsetnone.none.β𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.β).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.β).toFinsetnone.none.Λ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.Λ).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.Λ).toFinsetnone.none.W1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W1).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W1).toFinsetnone.none.W2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W2).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W2).toFinsetnone.none.W3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W3).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W3).toFinsetnone.none.W4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W4).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W4).toFinsetnone.none.K1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K1).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K1).toFinsetnone.none.K2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K2).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K2).toFinsetnone.none.topYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.topYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.topYukawa).toFinsetnone.none.bottomYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ((map f { qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := none, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinsetnone.some.μ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.μ).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.μ).toFinsetnone.some.β𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.β).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.β).toFinsetnone.some.Λ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.Λ).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.Λ).toFinsetnone.some.W1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W1).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W1).toFinsetnone.some.W2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W2).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W2).toFinsetnone.some.W3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W3).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W3).toFinsetnone.some.W4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W4).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W4).toFinsetnone.some.K1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K1).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K1).toFinsetnone.some.K2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K2).toFinset =
  Finset.image (⇑f) ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K2).toFinsetnone.some.topYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.topYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.topYukawa).toFinsetnone.some.bottomYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ((map f { qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := none, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinsetsome.none.μ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.μ).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.μ).toFinsetsome.none.β𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.β).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.β).toFinsetsome.none.Λ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.Λ).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.Λ).toFinsetsome.none.W1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W1).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W1).toFinsetsome.none.W2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W2).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W2).toFinsetsome.none.W3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W3).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W3).toFinsetsome.none.W4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W4).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W4).toFinsetsome.none.K1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K1).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K1).toFinsetsome.none.K2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K2).toFinset =
  Finset.image (⇑f) ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K2).toFinsetsome.none.topYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.topYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.topYukawa).toFinsetsome.none.bottomYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ((map f { qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := none, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinsetsome.some.μ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.μ).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.μ).toFinsetsome.some.β𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.β).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.β).toFinsetsome.some.Λ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.Λ).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.Λ).toFinsetsome.some.W1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W1).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W1).toFinsetsome.some.W2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W2).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W2).toFinsetsome.some.W3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W3).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W3).toFinsetsome.some.W4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.W4).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.W4).toFinsetsome.some.K1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K1).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K1).toFinsetsome.some.K2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.K2).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.K2).toFinsetsome.some.topYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm' PotentialTerm.topYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.topYukawa).toFinsetsome.some.bottomYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩qHu:𝓩⊢ ((map f { qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }).ofPotentialTerm'
      PotentialTerm.bottomYukawa).toFinset =
  Finset.image (⇑f)
    ({ qHd := some qHd, qHu := some qHu, Q5 := Q5, Q10 := Q10 }.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset
    simp only [ofPotentialTerm', map, Option.map_eq_map, Option.map_some, Option.map_none,
      Multiset.toFinset_map, Finset.val_toFinset, Function.comp_def, map_add, map_sub, map_neg,
      Multiset.toFinset_singleton, Finset.image_singleton, Finset.product_eq_sprod,
      ← Finset.prodMap_image_product, Finset.image_image, Prod.map_fst, Prod.map_snd]All goals completed! 🐙 <;>none.none.μ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.none.β𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.none.W2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.none.W3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.none.W4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.none.K2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.none.topYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.none.bottomYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.some.μ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.some.W2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.some.W4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.some.K2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetnone.some.bottomYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHu:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetsome.none.μ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetsome.none.β𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetsome.none.W3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetsome.none.W4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetsome.none.K2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinsetsome.none.topYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1heq:∀ {W : Type} [inst : AddCommGroup W] [inst_1 : DecidableEq W] (y : ChargeSpectrum W) (S : PotentialTerm),
  (y.ofPotentialTerm S).toFinset = (y.ofPotentialTerm' S).toFinsetQ5:Finset 𝓩Q10:Finset 𝓩qHd:𝓩⊢ ∅.toFinset = Finset.image ⇑f ∅.toFinset
    simpAll goals completed! 🐙

lemma mem_map_ofPotentialTerm_iff [DecidableEq 𝓩]
    (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩) (T : PotentialTerm) :
    i ∈ (ofPotentialTerm (map f x) T) ↔ i ∈ (ofPotentialTerm x T).map f := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ (map f x).ofPotentialTerm T ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm T)
  trans i ∈ (ofPotentialTerm (map f x) T).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ (map f x).ofPotentialTerm T ↔ i ∈ ((map f x).ofPotentialTerm T).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ ((map f x).ofPotentialTerm T).toFinset ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm T)
  ·𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ (map f x).ofPotentialTerm T ↔ i ∈ ((map f x).ofPotentialTerm T).toFinset simpAll goals completed! 🐙
  rw [map_ofPotentialTerm_toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ Finset.image (⇑f) (x.ofPotentialTerm T).toFinset ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm T)𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ Finset.image (⇑f) (x.ofPotentialTerm T).toFinset ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm T)]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ Finset.image (⇑f) (x.ofPotentialTerm T).toFinset ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm T)
  simpAll goals completed! 🐙

lemma mem_map_ofPotentialTerm'_iff[DecidableEq 𝓩]
    (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩) (T : PotentialTerm) :
    i ∈ (ofPotentialTerm' (map f x) T) ↔ i ∈ (ofPotentialTerm' x T).map f := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ (map f x).ofPotentialTerm' T ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T)
  rw [← mem_ofPotentialTerm_iff_mem_ofPotentialTerm𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ (map f x).ofPotentialTerm T ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T)𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ (map f x).ofPotentialTerm T ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T)]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ (map f x).ofPotentialTerm T ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T)
  rw [mem_map_ofPotentialTerm_iff𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm T) ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T)𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm T) ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T)]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm T) ↔ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T)
  simp only [Multiset.mem_map]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ (∃ a ∈ x.ofPotentialTerm T, f a = i) ↔ ∃ a ∈ x.ofPotentialTerm' T, f a = i
  constructormp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ (∃ a ∈ x.ofPotentialTerm T, f a = i) → ∃ a ∈ x.ofPotentialTerm' T, f a = impr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ (∃ a ∈ x.ofPotentialTerm' T, f a = i) → ∃ a ∈ x.ofPotentialTerm T, f a = i
  ·mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ (∃ a ∈ x.ofPotentialTerm T, f a = i) → ∃ a ∈ x.ofPotentialTerm' T, f a = i intro ⟨a, h, h1⟩mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerma:𝓩h:a ∈ x.ofPotentialTerm Th1:f a = i⊢ ∃ a ∈ x.ofPotentialTerm' T, f a = i
    refine ⟨a, ?_, h1⟩mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerma:𝓩h:a ∈ x.ofPotentialTerm Th1:f a = i⊢ a ∈ x.ofPotentialTerm' T
    exact mem_ofPotentialTerm_iff_mem_ofPotentialTerm.mp hAll goals completed! 🐙
  ·mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ (∃ a ∈ x.ofPotentialTerm' T, f a = i) → ∃ a ∈ x.ofPotentialTerm T, f a = i intro ⟨a, h, h1⟩mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerma:𝓩h:a ∈ x.ofPotentialTerm' Th1:f a = i⊢ ∃ a ∈ x.ofPotentialTerm T, f a = i
    refine ⟨a, ?_, h1⟩mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1i:𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerma:𝓩h:a ∈ x.ofPotentialTerm' Th1:f a = i⊢ a ∈ x.ofPotentialTerm T
    exact mem_ofPotentialTerm_iff_mem_ofPotentialTerm.mpr hAll goals completed! 🐙

lemma map_ofPotentialTerm'_toFinset [DecidableEq 𝓩]
    (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩) (T : PotentialTerm) :
    (ofPotentialTerm' (map f x) T).toFinset = (ofPotentialTerm' x T).toFinset.image f := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerm⊢ ((map f x).ofPotentialTerm' T).toFinset = Finset.image (⇑f) (x.ofPotentialTerm' T).toFinset
  ext i𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermi:𝓩1⊢ i ∈ ((map f x).ofPotentialTerm' T).toFinset ↔ i ∈ Finset.image (⇑f) (x.ofPotentialTerm' T).toFinset
  simp only [Multiset.mem_toFinset, Finset.mem_image]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermi:𝓩1⊢ i ∈ (map f x).ofPotentialTerm' T ↔ ∃ a ∈ x.ofPotentialTerm' T, f a = i
  rw [mem_map_ofPotentialTerm'_iff𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermi:𝓩1⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T) ↔ ∃ a ∈ x.ofPotentialTerm' T, f a = i𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermi:𝓩1⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T) ↔ ∃ a ∈ x.ofPotentialTerm' T, f a = i]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermi:𝓩1⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' T) ↔ ∃ a ∈ x.ofPotentialTerm' T, f a = i
  simpAll goals completed! 🐙

A.7. Mapping charge spectra of `allowsTermForm

lemma allowsTermForm_map {T} {f : 𝓩 →+ 𝓩1} {a b c : 𝓩} :
    (allowsTermForm a b c T).map f = allowsTermForm (f a) (f b) (f c) T := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩T:PotentialTermf:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c T) = allowsTermForm (f a) (f b) (f c) T
  cases Tμ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.μ) = allowsTermForm (f a) (f b) (f c) PotentialTerm.μβ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.β) = allowsTermForm (f a) (f b) (f c) PotentialTerm.βΛ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.Λ) = allowsTermForm (f a) (f b) (f c) PotentialTerm.ΛW1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.W1) = allowsTermForm (f a) (f b) (f c) PotentialTerm.W1W2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.W2) = allowsTermForm (f a) (f b) (f c) PotentialTerm.W2W3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.W3) = allowsTermForm (f a) (f b) (f c) PotentialTerm.W3W4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.W4) = allowsTermForm (f a) (f b) (f c) PotentialTerm.W4K1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.K1) = allowsTermForm (f a) (f b) (f c) PotentialTerm.K1K2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.K2) = allowsTermForm (f a) (f b) (f c) PotentialTerm.K2topYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.topYukawa) = allowsTermForm (f a) (f b) (f c) PotentialTerm.topYukawabottomYukawa𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1a:𝓩b:𝓩c:𝓩⊢ map f (allowsTermForm a b c PotentialTerm.bottomYukawa) = allowsTermForm (f a) (f b) (f c) PotentialTerm.bottomYukawa
  all_goals simp [allowsTermForm, map]All goals completed! 🐙

A.8. Mapping preserves whether a charge spectrum allows a potential term



lemma map_allowsTerm {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩} {T : PotentialTerm}
    (h : x.AllowsTerm T) : (map f x).AllowsTerm T := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermh:x.AllowsTerm T⊢ (map f x).AllowsTerm T
  rw [allowsTerm_iff_subset_allowsTermForm𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermh:∃ a b c, allowsTermForm a b c T ⊆ x⊢ ∃ a b c, allowsTermForm a b c T ⊆ map f x𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermh:∃ a b c, allowsTermForm a b c T ⊆ x⊢ ∃ a b c, allowsTermForm a b c T ⊆ map f x] at ⊢ h𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTermh:∃ a b c, allowsTermForm a b c T ⊆ x⊢ ∃ a b c, allowsTermForm a b c T ⊆ map f x
  obtain ⟨a, b, c, h1⟩ := h𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerma:𝓩b:𝓩c:𝓩h1:allowsTermForm a b c T ⊆ x⊢ ∃ a b c, allowsTermForm a b c T ⊆ map f x
  use f a, f b, f ch𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerma:𝓩b:𝓩c:𝓩h1:allowsTermForm a b c T ⊆ x⊢ allowsTermForm (f a) (f b) (f c) T ⊆ map f x
  rw [← allowsTermForm_maph𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerma:𝓩b:𝓩c:𝓩h1:allowsTermForm a b c T ⊆ x⊢ map f (allowsTermForm a b c T) ⊆ map f xh𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerma:𝓩b:𝓩c:𝓩h1:allowsTermForm a b c T ⊆ x⊢ map f (allowsTermForm a b c T) ⊆ map f x]h𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩T:PotentialTerma:𝓩b:𝓩c:𝓩h1:allowsTermForm a b c T ⊆ x⊢ map f (allowsTermForm a b c T) ⊆ map f x
  exact map_subset h1All goals completed! 🐙

A.9. Mapping preserves if a charge spectrum is pheno-constrained

lemma map_isPhenoConstrained (f : 𝓩 →+ 𝓩1) {x : ChargeSpectrum 𝓩}
    (h : x.IsPhenoConstrained) : (map f x).IsPhenoConstrained := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.IsPhenoConstrained⊢ (map f x).IsPhenoConstrained
  simp [IsPhenoConstrained] at ⊢ h𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.μ ∨
  x.AllowsTerm PotentialTerm.β ∨
    x.AllowsTerm PotentialTerm.Λ ∨
      x.AllowsTerm PotentialTerm.W2 ∨
        x.AllowsTerm PotentialTerm.W4 ∨
          x.AllowsTerm PotentialTerm.K1 ∨ x.AllowsTerm PotentialTerm.K2 ∨ x.AllowsTerm PotentialTerm.W1⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1
  rcases h with h | h | h | h | h | h | h | hinl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.μ⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.β⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.Λ⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1inr.inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.W2⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1inr.inr.inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.W4⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1inr.inr.inr.inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.K1⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1inr.inr.inr.inr.inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.K2⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1inr.inr.inr.inr.inr.inr.inr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.W1⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1
  ·inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.μ⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1 exact Or.inl (map_allowsTerm h)All goals completed! 🐙
  ·inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.β⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1 exact Or.inr (Or.inl (map_allowsTerm h))All goals completed! 🐙
  ·inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.Λ⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1 exact Or.inr (Or.inr (Or.inl (map_allowsTerm h)))All goals completed! 🐙
  ·inr.inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.W2⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1 exact Or.inr (Or.inr (Or.inr (Or.inl (map_allowsTerm h))))All goals completed! 🐙
  ·inr.inr.inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.W4⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1 exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inl (map_allowsTerm h)))))All goals completed! 🐙
  ·inr.inr.inr.inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.K1⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1 exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl (map_allowsTerm h))))))All goals completed! 🐙
  ·inr.inr.inr.inr.inr.inr.inl𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.K2⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1 exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inl (map_allowsTerm h)))))))All goals completed! 🐙
  ·inr.inr.inr.inr.inr.inr.inr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩h:x.AllowsTerm PotentialTerm.W1⊢ (map f x).AllowsTerm PotentialTerm.μ ∨
  (map f x).AllowsTerm PotentialTerm.β ∨
    (map f x).AllowsTerm PotentialTerm.Λ ∨
      (map f x).AllowsTerm PotentialTerm.W2 ∨
        (map f x).AllowsTerm PotentialTerm.W4 ∨
          (map f x).AllowsTerm PotentialTerm.K1 ∨
            (map f x).AllowsTerm PotentialTerm.K2 ∨ (map f x).AllowsTerm PotentialTerm.W1 exact Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr (Or.inr ((map_allowsTerm h))))))))All goals completed! 🐙

lemma not_isPhenoConstrained_of_map {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩}
    (h : ¬ (map f x).IsPhenoConstrained) : ¬ x.IsPhenoConstrained :=
  fun hn => h (map_isPhenoConstrained f hn)

A.10. Mapping preserves completeness of charge spectra

omit [DecidableEq 𝓩] in
lemma map_isComplete_iff {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩} :
    (map f x).IsComplete ↔ x.IsComplete := by𝓩:Type𝓩1:Typeinst✝²:AddCommGroup 𝓩inst✝¹:AddCommGroup 𝓩1inst✝:DecidableEq 𝓩1f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩⊢ (map f x).IsComplete ↔ x.IsComplete
  simp [IsComplete, map]All goals completed! 🐙

A.11. Mapping commutes with charges of Yukawa terms



lemma map_ofYukawaTerms_toFinset {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩} :
    (map f x).ofYukawaTerms.toFinset = x.ofYukawaTerms.toFinset.image f := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩⊢ (map f x).ofYukawaTerms.toFinset = Finset.image (⇑f) x.ofYukawaTerms.toFinset
  simp [ofYukawaTerms]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩⊢ ((map f x).ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪
    ((map f x).ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset =
  Finset.image (⇑f)
    ((x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪ (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset)
  ext i𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈
    ((map f x).ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪
      ((map f x).ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset ↔
  i ∈
    Finset.image (⇑f)
      ((x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪ (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset)
  rw [Finset.image_union𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈
    ((map f x).ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪
      ((map f x).ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset ↔
  i ∈
    Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪
      Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈
    ((map f x).ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪
      ((map f x).ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset ↔
  i ∈
    Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪
      Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈
    ((map f x).ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪
      ((map f x).ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset ↔
  i ∈
    Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∪
      Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset
  simp only [Finset.mem_union, Multiset.mem_toFinset]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ (map f x).ofPotentialTerm' PotentialTerm.topYukawa ∨ i ∈ (map f x).ofPotentialTerm' PotentialTerm.bottomYukawa ↔
  i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∨
    i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset
  rw [mem_map_ofPotentialTerm'_iff,𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa) ∨
    i ∈ (map f x).ofPotentialTerm' PotentialTerm.bottomYukawa ↔
  i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∨
    i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa) ∨
    i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa) ↔
  i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∨
    i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset mem_map_ofPotentialTerm'_iff𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa) ∨
    i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa) ↔
  i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∨
    i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa) ∨
    i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa) ↔
  i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∨
    i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa) ∨
    i ∈ Multiset.map (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa) ↔
  i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.topYukawa).toFinset ∨
    i ∈ Finset.image (⇑f) (x.ofPotentialTerm' PotentialTerm.bottomYukawa).toFinset
  simp [Multiset.mem_map]All goals completed! 🐙



lemma mem_map_ofYukawaTerms_iff {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩} {i} :
    i ∈ (map f x).ofYukawaTerms ↔ i ∈ x.ofYukawaTerms.map f := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ (map f x).ofYukawaTerms ↔ i ∈ Multiset.map (⇑f) x.ofYukawaTerms
  trans i ∈ (map f x).ofYukawaTerms.toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ (map f x).ofYukawaTerms ↔ i ∈ (map f x).ofYukawaTerms.toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ (map f x).ofYukawaTerms.toFinset ↔ i ∈ Multiset.map (⇑f) x.ofYukawaTerms
  ·𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ (map f x).ofYukawaTerms ↔ i ∈ (map f x).ofYukawaTerms.toFinset simpAll goals completed! 🐙
  rw [map_ofYukawaTerms_toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ Finset.image (⇑f) x.ofYukawaTerms.toFinset ↔ i ∈ Multiset.map (⇑f) x.ofYukawaTerms𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ Finset.image (⇑f) x.ofYukawaTerms.toFinset ↔ i ∈ Multiset.map (⇑f) x.ofYukawaTerms]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩i:𝓩1⊢ i ∈ Finset.image (⇑f) x.ofYukawaTerms.toFinset ↔ i ∈ Multiset.map (⇑f) x.ofYukawaTerms
  simpAll goals completed! 🐙

A.12. Mapping of charge spectra and regenerating dangerous Yukawa terms

              

lemma map_ofYukawaTermsNSum_toFinset {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩} {n : ℕ}:
    ((map f x).ofYukawaTermsNSum n).toFinset = (x.ofYukawaTermsNSum n).toFinset.image f:= by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕ⊢ ((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset
  induction n with
  | zero =>zero𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩⊢ ((map f x).ofYukawaTermsNSum 0).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum 0).toFinset simp [ofYukawaTermsNSum]All goals completed! 🐙
  | succ n ih =>succ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ((map f x).ofYukawaTermsNSum (n + 1)).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum (n + 1)).toFinset
    simp [ofYukawaTermsNSum]succ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ((map f x).ofYukawaTermsNSum n).toFinset ∪
    (((map f x).ofYukawaTermsNSum n).bind fun sSum =>
        Multiset.map (fun s => sSum + s) (map f x).ofYukawaTerms).toFinset =
  Finset.image (⇑f)
    ((x.ofYukawaTermsNSum n).toFinset ∪
      ((x.ofYukawaTermsNSum n).bind fun sSum => Multiset.map (fun s => sSum + s) x.ofYukawaTerms).toFinset)
    rw [Finset.image_unionsucc𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ((map f x).ofYukawaTermsNSum n).toFinset ∪
    (((map f x).ofYukawaTermsNSum n).bind fun sSum =>
        Multiset.map (fun s => sSum + s) (map f x).ofYukawaTerms).toFinset =
  Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ∪
    Finset.image (⇑f)
      ((x.ofYukawaTermsNSum n).bind fun sSum => Multiset.map (fun s => sSum + s) x.ofYukawaTerms).toFinsetsucc𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ((map f x).ofYukawaTermsNSum n).toFinset ∪
    (((map f x).ofYukawaTermsNSum n).bind fun sSum =>
        Multiset.map (fun s => sSum + s) (map f x).ofYukawaTerms).toFinset =
  Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ∪
    Finset.image (⇑f)
      ((x.ofYukawaTermsNSum n).bind fun sSum => Multiset.map (fun s => sSum + s) x.ofYukawaTerms).toFinset]succ𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ((map f x).ofYukawaTermsNSum n).toFinset ∪
    (((map f x).ofYukawaTermsNSum n).bind fun sSum =>
        Multiset.map (fun s => sSum + s) (map f x).ofYukawaTerms).toFinset =
  Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ∪
    Finset.image (⇑f)
      ((x.ofYukawaTermsNSum n).bind fun sSum => Multiset.map (fun s => sSum + s) x.ofYukawaTerms).toFinset
    congr 1e_a𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ (((map f x).ofYukawaTermsNSum n).bind fun sSum => Multiset.map (fun s => sSum + s) (map f x).ofYukawaTerms).toFinset =
  Finset.image (⇑f) ((x.ofYukawaTermsNSum n).bind fun sSum => Multiset.map (fun s => sSum + s) x.ofYukawaTerms).toFinset
    ext ie_a𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1⊢ i ∈
    (((map f x).ofYukawaTermsNSum n).bind fun sSum =>
        Multiset.map (fun s => sSum + s) (map f x).ofYukawaTerms).toFinset ↔
  i ∈
    Finset.image (⇑f)
      ((x.ofYukawaTermsNSum n).bind fun sSum => Multiset.map (fun s => sSum + s) x.ofYukawaTerms).toFinset
    simp only [Multiset.mem_toFinset, Multiset.mem_bind, Multiset.mem_map, Finset.mem_image,
      exists_exists_and_exists_and_eq_and, map_add]e_a𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1⊢ (∃ a ∈ (map f x).ofYukawaTermsNSum n, ∃ a_1 ∈ (map f x).ofYukawaTerms, a + a_1 = i) ↔
  ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i
    constructore_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1⊢ (∃ a ∈ (map f x).ofYukawaTermsNSum n, ∃ a_1 ∈ (map f x).ofYukawaTerms, a + a_1 = i) →
  ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = ie_a.mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1⊢ (∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i) →
  ∃ a ∈ (map f x).ofYukawaTermsNSum n, ∃ a_1 ∈ (map f x).ofYukawaTerms, a + a_1 = i
    ·e_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1⊢ (∃ a ∈ (map f x).ofYukawaTermsNSum n, ∃ a_1 ∈ (map f x).ofYukawaTerms, a + a_1 = i) →
  ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i intro he_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1h:∃ a ∈ (map f x).ofYukawaTermsNSum n, ∃ a_1 ∈ (map f x).ofYukawaTerms, a + a_1 = i⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i
      obtain ⟨a, a_mem, b, b_mem, h⟩ := he_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ (map f x).ofYukawaTermsh:a + b = i⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i
      have a_mem' : a ∈ ((map f x).ofYukawaTermsNSum n).toFinset := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕ⊢ ((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinsete_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ (map f x).ofYukawaTermsh:a + b = ia_mem':a ∈ ((map f x).ofYukawaTermsNSum n).toFinset⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i simpa using a_meme_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ (map f x).ofYukawaTermsh:a + b = ia_mem':a ∈ ((map f x).ofYukawaTermsNSum n).toFinset⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = ie_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ (map f x).ofYukawaTermsh:a + b = ia_mem':a ∈ ((map f x).ofYukawaTermsNSum n).toFinset⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i
      rw [ihe_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ (map f x).ofYukawaTermsh:a + b = ia_mem':a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = ie_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ (map f x).ofYukawaTermsh:a + b = ia_mem':a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i] at a_mem'e_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ (map f x).ofYukawaTermsh:a + b = ia_mem':a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i
      rw [mem_map_ofYukawaTerms_iffe_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ Multiset.map (⇑f) x.ofYukawaTermsh:a + b = ia_mem':a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = ie_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ Multiset.map (⇑f) x.ofYukawaTermsh:a + b = ia_mem':a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i] at b_meme_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1b_mem:b ∈ Multiset.map (⇑f) x.ofYukawaTermsh:a + b = ia_mem':a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i
      simp at a_mem' b_meme_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩1a_mem:a ∈ (map f x).ofYukawaTermsNSum nb:𝓩1h:a + b = ia_mem':∃ a_1 ∈ x.ofYukawaTermsNSum n, f a_1 = ab_mem:∃ a ∈ x.ofYukawaTerms, f a = b⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i
      obtain ⟨a, a_mem', rfl⟩ := a_mem'e_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1b:𝓩1b_mem:∃ a ∈ x.ofYukawaTerms, f a = ba:𝓩a_mem':a ∈ x.ofYukawaTermsNSum na_mem:f a ∈ (map f x).ofYukawaTermsNSum nh:f a + b = i⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i
      obtain ⟨b, b_mem', rfl⟩ := b_meme_a.mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem':a ∈ x.ofYukawaTermsNSum na_mem:f a ∈ (map f x).ofYukawaTermsNSum nb:𝓩b_mem':b ∈ x.ofYukawaTermsh:f a + f b = i⊢ ∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i
      exact ⟨a, a_mem', b, b_mem', h⟩All goals completed! 🐙
    ·e_a.mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1⊢ (∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i) →
  ∃ a ∈ (map f x).ofYukawaTermsNSum n, ∃ a_1 ∈ (map f x).ofYukawaTerms, a + a_1 = i intro he_a.mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1h:∃ a ∈ x.ofYukawaTermsNSum n, ∃ b ∈ x.ofYukawaTerms, f a + f b = i⊢ ∃ a ∈ (map f x).ofYukawaTermsNSum n, ∃ a_1 ∈ (map f x).ofYukawaTerms, a + a_1 = i
      obtain ⟨a, a_mem, b, b_mem, h⟩ := he_a.mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ ∃ a ∈ (map f x).ofYukawaTermsNSum n, ∃ a_1 ∈ (map f x).ofYukawaTerms, a + a_1 = i
      use f ah𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f a ∈ (map f x).ofYukawaTermsNSum n ∧ ∃ a_1 ∈ (map f x).ofYukawaTerms, f a + a_1 = i
      apply And.introh.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f a ∈ (map f x).ofYukawaTermsNSum nh.right𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ ∃ a_1 ∈ (map f x).ofYukawaTerms, f a + a_1 = i
      ·h.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f a ∈ (map f x).ofYukawaTermsNSum n rw [← Multiset.mem_toFinset,h.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f a ∈ ((map f x).ofYukawaTermsNSum n).toFinseth.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ihh.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseth.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset]h.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f a ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset
        simp only [Finset.mem_image, Multiset.mem_toFinset]h.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ ∃ a_1 ∈ x.ofYukawaTermsNSum n, f a_1 = f a
        use aAll goals completed! 🐙
      use f bh𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f b ∈ (map f x).ofYukawaTerms ∧ f a + f b = i
      apply And.introh.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f b ∈ (map f x).ofYukawaTermsh.right𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f a + f b = i
      ·h.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f b ∈ (map f x).ofYukawaTerms rw [mem_map_ofYukawaTerms_iffh.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f b ∈ Multiset.map (⇑f) x.ofYukawaTermsh.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f b ∈ Multiset.map (⇑f) x.ofYukawaTerms]h.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ f b ∈ Multiset.map (⇑f) x.ofYukawaTerms
        simp only [Multiset.mem_map]h.left𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕih:((map f x).ofYukawaTermsNSum n).toFinset = Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinseti:𝓩1a:𝓩a_mem:a ∈ x.ofYukawaTermsNSum nb:𝓩b_mem:b ∈ x.ofYukawaTermsh:f a + f b = i⊢ ∃ a ∈ x.ofYukawaTerms, f a = f b
        use bAll goals completed! 🐙
      exact hAll goals completed! 🐙

lemma mem_map_ofYukawaTermsNSum_iff {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩} {n i} :
    i ∈ (map f x).ofYukawaTermsNSum n ↔ i ∈ (x.ofYukawaTermsNSum n).map f := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕi:𝓩1⊢ i ∈ (map f x).ofYukawaTermsNSum n ↔ i ∈ Multiset.map (⇑f) (x.ofYukawaTermsNSum n)
  trans i ∈ ((map f x).ofYukawaTermsNSum n).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕi:𝓩1⊢ i ∈ (map f x).ofYukawaTermsNSum n ↔ i ∈ ((map f x).ofYukawaTermsNSum n).toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕi:𝓩1⊢ i ∈ ((map f x).ofYukawaTermsNSum n).toFinset ↔ i ∈ Multiset.map (⇑f) (x.ofYukawaTermsNSum n)
  ·𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕi:𝓩1⊢ i ∈ (map f x).ofYukawaTermsNSum n ↔ i ∈ ((map f x).ofYukawaTermsNSum n).toFinset simpAll goals completed! 🐙
  rw [map_ofYukawaTermsNSum_toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕi:𝓩1⊢ i ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ↔ i ∈ Multiset.map (⇑f) (x.ofYukawaTermsNSum n)𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕi:𝓩1⊢ i ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ↔ i ∈ Multiset.map (⇑f) (x.ofYukawaTermsNSum n)]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕi:𝓩1⊢ i ∈ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ↔ i ∈ Multiset.map (⇑f) (x.ofYukawaTermsNSum n)
  simpAll goals completed! 🐙lemma map_phenoConstrainingChargesSP_toFinset {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩} :
    (map f x).phenoConstrainingChargesSP.toFinset =
    x.phenoConstrainingChargesSP.toFinset.image f := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩⊢ (map f x).phenoConstrainingChargesSP.toFinset = Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset
  simp [phenoConstrainingChargesSP, map_ofPotentialTerm'_toFinset, Finset.image_union]All goals completed! 🐙

lemma map_yukawaGeneratesDangerousAtLevel (f : 𝓩 →+ 𝓩1) {x : ChargeSpectrum 𝓩} (n : ℕ)
    (h : x.YukawaGeneratesDangerousAtLevel n) : (map f x).YukawaGeneratesDangerousAtLevel n := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ (map f x).YukawaGeneratesDangerousAtLevel n
  rw [yukawaGeneratesDangerousAtLevel_iff_toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ((map f x).ofYukawaTermsNSum n).toFinset ∩ (map f x).phenoConstrainingChargesSP.toFinset ≠ ∅𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ((map f x).ofYukawaTermsNSum n).toFinset ∩ (map f x).phenoConstrainingChargesSP.toFinset ≠ ∅]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ((map f x).ofYukawaTermsNSum n).toFinset ∩ (map f x).phenoConstrainingChargesSP.toFinset ≠ ∅
  rw [map_phenoConstrainingChargesSP_toFinset,𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ((map f x).ofYukawaTermsNSum n).toFinset ∩ Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset ≠ ∅𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ∩ Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset ≠ ∅ map_ofYukawaTermsNSum_toFinset𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ∩ Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset ≠ ∅𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ∩ Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset ≠ ∅]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ∩ Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset ≠ ∅
  rw [← Finset.nonempty_iff_ne_empty,𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ (Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset ∩ Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset).Nonempty𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ¬Disjoint (Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset) (Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset) ← Finset.not_disjoint_iff_nonempty_inter𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ¬Disjoint (Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset) (Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset)𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ¬Disjoint (Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset) (Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset)]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ¬Disjoint (Finset.image (⇑f) (x.ofYukawaTermsNSum n).toFinset) (Finset.image (⇑f) x.phenoConstrainingChargesSP.toFinset)
  apply Disjoint.of_image_finset.mt𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ¬Disjoint (x.ofYukawaTermsNSum n).toFinset x.phenoConstrainingChargesSP.toFinset
  rw [Finset.not_disjoint_iff_nonempty_inter,𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ ((x.ofYukawaTermsNSum n).toFinset ∩ x.phenoConstrainingChargesSP.toFinset).Nonempty𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ (x.ofYukawaTermsNSum n).toFinset ∩ x.phenoConstrainingChargesSP.toFinset ≠ ∅ Finset.nonempty_iff_ne_empty𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ (x.ofYukawaTermsNSum n).toFinset ∩ x.phenoConstrainingChargesSP.toFinset ≠ ∅𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ (x.ofYukawaTermsNSum n).toFinset ∩ x.phenoConstrainingChargesSP.toFinset ≠ ∅]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩n:ℕh:x.YukawaGeneratesDangerousAtLevel n⊢ (x.ofYukawaTermsNSum n).toFinset ∩ x.phenoConstrainingChargesSP.toFinset ≠ ∅
  exact (yukawaGeneratesDangerousAtLevel_iff_toFinset _ _).mp hAll goals completed! 🐙lemma not_yukawaGeneratesDangerousAtLevel_of_map {f : 𝓩 →+ 𝓩1} {x : ChargeSpectrum 𝓩}
    (n : ℕ) (h : ¬ (map f x).YukawaGeneratesDangerousAtLevel n) :
    ¬ x.YukawaGeneratesDangerousAtLevel n :=
  fun hn => h (map_yukawaGeneratesDangerousAtLevel f n hn)

B. Preimage of a charge spectrum under a mapping

We give a computationally efficient way of calculating the preimage of a charge s : Charges 𝓩1 in a subset ofFinset S5 S10, and then show it is equal to the actual preimage.

The preimage of a charge Charges 𝓩1 in ofFinset S5 S10 ⊆ Charges 𝓩 under mapping charges through f : 𝓩 →+ 𝓩1.

def preimageOfFinset (S5 S10 : Finset 𝓩) (f : 𝓩 →+ 𝓩1)
    (x : ChargeSpectrum 𝓩1) : Finset (ChargeSpectrum 𝓩) :=
  let SHd := (S5.map ⟨Option.some, Option.some_injective 𝓩⟩ ∪ {none} : Finset (Option 𝓩)).filter
    fun y => f <$> y = x.qHd
  let SHu := (S5.map ⟨Option.some, Option.some_injective 𝓩⟩ ∪ {none} : Finset (Option 𝓩)).filter
    fun y => f <$> y = x.qHu
  let SQ5' := S5.filter fun y => f y ∈ x.Q5
  let SQ5 : Finset (Finset 𝓩) := SQ5'.powerset.filter fun y => y.image f = x.Q5
  let SQ10' := S10.filter fun y => f y ∈ x.Q10
  let SQ10 : Finset (Finset 𝓩) := SQ10'.powerset.filter fun y => y.image f = x.Q10
  (SHd.product <| SHu.product <| SQ5.product SQ10).map toProd.symm.toEmbedding

B.1. `preimageOfFinset` gives the actual preimage


lemma preimageOfFinset_eq (S5 S10 : Finset 𝓩) (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩1) :
    preimageOfFinset S5 S10 f x = {y : ChargeSpectrum 𝓩 | y.map f = x ∧ y ∈ ofFinset S5 S10} := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ ↑(preimageOfFinset S5 S10 f x) = {y | map f y = x ∧ y ∈ ofFinset S5 S10}
  ext y𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩⊢ y ∈ ↑(preimageOfFinset S5 S10 f x) ↔ y ∈ {y | map f y = x ∧ y ∈ ofFinset S5 S10}
  simp [preimageOfFinset, toProd]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩⊢ (∃ a a_1 a_2 b,
    (((a = none ∨ ∃ a_3 ∈ S5, some a_3 = a) ∧ Option.map (⇑f) a = x.qHd) ∧
        ((a_1 = none ∨ ∃ a ∈ S5, some a = a_1) ∧ Option.map (⇑f) a_1 = x.qHu) ∧
          (a_2 ⊆ {y ∈ S5 | f y ∈ x.Q5} ∧ Finset.image (⇑f) a_2 = x.Q5) ∧
            b ⊆ {y ∈ S10 | f y ∈ x.Q10} ∧ Finset.image (⇑f) b = x.Q10) ∧
      { qHd := a, qHu := a_1, Q5 := a_2, Q10 := b } = y) ↔
  map f y = x ∧ y ∈ ofFinset S5 S10
  match y, x with
  | ⟨yHd, yHu, y5, y10⟩, ⟨xHd, xHu, x5, x10⟩ =>𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1⊢ (∃ a a_1 a_2 b,
    (((a = none ∨ ∃ a_3 ∈ S5, some a_3 = a) ∧
          Option.map (⇑f) a = { qHd := xHd, qHu := xHu, Q5 := x5, Q10 := x10 }.qHd) ∧
        ((a_1 = none ∨ ∃ a ∈ S5, some a = a_1) ∧
            Option.map (⇑f) a_1 = { qHd := xHd, qHu := xHu, Q5 := x5, Q10 := x10 }.qHu) ∧
          (a_2 ⊆ {y ∈ S5 | f y ∈ { qHd := xHd, qHu := xHu, Q5 := x5, Q10 := x10 }.Q5} ∧
              Finset.image (⇑f) a_2 = { qHd := xHd, qHu := xHu, Q5 := x5, Q10 := x10 }.Q5) ∧
            b ⊆ {y ∈ S10 | f y ∈ { qHd := xHd, qHu := xHu, Q5 := x5, Q10 := x10 }.Q10} ∧
              Finset.image (⇑f) b = { qHd := xHd, qHu := xHu, Q5 := x5, Q10 := x10 }.Q10) ∧
      { qHd := a, qHu := a_1, Q5 := a_2, Q10 := b } = { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }) ↔
  map f { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } = { qHd := xHd, qHu := xHu, Q5 := x5, Q10 := x10 } ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10
  simp [map]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1⊢ ((yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧ Option.map (⇑f) yHd = xHd) ∧
    ((yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ Option.map (⇑f) yHu = xHu) ∧
      (y5 ⊆ {x ∈ S5 | f x ∈ x5} ∧ Finset.image (⇑f) y5 = x5) ∧
        y10 ⊆ {x ∈ S10 | f x ∈ x10} ∧ Finset.image (⇑f) y10 = x10 ↔
  (Option.map (⇑f) yHd = xHd ∧ Option.map (⇑f) yHu = xHu ∧ Finset.image (⇑f) y5 = x5 ∧ Finset.image (⇑f) y10 = x10) ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10
  constructormp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1⊢ ((yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧ Option.map (⇑f) yHd = xHd) ∧
    ((yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ Option.map (⇑f) yHu = xHu) ∧
      (y5 ⊆ {x ∈ S5 | f x ∈ x5} ∧ Finset.image (⇑f) y5 = x5) ∧
        y10 ⊆ {x ∈ S10 | f x ∈ x10} ∧ Finset.image (⇑f) y10 = x10 →
  (Option.map (⇑f) yHd = xHd ∧ Option.map (⇑f) yHu = xHu ∧ Finset.image (⇑f) y5 = x5 ∧ Finset.image (⇑f) y10 = x10) ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1⊢ (Option.map (⇑f) yHd = xHd ∧ Option.map (⇑f) yHu = xHu ∧ Finset.image (⇑f) y5 = x5 ∧ Finset.image (⇑f) y10 = x10) ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10 →
  ((yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧ Option.map (⇑f) yHd = xHd) ∧
    ((yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ Option.map (⇑f) yHu = xHu) ∧
      (y5 ⊆ {x ∈ S5 | f x ∈ x5} ∧ Finset.image (⇑f) y5 = x5) ∧ y10 ⊆ {x ∈ S10 | f x ∈ x10} ∧ Finset.image (⇑f) y10 = x10
  ·mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1⊢ ((yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧ Option.map (⇑f) yHd = xHd) ∧
    ((yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ Option.map (⇑f) yHu = xHu) ∧
      (y5 ⊆ {x ∈ S5 | f x ∈ x5} ∧ Finset.image (⇑f) y5 = x5) ∧
        y10 ⊆ {x ∈ S10 | f x ∈ x10} ∧ Finset.image (⇑f) y10 = x10 →
  (Option.map (⇑f) yHd = xHd ∧ Option.map (⇑f) yHu = xHu ∧ Finset.image (⇑f) y5 = x5 ∧ Finset.image (⇑f) y10 = x10) ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10 intro ⟨⟨h1, rfl⟩, ⟨h2, rfl⟩, ⟨h3, rfl⟩, ⟨h4, rfl⟩⟩mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ (Option.map (⇑f) yHd = Option.map (⇑f) yHd ∧
    Option.map (⇑f) yHu = Option.map (⇑f) yHu ∧
      Finset.image (⇑f) y5 = Finset.image (⇑f) y5 ∧ Finset.image (⇑f) y10 = Finset.image (⇑f) y10) ∧
  { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10
    simp only [true_and]mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10
    rw [mem_ofFinset_iffmp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHd.toFinset ⊆ S5 ∧
  { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHu.toFinset ⊆ S5 ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q5 ⊆ S5 ∧
      { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q10 ⊆ S10mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHd.toFinset ⊆ S5 ∧
  { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHu.toFinset ⊆ S5 ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q5 ⊆ S5 ∧
      { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q10 ⊆ S10]mp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHd.toFinset ⊆ S5 ∧
  { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHu.toFinset ⊆ S5 ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q5 ⊆ S5 ∧
      { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q10 ⊆ S10
    simp onlymp𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10
    refine ⟨?_, ?_, ?_, ?_⟩mp.refine_1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ yHd.toFinset ⊆ S5mp.refine_2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ yHu.toFinset ⊆ S5mp.refine_3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ y5 ⊆ S5mp.refine_4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ y10 ⊆ S10
    ·mp.refine_1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ yHd.toFinset ⊆ S5 match yHd with
      | some a =>𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}a:𝓩h1:some a = none ∨ ∃ a_1 ∈ S5, some a_1 = some a⊢ (some a).toFinset ⊆ S5 simpa using h1All goals completed! 🐙
      | none =>𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}h1:none = none ∨ ∃ a ∈ S5, some a = none⊢ none.toFinset ⊆ S5 simpAll goals completed! 🐙
    ·mp.refine_2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ yHu.toFinset ⊆ S5 match yHu with
      | some a =>𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}a:𝓩h2:some a = none ∨ ∃ a_1 ∈ S5, some a_1 = some a⊢ (some a).toFinset ⊆ S5 simpa using h2All goals completed! 🐙
      | none =>𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}h2:none = none ∨ ∃ a ∈ S5, some a = none⊢ none.toFinset ⊆ S5 simpAll goals completed! 🐙
    ·mp.refine_3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ y5 ⊆ S5 exact h3.trans <| Finset.filter_subset (fun y => f y ∈ Finset.image (⇑f) y5) S5All goals completed! 🐙
    ·mp.refine_4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h1:yHd = none ∨ ∃ a ∈ S5, some a = yHdh2:yHu = none ∨ ∃ a ∈ S5, some a = yHuh3:y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5}h4:y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10}⊢ y10 ⊆ S10 apply h4.trans <| Finset.filter_subset (fun y => f y ∈ Finset.image (⇑f) y10) S10All goals completed! 🐙
  ·mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1⊢ (Option.map (⇑f) yHd = xHd ∧ Option.map (⇑f) yHu = xHu ∧ Finset.image (⇑f) y5 = x5 ∧ Finset.image (⇑f) y10 = x10) ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10 →
  ((yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧ Option.map (⇑f) yHd = xHd) ∧
    ((yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ Option.map (⇑f) yHu = xHu) ∧
      (y5 ⊆ {x ∈ S5 | f x ∈ x5} ∧ Finset.image (⇑f) y5 = x5) ∧ y10 ⊆ {x ∈ S10 | f x ∈ x10} ∧ Finset.image (⇑f) y10 = x10 intro ⟨⟨rfl, rfl, rfl, rfl⟩, h2⟩mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:{ qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10⊢ ((yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧ Option.map (⇑f) yHd = Option.map (⇑f) yHd) ∧
  ((yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ Option.map (⇑f) yHu = Option.map (⇑f) yHu) ∧
    (y5 ⊆ {x ∈ S5 | f x ∈ Finset.image (⇑f) y5} ∧ Finset.image (⇑f) y5 = Finset.image (⇑f) y5) ∧
      y10 ⊆ {x ∈ S10 | f x ∈ Finset.image (⇑f) y10} ∧ Finset.image (⇑f) y10 = Finset.image (⇑f) y10
    simp only [and_true, Finset.mem_image]mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:{ qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 } ∈ ofFinset S5 S10⊢ (yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧
  (yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ y5 ⊆ {x ∈ S5 | ∃ a ∈ y5, f a = f x} ∧ y10 ⊆ {x ∈ S10 | ∃ a ∈ y10, f a = f x}
    rw [mem_ofFinset_iffmpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:{ qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHd.toFinset ⊆ S5 ∧
  { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHu.toFinset ⊆ S5 ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q5 ⊆ S5 ∧
      { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q10 ⊆ S10⊢ (yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧
  (yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ y5 ⊆ {x ∈ S5 | ∃ a ∈ y5, f a = f x} ∧ y10 ⊆ {x ∈ S10 | ∃ a ∈ y10, f a = f x}mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:{ qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHd.toFinset ⊆ S5 ∧
  { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHu.toFinset ⊆ S5 ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q5 ⊆ S5 ∧
      { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q10 ⊆ S10⊢ (yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧
  (yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ y5 ⊆ {x ∈ S5 | ∃ a ∈ y5, f a = f x} ∧ y10 ⊆ {x ∈ S10 | ∃ a ∈ y10, f a = f x}] at h2mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:{ qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHd.toFinset ⊆ S5 ∧
  { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.qHu.toFinset ⊆ S5 ∧
    { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q5 ⊆ S5 ∧
      { qHd := yHd, qHu := yHu, Q5 := y5, Q10 := y10 }.Q10 ⊆ S10⊢ (yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧
  (yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ y5 ⊆ {x ∈ S5 | ∃ a ∈ y5, f a = f x} ∧ y10 ⊆ {x ∈ S10 | ∃ a ∈ y10, f a = f x}
    simp at h2mpr𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ (yHd = none ∨ ∃ a ∈ S5, some a = yHd) ∧
  (yHu = none ∨ ∃ a ∈ S5, some a = yHu) ∧ y5 ⊆ {x ∈ S5 | ∃ a ∈ y5, f a = f x} ∧ y10 ⊆ {x ∈ S10 | ∃ a ∈ y10, f a = f x}
    refine ⟨?_, ?_, ?_, ?_⟩mpr.refine_1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ yHd = none ∨ ∃ a ∈ S5, some a = yHdmpr.refine_2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ yHu = none ∨ ∃ a ∈ S5, some a = yHumpr.refine_3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ y5 ⊆ {x ∈ S5 | ∃ a ∈ y5, f a = f x}mpr.refine_4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ y10 ⊆ {x ∈ S10 | ∃ a ∈ y10, f a = f x}
    ·mpr.refine_1𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ yHd = none ∨ ∃ a ∈ S5, some a = yHd match yHd with
      | some a =>𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1a:𝓩h2:(some a).toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ some a = none ∨ ∃ a_1 ∈ S5, some a_1 = some a
        simp at h2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1a:𝓩h2:a ∈ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ some a = none ∨ ∃ a_1 ∈ S5, some a_1 = some a
        simpa using h2.1All goals completed! 🐙
      | none =>𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:none.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ none = none ∨ ∃ a ∈ S5, some a = none simpAll goals completed! 🐙
    ·mpr.refine_2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ yHu = none ∨ ∃ a ∈ S5, some a = yHu match yHu with
      | some a =>𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1a:𝓩h2:yHd.toFinset ⊆ S5 ∧ (some a).toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ some a = none ∨ ∃ a_1 ∈ S5, some a_1 = some a
        simp at h2𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1a:𝓩h2:yHd.toFinset ⊆ S5 ∧ a ∈ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ some a = none ∨ ∃ a_1 ∈ S5, some a_1 = some a
        simpa using h2.2.1All goals completed! 🐙
      | none =>𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ none.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ none = none ∨ ∃ a ∈ S5, some a = none simpAll goals completed! 🐙
    ·mpr.refine_3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ y5 ⊆ {x ∈ S5 | ∃ a ∈ y5, f a = f x} refine Finset.subset_iff.mpr ?_mpr.refine_3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ ∀ ⦃x : 𝓩⦄, x ∈ y5 → x ∈ {x ∈ S5 | ∃ a ∈ y5, f a = f x}
      intro x hxmpr.refine_3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x✝:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10x:𝓩hx:x ∈ y5⊢ x ∈ {x ∈ S5 | ∃ a ∈ y5, f a = f x}
      simp only [Finset.mem_filter]mpr.refine_3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x✝:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10x:𝓩hx:x ∈ y5⊢ x ∈ S5 ∧ ∃ a ∈ y5, f a = f x
      refine ⟨h2.2.2.1 hx, ?_⟩mpr.refine_3𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x✝:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10x:𝓩hx:x ∈ y5⊢ ∃ a ∈ y5, f a = f x
      use xAll goals completed! 🐙
    ·mpr.refine_4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ y10 ⊆ {x ∈ S10 | ∃ a ∈ y10, f a = f x} refine Finset.subset_iff.mpr ?_mpr.refine_4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10⊢ ∀ ⦃x : 𝓩⦄, x ∈ y10 → x ∈ {x ∈ S10 | ∃ a ∈ y10, f a = f x}
      intro x hxmpr.refine_4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x✝:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10x:𝓩hx:x ∈ y10⊢ x ∈ {x ∈ S10 | ∃ a ∈ y10, f a = f x}
      simp only [Finset.mem_filter]mpr.refine_4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x✝:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10x:𝓩hx:x ∈ y10⊢ x ∈ S10 ∧ ∃ a ∈ y10, f a = f x
      refine ⟨h2.2.2.2 hx, ?_⟩mpr.refine_4𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x✝:ChargeSpectrum 𝓩1y:ChargeSpectrum 𝓩yHd:Option 𝓩yHu:Option 𝓩y5:Finset 𝓩y10:Finset 𝓩xHd:Option 𝓩1xHu:Option 𝓩1x5:Finset 𝓩1x10:Finset 𝓩1h2:yHd.toFinset ⊆ S5 ∧ yHu.toFinset ⊆ S5 ∧ y5 ⊆ S5 ∧ y10 ⊆ S10x:𝓩hx:x ∈ y10⊢ ∃ a ∈ y10, f a = f x
      use xAll goals completed! 🐙

B.2. Efficient definition for the cardinality of the preimage

The cardinality of the preimage of a charge Charges 𝓩1 in ofFinset S5 S10 ⊆ Charges 𝓩 under mapping charges through f : 𝓩 →+ 𝓩1.

def preimageOfFinsetCard (S5 S10 : Finset 𝓩) (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩1) : ℕ :=
  let SHd := (S5.map ⟨Option.some, Option.some_injective 𝓩⟩ ∪ {none} : Finset (Option 𝓩)).filter
    fun y => f <$> y = x.qHd
  let SHu := (S5.map ⟨Option.some, Option.some_injective 𝓩⟩ ∪ {none} : Finset (Option 𝓩)).filter
    fun y => f <$> y = x.qHu
  let SQ5' := S5.filter fun y => f y ∈ x.Q5
  let SQ5 : Finset (Finset 𝓩) := SQ5'.powerset.filter fun y => y.image f = x.Q5
  let SQ10' := S10.filter fun y => f y ∈ x.Q10
  let SQ10 : Finset (Finset 𝓩) := SQ10'.powerset.filter fun y => y.image f = x.Q10
  SHd.card * SHu.card * SQ5.card * SQ10.card

B.3. Definition for the cardinality equals cardinality of the preimage



lemma preimageOfFinset_card_eq (S5 S10 : Finset 𝓩) (f : 𝓩 →+ 𝓩1) (x : ChargeSpectrum 𝓩1) :
    preimageOfFinsetCard S5 S10 f x =
    (preimageOfFinset S5 S10 f x).card := by𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x = (preimageOfFinset S5 S10 f x).card
  rw [preimageOfFinset,𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  (Finset.map toProd.symm.toEmbedding
      ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHd}.product
        ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHu}.product
          ({y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5}.product
            ({y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10}))))).card𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHd}.product
      ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHu}.product
        ({y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5}.product
          ({y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10})))).card Finset.card_map𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHd}.product
      ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHu}.product
        ({y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5}.product
          ({y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10})))).card𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHd}.product
      ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHu}.product
        ({y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5}.product
          ({y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10})))).card]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHd}.product
      ({y ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | ⇑f <$> y = x.qHu}.product
        ({y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5}.product
          ({y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10})))).card
  simp only [Option.map_eq_map, Finset.product_eq_sprod]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  ({x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHd} ×ˢ
      {x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHu} ×ˢ
        {y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5} ×ˢ
          {y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10}).card
  repeat rw [Finset.card_product𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  {x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHd}.card *
    ({x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHu} ×ˢ
        {y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5} ×ˢ
          {y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10}).card𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  {x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHd}.card *
    ({x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHu}.card *
      ({y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5}.card *
        {y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10}.card))]𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  {x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHd}.card *
    ({x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHu}.card *
      ({y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5} ×ˢ
          {y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10}).card)𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  {x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHd}.card *
    ({x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHu}.card *
      ({y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5}.card *
        {y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10}.card))𝓩:Type𝓩1:Typeinst✝³:AddCommGroup 𝓩inst✝²:AddCommGroup 𝓩1inst✝¹:DecidableEq 𝓩1inst✝:DecidableEq 𝓩S5:Finset 𝓩S10:Finset 𝓩f:𝓩 →+ 𝓩1x:ChargeSpectrum 𝓩1⊢ preimageOfFinsetCard S5 S10 f x =
  {x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHd}.card *
    ({x_1 ∈ Finset.map { toFun := some, inj' := ⋯ } S5 ∪ {none} | Option.map (⇑f) x_1 = x.qHu}.card *
      ({y ∈ {y ∈ S5 | f y ∈ x.Q5}.powerset | Finset.image (⇑f) y = x.Q5}.card *
        {y ∈ {y ∈ S10 | f y ∈ x.Q10}.powerset | Finset.image (⇑f) y = x.Q10}.card))
  simp [preimageOfFinsetCard, mul_assoc]All goals completed! 🐙

Mapping charge spectra values

i. Overview

ii. Key results

iii. Table of contents

iv. References

A. The mapping of charge spectra

A.2. Mapping of charge spectra obeys composing maps

A.3. Mapping of charge spectra obeys the identity

A.4. The charges of a field label commute with mapping of charge spectra

A.5. Mappings of charge spectra preserve the subset relation

A.6. Mappings of charge spectra and charges of potential terms

A.7. Mapping charge spectra of `allowsTermForm

A.8. Mapping preserves whether a charge spectrum allows a potential term

A.9. Mapping preserves if a charge spectrum is pheno-constrained

A.10. Mapping preserves completeness of charge spectra

A.11. Mapping commutes with charges of Yukawa terms

A.12. Mapping of charge spectra and regenerating dangerous Yukawa terms

B. Preimage of a charge spectrum under a mapping

B.1. preimageOfFinset gives the actual preimage

B.2. Efficient definition for the cardinality of the preimage

B.3. Definition for the cardinality equals cardinality of the preimage

B.1. `preimageOfFinset` gives the actual preimage