Physlib.Relativity.PauliMatrices.Relations

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.Relativity.PauliMatrices.ToTensor
public import Physlib.Relativity.Tensors.ComplexTensor.Units.Basic

Contraction of indices of Pauli matrix.

The main result of this file is pauliMatrix_contract_pauliMatrix which states that η_{μν} σ^{μ α dot β} σ^{ν α' dot β'} = 2 ε^{αα'} ε^{dot β dot β'}.

The current way this result is proved is by using tensor tree manipulations. There is likely a more direct path to this result.

@[expose] public section

The statement that σᵥᵃᵇ σᵛᵃ'ᵇ' = 2 εᵃᵃ' εᵇᵇ'.


lemma pauliCo_contr_pauliContr :
    {σ_^^ | ν α β ⊗ σ^^^ | ν α' β' = (2 : ℂ) •ₜ εL | α α' ⊗ εR | β β'}ᵀ := by⊢ (contrT 4 0 3 ⋯) ((prodT σ_^^) (Tensorial.toTensor σ)) = (permT ![0, 2, 1, 3] ⋯) (2 • (prodT εL) εR)
  apply (Tensor.basis _).repr.injective⊢ (Tensor.basis
        (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)).repr
    ((contrT 4 0 3 ⋯) ((prodT σ_^^) (Tensorial.toTensor σ))) =
  (Tensor.basis
        (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)).repr
    ((permT ![0, 2, 1, 3] ⋯) (2 • (prodT εL) εR))
  ext bb:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)⊢ ((Tensor.basis
          (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘
            Fin.succSuccAbove 0 3)).repr
      ((contrT 4 0 3 ⋯) ((prodT σ_^^) (Tensorial.toTensor σ))))
    b =
  ((Tensor.basis
          (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘
            Fin.succSuccAbove 0 3)).repr
      ((permT ![0, 2, 1, 3] ⋯) (2 • (prodT εL) εR)))
    b
  conv_rhs =>
    rw [permT_basis_repr_symm_apply]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| ((Tensor.basis (Fin.append ![Color.upL, Color.upL] ![Color.upR, Color.upR])).repr (2 • (prodT εL) εR)) fun i =>
  (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))
    rw [_root_.map_smul]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| (2 • (Tensor.basis (Fin.append ![Color.upL, Color.upL] ![Color.upR, Color.upR])).repr ((prodT εL) εR)) fun i =>
  (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))
    simp only [Nat.reduceAdd, Nat.succ_eq_add_one, Fin.isValue, Fin.succAbove_zero,
      Function.comp_apply, Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| 2 *
  ((Tensor.basis (Fin.append ![Color.upL, Color.upL] ![Color.upR, Color.upR])).repr ((prodT εL) εR)) fun i =>
    (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))
    rw (transparency := .instances) [prodT_basis_repr_apply]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| 2 *
  (((Tensor.basis ![Color.upL, Color.upL]).repr εL)
      (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 *
    ((Tensor.basis ![Color.upR, Color.upR]).repr εR)
      (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2)
    simp only [Nat.reduceAdd, Nat.succ_eq_add_one, Fin.isValue, Fin.succAbove_zero,
      Function.comp_apply]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| 2 *
  (((Tensor.basis ![Color.upL, Color.upL]).repr εL)
      (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 *
    ((Tensor.basis ![Color.upR, Color.upR]).repr εR)
      (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2)
    rw [leftMetric_eq_ofRat, rightMetric_eq_ofRat]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| 2 *
  (((Tensor.basis ![Color.upL, Color.upL]).repr
        (ofRat fun f =>
          if f 0 = Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast leftMetric_eq_ofRat._proof_2 1 then -1
          else
            if f 1 = Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧ f 0 = Fin.cast leftMetric_eq_ofRat._proof_1 1 then 1
            else 0))
      (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 *
    ((Tensor.basis ![Color.upR, Color.upR]).repr
        (ofRat fun f =>
          if f 0 = Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast rightMetric_eq_ofRat._proof_2 1 then -1
          else
            if f 1 = Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧ f 0 = Fin.cast rightMetric_eq_ofRat._proof_1 1 then 1
            else 0))
      (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2)
    simp only [Nat.reduceAdd, Nat.succ_eq_add_one, Fin.isValue, Fin.succAbove_zero,
      Function.comp_apply, cons_val_zero, cons_val_one, head_cons, ofRat_basis_repr_apply]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| 2 *
  (Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
              Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
              Fin.cast leftMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 1 then
          1
        else 0) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
              Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
              Fin.cast rightMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 1 then
          1
        else 0))
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| 2 *
  Physlib.RatComplexNum.toComplexNum
    ((if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
              Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
              Fin.cast leftMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 1 then
          1
        else 0) *
      if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
              Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
              Fin.cast rightMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 1 then
          1
        else 0)
    change (2 : ℕ) * _b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| ↑2 *
  Physlib.RatComplexNum.toComplexNum
    ((if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
              Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
              Fin.cast leftMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 1 then
          1
        else 0) *
      if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
              Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
              Fin.cast rightMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 1 then
          1
        else 0)
    rw [Physlib.RatComplexNum.ofNat_mul_toComplexNum 2]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)| Physlib.RatComplexNum.toComplexNum
  (↑2 *
    ((if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
              Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
              Fin.cast leftMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 1 then
          1
        else 0) *
      if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
              Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
              Fin.cast rightMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 1 then
          1
        else 0))
  rw [contrT_basis_repr_applyb:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)⊢ ∑ b',
    ((Tensor.basis (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR])).repr
          ((prodT σ_^^) (Tensorial.toTensor σ)))
        ↑b' *
      (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0))
        ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => Lorentz.complexContrBasisFin4
            | Color.down => Lorentz.complexCoBasisFin4)
            (↑b' 0) ⊗ₜ[ℂ]
          (match
              complexLorentzTensor.τ
                (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => Lorentz.complexContrBasisFin4
            | Color.down => Lorentz.complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 3))) =
  Physlib.RatComplexNum.toComplexNum
    (↑2 *
      ((if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                  Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                  Fin.cast leftMetric_eq_ofRat._proof_1 1 then
            1
          else 0) *
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                  Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                  Fin.cast rightMetric_eq_ofRat._proof_1 1 then
            1
          else 0))b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)⊢ ∑ b',
    ((Tensor.basis (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR])).repr
          ((prodT σ_^^) (Tensorial.toTensor σ)))
        ↑b' *
      (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0))
        ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => Lorentz.complexContrBasisFin4
            | Color.down => Lorentz.complexCoBasisFin4)
            (↑b' 0) ⊗ₜ[ℂ]
          (match
              complexLorentzTensor.τ
                (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => Lorentz.complexContrBasisFin4
            | Color.down => Lorentz.complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 3))) =
  Physlib.RatComplexNum.toComplexNum
    (↑2 *
      ((if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                  Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                  Fin.cast leftMetric_eq_ofRat._proof_1 1 then
            1
          else 0) *
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                  Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                  Fin.cast rightMetric_eq_ofRat._proof_1 1 then
            1
          else 0))]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)⊢ ∑ b',
    ((Tensor.basis (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR])).repr
          ((prodT σ_^^) (Tensorial.toTensor σ)))
        ↑b' *
      (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0))
        ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0 with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => Lorentz.complexContrBasisFin4
            | Color.down => Lorentz.complexCoBasisFin4)
            (↑b' 0) ⊗ₜ[ℂ]
          (match
              complexLorentzTensor.τ
                (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0) with
            | Color.upL => LeftHandedWeyl.basis
            | Color.downL => DualLeftHandedWeyl.basis
            | Color.upR => RightHandedWeyl.basis
            | Color.downR => DualRightHandedWeyl.basis
            | Color.up => Lorentz.complexContrBasisFin4
            | Color.down => Lorentz.complexCoBasisFin4)
            ((basisIdxCongr ⋯) (↑b' 3))) =
  Physlib.RatComplexNum.toComplexNum
    (↑2 *
      ((if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                  Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                  Fin.cast leftMetric_eq_ofRat._proof_1 1 then
            1
          else 0) *
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                  Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                  Fin.cast rightMetric_eq_ofRat._proof_1 1 then
            1
          else 0))
  conv_lhs =>
    enter [2, x]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| ((Tensor.basis (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR])).repr
      ((prodT σ_^^) (Tensorial.toTensor σ)))
    ↑x *
  (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0))
    ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        (↑x 0) ⊗ₜ[ℂ]
      (match
          complexLorentzTensor.τ
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑x 3)))
    rw [prodT_basis_repr_apply]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr σ_^^) (ComponentIdx.prod ↑x).1 *
    ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr (Tensorial.toTensor σ)) (ComponentIdx.prod ↑x).2 *
  (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0))
    ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        (↑x 0) ⊗ₜ[ℂ]
      (match
          complexLorentzTensor.τ
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑x 3)))
    simp only [pauliCo_eq_ofRat, toTensor_eq_ofRat]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| ((Tensor.basis ![Color.down, Color.upL, Color.upR]).repr
        (ofRat fun b =>
          if b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
            else
              if
                  b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    b 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧ b 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      b 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧ b 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        b 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧ b 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          b 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧ b 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }))
      (ComponentIdx.prod ↑x).1 *
    ((Tensor.basis ![Color.up, Color.upL, Color.upR]).repr
        (ofRat fun b =>
          if b 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
            else
              if
                  b 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    b 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧ b 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    b 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      b 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧ b 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      b 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        b 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧ b 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        b 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          b 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧ b 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else 0))
      (ComponentIdx.prod ↑x).2 *
  (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0))
    ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        (↑x 0) ⊗ₜ[ℂ]
      (match
          complexLorentzTensor.τ
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑x 3)))
    simp only [Fin.isValue, Fin.cast_eq_self, ofRat_basis_repr_apply]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
              (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
          { fst := -1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
            { fst := 0, snd := 1 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                    (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) *
    Physlib.RatComplexNum.toComplexNum
      (if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
              (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                    (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := -1, snd := 0 }
                else 0) *
  (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0))
    ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        (↑x 0) ⊗ₜ[ℂ]
      (match
          complexLorentzTensor.τ
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑x 3)))
    leftb:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| Physlib.RatComplexNum.toComplexNum
    (if
        (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
          (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
      { fst := 1, snd := 0 }
    else
      if
          (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
            (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
        { fst := -1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
              (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
          { fst := 0, snd := 1 }
        else
          if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                  (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                  (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }) *
  Physlib.RatComplexNum.toComplexNum
    (if
        (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
          (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
      { fst := 1, snd := 0 }
    else
      if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
            (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
              (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
          { fst := 0, snd := -1 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                  (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
            { fst := 0, snd := 1 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := -1, snd := 0 }
              else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| Physlib.RatComplexNum.toComplexNum
  ((if
        (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
          (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
      { fst := 1, snd := 0 }
    else
      if
          (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
            (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
        { fst := -1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
              (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
          { fst := 0, snd := 1 }
        else
          if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                  (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                  (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 1, snd := 0 }
              else { fst := 0, snd := 0 }) *
    if
        (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
          (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
      { fst := 1, snd := 0 }
    else
      if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
            (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
              (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
          { fst := 0, snd := -1 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                  (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
            { fst := 0, snd := 1 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := -1, snd := 0 }
              else 0)
  conv_lhs =>
    enter [2, x]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| Physlib.RatComplexNum.toComplexNum
    ((if
          (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
              (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
          { fst := -1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
            { fst := 0, snd := 1 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                    (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) *
      if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
              (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                    (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := -1, snd := 0 }
                else 0) *
  (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0))
    ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0 with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        (↑x 0) ⊗ₜ[ℂ]
      (match
          complexLorentzTensor.τ
            (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0) with
        | Color.upL => LeftHandedWeyl.basis
        | Color.downL => DualLeftHandedWeyl.basis
        | Color.upR => RightHandedWeyl.basis
        | Color.downR => DualRightHandedWeyl.basis
        | Color.up => Lorentz.complexContrBasisFin4
        | Color.down => Lorentz.complexCoBasisFin4)
        ((basisIdxCongr ⋯) (↑x 3)))
    rightb:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| (complexLorentzTensor.contr (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0))
  ((match Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0 with
      | Color.upL => LeftHandedWeyl.basis
      | Color.downL => DualLeftHandedWeyl.basis
      | Color.upR => RightHandedWeyl.basis
      | Color.downR => DualRightHandedWeyl.basis
      | Color.up => Lorentz.complexContrBasisFin4
      | Color.down => Lorentz.complexCoBasisFin4)
      (↑x 0) ⊗ₜ[ℂ]
    (match
        complexLorentzTensor.τ (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] 0) with
      | Color.upL => LeftHandedWeyl.basis
      | Color.downL => DualLeftHandedWeyl.basis
      | Color.upR => RightHandedWeyl.basis
      | Color.downR => DualRightHandedWeyl.basis
      | Color.up => Lorentz.complexContrBasisFin4
      | Color.down => Lorentz.complexCoBasisFin4)
      ((basisIdxCongr ⋯) (↑x 3)))
    rw (transparency := .instances) [contr_basis_ratComplexNum]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| Physlib.RatComplexNum.toComplexNum (if ↑(↑x 0) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
  conv_lhs =>
    enter [2, x]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| Physlib.RatComplexNum.toComplexNum
    ((if
          (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
              (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
          { fst := -1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
            { fst := 0, snd := 1 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                    (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) *
      if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
              (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                    (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := -1, snd := 0 }
                else 0) *
  Physlib.RatComplexNum.toComplexNum (if ↑(↑x 0) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
    rw [← Physlib.RatComplexNum.toComplexNum.map_mul]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)x:↥b.DropPairSection| Physlib.RatComplexNum.toComplexNum
  (((if
          (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
              (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
          { fst := -1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
            { fst := 0, snd := 1 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                    (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                  { fst := 1, snd := 0 }
                else { fst := 0, snd := 0 }) *
      if
          (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
        { fst := 1, snd := 0 }
      else
        if
            (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
              (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                  (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
            { fst := 0, snd := -1 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                    (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := -1, snd := 0 }
                else 0) *
    if ↑(↑x 0) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0)
  rw [← map_sum Physlib.RatComplexNum.toComplexNumb:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 0) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (↑2 *
      ((if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                  Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                  Fin.cast leftMetric_eq_ofRat._proof_1 1 then
            1
          else 0) *
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                  Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                  Fin.cast rightMetric_eq_ofRat._proof_1 1 then
            1
          else 0))b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 0) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (↑2 *
      ((if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                  Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                  Fin.cast leftMetric_eq_ofRat._proof_1 1 then
            1
          else 0) *
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                  Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                  Fin.cast rightMetric_eq_ofRat._proof_1 1 then
            1
          else 0))]b:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 0) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  Physlib.RatComplexNum.toComplexNum
    (↑2 *
      ((if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                  Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                  Fin.cast leftMetric_eq_ofRat._proof_1 1 then
            1
          else 0) *
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                  Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                  Fin.cast rightMetric_eq_ofRat._proof_1 1 then
            1
          else 0))
  apply (Function.Injective.eq_iff Physlib.RatComplexNum.toComplexNum_injective).mprb:ComponentIdx (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)⊢ (∑ x,
    ((if
            (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
            { fst := -1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) *
        if
            (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                    { fst := -1, snd := 0 }
                  else 0) *
      if ↑(↑x 0) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
  ↑2 *
    ((if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
              Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
              Fin.cast leftMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 1 then
          1
        else 0) *
      if
          (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
              Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
              Fin.cast rightMetric_eq_ofRat._proof_2 1 then
        -1
      else
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 1 then
          1
        else 0)
  revert b⊢ ∀
  (b :
    ComponentIdx
      (Fin.append ![Color.down, Color.upL, Color.upR] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 0 3)),
  (∑ x,
      ((if
              (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).1 1 = (ComponentIdx.prod ↑x).1 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑x).1 1 ≠ (ComponentIdx.prod ↑x).1 2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).1 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑x).1 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑x).1 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) *
          if
              (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑x).2 1 = (ComponentIdx.prod ↑x).2 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑x).2 1 ≠ (ComponentIdx.prod ↑x).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑x).2 0 = Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑x).2 1 = Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑x).2 2 = Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
        if ↑(↑x 0) = ↑((basisIdxCongr ⋯) (↑x 3)) then 1 else 0) =
    ↑2 *
      ((if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                Fin.cast leftMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                Fin.cast leftMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 1 =
                  Fin.cast leftMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).1 0 =
                  Fin.cast leftMetric_eq_ofRat._proof_1 1 then
            1
          else 0) *
        if
            (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                Fin.cast rightMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                Fin.cast rightMetric_eq_ofRat._proof_2 1 then
          -1
        else
          if
              (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 1 =
                  Fin.cast rightMetric_eq_ofRat._proof_2 0 ∧
                (ComponentIdx.prod fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))).2 0 =
                  Fin.cast rightMetric_eq_ofRat._proof_1 1 then
            1
          else 0)
  decide +kernelAll goals completed! 🐙

              lemma pauliCoDown_trace_pauliCo : {(σ___ | μ β α ⊗ σ_^^ | ν α β) = (2 •ₜ η' | μ ν)}ᵀ := by⊢ (contrT 2 1 3 ⋯) ((contrT 4 2 4 ⋯) ((prodT σ___) σ_^^)) = (permT ![0, 1] ⋯) (2 • η')
  conv_lhs =>
    rw [pauliCoDown_eq_ofRat, pauliCo_eq_ofRat, prodT_ofRat_ofRat,
      contrT_ofRat, contrT_ofRat]| ofRat fun b =>
  ∑ x,
    ∑ x_1,
      (if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  0 =
                Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    0 =
                  Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      0 =
                    Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        1 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        2 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          1 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          2 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            1 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            2 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) *
        if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  0 =
                Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    0 =
                  Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    2 then
            { fst := -1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      0 =
                    Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        1 =
                      Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        2 =
                      Fin.cast pauliCo_eq_ofRat._proof_3 1 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          1 =
                        Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          2 =
                        Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            1 =
                          Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            2 =
                          Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }
  conv_rhs =>
    rw [coMetric_eq_ofRat]| (permT ![0, 1] ⋯)
  (2 •
    ofRat fun f =>
      if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
      else if f 0 = f 1 then -1 else 0)
    rw [← map_nsmul]| (permT ![0, 1] ⋯)
  (ofRat
    (2 • fun f =>
      if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
      else if f 0 = f 1 then -1 else 0))
  apply (Tensor.basis _).repr.injective⊢ (Tensor.basis
        ((Fin.append ![Color.down, Color.downR, Color.downL] ![Color.down, Color.upL, Color.upR] ∘
            Fin.succSuccAbove 2 4) ∘
          Fin.succSuccAbove 1 3)).repr
    (ofRat fun b =>
      ∑ x,
        ∑ x_1,
          (if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      0 =
                    Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            1 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            2 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              0 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                1 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                2 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                0 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  1 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  2 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) *
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      0 =
                    Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            1 =
                          Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            2 =
                          Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              0 =
                            Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                1 =
                              Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                2 =
                              Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                      { fst := -1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                0 =
                              Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  1 =
                                Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  2 =
                                Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) =
  (Tensor.basis
        ((Fin.append ![Color.down, Color.downR, Color.downL] ![Color.down, Color.upL, Color.upR] ∘
            Fin.succSuccAbove 2 4) ∘
          Fin.succSuccAbove 1 3)).repr
    ((permT ![0, 1] ⋯)
      (ofRat
        (2 • fun f =>
          if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
          else if f 0 = f 1 then -1 else 0)))
  ext bb:ComponentIdx
  ((Fin.append ![Color.down, Color.downR, Color.downL] ![Color.down, Color.upL, Color.upR] ∘ Fin.succSuccAbove 2 4) ∘
    Fin.succSuccAbove 1 3)⊢ ((Tensor.basis
          ((Fin.append ![Color.down, Color.downR, Color.downL] ![Color.down, Color.upL, Color.upR] ∘
              Fin.succSuccAbove 2 4) ∘
            Fin.succSuccAbove 1 3)).repr
      (ofRat fun b =>
        ∑ x,
          ∑ x_1,
            (if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        1 =
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          1 ≠
                        (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              0 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                1 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                2 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                0 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  1 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  2 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                        { fst := -1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  0 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                            (x_1, Fin.cast ⋯ x_1))).1
                                    1 =
                                  Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                            (x_1, Fin.cast ⋯ x_1))).1
                                    2 =
                                  Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                          { fst := 1, snd := 0 }
                        else { fst := 0, snd := 0 }) *
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        1 =
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          1 ≠
                        (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          2 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              0 =
                            Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                1 =
                              Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                2 =
                              Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                0 =
                              Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  1 =
                                Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  2 =
                                Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                        { fst := -1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  0 =
                                Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                            (x_1, Fin.cast ⋯ x_1))).2
                                    1 =
                                  Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                            (x_1, Fin.cast ⋯ x_1))).2
                                    2 =
                                  Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                          { fst := 1, snd := 0 }
                        else { fst := 0, snd := 0 }))
    b =
  ((Tensor.basis
          ((Fin.append ![Color.down, Color.downR, Color.downL] ![Color.down, Color.upL, Color.upR] ∘
              Fin.succSuccAbove 2 4) ∘
            Fin.succSuccAbove 1 3)).repr
      ((permT ![0, 1] ⋯)
        (ofRat
          (2 • fun f =>
            if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
            else if f 0 = f 1 then -1 else 0))))
    b
  conv_rhs => rw [permT_basis_repr_symm_apply]b:ComponentIdx
  ((Fin.append ![Color.down, Color.downR, Color.downL] ![Color.down, Color.upL, Color.upR] ∘ Fin.succSuccAbove 2 4) ∘
    Fin.succSuccAbove 1 3)| ((Tensor.basis ![Color.down, Color.down]).repr
    (ofRat
      (2 • fun f =>
        if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
        else if f 0 = f 1 then -1 else 0)))
  fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 1] ⋯ i))
  simp only [ofRat_basis_repr_apply]b:ComponentIdx
  ((Fin.append ![Color.down, Color.downR, Color.downL] ![Color.down, Color.upL, Color.upR] ∘ Fin.succSuccAbove 2 4) ∘
    Fin.succSuccAbove 1 3)⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ∑ x_1,
        (if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    0 =
                  Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      0 =
                    Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          1 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          2 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            1 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            2 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              0 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                1 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                2 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) *
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    0 =
                  Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      0 =
                    Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          1 =
                        Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          2 =
                        Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            1 =
                          Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            2 =
                          Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              0 =
                            Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                1 =
                              Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                2 =
                              Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) =
  Physlib.RatComplexNum.toComplexNum
    ((2 • fun f =>
        if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
        else if f 0 = f 1 then -1 else 0)
      fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 1] ⋯ i)))
  apply (Function.Injective.eq_iff Physlib.RatComplexNum.toComplexNum_injective).mprb:ComponentIdx
  ((Fin.append ![Color.down, Color.downR, Color.downL] ![Color.down, Color.upL, Color.upR] ∘ Fin.succSuccAbove 2 4) ∘
    Fin.succSuccAbove 1 3)⊢ (∑ x,
    ∑ x_1,
      (if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  0 =
                Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    0 =
                  Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      0 =
                    Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        1 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        2 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          1 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          2 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            1 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            2 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) *
        if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  0 =
                Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    0 =
                  Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    2 then
            { fst := -1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      0 =
                    Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        1 =
                      Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        2 =
                      Fin.cast pauliCo_eq_ofRat._proof_3 1 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          1 =
                        Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          2 =
                        Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            1 =
                          Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            2 =
                          Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) =
  (2 • fun f =>
      if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
      else if f 0 = f 1 then -1 else 0)
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 1] ⋯ i))
  revert b⊢ ∀
  (b :
    ComponentIdx
      ((Fin.append ![Color.down, Color.downR, Color.downL] ![Color.down, Color.upL, Color.upR] ∘
          Fin.succSuccAbove 2 4) ∘
        Fin.succSuccAbove 1 3)),
  (∑ x,
      ∑ x_1,
        (if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    0 =
                  Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      0 =
                    Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          1 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          2 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            1 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            2 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              0 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                1 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                2 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) *
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    0 =
                  Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      0 =
                    Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          1 =
                        Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          2 =
                        Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            1 =
                          Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            2 =
                          Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              0 =
                            Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                1 =
                              Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                2 =
                              Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) =
    (2 • fun f =>
        if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
        else if f 0 = f 1 then -1 else 0)
      fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 1] ⋯ i))
  decide +kernelAll goals completed! 🐙lemma pauliCo_trace_pauliCoDown: {σ_^^ | μ α β ⊗ σ___ | ν β α = 2 •ₜ η' | μ ν}ᵀ := by⊢ (contrT 2 1 3 ⋯) ((contrT 4 2 4 ⋯) ((prodT σ_^^) σ___)) = (permT ![0, 1] ⋯) (2 • η')
  conv_lhs =>
    rw [pauliCoDown_eq_ofRat, pauliCo_eq_ofRat]| (contrT 2 1 3 ⋯)
  ((contrT 4 2 4 ⋯)
    ((prodT
        (ofRat fun b =>
          if b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
          else
            if b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧ b 1 ≠ b 2 then { fst := -1, snd := 0 }
            else
              if
                  b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    b 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧ b 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      b 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧ b 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        b 1 = Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧ b 2 = Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        b 0 = Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          b 1 = Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧ b 2 = Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }))
      (ofRat fun b =>
        if b 0 = Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧ b 1 = b 2 then { fst := 1, snd := 0 }
        else
          if b 0 = Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧ b 1 ≠ b 2 then { fst := 1, snd := 0 }
          else
            if
                b 0 = Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                  b 1 = Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧ b 2 = Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
              { fst := 0, snd := -1 }
            else
              if
                  b 0 = Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                    b 1 = Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧ b 2 = Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                { fst := 0, snd := 1 }
              else
                if
                    b 0 = Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                      b 1 = Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                        b 2 = Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                  { fst := -1, snd := 0 }
                else
                  if
                      b 0 = Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                        b 1 = Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                          b 2 = Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 })))
    rw [prodT_ofRat_ofRat,
      contrT_ofRat, contrT_ofRat]| ofRat fun b =>
  ∑ x,
    ∑ x_1,
      (if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  0 =
                Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    0 =
                  Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    2 then
            { fst := -1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      0 =
                    Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        1 =
                      Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        2 =
                      Fin.cast pauliCo_eq_ofRat._proof_3 1 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          1 =
                        Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          2 =
                        Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            1 =
                          Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            2 =
                          Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) *
        if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  0 =
                Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    0 =
                  Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      0 =
                    Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        1 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        2 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          1 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          2 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            1 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            2 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }
  conv_rhs =>
    rw [coMetric_eq_ofRat]| (permT ![0, 1] ⋯)
  (2 •
    ofRat fun f =>
      if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
      else if f 0 = f 1 then -1 else 0)
    rw [← map_nsmul]| (permT ![0, 1] ⋯)
  (ofRat
    (2 • fun f =>
      if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
      else if f 0 = f 1 then -1 else 0))
  apply (Tensor.basis _).repr.injective⊢ (Tensor.basis
        ((Fin.append ![Color.down, Color.upL, Color.upR] ![Color.down, Color.downR, Color.downL] ∘
            Fin.succSuccAbove 2 4) ∘
          Fin.succSuccAbove 1 3)).repr
    (ofRat fun b =>
      ∑ x,
        ∑ x_1,
          (if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      0 =
                    Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            1 =
                          Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            2 =
                          Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              0 =
                            Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                1 =
                              Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                2 =
                              Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                      { fst := -1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                0 =
                              Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  1 =
                                Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  2 =
                                Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) *
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      0 =
                    Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            1 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            2 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              0 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                1 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                2 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                0 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  1 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  2 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                        { fst := 1, snd := 0 }
                      else { fst := 0, snd := 0 }) =
  (Tensor.basis
        ((Fin.append ![Color.down, Color.upL, Color.upR] ![Color.down, Color.downR, Color.downL] ∘
            Fin.succSuccAbove 2 4) ∘
          Fin.succSuccAbove 1 3)).repr
    ((permT ![0, 1] ⋯)
      (ofRat
        (2 • fun f =>
          if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
          else if f 0 = f 1 then -1 else 0)))
  ext bb:ComponentIdx
  ((Fin.append ![Color.down, Color.upL, Color.upR] ![Color.down, Color.downR, Color.downL] ∘ Fin.succSuccAbove 2 4) ∘
    Fin.succSuccAbove 1 3)⊢ ((Tensor.basis
          ((Fin.append ![Color.down, Color.upL, Color.upR] ![Color.down, Color.downR, Color.downL] ∘
              Fin.succSuccAbove 2 4) ∘
            Fin.succSuccAbove 1 3)).repr
      (ofRat fun b =>
        ∑ x,
          ∑ x_1,
            (if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        1 =
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          1 ≠
                        (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          2 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              0 =
                            Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                1 =
                              Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                2 =
                              Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                0 =
                              Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  1 =
                                Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  2 =
                                Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                        { fst := -1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).1
                                  0 =
                                Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                            (x_1, Fin.cast ⋯ x_1))).1
                                    1 =
                                  Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                            (x_1, Fin.cast ⋯ x_1))).1
                                    2 =
                                  Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                          { fst := 1, snd := 0 }
                        else { fst := 0, snd := 0 }) *
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        1 =
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          1 ≠
                        (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              0 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                1 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                2 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                0 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  1 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  2 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                        { fst := -1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                          (x_1, Fin.cast ⋯ x_1))).2
                                  0 =
                                Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                            (x_1, Fin.cast ⋯ x_1))).2
                                    1 =
                                  Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                            (x_1, Fin.cast ⋯ x_1))).2
                                    2 =
                                  Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                          { fst := 1, snd := 0 }
                        else { fst := 0, snd := 0 }))
    b =
  ((Tensor.basis
          ((Fin.append ![Color.down, Color.upL, Color.upR] ![Color.down, Color.downR, Color.downL] ∘
              Fin.succSuccAbove 2 4) ∘
            Fin.succSuccAbove 1 3)).repr
      ((permT ![0, 1] ⋯)
        (ofRat
          (2 • fun f =>
            if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
            else if f 0 = f 1 then -1 else 0))))
    b
  conv_rhs => rw [permT_basis_repr_symm_apply]b:ComponentIdx
  ((Fin.append ![Color.down, Color.upL, Color.upR] ![Color.down, Color.downR, Color.downL] ∘ Fin.succSuccAbove 2 4) ∘
    Fin.succSuccAbove 1 3)| ((Tensor.basis ![Color.down, Color.down]).repr
    (ofRat
      (2 • fun f =>
        if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
        else if f 0 = f 1 then -1 else 0)))
  fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 1] ⋯ i))
  simp only [ofRat_basis_repr_apply]b:ComponentIdx
  ((Fin.append ![Color.down, Color.upL, Color.upR] ![Color.down, Color.downR, Color.downL] ∘ Fin.succSuccAbove 2 4) ∘
    Fin.succSuccAbove 1 3)⊢ Physlib.RatComplexNum.toComplexNum
    (∑ x,
      ∑ x_1,
        (if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    0 =
                  Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      0 =
                    Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          1 =
                        Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          2 =
                        Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            1 =
                          Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            2 =
                          Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              0 =
                            Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                1 =
                              Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).1
                                2 =
                              Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) *
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    0 =
                  Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      0 =
                    Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          1 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          2 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            1 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            2 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              0 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                1 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                        (x_1, Fin.cast ⋯ x_1))).2
                                2 =
                              Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                      { fst := 1, snd := 0 }
                    else { fst := 0, snd := 0 }) =
  Physlib.RatComplexNum.toComplexNum
    ((2 • fun f =>
        if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
        else if f 0 = f 1 then -1 else 0)
      fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 1] ⋯ i)))
  apply (Function.Injective.eq_iff Physlib.RatComplexNum.toComplexNum_injective).mprb:ComponentIdx
  ((Fin.append ![Color.down, Color.upL, Color.upR] ![Color.down, Color.downR, Color.downL] ∘ Fin.succSuccAbove 2 4) ∘
    Fin.succSuccAbove 1 3)⊢ (∑ x,
    ∑ x_1,
      (if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  0 =
                Fin.cast pauliCo_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).1
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    0 =
                  Fin.cast pauliCo_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).1
                    2 then
            { fst := -1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).1
                      0 =
                    Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        1 =
                      Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        2 =
                      Fin.cast pauliCo_eq_ofRat._proof_3 1 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).1
                        0 =
                      Fin.cast pauliCo_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          1 =
                        Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          2 =
                        Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).1
                          0 =
                        Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            1 =
                          Fin.cast pauliCo_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            2 =
                          Fin.cast pauliCo_eq_ofRat._proof_3 0 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).1
                            0 =
                          Fin.cast pauliCo_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              1 =
                            Fin.cast pauliCo_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).1
                              2 =
                            Fin.cast pauliCo_eq_ofRat._proof_3 1 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) *
        if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  0 =
                Fin.cast pauliCoDown_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                          (x_1, Fin.cast ⋯ x_1))).2
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    0 =
                  Fin.cast pauliCoDown_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                            (x_1, Fin.cast ⋯ x_1))).2
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                              (x_1, Fin.cast ⋯ x_1))).2
                      0 =
                    Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        1 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        2 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                (x_1, Fin.cast ⋯ x_1))).2
                        0 =
                      Fin.cast pauliCoDown_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          1 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          2 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                  (x_1, Fin.cast ⋯ x_1))).2
                          0 =
                        Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            1 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            2 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_3 1 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                    (x_1, Fin.cast ⋯ x_1))).2
                            0 =
                          Fin.cast pauliCoDown_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              1 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x)))
                                      (x_1, Fin.cast ⋯ x_1))).2
                              2 =
                            Fin.cast pauliCoDown_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else { fst := 0, snd := 0 }) =
  (2 • fun f =>
      if f 0 = Fin.cast coMetric_eq_ofRat._proof_1 0 ∧ f 1 = Fin.cast coMetric_eq_ofRat._proof_2 0 then 1
      else if f 0 = f 1 then -1 else 0)
    fun i => (basisIdxCongr ⋯) (b (IsReindexing.inv ![0, 1] ⋯ i))
  decide +revert +kernelAll goals completed! 🐙lemma pauliContr_mul_pauliContrDown_add :
    {((σ^^^ | μ α β ⊗ σ^__ | ν β α') + (σ^^^ | ν α β ⊗ σ^__ | μ β α')) =
    2 •ₜ η | μ ν ⊗ δL | α α'}ᵀ := by⊢ (contrT 4 2 4 ⋯) ((prodT (Tensorial.toTensor σ)) σ^__) +
    (permT ![2, 1, 0, 3] ⋯) ((contrT 4 2 4 ⋯) ((prodT (Tensorial.toTensor σ)) σ^__)) =
  (permT ![0, 2, 1, 3] ⋯) (2 • (prodT η) δL)
  conv_lhs =>
    rw [pauliContrDown_ofRat, toTensor_eq_ofRat, prodT_ofRat_ofRat,
      contrT_ofRat, permT_ofRat, ← map_add]| ofRat
  ((fun b =>
      ∑ x,
        (if
              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                  Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                        Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                        Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
          if
              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                  Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                        Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                        Fin.cast pauliContrDown_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := -1, snd := 0 }
                    else 0) +
    fun b =>
    ∑ x,
      (if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).1
                  0 =
                Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).1
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).1
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    0 =
                  Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        1 =
                      Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        2 =
                      Fin.cast toTensor_eq_ofRat._proof_3 1 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          1 =
                        Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          2 =
                        Fin.cast toTensor_eq_ofRat._proof_3 0 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 1 then
                    { fst := -1, snd := 0 }
                  else 0) *
        if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).2
                  0 =
                Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).2
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).2
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    0 =
                  Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    2 then
            { fst := -1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        1 =
                      Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        2 =
                      Fin.cast pauliContrDown_ofRat._proof_3 1 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          1 =
                        Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          2 =
                        Fin.cast pauliContrDown_ofRat._proof_3 0 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 1 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else 0)
  conv_rhs =>
    rw [leftDualLeftUnit_eq_ofRat, contrMetric_eq_ofRat, prodT_ofRat_ofRat, ← map_nsmul,
      permT_ofRat]| ofRat fun b =>
  (2 • fun b =>
      (if
            (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
          1
        else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
        if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
    fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))
  apply (Tensor.basis _).repr.injective⊢ (Tensor.basis
        (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.up, Color.downR, Color.downL] ∘
          Fin.succSuccAbove 2 4)).repr
    (ofRat
      ((fun b =>
          ∑ x,
            (if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 0 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                                Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    1 =
                                  Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    2 =
                                  Fin.cast toTensor_eq_ofRat._proof_3 1 then
                          { fst := -1, snd := 0 }
                        else 0) *
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 1 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                                Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    1 =
                                  Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    2 =
                                  Fin.cast pauliContrDown_ofRat._proof_3 0 then
                          { fst := -1, snd := 0 }
                        else 0) +
        fun b =>
        ∑ x,
          (if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 1 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 0 then
                        { fst := -1, snd := 0 }
                      else 0)) =
  (Tensor.basis
        (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.up, Color.downR, Color.downL] ∘
          Fin.succSuccAbove 2 4)).repr
    (ofRat fun b =>
      (2 • fun b =>
          (if
                (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
              1
            else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
            if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
        fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i)))
  ext bb:ComponentIdx
  (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.up, Color.downR, Color.downL] ∘ Fin.succSuccAbove 2 4)⊢ ((Tensor.basis
          (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.up, Color.downR, Color.downL] ∘
            Fin.succSuccAbove 2 4)).repr
      (ofRat
        ((fun b =>
            ∑ x,
              (if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                            2 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 0 then
                        { fst := 0, snd := 1 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                                Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    1 =
                                  Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    2 =
                                  Fin.cast toTensor_eq_ofRat._proof_3 0 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    0 =
                                  Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                      1 =
                                    Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                                  (ComponentIdx.prod
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                      2 =
                                    Fin.cast toTensor_eq_ofRat._proof_3 1 then
                            { fst := -1, snd := 0 }
                          else 0) *
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                            2 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 1 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 0 then
                        { fst := 0, snd := -1 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                                Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    1 =
                                  Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    2 =
                                  Fin.cast pauliContrDown_ofRat._proof_3 1 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    0 =
                                  Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                      1 =
                                    Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                                  (ComponentIdx.prod
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                      2 =
                                    Fin.cast pauliContrDown_ofRat._proof_3 0 then
                            { fst := -1, snd := 0 }
                          else 0) +
          fun b =>
          ∑ x,
            (if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        1 =
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          1 ≠
                        (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 0 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  0 =
                                Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                            (x, Fin.cast ⋯ x))).1
                                    1 =
                                  Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                            (x, Fin.cast ⋯ x))).1
                                    2 =
                                  Fin.cast toTensor_eq_ofRat._proof_3 1 then
                          { fst := -1, snd := 0 }
                        else 0) *
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        1 =
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          1 ≠
                        (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          2 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 1 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  0 =
                                Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                            (x, Fin.cast ⋯ x))).2
                                    1 =
                                  Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                            (x, Fin.cast ⋯ x))).2
                                    2 =
                                  Fin.cast pauliContrDown_ofRat._proof_3 0 then
                          { fst := -1, snd := 0 }
                        else 0)))
    b =
  ((Tensor.basis
          (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.up, Color.downR, Color.downL] ∘
            Fin.succSuccAbove 2 4)).repr
      (ofRat fun b =>
        (2 • fun b =>
            (if
                  (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
                1
              else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
              if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
          fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))))
    b
  simp only [ofRat_basis_repr_apply]b:ComponentIdx
  (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.up, Color.downR, Color.downL] ∘ Fin.succSuccAbove 2 4)⊢ Physlib.RatComplexNum.toComplexNum
    (((fun b =>
          ∑ x,
            (if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 0 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                                Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    1 =
                                  Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    2 =
                                  Fin.cast toTensor_eq_ofRat._proof_3 1 then
                          { fst := -1, snd := 0 }
                        else 0) *
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 1 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                                Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    1 =
                                  Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    2 =
                                  Fin.cast pauliContrDown_ofRat._proof_3 0 then
                          { fst := -1, snd := 0 }
                        else 0) +
        fun b =>
        ∑ x,
          (if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 1 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 0 then
                        { fst := -1, snd := 0 }
                      else 0)
      b) =
  Physlib.RatComplexNum.toComplexNum
    ((2 • fun b =>
        (if
              (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
            1
          else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
          if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
      fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i)))
  apply (Function.Injective.eq_iff Physlib.RatComplexNum.toComplexNum_injective).mprb:ComponentIdx
  (Fin.append ![Color.up, Color.upL, Color.upR] ![Color.up, Color.downR, Color.downL] ∘ Fin.succSuccAbove 2 4)⊢ ((fun b =>
        ∑ x,
          (if
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 1 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 0 then
                        { fst := -1, snd := 0 }
                      else 0) +
      fun b =>
      ∑ x,
        (if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    0 =
                  Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          1 =
                        Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          2 =
                        Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else 0) *
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    0 =
                  Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          1 =
                        Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          2 =
                        Fin.cast pauliContrDown_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := -1, snd := 0 }
                    else 0)
    b =
  (2 • fun b =>
      (if
            (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
          1
        else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
        if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
    fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))
  decide +revert +kernelAll goals completed! 🐙lemma auliContrDown_pauliContr_mul_add :
    {((σ^__ | μ β α ⊗ σ^^^ | ν α β') + (σ^__ | ν β α ⊗ σ^^^ | μ α β')) =
    2 •ₜ η | μ ν ⊗ δR' | β β'}ᵀ := by⊢ (contrT 4 2 4 ⋯) ((prodT σ^__) (Tensorial.toTensor σ)) +
    (permT ![2, 1, 0, 3] ⋯) ((contrT 4 2 4 ⋯) ((prodT σ^__) (Tensorial.toTensor σ))) =
  (permT ![0, 2, 1, 3] ⋯) (2 • (prodT η) δR')
  conv_lhs =>
    rw [pauliContrDown_ofRat, toTensor_eq_ofRat, prodT_ofRat_ofRat,
      contrT_ofRat, permT_ofRat, ← map_add]| ofRat
  ((fun b =>
      ∑ x,
        (if
              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                  Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                        Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                        Fin.cast pauliContrDown_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := -1, snd := 0 }
                    else 0) *
          if
              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                  Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                        Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                        Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else 0) +
    fun b =>
    ∑ x,
      (if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).1
                  0 =
                Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).1
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).1
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    0 =
                  Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    2 then
            { fst := -1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        1 =
                      Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        2 =
                      Fin.cast pauliContrDown_ofRat._proof_3 1 then
              { fst := 0, snd := 1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          1 =
                        Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          2 =
                        Fin.cast pauliContrDown_ofRat._proof_3 0 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 1 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 0 then
                    { fst := -1, snd := 0 }
                  else 0) *
        if
            (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).2
                  0 =
                Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).2
                  1 =
                (ComponentIdx.prod
                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                          (x, Fin.cast ⋯ x))).2
                  2 then
          { fst := 1, snd := 0 }
        else
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    0 =
                  Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    1 ≠
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        1 =
                      Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        2 =
                      Fin.cast toTensor_eq_ofRat._proof_3 1 then
              { fst := 0, snd := -1 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          1 =
                        Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          2 =
                        Fin.cast toTensor_eq_ofRat._proof_3 0 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 1 then
                    { fst := -1, snd := 0 }
                  else 0)
  conv_rhs =>
    rw [dualRightRightUnit_eq_ofRat, contrMetric_eq_ofRat, prodT_ofRat_ofRat, ← map_nsmul,
      permT_ofRat]| ofRat fun b =>
  (2 • fun b =>
      (if
            (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
          1
        else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
        if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
    fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))
  apply (Tensor.basis _).repr.injective⊢ (Tensor.basis
        (Fin.append ![Color.up, Color.downR, Color.downL] ![Color.up, Color.upL, Color.upR] ∘
          Fin.succSuccAbove 2 4)).repr
    (ofRat
      ((fun b =>
          ∑ x,
            (if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 1 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                                Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    1 =
                                  Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    2 =
                                  Fin.cast pauliContrDown_ofRat._proof_3 0 then
                          { fst := -1, snd := 0 }
                        else 0) *
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 0 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                                Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    1 =
                                  Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    2 =
                                  Fin.cast toTensor_eq_ofRat._proof_3 1 then
                          { fst := -1, snd := 0 }
                        else 0) +
        fun b =>
        ∑ x,
          (if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 0 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 1 then
                        { fst := -1, snd := 0 }
                      else 0)) =
  (Tensor.basis
        (Fin.append ![Color.up, Color.downR, Color.downL] ![Color.up, Color.upL, Color.upR] ∘
          Fin.succSuccAbove 2 4)).repr
    (ofRat fun b =>
      (2 • fun b =>
          (if
                (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
              1
            else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
            if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
        fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i)))
  ext bb:ComponentIdx
  (Fin.append ![Color.up, Color.downR, Color.downL] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 2 4)⊢ ((Tensor.basis
          (Fin.append ![Color.up, Color.downR, Color.downL] ![Color.up, Color.upL, Color.upR] ∘
            Fin.succSuccAbove 2 4)).repr
      (ofRat
        ((fun b =>
            ∑ x,
              (if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                            2 then
                    { fst := -1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 1 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 0 then
                        { fst := 0, snd := -1 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                                Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    1 =
                                  Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    2 =
                                  Fin.cast pauliContrDown_ofRat._proof_3 1 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    0 =
                                  Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                      1 =
                                    Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                                  (ComponentIdx.prod
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                      2 =
                                    Fin.cast pauliContrDown_ofRat._proof_3 0 then
                            { fst := -1, snd := 0 }
                          else 0) *
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                            2 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 0 then
                        { fst := 0, snd := 1 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                                Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    1 =
                                  Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    2 =
                                  Fin.cast toTensor_eq_ofRat._proof_3 0 then
                          { fst := 1, snd := 0 }
                        else
                          if
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    0 =
                                  Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                      1 =
                                    Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                                  (ComponentIdx.prod
                                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                      2 =
                                    Fin.cast toTensor_eq_ofRat._proof_3 1 then
                            { fst := -1, snd := 0 }
                          else 0) +
          fun b =>
          ∑ x,
            (if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        1 =
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          1 ≠
                        (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          2 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 1 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  0 =
                                Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                            (x, Fin.cast ⋯ x))).1
                                    1 =
                                  Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                            (x, Fin.cast ⋯ x))).1
                                    2 =
                                  Fin.cast pauliContrDown_ofRat._proof_3 0 then
                          { fst := -1, snd := 0 }
                        else 0) *
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        1 =
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          1 ≠
                        (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 0 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  0 =
                                Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                            (x, Fin.cast ⋯ x))).2
                                    1 =
                                  Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod
                                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                            (x, Fin.cast ⋯ x))).2
                                    2 =
                                  Fin.cast toTensor_eq_ofRat._proof_3 1 then
                          { fst := -1, snd := 0 }
                        else 0)))
    b =
  ((Tensor.basis
          (Fin.append ![Color.up, Color.downR, Color.downL] ![Color.up, Color.upL, Color.upR] ∘
            Fin.succSuccAbove 2 4)).repr
      (ofRat fun b =>
        (2 • fun b =>
            (if
                  (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
                1
              else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
              if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
          fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))))
    b
  simp only [ofRat_basis_repr_apply]b:ComponentIdx
  (Fin.append ![Color.up, Color.downR, Color.downL] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 2 4)⊢ Physlib.RatComplexNum.toComplexNum
    (((fun b =>
          ∑ x,
            (if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                  { fst := -1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := 0, snd := -1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 1 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                                Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    1 =
                                  Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                    2 =
                                  Fin.cast pauliContrDown_ofRat._proof_3 0 then
                          { fst := -1, snd := 0 }
                        else 0) *
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                  { fst := 1, snd := 0 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 1 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 0, snd := 1 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 0 then
                        { fst := 1, snd := 0 }
                      else
                        if
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                                Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    1 =
                                  Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                    2 =
                                  Fin.cast toTensor_eq_ofRat._proof_3 1 then
                          { fst := -1, snd := 0 }
                        else 0) +
        fun b =>
        ∑ x,
          (if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 0 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      1 =
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        1 ≠
                      (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod
                                      ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                            Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                          (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 1 then
                        { fst := -1, snd := 0 }
                      else 0)
      b) =
  Physlib.RatComplexNum.toComplexNum
    ((2 • fun b =>
        (if
              (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
            1
          else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
          if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
      fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i)))
  apply (Function.Injective.eq_iff Physlib.RatComplexNum.toComplexNum_injective).mprb:ComponentIdx
  (Fin.append ![Color.up, Color.downR, Color.downL] ![Color.up, Color.upL, Color.upR] ∘ Fin.succSuccAbove 2 4)⊢ ((fun b =>
        ∑ x,
          (if
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 ≠
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 then
                { fst := -1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 1 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 0 then
                    { fst := 0, snd := -1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 1 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 0 =
                              Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1 1 =
                                Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).1
                                  2 =
                                Fin.cast pauliContrDown_ofRat._proof_3 0 then
                        { fst := -1, snd := 0 }
                      else 0) *
            if
                (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 ≠
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 then
                { fst := 1, snd := 0 }
              else
                if
                    (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 1 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 0, snd := 1 }
                  else
                    if
                        (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 0 then
                      { fst := 1, snd := 0 }
                    else
                      if
                          (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 0 =
                              Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                            (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2 1 =
                                Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                              (ComponentIdx.prod ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ b) (x, Fin.cast ⋯ x))).2
                                  2 =
                                Fin.cast toTensor_eq_ofRat._proof_3 1 then
                        { fst := -1, snd := 0 }
                      else 0) +
      fun b =>
      ∑ x,
        (if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    0 =
                  Fin.cast pauliContrDown_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).1
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      0 =
                    Fin.cast pauliContrDown_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).1
                      2 then
              { fst := -1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).1
                        0 =
                      Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          1 =
                        Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          2 =
                        Fin.cast pauliContrDown_ofRat._proof_3 1 then
                { fst := 0, snd := 1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).1
                          0 =
                        Fin.cast pauliContrDown_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            1 =
                          Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            2 =
                          Fin.cast pauliContrDown_ofRat._proof_3 0 then
                  { fst := 0, snd := -1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).1
                            0 =
                          Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              1 =
                            Fin.cast pauliContrDown_ofRat._proof_2 1 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              2 =
                            Fin.cast pauliContrDown_ofRat._proof_3 1 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).1
                              0 =
                            Fin.cast pauliContrDown_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                1 =
                              Fin.cast pauliContrDown_ofRat._proof_2 0 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).1
                                2 =
                              Fin.cast pauliContrDown_ofRat._proof_3 0 then
                      { fst := -1, snd := 0 }
                    else 0) *
          if
              (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    0 =
                  Fin.cast toTensor_eq_ofRat._proof_1 0 ∧
                (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    1 =
                  (ComponentIdx.prod
                        ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                              Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                            (x, Fin.cast ⋯ x))).2
                    2 then
            { fst := 1, snd := 0 }
          else
            if
                (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      0 =
                    Fin.cast toTensor_eq_ofRat._proof_1 1 ∧
                  (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      1 ≠
                    (ComponentIdx.prod
                          ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                              (x, Fin.cast ⋯ x))).2
                      2 then
              { fst := 1, snd := 0 }
            else
              if
                  (ComponentIdx.prod
                            ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                  Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                (x, Fin.cast ⋯ x))).2
                        0 =
                      Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          1 =
                        Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                      (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          2 =
                        Fin.cast toTensor_eq_ofRat._proof_3 1 then
                { fst := 0, snd := -1 }
              else
                if
                    (ComponentIdx.prod
                              ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                    Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                  (x, Fin.cast ⋯ x))).2
                          0 =
                        Fin.cast toTensor_eq_ofRat._proof_1 2 ∧
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            1 =
                          Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                        (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            2 =
                          Fin.cast toTensor_eq_ofRat._proof_3 0 then
                  { fst := 0, snd := 1 }
                else
                  if
                      (ComponentIdx.prod
                                ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                      Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                    (x, Fin.cast ⋯ x))).2
                            0 =
                          Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              1 =
                            Fin.cast toTensor_eq_ofRat._proof_2 0 ∧
                          (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              2 =
                            Fin.cast toTensor_eq_ofRat._proof_3 0 then
                    { fst := 1, snd := 0 }
                  else
                    if
                        (ComponentIdx.prod
                                  ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                        Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                      (x, Fin.cast ⋯ x))).2
                              0 =
                            Fin.cast toTensor_eq_ofRat._proof_1 3 ∧
                          (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                1 =
                              Fin.cast toTensor_eq_ofRat._proof_2 1 ∧
                            (ComponentIdx.prod
                                    ↑((ComponentIdx.DropPairSection.ofFinEquiv ⋯ fun i =>
                                          Fin.cast ⋯ (b (IsReindexing.inv ![2, 1, 0, 3] ⋯ i)))
                                        (x, Fin.cast ⋯ x))).2
                                2 =
                              Fin.cast toTensor_eq_ofRat._proof_3 1 then
                      { fst := -1, snd := 0 }
                    else 0)
    b =
  (2 • fun b =>
      (if
            (ComponentIdx.prod b).1 0 = Fin.cast contrMetric_eq_ofRat._proof_1 0 ∧
              (ComponentIdx.prod b).1 1 = Fin.cast contrMetric_eq_ofRat._proof_2 0 then
          1
        else if (ComponentIdx.prod b).1 0 = (ComponentIdx.prod b).1 1 then -1 else 0) *
        if (ComponentIdx.prod b).2 0 = (ComponentIdx.prod b).2 1 then 1 else 0)
    fun i => Fin.cast ⋯ (b (IsReindexing.inv ![0, 2, 1, 3] ⋯ i))
  decide +revert +kernelAll goals completed! 🐙