Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: case inr.«0»
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
hx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)
w : { w // w.IsComplex }
⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) (Sum.inr (w, (fun i ↦ i) ⟨0, ⋯⟩))) =
    ∑ x_1,
        fromBlocks 1 0 0 ((blockDiagonal fun x ↦ !![2⁻¹, 2⁻¹; 2⁻¹ * -I, 2⁻¹ * I]).submatrix Prod.swap Prod.swap)
            (Sum.inr (w, (fun i ↦ i) ⟨0, ⋯⟩)) (Sum.inl x_1) *
          x (↑x_1).embedding +
      ∑ x_1,
        (fromBlocks 1 0 0 ((blockDiagonal fun x ↦ !![2⁻¹, 2⁻¹; 2⁻¹ * -I, 2⁻¹ * I]).submatrix Prod.swap Prod.swap)
              (Sum.inr (w, (fun i ↦ i) ⟨0, ⋯⟩)) (Sum.inr (x_1, 0)) *
            x (↑x_1).embedding +
          fromBlocks 1 0 0 ((blockDiagonal fun x ↦ !![2⁻¹, 2⁻¹; 2⁻¹ * -I, 2⁻¹ * I]).submatrix Prod.swap Prod.swap)
              (Sum.inr (w, (fun i ↦ i) ⟨0, ⋯⟩)) (Sum.inr (x_1, 1)) *
            x (ComplexEmbedding.conjugate (↑x_1).embedding))

pp: case inr.«1»
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
hx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)
w : { w // w.IsComplex }
⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) (Sum.inr (w, (fun i ↦ i) ⟨1, ⋯⟩))) =
    ∑ x_1,
        fromBlocks 1 0 0 ((blockDiagonal fun x ↦ !![2⁻¹, 2⁻¹; 2⁻¹ * -I, 2⁻¹ * I]).submatrix Prod.swap Prod.swap)
            (Sum.inr (w, (fun i ↦ i) ⟨1, ⋯⟩)) (Sum.inl x_1) *
          x (↑x_1).embedding +
      ∑ x_1,
        (fromBlocks 1 0 0 ((blockDiagonal fun x ↦ !![2⁻¹, 2⁻¹; 2⁻¹ * -I, 2⁻¹ * I]).submatrix Prod.swap Prod.swap)
              (Sum.inr (w, (fun i ↦ i) ⟨1, ⋯⟩)) (Sum.inr (x_1, 0)) *
            x (↑x_1).embedding +
          fromBlocks 1 0 0 ((blockDiagonal fun x ↦ !![2⁻¹, 2⁻¹; 2⁻¹ * -I, 2⁻¹ * I]).submatrix Prod.swap Prod.swap)
              (Sum.inr (w, (fun i ↦ i) ⟨1, ⋯⟩)) (Sum.inr (x_1, 1)) *
            x (ComplexEmbedding.conjugate (↑x_1).embedding))

Tactic:
· simp only [Fin.zero_eta, Fin.isValue, stdBasis_apply_isComplex_fst, re_eq_add_conj,
        mul_neg, fromBlocks_apply₂₁, Matrix.zero_apply, zero_mul, sum_const_zero,
        fromBlocks_apply₂₂, submatrix_apply, Prod.swap_prod_mk, blockDiagonal_apply, of_apply,
        cons_val', cons_val_zero, empty_val', cons_val_fin_one, ite_mul, cons_val_one,
        sum_add_distrib, sum_ite_eq, mem_univ, ↓reduceIte, ← hx (embedding w), zero_add]
      ring

Goal state after:
case inr.«1»
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
hx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)
w : { w // w.IsComplex }
⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) (Sum.inr (w, (fun i ↦ i) ⟨1, ⋯⟩))) =
    ∑ x_1,
        fromBlocks 1 0 0 ((blockDiagonal fun x ↦ !![2⁻¹, 2⁻¹; 2⁻¹ * -I, 2⁻¹ * I]).submatrix Prod.swap Prod.swap)
            (Sum.inr (w, (fun i ↦ i) ⟨1, ⋯⟩)) (Sum.inl x_1) *
          x (↑x_1).embedding +
      ∑ x_1,
        (fromBlocks 1 0 0 ((blockDiagonal fun x ↦ !![2⁻¹, 2⁻¹; 2⁻¹ * -I, 2⁻¹ * I]).submatrix Prod.swap Prod.swap)
              (Sum.inr (w, (fun i ↦ i) ⟨1, ⋯⟩)) (Sum.inr (x_1, 0)) *
            x (↑x_1).embedding +
          fromBlocks 1 0 0 ((blockDiagonal fun x ↦ !![2⁻¹, 2⁻¹; 2⁻¹ * -I, 2⁻¹ * I]).submatrix Prod.swap Prod.swap)
              (Sum.inr (w, (fun i ↦ i) ⟨1, ⋯⟩)) (Sum.inr (x_1, 1)) *
            x (ComplexEmbedding.conjugate (↑x_1).embedding))