`fromComplex`

definition

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module: IndisputableMonolith.Foundation.ComplexFromLogic
domain: Foundation
line: 38 · github
papers citing: none yet

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IndisputableMonolith.Foundation.ComplexFromLogic on GitHub at line 38.

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depends on

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formal source

  35  ⟨toReal z.re, toReal z.im⟩
  36
  37/-- Transport a Mathlib complex number to the recovered complex line. -/
  38def fromComplex (z : ℂ) : LogicComplex where
  39  re := fromReal z.re
  40  im := fromReal z.im
  41
  42@[simp] theorem toComplex_re (z : LogicComplex) :
  43    (toComplex z).re = toReal z.re := rfl
  44
  45@[simp] theorem toComplex_im (z : LogicComplex) :
  46    (toComplex z).im = toReal z.im := rfl
  47
  48@[simp] theorem toComplex_fromComplex (z : ℂ) :
  49    toComplex (fromComplex z) = z := by
  50  apply Complex.ext <;> simp [toComplex, fromComplex, toReal_fromReal]
  51
  52@[simp] theorem fromComplex_toComplex (z : LogicComplex) :
  53    fromComplex (toComplex z) = z := by
  54  cases z with
  55  | mk re im =>
  56    simp [toComplex, fromComplex, fromReal_toReal]
  57
  58/-- Carrier equivalence between recovered complex numbers and Mathlib `ℂ`. -/
  59def equivComplex : LogicComplex ≃ ℂ where
  60  toFun := toComplex
  61  invFun := fromComplex
  62  left_inv := fromComplex_toComplex
  63  right_inv := toComplex_fromComplex
  64
  65/-- Equality transfer for recovered complex numbers. -/
  66theorem eq_iff_toComplex_eq {z w : LogicComplex} :
  67    z = w ↔ toComplex z = toComplex w := by
  68  constructor

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