Foundation Complex From Logic Of Logic Rat

Recovered complex numbers

Complex numbers are pairs of real numbers with a multiplication rule that gives the imaginary unit i its defining property i squared equals negative one. They are the natural arena for waves, rotations, and the algebra of oscillation. The standard complex plane is denoted ℂ, and its points are written as a + bi, where a and b are ordinary real numbers.

The framework's declaration recognition, the discrete record of events from which its mathematics grows, builds a recovered complex number as a pair of recovered reals, one for the real part and one for the imaginary part. Two functions move between this recovered carrier and Mathlib's standard ℂ: one sends a recovered pair to its ordinary complex counterpart, and the other sends an ordinary complex number back to a recovered pair. The library proves these two functions are mutual inverses, so the recovered complex numbers and the standard complex numbers are the same collection of objects under a perfect translation.

That equivalence is not a loose analogy. The machine-checked library of formal theorems proves the recovered complex numbers carry all the usual algebraic operations, addition, negation, subtraction, multiplication, inverse, and division, and that each operation matches its standard counterpart exactly. The recovered real line embeds into the recovered complex numbers as the horizontal axis, and recovered rational numbers embed through the reals into the same axis. The theorem logicComplex_recovered_from_mathlib states the round trip in both directions returns the starting point, which is the precise sense in which the recovered complex plane is the ordinary one.

What the declaration does not claim is just as important. It does not redevelop complex analysis, the theory of holomorphic functions, contour integrals, or Cauchy's theorem. The file explicitly says later analytic modules will state when they use standard ℂ through this equivalence. The recovered complex numbers are a carrier, a stage, not a new theory of analytic functions. The framework proves the stage is identical to the familiar one, and then hands the stage to future work.

/-- Carrier equivalence between recovered complex numbers and Mathlib `ℂ`. -/
def equivComplex : LogicComplex ≃ ℂ where
  toFun := toComplex
  invFun := fromComplex
  left_inv := fromComplex_toComplex
  right_inv := toComplex_fromComplex

@[simp] theorem toComplex_add (z w : LogicComplex) :
    toComplex (z + w) = toComplex z + toComplex w := by
  simp [HAdd.hAdd, Add.add]

@[simp] theorem toComplex_mul (z w : LogicComplex) :
    toComplex (z * w) = toComplex z * toComplex w := by
  simp [HMul.hMul, Mul.mul]

/-- The recovered complex carrier is exactly Mathlib `ℂ`, by transport. -/
theorem logicComplex_recovered_from_mathlib :
    (∀ z : LogicComplex, fromComplex (toComplex z) = z) ∧
    (∀ z : ℂ, toComplex (fromComplex z) = z) :=
  ⟨fromComplex_toComplex, toComplex_fromComplex⟩

What this page does not claim

The declaration does not redevelop complex analysis, including holomorphy and contour integration. The declaration does not prove any new property of the complex numbers beyond their identity with the standard carrier. The declaration does not derive the recovered real line from the recognition ledger; it takes that line as given.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/ComplexFromLogic.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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