Foundation Complex From Logic Eq Iff To Complex Eq

Equality by transport

The complex numbers are the usual extension of the real line where every polynomial has a root, written as a pair of real coordinates (a, b) and combined by the rule (a, b) × (c, d) = (ac − bd, ad + bc). The framework builds its own complex carrier, LogicComplex, as pairs of its recovered reals, then proves a transport theorem: two such recovered complex numbers are equal if and only if their images under the standard map to the usual complex numbers are equal. The declaration eq_iff_toComplex_eq states this equivalence directly.

The proof is short and mechanical. If the recovered numbers are equal, applying the transport map gives equal standard complex numbers. Conversely, if their standard images are equal, applying the inverse map and using the round-trip identities recovers the original equality. The two round-trip properties, fromComplex (toComplex z) = z and toComplex (fromComplex z) = z, are proved separately and together form an equivalence of carriers.

In Recognition Science, the recovered reals come from a forced recognition cost, and this theorem extends that construction to the complex plane. The framework models the recovered complex numbers as a faithful copy of the standard complex numbers, not a new analytic structure. The equivalence means any statement about the standard complex numbers can be transported to the recovered ones, and vice versa, without changing truth values.

The theorem does not develop complex analysis. It does not prove the fundamental theorem of algebra, Cauchy's theorem, or any property of holomorphic functions. It only establishes that the carrier sets and their equality relations match. Later modules can state when they work in the standard complex numbers via this equivalence, but the analytic content itself remains to be built.

/-- Equality transfer for recovered complex numbers. -/
theorem eq_iff_toComplex_eq {z w : LogicComplex} :
    z = w ↔ toComplex z = toComplex w := by
  constructor
  · exact congrArg toComplex
  · intro h
    have := congrArg fromComplex h
    rw [fromComplex_toComplex, fromComplex_toComplex] at this
    exact this

/-- 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

/-- 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

No complex analysis is developed in this file. No theorem about holomorphy or contour integration is proved here. The recovered complex numbers are not a new analytic structure, only a faithful copy.

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)

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