Lax Lean Archive

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Proof of `Regular functions are continuous`

groundedproofs/Lax916827Proofs/Results.lean · lax-916827

What this proof establishes

no assumptions

thm✓Lax916827.RegularContinuity

Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.

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In the paper

Description

Regular functions are continuous (Theorem C.1.1, the continuity half): induction on the composition tree, with Theorem B.1.5 for the rational primes and Lemmas C.1.2–C.1.3 for map reverse and map duplicate (Transducers.regularcontinuousTransducers.regular_continuous).

Proof strategy

The concept's IsRegularFunIsRegularFun is the source's through isRegularFuniffisRegularFun_iff (Part A's compClosureiffcompClosure_iff on the family, Part B's isRationalFuniffisRationalFun_iff and the map lifting bridge for the primes); ContinuousContinuous unfolds identically.

Attribution

Theorem C.1.1 of Transducers, Part C; formalised by Aristotle (Harmonic), PartC/Statements.leanPartC/Statements.lean, PartC/ContAux.leanPartC/ContAux.lean.