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Project-declaredLean 4.33.0-rc1 · mathlib@169c26b52a38

Is Regular mul

Cslib.Language.IsRegular.mul

Plain-language statement

The concatenation of two regular languages is regular.

Exact Lean statement

@[simp]
theorem IsRegular.mul {l1 l2 : Language Symbol}
    (h1 : l1.IsRegular) (h2 : l2.IsRegular) : (l1 * l2).IsRegular

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
@[simp]theorem IsRegular.mul {l1 l2 : Language Symbol}    (h1 : l1.IsRegular) (h2 : l2.IsRegular) : (l1 * l2).IsRegular := by  obtain (he | hne) := isEmpty_or_nonempty Symbol  · obtain (rfl | rfl) := Language.eq_zero_or_one_ofIsEmpty l1 <;>    obtain (rfl | rfl) := Language.eq_zero_or_one_ofIsEmpty l2 <;> simp  · have := Classical.inhabited_of_nonempty hne    rw [IsRegular.iff_nfa] at h1 h2     obtain State1, h_fin1, nfa1, rfl := h1    obtain State2, h_fin1, nfa2, rfl := h2    use Option State1Option State2, inferInstance,      finConcat nfa1 nfa2, inr '' (some '' nfa2.accept)    exact finConcat_language_eq
Project
Lean Computer Science Library
License
Apache-2.0
Commit
f36649cff2c9
Source
Cslib/Computability/Languages/RegularLanguage.lean:156-168

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Related declarations

Project-declaredLean 4.33.0-rc1

Unique minimal

Cslib.Automata.DA.FinAcc.unique_minimal

Plain-language statement

The minimal DFA M accepting the language l is unique up to unique isomorphism.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

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Project-declaredLean 4.33.0-rc1

Buchi Family cover

Cslib.Automata.NA.Buchi.buchiFamily_cover

Project documentation

na.buchiFamily is a cover if na has only finitely many states. This theorem uses the Ramsey theorem for infinite graphs and does not depend on any details of na.BuchiCongruence other than that it is of finite index.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

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