Concat language eq
Cslib.Automata.NA.Buchi.concat_language_eq
Plain-language statement
The Buchi automaton formed from concat na1 na2 accepts the ω-language that is the concatenation of the language of na1 and the ω-language of na2.
Exact Lean statement
@[simp]
theorem concat_language_eq {acc2 : Set State2} :
language (Buchi.mk (concat na1 na2) (inr '' acc2)) =
language na1 * language (Buchi.mk na2 acc2)Formal artifact
Lean source
@[simp]theorem concat_language_eq {acc2 : Set State2} : language (Buchi.mk (concat na1 na2) (inr '' acc2)) = language na1 * language (Buchi.mk na2 acc2) := by ext xs constructor · rintro ⟨ss, h_run, h_acc⟩ obtain ⟨k, h_k⟩ : ∃ k, (ss k).isRight := by grind [Frequently.exists h_acc] obtain ⟨n, _, h_acc1, ss2, h_run2, h_map2⟩ := concat_run_proj h_run h_k use xs.take n, h_acc1, xs.drop n, ?_, by simp use ss2, h_run2 grind [(drop_frequently_iff_frequently n).mpr h_acc] · rintro ⟨xs1, h_xs1, xs2, ⟨ss2, h_run2, h_acc2⟩, rfl⟩ obtain ⟨ss, h_run, h_map⟩ := concat_run_exists h_xs1 h_run2 use ss, h_run apply (drop_frequently_iff_frequently xs1.length).mp grind- Project
- Lean Computer Science Library
- License
- Apache-2.0
- Commit
- f36649cff2c9
- Source
- Cslib/Computability/Automata/NA/Concat.lean:124-140
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Person-level attribution pending.
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Person-level attribution pending.
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Plain-language statement
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Person-level attribution pending.