Acc lang concat e
Automata.acc_lang_concat_e
Plain-language statement
The language of the concatenation NA that is accepted by M0's accepting states is the language accepted by M0.
Exact Lean statement
theorem acc_lang_concat_e :
(M0.Concat acc0 M1).AcceptedLang (inl '' acc0) = M0.AcceptedLang acc0Formal artifact
Lean source
theorem acc_lang_concat_e : (M0.Concat acc0 M1).AcceptedLang (inl '' acc0) = M0.AcceptedLang acc0 := by ext al ; constructor · rintro ⟨m, as, ⟨ss, h_run, h_acc⟩, h_al⟩ have h_m : ∃ s0, ss m = inl s0 := by obtain ⟨s0, _, h_s0⟩ := h_acc use s0 ; simp [h_s0] obtain ⟨ss0, h_run0,h_ss0⟩ := na_concat_fin_run_0.mp ⟨h_run, h_m⟩ use m, as ; simp [h_al] use ss0 ; simp [h_run0] obtain ⟨sm, h_sm_acc, h_sm⟩ := h_acc rw [h_ss0 m (by omega), inl.inj_iff] at h_sm simpa [← h_sm] · rintro ⟨m, as, ⟨ss0, h_run0, h_acc0⟩, h_al⟩ use m, as ; simp [h_al] let ss : Stream' (M0.Concat acc0 M1).State := inl ∘ ss0 use ss ; constructor · suffices (M0.Concat acc0 M1).FinRun m as ss ∧ (∃ s0, ss m = inl s0) by tauto apply na_concat_fin_run_0.mpr use ss0 ; simp [h_run0] intro k h_k ; simp [ss] · simp [ss] ; use (ss0 m)- Project
- Automata Theory
- License
- Apache-2.0
- Commit
- f196548710ce
- Source
- AutomataTheory/Automata/Concat.lean:287-308
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Related declarations
Acc lang congr
acc_lang_congr
Plain-language statement
The language accepted by c.toDA with a unique accepting state s is exactly the equivalence class of s.
Source project: Automata Theory
Person-level attribution pending.
Acc lang concat ne
Automata.acc_lang_concat_ne
Plain-language statement
The language of the concatenation NA that is accepted by M1's accepting states is the language accepted by M0 concatenated with the language accepted by M1 minus the empty word.
Source project: Automata Theory
Person-level attribution pending.
Acc lang inter
Automata.acc_lang_inter
Plain-language statement
The language accepted by the product NA is the intersection of the languages accepted by the component automata.
Source project: Automata Theory
Person-level attribution pending.