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Project-declaredLean 4.24.0-rc1 · mathlib@aad26963

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 acc0

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
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

Project-declaredLean 4.24.0-rc1

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.

automata theoryformal languagescomputer science

Source project: Automata Theory

Person-level attribution pending.

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Project-declaredLean 4.24.0-rc1

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.

automata theoryformal languagescomputer science

Source project: Automata Theory

Person-level attribution pending.

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Project-declaredLean 4.24.0-rc1

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.

automata theoryformal languagescomputer science

Source project: Automata Theory

Person-level attribution pending.

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