| import OAI.Computability.StarHeight.Model | |
| namespace OAI | |
| namespace GeneralizedStarHeight | |
| universe u | |
| universe uι uι2 | |
| namespace HasHeightAtMost | |
| open scoped Computability | |
| variable {Alphabet : Type u} {L M : Language Alphabet} {n k : ℕ} | |
| lemma mono (h : HasHeightAtMost L n) (hn : n ≤ k) : HasHeightAtMost L k := by | |
| obtain ⟨P, rfl, hP⟩ := h | |
| exact ⟨P, rfl, hP.trans hn⟩ | |
| lemma zero (n : ℕ) : HasHeightAtMost (0 : Language Alphabet) n := | |
| ⟨Expression.zero, rfl, Nat.zero_le _⟩ | |
| lemma one (n : ℕ) : HasHeightAtMost (1 : Language Alphabet) n := | |
| ⟨Expression.one, rfl, Nat.zero_le _⟩ | |
| lemma letter (a : Alphabet) (n : ℕ) : HasHeightAtMost ({[a]} : Language Alphabet) n := | |
| ⟨Expression.letter a, rfl, Nat.zero_le _⟩ | |
| lemma union (hL : HasHeightAtMost L n) (hM : HasHeightAtMost M n) : | |
| HasHeightAtMost (L + M) n := by | |
| obtain ⟨P, rfl, hP⟩ := hL | |
| obtain ⟨Q, rfl, hQ⟩ := hM | |
| exact ⟨P.union Q, rfl, max_le hP hQ⟩ | |
| lemma concat (hL : HasHeightAtMost L n) (hM : HasHeightAtMost M n) : | |
| HasHeightAtMost (L * M) n := by | |
| obtain ⟨P, rfl, hP⟩ := hL | |
| obtain ⟨Q, rfl, hQ⟩ := hM | |
| exact ⟨P.concat Q, rfl, max_le hP hQ⟩ | |
| lemma compl (hL : HasHeightAtMost L n) : HasHeightAtMost Lᶜ n := by | |
| obtain ⟨P, rfl, hP⟩ := hL | |
| exact ⟨P.compl, rfl, hP⟩ | |
| lemma inter (hL : HasHeightAtMost L n) (hM : HasHeightAtMost M n) : | |
| HasHeightAtMost (L ⊓ M) n := by | |
| have h := (hL.compl.union hM.compl).compl | |
| have heq : ((Lᶜ + Mᶜ)ᶜ : Language Alphabet) = L ⊓ M := by | |
| ext x | |
| change (¬(x ∉ L ∨ x ∉ M)) ↔ x ∈ L ∧ x ∈ M | |
| tauto | |
| exact heq ▸ h | |
| lemma star (hL : HasHeightAtMost L n) : HasHeightAtMost L∗ (1 + n) := by | |
| obtain ⟨P, rfl, hP⟩ := hL | |
| exact ⟨P.star, rfl, Nat.add_le_add_left hP 1⟩ | |
| lemma finset_iSup {ι : Type uι} (s : Finset ι) (L : ι → Language Alphabet) | |
| (h : ∀ i ∈ s, HasHeightAtMost (L i) n) : | |
| HasHeightAtMost (⨆ i ∈ s, L i) n := by | |
| classical | |
| induction s using Finset.induction_on with | |
| | empty => | |
| simp only [Finset.notMem_empty, iSup_false, iSup_bot] | |
| exact zero n | |
| | @insert i s hi ih => | |
| rw [Finset.iSup_insert] | |
| exact (h i (Finset.mem_insert_self _ _)).union | |
| (ih (fun j hj => h j (Finset.mem_insert_of_mem hj))) | |
| lemma iSup {ι : Type uι2} [Finite ι] (L : ι → Language Alphabet) | |
| (h : ∀ i, HasHeightAtMost (L i) n) : HasHeightAtMost (⨆ i, L i) n := by | |
| classical | |
| let : Fintype ι := Fintype.ofFinite ι | |
| simpa using finset_iSup Finset.univ L (fun i _ => h i) | |
| lemma singleton (w : List Alphabet) (n : ℕ) : | |
| HasHeightAtMost ({w} : Language Alphabet) n := by | |
| induction w with | |
| | nil => exact one n | |
| | cons a w ih => | |
| have h := (letter a n).concat ih | |
| convert h using 1 | |
| apply Language.ext | |
| intro x | |
| change (x = a :: w) ↔ _ | |
| rw [Language.mem_mul] | |
| change (x = a :: w) ↔ ∃ s, s = [a] ∧ ∃ t, t = w ∧ s ++ t = x | |
| simp [eq_comm] | |
| lemma finite (L : Language Alphabet) (hL : (L : Set (List Alphabet)).Finite) (n : ℕ) : | |
| HasHeightAtMost L n := by | |
| have heq : L = ⨆ w : {w : List Alphabet // w ∈ L}, ({w.1} : Language Alphabet) := by | |
| ext x | |
| rw [Language.mem_iSup] | |
| change (x ∈ L) ↔ ∃ w : {w : List Alphabet // w ∈ L}, x = w.1 | |
| simp | |
| let : Finite {w : List Alphabet // w ∈ L} := hL.to_subtype | |
| rw [heq] | |
| exact iSup _ (fun w => singleton w.1 n) | |
| end HasHeightAtMost | |
| end GeneralizedStarHeight | |
| end OAI | |