118 lines
3 KiB
Text
118 lines
3 KiB
Text
import Mathlib.Data.List.Sort
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abbrev NatSet := { xs : List ℕ // List.Sorted (·<·) xs }
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namespace NatSet
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def empty : NatSet := ⟨[], List.sorted_nil⟩
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def singleton (x : ℕ) : NatSet := ⟨[x], List.sorted_singleton x⟩
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def remove (x : ℕ) : NatSet → NatSet
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| ⟨[], _⟩ => empty
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| ⟨n::ns, h⟩ =>
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let tail := ⟨ns, List.Sorted.of_cons h⟩
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if n = x
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then tail
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else ⟨n::tail, h⟩
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def list_insert (x : ℕ) : List ℕ → List ℕ
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| [] => [x]
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| a::as =>
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if x = a
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then as
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else if x < a
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then x::a::as
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else a::(list_insert x as)
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theorem list_insert_mem : b ∈ list_insert x xs → b = x ∨ b ∈ xs := by
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induction xs with simp [list_insert]
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| cons a as ih =>
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split
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next heq =>
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intro hmem
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right; right
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assumption
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next hne =>
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split
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· simp
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· next hnlt =>
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simp [*]
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intro h
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cases h
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· right; left; assumption
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cases ih ‹_›
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· left; assumption
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· right; right; assumption
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theorem list_insert_sorted (hs : List.Sorted (·<·) xs) : List.Sorted (·<·) (list_insert x xs) := by
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induction xs with simp [list_insert]
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| cons a as ih =>
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split
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· exact List.Sorted.of_cons hs
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· split
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· exact List.Sorted.cons ‹_› hs
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· have : a < x := by
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rename_i hne hnlt
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apply Nat.lt_of_le_of_ne
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exact Nat.le_of_not_lt hnlt
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exact fun a_1 => hne (Eq.symm a_1)
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simp only [List.sorted_cons]
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constructor
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· intro b hmem
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cases list_insert_mem hmem
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<;> simp_all only [imp_self, List.sorted_cons, and_true, not_lt]
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· simp_all only [imp_self, List.sorted_cons, and_true, not_lt]
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def insert (x : ℕ) : NatSet → NatSet
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| ⟨[], _⟩ => singleton x
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| ⟨n::ns, h⟩ =>
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if heq : x = n
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then ⟨n::ns, h⟩
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else if hlt : x < n
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then ⟨x::n::ns, List.Sorted.cons hlt h⟩
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else
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let tail : NatSet := ⟨ns, List.Sorted.of_cons h⟩
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let ⟨tail, h₂⟩ := tail.insert x
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⟨
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n::tail,
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by
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have : ∀b ∈ tail, b = x ∨ b ∈ ns := by
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intro b h
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induction tail
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· contradiction
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· simp [*] at *
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sorry
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have : ∀b ∈ ns, n < b := by
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simp only [List.sorted_cons] at h
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exact h.left
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simp only [List.sorted_cons]
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constructor
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· assumption
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-- have : n < x := by
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-- rw [Nat.lt_iff_le_and_ne]
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-- constructor
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-- · simp at hlt
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-- assumption
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-- · intro h
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-- exact heq h.symm
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sorry
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⟩
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instance : EmptyCollection NatSet where
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emptyCollection := empty
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instance : Singleton ℕ NatSet where
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singleton := singleton
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instance : Insert ℕ NatSet where
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insert := insert
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instance : LawfulSingleton ℕ NatSet where
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insert_empty_eq := by simp [Insert.insert, EmptyCollection.emptyCollection, empty, insert,
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Singleton.singleton, implies_true]
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