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Diffstat (limited to 'LeanAStar/HashSet.lean')
| -rw-r--r-- | LeanAStar/HashSet.lean | 122 |
1 files changed, 122 insertions, 0 deletions
diff --git a/LeanAStar/HashSet.lean b/LeanAStar/HashSet.lean new file mode 100644 index 0000000..33ef316 --- /dev/null +++ b/LeanAStar/HashSet.lean @@ -0,0 +1,122 @@ +import Std.Data.HashSet +import LeanAStar.Finite + +open Std.HashSet +namespace Std.HashSet + +theorem erase_not_mem {α : Type u} [BEq α] [Hashable α] [LawfulBEq α] (set : HashSet α) (a : α) : a ∉ (set.erase a) := by + simp only [HashSet.mem_iff_contains, HashSet.contains_erase, beq_self_eq_true, Bool.not_true, Bool.false_and, Bool.false_eq_true, not_false_eq_true] + +theorem erase_not_mem_of_not_mem {α : Type u} [BEq α] [Hashable α] [LawfulBEq α] (set : HashSet α) {a : α} (b : α) (h₁ : a ∉ set) : a ∉ (set.erase b) := by + intro h₂ + rw[HashSet.mem_erase] at h₂ + exact absurd h₂.right h₁ + +protected theorem size_le_finite_worker_size (α : Type u) [Finite α] [BEq α] [Hashable α] [LawfulBEq α] : ∀ (set : Std.HashSet α) (n : Fin (Finite.cardinality α)) (_h₁ : ∀(o : Fin (Finite.cardinality α)) (_h : o > n), Finite.nth o ∉ set), set.size ≤ (Finite.set.set_worker α HashSet.empty n).size := by + intros set n h₁ + cases n + case mk n h₂ => + cases n + case zero => + have h₃ : (Finite.set.set_worker α empty ⟨0,h₂⟩).size = 1 := by simp[Finite.set.set_worker, HashSet.size_insert] + rw[h₃] + cases h₄ : set.contains (Finite.nth ⟨0,h₂⟩) + case false => + have h₅ : ∀(o : Fin (Finite.cardinality α)), Finite.nth o ∉ set := by + simp[←Bool.not_eq_true, ←HashSet.mem_iff_contains] at h₄ + intro o + cases o + case mk o ho => + cases o + case zero => exact h₄ + case succ oo => exact h₁ ⟨oo+1, ho⟩ (Nat.succ_pos oo) + have h₆ : ∀(a : α), a∉set := Finite.forall_nth (λx ↦ x∉set) h₅ + have h₇ : set.isEmpty = true := HashSet.isEmpty_iff_forall_not_mem.mpr h₆ + have h₈ : true = (set.size == 0) := HashSet.isEmpty_eq_size_eq_zero.subst h₇.symm + have h₈ : set.size = 0 := beq_iff_eq.mp h₈.symm + rw[h₈] + exact Nat.zero_le _ + case true => + --show that the set from which we erase (nth 0) has size 0, just like in case false. + --then show that, since we removed one element, set.size equals 1. + let rset := set.erase $ Finite.nth ⟨0,h₂⟩ + have h₅ : (Finite.nth ⟨0,h₂⟩) ∉ rset := erase_not_mem set (Finite.nth ⟨0,h₂⟩) + have h₆ : ∀(o : Fin (Finite.cardinality α)), Finite.nth o ∉ rset := by + intro o + cases o + case mk o ho => + cases o + case zero => exact h₅ + case succ oo => exact erase_not_mem_of_not_mem set _ $ h₁ ⟨oo+1, ho⟩ (Nat.succ_pos oo) + have h₇ : ∀(a : α), a∉rset := Finite.forall_nth (λx ↦ x∉rset) h₆ + have h₈ : rset.isEmpty = true := HashSet.isEmpty_iff_forall_not_mem.mpr h₇ + have h₉ : true = (rset.size == 0) := HashSet.isEmpty_eq_size_eq_zero.subst h₈.symm + have h₉ : rset.size = 0 := beq_iff_eq.mp h₉.symm + have h₁₀ : set.size ≤ rset.size + 1 := HashSet.size_le_size_erase + rw[h₉, Nat.zero_add] at h₁₀ + assumption + case succ m => + --erase (nth m) from the set and recurse. + --then show that the current set can only be larger by 1, and therefore fulfills the goal + --HashSet.size_le_size_erase + let rset := set.erase $ Finite.nth ⟨m+1,h₂⟩ + have h₃ : (Finite.nth ⟨m+1,h₂⟩) ∉ rset := erase_not_mem set (Finite.nth ⟨m+1,h₂⟩) + have h₄ : ∀(o : Fin (Finite.cardinality α)) (h₄ : o > ⟨m,Nat.lt_of_succ_lt h₂⟩), Finite.nth o ∉ rset := by + intros o h₄ + cases o + case mk o h₅ => + cases h₆ : o == (m+1) <;> simp at h₆ + case true => + simp only[h₆] + exact erase_not_mem set $ Finite.nth ⟨m+1, h₂⟩ + case false => + have h₆ : o > m+1 := by simp at h₄; omega + exact erase_not_mem_of_not_mem _ _ (h₁ ⟨o,h₅⟩ h₆) + have h₅ := Std.HashSet.size_le_finite_worker_size α rset ⟨m, Nat.lt_of_succ_lt h₂⟩ h₄ + have h₆ : (Finite.set.set_worker α empty ⟨m, Nat.lt_of_succ_lt h₂⟩).size = empty.size + m + 1 := + Finite.set_worker_size α empty ⟨m, Nat.lt_of_succ_lt h₂⟩ (λa _ ↦Bool.eq_false_iff.mp $ HashSet.contains_empty (a := Finite.nth a)) + have h₇ : (Finite.set.set_worker α empty ⟨m+1,h₂⟩).size = empty.size + (m + 1) + 1 := + Finite.set_worker_size α empty ⟨m+1,h₂⟩ (λa _ ↦Bool.eq_false_iff.mp $ HashSet.contains_empty (a := Finite.nth a)) + have h₈ := HashSet.size_le_size_erase (m := set) (k:=Finite.nth ⟨m+1,h₂⟩) + simp[h₆] at h₅ + simp[h₇] + exact Nat.le_trans h₈ (Nat.succ_le_succ h₅) + +theorem size_le_finite_set_size {α : Type u} [Finite α] [BEq α] [LawfulBEq α] [Hashable α] (set : Std.HashSet α) : set.size ≤ (Finite.set α).size := by + unfold Finite.set + split + case h_1 h₁ => + if h : set.isEmpty then + have := HashSet.isEmpty_eq_size_eq_zero (m := set) + simp[h] at this + simp[this] + else + have := HashSet.isEmpty_eq_false_iff_exists_mem.mp (Bool.eq_false_iff.mpr h) + cases this + case intro x hx => + let xx := Finite.enumerate x + exact (Fin.cast h₁ xx).elim0 + case h_2 l h => + exact Std.HashSet.size_le_finite_worker_size α set ⟨l,h.substr $ Nat.lt_succ_self l⟩ (λa hx ↦ + by + have h₁ := a.isLt + simp_arith[h] at h₁ + have h₂ : a > l := Fin.lt_iff_val_lt_val.mp hx + exact absurd h₁ (Nat.not_le.mpr h₂) + ) + +theorem size_le_finite_cardinality {α : Type u} [Finite α] [BEq α] [LawfulBEq α] [Hashable α] (set : Std.HashSet α) : set.size ≤ Finite.cardinality α := + (Finite.set_size_eq_cardinality α).subst (size_le_finite_set_size set) + +theorem size_lt_finite_cardinality_of_not_mem {α : Type u} [Finite α] [BEq α] [LawfulBEq α] [Hashable α] (set : Std.HashSet α) (h₁ : ∃(a : α), a ∉ set) : set.size < Finite.cardinality α := by + have h₂ : set.size ≤ Finite.cardinality α := size_le_finite_cardinality set + cases h₁ + case intro e h₁ => + let iset := set.insert e + have h₂ : iset.size = set.size + 1 := by + have := HashSet.size_insert (m := set) (k := e) + simp[h₁] at this + assumption + have h₃ : iset.size ≤ Finite.cardinality α := size_le_finite_cardinality iset + rw[h₂] at h₃ + exact Nat.lt_of_succ_le h₃ |