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Diffstat (limited to 'Common/HashSet.lean')
| -rw-r--r-- | Common/HashSet.lean | 123 |
1 files changed, 0 insertions, 123 deletions
diff --git a/Common/HashSet.lean b/Common/HashSet.lean deleted file mode 100644 index 3271121..0000000 --- a/Common/HashSet.lean +++ /dev/null @@ -1,123 +0,0 @@ -import Std.Data.HashSet -import Common.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 |