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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
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