300 lines
7.8 KiB
Plaintext
300 lines
7.8 KiB
Plaintext
import Common.Logic.Basic
|
||
import Mathlib.Data.Set.Basic
|
||
import Mathlib.Data.Set.Prod
|
||
import Mathlib.Tactic.LibrarySearch
|
||
|
||
/-! # Common.Set.Basic
|
||
|
||
Additional theorems and definitions useful in the context of `Set`s.
|
||
-/
|
||
|
||
namespace Set
|
||
|
||
/-! ## Minkowski Sum -/
|
||
|
||
/-
|
||
The Minkowski sum of two `Set`s `s` and `t` is the set
|
||
`s + t = { a + b : a ∈ s, b ∈ t }`.
|
||
-/
|
||
def minkowskiSum {α : Type u} [Add α] (s t : Set α) :=
|
||
{ x | ∃ a ∈ s, ∃ b ∈ t, x = a + b }
|
||
|
||
/--
|
||
The sum of two `Set`s is nonempty **iff** the summands are nonempty.
|
||
-/
|
||
theorem nonempty_minkowski_sum_iff_nonempty_add_nonempty {α : Type u} [Add α]
|
||
{s t : Set α}
|
||
: (minkowskiSum s t).Nonempty ↔ s.Nonempty ∧ t.Nonempty := by
|
||
apply Iff.intro
|
||
· intro h
|
||
have ⟨x, hx⟩ := h
|
||
have ⟨a, ⟨ha, ⟨b, ⟨hb, _⟩⟩⟩⟩ := hx
|
||
apply And.intro
|
||
· exact ⟨a, ha⟩
|
||
· exact ⟨b, hb⟩
|
||
· intro ⟨⟨a, ha⟩, ⟨b, hb⟩⟩
|
||
exact ⟨a + b, ⟨a, ⟨ha, ⟨b, ⟨hb, rfl⟩⟩⟩⟩⟩
|
||
|
||
/-! ## Pair Sets -/
|
||
|
||
/--
|
||
If `{x, y} = {x}` then `x = y`.
|
||
-/
|
||
theorem pair_eq_singleton_mem_imp_eq_self {x y : α}
|
||
(h : {x, y} = ({x} : Set α)) : y = x := by
|
||
rw [Set.ext_iff] at h
|
||
have := h y
|
||
simp at this
|
||
exact this
|
||
|
||
/--
|
||
If `{x, y} = {z}` then `x = y = z`.
|
||
-/
|
||
theorem pair_eq_singleton_mem_imp_eq_all {x y z : α}
|
||
(h : {x, y} = ({z} : Set α)) : x = z ∧ y = z := by
|
||
have h' := h
|
||
rw [Set.ext_iff] at h'
|
||
have hz := h' z
|
||
simp at hz
|
||
apply Or.elim hz
|
||
· intro hzx
|
||
rw [← hzx] at h
|
||
have := pair_eq_singleton_mem_imp_eq_self h
|
||
exact ⟨hzx.symm, this⟩
|
||
· intro hzy
|
||
rw [← hzy, Set.pair_comm] at h
|
||
have := pair_eq_singleton_mem_imp_eq_self h
|
||
exact ⟨this, hzy.symm⟩
|
||
|
||
/-! ## Subsets -/
|
||
|
||
/--
|
||
There exists no proper subset of `∅`.
|
||
-/
|
||
theorem ssubset_empty_iff_false (S : Set α)
|
||
: S ⊂ ∅ ↔ False := by
|
||
apply Iff.intro
|
||
· intro h
|
||
rw [ssubset_iff_subset_ne, subset_empty_iff] at h
|
||
exact absurd h.left h.right
|
||
· simp only [IsEmpty.forall_iff]
|
||
|
||
/--
|
||
If `Set` `A` is a subset of `Set` `B`, then `A ∪ B = B`.
|
||
-/
|
||
theorem left_subset_union_eq_self {A B : Set α} (h : A ⊆ B)
|
||
: A ∪ B = B := by
|
||
rw [Set.ext_iff]
|
||
intro x
|
||
apply Iff.intro
|
||
· intro hU
|
||
apply Or.elim hU
|
||
· intro hA
|
||
exact h hA
|
||
· simp
|
||
· intro hB
|
||
exact Or.inr hB
|
||
|
||
/--
|
||
If `Set` `B` is a subset of `Set` `A`, then `A ∪ B = B`.
|
||
-/
|
||
theorem right_subset_union_eq_self {A B : Set α} (h : B ⊆ A)
|
||
: A ∪ B = A := by
|
||
rw [Set.union_comm]
|
||
exact left_subset_union_eq_self h
|
||
|
||
/--
|
||
If `x` and `y` are members of `Set` `A`, it follows `{x, y}` is a subset of `A`.
|
||
-/
|
||
theorem mem_mem_imp_pair_subset {x y : α}
|
||
(hx : x ∈ A) (hy : y ∈ A) : ({x, y} : Set α) ⊆ A := by
|
||
intro a ha
|
||
apply Or.elim ha
|
||
· intro hx'
|
||
rwa [hx']
|
||
· intro hy'
|
||
rwa [hy']
|
||
|
||
/-! ## Powerset -/
|
||
|
||
/--
|
||
Every `Set` is a member of its own powerset.
|
||
-/
|
||
theorem self_mem_powerset_self {A : Set α}
|
||
: A ∈ 𝒫 A := fun _ => mem_preimage.mp
|
||
|
||
/-! ## Cartesian Product -/
|
||
|
||
/--
|
||
For any `Set` `A`, `∅ × A = ∅`.
|
||
-/
|
||
theorem prod_left_emptyset_eq_emptyset {A : Set α}
|
||
: Set.prod (∅ : Set β) A = ∅ := by
|
||
unfold prod
|
||
simp only [mem_empty_iff_false, false_and, setOf_false]
|
||
|
||
/--
|
||
For any `Set` `A`, `A × ∅ = ∅`.
|
||
-/
|
||
theorem prod_right_emptyset_eq_emptyset {A : Set α}
|
||
: Set.prod A (∅ : Set β) = ∅ := by
|
||
unfold prod
|
||
simp only [mem_empty_iff_false, and_false, setOf_false]
|
||
|
||
/--
|
||
For any `Set`s `A` and `B`, if both `A` and `B` are nonempty, then `A × B` is
|
||
also nonempty.
|
||
-/
|
||
theorem prod_nonempty_nonempty_imp_nonempty_prod {A : Set α} {B : Set β}
|
||
: A ≠ ∅ ∧ B ≠ ∅ ↔ Set.prod A B ≠ ∅ := by
|
||
apply Iff.intro
|
||
· intro nAB h
|
||
have ⟨a, ha⟩ := nonempty_iff_ne_empty.mpr nAB.left
|
||
have ⟨b, hb⟩ := nonempty_iff_ne_empty.mpr nAB.right
|
||
rw [Set.ext_iff] at h
|
||
exact (h (a, b)).mp ⟨ha, hb⟩
|
||
· intro h
|
||
rw [← nonempty_iff_ne_empty] at h
|
||
have ⟨(a, b), ⟨ha, hb⟩⟩ := h
|
||
rw [← nonempty_iff_ne_empty, ← nonempty_iff_ne_empty]
|
||
exact ⟨⟨a, ha⟩, ⟨b, hb⟩⟩
|
||
|
||
/-! ## Difference -/
|
||
|
||
/--
|
||
For any sets `A ⊂ B`, if `x ∈ A` then `A - {x} ⊂ B - {x}`.
|
||
-/
|
||
theorem diff_ssubset_diff_left {A B : Set α} (h : A ⊂ B)
|
||
: x ∈ A → A \ {x} ⊂ B \ {x} := by
|
||
intro hx
|
||
rw [Set.ssubset_def]
|
||
apply And.intro
|
||
· exact diff_subset_diff_left (subset_of_ssubset h)
|
||
· by_contra nh
|
||
have : {x} ⊆ A := singleton_subset_iff.mpr hx
|
||
rw [diff_subset_iff, union_diff_cancel this] at nh
|
||
exact LT.lt.false (Set.ssubset_of_ssubset_of_subset h nh)
|
||
|
||
/--
|
||
For any sets `A ⊂ B`, `B \ A` is nonempty.
|
||
-/
|
||
theorem diff_ssubset_nonempty {A B : Set α} (h : A ⊂ B)
|
||
: Set.Nonempty (B \ A) := by
|
||
have : B = A ∪ (B \ A) := by
|
||
simp only [Set.union_diff_self]
|
||
exact (Set.left_subset_union_eq_self (subset_of_ssubset h)).symm
|
||
rw [this, Set.ssubset_def] at h
|
||
have : ¬ ∀ x, x ∈ A ∪ (B \ A) → x ∈ A := h.right
|
||
simp only [Set.mem_union, not_forall, exists_prop] at this
|
||
have ⟨x, hx⟩ := this
|
||
apply Or.elim hx.left
|
||
· intro nx
|
||
exact absurd nx hx.right
|
||
· intro hx
|
||
exact ⟨x, hx⟩
|
||
|
||
/--
|
||
For any set `A`, the difference between the sample space and `A` is the
|
||
complement of `A`.
|
||
-/
|
||
theorem univ_diff_self_eq_compl (A : Set α) : Set.univ \ A = A.compl := by
|
||
unfold Set.compl SDiff.sdiff instSDiffSet Set.diff
|
||
simp only [mem_univ, true_and]
|
||
|
||
/--
|
||
For any set `A`, the difference between the sample space and the complement of
|
||
`A` is `A`.
|
||
-/
|
||
theorem univ_diff_compl_eq_self (A : Set α) : Set.univ \ A.compl = A := by
|
||
unfold Set.compl SDiff.sdiff instSDiffSet Set.diff
|
||
simp only [mem_univ, mem_setOf_eq, not_not, true_and, setOf_mem_eq]
|
||
|
||
/-! ## Symmetric Difference -/
|
||
|
||
/--
|
||
If `x ∈ A` and `x ∉ B`, then `x ∈ A ∆ B`.
|
||
-/
|
||
theorem symm_diff_mem_left {A B : Set α} (hA : x ∈ A) (hB : x ∉ B)
|
||
: x ∈ A ∆ B := by
|
||
left
|
||
exact ⟨hA, hB⟩
|
||
|
||
/--
|
||
If `x ∉ A` and `x ∈ B`, then `x ∈ A ∆ B`.
|
||
-/
|
||
theorem symm_diff_mem_right {A B : Set α} (hA : x ∉ A) (hB : x ∈ B)
|
||
: x ∈ A ∆ B := by
|
||
right
|
||
exact ⟨hB, hA⟩
|
||
|
||
/--
|
||
If `x ∈ A` and `x ∈ B`, then `x ∉ A ∆ B`.
|
||
-/
|
||
theorem symm_diff_mem_both_not_mem {A B : Set α} (hA : x ∈ A) (hB : x ∈ B)
|
||
: x ∉ A ∆ B := by
|
||
intro h
|
||
apply Or.elim h
|
||
· intro ⟨_, nB⟩
|
||
exact absurd hB nB
|
||
· intro ⟨_, nA⟩
|
||
exact absurd hA nA
|
||
|
||
/--
|
||
If `x ∉ A` and `x ∉ B`, then `x ∉ A ∆ B`.
|
||
-/
|
||
theorem symm_diff_not_mem_both_not_mem {A B : Set α} (nA : x ∉ A) (nB : x ∉ B)
|
||
: x ∉ A ∆ B := by
|
||
intro h
|
||
apply Or.elim h
|
||
· intro ⟨hA, _⟩
|
||
exact absurd hA nA
|
||
· intro ⟨hB, _⟩
|
||
exact absurd hB nB
|
||
|
||
/--
|
||
`x` is a member of the `symmDiff` of `A` and `B` **iff** `x ∈ A ∧ x ∉ B` or
|
||
`x ∉ A ∧ x ∈ B`.
|
||
-/
|
||
theorem mem_symm_diff_iff_exclusive_mem {A B : Set α}
|
||
: x ∈ (A ∆ B) ↔ (x ∈ A ∧ x ∉ B) ∨ (x ∉ A ∧ x ∈ B) := by
|
||
unfold symmDiff
|
||
apply Iff.intro
|
||
· intro hx
|
||
simp at hx
|
||
conv => arg 2; rw [and_comm]
|
||
exact hx
|
||
· intro hx
|
||
simp
|
||
conv => arg 2; rw [and_comm]
|
||
exact hx
|
||
|
||
/--
|
||
`x` is not a member of the `symmDiff` of `A` and `B` **iff** `x ∈ A ∩ B` or
|
||
`x ∉ A ∪ B`.
|
||
|
||
This is the contraposition of `mem_symm_diff_iff_exclusive_mem`.
|
||
-/
|
||
theorem not_mem_symm_diff_inter_or_not_union {A B : Set α}
|
||
: x ∉ (A ∆ B) ↔ (x ∈ A ∩ B) ∨ (x ∉ A ∪ B) := by
|
||
show ¬(x ∈ A ∧ ¬x ∈ B ∨ x ∈ B ∧ ¬x ∈ A) ↔ x ∈ A ∧ x ∈ B ∨ ¬(x ∈ A ∨ x ∈ B)
|
||
rw [
|
||
not_or_de_morgan,
|
||
not_and_de_morgan, not_and_de_morgan,
|
||
not_not, not_not,
|
||
not_or_de_morgan
|
||
]
|
||
apply Iff.intro
|
||
· intro nx
|
||
apply Or.elim nx.left
|
||
· intro nA
|
||
exact Or.elim nx.right (Or.inr ⟨nA, ·⟩) (absurd · nA)
|
||
· intro hB
|
||
exact Or.elim nx.right (absurd hB ·) (Or.inl ⟨·, hB⟩)
|
||
· intro hx
|
||
apply Or.elim hx
|
||
· intro hy
|
||
exact ⟨Or.inr hy.right, Or.inr hy.left⟩
|
||
· intro hy
|
||
exact ⟨Or.inl hy.left, Or.inl hy.right⟩
|
||
|
||
end Set |