|
|
|
|
@ -1,10 +1,10 @@
|
|
|
|
|
import Mathlib
|
|
|
|
|
|
|
|
|
|
lemma not_odd {n : ℕ} : ¬ Odd n ↔ Even n := by
|
|
|
|
|
sorry
|
|
|
|
|
-- lemma not_odd {n : ℕ} : ¬ Odd n ↔ Even n := by
|
|
|
|
|
-- sorry
|
|
|
|
|
|
|
|
|
|
lemma not_even {n : ℕ} : ¬ Even n ↔ Odd n := by
|
|
|
|
|
sorry
|
|
|
|
|
-- lemma not_even {n : ℕ} : ¬ Even n ↔ Odd n := by
|
|
|
|
|
-- sorry
|
|
|
|
|
|
|
|
|
|
lemma even_square (n : ℕ) : Even n → Even (n ^ 2) := by
|
|
|
|
|
intro ⟨x, hx⟩
|
|
|
|
|
@ -13,72 +13,58 @@ lemma even_square (n : ℕ) : Even n → Even (n ^ 2) := by
|
|
|
|
|
rw [hx]
|
|
|
|
|
ring
|
|
|
|
|
|
|
|
|
|
section powerset
|
|
|
|
|
|
|
|
|
|
open Set
|
|
|
|
|
|
|
|
|
|
namespace Finset
|
|
|
|
|
|
|
|
|
|
theorem powerset_singleton {U : Type _} [DecidableEq U] (x : U) :
|
|
|
|
|
Finset.powerset {x} = {∅, {x}} := by
|
|
|
|
|
ext y
|
|
|
|
|
rw [mem_powerset, subset_singleton_iff, mem_insert, mem_singleton]
|
|
|
|
|
|
|
|
|
|
end Finset
|
|
|
|
|
|
|
|
|
|
/- The powerset of a singleton contains only `∅` and the singleton. -/
|
|
|
|
|
theorem powerset_singleton {U : Type _} (x : U) :
|
|
|
|
|
𝒫 ({x} : Set U) = {∅, {x}} := by
|
|
|
|
|
ext y
|
|
|
|
|
rw [mem_powerset_iff, subset_singleton_iff_eq, mem_insert_iff, mem_singleton_iff]
|
|
|
|
|
|
|
|
|
|
theorem subset_insert_iff_of_not_mem {U : Type _ } {s t : Set U} {a : U} (h : a ∉ s)
|
|
|
|
|
: s ⊆ insert a t ↔ s ⊆ t := by
|
|
|
|
|
constructor
|
|
|
|
|
· intro g y hy
|
|
|
|
|
specialize g hy
|
|
|
|
|
rw [mem_insert_iff] at g
|
|
|
|
|
rcases g with g | g
|
|
|
|
|
· rw [g] at hy
|
|
|
|
|
contradiction
|
|
|
|
|
· assumption
|
|
|
|
|
· intro g y hy
|
|
|
|
|
specialize g hy
|
|
|
|
|
exact mem_insert_of_mem _ g
|
|
|
|
|
|
|
|
|
|
theorem subset_insert_iff_of_not_mem' {U : Type _ } {s t : Set U} {a : U} (h : a ∉ s)
|
|
|
|
|
(g : s ⊆ t) : s ⊆ insert a t := by
|
|
|
|
|
intro y hy
|
|
|
|
|
specialize g hy
|
|
|
|
|
exact mem_insert_of_mem _ g
|
|
|
|
|
|
|
|
|
|
lemma mem_powerset_insert_iff {U : Type _} (A S : Set U) (x : U) :
|
|
|
|
|
S ∈ 𝒫 (insert x A) ↔ S ∈ 𝒫 A ∨ ∃ B ∈ 𝒫 A , insert x B = S := by
|
|
|
|
|
simp_rw [mem_powerset_iff]
|
|
|
|
|
constructor
|
|
|
|
|
· intro h
|
|
|
|
|
by_cases hs : x ∈ S
|
|
|
|
|
· right
|
|
|
|
|
use S \ {x}
|
|
|
|
|
rw [insert_diff_singleton, insert_eq_of_mem hs, diff_singleton_subset_iff]
|
|
|
|
|
exact ⟨h, rfl⟩
|
|
|
|
|
· left
|
|
|
|
|
exact (subset_insert_iff_of_not_mem hs).mp h
|
|
|
|
|
· intro h
|
|
|
|
|
rcases h with h | ⟨B, h₁, h₂⟩
|
|
|
|
|
· exact le_trans h (subset_insert x A)
|
|
|
|
|
· rw [←h₂]
|
|
|
|
|
exact insert_subset_insert h₁
|
|
|
|
|
|
|
|
|
|
lemma mem_powerset_insert_iff' {U : Type _} (A S : Set U) (x : U) :
|
|
|
|
|
S ∈ 𝒫 (insert x A) ↔ S \ {x} ∈ 𝒫 A := by
|
|
|
|
|
rw [mem_powerset_iff, mem_powerset_iff, diff_singleton_subset_iff]
|
|
|
|
|
|
|
|
|
|
lemma powerset_insert {U : Type _} (A : Set U) (x : U) :
|
|
|
|
|
𝒫 (insert x A) = A.powerset ∪ A.powerset.image (insert x) := by
|
|
|
|
|
ext y
|
|
|
|
|
rw [mem_powerset_insert_iff, mem_union, mem_image]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
end powerset
|
|
|
|
|
-- section powerset
|
|
|
|
|
|
|
|
|
|
-- open Set
|
|
|
|
|
|
|
|
|
|
-- namespace Finset
|
|
|
|
|
|
|
|
|
|
-- theorem powerset_singleton {U : Type _} [DecidableEq U] (x : U) :
|
|
|
|
|
-- Finset.powerset {x} = {∅, {x}} := by
|
|
|
|
|
-- ext y
|
|
|
|
|
-- rw [mem_powerset, subset_singleton_iff, mem_insert, mem_singleton]
|
|
|
|
|
|
|
|
|
|
-- end Finset
|
|
|
|
|
|
|
|
|
|
-- /- The powerset of a singleton contains only `∅` and the singleton. -/
|
|
|
|
|
-- theorem powerset_singleton {U : Type _} (x : U) :
|
|
|
|
|
-- 𝒫 ({x} : Set U) = {∅, {x}} := by
|
|
|
|
|
-- ext y
|
|
|
|
|
-- rw [mem_powerset_iff, subset_singleton_iff_eq, mem_insert_iff, mem_singleton_iff]
|
|
|
|
|
|
|
|
|
|
-- theorem subset_insert_iff_of_not_mem' {U : Type _ } {s t : Set U} {a : U} (h : a ∉ s)
|
|
|
|
|
-- (g : s ⊆ t) : s ⊆ insert a t := by
|
|
|
|
|
-- intro y hy
|
|
|
|
|
-- specialize g hy
|
|
|
|
|
-- exact mem_insert_of_mem _ g
|
|
|
|
|
|
|
|
|
|
-- lemma mem_powerset_insert_iff {U : Type _} (A S : Set U) (x : U) :
|
|
|
|
|
-- S ∈ 𝒫 (insert x A) ↔ S ∈ 𝒫 A ∨ ∃ B ∈ 𝒫 A , insert x B = S := by
|
|
|
|
|
-- simp_rw [mem_powerset_iff]
|
|
|
|
|
-- constructor
|
|
|
|
|
-- · intro h
|
|
|
|
|
-- by_cases hs : x ∈ S
|
|
|
|
|
-- · right
|
|
|
|
|
-- use S \ {x}
|
|
|
|
|
-- rw [insert_diff_singleton, insert_eq_of_mem hs, diff_singleton_subset_iff]
|
|
|
|
|
-- exact ⟨h, rfl⟩
|
|
|
|
|
-- · left
|
|
|
|
|
-- exact (subset_insert_iff_of_not_mem hs).mp h
|
|
|
|
|
-- · intro h
|
|
|
|
|
-- rcases h with h | ⟨B, h₁, h₂⟩
|
|
|
|
|
-- · exact le_trans h (subset_insert x A)
|
|
|
|
|
-- · rw [←h₂]
|
|
|
|
|
-- exact insert_subset_insert h₁
|
|
|
|
|
|
|
|
|
|
-- lemma mem_powerset_insert_iff' {U : Type _} (A S : Set U) (x : U) :
|
|
|
|
|
-- S ∈ 𝒫 (insert x A) ↔ S \ {x} ∈ 𝒫 A := by
|
|
|
|
|
-- rw [mem_powerset_iff, mem_powerset_iff, diff_singleton_subset_iff]
|
|
|
|
|
|
|
|
|
|
-- lemma powerset_insert {U : Type _} (A : Set U) (x : U) :
|
|
|
|
|
-- 𝒫 (insert x A) = A.powerset ∪ A.powerset.image (insert x) := by
|
|
|
|
|
-- ext y
|
|
|
|
|
-- rw [mem_powerset_insert_iff, mem_union, mem_image]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
-- end powerset
|
|
|
|
|
|
Jon Eugster
2 years ago