r
/
bookshelf
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Enderton. Fixup equivalence class def'ns and additional exercises.

finite-set-exercises
Joshua Potter 2023-07-10 16:08:48 -06:00
parent 98ecc12995
commit 2199503f70
3 changed files with 511 additions and 374 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

598
Bookshelf/Enderton/Set.tex
View File

File diff suppressed because it is too large Load Diff

177
Bookshelf/Enderton/Set/Chapter_3.lean
View File

@ -511,22 +511,22 @@ Assume that `F : A → B`, and that `A` is nonempty. There exists a function
theorem theorem_3j_a {F : Set.HRelation α β} {A : Set α} {B : Set β}
(hF : mapsInto F A B) (hA : Set.Nonempty A)
: (∃ G : Set.HRelation β α,
isSingleValued G ∧ mapsInto G B A ∧
mapsInto G B A ∧
(comp G F = { p | p.1 ∈ A ∧ p.1 = p.2 })) ↔ isOneToOne F := by
apply Iff.intro
· intro ⟨G, hG⟩
refine ⟨hF.left, ?_⟩
refine ⟨hF.is_func, ?_⟩
intro y hy
have ⟨x₁, hx₁⟩ := ran_exists hy
refine ⟨x₁, ⟨mem_pair_imp_fst_mem_dom hx₁, hx₁⟩, ?_⟩
intro x₂ hx₂
have hG' : y ∈ dom G := by
rw [hG.right.left.right.left]
exact hF.right.right hy
rw [hG.left.dom_eq]
exact hF.ran_ss hy
have ⟨z, hz⟩ := dom_exists hG'
have := hG.right.right
have := hG.right
unfold comp at this
rw [Set.ext_iff] at this
have h₁ := (this (x₁, z)).mp ⟨y, hx₁, hz⟩
@ -544,7 +544,7 @@ Assume that `F : A → B`, and that `A` is nonempty. There exists a function
theorem theorem_3j_b {F : Set.HRelation α β} {A : Set α} {B : Set β}
(hF : mapsInto F A B) (hA : Set.Nonempty A)
: (∃ H : Set.HRelation β α,
isSingleValued H ∧ mapsInto H B A ∧
mapsInto H B A ∧
(comp F H = { p | p.1 ∈ B ∧ p.1 = p.2 })) ↔ mapsOnto F A B := by
sorry
@ -1595,7 +1595,7 @@ theorem exercise_3_28 {A : Set α} {B : Set β}
simp only [Set.mem_powerset_iff] at hX₂
have ht' := hX₂.left ht
rw [← hf.right.right.left] at ht'
rw [← hf.right.dom_eq] at ht'
have ⟨ft, hft⟩ := dom_exists ht'
have hft' : ft ∈ image f X₂ := ⟨t, ht, hft⟩
@ -1610,7 +1610,7 @@ theorem exercise_3_28 {A : Set α} {B : Set β}
simp only [Set.mem_powerset_iff] at hX₁sub
have ht' := hX₁sub ht
rw [← hf.right.right.left] at ht'
rw [← hf.right.dom_eq] at ht'
have ⟨ft, hft⟩ := dom_exists ht'
have hft' : ft ∈ image f X₁ := ⟨t, ht, hft⟩
@ -1637,7 +1637,7 @@ theorem exercise_3_28 {A : Set α} {B : Set β}
show ∀ y, y ∈ image f a → y ∈ B
intro y ⟨b, hb⟩
have hz := mem_pair_imp_snd_mem_ran hb.right
exact hf.right.right.right hz
exact hf.right.ran_ss hz
/-- #### Exercise 3.29
@ -1653,34 +1653,34 @@ theorem exercise_3_29 {f : Set.HRelation α β} {G : Set.HRelation β (Set α)}
(hG : mapsInto G B (𝒫 A) ∧ G = { p | p.1 ∈ B ∧ p.2 = {x ∈ A | (x, p.1) ∈ f} })
: isOneToOne G := by
unfold isOneToOne
refine ⟨hG.left.left, ?_⟩
refine ⟨hG.left.is_func, ?_⟩
intro y hy
have ⟨x₁, hx₁⟩ := ran_exists hy
refine ⟨x₁, ⟨mem_pair_imp_fst_mem_dom hx₁, hx₁⟩, ?_⟩
intro x₂ hx₂
have hG₁ : (x₁, {x ∈ A | (x, x₁) ∈ f}) ∈ G := by
rw [hG.right, ← hG.left.right.left]
rw [hG.right, ← hG.left.dom_eq]
simp only [Set.mem_setOf_eq, and_true]
exact mem_pair_imp_fst_mem_dom hx₁
have hG₂ : (x₂, {x ∈ A | (x, x₂) ∈ f}) ∈ G := by
rw [hG.right, ← hG.left.right.left]
rw [hG.right, ← hG.left.dom_eq]
simp only [Set.mem_setOf_eq, and_true]
exact hx₂.left
have heq : {x ∈ A | (x, x₁) ∈ f} = {x ∈ A | (x, x₂) ∈ f} := by
have h₁ := single_valued_eq_unique hG.left.left hx₁ hG₁
have h₂ := single_valued_eq_unique hG.left.left hx₂.right hG₂
have h₁ := single_valued_eq_unique hG.left.is_func hx₁ hG₁
have h₂ := single_valued_eq_unique hG.left.is_func hx₂.right hG₂
rw [← h₁, ← h₂]
rw [hG.right, ← hf.right.right] at hG₁
rw [hG.right, ← hf.ran_eq] at hG₁
simp only [Set.mem_setOf_eq, and_true] at hG₁
have ⟨t, ht⟩ := ran_exists hG₁
have : t ∈ {x ∈ A | (x, x₁) ∈ f} := by
refine ⟨?_, ht⟩
rw [← hf.right.left]
rw [← hf.dom_eq]
exact mem_pair_imp_fst_mem_dom ht
rw [heq] at this
exact single_valued_eq_unique hf.left this.right ht
exact single_valued_eq_unique hf.is_func this.right ht
/-- #### Theorem 3M
@ -1690,8 +1690,7 @@ relation on `fld R`.
theorem theorem_3m {R : Set.Relation α}
(hS : R.isSymmetric) (hT : R.isTransitive)
: R.isEquivalence (fld R) := by
refine ⟨?_, hS, hT⟩
unfold isReflexive fld
refine ⟨Eq.subset rfl, ?_, hS, hT⟩
intro x hx
apply Or.elim hx
· intro h
@ -1703,43 +1702,31 @@ theorem theorem_3m {R : Set.Relation α}
have := hS ht
exact hT this ht
/-- #### Lemma 3N
Assume that `R` is an equivalence relation on `A` and that `x` and `y` belong
to `A`. Then `[x]_R = [y]_R ↔ xRy`.
-/
theorem lemma_3n {R : Set.Relation α} {A : Set α} {x y : α}
(hR : R.isEquivalence A) (_ : x ∈ A) (hy : y ∈ A)
: cell R x = cell R y ↔ (x, y) ∈ R := by
apply Iff.intro
· intro h
have : y ∈ cell R y := hR.left y hy
rwa [← h] at this
· intro h
rw [Set.ext_iff]
intro t
apply Iff.intro
· intro ht
have := hR.right.left h
exact hR.right.right this ht
· intro ht
exact hR.right.right h ht
/-- #### Theorem 3P
Assume that `R` is an equivalence relation on `A`. Then the set
`{[x]_R | x ∈ A}` of all equivalence classes is a partition of `P`.
-/
theorem theorem_3p {R : Set.Relation α} {A : Set α} {P : Set (Set α)}
(hR : R.isEquivalence A) (hP : P = {cell R x | x ∈ A})
(hR : R.isEquivalence A) (hP : P = {neighborhood R x | x ∈ A})
: isPartition P A := by
refine ⟨?_, ?_, ?_⟩
refine ⟨?_, ?_, ?_, ?_⟩
· -- Every member is a subset of `A`.
intro p hp
rw [hP] at hp
simp only [Set.mem_setOf_eq] at hp
have ⟨x, hx⟩ := hp
rw [← hx.right]
show ∀ t, t ∈ neighborhood R x → t ∈ A
intro t ht
exact hR.b_on (Or.inr (mem_pair_imp_snd_mem_ran ht))
· -- Every member is nonempty.
intro p hp
rw [hP] at hp
have ⟨x, hx⟩ := hp
rw [← hx.right]
exact ⟨x, hR.left x hx.left⟩
exact ⟨x, hR.refl x hx.left⟩
· -- Every pair of members is disjoint.
intro xR hxR yR hyR h
@ -1752,18 +1739,18 @@ theorem theorem_3p {R : Set.Relation α} {A : Set α} {P : Set (Set α)}
have ⟨x, hx⟩ := hxR
have ⟨y, hy⟩ := hyR
rw [← hx.right, ← hy.right] at hz
unfold cell at hz
unfold neighborhood at hz
simp only [Set.mem_inter_iff, Set.mem_setOf_eq] at hz
have hzy : (z, y) ∈ R := hR.right.left hz.right
have hxy : (x, y) ∈ R := hR.right.right hz.left hzy
have := (lemma_3n hR hx.left hy.left).mpr hxy
have hzy : (z, y) ∈ R := hR.symm hz.right
have hxy : (x, y) ∈ R := hR.trans hz.left hzy
have := (neighborhood_iff_mem hR hx.left hy.left).mpr hxy
rw [hx.right, hy.right] at this
exact absurd this h
· -- Every element of `A` is in `P`.
intro x hx
have := hR.left x hx
refine ⟨cell R x, ?_, this⟩
have := hR.refl x hx
refine ⟨neighborhood R x, ?_, this⟩
· rw [hP]
exact ⟨x, hx, rfl⟩
@ -1919,8 +1906,8 @@ theorem exercise_3_34_b {𝓐 : Set (Set.Relation ℕ)}
Show that for any `R` and `x`, we have `[x]_R = R⟦{x}⟧`.
-/
theorem exercise_3_35 {R : Set.Relation α} {x : α}
: cell R x = image R {x} := by
calc cell R x
: neighborhood R x = image R {x} := by
calc neighborhood R x
_ = { t | (x, t) ∈ R } := rfl
_ = { t | ∃ u ∈ ({x} : Set α), (u, t) ∈ R } := by simp
_ = image R {x} := rfl
@ -1933,40 +1920,63 @@ equivalence relation on `A`.
-/
theorem exercise_3_36 {f : Set.HRelation α β}
{Q : Set.Relation α} {R : Set.Relation β} {A : Set α} {B : Set β}
(hf : mapsInto f A B)
(hR : isEquivalence R B)
(hf : mapsInto f A B) (hR : isEquivalence R B)
(hQ : Q = { p | ∃ fx fy : β, (p.1, fx) ∈ f ∧ (p.2, fy) ∈ f ∧ (fx, fy) ∈ R })
: isEquivalence Q A := by
refine ⟨?_, ?_, ?_⟩
· unfold isReflexive
refine ⟨?_, ?_, ?_, ?_⟩
· -- `fld Q ⊆ A`
show ∀ x, x ∈ fld Q → x ∈ A
intro x hx
unfold mapsInto at hf
rw [← hf.right.left] at hx
rw [hQ] at hx
unfold fld dom ran at hx
simp only [
exists_and_left,
Set.mem_union,
Set.mem_image,
Set.mem_setOf_eq,
Prod.exists,
exists_and_right,
exists_eq_right
] at hx
apply Or.elim hx
· intro ⟨_, _, hx₁⟩
rw [← hf.dom_eq]
exact mem_pair_imp_fst_mem_dom hx₁.left
· intro ⟨_, _, _, _, hx₂⟩
rw [← hf.dom_eq]
exact mem_pair_imp_fst_mem_dom hx₂.left
· -- `isReflexive Q A`
intro x hx
rw [← hf.dom_eq] at hx
have ⟨fx, hfx⟩ := dom_exists hx
have := hR.left fx (hf.right.right $ mem_pair_imp_snd_mem_ran hfx)
have := hR.refl fx (hf.ran_ss $ mem_pair_imp_snd_mem_ran hfx)
rw [hQ]
simp only [exists_and_left, Set.mem_setOf_eq]
exact ⟨fx, hfx, fx, hfx, this⟩
· unfold isSymmetric
· -- `isSymmetric Q`
intro x y h
rw [hQ] at h
simp only [exists_and_left, Set.mem_setOf_eq] at h
have ⟨fx, hfx, fy, hfy, h'⟩ := h
have := hR.right.left h'
have := hR.symm h'
rw [hQ]
simp only [exists_and_left, Set.mem_setOf_eq]
exact ⟨fy, hfy, fx, hfx, this⟩
· unfold isTransitive
· -- `isTransitive Q`
intro x y z hx hy
rw [hQ] at hx hy
simp only [exists_and_left, Set.mem_setOf_eq] at hx hy
have ⟨fx, hfx, fy, hfy, h₁⟩ := hx
have ⟨fy₁, hfy₁, fz, hfz, h₂⟩ := hy
have hfy' : fy = fy₁ := single_valued_eq_unique hf.left hfy hfy₁
have hfy' : fy = fy₁ := single_valued_eq_unique hf.is_func hfy hfy₁
rw [hfy'] at h₁
rw [hQ]
simp only [exists_and_left, Set.mem_setOf_eq]
exact ⟨fx, hfx, fz, hfz, hR.right.right h₁ h₂⟩
exact ⟨fx, hfx, fz, hfz, hR.trans h₁ h₂⟩
/-- #### Exercise 3.37
@ -1985,13 +1995,36 @@ theorem exercise_3_37 {P : Set (Set α)} {A : Set α}
ext p
have (x, y) := p
exact hR x y
refine ⟨?_, ?_, ?_⟩
· unfold isReflexive
refine ⟨?_, ?_, ?_, ?_⟩
· -- `fld R ⊆ A`
show ∀ x, x ∈ fld R → x ∈ A
rw [hR']
intro x hx
unfold fld dom ran at hx
simp only [
Set.mem_union,
Set.mem_image,
Set.mem_setOf_eq,
Prod.exists,
exists_and_right,
exists_eq_right
] at hx
apply Or.elim hx
· intro ⟨t, B, hB⟩
have := hP.p_subset B hB.left
exact this hB.right.left
· intro ⟨a, B, hB⟩
have := hP.p_subset B hB.left
exact this hB.right.right
· -- `isReflexive R A`
intro x hx
rw [hR']
simp only [Set.mem_setOf_eq, and_self]
exact hP.right.right x hx
· unfold isSymmetric
exact hP.exhaustive x hx
· -- `isSymmetric R`
intro x y h
rw [hR'] at h
simp only [Set.mem_setOf_eq] at h
@ -2000,17 +2033,17 @@ theorem exercise_3_37 {P : Set (Set α)} {A : Set α}
simp only [Set.mem_setOf_eq]
conv at hB => right; rw [and_comm]
exact ⟨B, hB⟩
· unfold isTransitive
· -- `isTransitive R`
intro x y z hx hy
rw [hR'] at hx hy
simp only [Set.mem_setOf_eq] at hx hy
have ⟨B₁, hB₁⟩ := hx
have ⟨B₂, hB₂⟩ := hy
unfold isPartition at hP
have hB : B₁ = B₂ := by
have hy₁ : y ∈ B₁ := hB₁.right.right
have hy₂ : y ∈ B₂ := hB₂.right.left
have hy := hP.right.left B₁ hB₁.left B₂ hB₂.left
have hy := hP.disjoint B₁ hB₁.left B₂ hB₂.left
rw [contraposition] at hy
simp at hy
suffices B₁ ∩ B₂ ≠ ∅ from hy this
@ -2020,7 +2053,7 @@ theorem exercise_3_37 {P : Set (Set α)} {A : Set α}
rw [hR']
simp only [Set.mem_setOf_eq]
exact ⟨B₁, hB₁.left, hB₁.right.left, by rw [hB]; exact hB₂.right.right⟩
lemma test (h : ¬ p = q) : p ≠ q := by exact?
end Relation
end Enderton.Set.Chapter_3

108
Bookshelf/Enderton/Set/Relation.lean
View File

@ -314,14 +314,18 @@ Indicates `Relation` `F` is a function from `A` to `B`.
This is usually denoted as `F : A → B`.
-/
def mapsInto (F : HRelation α β) (A : Set α) (B : Set β) :=
isSingleValued F ∧ dom F = A ∧ ran F ⊆ B
structure mapsInto (F : HRelation α β) (A : Set α) (B : Set β) : Prop where
is_func : isSingleValued F
dom_eq : dom F = A
ran_ss : ran F ⊆ B
/--
Indicates `Relation` `F` is a function from `A` to `ran F = B`.
-/
def mapsOnto (F : HRelation α β) (A : Set α) (B : Set β) :=
isSingleValued F ∧ dom F = A ∧ ran F = B
structure mapsOnto (F : HRelation α β) (A : Set α) (B : Set β) : Prop where
is_func : isSingleValued F
dom_eq : dom F = A
ran_eq : ran F = B
/-! ## Composition -/
@ -493,7 +497,7 @@ def toOrderedPairs (R : Relation α) : Set (Set (Set α)) :=
/--
A binary `Relation` `R` is **reflexive** on `A` **iff** `xRx` for all `x ∈ A`.
-/
def isReflexive (R : Relation α) (A : Set α) := ∀ a ∈ A, (a, a) ∈ R
def isReflexive (R : Relation α) (A : Set α) := ∀ x ∈ A, (x, x) ∈ R
/--
A binary `Relation` `R` is **symmetric** **iff** whenever `xRy` then `yRx`.
@ -509,30 +513,100 @@ def isTransitive (R : Relation α) :=
/--
`Relation` `R` is an **equivalence relation** on set `A` **iff** `R` is a binary
relation that is relexive on `A`, symmetric, and transitive.
relation on `A` that is relexive on `A`, symmetric, and transitive.
-/
def isEquivalence (R : Relation α) (A : Set α) :=
isReflexive R A ∧ isSymmetric R ∧ isTransitive R
structure isEquivalence (R : Relation α) (A : Set α) : Prop where
b_on : fld R ⊆ A
refl : isReflexive R A
symm : isSymmetric R
trans : isTransitive R
/--
A set of members related to `x` via `Relation` `R`.
The term "neighborhood" here was chosen to reflect this relationship between `x`
and the members of the set. It isn't standard in anyway.
-/
def neighborhood (R : Relation α) (x : α) := { t | (x, t) ∈ R }
/--
Assume that `R` is an equivalence relation on `A` and that `x` and `y` belong
to `A`. Then `[x]_R = [y]_R ↔ xRy`.
-/
theorem neighborhood_iff_mem {R : Set.Relation α} {A : Set α} {x y : α}
(hR : isEquivalence R A) (_ : x ∈ A) (hy : y ∈ A)
: neighborhood R x = neighborhood R y ↔ (x, y) ∈ R := by
apply Iff.intro
· intro h
have : y ∈ neighborhood R y := hR.refl y hy
rwa [← h] at this
· intro h
rw [Set.ext_iff]
intro t
apply Iff.intro
· intro ht
have := hR.symm h
exact hR.trans this ht
· intro ht
exact hR.trans h ht
/--
A **partition** `Π` of a set `A` is a set of nonempty subsets of `A` that is
disjoint and exhaustive.
-/
def isPartition (P : Set (Set α)) (A : Set α) :=
(∀ p ∈ P, Set.Nonempty p) ∧
(∀ a ∈ P, ∀ b, b ∈ P → a ≠ b → a ∩ b = ∅) ∧
(∀ a ∈ A, ∃ p, p ∈ P ∧ a ∈ p)
structure isPartition (P : Set (Set α)) (A : Set α) : Prop where
p_subset : ∀ p ∈ P, p ⊆ A
nonempty : ∀ p ∈ P, Set.Nonempty p
disjoint : ∀ a ∈ P, ∀ b, b ∈ P → a ≠ b → a ∩ b = ∅
exhaustive : ∀ a ∈ A, ∃ p, p ∈ P ∧ a ∈ p
/--
A cell of some partition induced by `Relation` `R`.
The partition `A / R` induced by an equivalence relation `R`.
-/
def cell (R : Relation α) (x : α) := { t | (x, t) ∈ R }
def modEquiv {A : Set α} {R : Relation α} (_ : isEquivalence R A) :=
{neighborhood R x | x ∈ A}
/--
The equivalence class of `x` modulo `R`.
Show the sets formed by `modEquiv` do indeed form a `partition`.
-/
def isEquivClass (R : Relation α) (A : Set α) (x : α) :=
isEquivalence R A ∧ x ∈ fld R
theorem modEquiv_partition {A : Set α} {R : Relation α} (hR : isEquivalence R A)
: isPartition (modEquiv hR) A := by
refine ⟨?_, ?_, ?_, ?_⟩
· intro p hp
have ⟨x, hx⟩ := hp
rw [← hx.right]
show ∀ t, t ∈ neighborhood R x → t ∈ A
intro t ht
have : t ∈ fld R := Or.inr (mem_pair_imp_snd_mem_ran ht)
exact hR.b_on this
· intro p hp
have ⟨x, hx⟩ := hp
refine ⟨x, ?_⟩
rw [← hx.right]
exact hR.refl x hx.left
· intro X hX Y hY nXY
by_contra nh
have nh' : Set.Nonempty (X ∩ Y) := by
rw [← Set.nmem_singleton_empty]
exact nh
have ⟨x, hx⟩ := hX
have ⟨y, hy⟩ := hY
have ⟨z, hz⟩ := nh'
rw [← hx.right, ← hy.right] at hz
unfold neighborhood at hz
simp only [mem_inter_iff, mem_setOf_eq] at hz
have hz_mem : z ∈ A := by
have : z ∈ fld R := Or.inr (mem_pair_imp_snd_mem_ran hz.left)
exact hR.b_on this
rw [
← neighborhood_iff_mem hR hx.left hz_mem,
← neighborhood_iff_mem hR hy.left hz_mem,
hx.right, hy.right
] at hz
rw [hz.left, hz.right] at nXY
simp only [ne_eq, not_true] at nXY
· intro x hx
exact ⟨neighborhood R x, ⟨x, hx, rfl⟩, hR.refl x hx⟩
end Relation