Enderton. Formally verify exercises 3.12 through 3.15.

finite-set-exercises
Joshua Potter 2023-07-05 12:45:51 -06:00
parent 993e9fe981
commit d3f242cf07
3 changed files with 212 additions and 19 deletions

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@ -3949,7 +3949,7 @@ Then $F = G$.
\end{proof}
\subsection{\pending{Exercise 3.12}}%
\subsection{\verified{Exercise 3.12}}%
\label{sub:exercise-3.12}
Assume that $f$ and $g$ are functions and show that
@ -3958,6 +3958,9 @@ Assume that $f$ and $g$ are functions and show that
\begin{proof}
\lean{Bookshelf/Enderton/Set/Chapter\_3}
{Enderton.Set.Chapter\_3.exercise\_3\_12}
Let $f$ and $g$ be \nameref{ref:function}s.
\paragraph{($\Rightarrow$)}%
@ -3982,7 +3985,7 @@ Assume that $f$ and $g$ are functions and show that
\end{proof}
\subsection{\pending{Exercise 3.13}}%
\subsection{\verified{Exercise 3.13}}%
\label{sub:exercise-3.13}
Assume that $f$ and $g$ are functions with $f \subseteq g$ and
@ -3991,6 +3994,9 @@ Show that $f = g$.
\begin{proof}
\lean{Bookshelf/Enderton/Set/Chapter\_3}
{Enderton.Set.Chapter\_3.exercise\_3\_13}
Let $f$ and $g$ be functions such that $f \subseteq g$ and
$\dom{g} \subseteq \dom{f}$.
By \nameref{sub:exercise-3.12}, it follows that $\dom{f} \subseteq \dom{g}$
@ -4001,7 +4007,7 @@ Show that $f = g$.
\end{proof}
\subsection{\pending{Exercise 3.14}}%
\subsection{\verified{Exercise 3.14}}%
\label{sub:exercise-3.14}
Assume that $f$ and $g$ are functions.
@ -4014,17 +4020,22 @@ Assume that $f$ and $g$ are functions.
\begin{proof}
Assume $f$ and $g$ are functions.
\statementpadding
\lean*{Bookshelf/Enderton/Set/Chapter\_3}
{Enderton.Set.Chapter\_3.exercise\_3\_14\_a}
\lean{Bookshelf/Enderton/Set/Chapter\_3}
{Enderton.Set.Chapter\_3.exercise\_3\_14\_b}
Assume $f$ and $g$ are \nameref{ref:function}s.
\paragraph{(a)}%
Consider $f \cap g$.
By definition of the intersection of sets, $f \cap g \subseteq f$.
By \nameref{sub:exercise-3.12}, $\dom{(f \cap g)} = \dom{f}$ and
$(\forall x \in \dom{(f \cap g)} (f \cap g)(x) = f(x)$.
The latter conjunct shows that, since $f$ is single-valued, $f \cap g$ must
also be single-valued.
In other words, $f \cap g$ is a function.
Since $f$ is single-valued, it trivially follows that so must $f \cap g$.
Therefore $f \cap g$ is a function.
\paragraph{(b)}%
@ -4064,7 +4075,7 @@ Assume that $f$ and $g$ are functions.
\end{proof}
\subsection{\pending{Exercise 3.15}}%
\subsection{\verified{Exercise 3.15}}%
\label{sub:exercise-3.15}
Let $\mathscr{A}$ be a set of functions such that for any $f$ and $g$ in
@ -4073,6 +4084,9 @@ Show that $\bigcup{\mathscr{A}}$ is a function.
\begin{proof}
\lean{Bookshelf/Enderton/Set/Chapter\_3}
{Enderton.Set.Chapter\_3.exercise\_3\_15}
Let $\mathscr{A}$ be a set of \nameref{ref:function}s such that for any $f$
and $g$ in $\mathscr{A}$, either $f \subseteq g$ or $g \subseteq f$.
Let $x \in \dom{\bigcup{\mathscr{A}}}$.

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@ -325,7 +325,7 @@ theorem exercise_3_7 {R : Set.Relation α}
simp only [Set.mem_singleton_iff, Set.mem_insert_iff] at this
exact hxy_mem this
section
section Relation
open Set.Relation
@ -678,11 +678,8 @@ theorem theorem_3k_c_ii {F : Set.Relation α} {A B : Set α}
unfold image at nv
simp only [Set.mem_setOf_eq] at nv
have ⟨u₁, hu₁⟩ := nv
have ⟨x, hx⟩ := hF v (mem_pair_imp_snd_mem_ran hu.right)
simp only [and_imp] at hx
have hr₁ := hx.right u (mem_pair_imp_fst_mem_dom hu.right) hu.right
have hr₂ := hx.right u₁ (mem_pair_imp_fst_mem_dom hu₁.right) hu₁.right
rw [hr₂, ← hr₁] at hu₁
have := single_rooted_eq_unique hF hu.right hu₁.right
rw [← this] at hu₁
exact absurd hu₁.left hu.left.right
exact ⟨hv₁, hv₂⟩
@ -714,6 +711,173 @@ theorem corollary_3l_iii {G : Set.Relation α} {A B : Set α}
single_valued_self_iff_single_rooted_inv.mp hG
exact (theorem_3k_c_ii hG').symm
end
/-- #### Exercise 3.12
Assume that `f` and `g` are functions and show that
```
f ⊆ g ↔ dom f ⊆ dom g ∧ (∀ x ∈ dom f) f(x) = g(x).
```
-/
theorem exercise_3_12 {f g : Set.Relation α}
(hf : f.isSingleValued) (_ : g.isSingleValued)
: f ⊆ g ↔ dom f ⊆ dom g ∧ (∀ x ∈ dom f, ∃! y : α, (x, y) ∈ f ∧ (x, y) ∈ g) := by
apply Iff.intro
· intro h
apply And.intro
· show ∀ x, x ∈ dom f → x ∈ dom g
intro x hx
have ⟨y, hy⟩ := dom_exists hx
exact mem_pair_imp_fst_mem_dom (h hy)
· intro x hx
have ⟨y, hy⟩ := dom_exists hx
refine ⟨y, ⟨hy, h hy⟩, ?_⟩
intro y₁ hy₁
exact single_valued_eq_unique hf hy₁.left hy
· intro ⟨_, hx⟩
show ∀ p, p ∈ f → p ∈ g
intro (x, y) hp
have ⟨y₁, hy₁⟩ := hx x (mem_pair_imp_fst_mem_dom hp)
rw [single_valued_eq_unique hf hp hy₁.left.left]
exact hy₁.left.right
/-- #### Exercise 3.13
Assume that `f` and `g` are functions with `f ⊆ g` and `dom g ⊆ dom f`. Show
that `f = g`.
-/
theorem exercise_3_13 {f g : Set.Relation α}
(hf : f.isSingleValued) (hg : g.isSingleValued)
(h : f ⊆ g) (h₁ : dom g ⊆ dom f)
: f = g := by
have h₂ := (exercise_3_12 hf hg).mp h
have hfg := Set.Subset.antisymm_iff.mpr ⟨h₁, h₂.left⟩
ext p
have (a, b) := p
apply Iff.intro
· intro hp
have ⟨x, hx⟩ := h₂.right a (mem_pair_imp_fst_mem_dom hp)
rw [single_valued_eq_unique hf hp hx.left.left]
exact hx.left.right
· intro hp
rw [← hfg] at h₂
have ⟨x, hx⟩ := h₂.right a (mem_pair_imp_fst_mem_dom hp)
rw [single_valued_eq_unique hg hp hx.left.right]
exact hx.left.left
/-- #### Exercise 3.14 (a)
Assume that `f` and `g` are functions. Show that `f ∩ g` is a function.
-/
theorem exercise_3_14_a {f g : Set.Relation α}
(hf : f.isSingleValued) (_ : g.isSingleValued)
: (f ∩ g).isSingleValued :=
single_valued_subset hf (Set.inter_subset_left f g)
/-- #### Exercise 3.14 (b)
Assume that `f` and `g` are functions. Show that `f g` is a function **iff**
`f(x) = g(x)` for every `x` in `(dom f) ∩ (dom g)`.
-/
theorem exercise_3_14_b {f g : Set.Relation α}
(hf : f.isSingleValued) (hg : g.isSingleValued)
: (f g).isSingleValued ↔
(∀ x ∈ dom f ∩ dom g, ∃! y, (x, y) ∈ f ∧ (x, y) ∈ g) := by
apply Iff.intro
· intro h x hx
have ⟨y₁, hy₁⟩ := hf x hx.left
have ⟨y₂, hy₂⟩ := hg x hx.right
have : y₁ = y₂ := by
have hf' : (x, y₁) ∈ f g := Or.inl hy₁.left.right
have hg' : (x, y₂) ∈ f g := Or.inr hy₂.left.right
exact single_valued_eq_unique h hf' hg'
rw [← this] at hy₂
refine ⟨y₁, ⟨hy₁.left.right, hy₂.left.right⟩, ?_⟩
intro y₃ hfy₃
exact single_valued_eq_unique hf hfy₃.left hy₁.left.right
· intro h x hx
by_cases hfx : x ∈ dom f
· by_cases hgx : x ∈ dom g
· -- `x ∈ dom f ∧ x ∈ dom g`
have ⟨y₁, hy₁⟩ := hf x hfx
have ⟨y₂, hy₂⟩ := hg x hgx
refine ⟨y₁, ⟨?_, Or.inl hy₁.left.right⟩, ?_⟩
· unfold ran
simp only [Set.mem_image, Set.mem_union, Prod.exists, exists_eq_right]
exact ⟨x, Or.inl hy₁.left.right⟩
· intro y₃ hy₃
apply Or.elim hy₃.right
· intro hxy
exact single_valued_eq_unique hf hxy hy₁.left.right
· refine fun hxy => single_valued_eq_unique hg hxy ?_
have : y₁ = y₂ := by
have ⟨z, ⟨hz, _⟩⟩ := h x ⟨hfx, hgx⟩
rw [
single_valued_eq_unique hf hy₁.left.right hz.left,
single_valued_eq_unique hg hy₂.left.right hz.right
]
rw [this]
exact hy₂.left.right
· -- `x ∈ dom f ∧ x ∉ dom g`
have ⟨y, hy⟩ := dom_exists hfx
have hxy : (x, y) ∈ f g := (Set.subset_union_left f g) hy
refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hxy, hxy⟩, ?_⟩
intro y₁ hy₁
apply Or.elim hy₁.right
· intro hx'
exact single_valued_eq_unique hf hx' hy
· intro hx'
exact absurd (mem_pair_imp_fst_mem_dom hx') hgx
· by_cases hgx : x ∈ dom g
· -- `x ∉ dom f ∧ x ∈ dom g`
have ⟨y, hy⟩ := dom_exists hgx
have hxy : (x, y) ∈ f g := (Set.subset_union_right f g) hy
refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hxy, hxy⟩, ?_⟩
intro y₁ hy₁
apply Or.elim hy₁.right
· intro hx'
exact absurd (mem_pair_imp_fst_mem_dom hx') hfx
· intro hx'
exact single_valued_eq_unique hg hx' hy
· -- `x ∉ dom f ∧ x ∉ dom g`
exfalso
unfold dom at hx
simp only [
Set.mem_image,
Set.mem_union,
Prod.exists,
exists_and_right,
exists_eq_right
] at hx
have ⟨_, hy⟩ := hx
apply Or.elim hy
· intro hz
exact absurd (mem_pair_imp_fst_mem_dom hz) hfx
· intro hz
exact absurd (mem_pair_imp_fst_mem_dom hz) hgx
/-- #### Exercise 3.15
Let `𝓐` be a set of functions such that for any `f` and `g` in `𝓐`, either
`f ⊆ g` or `g ⊆ f`. Show that ` 𝓐` is a function.
-/
theorem exercise_3_15 {𝓐 : Set (Set.Relation α)}
(h𝓐 : ∀ F ∈ 𝓐, F.isSingleValued)
(h : ∀ F, ∀ G, F ∈ 𝓐 → G ∈ 𝓐 → F ⊆ G G ⊆ F)
: isSingleValued (⋃₀ 𝓐) := by
intro x hx
have ⟨y₁, hy₁⟩ := dom_exists hx
refine ⟨y₁, ⟨mem_pair_imp_snd_mem_ran hy₁, hy₁⟩, ?_⟩
intro y₂ hy₂
have ⟨f, hf⟩ : ∃ f : Set.Relation α, f ∈ 𝓐 ∧ (x, y₁) ∈ f := hy₁
have ⟨g, hg⟩ : ∃ g : Set.Relation α, g ∈ 𝓐 ∧ (x, y₂) ∈ g := hy₂.right
apply Or.elim (h f g hf.left hg.left)
· intro hf'
have := hf' hf.right
exact single_valued_eq_unique (h𝓐 g hg.left) hg.right this
· intro hg'
have := hg' hg.right
exact single_valued_eq_unique (h𝓐 f hf.left) this hf.right
end Relation
end Enderton.Set.Chapter_3

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@ -180,7 +180,7 @@ A `Relation` `R` is said to be single-rooted **iff** for all `y ∈ ran R`, ther
exists exactly one `x` such that `⟨x, y⟩ ∈ R`.
-/
def isSingleRooted (R : Relation α) : Prop :=
∀ y ∈ R.ran, ∃! x, x ∈ R.dom ∧ (x, y) ∈ R
∀ y ∈ ran R, ∃! x, x ∈ dom R ∧ (x, y) ∈ R
/--
A single-rooted `Relation` should map the same output to the same input.
@ -204,7 +204,7 @@ exists exactly one `y` such that `⟨x, y⟩ ∈ R`.
Notice, a `Relation` that is single-valued is a function.
-/
def isSingleValued (R : Relation α) : Prop :=
∀ x ∈ R.dom, ∃! y, y ∈ R.ran ∧ (x, y) ∈ R
∀ x ∈ dom R, ∃! y, y ∈ ran R ∧ (x, y) ∈ R
/--
A single-valued `Relation` should map the same input to the same output.
@ -263,6 +263,21 @@ theorem single_valued_self_iff_single_rooted_inv {F : Set.Relation α}
conv => lhs; rw [← inv_inv_eq_self F]
rw [single_valued_inv_iff_single_rooted_self]
/--
The subset of a function must also be a function.
-/
theorem single_valued_subset {F G : Set.Relation α}
(hG : G.isSingleValued) (h : F ⊆ G)
: F.isSingleValued := by
unfold isSingleValued
intro x hx
have ⟨y, hy⟩ := dom_exists hx
unfold ExistsUnique
simp only
refine ⟨y, ⟨mem_pair_imp_snd_mem_ran hy, hy⟩, ?_⟩
intro y₁ hy₁
exact single_valued_eq_unique hG (h hy₁.right) (h hy)
/--
A `Relation` `R` is one-to-one if it is a single-rooted function.
-/