Enderton (set). Theorem 4N, Corollary 4P.

finite-set-exercises
Joshua Potter 2023-08-11 08:17:30 -06:00
parent aae9eac4d0
commit 656530fed9
2 changed files with 136 additions and 28 deletions

View File

@ -6895,7 +6895,7 @@
\end{proof} \end{proof}
\subsection{\verified{% \subsection{\unverified{%
Trichotomy Law for \texorpdfstring{$\omega$}{Natural Numbers}}}% Trichotomy Law for \texorpdfstring{$\omega$}{Natural Numbers}}}%
\hyperlabel{sub:trichotomy-law-natural-numbers} \hyperlabel{sub:trichotomy-law-natural-numbers}
@ -6904,8 +6904,9 @@
$$m \in n, \quad m = n, \quad n \in m$$ holds. $$m \in n, \quad m = n, \quad n \in m$$ holds.
\end{theorem} \end{theorem}
\code{Bookshelf/Enderton/Set/Chapter\_4} \lean{Mathlib/Order/RelClasses}{IsAsymm}
{Enderton.Set.Chapter\_4.trichotomy\_law\_for\_nat}
\lean{Mathlib/Init/Algebra/Classes}{IsTrichotomous}
\begin{proof} \begin{proof}
@ -6974,7 +6975,7 @@
\end{proof} \end{proof}
\subsection{\verified{% \subsection{\unverified{%
Linear Ordering on \texorpdfstring{$\omega$}{Natural Numbers}}}% Linear Ordering on \texorpdfstring{$\omega$}{Natural Numbers}}}%
\hyperlabel{sub:linear-ordering-natural-numbers} \hyperlabel{sub:linear-ordering-natural-numbers}
@ -6987,8 +6988,8 @@
is a linear ordering on $\omega$. is a linear ordering on $\omega$.
\end{theorem} \end{theorem}
\code{Bookshelf/Enderton/Set/Chapter\_4} \lean{Mathlib/Init/Algebra/Order}
{Enderton.Set.Chapter\_4.linear\_ordering\_on\_nat} {LinearOrder.isStrictTotalOrder\_of\_linearOrder}
\begin{proof} \begin{proof}
@ -7083,23 +7084,32 @@
\end{proof} \end{proof}
\subsection{\pending{Theorem 4N}}% \subsection{\verified{Theorem 4N}}%
\hyperlabel{sub:theorem-4n} \hyperlabel{sub:theorem-4n}
\begin{theorem}[4N] \begin{theorem}[4N]
For any natural numbers $n$, $m$, and $p$, $$m \in n \iff m + p \in n + p.$$ For any natural numbers $n$, $m$, and $p$,
If, in addition, $p \neq 0$, then $$m \in n \iff m \cdot p \in n \cdot p.$$ \begin{equation}
m \in n \iff m + p \in n + p. \tag{i}
\end{equation}
If, in addition, $p \neq 0$, then
\begin{equation}
m \in n \iff m \cdot p \in n \cdot p. \tag{ii}
\end{equation}
\end{theorem} \end{theorem}
\code{Enderton/Set/Chapter\_4}
{Enderton.Set.Chapter\_4.theorem\_4n\_i}
\lean{Std/Data/Nat/Lemmas}{Nat.add\_lt\_add\_iff\_right} \lean{Std/Data/Nat/Lemmas}{Nat.add\_lt\_add\_iff\_right}
\code{Enderton/Set/Chapter\_4}
{Enderton.Set.Chapter\_4.theorem\_4n\_ii}
\lean{Init/Data/Nat/Basic}{Nat.mul\_lt\_mul\_of\_pos\_right} \lean{Init/Data/Nat/Basic}{Nat.mul\_lt\_mul\_of\_pos\_right}
\begin{proof} \begin{proof}
We prove that (i) $m \in n$ iff $m + p \in n + p$ and,
(ii) if $p \neq 0$, then $m \in n$ iff $m \cdot p \in n \cdot p$.
\paragraph{(i)}% \paragraph{(i)}%
\hyperlabel{par:theorem-4n-i} \hyperlabel{par:theorem-4n-i}
@ -7197,7 +7207,7 @@
\end{proof} \end{proof}
\subsection{\pending{Corollary 4P}}% \subsection{\verified{Corollary 4P}}%
\hyperlabel{sub:corollary-4p} \hyperlabel{sub:corollary-4p}
\begin{corollary}[4P] \begin{corollary}[4P]
@ -7210,10 +7220,13 @@
\end{align} \end{align}
\end{corollary} \end{corollary}
\code{Common/Nat/Basic}{Nat.mul\_right\_cancel} \code{Enderton/Set/Chapter\_4}
{Enderton.Set.Chapter\_4.corollary\_4p\_i}
\lean{Init/Data/Nat/Basic}{Nat.add\_right\_cancel} \lean{Init/Data/Nat/Basic}{Nat.add\_right\_cancel}
\code{Common/Nat/Basic}{Nat.mul\_right\_cancel}
\begin{proof} \begin{proof}
\paragraph{\eqref{sub:corollary-4p-eq1}}% \paragraph{\eqref{sub:corollary-4p-eq1}}%
@ -7320,7 +7333,7 @@
Therefore it isn't possible $f(n^+) \in f(n)$ for all $n \in \omega$. Therefore it isn't possible $f(n^+) \in f(n)$ for all $n \in \omega$.
\end{proof} \end{proof}
\subsection{\sorry{% \subsection{\pending{%
Strong Induction Principle for \texorpdfstring{$\omega$}{Natural Numbers}}}% Strong Induction Principle for \texorpdfstring{$\omega$}{Natural Numbers}}}%
\hyperlabel{sub:strong-induction-principle-natural-numbers} \hyperlabel{sub:strong-induction-principle-natural-numbers}

View File

@ -280,28 +280,123 @@ theorem zero_least_nat (n : )
rw [hm] rw [hm]
exact Nat.succ_pos m exact Nat.succ_pos m
/-- #### Trichotomy Law for ω /-! #### Theorem 4N
For any natural numbers `m` and `n`, exactly one of the three conditions For any natural numbers `n`, `m`, and `p`,
``` ```
m ∈ n, m = n, n ∈ m m ∈ n ↔ m ⬝ p ∈ n ⬝ p.
```
If, in addition, `p ≠ 0`, then
```
m ∈ n ↔ m ⬝ p ∈ n ⬝ p.
``` ```
holds.
-/ -/
theorem trichotomy_law_for_nat
: IsAsymm LT.lt ∧ IsTrichotomous LT.lt :=
⟨instIsAsymmLtToLT, instIsTrichotomousLtToLTToPreorderToPartialOrder⟩
/-- #### Linear Ordering on ω theorem theorem_4n_i (m n p : )
: m < n ↔ m + p < n + p := by
have hf : ∀ m n : , m < n → m + p < n + p := by
induction p with
| zero => simp
| succ p ih =>
intro m n hp
have := ih m n hp
rw [← Nat.succ_lt_succ_iff] at this
have h₁ : (m + p).succ = m + p.succ := rfl
have h₂ : (n + p).succ = n + p.succ := rfl
rwa [← h₁, ← h₂]
apply Iff.intro
· exact hf m n
· intro h
match @trichotomous LT.lt _ m n with
| Or.inl h₁ =>
exact h₁
| Or.inr (Or.inl h₁) =>
rw [← h₁] at h
exact absurd h (lemma_4l_b (m + p))
| Or.inr (Or.inr h₁) =>
have := hf n m h₁
exact absurd this (Nat.lt_asymm h)
Relation #check Nat.add_lt_add_iff_right
theorem theorem_4n_ii (m n p : )
: m < n ↔ m * p.succ < n * p.succ := by
have hf : ∀ m n : , m < n → m * p.succ < n * p.succ := by
intro m n hp₁
induction p with
| zero =>
simp only [Nat.mul_one]
exact hp₁
| succ p ih =>
have hp₂ : m * p.succ < n * p.succ := by
by_cases hp₃ : p = 0
· rw [hp₃] at *
simp only [Nat.mul_one] at *
exact hp₁
· exact ih
calc m * p.succ.succ
_ = m * p.succ + m := rfl
_ < n * p.succ + m := (theorem_4n_i (m * p.succ) (n * p.succ) m).mp hp₂
_ = m + n * p.succ := by rw [theorem_4k_2]
_ < n + n * p.succ := (theorem_4n_i m n (n * p.succ)).mp hp₁
_ = n * p.succ + n := by rw [theorem_4k_2]
_ = n * p.succ.succ := rfl
apply Iff.intro
· exact hf m n
· intro hp
match @trichotomous LT.lt _ m n with
| Or.inl h₁ =>
exact h₁
| Or.inr (Or.inl h₁) =>
rw [← h₁] at hp
exact absurd hp (lemma_4l_b (m * p.succ))
| Or.inr (Or.inr h₁) =>
have := hf n m h₁
exact absurd this (Nat.lt_asymm hp)
#check Nat.mul_lt_mul_of_pos_right
/-! #### Corollary 4P
The following cancellation laws hold for `m`, `n`, and `p` in `ω`:
``` ```
∈_ω = {⟨m, n⟩ ∈ ω × ω | m ∈ n} m + p = n + p ⇒ m = n,
m ⬝ p = n ⬝ p ∧ p ≠ 0 ⇒ m = n.
``` ```
is a linear ordering on `ω`.
-/ -/
theorem linear_ordering_on_nat
: IsStrictTotalOrder LT.lt := isStrictTotalOrder_of_linearOrder theorem corollary_4p_i (m n p : ) (h : m + p = n + p)
: m = n := by
match @trichotomous LT.lt _ m n with
| Or.inl h₁ =>
rw [theorem_4n_i m n p, h] at h₁
exact absurd h₁ (lemma_4l_b (n + p))
| Or.inr (Or.inl h₁) =>
exact h₁
| Or.inr (Or.inr h₁) =>
rw [theorem_4n_i n m p, h] at h₁
exact absurd h₁ (lemma_4l_b (n + p))
#check Nat.add_right_cancel
/-- #### Well Ordering of ω
Let `A` be a nonempty subset of `ω`. Then there is some `m ∈ A` such that
`m ≤ n` for all `n ∈ A`.
-/
theorem well_ordering_nat (A : Set ) (hA : Set.Nonempty A)
: ∃ m ∈ A, ∀ n, n ∈ A → m ≤ n := by
sorry
/-- #### Strong Induction Principle for ω
Let `A` be a subset of `ω`, and assume that for every `n ∈ ω`, if every number
less than `n` is in `A`, then `n ∈ A`. Then `A = ω`.
-/
theorem strong_induction_principle_nat (A : Set )
(h : ∀ n : , (∀ m : , m < n → m ∈ A) → n ∈ A)
: A = Set.univ := by
sorry
/-- #### Exercise 4.1 /-- #### Exercise 4.1