Enderton (set). Theorem 4N, Corollary 4P.
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@ -6895,7 +6895,7 @@
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\end{proof}
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\subsection{\verified{%
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\subsection{\unverified{%
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Trichotomy Law for \texorpdfstring{$\omega$}{Natural Numbers}}}%
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\hyperlabel{sub:trichotomy-law-natural-numbers}
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@ -6904,8 +6904,9 @@
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$$m \in n, \quad m = n, \quad n \in m$$ holds.
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\end{theorem}
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\code{Bookshelf/Enderton/Set/Chapter\_4}
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{Enderton.Set.Chapter\_4.trichotomy\_law\_for\_nat}
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\lean{Mathlib/Order/RelClasses}{IsAsymm}
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\lean{Mathlib/Init/Algebra/Classes}{IsTrichotomous}
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\begin{proof}
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@ -6974,7 +6975,7 @@
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\end{proof}
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\subsection{\verified{%
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\subsection{\unverified{%
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Linear Ordering on \texorpdfstring{$\omega$}{Natural Numbers}}}%
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\hyperlabel{sub:linear-ordering-natural-numbers}
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@ -6987,8 +6988,8 @@
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is a linear ordering on $\omega$.
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\end{theorem}
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\code{Bookshelf/Enderton/Set/Chapter\_4}
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{Enderton.Set.Chapter\_4.linear\_ordering\_on\_nat}
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\lean{Mathlib/Init/Algebra/Order}
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{LinearOrder.isStrictTotalOrder\_of\_linearOrder}
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\begin{proof}
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@ -7083,23 +7084,32 @@
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\end{proof}
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\subsection{\pending{Theorem 4N}}%
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\subsection{\verified{Theorem 4N}}%
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\hyperlabel{sub:theorem-4n}
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\begin{theorem}[4N]
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For any natural numbers $n$, $m$, and $p$, $$m \in n \iff m + p \in n + p.$$
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If, in addition, $p \neq 0$, then $$m \in n \iff m \cdot p \in n \cdot p.$$
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For any natural numbers $n$, $m$, and $p$,
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\begin{equation}
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m \in n \iff m + p \in n + p. \tag{i}
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\end{equation}
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If, in addition, $p \neq 0$, then
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\begin{equation}
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m \in n \iff m \cdot p \in n \cdot p. \tag{ii}
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\end{equation}
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\end{theorem}
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\code{Enderton/Set/Chapter\_4}
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{Enderton.Set.Chapter\_4.theorem\_4n\_i}
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\lean{Std/Data/Nat/Lemmas}{Nat.add\_lt\_add\_iff\_right}
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\code{Enderton/Set/Chapter\_4}
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{Enderton.Set.Chapter\_4.theorem\_4n\_ii}
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\lean{Init/Data/Nat/Basic}{Nat.mul\_lt\_mul\_of\_pos\_right}
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\begin{proof}
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We prove that (i) $m \in n$ iff $m + p \in n + p$ and,
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(ii) if $p \neq 0$, then $m \in n$ iff $m \cdot p \in n \cdot p$.
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\paragraph{(i)}%
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\hyperlabel{par:theorem-4n-i}
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@ -7197,7 +7207,7 @@
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\end{proof}
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\subsection{\pending{Corollary 4P}}%
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\subsection{\verified{Corollary 4P}}%
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\hyperlabel{sub:corollary-4p}
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\begin{corollary}[4P]
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@ -7210,10 +7220,13 @@
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\end{align}
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\end{corollary}
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\code{Common/Nat/Basic}{Nat.mul\_right\_cancel}
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\code{Enderton/Set/Chapter\_4}
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{Enderton.Set.Chapter\_4.corollary\_4p\_i}
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\lean{Init/Data/Nat/Basic}{Nat.add\_right\_cancel}
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\code{Common/Nat/Basic}{Nat.mul\_right\_cancel}
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\begin{proof}
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\paragraph{\eqref{sub:corollary-4p-eq1}}%
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@ -7320,7 +7333,7 @@
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Therefore it isn't possible $f(n^+) \in f(n)$ for all $n \in \omega$.
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\end{proof}
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\subsection{\sorry{%
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\subsection{\pending{%
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Strong Induction Principle for \texorpdfstring{$\omega$}{Natural Numbers}}}%
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\hyperlabel{sub:strong-induction-principle-natural-numbers}
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@ -280,28 +280,123 @@ theorem zero_least_nat (n : ℕ)
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rw [hm]
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exact Nat.succ_pos m
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/-- #### Trichotomy Law for ω
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/-! #### Theorem 4N
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For any natural numbers `m` and `n`, exactly one of the three conditions
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For any natural numbers `n`, `m`, and `p`,
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```
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m ∈ n, m = n, n ∈ m
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m ∈ n ↔ m ⬝ p ∈ n ⬝ p.
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```
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If, in addition, `p ≠ 0`, then
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```
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m ∈ n ↔ m ⬝ p ∈ n ⬝ p.
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```
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holds.
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-/
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theorem trichotomy_law_for_nat
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: IsAsymm ℕ LT.lt ∧ IsTrichotomous ℕ LT.lt :=
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⟨instIsAsymmLtToLT, instIsTrichotomousLtToLTToPreorderToPartialOrder⟩
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/-- #### Linear Ordering on ω
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theorem theorem_4n_i (m n p : ℕ)
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: m < n ↔ m + p < n + p := by
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have hf : ∀ m n : ℕ, m < n → m + p < n + p := by
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induction p with
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| zero => simp
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| succ p ih =>
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intro m n hp
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have := ih m n hp
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rw [← Nat.succ_lt_succ_iff] at this
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have h₁ : (m + p).succ = m + p.succ := rfl
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have h₂ : (n + p).succ = n + p.succ := rfl
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rwa [← h₁, ← h₂]
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apply Iff.intro
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· exact hf m n
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· intro h
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match @trichotomous ℕ LT.lt _ m n with
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| Or.inl h₁ =>
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exact h₁
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| Or.inr (Or.inl h₁) =>
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rw [← h₁] at h
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exact absurd h (lemma_4l_b (m + p))
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| Or.inr (Or.inr h₁) =>
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have := hf n m h₁
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exact absurd this (Nat.lt_asymm h)
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Relation
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#check Nat.add_lt_add_iff_right
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theorem theorem_4n_ii (m n p : ℕ)
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: m < n ↔ m * p.succ < n * p.succ := by
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have hf : ∀ m n : ℕ, m < n → m * p.succ < n * p.succ := by
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intro m n hp₁
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induction p with
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| zero =>
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simp only [Nat.mul_one]
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exact hp₁
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| succ p ih =>
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have hp₂ : m * p.succ < n * p.succ := by
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by_cases hp₃ : p = 0
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· rw [hp₃] at *
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simp only [Nat.mul_one] at *
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exact hp₁
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· exact ih
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calc m * p.succ.succ
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_ = m * p.succ + m := rfl
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_ < n * p.succ + m := (theorem_4n_i (m * p.succ) (n * p.succ) m).mp hp₂
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_ = m + n * p.succ := by rw [theorem_4k_2]
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_ < n + n * p.succ := (theorem_4n_i m n (n * p.succ)).mp hp₁
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_ = n * p.succ + n := by rw [theorem_4k_2]
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_ = n * p.succ.succ := rfl
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apply Iff.intro
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· exact hf m n
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· intro hp
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match @trichotomous ℕ LT.lt _ m n with
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| Or.inl h₁ =>
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exact h₁
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| Or.inr (Or.inl h₁) =>
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rw [← h₁] at hp
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exact absurd hp (lemma_4l_b (m * p.succ))
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| Or.inr (Or.inr h₁) =>
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have := hf n m h₁
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exact absurd this (Nat.lt_asymm hp)
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#check Nat.mul_lt_mul_of_pos_right
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/-! #### Corollary 4P
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The following cancellation laws hold for `m`, `n`, and `p` in `ω`:
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```
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∈_ω = {⟨m, n⟩ ∈ ω × ω | m ∈ n}
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m + p = n + p ⇒ m = n,
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m ⬝ p = n ⬝ p ∧ p ≠ 0 ⇒ m = n.
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```
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is a linear ordering on `ω`.
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-/
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theorem linear_ordering_on_nat
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: IsStrictTotalOrder ℕ LT.lt := isStrictTotalOrder_of_linearOrder
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theorem corollary_4p_i (m n p : ℕ) (h : m + p = n + p)
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: m = n := by
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match @trichotomous ℕ LT.lt _ m n with
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| Or.inl h₁ =>
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rw [theorem_4n_i m n p, h] at h₁
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exact absurd h₁ (lemma_4l_b (n + p))
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| Or.inr (Or.inl h₁) =>
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exact h₁
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| Or.inr (Or.inr h₁) =>
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rw [theorem_4n_i n m p, h] at h₁
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exact absurd h₁ (lemma_4l_b (n + p))
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#check Nat.add_right_cancel
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/-- #### Well Ordering of ω
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Let `A` be a nonempty subset of `ω`. Then there is some `m ∈ A` such that
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`m ≤ n` for all `n ∈ A`.
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-/
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theorem well_ordering_nat (A : Set ℕ) (hA : Set.Nonempty A)
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: ∃ m ∈ A, ∀ n, n ∈ A → m ≤ n := by
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sorry
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/-- #### Strong Induction Principle for ω
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Let `A` be a subset of `ω`, and assume that for every `n ∈ ω`, if every number
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less than `n` is in `A`, then `n ∈ A`. Then `A = ω`.
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-/
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theorem strong_induction_principle_nat (A : Set ℕ)
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(h : ∀ n : ℕ, (∀ m : ℕ, m < n → m ∈ A) → n ∈ A)
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: A = Set.univ := by
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sorry
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/-- #### Exercise 4.1
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