Enderton (set). Add Lean scaffolding for finite set theorems/bijections.
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@ -3171,8 +3171,8 @@
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\end{proof}
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\end{proof}
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\subsection{\unverified{Bijections are Two-Sided Inverses}}%
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\subsection{\unverified{Bijections and Inverses}}%
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\hyperlabel{sub:bijections-two-sided-inverses}
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\hyperlabel{sub:bijections-inverses}
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\begin{corollary}
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\begin{corollary}
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A function $f$ is a one-to-one correspondence if and only if it has a left
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A function $f$ is a one-to-one correspondence if and only if it has a left
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@ -3188,6 +3188,30 @@
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inverse.
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inverse.
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\end{proof}
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\end{proof}
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\subsection{\unverified{Left and Right Inverses and Two-Sided Inverses}}%
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\hyperlabel{sub:left-right-inverse-two-sided-inverse}
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\begin{lemma}
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Let $f$ be a function with left inverse $g_1$ and right inverse $g_2$.
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Then $g_1 = g_2 = f^{-1}$.
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\end{lemma}
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\begin{proof}
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Let $I$ denote the identity map with appropriate domain and codomain
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depending on placement in the following:
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\begin{align*}
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g_1
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& = g_1 \circ I \\
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& = g_1 \circ (f \circ g_2) \\
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& = (g_1 \circ f) \circ g_2 \\
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& = I \circ g_2 \\
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& = g_2.
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\end{align*}
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By \nameref{sub:bijections-inverses}, $f$ is a bijection meaning $f^{-1}$
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is both a left and right inverse.
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Hence $g_1 = g_2 = f^{-1}$.
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\end{proof}
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\subsection{\verified{Theorem 3K(a)}}%
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\subsection{\verified{Theorem 3K(a)}}%
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\hyperlabel{sub:theorem-3k-a}
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\hyperlabel{sub:theorem-3k-a}
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@ -67,7 +67,12 @@ No natural number is equinumerous to a proper subset of itself.
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-/
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-/
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theorem pigeonhole_principle (m n : ℕ) (hm : m < n)
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theorem pigeonhole_principle (m n : ℕ) (hm : m < n)
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: ∀ f : Fin m → Fin n, ¬ Function.Bijective f := by
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: ∀ f : Fin m → Fin n, ¬ Function.Bijective f := by
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sorry
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induction n with
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| zero =>
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intro f hf
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simp at hm
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| succ n ih =>
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sorry
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/-- #### Corollary 6C
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/-- #### Corollary 6C
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@ -97,8 +102,8 @@ theorem corollary_6d_b
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Any finite set is equinumerous to a unique natural number.
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Any finite set is equinumerous to a unique natural number.
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-/
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-/
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theorem corollary_6e (S : Set α) (f : S → Fin n) (hf : Function.Bijective f)
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theorem corollary_6e (S : Set α) (hn : S ≃ Fin n) (hm : S ≃ Fin m)
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: S ≃ Fin m → m = n := by
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: m = n := by
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sorry
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sorry
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/-- #### Lemma 6F
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/-- #### Lemma 6F
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@ -106,10 +111,14 @@ theorem corollary_6e (S : Set α) (f : S → Fin n) (hf : Function.Bijective f)
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If `C` is a proper subset of a natural number `n`, then `C ≈ m` for some `m`
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If `C` is a proper subset of a natural number `n`, then `C ≈ m` for some `m`
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less than `n`.
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less than `n`.
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-/
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-/
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lemma lemma_6f {n : ℕ} (C S : Finset ℕ) (hC : C ⊂ S) (hS : S ≃ Fin n)
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lemma lemma_6f {n : ℕ} (hC : C ⊂ Finset.range n)
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: ∃ m : ℕ, m < n ∧ ∃ f : C → Fin m, Function.Bijective f := by
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: ∃ m : ℕ, m < n ∧ ∃ f : C → Fin m, Function.Bijective f := by
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sorry
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sorry
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theorem corollary_6g (S S' : Set α) (hS : Finite S) (hS' : S' ⊆ S)
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: Finite S' := by
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sorry
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/-- #### Exercise 6.1
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/-- #### Exercise 6.1
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Show that the equation
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Show that the equation
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