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Joshua Potter 4fa4e8dc07 Annotate the pigeonhole principle with latex proof. 2023-11-11 09:06:40 -07:00
2 changed files with 110 additions and 90 deletions

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@ -72,28 +72,72 @@ lemma pigeonhole_principle_aux (n : )
∀ f : , ∀ f : ,
Set.MapsTo f M (Set.Iio n) ∧ Set.InjOn f M → Set.MapsTo f M (Set.Iio n) ∧ Set.InjOn f M →
¬ Set.SurjOn f M (Set.Iio n) := by ¬ Set.SurjOn f M (Set.Iio n) := by
/-
> Let
>
> `S = {n ∈ ω | ∀ M ⊂ n, every one-to-one function f: M → n is not onto}`. (1)
>
> We show that (i) `0 ∈ S` and (ii) if `n ∈ S`, then so is `n⁺`. Afterward we
> prove (iii) the theorem statement.
-/
induction n with induction n with
/-
## (i)
> By definition, `0 = ∅`.
-/
| zero => | zero =>
intro _ hM intro _ hM
unfold Set.Iio at hM unfold Set.Iio at hM
simp only [Nat.zero_eq, not_lt_zero', Set.setOf_false] at hM simp only [Nat.zero_eq, not_lt_zero', Set.setOf_false] at hM
/-
> Then `0` has no proper subsets.
-/
rw [Set.ssubset_empty_iff_false] at hM rw [Set.ssubset_empty_iff_false] at hM
/-
> Hence `0 ∈ S` vacuously.
-/
exact False.elim hM exact False.elim hM
/-
## (ii)
> Suppose `n ∈ S` and `M ⊂ n⁺`. Furthermore, let `f: M → n⁺` be a one-to-one
> function.
-/
| succ n ih => | succ n ih =>
intro M hM f ⟨hf_maps, hf_inj⟩ hf_surj intro M hM f ⟨hf_maps, hf_inj⟩ hf_surj
/-
> If `M = ∅`, it vacuously holds that `f` is not onto `n⁺`.
-/
by_cases hM' : M = ∅ by_cases hM' : M = ∅
· rw [hM', Set.SurjOn_emptyset_Iio_iff_eq_zero] at hf_surj · rw [hM', Set.SurjOn_emptyset_Iio_iff_eq_zero] at hf_surj
simp at hf_surj simp at hf_surj
/-
> Otherwise `M ≠ 0`. Because `M` is finite, the trichotomy law for `ω` implies
> the existence of a largest member `p ∈ M`. There are two cases to consider:
-/
by_cases h : ¬ ∃ t, t ∈ M ∧ f t = n by_cases h : ¬ ∃ t, t ∈ M ∧ f t = n
-- Trivial case. `f` must not be onto if this is the case. /-
### Case 1
> `n ∉ ran f`.
> Then `f` is not onto `n⁺`.
-/
· have ⟨t, ht⟩ := hf_surj (show n ∈ _ by simp) · have ⟨t, ht⟩ := hf_surj (show n ∈ _ by simp)
exact absurd ⟨t, ht⟩ h exact absurd ⟨t, ht⟩ h
/-
-- Continue under the assumption `n ∈ ran f`. ### Case 2
> `n ∈ ran f`.
> Then there exists some `t ∈ M` such that `⟨t, n⟩ ∈ f`.
-/
have ⟨t, ht₁, ht₂⟩ := not_not.mp h have ⟨t, ht₁, ht₂⟩ := not_not.mp h
/-
> Define `f': M → n⁺` given by
>
> `f'(p) = f(t) = n`
> `f'(t) = f(p)`
> `f'(x) = f(x)` for all other `x`.
>
> That is, `f'` is a variant of `f` in which the largest element of its domain
> (i.e. `p`) corresponds to value `n`.
-/
-- `M ≠ ∅` so `∃ p, ∀ x ∈ M, p ≥ x`, i.e. a maximum member. -- `M ≠ ∅` so `∃ p, ∀ x ∈ M, p ≥ x`, i.e. a maximum member.
have ⟨p, hp₁, hp₂⟩ : ∃ p ∈ M, ∀ x, x ∈ M → p ≥ x := by have ⟨p, hp₁, hp₂⟩ : ∃ p ∈ M, ∀ x, x ∈ M → p ≥ x := by
refine subset_finite_max_nat (show Set.Finite M from ?_) ?_ ?_ refine subset_finite_max_nat (show Set.Finite M from ?_) ?_ ?_
@ -104,69 +148,19 @@ lemma pigeonhole_principle_aux (n : )
exact Set.nmem_singleton_empty.mp hM' exact Set.nmem_singleton_empty.mp hM'
· show M ⊆ M · show M ⊆ M
exact Eq.subset rfl exact Eq.subset rfl
/-
-- `g` is a variant of `f` in which the largest element of its domain > Next define `g = f' - {⟨p, n⟩}`. Then `g` is a function mapping `M - {p}` to
-- (i.e. `p`) corresponds to value `n`. > `n`.
let g x := if x = p then n else if x = t then f p else f x -/
let g := Set.Function.swap f p t
have hg_maps : Set.MapsTo g M (Set.Iio (n + 1)) := by /-
intro x hx > Since `f` is one-to-one, `f'` and `g` are also one-to-one.
dsimp only -/
by_cases hx₁ : x = p have hg_maps := Set.Function.swap_MapsTo_self hp₁ ht₁ hf_maps
· rw [hx₁] have hg_inj := Set.Function.swap_InjOn_self hp₁ ht₁ hf_inj
simp /-
· rw [if_neg hx₁] > Then (1) indicates `g` must not be onto `n`.
by_cases hx₂ : x = t -/
· rw [hx₂]
simp only [ite_true, Set.mem_Iio]
exact hf_maps hp₁
· rw [if_neg hx₂]
simp only [Set.mem_Iio]
exact hf_maps hx
have hg_inj : Set.InjOn g M := by
intro x₁ hx₁ x₂ hx₂ hf'
by_cases hc₁ : x₁ = p
· by_cases hc₂ : x₂ = p
· rw [hc₁, hc₂]
· dsimp at hf'
rw [hc₁] at hf'
simp only [ite_self, ite_true] at hf'
by_cases hc₃ : x₂ = t
· rw [if_neg hc₂, if_pos hc₃, ← ht₂] at hf'
rw [hc₁] at hx₁ ⊢
rw [hc₃] at hx₂ ⊢
exact hf_inj hx₁ hx₂ hf'.symm
· rw [if_neg hc₂, if_neg hc₃, ← ht₂] at hf'
have := hf_inj ht₁ hx₂ hf'
exact absurd this.symm hc₃
· by_cases hc₂ : x₂ = p
· rw [hc₂] at hf'
simp only [ite_self, ite_true] at hf'
by_cases hc₃ : x₁ = t
· rw [if_neg hc₁, if_pos hc₃, ← ht₂] at hf'
rw [hc₃] at hx₁ ⊢
rw [hc₂] at hx₂ ⊢
have := hf_inj hx₂ hx₁ hf'
exact this.symm
· rw [if_neg hc₁, if_neg hc₃, ← ht₂] at hf'
have := hf_inj hx₁ ht₁ hf'
exact absurd this hc₃
· dsimp only at hf'
rw [if_neg hc₁, if_neg hc₂] at hf'
by_cases hc₃ : x₁ = t
· by_cases hc₄ : x₂ = t
· rw [hc₃, hc₄]
· rw [if_pos hc₃, if_neg hc₄] at hf'
have := hf_inj hp₁ hx₂ hf'
exact absurd this.symm hc₂
· by_cases hc₄ : x₂ = t
· rw [if_neg hc₃, if_pos hc₄] at hf'
have := hf_inj hx₁ hp₁ hf'
exact absurd this hc₁
· rw [if_neg hc₃, if_neg hc₄] at hf'
exact hf_inj hx₁ hx₂ hf'
let M' := M \ {p} let M' := M \ {p}
have hM' : M' ⊂ Set.Iio n := by have hM' : M' ⊂ Set.Iio n := by
by_cases hc : p = n by_cases hc : p = n
@ -209,18 +203,28 @@ lemma pigeonhole_principle_aux (n : )
· -- `Set.MapsTo g M' (Set.Iio n)` · -- `Set.MapsTo g M' (Set.Iio n)`
intro x hx intro x hx
have hx₁ : x ∈ M := Set.mem_of_mem_diff hx have hx₁ : x ∈ M := Set.mem_of_mem_diff hx
apply Or.elim (Nat.lt_or_eq_of_lt $ hg_maps hx₁) apply Or.elim (Nat.lt_or_eq_of_lt $ hg_maps hx₁) id
· exact id intro hx₂
· intro hx₂ unfold Set.Function.swap at hx₂
rw [← show g p = n by simp] at hx₂ by_cases hc₁ : x = p
exact absurd (hg_inj hx₁ hp₁ hx₂) hx.right · exact absurd hc₁ hx.right
· rw [if_neg hc₁] at hx₂
by_cases hc₂ : x = t
· rw [if_pos hc₂, ← ht₂] at hx₂
have := hf_inj hp₁ ht₁ hx₂
rw [← hc₂] at this
exact absurd this.symm hc₁
· rw [if_neg hc₂, ← ht₂] at hx₂
have := hf_inj hx₁ ht₁ hx₂
exact absurd this hc₂
· -- `Set.InjOn g M'` · -- `Set.InjOn g M'`
intro x₁ hx₁ x₂ hx₂ hg intro x₁ hx₁ x₂ hx₂ hg
have hx₁' : x₁ ∈ M := (Set.diff_subset M {p}) hx₁ have hx₁' : x₁ ∈ M := (Set.diff_subset M {p}) hx₁
have hx₂' : x₂ ∈ M := (Set.diff_subset M {p}) hx₂ have hx₂' : x₂ ∈ M := (Set.diff_subset M {p}) hx₂
exact hg_inj hx₁' hx₂' hg exact hg_inj hx₁' hx₂' hg
/-
-- We have shown `g` isn't surjective. This is another way of saying that. > That is, there exists some `a ∈ n` such that `a ∉ ran g`.
-/
have ⟨a, ha₁, ha₂⟩ : ∃ a, a < n ∧ a ∉ g '' M' := by have ⟨a, ha₁, ha₂⟩ : ∃ a, a < n ∧ a ∉ g '' M' := by
unfold Set.SurjOn at ng_surj unfold Set.SurjOn at ng_surj
rw [Set.subset_def] at ng_surj rw [Set.subset_def] at ng_surj
@ -235,8 +239,11 @@ lemma pigeonhole_principle_aux (n : )
unfold Set.image unfold Set.image
simp only [Set.mem_Iio, Set.mem_setOf_eq, not_exists, not_and] simp only [Set.mem_Iio, Set.mem_setOf_eq, not_exists, not_and]
exact ng_surj exact ng_surj
/-
-- If `g` isn't surjective then neither is `f`. > By the trichotomy law for `ω`, `a ≠ n`. Therefore `a ∉ ran f'`.
> `ran f' = ran f` meaning `a ∉ ran f`. Because `a ∈ n ∈n⁺`, Theorem 4F implies
> `a ∈ n⁺`. Hence `f` is not onto `n⁺`.
-/
refine absurd (hf_surj $ calc a refine absurd (hf_surj $ calc a
_ < n := ha₁ _ < n := ha₁
_ < n + 1 := by simp) (show ↑a ∉ f '' M from ?_) _ < n + 1 := by simp) (show ↑a ∉ f '' M from ?_)
@ -248,17 +255,20 @@ lemma pigeonhole_principle_aux (n : )
simp only [Set.mem_Iio, Set.mem_setOf_eq, not_exists, not_and] at ha₂ ⊢ simp only [Set.mem_Iio, Set.mem_setOf_eq, not_exists, not_and] at ha₂ ⊢
intro x hx intro x hx
by_cases hxp : x = p by_cases hxp : x = p
· rw [if_pos hxp] · unfold Set.Function.swap
rw [if_pos hxp, ht₂]
exact (Nat.ne_of_lt ha₁).symm exact (Nat.ne_of_lt ha₁).symm
· refine ha₂ x ?_ · refine ha₂ x ?_
exact Set.mem_diff_of_mem hx hxp exact Set.mem_diff_of_mem hx hxp
ext x ext x
dsimp only
unfold Set.Function.swap
simp only [Set.mem_image, Set.mem_Iio] simp only [Set.mem_image, Set.mem_Iio]
apply Iff.intro apply Iff.intro
· intro ⟨y, hy₁, hy₂⟩ · intro ⟨y, hy₁, hy₂⟩
by_cases hc₁ : y = p by_cases hc₁ : y = p
· rw [if_pos hc₁] at hy₂ · rw [if_pos hc₁, ht₂] at hy₂
rw [hy₂] at ht₂ rw [hy₂] at ht₂
exact ⟨t, ht₁, ht₂⟩ exact ⟨t, ht₁, ht₂⟩
· rw [if_neg hc₁] at hy₂ · rw [if_neg hc₁] at hy₂
@ -274,15 +284,25 @@ lemma pigeonhole_principle_aux (n : )
· rw [hc₂, ht₂] at hy₂ · rw [hc₂, ht₂] at hy₂
rw [← hc₁, ← hc₂] rw [← hc₁, ← hc₂]
simp only [ite_self, ite_true] simp only [ite_self, ite_true]
exact hy₂ rwa [hc₂, ht₂]
· rw [hc₁, ← Ne.def] at hc₂ · rw [hc₁, ← Ne.def] at hc₂
rwa [if_neg hc₂.symm, if_pos rfl, ← hc₁] rwa [if_neg hc₂.symm, if_pos rfl, ← hc₁]
· by_cases hc₂ : y = t · by_cases hc₂ : y = t
· refine ⟨p, hp₁, ?_⟩ · refine ⟨p, hp₁, ?_⟩
simp only [ite_self, ite_true] simp only [ite_self, ite_true]
rwa [hc₂, ht₂] at hy₂ rwa [hc₂] at hy₂
· refine ⟨y, hy₁, ?_⟩ · refine ⟨y, hy₁, ?_⟩
rwa [if_neg hc₁, if_neg hc₂] rwa [if_neg hc₁, if_neg hc₂]
/-
### Subconclusion
> The foregoing cases are exhaustive. Hence `n⁺ ∈ S`.
## (iii)
> By (i) and (ii), `S` is an inductive set. By Theorem 4B, `S = ω`. Thus for all
> natural numbers `n`, there is no one-to-one correspondence between `n` and a
> proper subset of `n`. In other words, no natural number is equinumerous to a
> proper subset of itself.
-/
/-- /--
No natural number is equinumerous to a proper subset of itself. No natural number is equinumerous to a proper subset of itself.

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@ -5,7 +5,7 @@ import Mathlib.Data.Set.Function
Additional theorems around functions defined on sets. Additional theorems around functions defined on sets.
-/ -/
namespace Set namespace Set.Function
/-- /--
Produce a new function that swaps the outputs of the two specified inputs. Produce a new function that swaps the outputs of the two specified inputs.
@ -232,4 +232,4 @@ theorem self_iff_swap_BijOn [DecidableEq α]
: BijOn (swap f a₁ a₂) A B ↔ BijOn f A B := : BijOn (swap f a₁ a₂) A B ↔ BijOn f A B :=
⟨self_BijOn_swap ha₁ ha₂, swap_BijOn_self ha₁ ha₂⟩ ⟨self_BijOn_swap ha₁ ha₂, swap_BijOn_self ha₁ ha₂⟩
end Set end Set.Function