224 lines
8.3 KiB
Text
224 lines
8.3 KiB
Text
/-
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Copyright (c) 2015 Jeremy Avigad. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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Binomial coefficients, "n choose k".
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-/
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import data.nat.div data.nat.fact data.finset
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open decidable
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namespace nat
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/- choose -/
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definition choose : ℕ → ℕ → ℕ
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| 0 0 := 1
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| 0 (succ k) := 0
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| (succ n) 0 := 1
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| (succ n) (succ k) := choose n (succ k) + choose n k
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theorem choose_zero_right (n : ℕ) : choose n 0 = 1 :=
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nat.cases_on n rfl (take m, rfl)
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theorem choose_zero_succ (k : ℕ) : choose 0 (succ k) = 0 := rfl
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theorem choose_succ_succ (n k : ℕ) : choose (succ n) (succ k) = choose n (succ k) + choose n k :=
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rfl
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theorem choose_eq_zero_of_lt {n : ℕ} : ∀{k : ℕ}, n < k → choose n k = 0 :=
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nat.induction_on n
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(take k, assume H : 0 < k,
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obtain k' (H : k = succ k'), from exists_eq_succ_of_pos H,
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by rewrite H)
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(take n',
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assume IH: ∀ k, n' < k → choose n' k = 0,
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take k,
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suppose succ n' < k,
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obtain k' (keq : k = succ k'), from exists_eq_succ_of_lt this,
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have n' < k', by rewrite keq at this; apply lt_of_succ_lt_succ this,
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by rewrite [keq, choose_succ_succ, IH _ this, IH _ (lt.trans this !lt_succ_self)])
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theorem choose_self (n : ℕ) : choose n n = 1 :=
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begin
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induction n with [n, ih],
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{apply rfl},
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rewrite [choose_succ_succ, ih, choose_eq_zero_of_lt !lt_succ_self]
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end
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theorem choose_succ_self (n : ℕ) : choose (succ n) n = succ n :=
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begin
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induction n with [n, ih],
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{apply rfl},
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rewrite [choose_succ_succ, ih, choose_self, add.comm]
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end
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theorem choose_one_right (n : ℕ) : choose n 1 = n :=
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begin
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induction n with [n, ih],
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{apply rfl},
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change choose (succ n) (succ 0) = succ n,
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krewrite [choose_succ_succ, ih, choose_zero_right]
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end
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theorem choose_pos {n : ℕ} : ∀ {k : ℕ}, k ≤ n → choose n k > 0 :=
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begin
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induction n with [n, ih],
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{intros [k, H],
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have k = 0, from eq_of_le_of_ge H !zero_le,
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subst k, rewrite choose_zero_right; apply zero_lt_one},
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intro k,
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cases k with k,
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{intros, rewrite [choose_zero_right], apply zero_lt_one},
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suppose succ k ≤ succ n,
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have k ≤ n, from le_of_succ_le_succ this,
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by rewrite [choose_succ_succ]; apply add_pos_right (ih this)
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end
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-- A key identity. The proof is subtle.
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theorem succ_mul_choose_eq (n : ℕ) :
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∀ k, succ n * (choose n k) = choose (succ n) (succ k) * succ k :=
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begin
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induction n with [n, ih],
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{intro k,
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cases k with k',
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{krewrite [*choose_self, one_mul, mul_one]},
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{have H : 1 < succ (succ k'), from succ_lt_succ !zero_lt_succ,
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krewrite [one_mul, choose_zero_succ, choose_eq_zero_of_lt H, zero_mul]}},
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intro k,
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cases k with k',
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{krewrite [choose_zero_right, choose_one_right]},
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rewrite [choose_succ_succ (succ n), right_distrib, -ih (succ k')],
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rewrite [choose_succ_succ at {1}, left_distrib, *succ_mul (succ n), mul_succ, -ih k'],
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rewrite [*add.assoc, add.left_comm (choose n _)]
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end
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theorem choose_mul_fact_mul_fact {n : ℕ} :
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∀ {k : ℕ}, k ≤ n → choose n k * fact k * fact (n - k) = fact n :=
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begin
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induction n using nat.strong_induction_on with [n, ih],
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cases n with n,
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{intro k H, have k = 0, from eq_zero_of_le_zero H, rewrite this},
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intro k,
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intro H, -- k ≤ n,
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cases k with k,
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{rewrite [choose_zero_right, fact_zero, *one_mul]},
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have k ≤ n, from le_of_succ_le_succ H,
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show choose (succ n) (succ k) * fact (succ k) * fact (succ n - succ k) = fact (succ n), from
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begin
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rewrite [succ_sub_succ, fact_succ, -mul.assoc, -succ_mul_choose_eq],
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rewrite [fact_succ n, -ih n !lt_succ_self this, *mul.assoc]
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end
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end
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theorem choose_def_alt {n k : ℕ} (H : k ≤ n) : choose n k = fact n / (fact k * fact (n -k)) :=
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eq.symm (nat.div_eq_of_eq_mul_left (mul_pos !fact_pos !fact_pos)
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(by rewrite [-mul.assoc, choose_mul_fact_mul_fact H]))
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theorem fact_mul_fact_dvd_fact {n k : ℕ} (H : k ≤ n) : fact k * fact (n - k) ∣ fact n :=
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by rewrite [-choose_mul_fact_mul_fact H, mul.assoc]; apply dvd_mul_left
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open finset
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/- the number of subsets of s of size k is n choose k -/
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section card_subsets
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variables {A : Type} [deceqA : decidable_eq A]
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include deceqA
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private theorem aux₀ (s : finset A) : {t ∈ powerset s | card t = 0} = '{∅} :=
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ext (take t, iff.intro
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(assume H,
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have t = ∅, from eq_empty_of_card_eq_zero (of_mem_sep H),
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show t ∈ '{ ∅ }, by rewrite [this, mem_singleton_iff])
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(assume H,
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have t = ∅, by rewrite mem_singleton_iff at H; assumption,
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by substvars; exact mem_sep_of_mem !empty_mem_powerset rfl))
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private theorem aux₁ (k : ℕ) : {t ∈ powerset (∅ : finset A) | card t = succ k} = ∅ :=
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eq_empty_of_forall_not_mem (take t, assume H,
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have t ∈ powerset ∅, from mem_of_mem_sep H,
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have t = ∅, by rewrite [powerset_empty at this, mem_singleton_iff at this]; assumption,
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have card (∅ : finset A) = succ k, by rewrite -this; apply of_mem_sep H,
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nat.no_confusion this)
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private theorem aux₂ {a : A} {s t : finset A} (anins : a ∉ s) (tpows : t ∈ powerset s) : a ∉ t :=
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suppose a ∈ t,
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have a ∈ s, from mem_of_subset_of_mem (subset_of_mem_powerset tpows) this,
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anins this
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private theorem aux₃ {a : A} {s t : finset A} (anins : a ∉ s) (k : ℕ) :
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t ∈ (insert a) ' (powerset s) ∧ card t = succ k ↔
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t ∈ (insert a) ' {t' ∈ powerset s | card t' = k} :=
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iff.intro
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(assume H,
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obtain H' cardt, from H,
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obtain t' [(t'pows : t' ∈ powerset s) (teq : insert a t' = t)], from exists_of_mem_image H',
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have aint : a ∈ t, by rewrite -teq; apply mem_insert,
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have anint' : a ∉ t', from
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(assume aint',
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have a ∈ s, from mem_of_subset_of_mem (subset_of_mem_powerset t'pows) aint',
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anins this),
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have t' = erase a t, by rewrite [-teq, erase_insert (aux₂ anins t'pows)],
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have card t' = k, by rewrite [this, card_erase_of_mem aint, cardt],
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mem_image (mem_sep_of_mem t'pows this) teq)
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(assume H,
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obtain t' [Ht' (teq : insert a t' = t)], from exists_of_mem_image H,
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have t'pows : t' ∈ powerset s, from mem_of_mem_sep Ht',
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have cardt' : card t' = k, from of_mem_sep Ht',
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and.intro
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(show t ∈ (insert a) ' (powerset s), from mem_image t'pows teq)
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(show card t = succ k,
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by rewrite [-teq, card_insert_of_not_mem (aux₂ anins t'pows), cardt']))
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private theorem aux₄ {a : A} {s : finset A} (anins : a ∉ s) (k : ℕ) :
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{t ∈ powerset (insert a s)| card t = succ k} =
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{t ∈ powerset s | card t = succ k} ∪ (insert a) ' {t ∈ powerset s | card t = k} :=
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begin
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apply ext, intro t,
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rewrite [powerset_insert anins, mem_union_iff, *mem_sep_iff, mem_union_iff, and.right_distrib,
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aux₃ anins]
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end
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private theorem aux₅ {a : A} {s : finset A} (anins : a ∉ s) (k : ℕ) :
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{t ∈ powerset s | card t = succ k} ∩ (insert a) ' {t ∈ powerset s | card t = k} = ∅ :=
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inter_eq_empty
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(take t, assume Ht₁ Ht₂,
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have tpows : t ∈ powerset s, from mem_of_mem_sep Ht₁,
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have anint : a ∉ t, from aux₂ anins tpows,
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obtain t' [Ht' (teq : insert a t' = t)], from exists_of_mem_image Ht₂,
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have aint : a ∈ t, by rewrite -teq; apply mem_insert,
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show false, from anint aint)
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private theorem aux₆ {a : A} {s : finset A} (anins : a ∉ s) (k : ℕ) :
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card ((insert a) ' {t ∈ powerset s | card t = k}) = card {t ∈ powerset s | card t = k} :=
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have set.inj_on (insert a) (ts {t ∈ powerset s| card t = k}), from
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take t₁ t₂, assume Ht₁ Ht₂,
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assume Heq : insert a t₁ = insert a t₂,
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have t₁ ∈ powerset s, from mem_of_mem_sep Ht₁,
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have anint₁ : a ∉ t₁, from aux₂ anins this,
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have t₂ ∈ powerset s, from mem_of_mem_sep Ht₂,
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have anint₂ : a ∉ t₂, from aux₂ anins this,
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calc
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t₁ = erase a (insert a t₁) : by rewrite (erase_insert anint₁)
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... = erase a (insert a t₂) : Heq
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... = t₂ : by rewrite (erase_insert anint₂),
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card_image_eq_of_inj_on this
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theorem card_subsets (s : finset A) : ∀k, card {t ∈ powerset s | card t = k} = choose (card s) k :=
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begin
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induction s with a s anins ih,
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{intro k,
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cases k with k,
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{rewrite aux₀},
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rewrite aux₁},
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intro k,
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cases k with k,
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{rewrite [aux₀, choose_zero_right]},
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rewrite [*(card_insert_of_not_mem anins), aux₄ anins, card_union_of_disjoint (aux₅ anins k),
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aux₆ anins k, *ih]
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end
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end card_subsets
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end nat
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