72 lines
2.3 KiB
Text
72 lines
2.3 KiB
Text
/-
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Copyright (c) 2015 Haitao Zhang. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Haitao Zhang
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Finite ordinal types.
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-/
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import data.list.basic data.finset.basic data.fintype.card
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open eq.ops nat function list finset fintype
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structure less_than (n : nat) := (val : nat) (is_lt : val < n)
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attribute less_than.val [coercion]
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namespace less_than
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section
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open decidable
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protected definition has_decidable_eq [instance] (n : nat) : ∀ (i j : less_than n), decidable (i = j)
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| (mk ival ilt) (mk jval jlt) :=
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match nat.has_decidable_eq ival jval with
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| inl veq := inl (by substvars)
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| inr vne := inr (by intro h; injection h; contradiction)
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end
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end
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lemma dinj_lt (n : nat) : dinj (λ i, i < n) less_than.mk :=
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take a1 a2 Pa1 Pa2 Pmkeq, less_than.no_confusion Pmkeq (λ Pe Pqe, Pe)
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lemma val_mk (n i : nat) (Plt : i < n) : less_than.val (less_than.mk i Plt) = i := rfl
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definition upto [reducible] (n : nat) : list (less_than n) :=
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dmap (λ i, i < n) less_than.mk (list.upto n)
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lemma nodup_upto (n : nat) : nodup (upto n) :=
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dmap_nodup_of_dinj (dinj_lt n) (list.nodup_upto n)
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lemma mem_upto (n : nat) : ∀ (i : less_than n), i ∈ upto n :=
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take i, less_than.destruct i
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(take ival Piltn,
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assert Pin : ival ∈ list.upto n, from mem_upto_of_lt Piltn,
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mem_of_dmap Piltn Pin)
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lemma upto_zero : upto 0 = [] :=
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by rewrite [↑upto, list.upto_nil, dmap_nil]
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lemma map_val_upto (n : nat) : map less_than.val (upto n) = list.upto n :=
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map_of_dmap_inv_pos (val_mk n) (@lt_of_mem_upto n)
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lemma length_upto (n : nat) : length (upto n) = n :=
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calc
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length (upto n) = length (list.upto n) : (map_val_upto n ▸ length_map less_than.val (upto n))⁻¹
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... = n : list.length_upto n
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definition fintype_less_than [instance] (n : nat) : fintype (less_than n) :=
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fintype.mk (upto n) (nodup_upto n) (mem_upto n)
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section pigeonhole
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open fintype
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lemma card_less_than (n : nat) : card (less_than n) = n := length_upto n
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theorem pigeonhole {n m : nat} (Pmltn : m < n) : ¬ (∃ f : less_than n → less_than m, injective f) :=
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not.intro
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(assume Pex, absurd Pmltn (not_lt_of_ge
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(calc
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n = card (less_than n) : card_less_than
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... ≤ card (less_than m) : card_le_of_inj (less_than n) (less_than m) Pex
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... = m : card_less_than)))
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end pigeonhole
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end less_than
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