351 lines
12 KiB
Text
351 lines
12 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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Authors: Haitao Zhang, Leonardo de Moura
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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 algebra.group
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open eq.ops nat function list finset fintype
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structure fin (n : nat) := (val : nat) (is_lt : val < n)
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definition less_than [reducible] := fin
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namespace fin
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attribute fin.val [coercion]
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section def_equal
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variable {n : nat}
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lemma eq_of_veq : ∀ {i j : fin n}, (val i) = j → i = j
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| (mk iv ilt) (mk jv jlt) := assume (veq : iv = jv), begin congruence, assumption end
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lemma veq_of_eq : ∀ {i j : fin n}, i = j → (val i) = j
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| (mk iv ilt) (mk jv jlt) := assume Peq,
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show iv = jv, from fin.no_confusion Peq (λ Pe Pqe, Pe)
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lemma eq_iff_veq : ∀ {i j : fin n}, (val i) = j ↔ i = j :=
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take i j, iff.intro eq_of_veq veq_of_eq
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definition val_inj := @eq_of_veq n
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end def_equal
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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 : fin 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) fin.mk :=
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take a1 a2 Pa1 Pa2 Pmkeq, fin.no_confusion Pmkeq (λ Pe Pqe, Pe)
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lemma val_mk (n i : nat) (Plt : i < n) : fin.val (fin.mk i Plt) = i := rfl
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definition upto [reducible] (n : nat) : list (fin n) :=
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dmap (λ i, i < n) fin.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 : fin n), i ∈ upto n :=
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take i, fin.destruct i
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(take ival Piltn,
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assert ival ∈ list.upto n, from mem_upto_of_lt Piltn,
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mem_dmap Piltn this)
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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 fin.val (upto n) = list.upto n :=
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map_dmap_of_inv_of_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 fin.val (upto n))⁻¹
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... = n : list.length_upto n
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definition is_fintype [instance] (n : nat) : fintype (fin 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_fin (n : nat) : card (fin n) = n := length_upto n
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theorem pigeonhole {n m : nat} (Pmltn : m < n) : ¬∃ f : fin n → fin m, injective f :=
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assume Pex, absurd Pmltn (not_lt_of_ge
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(calc
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n = card (fin n) : card_fin
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... ≤ card (fin m) : card_le_of_inj (fin n) (fin m) Pex
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... = m : card_fin))
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end pigeonhole
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definition zero (n : nat) : fin (succ n) :=
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mk 0 !zero_lt_succ
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definition mk_mod [reducible] (n i : nat) : fin (succ n) :=
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mk (i mod (succ n)) (mod_lt _ !zero_lt_succ)
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variable {n : nat}
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theorem val_lt : ∀ i : fin n, val i < n
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| (mk v h) := h
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lemma max_lt (i j : fin n) : max i j < n :=
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max_lt (is_lt i) (is_lt j)
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definition lift : fin n → Π m, fin (n + m)
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| (mk v h) m := mk v (lt_add_of_lt_right h m)
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definition lift_succ (i : fin n) : fin (nat.succ n) :=
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lift i 1
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definition maxi [reducible] : fin (succ n) :=
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mk n !lt_succ_self
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theorem val_lift : ∀ (i : fin n) (m : nat), val i = val (lift i m)
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| (mk v h) m := rfl
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lemma mk_succ_ne_zero {i : nat} : ∀ {P}, mk (succ i) P ≠ zero n :=
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assume P Pe, absurd (veq_of_eq Pe) !succ_ne_zero
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lemma mk_mod_eq {i : fin (succ n)} : i = mk_mod n i :=
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eq_of_veq begin rewrite [↑mk_mod, mod_eq_of_lt !is_lt] end
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lemma mk_mod_of_lt {i : nat} (Plt : i < succ n) : mk_mod n i = mk i Plt :=
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begin esimp [mk_mod], congruence, exact mod_eq_of_lt Plt end
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section lift_lower
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lemma lift_zero : lift_succ (zero n) = zero (succ n) := rfl
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lemma ne_max_of_lt_max {i : fin (succ n)} : i < n → i ≠ maxi :=
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by intro hlt he; substvars; exact absurd hlt (lt.irrefl n)
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lemma lt_max_of_ne_max {i : fin (succ n)} : i ≠ maxi → i < n :=
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assume hne : i ≠ maxi,
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assert vne : val i ≠ n, from
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assume he,
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have val (@maxi n) = n, from rfl,
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have val i = val (@maxi n), from he ⬝ this⁻¹,
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absurd (eq_of_veq this) hne,
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have val i < nat.succ n, from val_lt i,
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lt_of_le_of_ne (le_of_lt_succ this) vne
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lemma lift_succ_ne_max {i : fin n} : lift_succ i ≠ maxi :=
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begin
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cases i with v hlt, esimp [lift_succ, lift, max], intro he,
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injection he, substvars,
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exact absurd hlt (lt.irrefl v)
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end
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lemma lift_succ_inj : injective (@lift_succ n) :=
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take i j, destruct i (destruct j (take iv ilt jv jlt Pmkeq,
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begin congruence, apply fin.no_confusion Pmkeq, intros, assumption end))
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lemma lt_of_inj_of_max (f : fin (succ n) → fin (succ n)) :
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injective f → (f maxi = maxi) → ∀ i, i < n → f i < n :=
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assume Pinj Peq, take i, assume Pilt,
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assert P1 : f i = f maxi → i = maxi, from assume Peq, Pinj i maxi Peq,
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have f i ≠ maxi, from
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begin rewrite -Peq, intro P2, apply absurd (P1 P2) (ne_max_of_lt_max Pilt) end,
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lt_max_of_ne_max this
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definition lift_fun : (fin n → fin n) → (fin (succ n) → fin (succ n)) :=
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λ f i, dite (i = maxi) (λ Pe, maxi) (λ Pne, lift_succ (f (mk i (lt_max_of_ne_max Pne))))
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definition lower_inj (f : fin (succ n) → fin (succ n)) (inj : injective f) :
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f maxi = maxi → fin n → fin n :=
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assume Peq, take i, mk (f (lift_succ i)) (lt_of_inj_of_max f inj Peq (lift_succ i) (lt_max_of_ne_max lift_succ_ne_max))
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lemma lift_fun_max {f : fin n → fin n} : lift_fun f maxi = maxi :=
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begin rewrite [↑lift_fun, dif_pos rfl] end
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lemma lift_fun_of_ne_max {f : fin n → fin n} {i} (Pne : i ≠ maxi) :
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lift_fun f i = lift_succ (f (mk i (lt_max_of_ne_max Pne))) :=
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begin rewrite [↑lift_fun, dif_neg Pne] end
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lemma lift_fun_eq {f : fin n → fin n} {i : fin n} :
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lift_fun f (lift_succ i) = lift_succ (f i) :=
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begin
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rewrite [lift_fun_of_ne_max lift_succ_ne_max], congruence, congruence,
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rewrite [-eq_iff_veq], esimp, rewrite [↑lift_succ, -val_lift]
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end
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lemma lift_fun_of_inj {f : fin n → fin n} : injective f → injective (lift_fun f) :=
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assume Pinj, take i j,
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assert Pdi : decidable (i = maxi), from _, assert Pdj : decidable (j = maxi), from _,
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begin
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cases Pdi with Pimax Pinmax,
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cases Pdj with Pjmax Pjnmax,
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substvars, intros, exact rfl,
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substvars, rewrite [lift_fun_max, lift_fun_of_ne_max Pjnmax],
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intro Plmax, apply absurd Plmax⁻¹ lift_succ_ne_max,
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cases Pdj with Pjmax Pjnmax,
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substvars, rewrite [lift_fun_max, lift_fun_of_ne_max Pinmax],
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intro Plmax, apply absurd Plmax lift_succ_ne_max,
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rewrite [lift_fun_of_ne_max Pinmax, lift_fun_of_ne_max Pjnmax],
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intro Peq, rewrite [-eq_iff_veq],
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exact veq_of_eq (Pinj (lift_succ_inj Peq))
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end
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lemma lift_fun_inj : injective (@lift_fun n) :=
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take f₁ f₂ Peq, funext (λ i,
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assert lift_fun f₁ (lift_succ i) = lift_fun f₂ (lift_succ i), from congr_fun Peq _,
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begin revert this, rewrite [*lift_fun_eq], apply lift_succ_inj end)
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lemma lower_inj_apply {f Pinj Pmax} (i : fin n) :
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val (lower_inj f Pinj Pmax i) = val (f (lift_succ i)) :=
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by rewrite [↑lower_inj]
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end lift_lower
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section madd
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definition madd (i j : fin (succ n)) : fin (succ n) :=
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mk ((i + j) mod (succ n)) (mod_lt _ !zero_lt_succ)
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definition minv : ∀ i : fin (succ n), fin (succ n)
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| (mk iv ilt) := mk ((succ n - iv) mod succ n) (mod_lt _ !zero_lt_succ)
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lemma val_madd : ∀ i j : fin (succ n), val (madd i j) = (i + j) mod (succ n)
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| (mk iv ilt) (mk jv jlt) := by esimp
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lemma madd_inj : ∀ {i : fin (succ n)}, injective (madd i)
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| (mk iv ilt) :=
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take j₁ j₂, fin.destruct j₁ (fin.destruct j₂ (λ jv₁ jlt₁ jv₂ jlt₂, begin
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rewrite [↑madd, -eq_iff_veq],
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intro Peq, congruence,
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rewrite [-(mod_eq_of_lt jlt₁), -(mod_eq_of_lt jlt₂)],
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apply mod_eq_mod_of_add_mod_eq_add_mod_left Peq
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end))
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lemma madd_mk_mod {i j : nat} : madd (mk_mod n i) (mk_mod n j) = mk_mod n (i+j) :=
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eq_of_veq begin esimp [madd, mk_mod], rewrite [ mod_add_mod, add_mod_mod ] end
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lemma val_mod : ∀ i : fin (succ n), (val i) mod (succ n) = val i
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| (mk iv ilt) := by esimp; rewrite [(mod_eq_of_lt ilt)]
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lemma madd_comm (i j : fin (succ n)) : madd i j = madd j i :=
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by apply eq_of_veq; rewrite [*val_madd, add.comm (val i)]
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lemma zero_madd (i : fin (succ n)) : madd (zero n) i = i :=
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by apply eq_of_veq; rewrite [val_madd, ↑zero, nat.zero_add, mod_eq_of_lt (is_lt i)]
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lemma madd_zero (i : fin (succ n)) : madd i (zero n) = i :=
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!madd_comm ▸ zero_madd i
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lemma madd_assoc (i j k : fin (succ n)) : madd (madd i j) k = madd i (madd j k) :=
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by apply eq_of_veq; rewrite [*val_madd, mod_add_mod, add_mod_mod, add.assoc (val i)]
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lemma madd_left_inv : ∀ i : fin (succ n), madd (minv i) i = zero n
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| (mk iv ilt) := eq_of_veq (by
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rewrite [val_madd, ↑minv, ↑zero, mod_add_mod, sub_add_cancel (le_of_lt ilt), mod_self])
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open algebra
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definition madd_is_comm_group [instance] : add_comm_group (fin (succ n)) :=
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add_comm_group.mk madd madd_assoc (zero n) zero_madd madd_zero minv madd_left_inv madd_comm
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end madd
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definition pred : fin n → fin n
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| (mk v h) := mk (nat.pred v) (pre_lt_of_lt h)
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lemma val_pred : ∀ (i : fin n), val (pred i) = nat.pred (val i)
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| (mk v h) := rfl
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lemma pred_zero : pred (zero n) = zero n :=
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rfl
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definition mk_pred (i : nat) (h : succ i < succ n) : fin n :=
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mk i (lt_of_succ_lt_succ h)
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definition succ : fin n → fin (succ n)
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| (mk v h) := mk (nat.succ v) (succ_lt_succ h)
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lemma val_succ : ∀ (i : fin n), val (succ i) = nat.succ (val i)
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| (mk v h) := rfl
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lemma succ_max : fin.succ maxi = (@maxi (nat.succ n)) := rfl
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lemma lift_succ.comm : lift_succ ∘ (@succ n) = succ ∘ lift_succ :=
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funext take i, eq_of_veq (begin rewrite [↑lift_succ, -val_lift, *val_succ, -val_lift] end)
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definition elim0 {C : fin 0 → Type} : Π i : fin 0, C i
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| (mk v h) := absurd h !not_lt_zero
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definition zero_succ_cases {C : fin (nat.succ n) → Type} :
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C (zero n) → (Π j : fin n, C (succ j)) → (Π k : fin (nat.succ n), C k) :=
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begin
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intros CO CS k,
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induction k with [vk, pk],
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induction (nat.decidable_lt 0 vk) with [HT, HF],
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{ show C (mk vk pk), from
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let vj := nat.pred vk in
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have vk = vj+1, from
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eq.symm (succ_pred_of_pos HT),
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assert vj < n, from
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lt_of_succ_lt_succ (eq.subst `vk = vj+1` pk),
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have succ (mk vj `vj < n`) = mk vk pk, from
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val_inj (eq.symm `vk = vj+1`),
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eq.rec_on this (CS (mk vj `vj < n`)) },
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{ show C (mk vk pk), from
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have vk = 0, from
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eq_zero_of_le_zero (le_of_not_gt HF),
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have zero n = mk vk pk, from
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val_inj (eq.symm this),
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eq.rec_on this CO }
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end
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theorem choice {C : fin n → Type} :
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(∀ i : fin n, nonempty (C i)) → nonempty (Π i : fin n, C i) :=
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begin
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revert C,
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induction n with [n, IH],
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{ intros C H,
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apply nonempty.intro,
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exact elim0 },
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{ intros C H,
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fapply nonempty.elim (H (zero n)),
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intro CO,
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fapply nonempty.elim (IH (λ i, C (succ i)) (λ i, H (succ i))),
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intro CS,
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apply nonempty.intro,
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exact zero_succ_cases CO CS }
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end
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section
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open list
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local postfix `+1`:100 := nat.succ
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lemma dmap_map_lift {n : nat} : ∀ l : list nat, (∀ i, i ∈ l → i < n) → dmap (λ i, i < n +1) mk l = map lift_succ (dmap (λ i, i < n) mk l)
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| [] := assume Plt, rfl
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| (i::l) := assume Plt, begin
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rewrite [@dmap_cons_of_pos _ _ (λ i, i < n +1) _ _ _ (lt_succ_of_lt (Plt i !mem_cons)), @dmap_cons_of_pos _ _ (λ i, i < n) _ _ _ (Plt i !mem_cons), map_cons],
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congruence,
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apply dmap_map_lift,
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intro j Pjinl, apply Plt, apply mem_cons_of_mem, assumption end
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lemma upto_succ (n : nat) : upto (n +1) = maxi :: map lift_succ (upto n) :=
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begin
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rewrite [↑fin.upto, list.upto_succ, @dmap_cons_of_pos _ _ (λ i, i < n +1) _ _ _ (nat.self_lt_succ n)],
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congruence,
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apply dmap_map_lift, apply @list.lt_of_mem_upto
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end
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definition upto_step : ∀ {n : nat}, fin.upto (n +1) = (map succ (upto n))++[zero n]
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| 0 := rfl
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| (i +1) := begin rewrite [upto_succ i, map_cons, append_cons, succ_max, upto_succ, -lift_zero],
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congruence, rewrite [map_map, -lift_succ.comm, -map_map, -(map_singleton _ (zero i)), -map_append, -upto_step] end
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end
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end fin
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