feat(hott): port finite ordinal sets from the std library, with all things related to nat.mod and to fintype still missing.
create a logic.hlean file for further extension of the logic theory in the prelude. add distributivity lemmas for products and sums.
This commit is contained in:
Jakob von Raumer
Leonardo de Moura
parent
cb3bc1a311
commit
1042f6c29d
4 changed files with 596 additions and 1 deletions
18
hott/logic.hlean
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18
hott/logic.hlean
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/-
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Copyright (c) Jakob von Raumer. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jakob von Raumer
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Logic lemmas we don't want/need in the prelude.
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-/
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import types.pi
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open eq is_trunc decidable
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theorem dif_pos {c : Type} [H : decidable c] [P : is_hprop c] (Hc : c)
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{A : Type} {t : c → A} {e : ¬ c → A} : dite c t e = t Hc :=
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by induction H with Hc Hnc; apply ap t; apply is_hprop.elim; apply absurd Hc Hnc
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theorem dif_neg {c : Type} [H : decidable c] (Hnc : ¬c)
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{A : Type} {t : c → A} {e : ¬ c → A} : dite c t e = e Hnc :=
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by induction H with Hc Hnc; apply absurd Hc Hnc; apply ap e; apply is_hprop.elim
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541
hott/types/fin.hlean
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541
hott/types/fin.hlean
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/-
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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, Jakob von Raumer
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Finite ordinal types.
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-/
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import types.list algebra.group function logic types.prod types.sum
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open eq nat function list equiv is_trunc algebra sigma sum
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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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definition sigma_char : fin n ≃ Σ (val : nat), val < n :=
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begin
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fapply equiv.MK,
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intro i, cases i with i ilt, apply dpair i ilt,
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intro s, cases s with i ilt, apply fin.mk i ilt,
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intro s, cases s with i ilt, reflexivity,
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intro i, cases i with i ilt, reflexivity
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end
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definition is_hset_fin [instance] : is_hset (fin n) :=
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begin
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apply is_trunc_equiv_closed, apply equiv.symm, apply sigma_char,
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end
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definition eq_of_veq : Π {i j : fin n}, (val i) = j → i = j :=
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begin
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intro i j, cases i with i ilt, cases j with j jlt, esimp,
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intro p, induction p, apply ap (mk i), apply !is_hprop.elim
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end
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definition eq_of_veq_refl (i : fin n) : eq_of_veq (refl (val i)) = idp :=
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!is_hprop.elim
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definition veq_of_eq : Π {i j : fin n}, i = j → (val i) = j :=
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by intro i j P; apply ap val; exact P
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definition eq_iff_veq {i j : fin n} : (val i) = j ↔ i = j :=
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pair 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 decidable
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open decidable
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protected definition has_decidable_eq [instance] (n : nat) :
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Π (i j : fin n), decidable (i = j) :=
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begin
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intros i j, apply decidable_of_decidable_of_iff,
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apply nat.has_decidable_eq i j, apply eq_iff_veq,
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end
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end decidable
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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) : Π (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) : ¬Σ 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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protected definition zero [constructor] (n : nat) : fin (succ n) :=
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mk 0 !zero_lt_succ
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definition fin_has_zero [instance] [reducible] (n : nat) : has_zero (fin (succ n)) :=
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has_zero.mk (fin.zero n)
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definition val_zero (n : nat) : val (0 : fin (succ n)) = 0 := rfl
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/-definition mk_mod [reducible] (n i : nat) : fin (succ n) :=
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mk (i % (succ n)) (mod_lt _ !zero_lt_succ)-/
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/-theorem mk_mod_zero_eq (n : nat) : mk_mod n 0 = 0 :=
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rfl-/
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variable {n : nat}
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theorem val_lt : Π 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 [constructor] : fin n → Π m : nat, 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 [constructor] (i : fin n) : fin (nat.succ n) :=
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have r : fin (n+1), from lift i 1,
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r
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definition maxi [reducible] : fin (succ n) :=
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mk n !lt_succ_self
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definition val_lift : Π (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} : Π {P}, mk (succ i) P ≠ (0 : fin (succ 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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-/
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section lift_lower
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lemma lift_zero : lift_succ (0 : fin (succ n)) = (0 : fin (succ (succ n))) :=
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by apply eq_of_veq; reflexivity
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lemma ne_max_of_lt_max {i : fin (succ n)} : i < n → i ≠ maxi :=
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begin
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intro hlt he, assert he' : maxi = i, apply he⁻¹, induction he', apply nat.lt_irrefl n hlt,
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end
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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 [instance] : is_embedding (@lift_succ n) :=
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begin
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apply is_embedding_of_is_injective, intro i j,
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induction i with iv ilt, induction j with jv jlt, intro Pmkeq,
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apply eq_of_veq, apply veq_of_eq Pmkeq
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end
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definition lt_of_inj_of_max (f : fin (succ n) → fin (succ n)) :
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is_embedding f → (f maxi = maxi) → Π i : fin (succ n), 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, is_injective_of_is_embedding 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 : is_embedding 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], do 2 congruence,
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apply eq_of_veq, esimp, rewrite -val_lift,
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end
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lemma lift_fun_of_inj {f : fin n → fin n} : is_embedding f → is_embedding (lift_fun f) :=
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begin
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intro Pemb, apply is_embedding_of_is_injective, intro i j,
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assert Pdi : decidable (i = maxi), apply _,
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assert Pdj : decidable (j = maxi), apply _,
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cases Pdi with Pimax Pinmax,
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cases Pdj with Pjmax Pjnmax,
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substvars, intros, reflexivity,
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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, apply eq_of_veq,
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cases i with i ilt, cases j with j jlt, esimp at *,
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fapply veq_of_eq, apply is_injective_of_is_embedding,
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apply @is_injective_of_is_embedding _ _ lift_succ _ _ _ Peq,
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end
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lemma lift_fun_inj : is_embedding (@lift_fun n) :=
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begin
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apply is_embedding_of_is_injective, intro f f' Peq, apply eq_of_homotopy, intro i,
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assert H : lift_fun f (lift_succ i) = lift_fun f' (lift_succ i), apply congr_fun Peq _,
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revert H, rewrite [*lift_fun_eq], apply is_injective_of_is_embedding,
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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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/-
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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) % (succ n)) (mod_lt _ !zero_lt_succ)
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definition minv : Π i : fin (succ n), fin (succ n)
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| (mk iv ilt) := mk ((succ n - iv) % succ n) (mod_lt _ !zero_lt_succ)
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lemma val_madd : Π i j : fin (succ n), val (madd i j) = (i + j) % (succ n)
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| (mk iv ilt) (mk jv jlt) := by esimp
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lemma madd_inj : Π {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 : Π i : fin (succ n), (val i) % (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 0 i = i :=
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have H : madd (fin.zero n) i = i,
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by apply eq_of_veq; rewrite [val_madd, ↑fin.zero, nat.zero_add, mod_eq_of_lt (is_lt i)],
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H
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lemma madd_zero (i : fin (succ n)) : madd i (fin.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 : Π i : fin (succ n), madd (minv i) i = fin.zero n
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| (mk iv ilt) := eq_of_veq (by
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rewrite [val_madd, ↑minv, ↑fin.zero, mod_add_mod, nat.sub_add_cancel (le_of_lt ilt), mod_self])
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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 (fin.zero n) zero_madd madd_zero minv madd_left_inv madd_comm
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end madd-/
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definition pred [constructor] : 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 : Π (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 (fin.zero n) = fin.zero n :=
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begin
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induction n, reflexivity, apply eq_of_veq, reflexivity,
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end
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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 : Π (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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eq_of_homotopy take i,
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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 (fin.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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inverse (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 (inverse `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 fin.zero n = mk vk pk, from
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val_inj (inverse this),
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eq.rec_on this CO }
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end
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definition succ_maxi_cases {C : fin (nat.succ n) → Type} :
|
||||
(Π j : fin n, C (lift_succ j)) → C maxi → (Π k : fin (nat.succ n), C k) :=
|
||||
begin
|
||||
intros CL CM k,
|
||||
induction k with [vk, pk],
|
||||
induction (nat.decidable_lt vk n) with [HT, HF],
|
||||
{ show C (mk vk pk), from
|
||||
have HL : lift_succ (mk vk HT) = mk vk pk, from
|
||||
val_inj rfl,
|
||||
eq.rec_on HL (CL (mk vk HT)) },
|
||||
{ show C (mk vk pk), from
|
||||
have HMv : vk = n, from
|
||||
le.antisymm (le_of_lt_succ pk) (le_of_not_gt HF),
|
||||
have HM : maxi = mk vk pk, from
|
||||
val_inj (inverse HMv),
|
||||
eq.rec_on HM CM }
|
||||
end
|
||||
|
||||
open decidable
|
||||
|
||||
-- TODO there has to be a less painful way to do this
|
||||
definition elim_succ_maxi_cases_lift_succ {C : fin (nat.succ n) → Type}
|
||||
{Cls : Π j : fin n, C (lift_succ j)} {Cm : C maxi} (i : fin n) :
|
||||
succ_maxi_cases Cls Cm (lift_succ i) = Cls i :=
|
||||
begin
|
||||
esimp[succ_maxi_cases], cases i with i ilt, esimp,
|
||||
apply decidable.rec,
|
||||
{ intro ilt', esimp[val_inj], apply concat,
|
||||
apply ap (λ x, eq.rec_on x _), esimp[eq_of_veq, rfl], reflexivity,
|
||||
assert H : ilt = ilt', apply is_hprop.elim, cases H,
|
||||
assert H' : is_hprop.elim (lt_add_of_lt_right ilt 1) (lt_add_of_lt_right ilt 1) = idp,
|
||||
apply is_hprop.elim,
|
||||
krewrite H' },
|
||||
{ intro a, contradiction },
|
||||
end
|
||||
|
||||
definition elim_succ_maxi_cases_maxi {C : fin (nat.succ n) → Type}
|
||||
{Cls : Π j : fin n, C (lift_succ j)} {Cm : C maxi} :
|
||||
succ_maxi_cases Cls Cm maxi = Cm :=
|
||||
begin
|
||||
esimp[succ_maxi_cases, maxi],
|
||||
apply decidable.rec,
|
||||
{ intro a, apply absurd a !nat.lt_irrefl },
|
||||
{ intro a, esimp[val_inj], apply concat,
|
||||
assert H : (le.antisymm (le_of_lt_succ (lt_succ_self n)) (le_of_not_gt a))⁻¹ = idp,
|
||||
apply is_hprop.elim,
|
||||
apply ap _ H, krewrite eq_of_veq_refl },
|
||||
end
|
||||
|
||||
definition foldr {A B : Type} (m : A → B → B) (b : B) : Π {n : nat}, (fin n → A) → B :=
|
||||
nat.rec (λ f, b) (λ n IH f, m (f (fin.zero n)) (IH (λ i : fin n, f (succ i))))
|
||||
|
||||
definition foldl {A B : Type} (m : B → A → B) (b : B) : Π {n : nat}, (fin n → A) → B :=
|
||||
nat.rec (λ f, b) (λ n IH f, m (IH (λ i : fin n, f (lift_succ i))) (f maxi))
|
||||
|
||||
theorem choice {C : fin n → Type} :
|
||||
(Π i : fin n, nonempty (C i)) → nonempty (Π i : fin n, C i) :=
|
||||
begin
|
||||
revert C,
|
||||
induction n with [n, IH],
|
||||
{ intros C H,
|
||||
apply nonempty.intro,
|
||||
exact elim0 },
|
||||
{ intros C H,
|
||||
fapply nonempty.elim (H (fin.zero n)),
|
||||
intro CO,
|
||||
fapply nonempty.elim (IH (λ i, C (succ i)) (λ i, H (succ i))),
|
||||
intro CS,
|
||||
apply nonempty.intro,
|
||||
exact zero_succ_cases CO CS }
|
||||
end
|
||||
|
||||
/-section
|
||||
open list
|
||||
local postfix `+1`:100 := nat.succ
|
||||
|
||||
lemma dmap_map_lift {n : nat} : Π l : list nat, (Π i, i ∈ l → i < n) → dmap (λ i, i < n +1) mk l = map lift_succ (dmap (λ i, i < n) mk l)
|
||||
| [] := assume Plt, rfl
|
||||
| (i::l) := assume Plt, begin
|
||||
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],
|
||||
congruence,
|
||||
apply dmap_map_lift,
|
||||
intro j Pjinl, apply Plt, apply mem_cons_of_mem, assumption end
|
||||
|
||||
lemma upto_succ (n : nat) : upto (n +1) = maxi :: map lift_succ (upto n) :=
|
||||
begin
|
||||
rewrite [↑fin.upto, list.upto_succ, @dmap_cons_of_pos _ _ (λ i, i < n +1) _ _ _ (nat.self_lt_succ n)],
|
||||
congruence,
|
||||
apply dmap_map_lift, apply @list.lt_of_mem_upto
|
||||
end
|
||||
|
||||
definition upto_step : Π {n : nat}, fin.upto (n +1) = (map succ (upto n))++[0]
|
||||
| 0 := rfl
|
||||
| (i +1) := begin rewrite [upto_succ i, map_cons, append_cons, succ_max, upto_succ, -lift_zero],
|
||||
congruence, rewrite [map_map, -lift_succ.comm, -map_map, -(map_singleton _ 0), -map_append, -upto_step] end
|
||||
end-/
|
||||
|
||||
open sum equiv decidable
|
||||
|
||||
definition fin_zero_equiv_empty : fin 0 ≃ empty :=
|
||||
begin
|
||||
fapply equiv.MK, rotate 1, do 2 (intro x; contradiction),
|
||||
rotate 1, do 2 (intro x; apply elim0 x)
|
||||
end
|
||||
|
||||
definition is_contr_fin_one [instance] : is_contr (fin 1) :=
|
||||
begin
|
||||
fapply is_contr.mk, exact 0,
|
||||
intro x, induction x with v vlt,
|
||||
apply eq_of_veq, rewrite val_zero,
|
||||
apply inverse, apply eq_zero_of_le_zero, apply le_of_succ_le_succ, exact vlt,
|
||||
end
|
||||
|
||||
definition fin_sum_equiv (n m : nat) : (fin n + fin m) ≃ fin (n+m) :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
{ intro s, induction s with l r,
|
||||
induction l with v vlt, apply mk v, apply lt_add_of_lt_right, exact vlt,
|
||||
induction r with v vlt, apply mk (v + n), rewrite {n + m}add.comm,
|
||||
apply add_lt_add_of_lt_of_le vlt, apply nat.le_refl },
|
||||
{ intro f, induction f with v vlt,
|
||||
exact if h : v < n
|
||||
then sum.inl (mk v h)
|
||||
else sum.inr (mk (v-n) (nat.sub_lt_of_lt_add vlt (le_of_not_gt h))) },
|
||||
{ intro f, cases f with v vlt, esimp, apply @by_cases (v < n),
|
||||
intro vltn, rewrite [dif_pos vltn], apply eq_of_veq, reflexivity,
|
||||
intro nvltn, rewrite [dif_neg nvltn], apply eq_of_veq, esimp,
|
||||
apply nat.sub_add_cancel, apply le_of_not_gt, apply nvltn },
|
||||
{ intro s, cases s with f g,
|
||||
cases f with v vlt, rewrite [dif_pos vlt],
|
||||
cases g with v vlt, esimp, have ¬ v + n < n, from
|
||||
suppose v + n < n,
|
||||
assert v < n - n, from nat.lt_sub_of_add_lt this !le.refl,
|
||||
have v < 0, by rewrite [nat.sub_self at this]; exact this,
|
||||
absurd this !not_lt_zero,
|
||||
apply concat, apply dif_neg this, apply ap inr, apply eq_of_veq, esimp,
|
||||
apply nat.add_sub_cancel },
|
||||
end
|
||||
|
||||
definition fin_succ_equiv (n : nat) : fin (n + 1) ≃ fin n + unit :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
{ apply succ_maxi_cases, esimp, apply inl, apply inr unit.star },
|
||||
{ intro d, cases d, apply lift_succ a, apply maxi },
|
||||
{ intro d, cases d,
|
||||
cases a with a alt, esimp, apply elim_succ_maxi_cases_lift_succ,
|
||||
cases a, apply elim_succ_maxi_cases_maxi },
|
||||
{ intro a, apply succ_maxi_cases, esimp,
|
||||
intro j, krewrite elim_succ_maxi_cases_lift_succ,
|
||||
krewrite elim_succ_maxi_cases_maxi },
|
||||
end
|
||||
|
||||
open prod
|
||||
|
||||
definition fin_prod_equiv (n m : nat) : (fin n × fin m) ≃ fin (n*m) :=
|
||||
begin
|
||||
induction n,
|
||||
{ rewrite nat.zero_mul,
|
||||
calc fin 0 × fin m ≃ empty × fin m : fin_zero_equiv_empty
|
||||
... ≃ fin m × empty : prod_comm_equiv
|
||||
... ≃ empty : prod_empty_equiv
|
||||
... ≃ fin 0 : fin_zero_equiv_empty },
|
||||
{ assert H : (a + 1) * m = a * m + m, rewrite [nat.right_distrib, one_mul],
|
||||
calc fin (a + 1) × fin m
|
||||
≃ (fin a + unit) × fin m : prod.prod_equiv_prod_right !fin_succ_equiv
|
||||
... ≃ (fin a × fin m) + (unit × fin m) : sum_prod_right_distrib
|
||||
... ≃ (fin a × fin m) + (fin m × unit) : prod_comm_equiv
|
||||
... ≃ fin (a * m) + (fin m × unit) : v_0
|
||||
... ≃ fin (a * m) + fin m : prod_unit_equiv
|
||||
... ≃ fin (a * m + m) : fin_sum_equiv
|
||||
... ≃ fin ((a + 1) * m) : equiv_of_eq (ap fin H) },
|
||||
end
|
||||
|
||||
definition fin_two_equiv_bool : fin 2 ≃ bool :=
|
||||
let H := equiv_unit_of_is_contr (fin 1) in
|
||||
calc
|
||||
fin 2 ≃ fin (1 + 1) : equiv.refl
|
||||
... ≃ fin 1 + fin 1 : fin_sum_equiv
|
||||
... ≃ unit + unit : H
|
||||
... ≃ bool : bool_equiv_unit_sum_unit
|
||||
|
||||
definition fin_sum_unit_equiv (n : nat) : fin n + unit ≃ fin (n+1) :=
|
||||
let H := equiv_unit_of_is_contr (fin 1) in
|
||||
calc
|
||||
fin n + unit ≃ fin n + fin 1 : H
|
||||
... ≃ fin (n+1) : fin_sum_equiv
|
||||
|
||||
end fin
|
||||
|
|
@ -1,7 +1,7 @@
|
|||
/-
|
||||
Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Author: Floris van Doorn
|
||||
Authors: Floris van Doorn, Jakob von Raumer
|
||||
|
||||
Ported from Coq HoTT
|
||||
Theorems about products
|
||||
|
|
@ -182,6 +182,15 @@ namespace prod
|
|||
definition prod_unit_equiv [constructor] (A : Type) : A × unit ≃ A :=
|
||||
!prod_contr_equiv
|
||||
|
||||
definition prod_empty_equiv (A : Type) : A × empty ≃ empty :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
{ intro x, cases x with a e, cases e },
|
||||
{ intro e, cases e },
|
||||
{ intro e, cases e },
|
||||
{ intro x, cases x with a e, cases e }
|
||||
end
|
||||
|
||||
/- Universal mapping properties -/
|
||||
definition is_equiv_prod_rec [instance] [constructor] (P : A × B → Type)
|
||||
: is_equiv (prod.rec : (Πa b, P (a, b)) → Πu, P u) :=
|
||||
|
|
|
|||
|
|
@ -168,6 +168,33 @@ namespace sum
|
|||
definition empty_sum_equiv (A : Type) : empty + A ≃ A :=
|
||||
!sum_comm_equiv ⬝e !sum_empty_equiv
|
||||
|
||||
definition bool_equiv_unit_sum_unit : bool ≃ unit + unit :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
{ intro b, cases b, exact inl unit.star, exact inr unit.star },
|
||||
{ intro s, cases s, exact bool.ff, exact bool.tt },
|
||||
{ intro s, cases s, do 2 (cases a; reflexivity) },
|
||||
{ intro b, cases b, do 2 reflexivity },
|
||||
end
|
||||
|
||||
definition sum_prod_right_distrib [constructor] (A B C : Type) :
|
||||
(A + B) × C ≃ (A × C) + (B × C) :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
{ intro x, cases x with ab c, cases ab with a b, exact inl (a, c), exact inr (b, c) },
|
||||
{ intro x, cases x with ac bc, cases ac with a c, exact (inl a, c),
|
||||
cases bc with b c, exact (inr b, c) },
|
||||
{ intro x, cases x with ac bc, cases ac with a c, reflexivity, cases bc, reflexivity },
|
||||
{ intro x, cases x with ab c, cases ab with a b, do 2 reflexivity }
|
||||
end
|
||||
|
||||
definition sum_prod_left_distrib [constructor] (A B C : Type) :
|
||||
A × (B + C) ≃ (A × B) + (A × C) :=
|
||||
calc A × (B + C) ≃ (B + C) × A : prod_comm_equiv
|
||||
... ≃ (B × A) + (C × A) : sum_prod_right_distrib
|
||||
... ≃ (A × B) + (C × A) : prod_comm_equiv
|
||||
... ≃ (A × B) + (A × C) : prod_comm_equiv
|
||||
|
||||
/- universal property -/
|
||||
|
||||
definition sum_rec_unc [unfold 5] {P : A + B → Type} (fg : (Πa, P (inl a)) × (Πb, P (inr b)))
|
||||
|
|
|
|||
Loading…
Reference in a new issue