lean2/library/data/nat/order.lean
2015-11-08 14:04:54 -08:00

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/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad
The order relation on the natural numbers.
-/
import data.nat.basic algebra.ordered_ring
open eq.ops
namespace nat
/- lt and le -/
theorem le_of_lt_or_eq {m n : } (H : m < n m = n) : m ≤ n :=
le_of_eq_or_lt (or.swap H)
theorem lt_or_eq_of_le {m n : } (H : m ≤ n) : m < n m = n :=
or.swap (eq_or_lt_of_le H)
theorem le_iff_lt_or_eq (m n : ) : m ≤ n ↔ m < n m = n :=
iff.intro lt_or_eq_of_le le_of_lt_or_eq
theorem lt_of_le_and_ne {m n : } (H1 : m ≤ n) : m ≠ n → m < n :=
or_resolve_right (eq_or_lt_of_le H1)
theorem lt_iff_le_and_ne (m n : ) : m < n ↔ m ≤ n ∧ m ≠ n :=
iff.intro
(take H, and.intro (le_of_lt H) (take H1, !lt.irrefl (H1 ▸ H)))
(and.rec lt_of_le_and_ne)
theorem le_add_right (n k : ) : n ≤ n + k :=
nat.rec !le.refl (λ k, le_succ_of_le) k
theorem le_add_left (n m : ): n ≤ m + n :=
!add.comm ▸ !le_add_right
theorem le.intro {n m k : } (h : n + k = m) : n ≤ m :=
h ▸ !le_add_right
theorem le.elim {n m : } : n ≤ m → ∃k, n + k = m :=
le.rec (exists.intro 0 rfl) (λm h, Exists.rec
(λ k H, exists.intro (succ k) (H ▸ rfl)))
theorem le.total {m n : } : m ≤ n n ≤ m :=
or.imp_left le_of_lt !lt_or_ge
/- addition -/
theorem add_le_add_left {n m : } (H : n ≤ m) (k : ) : k + n ≤ k + m :=
obtain l Hl, from le.elim H, le.intro (Hl ▸ !add.assoc)
theorem add_le_add_right {n m : } (H : n ≤ m) (k : ) : n + k ≤ m + k :=
!add.comm ▸ !add.comm ▸ add_le_add_left H k
theorem le_of_add_le_add_left {k n m : } (H : k + n ≤ k + m) : n ≤ m :=
obtain l Hl, from le.elim H, le.intro (add.cancel_left (!add.assoc⁻¹ ⬝ Hl))
theorem lt_of_add_lt_add_left {k n m : } (H : k + n < k + m) : n < m :=
let H' := le_of_lt H in
lt_of_le_and_ne (le_of_add_le_add_left H') (assume Heq, !lt.irrefl (Heq ▸ H))
theorem add_lt_add_left {n m : } (H : n < m) (k : ) : k + n < k + m :=
lt_of_succ_le (!add_succ ▸ add_le_add_left (succ_le_of_lt H) k)
theorem add_lt_add_right {n m : } (H : n < m) (k : ) : n + k < m + k :=
!add.comm ▸ !add.comm ▸ add_lt_add_left H k
theorem lt_add_of_pos_right {n k : } (H : k > 0) : n < n + k :=
!add_zero ▸ add_lt_add_left H n
/- multiplication -/
theorem mul_le_mul_left {n m : } (k : ) (H : n ≤ m) : k * n ≤ k * m :=
obtain (l : ) (Hl : n + l = m), from le.elim H,
have k * n + k * l = k * m, by rewrite [-mul.left_distrib, Hl],
le.intro this
theorem mul_le_mul_right {n m : } (k : ) (H : n ≤ m) : n * k ≤ m * k :=
!mul.comm ▸ !mul.comm ▸ !mul_le_mul_left H
theorem mul_le_mul {n m k l : } (H1 : n ≤ k) (H2 : m ≤ l) : n * m ≤ k * l :=
le.trans (!mul_le_mul_right H1) (!mul_le_mul_left H2)
theorem mul_lt_mul_of_pos_left {n m k : } (H : n < m) (Hk : k > 0) : k * n < k * m :=
lt_of_lt_of_le (lt_add_of_pos_right Hk) (!mul_succ ▸ mul_le_mul_left k (succ_le_of_lt H))
theorem mul_lt_mul_of_pos_right {n m k : } (H : n < m) (Hk : k > 0) : n * k < m * k :=
!mul.comm ▸ !mul.comm ▸ mul_lt_mul_of_pos_left H Hk
/- min and max -/
/-
definition max (a b : ) : := if a < b then b else a
definition min (a b : ) : := if a < b then a else b
theorem max_self [simp] (a : ) : max a a = a :=
eq.rec_on !if_t_t rfl
theorem max_le {n m k : } (H₁ : n ≤ k) (H₂ : m ≤ k) : max n m ≤ k :=
if H : n < m then by rewrite [↑max, if_pos H]; apply H₂
else by rewrite [↑max, if_neg H]; apply H₁
theorem min_le_left (n m : ) : min n m ≤ n :=
if H : n < m then by rewrite [↑min, if_pos H]
else assert H' : m ≤ n, from or_resolve_right !lt_or_ge H,
by rewrite [↑min, if_neg H]; apply H'
theorem min_le_right (n m : ) : min n m ≤ m :=
if H : n < m then by rewrite [↑min, if_pos H]; apply le_of_lt H
else assert H' : m ≤ n, from or_resolve_right !lt_or_ge H,
by rewrite [↑min, if_neg H]
theorem le_min {n m k : } (H₁ : k ≤ n) (H₂ : k ≤ m) : k ≤ min n m :=
if H : n < m then by rewrite [↑min, if_pos H]; apply H₁
else by rewrite [↑min, if_neg H]; apply H₂
theorem eq_max_right {a b : } (H : a < b) : b = max a b :=
(if_pos H)⁻¹
theorem eq_max_left {a b : } (H : ¬ a < b) : a = max a b :=
(if_neg H)⁻¹
open decidable
theorem le_max_right (a b : ) : b ≤ max a b :=
lt.by_cases
(suppose a < b, (eq_max_right this) ▸ !le.refl)
(suppose a = b, this ▸ !max_self⁻¹ ▸ !le.refl)
(suppose b < a, (eq_max_left (lt.asymm this)) ▸ (le_of_lt this))
theorem le_max_left (a b : ) : a ≤ max a b :=
if h : a < b then le_of_lt (eq.rec_on (eq_max_right h) h)
else (eq_max_left h) ▸ !le.refl
-/
/- nat is an instance of a linearly ordered semiring and a lattice -/
open - [notations] algebra
protected definition decidable_linear_ordered_semiring [reducible] [instance] :
algebra.decidable_linear_ordered_semiring nat :=
⦃ algebra.decidable_linear_ordered_semiring, nat.comm_semiring,
add_left_cancel := @add.cancel_left,
add_right_cancel := @add.cancel_right,
lt := nat.lt,
le := nat.le,
le_refl := le.refl,
le_trans := @le.trans,
le_antisymm := @le.antisymm,
le_total := @le.total,
le_iff_lt_or_eq := @le_iff_lt_or_eq,
le_of_lt := @le_of_lt,
lt_irrefl := @lt.irrefl,
lt_of_lt_of_le := @lt_of_lt_of_le,
lt_of_le_of_lt := @lt_of_le_of_lt,
lt_of_add_lt_add_left := @lt_of_add_lt_add_left,
add_lt_add_left := @add_lt_add_left,
add_le_add_left := @add_le_add_left,
le_of_add_le_add_left := @le_of_add_le_add_left,
zero_lt_one := zero_lt_succ 0,
mul_le_mul_of_nonneg_left := (take a b c H1 H2, mul_le_mul_left c H1),
mul_le_mul_of_nonneg_right := (take a b c H1 H2, mul_le_mul_right c H1),
mul_lt_mul_of_pos_left := @mul_lt_mul_of_pos_left,
mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right,
decidable_lt := nat.decidable_lt ⦄
definition nat_has_dvd [reducible] [instance] [priority nat.prio] : has_dvd nat :=
has_dvd.mk has_dvd.dvd
theorem add_pos_left {a : } (H : 0 < a) (b : ) : 0 < a + b :=
@algebra.add_pos_of_pos_of_nonneg _ _ a b H !zero_le
theorem add_pos_right {a : } (H : 0 < a) (b : ) : 0 < b + a :=
by rewrite add.comm; apply add_pos_left H b
theorem add_eq_zero_iff_eq_zero_and_eq_zero {a b : } :
a + b = 0 ↔ a = 0 ∧ b = 0 :=
@algebra.add_eq_zero_iff_eq_zero_and_eq_zero_of_nonneg_of_nonneg _ _ a b !zero_le !zero_le
theorem le_add_of_le_left {a b c : } (H : b ≤ c) : b ≤ a + c :=
@algebra.le_add_of_nonneg_of_le _ _ a b c !zero_le H
theorem le_add_of_le_right {a b c : } (H : b ≤ c) : b ≤ c + a :=
@algebra.le_add_of_le_of_nonneg _ _ a b c H !zero_le
theorem lt_add_of_lt_left {b c : } (H : b < c) (a : ) : b < a + c :=
@algebra.lt_add_of_nonneg_of_lt _ _ a b c !zero_le H
theorem lt_add_of_lt_right {b c : } (H : b < c) (a : ) : b < c + a :=
@algebra.lt_add_of_lt_of_nonneg _ _ a b c H !zero_le
theorem lt_of_mul_lt_mul_left {a b c : } (H : c * a < c * b) : a < b :=
@algebra.lt_of_mul_lt_mul_left _ _ a b c H !zero_le
theorem lt_of_mul_lt_mul_right {a b c : } (H : a * c < b * c) : a < b :=
@algebra.lt_of_mul_lt_mul_right _ _ a b c H !zero_le
theorem pos_of_mul_pos_left {a b : } (H : 0 < a * b) : 0 < b :=
@algebra.pos_of_mul_pos_left _ _ a b H !zero_le
theorem pos_of_mul_pos_right {a b : } (H : 0 < a * b) : 0 < a :=
@algebra.pos_of_mul_pos_right _ _ a b H !zero_le
theorem zero_le_one : 0 ≤ 1 := dec_trivial
/- properties specific to nat -/
theorem lt_intro {n m k : } (H : succ n + k = m) : n < m :=
lt_of_succ_le (le.intro H)
theorem lt_elim {n m : } (H : n < m) : ∃k, succ n + k = m :=
le.elim (succ_le_of_lt H)
theorem lt_add_succ (n m : ) : n < n + succ m :=
lt_intro !succ_add_eq_succ_add
theorem eq_zero_of_le_zero {n : } (H : n ≤ 0) : n = 0 :=
obtain (k : ) (Hk : n + k = 0), from le.elim H,
eq_zero_of_add_eq_zero_right Hk
/- succ and pred -/
theorem le_of_lt_succ {m n : nat} : m < succ n → m ≤ n :=
le_of_succ_le_succ
theorem lt_iff_succ_le (m n : nat) : m < n ↔ succ m ≤ n :=
iff.rfl
theorem lt_succ_iff_le (m n : nat) : m < succ n ↔ m ≤ n :=
iff.intro le_of_lt_succ lt_succ_of_le
theorem self_le_succ (n : ) : n ≤ succ n :=
le.intro !add_one
theorem succ_le_or_eq_of_le {n m : } : n ≤ m → succ n ≤ m n = m :=
lt_or_eq_of_le
theorem pred_le_of_le_succ {n m : } : n ≤ succ m → pred n ≤ m :=
pred_le_pred
theorem succ_le_of_le_pred {n m : } : succ n ≤ m → n ≤ pred m :=
pred_le_pred
theorem pred_le_pred_of_le {n m : } : n ≤ m → pred n ≤ pred m :=
pred_le_pred
theorem pre_lt_of_lt {n m : } : n < m → pred n < m :=
lt_of_le_of_lt !pred_le
theorem lt_of_pred_lt_pred {n m : } (H : pred n < pred m) : n < m :=
lt_of_not_ge
(suppose m ≤ n,
not_lt_of_ge (pred_le_pred_of_le this) H)
theorem le_or_eq_succ_of_le_succ {n m : } (H : n ≤ succ m) : n ≤ m n = succ m :=
or.imp_left le_of_succ_le_succ (succ_le_or_eq_of_le H)
theorem le_pred_self (n : ) : pred n ≤ n :=
!pred_le
theorem succ_pos (n : ) : 0 < succ n :=
!zero_lt_succ
theorem succ_pred_of_pos {n : } (H : n > 0) : succ (pred n) = n :=
(or_resolve_right (eq_zero_or_eq_succ_pred n) (ne.symm (ne_of_lt H)))⁻¹
theorem exists_eq_succ_of_lt {n : } : Π {m : }, n < m → ∃k, m = succ k
| 0 H := absurd H !not_lt_zero
| (succ k) H := exists.intro k rfl
theorem lt_succ_self (n : ) : n < succ n :=
lt.base n
lemma lt_succ_of_lt {i j : nat} : i < j → i < succ j :=
assume Plt, lt.trans Plt (self_lt_succ j)
/- other forms of induction -/
protected definition strong_rec_on {P : nat → Type} (n : ) (H : ∀n, (∀m, m < n → P m) → P n) : P n :=
nat.rec (λm h, absurd h !not_lt_zero)
(λn' (IH : ∀ {m : }, m < n' → P m) m l,
or.by_cases (lt_or_eq_of_le (le_of_lt_succ l))
IH (λ e, eq.rec (H n' @IH) e⁻¹)) (succ n) n !lt_succ_self
protected theorem strong_induction_on {P : nat → Prop} (n : ) (H : ∀n, (∀m, m < n → P m) → P n) :
P n :=
nat.strong_rec_on n H
protected theorem case_strong_induction_on {P : nat → Prop} (a : nat) (H0 : P 0)
(Hind : ∀(n : nat), (∀m, m ≤ n → P m) → P (succ n)) : P a :=
nat.strong_induction_on a
(take n,
show (∀ m, m < n → P m) → P n, from
nat.cases_on n
(suppose (∀ m, m < 0 → P m), show P 0, from H0)
(take n,
suppose (∀ m, m < succ n → P m),
show P (succ n), from
Hind n (take m, assume H1 : m ≤ n, this _ (lt_succ_of_le H1))))
/- pos -/
theorem by_cases_zero_pos {P : → Prop} (y : ) (H0 : P 0) (H1 : ∀ {y : nat}, y > 0 → P y) :
P y :=
nat.cases_on y H0 (take y, H1 !succ_pos)
theorem eq_zero_or_pos (n : ) : n = 0 n > 0 :=
or_of_or_of_imp_left
(or.swap (lt_or_eq_of_le !zero_le))
(suppose 0 = n, by subst n)
theorem pos_of_ne_zero {n : } (H : n ≠ 0) : n > 0 :=
or.elim !eq_zero_or_pos (take H2 : n = 0, by contradiction) (take H2 : n > 0, H2)
theorem ne_zero_of_pos {n : } (H : n > 0) : n ≠ 0 :=
ne.symm (ne_of_lt H)
theorem exists_eq_succ_of_pos {n : } (H : n > 0) : exists l, n = succ l :=
exists_eq_succ_of_lt H
theorem pos_of_dvd_of_pos {m n : } (H1 : m n) (H2 : n > 0) : m > 0 :=
pos_of_ne_zero
(suppose m = 0,
assert n = 0, from eq_zero_of_zero_dvd (this ▸ H1),
ne_of_lt H2 (by subst n))
/- multiplication -/
theorem mul_lt_mul_of_le_of_lt {n m k l : } (Hk : k > 0) (H1 : n ≤ k) (H2 : m < l) :
n * m < k * l :=
lt_of_le_of_lt (mul_le_mul_right m H1) (mul_lt_mul_of_pos_left H2 Hk)
theorem mul_lt_mul_of_lt_of_le {n m k l : } (Hl : l > 0) (H1 : n < k) (H2 : m ≤ l) :
n * m < k * l :=
lt_of_le_of_lt (mul_le_mul_left n H2) (mul_lt_mul_of_pos_right H1 Hl)
theorem mul_lt_mul_of_le_of_le {n m k l : } (H1 : n < k) (H2 : m < l) : n * m < k * l :=
have H3 : n * m ≤ k * m, from mul_le_mul_right m (le_of_lt H1),
have H4 : k * m < k * l, from mul_lt_mul_of_pos_left H2 (lt_of_le_of_lt !zero_le H1),
lt_of_le_of_lt H3 H4
theorem eq_of_mul_eq_mul_left {m k n : } (Hn : n > 0) (H : n * m = n * k) : m = k :=
have n * m ≤ n * k, by rewrite H,
have m ≤ k, from le_of_mul_le_mul_left this Hn,
have n * k ≤ n * m, by rewrite H,
have k ≤ m, from le_of_mul_le_mul_left this Hn,
le.antisymm `m ≤ k` this
theorem eq_of_mul_eq_mul_right {n m k : } (Hm : m > 0) (H : n * m = k * m) : n = k :=
eq_of_mul_eq_mul_left Hm (!mul.comm ▸ !mul.comm ▸ H)
theorem eq_zero_or_eq_of_mul_eq_mul_left {n m k : } (H : n * m = n * k) : n = 0 m = k :=
or_of_or_of_imp_right !eq_zero_or_pos
(assume Hn : n > 0, eq_of_mul_eq_mul_left Hn H)
theorem eq_zero_or_eq_of_mul_eq_mul_right {n m k : } (H : n * m = k * m) : m = 0 n = k :=
eq_zero_or_eq_of_mul_eq_mul_left (!mul.comm ▸ !mul.comm ▸ H)
theorem eq_one_of_mul_eq_one_right {n m : } (H : n * m = 1) : n = 1 :=
have H2 : n * m > 0, by rewrite H; apply succ_pos,
or.elim (le_or_gt n 1)
(suppose n ≤ 1,
have n > 0, from pos_of_mul_pos_right H2,
show n = 1, from le.antisymm `n ≤ 1` (succ_le_of_lt this))
(suppose n > 1,
have m > 0, from pos_of_mul_pos_left H2,
have n * m ≥ 2 * 1, from mul_le_mul (succ_le_of_lt `n > 1`) (succ_le_of_lt this),
have 1 ≥ 2, from !mul_one ▸ H ▸ this,
absurd !lt_succ_self (not_lt_of_ge this))
theorem eq_one_of_mul_eq_one_left {n m : } (H : n * m = 1) : m = 1 :=
eq_one_of_mul_eq_one_right (!mul.comm ▸ H)
theorem eq_one_of_mul_eq_self_left {n m : } (Hpos : n > 0) (H : m * n = n) : m = 1 :=
eq_of_mul_eq_mul_right Hpos (H ⬝ !one_mul⁻¹)
theorem eq_one_of_mul_eq_self_right {n m : } (Hpos : m > 0) (H : m * n = m) : n = 1 :=
eq_one_of_mul_eq_self_left Hpos (!mul.comm ▸ H)
theorem eq_one_of_dvd_one {n : } (H : n 1) : n = 1 :=
dvd.elim H
(take m, suppose 1 = n * m,
eq_one_of_mul_eq_one_right this⁻¹)
/- min and max -/
open decidable
theorem le_max_left_iff_true [simp] (a b : ) : a ≤ max a b ↔ true :=
iff_true_intro (le_max_left a b)
theorem le_max_right_iff_true [simp] (a b : ) : b ≤ max a b ↔ true :=
iff_true_intro (le_max_right a b)
theorem min_zero [simp] (a : ) : min a 0 = 0 :=
by rewrite [min_eq_right !zero_le]
theorem zero_min [simp] (a : ) : min 0 a = 0 :=
by rewrite [min_eq_left !zero_le]
theorem max_zero [simp] (a : ) : max a 0 = a :=
by rewrite [max_eq_left !zero_le]
theorem zero_max [simp] (a : ) : max 0 a = a :=
by rewrite [max_eq_right !zero_le]
theorem min_succ_succ [simp] (a b : ) : min (succ a) (succ b) = succ (min a b) :=
or.elim !lt_or_ge
(suppose a < b, by rewrite [min_eq_left_of_lt this, min_eq_left_of_lt (succ_lt_succ this)])
(suppose a ≥ b, by rewrite [min_eq_right this, min_eq_right (succ_le_succ this)])
theorem max_succ_succ [simp] (a b : ) : max (succ a) (succ b) = succ (max a b) :=
or.elim !lt_or_ge
(suppose a < b, by rewrite [max_eq_right_of_lt this, max_eq_right_of_lt (succ_lt_succ this)])
(suppose a ≥ b, by rewrite [max_eq_left this, max_eq_left (succ_le_succ this)])
/- In algebra.ordered_group, these next four are only proved for additive groups, not additive
semigroups. -/
theorem min_add_add_left (a b c : ) : min (a + b) (a + c) = a + min b c :=
decidable.by_cases
(suppose b ≤ c,
assert a + b ≤ a + c, from add_le_add_left this _,
by rewrite [min_eq_left `b ≤ c`, min_eq_left this])
(suppose ¬ b ≤ c,
assert c ≤ b, from le_of_lt (lt_of_not_ge this),
assert a + c ≤ a + b, from add_le_add_left this _,
by rewrite [min_eq_right `c ≤ b`, min_eq_right this])
theorem min_add_add_right (a b c : ) : min (a + c) (b + c) = min a b + c :=
by rewrite [add.comm a c, add.comm b c, add.comm _ c]; apply min_add_add_left
theorem max_add_add_left (a b c : ) : max (a + b) (a + c) = a + max b c :=
decidable.by_cases
(suppose b ≤ c,
assert a + b ≤ a + c, from add_le_add_left this _,
by rewrite [max_eq_right `b ≤ c`, max_eq_right this])
(suppose ¬ b ≤ c,
assert c ≤ b, from le_of_lt (lt_of_not_ge this),
assert a + c ≤ a + b, from add_le_add_left this _,
by rewrite [max_eq_left `c ≤ b`, max_eq_left this])
theorem max_add_add_right (a b c : ) : max (a + c) (b + c) = max a b + c :=
by rewrite [add.comm a c, add.comm b c, add.comm _ c]; apply max_add_add_left
/- least and greatest -/
section least_and_greatest
variable (P : → Prop)
variable [decP : ∀ n, decidable (P n)]
include decP
-- returns the least i < n satisfying P, or n if there is none
definition least :
| 0 := 0
| (succ n) := if P (least n) then least n else succ n
theorem least_of_bound {n : } (H : P n) : P (least P n) :=
begin
induction n with [m, ih],
rewrite ↑least,
apply H,
rewrite ↑least,
cases decidable.em (P (least P m)) with [Hlp, Hlp],
rewrite [if_pos Hlp],
apply Hlp,
rewrite [if_neg Hlp],
apply H
end
theorem least_le (n : ) : least P n ≤ n:=
begin
induction n with [m, ih],
{rewrite ↑least},
rewrite ↑least,
cases decidable.em (P (least P m)) with [Psm, Pnsm],
rewrite [if_pos Psm],
apply le.trans ih !le_succ,
rewrite [if_neg Pnsm]
end
theorem least_of_lt {i n : } (ltin : i < n) (H : P i) : P (least P n) :=
begin
induction n with [m, ih],
exact absurd ltin !not_lt_zero,
rewrite ↑least,
cases decidable.em (P (least P m)) with [Psm, Pnsm],
rewrite [if_pos Psm],
apply Psm,
rewrite [if_neg Pnsm],
cases (lt_or_eq_of_le (le_of_lt_succ ltin)) with [Hlt, Heq],
exact absurd (ih Hlt) Pnsm,
rewrite Heq at H,
exact absurd (least_of_bound P H) Pnsm
end
theorem ge_least_of_lt {i n : } (ltin : i < n) (Hi : P i) : i ≥ least P n :=
begin
induction n with [m, ih],
exact absurd ltin !not_lt_zero,
rewrite ↑least,
cases decidable.em (P (least P m)) with [Psm, Pnsm],
rewrite [if_pos Psm],
cases (lt_or_eq_of_le (le_of_lt_succ ltin)) with [Hlt, Heq],
apply ih Hlt,
rewrite Heq,
apply least_le,
rewrite [if_neg Pnsm],
cases (lt_or_eq_of_le (le_of_lt_succ ltin)) with [Hlt, Heq],
apply absurd (least_of_lt P Hlt Hi) Pnsm,
rewrite Heq at Hi,
apply absurd (least_of_bound P Hi) Pnsm
end
theorem least_lt {n i : } (ltin : i < n) (Hi : P i) : least P n < n :=
lt_of_le_of_lt (ge_least_of_lt P ltin Hi) ltin
-- returns the largest i < n satisfying P, or n if there is none.
definition greatest :
| 0 := 0
| (succ n) := if P n then n else greatest n
theorem greatest_of_lt {i n : } (ltin : i < n) (Hi : P i) : P (greatest P n) :=
begin
induction n with [m, ih],
{exact absurd ltin !not_lt_zero},
{cases (decidable.em (P m)) with [Psm, Pnsm],
{rewrite [↑greatest, if_pos Psm]; exact Psm},
{rewrite [↑greatest, if_neg Pnsm],
have neim : i ≠ m, from assume H : i = m, absurd (H ▸ Hi) Pnsm,
have ltim : i < m, from lt_of_le_of_ne (le_of_lt_succ ltin) neim,
apply ih ltim}}
end
theorem le_greatest_of_lt {i n : } (ltin : i < n) (Hi : P i) : i ≤ greatest P n :=
begin
induction n with [m, ih],
{exact absurd ltin !not_lt_zero},
{cases (decidable.em (P m)) with [Psm, Pnsm],
{rewrite [↑greatest, if_pos Psm], apply le_of_lt_succ ltin},
{rewrite [↑greatest, if_neg Pnsm],
have neim : i ≠ m, from assume H : i = m, absurd (H ▸ Hi) Pnsm,
have ltim : i < m, from lt_of_le_of_ne (le_of_lt_succ ltin) neim,
apply ih ltim}}
end
end least_and_greatest
end nat