lean2/library/init/nat.lean

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/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Leonardo de Moura
-/
prelude
import init.wf init.tactic init.num
open eq.ops decidable or
namespace nat
notation `` := nat
/- basic definitions on natural numbers -/
inductive le (a : ) : → Prop :=
| refl : le a a
| step : Π {b}, le a b → le a (succ b)
infix `≤` := le
attribute le.refl [refl]
definition lt [reducible] (n m : ) := succ n ≤ m
definition ge [reducible] (n m : ) := m ≤ n
definition gt [reducible] (n m : ) := succ m ≤ n
infix `<` := lt
infix `≥` := ge
infix `>` := gt
definition pred [unfold 1] (a : nat) : nat :=
nat.cases_on a zero (λ a₁, a₁)
-- add is defined in init.num
definition sub (a b : nat) : nat :=
nat.rec_on b a (λ b₁, pred)
definition mul (a b : nat) : nat :=
nat.rec_on b zero (λ b₁ r, r + a)
notation a - b := sub a b
notation a * b := mul a b
/- properties of -/
protected definition is_inhabited [instance] : inhabited nat :=
inhabited.mk zero
protected definition has_decidable_eq [instance] : ∀ x y : nat, decidable (x = y)
| has_decidable_eq zero zero := inl rfl
| has_decidable_eq (succ x) zero := inr (by contradiction)
| has_decidable_eq zero (succ y) := inr (by contradiction)
| has_decidable_eq (succ x) (succ y) :=
match has_decidable_eq x y with
| inl xeqy := inl (by rewrite xeqy)
| inr xney := inr (λ h : succ x = succ y, by injection h with xeqy; exact absurd xeqy xney)
end
/- properties of inequality -/
theorem le_of_eq {n m : } (p : n = m) : n ≤ m := p ▸ !le.refl
theorem le_succ (n : ) : n ≤ succ n := le.step !le.refl
theorem pred_le (n : ) : pred n ≤ n := by cases n;repeat constructor
theorem le_succ_iff_true [simp] (n : ) : n ≤ succ n ↔ true :=
iff_true_intro (le_succ n)
theorem pred_le_iff_true [simp] (n : ) : pred n ≤ n ↔ true :=
iff_true_intro (pred_le n)
theorem le.trans [trans] {n m k : } (H1 : n ≤ m) : m ≤ k → n ≤ k :=
le.rec H1 (λp H2, le.step)
theorem le_succ_of_le {n m : } (H : n ≤ m) : n ≤ succ m := le.trans H !le_succ
theorem le_of_succ_le {n m : } (H : succ n ≤ m) : n ≤ m := le.trans !le_succ H
theorem le_of_lt {n m : } (H : n < m) : n ≤ m := le_of_succ_le H
theorem succ_le_succ {n m : } : n ≤ m → succ n ≤ succ m :=
le.rec !le.refl (λa b, le.step)
theorem pred_le_pred {n m : } : n ≤ m → pred n ≤ pred m :=
le.rec !le.refl (nat.rec (λa b, b) (λa b c, le.step))
theorem le_of_succ_le_succ {n m : } : succ n ≤ succ m → n ≤ m :=
pred_le_pred
theorem le_succ_of_pred_le {n m : } : pred n ≤ m → n ≤ succ m :=
nat.cases_on n le.step (λa, succ_le_succ)
theorem not_succ_le_zero (n : ) : ¬succ n ≤ zero :=
by intro H; cases H
theorem succ_le_zero_iff_false (n : ) : succ n ≤ zero ↔ false :=
iff_false_intro !not_succ_le_zero
theorem not_succ_le_self : Π {n : }, ¬succ n ≤ n :=
nat.rec !not_succ_le_zero (λa b c, b (le_of_succ_le_succ c))
theorem succ_le_self_iff_false [simp] (n : ) : succ n ≤ n ↔ false :=
iff_false_intro not_succ_le_self
theorem zero_le : ∀ (n : ), 0 ≤ n :=
nat.rec !le.refl (λa, le.step)
theorem zero_le_iff_true [simp] (n : ) : 0 ≤ n ↔ true :=
iff_true_intro !zero_le
theorem lt.step {n m : } : n < m → n < succ m := le.step
theorem zero_lt_succ (n : ) : 0 < succ n :=
succ_le_succ !zero_le
theorem zero_lt_succ_iff_true [simp] (n : ) : 0 < succ n ↔ true :=
iff_true_intro (zero_lt_succ n)
theorem lt.trans [trans] {n m k : } (H1 : n < m) : m < k → n < k :=
le.trans (le.step H1)
theorem lt_of_le_of_lt [trans] {n m k : } (H1 : n ≤ m) : m < k → n < k :=
le.trans (succ_le_succ H1)
theorem lt_of_lt_of_le [trans] {n m k : } : n < m → m ≤ k → n < k := le.trans
theorem lt.irrefl (n : ) : ¬n < n := not_succ_le_self
theorem lt_self_iff_false [simp] (n : ) : n < n ↔ false :=
iff_false_intro (lt.irrefl n)
theorem self_lt_succ (n : ) : n < succ n := !le.refl
theorem self_lt_succ_iff_true [simp] (n : ) : n < succ n ↔ true :=
iff_true_intro (self_lt_succ n)
theorem lt.base (n : ) : n < succ n := !le.refl
theorem le_lt_antisymm {n m : } (H1 : n ≤ m) (H2 : m < n) : false :=
!lt.irrefl (lt_of_le_of_lt H1 H2)
theorem le.antisymm {n m : } (H1 : n ≤ m) : m ≤ n → n = m :=
le.cases_on H1 (λa, rfl) (λa b c, absurd (lt_of_le_of_lt b c) !lt.irrefl)
theorem lt_le_antisymm {n m : } (H1 : n < m) (H2 : m ≤ n) : false :=
le_lt_antisymm H2 H1
theorem lt.asymm {n m : } (H1 : n < m) : ¬ m < n :=
le_lt_antisymm (le_of_lt H1)
theorem not_lt_zero (a : ) : ¬ a < zero := !not_succ_le_zero
theorem lt_zero_iff_false [simp] (a : ) : a < zero ↔ false :=
iff_false_intro (not_lt_zero a)
theorem eq_or_lt_of_le {a b : } (H : a ≤ b) : a = b a < b :=
le.cases_on H (inl rfl) (λn h, inr (succ_le_succ h))
theorem le_of_eq_or_lt {a b : } (H : a = b a < b) : a ≤ b :=
or.elim H !le_of_eq !le_of_lt
-- less-than is well-founded
definition lt.wf [instance] : well_founded lt :=
well_founded.intro (nat.rec
(!acc.intro (λn H, absurd H (not_lt_zero n)))
(λn IH, !acc.intro (λm H,
elim (eq_or_lt_of_le (le_of_succ_le_succ H))
(λe, eq.substr e IH) (acc.inv IH))))
definition measure {A : Type} : (A → ) → A → A → Prop :=
inv_image lt
definition measure.wf {A : Type} (f : A → ) : well_founded (measure f) :=
inv_image.wf f lt.wf
theorem succ_lt_succ {a b : } : a < b → succ a < succ b :=
succ_le_succ
theorem lt_of_succ_lt {a b : } : succ a < b → a < b :=
le_of_succ_le
theorem lt_of_succ_lt_succ {a b : } : succ a < succ b → a < b :=
le_of_succ_le_succ
definition decidable_le [instance] : decidable_rel le :=
nat.rec (λm, (decidable.inl !zero_le))
(λn IH m, !nat.cases_on (decidable.inr (not_succ_le_zero n))
(λm, decidable.rec (λH, inl (succ_le_succ H))
(λH, inr (λa, H (le_of_succ_le_succ a))) (IH m)))
definition decidable_lt [instance] : decidable_rel lt := _
definition decidable_gt [instance] : decidable_rel gt := _
definition decidable_ge [instance] : decidable_rel ge := _
theorem lt_or_ge (a b : ) : a < b a ≥ b :=
nat.rec (inr !zero_le) (λn, or.rec
(λh, inl (le_succ_of_le h))
(λh, elim (eq_or_lt_of_le h) (λe, inl (eq.subst e !le.refl)) inr)) b
definition lt_ge_by_cases {a b : } {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
by_cases H1 (λh, H2 (elim !lt_or_ge (λa, absurd a h) (λa, a)))
definition lt.by_cases {a b : } {P : Type} (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
lt_ge_by_cases H1 (λh₁,
lt_ge_by_cases H3 (λh₂, H2 (le.antisymm h₂ h₁)))
theorem lt.trichotomy (a b : ) : a < b a = b b < a :=
lt.by_cases (λH, inl H) (λH, inr (inl H)) (λH, inr (inr H))
theorem eq_or_lt_of_not_lt {a b : } (hnlt : ¬ a < b) : a = b b < a :=
or.rec_on (lt.trichotomy a b)
(λ hlt, absurd hlt hnlt)
(λ h, h)
theorem lt_succ_of_le {a b : } : a ≤ b → a < succ b :=
succ_le_succ
theorem lt_of_succ_le {a b : } (h : succ a ≤ b) : a < b := h
theorem succ_le_of_lt {a b : } (h : a < b) : succ a ≤ b := h
theorem succ_sub_succ_eq_sub [simp] (a b : ) : succ a - succ b = a - b :=
nat.rec rfl (λ b, congr_arg pred) b
theorem sub_eq_succ_sub_succ (a b : ) : a - b = succ a - succ b :=
eq.symm !succ_sub_succ_eq_sub
theorem zero_sub_eq_zero [simp] (a : ) : zero - a = zero :=
nat.rec rfl (λ a, congr_arg pred) a
theorem zero_eq_zero_sub (a : ) : zero = zero - a :=
eq.symm !zero_sub_eq_zero
theorem sub_le (a b : ) : a - b ≤ a :=
nat.rec_on b !le.refl (λ b₁, le.trans !pred_le)
theorem sub_le_iff_true [simp] (a b : ) : a - b ≤ a ↔ true :=
iff_true_intro (sub_le a b)
theorem sub_lt {a b : } (H1 : zero < a) (H2 : zero < b) : a - b < a :=
!nat.cases_on (λh, absurd h !lt.irrefl)
(λa h, succ_le_succ (!nat.cases_on (λh, absurd h !lt.irrefl)
(λb c, eq.substr !succ_sub_succ_eq_sub !sub_le) H2)) H1
theorem sub_lt_succ (a b : ) : a - b < succ a :=
lt_succ_of_le !sub_le
theorem sub_lt_succ_iff_true [simp] (a b : ) : a - b < succ a ↔ true :=
iff_true_intro !sub_lt_succ
end nat