595 lines
19 KiB
Text
595 lines
19 KiB
Text
--- 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
|
||
|
||
-- data.nat.order
|
||
-- ==============
|
||
--
|
||
-- The ordering on the natural numbers
|
||
|
||
import .basic logic.classes.decidable
|
||
import tools.fake_simplifier
|
||
|
||
using nat eq_ops tactic
|
||
using fake_simplifier
|
||
using decidable (decidable inl inr)
|
||
|
||
namespace nat
|
||
|
||
|
||
-- Less than or equal
|
||
-- ------------------
|
||
|
||
definition le (n m : ℕ) : Prop := exists k : nat, n + k = m
|
||
|
||
infix `<=` := le
|
||
infix `≤` := le
|
||
|
||
theorem le_intro {n m k : ℕ} (H : n + k = m) : n ≤ m :=
|
||
exists_intro k H
|
||
|
||
theorem le_elim {n m : ℕ} (H : n ≤ m) : ∃k, n + k = m :=
|
||
H
|
||
|
||
opaque_hint (hiding le)
|
||
|
||
-- ### partial order (totality is part of less than)
|
||
|
||
theorem le_refl {n : ℕ} : n ≤ n :=
|
||
le_intro add_zero_right
|
||
|
||
theorem zero_le {n : ℕ} : 0 ≤ n :=
|
||
le_intro add_zero_left
|
||
|
||
theorem le_zero {n : ℕ} (H : n ≤ 0) : n = 0 :=
|
||
obtain (k : ℕ) (Hk : n + k = 0), from le_elim H,
|
||
add_eq_zero_left Hk
|
||
|
||
theorem not_succ_zero_le {n : ℕ} : ¬ succ n ≤ 0 :=
|
||
not_intro
|
||
(assume H : succ n ≤ 0,
|
||
have H2 : succ n = 0, from le_zero H,
|
||
absurd H2 succ_ne_zero)
|
||
|
||
theorem le_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
|
||
obtain (l1 : ℕ) (Hl1 : n + l1 = m), from le_elim H1,
|
||
obtain (l2 : ℕ) (Hl2 : m + l2 = k), from le_elim H2,
|
||
le_intro
|
||
(calc
|
||
n + (l1 + l2) = n + l1 + l2 : add_assoc⁻¹
|
||
... = m + l2 : {Hl1}
|
||
... = k : Hl2)
|
||
|
||
theorem le_antisym {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m :=
|
||
obtain (k : ℕ) (Hk : n + k = m), from (le_elim H1),
|
||
obtain (l : ℕ) (Hl : m + l = n), from (le_elim H2),
|
||
have L1 : k + l = 0, from
|
||
add_cancel_left
|
||
(calc
|
||
n + (k + l) = n + k + l : add_assoc⁻¹
|
||
... = m + l : {Hk}
|
||
... = n : Hl
|
||
... = n + 0 : add_zero_right⁻¹),
|
||
have L2 : k = 0, from add_eq_zero_left L1,
|
||
calc
|
||
n = n + 0 : add_zero_right⁻¹
|
||
... = n + k : {L2⁻¹}
|
||
... = m : Hk
|
||
|
||
-- ### interaction with addition
|
||
|
||
theorem le_add_right {n m : ℕ} : n ≤ n + m :=
|
||
le_intro rfl
|
||
|
||
theorem le_add_left {n m : ℕ} : n ≤ m + n :=
|
||
le_intro add_comm
|
||
|
||
theorem add_le_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k + n ≤ k + m :=
|
||
obtain (l : ℕ) (Hl : n + l = m), from (le_elim H),
|
||
le_intro
|
||
(calc
|
||
k + n + l = k + (n + l) : add_assoc
|
||
... = k + m : {Hl})
|
||
|
||
theorem add_le_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n + k ≤ m + k :=
|
||
add_comm ▸ add_comm ▸ add_le_left H k
|
||
|
||
theorem add_le {n m k l : ℕ} (H1 : n ≤ k) (H2 : m ≤ l) : n + m ≤ k + l :=
|
||
le_trans (add_le_right H1 m) (add_le_left H2 k)
|
||
|
||
|
||
theorem add_le_cancel_left {n m k : ℕ} (H : k + n ≤ k + m) : n ≤ m :=
|
||
obtain (l : ℕ) (Hl : k + n + l = k + m), from (le_elim H),
|
||
le_intro (add_cancel_left
|
||
(calc
|
||
k + (n + l) = k + n + l : add_assoc⁻¹
|
||
... = k + m : Hl))
|
||
|
||
theorem add_le_cancel_right {n m k : ℕ} (H : n + k ≤ m + k) : n ≤ m :=
|
||
add_le_cancel_left (add_comm ▸ add_comm ▸ H)
|
||
|
||
theorem add_le_inv {n m k l : ℕ} (H1 : n + m ≤ k + l) (H2 : k ≤ n) : m ≤ l :=
|
||
obtain (a : ℕ) (Ha : k + a = n), from le_elim H2,
|
||
have H3 : k + (a + m) ≤ k + l, from add_assoc ▸ Ha⁻¹ ▸ H1,
|
||
have H4 : a + m ≤ l, from add_le_cancel_left H3,
|
||
show m ≤ l, from le_trans le_add_left H4
|
||
|
||
-- add_rewrite le_add_right le_add_left
|
||
|
||
-- ### interaction with successor and predecessor
|
||
|
||
theorem succ_le {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
|
||
add_one ▸ add_one ▸ add_le_right H 1
|
||
|
||
theorem succ_le_cancel {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
|
||
add_le_cancel_right (add_one⁻¹ ▸ add_one⁻¹ ▸ H)
|
||
|
||
theorem self_le_succ {n : ℕ} : n ≤ succ n :=
|
||
le_intro add_one
|
||
|
||
theorem le_imp_le_succ {n m : ℕ} (H : n ≤ m) : n ≤ succ m :=
|
||
le_trans H self_le_succ
|
||
|
||
theorem le_imp_succ_le_or_eq {n m : ℕ} (H : n ≤ m) : succ n ≤ m ∨ n = m :=
|
||
obtain (k : ℕ) (Hk : n + k = m), from (le_elim H),
|
||
discriminate
|
||
(assume H3 : k = 0,
|
||
have Heq : n = m,
|
||
from calc
|
||
n = n + 0 : add_zero_right⁻¹
|
||
... = n + k : {H3⁻¹}
|
||
... = m : Hk,
|
||
or_inr Heq)
|
||
(take l : nat,
|
||
assume H3 : k = succ l,
|
||
have Hlt : succ n ≤ m, from
|
||
(le_intro
|
||
(calc
|
||
succ n + l = n + succ l : add_move_succ
|
||
... = n + k : {H3⁻¹}
|
||
... = m : Hk)),
|
||
or_inl Hlt)
|
||
|
||
theorem le_ne_imp_succ_le {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : succ n ≤ m :=
|
||
resolve_left (le_imp_succ_le_or_eq H1) H2
|
||
|
||
theorem le_succ_imp_le_or_eq {n m : ℕ} (H : n ≤ succ m) : n ≤ m ∨ n = succ m :=
|
||
or_imp_or_left (le_imp_succ_le_or_eq H)
|
||
(take H2 : succ n ≤ succ m, show n ≤ m, from succ_le_cancel H2)
|
||
|
||
theorem succ_le_imp_le_and_ne {n m : ℕ} (H : succ n ≤ m) : n ≤ m ∧ n ≠ m :=
|
||
obtain (k : ℕ) (H2 : succ n + k = m), from (le_elim H),
|
||
and_intro
|
||
(have H3 : n + succ k = m,
|
||
from calc
|
||
n + succ k = succ n + k : add_move_succ⁻¹
|
||
... = m : H2,
|
||
show n ≤ m, from le_intro H3)
|
||
(assume H3 : n = m,
|
||
have H4 : succ n ≤ n, from H3⁻¹ ▸ H,
|
||
have H5 : succ n = n, from le_antisym H4 self_le_succ,
|
||
show false, from absurd H5 succ_ne_self)
|
||
|
||
theorem le_pred_self {n : ℕ} : pred n ≤ n :=
|
||
case n
|
||
(pred_zero⁻¹ ▸ le_refl)
|
||
(take k : ℕ, pred_succ⁻¹ ▸ self_le_succ)
|
||
|
||
theorem pred_le {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m :=
|
||
discriminate
|
||
(take Hn : n = 0,
|
||
have H2 : pred n = 0,
|
||
from calc
|
||
pred n = pred 0 : {Hn}
|
||
... = 0 : pred_zero,
|
||
H2⁻¹ ▸ zero_le)
|
||
(take k : ℕ,
|
||
assume Hn : n = succ k,
|
||
obtain (l : ℕ) (Hl : n + l = m), from le_elim H,
|
||
have H2 : pred n + l = pred m,
|
||
from calc
|
||
pred n + l = pred (succ k) + l : {Hn}
|
||
... = k + l : {pred_succ}
|
||
... = pred (succ (k + l)) : pred_succ⁻¹
|
||
... = pred (succ k + l) : {add_succ_left⁻¹}
|
||
... = pred (n + l) : {Hn⁻¹}
|
||
... = pred m : {Hl},
|
||
le_intro H2)
|
||
|
||
theorem pred_le_imp_le_or_eq {n m : ℕ} (H : pred n ≤ m) : n ≤ m ∨ n = succ m :=
|
||
discriminate
|
||
(take Hn : n = 0,
|
||
or_inl (Hn⁻¹ ▸ zero_le))
|
||
(take k : ℕ,
|
||
assume Hn : n = succ k,
|
||
have H2 : pred n = k,
|
||
from calc
|
||
pred n = pred (succ k) : {Hn}
|
||
... = k : pred_succ,
|
||
have H3 : k ≤ m, from subst H2 H,
|
||
have H4 : succ k ≤ m ∨ k = m, from le_imp_succ_le_or_eq H3,
|
||
show n ≤ m ∨ n = succ m, from
|
||
or_imp_or H4
|
||
(take H5 : succ k ≤ m, show n ≤ m, from Hn⁻¹ ▸ H5)
|
||
(take H5 : k = m, show n = succ m, from H5 ▸ Hn))
|
||
|
||
-- ### interaction with multiplication
|
||
|
||
theorem mul_le_left {n m : ℕ} (H : n ≤ m) (k : ℕ) : k * n ≤ k * m :=
|
||
obtain (l : ℕ) (Hl : n + l = m), from (le_elim H),
|
||
have H2 : k * n + k * l = k * m, from
|
||
calc
|
||
k * n + k * l = k * (n + l) : by simp
|
||
... = k * m : {Hl},
|
||
le_intro H2
|
||
|
||
theorem mul_le_right {n m : ℕ} (H : n ≤ m) (k : ℕ) : n * k ≤ m * k :=
|
||
mul_comm ▸ mul_comm ▸ (mul_le_left H k)
|
||
|
||
theorem mul_le {n m k l : ℕ} (H1 : n ≤ k) (H2 : m ≤ l) : n * m ≤ k * l :=
|
||
le_trans (mul_le_right H1 m) (mul_le_left H2 k)
|
||
|
||
-- mul_le_[left|right]_inv below
|
||
|
||
theorem le_decidable [instance] (n m : ℕ) : decidable (n ≤ m) :=
|
||
have general : ∀n, decidable (n ≤ m), from
|
||
rec_on m
|
||
(take n,
|
||
rec_on n
|
||
(inl le_refl)
|
||
(take m iH, inr not_succ_zero_le))
|
||
(take (m' : ℕ) (iH1 : ∀n, decidable (n ≤ m')) (n : ℕ),
|
||
rec_on n
|
||
(inl zero_le)
|
||
(take (n' : ℕ) (iH2 : decidable (n' ≤ succ m')),
|
||
have d1 : decidable (n' ≤ m'), from iH1 n',
|
||
decidable.rec_on d1
|
||
(assume Hp : n' ≤ m', inl (succ_le Hp))
|
||
(assume Hn : ¬ n' ≤ m',
|
||
have H : ¬ succ n' ≤ succ m', from
|
||
assume Hle : succ n' ≤ succ m',
|
||
absurd (succ_le_cancel Hle) Hn,
|
||
inr H))),
|
||
general n
|
||
|
||
-- Less than, Greater than, Greater than or equal
|
||
-- ----------------------------------------------
|
||
|
||
-- ge and gt will be transparent, so we don't need to reprove theorems for le and lt for them
|
||
|
||
definition lt (n m : ℕ) := succ n ≤ m
|
||
infix `<` := lt
|
||
|
||
abbreviation ge (n m : ℕ) := m ≤ n
|
||
infix `>=` := ge
|
||
infix `≥` := ge
|
||
|
||
abbreviation gt (n m : ℕ) := m < n
|
||
infix `>` := gt
|
||
|
||
theorem lt_def (n m : ℕ) : (n < m) = (succ n ≤ m) := refl (n < m)
|
||
|
||
-- add_rewrite gt_def ge_def --it might be possible to remove this in Lean 0.2
|
||
|
||
theorem lt_intro {n m k : ℕ} (H : succ n + k = m) : n < m :=
|
||
le_intro H
|
||
|
||
theorem lt_elim {n m : ℕ} (H : n < m) : ∃ k, succ n + k = m :=
|
||
le_elim H
|
||
|
||
theorem lt_add_succ (n m : ℕ) : n < n + succ m :=
|
||
lt_intro add_move_succ
|
||
|
||
-- ### basic facts
|
||
|
||
theorem lt_imp_ne {n m : ℕ} (H : n < m) : n ≠ m :=
|
||
and_elim_right (succ_le_imp_le_and_ne H)
|
||
|
||
theorem lt_irrefl {n : ℕ} : ¬ n < n :=
|
||
not_intro (assume H : n < n, absurd rfl (lt_imp_ne H))
|
||
|
||
theorem succ_pos {n : ℕ} : 0 < succ n :=
|
||
succ_le zero_le
|
||
|
||
theorem not_lt_zero {n : ℕ} : ¬ n < 0 :=
|
||
not_succ_zero_le
|
||
|
||
theorem lt_imp_eq_succ {n m : ℕ} (H : n < m) : exists k, m = succ k :=
|
||
discriminate
|
||
(take (Hm : m = 0), absurd (Hm ▸ H) not_lt_zero)
|
||
(take (l : ℕ) (Hm : m = succ l), exists_intro l Hm)
|
||
|
||
-- ### interaction with le
|
||
|
||
theorem lt_imp_le_succ {n m : ℕ} (H : n < m) : succ n ≤ m :=
|
||
H
|
||
|
||
theorem le_succ_imp_lt {n m : ℕ} (H : succ n ≤ m) : n < m :=
|
||
H
|
||
|
||
theorem self_lt_succ {n : ℕ} : n < succ n :=
|
||
le_refl
|
||
|
||
theorem lt_imp_le {n m : ℕ} (H : n < m) : n ≤ m :=
|
||
and_elim_left (succ_le_imp_le_and_ne H)
|
||
|
||
theorem le_imp_lt_or_eq {n m : ℕ} (H : n ≤ m) : n < m ∨ n = m :=
|
||
le_imp_succ_le_or_eq H
|
||
|
||
theorem le_ne_imp_lt {n m : ℕ} (H1 : n ≤ m) (H2 : n ≠ m) : n < m :=
|
||
le_ne_imp_succ_le H1 H2
|
||
|
||
theorem le_imp_lt_succ {n m : ℕ} (H : n ≤ m) : n < succ m :=
|
||
succ_le H
|
||
|
||
theorem lt_succ_imp_le {n m : ℕ} (H : n < succ m) : n ≤ m :=
|
||
succ_le_cancel H
|
||
|
||
-- ### transitivity, antisymmmetry
|
||
|
||
theorem lt_le_trans {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k :=
|
||
le_trans H1 H2
|
||
|
||
theorem le_lt_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k :=
|
||
le_trans (succ_le H1) H2
|
||
|
||
theorem lt_trans {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
|
||
lt_le_trans H1 (lt_imp_le H2)
|
||
|
||
theorem le_imp_not_gt {n m : ℕ} (H : n ≤ m) : ¬ n > m :=
|
||
not_intro (assume H2 : m < n, absurd (le_lt_trans H H2) lt_irrefl)
|
||
|
||
theorem lt_imp_not_ge {n m : ℕ} (H : n < m) : ¬ n ≥ m :=
|
||
not_intro (assume H2 : m ≤ n, absurd (lt_le_trans H H2) lt_irrefl)
|
||
|
||
theorem lt_antisym {n m : ℕ} (H : n < m) : ¬ m < n :=
|
||
le_imp_not_gt (lt_imp_le H)
|
||
|
||
-- ### interaction with addition
|
||
|
||
theorem add_lt_left {n m : ℕ} (H : n < m) (k : ℕ) : k + n < k + m :=
|
||
add_succ_right ▸ add_le_left H k
|
||
|
||
theorem add_lt_right {n m : ℕ} (H : n < m) (k : ℕ) : n + k < m + k :=
|
||
add_comm ▸ add_comm ▸ add_lt_left H k
|
||
|
||
theorem add_le_lt {n m k l : ℕ} (H1 : n ≤ k) (H2 : m < l) : n + m < k + l :=
|
||
le_lt_trans (add_le_right H1 m) (add_lt_left H2 k)
|
||
|
||
theorem add_lt_le {n m k l : ℕ} (H1 : n < k) (H2 : m ≤ l) : n + m < k + l :=
|
||
lt_le_trans (add_lt_right H1 m) (add_le_left H2 k)
|
||
|
||
theorem add_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n + m < k + l :=
|
||
add_lt_le H1 (lt_imp_le H2)
|
||
|
||
theorem add_lt_cancel_left {n m k : ℕ} (H : k + n < k + m) : n < m :=
|
||
add_le_cancel_left (add_succ_right⁻¹ ▸ H)
|
||
|
||
theorem add_lt_cancel_right {n m k : ℕ} (H : n + k < m + k) : n < m :=
|
||
add_lt_cancel_left (add_comm ▸ add_comm ▸ H)
|
||
|
||
-- ### interaction with successor (see also the interaction with le)
|
||
|
||
theorem succ_lt {n m : ℕ} (H : n < m) : succ n < succ m :=
|
||
add_one ▸ add_one ▸ add_lt_right H 1
|
||
|
||
theorem succ_lt_cancel {n m : ℕ} (H : succ n < succ m) : n < m :=
|
||
add_lt_cancel_right (add_one⁻¹ ▸ add_one⁻¹ ▸ H)
|
||
|
||
theorem lt_imp_lt_succ {n m : ℕ} (H : n < m) : n < succ m
|
||
:= lt_trans H self_lt_succ
|
||
|
||
-- ### totality of lt and le
|
||
|
||
theorem le_or_gt {n m : ℕ} : n ≤ m ∨ n > m :=
|
||
induction_on n
|
||
(or_inl zero_le)
|
||
(take (k : ℕ),
|
||
assume IH : k ≤ m ∨ m < k,
|
||
or_elim IH
|
||
(assume H : k ≤ m,
|
||
obtain (l : ℕ) (Hl : k + l = m), from le_elim H,
|
||
discriminate
|
||
(assume H2 : l = 0,
|
||
have H3 : m = k,
|
||
from calc
|
||
m = k + l : Hl⁻¹
|
||
... = k + 0 : {H2}
|
||
... = k : add_zero_right,
|
||
have H4 : m < succ k, from subst H3 self_lt_succ,
|
||
or_inr H4)
|
||
(take l2 : ℕ,
|
||
assume H2 : l = succ l2,
|
||
have H3 : succ k + l2 = m,
|
||
from calc
|
||
succ k + l2 = k + succ l2 : add_move_succ
|
||
... = k + l : {H2⁻¹}
|
||
... = m : Hl,
|
||
or_inl (le_intro H3)))
|
||
(assume H : m < k, or_inr (lt_imp_lt_succ H)))
|
||
|
||
theorem trichotomy_alt {n m : ℕ} : (n < m ∨ n = m) ∨ n > m :=
|
||
or_imp_or_left le_or_gt (assume H : n ≤ m, le_imp_lt_or_eq H)
|
||
|
||
theorem trichotomy {n m : ℕ} : n < m ∨ n = m ∨ n > m :=
|
||
iff_elim_left or_assoc trichotomy_alt
|
||
|
||
theorem le_total {n m : ℕ} : n ≤ m ∨ m ≤ n :=
|
||
or_imp_or_right le_or_gt (assume H : m < n, lt_imp_le H)
|
||
|
||
theorem not_lt_imp_ge {n m : ℕ} (H : ¬ n < m) : n ≥ m :=
|
||
resolve_left le_or_gt H
|
||
|
||
theorem not_le_imp_gt {n m : ℕ} (H : ¬ n ≤ m) : n > m :=
|
||
resolve_right le_or_gt H
|
||
|
||
-- The following three theorems are automatically proved using the instance le_decidable
|
||
theorem lt_decidable [instance] (n m : ℕ) : decidable (n < m)
|
||
theorem gt_decidable [instance] (n m : ℕ) : decidable (n > m)
|
||
theorem ge_decidable [instance] (n m : ℕ) : decidable (n ≥ m)
|
||
|
||
-- Note: interaction with multiplication under "positivity"
|
||
|
||
-- ### misc
|
||
|
||
theorem strong_induction_on {P : nat → Prop} (n : ℕ) (H : ∀n, (∀m, m < n → P m) → P n) : P n :=
|
||
have H1 : ∀ {n m : nat}, m < n → P m, from
|
||
take n,
|
||
induction_on n
|
||
(show ∀m, m < 0 → P m, from take m H, absurd H not_lt_zero)
|
||
(take n',
|
||
assume IH : ∀ {m : nat}, m < n' → P m,
|
||
have H2: P n', from H n' @IH,
|
||
show ∀m, m < succ n' → P m, from
|
||
take m,
|
||
assume H3 : m < succ n',
|
||
or_elim (le_imp_lt_or_eq (lt_succ_imp_le H3))
|
||
(assume H4: m < n', IH H4)
|
||
(assume H4: m = n', H4⁻¹ ▸ H2)),
|
||
H1 self_lt_succ
|
||
|
||
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 :=
|
||
strong_induction_on a (
|
||
take n,
|
||
show (∀m, m < n → P m) → P n, from
|
||
case n
|
||
(assume H : (∀m, m < 0 → P m), show P 0, from H0)
|
||
(take n,
|
||
assume H : (∀m, m < succ n → P m),
|
||
show P (succ n), from
|
||
Hind n (take m, assume H1 : m ≤ n, H _ (le_imp_lt_succ H1))))
|
||
|
||
-- Positivity
|
||
-- ---------
|
||
--
|
||
-- Writing "t > 0" is the preferred way to assert that a natural number is positive.
|
||
|
||
-- ### basic
|
||
|
||
theorem case_zero_pos {P : ℕ → Prop} (y : ℕ) (H0 : P 0) (H1 : ∀ {y : nat}, y > 0 → P y) : P y :=
|
||
case y H0 (take y, H1 succ_pos)
|
||
|
||
theorem zero_or_pos {n : ℕ} : n = 0 ∨ n > 0 :=
|
||
or_imp_or_left
|
||
(or_swap (le_imp_lt_or_eq zero_le))
|
||
(take H : 0 = n, H⁻¹)
|
||
|
||
theorem succ_imp_pos {n m : ℕ} (H : n = succ m) : n > 0 :=
|
||
H⁻¹ ▸ succ_pos
|
||
|
||
theorem ne_zero_imp_pos {n : ℕ} (H : n ≠ 0) : n > 0 :=
|
||
or_elim zero_or_pos (take H2 : n = 0, absurd H2 H) (take H2 : n > 0, H2)
|
||
|
||
theorem pos_imp_ne_zero {n : ℕ} (H : n > 0) : n ≠ 0 :=
|
||
ne_symm (lt_imp_ne H)
|
||
|
||
theorem pos_imp_eq_succ {n : ℕ} (H : n > 0) : exists l, n = succ l :=
|
||
lt_imp_eq_succ H
|
||
|
||
theorem add_pos_right {n k : ℕ} (H : k > 0) : n + k > n :=
|
||
add_zero_right ▸ add_lt_left H n
|
||
|
||
theorem add_pos_left {n : ℕ} {k : ℕ} (H : k > 0) : k + n > n :=
|
||
add_comm ▸ add_pos_right H
|
||
|
||
-- ### multiplication
|
||
|
||
theorem mul_pos {n m : ℕ} (Hn : n > 0) (Hm : m > 0) : n * m > 0 :=
|
||
obtain (k : ℕ) (Hk : n = succ k), from pos_imp_eq_succ Hn,
|
||
obtain (l : ℕ) (Hl : m = succ l), from pos_imp_eq_succ Hm,
|
||
succ_imp_pos (calc
|
||
n * m = succ k * m : {Hk}
|
||
... = succ k * succ l : {Hl}
|
||
... = succ k * l + succ k : mul_succ_right
|
||
... = succ (succ k * l + k) : add_succ_right)
|
||
|
||
theorem mul_pos_imp_pos_left {n m : ℕ} (H : n * m > 0) : n > 0 :=
|
||
discriminate
|
||
(assume H2 : n = 0,
|
||
have H3 : n * m = 0,
|
||
from calc
|
||
n * m = 0 * m : {H2}
|
||
... = 0 : mul_zero_left,
|
||
have H4 : 0 > 0, from H3 ▸ H,
|
||
absurd H4 lt_irrefl)
|
||
(take l : nat,
|
||
assume Hl : n = succ l,
|
||
Hl⁻¹ ▸ succ_pos)
|
||
|
||
theorem mul_pos_imp_pos_right {m n : ℕ} (H : n * m > 0) : m > 0 :=
|
||
mul_pos_imp_pos_left (mul_comm ▸ H)
|
||
|
||
-- ### interaction of mul with le and lt
|
||
|
||
theorem mul_lt_left {n m k : ℕ} (Hk : k > 0) (H : n < m) : k * n < k * m :=
|
||
have H2 : k * n < k * n + k, from add_pos_right Hk,
|
||
have H3 : k * n + k ≤ k * m, from mul_succ_right ▸ mul_le_left H k,
|
||
lt_le_trans H2 H3
|
||
|
||
theorem mul_lt_right {n m k : ℕ} (Hk : k > 0) (H : n < m) : n * k < m * k :=
|
||
mul_comm ▸ mul_comm ▸ mul_lt_left Hk H
|
||
|
||
theorem mul_le_lt {n m k l : ℕ} (Hk : k > 0) (H1 : n ≤ k) (H2 : m < l) : n * m < k * l :=
|
||
le_lt_trans (mul_le_right H1 m) (mul_lt_left Hk H2)
|
||
|
||
theorem mul_lt_le {n m k l : ℕ} (Hl : l > 0) (H1 : n < k) (H2 : m ≤ l) : n * m < k * l :=
|
||
le_lt_trans (mul_le_left H2 n) (mul_lt_right Hl H1)
|
||
|
||
theorem mul_lt {n m k l : ℕ} (H1 : n < k) (H2 : m < l) : n * m < k * l :=
|
||
have H3 : n * m ≤ k * m, from mul_le_right (lt_imp_le H1) m,
|
||
have H4 : k * m < k * l, from mul_lt_left (le_lt_trans zero_le H1) H2,
|
||
le_lt_trans H3 H4
|
||
|
||
theorem mul_lt_cancel_left {n m k : ℕ} (H : k * n < k * m) : n < m :=
|
||
or_elim le_or_gt
|
||
(assume H2 : m ≤ n,
|
||
have H3 : k * m ≤ k * n, from mul_le_left H2 k,
|
||
absurd H3 (lt_imp_not_ge H))
|
||
(assume H2 : n < m, H2)
|
||
|
||
theorem mul_lt_cancel_right {n m k : ℕ} (H : n * k < m * k) : n < m :=
|
||
mul_lt_cancel_left (mul_comm ▸ mul_comm ▸ H)
|
||
|
||
theorem mul_le_cancel_left {n m k : ℕ} (Hk : k > 0) (H : k * n ≤ k * m) : n ≤ m :=
|
||
have H2 : k * n < k * m + k, from le_lt_trans H (add_pos_right Hk),
|
||
have H3 : k * n < k * succ m, from mul_succ_right⁻¹ ▸ H2,
|
||
have H4 : n < succ m, from mul_lt_cancel_left H3,
|
||
show n ≤ m, from lt_succ_imp_le H4
|
||
|
||
theorem mul_le_cancel_right {n k m : ℕ} (Hm : m > 0) (H : n * m ≤ k * m) : n ≤ k :=
|
||
mul_le_cancel_left Hm (mul_comm ▸ mul_comm ▸ H)
|
||
|
||
theorem mul_cancel_left {m k n : ℕ} (Hn : n > 0) (H : n * m = n * k) : m = k :=
|
||
have H2 : n * m ≤ n * k, from H ▸ le_refl,
|
||
have H3 : n * k ≤ n * m, from H ▸ le_refl,
|
||
have H4 : m ≤ k, from mul_le_cancel_left Hn H2,
|
||
have H5 : k ≤ m, from mul_le_cancel_left Hn H3,
|
||
le_antisym H4 H5
|
||
|
||
theorem mul_cancel_left_or {n m k : ℕ} (H : n * m = n * k) : n = 0 ∨ m = k :=
|
||
or_imp_or_right zero_or_pos
|
||
(assume Hn : n > 0, mul_cancel_left Hn H)
|
||
|
||
theorem mul_cancel_right {n m k : ℕ} (Hm : m > 0) (H : n * m = k * m) : n = k :=
|
||
mul_cancel_left Hm (mul_comm ▸ mul_comm ▸ H)
|
||
|
||
theorem mul_cancel_right_or {n m k : ℕ} (H : n * m = k * m) : m = 0 ∨ n = k :=
|
||
mul_cancel_left_or (mul_comm ▸ mul_comm ▸ H)
|
||
|
||
theorem mul_eq_one_left {n m : ℕ} (H : n * m = 1) : n = 1 :=
|
||
have H2 : n * m > 0, from H⁻¹ ▸ succ_pos,
|
||
have H3 : n > 0, from mul_pos_imp_pos_left H2,
|
||
have H4 : m > 0, from mul_pos_imp_pos_right H2,
|
||
or_elim le_or_gt
|
||
(assume H5 : n ≤ 1,
|
||
show n = 1, from le_antisym H5 H3)
|
||
(assume H5 : n > 1,
|
||
have H6 : n * m ≥ 2 * 1, from mul_le H5 H4,
|
||
have H7 : 1 ≥ 2, from mul_one_right ▸ H ▸ H6,
|
||
absurd self_lt_succ (le_imp_not_gt H7))
|
||
|
||
theorem mul_eq_one_right {n m : ℕ} (H : n * m = 1) : m = 1 :=
|
||
mul_eq_one_left (mul_comm ▸ H)
|
||
|
||
end nat
|