feat(library/data/{nat,int}/div.lean,*): improve and extend div in nat and int

This commit is contained in:
Jeremy Avigad 2015-05-30 22:09:34 +10:00
parent c986ee305b
commit b76445df39
7 changed files with 318 additions and 233 deletions

View file

@ -120,7 +120,7 @@ section comm_semiring
theorem dvd_mul_of_dvd_right {a b : A} (H : a b) (c : A) : a c * b := theorem dvd_mul_of_dvd_right {a b : A} (H : a b) (c : A) : a c * b :=
!mul.comm ▸ (dvd_mul_of_dvd_left H _) !mul.comm ▸ (dvd_mul_of_dvd_left H _)
theorem mul_dvd_mul {a b c d : A} (dvd_ab : (a b)) (dvd_cd : c d) : a * c b * d := theorem mul_dvd_mul {a b c d : A} (dvd_ab : a b) (dvd_cd : c d) : a * c b * d :=
dvd.elim dvd_ab dvd.elim dvd_ab
(take e, assume Haeb : b = a * e, (take e, assume Haeb : b = a * e,
dvd.elim dvd_cd dvd.elim dvd_cd

View file

@ -76,7 +76,7 @@ theorem decode_encode_sum : ∀ s : sum A B, decode_sum (encode_sum s) = some s
end end
| (sum.inr b) := | (sum.inr b) :=
assert aux₁ : 2 > 0, from dec_trivial, assert aux₁ : 2 > 0, from dec_trivial,
assert aux₂ : 1 mod 2 = 1, by rewrite [modulo_def], assert aux₂ : 1 mod 2 = 1, by rewrite [nat.modulo_def],
assert aux₃ : 1 ≠ 0, from dec_trivial, assert aux₃ : 1 ≠ 0, from dec_trivial,
begin begin
esimp [encode_sum, decode_sum], esimp [encode_sum, decode_sum],

View file

@ -95,6 +95,9 @@ infix * := int.mul
theorem of_nat.inj {m n : } (H : of_nat m = of_nat n) : m = n := theorem of_nat.inj {m n : } (H : of_nat m = of_nat n) : m = n :=
by injection H; assumption by injection H; assumption
theorem of_nat_eq_of_nat (m n : ) : of_nat m = of_nat n ↔ m = n :=
iff.intro of_nat.inj !congr_arg
theorem neg_succ_of_nat.inj {m n : } (H : neg_succ_of_nat m = neg_succ_of_nat n) : m = n := theorem neg_succ_of_nat.inj {m n : } (H : neg_succ_of_nat m = neg_succ_of_nat n) : m = n :=
by injection H; assumption by injection H; assumption

View file

@ -3,7 +3,7 @@ Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE. Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad Author: Jeremy Avigad
Definitions and properties of div, mod, gcd, lcm, coprime, following the SSReflect library. Definitions and properties of div and mod, following the SSReflect library.
Following SSReflect and the SMTlib standard, we define a mod b so that 0 ≤ a mod b < |b| when b ≠ 0. Following SSReflect and the SMTlib standard, we define a mod b so that 0 ≤ a mod b < |b| when b ≠ 0.
-/ -/
@ -13,8 +13,6 @@ open [declarations] nat (succ)
open eq.ops open eq.ops
notation `` := nat notation `` := nat
set_option pp.beta true
namespace int namespace int
/- definitions -/ /- definitions -/
@ -25,14 +23,11 @@ sign b *
| of_nat m := #nat m div (nat_abs b) | of_nat m := #nat m div (nat_abs b)
| -[ m +1] := -[ (#nat m div (nat_abs b)) +1] | -[ m +1] := -[ (#nat m div (nat_abs b)) +1]
end) end)
notation a div b := divide a b notation a div b := divide a b
definition modulo (a b : ) : := a - a div b * b definition modulo (a b : ) : := a - a div b * b
notation a mod b := modulo a b notation a mod b := modulo a b
notation a `≡` b `[mod`:100 c `]`:0 := a mod c = b mod c
notation a = b [mod c] := a mod c = b mod c
/- div -/ /- div -/
@ -125,95 +120,6 @@ lt.by_cases
have H3 : a < b, from abs_of_pos H ▸ H2, have H3 : a < b, from abs_of_pos H ▸ H2,
div_eq_zero_of_lt H1 H3) div_eq_zero_of_lt H1 H3)
/- mod -/
theorem of_nat_mod_of_nat (m n : nat) : m mod n = (#nat m mod n) :=
have H : m = (#nat m mod n) + m div n * n, from calc
m = of_nat (#nat m div n * n + m mod n) : nat.eq_div_mul_add_mod
... = (#nat m div n) * n + (#nat m mod n) : rfl
... = m div n * n + (#nat m mod n) : of_nat_div_of_nat
... = (#nat m mod n) + m div n * n : add.comm,
calc
m mod n = m - m div n * n : rfl
... = (#nat m mod n) : sub_eq_of_eq_add H
theorem neg_succ_of_nat_mod (m : ) {b : } (bpos : b > 0) :
-[m +1] mod b = b - 1 - m mod b :=
calc
-[m +1] mod b = -(m + 1) - -[m +1] div b * b : rfl
... = -(m + 1) - -(m div b + 1) * b : neg_succ_of_nat_div _ bpos
... = -m + -1 + (b + m div b * b) :
by rewrite [neg_add, -neg_mul_eq_neg_mul, sub_neg_eq_add, mul.right_distrib,
one_mul, (add.comm b)]
... = b + -1 + (-m + m div b * b) :
by rewrite [-*add.assoc, add.comm (-m), add.right_comm (-1), (add.comm b)]
... = b - 1 - m mod b :
by rewrite [↑modulo, *sub_eq_add_neg, neg_add, neg_neg]
theorem mod_neg (a b : ) : a mod -b = a mod b :=
calc
a mod -b = a - (a div -b) * -b : rfl
... = a - -(a div b) * -b : div_neg
... = a - a div b * b : neg_mul_neg
... = a mod b : rfl
theorem mod_abs (a b : ) : a mod (abs b) = a mod b :=
abs.by_cases rfl !mod_neg
theorem zero_mod (b : ) : 0 mod b = 0 :=
by rewrite [↑modulo, zero_div, zero_mul, sub_zero]
theorem mod_zero (a : ) : a mod 0 = a :=
by rewrite [↑modulo, mul_zero, sub_zero]
theorem mod_one (a : ) : a mod 1 = 0 :=
calc
a mod 1 = a - a div 1 * 1 : rfl
... = 0 : by rewrite [mul_one, div_one, sub_self]
private lemma of_nat_mod_abs (m : ) (b : ) : m mod (abs b) = (#nat m mod (nat_abs b)) :=
calc
m mod (abs b) = m mod (nat_abs b) : of_nat_nat_abs
... = (#nat m mod (nat_abs b)) : of_nat_mod_of_nat
private lemma of_nat_mod_abs_lt (m : ) {b : } (H : b ≠ 0) : m mod (abs b) < (abs b) :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : (#nat nat_abs b > 0), from lt_of_of_nat_lt_of_nat (!of_nat_nat_abs⁻¹ ▸ H1),
calc
m mod (abs b) = (#nat m mod (nat_abs b)) : of_nat_mod_abs m b
... < nat_abs b : of_nat_lt_of_nat_of_lt (!nat.mod_lt H2)
... = abs b : of_nat_nat_abs _
theorem mod_nonneg (a : ) {b : } (H : b ≠ 0) : a mod b ≥ 0 :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : a mod (abs b) ≥ 0, from
int.cases_on a
(take m, (of_nat_mod_abs m b)⁻¹ ▸ of_nat_nonneg (nat.modulo m (nat_abs b)))
(take m,
have H3 : 1 + m mod (abs b) ≤ (abs b),
from (!add.comm ▸ add_one_le_of_lt (of_nat_mod_abs_lt m H)),
calc
-[ m +1] mod (abs b) = abs b - 1 - m mod (abs b) : neg_succ_of_nat_mod _ H1
... = abs b - (1 + m mod (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
... ≥ 0 : iff.mp' !sub_nonneg_iff_le H3),
!mod_abs ▸ H2
theorem mod_lt (a : ) {b : } (H : b ≠ 0) : a mod b < (abs b) :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : a mod (abs b) < abs b, from
int.cases_on a
(take m, of_nat_mod_abs_lt m H)
(take m,
have H3 : abs b ≠ 0, from assume H', H (eq_zero_of_abs_eq_zero H'),
have H4 : 1 + m mod (abs b) > 0, from add_pos_of_pos_of_nonneg dec_trivial (mod_nonneg _ H3),
calc
-[ m +1] mod (abs b) = abs b - 1 - m mod (abs b) : neg_succ_of_nat_mod _ H1
... = abs b - (1 + m mod (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
... < abs b : sub_lt_self _ H4),
!mod_abs ▸ H2
/- both div and mod -/
private theorem add_mul_div_self_aux1 {a : } {k : } (n : ) private theorem add_mul_div_self_aux1 {a : } {k : } (n : )
(H1 : a ≥ 0) (H2 : #nat k > 0) : (H1 : a ≥ 0) (H2 : #nat k > 0) :
(a + n * k) div k = a div k + n := (a + n * k) div k = a div k + n :=
@ -233,7 +139,7 @@ or.elim (nat.lt_or_ge m (#nat n * k))
(assume m_lt_nk : #nat m < n * k, (assume m_lt_nk : #nat m < n * k,
have H3 : #nat (m + 1 ≤ n * k), from nat.succ_le_of_lt m_lt_nk, have H3 : #nat (m + 1 ≤ n * k), from nat.succ_le_of_lt m_lt_nk,
have H4 : #nat m div k + 1 ≤ n, have H4 : #nat m div k + 1 ≤ n,
from nat.succ_le_of_lt (nat.div_lt_of_lt_mul (!nat.mul.comm ▸ m_lt_nk)), from nat.succ_le_of_lt (nat.div_lt_of_lt_mul m_lt_nk),
Hm⁻¹ ▸ (calc Hm⁻¹ ▸ (calc
(-[m +1] + n * k) div k = (n * k - (m + 1)) div k : by rewrite [add.comm, neg_succ_of_nat_eq] (-[m +1] + n * k) div k = (n * k - (m + 1)) div k : by rewrite [add.comm, neg_succ_of_nat_eq]
... = ((#nat n * k) - (#nat m + 1)) div k : rfl ... = ((#nat n * k) - (#nat m + 1)) div k : rfl
@ -314,6 +220,105 @@ calc
theorem mul_div_cancel_left {a : } (b : ) (H : a ≠ 0) : a * b div a = b := theorem mul_div_cancel_left {a : } (b : ) (H : a ≠ 0) : a * b div a = b :=
!mul.comm ▸ mul_div_cancel b H !mul.comm ▸ mul_div_cancel b H
theorem div_self {a : } (H : a ≠ 0) : a div a = 1 :=
!mul_one ▸ !mul_div_cancel_left H
/- mod -/
theorem of_nat_mod_of_nat (m n : nat) : m mod n = (#nat m mod n) :=
have H : m = (#nat m mod n) + m div n * n, from calc
m = of_nat (#nat m div n * n + m mod n) : nat.eq_div_mul_add_mod
... = (#nat m div n) * n + (#nat m mod n) : rfl
... = m div n * n + (#nat m mod n) : of_nat_div_of_nat
... = (#nat m mod n) + m div n * n : add.comm,
calc
m mod n = m - m div n * n : rfl
... = (#nat m mod n) : sub_eq_of_eq_add H
theorem neg_succ_of_nat_mod (m : ) {b : } (bpos : b > 0) :
-[m +1] mod b = b - 1 - m mod b :=
calc
-[m +1] mod b = -(m + 1) - -[m +1] div b * b : rfl
... = -(m + 1) - -(m div b + 1) * b : neg_succ_of_nat_div _ bpos
... = -m + -1 + (b + m div b * b) :
by rewrite [neg_add, -neg_mul_eq_neg_mul, sub_neg_eq_add, mul.right_distrib,
one_mul, (add.comm b)]
... = b + -1 + (-m + m div b * b) :
by rewrite [-*add.assoc, add.comm (-m), add.right_comm (-1), (add.comm b)]
... = b - 1 - m mod b :
by rewrite [↑modulo, *sub_eq_add_neg, neg_add, neg_neg]
theorem mod_neg (a b : ) : a mod -b = a mod b :=
calc
a mod -b = a - (a div -b) * -b : rfl
... = a - -(a div b) * -b : div_neg
... = a - a div b * b : neg_mul_neg
... = a mod b : rfl
theorem mod_abs (a b : ) : a mod (abs b) = a mod b :=
abs.by_cases rfl !mod_neg
theorem zero_mod (b : ) : 0 mod b = 0 :=
by rewrite [↑modulo, zero_div, zero_mul, sub_zero]
theorem mod_zero (a : ) : a mod 0 = a :=
by rewrite [↑modulo, mul_zero, sub_zero]
theorem mod_one (a : ) : a mod 1 = 0 :=
calc
a mod 1 = a - a div 1 * 1 : rfl
... = 0 : by rewrite [mul_one, div_one, sub_self]
private lemma of_nat_mod_abs (m : ) (b : ) : m mod (abs b) = (#nat m mod (nat_abs b)) :=
calc
m mod (abs b) = m mod (nat_abs b) : of_nat_nat_abs
... = (#nat m mod (nat_abs b)) : of_nat_mod_of_nat
private lemma of_nat_mod_abs_lt (m : ) {b : } (H : b ≠ 0) : m mod (abs b) < (abs b) :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : (#nat nat_abs b > 0), from lt_of_of_nat_lt_of_nat (!of_nat_nat_abs⁻¹ ▸ H1),
calc
m mod (abs b) = (#nat m mod (nat_abs b)) : of_nat_mod_abs m b
... < nat_abs b : of_nat_lt_of_nat_of_lt (!nat.mod_lt H2)
... = abs b : of_nat_nat_abs _
theorem mod_eq_of_lt {a b : } (H1 : 0 ≤ a) (H2 : a < b) : a mod b = a :=
obtain m (Hm : a = of_nat m), from exists_eq_of_nat H1,
obtain n (Hn : b = of_nat n), from exists_eq_of_nat (le_of_lt (lt_of_le_of_lt H1 H2)),
begin
revert H2,
rewrite [Hm, Hn, of_nat_mod_of_nat, of_nat_lt_of_nat, of_nat_eq_of_nat],
apply nat.mod_eq_of_lt
end
theorem mod_nonneg (a : ) {b : } (H : b ≠ 0) : a mod b ≥ 0 :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : a mod (abs b) ≥ 0, from
int.cases_on a
(take m, (of_nat_mod_abs m b)⁻¹ ▸ of_nat_nonneg (nat.modulo m (nat_abs b)))
(take m,
have H3 : 1 + m mod (abs b) ≤ (abs b),
from (!add.comm ▸ add_one_le_of_lt (of_nat_mod_abs_lt m H)),
calc
-[ m +1] mod (abs b) = abs b - 1 - m mod (abs b) : neg_succ_of_nat_mod _ H1
... = abs b - (1 + m mod (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
... ≥ 0 : iff.mp' !sub_nonneg_iff_le H3),
!mod_abs ▸ H2
theorem mod_lt (a : ) {b : } (H : b ≠ 0) : a mod b < (abs b) :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : a mod (abs b) < abs b, from
int.cases_on a
(take m, of_nat_mod_abs_lt m H)
(take m,
have H3 : abs b ≠ 0, from assume H', H (eq_zero_of_abs_eq_zero H'),
have H4 : 1 + m mod (abs b) > 0, from add_pos_of_pos_of_nonneg dec_trivial (mod_nonneg _ H3),
calc
-[ m +1] mod (abs b) = abs b - 1 - m mod (abs b) : neg_succ_of_nat_mod _ H1
... = abs b - (1 + m mod (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
... < abs b : sub_lt_self _ H4),
!mod_abs ▸ H2
theorem add_mul_mod_self {a b c : } : (a + b * c) mod c = a mod c := theorem add_mul_mod_self {a b c : } : (a + b * c) mod c = a mod c :=
decidable.by_cases decidable.by_cases
(assume cz : c = 0, by rewrite [cz, mul_zero, add_zero]) (assume cz : c = 0, by rewrite [cz, mul_zero, add_zero])
@ -323,15 +328,18 @@ decidable.by_cases
theorem add_mul_mod_self_left (a b c : ) : (a + b * c) mod b = a mod b := theorem add_mul_mod_self_left (a b c : ) : (a + b * c) mod b = a mod b :=
!mul.comm ▸ !add_mul_mod_self !mul.comm ▸ !add_mul_mod_self
theorem add_mod_self {a b : } : (a + b) mod b = a mod b :=
by rewrite -(int.mul_one b) at {1}; apply add_mul_mod_self_left
theorem add_mod_self_left {a b : } : (a + b) mod a = b mod a :=
!add.comm ▸ !add_mod_self
theorem mul_mod_left (a b : ) : (a * b) mod b = 0 := theorem mul_mod_left (a b : ) : (a * b) mod b = 0 :=
by rewrite [-zero_add (a * b), add_mul_mod_self, zero_mod] by rewrite [-zero_add (a * b), add_mul_mod_self, zero_mod]
theorem mul_mod_right (a b : ) : (a * b) mod a = 0 := theorem mul_mod_right (a b : ) : (a * b) mod a = 0 :=
!mul.comm ▸ !mul_mod_left !mul.comm ▸ !mul_mod_left
theorem div_self {a : } (H : a ≠ 0) : a div a = 1 :=
!mul_one ▸ !mul_div_cancel_left H
theorem mod_self {a : } : a mod a = 0 := theorem mod_self {a : } : a mod a = 0 :=
decidable.by_cases decidable.by_cases
(assume H : a = 0, H⁻¹ ▸ !mod_zero) (assume H : a = 0, H⁻¹ ▸ !mod_zero)
@ -343,6 +351,8 @@ decidable.by_cases
theorem mod_lt_of_pos (a : ) {b : } (H : b > 0) : a mod b < b := theorem mod_lt_of_pos (a : ) {b : } (H : b > 0) : a mod b < b :=
!abs_of_pos H ▸ !mod_lt (ne.symm (ne_of_lt H)) !abs_of_pos H ▸ !mod_lt (ne.symm (ne_of_lt H))
/- properties of div and mod -/
theorem mul_div_mul_of_pos_aux {a : } (b : ) {c : } theorem mul_div_mul_of_pos_aux {a : } (b : ) {c : }
(H1 : a > 0) (H2 : c > 0) : a * b div (a * c) = b div c := (H1 : a > 0) (H2 : c > 0) : a * b div (a * c) = b div c :=
have H3 : a * c ≠ 0, from ne.symm (ne_of_lt (mul_pos H1 H2)), have H3 : a * c ≠ 0, from ne.symm (ne_of_lt (mul_pos H1 H2)),
@ -377,10 +387,8 @@ theorem mul_div_mul_of_pos_left (a : ) {b : } (c : ) (H : b > 0) :
a * b div (c * b) = a div c := a * b div (c * b) = a div c :=
!mul.comm ▸ !mul.comm ▸ !mul_div_mul_of_pos H !mul.comm ▸ !mul.comm ▸ !mul_div_mul_of_pos H
theorem div_mul_le (a : ) {b : } (H : b ≠ 0) : a div b * b ≤ a := theorem mul_mod_mul_of_pos {a : } (b c : ) (H : a > 0) : a * b mod (a * c) = a * (b mod c) :=
calc by rewrite [↑modulo, !mul_div_mul_of_pos H, mul_sub_left_distrib, mul.left_comm]
a = a div b * b + a mod b : eq_div_mul_add_mod
... ≥ a div b * b : le_add_of_nonneg_right (!mod_nonneg H)
theorem lt_div_add_one_mul_self (a : ) {b : } (H : b > 0) : a < (a div b + 1) * b := theorem lt_div_add_one_mul_self (a : ) {b : } (H : b > 0) : a < (a div b + 1) * b :=
have H : a - a div b * b < b, from !mod_lt_of_pos H, have H : a - a div b * b < b, from !mod_lt_of_pos H,
@ -393,7 +401,7 @@ obtain (m : ) (Hm : a = m), from exists_eq_of_nat Ha,
obtain (n : ) (Hn : b = n), from exists_eq_of_nat Hb, obtain (n : ) (Hn : b = n), from exists_eq_of_nat Hb,
calc calc
a div b = #nat m div n : by rewrite [Hm, Hn, of_nat_div_of_nat] a div b = #nat m div n : by rewrite [Hm, Hn, of_nat_div_of_nat]
... ≤ m : of_nat_le_of_nat_of_le !nat.div_le ... ≤ m : of_nat_le_of_nat_of_le !nat.div_le_self
... = a : Hm ... = a : Hm
theorem abs_div_le_abs (a b : ) : abs (a div b) ≤ abs a := theorem abs_div_le_abs (a b : ) : abs (a div b) ≤ abs a :=
@ -434,7 +442,7 @@ by rewrite [eq_div_mul_add_mod a b at {2}, H, add_zero]
theorem mul_div_cancel_of_mod_eq_zero {a b : } (H : a mod b = 0) : b * (a div b) = a := theorem mul_div_cancel_of_mod_eq_zero {a b : } (H : a mod b = 0) : b * (a div b) = a :=
!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H !mul.comm ▸ div_mul_cancel_of_mod_eq_zero H
/- divides -/ /- dvd -/
theorem dvd_of_mod_eq_zero {a b : } (H : b mod a = 0) : a b := theorem dvd_of_mod_eq_zero {a b : } (H : b mod a = 0) : a b :=
dvd.intro (!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H) dvd.intro (!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H)
@ -487,32 +495,82 @@ theorem eq_mul_of_div_eq_left {a b c : } (H1 : b a) (H2 : a div b = c) :
a = c * b := a = c * b :=
!mul.comm ▸ !eq_mul_of_div_eq_right H1 H2 !mul.comm ▸ !eq_mul_of_div_eq_right H1 H2
theorem div_eq_of_eq_mul_left {a b c : } (H1 : b ≠ 0) (H2 : b a) (H3 : a = c * b) : theorem div_eq_of_eq_mul_left {a b c : } (H1 : b ≠ 0) (H2 : a = c * b) :
a div b = c := a div b = c :=
iff.mp' (!div_eq_iff_eq_mul_left H1 H2) H3 div_eq_of_eq_mul_right H1 (!mul.comm ▸ H2)
theorem div_le_iff_le_mul_right {a b : } (c : ) (H : b > 0) (H' : b a) : /- div and ordering -/
theorem div_mul_le (a : ) {b : } (H : b ≠ 0) : a div b * b ≤ a :=
calc
a = a div b * b + a mod b : eq_div_mul_add_mod
... ≥ a div b * b : le_add_of_nonneg_right (!mod_nonneg H)
theorem div_le_of_le_mul {a b c : } (H : c > 0) (H' : a ≤ b * c) : a div c ≤ b :=
le_of_mul_le_mul_right (calc
a div c * c = a div c * c + 0 : add_zero
... ≤ a div c * c + a mod c : add_le_add_left (!mod_nonneg (ne_of_gt H))
... = a : eq_div_mul_add_mod
... ≤ b * c : H') H
theorem div_le_self (a : ) {b : } (H1 : a ≥ 0) (H2 : b ≥ 0) : a div b ≤ a :=
or.elim (lt_or_eq_of_le H2)
(assume H3 : b > 0,
have H4 : b ≥ 1, from add_one_le_of_lt H3,
have H5 : a ≤ a * b, from calc
a = a * 1 : mul_one
... ≤ a * b : !mul_le_mul_of_nonneg_left H4 H1,
div_le_of_le_mul H3 H5)
(assume H3 : 0 = b,
by rewrite [-H3, div_zero]; apply H1)
theorem mul_le_of_le_div {a b c : } (H1 : c > 0) (H2 : a ≤ b div c) : a * c ≤ b :=
calc
a * c ≤ b div c * c : !mul_le_mul_of_nonneg_right H2 (le_of_lt H1)
... ≤ b : !div_mul_le (ne_of_gt H1)
theorem le_div_of_mul_le {a b c : } (H1 : c > 0) (H2 : a * c ≤ b) : a ≤ b div c :=
have H3 : a * c < (b div c + 1) * c, from
calc
a * c ≤ b : H2
... = b div c * c + b mod c : eq_div_mul_add_mod
... < b div c * c + c : add_lt_add_left (!mod_lt_of_pos H1)
... = (b div c + 1) * c : by rewrite [mul.right_distrib, one_mul],
le_of_lt_add_one (lt_of_mul_lt_mul_right H3 (le_of_lt H1))
theorem le_div_iff_mul_le {a b c : } (H : c > 0) : a ≤ b div c ↔ a * c ≤ b :=
iff.intro (!mul_le_of_le_div H) (!le_div_of_mul_le H)
theorem div_le_div {a b c : } (H : c > 0) (H' : a ≤ b) : a div c ≤ b div c :=
le_div_of_mul_le H (le.trans (!div_mul_le (ne_of_gt H)) H')
theorem div_lt_of_lt_mul {a b c : } (H : c > 0) (H' : a < b * c) : a div c < b :=
lt_of_mul_lt_mul_right
(calc
a div c * c = a div c * c + 0 : add_zero
... ≤ a div c * c + a mod c : add_le_add_left (!mod_nonneg (ne_of_gt H))
... = a : eq_div_mul_add_mod
... < b * c : H')
(le_of_lt H)
theorem lt_mul_of_div_lt {a b c : } (H1 : c > 0) (H2 : a div c < b) : a < b * c :=
assert H3 : (a div c + 1) * c ≤ b * c,
from !mul_le_mul_of_nonneg_right (add_one_le_of_lt H2) (le_of_lt H1),
have H4 : a div c * c + c ≤ b * c, by rewrite [mul.right_distrib at H3, one_mul at H3]; apply H3,
calc
a = a div c * c + a mod c : eq_div_mul_add_mod
... < a div c * c + c : add_lt_add_left (!mod_lt_of_pos H1)
... ≤ b * c : H4
theorem div_lt_iff_lt_mul {a b c : } (H : c > 0) : a div c < b ↔ a < b * c :=
iff.intro (!lt_mul_of_div_lt H) (!div_lt_of_lt_mul H)
theorem div_le_iff_le_mul_of_div {a b : } (c : ) (H : b > 0) (H' : b a) :
a div b ≤ c ↔ a ≤ c * b := a div b ≤ c ↔ a ≤ c * b :=
by rewrite [propext (!le_iff_mul_le_mul_right H), !div_mul_cancel H'] by rewrite [propext (!le_iff_mul_le_mul_right H), !div_mul_cancel H']
theorem div_le_iff_le_mul_left {a b : } (c : ) (H : b > 0) (H' : b a) : theorem le_mul_of_div_le_of_div {a b c : } (H1 : b > 0) (H2 : b a) (H3 : a div b ≤ c) :
a div b ≤ c ↔ a ≤ b * c :=
!mul.comm ▸ !div_le_iff_le_mul_right H H'
theorem eq_mul_of_div_le_right {a b c : } (H1 : b > 0) (H2 : b a) (H3 : a div b ≤ c) :
a ≤ c * b := a ≤ c * b :=
iff.mp (!div_le_iff_le_mul_right H1 H2) H3 iff.mp (!div_le_iff_le_mul_of_div H1 H2) H3
theorem div_le_of_eq_mul_right {a b c : } (H1 : b > 0) (H2 : b a) (H3 : a ≤ c * b) :
a div b ≤ c :=
iff.mp' (!div_le_iff_le_mul_right H1 H2) H3
theorem eq_mul_of_div_le_left {a b c : } (H1 : b > 0) (H2 : b a) (H3 : a div b ≤ c) :
a ≤ b * c :=
iff.mp (!div_le_iff_le_mul_left H1 H2) H3
theorem div_le_of_eq_mul_left {a b c : } (H1 : b > 0) (H2 : b a) (H3 : a ≤ b * c) :
a div b ≤ c :=
iff.mp' (!div_le_iff_le_mul_left H1 H2) H3
end int end int

View file

@ -10,7 +10,7 @@ open eq.ops well_founded decidable fake_simplifier prod
namespace nat namespace nat
/- div and mod -/ /- div -/
-- auxiliary lemma used to justify div -- auxiliary lemma used to justify div
private definition div_rec_lemma {x y : nat} (H : 0 < y ∧ y ≤ x) : x - y < x := private definition div_rec_lemma {x y : nat} (H : 0 < y ∧ y ≤ x) : x - y < x :=
@ -20,12 +20,11 @@ private definition div.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y :
if H : 0 < y ∧ y ≤ x then f (x - y) (div_rec_lemma H) y + 1 else zero if H : 0 < y ∧ y ≤ x then f (x - y) (div_rec_lemma H) y + 1 else zero
definition divide (x y : nat) := fix div.F x y definition divide (x y : nat) := fix div.F x y
notation a div b := divide a b
theorem divide_def (x y : nat) : divide x y = if 0 < y ∧ y ≤ x then divide (x - y) y + 1 else 0 := theorem divide_def (x y : nat) : divide x y = if 0 < y ∧ y ≤ x then divide (x - y) y + 1 else 0 :=
congr_fun (fix_eq div.F x) y congr_fun (fix_eq div.F x) y
notation a div b := divide a b
theorem div_zero (a : ) : a div 0 = 0 := theorem div_zero (a : ) : a div 0 = 0 :=
divide_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0)) divide_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0))
@ -71,12 +70,14 @@ calc
theorem mul_div_cancel_left {m : } (n : ) (H : m > 0) : m * n div m = n := theorem mul_div_cancel_left {m : } (n : ) (H : m > 0) : m * n div m = n :=
!mul.comm ▸ !mul_div_cancel H !mul.comm ▸ !mul_div_cancel H
/- mod -/
private definition mod.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y : nat) : nat := private definition mod.F (x : nat) (f : Π x₁, x₁ < x → nat → nat) (y : nat) : nat :=
if H : 0 < y ∧ y ≤ x then f (x - y) (div_rec_lemma H) y else x if H : 0 < y ∧ y ≤ x then f (x - y) (div_rec_lemma H) y else x
definition modulo (x y : nat) := fix mod.F x y definition modulo (x y : nat) := fix mod.F x y
notation a mod b := modulo a b notation a mod b := modulo a b
notation a `≡` b `[mod`:100 c `]`:0 := a mod c = b mod c
theorem modulo_def (x y : nat) : modulo x y = if 0 < y ∧ y ≤ x then modulo (x - y) y else x := theorem modulo_def (x y : nat) : modulo x y = if 0 < y ∧ y ≤ x then modulo (x - y) y else x :=
congr_fun (fix_eq mod.F x) y congr_fun (fix_eq mod.F x) y
@ -143,7 +144,11 @@ nat.case_strong_induction_on x
have H5 : succ x - y ≤ x, from le_of_lt_succ H4, have H5 : succ x - y ≤ x, from le_of_lt_succ H4,
show succ x mod y < y, from H3⁻¹ ▸ IH _ H5)) show succ x mod y < y, from H3⁻¹ ▸ IH _ H5))
/- properties of div and mod together -/ theorem mod_one (n : ) : n mod 1 = 0 :=
have H1 : n mod 1 < 1, from !mod_lt !succ_pos,
eq_zero_of_le_zero (le_of_lt_succ H1)
/- properties of div and mod -/
-- the quotient / remainder theorem -- the quotient / remainder theorem
theorem eq_div_mul_add_mod (x y : ) : x = x div y * y + x mod y := theorem eq_div_mul_add_mod (x y : ) : x = x div y * y + x mod y :=
@ -187,12 +192,12 @@ theorem mod_le {x y : } : x mod y ≤ x :=
theorem eq_remainder {q1 r1 q2 r2 y : } (H1 : r1 < y) (H2 : r2 < y) theorem eq_remainder {q1 r1 q2 r2 y : } (H1 : r1 < y) (H2 : r2 < y)
(H3 : q1 * y + r1 = q2 * y + r2) : r1 = r2 := (H3 : q1 * y + r1 = q2 * y + r2) : r1 = r2 :=
calc calc
r1 = r1 mod y : by simp r1 = r1 mod y : mod_eq_of_lt H1
... = (r1 + q1 * y) mod y : !add_mul_mod_self⁻¹ ... = (r1 + q1 * y) mod y : !add_mul_mod_self⁻¹
... = (q1 * y + r1) mod y : add.comm ... = (q1 * y + r1) mod y : add.comm
... = (r2 + q2 * y) mod y : by simp ... = (r2 + q2 * y) mod y : by rewrite [H3, add.comm]
... = r2 mod y : !add_mul_mod_self ... = r2 mod y : !add_mul_mod_self
... = r2 : by simp ... = r2 : mod_eq_of_lt H2
theorem eq_quotient {q1 r1 q2 r2 y : } (H1 : r1 < y) (H2 : r2 < y) theorem eq_quotient {q1 r1 q2 r2 y : } (H1 : r1 < y) (H2 : r2 < y)
(H3 : q1 * y + r1 = q2 * y + r2) : q1 = q2 := (H3 : q1 * y + r1 = q2 * y + r2) : q1 = q2 :=
@ -245,10 +250,6 @@ or.elim (eq_zero_or_pos z)
theorem mul_mod_mul_right (x z y : ) : (x * z) mod (y * z) = (x mod y) * z := theorem mul_mod_mul_right (x z y : ) : (x * z) mod (y * z) = (x mod y) * z :=
mul.comm z x ▸ mul.comm z y ▸ !mul.comm ▸ !mul_mod_mul_left mul.comm z x ▸ mul.comm z y ▸ !mul.comm ▸ !mul_mod_mul_left
theorem mod_one (n : ) : n mod 1 = 0 :=
have H1 : n mod 1 < 1, from !mod_lt !succ_pos,
eq_zero_of_le_zero (le_of_lt_succ H1)
theorem mod_self (n : ) : n mod n = 0 := theorem mod_self (n : ) : n mod n = 0 :=
nat.cases_on n (by simp) nat.cases_on n (by simp)
(take n, (take n,
@ -279,60 +280,7 @@ by rewrite [eq_div_mul_add_mod m n at {2}, H, add_zero]
theorem mul_div_cancel_of_mod_eq_zero {m n : } (H : m mod n = 0) : n * (m div n) = m := theorem mul_div_cancel_of_mod_eq_zero {m n : } (H : m mod n = 0) : n * (m div n) = m :=
!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H !mul.comm ▸ div_mul_cancel_of_mod_eq_zero H
theorem div_lt_of_lt_mul {m n k : } (H : m < k * n) : m div k < n := /- dvd -/
lt_of_mul_lt_mul_right (calc
m div k * k ≤ m div k * k + m mod k : le_add_right
... = m : eq_div_mul_add_mod
... < k * n : H
... = n * k : nat.mul.comm)
theorem div_le_of_le_mul {m n k : } (H : m ≤ k * n) : m div k ≤ n :=
or.elim (eq_zero_or_pos k)
(assume H1 : k = 0,
calc
m div k = m div 0 : H1
... = 0 : div_zero
... ≤ n : zero_le)
(assume H1 : k > 0,
le_of_mul_le_mul_right (calc
m div k * k ≤ m div k * k + m mod k : le_add_right
... = m : eq_div_mul_add_mod
... ≤ k * n : H
... = n * k : nat.mul.comm) H1)
theorem div_le (m n : ) : m div n ≤ m :=
nat.cases_on n (!div_zero⁻¹ ▸ !zero_le)
take n,
have H : m ≤ succ n * m, from calc
m = 1 * m : one_mul
... ≤ succ n * m : mul_le_mul_right (succ_le_succ !zero_le),
div_le_of_le_mul H
theorem mul_sub_div_of_lt {m n k : } (H : k < m * n) :
(m * n - (k + 1)) div m = n - k div m - 1 :=
have H1 : k div m < n, from div_lt_of_lt_mul H,
have H2 : n - k div m ≥ 1, from
le_sub_of_add_le (calc
1 + k div m = succ (k div m) : add.comm
... ≤ n : succ_le_of_lt H1),
assert H3 : n - k div m = n - k div m - 1 + 1, from (sub_add_cancel H2)⁻¹,
assert H4 : m > 0, from pos_of_ne_zero (assume H': m = 0, not_lt_zero _ (!zero_mul ▸ H' ▸ H)),
have H5 : k mod m + 1 ≤ m, from succ_le_of_lt (!mod_lt H4),
assert H6 : m - (k mod m + 1) < m, from sub_lt_self H4 !succ_pos,
calc
(m * n - (k + 1)) div m = (m * n - (k div m * m + k mod m + 1)) div m : eq_div_mul_add_mod
... = (m * n - k div m * m - (k mod m + 1)) div m : by rewrite [*sub_sub]
... = ((n - k div m) * m - (k mod m + 1)) div m :
by rewrite [mul.comm m, mul_sub_right_distrib]
... = ((n - k div m - 1) * m + m - (k mod m + 1)) div m :
by rewrite [H3 at {1}, mul.right_distrib, nat.one_mul]
... = ((n - k div m - 1) * m + (m - (k mod m + 1))) div m : {add_sub_assoc H5 _}
... = (m - (k mod m + 1)) div m + (n - k div m - 1) :
by rewrite [add.comm, (add_mul_div_self H4)]
... = n - k div m - 1 :
by rewrite [div_eq_zero_of_lt H6, zero_add]
/- divides -/
theorem dvd_of_mod_eq_zero {m n : } (H : n mod m = 0) : m n := theorem dvd_of_mod_eq_zero {m n : } (H : n mod m = 0) : m n :=
dvd.intro (!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H) dvd.intro (!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H)
@ -362,17 +310,17 @@ have aux : m * (c₁ - c₂) = n₂, from calc
... = n₂ : add_sub_cancel_left, ... = n₂ : add_sub_cancel_left,
dvd.intro aux dvd.intro aux
theorem dvd_of_dvd_add_right {m n1 n2 : } (H : m (n1 + n2)) : m n2 → m n1 := theorem dvd_of_dvd_add_right {m n₁ n₂ : } (H : m n₁ + n₂) : m n₂ → m n₁ :=
dvd_of_dvd_add_left (!add.comm ▸ H) dvd_of_dvd_add_left (!add.comm ▸ H)
theorem dvd_sub {m n1 n2 : } (H1 : m n1) (H2 : m n2) : m n1 - n2 := theorem dvd_sub {m n₁ n₂ : } (H1 : m n₁) (H2 : m n₂) : m n₁ - n₂ :=
by_cases by_cases
(assume H3 : n1 ≥ n2, (assume H3 : n₁ ≥ n₂,
have H4 : n1 = n1 - n2 + n2, from (sub_add_cancel H3)⁻¹, have H4 : n₁ = n₁ - n₂ + n₂, from (sub_add_cancel H3)⁻¹,
show m n1 - n2, from dvd_of_dvd_add_right (H4 ▸ H1) H2) show m n₁ - n₂, from dvd_of_dvd_add_right (H4 ▸ H1) H2)
(assume H3 : ¬ (n1 ≥ n2), (assume H3 : ¬ (n₁ ≥ n₂),
have H4 : n1 - n2 = 0, from sub_eq_zero_of_le (le_of_lt (lt_of_not_ge H3)), have H4 : n₁ - n₂ = 0, from sub_eq_zero_of_le (le_of_lt (lt_of_not_ge H3)),
show m n1 - n2, from H4⁻¹ ▸ dvd_zero _) show m n₁ - n₂, from H4⁻¹ ▸ dvd_zero _)
theorem dvd.antisymm {m n : } : m n → n m → m = n := theorem dvd.antisymm {m n : } : m n → n m → m = n :=
by_cases_zero_pos n by_cases_zero_pos n
@ -454,28 +402,104 @@ theorem div_eq_of_eq_mul_left {m n k : } (H1 : n > 0) (H2 : m = k * n) :
m div n = k := m div n = k :=
!div_eq_of_eq_mul_right H1 (!mul.comm ▸ H2) !div_eq_of_eq_mul_right H1 (!mul.comm ▸ H2)
theorem div_le_iff_le_mul_right {m n : } (k : ) (H : n > 0) (H' : n m) : /- div and ordering -/
theorem div_mul_le (m n : ) : m div n * n ≤ m :=
calc
m = m div n * n + m mod n : eq_div_mul_add_mod
... ≥ m div n * n : le_add_right
theorem div_le_of_le_mul {m n k : } (H : m ≤ n * k) : m div k ≤ n :=
or.elim (eq_zero_or_pos k)
(assume H1 : k = 0,
calc
m div k = m div 0 : H1
... = 0 : div_zero
... ≤ n : zero_le)
(assume H1 : k > 0,
le_of_mul_le_mul_right (calc
m div k * k ≤ m div k * k + m mod k : le_add_right
... = m : eq_div_mul_add_mod
... ≤ n * k : H) H1)
theorem div_le_self (m n : ) : m div n ≤ m :=
nat.cases_on n (!div_zero⁻¹ ▸ !zero_le)
take n,
have H : m ≤ m * succ n, from calc
m = m * 1 : mul_one
... ≤ m * succ n : !mul_le_mul_left (succ_le_succ !zero_le),
div_le_of_le_mul H
theorem mul_le_of_le_div {m n k : } (H : m ≤ n div k) : m * k ≤ n :=
calc
m * k ≤ n div k * k : !mul_le_mul_right H
... ≤ n : div_mul_le
theorem le_div_of_mul_le {m n k : } (H1 : k > 0) (H2 : m * k ≤ n) : m ≤ n div k :=
have H3 : m * k < (succ (n div k)) * k, from
calc
m * k ≤ n : H2
... = n div k * k + n mod k : eq_div_mul_add_mod
... < n div k * k + k : add_lt_add_left (!mod_lt H1)
... = (succ (n div k)) * k : succ_mul,
lt_of_mul_lt_mul_right H3
theorem le_div_iff_mul_le {m n k : } (H : k > 0) : m ≤ n div k ↔ m * k ≤ n :=
iff.intro !mul_le_of_le_div (!le_div_of_mul_le H)
theorem div_le_div {m n : } (k : ) (H : m ≤ n) : m div k ≤ n div k :=
by_cases_zero_pos k
(by rewrite [*div_zero])
(take k, assume H1 : k > 0, le_div_of_mul_le H1 (le.trans !div_mul_le H))
theorem div_lt_of_lt_mul {m n k : } (H : m < n * k) : m div k < n :=
lt_of_mul_lt_mul_right (calc
m div k * k ≤ m div k * k + m mod k : le_add_right
... = m : eq_div_mul_add_mod
... < n * k : H)
theorem lt_mul_of_div_lt {m n k : } (H1 : k > 0) (H2 : m div k < n) : m < n * k :=
assert H3 : succ (m div k) * k ≤ n * k, from !mul_le_mul_right (succ_le_of_lt H2),
have H4 : m div k * k + k ≤ n * k, by rewrite [succ_mul at H3]; apply H3,
calc
m = m div k * k + m mod k : eq_div_mul_add_mod
... < m div k * k + k : add_lt_add_left (!mod_lt H1)
... ≤ n * k : H4
theorem div_lt_iff_lt_mul {m n k : } (H : k > 0) : m div k < n ↔ m < n * k :=
iff.intro (!lt_mul_of_div_lt H) !div_lt_of_lt_mul
theorem div_le_iff_le_mul_of_div {m n : } (k : ) (H : n > 0) (H' : n m) :
m div n ≤ k ↔ m ≤ k * n := m div n ≤ k ↔ m ≤ k * n :=
by rewrite [propext (!le_iff_mul_le_mul_right H), !div_mul_cancel H'] by rewrite [propext (!le_iff_mul_le_mul_right H), !div_mul_cancel H']
theorem div_le_iff_le_mul_left {m n : } (k : ) (H : n > 0) (H' : n m) : theorem le_mul_of_div_le_of_div {m n k : } (H1 : n > 0) (H2 : n m) (H3 : m div n ≤ k) :
m div n ≤ k ↔ m ≤ n * k :=
!mul.comm ▸ !div_le_iff_le_mul_right H H'
theorem eq_mul_of_div_le_right {m n k : } (H1 : n > 0) (H2 : n m) (H3 : m div n ≤ k) :
m ≤ k * n := m ≤ k * n :=
iff.mp (!div_le_iff_le_mul_right H1 H2) H3 iff.mp (!div_le_iff_le_mul_of_div H1 H2) H3
theorem div_le_of_eq_mul_right {m n k : } (H1 : n > 0) (H2 : n m) (H3 : m ≤ k * n) : -- needed for integer division
m div n ≤ k := theorem mul_sub_div_of_lt {m n k : } (H : k < m * n) :
iff.mp' (!div_le_iff_le_mul_right H1 H2) H3 (m * n - (k + 1)) div m = n - k div m - 1 :=
have H1 : k div m < n, from div_lt_of_lt_mul (!mul.comm ▸ H),
theorem eq_mul_of_div_le_left {m n k : } (H1 : n > 0) (H2 : n m) (H3 : m div n ≤ k) : have H2 : n - k div m ≥ 1, from
m ≤ n * k := le_sub_of_add_le (calc
iff.mp (!div_le_iff_le_mul_left H1 H2) H3 1 + k div m = succ (k div m) : add.comm
... ≤ n : succ_le_of_lt H1),
theorem div_le_of_eq_mul_left {m n k : } (H1 : n > 0) (H2 : n m) (H3 : m ≤ n * k) : assert H3 : n - k div m = n - k div m - 1 + 1, from (sub_add_cancel H2)⁻¹,
m div n ≤ k := assert H4 : m > 0, from pos_of_ne_zero (assume H': m = 0, not_lt_zero _ (!zero_mul ▸ H' ▸ H)),
iff.mp' (!div_le_iff_le_mul_left H1 H2) H3 have H5 : k mod m + 1 ≤ m, from succ_le_of_lt (!mod_lt H4),
assert H6 : m - (k mod m + 1) < m, from sub_lt_self H4 !succ_pos,
calc
(m * n - (k + 1)) div m = (m * n - (k div m * m + k mod m + 1)) div m : eq_div_mul_add_mod
... = (m * n - k div m * m - (k mod m + 1)) div m : by rewrite [*sub_sub]
... = ((n - k div m) * m - (k mod m + 1)) div m :
by rewrite [mul.comm m, mul_sub_right_distrib]
... = ((n - k div m - 1) * m + m - (k mod m + 1)) div m :
by rewrite [H3 at {1}, mul.right_distrib, nat.one_mul]
... = ((n - k div m - 1) * m + (m - (k mod m + 1))) div m : {add_sub_assoc H5 _}
... = (m - (k mod m + 1)) div m + (n - k div m - 1) :
by rewrite [add.comm, (add_mul_div_self H4)]
... = n - k div m - 1 :
by rewrite [div_eq_zero_of_lt H6, zero_add]
end nat end nat

View file

@ -105,20 +105,20 @@ theorem lt_add_of_pos_right {n k : } (H : k > 0) : n < n + k :=
/- multiplication -/ /- multiplication -/
theorem mul_le_mul_left {n m : } (H : n ≤ m) (k : ) : k * n ≤ k * m := 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, obtain (l : ) (Hl : n + l = m), from le.elim H,
have H2 : k * n + k * l = k * m, by rewrite [-mul.left_distrib, Hl], have H2 : k * n + k * l = k * m, by rewrite [-mul.left_distrib, Hl],
le.intro H2 le.intro H2
theorem mul_le_mul_right {n m : } (H : n ≤ m) (k : ) : n * k ≤ m * k := theorem mul_le_mul_right {n m : } (k : ) (H : n ≤ m) : n * k ≤ m * k :=
!mul.comm ▸ !mul.comm ▸ (mul_le_mul_left H 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 := 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 m) (mul_le_mul_left H2 k) 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 := theorem mul_lt_mul_of_pos_left {n m k : } (H : n < m) (Hk : k > 0) : k * n < k * m :=
have H2 : k * n < k * n + k, from lt_add_of_pos_right Hk, have H2 : k * n < k * n + k, from lt_add_of_pos_right Hk,
have H3 : k * n + k ≤ k * m, from !mul_succ ▸ mul_le_mul_left (succ_le_of_lt H) k, have H3 : k * n + k ≤ k * m, from !mul_succ ▸ mul_le_mul_left k (succ_le_of_lt H),
lt_of_lt_of_le H2 H3 lt_of_lt_of_le H2 H3
theorem mul_lt_mul_of_pos_right {n m k : } (H : n < m) (Hk : k > 0) : n * k < m * k := theorem mul_lt_mul_of_pos_right {n m k : } (H : n < m) (Hk : k > 0) : n * k < m * k :=
@ -170,8 +170,8 @@ section migrate_algebra
add_le_add_left := @add_le_add_left, add_le_add_left := @add_le_add_left,
le_of_add_le_add_left := @le_of_add_le_add_left, le_of_add_le_add_left := @le_of_add_le_add_left,
zero_ne_one := ne.symm (succ_ne_zero zero), zero_ne_one := ne.symm (succ_ne_zero zero),
mul_le_mul_of_nonneg_left := (take a b c H1 H2, mul_le_mul_left H1 c), 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 H1 c), 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_left := @mul_lt_mul_of_pos_left,
mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right ⦄ mul_lt_mul_of_pos_right := @mul_lt_mul_of_pos_right ⦄
@ -379,14 +379,14 @@ pos_of_ne_zero
theorem mul_lt_mul_of_le_of_lt {n m k l : } (Hk : k > 0) (H1 : n ≤ k) (H2 : m < l) : 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 := n * m < k * l :=
lt_of_le_of_lt (mul_le_mul_right H1 m) (mul_lt_mul_of_pos_left H2 Hk) 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) : 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 := n * m < k * l :=
lt_of_le_of_lt (mul_le_mul_left H2 n) (mul_lt_mul_of_pos_right H1 Hl) 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 := 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 (le_of_lt H1) m, 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), 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 lt_of_le_of_lt H3 H4

View file

@ -444,13 +444,13 @@ theorem dist_mul_dist (n m k l : ) : dist n m * dist k l = dist (n * k + m *
have aux : ∀k l, k ≥ l → dist n m * dist k l = dist (n * k + m * l) (n * l + m * k), from have aux : ∀k l, k ≥ l → dist n m * dist k l = dist (n * k + m * l) (n * l + m * k), from
take k l : , take k l : ,
assume H : k ≥ l, assume H : k ≥ l,
have H2 : m * k ≥ m * l, from mul_le_mul_left H m, have H2 : m * k ≥ m * l, from !mul_le_mul_left H,
have H3 : n * l + m * k ≥ m * l, from le.trans H2 !le_add_left, have H3 : n * l + m * k ≥ m * l, from le.trans H2 !le_add_left,
calc calc
dist n m * dist k l = dist n m * (k - l) : dist_eq_sub_of_ge H dist n m * dist k l = dist n m * (k - l) : dist_eq_sub_of_ge H
... = dist (n * (k - l)) (m * (k - l)) : dist_mul_right ... = dist (n * (k - l)) (m * (k - l)) : dist_mul_right
... = dist (n * k - n * l) (m * k - m * l) : by simp ... = dist (n * k - n * l) (m * k - m * l) : by simp
... = dist (n * k) (m * k - m * l + n * l) : dist_sub_eq_dist_add_left (mul_le_mul_left H n) ... = dist (n * k) (m * k - m * l + n * l) : dist_sub_eq_dist_add_left (!mul_le_mul_left H)
... = dist (n * k) (n * l + (m * k - m * l)) : add.comm ... = dist (n * k) (n * l + (m * k - m * l)) : add.comm
... = dist (n * k) (n * l + m * k - m * l) : add_sub_assoc H2 (n * l) ... = dist (n * k) (n * l + m * k - m * l) : add_sub_assoc H2 (n * l)
... = dist (n * k + m * l) (n * l + m * k) : dist_sub_eq_dist_add_right H3 _, ... = dist (n * k + m * l) (n * l + m * k) : dist_sub_eq_dist_add_right H3 _,