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refactor(library/data/int/div): reorient of_nat_div and of_nat_mod

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Jeremy Avigad 2015-06-27 18:47:36 +10:00
parent 7c118f40fe
commit 829c3fb22c
2 changed files with 17 additions and 17 deletions
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32
library/data/int/div.lean
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@ -31,7 +31,7 @@ notation a `≡` b `[mod`:100 c `]`:0 := a mod c = b mod c
/- div -/ /- div -/
theorem of_nat_div_of_nat (m n : nat) : m div n = of_nat (#nat m div n) := theorem of_nat_div (m n : nat) : of_nat (#nat m div n) = m div n :=
nat.cases_on n nat.cases_on n
(by rewrite [↑divide, sign_zero, zero_mul, nat.div_zero]) (by rewrite [↑divide, sign_zero, zero_mul, nat.div_zero])
(take n, by rewrite [↑divide, sign_of_succ, one_mul]) (take n, by rewrite [↑divide, sign_of_succ, one_mul])
@ -57,7 +57,7 @@ theorem div_nonneg {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a div b ≥ 0 :=
obtain (m : ℕ) (Hm : a = m), from exists_eq_of_nat Ha, obtain (m : ℕ) (Hm : a = m), from exists_eq_of_nat Ha,
obtain (n : ℕ) (Hn : b = n), from exists_eq_of_nat Hb, obtain (n : 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]
... ≥ 0 : begin change (0 ≤ #nat m div n), apply trivial end ... ≥ 0 : begin change (0 ≤ #nat m div n), apply trivial end
theorem div_nonpos {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≤ 0) : a div b ≤ 0 := theorem div_nonpos {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≤ 0) : a div b ≤ 0 :=
@ -84,8 +84,8 @@ by rewrite [↑divide, sign_zero, zero_mul]
theorem div_one (a : ℤ) :a div 1 = a := theorem div_one (a : ℤ) :a div 1 = a :=
assert H : 1 > 0, from dec_trivial, assert H : 1 > 0, from dec_trivial,
int.cases_on a int.cases_on a
(take m, by rewrite [of_nat_div_of_nat, nat.div_one]) (take m, by rewrite [-of_nat_div, nat.div_one])
(take m, by rewrite [!neg_succ_of_nat_div H, of_nat_div_of_nat, nat.div_one]) (take m, by rewrite [!neg_succ_of_nat_div H, -of_nat_div, nat.div_one])
theorem eq_div_mul_add_mod (a b : ℤ) : a = a div b * b + a mod b := theorem eq_div_mul_add_mod (a b : ℤ) : a = a div b * b + a mod b :=
!add.comm ▸ eq_add_of_sub_eq rfl !add.comm ▸ eq_add_of_sub_eq rfl
@ -97,7 +97,7 @@ int.cases_on a
(take n, (take n,
assume H : m < n, assume H : m < n,
calc calc
m div n = #nat m div n : of_nat_div_of_nat m div n = #nat m div n : of_nat_div
... = 0 : nat.div_eq_zero_of_lt (lt_of_of_nat_lt_of_nat H)) ... = 0 : nat.div_eq_zero_of_lt (lt_of_of_nat_lt_of_nat H))
(take n, (take n,
assume H : m < -[ n +1], assume H : m < -[ n +1],
@ -126,10 +126,10 @@ private theorem add_mul_div_self_aux1 {a : ℤ} {k : ℕ} (n : ℕ)
obtain m (Hm : a = of_nat m), from exists_eq_of_nat H1, obtain m (Hm : a = of_nat m), from exists_eq_of_nat H1,
Hm⁻¹ ▸ (calc Hm⁻¹ ▸ (calc
(m + n * k) div k = (#nat (m + n * k)) div k : rfl (m + n * k) div k = (#nat (m + n * k)) div k : rfl
... = (#nat (m + n * k) div k) : of_nat_div_of_nat ... = (#nat (m + n * k) div k) : of_nat_div
... = (#nat m div k + n) : !nat.add_mul_div_self H2 ... = (#nat m div k + n) : !nat.add_mul_div_self H2
... = (#nat m div k) + n : rfl ... = (#nat m div k) + n : rfl
... = m div k + n : of_nat_div_of_nat) ... = m div k + n : of_nat_div)
private theorem add_mul_div_self_aux2 {a : ℤ} {k : ℕ} (n : ℕ) private theorem add_mul_div_self_aux2 {a : ℤ} {k : ℕ} (n : ℕ)
(H1 : a < 0) (H2 : #nat k > 0) : (H1 : a < 0) (H2 : #nat k > 0) :
@ -144,26 +144,26 @@ or.elim (nat.lt_or_ge m (#nat n * k))
(-[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
... = (#nat n * k - (m + 1)) div k : {(of_nat_sub H3)⁻¹} ... = (#nat n * k - (m + 1)) div k : {(of_nat_sub H3)⁻¹}
... = #nat (n * k - (m + 1)) div k : of_nat_div_of_nat ... = #nat (n * k - (m + 1)) div k : of_nat_div
... = #nat (k * n - (m + 1)) div k : nat.mul.comm ... = #nat (k * n - (m + 1)) div k : nat.mul.comm
... = #nat n - m div k - 1 : ... = #nat n - m div k - 1 :
nat.mul_sub_div_of_lt (!nat.mul.comm ▸ m_lt_nk) nat.mul_sub_div_of_lt (!nat.mul.comm ▸ m_lt_nk)
... = #nat n - (m div k + 1) : nat.sub_sub ... = #nat n - (m div k + 1) : nat.sub_sub
... = n - (#nat m div k + 1) : of_nat_sub H4 ... = n - (#nat m div k + 1) : of_nat_sub H4
... = -(m div k + 1) + n : ... = -(m div k + 1) + n :
by rewrite [add.comm, -sub_eq_add_neg, of_nat_add, of_nat_div_of_nat] by rewrite [add.comm, -sub_eq_add_neg, of_nat_add, of_nat_div]
... = -[m +1] div k + n : ... = -[m +1] div k + n :
neg_succ_of_nat_div m (of_nat_lt_of_nat_of_lt H2))) neg_succ_of_nat_div m (of_nat_lt_of_nat_of_lt H2)))
(assume nk_le_m : #nat n * k ≤ m, (assume nk_le_m : #nat n * k ≤ m,
eq.symm (Hm⁻¹ ▸ (calc eq.symm (Hm⁻¹ ▸ (calc
-[m +1] div k + n = -(m div k + 1) + n : -[m +1] div k + n = -(m div k + 1) + n :
neg_succ_of_nat_div m (of_nat_lt_of_nat_of_lt H2) neg_succ_of_nat_div m (of_nat_lt_of_nat_of_lt H2)
... = -((#nat m div k) + 1) + n : of_nat_div_of_nat ... = -((#nat m div k) + 1) + n : of_nat_div
... = -((#nat (m - n * k + n * k) div k) + 1) + n : nat.sub_add_cancel nk_le_m ... = -((#nat (m - n * k + n * k) div k) + 1) + n : nat.sub_add_cancel nk_le_m
... = -((#nat (m - n * k) div k + n) + 1) + n : nat.add_mul_div_self H2 ... = -((#nat (m - n * k) div k + n) + 1) + n : nat.add_mul_div_self H2
... = -((#nat m - n * k) div k + 1) : ... = -((#nat m - n * k) div k + 1) :
by rewrite [of_nat_add, *neg_add, add.right_comm, neg_add_cancel_right, by rewrite [of_nat_add, *neg_add, add.right_comm, neg_add_cancel_right,
of_nat_div_of_nat] of_nat_div]
... = -[(#nat m - n * k) +1] div k : ... = -[(#nat m - n * k) +1] div k :
neg_succ_of_nat_div _ (of_nat_lt_of_nat_of_lt H2) neg_succ_of_nat_div _ (of_nat_lt_of_nat_of_lt H2)
... = -((#nat m - n * k) + 1) div k : rfl ... = -((#nat m - n * k) + 1) div k : rfl
@ -225,11 +225,11 @@ theorem div_self {a : ℤ} (H : a ≠ 0) : a div a = 1 :=
/- mod -/ /- mod -/
theorem of_nat_mod_of_nat (m n : nat) : m mod n = (#nat m mod n) := theorem of_nat_mod (m n : nat) : m mod n = (#nat m mod n) :=
have H : m = (#nat m mod n) + m div n * n, from calc 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 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 ... = (#nat m div n) * n + (#nat m mod n) : rfl
... = m div n * n + (#nat m mod n) : of_nat_div_of_nat ... = m div n * n + (#nat m mod n) : of_nat_div
... = (#nat m mod n) + m div n * n : add.comm, ... = (#nat m mod n) + m div n * n : add.comm,
calc calc
m mod n = m - m div n * n : rfl m mod n = m - m div n * n : rfl
@ -272,7 +272,7 @@ calc
private lemma of_nat_mod_abs (m : ℕ) (b : ℤ) : m mod (abs b) = (#nat m mod (nat_abs b)) := private lemma of_nat_mod_abs (m : ℕ) (b : ℤ) : m mod (abs b) = (#nat m mod (nat_abs b)) :=
calc calc
m mod (abs b) = m mod (nat_abs b) : of_nat_nat_abs m mod (abs b) = m mod (nat_abs b) : of_nat_nat_abs
... = (#nat m mod (nat_abs b)) : of_nat_mod_of_nat ... = (#nat m mod (nat_abs b)) : of_nat_mod
private lemma of_nat_mod_abs_lt (m : ℕ) {b : ℤ} (H : b ≠ 0) : m mod (abs b) < (abs b) := private lemma of_nat_mod_abs_lt (m : ℕ) {b : ℤ} (H : b ≠ 0) : m mod (abs b) < (abs b) :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H, have H1 : abs b > 0, from abs_pos_of_ne_zero H,
@ -287,7 +287,7 @@ 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)), obtain n (Hn : b = of_nat n), from exists_eq_of_nat (le_of_lt (lt_of_le_of_lt H1 H2)),
begin begin
revert H2, revert H2,
rewrite [Hm, Hn, of_nat_mod_of_nat, of_nat_lt_of_nat, of_nat_eq_of_nat], rewrite [Hm, Hn, of_nat_mod, of_nat_lt_of_nat, of_nat_eq_of_nat],
apply nat.mod_eq_of_lt apply nat.mod_eq_of_lt
end end
@ -424,7 +424,7 @@ theorem div_le_of_nonneg_of_nonneg {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a
obtain (m : ℕ) (Hm : a = m), from exists_eq_of_nat Ha, obtain (m : ℕ) (Hm : a = m), from exists_eq_of_nat Ha,
obtain (n : ℕ) (Hn : b = n), from exists_eq_of_nat Hb, obtain (n : 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]
... ≤ m : of_nat_le_of_nat_of_le !nat.div_le_self ... ≤ m : of_nat_le_of_nat_of_le !nat.div_le_self
... = a : Hm ... = a : Hm

2
library/data/int/gcd.lean
View file

@ -49,7 +49,7 @@ assert H3 : nat.gcd (nat_abs a) (nat_abs b) = (#nat nat.gcd (nat_abs b) (nat_abs
calc calc
gcd a b = nat.gcd (nat_abs b) (#nat nat_abs a mod nat_abs b) : by rewrite [↑gcd, H3] gcd a b = nat.gcd (nat_abs b) (#nat nat_abs a mod nat_abs b) : by rewrite [↑gcd, H3]
... = gcd (abs b) (abs a mod abs b) : ... = gcd (abs b) (abs a mod abs b) :
by rewrite [↑gcd, -*of_nat_nat_abs, of_nat_mod_of_nat] by rewrite [↑gcd, -*of_nat_nat_abs, of_nat_mod]
... = gcd b (abs a mod abs b) : by rewrite [↑gcd, *nat_abs_abs] ... = gcd b (abs a mod abs b) : by rewrite [↑gcd, *nat_abs_abs]
theorem gcd_of_pos (a : ℤ) {b : ℤ} (H : b > 0) : gcd a b = gcd b (abs a mod b) := theorem gcd_of_pos (a : ℤ) {b : ℤ} (H : b > 0) : gcd a b = gcd b (abs a mod b) :=