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refactor(library/*): use type classes for div and mod

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This commit is contained in:
Jeremy Avigad 2015-10-29 15:36:26 -04:00 committed by Leonardo de Moura
parent eea4e4ec55
commit 697df0e68c
27 changed files with 809 additions and 814 deletions
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8
library/algebra/field.lean
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@ -14,20 +14,18 @@ namespace algebra
variable {A : Type}
-- in division rings, 1 / 0 = 0
structure division_ring [class] (A : Type) extends ring A, has_inv A, zero_ne_one_class A :=
(mul_inv_cancel : ∀{a}, a ≠ zero → mul a (inv a) = one)
(inv_mul_cancel : ∀{a}, a ≠ zero → mul (inv a) a = one)
--(inv_zero : inv zero = zero)
section division_ring
variables [s : division_ring A] {a b c : A}
include s
protected definition division (a b : A) : A := a * b⁻¹
protected definition div (a b : A) : A := a * b⁻¹
definition division_ring_has_division [reducible] [instance] : has_division A :=
has_division.mk algebra.division
definition division_ring_has_div [reducible] [instance] : has_div A :=
has_div.mk algebra.div
lemma division.def (a b : A) : a / b = a * b⁻¹ :=
rfl

8
library/data/encodable.lean
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@ -57,13 +57,13 @@ private definition encode_sum : sum A B → nat
| (sum.inr b) := 2 * encode b + 1
private definition decode_sum (n : nat) : option (sum A B) :=
if n mod 2 = 0 then
match decode A (n div 2) with
if n % 2 = 0 then
match decode A (n / 2) with
| some a := some (sum.inl a)
| none := none
end
else
match decode B ((n - 1) div 2) with
match decode B ((n - 1) / 2) with
| some b := some (sum.inr b)
| none := none
end
@ -79,7 +79,7 @@ private theorem decode_encode_sum : ∀ s : sum A B, decode_sum (encode_sum s) =
end
| (sum.inr b) :=
assert aux₁ : 2 > (0:nat), from dec_trivial,
assert aux₂ : 1 mod 2 = (1:nat), by rewrite [nat.modulo_def],
assert aux₂ : 1 % 2 = (1:nat), by rewrite [nat.mod_def],
assert aux₃ : 1 ≠ (0:nat), from dec_trivial,
begin
esimp [encode_sum, decode_sum],

2
library/data/equiv.lean
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@ -263,7 +263,7 @@ calc (nat + bool) ≃ (nat + (unit + unit)) : sum_congr !equiv.refl bool_equiv_u
open decidable
definition nat_sum_nat_equiv_nat [simp] : (nat + nat) ≃ nat :=
mk (λ s, match s with inl n := 2*n | inr n := 2*n+1 end)
(λ n, if even n then inl (n div 2) else inr ((n - 1) div 2))
(λ n, if even n then inl (n / 2) else inr ((n - 1) / 2))
(λ s, begin
have two_gt_0 : 2 > zero, from dec_trivial,
cases s,

16
library/data/fin.lean
View file

@ -95,7 +95,7 @@ has_zero.mk (fin.zero n)
theorem val_zero (n : nat) : val (0 : fin (succ n)) = 0 := rfl
definition mk_mod [reducible] (n i : nat) : fin (succ n) :=
mk (i mod (succ n)) (mod_lt _ !zero_lt_succ)
mk (i % (succ n)) (mod_lt _ !zero_lt_succ)
theorem mk_mod_zero_eq (n : nat) : mk_mod n 0 = 0 :=
rfl
@ -218,12 +218,12 @@ end lift_lower
section madd
definition madd (i j : fin (succ n)) : fin (succ n) :=
mk ((i + j) mod (succ n)) (mod_lt _ !zero_lt_succ)
mk ((i + j) % (succ n)) (mod_lt _ !zero_lt_succ)
definition minv : ∀ i : fin (succ n), fin (succ n)
| (mk iv ilt) := mk ((succ n - iv) mod succ n) (mod_lt _ !zero_lt_succ)
| (mk iv ilt) := mk ((succ n - iv) % succ n) (mod_lt _ !zero_lt_succ)
lemma val_madd : ∀ i j : fin (succ n), val (madd i j) = (i + j) mod (succ n)
lemma val_madd : ∀ i j : fin (succ n), val (madd i j) = (i + j) % (succ n)
| (mk iv ilt) (mk jv jlt) := by esimp
lemma madd_inj : ∀ {i : fin (succ n)}, injective (madd i)
@ -238,7 +238,7 @@ end))
lemma madd_mk_mod {i j : nat} : madd (mk_mod n i) (mk_mod n j) = mk_mod n (i+j) :=
eq_of_veq begin esimp [madd, mk_mod], rewrite [ mod_add_mod, add_mod_mod ] end
lemma val_mod : ∀ i : fin (succ n), (val i) mod (succ n) = val i
lemma val_mod : ∀ i : fin (succ n), (val i) % (succ n) = val i
| (mk iv ilt) := by esimp; rewrite [(mod_eq_of_lt ilt)]
lemma madd_comm (i j : fin (succ n)) : madd i j = madd j i :=
@ -450,13 +450,13 @@ assert aux₁ : ∀ {v₁ v₂}, v₁ < n → v₂ < m → v₁ + v₂ * n < n*m
assert v₁ + (v₂ * n + n) < n + n * m, from add_lt_add_of_lt_of_le h₁ this,
have v₁ + v₂ * n + n < n * m + n, by rewrite [add.assoc, add.comm (n*m) n]; exact this,
lt_of_add_lt_add_right this,
assert aux₂ : ∀ v, v mod n < n, from
assert aux₂ : ∀ v, v % n < n, from
take v, mod_lt _ `n > 0`,
assert aux₃ : ∀ {v}, v < n * m → v div n < m, from
assert aux₃ : ∀ {v}, v < n * m → v / n < m, from
take v, assume h, by rewrite mul.comm at h; exact nat.div_lt_of_lt_mul h,
⦃ equiv,
to_fun := λ p : (fin n × fin m), match p with (mk v₁ hlt₁, mk v₂ hlt₂) := mk (v₁ + v₂ * n) (aux₁ hlt₁ hlt₂) end,
inv_fun := λ f : fin (n*m), match f with (mk v hlt) := (mk (v mod n) (aux₂ v), mk (v div n) (aux₃ hlt)) end,
inv_fun := λ f : fin (n*m), match f with (mk v hlt) := (mk (v % n) (aux₂ v), mk (v / n) (aux₃ hlt)) end,
left_inv := begin
intro p, cases p with f₁ f₂, cases f₁ with v₁ hlt₁, cases f₂ with v₂ hlt₂, esimp,
congruence,

69
library/data/finset/equiv.lean
View file

@ -22,22 +22,22 @@ lemma to_nat_insert {n : nat} {s : finset nat} : n ∉ s → to_nat (insert n s)
assume h, Sum_insert_of_not_mem _ h
protected definition of_nat (s : nat) : finset nat :=
{ n ∈ upto (succ s) | odd (s div 2^n) }
{ n ∈ upto (succ s) | odd (s / 2^n) }
open finset (of_nat)
private lemma of_nat_zero : of_nat 0 = ∅ :=
rfl
private lemma odd_of_mem_of_nat {n : nat} {s : nat} : n ∈ of_nat s → odd (s div 2^n) :=
private lemma odd_of_mem_of_nat {n : nat} {s : nat} : n ∈ of_nat s → odd (s / 2^n) :=
assume h, of_mem_sep h
private lemma mem_of_nat_of_odd {n : nat} {s : nat} : odd (s div 2^n) → n ∈ of_nat s :=
private lemma mem_of_nat_of_odd {n : nat} {s : nat} : odd (s / 2^n) → n ∈ of_nat s :=
assume h,
have 2^n < succ s, from by_contradiction
(suppose ¬(2^n < succ s),
assert 2^n > s, from lt_of_succ_le (le_of_not_gt this),
assert s div 2^n = 0, from div_eq_zero_of_lt this,
assert s / 2^n = 0, from div_eq_zero_of_lt this,
by rewrite this at h; exact absurd h dec_trivial),
have n < succ s, from calc
n ≤ 2^n : le_pow_self dec_trivial n
@ -45,15 +45,15 @@ have n < succ s, from calc
have n ∈ upto (succ s), from mem_upto_of_lt this,
mem_sep_of_mem this h
private lemma succ_mem_of_nat (n : nat) (s : nat) : succ n ∈ of_nat s ↔ n ∈ of_nat (s div 2) :=
private lemma succ_mem_of_nat (n : nat) (s : nat) : succ n ∈ of_nat s ↔ n ∈ of_nat (s / 2) :=
iff.intro
(suppose succ n ∈ of_nat s,
assert odd (s div 2^(succ n)), from odd_of_mem_of_nat this,
have odd ((s div 2) div (2 ^ n)), by rewrite [pow_succ' at this, nat.div_div_eq_div_mul, mul.comm]; assumption,
show n ∈ of_nat (s div 2), from mem_of_nat_of_odd this)
(suppose n ∈ of_nat (s div 2),
assert odd ((s div 2) div (2 ^ n)), from odd_of_mem_of_nat this,
assert odd (s div 2^(succ n)), by rewrite [pow_succ', mul.comm, -nat.div_div_eq_div_mul]; assumption,
assert odd (s / 2^(succ n)), from odd_of_mem_of_nat this,
have odd ((s / 2) / (2 ^ n)), by rewrite [pow_succ' at this, nat.div_div_eq_div_mul, mul.comm]; assumption,
show n ∈ of_nat (s / 2), from mem_of_nat_of_odd this)
(suppose n ∈ of_nat (s / 2),
assert odd ((s / 2) / (2 ^ n)), from odd_of_mem_of_nat this,
assert odd (s / 2^(succ n)), by rewrite [pow_succ', mul.comm, -nat.div_div_eq_div_mul]; assumption,
show succ n ∈ of_nat s, from mem_of_nat_of_odd this)
private lemma odd_of_zero_mem (s : nat) : 0 ∈ of_nat s ↔ odd s :=
@ -123,12 +123,12 @@ begin
match n with
| 0 := iff.intro (λ h, !mem_insert) (λ h, by rewrite [hw at zmem]; exact zmem)
| succ m :=
assert d₁ : 1 div 2 = (0:nat), from dec_trivial,
assert d₁ : 1 / 2 = (0:nat), from dec_trivial,
assert aux : _, from calc
succ m ∈ of_nat (2 * w + 1) ↔ m ∈ of_nat ((2*w+1) div 2) : succ_mem_of_nat
... ↔ m ∈ of_nat w : by rewrite [add.comm, add_mul_div_self_left _ _ (dec_trivial : 2 > 0), d₁, zero_add]
... ↔ m ∈ of_nat (2*w div 2) : by rewrite [mul.comm, nat.mul_div_cancel _ (dec_trivial : 2 > 0)]
... ↔ succ m ∈ of_nat (2*w) : succ_mem_of_nat,
succ m ∈ of_nat (2 * w + 1) ↔ m ∈ of_nat ((2*w+1) / 2) : succ_mem_of_nat
... ↔ m ∈ of_nat w : by rewrite [add.comm, add_mul_div_self_left _ _ (dec_trivial : 2 > 0), d₁, zero_add]
... ↔ m ∈ of_nat (2*w / 2) : by rewrite [mul.comm, nat.mul_div_cancel _ (dec_trivial : 2 > 0)]
... ↔ succ m ∈ of_nat (2*w) : succ_mem_of_nat,
iff.intro
(λ hl, finset.mem_insert_of_mem _ (iff.mp aux hl))
(λ hr, or.elim (eq_or_mem_of_mem_insert hr)
@ -140,8 +140,9 @@ end
private lemma of_nat_eq_insert : ∀ {n s : nat}, n ∉ of_nat s → of_nat (2^n + s) = insert n (of_nat s)
| 0 s h := of_nat_eq_insert_zero h
| (succ n) s h :=
have n ∉ of_nat (s div 2), from iff.mp (not_iff_not_of_iff !succ_mem_of_nat) h,
assert ih : of_nat (2^n + s div 2) = insert n (of_nat (s div 2)), from of_nat_eq_insert this,
have n ∉ of_nat (s / 2),
from iff.mp (not_iff_not_of_iff !succ_mem_of_nat) h,
assert ih : of_nat (2^n + s / 2) = insert n (of_nat (s / 2)), from of_nat_eq_insert this,
finset.ext (λ x,
have gen : ∀ m, m ∈ of_nat (2^(succ n) + s) ↔ m ∈ insert (succ n) (of_nat s)
| zero :=
@ -163,10 +164,10 @@ private lemma of_nat_eq_insert : ∀ {n s : nat}, n ∉ of_nat s → of_nat (2^n
... ↔ odd s : aux₁
... ↔ 0 ∈ insert (succ n) (of_nat s) : aux₂
| (succ m) :=
assert aux : m ∈ insert n (of_nat (s div 2)) ↔ succ m ∈ insert (succ n) (of_nat s), from iff.intro
assert aux : m ∈ insert n (of_nat (s / 2)) ↔ succ m ∈ insert (succ n) (of_nat s), from iff.intro
(assume hl, or.elim (eq_or_mem_of_mem_insert hl)
(suppose m = n, by subst m; apply mem_insert)
(suppose m ∈ of_nat (s div 2), finset.mem_insert_of_mem _ (iff.mpr !succ_mem_of_nat this)))
(suppose m ∈ of_nat (s / 2), finset.mem_insert_of_mem _ (iff.mpr !succ_mem_of_nat this)))
(assume hr, or.elim (eq_or_mem_of_mem_insert hr)
(suppose succ m = succ n,
assert m = n, by injection this; assumption,
@ -174,9 +175,9 @@ private lemma of_nat_eq_insert : ∀ {n s : nat}, n ∉ of_nat s → of_nat (2^n
(suppose succ m ∈ of_nat s, finset.mem_insert_of_mem _ (iff.mp !succ_mem_of_nat this))),
calc
succ m ∈ of_nat (2^(succ n) + s) ↔ succ m ∈ of_nat (2^n * 2 + s) : by rewrite pow_succ'
... ↔ m ∈ of_nat ((2^n * 2 + s) div 2) : succ_mem_of_nat
... ↔ m ∈ of_nat (2^n + s div 2) : by rewrite [add.comm, add_mul_div_self (dec_trivial : 2 > 0), add.comm]
... ↔ m ∈ insert n (of_nat (s div 2)) : by rewrite ih
... ↔ m ∈ of_nat ((2^n * 2 + s) / 2) : succ_mem_of_nat
... ↔ m ∈ of_nat (2^n + s / 2) : by rewrite [add.comm, add_mul_div_self (dec_trivial : 2 > 0), add.comm]
... ↔ m ∈ insert n (of_nat (s / 2)) : by rewrite ih
... ↔ succ m ∈ insert (succ n) (of_nat s) : aux,
gen x)
@ -233,16 +234,16 @@ finset.ext (λ m,
and.intro (or.inr h₁) (or.inr h₂))
end)
private lemma of_nat_div2 (s : nat) : of_nat (s div 2) = predimage (of_nat s) :=
private lemma of_nat_div2 (s : nat) : of_nat (s / 2) = predimage (of_nat s) :=
finset.ext (λ n, iff.intro
(suppose n ∈ of_nat (s div 2),
(suppose n ∈ of_nat (s / 2),
have succ n ∈ of_nat s, from iff.mpr !succ_mem_of_nat this,
mem_predimage_of_succ_mem this)
(suppose n ∈ predimage (of_nat s),
have succ n ∈ of_nat s, from succ_mem_of_mem_predimage this,
iff.mp !succ_mem_of_nat this))
private lemma to_nat_predimage (s : finset nat) : to_nat (predimage s) = (to_nat s) div 2 :=
private lemma to_nat_predimage (s : finset nat) : to_nat (predimage s) = (to_nat s) / 2 :=
begin
induction s with a s nains ih,
reflexivity,
@ -253,8 +254,8 @@ begin
obtain (w : nat) (hw : to_nat s = 2*w), from exists_of_even this,
begin
rewrite hw,
have d₁ : 1 div 2 = (0:nat), from dec_trivial,
show 2 * w div 2 = (1 + 2 * w) div 2, by
have d₁ : 1 / 2 = (0:nat), from dec_trivial,
show 2 * w / 2 = (1 + 2 * w) / 2, by
rewrite [add_mul_div_self_left _ _ (dec_trivial : 2 > 0), mul.comm,
nat.mul_div_cancel _ (dec_trivial : 2 > 0), d₁, zero_add]
end },
@ -268,12 +269,12 @@ nat.strong_induction_on s
(λ n ih, by_cases
(suppose n = 0, by rewrite this)
(suppose n ≠ 0,
have n div 2 < n, from div_lt_of_ne_zero this,
have to_nat (of_nat (n div 2)) = n div 2, from ih _ this,
have e₁ : to_nat (of_nat n) div 2 = n div 2, from calc
to_nat (of_nat n) div 2 = to_nat (predimage (of_nat n)) : by rewrite to_nat_predimage
... = to_nat (of_nat (n div 2)) : by rewrite of_nat_div2
... = n div 2 : this,
have n / 2 < n, from div_lt_of_ne_zero this,
have to_nat (of_nat (n / 2)) = n / 2, from ih _ this,
have e₁ : to_nat (of_nat n) / 2 = n / 2, from calc
to_nat (of_nat n) / 2 = to_nat (predimage (of_nat n)) : by rewrite to_nat_predimage
... = to_nat (of_nat (n / 2)) : by rewrite of_nat_div2
... = n / 2 : this,
have e₂ : even (to_nat (of_nat n)) ↔ even n, from calc
even (to_nat (of_nat n)) ↔ 0 ∉ of_nat n : even_to_nat
... ↔ even n : even_of_not_zero_mem,

498
library/data/int/div.lean
View file

@ -5,7 +5,7 @@ Author: Jeremy Avigad
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 % b so that 0 ≤ a % b < |b| when b ≠ 0.
-/
import data.int.order data.nat.div
open [coercions] [reduce_hints] nat
@ -17,110 +17,110 @@ namespace int
/- definitions -/
protected definition divide (a b : ℤ) : ℤ :=
protected definition div (a b : ℤ) : ℤ :=
sign b *
(match a with
| of_nat m := of_nat (m div (nat_abs b))
| -[1+m] := -[1+ ((m:nat) div (nat_abs b))]
| of_nat m := of_nat (m / (nat_abs b))
| -[1+m] := -[1+ ((m:nat) / (nat_abs b))]
end)
definition int_has_divide [reducible] [instance] [priority int.prio] : has_divide int :=
has_divide.mk int.divide
definition int_has_div [reducible] [instance] [priority int.prio] : has_div int :=
has_div.mk int.div
lemma of_nat_div_eq (m : nat) (b : ℤ) : (of_nat m) div b = sign b * of_nat (m div (nat_abs b)) :=
lemma of_nat_div_eq (m : nat) (b : ℤ) : (of_nat m) / b = sign b * of_nat (m / (nat_abs b)) :=
rfl
lemma neg_succ_div_eq (m: nat) (b : ℤ) : -[1+m] div b = sign b * -[1+ (m div (nat_abs b))] :=
lemma neg_succ_div_eq (m: nat) (b : ℤ) : -[1+m] / b = sign b * -[1+ (m / (nat_abs b))] :=
rfl
lemma divide.def (a b : ℤ) : a div b =
lemma div_def (a b : ℤ) : a / b =
sign b *
(match a with
| of_nat m := of_nat (m div (nat_abs b))
| -[1+m] := -[1+ ((m:nat) div (nat_abs b))]
| of_nat m := of_nat (m / (nat_abs b))
| -[1+m] := -[1+ ((m:nat) / (nat_abs b))]
end) :=
rfl
protected definition modulo (a b : ℤ) : ℤ := a - a div b * b
protected definition mod (a b : ℤ) : ℤ := a - a / b * b
definition int_has_modulo [reducible] [instance] [priority int.prio] : has_modulo int :=
has_modulo.mk int.modulo
definition int_has_mod [reducible] [instance] [priority int.prio] : has_mod int :=
has_mod.mk int.mod
lemma modulo.def (a b : ℤ) : a mod b = a - a div b * b :=
lemma mod_def (a b : ℤ) : a % b = a - a / b * b :=
rfl
notation [priority int.prio] a ≡ b `[mod `:0 c:0 `]` := a mod c = b mod c
notation [priority int.prio] a ≡ b `[mod `:0 c:0 `]` := a % c = b % c
/- div -/
/- / -/
theorem of_nat_div (m n : nat) : of_nat (m div n) = (of_nat m) div (of_nat n) :=
theorem of_nat_div (m n : nat) : of_nat (m / n) = (of_nat m) / (of_nat n) :=
nat.cases_on n
(begin rewrite [of_nat_div_eq, of_nat_zero, sign_zero, zero_mul, nat.div_zero] end)
(take (n : nat), by rewrite [of_nat_div_eq, sign_of_succ, one_mul])
theorem neg_succ_of_nat_div (m : nat) {b : ℤ} (H : b > 0) :
-[1+m] div b = -(m div b + 1) :=
-[1+m] / b = -(m / b + 1) :=
calc
-[1+m] div b = sign b * _ : rfl
... = -[1+(m div (nat_abs b))] : by rewrite [sign_of_pos H, one_mul]
... = -(m div b + 1) : by rewrite [of_nat_div_eq, sign_of_pos H, one_mul]
-[1+m] / b = sign b * _ : rfl
... = -[1+(m / (nat_abs b))] : by rewrite [sign_of_pos H, one_mul]
... = -(m / b + 1) : by rewrite [of_nat_div_eq, sign_of_pos H, one_mul]
protected theorem div_neg (a b : ℤ) : a div -b = -(a div b) :=
protected theorem div_neg (a b : ℤ) : a / -b = -(a / b) :=
begin
induction a,
rewrite [*of_nat_div_eq, sign_neg, neg_mul_eq_neg_mul, nat_abs_neg],
rewrite [*of_nat_div_eq, sign_neg, neg_mul_eq_neg_mul, nat_abs_neg],
rewrite [*neg_succ_div_eq, sign_neg, neg_mul_eq_neg_mul, nat_abs_neg],
end
theorem div_of_neg_of_pos {a b : ℤ} (Ha : a < 0) (Hb : b > 0) : a div b = -((-a - 1) div b + 1) :=
theorem div_of_neg_of_pos {a b : ℤ} (Ha : a < 0) (Hb : b > 0) : a / b = -((-a - 1) / b + 1) :=
obtain (m : nat) (H1 : a = -[1+m]), from exists_eq_neg_succ_of_nat Ha,
calc
a div b = -(m div b + 1) : by rewrite [H1, neg_succ_of_nat_div _ Hb]
... = -((-a -1) div b + 1) : by rewrite [H1, neg_succ_of_nat_eq', neg_sub, sub_neg_eq_add,
a / b = -(m / b + 1) : by rewrite [H1, neg_succ_of_nat_div _ Hb]
... = -((-a -1) / b + 1) : by rewrite [H1, neg_succ_of_nat_eq', neg_sub, sub_neg_eq_add,
add.comm 1, add_sub_cancel]
protected theorem div_nonneg {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a div b ≥ 0 :=
protected theorem div_nonneg {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a / b ≥ 0 :=
obtain (m : ℕ) (Hm : a = m), from exists_eq_of_nat Ha,
obtain (n : ℕ) (Hn : b = n), from exists_eq_of_nat Hb,
calc
a div b = m div n : by rewrite [Hm, Hn]
a / b = m / n : by rewrite [Hm, Hn]
... ≥ 0 : by rewrite -of_nat_div; apply trivial
protected theorem div_nonpos {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≤ 0) : a div b ≤ 0 :=
protected theorem div_nonpos {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≤ 0) : a / b ≤ 0 :=
calc
a div b = -(a div -b) : by rewrite [int.div_neg, neg_neg]
... ≤ 0 : neg_nonpos_of_nonneg (int.div_nonneg Ha (neg_nonneg_of_nonpos Hb))
a / b = -(a / -b) : by rewrite [int.div_neg, neg_neg]
... ≤ 0 : neg_nonpos_of_nonneg (int.div_nonneg Ha (neg_nonneg_of_nonpos Hb))
theorem div_neg' {a b : ℤ} (Ha : a < 0) (Hb : b > 0) : a div b < 0 :=
theorem div_neg' {a b : ℤ} (Ha : a < 0) (Hb : b > 0) : a / b < 0 :=
have -a - 1 ≥ 0, from le_sub_one_of_lt (neg_pos_of_neg Ha),
have (-a - 1) div b + 1 > 0, from lt_add_one_of_le (int.div_nonneg this (le_of_lt Hb)),
have (-a - 1) / b + 1 > 0, from lt_add_one_of_le (int.div_nonneg this (le_of_lt Hb)),
calc
a div b = -((-a - 1) div b + 1) : div_of_neg_of_pos Ha Hb
... < 0 : neg_neg_of_pos this
a / b = -((-a - 1) / b + 1) : div_of_neg_of_pos Ha Hb
... < 0 : neg_neg_of_pos this
protected theorem zero_div (b : ℤ) : 0 div b = 0 :=
protected theorem zero_div (b : ℤ) : 0 / b = 0 :=
by rewrite [of_nat_div_eq, nat.zero_div, of_nat_zero, mul_zero]
protected theorem div_zero (a : ℤ) : a div 0 = 0 :=
by rewrite [divide.def, sign_zero, zero_mul]
protected theorem div_zero (a : ℤ) : a / 0 = 0 :=
by rewrite [div_def, sign_zero, zero_mul]
protected theorem div_one (a : ℤ) : a div 1 = a :=
protected theorem div_one (a : ℤ) : a / 1 = a :=
assert (1 : int) > 0, from dec_trivial,
int.cases_on a
(take m : nat, by rewrite [-of_nat_one, -of_nat_div, nat.div_one])
(take m : nat, by rewrite [!neg_succ_of_nat_div this, -of_nat_one, -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 / b * b + a % b :=
!add.comm ▸ eq_add_of_sub_eq rfl
theorem div_eq_zero_of_lt {a b : ℤ} : 0 ≤ a → a < b → a div b = 0 :=
theorem div_eq_zero_of_lt {a b : ℤ} : 0 ≤ a → a < b → a / b = 0 :=
int.cases_on a
(take (m : nat), assume H,
int.cases_on b
(take (n : nat),
assume H : m < n,
show m div n = 0,
show m / n = 0,
by rewrite [-of_nat_div, nat.div_eq_zero_of_lt (lt_of_of_nat_lt_of_nat H)])
(take (n : nat),
assume H : m < -[1+n],
@ -131,20 +131,20 @@ int.cases_on a
have ¬ (0 ≤ -[1+m]), from dec_trivial,
absurd H this)
theorem div_eq_zero_of_lt_abs {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < abs b) : a div b = 0 :=
theorem div_eq_zero_of_lt_abs {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < abs b) : a / b = 0 :=
lt.by_cases
(suppose b < 0,
assert a < -b, from abs_of_neg this ▸ H2,
calc
a div b = - (a div -b) : by rewrite [int.div_neg, neg_neg]
... = 0 : by rewrite [div_eq_zero_of_lt H1 this, neg_zero])
a / b = - (a / -b) : by rewrite [int.div_neg, neg_neg]
... = 0 : by rewrite [div_eq_zero_of_lt H1 this, neg_zero])
(suppose b = 0, this⁻¹ ▸ !int.div_zero)
(suppose b > 0,
have a < b, from abs_of_pos this ▸ H2,
div_eq_zero_of_lt H1 this)
private theorem add_mul_div_self_aux1 {a : ℤ} {k : ℕ} (n : ℕ) (H1 : a ≥ 0) (H2 : k > 0) :
(a + n * k) div k = a div k + n :=
(a + n * k) / k = a / k + n :=
obtain (m : nat) (Hm : a = of_nat m), from exists_eq_of_nat H1,
begin
subst Hm,
@ -152,156 +152,157 @@ begin
end
private theorem add_mul_div_self_aux2 {a : ℤ} {k : ℕ} (n : ℕ) (H1 : a < 0) (H2 : k > 0) :
(a + n * k) div k = a div k + n :=
(a + n * k) / k = a / k + n :=
obtain m (Hm : a = -[1+m]), from exists_eq_neg_succ_of_nat H1,
or.elim (nat.lt_or_ge m (n * k))
(assume m_lt_nk : m < n * k,
assert H3 : m + 1 ≤ n * k, from nat.succ_le_of_lt m_lt_nk,
assert H4 : m div k + 1 ≤ n,
assert H4 : m / k + 1 ≤ n,
from nat.succ_le_of_lt (nat.div_lt_of_lt_mul m_lt_nk),
have (-[1+m] + n * k) div k = -[1+m] div k + n, from calc
(-[1+m] + n * k) div k
= of_nat ((k * n - (m + 1)) div k) :
have (-[1+m] + n * k) / k = -[1+m] / k + n, from calc
(-[1+m] + n * k) / k
= of_nat ((k * n - (m + 1)) / k) :
by rewrite [add.comm, neg_succ_of_nat_eq, of_nat_div, algebra.mul.comm k n,
of_nat_sub H3]
... = of_nat (n - m div k - 1) :
... = of_nat (n - m / k - 1) :
nat.mul_sub_div_of_lt (!nat.mul_comm ▸ m_lt_nk)
... = -[1+m] div k + n :
... = -[1+m] / k + n :
by rewrite [nat.sub_sub, of_nat_sub H4, int.add_comm, sub_eq_add_neg,
!neg_succ_of_nat_div (of_nat_lt_of_nat_of_lt H2),
of_nat_add, of_nat_div],
Hm⁻¹ ▸ this)
(assume nk_le_m : n * k ≤ m,
have -[1+m] div k + n = (-[1+m] + n * k) div k, from calc
-[1+m] div k + n
= -(of_nat ((m - n * k + n * k) div k) + 1) + n :
have -[1+m] / k + n = (-[1+m] + n * k) / k, from calc
-[1+m] / k + n
= -(of_nat ((m - n * k + n * k) / k) + 1) + n :
by rewrite [neg_succ_of_nat_div m (of_nat_lt_of_nat_of_lt H2),
nat.sub_add_cancel nk_le_m, of_nat_div]
... = -(of_nat ((m - n * k) div k + n) + 1) + n : nat.add_mul_div_self H2
... = -(of_nat (m - n * k) div k + 1) :
... = -(of_nat ((m - n * k) / k + n) + 1) + n : nat.add_mul_div_self H2
... = -(of_nat (m - n * k) / k + 1) :
by rewrite [of_nat_add, *neg_add, add.right_comm, neg_add_cancel_right,
of_nat_div]
... = -[1+(m - n * k)] div k :
... = -[1+(m - n * k)] / k :
neg_succ_of_nat_div _ (of_nat_lt_of_nat_of_lt H2)
... = -(of_nat(m - n * k) + 1) div k : rfl
... = -(of_nat m - of_nat(n * k) + 1) div k : of_nat_sub nk_le_m
... = (-(of_nat m + 1) + n * k) div k :
... = -(of_nat(m - n * k) + 1) / k : rfl
... = -(of_nat m - of_nat(n * k) + 1) / k : of_nat_sub nk_le_m
... = (-(of_nat m + 1) + n * k) / k :
by rewrite [sub_eq_add_neg, -*add.assoc, *neg_add, neg_neg, add.right_comm]
... = (-[1+m] + n * k) div k : rfl,
... = (-[1+m] + n * k) / k : rfl,
Hm⁻¹ ▸ this⁻¹)
private theorem add_mul_div_self_aux3 (a : ℤ) {b c : ℤ} (H1 : b ≥ 0) (H2 : c > 0) :
(a + b * c) div c = a div c + b :=
(a + b * c) / c = a / c + b :=
obtain (n : nat) (Hn : b = of_nat n), from exists_eq_of_nat H1,
obtain (k : nat) (Hk : c = of_nat k), from exists_eq_of_nat (le_of_lt H2),
have knz : k ≠ 0, from assume kz, !lt.irrefl (kz ▸ Hk ▸ H2),
have kgt0 : (#nat k > 0), from nat.pos_of_ne_zero knz,
have H3 : (a + n * k) div k = a div k + n, from
have H3 : (a + n * k) / k = a / k + n, from
or.elim (lt_or_ge a 0)
(assume Ha : a < 0, add_mul_div_self_aux2 _ Ha kgt0)
(assume Ha : a ≥ 0, add_mul_div_self_aux1 _ Ha kgt0),
Hn⁻¹ ▸ Hk⁻¹ ▸ H3
private theorem add_mul_div_self_aux4 (a b : ℤ) {c : ℤ} (H : c > 0) :
(a + b * c) div c = a div c + b :=
(a + b * c) / c = a / c + b :=
or.elim (le.total 0 b)
(assume H1 : 0 ≤ b, add_mul_div_self_aux3 _ H1 H)
(assume H1 : 0 ≥ b,
eq.symm (calc
a div c + b = (a + b * c + -b * c) div c + b :
a / c + b = (a + b * c + -b * c) / c + b :
by rewrite [-neg_mul_eq_neg_mul, add_neg_cancel_right]
... = (a + b * c) div c + - b + b :
... = (a + b * c) / c + - b + b :
add_mul_div_self_aux3 _ (neg_nonneg_of_nonpos H1) H
... = (a + b * c) div c : neg_add_cancel_right))
... = (a + b * c) / c : neg_add_cancel_right))
protected theorem add_mul_div_self (a b : ℤ) {c : ℤ} (H : c ≠ 0) :
(a + b * c) div c = a div c + b :=
(a + b * c) / c = a / c + b :=
lt.by_cases
(assume H1 : 0 < c, !add_mul_div_self_aux4 H1)
(assume H1 : 0 = c, absurd H1⁻¹ H)
(assume H1 : 0 > c,
have H2 : -c > 0, from neg_pos_of_neg H1,
calc
(a + b * c) div c = - ((a + -b * -c) div -c) : by rewrite [int.div_neg, neg_mul_neg, neg_neg]
... = -(a div -c + -b) : !add_mul_div_self_aux4 H2
... = a div c + b : by rewrite [int.div_neg, neg_add, *neg_neg])
(a + b * c) / c = - ((a + -b * -c) / -c) : by rewrite [int.div_neg, neg_mul_neg, neg_neg]
... = -(a / -c + -b) : !add_mul_div_self_aux4 H2
... = a / c + b : by rewrite [int.div_neg, neg_add, *neg_neg])
protected theorem add_mul_div_self_left (a : ℤ) {b : ℤ} (c : ℤ) (H : b ≠ 0) :
(a + b * c) div b = a div b + c :=
(a + b * c) / b = a / b + c :=
!mul.comm ▸ !int.add_mul_div_self H
protected theorem mul_div_cancel (a : ℤ) {b : ℤ} (H : b ≠ 0) : a * b div b = a :=
protected theorem mul_div_cancel (a : ℤ) {b : ℤ} (H : b ≠ 0) : a * b / b = a :=
calc
a * b div b = (0 + a * b) div b : zero_add
... = 0 div b + a : !int.add_mul_div_self H
... = a : by rewrite [int.zero_div, zero_add]
a * b / b = (0 + a * b) / b : zero_add
... = 0 / b + a : !int.add_mul_div_self H
... = a : by rewrite [int.zero_div, zero_add]
protected theorem mul_div_cancel_left {a : ℤ} (b : ℤ) (H : a ≠ 0) : a * b div a = b :=
protected theorem mul_div_cancel_left {a : ℤ} (b : ℤ) (H : a ≠ 0) : a * b / a = b :=
!mul.comm ▸ int.mul_div_cancel b H
protected theorem div_self {a : ℤ} (H : a ≠ 0) : a div a = 1 :=
protected theorem div_self {a : ℤ} (H : a ≠ 0) : a / a = 1 :=
!mul_one ▸ !int.mul_div_cancel_left H
/- mod -/
theorem of_nat_mod (m n : nat) : m mod n = of_nat (m mod n) :=
have H : m = of_nat (m mod n) + m div n * n, from calc
m = of_nat (m div n * n + m mod n) : nat.eq_div_mul_add_mod
... = of_nat (m div n) * n + of_nat (m mod n) : rfl
... = m div n * n + of_nat (m mod n) : of_nat_div
... = of_nat (m mod n) + m div n * n : add.comm,
theorem of_nat_mod (m n : nat) : m % n = of_nat (m % n) :=
have H : m = of_nat (m % n) + m / n * n, from calc
m = of_nat (m / n * n + m % n) : nat.eq_div_mul_add_mod
... = of_nat (m / n) * n + of_nat (m % n) : rfl
... = m / n * n + of_nat (m % n) : of_nat_div
... = of_nat (m % n) + m / n * n : add.comm,
calc
m mod n = m - m div n * n : rfl
... = of_nat (m mod n) : sub_eq_of_eq_add H
m % n = m - m / n * n : rfl
... = of_nat (m % n) : sub_eq_of_eq_add H
theorem neg_succ_of_nat_mod (m : ℕ) {b : ℤ} (bpos : b > 0) :
-[1+m] mod b = b - 1 - m mod b :=
-[1+m] % b = b - 1 - m % b :=
calc
-[1+m] mod b = -(m + 1) - -[1+m] div b * b : rfl
... = -(m + 1) - -(m div b + 1) * b : neg_succ_of_nat_div _ bpos
... = -m + -1 + (b + m div b * b) :
-[1+m] % b = -(m + 1) - -[1+m] / b * b : rfl
... = -(m + 1) - -(m / b + 1) * b : neg_succ_of_nat_div _ bpos
... = -m + -1 + (b + m / b * b) :
by rewrite [neg_add, -neg_mul_eq_neg_mul, sub_neg_eq_add, right_distrib,
one_mul, (add.comm b)]
... = b + -1 + (-m + m div b * b) :
... = b + -1 + (-m + m / b * b) :
by rewrite [-*add.assoc, add.comm (-m), add.right_comm (-1), (add.comm b)]
... = b - 1 - m mod b :
by rewrite [(modulo.def), *sub_eq_add_neg, neg_add, neg_neg]
... = b - 1 - m % b :
by rewrite [(mod_def), *sub_eq_add_neg, neg_add, neg_neg]
-- it seems the parser has difficulty here, because "mod" is a token?
theorem mod_neg (a b : ℤ) : a mod -b = a mod b :=
theorem mod_neg (a b : ℤ) : a % -b = a % b :=
calc
a mod -b = a - (a div -b) * -b : rfl
... = a - -(a div b) * -b : int.div_neg
... = a - a div b * b : neg_mul_neg
... = a mod b : rfl
a % -b = a - (a / -b) * -b : rfl
... = a - -(a / b) * -b : int.div_neg
... = a - a / b * b : neg_mul_neg
... = a % b : rfl
theorem mod_abs (a b : ℤ) : a mod (abs b) = a mod b :=
theorem mod_abs (a b : ℤ) : a % (abs b) = a % b :=
abs.by_cases rfl !mod_neg
theorem zero_mod (b : ℤ) : 0 mod b = 0 :=
by rewrite [(modulo.def), int.zero_div, zero_mul, sub_zero]
theorem zero_mod (b : ℤ) : 0 % b = 0 :=
by rewrite [(mod_def), int.zero_div, zero_mul, sub_zero]
theorem mod_zero (a : ℤ) : a mod 0 = a :=
by rewrite [(modulo.def), mul_zero, sub_zero]
theorem mod_zero (a : ℤ) : a % 0 = a :=
by rewrite [(mod_def), mul_zero, sub_zero]
theorem mod_one (a : ℤ) : a mod 1 = 0 :=
theorem mod_one (a : ℤ) : a % 1 = 0 :=
calc
a mod 1 = a - a div 1 * 1 : rfl
a % 1 = a - a / 1 * 1 : rfl
... = 0 : by rewrite [mul_one, int.div_one, sub_self]
private lemma of_nat_mod_abs (m : ℕ) (b : ℤ) : m mod (abs b) = of_nat (m mod (nat_abs b)) :=
private lemma of_nat_mod_abs (m : ℕ) (b : ℤ) : m % (abs b) = of_nat (m % (nat_abs b)) :=
calc
m mod (abs b) = m mod (nat_abs b) : of_nat_nat_abs
... = of_nat (m mod (nat_abs b)) : of_nat_mod
m % (abs b) = m % (nat_abs b) : of_nat_nat_abs
... = of_nat (m % (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 % (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) = of_nat (m mod (nat_abs b)) : of_nat_mod_abs m b
m % (abs b) = of_nat (m % (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 : ℤ} (H1 : 0 ≤ a) (H2 : a < b) : a mod b = a :=
theorem mod_eq_of_lt {a b : ℤ} (H1 : 0 ≤ a) (H2 : a < b) : a % b = a :=
obtain (m : nat) (Hm : a = of_nat m), from exists_eq_of_nat H1,
obtain (n : nat) (Hn : b = of_nat n), from exists_eq_of_nat (le_of_lt (lt_of_le_of_lt H1 H2)),
begin
@ -310,181 +311,180 @@ begin
apply nat.mod_eq_of_lt
end
theorem mod_nonneg (a : ℤ) {b : ℤ} (H : b ≠ 0) : a mod b ≥ 0 :=
theorem mod_nonneg (a : ℤ) {b : ℤ} (H : b ≠ 0) : a % b ≥ 0 :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : a mod (abs b) ≥ 0, from
have H2 : a % (abs b) ≥ 0, from
int.cases_on a
(take m : nat, (of_nat_mod_abs m b)⁻¹ ▸ of_nat_nonneg (nat.modulo m (nat_abs b)))
(take m : nat, (of_nat_mod_abs m b)⁻¹ ▸ of_nat_nonneg (nat.mod m (nat_abs b)))
(take m : nat,
have H3 : 1 + m mod (abs b) ≤ (abs b),
have H3 : 1 + m % (abs b) ≤ (abs b),
from (!add.comm ▸ add_one_le_of_lt (of_nat_mod_abs_lt m H)),
calc
-[1+m] 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.mpr !sub_nonneg_iff_le H3),
-[1+m] % (abs b) = abs b - 1 - m % (abs b) : neg_succ_of_nat_mod _ H1
... = abs b - (1 + m % (abs b)) : by rewrite [*sub_eq_add_neg, neg_add, add.assoc]
... ≥ 0 : iff.mpr !sub_nonneg_iff_le H3),
!mod_abs ▸ H2
theorem mod_lt (a : ℤ) {b : ℤ} (H : b ≠ 0) : a mod b < (abs b) :=
theorem mod_lt (a : ℤ) {b : ℤ} (H : b ≠ 0) : a % b < (abs b) :=
have H1 : abs b > 0, from abs_pos_of_ne_zero H,
have H2 : a mod (abs b) < abs b, from
have H2 : a % (abs b) < abs b, from
int.cases_on a
(take m, of_nat_mod_abs_lt m H)
(take m : nat,
have H3 : abs b ≠ 0, from assume H', H (eq_zero_of_abs_eq_zero H'),
have H4 : 1 + m mod (abs b) > 0,
have H4 : 1 + m % (abs b) > 0,
from add_pos_of_pos_of_nonneg dec_trivial (mod_nonneg _ H3),
calc
-[1+m] 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]
-[1+m] % (abs b) = abs b - 1 - m % (abs b) : neg_succ_of_nat_mod _ H1
... = abs b - (1 + m % (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 : ℤ} : (a + b * c) mod c = a mod c :=
theorem add_mul_mod_self {a b c : ℤ} : (a + b * c) % c = a % c :=
decidable.by_cases
(assume cz : c = 0, by rewrite [cz, mul_zero, add_zero])
(assume cnz, by rewrite [(modulo.def), !int.add_mul_div_self cnz, right_distrib,
(assume cnz, by rewrite [(mod_def), !int.add_mul_div_self cnz, right_distrib,
sub_add_eq_sub_sub_swap, add_sub_cancel])
theorem add_mul_mod_self_left (a b c : ℤ) : (a + b * c) mod b = a mod b :=
theorem add_mul_mod_self_left (a b c : ℤ) : (a + b * c) % b = a % b :=
!mul.comm ▸ !add_mul_mod_self
theorem add_mod_self {a b : ℤ} : (a + b) mod b = a mod b :=
theorem add_mod_self {a b : ℤ} : (a + b) % b = a % b :=
by rewrite -(int.mul_one b) at {1}; apply add_mul_mod_self_left
theorem add_mod_self_left {a b : ℤ} : (a + b) mod a = b mod a :=
theorem add_mod_self_left {a b : ℤ} : (a + b) % a = b % a :=
!add.comm ▸ !add_mod_self
theorem mod_add_mod (m n k : ℤ) : (m mod n + k) mod n = (m + k) mod n :=
by rewrite [eq_div_mul_add_mod m n at {2}, add.assoc, add.comm (m div n * n), add_mul_mod_self]
theorem mod_add_mod (m n k : ℤ) : (m % n + k) % n = (m + k) % n :=
by rewrite [eq_div_mul_add_mod m n at {2}, add.assoc, add.comm (m / n * n), add_mul_mod_self]
theorem add_mod_mod (m n k : ℤ) : (m + n mod k) mod k = (m + n) mod k :=
theorem add_mod_mod (m n k : ℤ) : (m + n % k) % k = (m + n) % k :=
by rewrite [add.comm, mod_add_mod, add.comm]
theorem add_mod_eq_add_mod_right {m n k : ℤ} (i : ℤ) (H : m mod n = k mod n) :
(m + i) mod n = (k + i) mod n :=
theorem add_mod_eq_add_mod_right {m n k : ℤ} (i : ℤ) (H : m % n = k % n) :
(m + i) % n = (k + i) % n :=
by rewrite [-mod_add_mod, -mod_add_mod k, H]
theorem add_mod_eq_add_mod_left {m n k : ℤ} (i : ℤ) (H : m mod n = k mod n) :
(i + m) mod n = (i + k) mod n :=
theorem add_mod_eq_add_mod_left {m n k : ℤ} (i : ℤ) (H : m % n = k % n) :
(i + m) % n = (i + k) % n :=
by rewrite [add.comm, add_mod_eq_add_mod_right _ H, add.comm]
theorem mod_eq_mod_of_add_mod_eq_add_mod_right {m n k i : ℤ}
(H : (m + i) mod n = (k + i) mod n) :
m mod n = k mod n :=
assert H1 : (m + i + (-i)) mod n = (k + i + (-i)) mod n, from add_mod_eq_add_mod_right _ H,
(H : (m + i) % n = (k + i) % n) :
m % n = k % n :=
assert H1 : (m + i + (-i)) % n = (k + i + (-i)) % n, from add_mod_eq_add_mod_right _ H,
by rewrite [*add_neg_cancel_right at H1]; apply H1
theorem mod_eq_mod_of_add_mod_eq_add_mod_left {m n k i : ℤ} :
(i + m) mod n = (i + k) mod n → m mod n = k mod n :=
(i + m) % n = (i + k) % n → m % n = k % n :=
by rewrite [add.comm i m, add.comm i k]; apply mod_eq_mod_of_add_mod_eq_add_mod_right
theorem mul_mod_left (a b : ℤ) : (a * b) mod b = 0 :=
theorem mul_mod_left (a b : ℤ) : (a * b) % b = 0 :=
by rewrite [-zero_add (a * b), add_mul_mod_self, zero_mod]
theorem mul_mod_right (a b : ℤ) : (a * b) mod a = 0 :=
theorem mul_mod_right (a b : ℤ) : (a * b) % a = 0 :=
!mul.comm ▸ !mul_mod_left
theorem mod_self {a : ℤ} : a mod a = 0 :=
theorem mod_self {a : ℤ} : a % a = 0 :=
decidable.by_cases
(assume H : a = 0, H⁻¹ ▸ !mod_zero)
(assume H : a ≠ 0,
calc
a mod a = a - a div a * a : rfl
a % a = a - a / a * a : rfl
... = 0 : by rewrite [!int.div_self H, one_mul, sub_self])
theorem mod_lt_of_pos (a : ℤ) {b : ℤ} (H : b > 0) : a mod b < b :=
theorem mod_lt_of_pos (a : ℤ) {b : ℤ} (H : b > 0) : a % b < b :=
!abs_of_pos H ▸ !mod_lt (ne.symm (ne_of_lt H))
/- properties of div and mod -/
/- properties of / and % -/
theorem mul_div_mul_of_pos_aux {a : ℤ} (b : ℤ) {c : ℤ}
(H1 : a > 0) (H2 : c > 0) : a * b div (a * c) = b div c :=
(H1 : a > 0) (H2 : c > 0) : a * b / (a * c) = b / c :=
have H3 : a * c ≠ 0, from ne.symm (ne_of_lt (mul_pos H1 H2)),
have H4 : a * (b mod c) < a * c, from mul_lt_mul_of_pos_left (!mod_lt_of_pos H2) H1,
have H5 : a * (b mod c) ≥ 0, from mul_nonneg (le_of_lt H1) (!mod_nonneg (ne.symm (ne_of_lt H2))),
have H4 : a * (b % c) < a * c, from mul_lt_mul_of_pos_left (!mod_lt_of_pos H2) H1,
have H5 : a * (b % c) ≥ 0, from mul_nonneg (le_of_lt H1) (!mod_nonneg (ne.symm (ne_of_lt H2))),
calc
a * b div (a * c) = a * (b div c * c + b mod c) div (a * c) : eq_div_mul_add_mod
a * b / (a * c) = a * (b / c * c + b % c) / (a * c) : eq_div_mul_add_mod
... = (a * (b mod c) + a * c * (b div c)) div (a * c) :
... = (a * (b % c) + a * c * (b / c)) / (a * c) :
by rewrite [!add.comm, int.left_distrib, mul.comm _ c, -!mul.assoc]
... = a * (b mod c) div (a * c) + b div c : !int.add_mul_div_self_left H3
... = 0 + b div c : {!div_eq_zero_of_lt H5 H4}
... = b div c : zero_add
... = a * (b % c) / (a * c) + b / c : !int.add_mul_div_self_left H3
... = 0 + b / c : {!div_eq_zero_of_lt H5 H4}
... = b / c : zero_add
theorem mul_div_mul_of_pos {a : ℤ} (b c : ℤ) (H : a > 0) : a * b div (a * c) = b div c :=
theorem mul_div_mul_of_pos {a : ℤ} (b c : ℤ) (H : a > 0) : a * b / (a * c) = b / c :=
lt.by_cases
(assume H1 : c < 0,
have H2 : -c > 0, from neg_pos_of_neg H1,
calc
a * b div (a * c) = - (a * b div (a * -c)) :
a * b / (a * c) = - (a * b / (a * -c)) :
by rewrite [-neg_mul_eq_mul_neg, int.div_neg, neg_neg]
... = - (b div -c) : mul_div_mul_of_pos_aux _ H H2
... = b div c : by rewrite [int.div_neg, neg_neg])
... = - (b / -c) : mul_div_mul_of_pos_aux _ H H2
... = b / c : by rewrite [int.div_neg, neg_neg])
(assume H1 : c = 0,
calc
a * b div (a * c) = 0 : by rewrite [H1, mul_zero, int.div_zero]
... = b div c : by rewrite [H1, int.div_zero])
a * b / (a * c) = 0 : by rewrite [H1, mul_zero, int.div_zero]
... = b / c : by rewrite [H1, int.div_zero])
(assume H1 : c > 0,
mul_div_mul_of_pos_aux _ H H1)
theorem mul_div_mul_of_pos_left (a : ℤ) {b : ℤ} (c : ℤ) (H : b > 0) :
a * b div (c * b) = a div c :=
a * b / (c * b) = a / c :=
!mul.comm ▸ !mul.comm ▸ !mul_div_mul_of_pos H
-- TODO: something strange here: why doesn't !modulo.def or !(modulo.def) work?
theorem mul_mod_mul_of_pos {a : ℤ} (b c : ℤ) (H : a > 0) : a * b mod (a * c) = a * (b mod c) :=
by rewrite [(modulo.def), modulo.def, !mul_div_mul_of_pos H, mul_sub_left_distrib, mul.left_comm]
theorem mul_mod_mul_of_pos {a : ℤ} (b c : ℤ) (H : a > 0) : a * b % (a * c) = a * (b % c) :=
by rewrite [(mod_def), mod_def, !mul_div_mul_of_pos H, mul_sub_left_distrib, mul.left_comm]
theorem lt_div_add_one_mul_self (a : ℤ) {b : ℤ} (H : b > 0) : a < (a div b + 1) * b :=
have H : a - a div b * b < b, from !mod_lt_of_pos H,
theorem lt_div_add_one_mul_self (a : ℤ) {b : ℤ} (H : b > 0) : a < (a / b + 1) * b :=
have H : a - a / b * b < b, from !mod_lt_of_pos H,
calc
a < a div b * b + b : iff.mpr !lt_add_iff_sub_lt_left H
... = (a div b + 1) * b : by rewrite [right_distrib, one_mul]
a < a / b * b + b : iff.mpr !lt_add_iff_sub_lt_left H
... = (a / b + 1) * b : by rewrite [right_distrib, one_mul]
theorem div_le_of_nonneg_of_nonneg {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a div b ≤ a :=
theorem div_le_of_nonneg_of_nonneg {a b : ℤ} (Ha : a ≥ 0) (Hb : b ≥ 0) : a / b ≤ a :=
obtain (m : ℕ) (Hm : a = m), from exists_eq_of_nat Ha,
obtain (n : ℕ) (Hn : b = n), from exists_eq_of_nat Hb,
calc
a div b = of_nat (m div n) : by rewrite [Hm, Hn, of_nat_div]
... ≤ m : of_nat_le_of_nat_of_le !nat.div_le_self
... = a : Hm
a / b = of_nat (m / n) : by rewrite [Hm, Hn, of_nat_div]
... ≤ m : of_nat_le_of_nat_of_le !nat.div_le_self
... = a : Hm
theorem abs_div_le_abs (a b : ℤ) : abs (a div b) ≤ abs a :=
have H : ∀a b, b > 0 → abs (a div b) ≤ abs a, from
theorem abs_div_le_abs (a b : ℤ) : abs (a / b) ≤ abs a :=
have H : ∀a b, b > 0 → abs (a / b) ≤ abs a, from
take a b,
assume H1 : b > 0,
or.elim (le_or_gt 0 a)
(assume H2 : 0 ≤ a,
have H3 : 0 ≤ b, from le_of_lt H1,
calc
abs (a div b) = a div b : abs_of_nonneg (int.div_nonneg H2 H3)
... ≤ a : div_le_of_nonneg_of_nonneg H2 H3
... = abs a : abs_of_nonneg H2)
abs (a / b) = a / b : abs_of_nonneg (int.div_nonneg H2 H3)
... ≤ a : div_le_of_nonneg_of_nonneg H2 H3
... = abs a : abs_of_nonneg H2)
(assume H2 : a < 0,
have H3 : -a - 1 ≥ 0, from le_sub_one_of_lt (neg_pos_of_neg H2),
have H4 : (-a - 1) div b + 1 ≥ 0,
have H4 : (-a - 1) / b + 1 ≥ 0,
from add_nonneg (int.div_nonneg H3 (le_of_lt H1)) (of_nat_le_of_nat_of_le !nat.zero_le),
have H5 : (-a - 1) div b ≤ -a - 1, from div_le_of_nonneg_of_nonneg H3 (le_of_lt H1),
have H5 : (-a - 1) / b ≤ -a - 1, from div_le_of_nonneg_of_nonneg H3 (le_of_lt H1),
calc
abs (a div b) = abs ((-a - 1) div b + 1) : by rewrite [div_of_neg_of_pos H2 H1, abs_neg]
... = (-a - 1) div b + 1 : abs_of_nonneg H4
... ≤ -a - 1 + 1 : add_le_add_right H5 _
... = abs a : by rewrite [sub_add_cancel, abs_of_neg H2]),
abs (a / b) = abs ((-a - 1) / b + 1) : by rewrite [div_of_neg_of_pos H2 H1, abs_neg]
... = (-a - 1) / b + 1 : abs_of_nonneg H4
... ≤ -a - 1 + 1 : add_le_add_right H5 _
... = abs a : by rewrite [sub_add_cancel, abs_of_neg H2]),
lt.by_cases
(assume H1 : b < 0,
calc
abs (a div b) = abs (a div -b) : by rewrite [int.div_neg, abs_neg]
... ≤ abs a : H _ _ (neg_pos_of_neg H1))
abs (a / b) = abs (a / -b) : by rewrite [int.div_neg, abs_neg]
... ≤ abs a : H _ _ (neg_pos_of_neg H1))
(assume H1 : b = 0,
calc
abs (a div b) = 0 : by rewrite [H1, int.div_zero, abs_zero]
abs (a / b) = 0 : by rewrite [H1, int.div_zero, abs_zero]
... ≤ abs a : abs_nonneg)
(assume H1 : b > 0, H _ _ H1)
theorem div_mul_cancel_of_mod_eq_zero {a b : ℤ} (H : a mod b = 0) : a div b * b = a :=
theorem div_mul_cancel_of_mod_eq_zero {a b : ℤ} (H : a % b = 0) : a / b * b = a :=
by rewrite [eq_div_mul_add_mod a b at {2}, H, add_zero]
theorem mul_div_cancel_of_mod_eq_zero {a b : ℤ} (H : a mod b = 0) : b * (a div b) = a :=
theorem mul_div_cancel_of_mod_eq_zero {a b : ℤ} (H : a % b = 0) : b * (a / b) = a :=
!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H
/- dvd -/
@ -519,72 +519,72 @@ begin
apply nat.dvd.antisymm
end
theorem dvd_of_mod_eq_zero {a b : ℤ} (H : b mod a = 0) : a ∣ b :=
theorem dvd_of_mod_eq_zero {a b : ℤ} (H : b % a = 0) : a ∣ b :=
dvd.intro (!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H)
theorem mod_eq_zero_of_dvd {a b : ℤ} (H : a ∣ b) : b mod a = 0 :=
theorem mod_eq_zero_of_dvd {a b : ℤ} (H : a ∣ b) : b % a = 0 :=
dvd.elim H (take z, assume H1 : b = a * z, H1⁻¹ ▸ !mul_mod_right)
theorem dvd_iff_mod_eq_zero (a b : ℤ) : a ∣ b ↔ b mod a = 0 :=
theorem dvd_iff_mod_eq_zero (a b : ℤ) : a ∣ b ↔ b % a = 0 :=
iff.intro mod_eq_zero_of_dvd dvd_of_mod_eq_zero
definition dvd.decidable_rel [instance] : decidable_rel dvd :=
take a n, decidable_of_decidable_of_iff _ (iff.symm !dvd_iff_mod_eq_zero)
protected theorem div_mul_cancel {a b : ℤ} (H : b ∣ a) : a div b * b = a :=
protected theorem div_mul_cancel {a b : ℤ} (H : b ∣ a) : a / b * b = a :=
div_mul_cancel_of_mod_eq_zero (mod_eq_zero_of_dvd H)
protected theorem mul_div_cancel' {a b : ℤ} (H : a ∣ b) : a * (b div a) = b :=
protected theorem mul_div_cancel' {a b : ℤ} (H : a ∣ b) : a * (b / a) = b :=
!mul.comm ▸ !int.div_mul_cancel H
protected theorem mul_div_assoc (a : ℤ) {b c : ℤ} (H : c ∣ b) : (a * b) div c = a * (b div c) :=
protected theorem mul_div_assoc (a : ℤ) {b c : ℤ} (H : c ∣ b) : (a * b) / c = a * (b / c) :=
decidable.by_cases
(assume cz : c = 0, by rewrite [cz, *int.div_zero, mul_zero])
(assume cnz : c ≠ 0,
obtain d (H' : b = d * c), from exists_eq_mul_left_of_dvd H,
by rewrite [H', -mul.assoc, *(!int.mul_div_cancel cnz)])
theorem div_dvd_div {a b c : ℤ} (H1 : a ∣ b) (H2 : b ∣ c) : b div a ∣ c div a :=
have H3 : b = b div a * a, from (int.div_mul_cancel H1)⁻¹,
have H4 : c = c div a * a, from (int.div_mul_cancel (dvd.trans H1 H2))⁻¹,
theorem div_dvd_div {a b c : ℤ} (H1 : a ∣ b) (H2 : b ∣ c) : b / a ∣ c / a :=
have H3 : b = b / a * a, from (int.div_mul_cancel H1)⁻¹,
have H4 : c = c / a * a, from (int.div_mul_cancel (dvd.trans H1 H2))⁻¹,
decidable.by_cases
(assume H5 : a = 0,
have H6: c div a = 0, from (congr_arg _ H5 ⬝ !int.div_zero),
have H6: c / a = 0, from (congr_arg _ H5 ⬝ !int.div_zero),
H6⁻¹ ▸ !dvd_zero)
(assume H5 : a ≠ 0,
dvd_of_mul_dvd_mul_right H5 (H3 ▸ H4 ▸ H2))
protected theorem div_eq_iff_eq_mul_right {a b : ℤ} (c : ℤ) (H : b ≠ 0) (H' : b ∣ a) :
a div b = c ↔ a = b * c :=
a / b = c ↔ a = b * c :=
iff.intro
(assume H1, by rewrite [-H1, int.mul_div_cancel' H'])
(assume H1, by rewrite [H1, !int.mul_div_cancel_left H])
protected theorem div_eq_iff_eq_mul_left {a b : ℤ} (c : ℤ) (H : b ≠ 0) (H' : b ∣ a) :
a div b = c ↔ a = c * b :=
a / b = c ↔ a = c * b :=
!mul.comm ▸ !int.div_eq_iff_eq_mul_right H H'
protected theorem eq_mul_of_div_eq_right {a b c : ℤ} (H1 : b ∣ a) (H2 : a div b = c) :
protected theorem eq_mul_of_div_eq_right {a b c : ℤ} (H1 : b ∣ a) (H2 : a / b = c) :
a = b * c :=
calc
a = b * (a div b) : int.mul_div_cancel' H1
... = b * c : H2
a = b * (a / b) : int.mul_div_cancel' H1
... = b * c : H2
protected theorem div_eq_of_eq_mul_right {a b c : ℤ} (H1 : b ≠ 0) (H2 : a = b * c) :
a div b = c :=
a / b = c :=
calc
a div b = b * c div b : H2
... = c : !int.mul_div_cancel_left H1
a / b = b * c / b : H2
... = c : !int.mul_div_cancel_left H1
protected theorem eq_mul_of_div_eq_left {a b c : ℤ} (H1 : b ∣ a) (H2 : a div b = c) :
protected theorem eq_mul_of_div_eq_left {a b c : ℤ} (H1 : b ∣ a) (H2 : a / b = c) :
a = c * b :=
!mul.comm ▸ !int.eq_mul_of_div_eq_right H1 H2
protected theorem div_eq_of_eq_mul_left {a b c : ℤ} (H1 : b ≠ 0) (H2 : a = c * b) :
a div b = c :=
a / b = c :=
int.div_eq_of_eq_mul_right H1 (!mul.comm ▸ H2)
theorem neg_div_of_dvd {a b : ℤ} (H : b ∣ a) : -a div b = -(a div b) :=
theorem neg_div_of_dvd {a b : ℤ} (H : b ∣ a) : -a / b = -(a / b) :=
decidable.by_cases
(assume H1 : b = 0, by rewrite [H1, *int.div_zero, neg_zero])
(assume H1 : b ≠ 0,
@ -592,7 +592,7 @@ decidable.by_cases
(take c, assume H' : a = b * c,
by rewrite [H', neg_mul_eq_mul_neg, *!int.mul_div_cancel_left H1]))
protected theorem sign_eq_div_abs (a : ℤ) : sign a = a div (abs a) :=
protected theorem sign_eq_div_abs (a : ℤ) : sign a = a / (abs a) :=
decidable.by_cases
(suppose a = 0, by subst a)
(suppose a ≠ 0,
@ -612,21 +612,21 @@ or.elim !le_or_gt
... ≤ a * c : mul_le_mul_of_nonneg_left (add_one_le_of_lt `c > 0`) (le_of_lt `a > 0`)
... = b : Hc)
/- div and ordering -/
/- / and ordering -/
protected theorem div_mul_le (a : ℤ) {b : ℤ} (H : b ≠ 0) : a div b * b ≤ a :=
protected theorem div_mul_le (a : ℤ) {b : ℤ} (H : b ≠ 0) : a / 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)
a = a / b * b + a % b : eq_div_mul_add_mod
... ≥ a / b * b : le_add_of_nonneg_right (!mod_nonneg H)
protected theorem div_le_of_le_mul {a b c : ℤ} (H : c > 0) (H' : a ≤ b * c) : a div c ≤ b :=
protected theorem div_le_of_le_mul {a b c : ℤ} (H : c > 0) (H' : a ≤ b * c) : a / 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
a / c * c = a / c * c + 0 : add_zero
... ≤ a / c * c + a % c : add_le_add_left (!mod_nonneg (ne_of_gt H))
... = a : eq_div_mul_add_mod
... ≤ b * c : H') H
protected theorem div_le_self (a : ℤ) {b : ℤ} (H1 : a ≥ 0) (H2 : b ≥ 0) : a div b ≤ a :=
protected theorem div_le_self (a : ℤ) {b : ℤ} (H1 : a ≥ 0) (H2 : b ≥ 0) : a / 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,
@ -637,66 +637,66 @@ or.elim (lt_or_eq_of_le H2)
(assume H3 : 0 = b,
by rewrite [-H3, int.div_zero]; apply H1)
protected theorem mul_le_of_le_div {a b c : ℤ} (H1 : c > 0) (H2 : a ≤ b div c) : a * c ≤ b :=
protected theorem mul_le_of_le_div {a b c : ℤ} (H1 : c > 0) (H2 : a ≤ b / c) : a * c ≤ b :=
calc
a * c ≤ b div c * c : !mul_le_mul_of_nonneg_right H2 (le_of_lt H1)
a * c ≤ b / c * c : !mul_le_mul_of_nonneg_right H2 (le_of_lt H1)
... ≤ b : !int.div_mul_le (ne_of_gt H1)
protected theorem le_div_of_mul_le {a b c : ℤ} (H1 : c > 0) (H2 : a * c ≤ b) : a ≤ b div c :=
have H3 : a * c < (b div c + 1) * c, from
protected theorem le_div_of_mul_le {a b c : ℤ} (H1 : c > 0) (H2 : a * c ≤ b) : a ≤ b / c :=
have H3 : a * c < (b / 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 [right_distrib, one_mul],
a * c ≤ b : H2
... = b / c * c + b % c : eq_div_mul_add_mod
... < b / c * c + c : add_lt_add_left (!mod_lt_of_pos H1)
... = (b / c + 1) * c : by rewrite [right_distrib, one_mul],
le_of_lt_add_one (lt_of_mul_lt_mul_right H3 (le_of_lt H1))
protected theorem le_div_iff_mul_le {a b c : ℤ} (H : c > 0) : a ≤ b div c ↔ a * c ≤ b :=
protected theorem le_div_iff_mul_le {a b c : ℤ} (H : c > 0) : a ≤ b / c ↔ a * c ≤ b :=
iff.intro (!int.mul_le_of_le_div H) (!int.le_div_of_mul_le H)
protected theorem div_le_div {a b c : ℤ} (H : c > 0) (H' : a ≤ b) : a div c ≤ b div c :=
protected theorem div_le_div {a b c : ℤ} (H : c > 0) (H' : a ≤ b) : a / c ≤ b / c :=
int.le_div_of_mul_le H (le.trans (!int.div_mul_le (ne_of_gt H)) H')
protected theorem div_lt_of_lt_mul {a b c : ℤ} (H : c > 0) (H' : a < b * c) : a div c < b :=
protected theorem div_lt_of_lt_mul {a b c : ℤ} (H : c > 0) (H' : a < b * c) : a / 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')
a / c * c = a / c * c + 0 : add_zero
... ≤ a / c * c + a % c : add_le_add_left (!mod_nonneg (ne_of_gt H))
... = a : eq_div_mul_add_mod
... < b * c : H')
(le_of_lt H)
protected theorem lt_mul_of_div_lt {a b c : ℤ} (H1 : c > 0) (H2 : a div c < b) : a < b * c :=
assert H3 : (a div c + 1) * c ≤ b * c,
protected theorem lt_mul_of_div_lt {a b c : ℤ} (H1 : c > 0) (H2 : a / c < b) : a < b * c :=
assert H3 : (a / 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 [right_distrib at H3, one_mul at H3]; apply H3,
have H4 : a / c * c + c ≤ b * c, by rewrite [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
a = a / c * c + a % c : eq_div_mul_add_mod
... < a / c * c + c : add_lt_add_left (!mod_lt_of_pos H1)
... ≤ b * c : H4
protected theorem div_lt_iff_lt_mul {a b c : ℤ} (H : c > 0) : a div c < b ↔ a < b * c :=
protected theorem div_lt_iff_lt_mul {a b c : ℤ} (H : c > 0) : a / c < b ↔ a < b * c :=
iff.intro (!int.lt_mul_of_div_lt H) (!int.div_lt_of_lt_mul H)
protected theorem div_le_iff_le_mul_of_div {a b : ℤ} (c : ℤ) (H : b > 0) (H' : b ∣ a) :
a div b ≤ c ↔ a ≤ c * b :=
a / b ≤ c ↔ a ≤ c * b :=
by rewrite [propext (!le_iff_mul_le_mul_right H), !int.div_mul_cancel H']
protected theorem le_mul_of_div_le_of_div {a b c : ℤ} (H1 : b > 0) (H2 : b ∣ a) (H3 : a div b ≤ c) :
protected theorem le_mul_of_div_le_of_div {a b c : ℤ} (H1 : b > 0) (H2 : b ∣ a) (H3 : a / b ≤ c) :
a ≤ c * b :=
iff.mp (!int.div_le_iff_le_mul_of_div H1 H2) H3
theorem div_pos_of_pos_of_dvd {a b : ℤ} (H1 : a > 0) (H2 : b ≥ 0) (H3 : b ∣ a) : a div b > 0 :=
theorem div_pos_of_pos_of_dvd {a b : ℤ} (H1 : a > 0) (H2 : b ≥ 0) (H3 : b ∣ a) : a / b > 0 :=
have H4 : b ≠ 0, from
(assume H5 : b = 0,
have H6 : a = 0, from eq_zero_of_zero_dvd (H5 ▸ H3),
ne_of_gt H1 H6),
have H6 : (a div b) * b > 0, by rewrite (int.div_mul_cancel H3); apply H1,
have H6 : (a / b) * b > 0, by rewrite (int.div_mul_cancel H3); apply H1,
pos_of_mul_pos_right H6 H2
theorem div_eq_div_of_dvd_of_dvd {a b c d : ℤ} (H1 : b ∣ a) (H2 : d ∣ c) (H3 : b ≠ 0)
(H4 : d ≠ 0) (H5 : a * d = b * c) :
a div b = c div d :=
a / b = c / d :=
begin
apply int.div_eq_of_eq_mul_right H3,
rewrite [-!int.mul_div_assoc H2],

74
library/data/int/gcd.lean
View file

@ -44,21 +44,21 @@ by rewrite [↑gcd, *nat_abs_abs]
section
open nat
theorem gcd_of_ne_zero (a : ℤ) {b : ℤ} (H : b ≠ 0) : gcd a b = gcd b (abs a mod abs b) :=
theorem gcd_of_ne_zero (a : ℤ) {b : ℤ} (H : b ≠ 0) : gcd a b = gcd b (abs a % abs b) :=
have nat_abs b ≠ 0, from assume H', H (eq_zero_of_nat_abs_eq_zero H'),
have nat_abs b > 0, from pos_of_ne_zero this,
assert nat.gcd (nat_abs a) (nat_abs b) = (nat.gcd (nat_abs b) (nat_abs a mod nat_abs b)),
assert nat.gcd (nat_abs a) (nat_abs b) = (nat.gcd (nat_abs b) (nat_abs a % nat_abs b)),
from @nat.gcd_of_pos (nat_abs a) (nat_abs b) this,
calc
gcd a b = nat.gcd (nat_abs b) (nat_abs a mod nat_abs b) : by rewrite [↑gcd, this]
... = gcd (abs b) (abs a mod abs b) : by rewrite [↑gcd, -*of_nat_nat_abs, of_nat_mod]
... = gcd b (abs a mod abs b) : by rewrite [↑gcd, *nat_abs_abs]
gcd a b = nat.gcd (nat_abs b) (nat_abs a % nat_abs b) : by rewrite [↑gcd, this]
... = gcd (abs b) (abs a % abs b) : by rewrite [↑gcd, -*of_nat_nat_abs, of_nat_mod]
... = gcd b (abs a % abs b) : by rewrite [↑gcd, *nat_abs_abs]
end
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 % b) :=
by rewrite [!gcd_of_ne_zero (ne_of_gt H), abs_of_pos H]
theorem gcd_of_nonneg_of_pos {a b : ℤ} (H1 : a ≥ 0) (H2 : b > 0) : gcd a b = gcd b (a mod b) :=
theorem gcd_of_nonneg_of_pos {a b : ℤ} (H1 : a ≥ 0) (H2 : b > 0) : gcd a b = gcd b (a % b) :=
by rewrite [!gcd_of_pos H2, abs_of_nonneg H1]
theorem gcd_self (a : ℤ) : gcd a a = abs a :=
@ -114,21 +114,21 @@ theorem eq_zero_of_gcd_eq_zero_right {a b : ℤ} (H : gcd a b = 0) : b = 0 :=
by rewrite gcd.comm at H; apply !eq_zero_of_gcd_eq_zero_left H
theorem gcd_div {a b c : ℤ} (H1 : c ∣ a) (H2 : c ∣ b) :
gcd (a div c) (b div c) = gcd a b div (abs c) :=
gcd (a / c) (b / c) = gcd a b / (abs c) :=
decidable.by_cases
(suppose c = 0,
calc
gcd (a div c) (b div c) = gcd 0 0 : by subst c; rewrite *int.div_zero
... = 0 : gcd_zero_left
... = gcd a b div 0 : int.div_zero
... = gcd a b div (abs c) : by subst c)
gcd (a / c) (b / c) = gcd 0 0 : by subst c; rewrite *int.div_zero
... = 0 : gcd_zero_left
... = gcd a b / 0 : int.div_zero
... = gcd a b / (abs c) : by subst c)
(suppose c ≠ 0,
have abs c ≠ 0, from assume H', this (eq_zero_of_abs_eq_zero H'),
eq.symm (int.div_eq_of_eq_mul_left this
(eq.symm (calc
gcd (a div c) (b div c) * abs c = gcd (a div c * c) (b div c * c) : gcd_mul_right
... = gcd a (b div c * c) : int.div_mul_cancel H1
... = gcd a b : int.div_mul_cancel H2))))
gcd (a / c) (b / c) * abs c = gcd (a / c * c) (b / c * c) : gcd_mul_right
... = gcd a (b / c * c) : int.div_mul_cancel H1
... = gcd a b : int.div_mul_cancel H2))))
theorem gcd_dvd_gcd_mul_left (a b c : ℤ) : gcd a b ∣ gcd (c * a) b :=
dvd_gcd (dvd.trans !gcd_dvd_left !dvd_mul_left) !gcd_dvd_right
@ -138,7 +138,7 @@ theorem gcd_dvd_gcd_mul_right (a b c : ℤ) : gcd a b ∣ gcd (a * c) b :=
theorem div_gcd_eq_div_gcd_of_nonneg {a₁ b₁ a₂ b₂ : ℤ} (H : a₁ * b₂ = a₂ * b₁)
(H1 : b₁ ≠ 0) (H2 : b₂ ≠ 0) (H3 : a₁ ≥ 0) (H4 : a₂ ≥ 0) :
a₁ div (gcd a₁ b₁) = a₂ div (gcd a₂ b₂) :=
a₁ / (gcd a₁ b₁) = a₂ / (gcd a₂ b₂) :=
begin
apply div_eq_div_of_dvd_of_dvd,
repeat (apply gcd_dvd_left),
@ -149,7 +149,7 @@ begin
end
theorem div_gcd_eq_div_gcd {a₁ b₁ a₂ b₂ : ℤ} (H : a₁ * b₂ = a₂ * b₁) (H1 : b₁ > 0) (H2 : b₂ > 0) :
a₁ div (gcd a₁ b₁) = a₂ div (gcd a₂ b₂) :=
a₁ / (gcd a₁ b₁) = a₂ / (gcd a₂ b₂) :=
or.elim (le_or_gt 0 a₁)
(assume H3 : a₁ ≥ 0,
have H4 : a₂ * b₁ ≥ 0, by rewrite -H; apply mul_nonneg H3 (le_of_lt H2),
@ -158,7 +158,7 @@ or.elim (le_or_gt 0 a₁)
(assume H3 : a₁ < 0,
have H4 : a₂ * b₁ < 0, by rewrite -H; apply mul_neg_of_neg_of_pos H3 H2,
assert H5 : a₂ < 0, from neg_of_mul_neg_right H4 (le_of_lt H1),
assert H6 : abs a₁ div (gcd (abs a₁) (abs b₁)) = abs a₂ div (gcd (abs a₂) (abs b₂)),
assert H6 : abs a₁ / (gcd (abs a₁) (abs b₁)) = abs a₂ / (gcd (abs a₂) (abs b₂)),
begin
apply div_gcd_eq_div_gcd_of_nonneg,
rewrite [abs_of_pos H1, abs_of_pos H2, abs_of_neg H3, abs_of_neg H5],
@ -167,17 +167,17 @@ or.elim (le_or_gt 0 a₁)
apply ne_of_gt (abs_pos_of_pos H2),
repeat (apply abs_nonneg)
end,
have H7 : -a₁ div (gcd a₁ b₁) = -a₂ div (gcd a₂ b₂),
have H7 : -a₁ / (gcd a₁ b₁) = -a₂ / (gcd a₂ b₂),
begin
rewrite [-abs_of_neg H3, -abs_of_neg H5, -gcd_abs_abs a₁],
rewrite [-gcd_abs_abs a₂ b₂],
exact H6
end,
calc
a₁ div (gcd a₁ b₁) = -(-a₁ div (gcd a₁ b₁)) :
a₁ / (gcd a₁ b₁) = -(-a₁ / (gcd a₁ b₁)) :
by rewrite [neg_div_of_dvd !gcd_dvd_left, neg_neg]
... = -(-a₂ div (gcd a₂ b₂)) : H7
... = a₂ div (gcd a₂ b₂) :
... = -(-a₂ / (gcd a₂ b₂)) : H7
... = a₂ / (gcd a₂ b₂) :
by rewrite [neg_div_of_dvd !gcd_dvd_left, neg_neg])
/- lcm -/
@ -283,11 +283,11 @@ theorem gcd_mul_right_cancel_of_coprime_right {c a : ℤ} (b : ℤ) (H : coprime
!gcd.comm ▸ !gcd.comm ▸ !gcd_mul_right_cancel_of_coprime H
theorem coprime_div_gcd_div_gcd {a b : ℤ} (H : gcd a b ≠ 0) :
coprime (a div gcd a b) (b div gcd a b) :=
coprime (a / gcd a b) (b / gcd a b) :=
calc
gcd (a div gcd a b) (b div gcd a b)
= gcd a b div abs (gcd a b) : gcd_div !gcd_dvd_left !gcd_dvd_right
... = 1 : by rewrite [abs_of_nonneg !gcd_nonneg, int.div_self H]
gcd (a / gcd a b) (b / gcd a b)
= gcd a b / abs (gcd a b) : gcd_div !gcd_dvd_left !gcd_dvd_right
... = 1 : by rewrite [abs_of_nonneg !gcd_nonneg, int.div_self H]
theorem not_coprime_of_dvd_of_dvd {m n d : ℤ} (dgt1 : d > 1) (Hm : d ∣ m) (Hn : d ∣ n) :
¬ coprime m n :=
@ -299,8 +299,8 @@ show false, from not_lt_of_ge `d ≤ 1` `d > 1`
theorem exists_coprime {a b : ℤ} (H : gcd a b ≠ 0) :
exists a' b', coprime a' b' ∧ a = a' * gcd a b ∧ b = b' * gcd a b :=
have H1 : a = (a div gcd a b) * gcd a b, from (int.div_mul_cancel !gcd_dvd_left)⁻¹,
have H2 : b = (b div gcd a b) * gcd a b, from (int.div_mul_cancel !gcd_dvd_right)⁻¹,
have H1 : a = (a / gcd a b) * gcd a b, from (int.div_mul_cancel !gcd_dvd_left)⁻¹,
have H2 : b = (b / gcd a b) * gcd a b, from (int.div_mul_cancel !gcd_dvd_right)⁻¹,
exists.intro _ (exists.intro _ (and.intro (coprime_div_gcd_div_gcd H) (and.intro H1 H2)))
theorem coprime_mul {a b c : ℤ} (H1 : coprime a c) (H2 : coprime b c) : coprime (a * b) c :=
@ -336,16 +336,16 @@ decidable.by_cases
exists.intro _ (exists.intro _ (and.intro `c = 0 * b` (and.intro `0 ∣ a` `b ∣ b`))))
(suppose gcd c a ≠ 0,
have gcd c a ∣ c, from !gcd_dvd_left,
have H3 : c div gcd c a ∣ (a * b) div gcd c a, from div_dvd_div this H,
have H4 : (a * b) div gcd c a = (a div gcd c a) * b, from
have H3 : c / gcd c a ∣ (a * b) / gcd c a, from div_dvd_div this H,
have H4 : (a * b) / gcd c a = (a / gcd c a) * b, from
calc
a * b div gcd c a = b * a div gcd c a : mul.comm
... = b * (a div gcd c a) : !int.mul_div_assoc !gcd_dvd_right
... = a div gcd c a * b : mul.comm,
have H5 : c div gcd c a ∣ (a div gcd c a) * b, from H4 ▸ H3,
have H6 : coprime (c div gcd c a) (a div gcd c a), from coprime_div_gcd_div_gcd `gcd c a ≠ 0`,
have H7 : c div gcd c a ∣ b, from dvd_of_coprime_of_dvd_mul_left H6 H5,
have H8 : c = gcd c a * (c div gcd c a), from (int.mul_div_cancel' `gcd c a ∣ c`)⁻¹,
a * b / gcd c a = b * a / gcd c a : mul.comm
... = b * (a / gcd c a) : !int.mul_div_assoc !gcd_dvd_right
... = a / gcd c a * b : mul.comm,
have H5 : c / gcd c a ∣ (a / gcd c a) * b, from H4 ▸ H3,
have H6 : coprime (c / gcd c a) (a / gcd c a), from coprime_div_gcd_div_gcd `gcd c a ≠ 0`,
have H7 : c / gcd c a ∣ b, from dvd_of_coprime_of_dvd_mul_left H6 H5,
have H8 : c = gcd c a * (c / gcd c a), from (int.mul_div_cancel' `gcd c a ∣ c`)⁻¹,
exists.intro _ (exists.intro _ (and.intro H8 (and.intro !gcd_dvd_right H7))))
end int

470
library/data/nat/div.lean
View file

@ -20,58 +20,58 @@ and.rec (λ ypos ylex, sub_lt (lt_of_lt_of_le ypos ylex) ypos)
private definition div.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 + 1 else zero
protected definition divide := fix div.F
protected definition div := fix div.F
definition nat_has_divide [reducible] [instance] [priority nat.prio] : has_divide nat :=
has_divide.mk nat.divide
definition nat_has_divide [reducible] [instance] [priority nat.prio] : has_div nat :=
has_div.mk nat.div
theorem divide_def (x y : nat) : divide x y = if 0 < y ∧ y ≤ x then divide (x - y) y + 1 else 0 :=
theorem div_def (x y : nat) : div x y = if 0 < y ∧ y ≤ x then div (x - y) y + 1 else 0 :=
congr_fun (fix_eq div.F x) y
protected theorem div_zero (a : ℕ) : a div 0 = 0 :=
divide_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0))
protected theorem div_zero (a : ℕ) : a / 0 = 0 :=
div_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0))
theorem div_eq_zero_of_lt {a b : ℕ} (h : a < b) : a div b = 0 :=
divide_def a b ⬝ if_neg (!not_and_of_not_right (not_le_of_gt h))
theorem div_eq_zero_of_lt {a b : ℕ} (h : a < b) : a / b = 0 :=
div_def a b ⬝ if_neg (!not_and_of_not_right (not_le_of_gt h))
protected theorem zero_div (b : ℕ) : 0 div b = 0 :=
divide_def 0 b ⬝ if_neg (and.rec not_le_of_gt)
protected theorem zero_div (b : ℕ) : 0 / b = 0 :=
div_def 0 b ⬝ if_neg (and.rec not_le_of_gt)
theorem div_eq_succ_sub_div {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a div b = succ ((a - b) div b) :=
divide_def a b ⬝ if_pos (and.intro h₁ h₂)
theorem div_eq_succ_sub_div {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a / b = succ ((a - b) / b) :=
div_def a b ⬝ if_pos (and.intro h₁ h₂)
theorem add_div_self (x : ℕ) {z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) :=
theorem add_div_self (x : ℕ) {z : ℕ} (H : z > 0) : (x + z) / z = succ (x / z) :=
calc
(x + z) div z = if 0 < z ∧ z ≤ x + z then (x + z - z) div z + 1 else 0 : !divide_def
... = (x + z - z) div z + 1 : if_pos (and.intro H (le_add_left z x))
... = succ (x div z) : {!nat.add_sub_cancel}
(x + z) / z = if 0 < z ∧ z ≤ x + z then (x + z - z) / z + 1 else 0 : !div_def
... = (x + z - z) / z + 1 : if_pos (and.intro H (le_add_left z x))
... = succ (x / z) : {!nat.add_sub_cancel}
theorem add_div_self_left {x : ℕ} (z : ℕ) (H : x > 0) : (x + z) div x = succ (z div x) :=
theorem add_div_self_left {x : ℕ} (z : ℕ) (H : x > 0) : (x + z) / x = succ (z / x) :=
!add.comm ▸ !add_div_self H
theorem add_mul_div_self {x y z : ℕ} (H : z > 0) : (x + y * z) div z = x div z + y :=
theorem add_mul_div_self {x y z : ℕ} (H : z > 0) : (x + y * z) / z = x / z + y :=
nat.induction_on y
(calc (x + 0 * z) div z = (x + 0) div z : zero_mul
... = x div z : add_zero
... = x div z + 0 : add_zero)
(calc (x + 0 * z) / z = (x + 0) / z : zero_mul
... = x / z : add_zero
... = x / z + 0 : add_zero)
(take y,
assume IH : (x + y * z) div z = x div z + y, calc
(x + succ y * z) div z = (x + (y * z + z)) div z : succ_mul
... = (x + y * z + z) div z : add.assoc
... = succ ((x + y * z) div z) : !add_div_self H
... = succ (x div z + y) : IH)
assume IH : (x + y * z) / z = x / z + y, calc
(x + succ y * z) / z = (x + (y * z + z)) / z : succ_mul
... = (x + y * z + z) / z : add.assoc
... = succ ((x + y * z) / z) : !add_div_self H
... = succ (x / z + y) : IH)
theorem add_mul_div_self_left (x z : ℕ) {y : ℕ} (H : y > 0) : (x + y * z) div y = x div y + z :=
theorem add_mul_div_self_left (x z : ℕ) {y : ℕ} (H : y > 0) : (x + y * z) / y = x / y + z :=
!mul.comm ▸ add_mul_div_self H
protected theorem mul_div_cancel (m : ℕ) {n : ℕ} (H : n > 0) : m * n div n = m :=
protected theorem mul_div_cancel (m : ℕ) {n : ℕ} (H : n > 0) : m * n / n = m :=
calc
m * n div n = (0 + m * n) div n : zero_add
... = 0 div n + m : add_mul_div_self H
... = 0 + m : nat.zero_div
... = m : zero_add
m * n / n = (0 + m * n) / n : zero_add
... = 0 / n + m : add_mul_div_self H
... = 0 + m : nat.zero_div
... = m : zero_add
protected theorem mul_div_cancel_left {m : ℕ} (n : ℕ) (H : m > 0) : m * n div m = n :=
protected theorem mul_div_cancel_left {m : ℕ} (n : ℕ) (H : m > 0) : m * n / m = n :=
!mul.comm ▸ !nat.mul_div_cancel H
/- mod -/
@ -79,147 +79,147 @@ protected theorem mul_div_cancel_left {m : ℕ} (n : ℕ) (H : m > 0) : m * n di
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
protected definition modulo := fix mod.F
protected definition mod := fix mod.F
definition nat_has_modulo [reducible] [instance] [priority nat.prio] : has_modulo nat :=
has_modulo.mk nat.modulo
definition nat_has_mod [reducible] [instance] [priority nat.prio] : has_mod nat :=
has_mod.mk nat.mod
notation [priority nat.prio] a ≡ b `[mod `:0 c:0 `]` := a mod c = b mod c
notation [priority nat.prio] a ≡ b `[mod `:0 c:0 `]` := a % c = b % c
theorem modulo_def (x y : nat) : modulo x y = if 0 < y ∧ y ≤ x then modulo (x - y) y else x :=
theorem mod_def (x y : nat) : mod x y = if 0 < y ∧ y ≤ x then mod (x - y) y else x :=
congr_fun (fix_eq mod.F x) y
theorem mod_zero (a : ℕ) : a mod 0 = a :=
modulo_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0))
theorem mod_zero (a : ℕ) : a % 0 = a :=
mod_def a 0 ⬝ if_neg (!not_and_of_not_left (lt.irrefl 0))
theorem mod_eq_of_lt {a b : ℕ} (h : a < b) : a mod b = a :=
modulo_def a b ⬝ if_neg (!not_and_of_not_right (not_le_of_gt h))
theorem mod_eq_of_lt {a b : ℕ} (h : a < b) : a % b = a :=
mod_def a b ⬝ if_neg (!not_and_of_not_right (not_le_of_gt h))
theorem zero_mod (b : ℕ) : 0 mod b = 0 :=
modulo_def 0 b ⬝ if_neg (λ h, and.rec_on h (λ l r, absurd (lt_of_lt_of_le l r) (lt.irrefl 0)))
theorem zero_mod (b : ℕ) : 0 % b = 0 :=
mod_def 0 b ⬝ if_neg (λ h, and.rec_on h (λ l r, absurd (lt_of_lt_of_le l r) (lt.irrefl 0)))
theorem mod_eq_sub_mod {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a mod b = (a - b) mod b :=
modulo_def a b ⬝ if_pos (and.intro h₁ h₂)
theorem mod_eq_sub_mod {a b : ℕ} (h₁ : b > 0) (h₂ : a ≥ b) : a % b = (a - b) % b :=
mod_def a b ⬝ if_pos (and.intro h₁ h₂)
theorem add_mod_self (x z : ℕ) : (x + z) mod z = x mod z :=
theorem add_mod_self (x z : ℕ) : (x + z) % z = x % z :=
by_cases_zero_pos z
(by rewrite add_zero)
(take z, assume H : z > 0,
calc
(x + z) mod z = if 0 < z ∧ z ≤ x + z then (x + z - z) mod z else _ : modulo_def
... = (x + z - z) mod z : if_pos (and.intro H (le_add_left z x))
... = x mod z : nat.add_sub_cancel)
(x + z) % z = if 0 < z ∧ z ≤ x + z then (x + z - z) % z else _ : mod_def
... = (x + z - z) % z : if_pos (and.intro H (le_add_left z x))
... = x % z : nat.add_sub_cancel)
theorem add_mod_self_left (x z : ℕ) : (x + z) mod x = z mod x :=
theorem add_mod_self_left (x z : ℕ) : (x + z) % x = z % x :=
!add.comm ▸ !add_mod_self
theorem add_mul_mod_self (x y z : ℕ) : (x + y * z) mod z = x mod z :=
theorem add_mul_mod_self (x y z : ℕ) : (x + y * z) % z = x % z :=
nat.induction_on y
(calc (x + 0 * z) mod z = (x + 0) mod z : zero_mul
... = x mod z : add_zero)
(calc (x + 0 * z) % z = (x + 0) % z : zero_mul
... = x % z : add_zero)
(take y,
assume IH : (x + y * z) mod z = x mod z,
assume IH : (x + y * z) % z = x % z,
calc
(x + succ y * z) mod z = (x + (y * z + z)) mod z : succ_mul
... = (x + y * z + z) mod z : add.assoc
... = (x + y * z) mod z : !add_mod_self
... = x mod z : IH)
(x + succ y * z) % z = (x + (y * z + z)) % z : succ_mul
... = (x + y * z + z) % z : add.assoc
... = (x + y * z) % z : !add_mod_self
... = x % z : IH)
theorem add_mul_mod_self_left (x y z : ℕ) : (x + y * z) mod y = x mod y :=
theorem add_mul_mod_self_left (x y z : ℕ) : (x + y * z) % y = x % y :=
!mul.comm ▸ !add_mul_mod_self
theorem mul_mod_left (m n : ℕ) : (m * n) mod n = 0 :=
theorem mul_mod_left (m n : ℕ) : (m * n) % n = 0 :=
by rewrite [-zero_add (m * n), add_mul_mod_self, zero_mod]
theorem mul_mod_right (m n : ℕ) : (m * n) mod m = 0 :=
theorem mul_mod_right (m n : ℕ) : (m * n) % m = 0 :=
!mul.comm ▸ !mul_mod_left
theorem mod_lt (x : ℕ) {y : ℕ} (H : y > 0) : x mod y < y :=
theorem mod_lt (x : ℕ) {y : ℕ} (H : y > 0) : x % y < y :=
nat.case_strong_induction_on x
(show 0 mod y < y, from !zero_mod⁻¹ ▸ H)
(show 0 % y < y, from !zero_mod⁻¹ ▸ H)
(take x,
assume IH : ∀x', x' ≤ x → x' mod y < y,
show succ x mod y < y, from
assume IH : ∀x', x' ≤ x → x' % y < y,
show succ x % y < y, from
by_cases -- (succ x < y)
(assume H1 : succ x < y,
have succ x mod y = succ x, from mod_eq_of_lt H1,
show succ x mod y < y, from this⁻¹ ▸ H1)
have succ x % y = succ x, from mod_eq_of_lt H1,
show succ x % y < y, from this⁻¹ ▸ H1)
(assume H1 : ¬ succ x < y,
have y ≤ succ x, from le_of_not_gt H1,
have h : succ x mod y = (succ x - y) mod y, from mod_eq_sub_mod H this,
have h : succ x % y = (succ x - y) % y, from mod_eq_sub_mod H this,
have succ x - y < succ x, from sub_lt !succ_pos H,
have succ x - y ≤ x, from le_of_lt_succ this,
show succ x mod y < y, from h⁻¹ ▸ IH _ this))
show succ x % y < y, from h⁻¹ ▸ IH _ this))
theorem mod_one (n : ℕ) : n mod 1 = 0 :=
have H1 : n mod 1 < 1, from !mod_lt !succ_pos,
theorem mod_one (n : ℕ) : n % 1 = 0 :=
have H1 : n % 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
theorem eq_div_mul_add_mod (x y : ℕ) : x = x div y * y + x mod y :=
-- the quotient - remainder theorem
theorem eq_div_mul_add_mod (x y : ℕ) : x = x / y * y + x % y :=
begin
eapply by_cases_zero_pos y,
show x = x div 0 * 0 + x mod 0, from
show x = x / 0 * 0 + x % 0, from
(calc
x div 0 * 0 + x mod 0 = 0 + x mod 0 : mul_zero
... = x mod 0 : zero_add
... = x : mod_zero)⁻¹,
x / 0 * 0 + x % 0 = 0 + x % 0 : mul_zero
... = x % 0 : zero_add
... = x : mod_zero)⁻¹,
intro y H,
show x = x div y * y + x mod y,
show x = x / y * y + x % y,
begin
eapply nat.case_strong_induction_on x,
show 0 = (0 div y) * y + 0 mod y, by rewrite [zero_mod, add_zero, nat.zero_div, zero_mul],
show 0 = (0 / y) * y + 0 % y, by rewrite [zero_mod, add_zero, nat.zero_div, zero_mul],
intro x IH,
show succ x = succ x div y * y + succ x mod y, from
show succ x = succ x / y * y + succ x % y, from
if H1 : succ x < y then
assert H2 : succ x div y = 0, from div_eq_zero_of_lt H1,
assert H3 : succ x mod y = succ x, from mod_eq_of_lt H1,
assert H2 : succ x / y = 0, from div_eq_zero_of_lt H1,
assert H3 : succ x % y = succ x, from mod_eq_of_lt H1,
begin rewrite [H2, H3, zero_mul, zero_add] end
else
have H2 : y ≤ succ x, from le_of_not_gt H1,
assert H3 : succ x div y = succ ((succ x - y) div y), from div_eq_succ_sub_div H H2,
assert H4 : succ x mod y = (succ x - y) mod y, from mod_eq_sub_mod H H2,
assert H3 : succ x / y = succ ((succ x - y) / y), from div_eq_succ_sub_div H H2,
assert H4 : succ x % y = (succ x - y) % y, from mod_eq_sub_mod H H2,
have H5 : succ x - y < succ x, from sub_lt !succ_pos H,
assert H6 : succ x - y ≤ x, from le_of_lt_succ H5,
(calc
succ x div y * y + succ x mod y =
succ ((succ x - y) div y) * y + succ x mod y : by rewrite H3
... = ((succ x - y) div y) * y + y + succ x mod y : by rewrite succ_mul
... = ((succ x - y) div y) * y + y + (succ x - y) mod y : by rewrite H4
... = ((succ x - y) div y) * y + (succ x - y) mod y + y : add.right_comm
... = succ x - y + y : by rewrite -(IH _ H6)
... = succ x : nat.sub_add_cancel H2)⁻¹
succ x / y * y + succ x % y =
succ ((succ x - y) / y) * y + succ x % y : by rewrite H3
... = ((succ x - y) / y) * y + y + succ x % y : by rewrite succ_mul
... = ((succ x - y) / y) * y + y + (succ x - y) % y : by rewrite H4
... = ((succ x - y) / y) * y + (succ x - y) % y + y : add.right_comm
... = succ x - y + y : by rewrite -(IH _ H6)
... = succ x : nat.sub_add_cancel H2)⁻¹
end
end
theorem mod_eq_sub_div_mul (x y : ℕ) : x mod y = x - x div y * y :=
theorem mod_eq_sub_div_mul (x y : ℕ) : x % y = x - x / y * y :=
nat.eq_sub_of_add_eq (!add.comm ▸ !eq_div_mul_add_mod)⁻¹
theorem mod_add_mod (m n k : ℕ) : (m mod n + k) mod n = (m + k) mod n :=
by rewrite [eq_div_mul_add_mod m n at {2}, add.assoc, add.comm (m div n * n), add_mul_mod_self]
theorem mod_add_mod (m n k : ℕ) : (m % n + k) % n = (m + k) % n :=
by rewrite [eq_div_mul_add_mod m n at {2}, add.assoc, add.comm (m / n * n), add_mul_mod_self]
theorem add_mod_mod (m n k : ℕ) : (m + n mod k) mod k = (m + n) mod k :=
theorem add_mod_mod (m n k : ℕ) : (m + n % k) % k = (m + n) % k :=
by rewrite [add.comm, mod_add_mod, add.comm]
theorem add_mod_eq_add_mod_right {m n k : ℕ} (i : ℕ) (H : m mod n = k mod n) :
(m + i) mod n = (k + i) mod n :=
theorem add_mod_eq_add_mod_right {m n k : ℕ} (i : ℕ) (H : m % n = k % n) :
(m + i) % n = (k + i) % n :=
by rewrite [-mod_add_mod, -mod_add_mod k, H]
theorem add_mod_eq_add_mod_left {m n k : ℕ} (i : ℕ) (H : m mod n = k mod n) :
(i + m) mod n = (i + k) mod n :=
theorem add_mod_eq_add_mod_left {m n k : ℕ} (i : ℕ) (H : m % n = k % n) :
(i + m) % n = (i + k) % n :=
by rewrite [add.comm, add_mod_eq_add_mod_right _ H, add.comm]
theorem mod_eq_mod_of_add_mod_eq_add_mod_right {m n k i : ℕ} :
(m + i) mod n = (k + i) mod n → m mod n = k mod n :=
(m + i) % n = (k + i) % n → m % n = k % n :=
by_cases_zero_pos n
(by rewrite [*mod_zero]; apply eq_of_add_eq_add_right)
(take n,
assume npos : n > 0,
assume H1 : (m + i) mod n = (k + i) mod n,
have H2 : (m + i mod n) mod n = (k + i mod n) mod n, by rewrite [*add_mod_mod, H1],
assert H3 : (m + i mod n + (n - i mod n)) mod n = (k + i mod n + (n - i mod n)) mod n,
assume H1 : (m + i) % n = (k + i) % n,
have H2 : (m + i % n) % n = (k + i % n) % n, by rewrite [*add_mod_mod, H1],
assert H3 : (m + i % n + (n - i % n)) % n = (k + i % n + (n - i % n)) % n,
from add_mod_eq_add_mod_right _ H2,
begin
revert H3,
@ -228,21 +228,21 @@ by_cases_zero_pos n
end)
theorem mod_eq_mod_of_add_mod_eq_add_mod_left {m n k i : ℕ} :
(i + m) mod n = (i + k) mod n → m mod n = k mod n :=
(i + m) % n = (i + k) % n → m % n = k % n :=
by rewrite [add.comm i m, add.comm i k]; apply mod_eq_mod_of_add_mod_eq_add_mod_right
theorem mod_le {x y : ℕ} : x mod y ≤ x :=
theorem mod_le {x y : ℕ} : x % y ≤ x :=
!eq_div_mul_add_mod⁻¹ ▸ !le_add_left
theorem eq_remainder {q1 r1 q2 r2 y : ℕ} (H1 : r1 < y) (H2 : r2 < y)
(H3 : q1 * y + r1 = q2 * y + r2) : r1 = r2 :=
calc
r1 = r1 mod y : mod_eq_of_lt H1
... = (r1 + q1 * y) mod y : !add_mul_mod_self⁻¹
... = (q1 * y + r1) mod y : add.comm
... = (r2 + q2 * y) mod y : by rewrite [H3, add.comm]
... = r2 mod y : !add_mul_mod_self
... = r2 : mod_eq_of_lt H2
r1 = r1 % y : mod_eq_of_lt H1
... = (r1 + q1 * y) % y : !add_mul_mod_self⁻¹
... = (q1 * y + r1) % y : add.comm
... = (r2 + q2 * y) % y : by rewrite [H3, add.comm]
... = r2 % y : !add_mul_mod_self
... = r2 : mod_eq_of_lt H2
theorem eq_quotient {q1 r1 q2 r2 y : ℕ} (H1 : r1 < y) (H2 : r2 < y)
(H3 : q1 * y + r1 = q2 * y + r2) : q1 = q2 :=
@ -252,92 +252,92 @@ have H6 : y > 0, from lt_of_le_of_lt !zero_le H1,
show q1 = q2, from eq_of_mul_eq_mul_right H6 H5
protected theorem mul_div_mul_left {z : ℕ} (x y : ℕ) (zpos : z > 0) :
(z * x) div (z * y) = x div y :=
(z * x) / (z * y) = x / y :=
if H : y = 0 then
by rewrite [H, mul_zero, *nat.div_zero]
else
have ypos : y > 0, from pos_of_ne_zero H,
have zypos : z * y > 0, from mul_pos zpos ypos,
have H1 : (z * x) mod (z * y) < z * y, from !mod_lt zypos,
have H2 : z * (x mod y) < z * y, from mul_lt_mul_of_pos_left (!mod_lt ypos) zpos,
have H1 : (z * x) % (z * y) < z * y, from !mod_lt zypos,
have H2 : z * (x % y) < z * y, from mul_lt_mul_of_pos_left (!mod_lt ypos) zpos,
eq_quotient H1 H2
(calc
((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : eq_div_mul_add_mod
... = z * (x div y * y + x mod y) : eq_div_mul_add_mod
... = z * (x div y * y) + z * (x mod y) : left_distrib
... = (x div y) * (z * y) + z * (x mod y) : mul.left_comm)
((z * x) / (z * y)) * (z * y) + (z * x) % (z * y) = z * x : eq_div_mul_add_mod
... = z * (x / y * y + x % y) : eq_div_mul_add_mod
... = z * (x / y * y) + z * (x % y) : left_distrib
... = (x / y) * (z * y) + z * (x % y) : mul.left_comm)
protected theorem mul_div_mul_right {x z y : ℕ} (zpos : z > 0) : (x * z) div (y * z) = x div y :=
protected theorem mul_div_mul_right {x z y : ℕ} (zpos : z > 0) : (x * z) / (y * z) = x / y :=
!mul.comm ▸ !mul.comm ▸ !nat.mul_div_mul_left zpos
theorem mul_mod_mul_left (z x y : ℕ) : (z * x) mod (z * y) = z * (x mod y) :=
theorem mul_mod_mul_left (z x y : ℕ) : (z * x) % (z * y) = z * (x % y) :=
or.elim (eq_zero_or_pos z)
(assume H : z = 0, H⁻¹ ▸ calc
(0 * x) mod (z * y) = 0 mod (z * y) : zero_mul
... = 0 : zero_mod
... = 0 * (x mod y) : zero_mul)
(0 * x) % (z * y) = 0 % (z * y) : zero_mul
... = 0 : zero_mod
... = 0 * (x % y) : zero_mul)
(assume zpos : z > 0,
or.elim (eq_zero_or_pos y)
(assume H : y = 0, by rewrite [H, mul_zero, *mod_zero])
(assume ypos : y > 0,
have zypos : z * y > 0, from mul_pos zpos ypos,
have H1 : (z * x) mod (z * y) < z * y, from !mod_lt zypos,
have H2 : z * (x mod y) < z * y, from mul_lt_mul_of_pos_left (!mod_lt ypos) zpos,
have H1 : (z * x) % (z * y) < z * y, from !mod_lt zypos,
have H2 : z * (x % y) < z * y, from mul_lt_mul_of_pos_left (!mod_lt ypos) zpos,
eq_remainder H1 H2
(calc
((z * x) div (z * y)) * (z * y) + (z * x) mod (z * y) = z * x : eq_div_mul_add_mod
... = z * (x div y * y + x mod y) : eq_div_mul_add_mod
... = z * (x div y * y) + z * (x mod y) : left_distrib
... = (x div y) * (z * y) + z * (x mod y) : mul.left_comm)))
((z * x) / (z * y)) * (z * y) + (z * x) % (z * y) = z * x : eq_div_mul_add_mod
... = z * (x / y * y + x % y) : eq_div_mul_add_mod
... = z * (x / y * y) + z * (x % y) : left_distrib
... = (x / y) * (z * y) + z * (x % y) : mul.left_comm)))
theorem mul_mod_mul_right (x z y : ℕ) : (x * z) mod (y * z) = (x mod y) * z :=
theorem mul_mod_mul_right (x z y : ℕ) : (x * z) % (y * z) = (x % y) * z :=
mul.comm z x ▸ mul.comm z y ▸ !mul.comm ▸ !mul_mod_mul_left
theorem mod_self (n : ℕ) : n mod n = 0 :=
theorem mod_self (n : ℕ) : n % n = 0 :=
nat.cases_on n (by rewrite zero_mod)
(take n, by rewrite [-zero_add (succ n) at {1}, add_mod_self])
theorem mul_mod_eq_mod_mul_mod (m n k : nat) : (m * n) mod k = ((m mod k) * n) mod k :=
theorem mul_mod_eq_mod_mul_mod (m n k : nat) : (m * n) % k = ((m % k) * n) % k :=
calc
(m * n) mod k = (((m div k) * k + m mod k) * n) mod k : eq_div_mul_add_mod
... = ((m mod k) * n) mod k :
(m * n) % k = (((m / k) * k + m % k) * n) % k : eq_div_mul_add_mod
... = ((m % k) * n) % k :
by rewrite [right_distrib, mul.right_comm, add.comm, add_mul_mod_self]
theorem mul_mod_eq_mul_mod_mod (m n k : nat) : (m * n) mod k = (m * (n mod k)) mod k :=
theorem mul_mod_eq_mul_mod_mod (m n k : nat) : (m * n) % k = (m * (n % k)) % k :=
!mul.comm ▸ !mul.comm ▸ !mul_mod_eq_mod_mul_mod
protected theorem div_one (n : ℕ) : n div 1 = n :=
assert n div 1 * 1 + n mod 1 = n, from !eq_div_mul_add_mod⁻¹,
protected theorem div_one (n : ℕ) : n / 1 = n :=
assert n / 1 * 1 + n % 1 = n, from !eq_div_mul_add_mod⁻¹,
begin rewrite [-this at {2}, mul_one, mod_one] end
protected theorem div_self {n : ℕ} (H : n > 0) : n div n = 1 :=
assert (n * 1) div (n * 1) = 1 div 1, from !nat.mul_div_mul_left H,
protected theorem div_self {n : ℕ} (H : n > 0) : n / n = 1 :=
assert (n * 1) / (n * 1) = 1 / 1, from !nat.mul_div_mul_left H,
by rewrite [nat.div_one at this, -this, *mul_one]
theorem div_mul_cancel_of_mod_eq_zero {m n : ℕ} (H : m mod n = 0) : m div n * n = m :=
theorem div_mul_cancel_of_mod_eq_zero {m n : ℕ} (H : m % n = 0) : m / n * n = m :=
by rewrite [eq_div_mul_add_mod m n at {2}, H, add_zero]
theorem mul_div_cancel_of_mod_eq_zero {m n : ℕ} (H : m mod n = 0) : n * (m div n) = m :=
theorem mul_div_cancel_of_mod_eq_zero {m n : ℕ} (H : m % n = 0) : n * (m / n) = m :=
!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H
/- dvd -/
theorem dvd_of_mod_eq_zero {m n : ℕ} (H : n mod m = 0) : m ∣ n :=
theorem dvd_of_mod_eq_zero {m n : ℕ} (H : n % m = 0) : m ∣ n :=
dvd.intro (!mul.comm ▸ div_mul_cancel_of_mod_eq_zero H)
theorem mod_eq_zero_of_dvd {m n : ℕ} (H : m ∣ n) : n mod m = 0 :=
theorem mod_eq_zero_of_dvd {m n : ℕ} (H : m ∣ n) : n % m = 0 :=
dvd.elim H (take z, assume H1 : n = m * z, H1⁻¹ ▸ !mul_mod_right)
theorem dvd_iff_mod_eq_zero (m n : ℕ) : m ∣ n ↔ n mod m = 0 :=
theorem dvd_iff_mod_eq_zero (m n : ℕ) : m ∣ n ↔ n % m = 0 :=
iff.intro mod_eq_zero_of_dvd dvd_of_mod_eq_zero
definition dvd.decidable_rel [instance] : decidable_rel dvd :=
take m n, decidable_of_decidable_of_iff _ (iff.symm !dvd_iff_mod_eq_zero)
protected theorem div_mul_cancel {m n : ℕ} (H : n ∣ m) : m div n * n = m :=
protected theorem div_mul_cancel {m n : ℕ} (H : n ∣ m) : m / n * n = m :=
div_mul_cancel_of_mod_eq_zero (mod_eq_zero_of_dvd H)
protected theorem mul_div_cancel' {m n : ℕ} (H : n ∣ m) : n * (m div n) = m :=
protected theorem mul_div_cancel' {m n : ℕ} (H : n ∣ m) : n * (m / n) = m :=
!mul.comm ▸ nat.div_mul_cancel H
theorem dvd_of_dvd_add_left {m n₁ n₂ : ℕ} (H₁ : m ∣ n₁ + n₂) (H₂ : m ∣ n₁) : m ∣ n₂ :=
@ -376,21 +376,21 @@ by_cases_zero_pos n
have k = 1, from eq_one_of_mul_eq_one_left this,
show m = n, from (mul_one m)⁻¹ ⬝ (this ▸ Hk⁻¹))
protected theorem mul_div_assoc (m : ℕ) {n k : ℕ} (H : k ∣ n) : m * n div k = m * (n div k) :=
protected theorem mul_div_assoc (m : ℕ) {n k : ℕ} (H : k ∣ n) : m * n / k = m * (n / k) :=
or.elim (eq_zero_or_pos k)
(assume H1 : k = 0,
calc
m * n div k = m * n div 0 : H1
... = 0 : nat.div_zero
... = m * 0 : mul_zero m
... = m * (n div 0) : nat.div_zero
... = m * (n div k) : H1)
m * n / k = m * n / 0 : H1
... = 0 : nat.div_zero
... = m * 0 : mul_zero m
... = m * (n / 0) : nat.div_zero
... = m * (n / k) : H1)
(assume H1 : k > 0,
have H2 : n = n div k * k, from (nat.div_mul_cancel H)⁻¹,
have H2 : n = n / k * k, from (nat.div_mul_cancel H)⁻¹,
calc
m * n div k = m * (n div k * k) div k : H2
... = m * (n div k) * k div k : mul.assoc
... = m * (n div k) : nat.mul_div_cancel _ H1)
m * n / k = m * (n / k * k) / k : H2
... = m * (n / k) * k / k : mul.assoc
... = m * (n / k) : nat.mul_div_cancel _ H1)
theorem dvd_of_mul_dvd_mul_left {m n k : ℕ} (kpos : k > 0) (H : k * m ∣ k * n) : m ∣ n :=
dvd.elim H
@ -405,47 +405,47 @@ nat.dvd_of_mul_dvd_mul_left kpos (!mul.comm ▸ !mul.comm ▸ H)
lemma dvd_of_eq_mul (i j n : nat) : n = j*i → j ∣ n :=
begin intros, subst n, apply dvd_mul_right end
theorem div_dvd_div {k m n : ℕ} (H1 : k ∣ m) (H2 : m ∣ n) : m div k ∣ n div k :=
have H3 : m = m div k * k, from (nat.div_mul_cancel H1)⁻¹,
have H4 : n = n div k * k, from (nat.div_mul_cancel (dvd.trans H1 H2))⁻¹,
theorem div_dvd_div {k m n : ℕ} (H1 : k ∣ m) (H2 : m ∣ n) : m / k ∣ n / k :=
have H3 : m = m / k * k, from (nat.div_mul_cancel H1)⁻¹,
have H4 : n = n / k * k, from (nat.div_mul_cancel (dvd.trans H1 H2))⁻¹,
or.elim (eq_zero_or_pos k)
(assume H5 : k = 0,
have H6: n div k = 0, from (congr_arg _ H5 ⬝ !nat.div_zero),
have H6: n / k = 0, from (congr_arg _ H5 ⬝ !nat.div_zero),
H6⁻¹ ▸ !dvd_zero)
(assume H5 : k > 0,
nat.dvd_of_mul_dvd_mul_right H5 (H3 ▸ H4 ▸ H2))
protected theorem div_eq_iff_eq_mul_right {m n : ℕ} (k : ℕ) (H : n > 0) (H' : n ∣ m) :
m div n = k ↔ m = n * k :=
m / n = k ↔ m = n * k :=
iff.intro
(assume H1, by rewrite [-H1, nat.mul_div_cancel' H'])
(assume H1, by rewrite [H1, !nat.mul_div_cancel_left H])
protected theorem div_eq_iff_eq_mul_left {m n : ℕ} (k : ℕ) (H : n > 0) (H' : n ∣ m) :
m div n = k ↔ m = k * n :=
m / n = k ↔ m = k * n :=
!mul.comm ▸ !nat.div_eq_iff_eq_mul_right H H'
protected theorem eq_mul_of_div_eq_right {m n k : ℕ} (H1 : n ∣ m) (H2 : m div n = k) :
protected theorem eq_mul_of_div_eq_right {m n k : ℕ} (H1 : n ∣ m) (H2 : m / n = k) :
m = n * k :=
calc
m = n * (m div n) : nat.mul_div_cancel' H1
... = n * k : H2
m = n * (m / n) : nat.mul_div_cancel' H1
... = n * k : H2
protected theorem div_eq_of_eq_mul_right {m n k : ℕ} (H1 : n > 0) (H2 : m = n * k) :
m div n = k :=
m / n = k :=
calc
m div n = n * k div n : H2
... = k : !nat.mul_div_cancel_left H1
m / n = n * k / n : H2
... = k : !nat.mul_div_cancel_left H1
protected theorem eq_mul_of_div_eq_left {m n k : ℕ} (H1 : n ∣ m) (H2 : m div n = k) :
protected theorem eq_mul_of_div_eq_left {m n k : ℕ} (H1 : n ∣ m) (H2 : m / n = k) :
m = k * n :=
!mul.comm ▸ !nat.eq_mul_of_div_eq_right H1 H2
protected theorem div_eq_of_eq_mul_left {m n k : ℕ} (H1 : n > 0) (H2 : m = k * n) :
m div n = k :=
m / n = k :=
!nat.div_eq_of_eq_mul_right H1 (!mul.comm ▸ H2)
lemma add_mod_eq_of_dvd (i j n : nat) : n ∣ j → (i + j) mod n = i mod n :=
lemma add_mod_eq_of_dvd (i j n : nat) : n ∣ j → (i + j) % n = i % n :=
assume h,
obtain k (hk : j = n * k), from exists_eq_mul_right_of_dvd h,
begin
@ -453,39 +453,39 @@ begin
apply add_mul_mod_self
end
/- div and ordering -/
/- / and ordering -/
lemma le_of_dvd {m n : nat} : n > 0 → m ∣ n → m ≤ n :=
assume (h₁ : n > 0) (h₂ : m ∣ n),
assert h₃ : n mod m = 0, from mod_eq_zero_of_dvd h₂,
assert h₃ : n % m = 0, from mod_eq_zero_of_dvd h₂,
by_contradiction
(λ nle : ¬ m ≤ n,
have h₄ : m > n, from lt_of_not_ge nle,
assert h₅ : n mod m = n, from mod_eq_of_lt h₄,
assert h₅ : n % m = n, from mod_eq_of_lt h₄,
begin
rewrite h₃ at h₅, subst n,
exact absurd h₁ (lt.irrefl 0)
end)
theorem div_mul_le (m n : ℕ) : m div n * n ≤ m :=
theorem div_mul_le (m n : ℕ) : m / n * n ≤ m :=
calc
m = m div n * n + m mod n : eq_div_mul_add_mod
... ≥ m div n * n : le_add_right
m = m / n * n + m % n : eq_div_mul_add_mod
... ≥ m / n * n : le_add_right
protected theorem div_le_of_le_mul {m n k : ℕ} (H : m ≤ n * k) : m div k ≤ n :=
protected theorem div_le_of_le_mul {m n k : ℕ} (H : m ≤ n * k) : m / k ≤ n :=
or.elim (eq_zero_or_pos k)
(assume H1 : k = 0,
calc
m div k = m div 0 : H1
m / k = m / 0 : H1
... = 0 : nat.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)
m / k * k ≤ m / k * k + m % k : le_add_right
... = m : eq_div_mul_add_mod
... ≤ n * k : H) H1)
protected theorem div_le_self (m n : ℕ) : m div n ≤ m :=
protected theorem div_le_self (m n : ℕ) : m / n ≤ m :=
nat.cases_on n (!nat.div_zero⁻¹ ▸ !zero_le)
take n,
have H : m ≤ m * succ n, from calc
@ -493,86 +493,86 @@ nat.cases_on n (!nat.div_zero⁻¹ ▸ !zero_le)
... ≤ m * succ n : !mul_le_mul_left (succ_le_succ !zero_le),
nat.div_le_of_le_mul H
protected theorem mul_le_of_le_div {m n k : ℕ} (H : m ≤ n div k) : m * k ≤ n :=
protected theorem mul_le_of_le_div {m n k : ℕ} (H : m ≤ n / k) : m * k ≤ n :=
calc
m * k ≤ n div k * k : !mul_le_mul_right H
m * k ≤ n / k * k : !mul_le_mul_right H
... ≤ n : div_mul_le
protected theorem le_div_of_mul_le {m n k : ℕ} (H1 : k > 0) (H2 : m * k ≤ n) : m ≤ n div k :=
have H3 : m * k < (succ (n div k)) * k, from
protected theorem le_div_of_mul_le {m n k : ℕ} (H1 : k > 0) (H2 : m * k ≤ n) : m ≤ n / k :=
have H3 : m * k < (succ (n / 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,
m * k ≤ n : H2
... = n / k * k + n % k : eq_div_mul_add_mod
... < n / k * k + k : add_lt_add_left (!mod_lt H1)
... = (succ (n / k)) * k : succ_mul,
le_of_lt_succ (lt_of_mul_lt_mul_right H3)
protected theorem le_div_iff_mul_le {m n k : ℕ} (H : k > 0) : m ≤ n div k ↔ m * k ≤ n :=
protected theorem le_div_iff_mul_le {m n k : ℕ} (H : k > 0) : m ≤ n / k ↔ m * k ≤ n :=
iff.intro !nat.mul_le_of_le_div (!nat.le_div_of_mul_le H)
protected theorem div_le_div {m n : ℕ} (k : ℕ) (H : m ≤ n) : m div k ≤ n div k :=
protected theorem div_le_div {m n : ℕ} (k : ℕ) (H : m ≤ n) : m / k ≤ n / k :=
by_cases_zero_pos k
(by rewrite [*nat.div_zero])
(take k, assume H1 : k > 0, nat.le_div_of_mul_le H1 (le.trans !div_mul_le H))
protected theorem div_lt_of_lt_mul {m n k : ℕ} (H : m < n * k) : m div k < n :=
protected theorem div_lt_of_lt_mul {m n k : ℕ} (H : m < n * k) : m / 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)
m / k * k ≤ m / k * k + m % k : le_add_right
... = m : eq_div_mul_add_mod
... < n * k : H)
protected theorem lt_mul_of_div_lt {m n k : ℕ} (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,
protected theorem lt_mul_of_div_lt {m n k : ℕ} (H1 : k > 0) (H2 : m / k < n) : m < n * k :=
assert H3 : succ (m / k) * k ≤ n * k, from !mul_le_mul_right (succ_le_of_lt H2),
have H4 : m / 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
m = m / k * k + m % k : eq_div_mul_add_mod
... < m / k * k + k : add_lt_add_left (!mod_lt H1)
... ≤ n * k : H4
protected theorem div_lt_iff_lt_mul {m n k : ℕ} (H : k > 0) : m div k < n ↔ m < n * k :=
protected theorem div_lt_iff_lt_mul {m n k : ℕ} (H : k > 0) : m / k < n ↔ m < n * k :=
iff.intro (!nat.lt_mul_of_div_lt H) !nat.div_lt_of_lt_mul
protected theorem div_le_iff_le_mul_of_div {m n : ℕ} (k : ℕ) (H : n > 0) (H' : n ∣ m) :
m div n ≤ k ↔ m ≤ k * n :=
m / n ≤ k ↔ m ≤ k * n :=
by rewrite [propext (!le_iff_mul_le_mul_right H), !nat.div_mul_cancel H']
protected theorem le_mul_of_div_le_of_div {m n k : ℕ} (H1 : n > 0) (H2 : n ∣ m) (H3 : m div n ≤ k) :
protected theorem le_mul_of_div_le_of_div {m n k : ℕ} (H1 : n > 0) (H2 : n ∣ m) (H3 : m / n ≤ k) :
m ≤ k * n :=
iff.mp (!nat.div_le_iff_le_mul_of_div H1 H2) H3
-- needed for integer division
theorem mul_sub_div_of_lt {m n k : ℕ} (H : k < m * n) :
(m * n - (k + 1)) div m = n - k div m - 1 :=
(m * n - (k + 1)) / m = n - k / m - 1 :=
begin
have H1 : k div m < n, from nat.div_lt_of_lt_mul (!mul.comm ▸ H),
have H2 : n - k div m ≥ 1, from
have H1 : k / m < n, from nat.div_lt_of_lt_mul (!mul.comm ▸ H),
have H2 : n - k / m ≥ 1, from
nat.le_sub_of_add_le (calc
1 + k div m = succ (k div m) : add.comm
... ≤ n : succ_le_of_lt H1),
have H3 : n - k div m = n - k div m - 1 + 1, from (nat.sub_add_cancel H2)⁻¹,
1 + k / m = succ (k / m) : add.comm
... ≤ n : succ_le_of_lt H1),
have H3 : n - k / m = n - k / m - 1 + 1, from (nat.sub_add_cancel H2)⁻¹,
have H4 : m > 0, from pos_of_ne_zero (assume H': m = 0, not_lt_zero k (begin rewrite [H' at H, zero_mul at H], exact H end)),
have H5 : k mod m + 1 ≤ m, from succ_le_of_lt (!mod_lt H4),
have H6 : m - (k mod m + 1) < m, from nat.sub_lt_self H4 !succ_pos,
have H5 : k % m + 1 ≤ m, from succ_le_of_lt (!mod_lt H4),
have H6 : m - (k % m + 1) < m, from nat.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 [*nat.sub_sub]
... = ((n - k div m) * m - (k mod m + 1)) div m :
(m * n - (k + 1)) / m = (m * n - (k / m * m + k % m + 1)) / m : eq_div_mul_add_mod
... = (m * n - k / m * m - (k % m + 1)) / m : by rewrite [*nat.sub_sub]
... = ((n - k / m) * m - (k % m + 1)) / m :
by rewrite [mul.comm m, nat.mul_sub_right_distrib]
... = ((n - k div m - 1) * m + m - (k mod m + 1)) div m :
... = ((n - k / m - 1) * m + m - (k % m + 1)) / m :
by rewrite [H3 at {1}, right_distrib, nat.one_mul]
... = ((n - k div m - 1) * m + (m - (k mod m + 1))) div m : {nat.add_sub_assoc H5 _}
... = (m - (k mod m + 1)) div m + (n - k div m - 1) :
... = ((n - k / m - 1) * m + (m - (k % m + 1))) / m : {nat.add_sub_assoc H5 _}
... = (m - (k % m + 1)) / m + (n - k / m - 1) :
by rewrite [add.comm, (add_mul_div_self H4)]
... = n - k div m - 1 :
... = n - k / m - 1 :
by rewrite [div_eq_zero_of_lt H6, zero_add]
end
private lemma div_div_aux (a b c : nat) : b > 0 → c > 0 → (a div b) div c = a div (b * c) :=
private lemma div_div_aux (a b c : nat) : b > 0 → c > 0 → (a / b) / c = a / (b * c) :=
suppose b > 0, suppose c > 0,
nat.strong_induction_on a
(λ a ih,
let k₁ := a div (b*c) in
let k₂ := a mod (b*c) in
let k₁ := a / (b*c) in
let k₂ := a %(b*c) in
assert bc_pos : b*c > 0, from mul_pos `b > 0` `c > 0`,
assert k₂ < b * c, from mod_lt _ bc_pos,
assert k₂ ≤ a, from !mod_le,
@ -580,24 +580,24 @@ nat.strong_induction_on a
(suppose k₂ = a,
assert i₁ : a < b * c, by rewrite -this; assumption,
assert k₁ = 0, from div_eq_zero_of_lt i₁,
assert a div b < c, by rewrite [mul.comm at i₁]; exact nat.div_lt_of_lt_mul i₁,
assert a / b < c, by rewrite [mul.comm at i₁]; exact nat.div_lt_of_lt_mul i₁,
begin
rewrite [`k₁ = 0`],
show (a div b) div c = 0, from div_eq_zero_of_lt `a div b < c`
show (a / b) / c = 0, from div_eq_zero_of_lt `a / b < c`
end)
(suppose k₂ < a,
assert a = k₁*(b*c) + k₂, from eq_div_mul_add_mod a (b*c),
assert a div b = k₁*c + k₂ div b, by
assert a / b = k₁*c + k₂ / b, by
rewrite [this at {1}, mul.comm b c at {2}, -mul.assoc,
add.comm, add_mul_div_self `b > 0`, add.comm],
assert e₁ : (a div b) div c = k₁ + (k₂ div b) div c, by
assert e₁ : (a / b) / c = k₁ + (k₂ / b) / c, by
rewrite [this, add.comm, add_mul_div_self `c > 0`, add.comm],
assert e₂ : (k₂ div b) div c = k₂ div (b * c), from ih k₂ `k₂ < a`,
assert e₃ : k₂ div (b * c) = 0, from div_eq_zero_of_lt `k₂ < b * c`,
assert (k₂ div b) div c = 0, by rewrite [e₂, e₃],
show (a div b) div c = k₁, by rewrite [e₁, this]))
assert e₂ : (k₂ / b) / c = k₂ / (b * c), from ih k₂ `k₂ < a`,
assert e₃ : k₂ / (b * c) = 0, from div_eq_zero_of_lt `k₂ < b * c`,
assert (k₂ / b) / c = 0, by rewrite [e₂, e₃],
show (a / b) / c = k₁, by rewrite [e₁, this]))
protected lemma div_div_eq_div_mul (a b c : nat) : (a div b) div c = a div (b * c) :=
protected lemma div_div_eq_div_mul (a b c : nat) : (a / b) / c = a / (b * c) :=
begin
cases b with b,
rewrite [zero_mul, *nat.div_zero, nat.zero_div],
@ -606,7 +606,7 @@ begin
apply div_div_aux a (succ b) (succ c) dec_trivial dec_trivial
end
lemma div_lt_of_ne_zero : ∀ {n : nat}, n ≠ 0 → n div 2 < n
lemma div_lt_of_ne_zero : ∀ {n : nat}, n ≠ 0 → n / 2 < n
| 0 h := absurd rfl h
| (succ n) h :=
begin

4
library/data/nat/examples/partial_sum.lean
View file

@ -26,8 +26,8 @@ lemma two_mul_partial_sum_eq : ∀ n, 2 * partial_sum n = (succ n) * n
... = (n + 2) * succ n : add.comm
... = (succ (succ n)) * succ n : rfl
theorem partial_sum_eq : ∀ n, partial_sum n = ((n + 1) * n) div 2 :=
theorem partial_sum_eq : ∀ n, partial_sum n = ((n + 1) * n) / 2 :=
take n,
assert h₁ : (2 * partial_sum n) div 2 = ((succ n) * n) div 2, by rewrite two_mul_partial_sum_eq,
assert h₁ : (2 * partial_sum n) / 2 = ((succ n) * n) / 2, by rewrite two_mul_partial_sum_eq,
assert h₂ : (2:nat) > 0, from dec_trivial,
by rewrite [nat.mul_div_cancel_left _ h₂ at h₁]; exact h₁

108
library/data/nat/gcd.lean
View file

@ -20,25 +20,25 @@ intro_k (measure.wf pr₂) 20 -- we use intro_k to be able to execute gcd effic
local attribute pair_nat.lt.wf [instance] -- instance will not be saved in .olean
local infixl ` ≺ `:50 := pair_nat.lt
private definition gcd.lt.dec (x y₁ : nat) : (succ y₁, x mod succ y₁) ≺ (x, succ y₁) :=
private definition gcd.lt.dec (x y₁ : nat) : (succ y₁, x % succ y₁) ≺ (x, succ y₁) :=
!mod_lt (succ_pos y₁)
definition gcd.F : Π (p₁ : nat × nat), (Π p₂ : nat × nat, p₂ ≺ p₁ → nat) → nat
| (x, 0) f := x
| (x, succ y) f := f (succ y, x mod succ y) !gcd.lt.dec
| (x, succ y) f := f (succ y, x % succ y) !gcd.lt.dec
definition gcd (x y : nat) := fix gcd.F (x, y)
theorem gcd_zero_right (x : nat) : gcd x 0 = x := rfl
theorem gcd_succ (x y : nat) : gcd x (succ y) = gcd (succ y) (x mod succ y) :=
theorem gcd_succ (x y : nat) : gcd x (succ y) = gcd (succ y) (x % succ y) :=
well_founded.fix_eq gcd.F (x, succ y)
theorem gcd_one_right (n : ℕ) : gcd n 1 = 1 :=
calc gcd n 1 = gcd 1 (n mod 1) : gcd_succ
... = gcd 1 0 : mod_one
calc gcd n 1 = gcd 1 (n % 1) : gcd_succ
... = gcd 1 0 : mod_one
theorem gcd_def (x : ℕ) : Π (y : ℕ), gcd x y = if y = 0 then x else gcd y (x mod y)
theorem gcd_def (x : ℕ) : Π (y : ℕ), gcd x y = if y = 0 then x else gcd y (x % y)
| 0 := !gcd_zero_right
| (succ y) := !gcd_succ ⬝ (if_neg !succ_ne_zero)⁻¹
@ -46,29 +46,29 @@ theorem gcd_def (x : ℕ) : Π (y : ℕ), gcd x y = if y = 0 then x else gcd y (
theorem gcd_self : Π (n : ℕ), gcd n n = n
| 0 := rfl
| (succ n₁) := calc
gcd (succ n₁) (succ n₁) = gcd (succ n₁) (succ n₁ mod succ n₁) : gcd_succ
gcd (succ n₁) (succ n₁) = gcd (succ n₁) (succ n₁ % succ n₁) : gcd_succ
... = gcd (succ n₁) 0 : mod_self
theorem gcd_zero_left : Π (n : ℕ), gcd 0 n = n
| 0 := rfl
| (succ n₁) := calc
gcd 0 (succ n₁) = gcd (succ n₁) (0 mod succ n₁) : gcd_succ
gcd 0 (succ n₁) = gcd (succ n₁) (0 % succ n₁) : gcd_succ
... = gcd (succ n₁) 0 : zero_mod
theorem gcd_of_pos (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m mod n) :=
theorem gcd_of_pos (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m % n) :=
gcd_def m n ⬝ if_neg (ne_zero_of_pos H)
theorem gcd_rec (m n : ℕ) : gcd m n = gcd n (m mod n) :=
theorem gcd_rec (m n : ℕ) : gcd m n = gcd n (m % n) :=
by_cases_zero_pos n
(calc
m = gcd 0 m : gcd_zero_left
... = gcd 0 (m mod 0) : mod_zero)
m = gcd 0 m : gcd_zero_left
... = gcd 0 (m % 0) : mod_zero)
(take n, assume H : 0 < n, gcd_of_pos m H)
theorem gcd.induction {P : ℕ → ℕ → Prop}
(m n : ℕ)
(H0 : ∀m, P m 0)
(H1 : ∀m n, 0 < n → P n (m mod n) → P m n) :
(H1 : ∀m n, 0 < n → P n (m % n) → P m n) :
P m n :=
induction (m, n) (prod.rec (λm, nat.rec (λ IH, H0 m)
(λ n₁ v (IH : ∀p₂, p₂ ≺ (m, succ n₁) → P (pr₁ p₂) (pr₂ p₂)),
@ -78,10 +78,10 @@ theorem gcd_dvd (m n : ℕ) : (gcd m n ∣ m) ∧ (gcd m n ∣ n) :=
gcd.induction m n
(take m, and.intro (!one_mul ▸ !dvd_mul_left) !dvd_zero)
(take m n (npos : 0 < n), and.rec
(assume (IH₁ : gcd n (m mod n) ∣ n) (IH₂ : gcd n (m mod n) ∣ (m mod n)),
have H : (gcd n (m mod n) ∣ (m div n * n + m mod n)), from
(assume (IH₁ : gcd n (m % n) ∣ n) (IH₂ : gcd n (m % n) ∣ (m % n)),
have H : (gcd n (m % n) ∣ (m / n * n + m % n)), from
dvd_add (dvd.trans IH₁ !dvd_mul_left) IH₂,
have H1 : (gcd n (m mod n) ∣ m), from !eq_div_mul_add_mod⁻¹ ▸ H,
have H1 : (gcd n (m % n) ∣ m), from !eq_div_mul_add_mod⁻¹ ▸ H,
show (gcd m n ∣ m) ∧ (gcd m n ∣ n), from !gcd_rec⁻¹ ▸ (and.intro H1 IH₁)))
theorem gcd_dvd_left (m n : ℕ) : gcd m n ∣ m := and.left !gcd_dvd
@ -91,10 +91,10 @@ theorem gcd_dvd_right (m n : ℕ) : gcd m n ∣ n := and.right !gcd_dvd
theorem dvd_gcd {m n k : ℕ} : k ∣ m → k ∣ n → k ∣ gcd m n :=
gcd.induction m n (take m, imp.intro)
(take m n (npos : n > 0)
(IH : k ∣ n → k ∣ m mod n → k ∣ gcd n (m mod n))
(IH : k ∣ n → k ∣ m % n → k ∣ gcd n (m % n))
(H1 : k ∣ m) (H2 : k ∣ n),
have H3 : k ∣ m div n * n + m mod n, from !eq_div_mul_add_mod ▸ H1,
have H4 : k ∣ m mod n, from nat.dvd_of_dvd_add_left H3 (dvd.trans H2 !dvd_mul_left),
have H3 : k ∣ m / n * n + m % n, from !eq_div_mul_add_mod ▸ H1,
have H4 : k ∣ m % n, from nat.dvd_of_dvd_add_left H3 (dvd.trans H2 !dvd_mul_left),
!gcd_rec⁻¹ ▸ IH H2 H4)
theorem gcd.comm (m n : ℕ) : gcd m n = gcd n m :=
@ -119,12 +119,12 @@ gcd.induction n k
(take n, calc gcd (m * n) (m * 0) = gcd (m * n) 0 : mul_zero)
(take n k,
assume H : 0 < k,
assume IH : gcd (m * k) (m * (n mod k)) = m * gcd k (n mod k),
assume IH : gcd (m * k) (m * (n % k)) = m * gcd k (n % k),
calc
gcd (m * n) (m * k) = gcd (m * k) (m * n mod (m * k)) : !gcd_rec
... = gcd (m * k) (m * (n mod k)) : mul_mod_mul_left
... = m * gcd k (n mod k) : IH
... = m * gcd n k : !gcd_rec)
gcd (m * n) (m * k) = gcd (m * k) (m * n % (m * k)) : !gcd_rec
... = gcd (m * k) (m * (n % k)) : mul_mod_mul_left
... = m * gcd k (n % k) : IH
... = m * gcd n k : !gcd_rec)
theorem gcd_mul_right (m n k : ℕ) : gcd (m * n) (k * n) = gcd m k * n :=
calc
@ -148,13 +148,13 @@ theorem eq_zero_of_gcd_eq_zero_right {m n : ℕ} (H : gcd m n = 0) : n = 0 :=
eq_zero_of_gcd_eq_zero_left (!gcd.comm ▸ H)
theorem gcd_div {m n k : ℕ} (H1 : k ∣ m) (H2 : k ∣ n) :
gcd (m div k) (n div k) = gcd m n div k :=
gcd (m / k) (n / k) = gcd m n / k :=
or.elim (eq_zero_or_pos k)
(assume H3 : k = 0, by subst k; rewrite *nat.div_zero)
(assume H3 : k > 0, (nat.div_eq_of_eq_mul_left H3 (calc
gcd m n = gcd m (n div k * k) : nat.div_mul_cancel H2
... = gcd (m div k * k) (n div k * k) : nat.div_mul_cancel H1
... = gcd (m div k) (n div k) * k : gcd_mul_right))⁻¹)
gcd m n = gcd m (n / k * k) : nat.div_mul_cancel H2
... = gcd (m / k * k) (n / k * k) : nat.div_mul_cancel H1
... = gcd (m / k) (n / k) * k : gcd_mul_right))⁻¹)
theorem gcd_dvd_gcd_mul_left (m n k : ℕ) : gcd m n ∣ gcd (k * m) n :=
dvd_gcd (dvd.trans !gcd_dvd_left !dvd_mul_left) !gcd_dvd_right
@ -170,42 +170,42 @@ theorem gcd_dvd_gcd_mul_right_right (m n k : ℕ) : gcd m n ∣ gcd m (n * k) :=
/- lcm -/
definition lcm (m n : ℕ) : ℕ := m * n div (gcd m n)
definition lcm (m n : ℕ) : ℕ := m * n / (gcd m n)
theorem lcm.comm (m n : ℕ) : lcm m n = lcm n m :=
calc
lcm m n = m * n div gcd m n : rfl
... = n * m div gcd m n : mul.comm
... = n * m div gcd n m : gcd.comm
lcm m n = m * n / gcd m n : rfl
... = n * m / gcd m n : mul.comm
... = n * m / gcd n m : gcd.comm
... = lcm n m : rfl
theorem lcm_zero_left (m : ℕ) : lcm 0 m = 0 :=
calc
lcm 0 m = 0 * m div gcd 0 m : rfl
... = 0 div gcd 0 m : zero_mul
lcm 0 m = 0 * m / gcd 0 m : rfl
... = 0 / gcd 0 m : zero_mul
... = 0 : nat.zero_div
theorem lcm_zero_right (m : ℕ) : lcm m 0 = 0 := !lcm.comm ▸ !lcm_zero_left
theorem lcm_one_left (m : ℕ) : lcm 1 m = m :=
calc
lcm 1 m = 1 * m div gcd 1 m : rfl
... = m div gcd 1 m : one_mul
... = m div 1 : gcd_one_left
lcm 1 m = 1 * m / gcd 1 m : rfl
... = m / gcd 1 m : one_mul
... = m / 1 : gcd_one_left
... = m : nat.div_one
theorem lcm_one_right (m : ℕ) : lcm m 1 = m := !lcm.comm ▸ !lcm_one_left
theorem lcm_self (m : ℕ) : lcm m m = m :=
have H : m * m div m = m, from
have H : m * m / m = m, from
by_cases_zero_pos m !nat.div_zero (take m, assume H1 : m > 0, !nat.mul_div_cancel H1),
calc
lcm m m = m * m div gcd m m : rfl
... = m * m div m : gcd_self
lcm m m = m * m / gcd m m : rfl
... = m * m / m : gcd_self
... = m : H
theorem dvd_lcm_left (m n : ℕ) : m ∣ lcm m n :=
have H : lcm m n = m * (n div gcd m n), from nat.mul_div_assoc _ !gcd_dvd_right,
have H : lcm m n = m * (n / gcd m n), from nat.mul_div_assoc _ !gcd_dvd_right,
dvd.intro H⁻¹
theorem dvd_lcm_right (m n : ℕ) : n ∣ lcm m n :=
@ -299,9 +299,9 @@ theorem gcd_mul_right_cancel_of_coprime_right {k m : ℕ} (n : ℕ) (H : coprime
!gcd.comm ▸ !gcd.comm ▸ !gcd_mul_right_cancel_of_coprime H
theorem coprime_div_gcd_div_gcd {m n : ℕ} (H : gcd m n > 0) :
coprime (m div gcd m n) (n div gcd m n) :=
coprime (m / gcd m n) (n / gcd m n) :=
calc
gcd (m div gcd m n) (n div gcd m n) = gcd m n div gcd m n : gcd_div !gcd_dvd_left !gcd_dvd_right
gcd (m / gcd m n) (n / gcd m n) = gcd m n / gcd m n : gcd_div !gcd_dvd_left !gcd_dvd_right
... = 1 : nat.div_self H
theorem not_coprime_of_dvd_of_dvd {m n d : ℕ} (dgt1 : d > 1) (Hm : d ∣ m) (Hn : d ∣ n) :
@ -314,8 +314,8 @@ show false, from not_lt_of_ge `d ≤ 1` `d > 1`
theorem exists_coprime {m n : ℕ} (H : gcd m n > 0) :
exists m' n', coprime m' n' ∧ m = m' * gcd m n ∧ n = n' * gcd m n :=
have H1 : m = (m div gcd m n) * gcd m n, from (nat.div_mul_cancel !gcd_dvd_left)⁻¹,
have H2 : n = (n div gcd m n) * gcd m n, from (nat.div_mul_cancel !gcd_dvd_right)⁻¹,
have H1 : m = (m / gcd m n) * gcd m n, from (nat.div_mul_cancel !gcd_dvd_left)⁻¹,
have H2 : n = (n / gcd m n) * gcd m n, from (nat.div_mul_cancel !gcd_dvd_right)⁻¹,
exists.intro _ (exists.intro _ (and.intro (coprime_div_gcd_div_gcd H) (and.intro H1 H2)))
theorem coprime_mul {m n k : ℕ} (H1 : coprime m k) (H2 : coprime n k) : coprime (m * n) k :=
@ -357,16 +357,16 @@ or.elim (eq_zero_or_pos (gcd k m))
exists.intro _ (exists.intro _ (and.intro H4 (and.intro H5 H6))))
(assume H1 : gcd k m > 0,
have H2 : gcd k m ∣ k, from !gcd_dvd_left,
have H3 : k div gcd k m ∣ (m * n) div gcd k m, from nat.div_dvd_div H2 H,
have H4 : (m * n) div gcd k m = (m div gcd k m) * n, from
have H3 : k / gcd k m ∣ (m * n) / gcd k m, from nat.div_dvd_div H2 H,
have H4 : (m * n) / gcd k m = (m / gcd k m) * n, from
calc
m * n div gcd k m = n * m div gcd k m : mul.comm
... = n * (m div gcd k m) : !nat.mul_div_assoc !gcd_dvd_right
... = m div gcd k m * n : mul.comm,
have H5 : k div gcd k m ∣ (m div gcd k m) * n, from H4 ▸ H3,
have H6 : coprime (k div gcd k m) (m div gcd k m), from coprime_div_gcd_div_gcd H1,
have H7 : k div gcd k m ∣ n, from dvd_of_coprime_of_dvd_mul_left H6 H5,
have H8 : k = gcd k m * (k div gcd k m), from (nat.mul_div_cancel' H2)⁻¹,
m * n / gcd k m = n * m / gcd k m : mul.comm
... = n * (m / gcd k m) : !nat.mul_div_assoc !gcd_dvd_right
... = m / gcd k m * n : mul.comm,
have H5 : k / gcd k m ∣ (m / gcd k m) * n, from H4 ▸ H3,
have H6 : coprime (k / gcd k m) (m / gcd k m), from coprime_div_gcd_div_gcd H1,
have H7 : k / gcd k m ∣ n, from dvd_of_coprime_of_dvd_mul_left H6 H5,
have H8 : k = gcd k m * (k / gcd k m), from (nat.mul_div_cancel' H2)⁻¹,
exists.intro _ (exists.intro _ (and.intro H8 (and.intro !gcd_dvd_right H7))))
end nat

37
library/data/nat/parity.lean
View file

@ -3,7 +3,7 @@ Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
Parity
Parity.
-/
import data.nat.power logic.identities
@ -11,7 +11,7 @@ namespace nat
open decidable
open algebra
definition even (n : nat) := n mod 2 = 0
definition even (n : nat) := n % 2 = 0
definition decidable_even [instance] : ∀ n, decidable (even n) :=
take n, !nat.has_decidable_eq
@ -77,22 +77,22 @@ lemma eq_1_of_ne_0_lt_2 : ∀ {n : nat}, n ≠ 0 → n < 2 → n = 1
| 1 h₁ h₂ := rfl
| (n+2) h₁ h₂ := absurd (lt_of_succ_lt_succ (lt_of_succ_lt_succ h₂)) !not_lt_zero
lemma mod_eq_of_odd {n} : odd n → n mod 2 = 1 :=
lemma mod_eq_of_odd {n} : odd n → n % 2 = 1 :=
suppose odd n,
have ¬ n mod 2 = 0, from this,
have n mod 2 < 2, from mod_lt n dec_trivial,
eq_1_of_ne_0_lt_2 `¬ n mod 2 = 0` `n mod 2 < 2`
have ¬ n % 2 = 0, from this,
have n % 2 < 2, from mod_lt n dec_trivial,
eq_1_of_ne_0_lt_2 `¬ n % 2 = 0` `n % 2 < 2`
lemma odd_of_mod_eq {n} : n mod 2 = 1 → odd n :=
suppose n mod 2 = 1,
lemma odd_of_mod_eq {n} : n % 2 = 1 → odd n :=
suppose n % 2 = 1,
by_contradiction (suppose ¬ odd n,
assert n mod 2 = 0, from even_of_not_odd this,
assert n % 2 = 0, from even_of_not_odd this,
by rewrite this at *; contradiction)
lemma even_succ_of_odd {n} : odd n → even (succ n) :=
suppose odd n,
assert n mod 2 = 1 mod 2, from mod_eq_of_odd this,
assert (n+1) mod 2 = 2 mod 2, from add_mod_eq_add_mod_right 1 this,
assert n % 2 = 1 % 2, from mod_eq_of_odd this,
assert (n+1) % 2 = 2 % 2, from add_mod_eq_add_mod_right 1 this,
by rewrite mod_self at this; exact this
lemma odd_succ_succ_of_odd {n} : odd n → odd (succ (succ n)) :=
@ -175,7 +175,7 @@ assume h, by_contradiction (λ hn,
have ∃ k, n = 2 * k, from exists_of_even this,
obtain k₁ (hk₁ : n = 2 * k₁ + 1), from h,
obtain k₂ (hk₂ : n = 2 * k₂), from this,
assert (2 * k₁ + 1) mod 2 = (2 * k₂) mod 2, by rewrite [-hk₁, -hk₂],
assert (2 * k₁ + 1) % 2 = (2 * k₂) % 2, by rewrite [-hk₁, -hk₂],
begin
rewrite [mul_mod_right at this, add.comm at this, add_mul_mod_self_left at this],
contradiction
@ -189,7 +189,8 @@ even_of_exists (exists.intro (k₁+k₂) (by rewrite [hk₁, hk₂, left_distrib
lemma even_add_of_odd_of_odd {n m} : odd n → odd m → even (n+m) :=
suppose odd n, suppose odd m,
assert even (succ n + succ m), from even_add_of_even_of_even (even_succ_of_odd `odd n`) (even_succ_of_odd `odd m`),
assert even (succ n + succ m),
from even_add_of_even_of_even (even_succ_of_odd `odd n`) (even_succ_of_odd `odd m`),
have even(succ (succ (n + m))), by rewrite [add_succ at this, succ_add at this]; exact this,
even_of_even_succ_succ this
@ -259,7 +260,7 @@ by_contradiction
have odd (n^m), from odd_pow this,
show false, from this `even (n^m)`)
lemma eq_of_div2_of_even {n m : nat} : n div 2 = m div 2 → (even n ↔ even m) → n = m :=
lemma eq_of_div2_of_even {n m : nat} : n / 2 = m / 2 → (even n ↔ even m) → n = m :=
assume h₁ h₂,
or.elim (em (even n))
(suppose even n, or.elim (em (even m))
@ -274,13 +275,15 @@ assume h₁ h₂,
(suppose odd n, or.elim (em (even m))
(suppose even m, absurd `odd n` (not_odd_of_even (iff.mpr h₂ `even m`)))
(suppose odd m,
assert d : 1 div 2 = (0:nat), from dec_trivial,
assert d : 1 / 2 = (0:nat), from dec_trivial,
obtain w₁ (hw₁ : n = 2*w₁ + 1), from exists_of_odd `odd n`,
obtain w₂ (hw₂ : m = 2*w₂ + 1), from exists_of_odd `odd m`,
begin
substvars,
rewrite [add.comm at h₁, add_mul_div_self_left _ _ (dec_trivial : 2 > 0) at h₁, d at h₁, zero_add at h₁],
rewrite [add.comm at h₁, add_mul_div_self_left _ _ (dec_trivial : 2 > 0) at h₁, d at h₁, zero_add at h₁],
rewrite [add.comm at h₁, add_mul_div_self_left _ _ (dec_trivial : 2 > 0) at h₁, d at h₁,
zero_add at h₁],
rewrite [add.comm at h₁, add_mul_div_self_left _ _ (dec_trivial : 2 > 0) at h₁, d at h₁,
zero_add at h₁],
rewrite h₁
end))
end nat

6
library/data/nat/power.lean
View file

@ -56,7 +56,7 @@ theorem mul_self_eq_pow_2 (a : nat) : a * a = a ^ 2 :=
show a * a = a ^ (succ (succ zero)), from
by rewrite [*pow_succ, *pow_zero, mul_one]
theorem pow_cancel_left : ∀ {a b c : nat}, a > 1 → a^b = a^c → b = c
theorem pow_cancel_left : ∀ {a b c : nat}, a > 1 → a ^ b = a ^ c → b = c
| a 0 0 h₁ h₂ := rfl
| a (succ b) 0 h₁ h₂ :=
assert a = 1, by rewrite [pow_succ at h₂, pow_zero at h₂]; exact (eq_one_of_mul_eq_one_right h₂),
@ -71,7 +71,7 @@ theorem pow_cancel_left : ∀ {a b c : nat}, a > 1 → a^b = a^c → b = c
assert a^b = a^c, by rewrite [*pow_succ at h₂]; exact (eq_of_mul_eq_mul_left (pos_of_ne_zero this) h₂),
by rewrite [pow_cancel_left h₁ this]
theorem pow_div_cancel : ∀ {a b : nat}, a ≠ 0 → (a ^ succ b) div a = a ^ b
theorem pow_div_cancel : ∀ {a b : nat}, a ≠ 0 → (a ^ succ b) / a = a ^ b
| a 0 h := by rewrite [pow_succ, pow_zero, mul_one, nat.div_self (pos_of_ne_zero h)]
| a (succ b) h := by rewrite [pow_succ, nat.mul_div_cancel_left _ (pos_of_ne_zero h)]
@ -83,7 +83,7 @@ lemma dvd_pow_of_dvd_of_pos : ∀ {i j n : nat}, i ∣ j → n > 0 → i ∣ j^n
| i j 0 h₁ h₂ := absurd h₂ !lt.irrefl
| i j (succ n) h₁ h₂ := by rewrite [pow_succ']; apply dvd_mul_of_dvd_right h₁
lemma pow_mod_eq_zero (i : nat) {n : nat} (h : n > 0) : (i^n) mod i = 0 :=
lemma pow_mod_eq_zero (i : nat) {n : nat} (h : n > 0) : (i ^ n) % i = 0 :=
iff.mp !dvd_iff_mod_eq_zero (dvd_pow i h)
lemma pow_dvd_of_pow_succ_dvd {p i n : nat} : p^(succ i) ∣ n → p^i ∣ n :=

16
library/data/rat/basic.lean
View file

@ -286,9 +286,9 @@ by esimp [mul, of_int, equiv]; rewrite [*int.mul_one]
definition reduce : prerat → prerat
| (mk an ad adpos) :=
have pos : ad div gcd an ad > 0, from div_pos_of_pos_of_dvd adpos !gcd_nonneg !gcd_dvd_right,
have pos : ad / gcd an ad > 0, from div_pos_of_pos_of_dvd adpos !gcd_nonneg !gcd_dvd_right,
if an = 0 then prerat.zero
else mk (an div gcd an ad) (ad div gcd an ad) pos
else mk (an / gcd an ad) (ad / gcd an ad) pos
protected theorem eq {a b : prerat} (Hn : num a = num b) (Hd : denom a = denom b) : a = b :=
begin
@ -309,7 +309,7 @@ theorem reduce_equiv : ∀ a : prerat, reduce a ≡ a
by rewrite [↑reduce, if_neg annz, ↑equiv, algebra.mul.comm, -!int.mul_div_assoc
!gcd_dvd_left, -!int.mul_div_assoc !gcd_dvd_right, algebra.mul.comm])
theorem reduce_eq_reduce : ∀{a b : prerat}, a ≡ b → reduce a = reduce b
theorem reduce_eq_reduce : ∀ {a b : prerat}, a ≡ b → reduce a = reduce b
| (mk an ad adpos) (mk bn bd bdpos) :=
assume H : an * bd = bn * ad,
decidable.by_cases
@ -575,8 +575,8 @@ protected definition discrete_field [reducible] [trans_instance] : algebra.discr
inv_zero := rat.inv_zero,
has_decidable_eq := has_decidable_eq⦄
definition rat_has_division [instance] [reducible] [priority rat.prio] : has_division rat :=
has_division.mk has_division.division
definition rat_has_div [instance] [reducible] [priority rat.prio] : has_div rat :=
has_div.mk has_div.div
definition rat_has_pow_nat [instance] [reducible] [priority rat.prio] : has_pow_nat rat :=
has_pow_nat.mk has_pow_nat.pow_nat
@ -585,19 +585,19 @@ theorem eq_num_div_denom (a : ℚ) : a = num a / denom a :=
have H : of_int (denom a) ≠ 0, from assume H', ne_of_gt (denom_pos a) (of_int.inj H'),
iff.mpr (!eq_div_iff_mul_eq H) (mul_denom a)
theorem of_int_div {a b : ℤ} (H : b ∣ a) : of_int (a div b) = of_int a / of_int b :=
theorem of_int_div {a b : ℤ} (H : b ∣ a) : of_int (a / b) = of_int a / of_int b :=
decidable.by_cases
(assume bz : b = 0,
by rewrite [bz, int.div_zero, of_int_zero, algebra.div_zero])
(assume bnz : b ≠ 0,
have bnz' : of_int b ≠ 0, from assume oibz, bnz (of_int.inj oibz),
have H' : of_int (a div b) * of_int b = of_int a, from
have H' : of_int (a / b) * of_int b = of_int a, from
dvd.elim H
(take c, assume Hc : a = b * c,
by rewrite [Hc, !int.mul_div_cancel_left bnz, mul.comm]),
iff.mpr (!eq_div_iff_mul_eq bnz') H')
theorem of_nat_div {a b : ℕ} (H : b ∣ a) : of_nat (a div b) = of_nat a / of_nat b :=
theorem of_nat_div {a b : ℕ} (H : b ∣ a) : of_nat (a / b) = of_nat a / of_nat b :=
have H' : (int.of_nat b ∣ int.of_nat a), by rewrite [int.of_nat_dvd_of_nat_iff]; exact H,
by+ rewrite [of_nat_eq, int.of_nat_div, of_int_div H']

11
library/data/real/division.lean
View file

@ -590,12 +590,11 @@ noncomputable protected definition inv (x : ℝ) : ℝ :=
noncomputable definition real_has_inv [instance] [reducible] [priority real.prio] : has_inv real :=
has_inv.mk real.inv
noncomputable protected definition division (x y : ℝ) : ℝ :=
noncomputable protected definition div (x y : ℝ) : ℝ :=
x * y⁻¹
noncomputable definition real_has_division [instance] [reducible] [priority real.prio] :
has_division real :=
has_division.mk real.division
noncomputable definition real_has_div [instance] [reducible] [priority real.prio] : has_div real :=
has_div.mk real.div
protected theorem le_total (x y : ℝ) : x ≤ y ∨ y ≤ x :=
quot.induction_on₂ x y (λ s t, rat_seq.r_le_total s t)
@ -663,10 +662,10 @@ by_cases
open int
theorem of_int_div (x y : ℤ) (H : (#int y ∣ x)) : of_int ((x div y)) = of_int x / of_int y :=
theorem of_int_div (x y : ℤ) (H : y ∣ x) : of_int (x / y) = of_int x / of_int y :=
by rewrite [of_int_eq, rat.of_int_div H, of_rat_divide]
theorem of_nat_div (x y : ℕ) (H : (#nat y ∣ x)) : of_nat (x div y) = of_nat x / of_nat y :=
theorem of_nat_div (x y : ℕ) (H : y ∣ x) : of_nat (x / y) = of_nat x / of_nat y :=
by rewrite [of_nat_eq, rat.of_nat_div H, of_rat_divide]
/- useful for proving equalities -/

77
library/init/reserved_notation.lean
View file

@ -10,32 +10,31 @@ notation `assume` binders `,` r:(scoped f, f) := r
notation `take` binders `,` r:(scoped f, f) := r
structure has_zero [class] (A : Type) := (zero : A)
structure has_one [class] (A : Type) := (one : A)
structure has_add [class] (A : Type) := (add : A → A → A)
structure has_mul [class] (A : Type) := (mul : A → A → A)
structure has_inv [class] (A : Type) := (inv : A → A)
structure has_neg [class] (A : Type) := (neg : A → A)
structure has_sub [class] (A : Type) := (sub : A → A → A)
structure has_division [class] (A : Type) := (division : A → A → A)
structure has_divide [class] (A : Type) := (divide : A → A → A)
structure has_modulo [class] (A : Type) := (modulo : A → A → A)
structure has_dvd [class] (A : Type) := (dvd : A → A → Prop)
structure has_le [class] (A : Type) := (le : A → A → Prop)
structure has_lt [class] (A : Type) := (lt : A → A → Prop)
structure has_one [class] (A : Type) := (one : A)
structure has_add [class] (A : Type) := (add : A → A → A)
structure has_mul [class] (A : Type) := (mul : A → A → A)
structure has_inv [class] (A : Type) := (inv : A → A)
structure has_neg [class] (A : Type) := (neg : A → A)
structure has_sub [class] (A : Type) := (sub : A → A → A)
structure has_div [class] (A : Type) := (div : A → A → A)
structure has_dvd [class] (A : Type) := (dvd : A → A → Prop)
structure has_mod [class] (A : Type) := (mod : A → A → A)
structure has_le [class] (A : Type) := (le : A → A → Prop)
structure has_lt [class] (A : Type) := (lt : A → A → Prop)
definition zero {A : Type} [s : has_zero A] : A := has_zero.zero A
definition one {A : Type} [s : has_one A] : A := has_one.one A
definition add {A : Type} [s : has_add A] : A → A → A := has_add.add
definition mul {A : Type} [s : has_mul A] : A → A → A := has_mul.mul
definition sub {A : Type} [s : has_sub A] : A → A → A := has_sub.sub
definition div {A : Type} [s : has_div A] : A → A → A := has_div.div
definition dvd {A : Type} [s : has_dvd A] : A → A → Prop := has_dvd.dvd
definition mod {A : Type} [s : has_mod A] : A → A → A := has_mod.mod
definition neg {A : Type} [s : has_neg A] : A → A := has_neg.neg
definition inv {A : Type} [s : has_inv A] : A → A := has_inv.inv
definition le {A : Type} [s : has_le A] : A → A → Prop := has_le.le
definition lt {A : Type} [s : has_lt A] : A → A → Prop := has_lt.lt
definition zero {A : Type} [s : has_zero A] : A := has_zero.zero A
definition one {A : Type} [s : has_one A] : A := has_one.one A
definition add {A : Type} [s : has_add A] : A → A → A := has_add.add
definition mul {A : Type} [s : has_mul A] : A → A → A := has_mul.mul
definition sub {A : Type} [s : has_sub A] : A → A → A := has_sub.sub
definition division {A : Type} [s : has_division A] : A → A → A := has_division.division
definition divide {A : Type} [s : has_divide A] : A → A → A := has_divide.divide
definition modulo {A : Type} [s : has_modulo A] : A → A → A := has_modulo.modulo
definition dvd {A : Type} [s : has_dvd A] : A → A → Prop := has_dvd.dvd
definition neg {A : Type} [s : has_neg A] : A → A := has_neg.neg
definition inv {A : Type} [s : has_inv A] : A → A := has_inv.inv
definition le {A : Type} [s : has_le A] : A → A → Prop := has_le.le
definition lt {A : Type} [s : has_lt A] : A → A → Prop := has_lt.lt
definition ge [reducible] {A : Type} [s : has_le A] (a b : A) : Prop := le b a
definition gt [reducible] {A : Type} [s : has_lt A] (a b : A) : Prop := lt b a
definition bit0 {A : Type} [s : has_add A] (a : A) : A := add a a
@ -164,9 +163,8 @@ reserve infixl ` × `:30
reserve infixl ` + `:65
reserve infixl ` - `:65
reserve infixl ` * `:70
reserve infixl ` div `:70
reserve infixl ` mod `:70
reserve infixl ` / `:70
reserve infixl ` % `:70
reserve prefix `-`:100
reserve infix ` ^ `:80
@ -200,10 +198,9 @@ reserve infixr ` :: `:65
infix + := add
infix * := mul
infix - := sub
infix / := division
infix div := divide
infix / := div
infix ∣ := dvd
infix mod := modulo
infix % := mod
prefix - := neg
postfix ⁻¹ := inv
infix ≤ := le
@ -211,13 +208,13 @@ infix ≥ := ge
infix < := lt
infix > := gt
notation [parsing_only] x ` +[`:65 A:0 `] `:0 y:65 := @add A _ x y
notation [parsing_only] x ` -[`:65 A:0 `] `:0 y:65 := @sub A _ x y
notation [parsing_only] x ` *[`:70 A:0 `] `:0 y:70 := @mul A _ x y
notation [parsing_only] x ` /[`:70 A:0 `] `:0 y:70 := @division A _ x y
notation [parsing_only] x ` div[`:70 A:0 `] `:0 y:70 := @divide A _ x y
notation [parsing_only] x ` mod[`:70 A:0 `] `:0 y:70 := @modulo A _ x y
notation [parsing_only] x ` ≤[`:50 A:0 `] `:0 y:50 := @le A _ x y
notation [parsing_only] x ` ≥[`:50 A:0 `] `:0 y:50 := @ge A _ x y
notation [parsing_only] x ` <[`:50 A:0 `] `:0 y:50 := @lt A _ x y
notation [parsing_only] x ` >[`:50 A:0 `] `:0 y:50 := @gt A _ x y
notation [parsing_only] x ` +[`:65 A:0 `] `:0 y:65 := @add A _ x y
notation [parsing_only] x ` -[`:65 A:0 `] `:0 y:65 := @sub A _ x y
notation [parsing_only] x ` *[`:70 A:0 `] `:0 y:70 := @mul A _ x y
notation [parsing_only] x ` /[`:70 A:0 `] `:0 y:70 := @div A _ x y
notation [parsing_only] x ` ∣[`:70 A:0 `] `:0 y:70 := @dvd A _ x y
notation [parsing_only] x ` %[`:70 A:0 `] `:0 y:70 := @mod A _ x y
notation [parsing_only] x ` ≤[`:50 A:0 `] `:0 y:50 := @le A _ x y
notation [parsing_only] x ` ≥[`:50 A:0 `] `:0 y:50 := @ge A _ x y
notation [parsing_only] x ` <[`:50 A:0 `] `:0 y:50 := @lt A _ x y
notation [parsing_only] x ` >[`:50 A:0 `] `:0 y:50 := @gt A _ x y

2
library/theories/combinatorics/choose.lean
View file

@ -112,7 +112,7 @@ begin
end
end
theorem choose_def_alt {n k : ℕ} (H : k ≤ n) : choose n k = fact n div (fact k * fact (n -k)) :=
theorem choose_def_alt {n k : ℕ} (H : k ≤ n) : choose n k = fact n / (fact k * fact (n -k)) :=
eq.symm (nat.div_eq_of_eq_mul_left (mul_pos !fact_pos !fact_pos)
(by rewrite [-mul.assoc, choose_mul_fact_mul_fact H]))

38
library/theories/group_theory/cyclic.lean
View file

@ -20,17 +20,17 @@ variable {A : Type}
variable [ambG : group A]
include ambG
lemma pow_mod {a : A} {n m : nat} : a ^ m = 1 → a ^ n = a ^ (n mod m) :=
lemma pow_mod {a : A} {n m : nat} : a ^ m = 1 → a ^ n = a ^ (n % m) :=
assume Pid,
assert a ^ (n div m * m) = 1, from calc
a ^ (n div m * m) = a ^ (m * (n div m)) : by rewrite (mul.comm (n div m) m)
... = (a ^ m) ^ (n div m) : by rewrite pow_mul
... = 1 ^ (n div m) : by rewrite Pid
... = 1 : one_pow (n div m),
calc a ^ n = a ^ (n div m * m + n mod m) : by rewrite -(eq_div_mul_add_mod n m)
... = a ^ (n div m * m) * a ^ (n mod m) : by rewrite pow_add
... = 1 * a ^ (n mod m) : by rewrite this
... = a ^ (n mod m) : by rewrite one_mul
assert a ^ (n / m * m) = 1, from calc
a ^ (n / m * m) = a ^ (m * (n / m)) : by rewrite (mul.comm (n / m) m)
... = (a ^ m) ^ (n / m) : by rewrite pow_mul
... = 1 ^ (n / m) : by rewrite Pid
... = 1 : one_pow (n / m),
calc a ^ n = a ^ (n / m * m + n % m) : by rewrite -(eq_div_mul_add_mod n m)
... = a ^ (n / m * m) * a ^ (n % m) : by rewrite pow_add
... = 1 * a ^ (n % m) : by rewrite this
... = a ^ (n % m) : by rewrite one_mul
lemma pow_sub_eq_one_of_pow_eq {a : A} {i j : nat} :
a^i = a^j → a^(i - j) = 1 :=
@ -47,9 +47,9 @@ assume Pe, or.elim (lt_or_ge i j)
lemma pow_madd {a : A} {n : nat} {i j : fin (succ n)} :
a^(succ n) = 1 → a^(val (i + j)) = a^i * a^j :=
assume Pe, calc
a^(val (i + j)) = a^((i + j) mod (succ n)) : rfl
... = a^(val i + val j) : by rewrite [-pow_mod Pe]
... = a^i * a^j : by rewrite pow_add
a^(val (i + j)) = a^((i + j) % (succ n)) : rfl
... = a^(val i + val j) : by rewrite [-pow_mod Pe]
... = a^i * a^j : by rewrite pow_add
lemma mk_pow_mod {a : A} {n m : nat} : a ^ (succ m) = 1 → a ^ n = a ^ (mk_mod m n) :=
assume Pe, pow_mod Pe
@ -113,7 +113,7 @@ obtain j Pjlt Pjh, from of_mem_sep Phin,
begin
rewrite [-Pig, -Pjh, -pow_add, pow_mod Pe],
apply mem_sep_of_mem !mem_univ,
existsi ((i + j) mod (succ n)), apply and.intro,
existsi ((i + j) % (succ n)), apply and.intro,
apply nat.lt_trans (mod_lt (i+j) !zero_lt_succ) (succ_lt_succ Plt),
apply rfl
end
@ -200,7 +200,7 @@ or.elim (eq_or_lt_of_le (order_le Pone)) (λ P, P)
exact absurd (pow_order a) Pn end)
lemma order_dvd_of_pow_eq_one {a : A} {n : nat} (Pone : a^n = 1) : order a ∣ n :=
assert Pe : a^(n mod order a) = 1, from
assert Pe : a^(n % order a) = 1, from
begin
revert Pone,
rewrite [eq_div_mul_add_mod n (order a) at {1}, pow_add, mul.comm _ (order a), pow_mul, pow_order, one_pow, one_mul],
@ -373,10 +373,10 @@ lemma rotl_perm_pow_eq : ∀ {i : nat}, (rotl_perm A n 1) ^ i = rotl_perm A n i
lemma rotl_perm_pow_eq_one : (rotl_perm A n 1) ^ n = 1 :=
eq.trans rotl_perm_pow_eq (eq_of_feq begin esimp [rotl_perm], rewrite [↑rotl_fun, rotl_id] end)
lemma rotl_perm_mod {i : nat} : rotl_perm A n i = rotl_perm A n (i mod n) :=
calc rotl_perm A n i = (rotl_perm A n 1) ^ i : by rewrite rotl_perm_pow_eq
... = (rotl_perm A n 1) ^ (i mod n) : by rewrite (pow_mod rotl_perm_pow_eq_one)
... = rotl_perm A n (i mod n) : by rewrite rotl_perm_pow_eq
lemma rotl_perm_mod {i : nat} : rotl_perm A n i = rotl_perm A n (i % n) :=
calc rotl_perm A n i = (rotl_perm A n 1) ^ i : by rewrite rotl_perm_pow_eq
... = (rotl_perm A n 1) ^ (i % n) : by rewrite (pow_mod rotl_perm_pow_eq_one)
... = rotl_perm A n (i % n) : by rewrite rotl_perm_pow_eq
-- needs A to have at least two elements!
lemma rotl_perm_pow_ne_one (Pex : ∃ a b : A, a ≠ b) : ∀ i, i < n → (rotl_perm A (succ n) 1)^(succ i) ≠ 1 :=

13
library/theories/group_theory/pgroup.lean
View file

@ -3,7 +3,6 @@ Copyright (c) 2015 Haitao Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author : Haitao Zhang
-/
import theories.number_theory.primes data algebra.group algebra.group_power algebra.group_bigops
import .cyclic .finsubg .hom .perm .action
@ -29,7 +28,7 @@ include Hom
open finset.partition
lemma card_mod_eq_of_action_by_psubg {p : nat} :
∀ {m : nat}, psubg H p m → (card S) mod p = (card (fixed_points hom H)) mod p
∀ {m : nat}, psubg H p m → (card S) % p = (card (fixed_points hom H)) % p
| 0 := by rewrite [↑psubg, pow_zero]; intro Psubg;
rewrite [finsubg_eq_singleton_one_of_card_one (and.right Psubg), fixed_points_of_one]
| (succ m) := take Ppsubg, begin
@ -64,7 +63,7 @@ variables {H : finset G} [finsubgH : is_finsubg H]
include finsubgH
lemma card_psubg_cosets_mod_eq {p : nat} {m : nat} :
psubg H p m → (card (lcoset_type univ H)) mod p = card (lcoset_type (normalizer H) H) mod p :=
psubg H p m → (card (lcoset_type univ H)) % p = card (lcoset_type (normalizer H) H) % p :=
assume Psubg, by rewrite [-card_aol_fixed_points_eq_card_cosets]; exact card_mod_eq_of_action_by_psubg Psubg
end psubg_cosets
@ -258,7 +257,7 @@ lemma rotl_peo_seq_compose {n i j : nat} :
(rotl_peo_seq A n i) ∘ (rotl_peo_seq A n j) = rotl_peo_seq A n (j + i) :=
funext take s, subtype.eq begin rewrite [↑rotl_peo_seq, ↑rotl_perm, ↑rotl_fun, compose.assoc, rotl_compose] end
lemma rotl_peo_seq_mod {n i : nat} : rotl_peo_seq A n i = rotl_peo_seq A n (i mod succ n) :=
lemma rotl_peo_seq_mod {n i : nat} : rotl_peo_seq A n i = rotl_peo_seq A n (i % succ n) :=
funext take s, subtype.eq begin rewrite [↑rotl_peo_seq, rotl_perm_mod] end
lemma rotl_peo_seq_inj {n m : nat} : injective (rotl_peo_seq A n m) :=
@ -323,11 +322,11 @@ let pp := nat.pred p, spp := nat.succ pp in
assert Peq : spp = p, from succ_pred_prime Pprime,
assert Ppsubg : psubg (@univ (fin spp) _) spp 1,
from and.intro (eq.symm Peq ▸ Pprime) (by rewrite [Peq, card_fin, pow_one]),
have (pow_nat (card A) pp) mod spp = (card (fixed_points (rotl_perm_ps A pp) univ)) mod spp,
have (pow_nat (card A) pp) % spp = (card (fixed_points (rotl_perm_ps A pp) univ)) % spp,
by rewrite -card_peo_seq; apply card_mod_eq_of_action_by_psubg Ppsubg,
have Pcardmod : (pow_nat (card A) pp) mod p = (card (fixed_points (rotl_perm_ps A pp) univ)) mod p,
have Pcardmod : (pow_nat (card A) pp) % p = (card (fixed_points (rotl_perm_ps A pp) univ)) % p,
from Peq ▸ this,
have Pfpcardmod : (card (fixed_points (rotl_perm_ps A pp) univ)) mod p = 0,
have Pfpcardmod : (card (fixed_points (rotl_perm_ps A pp) univ)) % p = 0,
from eq.trans (eq.symm Pcardmod) (mod_eq_zero_of_dvd (dvd_pow_of_dvd_of_pos Pdvd (pred_prime_pos Pprime))),
have Pfpcardpos : card (fixed_points (rotl_perm_ps A pp) univ) > 0,
from card_pos_of_mem peo_seq_one_mem_fixed_points,

18
library/theories/number_theory/bezout.lean
View file

@ -20,7 +20,7 @@ private definition pair_nat.lt.wf : well_founded pair_nat.lt := intro_k (measure
local attribute pair_nat.lt.wf [instance]
local infixl `≺`:50 := pair_nat.lt
private definition gcd.lt.dec (x y₁ : ℕ) : (succ y₁, x mod succ y₁) ≺ (x, succ y₁) :=
private definition gcd.lt.dec (x y₁ : ℕ) : (succ y₁, x % succ y₁) ≺ (x, succ y₁) :=
!nat.mod_lt (succ_pos y₁)
private definition egcd_rec_f (z : ℤ) : ℤ → ℤ → ℤ × ℤ := λ s t, (t, s - t * z)
@ -29,8 +29,8 @@ definition egcd.F : Π (p₁ : ℕ × ℕ), (Π p₂ : ℕ × ℕ, p₂ ≺ p₁
| (x, y) := nat.cases_on y
(λ f, (1, 0) )
(λ y₁ (f : Π p₂, p₂ ≺ (x, succ y₁) → ℤ × ℤ),
let bz := f (succ y₁, x mod succ y₁) !gcd.lt.dec in
prod.cases_on bz (egcd_rec_f (x div succ y₁)))
let bz := f (succ y₁, x % succ y₁) !gcd.lt.dec in
prod.cases_on bz (egcd_rec_f (x / succ y₁)))
definition egcd (x y : ℕ) := fix egcd.F (pair x y)
@ -38,12 +38,12 @@ theorem egcd_zero (x : ℕ) : egcd x 0 = (1, 0) :=
well_founded.fix_eq egcd.F (x, 0)
theorem egcd_succ (x y : ℕ) :
egcd x (succ y) = prod.cases_on (egcd (succ y) (x mod succ y)) (egcd_rec_f (x div succ y)) :=
egcd x (succ y) = prod.cases_on (egcd (succ y) (x % succ y)) (egcd_rec_f (x / succ y)) :=
well_founded.fix_eq egcd.F (x, succ y)
theorem egcd_of_pos (x : ℕ) {y : ℕ} (ypos : y > 0) :
let erec := egcd y (x mod y), u := pr₁ erec, v := pr₂ erec in
egcd x y = (v, u - v * (x div y)) :=
let erec := egcd y (x % y), u := pr₁ erec, v := pr₂ erec in
egcd x y = (v, u - v * (x / y)) :=
obtain (y' : nat) (yeq : y = succ y'), from exists_eq_succ_of_pos ypos,
begin
rewrite [yeq, egcd_succ, -prod.eta (egcd _ _)],
@ -64,11 +64,11 @@ gcd.induction x y
rewrite [gcd_rec, -IH],
rewrite [algebra.add.comm],
rewrite [-of_nat_mod],
rewrite [int.modulo.def],
rewrite [int.mod_def],
rewrite [+algebra.mul_sub_right_distrib],
rewrite [+algebra.mul_sub_left_distrib, *left_distrib],
rewrite [*sub_eq_add_neg, {pr₂ (egcd n (m mod n)) * of_nat m + - _}algebra.add.comm, -algebra.add.assoc],
rewrite [algebra.mul.assoc]
rewrite [*sub_eq_add_neg, {pr₂ (egcd n (m % n)) * of_nat m + - _}algebra.add.comm],
rewrite [-algebra.add.assoc ,algebra.mul.assoc]
end)
theorem Bezout_aux (x y : ℕ) : ∃ a b : ℤ, a * x + b * y = gcd x y :=

6
library/theories/number_theory/irrational_roots.lean
View file

@ -203,11 +203,11 @@ section
from gcd_mul_right_cancel_of_coprime _ (coprime_swap co),
have a * a = c * b * b,
by rewrite -mul.assoc at H; apply H,
have a div (gcd a b) = c * b div gcd (c * b) a,
have a / (gcd a b) = c * b / gcd (c * b) a,
from div_gcd_eq_div_gcd this bpos apos,
have a = c * b div gcd c a, using this,
have a = c * b / gcd c a, using this,
by revert this; rewrite [↑coprime at co, co, int.div_one, H₁]; intros; assumption,
have a = b * (c div gcd c a), using this,
have a = b * (c / gcd c a), using this,
by revert this; rewrite [mul.comm, !int.mul_div_assoc !gcd_dvd_left]; intros; assumption,
have b ∣ a,
from dvd_of_mul_right_eq this⁻¹,

20
library/theories/number_theory/prime_factorization.lean
View file

@ -28,7 +28,7 @@ end
/- multiplicity -/
theorem mult_rec_decreasing {p n : ℕ} (Hp : p > 1) (Hn : n > 0) : n div p < n :=
theorem mult_rec_decreasing {p n : ℕ} (Hp : p > 1) (Hn : n > 0) : n / p < n :=
have H' : n < n * p,
by rewrite [-mul_one n at {1}]; apply mul_lt_mul_of_pos_left Hp Hn,
nat.div_lt_of_lt_mul H'
@ -41,7 +41,7 @@ else 0
definition mult (p n : ℕ) : ℕ := fix (mult.F p) n
theorem mult_rec {p n : ℕ} (pgt1 : p > 1) (ngt0 : n > 0) (pdivn : p ∣ n) :
mult p n = succ (mult p (n div p)) :=
mult p n = succ (mult p (n / p)) :=
have (p > 1 ∧ n > 0) ∧ p ∣ n, from and.intro (and.intro pgt1 ngt0) pdivn,
eq.trans (well_founded.fix_eq (mult.F p) n) (dif_pos this)
@ -95,11 +95,11 @@ begin
(take n',
suppose n = p * n',
have p > 0, from lt.trans zero_lt_one pgt1,
assert n div p = n', from !nat.div_eq_of_eq_mul_right this `n = p * n'`,
assert n / p = n', from !nat.div_eq_of_eq_mul_right this `n = p * n'`,
assert n' < n,
by rewrite -this; apply mult_rec_decreasing pgt1 npos,
begin
rewrite [mult_rec pgt1 npos pdvdn, `n div p = n'`, pow_succ], subst n,
rewrite [mult_rec pgt1 npos pdvdn, `n / p = n'`, pow_succ], subst n,
apply mul_dvd_mul !dvd.refl,
apply ih _ this
end)
@ -141,12 +141,12 @@ dvd.elim pidvd
assert m > 0, from pos_of_mul_pos_left (this ▸ npos),
by subst n; rewrite [mult_pow_mul i pgt1 this]; apply le_add_right)
theorem not_dvd_div_pow_mult {p n : ℕ} (pgt1 : p > 1) (npos : n > 0) : ¬ p ∣ n div p^(mult p n) :=
assume pdvd : p ∣ n div p^(mult p n),
obtain m (H : n div p^(mult p n) = p * m), from exists_eq_mul_right_of_dvd pdvd,
theorem not_dvd_div_pow_mult {p n : ℕ} (pgt1 : p > 1) (npos : n > 0) : ¬ p ∣ n / p^(mult p n) :=
assume pdvd : p ∣ n / p^(mult p n),
obtain m (H : n / p^(mult p n) = p * m), from exists_eq_mul_right_of_dvd pdvd,
assert n = p^(succ (mult p n)) * m, from
calc
n = p^mult p n * (n div p^mult p n) : by rewrite (nat.mul_div_cancel' !pow_mult_dvd)
n = p^mult p n * (n / p^mult p n) : by rewrite (nat.mul_div_cancel' !pow_mult_dvd)
... = p^(succ (mult p n)) * m : by rewrite [H, pow_succ', mul.assoc],
have p^(succ (mult p n)) ∣ n, by rewrite this at {2}; apply dvd_mul_right,
have succ (mult p n) ≤ mult p n, from le_mult pgt1 npos this,
@ -154,7 +154,7 @@ show false, from !not_succ_le_self this
theorem mult_mul {p m n : ℕ} (primep : prime p) (mpos : m > 0) (npos : n > 0) :
mult p (m * n) = mult p m + mult p n :=
let m' := m div p^mult p m, n' := n div p^mult p n in
let m' := m / p^mult p m, n' := n / p^mult p n in
assert p > 1, from gt_one_of_prime primep,
assert meq : m = p^mult p m * m', by rewrite (nat.mul_div_cancel' !pow_mult_dvd),
assert neq : n = p^mult p n * n', by rewrite (nat.mul_div_cancel' !pow_mult_dvd),
@ -168,7 +168,7 @@ assert multm'n' : mult p (m' * n') = 0, from mult_eq_zero_of_not_dvd npdvdm'n',
calc
mult p (m * n) = mult p (p^(mult p m + mult p n) * (m' * n')) :
by rewrite [pow_add, mul.right_comm, -mul.assoc, -meq, mul.assoc,
mul.comm (n div _), -neq]
mul.comm (n / _), -neq]
... = mult p m + mult p n :
by rewrite [!mult_pow_mul `p > 1` m'n'pos, multm'n']

2
library/theories/number_theory/primes.lean
View file

@ -63,7 +63,7 @@ assume h, succ_pred_of_pos (pos_of_prime h)
lemma eq_one_or_eq_self_of_prime_of_dvd {p m : nat} : prime p → m ∣ p → m = 1 ∨ m = p :=
assume h d, obtain h₁ h₂, from h, h₂ m d
lemma gt_one_of_pos_of_prime_dvd {i p : nat} : prime p → 0 < i → i mod p = 0 → 1 < i :=
lemma gt_one_of_pos_of_prime_dvd {i p : nat} : prime p → 0 < i → i % p = 0 → 1 < i :=
assume ipp pos h,
have p ≥ 2, from ge_two_of_prime ipp,
have p ∣ i, from dvd_of_mod_eq_zero h,

4
tests/lean/local_notation_bug2.lean
View file

@ -2,6 +2,6 @@ open nat
section
parameters (b : ℕ)
definition add_b (n : ℕ) := n + b
local postfix `%`:max := add_b
local postfix `%%`:max := add_b
end
eval 5% -- Error, unexpected token
eval 5%% -- Error, unexpected token

2
tests/lean/local_notation_bug2.lean.expected.out
View file

@ -1 +1 @@
local_notation_bug2.lean:7:6: error: unexpected token
local_notation_bug2.lean:7:7: error: invalid expression

4
tests/lean/noncomp_theory.lean.expected.out
View file

@ -1,5 +1,5 @@
noncomp_theory.lean:4:0: error: definition 'f' is noncomputable, it depends on 'real.real_has_division'
noncomp_theory.lean:4:0: error: definition 'f' is noncomputable, it depends on 'real.real_has_div'
noncomputable definition g : ℝ → ℝ → ℝ :=
λ (a : ℝ), division (a + a)
λ (a : ℝ), div (a + a)
definition r : ℕ → ℕ :=
λ (a : ℕ), a

2
tests/lean/run/occurs_check_bug1.lean
View file

@ -4,7 +4,7 @@ open nat prod
open decidable
constant modulo1 (x : ℕ) (y : ℕ) : ℕ
infixl `mod` := modulo1
infixl `mod`:70 := modulo1
constant gcd_aux : ℕ × ℕ → ℕ

10
tests/lean/struct_class.lean.expected.out
View file

@ -1,12 +1,11 @@
decidable : Prop → Type₁
has_add : Type → Type
has_divide : Type → Type
has_division : Type → Type
has_div : Type → Type
has_dvd : Type → Type
has_inv : Type → Type
has_le : Type → Type
has_lt : Type → Type
has_modulo : Type → Type
has_mod : Type → Type
has_mul : Type → Type
has_neg : Type → Type
has_one : Type → Type
@ -21,13 +20,12 @@ subsingleton : Type → Prop
well_founded : Π {A : Type}, (A → A → Prop) → Prop
decidable : Prop → Type₁
has_add : Type → Type
has_divide : Type → Type
has_division : Type → Type
has_div : Type → Type
has_dvd : Type → Type
has_inv : Type → Type
has_le : Type → Type
has_lt : Type → Type
has_modulo : Type → Type
has_mod : Type → Type
has_mul : Type → Type
has_neg : Type → Type
has_one : Type → Type