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fix(library/data): hf, int, nat, pnat

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Leonardo de Moura 2015-10-09 12:47:55 -07:00
parent 3369152559
commit 27b4eb2058
14 changed files with 82 additions and 58 deletions
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1
library/data/fintype/function.lean
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@ -7,6 +7,7 @@ Author : Haitao Zhang
import data
open nat function eq.ops
open - [notations] algebra
namespace list
-- this is in preparation for counting the number of finite functions

4
library/data/hf.lean
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@ -11,8 +11,8 @@ this bijection is implemeted using the Ackermann coding.
-/
import data.nat data.finset.equiv data.list
open nat binary
open -[notations] finset
open -[notations] algebra
open - [notations] finset
open - [notations] algebra
definition hf := nat

4
library/data/int/basic.lean
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@ -523,10 +523,10 @@ protected definition integral_domain [reducible] [instance] : algebra.integral_d
eq_zero_or_eq_zero_of_mul_eq_zero := @eq_zero_or_eq_zero_of_mul_eq_zero⦄
definition int_has_sub [reducible] [instance] [priority int.prio] : has_sub int :=
has_sub.mk algebra.sub
has_sub.mk has_sub.sub
definition int_has_dvd [reducible] [instance] [priority int.prio] : has_dvd int :=
has_dvd.mk algebra.dvd
has_dvd.mk has_dvd.dvd
set_option pp.coercions true

4
library/data/int/div.lean
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@ -24,8 +24,8 @@ sign b *
| -[1+m] := -[1+ ((m:nat) div (nat_abs b))]
end)
definition int_has_divides [reducible] [instance] [priority int.prio] : has_divides int :=
has_divides.mk int.divide
definition int_has_divide [reducible] [instance] [priority int.prio] : has_divide int :=
has_divide.mk int.divide
lemma divide_of_nat (a : nat) (b : ℤ) : (of_nat a) div b = sign b * (a div (nat_abs b) : nat) :=
rfl

38
library/data/int/order.lean
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@ -116,11 +116,11 @@ obtain (n : ℕ) (Hn : a + n = b), from le.elim H₁,
obtain (m : ℕ) (Hm : b + m = a), from le.elim H₂,
have a + of_nat (n + m) = a + 0, from
calc
a + of_nat (n + m) = a + (of_nat n + m) : of_nat_add
... = a + n + m : add.assoc
... = b + m : Hn
... = a : Hm
... = a + 0 : add_zero,
a + of_nat (n + m) = a + (of_nat n + m) : by rewrite of_nat_add
... = a + n + m : by rewrite add.assoc
... = b + m : by rewrite Hn
... = a : by rewrite Hm
... = a + 0 : by rewrite add_zero,
have of_nat (n + m) = of_nat 0, from add.left_cancel this,
have n + m = 0, from of_nat.inj this,
have n = 0, from nat.eq_zero_of_add_eq_zero_right this,
@ -193,11 +193,11 @@ obtain (m : ℕ) (Hm : 0 + m = b), from le.elim Hb,
le.intro
(eq.symm
(calc
a * b = (0 + n) * b : Hn
... = n * b : zero_add
... = n * (0 + m) : {Hm⁻¹}
... = n * m : zero_add
... = 0 + n * m : zero_add))
a * b = (0 + n) * b : by rewrite Hn
... = n * b : by rewrite zero_add
... = n * (0 + m) : by rewrite Hm
... = n * m : by rewrite zero_add
... = 0 + n * m : by rewrite zero_add))
theorem mul_pos {a b : ℤ} (Ha : 0 < a) (Hb : 0 < b) : 0 < a * b :=
obtain (n : ℕ) (Hn : 0 + nat.succ n = a), from lt.elim Ha,
@ -205,14 +205,14 @@ obtain (m : ℕ) (Hm : 0 + nat.succ m = b), from lt.elim Hb,
lt.intro
(eq.symm
(calc
a * b = (0 + nat.succ n) * b : Hn
... = nat.succ n * b : zero_add
... = nat.succ n * (0 + nat.succ m) : {Hm⁻¹}
... = nat.succ n * nat.succ m : zero_add
... = of_nat (nat.succ n * nat.succ m) : of_nat_mul
... = of_nat (nat.succ n * m + nat.succ n) : nat.mul_succ
... = of_nat (nat.succ (nat.succ n * m + n)) : nat.add_succ
... = 0 + nat.succ (nat.succ n * m + n) : zero_add))
a * b = (0 + nat.succ n) * b : by rewrite Hn
... = nat.succ n * b : by rewrite zero_add
... = nat.succ n * (0 + nat.succ m) : by rewrite Hm
... = nat.succ n * nat.succ m : by rewrite zero_add
... = of_nat (nat.succ n * nat.succ m) : by rewrite of_nat_mul
... = of_nat (nat.succ n * m + nat.succ n) : by rewrite nat.mul_succ
... = of_nat (nat.succ (nat.succ n * m + n)) : by rewrite nat.add_succ
... = 0 + nat.succ (nat.succ n * m + n) : by rewrite zero_add))
theorem zero_lt_one : (0 : ℤ) < 1 := trivial
@ -321,7 +321,7 @@ theorem lt_of_sub_one_le {a b : ℤ} (H : a - 1 < b) : a ≤ b :=
!sub_add_cancel ▸ add_one_le_of_lt H
theorem le_sub_one_of_lt {a b : ℤ} (H : a < b) : a ≤ b - 1 :=
le_of_lt_add_one (!sub_add_cancel⁻¹ ▸ H)
le_of_lt_add_one begin rewrite algebra.sub_add_cancel, exact H end
theorem lt_of_le_sub_one {a b : ℤ} (H : a ≤ b - 1) : a < b :=
!sub_add_cancel ▸ (lt_add_one_of_le H)

2
library/data/int/power.lean
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@ -11,7 +11,7 @@ namespace int
open - [notations] algebra
definition int_has_pow_nat : has_pow_nat int :=
has_pow_nat.mk pow_nat
has_pow_nat.mk has_pow_nat.pow_nat
/-
definition nmul (n : ℕ) (a : ℤ) : ℤ := algebra.nmul n a

6
library/data/nat/div.lean
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@ -20,10 +20,10 @@ 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
definition divide := fix div.F
protected definition divide := fix div.F
definition nat_has_divides [reducible] [instance] [priority nat.prio] : has_divides nat :=
has_divides.mk divide
definition nat_has_divide [reducible] [instance] [priority nat.prio] : has_divide nat :=
has_divide.mk nat.divide
theorem divide_def (x y : nat) : divide x y = if 0 < y ∧ y ≤ x then divide (x - y) y + 1 else 0 :=
congr_fun (fix_eq div.F x) y

3
library/data/nat/gcd.lean
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@ -257,6 +257,9 @@ dvd.antisymm
definition coprime [reducible] (m n : ℕ) : Prop := gcd m n = 1
lemma gcd_eq_one_of_coprime {m n : ℕ} : coprime m n → gcd m n = 1 :=
λ h, h
theorem coprime_swap {m n : ℕ} (H : coprime n m) : coprime m n :=
!gcd.comm ▸ H

4
library/data/nat/order.lean
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@ -134,7 +134,7 @@ else (eq_max_left h) ▸ !le.refl
/- nat is an instance of a linearly ordered semiring and a lattice -/
open -[notations] algebra
open - [notations] algebra
protected definition decidable_linear_ordered_semiring [reducible] [instance] :
algebra.decidable_linear_ordered_semiring nat :=
@ -165,7 +165,7 @@ algebra.decidable_linear_ordered_semiring nat :=
definition nat_has_dvd [reducible] [instance] [priority nat.prio] : has_dvd nat :=
has_dvd.mk algebra.dvd
has_dvd.mk has_dvd.dvd
theorem add_pos_left {a : ℕ} (H : 0 < a) (b : ℕ) : 0 < a + b :=
@algebra.add_pos_of_pos_of_nonneg _ _ a b H !zero_le

9
library/data/nat/power.lean
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@ -10,23 +10,22 @@ open - [notations] algebra
namespace nat
definition nat_has_pow_nat : has_pow_nat nat :=
has_pow_nat.mk pow_nat
definition nat_has_pow_nat [instance] [reducible] [priority nat.prio] : has_pow_nat nat :=
has_pow_nat.mk has_pow_nat.pow_nat
theorem pow_le_pow_of_le {x y : ℕ} (i : ℕ) (H : x ≤ y) : x^i ≤ y^i :=
algebra.pow_le_pow_of_le i !zero_le H
theorem eq_zero_of_pow_eq_zero {a m : ℕ} (H : a^m = 0) : a = 0 :=
or.elim (eq_zero_or_pos m)
(suppose m = 0,
by rewrite [`m = 0` at H, pow_zero at H]; contradiction)
by krewrite [`m = 0` at H, pow_zero at H]; contradiction)
(suppose m > 0,
have h₁ : ∀ m, a^succ m = 0 → a = 0,
begin
intro m,
induction m with m ih,
{rewrite pow_one; intros; assumption},
{krewrite pow_one; intros; assumption},
rewrite pow_succ,
intro H,
cases eq_zero_or_eq_zero_of_mul_eq_zero H with h₃ h₄,

2
library/data/nat/sub.lean
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@ -289,7 +289,7 @@ sub.cases
... = k - n + n : sub_add_cancel H3,
le.intro (add.cancel_right H4))
open -[notations] algebra
open - [notations] algebra
theorem sub_pos_of_lt {m n : ℕ} (H : m < n) : n - m > 0 :=
assert H1 : n = n - m + m, from (sub_add_cancel (le_of_lt H))⁻¹,

1
library/data/pnat.lean
View file

@ -10,6 +10,7 @@ are those needed for that construction.
-/
import data.rat.order data.nat
open nat rat subtype eq.ops
open - [notations] algebra
namespace pnat

17
library/init/reserved_notation.lean
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@ -102,7 +102,7 @@ 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_divides [class] (A : Type) := (divides : 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)
@ -112,7 +112,7 @@ 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 divides {A : Type} [s : has_divides A] : A → A → A := has_divides.divides
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
@ -126,7 +126,7 @@ infix + := add
infix * := mul
infix - := sub
infix / := division
infix div := divides
infix div := divide
infix ∣ := dvd
infix mod := modulo
prefix - := neg
@ -135,3 +135,14 @@ infix ≤ := le
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

45
library/theories/number_theory/irrational_roots.lean
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@ -7,6 +7,7 @@ A proof that if n > 1 and a > 0, then the nth root of a is irrational, unless a
-/
import data.rat .prime_factorization
open eq.ops
open - [notations] algebra
/- First, a textbook proof that sqrt 2 is irrational. -/
@ -15,15 +16,15 @@ section
theorem sqrt_two_irrational {a b : ℕ} (co : coprime a b) : a^2 ≠ 2 * b^2 :=
assume H : a^2 = 2 * b^2,
have even (a^2), from even_of_exists (exists.intro _ H),
have even a, from even_of_even_pow this,
obtain c (aeq : a = 2 * c), from exists_of_even this,
have 2 * (2 * c^2) = 2 * b^2, by rewrite [-H, aeq, *pow_two, mul.assoc, mul.left_comm c],
have 2 * c^2 = b^2, from eq_of_mul_eq_mul_left dec_trivial this,
have even (b^2), from even_of_exists (exists.intro _ (eq.symm this)),
have even b, from even_of_even_pow this,
have 2 ∣ gcd a b, from dvd_gcd (dvd_of_even `even a`) (dvd_of_even `even b`),
have 2 ∣ 1, from co ▸ this,
have even (a^2), from even_of_exists (exists.intro _ H),
have even a, from even_of_even_pow this,
obtain (c : nat) (aeq : a = 2 * c), from exists_of_even this,
have 2 * (2 * c^2) = 2 * b^2, begin rewrite [-H, aeq, *pow_two, algebra.mul.assoc, algebra.mul.left_comm c] end,
have 2 * c^2 = b^2, from eq_of_mul_eq_mul_left dec_trivial this,
have even (b^2), from even_of_exists (exists.intro _ (eq.symm this)),
have even b, from even_of_even_pow this,
assert 2 ∣ gcd a b, from dvd_gcd (dvd_of_even `even a`) (dvd_of_even `even b`),
have 2 ∣ 1, begin rewrite [gcd_eq_one_of_coprime co at this], exact this end,
show false, from absurd `2 ∣ 1` dec_trivial
end
@ -41,7 +42,7 @@ section
b = 1 :=
have bpos : b > 0, from pos_of_ne_zero
(suppose b = 0,
have a^n = 0, by rewrite [H, this, zero_pow npos],
have a^n = 0, by krewrite [H, this, zero_pow npos],
assert a = 0, from eq_zero_of_pow_eq_zero this,
show false, from ne_of_lt `0 < a` this⁻¹),
have H₁ : ∀ p, prime p → ¬ p ∣ b, from
@ -51,12 +52,12 @@ section
have p ∣ a, from dvd_of_prime_of_dvd_pow `prime p` this,
have ¬ coprime a b, from not_coprime_of_dvd_of_dvd (gt_one_of_prime `prime p`) `p ∣ a` `p ∣ b`,
show false, from this `coprime a b`,
have b < 2, from by_contradiction
have blt2 : b < 2, from by_contradiction
(suppose ¬ b < 2,
have b ≥ 2, from le_of_not_gt this,
obtain p [primep pdvdb], from exists_prime_and_dvd this,
show false, from H₁ p primep pdvdb),
show b = 1, from (le.antisymm (le_of_lt_succ `b < 2`) (succ_le_of_lt `b > 0`))
show b = 1, from (le.antisymm (le_of_lt_succ blt2) (succ_le_of_lt bpos))
end
/-
@ -72,13 +73,13 @@ section
let a := num q, b := denom q in
have b ≠ 0, from ne_of_gt (denom_pos q),
have bnz : b ≠ (0 : ℚ), from assume H, `b ≠ 0` (of_int.inj H),
have bnnz : (#rat b^n ≠ 0), from assume bneqz, bnz (eq_zero_of_pow_eq_zero bneqz),
have a^n / b^n = c, using bnz, by rewrite [*of_int_pow, -div_pow, -eq_num_div_denom, -H],
have a^n = c * b^n, from eq.symm (!mul_eq_of_eq_div bnnz this⁻¹),
have bnnz : ((b : rat)^n ≠ 0), from assume bneqz, bnz (eq_zero_of_pow_eq_zero bneqz),
have a^n /[rat] b^n = c, using bnz, begin krewrite [*of_int_pow, -div_pow, -eq_num_div_denom, -H] end,
have (a^n : rat) = c *[rat] b^n, from eq.symm (!mul_eq_of_eq_div bnnz this⁻¹),
have a^n = c * b^n, -- int version
using this, by rewrite [-of_int_pow at this, -of_int_mul at this]; exact of_int.inj this,
have (abs a)^n = abs c * (abs b)^n,
using this, by rewrite [-int.abs_pow, this, int.abs_mul, int.abs_pow],
using this, by rewrite [-abs_pow, this, abs_mul, abs_pow],
have H₁ : (nat_abs a)^n = nat_abs c * (nat_abs b)^n,
using this,
by apply int.of_nat.inj; rewrite [int.of_nat_mul, +int.of_nat_pow, +of_nat_nat_abs]; assumption,
@ -87,11 +88,19 @@ section
by_cases
(suppose q = 0, by rewrite this)
(suppose q ≠ 0,
have a ≠ 0, from suppose a = 0, `q ≠ 0` (by rewrite [eq_num_div_denom, `a = 0`, zero_div]),
have nat_abs a ≠ 0, from suppose nat_abs a = 0, `a ≠ 0` (eq_zero_of_nat_abs_eq_zero this),
have ane0 : a ≠ 0, from
suppose aeq0 : a = 0,
have q = 0, begin rewrite eq_num_div_denom, rewrite aeq0, krewrite zero_div end,
absurd `q = 0` `q ≠ 0`,
have nat_abs a ≠ 0, from
suppose nat_abs a = 0,
have a = 0, from eq_zero_of_nat_abs_eq_zero this,
absurd `a = 0` `a ≠ 0`,
show nat_abs b = 1, from (root_irrational npos (pos_of_ne_zero this) H₂ H₁)),
show b = 1, using this, by rewrite [-of_nat_nat_abs_of_nonneg (le_of_lt !denom_pos), this]
exit
theorem eq_num_pow_of_pow_eq {q : ℚ} {n : ℕ} {c : ℤ} (npos : n > 0) (H : q^n = c) :
c = (num q)^n :=
have denom q = 1, from denom_eq_one_of_pow_eq npos H,