michael/lean2
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

refactor(library/data/nat): merge comm_semiring, make minor fixes

Browse source
This commit is contained in:
Jeremy Avigad 2015-02-01 09:55:12 -05:00 committed by Leonardo de Moura
parent 15716c1471
commit 95d79e7bda
6 changed files with 57 additions and 71 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

48
library/data/nat/basic.lean
View file

@ -7,7 +7,7 @@ Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad
Basic operations on the natural numbers.
-/
import logic.connectives data.num algebra.binary
import logic.connectives data.num algebra.binary algebra.ring
open binary eq.ops
namespace nat
@ -286,4 +286,50 @@ cases_on n
... = succ n' * m' + succ n' : mul_succ
... = succ (succ n' * m' + n') : add_succ)⁻¹)
!succ_ne_zero))
section
open [classes] algebra
protected definition comm_semiring [instance] [reducible] : algebra.comm_semiring nat :=
algebra.comm_semiring.mk add add.assoc zero zero_add add_zero add.comm
mul mul.assoc (succ zero) one_mul mul_one mul.left_distrib mul.right_distrib
zero_mul mul_zero (ne.symm (succ_ne_zero zero)) mul.comm
end
section port_algebra
theorem mul.left_comm : ∀a b c : ℕ, a * (b * c) = b * (a * c) := algebra.mul.left_comm
theorem mul.right_comm : ∀a b c : ℕ, (a * b) * c = (a * c) * b := algebra.mul.right_comm
definition dvd (a b : ℕ) : Prop := algebra.dvd a b
infix `|` := dvd
theorem dvd.intro : ∀{a b c : ℕ} (H : a * c = b), a | b := @algebra.dvd.intro _ _
theorem dvd.intro_left : ∀{a b c : ℕ} (H : c * a = b), a | b := @algebra.dvd.intro_left _ _
theorem exists_eq_mul_right_of_dvd : ∀{a b : ℕ} (H : a | b), ∃c, b = a * c :=
@algebra.exists_eq_mul_right_of_dvd _ _
theorem dvd.elim : ∀{P : Prop} {a b : ℕ} (H₁ : a | b) (H₂ : ∀c, b = a * c → P), P :=
@algebra.dvd.elim _ _
theorem exists_eq_mul_left_of_dvd : ∀{a b : ℕ} (H : a | b), ∃c, b = c * a :=
@algebra.exists_eq_mul_left_of_dvd _ _
theorem dvd.elim_left : ∀{P : Prop} {a b : ℕ} (H₁ : a | b) (H₂ : ∀c, b = c * a → P), P :=
@algebra.dvd.elim_left _ _
theorem dvd.refl : ∀a : ℕ, a | a := algebra.dvd.refl
theorem dvd.trans : ∀{a b c : ℕ} (H₁ : a | b) (H₂ : b | c), a | c := @algebra.dvd.trans _ _
theorem eq_zero_of_zero_dvd : ∀{a : ℕ} (H : 0 | a), a = 0 := @algebra.eq_zero_of_zero_dvd _ _
theorem dvd_zero : ∀a : ℕ, a | 0 := algebra.dvd_zero
theorem one_dvd : ∀a : ℕ, 1 | a := algebra.one_dvd
theorem dvd_mul_right : ∀a b : ℕ, a | a * b := algebra.dvd_mul_right
theorem dvd_mul_left : ∀a b : ℕ, a | b * a := algebra.dvd_mul_left
theorem dvd_mul_of_dvd_left : ∀{a b : ℕ} (H : a | b) (c : ℕ), a | b * c :=
@algebra.dvd_mul_of_dvd_left _ _
theorem dvd_mul_of_dvd_right : ∀{a b : ℕ} (H : a | b) (c : ℕ), a | c * b :=
@algebra.dvd_mul_of_dvd_right _ _
theorem mul_dvd_mul : ∀{a b c d : ℕ}, a | b → c | d → a * c | b * d :=
@algebra.mul_dvd_mul _ _
theorem dvd_of_mul_right_dvd : ∀{a b c : ℕ}, a * b | c → a | c :=
@algebra.dvd_of_mul_right_dvd _ _
theorem dvd_of_mul_left_dvd : ∀{a b c : ℕ}, a * b | c → b | c :=
@algebra.dvd_of_mul_left_dvd _ _
theorem dvd_add : ∀{a b c : ℕ}, a | b → a | c → a | b + c := @algebra.dvd_add _ _
end port_algebra
end nat

58
library/data/nat/comm_semiring.lean
View file

@ -1,58 +0,0 @@
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: data.nat.comm_semiring
Author: Jeremy Avigad
ℕ is a comm_semiring.
-/
import data.nat.basic algebra.binary algebra.ring
open binary
namespace nat
section
open [classes] algebra
protected definition comm_semiring [instance] [reducible] : algebra.comm_semiring nat :=
algebra.comm_semiring.mk add add.assoc zero zero_add add_zero add.comm
mul mul.assoc (succ zero) one_mul mul_one mul.left_distrib mul.right_distrib
zero_mul mul_zero (ne.symm (succ_ne_zero zero)) mul.comm
end
section port_algebra
theorem mul.left_comm : ∀a b c : ℕ, a * (b * c) = b * (a * c) := algebra.mul.left_comm
theorem mul.right_comm : ∀a b c : ℕ, (a * b) * c = (a * c) * b := algebra.mul.right_comm
definition dvd (a b : ℕ) : Prop := algebra.dvd a b
infix `|` := dvd
theorem dvd.intro : ∀{a b c : ℕ} (H : a * c = b), a | b := @algebra.dvd.intro _ _
theorem dvd.intro_left : ∀{a b c : ℕ} (H : c * a = b), a | b := @algebra.dvd.intro_left _ _
theorem exists_eq_mul_right_of_dvd : ∀{a b : ℕ} (H : a | b), ∃c, b = a * c :=
@algebra.exists_eq_mul_right_of_dvd _ _
theorem dvd.elim : ∀{P : Prop} {a b : ℕ} (H₁ : a | b) (H₂ : ∀c, b = a * c → P), P :=
@algebra.dvd.elim _ _
theorem exists_eq_mul_left_of_dvd : ∀{a b : ℕ} (H : a | b), ∃c, b = c * a :=
@algebra.exists_eq_mul_left_of_dvd _ _
theorem dvd.elim_left : ∀{P : Prop} {a b : ℕ} (H₁ : a | b) (H₂ : ∀c, b = c * a → P), P :=
@algebra.dvd.elim_left _ _
theorem dvd.refl : ∀a : ℕ, a | a := algebra.dvd.refl
theorem dvd.trans : ∀{a b c : ℕ} (H₁ : a | b) (H₂ : b | c), a | c := @algebra.dvd.trans _ _
theorem eq_zero_of_zero_dvd : ∀{a : ℕ} (H : 0 | a), a = 0 := @algebra.eq_zero_of_zero_dvd _ _
theorem dvd_zero : ∀a : ℕ, a | 0 := algebra.dvd_zero
theorem one_dvd : ∀a : ℕ, 1 | a := algebra.one_dvd
theorem dvd_mul_right : ∀a b : ℕ, a | a * b := algebra.dvd_mul_right
theorem dvd_mul_left : ∀a b : ℕ, a | b * a := algebra.dvd_mul_left
theorem dvd_mul_of_dvd_left : ∀{a b : ℕ} (H : a | b) (c : ℕ), a | b * c :=
@algebra.dvd_mul_of_dvd_left _ _
theorem dvd_mul_of_dvd_right : ∀{a b : ℕ} (H : a | b) (c : ℕ), a | c * b :=
@algebra.dvd_mul_of_dvd_right _ _
theorem mul_dvd_mul : ∀{a b c d : ℕ}, a | b → c | d → a * c | b * d :=
@algebra.mul_dvd_mul _ _
theorem dvd_of_mul_right_dvd : ∀{a b c : ℕ}, a * b | c → a | c :=
@algebra.dvd_of_mul_right_dvd _ _
theorem dvd_of_mul_left_dvd : ∀{a b c : ℕ}, a * b | c → b | c :=
@algebra.dvd_of_mul_left_dvd _ _
theorem dvd_add : ∀{a b c : ℕ}, a | b → a | c → a | b + c := @algebra.dvd_add _ _
end port_algebra
end nat

11
library/data/nat/default.lean
View file

@ -1,5 +1,6 @@
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Jeremy Avigad
import .basic .order .sub .div .bquant .comm_semiring
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Jeremy Avigad
-/
import .basic .order .sub .div .bquant

4
library/data/nat/div.lean
View file

@ -8,7 +8,7 @@ Authors: Jeremy Avigad, Leonardo de Moura
Definitions of div, mod, and gcd on the natural numbers.
-/
import data.nat.sub data.nat.comm_semiring tools.fake_simplifier
import data.nat.sub tools.fake_simplifier
open eq.ops well_founded decidable fake_simplifier prod
namespace nat
@ -357,7 +357,7 @@ or.elim (eq_zero_or_pos k)
calc
m * n div k = m * n div 0 : H1
... = 0 : div_zero
... = m * 0 : mul_zero m -- TODO: why do we have to specify m here?
... = m * 0 : mul_zero m
... = m * (n div 0) : div_zero
... = m * (n div k) : H1)
(assume H1 : k > 0,

3
library/data/nat/order.lean
View file

@ -7,8 +7,7 @@ Authors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad
The order relation on the natural numbers.
-/
import data.nat.basic data.nat.comm_semiring algebra.ordered_ring
import data.nat.basic algebra.ordered_ring
open eq.ops
namespace nat

4
library/data/nat/sub.lean
View file

@ -7,10 +7,8 @@ Module: data.nat.sub
Subtraction on the natural numbers, as well as min, max, and distance.
-/
import .order
import tools.fake_simplifier
open eq.ops
open fake_simplifier
@ -143,7 +141,7 @@ induction_on m
... = n * k - (l * k + k) : sub_sub
... = n * k - (succ l * k) : succ_mul)
theorem mul_sub_distr_left (n m k : ℕ) : n * (m - k) = n * m - n * k :=
theorem mul_sub_left_distrib (n m k : ℕ) : n * (m - k) = n * m - n * k :=
calc
n * (m - k) = (m - k) * n : !mul.comm
... = m * n - k * n : !mul_sub_right_distrib