86 lines
2.1 KiB
Text
86 lines
2.1 KiB
Text
-- Copyright (c) 2014 Jeremy Avigad. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Authors: Jeremy Avigad, Leonardo de Moura
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-- algebra.group
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-- =============
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-- Various structures with 1, *, inv, including groups.
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open eq
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namespace algebra
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-- classes for notation
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-- --------------------
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inductive [class] has_mul (A : Type) : Type | mk : (A → A → A) → has_mul
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inductive [class] has_one (A : Type) : Type | mk : A → has_one
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inductive [class] has_inv (A : Type) : Type | mk : (A → A) → has_inv
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definition mul {A : Type} [s : has_mul A] (a b : A) : A := has_mul.rec (λf, f) s a b
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definition one {A : Type} [s : has_one A] : A := has_one.rec (λo, o) s
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definition inv {A : Type} [s : has_inv A] (a : A) : A := has_inv.rec (λi, i) s a
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infix `*` := mul
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postfix `⁻¹` := inv
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notation 1 := one
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-- semigroup
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-- ---------
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inductive [class] semigroup (A : Type) : Type
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| mk : Π mul: A → A → A,
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(∀a b c : A, (mul (mul a b) c = mul a (mul b c))) →
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semigroup
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namespace semigroup
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section
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variables {A : Type} [s : semigroup A]
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variables a b c : A
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definition mul := semigroup.rec (λmul assoc, mul) s a b
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section
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infixl `*` := mul
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end
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end
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end semigroup
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section
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variables {A : Type} [s : semigroup A]
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include s
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attribute [instance]
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definition semigroup_has_mul : has_mul A := has_mul.mk semigroup.mul
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end
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-- comm_semigroup
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-- --------------
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inductive [class] comm_semigroup (A : Type) : Type
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| mk : Π (mul: A → A → A)
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(infixl `*` := mul),
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(∀a b c, (a * b) * c = a * (b * c)) →
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(∀a b, a * b = b * a) →
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comm_semigroup
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namespace comm_semigroup
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section
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variables {A : Type} [s : comm_semigroup A]
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variables a b c : A
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definition mul (a b : A) : A := comm_semigroup.rec (λmul assoc comm, mul) s a b
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end
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end comm_semigroup
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-- monoid
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-- ------
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inductive [class] monoid (A : Type) : Type
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| mk : Π (mul: A → A → A) (one : A)
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(infixl `*` := mul) (notation 1 := one),
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(∀a b c, (a * b) * c = a * (b * c)) →
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(∀a, a * 1 = a) →
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(∀a, 1 * a = a) →
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monoid
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end algebra
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