feat(library/algebra/homomorphism): add homomorphisms between algebraic structures
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Jeremy Avigad
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3 changed files with 188 additions and 66 deletions
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@ -24,4 +24,5 @@ Algebraic structures.
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* [ring_bigops](ring_bigops.lean) : products and sums in various structures
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* [order_bigops](order_bigops.lean) : min and max over finsets and finite sets
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* [bundled](bundled.lean) : bundled versions of the algebraic structures
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* [homomorphism](homomorphism.lean) : homomorphisms between algebraic structures
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* [category](category/category.md) : category theory (outdated, see HoTT category theory folder)
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185
library/algebra/homomorphism.lean
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185
library/algebra/homomorphism.lean
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@ -0,0 +1,185 @@
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/-
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Copyright (c) 2016 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
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Homomorphisms between structures:
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is_add_hom : structures with has_add
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is_mul_hom : structures with has_mul
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is_module_hom : structures with has_add, has_smul
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is_ring_hom : structures with has_add, has_mul
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If you are working with a one particular kind of homomorphism, e.g. multiplicative, we recommend
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local abbreviation is_hom := @is_mul_hom
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These are all tentatively declared as type classes. The theorems which infer id and compose
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as instances are *not*, however, declared as instances: the first is rarely useful and the
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second makes class inference loop.
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Type class inference is useful here because usually a hypothesis like is_hom f is in the context.
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If you need an instance that the system does not infer, simply put it in the context, e.g.
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assert is_hom f, from ...,
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...
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-/
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import algebra.module data.set
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open function set
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variables {A B C : Type}
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/- additive structures -/
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definition add_ker [has_zero B] (f : A → B) : set A := {a | f a = 0}
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proposition add_ker_eq [has_zero B] (f : A → B) : add_ker f = f '- '{0} :=
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ext (take x, iff.intro
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(assume H, mem_preimage (mem_singleton_of_eq H))
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(assume H, eq_of_mem_singleton (mem_of_mem_preimage H)))
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structure is_add_hom [class] [has_add A] [has_add B] (f : A → B) : Prop :=
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(hom_add : ∀ a₁ a₂, f (a₁ + a₂) = f a₁ + f a₂)
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proposition hom_add [has_add A] [has_add B] (f : A → B) [H : is_add_hom f] (a₁ a₂ : A) :
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f (a₁ + a₂) = f a₁ + f a₂ := is_add_hom.hom_add _ _ f a₁ a₂
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proposition is_add_hom_id [has_add A] : is_add_hom (@id A) :=
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is_add_hom.mk (take a₁ a₂, rfl)
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proposition is_add_hom_comp [has_add A] [has_add B] [has_add C]
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{f : B → C} {g : A → B} [is_add_hom f] [is_add_hom g] : is_add_hom (f ∘ g) :=
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is_add_hom.mk (take a₁ a₂, by esimp; rewrite *hom_add)
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section add_group_A_B
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variables [add_group A] [add_group B]
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proposition hom_zero (f : A → B) [is_add_hom f] :
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f (0 : A) = 0 :=
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have f 0 + f 0 = f 0 + 0, by rewrite [-hom_add f, +add_zero],
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eq_of_add_eq_add_left this
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proposition hom_neg (f : A → B) [is_add_hom f] (a : A) :
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f (- a) = - f a :=
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have f (- a) + f a = 0, by rewrite [-hom_add f, add.left_inv, hom_zero],
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eq_neg_of_add_eq_zero this
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proposition hom_sub (f : A → B) [is_add_hom f] (a₁ a₂ : A) :
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f (a₁ - a₂) = f a₁ - f a₂ :=
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by rewrite [*sub_eq_add_neg, *hom_add, hom_neg]
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proposition injective_hom_add [add_group B] {f : A → B} [is_add_hom f]
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(H : ∀ x, f x = 0 → x = 0) :
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injective f :=
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take x₁ x₂,
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suppose f x₁ = f x₂,
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have f (x₁ - x₂) = 0, using this, by rewrite [hom_sub, this, sub_self],
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have x₁ - x₂ = 0, from H _ this,
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eq_of_sub_eq_zero this
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proposition eq_zero_of_injective_hom [add_group B] {f : A → B} [is_add_hom f]
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(injf : injective f) {a : A} (fa0 : f a = 0) :
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a = 0 :=
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have f a = f 0, by rewrite [fa0, hom_zero],
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show a = 0, from injf this
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end add_group_A_B
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/- multiplicative structures -/
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definition mul_ker [has_one B] (f : A → B) : set A := {a | f a = 1}
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proposition mul_ker_eq [has_one B] (f : A → B) : mul_ker f = f '- '{1} :=
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ext (take x, iff.intro
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(assume H, mem_preimage (mem_singleton_of_eq H))
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(assume H, eq_of_mem_singleton (mem_of_mem_preimage H)))
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structure is_mul_hom [class] [has_mul A] [has_mul B] (f : A → B) : Prop :=
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(hom_mul : ∀ a₁ a₂, f (a₁ * a₂) = f a₁ * f a₂)
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proposition hom_mul [has_mul A] [has_mul B] (f : A → B) [H : is_mul_hom f] (a₁ a₂ : A) :
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f (a₁ * a₂) = f a₁ * f a₂ := is_mul_hom.hom_mul _ _ f a₁ a₂
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proposition is_mul_hom_id [has_mul A] : is_mul_hom (@id A) :=
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is_mul_hom.mk (take a₁ a₂, rfl)
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proposition is_mul_hom_comp [has_mul A] [has_mul B] [has_mul C]
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{f : B → C} {g : A → B} [is_mul_hom f] [is_mul_hom g] : is_mul_hom (f ∘ g) :=
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is_mul_hom.mk (take a₁ a₂, by esimp; rewrite *hom_mul)
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section group_A_B
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variables [group A] [group B]
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proposition hom_one (f : A → B) [is_mul_hom f] :
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f (1 : A) = 1 :=
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have f 1 * f 1 = f 1 * 1, by rewrite [-hom_mul f, *mul_one],
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eq_of_mul_eq_mul_left' this
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proposition hom_inv (f : A → B) [is_mul_hom f] (a : A) :
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f (a⁻¹) = (f a)⁻¹ :=
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have f (a⁻¹) * f a = 1, by rewrite [-hom_mul f, mul.left_inv, hom_one],
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eq_inv_of_mul_eq_one this
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proposition injective_hom_mul [group B] {f : A → B} [is_mul_hom f]
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(H : ∀ x, f x = 1 → x = 1) :
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injective f :=
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take x₁ x₂,
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suppose f x₁ = f x₂,
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have f (x₁ * x₂⁻¹) = 1, using this, by rewrite [hom_mul, hom_inv, this, mul.right_inv],
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have x₁ * x₂⁻¹ = 1, from H _ this,
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eq_of_mul_inv_eq_one this
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proposition eq_one_of_injective_hom [group B] {f : A → B} [is_mul_hom f]
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(injf : injective f) {a : A} (fa1 : f a = 1) :
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a = 1 :=
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have f a = f 1, by rewrite [fa1, hom_one],
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show a = 1, from injf this
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end group_A_B
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/- modules -/
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structure is_module_hom [class] (R : Type) {M₁ M₂ : Type}
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[has_scalar R M₁] [has_scalar R M₂] [has_add M₁] [has_add M₂]
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(f : M₁ → M₂) extends is_add_hom f :=
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(hom_smul : ∀ r : R, ∀ a : M₁, f (r • a) = r • f a)
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section module_hom
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variables {R : Type} {M₁ M₂ M₃ : Type}
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variables [has_scalar R M₁] [has_scalar R M₂] [has_scalar R M₃]
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variables [has_add M₁] [has_add M₂] [has_add M₃]
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variables (g : M₂ → M₃) (f : M₁ → M₂) [is_module_hom R g] [is_module_hom R f]
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proposition hom_smul (r : R) (a : M₁) : f (r • a) = r • f a :=
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is_module_hom.hom_smul _ _ _ _ f r a
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proposition is_module_hom_id : is_module_hom R (@id M₁) :=
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is_module_hom.mk (λ a₁ a₂, rfl) (λ r a, rfl)
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proposition is_module_hom_comp : is_module_hom R (g ∘ f) :=
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is_module_hom.mk
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(take a₁ a₂, by esimp; rewrite *hom_add)
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(take r a, by esimp; rewrite [hom_smul f, hom_smul g])
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proposition hom_smul_add_smul (a b : R) (u v : M₁) : f (a • u + b • v) = a • f u + b • f v :=
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by rewrite [hom_add, +hom_smul f]
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end module_hom
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/- rings -/
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structure is_ring_hom [class] {R₁ R₂ : Type} [has_mul R₁] [has_mul R₂] [has_add R₁] [has_add R₂]
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(f : R₁ → R₂) extends is_add_hom f, is_mul_hom f
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section semiring
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variables {R₁ R₂ R₃ : Type} [semiring R₁] [semiring R₂] [semiring R₃]
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variables (g : R₂ → R₃) (f : R₁ → R₂) [is_ring_hom g] [is_ring_hom f]
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proposition is_ring_hom_id : is_ring_hom (@id R₁) :=
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is_ring_hom.mk (λ a₁ a₂, rfl) (λ a₁ a₂, rfl)
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proposition is_ring_hom_comp : is_ring_hom (g ∘ f) :=
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is_ring_hom.mk
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(take a₁ a₂, by esimp; rewrite *hom_add)
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(take r a, by esimp; rewrite [hom_mul f, hom_mul g])
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proposition hom_mul_add_mul (a b c d : R₁) : f (a * b + c * d) = f a * f b + f c * f d :=
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by rewrite [hom_add, +hom_mul]
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end semiring
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@ -4,6 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Nathaniel Thomas, Jeremy Avigad
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Modules and vector spaces over a ring.
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(We use "left_module," which is more precise, because "module" is a keyword.)
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-/
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import algebra.field
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@ -64,73 +66,7 @@ section left_module
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by rewrite [sub_eq_add_neg, smul_right_distrib, neg_smul]
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end left_module
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/- linear maps -/
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structure is_linear_map [class] (R : Type) {M₁ M₂ : Type}
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[smul₁ : has_scalar R M₁] [smul₂ : has_scalar R M₂]
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[add₁ : has_add M₁] [add₂ : has_add M₂]
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(T : M₁ → M₂) :=
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(additive : ∀ u v : M₁, T (u + v) = T u + T v)
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(homogeneous : ∀ a : R, ∀ u : M₁, T (a • u) = a • T u)
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proposition linear_map_additive (R : Type) {M₁ M₂ : Type}
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[smul₁ : has_scalar R M₁] [smul₂ : has_scalar R M₂]
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[add₁ : has_add M₁] [add₂ : has_add M₂]
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(T : M₁ → M₂) [linT : is_linear_map R T] (u v : M₁) :
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T (u + v) = T u + T v :=
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is_linear_map.additive smul₁ smul₂ _ _ T u v
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proposition linear_map_homogeneous {R M₁ M₂ : Type}
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[smul₁ : has_scalar R M₁] [smul₂ : has_scalar R M₂]
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[add₁ : has_add M₁] [add₂ : has_add M₂]
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(T : M₁ → M₂) [linT : is_linear_map R T] (a : R) (u : M₁) :
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T (a • u) = a • T u :=
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is_linear_map.homogeneous smul₁ smul₂ _ _ T a u
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proposition is_linear_map_id [instance] (R : Type) {M : Type}
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[smulRM : has_scalar R M] [has_addM : has_add M] :
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is_linear_map R (id : M → M) :=
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is_linear_map.mk (take u v, rfl) (take a u, rfl)
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section is_linear_map
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variables {R M₁ M₂ : Type}
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variable [ringR : ring R]
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variable [moduleRM₁ : left_module R M₁]
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variable [moduleRM₂ : left_module R M₂]
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include ringR moduleRM₁ moduleRM₂
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variable T : M₁ → M₂
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variable [is_linear_mapT : is_linear_map R T]
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include is_linear_mapT
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proposition linear_map_zero : T 0 = 0 :=
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calc
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T 0 = T ((0 : R) • 0) : zero_smul
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... = (0 : R) • T 0 : linear_map_homogeneous T
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... = 0 : zero_smul
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proposition linear_map_neg (u : M₁) : T (-u) = -(T u) :=
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by rewrite [-neg_one_smul, linear_map_homogeneous T, neg_one_smul]
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proposition linear_map_smul_add_smul (a b : R) (u v : M₁) :
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T (a • u + b • v) = a • T u + b • T v :=
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by rewrite [linear_map_additive R T, *linear_map_homogeneous T]
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end is_linear_map
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/- vector spaces -/
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structure vector_space [class] (F V : Type) [fieldF : field F]
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extends left_module F V
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/- an example -/
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section
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variables (F V : Type)
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variables [field F] [vector_spaceFV : vector_space F V]
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variable T : V → V
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variable [is_linear_map F T]
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include vector_spaceFV
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example (a b : F) (u v : V) : T (a • u + b • v) = a • T u + b • T v :=
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!linear_map_smul_add_smul
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end
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