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chore(algebra/module/pi): split out
group_theory/group_action/pi to…
… match `group_theory/group_action/prod` (#10485) These declarations are copied without modification.
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| Original file line number | Diff line number | Diff line change |
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| /- | ||
| Copyright (c) 2018 Simon Hudon. All rights reserved. | ||
| Released under Apache 2.0 license as described in the file LICENSE. | ||
| Authors: Simon Hudon, Patrick Massot | ||
| -/ | ||
| import algebra.group.pi | ||
| import group_theory.group_action.defs | ||
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| /-! | ||
| # Pi instances for multiplicative actions | ||
| This file defines instances for mul_action and related structures on Pi Types | ||
| -/ | ||
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| universes u v w | ||
| variable {I : Type u} -- The indexing type | ||
| variable {f : I → Type v} -- The family of types already equipped with instances | ||
| variables (x y : Π i, f i) (i : I) | ||
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| namespace pi | ||
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| @[to_additive pi.has_vadd] | ||
| instance has_scalar {α : Type*} [Π i, has_scalar α $ f i] : | ||
| has_scalar α (Π i : I, f i) := | ||
| ⟨λ s x, λ i, s • (x i)⟩ | ||
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| @[to_additive] | ||
| lemma smul_def {α : Type*} [Π i, has_scalar α $ f i] (s : α) : s • x = λ i, s • x i := rfl | ||
| @[simp, to_additive] | ||
| lemma smul_apply {α : Type*} [Π i, has_scalar α $ f i] (s : α) : (s • x) i = s • x i := rfl | ||
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| @[to_additive pi.has_vadd'] | ||
| instance has_scalar' {g : I → Type*} [Π i, has_scalar (f i) (g i)] : | ||
| has_scalar (Π i, f i) (Π i : I, g i) := | ||
| ⟨λ s x, λ i, (s i) • (x i)⟩ | ||
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| @[simp, to_additive] | ||
| lemma smul_apply' {g : I → Type*} [∀ i, has_scalar (f i) (g i)] (s : Π i, f i) (x : Π i, g i) : | ||
| (s • x) i = s i • x i := | ||
| rfl | ||
| instance is_scalar_tower {α β : Type*} | ||
| [has_scalar α β] [Π i, has_scalar β $ f i] [Π i, has_scalar α $ f i] | ||
| [Π i, is_scalar_tower α β (f i)] : is_scalar_tower α β (Π i : I, f i) := | ||
| ⟨λ x y z, funext $ λ i, smul_assoc x y (z i)⟩ | ||
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| instance is_scalar_tower' {g : I → Type*} {α : Type*} | ||
| [Π i, has_scalar α $ f i] [Π i, has_scalar (f i) (g i)] [Π i, has_scalar α $ g i] | ||
| [Π i, is_scalar_tower α (f i) (g i)] : is_scalar_tower α (Π i : I, f i) (Π i : I, g i) := | ||
| ⟨λ x y z, funext $ λ i, smul_assoc x (y i) (z i)⟩ | ||
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| instance is_scalar_tower'' {g : I → Type*} {h : I → Type*} | ||
| [Π i, has_scalar (f i) (g i)] [Π i, has_scalar (g i) (h i)] [Π i, has_scalar (f i) (h i)] | ||
| [Π i, is_scalar_tower (f i) (g i) (h i)] : is_scalar_tower (Π i, f i) (Π i, g i) (Π i, h i) := | ||
| ⟨λ x y z, funext $ λ i, smul_assoc (x i) (y i) (z i)⟩ | ||
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| @[to_additive] | ||
| instance smul_comm_class {α β : Type*} | ||
| [Π i, has_scalar α $ f i] [Π i, has_scalar β $ f i] [∀ i, smul_comm_class α β (f i)] : | ||
| smul_comm_class α β (Π i : I, f i) := | ||
| ⟨λ x y z, funext $ λ i, smul_comm x y (z i)⟩ | ||
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| @[to_additive] | ||
| instance smul_comm_class' {g : I → Type*} {α : Type*} | ||
| [Π i, has_scalar α $ g i] [Π i, has_scalar (f i) (g i)] [∀ i, smul_comm_class α (f i) (g i)] : | ||
| smul_comm_class α (Π i : I, f i) (Π i : I, g i) := | ||
| ⟨λ x y z, funext $ λ i, smul_comm x (y i) (z i)⟩ | ||
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| @[to_additive] | ||
| instance smul_comm_class'' {g : I → Type*} {h : I → Type*} | ||
| [Π i, has_scalar (g i) (h i)] [Π i, has_scalar (f i) (h i)] | ||
| [∀ i, smul_comm_class (f i) (g i) (h i)] : smul_comm_class (Π i, f i) (Π i, g i) (Π i, h i) := | ||
| ⟨λ x y z, funext $ λ i, smul_comm (x i) (y i) (z i)⟩ | ||
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| /-- If `f i` has a faithful scalar action for a given `i`, then so does `Π i, f i`. This is | ||
| not an instance as `i` cannot be inferred. -/ | ||
| @[to_additive pi.has_faithful_vadd_at] | ||
| lemma has_faithful_scalar_at {α : Type*} | ||
| [Π i, has_scalar α $ f i] [Π i, nonempty (f i)] (i : I) [has_faithful_scalar α (f i)] : | ||
| has_faithful_scalar α (Π i, f i) := | ||
| ⟨λ x y h, eq_of_smul_eq_smul $ λ a : f i, begin | ||
| classical, | ||
| have := congr_fun (h $ function.update (λ j, classical.choice (‹Π i, nonempty (f i)› j)) i a) i, | ||
| simpa using this, | ||
| end⟩ | ||
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| @[to_additive pi.has_faithful_vadd] | ||
| instance has_faithful_scalar {α : Type*} | ||
| [nonempty I] [Π i, has_scalar α $ f i] [Π i, nonempty (f i)] [Π i, has_faithful_scalar α (f i)] : | ||
| has_faithful_scalar α (Π i, f i) := | ||
| let ⟨i⟩ := ‹nonempty I› in has_faithful_scalar_at i | ||
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| @[to_additive] | ||
| instance mul_action (α) {m : monoid α} [Π i, mul_action α $ f i] : | ||
| @mul_action α (Π i : I, f i) m := | ||
| { smul := (•), | ||
| mul_smul := λ r s f, funext $ λ i, mul_smul _ _ _, | ||
| one_smul := λ f, funext $ λ i, one_smul α _ } | ||
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| @[to_additive] | ||
| instance mul_action' {g : I → Type*} {m : Π i, monoid (f i)} [Π i, mul_action (f i) (g i)] : | ||
| @mul_action (Π i, f i) (Π i : I, g i) (@pi.monoid I f m) := | ||
| { smul := (•), | ||
| mul_smul := λ r s f, funext $ λ i, mul_smul _ _ _, | ||
| one_smul := λ f, funext $ λ i, one_smul _ _ } | ||
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| instance distrib_mul_action (α) {m : monoid α} {n : ∀ i, add_monoid $ f i} | ||
| [∀ i, distrib_mul_action α $ f i] : | ||
| @distrib_mul_action α (Π i : I, f i) m (@pi.add_monoid I f n) := | ||
| { smul_zero := λ c, funext $ λ i, smul_zero _, | ||
| smul_add := λ c f g, funext $ λ i, smul_add _ _ _, | ||
| ..pi.mul_action _ } | ||
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| instance distrib_mul_action' {g : I → Type*} {m : Π i, monoid (f i)} {n : Π i, add_monoid $ g i} | ||
| [Π i, distrib_mul_action (f i) (g i)] : | ||
| @distrib_mul_action (Π i, f i) (Π i : I, g i) (@pi.monoid I f m) (@pi.add_monoid I g n) := | ||
| { smul_add := by { intros, ext x, apply smul_add }, | ||
| smul_zero := by { intros, ext x, apply smul_zero } } | ||
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| lemma single_smul {α} [monoid α] [Π i, add_monoid $ f i] | ||
| [Π i, distrib_mul_action α $ f i] [decidable_eq I] (i : I) (r : α) (x : f i) : | ||
| single i (r • x) = r • single i x := | ||
| single_op (λ i : I, ((•) r : f i → f i)) (λ j, smul_zero _) _ _ | ||
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| /-- A version of `pi.single_smul` for non-dependent functions. It is useful in cases Lean fails | ||
| to apply `pi.single_smul`. -/ | ||
| lemma single_smul' {α β} [monoid α] [add_monoid β] | ||
| [distrib_mul_action α β] [decidable_eq I] (i : I) (r : α) (x : β) : | ||
| single i (r • x) = r • single i x := | ||
| single_smul i r x | ||
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| lemma single_smul₀ {g : I → Type*} [Π i, monoid_with_zero (f i)] [Π i, add_monoid (g i)] | ||
| [Π i, distrib_mul_action (f i) (g i)] [decidable_eq I] (i : I) (r : f i) (x : g i) : | ||
| single i (r • x) = single i r • single i x := | ||
| single_op₂ (λ i : I, ((•) : f i → g i → g i)) (λ j, smul_zero _) _ _ _ | ||
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| instance mul_distrib_mul_action (α) {m : monoid α} {n : Π i, monoid $ f i} | ||
| [Π i, mul_distrib_mul_action α $ f i] : | ||
| @mul_distrib_mul_action α (Π i : I, f i) m (@pi.monoid I f n) := | ||
| { smul_one := λ c, funext $ λ i, smul_one _, | ||
| smul_mul := λ c f g, funext $ λ i, smul_mul' _ _ _, | ||
| ..pi.mul_action _ } | ||
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| instance mul_distrib_mul_action' {g : I → Type*} {m : Π i, monoid (f i)} {n : Π i, monoid $ g i} | ||
| [Π i, mul_distrib_mul_action (f i) (g i)] : | ||
| @mul_distrib_mul_action (Π i, f i) (Π i : I, g i) (@pi.monoid I f m) (@pi.monoid I g n) := | ||
| { smul_mul := by { intros, ext x, apply smul_mul' }, | ||
| smul_one := by { intros, ext x, apply smul_one } } | ||
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| end pi | ||
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| namespace function | ||
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| @[to_additive] | ||
| lemma update_smul {α : Type*} [Π i, has_scalar α (f i)] [decidable_eq I] | ||
| (c : α) (f₁ : Π i, f i) (i : I) (x₁ : f i) : | ||
| update (c • f₁) i (c • x₁) = c • update f₁ i x₁ := | ||
| funext $ λ j, (apply_update (λ i, (•) c) f₁ i x₁ j).symm | ||
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| end function | ||
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| namespace set | ||
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| @[to_additive] | ||
| lemma piecewise_smul {α : Type*} [Π i, has_scalar α (f i)] (s : set I) [Π i, decidable (i ∈ s)] | ||
| (c : α) (f₁ g₁ : Π i, f i) : | ||
| s.piecewise (c • f₁) (c • g₁) = c • s.piecewise f₁ g₁ := | ||
| s.piecewise_op _ _ (λ _, (•) c) | ||
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| end set | ||
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| section extend | ||
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| @[to_additive] lemma function.extend_smul {R α β γ : Type*} [has_scalar R γ] | ||
| (r : R) (f : α → β) (g : α → γ) (e : β → γ) : | ||
| function.extend f (r • g) (r • e) = r • function.extend f g e := | ||
| funext $ λ _, by convert (apply_dite ((•) r) _ _ _).symm | ||
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| end extend |