/- 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.module.basic import algebra.regular.smul import algebra.ring.pi import group_theory.group_action.pi /-! # Pi instances for modules This file defines instances for module and related structures on Pi Types -/ 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) namespace pi lemma _root_.is_smul_regular.pi {α : Type*} [Π i, has_smul α $ f i] {k : α} (hk : Π i, is_smul_regular (f i) k) : is_smul_regular (Π i, f i) k := λ _ _ h, funext $ λ i, hk i (congr_fun h i : _) instance smul_with_zero (α) [has_zero α] [Π i, has_zero (f i)] [Π i, smul_with_zero α (f i)] : smul_with_zero α (Π i, f i) := { smul_zero := λ _, funext $ λ _, smul_zero' (f _) _, zero_smul := λ _, funext $ λ _, zero_smul _ _, ..pi.has_smul } instance smul_with_zero' {g : I → Type*} [Π i, has_zero (g i)] [Π i, has_zero (f i)] [Π i, smul_with_zero (g i) (f i)] : smul_with_zero (Π i, g i) (Π i, f i) := { smul_zero := λ _, funext $ λ _, smul_zero' (f _) _, zero_smul := λ _, funext $ λ _, zero_smul _ _, ..pi.has_smul' } instance mul_action_with_zero (α) [monoid_with_zero α] [Π i, has_zero (f i)] [Π i, mul_action_with_zero α (f i)] : mul_action_with_zero α (Π i, f i) := { ..pi.mul_action _, ..pi.smul_with_zero _ } instance mul_action_with_zero' {g : I → Type*} [Π i, monoid_with_zero (g i)] [Π i, has_zero (f i)] [Π i, mul_action_with_zero (g i) (f i)] : mul_action_with_zero (Π i, g i) (Π i, f i) := { ..pi.mul_action', ..pi.smul_with_zero' } variables (I f) instance module (α) {r : semiring α} {m : ∀ i, add_comm_monoid $ f i} [∀ i, module α $ f i] : @module α (Π i : I, f i) r (@pi.add_comm_monoid I f m) := { add_smul := λ c f g, funext $ λ i, add_smul _ _ _, zero_smul := λ f, funext $ λ i, zero_smul α _, ..pi.distrib_mul_action _ } /- Extra instance to short-circuit type class resolution. For unknown reasons, this is necessary for certain inference problems. E.g., for this to succeed: ```lean example (β X : Type*) [normed_add_comm_group β] [normed_space ℝ β] : module ℝ (X → β) := infer_instance ``` See: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Typeclass.20resolution.20under.20binders/near/281296989 -/ /-- A special case of `pi.module` for non-dependent types. Lean struggles to elaborate definitions elsewhere in the library without this. -/ instance _root_.function.module (α β : Type*) [semiring α] [add_comm_monoid β] [module α β] : module α (I → β) := pi.module _ _ _ variables {I f} instance module' {g : I → Type*} {r : Π i, semiring (f i)} {m : Π i, add_comm_monoid (g i)} [Π i, module (f i) (g i)] : module (Π i, f i) (Π i, g i) := { add_smul := by { intros, ext1, apply add_smul }, zero_smul := by { intros, ext1, apply zero_smul } } instance (α) {r : semiring α} {m : Π i, add_comm_monoid $ f i} [Π i, module α $ f i] [∀ i, no_zero_smul_divisors α $ f i] : no_zero_smul_divisors α (Π i : I, f i) := ⟨λ c x h, or_iff_not_imp_left.mpr (λ hc, funext (λ i, (smul_eq_zero.mp (congr_fun h i)).resolve_left hc))⟩ end pi