[Merged by Bors] - feat: port Algebra.Ring.Pi

Mathlib.lean

@@ -119,6 +119,7 @@ import Mathlib.Algebra.Ring.Idempotents
 import Mathlib.Algebra.Ring.InjSurj
 import Mathlib.Algebra.Ring.Opposite
 import Mathlib.Algebra.Ring.OrderSynonym
+import Mathlib.Algebra.Ring.Pi
 import Mathlib.Algebra.Ring.Regular
 import Mathlib.Algebra.Ring.Semiconj
 import Mathlib.Algebra.Ring.Units

Mathlib/Algebra/Ring/Pi.lean

@@ -0,0 +1,184 @@
+/-
+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
+
+! This file was ported from Lean 3 source module algebra.ring.pi
+! leanprover-community/mathlib commit ba2245edf0c8bb155f1569fd9b9492a9b384cde6
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Algebra.Group.Pi
+import Mathlib.Algebra.Hom.Ring
+
+/-!
+# Pi instances for ring
+
+This file defines instances for ring, semiring and related structures on Pi Types
+-/
+
+-- Porting notes: used to import `tactic.pi_instances`
+
+namespace Pi
+
+universe u v w
+
+variable {I : Type u}
+
+-- The indexing type
+variable {f : I → Type v}
+
+-- The family of types already equipped with instances
+variable (x y : ∀ i, f i) (i : I)
+
+-- Porting note: All these instances used `refine_struct` and `pi_instance_derive_field`
+
+instance distrib [∀ i, Distrib <| f i] : Distrib (∀ i : I, f i) :=
+  { add := (· + ·)
+    mul := (· * ·)
+    left_distrib := by intros; ext; exact mul_add _ _ _
+    right_distrib := by intros; ext; exact add_mul _ _ _}
+#align pi.distrib Pi.distrib
+
+instance nonUnitalNonAssocSemiring [∀ i, NonUnitalNonAssocSemiring <| f i] :
+    NonUnitalNonAssocSemiring (∀ i : I, f i) :=
+  { Pi.distrib, Pi.addCommMonoid, Pi.mulZeroClass with }
+#align pi.non_unital_non_assoc_semiring Pi.nonUnitalNonAssocSemiring
+
+instance nonUnitalSemiring [∀ i, NonUnitalSemiring <| f i] : NonUnitalSemiring (∀ i : I, f i) :=
+  { Pi.nonUnitalNonAssocSemiring, Pi.semigroupWithZero with }
+#align pi.non_unital_semiring Pi.nonUnitalSemiring
+
+instance nonAssocSemiring [∀ i, NonAssocSemiring <| f i] : NonAssocSemiring (∀ i : I, f i) :=
+  { Pi.nonUnitalNonAssocSemiring, Pi.mulZeroOneClass with }
+#align pi.non_assoc_semiring Pi.nonAssocSemiring
+
+instance semiring [∀ i, Semiring <| f i] : Semiring (∀ i : I, f i) :=
+  { Pi.nonUnitalSemiring, Pi.nonAssocSemiring, Pi.monoidWithZero with }
+#align pi.semiring Pi.semiring
+
+instance nonUnitalCommSemiring [∀ i, NonUnitalCommSemiring <| f i] :
+    NonUnitalCommSemiring (∀ i : I, f i) :=
+  { Pi.nonUnitalSemiring, Pi.commSemigroup with }
+#align pi.non_unital_comm_semiring Pi.nonUnitalCommSemiring
+
+instance commSemiring [∀ i, CommSemiring <| f i] : CommSemiring (∀ i : I, f i) :=
+  { Pi.semiring, Pi.commMonoid with }
+#align pi.comm_semiring Pi.commSemiring
+
+instance nonUnitalNonAssocRing [∀ i, NonUnitalNonAssocRing <| f i] :
+    NonUnitalNonAssocRing (∀ i : I, f i) :=
+  { Pi.addCommGroup, Pi.nonUnitalNonAssocSemiring with }
+#align pi.non_unital_non_assoc_ring Pi.nonUnitalNonAssocRing
+
+instance nonUnitalRing [∀ i, NonUnitalRing <| f i] : NonUnitalRing (∀ i : I, f i) :=
+  { Pi.nonUnitalNonAssocRing, Pi.nonUnitalSemiring with }
+#align pi.non_unital_ring Pi.nonUnitalRing
+
+instance nonAssocRing [∀ i, NonAssocRing <| f i] : NonAssocRing (∀ i : I, f i) :=
+  { Pi.nonUnitalNonAssocRing, Pi.nonAssocSemiring with }
+#align pi.non_assoc_ring Pi.nonAssocRing
+
+instance ring [∀ i, Ring <| f i] : Ring (∀ i : I, f i) :=
+  { Pi.semiring, Pi.addCommGroup, Pi.addGroupWithOne with }
+#align pi.ring Pi.ring
+
+instance nonUnitalCommRing [∀ i, NonUnitalCommRing <| f i] : NonUnitalCommRing (∀ i : I, f i) :=
+  { Pi.nonUnitalRing, Pi.commSemigroup with }
+#align pi.non_unital_comm_ring Pi.nonUnitalCommRing
+
+instance commRing [∀ i, CommRing <| f i] : CommRing (∀ i : I, f i) :=
+  { Pi.ring, Pi.commSemiring with }
+#align pi.comm_ring Pi.commRing
+
+/-- A family of non-unital ring homomorphisms `f a : γ →ₙ+* β a` defines a non-unital ring
+homomorphism `Pi.nonUnitalRingHom f : γ →+* Π a, β a` given by
+`Pi.nonUnitalRingHom f x b = f b x`. -/
+@[simps]
+protected def nonUnitalRingHom {γ : Type w} [∀ i, NonUnitalNonAssocSemiring (f i)]
+    [NonUnitalNonAssocSemiring γ] (g : ∀ i, γ →ₙ+* f i) : γ →ₙ+* ∀ i, f i :=
+  { Pi.mulHom fun i => (g i).toMulHom, Pi.addMonoidHom fun i => (g i).toAddMonoidHom with
+    toFun := fun x b => g b x }
+#align pi.non_unital_ring_hom Pi.nonUnitalRingHom
+
+theorem nonUnitalRingHom_injective {γ : Type w} [Nonempty I]
+    [∀ i, NonUnitalNonAssocSemiring (f i)] [NonUnitalNonAssocSemiring γ] (g : ∀ i, γ →ₙ+* f i)
+    (hg : ∀ i, Function.Injective (g i)) : Function.Injective (Pi.nonUnitalRingHom g) :=
+  mulHom_injective (fun i => (g i).toMulHom) hg
+#align pi.non_unital_ring_hom_injective Pi.nonUnitalRingHom_injective
+
+/-- A family of ring homomorphisms `f a : γ →+* β a` defines a ring homomorphism
+`Pi.ringHom f : γ →+* Π a, β a` given by `Pi.ringHom f x b = f b x`. -/
+@[simps]
+protected def ringHom {γ : Type w} [∀ i, NonAssocSemiring (f i)] [NonAssocSemiring γ]
+    (g : ∀ i, γ →+* f i) : γ →+* ∀ i, f i :=
+  { Pi.monoidHom fun i => (g i).toMonoidHom, Pi.addMonoidHom fun i => (g i).toAddMonoidHom with
+    toFun := fun x b => g b x }
+#align pi.ring_hom Pi.ringHom
+
+theorem ringHom_injective {γ : Type w} [Nonempty I] [∀ i, NonAssocSemiring (f i)]
+    [NonAssocSemiring γ] (g : ∀ i, γ →+* f i) (hg : ∀ i, Function.Injective (g i)) :
+    Function.Injective (Pi.ringHom g) :=
+  monoidHom_injective (fun i => (g i).toMonoidHom) hg
+#align pi.ring_hom_injective Pi.ringHom_injective
+
+end Pi
+
+section NonUnitalRingHom
+
+universe u v
+
+variable {I : Type u}
+
+/-- Evaluation of functions into an indexed collection of non-unital rings at a point is a
+non-unital ring homomorphism. This is `function.eval` as a `NonUnitalRingHom`. -/
+@[simps]
+def Pi.evalNonUnitalRingHom (f : I → Type v) [∀ i, NonUnitalNonAssocSemiring (f i)] (i : I) :
+    (∀ i, f i) →ₙ+* f i :=
+  { Pi.evalMulHom f i, Pi.evalAddMonoidHom f i with }
+#align pi.eval_non_unital_ring_hom Pi.evalNonUnitalRingHom
+
+/-- `function.const` as a `NonUnitalRingHom`. -/
+@[simps]
+def Pi.constNonUnitalRingHom (α β : Type _) [NonUnitalNonAssocSemiring β] : β →ₙ+* α → β :=
+  { Pi.nonUnitalRingHom fun _ => NonUnitalRingHom.id β with toFun := Function.const _ }
+#align pi.const_non_unital_ring_hom Pi.constNonUnitalRingHom
+
+/-- Non-unital ring homomorphism between the function spaces `I → α` and `I → β`, induced by a
+non-unital ring homomorphism `f` between `α` and `β`. -/
+@[simps]
+protected def NonUnitalRingHom.compLeft {α β : Type _} [NonUnitalNonAssocSemiring α]
+    [NonUnitalNonAssocSemiring β] (f : α →ₙ+* β) (I : Type _) : (I → α) →ₙ+* I → β :=
+  { f.toMulHom.compLeft I, f.toAddMonoidHom.compLeft I with toFun := fun h => f ∘ h }
+#align non_unital_ring_hom.comp_left NonUnitalRingHom.compLeft
+
+end NonUnitalRingHom
+
+section RingHom
+
+universe u v
+
+variable {I : Type u}
+
+/-- Evaluation of functions into an indexed collection of rings at a point is a ring
+homomorphism. This is `function.eval` as a `RingHom`. -/
+@[simps]
+def Pi.evalRingHom (f : I → Type v) [∀ i, NonAssocSemiring (f i)] (i : I) : (∀ i, f i) →+* f i :=
+  { Pi.evalMonoidHom f i, Pi.evalAddMonoidHom f i with }
+#align pi.eval_ring_hom Pi.evalRingHom
+
+/-- `function.const` as a `RingHom`. -/
+@[simps]
+def Pi.constRingHom (α β : Type _) [NonAssocSemiring β] : β →+* α → β :=
+  { Pi.ringHom fun _ => RingHom.id β with toFun := Function.const _ }
+#align pi.const_ring_hom Pi.constRingHom
+
+/-- Ring homomorphism between the function spaces `I → α` and `I → β`, induced by a ring
+homomorphism `f` between `α` and `β`. -/
+@[simps]
+protected def RingHom.compLeft {α β : Type _} [NonAssocSemiring α] [NonAssocSemiring β]
+    (f : α →+* β) (I : Type _) : (I → α) →+* I → β :=
+  { f.toMonoidHom.compLeft I, f.toAddMonoidHom.compLeft I with toFun := fun h => f ∘ h }
+#align ring_hom.comp_left RingHom.compLeft
+
+end RingHom