40 lines
1.2 KiB
Text
40 lines
1.2 KiB
Text
variable {α : Type}
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variable {R : Type}
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class Zip (α : Type) -- represents `Zero`
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class MulAction (R : Type) (β : Type) extends SMul R β
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class SMulZeroClass (R α : Type) extends SMul R α where
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smul_zero : ∀ r : R, ∀ a : α, r • a = a
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class MulActionWithZero (R α : Type) extends MulAction R α, SMulZeroClass R α
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class SMulWithZero (R α : Type) [Zip R] [Zip α] extends SMulZeroClass R α
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instance MulActionWithZero.toSMulWithZero (R M : Type)
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[Zip R] [Zip M] [m : MulActionWithZero R M] :
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SMulWithZero R M :=
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{ m with }
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namespace pi
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variable {I : Type}
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variable {f : I → Type}
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instance instZero [∀ i, Zip (f i)] : Zip (∀ i : I, f i) := ⟨⟩
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instance instSMul [∀ i, SMul R (f i)] : SMul R (∀ i : I, f i) :=
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⟨fun s x => fun i => s • x i⟩
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instance smulWithZero (R) [Zip R] [∀ i, Zip (f i)] [∀ i, SMulWithZero R (f i)] :
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SMulWithZero R (∀ i, f i) :=
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{ pi.instSMul with
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smul_zero := sorry }
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instance mulAction (R) [∀ i, MulAction R (f i)] : MulAction R (∀ i : I, f i) where
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smul := (· • ·)
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instance mulActionWithZero (R) [Zip R] [∀ i, Zip (f i)] [∀ i, MulActionWithZero R (f i)] :
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MulActionWithZero R (∀ i, f i) :=
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{ pi.mulAction _, pi.smulWithZero _ with }
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