# Documentation

Mathlib.Data.QPF.Multivariate.Constructions.Quot

# The quotient of QPF is itself a QPF #

The quotients are here defined using a surjective function and its right inverse. They are very similar to the abs and repr functions found in the definition of MvQPF

def MvQPF.quotientQPF {n : } {F : Type u} [] [q : ] {G : Type u} [] {FG_abs : {α : } → F αG α} {FG_repr : {α : } → G αF α} (FG_abs_repr : ∀ {α : } (x : G α), FG_abs α (FG_repr α x) = x) (FG_abs_map : ∀ {α β : } (f : ) (x : F α), FG_abs β () = MvFunctor.map f (FG_abs α x)) :

If F is a QPF then G is a QPF as well. Can be used to construct MvQPF instances by transporting them across surjective functions

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def MvQPF.Quot1 {n : } {F : Type u} (R : α : ⦄ → F αF αProp) (α : ) :

Functorial quotient type

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instance MvQPF.Quot1.inhabited {n : } {F : Type u} (R : α : ⦄ → F αF αProp) {α : } [Inhabited (F α)] :
def MvQPF.Quot1.map {n : } {F : Type u} (R : α : ⦄ → F αF αProp) [] (Hfunc : α β : ⦄ → (a b : F α) → (f : ) → R α a bR β () ()) ⦃α : ⦃β : (f : ) :

map of the Quot1 functor

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def MvQPF.Quot1.mvFunctor {n : } {F : Type u} (R : α : ⦄ → F αF αProp) [] (Hfunc : α β : ⦄ → (a b : F α) → (f : ) → R α a bR β () ()) :

mvFunctor instance for Quot1 with well-behaved R

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noncomputable def MvQPF.relQuot {n : } {F : Type u} (R : α : ⦄ → F αF αProp) [] [q : ] (Hfunc : α β : ⦄ → (a b : F α) → (f : ) → R α a bR β () ()) :

Quot1 is a QPF

Instances For