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Mathlib.GroupTheory.GroupAction.Support

Support of an element under an action action #

Given an action of a group G on a type α, we say that a set s : Set α supports an element a : α if, for all g that fix s pointwise, g fixes a.

This is crucial in Fourier-Motzkin constructions.

def AddAction.Supports (G : Type u_1) {α : Type u_3} {β : Type u_4} [VAdd G α] [VAdd G β] (s : Set α) (b : β) :

A set s supports b if g +ᵥ b = b whenever g +ᵥ a = a for all a ∈ s.

Instances For
    def MulAction.Supports (G : Type u_1) {α : Type u_3} {β : Type u_4} [SMul G α] [SMul G β] (s : Set α) (b : β) :

    A set s supports b if g • b = b whenever g • a = a for all a ∈ s.

    Instances For
      theorem AddAction.supports_of_mem (G : Type u_1) {α : Type u_3} [VAdd G α] {s : Set α} {a : α} (ha : a s) :
      theorem MulAction.supports_of_mem (G : Type u_1) {α : Type u_3} [SMul G α] {s : Set α} {a : α} (ha : a s) :
      theorem AddAction.Supports.mono {G : Type u_1} {α : Type u_3} {β : Type u_4} [VAdd G α] [VAdd G β] {s : Set α} {t : Set α} {b : β} (h : s t) (hs : AddAction.Supports G s b) :
      theorem MulAction.Supports.mono {G : Type u_1} {α : Type u_3} {β : Type u_4} [SMul G α] [SMul G β] {s : Set α} {t : Set α} {b : β} (h : s t) (hs : MulAction.Supports G s b) :
      theorem AddAction.Supports.vadd {G : Type u_1} {H : Type u_2} {α : Type u_3} {β : Type u_4} [AddGroup H] [VAdd G α] [VAdd G β] [AddAction H α] [VAdd H β] [VAddCommClass G H β] [VAddCommClass G H α] {s : Set α} {b : β} (g : H) (h : AddAction.Supports G s b) :
      theorem MulAction.Supports.smul {G : Type u_1} {H : Type u_2} {α : Type u_3} {β : Type u_4} [Group H] [SMul G α] [SMul G β] [MulAction H α] [SMul H β] [SMulCommClass G H β] [SMulCommClass G H α] {s : Set α} {b : β} (g : H) (h : MulAction.Supports G s b) :
      MulAction.Supports G (g s) (g b)