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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_2} {β : Type u_3} [inst : VAdd G α] [inst : 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∈ s.

Equations
def MulAction.Supports (G : Type u_1) {α : Type u_2} {β : Type u_3} [inst : SMul G α] [inst : 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∈ s.

Equations
theorem AddAction.supports_of_mem (G : Type u_2) {α : Type u_1} [inst : VAdd G α] {s : Set α} {a : α} (ha : a s) :
theorem MulAction.supports_of_mem (G : Type u_2) {α : Type u_1} [inst : SMul G α] {s : Set α} {a : α} (ha : a s) :
theorem AddAction.Supports.mono {G : Type u_2} {α : Type u_1} {β : Type u_3} [inst : VAdd G α] [inst : 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_2} {α : Type u_1} {β : Type u_3} [inst : SMul G α] [inst : 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_4} {α : Type u_2} {β : Type u_3} [inst : AddGroup H] [inst : VAdd G α] [inst : VAdd G β] [inst : AddAction H α] [inst : VAdd H β] [inst : VAddCommClass G H β] [inst : 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_4} {α : Type u_2} {β : Type u_3} [inst : Group H] [inst : SMul G α] [inst : SMul G β] [inst : MulAction H α] [inst : SMul H β] [inst : SMulCommClass G H β] [inst : SMulCommClass G H α] {s : Set α} {b : β} (g : H) (h : MulAction.Supports G s b) :
MulAction.Supports G (g s) (g b)