| {"name":"FiniteRange.pow","declaration":"/-- A function of finite range raised to a constant power, has finite range. -/\ninstance FiniteRange.pow {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [Group G] [hX : FiniteRange X] (c : ℤ) : FiniteRange (X ^ c)"} | |
| {"name":"FiniteRange.finite","declaration":"def FiniteRange.finite {Ω : Type u_1} {G : Type u_2} {X : Ω → G} [self : FiniteRange X] : Set.Finite (Set.range X)"} | |
| {"name":"FiniteRange.mk","declaration":"ctor FiniteRange.mk {Ω : Type u_1} {G : Type u_2} {X : Ω → G} (finite : Set.Finite (Set.range X)) : FiniteRange X"} | |
| {"name":"instFiniteRangeComp_1","declaration":"/-- If X has finite range, then X of any function has finite range. -/\ninstance instFiniteRangeComp_1 {Ω : Type u_1} {Ω' : Type u_2} {G : Type u_3} (X : Ω → G) (f : Ω' → Ω) [hX : FiniteRange X] : FiniteRange (X ∘ f)"} | |
| {"name":"FiniteRange.toFinset","declaration":"/-- The range of a finite range map, as a finset. -/\ndef FiniteRange.toFinset {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [hX : FiniteRange X] : Finset G"} | |
| {"name":"FiniteRange.mem","declaration":"theorem FiniteRange.mem {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [FiniteRange X] (ω : Ω) : X ω ∈ FiniteRange.toFinset X"} | |
| {"name":"FiniteRange.sub","declaration":"/-- The difference of functions of finite range, has finite range.-/\ninstance FiniteRange.sub {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [AddGroup G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X - Y)"} | |
| {"name":"instFiniteRange_1","declaration":"/-- Constants have finite range -/\ninstance instFiniteRange_1 {Ω : Type u_1} {G : Type u_2} (c : G) : FiniteRange fun x => c"} | |
| {"name":"instFiniteRangeProdProd","declaration":"/-- If X, Y have finite range, then so does the pair ⟨X, Y⟩. -/\ninstance instFiniteRangeProdProd {Ω : Type u_1} {G : Type u_2} {H : Type u_3} (X : Ω → G) (Y : Ω → H) [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (⟨X, Y⟩)"} | |
| {"name":"FiniteRange.null_of_compl","declaration":"theorem FiniteRange.null_of_compl {Ω : Type u_1} {G : Type u_2} [MeasurableSpace Ω] [MeasurableSpace G] [MeasurableSingletonClass G] (μ : MeasureTheory.Measure Ω) (X : Ω → G) [FiniteRange X] : ↑↑(MeasureTheory.Measure.map X μ) (↑(FiniteRange.toFinset X))ᶜ = 0"} | |
| {"name":"FiniteRange.range","declaration":"theorem FiniteRange.range {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [hX : FiniteRange X] : Set.range X = ↑(FiniteRange.toFinset X)"} | |
| {"name":"FiniteRange.nsmul","declaration":"/-- The multiple of a function of finite range by a constant, has finite range.-/\ninstance FiniteRange.nsmul {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [AddGroup G] [hX : FiniteRange X] (c : ℤ) : FiniteRange (c • X)"} | |
| {"name":"FiniteRange.sum","declaration":"/-- The sum of functions of finite range, has finite range.-/\ninstance FiniteRange.sum {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [AddGroup G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X + Y)"} | |
| {"name":"FiniteRange.neg","declaration":"/-- The negation of a function of finite range, has finite range.-/\ninstance FiniteRange.neg {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [AddGroup G] [hX : FiniteRange X] : FiniteRange (-X)"} | |
| {"name":"FiniteRange","declaration":"/-- The property of having a finite range. -/\nclass FiniteRange {Ω : Type u_1} {G : Type u_2} (X : Ω → G) : Prop"} | |
| {"name":"FiniteRange.fintype","declaration":"/-- fintype structure on the range of a finite range map. -/\ndef FiniteRange.fintype {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [hX : FiniteRange X] : Fintype ↑(Set.range X)"} | |
| {"name":"instFiniteRangeComp","declaration":"/-- If X has finite range, then any function of X has finite range. -/\ninstance instFiniteRangeComp {Ω : Type u_1} {G : Type u_2} {H : Type u_3} (X : Ω → G) (f : G → H) [hX : FiniteRange X] : FiniteRange (f ∘ X)"} | |
| {"name":"FiniteRange.mem_iff","declaration":"theorem FiniteRange.mem_iff {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [FiniteRange X] (x : G) : x ∈ FiniteRange.toFinset X ↔ ∃ ω, X ω = x"} | |
| {"name":"instFiniteRange","declaration":"/-- If the codomain of X is finite, then X has finite range. -/\ninstance instFiniteRange {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [Fintype G] : FiniteRange X"} | |
| {"name":"finiteRange_of_finset","declaration":"/-- Functions ranging in a Finset have finite range -/\ntheorem finiteRange_of_finset {Ω : Type u_1} {G : Type u_2} (f : Ω → G) (A : Finset G) (h : ∀ (ω : Ω), f ω ∈ A) : FiniteRange f"} | |
| {"name":"FiniteRange.full","declaration":"theorem FiniteRange.full {Ω : Type u_1} {G : Type u_2} [MeasurableSpace Ω] [MeasurableSpace G] [MeasurableSingletonClass G] {X : Ω → G} (hX : Measurable X) [FiniteRange X] (μ : MeasureTheory.Measure Ω) : ↑↑(MeasureTheory.Measure.map X μ) ↑(FiniteRange.toFinset X) = ↑↑μ Set.univ"} | |
| {"name":"FiniteRange.div","declaration":"/-- The quotient of two functions with finite range, has finite range. -/\ninstance FiniteRange.div {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [Group G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X / Y)"} | |
| {"name":"FiniteRange.prod","declaration":"/-- The product of functions of finite range, has finite range. -/\ninstance FiniteRange.prod {Ω : Type u_1} {G : Type u_2} (X : Ω → G) (Y : Ω → G) [Group G] [hX : FiniteRange X] [hY : FiniteRange Y] : FiniteRange (X * Y)"} | |
| {"name":"FiniteRange.inv","declaration":"/-- The inverse of a function of finite range, has finite range.-/\ninstance FiniteRange.inv {Ω : Type u_1} {G : Type u_2} (X : Ω → G) [Group G] [hX : FiniteRange X] : FiniteRange X⁻¹"} | |