File: Mathlib/Order/Filter/Partial.lean

full_name: Filter.rmap
code: /-- The forward map of a filter under a relation. Generalization of `Filter.map` to relations. Note
that `Rel.core` generalizes `Set.preimage`. -/
def rmap (r : Rel α β) (l : Filter α) : Filter β 
goal: Filter β

full_name: Filter.rmap_sets
code: theorem rmap_sets (r : Rel α β) (l : Filter α) : (l.rmap r).sets = r.core ⁻¹' l.sets 
goal: (Filter.rmap r l).sets = r.core ⁻¹' l.sets

full_name: Filter.mem_rmap
code: @[simp]
theorem mem_rmap (r : Rel α β) (l : Filter α) (s : Set β) : s ∈ l.rmap r ↔ r.core s ∈ l 
goal: s ∈ Filter.rmap r l ↔ r.core s ∈ l

full_name: Filter.rmap_rmap
code: @[simp]
theorem rmap_rmap (r : Rel α β) (s : Rel β γ) (l : Filter α) :
    rmap s (rmap r l) = rmap (r.comp s) l 
goal: Filter.rmap s (Filter.rmap r l) = Filter.rmap (r.comp s) l

full_name: Filter.rmap_compose
code: @[simp]
theorem rmap_compose (r : Rel α β) (s : Rel β γ) : rmap s ∘ rmap r = rmap (r.comp s) 
goal: Filter.rmap s ∘ Filter.rmap r = Filter.rmap (r.comp s)

full_name: Filter.RTendsto
code: /-- Generic "limit of a relation" predicate. `RTendsto r l₁ l₂` asserts that for every
`l₂`-neighborhood `a`, the `r`-core of `a` is an `l₁`-neighborhood. One generalization of
`Filter.Tendsto` to relations. -/
def RTendsto (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) 
goal: Prop

full_name: Filter.rtendsto_def
code: theorem rtendsto_def (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) :
    RTendsto r l₁ l₂ ↔ ∀ s ∈ l₂, r.core s ∈ l₁ 
goal: Filter.RTendsto r l₁ l₂ ↔ ∀ s ∈ l₂, r.core s ∈ l₁

full_name: Filter.rcomap
code: /-- One way of taking the inverse map of a filter under a relation. One generalization of
`Filter.comap` to relations. Note that `Rel.core` generalizes `Set.preimage`. -/
def rcomap (r : Rel α β) (f : Filter β) : Filter α 
goal: Filter α

full_name: Filter.rcomap_sets
code: theorem rcomap_sets (r : Rel α β) (f : Filter β) :
    (rcomap r f).sets = Rel.image (fun s t => r.core s ⊆ t) f.sets 
goal: (Filter.rcomap r f).sets = Rel.image (fun s t => r.core s ⊆ t) f.sets

full_name: Filter.rcomap_rcomap
code: theorem rcomap_rcomap (r : Rel α β) (s : Rel β γ) (l : Filter γ) :
    rcomap r (rcomap s l) = rcomap (r.comp s) l 
goal: Filter.rcomap r (Filter.rcomap s l) = Filter.rcomap (r.comp s) l

full_name: Filter.rcomap_compose
code: @[simp]
theorem rcomap_compose (r : Rel α β) (s : Rel β γ) : rcomap r ∘ rcomap s = rcomap (r.comp s) 
goal: Filter.rcomap r ∘ Filter.rcomap s = Filter.rcomap (r.comp s)

full_name: Filter.rtendsto_iff_le_rcomap
code: theorem rtendsto_iff_le_rcomap (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) :
    RTendsto r l₁ l₂ ↔ l₁ ≤ l₂.rcomap r 
goal: Filter.RTendsto r l₁ l₂ ↔ l₁ ≤ Filter.rcomap r l₂

full_name: Filter.rcomap'
code: /-- One way of taking the inverse map of a filter under a relation. Generalization of `Filter.comap`
to relations. -/
def rcomap' (r : Rel α β) (f : Filter β) : Filter α 
goal: Filter α

full_name: Filter.mem_rcomap'
code: @[simp]
theorem mem_rcomap' (r : Rel α β) (l : Filter β) (s : Set α) :
    s ∈ l.rcomap' r ↔ ∃ t ∈ l, r.preimage t ⊆ s 
goal: s ∈ Filter.rcomap' r l ↔ ∃ t ∈ l, r.preimage t ⊆ s

full_name: Filter.rcomap'_sets
code: theorem rcomap'_sets (r : Rel α β) (f : Filter β) :
    (rcomap' r f).sets = Rel.image (fun s t => r.preimage s ⊆ t) f.sets 
goal: (Filter.rcomap' r f).sets = Rel.image (fun s t => r.preimage s ⊆ t) f.sets

full_name: Filter.rcomap'_rcomap'
code: @[simp]
theorem rcomap'_rcomap' (r : Rel α β) (s : Rel β γ) (l : Filter γ) :
    rcomap' r (rcomap' s l) = rcomap' (r.comp s) l 
goal: Filter.rcomap' r (Filter.rcomap' s l) = Filter.rcomap' (r.comp s) l

full_name: Filter.rcomap'_compose
code: @[simp]
theorem rcomap'_compose (r : Rel α β) (s : Rel β γ) : rcomap' r ∘ rcomap' s = rcomap' (r.comp s) 
goal: Filter.rcomap' r ∘ Filter.rcomap' s = Filter.rcomap' (r.comp s)

full_name: Filter.RTendsto'
code: /-- Generic "limit of a relation" predicate. `RTendsto' r l₁ l₂` asserts that for every
`l₂`-neighborhood `a`, the `r`-preimage of `a` is an `l₁`-neighborhood. One generalization of
`Filter.Tendsto` to relations. -/
def RTendsto' (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) 
goal: Prop

full_name: Filter.rtendsto'_def
code: theorem rtendsto'_def (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) :
    RTendsto' r l₁ l₂ ↔ ∀ s ∈ l₂, r.preimage s ∈ l₁ 
goal: Filter.RTendsto' r l₁ l₂ ↔ ∀ s ∈ l₂, r.preimage s ∈ l₁

full_name: Filter.tendsto_iff_rtendsto
code: theorem tendsto_iff_rtendsto (l₁ : Filter α) (l₂ : Filter β) (f : α → β) :
    Tendsto f l₁ l₂ ↔ RTendsto (Function.graph f) l₁ l₂ 
goal: Filter.Tendsto f l₁ l₂ ↔ Filter.RTendsto (Function.graph f) l₁ l₂

full_name: Filter.tendsto_iff_rtendsto'
code: theorem tendsto_iff_rtendsto' (l₁ : Filter α) (l₂ : Filter β) (f : α → β) :
    Tendsto f l₁ l₂ ↔ RTendsto' (Function.graph f) l₁ l₂ 
goal: Filter.Tendsto f l₁ l₂ ↔ Filter.RTendsto' (Function.graph f) l₁ l₂

full_name: Filter.pmap
code: /-- The forward map of a filter under a partial function. Generalization of `Filter.map` to partial
functions. -/
def pmap (f : α →. β) (l : Filter α) : Filter β 
goal: Filter β

full_name: Filter.mem_pmap
code: @[simp]
theorem mem_pmap (f : α →. β) (l : Filter α) (s : Set β) : s ∈ l.pmap f ↔ f.core s ∈ l 
goal: s ∈ Filter.pmap f l ↔ f.core s ∈ l

full_name: Filter.PTendsto
code: /-- Generic "limit of a partial function" predicate. `PTendsto r l₁ l₂` asserts that for every
`l₂`-neighborhood `a`, the `p`-core of `a` is an `l₁`-neighborhood. One generalization of
`Filter.Tendsto` to partial function. -/
def PTendsto (f : α →. β) (l₁ : Filter α) (l₂ : Filter β) 
goal: Prop

full_name: Filter.ptendsto_def
code: theorem ptendsto_def (f : α →. β) (l₁ : Filter α) (l₂ : Filter β) :
    PTendsto f l₁ l₂ ↔ ∀ s ∈ l₂, f.core s ∈ l₁ 
goal: Filter.PTendsto f l₁ l₂ ↔ ∀ s ∈ l₂, f.core s ∈ l₁

full_name: Filter.ptendsto_iff_rtendsto
code: theorem ptendsto_iff_rtendsto (l₁ : Filter α) (l₂ : Filter β) (f : α →. β) :
    PTendsto f l₁ l₂ ↔ RTendsto f.graph' l₁ l₂ 
goal: Filter.PTendsto f l₁ l₂ ↔ Filter.RTendsto f.graph' l₁ l₂

full_name: Filter.pmap_res
code: theorem pmap_res (l : Filter α) (s : Set α) (f : α → β) :
    pmap (PFun.res f s) l = map f (l ⊓ 𝓟 s) 
goal: Filter.pmap (PFun.res f s) l = Filter.map f (l ⊓ Filter.principal s)

full_name: Filter.tendsto_iff_ptendsto
code: theorem tendsto_iff_ptendsto (l₁ : Filter α) (l₂ : Filter β) (s : Set α) (f : α → β) :
    Tendsto f (l₁ ⊓ 𝓟 s) l₂ ↔ PTendsto (PFun.res f s) l₁ l₂ 
goal: Filter.Tendsto f (l₁ ⊓ Filter.principal s) l₂ ↔ Filter.PTendsto (PFun.res f s) l₁ l₂

full_name: Filter.tendsto_iff_ptendsto_univ
code: theorem tendsto_iff_ptendsto_univ (l₁ : Filter α) (l₂ : Filter β) (f : α → β) :
    Tendsto f l₁ l₂ ↔ PTendsto (PFun.res f Set.univ) l₁ l₂ 
goal: Filter.Tendsto f l₁ l₂ ↔ Filter.PTendsto (PFun.res f Set.univ) l₁ l₂

full_name: Filter.pcomap'
code: /-- Inverse map of a filter under a partial function. One generalization of `Filter.comap` to
partial functions. -/
def pcomap' (f : α →. β) (l : Filter β) : Filter α 
goal: Filter α

full_name: Filter.PTendsto'
code: /-- Generic "limit of a partial function" predicate. `PTendsto' r l₁ l₂` asserts that for every
`l₂`-neighborhood `a`, the `p`-preimage of `a` is an `l₁`-neighborhood. One generalization of
`Filter.Tendsto` to partial functions. -/
def PTendsto' (f : α →. β) (l₁ : Filter α) (l₂ : Filter β) 
goal: Prop

full_name: Filter.ptendsto'_def
code: theorem ptendsto'_def (f : α →. β) (l₁ : Filter α) (l₂ : Filter β) :
    PTendsto' f l₁ l₂ ↔ ∀ s ∈ l₂, f.preimage s ∈ l₁ 
goal: Filter.PTendsto' f l₁ l₂ ↔ ∀ s ∈ l₂, f.preimage s ∈ l₁

full_name: Filter.ptendsto_of_ptendsto'
code: theorem ptendsto_of_ptendsto' {f : α →. β} {l₁ : Filter α} {l₂ : Filter β} :
    PTendsto' f l₁ l₂ → PTendsto f l₁ l₂ 
goal: Filter.PTendsto' f l₁ l₂ → Filter.PTendsto f l₁ l₂

full_name: Filter.ptendsto'_of_ptendsto
code: theorem ptendsto'_of_ptendsto {f : α →. β} {l₁ : Filter α} {l₂ : Filter β} (h : f.Dom ∈ l₁) :
    PTendsto f l₁ l₂ → PTendsto' f l₁ l₂ 
goal: Filter.PTendsto f l₁ l₂ → Filter.PTendsto' f l₁ l₂