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Mathlib.Data.Fintype.Pi

Fintype instances for pi types #

def Fintype.piFinset {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} (t : (a : α) → Finset (δ a)) :
Finset ((a : α) → δ a)

Given for all a : α a finset t a of δ a, then one can define the finset Fintype.piFinset t of all functions taking values in t a for all a. This is the analogue of Finset.pi where the base finset is univ (but formally they are not the same, as there is an additional condition i ∈ Finset.univ in the Finset.pi definition).

Equations
Instances For
    @[simp]
    theorem Fintype.mem_piFinset {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} {t : (a : α) → Finset (δ a)} {f : (a : α) → δ a} :
    f Fintype.piFinset t ∀ (a : α), f a t a
    @[simp]
    theorem Fintype.coe_piFinset {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} (t : (a : α) → Finset (δ a)) :
    (Fintype.piFinset t) = Set.univ.pi fun (a : α) => (t a)
    theorem Fintype.piFinset_subset {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} (t₁ : (a : α) → Finset (δ a)) (t₂ : (a : α) → Finset (δ a)) (h : ∀ (a : α), t₁ a t₂ a) :
    @[simp]
    theorem Fintype.piFinset_eq_empty {α : Type u_1} [DecidableEq α] [Fintype α] {γ : αType u_3} {s : (a : α) → Finset (γ a)} :
    Fintype.piFinset s = ∃ (i : α), s i =
    @[simp]
    theorem Fintype.piFinset_empty {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} [Nonempty α] :
    (Fintype.piFinset fun (x : α) => ) =
    @[simp]
    theorem Fintype.piFinset_nonempty {α : Type u_1} [DecidableEq α] [Fintype α] {γ : αType u_3} {s : (a : α) → Finset (γ a)} :
    (Fintype.piFinset s).Nonempty ∀ (a : α), (s a).Nonempty
    theorem Fintype.Aesop.piFinset_nonempty_of_forall_nonempty {α : Type u_1} [DecidableEq α] [Fintype α] {γ : αType u_3} {s : (a : α) → Finset (γ a)} :
    (∀ (a : α), (s a).Nonempty)(Fintype.piFinset s).Nonempty

    Alias of the reverse direction of Fintype.piFinset_nonempty.

    theorem Finset.Nonempty.piFinset_const {β : Type u_2} {ι : Type u_5} [Fintype ι] [DecidableEq ι] {s : Finset β} (hs : s.Nonempty) :
    (Fintype.piFinset fun (x : ι) => s).Nonempty
    @[simp]
    theorem Fintype.piFinset_of_isEmpty {α : Type u_1} [DecidableEq α] [Fintype α] {γ : αType u_3} [IsEmpty α] (s : (a : α) → Finset (γ a)) :
    Fintype.piFinset s = Finset.univ
    @[simp]
    theorem Fintype.piFinset_singleton {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} (f : (i : α) → δ i) :
    (Fintype.piFinset fun (i : α) => {f i}) = {f}
    theorem Fintype.piFinset_subsingleton {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} {f : (i : α) → Finset (δ i)} (hf : ∀ (i : α), (↑(f i)).Subsingleton) :
    (↑(Fintype.piFinset f)).Subsingleton
    theorem Fintype.piFinset_disjoint_of_disjoint {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} (t₁ : (a : α) → Finset (δ a)) (t₂ : (a : α) → Finset (δ a)) {a : α} (h : Disjoint (t₁ a) (t₂ a)) :
    theorem Fintype.piFinset_image {α : Type u_1} [DecidableEq α] [Fintype α] {γ : αType u_3} {δ : αType u_4} [(a : α) → DecidableEq (δ a)] (f : (a : α) → γ aδ a) (s : (a : α) → Finset (γ a)) :
    (Fintype.piFinset fun (a : α) => Finset.image (f a) (s a)) = Finset.image (fun (b : (a : α) → γ a) (a : α) => f a (b a)) (Fintype.piFinset s)
    theorem Fintype.eval_image_piFinset_subset {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} (t : (a : α) → Finset (δ a)) (a : α) [DecidableEq (δ a)] :
    Finset.image (fun (f : (a : α) → δ a) => f a) (Fintype.piFinset t) t a
    theorem Fintype.eval_image_piFinset {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} (t : (a : α) → Finset (δ a)) (a : α) [DecidableEq (δ a)] (ht : ∀ (b : α), a b(t b).Nonempty) :
    Finset.image (fun (f : (a : α) → δ a) => f a) (Fintype.piFinset t) = t a
    theorem Fintype.eval_image_piFinset_const {α : Type u_1} [DecidableEq α] [Fintype α] {β : Type u_5} [DecidableEq β] (t : Finset β) (a : α) :
    Finset.image (fun (f : αβ) => f a) (Fintype.piFinset fun (_i : α) => t) = t
    theorem Fintype.filter_piFinset_of_not_mem {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} [(a : α) → DecidableEq (δ a)] (t : (a : α) → Finset (δ a)) (a : α) (x : δ a) (hx : xt a) :
    Finset.filter (fun (f : (a : α) → δ a) => f a = x) (Fintype.piFinset t) =
    theorem Fintype.piFinset_update_eq_filter_piFinset_mem {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} [(a : α) → DecidableEq (δ a)] (s : (i : α) → Finset (δ i)) (i : α) {t : Finset (δ i)} (hts : t s i) :
    Fintype.piFinset (Function.update s i t) = Finset.filter (fun (f : (a : α) → δ a) => f i t) (Fintype.piFinset s)
    theorem Fintype.piFinset_update_singleton_eq_filter_piFinset_eq {α : Type u_1} [DecidableEq α] [Fintype α] {δ : αType u_4} [(a : α) → DecidableEq (δ a)] (s : (i : α) → Finset (δ i)) (i : α) {a : δ i} (ha : a s i) :
    Fintype.piFinset (Function.update s i {a}) = Finset.filter (fun (f : (a : α) → δ a) => f i = a) (Fintype.piFinset s)

    pi #

    instance Pi.fintype {α : Type u_3} {β : αType u_4} [DecidableEq α] [Fintype α] [(a : α) → Fintype (β a)] :
    Fintype ((a : α) → β a)

    A dependent product of fintypes, indexed by a fintype, is a fintype.

    Equations
    • Pi.fintype = { elems := Fintype.piFinset fun (x : α) => Finset.univ, complete := }
    @[simp]
    theorem Fintype.piFinset_univ {α : Type u_3} {β : αType u_4} [DecidableEq α] [Fintype α] [(a : α) → Fintype (β a)] :
    (Fintype.piFinset fun (a : α) => Finset.univ) = Finset.univ
    noncomputable instance Function.Embedding.fintype {α : Type u_3} {β : Type u_4} [Fintype α] [Fintype β] :
    Fintype (α β)
    Equations
    instance RelHom.instFintype {α : Type u_3} {β : Type u_4} [Fintype α] [Fintype β] [DecidableEq α] {r : ααProp} {s : ββProp} [DecidableRel r] [DecidableRel s] :
    Equations
    • One or more equations did not get rendered due to their size.
    noncomputable instance RelEmbedding.instFintype {α : Type u_3} {β : Type u_4} [Fintype α] [Fintype β] {r : ααProp} {s : ββProp} :
    Equations
    @[simp]
    theorem Finset.univ_pi_univ {α : Type u_3} {β : αType u_4} [DecidableEq α] [Fintype α] [(a : α) → Fintype (β a)] :
    (Finset.univ.pi fun (a : α) => Finset.univ) = Finset.univ

    Diagonal #

    theorem Finset.piFinset_filter_const {α : Type u_1} {ι : Type u_3} [DecidableEq (ια)] {s : Finset α} [DecidableEq ι] [Fintype ι] :
    Finset.filter (fun (f : ια) => as, Function.const ι a = f) (Fintype.piFinset fun (x : ι) => s) = s.piDiag ι
    theorem Finset.piDiag_subset_piFinset {α : Type u_1} {ι : Type u_3} [DecidableEq (ια)] {s : Finset α} [DecidableEq ι] [Fintype ι] :
    s.piDiag ι Fintype.piFinset fun (x : ι) => s