Multisets

These are implemented as the quotient of a list by permutations.

Notation

We define the global infix notation ::ₘ for Multiset.cons.

def Multiset (α : Type u) : Type u

Multiset α is the quotient of List α by list permutation. The result is a type of finite sets with duplicates allowed.

Equations

• Multiset α = Quotient (List.isSetoid α)

def Multiset.ofList {α : Type u_1} : List α → Multiset α

The quotient map from List α to Multiset α.

Equations

• Multiset.ofList = Quot.mk Setoid.r

instance Multiset.instCoeListMultiset {α : Type u_1} : Coe (List α) (Multiset α)

Equations

• Multiset.instCoeListMultiset = { coe := Multiset.ofList }

@[simp]

theorem Multiset.quot_mk_to_coe {α : Type u_1} (l : List α) : Quotient.mk (List.isSetoid α) l = ↑l

@[simp]

theorem Multiset.quot_mk_to_coe' {α : Type u_1} (l : List α) : Quot.mk (fun x x_1 => x ≈ x_1) l = ↑l

@[simp]

theorem Multiset.quot_mk_to_coe'' {α : Type u_1} (l : List α) : Quot.mk Setoid.r l = ↑l

@[simp]

theorem Multiset.coe_eq_coe {α : Type u_1} {l₁ : List α} {l₂ : List α} : ↑l₁ = ↑l₂ ↔ l₁ ~ l₂

instance Multiset.instDecidableEquivListInstHasEquivIsSetoid {α : Type u_1} [DecidableEq α] (l₁ : List α) (l₂ : List α) : Decidable (l₁ ≈ l₂)

Equations

• Multiset.instDecidableEquivListInstHasEquivIsSetoid l₁ l₂ = inferInstanceAs (Decidable (l₁ ~ l₂))

instance Multiset.decidableEq {α : Type u_1} [DecidableEq α] : DecidableEq (Multiset α)

Equations

• One or more equations did not get rendered due to their size.

def Multiset.sizeOf {α : Type u_1} [SizeOf α] (s : Multiset α) : ℕ

defines a size for a multiset by referring to the size of the underlying list

Equations

• Multiset.sizeOf s = Quot.liftOn s sizeOf (_ : ∀ (x x_1 : List α), x ~ x_1 → sizeOf x = sizeOf x_1)

instance Multiset.instSizeOfMultiset {α : Type u_1} [SizeOf α] : SizeOf (Multiset α)

Equations

• Multiset.instSizeOfMultiset = { sizeOf := Multiset.sizeOf }

Empty multiset

def Multiset.zero {α : Type u_1} : Multiset α

0 : Multiset α is the empty set

Equations

• Multiset.zero = ↑[]

instance Multiset.instZeroMultiset {α : Type u_1} : Zero (Multiset α)

Equations

• Multiset.instZeroMultiset = { zero := Multiset.zero }

instance Multiset.instEmptyCollectionMultiset {α : Type u_1} : EmptyCollection (Multiset α)

Equations

• Multiset.instEmptyCollectionMultiset = { emptyCollection := 0 }

instance Multiset.inhabitedMultiset {α : Type u_1} : Inhabited (Multiset α)

Equations

• Multiset.inhabitedMultiset = { default := 0 }

@[simp]

theorem Multiset.coe_nil {α : Type u_1} : ↑[] = 0

@[simp]

theorem Multiset.empty_eq_zero {α : Type u_1} : ∅ = 0

@[simp]

theorem Multiset.coe_eq_zero {α : Type u_1} (l : List α) : ↑l = 0 ↔ l = []

theorem Multiset.coe_eq_zero_iff_isEmpty {α : Type u_1} (l : List α) : ↑l = 0 ↔ List.isEmpty l = true

Multiset.cons

def Multiset.cons {α : Type u_1} (a : α) (s : Multiset α) : Multiset α

cons a s is the multiset which contains s plus one more instance of a.

Equations

• a ::ₘ s = Quot.liftOn s (fun l => ↑(a :: l)) (_ : ∀ (x x_1 : List α), Setoid.r x x_1 → Quot.mk Setoid.r (a :: x) = Quot.mk Setoid.r (a :: x_1))

def Multiset.«term_::ₘ_» : Lean.TrailingParserDescr

cons a s is the multiset which contains s plus one more instance of a.

Equations

• Multiset.«term_::ₘ_» = Lean.ParserDescr.trailingNode `Multiset.term_::ₘ_ 67 68 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol " ::ₘ ") (Lean.ParserDescr.cat `term 67))

instance Multiset.instInsertMultiset {α : Type u_1} : Insert α (Multiset α)

Equations

• Multiset.instInsertMultiset = { insert := Multiset.cons }

@[simp]

theorem Multiset.insert_eq_cons {α : Type u_1} (a : α) (s : Multiset α) : insert a s = a ::ₘ s

@[simp]

theorem Multiset.cons_coe {α : Type u_1} (a : α) (l : List α) : a ::ₘ ↑l = ↑(a :: l)

@[simp]

theorem Multiset.cons_inj_left {α : Type u_1} {a : α} {b : α} (s : Multiset α) : a ::ₘ s = b ::ₘ s ↔ a = b

@[simp]

theorem Multiset.cons_inj_right {α : Type u_1} (a : α) {s : Multiset α} {t : Multiset α} : a ::ₘ s = a ::ₘ t ↔ s = t

theorem Multiset.induction {α : Type u_1} {p : Multiset α → Prop} (empty : p 0) (cons : ⦃a : α⦄ → {s : Multiset α} → p s → p (a ::ₘ s)) (s : Multiset α) : p s

theorem Multiset.induction_on {α : Type u_1} {p : Multiset α → Prop} (s : Multiset α) (empty : p 0) (cons : ⦃a : α⦄ → {s : Multiset α} → p s → p (a ::ₘ s)) : p s

theorem Multiset.cons_swap {α : Type u_1} (a : α) (b : α) (s : Multiset α) : a ::ₘ b ::ₘ s = b ::ₘ a ::ₘ s

def Multiset.rec {α : Type u_1} {C : Multiset α → Sort u_4} (C_0 : C 0) (C_cons : (a : α) → (m : Multiset α) → C m → C (a ::ₘ m)) (C_cons_heq : ∀ (a a' : α) (m : Multiset α) (b : C m), HEq (C_cons a (a' ::ₘ m) (C_cons a' m b)) (C_cons a' (a ::ₘ m) (C_cons a m b))) (m : Multiset α) : C m

Dependent recursor on multisets. TODO: should be @[recursor 6], but then the definition of Multiset.pi fails with a stack overflow in whnf.

Equations

• One or more equations did not get rendered due to their size.

def Multiset.recOn {α : Type u_1} {C : Multiset α → Sort u_4} (m : Multiset α) (C_0 : C 0) (C_cons : (a : α) → (m : Multiset α) → C m → C (a ::ₘ m)) (C_cons_heq : ∀ (a a' : α) (m : Multiset α) (b : C m), HEq (C_cons a (a' ::ₘ m) (C_cons a' m b)) (C_cons a' (a ::ₘ m) (C_cons a m b))) : C m

Companion to Multiset.rec with more convenient argument order.

Equations

• Multiset.recOn m C_0 C_cons C_cons_heq = Multiset.rec C_0 C_cons C_cons_heq m

@[simp]

theorem Multiset.recOn_0 {α : Type u_1} {C : Multiset α → Sort u_4} {C_0 : C 0} {C_cons : (a : α) → (m : Multiset α) → C m → C (a ::ₘ m)} {C_cons_heq : ∀ (a a' : α) (m : Multiset α) (b : C m), HEq (C_cons a (a' ::ₘ m) (C_cons a' m b)) (C_cons a' (a ::ₘ m) (C_cons a m b))} : Multiset.recOn 0 C_0 C_cons C_cons_heq = C_0

@[simp]

theorem Multiset.recOn_cons {α : Type u_1} {C : Multiset α → Sort u_4} {C_0 : C 0} {C_cons : (a : α) → (m : Multiset α) → C m → C (a ::ₘ m)} {C_cons_heq : ∀ (a a' : α) (m : Multiset α) (b : C m), HEq (C_cons a (a' ::ₘ m) (C_cons a' m b)) (C_cons a' (a ::ₘ m) (C_cons a m b))} (a : α) (m : Multiset α) : Multiset.recOn (a ::ₘ m) C_0 C_cons C_cons_heq = C_cons a m (Multiset.recOn m C_0 C_cons C_cons_heq)

def Multiset.Mem {α : Type u_1} (a : α) (s : Multiset α) : Prop

a ∈ s means that a has nonzero multiplicity in s.

Equations

• Multiset.Mem a s = Quot.liftOn s (fun l => a ∈ l) (_ : ∀ (l₁ l₂ : List α), l₁ ~ l₂ → (fun l => a ∈ l) l₁ = (fun l => a ∈ l) l₂)

instance Multiset.instMembershipMultiset {α : Type u_1} : Membership α (Multiset α)

Equations

• Multiset.instMembershipMultiset = { mem := Multiset.Mem }

@[simp]

theorem Multiset.mem_coe {α : Type u_1} {a : α} {l : List α} : a ∈ ↑l ↔ a ∈ l

instance Multiset.decidableMem {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) : Decidable (a ∈ s)

Equations

• Multiset.decidableMem a s = Quot.recOnSubsingleton' s fun l => inferInstanceAs (Decidable (a ∈ l))

@[simp]

theorem Multiset.mem_cons {α : Type u_1} {a : α} {b : α} {s : Multiset α} : a ∈ b ::ₘ s ↔ a = b ∨ a ∈ s

theorem Multiset.mem_cons_of_mem {α : Type u_1} {a : α} {b : α} {s : Multiset α} (h : a ∈ s) : a ∈ b ::ₘ s

theorem Multiset.mem_cons_self {α : Type u_1} (a : α) (s : Multiset α) : a ∈ a ::ₘ s

theorem Multiset.forall_mem_cons {α : Type u_1} {p : α → Prop} {a : α} {s : Multiset α} : ((x : α) → x ∈ a ::ₘ s → p x) ↔ p a ∧ ((x : α) → x ∈ s → p x)

theorem Multiset.exists_cons_of_mem {α : Type u_1} {s : Multiset α} {a : α} : a ∈ s → ∃ t, s = a ::ₘ t

@[simp]

theorem Multiset.not_mem_zero {α : Type u_1} (a : α) : ¬a ∈ 0

theorem Multiset.eq_zero_of_forall_not_mem {α : Type u_1} {s : Multiset α} : (∀ (x : α), ¬x ∈ s) → s = 0

theorem Multiset.eq_zero_iff_forall_not_mem {α : Type u_1} {s : Multiset α} : s = 0 ↔ ∀ (a : α), ¬a ∈ s

theorem Multiset.exists_mem_of_ne_zero {α : Type u_1} {s : Multiset α} : s ≠ 0 → ∃ a, a ∈ s

theorem Multiset.empty_or_exists_mem {α : Type u_1} (s : Multiset α) : s = 0 ∨ ∃ a, a ∈ s

@[simp]

theorem Multiset.zero_ne_cons {α : Type u_1} {a : α} {m : Multiset α} : 0 ≠ a ::ₘ m

@[simp]

theorem Multiset.cons_ne_zero {α : Type u_1} {a : α} {m : Multiset α} : a ::ₘ m ≠ 0

theorem Multiset.cons_eq_cons {α : Type u_1} {a : α} {b : α} {as : Multiset α} {bs : Multiset α} : a ::ₘ as = b ::ₘ bs ↔ a = b ∧ as = bs ∨ a ≠ b ∧ ∃ cs, as = b ::ₘ cs ∧ bs = a ::ₘ cs

Singleton

instance Multiset.instSingletonMultiset {α : Type u_1} : Singleton α (Multiset α)

Equations

• Multiset.instSingletonMultiset = { singleton := fun a => a ::ₘ 0 }

instance Multiset.instIsLawfulSingletonMultisetInstEmptyCollectionMultisetInstInsertMultisetInstSingletonMultiset {α : Type u_1} : IsLawfulSingleton α (Multiset α)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.cons_zero {α : Type u_1} (a : α) : a ::ₘ 0 = {a}

@[simp]

theorem Multiset.coe_singleton {α : Type u_1} (a : α) : ↑[a] = {a}

@[simp]

theorem Multiset.mem_singleton {α : Type u_1} {a : α} {b : α} : b ∈ {a} ↔ b = a

theorem Multiset.mem_singleton_self {α : Type u_1} (a : α) : a ∈ {a}

@[simp]

theorem Multiset.singleton_inj {α : Type u_1} {a : α} {b : α} : {a} = {b} ↔ a = b

@[simp]

theorem Multiset.coe_eq_singleton {α : Type u_1} {l : List α} {a : α} : ↑l = {a} ↔ l = [a]

@[simp]

theorem Multiset.singleton_eq_cons_iff {α : Type u_1} {a : α} {b : α} (m : Multiset α) : {a} = b ::ₘ m ↔ a = b ∧ m = 0

theorem Multiset.pair_comm {α : Type u_1} (x : α) (y : α) : {x, y} = {y, x}

Multiset.Subset

def Multiset.Subset {α : Type u_1} (s : Multiset α) (t : Multiset α) : Prop

s ⊆ t is the lift of the list subset relation. It means that any element with nonzero multiplicity in s has nonzero multiplicity in t, but it does not imply that the multiplicity of a in s is less or equal than in t; see s ≤ t for this relation.

Equations

• Multiset.Subset s t = ∀ ⦃a : α⦄, a ∈ s → a ∈ t

instance Multiset.instHasSubsetMultiset {α : Type u_1} : HasSubset (Multiset α)

Equations

• Multiset.instHasSubsetMultiset = { Subset := Multiset.Subset }

instance Multiset.instHasSSubsetMultiset {α : Type u_1} : HasSSubset (Multiset α)

Equations

• Multiset.instHasSSubsetMultiset = { SSubset := fun s t => s ⊆ t ∧ ¬t ⊆ s }

@[simp]

theorem Multiset.coe_subset {α : Type u_1} {l₁ : List α} {l₂ : List α} : ↑l₁ ⊆ ↑l₂ ↔ l₁ ⊆ l₂

@[simp]

theorem Multiset.Subset.refl {α : Type u_1} (s : Multiset α) : s ⊆ s

theorem Multiset.Subset.trans {α : Type u_1} {s : Multiset α} {t : Multiset α} {u : Multiset α} : s ⊆ t → t ⊆ u → s ⊆ u

theorem Multiset.subset_iff {α : Type u_1} {s : Multiset α} {t : Multiset α} : s ⊆ t ↔ ∀ ⦃x : α⦄, x ∈ s → x ∈ t

theorem Multiset.mem_of_subset {α : Type u_1} {s : Multiset α} {t : Multiset α} {a : α} (h : s ⊆ t) : a ∈ s → a ∈ t

@[simp]

theorem Multiset.zero_subset {α : Type u_1} (s : Multiset α) : 0 ⊆ s

theorem Multiset.subset_cons {α : Type u_1} (s : Multiset α) (a : α) : s ⊆ a ::ₘ s

theorem Multiset.ssubset_cons {α : Type u_1} {s : Multiset α} {a : α} (ha : ¬a ∈ s) : s ⊂ a ::ₘ s

@[simp]

theorem Multiset.cons_subset {α : Type u_1} {a : α} {s : Multiset α} {t : Multiset α} : a ::ₘ s ⊆ t ↔ a ∈ t ∧ s ⊆ t

theorem Multiset.cons_subset_cons {α : Type u_1} {a : α} {s : Multiset α} {t : Multiset α} : s ⊆ t → a ::ₘ s ⊆ a ::ₘ t

theorem Multiset.eq_zero_of_subset_zero {α : Type u_1} {s : Multiset α} (h : s ⊆ 0) : s = 0

theorem Multiset.subset_zero {α : Type u_1} {s : Multiset α} : s ⊆ 0 ↔ s = 0

theorem Multiset.induction_on' {α : Type u_1} {p : Multiset α → Prop} (S : Multiset α) (h₁ : p 0) (h₂ : {a : α} → {s : Multiset α} → a ∈ S → s ⊆ S → p s → p (insert a s)) : p S

Multiset.toList

noncomputable def Multiset.toList {α : Type u_1} (s : Multiset α) : List α

Produces a list of the elements in the multiset using choice.

Equations

• Multiset.toList s = Quotient.out' s

@[simp]

theorem Multiset.coe_toList {α : Type u_1} (s : Multiset α) : ↑(Multiset.toList s) = s

@[simp]

theorem Multiset.toList_eq_nil {α : Type u_1} {s : Multiset α} : Multiset.toList s = [] ↔ s = 0

@[simp]

theorem Multiset.empty_toList {α : Type u_1} {s : Multiset α} : List.isEmpty (Multiset.toList s) = true ↔ s = 0

@[simp]

theorem Multiset.toList_zero {α : Type u_1} : Multiset.toList 0 = []

@[simp]

theorem Multiset.mem_toList {α : Type u_1} {a : α} {s : Multiset α} : a ∈ Multiset.toList s ↔ a ∈ s

@[simp]

theorem Multiset.toList_eq_singleton_iff {α : Type u_1} {a : α} {m : Multiset α} : Multiset.toList m = [a] ↔ m = {a}

@[simp]

theorem Multiset.toList_singleton {α : Type u_1} (a : α) : Multiset.toList {a} = [a]

Partial order on Multisets

def Multiset.Le {α : Type u_1} (s : Multiset α) (t : Multiset α) : Prop

s ≤ t means that s is a sublist of t (up to permutation). Equivalently, s ≤ t means that count a s ≤ count a t for all a.

Equations

• Multiset.Le s t = Quotient.liftOn₂ s t (fun x x_1 => x <+~ x_1) (_ : ∀ (x x_1 x_2 x_3 : List α), x ≈ x_2 → x_1 ≈ x_3 → (x <+~ x_1) = (x_2 <+~ x_3))

instance Multiset.instPartialOrderMultiset {α : Type u_1} : PartialOrder (Multiset α)

Equations

• Multiset.instPartialOrderMultiset = PartialOrder.mk (_ : ∀ (a b : Multiset α), a ≤ b → b ≤ a → a = b)

instance Multiset.decidableLE {α : Type u_1} [DecidableEq α] : DecidableRel fun x x_1 => x ≤ x_1

Equations

• Multiset.decidableLE s t = Quotient.recOnSubsingleton₂ s t List.decidableSubperm

theorem Multiset.subset_of_le {α : Type u_1} {s : Multiset α} {t : Multiset α} : s ≤ t → s ⊆ t

theorem Multiset.Le.subset {α : Type u_1} {s : Multiset α} {t : Multiset α} : s ≤ t → s ⊆ t

Alias of Multiset.subset_of_le.

theorem Multiset.mem_of_le {α : Type u_1} {s : Multiset α} {t : Multiset α} {a : α} (h : s ≤ t) : a ∈ s → a ∈ t

theorem Multiset.not_mem_mono {α : Type u_1} {s : Multiset α} {t : Multiset α} {a : α} (h : s ⊆ t) : ¬a ∈ t → ¬a ∈ s

@[simp]

theorem Multiset.coe_le {α : Type u_1} {l₁ : List α} {l₂ : List α} : ↑l₁ ≤ ↑l₂ ↔ l₁ <+~ l₂

theorem Multiset.leInductionOn {α : Type u_1} {C : Multiset α → Multiset α → Prop} {s : Multiset α} {t : Multiset α} (h : s ≤ t) (H : {l₁ l₂ : List α} → List.Sublist l₁ l₂ → C ↑l₁ ↑l₂) : C s t

theorem Multiset.zero_le {α : Type u_1} (s : Multiset α) : 0 ≤ s

instance Multiset.instOrderBotMultisetToLEToPreorderInstPartialOrderMultiset {α : Type u_1} : OrderBot (Multiset α)

Equations

• Multiset.instOrderBotMultisetToLEToPreorderInstPartialOrderMultiset = OrderBot.mk (_ : ∀ (s : Multiset α), 0 ≤ s)

@[simp]

theorem Multiset.bot_eq_zero {α : Type u_1} : ⊥ = 0

This is a rfl and simp version of bot_eq_zero.

theorem Multiset.le_zero {α : Type u_1} {s : Multiset α} : s ≤ 0 ↔ s = 0

theorem Multiset.lt_cons_self {α : Type u_1} (s : Multiset α) (a : α) : s < a ::ₘ s

theorem Multiset.le_cons_self {α : Type u_1} (s : Multiset α) (a : α) : s ≤ a ::ₘ s

theorem Multiset.cons_le_cons_iff {α : Type u_1} {s : Multiset α} {t : Multiset α} (a : α) : a ::ₘ s ≤ a ::ₘ t ↔ s ≤ t

theorem Multiset.cons_le_cons {α : Type u_1} {s : Multiset α} {t : Multiset α} (a : α) : s ≤ t → a ::ₘ s ≤ a ::ₘ t

theorem Multiset.le_cons_of_not_mem {α : Type u_1} {s : Multiset α} {t : Multiset α} {a : α} (m : ¬a ∈ s) : s ≤ a ::ₘ t ↔ s ≤ t

@[simp]

theorem Multiset.singleton_ne_zero {α : Type u_1} (a : α) : {a} ≠ 0

@[simp]

theorem Multiset.singleton_le {α : Type u_1} {a : α} {s : Multiset α} : {a} ≤ s ↔ a ∈ s

Additive monoid

def Multiset.add {α : Type u_1} (s₁ : Multiset α) (s₂ : Multiset α) : Multiset α

The sum of two multisets is the lift of the list append operation. This adds the multiplicities of each element, i.e. count a (s + t) = count a s + count a t.

Equations

• Multiset.add s₁ s₂ = Quotient.liftOn₂ s₁ s₂ (fun l₁ l₂ => ↑(l₁ ++ l₂)) (_ : ∀ (x x_1 x_2 x_3 : List α), x ≈ x_2 → x_1 ≈ x_3 → Quot.mk Setoid.r (x ++ x_1) = Quot.mk Setoid.r (x_2 ++ x_3))

instance Multiset.instAddMultiset {α : Type u_1} : Add (Multiset α)

Equations

• Multiset.instAddMultiset = { add := Multiset.add }

@[simp]

theorem Multiset.coe_add {α : Type u_1} (s : List α) (t : List α) : ↑s + ↑t = ↑(s ++ t)

@[simp]

theorem Multiset.singleton_add {α : Type u_1} (a : α) (s : Multiset α) : {a} + s = a ::ₘ s

instance Multiset.instCovariantClassMultisetHAddInstHAddInstAddMultisetLeToLEToPreorderInstPartialOrderMultiset {α : Type u_1} : CovariantClass (Multiset α) (Multiset α) (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

Equations

• One or more equations did not get rendered due to their size.

instance Multiset.instContravariantClassMultisetHAddInstHAddInstAddMultisetLeToLEToPreorderInstPartialOrderMultiset {α : Type u_1} : ContravariantClass (Multiset α) (Multiset α) (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1

Equations

• One or more equations did not get rendered due to their size.

instance Multiset.instOrderedCancelAddCommMonoidMultiset {α : Type u_1} : OrderedCancelAddCommMonoid (Multiset α)

Equations

• One or more equations did not get rendered due to their size.

theorem Multiset.le_add_right {α : Type u_1} (s : Multiset α) (t : Multiset α) : s ≤ s + t

theorem Multiset.le_add_left {α : Type u_1} (s : Multiset α) (t : Multiset α) : s ≤ t + s

theorem Multiset.le_iff_exists_add {α : Type u_1} {s : Multiset α} {t : Multiset α} : s ≤ t ↔ ∃ u, t = s + u

instance Multiset.instCanonicallyOrderedAddMonoidMultiset {α : Type u_1} : CanonicallyOrderedAddMonoid (Multiset α)

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.cons_add {α : Type u_1} (a : α) (s : Multiset α) (t : Multiset α) : a ::ₘ s + t = a ::ₘ (s + t)

@[simp]

theorem Multiset.add_cons {α : Type u_1} (a : α) (s : Multiset α) (t : Multiset α) : s + a ::ₘ t = a ::ₘ (s + t)

@[simp]

theorem Multiset.mem_add {α : Type u_1} {a : α} {s : Multiset α} {t : Multiset α} : a ∈ s + t ↔ a ∈ s ∨ a ∈ t

theorem Multiset.mem_of_mem_nsmul {α : Type u_1} {a : α} {s : Multiset α} {n : ℕ} (h : a ∈ n • s) : a ∈ s

@[simp]

theorem Multiset.mem_nsmul {α : Type u_1} {a : α} {s : Multiset α} {n : ℕ} (h0 : n ≠ 0) : a ∈ n • s ↔ a ∈ s

theorem Multiset.nsmul_cons {α : Type u_1} {s : Multiset α} (n : ℕ) (a : α) : n • (a ::ₘ s) = n • {a} + n • s

Cardinality

def Multiset.card {α : Type u_1} : Multiset α →+ ℕ

The cardinality of a multiset is the sum of the multiplicities of all its elements, or simply the length of the underlying list.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.coe_card {α : Type u_1} (l : List α) : ↑Multiset.card ↑l = List.length l

@[simp]

theorem Multiset.length_toList {α : Type u_1} (s : Multiset α) : List.length (Multiset.toList s) = ↑Multiset.card s

@[simp]

theorem Multiset.card_zero {α : Type u_1} : ↑Multiset.card 0 = 0

theorem Multiset.card_add {α : Type u_1} (s : Multiset α) (t : Multiset α) : ↑Multiset.card (s + t) = ↑Multiset.card s + ↑Multiset.card t

theorem Multiset.card_nsmul {α : Type u_1} (s : Multiset α) (n : ℕ) : ↑Multiset.card (n • s) = n * ↑Multiset.card s

@[simp]

theorem Multiset.card_cons {α : Type u_1} (a : α) (s : Multiset α) : ↑Multiset.card (a ::ₘ s) = ↑Multiset.card s + 1

@[simp]

theorem Multiset.card_singleton {α : Type u_1} (a : α) : ↑Multiset.card {a} = 1

theorem Multiset.card_pair {α : Type u_1} (a : α) (b : α) : ↑Multiset.card {a, b} = 2

theorem Multiset.card_eq_one {α : Type u_1} {s : Multiset α} : ↑Multiset.card s = 1 ↔ ∃ a, s = {a}

theorem Multiset.card_le_of_le {α : Type u_1} {s : Multiset α} {t : Multiset α} (h : s ≤ t) : ↑Multiset.card s ≤ ↑Multiset.card t

theorem Multiset.card_mono {α : Type u_1} : Monotone ↑Multiset.card

theorem Multiset.eq_of_le_of_card_le {α : Type u_1} {s : Multiset α} {t : Multiset α} (h : s ≤ t) : ↑Multiset.card t ≤ ↑Multiset.card s → s = t

theorem Multiset.card_lt_of_lt {α : Type u_1} {s : Multiset α} {t : Multiset α} (h : s < t) : ↑Multiset.card s < ↑Multiset.card t

theorem Multiset.lt_iff_cons_le {α : Type u_1} {s : Multiset α} {t : Multiset α} : s < t ↔ ∃ a, a ::ₘ s ≤ t

@[simp]

theorem Multiset.card_eq_zero {α : Type u_1} {s : Multiset α} : ↑Multiset.card s = 0 ↔ s = 0

theorem Multiset.card_pos {α : Type u_1} {s : Multiset α} : 0 < ↑Multiset.card s ↔ s ≠ 0

theorem Multiset.card_pos_iff_exists_mem {α : Type u_1} {s : Multiset α} : 0 < ↑Multiset.card s ↔ ∃ a, a ∈ s

theorem Multiset.card_eq_two {α : Type u_1} {s : Multiset α} : ↑Multiset.card s = 2 ↔ ∃ x y, s = {x, y}

theorem Multiset.card_eq_three {α : Type u_1} {s : Multiset α} : ↑Multiset.card s = 3 ↔ ∃ x y z, s = {x, y, z}

Induction principles

def Multiset.strongInductionOn {α : Type u_1} {p : Multiset α → Sort u_4} (s : Multiset α) (ih : (s : Multiset α) → ((t : Multiset α) → t < s → p t) → p s) : p s

The strong induction principle for multisets.

Equations

• Multiset.strongInductionOn s ih = ih s fun t _h => Multiset.strongInductionOn t ih

theorem Multiset.strongInductionOn_eq {α : Type u_1} {p : Multiset α → Sort u_4} (s : Multiset α) (H : (s : Multiset α) → ((t : Multiset α) → t < s → p t) → p s) : Multiset.strongInductionOn s H = H s fun t _h => Multiset.strongInductionOn t H

theorem Multiset.case_strongInductionOn {α : Type u_1} {p : Multiset α → Prop} (s : Multiset α) (h₀ : p 0) (h₁ : (a : α) → (s : Multiset α) → ((t : Multiset α) → t ≤ s → p t) → p (a ::ₘ s)) : p s

def Multiset.strongDownwardInduction {α : Type u_1} {p : Multiset α → Sort u_4} {n : ℕ} (H : (t₁ : Multiset α) → ({t₂ : Multiset α} → ↑Multiset.card t₂ ≤ n → t₁ < t₂ → p t₂) → ↑Multiset.card t₁ ≤ n → p t₁) (s : Multiset α) : ↑Multiset.card s ≤ n → p s

Suppose that, given that p t can be defined on all supersets of s of cardinality less than n, one knows how to define p s. Then one can inductively define p s for all multisets s of cardinality less than n, starting from multisets of card n and iterating. This can be used either to define data, or to prove properties.

Equations

• Multiset.strongDownwardInduction H s = H s fun {t} ht _h => Multiset.strongDownwardInduction H t ht

theorem Multiset.strongDownwardInduction_eq {α : Type u_1} {p : Multiset α → Sort u_4} {n : ℕ} (H : (t₁ : Multiset α) → ({t₂ : Multiset α} → ↑Multiset.card t₂ ≤ n → t₁ < t₂ → p t₂) → ↑Multiset.card t₁ ≤ n → p t₁) (s : Multiset α) : Multiset.strongDownwardInduction H s = H s fun {t₂} ht _hst => Multiset.strongDownwardInduction H t₂ ht

def Multiset.strongDownwardInductionOn {α : Type u_1} {p : Multiset α → Sort u_4} {n : ℕ} (s : Multiset α) : ((t₁ : Multiset α) → ({t₂ : Multiset α} → ↑Multiset.card t₂ ≤ n → t₁ < t₂ → p t₂) → ↑Multiset.card t₁ ≤ n → p t₁) → ↑Multiset.card s ≤ n → p s

Analogue of strongDownwardInduction with order of arguments swapped.

Equations

• Multiset.strongDownwardInductionOn s H = Multiset.strongDownwardInduction H s

theorem Multiset.strongDownwardInductionOn_eq {α : Type u_1} {p : Multiset α → Sort u_4} (s : Multiset α) {n : ℕ} (H : (t₁ : Multiset α) → ({t₂ : Multiset α} → ↑Multiset.card t₂ ≤ n → t₁ < t₂ → p t₂) → ↑Multiset.card t₁ ≤ n → p t₁) : (fun a => Multiset.strongDownwardInductionOn s H a) = H s fun {t} ht _h => Multiset.strongDownwardInductionOn t H ht

theorem Multiset.wellFounded_lt {α : Type u_1} : WellFounded fun x x_1 => x < x_1

Another way of expressing strongInductionOn: the (<) relation is well-founded.

instance Multiset.is_wellFounded_lt {α : Type u_1} : WellFoundedLT (Multiset α)

Equations

• @Multiset.is_wellFounded_lt = @Multiset.is_wellFounded_lt.proof_1

Multiset.replicate

def Multiset.replicate {α : Type u_1} (n : ℕ) (a : α) : Multiset α

replicate n a is the multiset containing only a with multiplicity n.

Equations

• Multiset.replicate n a = ↑(List.replicate n a)

theorem Multiset.coe_replicate {α : Type u_1} (n : ℕ) (a : α) : ↑(List.replicate n a) = Multiset.replicate n a

@[simp]

theorem Multiset.replicate_zero {α : Type u_1} (a : α) : Multiset.replicate 0 a = 0

@[simp]

theorem Multiset.replicate_succ {α : Type u_1} (a : α) (n : ℕ) : Multiset.replicate (n + 1) a = a ::ₘ Multiset.replicate n a

theorem Multiset.replicate_add {α : Type u_1} (m : ℕ) (n : ℕ) (a : α) : Multiset.replicate (m + n) a = Multiset.replicate m a + Multiset.replicate n a

@[simp]

theorem Multiset.replicateAddMonoidHom_apply {α : Type u_1} (a : α) (n : ℕ) : ↑(Multiset.replicateAddMonoidHom a) n = Multiset.replicate n a

def Multiset.replicateAddMonoidHom {α : Type u_1} (a : α) : ℕ →+ Multiset α

Multiset.replicate as an AddMonoidHom.

Equations

• One or more equations did not get rendered due to their size.

theorem Multiset.replicate_one {α : Type u_1} (a : α) : Multiset.replicate 1 a = {a}

@[simp]

theorem Multiset.card_replicate {α : Type u_1} (n : ℕ) (a : α) : ↑Multiset.card (Multiset.replicate n a) = n

theorem Multiset.mem_replicate {α : Type u_1} {a : α} {b : α} {n : ℕ} : b ∈ Multiset.replicate n a ↔ n ≠ 0 ∧ b = a

theorem Multiset.eq_of_mem_replicate {α : Type u_1} {a : α} {b : α} {n : ℕ} : b ∈ Multiset.replicate n a → b = a

theorem Multiset.eq_replicate_card {α : Type u_1} {a : α} {s : Multiset α} : s = Multiset.replicate (↑Multiset.card s) a ↔ ∀ (b : α), b ∈ s → b = a

theorem Multiset.eq_replicate_of_mem {α : Type u_1} {a : α} {s : Multiset α} : (∀ (b : α), b ∈ s → b = a) → s = Multiset.replicate (↑Multiset.card s) a

Alias of the reverse direction of Multiset.eq_replicate_card.

theorem Multiset.eq_replicate {α : Type u_1} {a : α} {n : ℕ} {s : Multiset α} : s = Multiset.replicate n a ↔ ↑Multiset.card s = n ∧ ∀ (b : α), b ∈ s → b = a

theorem Multiset.replicate_right_injective {α : Type u_1} {n : ℕ} (hn : n ≠ 0) : Function.Injective (Multiset.replicate n)

@[simp]

theorem Multiset.replicate_right_inj {α : Type u_1} {a : α} {b : α} {n : ℕ} (h : n ≠ 0) : Multiset.replicate n a = Multiset.replicate n b ↔ a = b

theorem Multiset.replicate_left_injective {α : Type u_1} (a : α) : Function.Injective fun x => Multiset.replicate x a

theorem Multiset.replicate_subset_singleton {α : Type u_1} (n : ℕ) (a : α) : Multiset.replicate n a ⊆ {a}

theorem Multiset.replicate_le_coe {α : Type u_1} {a : α} {n : ℕ} {l : List α} : Multiset.replicate n a ≤ ↑l ↔ List.Sublist (List.replicate n a) l

theorem Multiset.nsmul_replicate {α : Type u_1} {a : α} (n : ℕ) (m : ℕ) : n • Multiset.replicate m a = Multiset.replicate (n * m) a

theorem Multiset.nsmul_singleton {α : Type u_1} (a : α) (n : ℕ) : n • {a} = Multiset.replicate n a

theorem Multiset.replicate_le_replicate {α : Type u_1} (a : α) {k : ℕ} {n : ℕ} : Multiset.replicate k a ≤ Multiset.replicate n a ↔ k ≤ n

theorem Multiset.le_replicate_iff {α : Type u_1} {m : Multiset α} {a : α} {n : ℕ} : m ≤ Multiset.replicate n a ↔ ∃ k, k ≤ n ∧ m = Multiset.replicate k a

theorem Multiset.lt_replicate_succ {α : Type u_1} {m : Multiset α} {x : α} {n : ℕ} : m < Multiset.replicate (n + 1) x ↔ m ≤ Multiset.replicate n x

Erasing one copy of an element

def Multiset.erase {α : Type u_1} [DecidableEq α] (s : Multiset α) (a : α) : Multiset α

erase s a is the multiset that subtracts 1 from the multiplicity of a.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.coe_erase {α : Type u_1} [DecidableEq α] (l : List α) (a : α) : Multiset.erase (↑l) a = ↑(List.erase l a)

@[simp]

theorem Multiset.erase_zero {α : Type u_1} [DecidableEq α] (a : α) : Multiset.erase 0 a = 0

@[simp]

theorem Multiset.erase_cons_head {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) : Multiset.erase (a ::ₘ s) a = s

@[simp]

theorem Multiset.erase_cons_tail {α : Type u_1} [DecidableEq α] {a : α} {b : α} (s : Multiset α) (h : b ≠ a) : Multiset.erase (b ::ₘ s) a = b ::ₘ Multiset.erase s a

@[simp]

theorem Multiset.erase_singleton {α : Type u_1} [DecidableEq α] (a : α) : Multiset.erase {a} a = 0

@[simp]

theorem Multiset.erase_of_not_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : ¬a ∈ s → Multiset.erase s a = s

@[simp]

theorem Multiset.cons_erase {α : Type u_1} [DecidableEq α] {s : Multiset α} {a : α} : a ∈ s → a ::ₘ Multiset.erase s a = s

theorem Multiset.le_cons_erase {α : Type u_1} [DecidableEq α] (s : Multiset α) (a : α) : s ≤ a ::ₘ Multiset.erase s a

theorem Multiset.add_singleton_eq_iff {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {a : α} : s + {a} = t ↔ a ∈ t ∧ s = Multiset.erase t a

theorem Multiset.erase_add_left_pos {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} (t : Multiset α) : a ∈ s → Multiset.erase (s + t) a = Multiset.erase s a + t

theorem Multiset.erase_add_right_pos {α : Type u_1} [DecidableEq α] {a : α} (s : Multiset α) {t : Multiset α} (h : a ∈ t) : Multiset.erase (s + t) a = s + Multiset.erase t a

theorem Multiset.erase_add_right_neg {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} (t : Multiset α) : ¬a ∈ s → Multiset.erase (s + t) a = s + Multiset.erase t a

theorem Multiset.erase_add_left_neg {α : Type u_1} [DecidableEq α] {a : α} (s : Multiset α) {t : Multiset α} (h : ¬a ∈ t) : Multiset.erase (s + t) a = Multiset.erase s a + t

theorem Multiset.erase_le {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) : Multiset.erase s a ≤ s

@[simp]

theorem Multiset.erase_lt {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : Multiset.erase s a < s ↔ a ∈ s

theorem Multiset.erase_subset {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) : Multiset.erase s a ⊆ s

theorem Multiset.mem_erase_of_ne {α : Type u_1} [DecidableEq α] {a : α} {b : α} {s : Multiset α} (ab : a ≠ b) : a ∈ Multiset.erase s b ↔ a ∈ s

theorem Multiset.mem_of_mem_erase {α : Type u_1} [DecidableEq α] {a : α} {b : α} {s : Multiset α} : a ∈ Multiset.erase s b → a ∈ s

theorem Multiset.erase_comm {α : Type u_1} [DecidableEq α] (s : Multiset α) (a : α) (b : α) : Multiset.erase (Multiset.erase s a) b = Multiset.erase (Multiset.erase s b) a

theorem Multiset.erase_le_erase {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (a : α) (h : s ≤ t) : Multiset.erase s a ≤ Multiset.erase t a

theorem Multiset.erase_le_iff_le_cons {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {a : α} : Multiset.erase s a ≤ t ↔ s ≤ a ::ₘ t

@[simp]

theorem Multiset.card_erase_of_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : a ∈ s → ↑Multiset.card (Multiset.erase s a) = Nat.pred (↑Multiset.card s)

@[simp]

theorem Multiset.card_erase_add_one {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : a ∈ s → ↑Multiset.card (Multiset.erase s a) + 1 = ↑Multiset.card s

theorem Multiset.card_erase_lt_of_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : a ∈ s → ↑Multiset.card (Multiset.erase s a) < ↑Multiset.card s

theorem Multiset.card_erase_le {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : ↑Multiset.card (Multiset.erase s a) ≤ ↑Multiset.card s

theorem Multiset.card_erase_eq_ite {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : ↑Multiset.card (Multiset.erase s a) = if a ∈ s then Nat.pred (↑Multiset.card s) else ↑Multiset.card s

@[simp]

theorem Multiset.coe_reverse {α : Type u_1} (l : List α) : ↑(List.reverse l) = ↑l

Multiset.map

def Multiset.map {α : Type u_1} {β : Type u_2} (f : α → β) (s : Multiset α) : Multiset β

map f s is the lift of the list map operation. The multiplicity of b in map f s is the number of a ∈ s (counting multiplicity) such that f a = b.

Equations

• Multiset.map f s = Quot.liftOn s (fun l => ↑(List.map f l)) (_ : ∀ (_l₁ _l₂ : List α), Setoid.r _l₁ _l₂ → Quot.mk Setoid.r (List.map f _l₁) = Quot.mk Setoid.r (List.map f _l₂))

theorem Multiset.map_congr {α : Type u_1} {β : Type u_2} {f : α → β} {g : α → β} {s : Multiset α} {t : Multiset α} : s = t → (∀ (x : α), x ∈ t → f x = g x) → Multiset.map f s = Multiset.map g t

theorem Multiset.map_hcongr {α : Type u_1} {β : Type u_2} {β' : Type u_2} {m : Multiset α} {f : α → β} {f' : α → β'} (h : β = β') (hf : ∀ (a : α), a ∈ m → HEq (f a) (f' a)) : HEq (Multiset.map f m) (Multiset.map f' m)

theorem Multiset.forall_mem_map_iff {α : Type u_1} {β : Type u_2} {f : α → β} {p : β → Prop} {s : Multiset α} : ((y : β) → y ∈ Multiset.map f s → p y) ↔ (x : α) → x ∈ s → p (f x)

@[simp]

theorem Multiset.coe_map {α : Type u_1} {β : Type u_2} (f : α → β) (l : List α) : Multiset.map f ↑l = ↑(List.map f l)

@[simp]

theorem Multiset.map_zero {α : Type u_1} {β : Type u_2} (f : α → β) : Multiset.map f 0 = 0

@[simp]

theorem Multiset.map_cons {α : Type u_1} {β : Type u_2} (f : α → β) (a : α) (s : Multiset α) : Multiset.map f (a ::ₘ s) = f a ::ₘ Multiset.map f s

theorem Multiset.map_comp_cons {α : Type u_1} {β : Type u_2} (f : α → β) (t : α) : Multiset.map f ∘ Multiset.cons t = Multiset.cons (f t) ∘ Multiset.map f

@[simp]

theorem Multiset.map_singleton {α : Type u_1} {β : Type u_2} (f : α → β) (a : α) : Multiset.map f {a} = {f a}

@[simp]

theorem Multiset.map_replicate {α : Type u_1} {β : Type u_2} (f : α → β) (k : ℕ) (a : α) : Multiset.map f (Multiset.replicate k a) = Multiset.replicate k (f a)

@[simp]

theorem Multiset.map_add {α : Type u_1} {β : Type u_2} (f : α → β) (s : Multiset α) (t : Multiset α) : Multiset.map f (s + t) = Multiset.map f s + Multiset.map f t

instance Multiset.canLift {α : Type u_1} {β : Type u_2} (c : β → α) (p : α → Prop) [CanLift α β c p] : CanLift (Multiset α) (Multiset β) (Multiset.map c) fun s => (x : α) → x ∈ s → p x

If each element of s : Multiset α can be lifted to β, then s can be lifted to Multiset β.

Equations

• @Multiset.canLift = @Multiset.canLift.proof_1

def Multiset.mapAddMonoidHom {α : Type u_1} {β : Type u_2} (f : α → β) : Multiset α →+ Multiset β

Multiset.map as an AddMonoidHom.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.coe_mapAddMonoidHom {α : Type u_1} {β : Type u_2} (f : α → β) : ↑(Multiset.mapAddMonoidHom f) = Multiset.map f

theorem Multiset.map_nsmul {α : Type u_1} {β : Type u_2} (f : α → β) (n : ℕ) (s : Multiset α) : Multiset.map f (n • s) = n • Multiset.map f s

@[simp]

theorem Multiset.mem_map {α : Type u_1} {β : Type u_2} {f : α → β} {b : β} {s : Multiset α} : b ∈ Multiset.map f s ↔ ∃ a, a ∈ s ∧ f a = b

@[simp]

theorem Multiset.card_map {α : Type u_1} {β : Type u_2} (f : α → β) (s : Multiset α) : ↑Multiset.card (Multiset.map f s) = ↑Multiset.card s

@[simp]

theorem Multiset.map_eq_zero {α : Type u_1} {β : Type u_2} {s : Multiset α} {f : α → β} : Multiset.map f s = 0 ↔ s = 0

theorem Multiset.mem_map_of_mem {α : Type u_1} {β : Type u_2} (f : α → β) {a : α} {s : Multiset α} (h : a ∈ s) : f a ∈ Multiset.map f s

theorem Multiset.map_eq_singleton {α : Type u_1} {β : Type u_2} {f : α → β} {s : Multiset α} {b : β} : Multiset.map f s = {b} ↔ ∃ a, s = {a} ∧ f a = b

theorem Multiset.map_eq_cons {α : Type u_1} {β : Type u_2} [DecidableEq α] (f : α → β) (s : Multiset α) (t : Multiset β) (b : β) : (∃ a, a ∈ s ∧ f a = b ∧ Multiset.map f (Multiset.erase s a) = t) ↔ Multiset.map f s = b ::ₘ t

@[simp]

theorem Multiset.mem_map_of_injective {α : Type u_1} {β : Type u_2} {f : α → β} (H : Function.Injective f) {a : α} {s : Multiset α} : f a ∈ Multiset.map f s ↔ a ∈ s

@[simp]

theorem Multiset.map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} (g : β → γ) (f : α → β) (s : Multiset α) : Multiset.map g (Multiset.map f s) = Multiset.map (g ∘ f) s

theorem Multiset.map_id {α : Type u_1} (s : Multiset α) : Multiset.map id s = s

@[simp]

theorem Multiset.map_id' {α : Type u_1} (s : Multiset α) : Multiset.map (fun x => x) s = s

theorem Multiset.map_const {α : Type u_1} {β : Type u_2} (s : Multiset α) (b : β) : Multiset.map (Function.const α b) s = Multiset.replicate (↑Multiset.card s) b

@[simp]

theorem Multiset.map_const' {α : Type u_1} {β : Type u_2} (s : Multiset α) (b : β) : Multiset.map (fun x => b) s = Multiset.replicate (↑Multiset.card s) b

theorem Multiset.eq_of_mem_map_const {α : Type u_1} {β : Type u_2} {b₁ : β} {b₂ : β} {l : List α} (h : b₁ ∈ Multiset.map (Function.const α b₂) ↑l) : b₁ = b₂

@[simp]

theorem Multiset.map_le_map {α : Type u_1} {β : Type u_2} {f : α → β} {s : Multiset α} {t : Multiset α} (h : s ≤ t) : Multiset.map f s ≤ Multiset.map f t

@[simp]

theorem Multiset.map_lt_map {α : Type u_1} {β : Type u_2} {f : α → β} {s : Multiset α} {t : Multiset α} (h : s < t) : Multiset.map f s < Multiset.map f t

theorem Multiset.map_mono {α : Type u_1} {β : Type u_2} (f : α → β) : Monotone (Multiset.map f)

theorem Multiset.map_strictMono {α : Type u_1} {β : Type u_2} (f : α → β) : StrictMono (Multiset.map f)

@[simp]

theorem Multiset.map_subset_map {α : Type u_1} {β : Type u_2} {f : α → β} {s : Multiset α} {t : Multiset α} (H : s ⊆ t) : Multiset.map f s ⊆ Multiset.map f t

theorem Multiset.map_erase {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] (f : α → β) (hf : Function.Injective f) (x : α) (s : Multiset α) : Multiset.map f (Multiset.erase s x) = Multiset.erase (Multiset.map f s) (f x)

theorem Multiset.map_surjective_of_surjective {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Surjective f) : Function.Surjective (Multiset.map f)

Multiset.fold

def Multiset.foldl {α : Type u_1} {β : Type u_2} (f : β → α → β) (H : RightCommutative f) (b : β) (s : Multiset α) : β

foldl f H b s is the lift of the list operation foldl f b l, which folds f over the multiset. It is well defined when f is right-commutative, that is, f (f b a₁) a₂ = f (f b a₂) a₁.

Equations

• Multiset.foldl f H b s = Quot.liftOn s (fun l => List.foldl f b l) (_ : ∀ (_l₁ _l₂ : List α), Setoid.r _l₁ _l₂ → List.foldl f b _l₁ = List.foldl f b _l₂)

@[simp]

theorem Multiset.foldl_zero {α : Type u_1} {β : Type u_2} (f : β → α → β) (H : RightCommutative f) (b : β) : Multiset.foldl f H b 0 = b

@[simp]

theorem Multiset.foldl_cons {α : Type u_1} {β : Type u_2} (f : β → α → β) (H : RightCommutative f) (b : β) (a : α) (s : Multiset α) : Multiset.foldl f H b (a ::ₘ s) = Multiset.foldl f H (f b a) s

@[simp]

theorem Multiset.foldl_add {α : Type u_1} {β : Type u_2} (f : β → α → β) (H : RightCommutative f) (b : β) (s : Multiset α) (t : Multiset α) : Multiset.foldl f H b (s + t) = Multiset.foldl f H (Multiset.foldl f H b s) t

def Multiset.foldr {α : Type u_1} {β : Type u_2} (f : α → β → β) (H : LeftCommutative f) (b : β) (s : Multiset α) : β

foldr f H b s is the lift of the list operation foldr f b l, which folds f over the multiset. It is well defined when f is left-commutative, that is, f a₁ (f a₂ b) = f a₂ (f a₁ b).

Equations

• Multiset.foldr f H b s = Quot.liftOn s (fun l => List.foldr f b l) (_ : ∀ (_l₁ _l₂ : List α), Setoid.r _l₁ _l₂ → List.foldr f b _l₁ = List.foldr f b _l₂)

@[simp]

theorem Multiset.foldr_zero {α : Type u_1} {β : Type u_2} (f : α → β → β) (H : LeftCommutative f) (b : β) : Multiset.foldr f H b 0 = b

@[simp]

theorem Multiset.foldr_cons {α : Type u_1} {β : Type u_2} (f : α → β → β) (H : LeftCommutative f) (b : β) (a : α) (s : Multiset α) : Multiset.foldr f H b (a ::ₘ s) = f a (Multiset.foldr f H b s)

@[simp]

theorem Multiset.foldr_singleton {α : Type u_1} {β : Type u_2} (f : α → β → β) (H : LeftCommutative f) (b : β) (a : α) : Multiset.foldr f H b {a} = f a b

@[simp]

theorem Multiset.foldr_add {α : Type u_1} {β : Type u_2} (f : α → β → β) (H : LeftCommutative f) (b : β) (s : Multiset α) (t : Multiset α) : Multiset.foldr f H b (s + t) = Multiset.foldr f H (Multiset.foldr f H b t) s

@[simp]

theorem Multiset.coe_foldr {α : Type u_1} {β : Type u_2} (f : α → β → β) (H : LeftCommutative f) (b : β) (l : List α) : Multiset.foldr f H b ↑l = List.foldr f b l

@[simp]

theorem Multiset.coe_foldl {α : Type u_1} {β : Type u_2} (f : β → α → β) (H : RightCommutative f) (b : β) (l : List α) : Multiset.foldl f H b ↑l = List.foldl f b l

theorem Multiset.coe_foldr_swap {α : Type u_1} {β : Type u_2} (f : α → β → β) (H : LeftCommutative f) (b : β) (l : List α) : Multiset.foldr f H b ↑l = List.foldl (fun x y => f y x) b l

theorem Multiset.foldr_swap {α : Type u_1} {β : Type u_2} (f : α → β → β) (H : LeftCommutative f) (b : β) (s : Multiset α) : Multiset.foldr f H b s = Multiset.foldl (fun x y => f y x) (_ : ∀ (_x : β) (_y _z : α), f _z (f _y _x) = f _y (f _z _x)) b s

theorem Multiset.foldl_swap {α : Type u_1} {β : Type u_2} (f : β → α → β) (H : RightCommutative f) (b : β) (s : Multiset α) : Multiset.foldl f H b s = Multiset.foldr (fun x y => f y x) (_ : ∀ (_x _y : α) (_z : β), f (f _z _y) _x = f (f _z _x) _y) b s

theorem Multiset.foldr_induction' {α : Type u_1} {β : Type u_2} (f : α → β → β) (H : LeftCommutative f) (x : β) (q : α → Prop) (p : β → Prop) (s : Multiset α) (hpqf : (a : α) → (b : β) → q a → p b → p (f a b)) (px : p x) (q_s : (a : α) → a ∈ s → q a) : p (Multiset.foldr f H x s)

theorem Multiset.foldr_induction {α : Type u_1} (f : α → α → α) (H : LeftCommutative f) (x : α) (p : α → Prop) (s : Multiset α) (p_f : (a b : α) → p a → p b → p (f a b)) (px : p x) (p_s : (a : α) → a ∈ s → p a) : p (Multiset.foldr f H x s)

theorem Multiset.foldl_induction' {α : Type u_1} {β : Type u_2} (f : β → α → β) (H : RightCommutative f) (x : β) (q : α → Prop) (p : β → Prop) (s : Multiset α) (hpqf : (a : α) → (b : β) → q a → p b → p (f b a)) (px : p x) (q_s : (a : α) → a ∈ s → q a) : p (Multiset.foldl f H x s)

theorem Multiset.foldl_induction {α : Type u_1} (f : α → α → α) (H : RightCommutative f) (x : α) (p : α → Prop) (s : Multiset α) (p_f : (a b : α) → p a → p b → p (f b a)) (px : p x) (p_s : (a : α) → a ∈ s → p a) : p (Multiset.foldl f H x s)

Map for partial functions

def Multiset.pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (s : Multiset α) : ((a : α) → a ∈ s → p a) → Multiset β

Lift of the list pmap operation. Map a partial function f over a multiset s whose elements are all in the domain of f.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.coe_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (l : List α) (H : (a : α) → a ∈ l → p a) : Multiset.pmap f (↑l) H = ↑(List.pmap f l H)

@[simp]

theorem Multiset.pmap_zero {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (h : (a : α) → a ∈ 0 → p a) : Multiset.pmap f 0 h = 0

@[simp]

theorem Multiset.pmap_cons {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (a : α) (m : Multiset α) (h : (b : α) → b ∈ a ::ₘ m → p b) : Multiset.pmap f (a ::ₘ m) h = f a (h a (_ : a ∈ a ::ₘ m)) ::ₘ Multiset.pmap f m fun a ha => h a (_ : a ∈ a ::ₘ m)

def Multiset.attach {α : Type u_1} (s : Multiset α) : Multiset { x // x ∈ s }

"Attach" a proof that a ∈ s to each element a in s to produce a multiset on {x // x ∈ s}.

Equations

• Multiset.attach s = Multiset.pmap Subtype.mk s (_ : ∀ (_a : α), _a ∈ s → _a ∈ s)

@[simp]

theorem Multiset.coe_attach {α : Type u_1} (l : List α) : Multiset.attach ↑l = ↑(List.attach l)

theorem Multiset.sizeOf_lt_sizeOf_of_mem {α : Type u_1} [SizeOf α] {x : α} {s : Multiset α} (hx : x ∈ s) : sizeOf x < sizeOf s

theorem Multiset.pmap_eq_map {α : Type u_1} {β : Type u_2} (p : α → Prop) (f : α → β) (s : Multiset α) (H : (a : α) → a ∈ s → p a) : Multiset.pmap (fun a x => f a) s H = Multiset.map f s

theorem Multiset.pmap_congr {α : Type u_1} {β : Type u_2} {p : α → Prop} {q : α → Prop} {f : (a : α) → p a → β} {g : (a : α) → q a → β} (s : Multiset α) {H₁ : (a : α) → a ∈ s → p a} {H₂ : (a : α) → a ∈ s → q a} : (∀ (a : α), a ∈ s → ∀ (h₁ : p a) (h₂ : q a), f a h₁ = g a h₂) → Multiset.pmap f s H₁ = Multiset.pmap g s H₂

theorem Multiset.map_pmap {α : Type u_1} {β : Type u_2} {γ : Type u_3} {p : α → Prop} (g : β → γ) (f : (a : α) → p a → β) (s : Multiset α) (H : (a : α) → a ∈ s → p a) : Multiset.map g (Multiset.pmap f s H) = Multiset.pmap (fun a h => g (f a h)) s H

theorem Multiset.pmap_eq_map_attach {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (s : Multiset α) (H : (a : α) → a ∈ s → p a) : Multiset.pmap f s H = Multiset.map (fun x => f (↑x) (H ↑x (_ : ↑x ∈ s))) (Multiset.attach s)

theorem Multiset.attach_map_val' {α : Type u_1} {β : Type u_2} (s : Multiset α) (f : α → β) : Multiset.map (fun i => f ↑i) (Multiset.attach s) = Multiset.map f s

@[simp]

theorem Multiset.attach_map_val {α : Type u_1} (s : Multiset α) : Multiset.map Subtype.val (Multiset.attach s) = s

@[simp]

theorem Multiset.mem_attach {α : Type u_1} (s : Multiset α) (x : { x // x ∈ s }) : x ∈ Multiset.attach s

@[simp]

theorem Multiset.mem_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} {f : (a : α) → p a → β} {s : Multiset α} {H : (a : α) → a ∈ s → p a} {b : β} : b ∈ Multiset.pmap f s H ↔ ∃ a h, f a (H a h) = b

@[simp]

theorem Multiset.card_pmap {α : Type u_1} {β : Type u_2} {p : α → Prop} (f : (a : α) → p a → β) (s : Multiset α) (H : (a : α) → a ∈ s → p a) : ↑Multiset.card (Multiset.pmap f s H) = ↑Multiset.card s

@[simp]

theorem Multiset.card_attach {α : Type u_1} {m : Multiset α} : ↑Multiset.card (Multiset.attach m) = ↑Multiset.card m

@[simp]

theorem Multiset.attach_zero {α : Type u_1} : Multiset.attach 0 = 0

theorem Multiset.attach_cons {α : Type u_1} (a : α) (m : Multiset α) : Multiset.attach (a ::ₘ m) = { val := a, property := (_ : a ∈ a ::ₘ m) } ::ₘ Multiset.map (fun p => { val := ↑p, property := (_ : ↑p ∈ a ::ₘ m) }) (Multiset.attach m)

def Multiset.decidableForallMultiset {α : Type u_1} {m : Multiset α} {p : α → Prop} [hp : (a : α) → Decidable (p a)] : Decidable ((a : α) → a ∈ m → p a)

If p is a decidable predicate, so is the predicate that all elements of a multiset satisfy p.

Equations

• Multiset.decidableForallMultiset = Quotient.recOnSubsingleton m fun l => decidable_of_iff ((a : α) → a ∈ l → p a) (_ : ((a : α) → a ∈ l → p a) ↔ (a : α) → a ∈ ↑l → p a)

instance Multiset.decidableDforallMultiset {α : Type u_1} {m : Multiset α} {p : (a : α) → a ∈ m → Prop} [_hp : (a : α) → (h : a ∈ m) → Decidable (p a h)] : Decidable ((a : α) → (h : a ∈ m) → p a h)

Equations

• One or more equations did not get rendered due to their size.

instance Multiset.decidableEqPiMultiset {α : Type u_1} {m : Multiset α} {β : α → Type u_4} [h : (a : α) → DecidableEq (β a)] : DecidableEq ((a : α) → a ∈ m → β a)

decidable equality for functions whose domain is bounded by multisets

Equations

• Multiset.decidableEqPiMultiset f g = decidable_of_iff (∀ (a : α) (h : a ∈ m), f a h = g a h) (_ : (∀ (a : α) (h : a ∈ m), f a h = g a h) ↔ f = g)

def Multiset.decidableExistsMultiset {α : Type u_1} {m : Multiset α} {p : α → Prop} [DecidablePred p] : Decidable (∃ x, x ∈ m ∧ p x)

If p is a decidable predicate, so is the existence of an element in a multiset satisfying p.

Equations

• Multiset.decidableExistsMultiset = Quotient.recOnSubsingleton m fun l => decidable_of_iff (∃ a, a ∈ l ∧ p a) (_ : (∃ a, a ∈ l ∧ p a) ↔ ∃ x, x ∈ ↑l ∧ p x)

instance Multiset.decidableDexistsMultiset {α : Type u_1} {m : Multiset α} {p : (a : α) → a ∈ m → Prop} [_hp : (a : α) → (h : a ∈ m) → Decidable (p a h)] : Decidable (∃ a h, p a h)

Equations

• Multiset.decidableDexistsMultiset = decidable_of_iff (∃ x, x ∈ Multiset.attach m ∧ p ↑x (_ : ↑x ∈ m)) (_ : (∃ x, x ∈ Multiset.attach m ∧ p ↑x (_ : ↑x ∈ m)) ↔ ∃ a h, p a h)

Subtraction

def Multiset.sub {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : Multiset α

s - t is the multiset such that count a (s - t) = count a s - count a t for all a (note that it is truncated subtraction, so it is 0 if count a t ≥ count a s).

Equations

• One or more equations did not get rendered due to their size.

instance Multiset.instSubMultiset {α : Type u_1} [DecidableEq α] : Sub (Multiset α)

Equations

• Multiset.instSubMultiset = { sub := Multiset.sub }

@[simp]

theorem Multiset.coe_sub {α : Type u_1} [DecidableEq α] (s : List α) (t : List α) : ↑s - ↑t = ↑(List.diff s t)

theorem Multiset.sub_zero {α : Type u_1} [DecidableEq α] (s : Multiset α) : s - 0 = s

This is a special case of tsub_zero, which should be used instead of this. This is needed to prove OrderedSub (Multiset α).

@[simp]

theorem Multiset.sub_cons {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) (t : Multiset α) : s - a ::ₘ t = Multiset.erase s a - t

theorem Multiset.sub_le_iff_le_add {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {u : Multiset α} : s - t ≤ u ↔ s ≤ u + t

This is a special case of tsub_le_iff_right, which should be used instead of this. This is needed to prove OrderedSub (Multiset α).

instance Multiset.instOrderedSubMultisetToLEToPreorderInstPartialOrderMultisetInstAddMultisetInstSubMultiset {α : Type u_1} [DecidableEq α] : OrderedSub (Multiset α)

Equations

• One or more equations did not get rendered due to their size.

theorem Multiset.cons_sub_of_le {α : Type u_1} [DecidableEq α] (a : α) {s : Multiset α} {t : Multiset α} (h : t ≤ s) : a ::ₘ s - t = a ::ₘ (s - t)

theorem Multiset.sub_eq_fold_erase {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s - t = Multiset.foldl Multiset.erase (_ : ∀ (s : Multiset α) (a b : α), Multiset.erase (Multiset.erase s a) b = Multiset.erase (Multiset.erase s b) a) s t

@[simp]

theorem Multiset.card_sub {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (h : t ≤ s) : ↑Multiset.card (s - t) = ↑Multiset.card s - ↑Multiset.card t

Union

def Multiset.union {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : Multiset α

s ∪ t is the lattice join operation with respect to the multiset ≤. The multiplicity of a in s ∪ t is the maximum of the multiplicities in s and t.

Equations

• Multiset.union s t = s - t + t

instance Multiset.instUnionMultiset {α : Type u_1} [DecidableEq α] : Union (Multiset α)

Equations

• Multiset.instUnionMultiset = { union := Multiset.union }

theorem Multiset.union_def {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ∪ t = s - t + t

theorem Multiset.le_union_left {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ≤ s ∪ t

theorem Multiset.le_union_right {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : t ≤ s ∪ t

theorem Multiset.eq_union_left {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : t ≤ s → s ∪ t = s

theorem Multiset.union_le_union_right {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (h : s ≤ t) (u : Multiset α) : s ∪ u ≤ t ∪ u

theorem Multiset.union_le {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {u : Multiset α} (h₁ : s ≤ u) (h₂ : t ≤ u) : s ∪ t ≤ u

@[simp]

theorem Multiset.mem_union {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {a : α} : a ∈ s ∪ t ↔ a ∈ s ∨ a ∈ t

@[simp]

theorem Multiset.map_union {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] {f : α → β} (finj : Function.Injective f) {s : Multiset α} {t : Multiset α} : Multiset.map f (s ∪ t) = Multiset.map f s ∪ Multiset.map f t

@[simp]

theorem Multiset.zero_union {α : Type u_1} [DecidableEq α] {s : Multiset α} : 0 ∪ s = s

@[simp]

theorem Multiset.union_zero {α : Type u_1} [DecidableEq α] {s : Multiset α} : s ∪ 0 = s

Intersection

def Multiset.inter {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : Multiset α

s ∩ t is the lattice meet operation with respect to the multiset ≤. The multiplicity of a in s ∩ t is the minimum of the multiplicities in s and t.

Equations

• One or more equations did not get rendered due to their size.

instance Multiset.instInterMultiset {α : Type u_1} [DecidableEq α] : Inter (Multiset α)

Equations

• Multiset.instInterMultiset = { inter := Multiset.inter }

@[simp]

theorem Multiset.inter_zero {α : Type u_1} [DecidableEq α] (s : Multiset α) : s ∩ 0 = 0

@[simp]

theorem Multiset.zero_inter {α : Type u_1} [DecidableEq α] (s : Multiset α) : 0 ∩ s = 0

@[simp]

theorem Multiset.cons_inter_of_pos {α : Type u_1} [DecidableEq α] {a : α} (s : Multiset α) {t : Multiset α} : a ∈ t → (a ::ₘ s) ∩ t = a ::ₘ s ∩ Multiset.erase t a

@[simp]

theorem Multiset.cons_inter_of_neg {α : Type u_1} [DecidableEq α] {a : α} (s : Multiset α) {t : Multiset α} : ¬a ∈ t → (a ::ₘ s) ∩ t = s ∩ t

theorem Multiset.inter_le_left {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ∩ t ≤ s

theorem Multiset.inter_le_right {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ∩ t ≤ t

theorem Multiset.le_inter {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {u : Multiset α} (h₁ : s ≤ t) (h₂ : s ≤ u) : s ≤ t ∩ u

@[simp]

theorem Multiset.mem_inter {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {a : α} : a ∈ s ∩ t ↔ a ∈ s ∧ a ∈ t

instance Multiset.instLatticeMultiset {α : Type u_1} [DecidableEq α] : Lattice (Multiset α)

Equations

• Multiset.instLatticeMultiset = Lattice.mk (_ : ∀ (s t : Multiset α), s ∩ t ≤ s) (_ : ∀ (s t : Multiset α), s ∩ t ≤ t) (_ : ∀ {s t u : Multiset α}, s ≤ t → s ≤ u → s ≤ t ∩ u)

@[simp]

theorem Multiset.sup_eq_union {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ⊔ t = s ∪ t

@[simp]

theorem Multiset.inf_eq_inter {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ⊓ t = s ∩ t

@[simp]

theorem Multiset.le_inter_iff {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {u : Multiset α} : s ≤ t ∩ u ↔ s ≤ t ∧ s ≤ u

@[simp]

theorem Multiset.union_le_iff {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {u : Multiset α} : s ∪ t ≤ u ↔ s ≤ u ∧ t ≤ u

theorem Multiset.union_comm {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ∪ t = t ∪ s

theorem Multiset.inter_comm {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ∩ t = t ∩ s

theorem Multiset.eq_union_right {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (h : s ≤ t) : s ∪ t = t

theorem Multiset.union_le_union_left {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} (h : s ≤ t) (u : Multiset α) : u ∪ s ≤ u ∪ t

theorem Multiset.union_le_add {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ∪ t ≤ s + t

theorem Multiset.union_add_distrib {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) (u : Multiset α) : s ∪ t + u = s + u ∪ (t + u)

theorem Multiset.add_union_distrib {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) (u : Multiset α) : s + (t ∪ u) = s + t ∪ (s + u)

theorem Multiset.cons_union_distrib {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) (t : Multiset α) : a ::ₘ (s ∪ t) = a ::ₘ s ∪ a ::ₘ t

theorem Multiset.inter_add_distrib {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) (u : Multiset α) : s ∩ t + u = (s + u) ∩ (t + u)

theorem Multiset.add_inter_distrib {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) (u : Multiset α) : s + t ∩ u = (s + t) ∩ (s + u)

theorem Multiset.cons_inter_distrib {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) (t : Multiset α) : a ::ₘ s ∩ t = (a ::ₘ s) ∩ (a ::ₘ t)

theorem Multiset.union_add_inter {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s ∪ t + s ∩ t = s + t

theorem Multiset.sub_add_inter {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s - t + s ∩ t = s

theorem Multiset.sub_inter {α : Type u_1} [DecidableEq α] (s : Multiset α) (t : Multiset α) : s - s ∩ t = s - t

Multiset.filter

def Multiset.filter {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) : Multiset α

Filter p s returns the elements in s (with the same multiplicities) which satisfy p, and removes the rest.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.coe_filter {α : Type u_1} (p : α → Prop) [DecidablePred p] (l : List α) : Multiset.filter p ↑l = ↑(List.filter (fun b => decide (p b)) l)

@[simp]

theorem Multiset.filter_zero {α : Type u_1} (p : α → Prop) [DecidablePred p] : Multiset.filter p 0 = 0

theorem Multiset.filter_congr {α : Type u_1} {p : α → Prop} {q : α → Prop} [DecidablePred p] [DecidablePred q] {s : Multiset α} : (∀ (x : α), x ∈ s → (p x ↔ q x)) → Multiset.filter p s = Multiset.filter q s

@[simp]

theorem Multiset.filter_add {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) (t : Multiset α) : Multiset.filter p (s + t) = Multiset.filter p s + Multiset.filter p t

@[simp]

theorem Multiset.filter_le {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) : Multiset.filter p s ≤ s

@[simp]

theorem Multiset.filter_subset {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) : Multiset.filter p s ⊆ s

theorem Multiset.filter_le_filter {α : Type u_1} (p : α → Prop) [DecidablePred p] {s : Multiset α} {t : Multiset α} (h : s ≤ t) : Multiset.filter p s ≤ Multiset.filter p t

theorem Multiset.monotone_filter_left {α : Type u_1} (p : α → Prop) [DecidablePred p] : Monotone (Multiset.filter p)

theorem Multiset.monotone_filter_right {α : Type u_1} (s : Multiset α) ⦃p : α → Prop⦄ ⦃q : α → Prop⦄ [DecidablePred p] [DecidablePred q] (h : (b : α) → p b → q b) : Multiset.filter p s ≤ Multiset.filter q s

@[simp]

theorem Multiset.filter_cons_of_pos {α : Type u_1} {p : α → Prop} [DecidablePred p] {a : α} (s : Multiset α) : p a → Multiset.filter p (a ::ₘ s) = a ::ₘ Multiset.filter p s

@[simp]

theorem Multiset.filter_cons_of_neg {α : Type u_1} {p : α → Prop} [DecidablePred p] {a : α} (s : Multiset α) : ¬p a → Multiset.filter p (a ::ₘ s) = Multiset.filter p s

@[simp]

theorem Multiset.mem_filter {α : Type u_1} {p : α → Prop} [DecidablePred p] {a : α} {s : Multiset α} : a ∈ Multiset.filter p s ↔ a ∈ s ∧ p a

theorem Multiset.of_mem_filter {α : Type u_1} {p : α → Prop} [DecidablePred p] {a : α} {s : Multiset α} (h : a ∈ Multiset.filter p s) : p a

theorem Multiset.mem_of_mem_filter {α : Type u_1} {p : α → Prop} [DecidablePred p] {a : α} {s : Multiset α} (h : a ∈ Multiset.filter p s) : a ∈ s

theorem Multiset.mem_filter_of_mem {α : Type u_1} {p : α → Prop} [DecidablePred p] {a : α} {l : Multiset α} (m : a ∈ l) (h : p a) : a ∈ Multiset.filter p l

theorem Multiset.filter_eq_self {α : Type u_1} {p : α → Prop} [DecidablePred p] {s : Multiset α} : Multiset.filter p s = s ↔ (a : α) → a ∈ s → p a

theorem Multiset.filter_eq_nil {α : Type u_1} {p : α → Prop} [DecidablePred p] {s : Multiset α} : Multiset.filter p s = 0 ↔ ∀ (a : α), a ∈ s → ¬p a

theorem Multiset.le_filter {α : Type u_1} {p : α → Prop} [DecidablePred p] {s : Multiset α} {t : Multiset α} : s ≤ Multiset.filter p t ↔ s ≤ t ∧ ((a : α) → a ∈ s → p a)

theorem Multiset.filter_cons {α : Type u_1} {p : α → Prop} [DecidablePred p] {a : α} (s : Multiset α) : Multiset.filter p (a ::ₘ s) = (if p a then {a} else 0) + Multiset.filter p s

theorem Multiset.filter_singleton {α : Type u_1} {a : α} (p : α → Prop) [DecidablePred p] : Multiset.filter p {a} = if p a then {a} else ∅

theorem Multiset.filter_nsmul {α : Type u_1} {p : α → Prop} [DecidablePred p] (s : Multiset α) (n : ℕ) : Multiset.filter p (n • s) = n • Multiset.filter p s

@[simp]

theorem Multiset.filter_sub {α : Type u_1} (p : α → Prop) [DecidablePred p] [DecidableEq α] (s : Multiset α) (t : Multiset α) : Multiset.filter p (s - t) = Multiset.filter p s - Multiset.filter p t

@[simp]

theorem Multiset.filter_union {α : Type u_1} (p : α → Prop) [DecidablePred p] [DecidableEq α] (s : Multiset α) (t : Multiset α) : Multiset.filter p (s ∪ t) = Multiset.filter p s ∪ Multiset.filter p t

@[simp]

theorem Multiset.filter_inter {α : Type u_1} (p : α → Prop) [DecidablePred p] [DecidableEq α] (s : Multiset α) (t : Multiset α) : Multiset.filter p (s ∩ t) = Multiset.filter p s ∩ Multiset.filter p t

@[simp]

theorem Multiset.filter_filter {α : Type u_1} (p : α → Prop) [DecidablePred p] (q : α → Prop) [DecidablePred q] (s : Multiset α) : Multiset.filter p (Multiset.filter q s) = Multiset.filter (fun a => p a ∧ q a) s

theorem Multiset.filter_add_filter {α : Type u_1} (p : α → Prop) [DecidablePred p] (q : α → Prop) [DecidablePred q] (s : Multiset α) : Multiset.filter p s + Multiset.filter q s = Multiset.filter (fun a => p a ∨ q a) s + Multiset.filter (fun a => p a ∧ q a) s

theorem Multiset.filter_add_not {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) : Multiset.filter p s + Multiset.filter (fun a => ¬p a) s = s

theorem Multiset.map_filter {α : Type u_1} {β : Type u_2} (p : α → Prop) [DecidablePred p] (f : β → α) (s : Multiset β) : Multiset.filter p (Multiset.map f s) = Multiset.map f (Multiset.filter (p ∘ f) s)

Simultaneously filter and map elements of a multiset

def Multiset.filterMap {α : Type u_1} {β : Type u_2} (f : α → Option β) (s : Multiset α) : Multiset β

filterMap f s is a combination filter/map operation on s. The function f : α → Option β is applied to each element of s; if f a is some b then b is added to the result, otherwise a is removed from the resulting multiset.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.coe_filterMap {α : Type u_1} {β : Type u_2} (f : α → Option β) (l : List α) : Multiset.filterMap f ↑l = ↑(List.filterMap f l)

@[simp]

theorem Multiset.filterMap_zero {α : Type u_1} {β : Type u_2} (f : α → Option β) : Multiset.filterMap f 0 = 0

@[simp]

theorem Multiset.filterMap_cons_none {α : Type u_1} {β : Type u_2} {f : α → Option β} (a : α) (s : Multiset α) (h : f a = none) : Multiset.filterMap f (a ::ₘ s) = Multiset.filterMap f s

@[simp]

theorem Multiset.filterMap_cons_some {α : Type u_1} {β : Type u_2} (f : α → Option β) (a : α) (s : Multiset α) {b : β} (h : f a = some b) : Multiset.filterMap f (a ::ₘ s) = b ::ₘ Multiset.filterMap f s

theorem Multiset.filterMap_eq_map {α : Type u_1} {β : Type u_2} (f : α → β) : Multiset.filterMap (some ∘ f) = Multiset.map f

theorem Multiset.filterMap_eq_filter {α : Type u_1} (p : α → Prop) [DecidablePred p] : Multiset.filterMap (Option.guard p) = Multiset.filter p

theorem Multiset.filterMap_filterMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → Option β) (g : β → Option γ) (s : Multiset α) : Multiset.filterMap g (Multiset.filterMap f s) = Multiset.filterMap (fun x => Option.bind (f x) g) s

theorem Multiset.map_filterMap {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → Option β) (g : β → γ) (s : Multiset α) : Multiset.map g (Multiset.filterMap f s) = Multiset.filterMap (fun x => Option.map g (f x)) s

theorem Multiset.filterMap_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} (f : α → β) (g : β → Option γ) (s : Multiset α) : Multiset.filterMap g (Multiset.map f s) = Multiset.filterMap (g ∘ f) s

theorem Multiset.filter_filterMap {α : Type u_1} {β : Type u_2} (f : α → Option β) (p : β → Prop) [DecidablePred p] (s : Multiset α) : Multiset.filter p (Multiset.filterMap f s) = Multiset.filterMap (fun x => Option.filter (fun b => decide (p b)) (f x)) s

theorem Multiset.filterMap_filter {α : Type u_1} {β : Type u_2} (p : α → Prop) [DecidablePred p] (f : α → Option β) (s : Multiset α) : Multiset.filterMap f (Multiset.filter p s) = Multiset.filterMap (fun x => if p x then f x else none) s

@[simp]

theorem Multiset.filterMap_some {α : Type u_1} (s : Multiset α) : Multiset.filterMap some s = s

@[simp]

theorem Multiset.mem_filterMap {α : Type u_1} {β : Type u_2} (f : α → Option β) (s : Multiset α) {b : β} : b ∈ Multiset.filterMap f s ↔ ∃ a, a ∈ s ∧ f a = some b

theorem Multiset.map_filterMap_of_inv {α : Type u_1} {β : Type u_2} (f : α → Option β) (g : β → α) (H : ∀ (x : α), Option.map g (f x) = some x) (s : Multiset α) : Multiset.map g (Multiset.filterMap f s) = s

theorem Multiset.filterMap_le_filterMap {α : Type u_1} {β : Type u_2} (f : α → Option β) {s : Multiset α} {t : Multiset α} (h : s ≤ t) : Multiset.filterMap f s ≤ Multiset.filterMap f t

countP

def Multiset.countP {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) : ℕ

countP p s counts the number of elements of s (with multiplicity) that satisfy p.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.coe_countP {α : Type u_1} (p : α → Prop) [DecidablePred p] (l : List α) : Multiset.countP p ↑l = List.countP (fun b => decide (p b)) l

@[simp]

theorem Multiset.countP_zero {α : Type u_1} (p : α → Prop) [DecidablePred p] : Multiset.countP p 0 = 0

@[simp]

theorem Multiset.countP_cons_of_pos {α : Type u_1} {p : α → Prop} [DecidablePred p] {a : α} (s : Multiset α) : p a → Multiset.countP p (a ::ₘ s) = Multiset.countP p s + 1

@[simp]

theorem Multiset.countP_cons_of_neg {α : Type u_1} {p : α → Prop} [DecidablePred p] {a : α} (s : Multiset α) : ¬p a → Multiset.countP p (a ::ₘ s) = Multiset.countP p s

theorem Multiset.countP_cons {α : Type u_1} (p : α → Prop) [DecidablePred p] (b : α) (s : Multiset α) : Multiset.countP p (b ::ₘ s) = Multiset.countP p s + if p b then 1 else 0

theorem Multiset.countP_eq_card_filter {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) : Multiset.countP p s = ↑Multiset.card (Multiset.filter p s)

theorem Multiset.countP_le_card {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) : Multiset.countP p s ≤ ↑Multiset.card s

@[simp]

theorem Multiset.countP_add {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) (t : Multiset α) : Multiset.countP p (s + t) = Multiset.countP p s + Multiset.countP p t

@[simp]

theorem Multiset.countP_nsmul {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) (n : ℕ) : Multiset.countP p (n • s) = n * Multiset.countP p s

theorem Multiset.card_eq_countP_add_countP {α : Type u_1} (p : α → Prop) [DecidablePred p] (s : Multiset α) : ↑Multiset.card s = Multiset.countP p s + Multiset.countP (fun x => ¬p x) s

def Multiset.countPAddMonoidHom {α : Type u_1} (p : α → Prop) [DecidablePred p] : Multiset α →+ ℕ

countP p, the number of elements of a multiset satisfying p, promoted to an AddMonoidHom.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.coe_countPAddMonoidHom {α : Type u_1} (p : α → Prop) [DecidablePred p] : ↑(Multiset.countPAddMonoidHom p) = Multiset.countP p

@[simp]

theorem Multiset.countP_sub {α : Type u_1} (p : α → Prop) [DecidablePred p] [DecidableEq α] {s : Multiset α} {t : Multiset α} (h : t ≤ s) : Multiset.countP p (s - t) = Multiset.countP p s - Multiset.countP p t

theorem Multiset.countP_le_of_le {α : Type u_1} (p : α → Prop) [DecidablePred p] {s : Multiset α} {t : Multiset α} (h : s ≤ t) : Multiset.countP p s ≤ Multiset.countP p t

@[simp]

theorem Multiset.countP_filter {α : Type u_1} (p : α → Prop) [DecidablePred p] (q : α → Prop) [DecidablePred q] (s : Multiset α) : Multiset.countP p (Multiset.filter q s) = Multiset.countP (fun a => p a ∧ q a) s

theorem Multiset.countP_eq_countP_filter_add {α : Type u_1} (s : Multiset α) (p : α → Prop) (q : α → Prop) [DecidablePred p] [DecidablePred q] : Multiset.countP p s = Multiset.countP p (Multiset.filter q s) + Multiset.countP p (Multiset.filter (fun a => ¬q a) s)

@[simp]

theorem Multiset.countP_True {α : Type u_1} {s : Multiset α} : Multiset.countP (fun x => True) s = ↑Multiset.card s

@[simp]

theorem Multiset.countP_False {α : Type u_1} {s : Multiset α} : Multiset.countP (fun x => False) s = 0

theorem Multiset.countP_map {α : Type u_1} {β : Type u_2} (f : α → β) (s : Multiset α) (p : β → Prop) [DecidablePred p] : Multiset.countP p (Multiset.map f s) = ↑Multiset.card (Multiset.filter (fun a => p (f a)) s)

theorem Multiset.countP_pos {α : Type u_1} {p : α → Prop} [DecidablePred p] {s : Multiset α} : 0 < Multiset.countP p s ↔ ∃ a, a ∈ s ∧ p a

theorem Multiset.countP_eq_zero {α : Type u_1} {p : α → Prop} [DecidablePred p] {s : Multiset α} : Multiset.countP p s = 0 ↔ ∀ (a : α), a ∈ s → ¬p a

theorem Multiset.countP_eq_card {α : Type u_1} {p : α → Prop} [DecidablePred p] {s : Multiset α} : Multiset.countP p s = ↑Multiset.card s ↔ (a : α) → a ∈ s → p a

theorem Multiset.countP_pos_of_mem {α : Type u_1} {p : α → Prop} [DecidablePred p] {s : Multiset α} {a : α} (h : a ∈ s) (pa : p a) : 0 < Multiset.countP p s

theorem Multiset.countP_congr {α : Type u_1} {s : Multiset α} {s' : Multiset α} (hs : s = s') {p : α → Prop} {p' : α → Prop} [DecidablePred p] [DecidablePred p'] (hp : ∀ (x : α), x ∈ s → p x = p' x) : Multiset.countP p s = Multiset.countP p' s'

Multiplicity of an element

def Multiset.count {α : Type u_1} [DecidableEq α] (a : α) : Multiset α → ℕ

count a s is the multiplicity of a in s.

Equations

• Multiset.count a = Multiset.countP fun x => a = x

@[simp]

theorem Multiset.coe_count {α : Type u_1} [DecidableEq α] (a : α) (l : List α) : Multiset.count a ↑l = List.count a l

@[simp]

theorem Multiset.count_zero {α : Type u_1} [DecidableEq α] (a : α) : Multiset.count a 0 = 0

@[simp]

theorem Multiset.count_cons_self {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) : Multiset.count a (a ::ₘ s) = Multiset.count a s + 1

@[simp]

theorem Multiset.count_cons_of_ne {α : Type u_1} [DecidableEq α] {a : α} {b : α} (h : a ≠ b) (s : Multiset α) : Multiset.count a (b ::ₘ s) = Multiset.count a s

theorem Multiset.count_le_card {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) : Multiset.count a s ≤ ↑Multiset.card s

theorem Multiset.count_le_of_le {α : Type u_1} [DecidableEq α] (a : α) {s : Multiset α} {t : Multiset α} : s ≤ t → Multiset.count a s ≤ Multiset.count a t

theorem Multiset.count_le_count_cons {α : Type u_1} [DecidableEq α] (a : α) (b : α) (s : Multiset α) : Multiset.count a s ≤ Multiset.count a (b ::ₘ s)

theorem Multiset.count_cons {α : Type u_1} [DecidableEq α] (a : α) (b : α) (s : Multiset α) : Multiset.count a (b ::ₘ s) = Multiset.count a s + if a = b then 1 else 0

theorem Multiset.count_singleton_self {α : Type u_1} [DecidableEq α] (a : α) : Multiset.count a {a} = 1

theorem Multiset.count_singleton {α : Type u_1} [DecidableEq α] (a : α) (b : α) : Multiset.count a {b} = if a = b then 1 else 0

@[simp]

theorem Multiset.count_add {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) (t : Multiset α) : Multiset.count a (s + t) = Multiset.count a s + Multiset.count a t

def Multiset.countAddMonoidHom {α : Type u_1} [DecidableEq α] (a : α) : Multiset α →+ ℕ

count a, the multiplicity of a in a multiset, promoted to an AddMonoidHom.

Equations

• Multiset.countAddMonoidHom a = Multiset.countPAddMonoidHom fun x => a = x

@[simp]

theorem Multiset.coe_countAddMonoidHom {α : Type u_1} [DecidableEq α] {a : α} : ↑(Multiset.countAddMonoidHom a) = Multiset.count a

@[simp]

theorem Multiset.count_nsmul {α : Type u_1} [DecidableEq α] (a : α) (n : ℕ) (s : Multiset α) : Multiset.count a (n • s) = n * Multiset.count a s

theorem Multiset.count_pos {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : 0 < Multiset.count a s ↔ a ∈ s

theorem Multiset.one_le_count_iff_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : 1 ≤ Multiset.count a s ↔ a ∈ s

@[simp]

theorem Multiset.count_eq_zero_of_not_mem {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} (h : ¬a ∈ s) : Multiset.count a s = 0

@[simp]

theorem Multiset.count_eq_zero {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : Multiset.count a s = 0 ↔ ¬a ∈ s

theorem Multiset.count_ne_zero {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : Multiset.count a s ≠ 0 ↔ a ∈ s

theorem Multiset.count_eq_card {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} : Multiset.count a s = ↑Multiset.card s ↔ ∀ (x : α), x ∈ s → a = x

@[simp]

theorem Multiset.count_replicate_self {α : Type u_1} [DecidableEq α] (a : α) (n : ℕ) : Multiset.count a (Multiset.replicate n a) = n

theorem Multiset.count_replicate {α : Type u_1} [DecidableEq α] (a : α) (b : α) (n : ℕ) : Multiset.count a (Multiset.replicate n b) = if a = b then n else 0

@[simp]

theorem Multiset.count_erase_self {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) : Multiset.count a (Multiset.erase s a) = Multiset.count a s - 1

@[simp]

theorem Multiset.count_erase_of_ne {α : Type u_1} [DecidableEq α] {a : α} {b : α} (ab : a ≠ b) (s : Multiset α) : Multiset.count a (Multiset.erase s b) = Multiset.count a s

@[simp]

theorem Multiset.count_sub {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) (t : Multiset α) : Multiset.count a (s - t) = Multiset.count a s - Multiset.count a t

@[simp]

theorem Multiset.count_union {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) (t : Multiset α) : Multiset.count a (s ∪ t) = max (Multiset.count a s) (Multiset.count a t)

@[simp]

theorem Multiset.count_inter {α : Type u_1} [DecidableEq α] (a : α) (s : Multiset α) (t : Multiset α) : Multiset.count a (s ∩ t) = min (Multiset.count a s) (Multiset.count a t)

theorem Multiset.le_count_iff_replicate_le {α : Type u_1} [DecidableEq α] {a : α} {s : Multiset α} {n : ℕ} : n ≤ Multiset.count a s ↔ Multiset.replicate n a ≤ s

@[simp]

theorem Multiset.count_filter_of_pos {α : Type u_1} [DecidableEq α] {p : α → Prop} [DecidablePred p] {a : α} {s : Multiset α} (h : p a) : Multiset.count a (Multiset.filter p s) = Multiset.count a s

@[simp]

theorem Multiset.count_filter_of_neg {α : Type u_1} [DecidableEq α] {p : α → Prop} [DecidablePred p] {a : α} {s : Multiset α} (h : ¬p a) : Multiset.count a (Multiset.filter p s) = 0

theorem Multiset.count_filter {α : Type u_1} [DecidableEq α] {p : α → Prop} [DecidablePred p] {a : α} {s : Multiset α} : Multiset.count a (Multiset.filter p s) = if p a then Multiset.count a s else 0

theorem Multiset.ext {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : s = t ↔ ∀ (a : α), Multiset.count a s = Multiset.count a t

theorem Multiset.ext' {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : (∀ (a : α), Multiset.count a s = Multiset.count a t) → s = t

@[simp]

theorem Multiset.coe_inter {α : Type u_1} [DecidableEq α] (s : List α) (t : List α) : ↑s ∩ ↑t = ↑(List.bagInter s t)

theorem Multiset.le_iff_count {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : s ≤ t ↔ ∀ (a : α), Multiset.count a s ≤ Multiset.count a t

instance Multiset.instDistribLatticeMultiset {α : Type u_1} [DecidableEq α] : DistribLattice (Multiset α)

Equations

• Multiset.instDistribLatticeMultiset = DistribLattice.mk (_ : ∀ (s t u : Multiset α), (s ⊔ t) ⊓ (s ⊔ u) ≤ s ⊔ t ⊓ u)

theorem Multiset.count_map {α : Type u_4} {β : Type u_5} (f : α → β) (s : Multiset α) [DecidableEq β] (b : β) : Multiset.count b (Multiset.map f s) = ↑Multiset.card (Multiset.filter (fun a => b = f a) s)

theorem Multiset.count_map_eq_count {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] (f : α → β) (s : Multiset α) (hf : Set.InjOn f {x | x ∈ s}) (x : α) (H : x ∈ s) : Multiset.count (f x) (Multiset.map f s) = Multiset.count x s

Multiset.map f preserves count if f is injective on the set of elements contained in the multiset

theorem Multiset.count_map_eq_count' {α : Type u_1} {β : Type u_2} [DecidableEq α] [DecidableEq β] (f : α → β) (s : Multiset α) (hf : Function.Injective f) (x : α) : Multiset.count (f x) (Multiset.map f s) = Multiset.count x s

Multiset.map f preserves count if f is injective

@[simp]

theorem Multiset.attach_count_eq_count_coe {α : Type u_1} [DecidableEq α] (m : Multiset α) (a : { x // x ∈ m }) : Multiset.count a (Multiset.attach m) = Multiset.count (↑a) m

theorem Multiset.filter_eq' {α : Type u_1} [DecidableEq α] (s : Multiset α) (b : α) : Multiset.filter (fun x => x = b) s = Multiset.replicate (Multiset.count b s) b

theorem Multiset.filter_eq {α : Type u_1} [DecidableEq α] (s : Multiset α) (b : α) : Multiset.filter (Eq b) s = Multiset.replicate (Multiset.count b s) b

@[simp]

theorem Multiset.replicate_inter {α : Type u_1} [DecidableEq α] (n : ℕ) (x : α) (s : Multiset α) : Multiset.replicate n x ∩ s = Multiset.replicate (min n (Multiset.count x s)) x

@[simp]

theorem Multiset.inter_replicate {α : Type u_1} [DecidableEq α] (s : Multiset α) (n : ℕ) (x : α) : s ∩ Multiset.replicate n x = Multiset.replicate (min (Multiset.count x s) n) x

theorem Multiset.addHom_ext {α : Type u_1} {β : Type u_2} [AddZeroClass β] ⦃f : Multiset α →+ β⦄ ⦃g : Multiset α →+ β⦄ (h : ∀ (x : α), ↑f {x} = ↑g {x}) : f = g

@[simp]

theorem Multiset.map_le_map_iff {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) {s : Multiset α} {t : Multiset α} : Multiset.map f s ≤ Multiset.map f t ↔ s ≤ t

@[simp]

theorem Multiset.mapEmbedding_apply {α : Type u_1} {β : Type u_2} (f : α ↪ β) (s : Multiset α) : ↑(Multiset.mapEmbedding f) s = Multiset.map (↑f) s

def Multiset.mapEmbedding {α : Type u_1} {β : Type u_2} (f : α ↪ β) : Multiset α ↪o Multiset β

Associate to an embedding f from α to β the order embedding that maps a multiset to its image under f.

Equations

• Multiset.mapEmbedding f = OrderEmbedding.ofMapLEIff (Multiset.map ↑f) (_ : ∀ (x x_1 : Multiset α), Multiset.map f.toFun x ≤ Multiset.map f.toFun x_1 ↔ x ≤ x_1)

theorem Multiset.count_eq_card_filter_eq {α : Type u_1} [DecidableEq α] (s : Multiset α) (a : α) : Multiset.count a s = ↑Multiset.card (Multiset.filter (fun x => a = x) s)

@[simp]

theorem Multiset.map_count_True_eq_filter_card {α : Type u_1} (s : Multiset α) (p : α → Prop) [DecidablePred p] : Multiset.count True (Multiset.map p s) = ↑Multiset.card (Multiset.filter p s)

Mapping a multiset through a predicate and counting the Trues yields the cardinality of the set filtered by the predicate. Note that this uses the notion of a multiset of Props - due to the decidability requirements of count, the decidability instance on the LHS is different from the RHS. In particular, the decidability instance on the left leaks Classical.decEq. See here for more discussion.

Lift a relation to Multisets

theorem Multiset.Rel_iff {α : Type u_1} {β : Type u_2} (r : α → β → Prop) : ∀ (a : Multiset α) (a_1 : Multiset β), Multiset.Rel r a a_1 ↔ a = 0 ∧ a_1 = 0 ∨ ∃ a_2 b as bs, r a_2 b ∧ Multiset.Rel r as bs ∧ a = a_2 ::ₘ as ∧ a_1 = b ::ₘ bs

inductive Multiset.Rel {α : Type u_1} {β : Type u_2} (r : α → β → Prop) : Multiset α → Multiset β → Prop

• zero: ∀ {α : Type u_1} {β : Type u_2} {r : α → β → Prop}, Multiset.Rel r 0 0
• cons: ∀ {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {a : α} {b : β} {as : Multiset α} {bs : Multiset β}, r a b → Multiset.Rel r as bs → Multiset.Rel r (a ::ₘ as) (b ::ₘ bs)

Rel r s t -- lift the relation r between two elements to a relation between s and t, s.t. there is a one-to-one mapping between elements in s and t following r.

theorem Multiset.rel_flip {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {s : Multiset β} {t : Multiset α} : Multiset.Rel (flip r) s t ↔ Multiset.Rel r t s

theorem Multiset.rel_refl_of_refl_on {α : Type u_1} {m : Multiset α} {r : α → α → Prop} : ((x : α) → x ∈ m → r x x) → Multiset.Rel r m m

theorem Multiset.rel_eq_refl {α : Type u_1} {s : Multiset α} : Multiset.Rel (fun x x_1 => x = x_1) s s

theorem Multiset.rel_eq {α : Type u_1} {s : Multiset α} {t : Multiset α} : Multiset.Rel (fun x x_1 => x = x_1) s t ↔ s = t

theorem Multiset.Rel.mono {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {p : α → β → Prop} {s : Multiset α} {t : Multiset β} (hst : Multiset.Rel r s t) (h : (a : α) → a ∈ s → (b : β) → b ∈ t → r a b → p a b) : Multiset.Rel p s t

theorem Multiset.Rel.add {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {s : Multiset α} {t : Multiset β} {u : Multiset α} {v : Multiset β} (hst : Multiset.Rel r s t) (huv : Multiset.Rel r u v) : Multiset.Rel r (s + u) (t + v)

theorem Multiset.rel_flip_eq {α : Type u_1} {s : Multiset α} {t : Multiset α} : Multiset.Rel (fun a b => b = a) s t ↔ s = t

@[simp]

theorem Multiset.rel_zero_left {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {b : Multiset β} : Multiset.Rel r 0 b ↔ b = 0

@[simp]

theorem Multiset.rel_zero_right {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {a : Multiset α} : Multiset.Rel r a 0 ↔ a = 0

theorem Multiset.rel_cons_left {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {a : α} {as : Multiset α} {bs : Multiset β} : Multiset.Rel r (a ::ₘ as) bs ↔ ∃ b bs', r a b ∧ Multiset.Rel r as bs' ∧ bs = b ::ₘ bs'

theorem Multiset.rel_cons_right {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {as : Multiset α} {b : β} {bs : Multiset β} : Multiset.Rel r as (b ::ₘ bs) ↔ ∃ a as', r a b ∧ Multiset.Rel r as' bs ∧ as = a ::ₘ as'

theorem Multiset.rel_add_left {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {as₀ : Multiset α} {as₁ : Multiset α} {bs : Multiset β} : Multiset.Rel r (as₀ + as₁) bs ↔ ∃ bs₀ bs₁, Multiset.Rel r as₀ bs₀ ∧ Multiset.Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁

theorem Multiset.rel_add_right {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {as : Multiset α} {bs₀ : Multiset β} {bs₁ : Multiset β} : Multiset.Rel r as (bs₀ + bs₁) ↔ ∃ as₀ as₁, Multiset.Rel r as₀ bs₀ ∧ Multiset.Rel r as₁ bs₁ ∧ as = as₀ + as₁

theorem Multiset.rel_map_left {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → β → Prop} {s : Multiset γ} {f : γ → α} {t : Multiset β} : Multiset.Rel r (Multiset.map f s) t ↔ Multiset.Rel (fun a b => r (f a) b) s t

theorem Multiset.rel_map_right {α : Type u_1} {β : Type u_2} {γ : Type u_3} {r : α → β → Prop} {s : Multiset α} {t : Multiset γ} {f : γ → β} : Multiset.Rel r s (Multiset.map f t) ↔ Multiset.Rel (fun a b => r a (f b)) s t

theorem Multiset.rel_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {δ : Type u_4} {p : γ → δ → Prop} {s : Multiset α} {t : Multiset β} {f : α → γ} {g : β → δ} : Multiset.Rel p (Multiset.map f s) (Multiset.map g t) ↔ Multiset.Rel (fun a b => p (f a) (g b)) s t

theorem Multiset.card_eq_card_of_rel {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {s : Multiset α} {t : Multiset β} (h : Multiset.Rel r s t) : ↑Multiset.card s = ↑Multiset.card t

theorem Multiset.exists_mem_of_rel_of_mem {α : Type u_1} {β : Type u_2} {r : α → β → Prop} {s : Multiset α} {t : Multiset β} (h : Multiset.Rel r s t) {a : α} (_ha : a ∈ s) : ∃ b, b ∈ t ∧ r a b

theorem Multiset.rel_of_forall {α : Type u_1} {m1 : Multiset α} {m2 : Multiset α} {r : α → α → Prop} (h : (a b : α) → a ∈ m1 → b ∈ m2 → r a b) (hc : ↑Multiset.card m1 = ↑Multiset.card m2) : Multiset.Rel r m1 m2

theorem Multiset.rel_replicate_left {α : Type u_1} {m : Multiset α} {a : α} {r : α → α → Prop} {n : ℕ} : Multiset.Rel r (Multiset.replicate n a) m ↔ ↑Multiset.card m = n ∧ ((x : α) → x ∈ m → r a x)

theorem Multiset.rel_replicate_right {α : Type u_1} {m : Multiset α} {a : α} {r : α → α → Prop} {n : ℕ} : Multiset.Rel r m (Multiset.replicate n a) ↔ ↑Multiset.card m = n ∧ ((x : α) → x ∈ m → r x a)

theorem Multiset.Rel.trans {α : Type u_1} (r : α → α → Prop) [IsTrans α r] {s : Multiset α} {t : Multiset α} {u : Multiset α} (r1 : Multiset.Rel r s t) (r2 : Multiset.Rel r t u) : Multiset.Rel r s u

theorem Multiset.Rel.countP_eq {α : Type u_1} (r : α → α → Prop) [IsTrans α r] [IsSymm α r] {s : Multiset α} {t : Multiset α} (x : α) [DecidablePred (r x)] (h : Multiset.Rel r s t) : Multiset.countP (r x) s = Multiset.countP (r x) t

theorem Multiset.map_eq_map {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) {s : Multiset α} {t : Multiset α} : Multiset.map f s = Multiset.map f t ↔ s = t

theorem Multiset.map_injective {α : Type u_1} {β : Type u_2} {f : α → β} (hf : Function.Injective f) : Function.Injective (Multiset.map f)

theorem Multiset.map_mk_eq_map_mk_of_rel {α : Type u_1} {r : α → α → Prop} {s : Multiset α} {t : Multiset α} (hst : Multiset.Rel r s t) : Multiset.map (Quot.mk r) s = Multiset.map (Quot.mk r) t

theorem Multiset.exists_multiset_eq_map_quot_mk {α : Type u_1} {r : α → α → Prop} (s : Multiset (Quot r)) : ∃ t, s = Multiset.map (Quot.mk r) t

theorem Multiset.induction_on_multiset_quot {α : Type u_1} {r : α → α → Prop} {p : Multiset (Quot r) → Prop} (s : Multiset (Quot r)) : ((s : Multiset α) → p (Multiset.map (Quot.mk r) s)) → p s

Disjoint multisets

def Multiset.Disjoint {α : Type u_1} (s : Multiset α) (t : Multiset α) : Prop

Disjoint s t means that s and t have no elements in common.

Equations

• Multiset.Disjoint s t = ∀ ⦃a : α⦄, a ∈ s → a ∈ t → False

@[simp]

theorem Multiset.coe_disjoint {α : Type u_1} (l₁ : List α) (l₂ : List α) : Multiset.Disjoint ↑l₁ ↑l₂ ↔ List.Disjoint l₁ l₂

theorem Multiset.Disjoint.symm {α : Type u_1} {s : Multiset α} {t : Multiset α} (d : Multiset.Disjoint s t) : Multiset.Disjoint t s

theorem Multiset.disjoint_comm {α : Type u_1} {s : Multiset α} {t : Multiset α} : Multiset.Disjoint s t ↔ Multiset.Disjoint t s

theorem Multiset.disjoint_left {α : Type u_1} {s : Multiset α} {t : Multiset α} : Multiset.Disjoint s t ↔ ∀ {a : α}, a ∈ s → ¬a ∈ t

theorem Multiset.disjoint_right {α : Type u_1} {s : Multiset α} {t : Multiset α} : Multiset.Disjoint s t ↔ ∀ {a : α}, a ∈ t → ¬a ∈ s

theorem Multiset.disjoint_iff_ne {α : Type u_1} {s : Multiset α} {t : Multiset α} : Multiset.Disjoint s t ↔ ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b

theorem Multiset.disjoint_of_subset_left {α : Type u_1} {s : Multiset α} {t : Multiset α} {u : Multiset α} (h : s ⊆ u) (d : Multiset.Disjoint u t) : Multiset.Disjoint s t

theorem Multiset.disjoint_of_subset_right {α : Type u_1} {s : Multiset α} {t : Multiset α} {u : Multiset α} (h : t ⊆ u) (d : Multiset.Disjoint s u) : Multiset.Disjoint s t

theorem Multiset.disjoint_of_le_left {α : Type u_1} {s : Multiset α} {t : Multiset α} {u : Multiset α} (h : s ≤ u) : Multiset.Disjoint u t → Multiset.Disjoint s t

theorem Multiset.disjoint_of_le_right {α : Type u_1} {s : Multiset α} {t : Multiset α} {u : Multiset α} (h : t ≤ u) : Multiset.Disjoint s u → Multiset.Disjoint s t

@[simp]

theorem Multiset.zero_disjoint {α : Type u_1} (l : Multiset α) : Multiset.Disjoint 0 l

@[simp]

theorem Multiset.singleton_disjoint {α : Type u_1} {l : Multiset α} {a : α} : Multiset.Disjoint {a} l ↔ ¬a ∈ l

@[simp]

theorem Multiset.disjoint_singleton {α : Type u_1} {l : Multiset α} {a : α} : Multiset.Disjoint l {a} ↔ ¬a ∈ l

@[simp]

theorem Multiset.disjoint_add_left {α : Type u_1} {s : Multiset α} {t : Multiset α} {u : Multiset α} : Multiset.Disjoint (s + t) u ↔ Multiset.Disjoint s u ∧ Multiset.Disjoint t u

@[simp]

theorem Multiset.disjoint_add_right {α : Type u_1} {s : Multiset α} {t : Multiset α} {u : Multiset α} : Multiset.Disjoint s (t + u) ↔ Multiset.Disjoint s t ∧ Multiset.Disjoint s u

@[simp]

theorem Multiset.disjoint_cons_left {α : Type u_1} {a : α} {s : Multiset α} {t : Multiset α} : Multiset.Disjoint (a ::ₘ s) t ↔ ¬a ∈ t ∧ Multiset.Disjoint s t

@[simp]

theorem Multiset.disjoint_cons_right {α : Type u_1} {a : α} {s : Multiset α} {t : Multiset α} : Multiset.Disjoint s (a ::ₘ t) ↔ ¬a ∈ s ∧ Multiset.Disjoint s t

theorem Multiset.inter_eq_zero_iff_disjoint {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : s ∩ t = 0 ↔ Multiset.Disjoint s t

@[simp]

theorem Multiset.disjoint_union_left {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {u : Multiset α} : Multiset.Disjoint (s ∪ t) u ↔ Multiset.Disjoint s u ∧ Multiset.Disjoint t u

@[simp]

theorem Multiset.disjoint_union_right {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} {u : Multiset α} : Multiset.Disjoint s (t ∪ u) ↔ Multiset.Disjoint s t ∧ Multiset.Disjoint s u

theorem Multiset.add_eq_union_iff_disjoint {α : Type u_1} [DecidableEq α] {s : Multiset α} {t : Multiset α} : s + t = s ∪ t ↔ Multiset.Disjoint s t

theorem Multiset.disjoint_map_map {α : Type u_1} {β : Type u_2} {γ : Type u_3} {f : α → γ} {g : β → γ} {s : Multiset α} {t : Multiset β} : Multiset.Disjoint (Multiset.map f s) (Multiset.map g t) ↔ ∀ (a : α), a ∈ s → ∀ (b : β), b ∈ t → f a ≠ g b

def Multiset.Pairwise {α : Type u_1} (r : α → α → Prop) (m : Multiset α) : Prop

Pairwise r m states that there exists a list of the elements s.t. r holds pairwise on this list.

Equations

• Multiset.Pairwise r m = ∃ l, m = ↑l ∧ List.Pairwise r l

@[simp]

theorem Multiset.pairwise_zero {α : Type u_1} (r : α → α → Prop) : Multiset.Pairwise r 0

theorem Multiset.pairwise_coe_iff {α : Type u_1} {r : α → α → Prop} {l : List α} : Multiset.Pairwise r ↑l ↔ ∃ l', l ~ l' ∧ List.Pairwise r l'

theorem Multiset.pairwise_coe_iff_pairwise {α : Type u_1} {r : α → α → Prop} (hr : Symmetric r) {l : List α} : Multiset.Pairwise r ↑l ↔ List.Pairwise r l

theorem Multiset.map_set_pairwise {α : Type u_1} {β : Type u_2} {f : α → β} {r : β → β → Prop} {m : Multiset α} (h : Set.Pairwise {a | a ∈ m} fun a₁ a₂ => r (f a₁) (f a₂)) : Set.Pairwise {b | b ∈ Multiset.map f m} r

def Multiset.chooseX {α : Type u_1} (p : α → Prop) [DecidablePred p] (l : Multiset α) (_hp : ∃! a, a ∈ l ∧ p a) : { a // a ∈ l ∧ p a }

Given a proof hp that there exists a unique a ∈ l such that p a, chooseX p l hp returns that a together with proofs of a ∈ l and p a.

Equations

• One or more equations did not get rendered due to their size.

def Multiset.choose {α : Type u_1} (p : α → Prop) [DecidablePred p] (l : Multiset α) (hp : ∃! a, a ∈ l ∧ p a) : α

Given a proof hp that there exists a unique a ∈ l such that p a, choose p l hp returns that a.

Equations

• Multiset.choose p l hp = ↑(Multiset.chooseX p l hp)

theorem Multiset.choose_spec {α : Type u_1} (p : α → Prop) [DecidablePred p] (l : Multiset α) (hp : ∃! a, a ∈ l ∧ p a) : Multiset.choose p l hp ∈ l ∧ p (Multiset.choose p l hp)

theorem Multiset.choose_mem {α : Type u_1} (p : α → Prop) [DecidablePred p] (l : Multiset α) (hp : ∃! a, a ∈ l ∧ p a) : Multiset.choose p l hp ∈ l

theorem Multiset.choose_property {α : Type u_1} (p : α → Prop) [DecidablePred p] (l : Multiset α) (hp : ∃! a, a ∈ l ∧ p a) : p (Multiset.choose p l hp)

def Multiset.subsingletonEquiv (α : Type u_1) [Subsingleton α] : List α ≃ Multiset α

The equivalence between lists and multisets of a subsingleton type.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem Multiset.coe_subsingletonEquiv {α : Type u_1} [Subsingleton α] : ↑(Multiset.subsingletonEquiv α) = Multiset.ofList