Pointwise order on finitely supported functions #

This file lifts order structures on α to ι →₀ α.

Main declarations #

Finsupp.orderEmbeddingToFun: The order embedding from finitely supported functions to functions.

Order structures #

theorem Finsupp.sum_le_sum {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [OrderedAddCommMonoid β] {f : ι →₀ α} {h₁ h₂ : ι → α → β} (h : ∀ i ∈ f.support, h₁ i (f i) ≤ h₂ i (f i)) :

f.sum h₁ ≤ f.sum h₂

source

instance Finsupp.instLEFinsupp {ι : Type u_1} {α : Type u_3} [Zero α] [LE α] :

LE (ι →₀ α)

Equations

Finsupp.instLEFinsupp = { le := fun (f g : ι →₀ α) => ∀ (i : ι), f i ≤ g i }

source

theorem Finsupp.le_def {ι : Type u_1} {α : Type u_3} [Zero α] [LE α] {f g : ι →₀ α} :

f ≤ g ↔ ∀ (i : ι), f i ≤ g i

source

@[simp]

theorem Finsupp.coe_le_coe {ι : Type u_1} {α : Type u_3} [Zero α] [LE α] {f g : ι →₀ α} :

⇑f ≤ ⇑g ↔ f ≤ g

source

def Finsupp.orderEmbeddingToFun {ι : Type u_1} {α : Type u_3} [Zero α] [LE α] :

(ι →₀ α) ↪o (ι → α)

The order on Finsupps over a partial order embeds into the order on functions

Equations

Finsupp.orderEmbeddingToFun = { toFun := fun (f : ι →₀ α) => ⇑f, inj' := ⋯, map_rel_iff' := ⋯ }

Instances For

source

@[simp]

theorem Finsupp.orderEmbeddingToFun_apply {ι : Type u_1} {α : Type u_3} [Zero α] [LE α] {f : ι →₀ α} {i : ι} :

Finsupp.orderEmbeddingToFun f i = f i

source

instance Finsupp.preorder {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] :

Preorder (ι →₀ α)

Equations

Finsupp.preorder = Preorder.mk ⋯ ⋯ ⋯

source

theorem Finsupp.lt_def {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] {f g : ι →₀ α} :

f < g ↔ f ≤ g ∧ ∃ (i : ι), f i < g i

source

@[simp]

theorem Finsupp.coe_lt_coe {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] {f g : ι →₀ α} :

⇑f < ⇑g ↔ f < g

source

theorem Finsupp.coe_mono {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] :

Monotone Finsupp.toFun

source

theorem Finsupp.coe_strictMono {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] :

Monotone Finsupp.toFun

source

@[simp]

theorem Finsupp.single_le_single {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] {i : ι} {a b : α} :

Finsupp.single i a ≤ Finsupp.single i b ↔ a ≤ b

source

theorem Finsupp.single_mono {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] {i : ι} :

Monotone (Finsupp.single i)

source

theorem Finsupp.GCongr.single_mono {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] {i : ι} {a b : α} :

a ≤ b → Finsupp.single i a ≤ Finsupp.single i b

Alias of the reverse direction of Finsupp.single_le_single.

source

@[simp]

theorem Finsupp.single_nonneg {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] {i : ι} {a : α} :

0 ≤ Finsupp.single i a ↔ 0 ≤ a

source

@[simp]

theorem Finsupp.single_nonpos {ι : Type u_1} {α : Type u_3} [Zero α] [Preorder α] {i : ι} {a : α} :

Finsupp.single i a ≤ 0 ↔ a ≤ 0

source

theorem Finsupp.sum_le_sum_index {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [Preorder α] [OrderedAddCommMonoid β] [DecidableEq ι] {f₁ f₂ : ι →₀ α} {h : ι → α → β} (hf : f₁ ≤ f₂) (hh : ∀ i ∈ f₁.support ∪ f₂.support, Monotone (h i)) (hh₀ : ∀ i ∈ f₁.support ∪ f₂.support, h i 0 = 0) :

f₁.sum h ≤ f₂.sum h

source

instance Finsupp.partialorder {ι : Type u_1} {α : Type u_3} [Zero α] [PartialOrder α] :

PartialOrder (ι →₀ α)

Equations

Finsupp.partialorder = PartialOrder.mk ⋯

source

instance Finsupp.semilatticeInf {ι : Type u_1} {α : Type u_3} [Zero α] [SemilatticeInf α] :

SemilatticeInf (ι →₀ α)

Equations

Finsupp.semilatticeInf = SemilatticeInf.mk (Finsupp.zipWith (fun (x1 x2 : α) => x1 ⊓ x2) ⋯) ⋯ ⋯ ⋯

source

@[simp]

theorem Finsupp.inf_apply {ι : Type u_1} {α : Type u_3} [Zero α] [SemilatticeInf α] {i : ι} {f g : ι →₀ α} :

(f ⊓ g) i = f i ⊓ g i

source

instance Finsupp.semilatticeSup {ι : Type u_1} {α : Type u_3} [Zero α] [SemilatticeSup α] :

SemilatticeSup (ι →₀ α)

Equations

Finsupp.semilatticeSup = SemilatticeSup.mk (Finsupp.zipWith (fun (x1 x2 : α) => x1 ⊔ x2) ⋯) ⋯ ⋯ ⋯

source

@[simp]

theorem Finsupp.sup_apply {ι : Type u_1} {α : Type u_3} [Zero α] [SemilatticeSup α] {i : ι} {f g : ι →₀ α} :

(f ⊔ g) i = f i ⊔ g i

source

instance Finsupp.lattice {ι : Type u_1} {α : Type u_3} [Zero α] [Lattice α] :

Lattice (ι →₀ α)

Equations

Finsupp.lattice = Lattice.mk SemilatticeInf.inf ⋯ ⋯ ⋯

source

theorem Finsupp.support_inf_union_support_sup {ι : Type u_1} {α : Type u_3} [Zero α] [DecidableEq ι] [Lattice α] (f g : ι →₀ α) :

(f ⊓ g).support ∪ (f ⊔ g).support = f.support ∪ g.support

source

theorem Finsupp.support_sup_union_support_inf {ι : Type u_1} {α : Type u_3} [Zero α] [DecidableEq ι] [Lattice α] (f g : ι →₀ α) :

(f ⊔ g).support ∪ (f ⊓ g).support = f.support ∪ g.support

Algebraic order structures #

source

instance Finsupp.orderedAddCommMonoid {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] :

OrderedAddCommMonoid (ι →₀ α)

Equations

Finsupp.orderedAddCommMonoid = OrderedAddCommMonoid.mk ⋯

source

theorem Finsupp.mapDomain_mono {ι : Type u_1} {κ : Type u_2} {α : Type u_3} [OrderedAddCommMonoid α] {f : ι → κ} :

Monotone (Finsupp.mapDomain f)

source

theorem Finsupp.GCongr.mapDomain_mono {ι : Type u_1} {κ : Type u_2} {α : Type u_3} [OrderedAddCommMonoid α] {f : ι → κ} {g₁ g₂ : ι →₀ α} (hg : g₁ ≤ g₂) :

Finsupp.mapDomain f g₁ ≤ Finsupp.mapDomain f g₂

source

theorem Finsupp.mapDomain_nonneg {ι : Type u_1} {κ : Type u_2} {α : Type u_3} [OrderedAddCommMonoid α] {f : ι → κ} {g : ι →₀ α} (hg : 0 ≤ g) :

0 ≤ Finsupp.mapDomain f g

source

theorem Finsupp.mapDomain_nonpos {ι : Type u_1} {κ : Type u_2} {α : Type u_3} [OrderedAddCommMonoid α] {f : ι → κ} {g : ι →₀ α} (hg : g ≤ 0) :

Finsupp.mapDomain f g ≤ 0

source

instance Finsupp.orderedCancelAddCommMonoid {ι : Type u_1} {α : Type u_3} [OrderedCancelAddCommMonoid α] :

OrderedCancelAddCommMonoid (ι →₀ α)

Equations

Finsupp.orderedCancelAddCommMonoid = OrderedCancelAddCommMonoid.mk ⋯

source

instance Finsupp.addLeftReflectLE {ι : Type u_1} {α : Type u_3} [OrderedAddCommMonoid α] [AddLeftReflectLE α] :

AddLeftReflectLE (ι →₀ α)

Equations

⋯ = ⋯

source

instance Finsupp.instPosSMulMono {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [Preorder α] [Zero β] [Preorder β] [SMulZeroClass α β] [PosSMulMono α β] :

PosSMulMono α (ι →₀ β)

Equations

⋯ = ⋯

source

instance Finsupp.instSMulPosMono {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Zero β] [Preorder β] [SMulZeroClass α β] [SMulPosMono α β] :

SMulPosMono α (ι →₀ β)

Equations

⋯ = ⋯

source

instance Finsupp.instPosSMulReflectLE {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [Preorder α] [Zero β] [Preorder β] [SMulZeroClass α β] [PosSMulReflectLE α β] :

PosSMulReflectLE α (ι →₀ β)

Equations

⋯ = ⋯

source

instance Finsupp.instSMulPosReflectLE {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Preorder α] [Zero β] [Preorder β] [SMulZeroClass α β] [SMulPosReflectLE α β] :

SMulPosReflectLE α (ι →₀ β)

Equations

⋯ = ⋯

source

instance Finsupp.instPosSMulStrictMono {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [PartialOrder α] [Zero β] [PartialOrder β] [SMulWithZero α β] [PosSMulStrictMono α β] :

PosSMulStrictMono α (ι →₀ β)

Equations

⋯ = ⋯

source

instance Finsupp.instSMulPosStrictMono {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [PartialOrder α] [Zero β] [PartialOrder β] [SMulWithZero α β] [SMulPosStrictMono α β] :

SMulPosStrictMono α (ι →₀ β)

Equations

⋯ = ⋯

source

instance Finsupp.instSMulPosReflectLT {ι : Type u_1} {α : Type u_3} {β : Type u_4} [Zero α] [PartialOrder α] [Zero β] [PartialOrder β] [SMulWithZero α β] [SMulPosReflectLT α β] :

SMulPosReflectLT α (ι →₀ β)

Equations

⋯ = ⋯

source

instance Finsupp.orderBot {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] :

OrderBot (ι →₀ α)

Equations

Finsupp.orderBot = OrderBot.mk ⋯

source

theorem Finsupp.bot_eq_zero {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] :

⊥ = 0

source

@[simp]

theorem Finsupp.add_eq_zero_iff {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] (f g : ι →₀ α) :

f + g = 0 ↔ f = 0 ∧ g = 0

source

theorem Finsupp.le_iff' {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] (f g : ι →₀ α) {s : Finset ι} (hf : f.support ⊆ s) :

f ≤ g ↔ ∀ i ∈ s, f i ≤ g i

source

theorem Finsupp.le_iff {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] (f g : ι →₀ α) :

f ≤ g ↔ ∀ i ∈ f.support, f i ≤ g i

source

theorem Finsupp.support_monotone {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] :

Monotone Finsupp.support

source

theorem Finsupp.support_mono {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] {f g : ι →₀ α} (hfg : f ≤ g) :

f.support ⊆ g.support

source

instance Finsupp.decidableLE {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [DecidableRel LE.le] :

DecidableRel LE.le

Equations

f.decidableLE g = decidable_of_iff (∀ i ∈ f.support, f i ≤ g i) ⋯

source

instance Finsupp.decidableLT {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [DecidableRel LE.le] :

DecidableRel LT.lt

Equations

Finsupp.decidableLT = decidableLTOfDecidableLE

source

@[simp]

theorem Finsupp.single_le_iff {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] {i : ι} {x : α} {f : ι →₀ α} :

Finsupp.single i x ≤ f ↔ x ≤ f i

source

instance Finsupp.tsub {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] :

Sub (ι →₀ α)

This is called tsub for truncated subtraction, to distinguish it with subtraction in an additive group.

Equations

Finsupp.tsub = { sub := Finsupp.zipWith (fun (m n : α) => m - n) ⋯ }

source

instance Finsupp.orderedSub {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] :

OrderedSub (ι →₀ α)

Equations

⋯ = ⋯

source

instance Finsupp.instCanonicallyOrderedAddCommMonoid {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] :

CanonicallyOrderedAddCommMonoid (ι →₀ α)

Equations

Finsupp.instCanonicallyOrderedAddCommMonoid = CanonicallyOrderedAddCommMonoid.mk ⋯ ⋯

source

@[simp]

theorem Finsupp.coe_tsub {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] (f g : ι →₀ α) :

⇑(f - g) = ⇑f - ⇑g

source

theorem Finsupp.tsub_apply {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] (f g : ι →₀ α) (a : ι) :

(f - g) a = f a - g a

source

@[simp]

theorem Finsupp.single_tsub {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] {i : ι} {a b : α} :

Finsupp.single i (a - b) = Finsupp.single i a - Finsupp.single i b

source

theorem Finsupp.support_tsub {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] {f1 f2 : ι →₀ α} :

(f1 - f2).support ⊆ f1.support

source

theorem Finsupp.subset_support_tsub {ι : Type u_1} {α : Type u_3} [CanonicallyOrderedAddCommMonoid α] [Sub α] [OrderedSub α] [DecidableEq ι] {f1 f2 : ι →₀ α} :

f1.support \ f2.support ⊆ (f1 - f2).support

source

@[simp]

theorem Finsupp.support_inf {ι : Type u_1} {α : Type u_3} [CanonicallyLinearOrderedAddCommMonoid α] [DecidableEq ι] (f g : ι →₀ α) :

(f ⊓ g).support = f.support ∩ g.support

source

@[simp]

theorem Finsupp.support_sup {ι : Type u_1} {α : Type u_3} [CanonicallyLinearOrderedAddCommMonoid α] [DecidableEq ι] (f g : ι →₀ α) :

(f ⊔ g).support = f.support ∪ g.support

source

theorem Finsupp.disjoint_iff {ι : Type u_1} {α : Type u_3} [CanonicallyLinearOrderedAddCommMonoid α] {f g : ι →₀ α} :

Disjoint f g ↔ Disjoint f.support g.support

Some lemmas about `ℕ` #

source

theorem Finsupp.sub_single_one_add {ι : Type u_1} {a : ι} {u u' : ι →₀ ℕ} (h : u a ≠ 0) :

u - Finsupp.single a 1 + u' = u + u' - Finsupp.single a 1

source

theorem Finsupp.add_sub_single_one {ι : Type u_1} {a : ι} {u u' : ι →₀ ℕ} (h : u' a ≠ 0) :

u + (u' - Finsupp.single a 1) = u + u' - Finsupp.single a 1

Documentation

Mathlib.Data.Finsupp.Order

Pointwise order on finitely supported functions #

Main declarations #

Order structures #

Algebraic order structures #

Some lemmas about `ℕ` #

Pointwise order on finitely supported functions #

Main declarations #

Order structures #

Algebraic order structures #

Some lemmas about ℕ #

Some lemmas about `ℕ` #