Pointwise order on finitely supported dependent functions

This file lifts order structures on the α i to Π₀ i, α i.

Main declarations

• DFinsupp.orderEmbeddingToFun: The order embedding from finitely supported dependent functions to functions.

Order structures

instance DFinsupp.instLEDFinsupp {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → LE (α i)] : LE (Π₀ (i : ι), α i)

Equations

• DFinsupp.instLEDFinsupp = { le := fun (f g : Π₀ (i : ι), α i) => ∀ (i : ι), f i ≤ g i }

theorem DFinsupp.le_def {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → LE (α i)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} : f ≤ g ↔ ∀ (i : ι), f i ≤ g i

@[simp]

theorem DFinsupp.coe_le_coe {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → LE (α i)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} : ⇑f ≤ ⇑g ↔ f ≤ g

def DFinsupp.orderEmbeddingToFun {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → LE (α i)] : (Π₀ (i : ι), α i) ↪o ((i : ι) → α i)

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

Equations

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

@[simp]

theorem DFinsupp.coe_orderEmbeddingToFun {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → LE (α i)] : ⇑DFinsupp.orderEmbeddingToFun = DFunLike.coe

theorem DFinsupp.orderEmbeddingToFun_apply {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → LE (α i)] {f : Π₀ (i : ι), α i} {i : ι} : DFinsupp.orderEmbeddingToFun f i = f i

instance DFinsupp.instPreorderDFinsupp {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → Preorder (α i)] : Preorder (Π₀ (i : ι), α i)

Equations

• DFinsupp.instPreorderDFinsupp = let src := inferInstance; Preorder.mk (_ : ∀ (f : Π₀ (i : ι), α i) (i : ι), f i ≤ f i) (_ : ∀ (f g h : Π₀ (i : ι), α i), f ≤ g → g ≤ h → ∀ (i : ι), f i ≤ h i)

theorem DFinsupp.lt_def {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → Preorder (α i)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} : f < g ↔ f ≤ g ∧ ∃ (i : ι), f i < g i

@[simp]

theorem DFinsupp.coe_lt_coe {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → Preorder (α i)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} : ⇑f < ⇑g ↔ f < g

theorem DFinsupp.coe_mono {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → Preorder (α i)] : Monotone DFunLike.coe

theorem DFinsupp.coe_strictMono {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → Preorder (α i)] : Monotone DFunLike.coe

instance DFinsupp.instPartialOrderDFinsupp {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → PartialOrder (α i)] : PartialOrder (Π₀ (i : ι), α i)

Equations

• DFinsupp.instPartialOrderDFinsupp = let src := inferInstance; PartialOrder.mk (_ : ∀ (x x_1 : Π₀ (i : ι), α i), x ≤ x_1 → x_1 ≤ x → x = x_1)

instance DFinsupp.instSemilatticeInfDFinsupp {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → SemilatticeInf (α i)] : SemilatticeInf (Π₀ (i : ι), α i)

Equations

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

@[simp]

theorem DFinsupp.coe_inf {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → SemilatticeInf (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : ⇑(f ⊓ g) = ⇑f ⊓ ⇑g

theorem DFinsupp.inf_apply {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → SemilatticeInf (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) (i : ι) : (f ⊓ g) i = f i ⊓ g i

instance DFinsupp.instSemilatticeSupDFinsupp {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → SemilatticeSup (α i)] : SemilatticeSup (Π₀ (i : ι), α i)

Equations

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

@[simp]

theorem DFinsupp.coe_sup {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → SemilatticeSup (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : ⇑(f ⊔ g) = ⇑f ⊔ ⇑g

theorem DFinsupp.sup_apply {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → SemilatticeSup (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) (i : ι) : (f ⊔ g) i = f i ⊔ g i

instance DFinsupp.lattice {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → Lattice (α i)] : Lattice (Π₀ (i : ι), α i)

Equations

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

theorem DFinsupp.support_inf_union_support_sup {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → Lattice (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] : DFinsupp.support (f ⊓ g) ∪ DFinsupp.support (f ⊔ g) = DFinsupp.support f ∪ DFinsupp.support g

theorem DFinsupp.support_sup_union_support_inf {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → Zero (α i)] [(i : ι) → Lattice (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] : DFinsupp.support (f ⊔ g) ∪ DFinsupp.support (f ⊓ g) = DFinsupp.support f ∪ DFinsupp.support g

Algebraic order structures

instance DFinsupp.instOrderedAddCommMonoidDFinsuppToZeroToAddMonoidToAddCommMonoid {ι : Type u_1} (α : ι → Type u_3) [(i : ι) → OrderedAddCommMonoid (α i)] : OrderedAddCommMonoid (Π₀ (i : ι), α i)

Equations

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

instance DFinsupp.instOrderedCancelAddCommMonoidDFinsuppToZeroToAddRightCancelMonoidToAddCancelMonoidToCancelAddCommMonoid {ι : Type u_1} (α : ι → Type u_3) [(i : ι) → OrderedCancelAddCommMonoid (α i)] : OrderedCancelAddCommMonoid (Π₀ (i : ι), α i)

Equations

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

instance DFinsupp.instContravariantClassDFinsuppToZeroToAddMonoidToAddCommMonoidHAddInstHAddInstAddDFinsuppToZeroToAddZeroClassLeInstLEDFinsuppToLEToPreorderToPartialOrder {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → OrderedAddCommMonoid (α i)] [∀ (i : ι), ContravariantClass (α i) (α i) (fun (x x_1 : α i) => x + x_1) fun (x x_1 : α i) => x ≤ x_1] : ContravariantClass (Π₀ (i : ι), α i) (Π₀ (i : ι), α i) (fun (x x_1 : Π₀ (i : ι), α i) => x + x_1) fun (x x_1 : Π₀ (i : ι), α i) => x ≤ x_1

Equations

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

instance DFinsupp.instPosSMulMono {ι : Type u_1} {α : Type u_3} {β : ι → Type u_4} [Semiring α] [Preorder α] [(i : ι) → AddCommMonoid (β i)] [(i : ι) → Preorder (β i)] [(i : ι) → Module α (β i)] [∀ (i : ι), PosSMulMono α (β i)] : PosSMulMono α (Π₀ (i : ι), β i)

Equations

• (_ : PosSMulMono α (Π₀ (i : ι), β i)) = (_ : PosSMulMono α (Π₀ (i : ι), β i))

instance DFinsupp.instSMulPosMono {ι : Type u_1} {α : Type u_3} {β : ι → Type u_4} [Semiring α] [Preorder α] [(i : ι) → AddCommMonoid (β i)] [(i : ι) → Preorder (β i)] [(i : ι) → Module α (β i)] [∀ (i : ι), SMulPosMono α (β i)] : SMulPosMono α (Π₀ (i : ι), β i)

Equations

• (_ : SMulPosMono α (Π₀ (i : ι), β i)) = (_ : SMulPosMono α (Π₀ (i : ι), β i))

instance DFinsupp.instPosSMulReflectLE {ι : Type u_1} {α : Type u_3} {β : ι → Type u_4} [Semiring α] [Preorder α] [(i : ι) → AddCommMonoid (β i)] [(i : ι) → Preorder (β i)] [(i : ι) → Module α (β i)] [∀ (i : ι), PosSMulReflectLE α (β i)] : PosSMulReflectLE α (Π₀ (i : ι), β i)

Equations

• (_ : PosSMulReflectLE α (Π₀ (i : ι), β i)) = (_ : PosSMulReflectLE α (Π₀ (i : ι), β i))

instance DFinsupp.instSMulPosReflectLE {ι : Type u_1} {α : Type u_3} {β : ι → Type u_4} [Semiring α] [Preorder α] [(i : ι) → AddCommMonoid (β i)] [(i : ι) → Preorder (β i)] [(i : ι) → Module α (β i)] [∀ (i : ι), SMulPosReflectLE α (β i)] : SMulPosReflectLE α (Π₀ (i : ι), β i)

Equations

• (_ : SMulPosReflectLE α (Π₀ (i : ι), β i)) = (_ : SMulPosReflectLE α (Π₀ (i : ι), β i))

instance DFinsupp.instPosSMulStrictMono {ι : Type u_1} {α : Type u_3} {β : ι → Type u_4} [Semiring α] [PartialOrder α] [(i : ι) → AddCommMonoid (β i)] [(i : ι) → PartialOrder (β i)] [(i : ι) → Module α (β i)] [∀ (i : ι), PosSMulStrictMono α (β i)] : PosSMulStrictMono α (Π₀ (i : ι), β i)

Equations

• (_ : PosSMulStrictMono α (Π₀ (i : ι), β i)) = (_ : PosSMulStrictMono α (Π₀ (i : ι), β i))

instance DFinsupp.instSMulPosStrictMono {ι : Type u_1} {α : Type u_3} {β : ι → Type u_4} [Semiring α] [PartialOrder α] [(i : ι) → AddCommMonoid (β i)] [(i : ι) → PartialOrder (β i)] [(i : ι) → Module α (β i)] [∀ (i : ι), SMulPosStrictMono α (β i)] : SMulPosStrictMono α (Π₀ (i : ι), β i)

Equations

• (_ : SMulPosStrictMono α (Π₀ (i : ι), β i)) = (_ : SMulPosStrictMono α (Π₀ (i : ι), β i))

instance DFinsupp.instSMulPosReflectLT {ι : Type u_1} {α : Type u_3} {β : ι → Type u_4} [Semiring α] [PartialOrder α] [(i : ι) → AddCommMonoid (β i)] [(i : ι) → PartialOrder (β i)] [(i : ι) → Module α (β i)] [∀ (i : ι), SMulPosReflectLT α (β i)] : SMulPosReflectLT α (Π₀ (i : ι), β i)

Equations

• (_ : SMulPosReflectLT α (Π₀ (i : ι), β i)) = (_ : SMulPosReflectLT α (Π₀ (i : ι), β i))

instance DFinsupp.instOrderBotDFinsuppToZeroToAddMonoidToAddCommMonoidToOrderedAddCommMonoidInstLEDFinsuppToLEToPreorderToPartialOrder {ι : Type u_1} (α : ι → Type u_2) [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] : OrderBot (Π₀ (i : ι), α i)

Equations

• DFinsupp.instOrderBotDFinsuppToZeroToAddMonoidToAddCommMonoidToOrderedAddCommMonoidInstLEDFinsuppToLEToPreorderToPartialOrder α = OrderBot.mk (_ : ∀ (a : Π₀ (i : ι), α i), 0 ≤ a)

theorem DFinsupp.bot_eq_zero {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] : ⊥ = 0

@[simp]

theorem DFinsupp.add_eq_zero_iff {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : f + g = 0 ↔ f = 0 ∧ g = 0

theorem DFinsupp.le_iff' {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} {s : Finset ι} (hf : DFinsupp.support f ⊆ s) : f ≤ g ↔ ∀ i ∈ s, f i ≤ g i

theorem DFinsupp.le_iff {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} : f ≤ g ↔ ∀ i ∈ DFinsupp.support f, f i ≤ g i

theorem DFinsupp.support_monotone {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] : Monotone DFinsupp.support

theorem DFinsupp.support_mono {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} (hfg : f ≤ g) : DFinsupp.support f ⊆ DFinsupp.support g

instance DFinsupp.decidableLE {ι : Type u_1} (α : ι → Type u_2) [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] [(i : ι) → DecidableRel LE.le] : DecidableRel LE.le

Equations

• DFinsupp.decidableLE α x✝ x = decidable_of_iff (∀ i ∈ DFinsupp.support x✝, x✝ i ≤ x i) (_ : (∀ i ∈ DFinsupp.support x✝, x✝ i ≤ x i) ↔ x✝ ≤ x)

@[simp]

theorem DFinsupp.single_le_iff {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] {f : Π₀ (i : ι), α i} {i : ι} {a : α i} : DFinsupp.single i a ≤ f ↔ a ≤ f i

instance DFinsupp.tsub {ι : Type u_1} (α : ι → Type u_2) [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → Sub (α i)] [∀ (i : ι), OrderedSub (α i)] : Sub (Π₀ (i : ι), α i)

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

Equations

• DFinsupp.tsub α = { sub := DFinsupp.zipWith (fun (x : ι) (m n : α x) => m - n) (_ : ∀ (x : ι), 0 - 0 = 0) }

theorem DFinsupp.tsub_apply {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → Sub (α i)] [∀ (i : ι), OrderedSub (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) (i : ι) : (f - g) i = f i - g i

@[simp]

theorem DFinsupp.coe_tsub {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → Sub (α i)] [∀ (i : ι), OrderedSub (α i)] (f : Π₀ (i : ι), α i) (g : Π₀ (i : ι), α i) : ⇑(f - g) = ⇑f - ⇑g

instance DFinsupp.instOrderedSubDFinsuppToZeroToAddMonoidToAddCommMonoidToOrderedAddCommMonoidInstLEDFinsuppToLEToPreorderToPartialOrderInstAddDFinsuppToZeroToAddZeroClassTsub {ι : Type u_1} (α : ι → Type u_2) [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → Sub (α i)] [∀ (i : ι), OrderedSub (α i)] : OrderedSub (Π₀ (i : ι), α i)

Equations

• (_ : OrderedSub (Π₀ (i : ι), α i)) = (_ : OrderedSub (Π₀ (i : ι), α i))

instance DFinsupp.instCanonicallyOrderedAddCommMonoidDFinsuppToZeroToAddMonoidToAddCommMonoidToOrderedAddCommMonoid {ι : Type u_1} (α : ι → Type u_2) [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → Sub (α i)] [∀ (i : ι), OrderedSub (α i)] : CanonicallyOrderedAddCommMonoid (Π₀ (i : ι), α i)

Equations

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

@[simp]

theorem DFinsupp.single_tsub {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → Sub (α i)] [∀ (i : ι), OrderedSub (α i)] {i : ι} {a : α i} {b : α i} [DecidableEq ι] : DFinsupp.single i (a - b) = DFinsupp.single i a - DFinsupp.single i b

theorem DFinsupp.support_tsub {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → Sub (α i)] [∀ (i : ι), OrderedSub (α i)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] : DFinsupp.support (f - g) ⊆ DFinsupp.support f

theorem DFinsupp.subset_support_tsub {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyOrderedAddCommMonoid (α i)] [(i : ι) → Sub (α i)] [∀ (i : ι), OrderedSub (α i)] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} [DecidableEq ι] [(i : ι) → (x : α i) → Decidable (x ≠ 0)] : DFinsupp.support f \ DFinsupp.support g ⊆ DFinsupp.support (f - g)

@[simp]

theorem DFinsupp.support_inf {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyLinearOrderedAddCommMonoid (α i)] [DecidableEq ι] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} : DFinsupp.support (f ⊓ g) = DFinsupp.support f ∩ DFinsupp.support g

@[simp]

theorem DFinsupp.support_sup {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyLinearOrderedAddCommMonoid (α i)] [DecidableEq ι] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} : DFinsupp.support (f ⊔ g) = DFinsupp.support f ∪ DFinsupp.support g

theorem DFinsupp.disjoint_iff {ι : Type u_1} {α : ι → Type u_2} [(i : ι) → CanonicallyLinearOrderedAddCommMonoid (α i)] [DecidableEq ι] {f : Π₀ (i : ι), α i} {g : Π₀ (i : ι), α i} : Disjoint f g ↔ Disjoint (DFinsupp.support f) (DFinsupp.support g)